1,1,53,0,0.0290957,"\int \left(a \sin ^2(x)\right)^{5/2} \, dx","Int[(a*Sin[x]^2)^(5/2),x]","-\frac{8}{15} a^2 \cot (x) \sqrt{a \sin ^2(x)}-\frac{1}{5} \cot (x) \left(a \sin ^2(x)\right)^{5/2}-\frac{4}{15} a \cot (x) \left(a \sin ^2(x)\right)^{3/2}","-\frac{8}{15} a^2 \cot (x) \sqrt{a \sin ^2(x)}-\frac{1}{5} \cot (x) \left(a \sin ^2(x)\right)^{5/2}-\frac{4}{15} a \cot (x) \left(a \sin ^2(x)\right)^{3/2}",1,"(-8*a^2*Cot[x]*Sqrt[a*Sin[x]^2])/15 - (4*a*Cot[x]*(a*Sin[x]^2)^(3/2))/15 - (Cot[x]*(a*Sin[x]^2)^(5/2))/5","A",4,3,10,0.3000,1,"{3203, 3207, 2638}"
2,1,34,0,0.0185186,"\int \left(a \sin ^2(x)\right)^{3/2} \, dx","Int[(a*Sin[x]^2)^(3/2),x]","-\frac{1}{3} \cot (x) \left(a \sin ^2(x)\right)^{3/2}-\frac{2}{3} a \cot (x) \sqrt{a \sin ^2(x)}","-\frac{1}{3} \cot (x) \left(a \sin ^2(x)\right)^{3/2}-\frac{2}{3} a \cot (x) \sqrt{a \sin ^2(x)}",1,"(-2*a*Cot[x]*Sqrt[a*Sin[x]^2])/3 - (Cot[x]*(a*Sin[x]^2)^(3/2))/3","A",3,3,10,0.3000,1,"{3203, 3207, 2638}"
3,1,14,0,0.0098667,"\int \sqrt{a \sin ^2(x)} \, dx","Int[Sqrt[a*Sin[x]^2],x]","-\cot (x) \sqrt{a \sin ^2(x)}","-\cot (x) \sqrt{a \sin ^2(x)}",1,"-(Cot[x]*Sqrt[a*Sin[x]^2])","A",2,2,10,0.2000,1,"{3207, 2638}"
4,1,17,0,0.0113021,"\int \frac{1}{\sqrt{a \sin ^2(x)}} \, dx","Int[1/Sqrt[a*Sin[x]^2],x]","-\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{a \sin ^2(x)}}","-\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{a \sin ^2(x)}}",1,"-((ArcTanh[Cos[x]]*Sin[x])/Sqrt[a*Sin[x]^2])","A",2,2,10,0.2000,1,"{3207, 3770}"
5,1,42,0,0.0242362,"\int \frac{1}{\left(a \sin ^2(x)\right)^{3/2}} \, dx","Int[(a*Sin[x]^2)^(-3/2),x]","-\frac{\cot (x)}{2 a \sqrt{a \sin ^2(x)}}-\frac{\sin (x) \tanh ^{-1}(\cos (x))}{2 a \sqrt{a \sin ^2(x)}}","-\frac{\cot (x)}{2 a \sqrt{a \sin ^2(x)}}-\frac{\sin (x) \tanh ^{-1}(\cos (x))}{2 a \sqrt{a \sin ^2(x)}}",1,"-Cot[x]/(2*a*Sqrt[a*Sin[x]^2]) - (ArcTanh[Cos[x]]*Sin[x])/(2*a*Sqrt[a*Sin[x]^2])","A",3,3,10,0.3000,1,"{3204, 3207, 3770}"
6,1,61,0,0.0302122,"\int \frac{1}{\left(a \sin ^2(x)\right)^{5/2}} \, dx","Int[(a*Sin[x]^2)^(-5/2),x]","-\frac{3 \cot (x)}{8 a^2 \sqrt{a \sin ^2(x)}}-\frac{3 \sin (x) \tanh ^{-1}(\cos (x))}{8 a^2 \sqrt{a \sin ^2(x)}}-\frac{\cot (x)}{4 a \left(a \sin ^2(x)\right)^{3/2}}","-\frac{3 \cot (x)}{8 a^2 \sqrt{a \sin ^2(x)}}-\frac{3 \sin (x) \tanh ^{-1}(\cos (x))}{8 a^2 \sqrt{a \sin ^2(x)}}-\frac{\cot (x)}{4 a \left(a \sin ^2(x)\right)^{3/2}}",1,"-Cot[x]/(4*a*(a*Sin[x]^2)^(3/2)) - (3*Cot[x])/(8*a^2*Sqrt[a*Sin[x]^2]) - (3*ArcTanh[Cos[x]]*Sin[x])/(8*a^2*Sqrt[a*Sin[x]^2])","A",4,3,10,0.3000,1,"{3204, 3207, 3770}"
7,1,123,0,0.0407798,"\int \left(a \sin ^3(x)\right)^{5/2} \, dx","Int[(a*Sin[x]^3)^(5/2),x]","-\frac{2}{15} a^2 \sin ^5(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{26}{165} a^2 \sin ^3(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{78}{385} a^2 \sin (x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{26}{77} a^2 \cot (x) \sqrt{a \sin ^3(x)}-\frac{26 a^2 F\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right) \sqrt{a \sin ^3(x)}}{77 \sin ^{\frac{3}{2}}(x)}","-\frac{2}{15} a^2 \sin ^5(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{26}{165} a^2 \sin ^3(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{78}{385} a^2 \sin (x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{26}{77} a^2 \cot (x) \sqrt{a \sin ^3(x)}-\frac{26 a^2 F\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right) \sqrt{a \sin ^3(x)}}{77 \sin ^{\frac{3}{2}}(x)}",1,"(-26*a^2*Cot[x]*Sqrt[a*Sin[x]^3])/77 - (26*a^2*EllipticF[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(77*Sin[x]^(3/2)) - (78*a^2*Cos[x]*Sin[x]*Sqrt[a*Sin[x]^3])/385 - (26*a^2*Cos[x]*Sin[x]^3*Sqrt[a*Sin[x]^3])/165 - (2*a^2*Cos[x]*Sin[x]^5*Sqrt[a*Sin[x]^3])/15","A",6,3,10,0.3000,1,"{3207, 2635, 2641}"
8,1,73,0,0.0247842,"\int \left(a \sin ^3(x)\right)^{3/2} \, dx","Int[(a*Sin[x]^3)^(3/2),x]","-\frac{2}{9} a \sin ^2(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{14}{45} a \cos (x) \sqrt{a \sin ^3(x)}-\frac{14 a E\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right) \sqrt{a \sin ^3(x)}}{15 \sin ^{\frac{3}{2}}(x)}","-\frac{2}{9} a \sin ^2(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{14}{45} a \cos (x) \sqrt{a \sin ^3(x)}-\frac{14 a E\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right) \sqrt{a \sin ^3(x)}}{15 \sin ^{\frac{3}{2}}(x)}",1,"(-14*a*Cos[x]*Sqrt[a*Sin[x]^3])/45 - (14*a*EllipticE[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(15*Sin[x]^(3/2)) - (2*a*Cos[x]*Sin[x]^2*Sqrt[a*Sin[x]^3])/9","A",4,3,10,0.3000,1,"{3207, 2635, 2639}"
9,1,50,0,0.0176481,"\int \sqrt{a \sin ^3(x)} \, dx","Int[Sqrt[a*Sin[x]^3],x]","-\frac{2}{3} \cot (x) \sqrt{a \sin ^3(x)}-\frac{2 F\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right) \sqrt{a \sin ^3(x)}}{3 \sin ^{\frac{3}{2}}(x)}","-\frac{2}{3} \cot (x) \sqrt{a \sin ^3(x)}-\frac{2 F\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right) \sqrt{a \sin ^3(x)}}{3 \sin ^{\frac{3}{2}}(x)}",1,"(-2*Cot[x]*Sqrt[a*Sin[x]^3])/3 - (2*EllipticF[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(3*Sin[x]^(3/2))","A",3,3,10,0.3000,1,"{3207, 2635, 2641}"
10,1,48,0,0.0173165,"\int \frac{1}{\sqrt{a \sin ^3(x)}} \, dx","Int[1/Sqrt[a*Sin[x]^3],x]","\frac{2 \sin ^{\frac{3}{2}}(x) E\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right)}{\sqrt{a \sin ^3(x)}}-\frac{2 \sin (x) \cos (x)}{\sqrt{a \sin ^3(x)}}","\frac{2 \sin ^{\frac{3}{2}}(x) E\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right)}{\sqrt{a \sin ^3(x)}}-\frac{2 \sin (x) \cos (x)}{\sqrt{a \sin ^3(x)}}",1,"(-2*Cos[x]*Sin[x])/Sqrt[a*Sin[x]^3] + (2*EllipticE[Pi/4 - x/2, 2]*Sin[x]^(3/2))/Sqrt[a*Sin[x]^3]","A",3,3,10,0.3000,1,"{3207, 2636, 2639}"
11,1,77,0,0.0254569,"\int \frac{1}{\left(a \sin ^3(x)\right)^{3/2}} \, dx","Int[(a*Sin[x]^3)^(-3/2),x]","-\frac{10 \cos (x)}{21 a \sqrt{a \sin ^3(x)}}-\frac{10 \sin ^{\frac{3}{2}}(x) F\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right)}{21 a \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc (x)}{7 a \sqrt{a \sin ^3(x)}}","-\frac{10 \cos (x)}{21 a \sqrt{a \sin ^3(x)}}-\frac{10 \sin ^{\frac{3}{2}}(x) F\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right)}{21 a \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc (x)}{7 a \sqrt{a \sin ^3(x)}}",1,"(-10*Cos[x])/(21*a*Sqrt[a*Sin[x]^3]) - (2*Cot[x]*Csc[x])/(7*a*Sqrt[a*Sin[x]^3]) - (10*EllipticF[Pi/4 - x/2, 2]*Sin[x]^(3/2))/(21*a*Sqrt[a*Sin[x]^3])","A",4,3,10,0.3000,1,"{3207, 2636, 2641}"
12,1,123,0,0.0420212,"\int \frac{1}{\left(a \sin ^3(x)\right)^{5/2}} \, dx","Int[(a*Sin[x]^3)^(-5/2),x]","-\frac{154 \sin (x) \cos (x)}{195 a^2 \sqrt{a \sin ^3(x)}}-\frac{154 \cot (x)}{585 a^2 \sqrt{a \sin ^3(x)}}+\frac{154 \sin ^{\frac{3}{2}}(x) E\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right)}{195 a^2 \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt{a \sin ^3(x)}}-\frac{22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt{a \sin ^3(x)}}","-\frac{154 \sin (x) \cos (x)}{195 a^2 \sqrt{a \sin ^3(x)}}-\frac{154 \cot (x)}{585 a^2 \sqrt{a \sin ^3(x)}}+\frac{154 \sin ^{\frac{3}{2}}(x) E\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right)}{195 a^2 \sqrt{a \sin ^3(x)}}-\frac{2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt{a \sin ^3(x)}}-\frac{22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt{a \sin ^3(x)}}",1,"(-154*Cot[x])/(585*a^2*Sqrt[a*Sin[x]^3]) - (22*Cot[x]*Csc[x]^2)/(117*a^2*Sqrt[a*Sin[x]^3]) - (2*Cot[x]*Csc[x]^4)/(13*a^2*Sqrt[a*Sin[x]^3]) - (154*Cos[x]*Sin[x])/(195*a^2*Sqrt[a*Sin[x]^3]) + (154*EllipticE[Pi/4 - x/2, 2]*Sin[x]^(3/2))/(195*a^2*Sqrt[a*Sin[x]^3])","A",6,3,10,0.3000,1,"{3207, 2636, 2639}"
13,1,132,0,0.0438645,"\int \left(a \sin ^4(x)\right)^{5/2} \, dx","Int[(a*Sin[x]^4)^(5/2),x]","-\frac{1}{10} a^2 \sin ^7(x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{9}{80} a^2 \sin ^5(x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{21}{160} a^2 \sin ^3(x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{21}{128} a^2 \sin (x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{63}{256} a^2 \cot (x) \sqrt{a \sin ^4(x)}+\frac{63}{256} a^2 x \csc ^2(x) \sqrt{a \sin ^4(x)}","-\frac{1}{10} a^2 \sin ^7(x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{9}{80} a^2 \sin ^5(x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{21}{160} a^2 \sin ^3(x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{21}{128} a^2 \sin (x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{63}{256} a^2 \cot (x) \sqrt{a \sin ^4(x)}+\frac{63}{256} a^2 x \csc ^2(x) \sqrt{a \sin ^4(x)}",1,"(-63*a^2*Cot[x]*Sqrt[a*Sin[x]^4])/256 + (63*a^2*x*Csc[x]^2*Sqrt[a*Sin[x]^4])/256 - (21*a^2*Cos[x]*Sin[x]*Sqrt[a*Sin[x]^4])/128 - (21*a^2*Cos[x]*Sin[x]^3*Sqrt[a*Sin[x]^4])/160 - (9*a^2*Cos[x]*Sin[x]^5*Sqrt[a*Sin[x]^4])/80 - (a^2*Cos[x]*Sin[x]^7*Sqrt[a*Sin[x]^4])/10","A",7,3,10,0.3000,1,"{3207, 2635, 8}"
14,1,78,0,0.0277571,"\int \left(a \sin ^4(x)\right)^{3/2} \, dx","Int[(a*Sin[x]^4)^(3/2),x]","-\frac{1}{6} a \sin ^3(x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{5}{24} a \sin (x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{5}{16} a \cot (x) \sqrt{a \sin ^4(x)}+\frac{5}{16} a x \csc ^2(x) \sqrt{a \sin ^4(x)}","-\frac{1}{6} a \sin ^3(x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{5}{24} a \sin (x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{5}{16} a \cot (x) \sqrt{a \sin ^4(x)}+\frac{5}{16} a x \csc ^2(x) \sqrt{a \sin ^4(x)}",1,"(-5*a*Cot[x]*Sqrt[a*Sin[x]^4])/16 + (5*a*x*Csc[x]^2*Sqrt[a*Sin[x]^4])/16 - (5*a*Cos[x]*Sin[x]*Sqrt[a*Sin[x]^4])/24 - (a*Cos[x]*Sin[x]^3*Sqrt[a*Sin[x]^4])/6","A",5,3,10,0.3000,1,"{3207, 2635, 8}"
15,1,36,0,0.0131223,"\int \sqrt{a \sin ^4(x)} \, dx","Int[Sqrt[a*Sin[x]^4],x]","\frac{1}{2} x \csc ^2(x) \sqrt{a \sin ^4(x)}-\frac{1}{2} \cot (x) \sqrt{a \sin ^4(x)}","\frac{1}{2} x \csc ^2(x) \sqrt{a \sin ^4(x)}-\frac{1}{2} \cot (x) \sqrt{a \sin ^4(x)}",1,"-(Cot[x]*Sqrt[a*Sin[x]^4])/2 + (x*Csc[x]^2*Sqrt[a*Sin[x]^4])/2","A",3,3,10,0.3000,1,"{3207, 2635, 8}"
16,1,16,0,0.0136734,"\int \frac{1}{\sqrt{a \sin ^4(x)}} \, dx","Int[1/Sqrt[a*Sin[x]^4],x]","-\frac{\sin (x) \cos (x)}{\sqrt{a \sin ^4(x)}}","-\frac{\sin (x) \cos (x)}{\sqrt{a \sin ^4(x)}}",1,"-((Cos[x]*Sin[x])/Sqrt[a*Sin[x]^4])","A",3,3,10,0.3000,1,"{3207, 3767, 8}"
17,1,68,0,0.0195453,"\int \frac{1}{\left(a \sin ^4(x)\right)^{3/2}} \, dx","Int[(a*Sin[x]^4)^(-3/2),x]","-\frac{\sin (x) \cos (x)}{a \sqrt{a \sin ^4(x)}}-\frac{\cos ^2(x) \cot ^3(x)}{5 a \sqrt{a \sin ^4(x)}}-\frac{2 \cos ^2(x) \cot (x)}{3 a \sqrt{a \sin ^4(x)}}","-\frac{\sin (x) \cos (x)}{a \sqrt{a \sin ^4(x)}}-\frac{\cos ^2(x) \cot ^3(x)}{5 a \sqrt{a \sin ^4(x)}}-\frac{2 \cos ^2(x) \cot (x)}{3 a \sqrt{a \sin ^4(x)}}",1,"(-2*Cos[x]^2*Cot[x])/(3*a*Sqrt[a*Sin[x]^4]) - (Cos[x]^2*Cot[x]^3)/(5*a*Sqrt[a*Sin[x]^4]) - (Cos[x]*Sin[x])/(a*Sqrt[a*Sin[x]^4])","A",3,2,10,0.2000,1,"{3207, 3767}"
18,1,118,0,0.0282477,"\int \frac{1}{\left(a \sin ^4(x)\right)^{5/2}} \, dx","Int[(a*Sin[x]^4)^(-5/2),x]","-\frac{\sin (x) \cos (x)}{a^2 \sqrt{a \sin ^4(x)}}-\frac{\cos ^2(x) \cot ^7(x)}{9 a^2 \sqrt{a \sin ^4(x)}}-\frac{4 \cos ^2(x) \cot ^5(x)}{7 a^2 \sqrt{a \sin ^4(x)}}-\frac{6 \cos ^2(x) \cot ^3(x)}{5 a^2 \sqrt{a \sin ^4(x)}}-\frac{4 \cos ^2(x) \cot (x)}{3 a^2 \sqrt{a \sin ^4(x)}}","-\frac{\sin (x) \cos (x)}{a^2 \sqrt{a \sin ^4(x)}}-\frac{\cos ^2(x) \cot ^7(x)}{9 a^2 \sqrt{a \sin ^4(x)}}-\frac{4 \cos ^2(x) \cot ^5(x)}{7 a^2 \sqrt{a \sin ^4(x)}}-\frac{6 \cos ^2(x) \cot ^3(x)}{5 a^2 \sqrt{a \sin ^4(x)}}-\frac{4 \cos ^2(x) \cot (x)}{3 a^2 \sqrt{a \sin ^4(x)}}",1,"(-4*Cos[x]^2*Cot[x])/(3*a^2*Sqrt[a*Sin[x]^4]) - (6*Cos[x]^2*Cot[x]^3)/(5*a^2*Sqrt[a*Sin[x]^4]) - (4*Cos[x]^2*Cot[x]^5)/(7*a^2*Sqrt[a*Sin[x]^4]) - (Cos[x]^2*Cot[x]^7)/(9*a^2*Sqrt[a*Sin[x]^4]) - (Cos[x]*Sin[x])/(a^2*Sqrt[a*Sin[x]^4])","A",3,2,10,0.2000,1,"{3207, 3767}"
19,1,89,0,0.0390368,"\int \left(c \sin ^m(a+b x)\right)^{5/2} \, dx","Int[(c*Sin[a + b*x]^m)^(5/2),x]","\frac{2 c^2 \cos (a+b x) \sin ^{2 m+1}(a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left(\frac{1}{2},\frac{1}{4} (5 m+2);\frac{1}{4} (5 m+6);\sin ^2(a+b x)\right)}{b (5 m+2) \sqrt{\cos ^2(a+b x)}}","\frac{2 c^2 \cos (a+b x) \sin ^{2 m+1}(a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left(\frac{1}{2},\frac{1}{4} (5 m+2);\frac{1}{4} (5 m+6);\sin ^2(a+b x)\right)}{b (5 m+2) \sqrt{\cos ^2(a+b x)}}",1,"(2*c^2*Cos[a + b*x]*Hypergeometric2F1[1/2, (2 + 5*m)/4, (6 + 5*m)/4, Sin[a + b*x]^2]*Sin[a + b*x]^(1 + 2*m)*Sqrt[c*Sin[a + b*x]^m])/(b*(2 + 5*m)*Sqrt[Cos[a + b*x]^2])","A",2,2,14,0.1429,1,"{3208, 2643}"
20,1,83,0,0.0379316,"\int \left(c \sin ^m(a+b x)\right)^{3/2} \, dx","Int[(c*Sin[a + b*x]^m)^(3/2),x]","\frac{2 c \cos (a+b x) \sin ^{m+1}(a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left(\frac{1}{2},\frac{1}{4} (3 m+2);\frac{3 (m+2)}{4};\sin ^2(a+b x)\right)}{b (3 m+2) \sqrt{\cos ^2(a+b x)}}","\frac{2 c \cos (a+b x) \sin ^{m+1}(a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left(\frac{1}{2},\frac{1}{4} (3 m+2);\frac{3 (m+2)}{4};\sin ^2(a+b x)\right)}{b (3 m+2) \sqrt{\cos ^2(a+b x)}}",1,"(2*c*Cos[a + b*x]*Hypergeometric2F1[1/2, (2 + 3*m)/4, (3*(2 + m))/4, Sin[a + b*x]^2]*Sin[a + b*x]^(1 + m)*Sqrt[c*Sin[a + b*x]^m])/(b*(2 + 3*m)*Sqrt[Cos[a + b*x]^2])","A",2,2,14,0.1429,1,"{3208, 2643}"
21,1,74,0,0.0360372,"\int \sqrt{c \sin ^m(a+b x)} \, dx","Int[Sqrt[c*Sin[a + b*x]^m],x]","\frac{2 \sin (a+b x) \cos (a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left(\frac{1}{2},\frac{m+2}{4};\frac{m+6}{4};\sin ^2(a+b x)\right)}{b (m+2) \sqrt{\cos ^2(a+b x)}}","\frac{2 \sin (a+b x) \cos (a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left(\frac{1}{2},\frac{m+2}{4};\frac{m+6}{4};\sin ^2(a+b x)\right)}{b (m+2) \sqrt{\cos ^2(a+b x)}}",1,"(2*Cos[a + b*x]*Hypergeometric2F1[1/2, (2 + m)/4, (6 + m)/4, Sin[a + b*x]^2]*Sin[a + b*x]*Sqrt[c*Sin[a + b*x]^m])/(b*(2 + m)*Sqrt[Cos[a + b*x]^2])","A",2,2,14,0.1429,1,"{3208, 2643}"
22,1,80,0,0.041786,"\int \frac{1}{\sqrt{c \sin ^m(a+b x)}} \, dx","Int[1/Sqrt[c*Sin[a + b*x]^m],x]","\frac{2 \sin (a+b x) \cos (a+b x) \, _2F_1\left(\frac{1}{2},\frac{2-m}{4};\frac{6-m}{4};\sin ^2(a+b x)\right)}{b (2-m) \sqrt{\cos ^2(a+b x)} \sqrt{c \sin ^m(a+b x)}}","\frac{2 \sin (a+b x) \cos (a+b x) \, _2F_1\left(\frac{1}{2},\frac{2-m}{4};\frac{6-m}{4};\sin ^2(a+b x)\right)}{b (2-m) \sqrt{\cos ^2(a+b x)} \sqrt{c \sin ^m(a+b x)}}",1,"(2*Cos[a + b*x]*Hypergeometric2F1[1/2, (2 - m)/4, (6 - m)/4, Sin[a + b*x]^2]*Sin[a + b*x])/(b*(2 - m)*Sqrt[Cos[a + b*x]^2]*Sqrt[c*Sin[a + b*x]^m])","A",2,2,14,0.1429,1,"{3208, 2643}"
23,1,89,0,0.0405227,"\int \frac{1}{\left(c \sin ^m(a+b x)\right)^{3/2}} \, dx","Int[(c*Sin[a + b*x]^m)^(-3/2),x]","\frac{2 \cos (a+b x) \sin ^{1-m}(a+b x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2-3 m);\frac{3 (2-m)}{4};\sin ^2(a+b x)\right)}{b c (2-3 m) \sqrt{\cos ^2(a+b x)} \sqrt{c \sin ^m(a+b x)}}","\frac{2 \cos (a+b x) \sin ^{1-m}(a+b x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2-3 m);\frac{3 (2-m)}{4};\sin ^2(a+b x)\right)}{b c (2-3 m) \sqrt{\cos ^2(a+b x)} \sqrt{c \sin ^m(a+b x)}}",1,"(2*Cos[a + b*x]*Hypergeometric2F1[1/2, (2 - 3*m)/4, (3*(2 - m))/4, Sin[a + b*x]^2]*Sin[a + b*x]^(1 - m))/(b*c*(2 - 3*m)*Sqrt[Cos[a + b*x]^2]*Sqrt[c*Sin[a + b*x]^m])","A",2,2,14,0.1429,1,"{3208, 2643}"
24,1,89,0,0.0408314,"\int \frac{1}{\left(c \sin ^m(a+b x)\right)^{5/2}} \, dx","Int[(c*Sin[a + b*x]^m)^(-5/2),x]","\frac{2 \cos (a+b x) \sin ^{1-2 m}(a+b x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2-5 m);\frac{1}{4} (6-5 m);\sin ^2(a+b x)\right)}{b c^2 (2-5 m) \sqrt{\cos ^2(a+b x)} \sqrt{c \sin ^m(a+b x)}}","\frac{2 \cos (a+b x) \sin ^{1-2 m}(a+b x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2-5 m);\frac{1}{4} (6-5 m);\sin ^2(a+b x)\right)}{b c^2 (2-5 m) \sqrt{\cos ^2(a+b x)} \sqrt{c \sin ^m(a+b x)}}",1,"(2*Cos[a + b*x]*Hypergeometric2F1[1/2, (2 - 5*m)/4, (6 - 5*m)/4, Sin[a + b*x]^2]*Sin[a + b*x]^(1 - 2*m))/(b*c^2*(2 - 5*m)*Sqrt[Cos[a + b*x]^2]*Sqrt[c*Sin[a + b*x]^m])","A",2,2,14,0.1429,1,"{3208, 2643}"
25,1,77,0,0.0354642,"\int \left(b \sin ^n(c+d x)\right)^p \, dx","Int[(b*Sin[c + d*x]^n)^p,x]","\frac{\sin (c+d x) \cos (c+d x) \left(b \sin ^n(c+d x)\right)^p \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(c+d x)\right)}{d (n p+1) \sqrt{\cos ^2(c+d x)}}","\frac{\sin (c+d x) \cos (c+d x) \left(b \sin ^n(c+d x)\right)^p \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(c+d x)\right)}{d (n p+1) \sqrt{\cos ^2(c+d x)}}",1,"(Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[c + d*x]^2]*Sin[c + d*x]*(b*Sin[c + d*x]^n)^p)/(d*(1 + n*p)*Sqrt[Cos[c + d*x]^2])","A",2,2,12,0.1667,1,"{3208, 2643}"
26,1,77,0,0.0334552,"\int \left(c \sin ^2(a+b x)\right)^p \, dx","Int[(c*Sin[a + b*x]^2)^p,x]","\frac{\sin (a+b x) \cos (a+b x) \left(c \sin ^2(a+b x)\right)^p \, _2F_1\left(\frac{1}{2},\frac{1}{2} (2 p+1);\frac{1}{2} (2 p+3);\sin ^2(a+b x)\right)}{b (2 p+1) \sqrt{\cos ^2(a+b x)}}","\frac{\sin (a+b x) \cos (a+b x) \left(c \sin ^2(a+b x)\right)^p \, _2F_1\left(\frac{1}{2},\frac{1}{2} (2 p+1);\frac{1}{2} (2 p+3);\sin ^2(a+b x)\right)}{b (2 p+1) \sqrt{\cos ^2(a+b x)}}",1,"(Cos[a + b*x]*Hypergeometric2F1[1/2, (1 + 2*p)/2, (3 + 2*p)/2, Sin[a + b*x]^2]*Sin[a + b*x]*(c*Sin[a + b*x]^2)^p)/(b*(1 + 2*p)*Sqrt[Cos[a + b*x]^2])","A",2,2,12,0.1667,1,"{3207, 2643}"
27,1,75,0,0.0346214,"\int \left(c \sin ^3(a+b x)\right)^p \, dx","Int[(c*Sin[a + b*x]^3)^p,x]","\frac{\sin (a+b x) \cos (a+b x) \left(c \sin ^3(a+b x)\right)^p \, _2F_1\left(\frac{1}{2},\frac{1}{2} (3 p+1);\frac{3 (p+1)}{2};\sin ^2(a+b x)\right)}{b (3 p+1) \sqrt{\cos ^2(a+b x)}}","\frac{\sin (a+b x) \cos (a+b x) \left(c \sin ^3(a+b x)\right)^p \, _2F_1\left(\frac{1}{2},\frac{1}{2} (3 p+1);\frac{3 (p+1)}{2};\sin ^2(a+b x)\right)}{b (3 p+1) \sqrt{\cos ^2(a+b x)}}",1,"(Cos[a + b*x]*Hypergeometric2F1[1/2, (1 + 3*p)/2, (3*(1 + p))/2, Sin[a + b*x]^2]*Sin[a + b*x]*(c*Sin[a + b*x]^3)^p)/(b*(1 + 3*p)*Sqrt[Cos[a + b*x]^2])","A",2,2,12,0.1667,1,"{3207, 2643}"
28,1,77,0,0.0332862,"\int \left(c \sin ^4(a+b x)\right)^p \, dx","Int[(c*Sin[a + b*x]^4)^p,x]","\frac{\sin (a+b x) \cos (a+b x) \left(c \sin ^4(a+b x)\right)^p \, _2F_1\left(\frac{1}{2},\frac{1}{2} (4 p+1);\frac{1}{2} (4 p+3);\sin ^2(a+b x)\right)}{b (4 p+1) \sqrt{\cos ^2(a+b x)}}","\frac{\sin (a+b x) \cos (a+b x) \left(c \sin ^4(a+b x)\right)^p \, _2F_1\left(\frac{1}{2},\frac{1}{2} (4 p+1);\frac{1}{2} (4 p+3);\sin ^2(a+b x)\right)}{b (4 p+1) \sqrt{\cos ^2(a+b x)}}",1,"(Cos[a + b*x]*Hypergeometric2F1[1/2, (1 + 4*p)/2, (3 + 4*p)/2, Sin[a + b*x]^2]*Sin[a + b*x]*(c*Sin[a + b*x]^4)^p)/(b*(1 + 4*p)*Sqrt[Cos[a + b*x]^2])","A",2,2,12,0.1667,1,"{3207, 2643}"
29,1,25,0,0.0189031,"\int \left(c \sin ^n(a+b x)\right)^{\frac{1}{n}} \, dx","Int[(c*Sin[a + b*x]^n)^n^(-1),x]","-\frac{\cot (a+b x) \left(c \sin ^n(a+b x)\right)^{\frac{1}{n}}}{b}","-\frac{\cot (a+b x) \left(c \sin ^n(a+b x)\right)^{\frac{1}{n}}}{b}",1,"-((Cot[a + b*x]*(c*Sin[a + b*x]^n)^n^(-1))/b)","A",2,2,14,0.1429,1,"{3208, 2638}"
30,1,79,0,0.0370264,"\int \left(a (b \sin (c+d x))^p\right)^n \, dx","Int[(a*(b*Sin[c + d*x])^p)^n,x]","\frac{\sin (c+d x) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(c+d x)\right) \left(a (b \sin (c+d x))^p\right)^n}{d (n p+1) \sqrt{\cos ^2(c+d x)}}","\frac{\sin (c+d x) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(c+d x)\right) \left(a (b \sin (c+d x))^p\right)^n}{d (n p+1) \sqrt{\cos ^2(c+d x)}}",1,"(Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[c + d*x]^2]*Sin[c + d*x]*(a*(b*Sin[c + d*x])^p)^n)/(d*(1 + n*p)*Sqrt[Cos[c + d*x]^2])","A",2,2,14,0.1429,1,"{3208, 2643}"
31,1,16,0,0.0085589,"\int \left(a-a \sin ^2(x)\right) \, dx","Int[a - a*Sin[x]^2,x]","\frac{a x}{2}+\frac{1}{2} a \sin (x) \cos (x)","\frac{a x}{2}+\frac{1}{2} a \sin (x) \cos (x)",1,"(a*x)/2 + (a*Cos[x]*Sin[x])/2","A",3,2,9,0.2222,1,"{2635, 8}"
32,1,33,0,0.0256267,"\int \left(a-a \sin ^2(x)\right)^2 \, dx","Int[(a - a*Sin[x]^2)^2,x]","\frac{3 a^2 x}{8}+\frac{1}{4} a^2 \sin (x) \cos ^3(x)+\frac{3}{8} a^2 \sin (x) \cos (x)","\frac{3 a^2 x}{8}+\frac{1}{4} a^2 \sin (x) \cos ^3(x)+\frac{3}{8} a^2 \sin (x) \cos (x)",1,"(3*a^2*x)/8 + (3*a^2*Cos[x]*Sin[x])/8 + (a^2*Cos[x]^3*Sin[x])/4","A",4,3,11,0.2727,1,"{3175, 2635, 8}"
33,1,46,0,0.0324165,"\int \left(a-a \sin ^2(x)\right)^3 \, dx","Int[(a - a*Sin[x]^2)^3,x]","\frac{5 a^3 x}{16}+\frac{1}{6} a^3 \sin (x) \cos ^5(x)+\frac{5}{24} a^3 \sin (x) \cos ^3(x)+\frac{5}{16} a^3 \sin (x) \cos (x)","\frac{5 a^3 x}{16}+\frac{1}{6} a^3 \sin (x) \cos ^5(x)+\frac{5}{24} a^3 \sin (x) \cos ^3(x)+\frac{5}{16} a^3 \sin (x) \cos (x)",1,"(5*a^3*x)/16 + (5*a^3*Cos[x]*Sin[x])/16 + (5*a^3*Cos[x]^3*Sin[x])/24 + (a^3*Cos[x]^5*Sin[x])/6","A",5,3,11,0.2727,1,"{3175, 2635, 8}"
34,1,59,0,0.0415468,"\int \left(a-a \sin ^2(x)\right)^4 \, dx","Int[(a - a*Sin[x]^2)^4,x]","\frac{35 a^4 x}{128}+\frac{1}{8} a^4 \sin (x) \cos ^7(x)+\frac{7}{48} a^4 \sin (x) \cos ^5(x)+\frac{35}{192} a^4 \sin (x) \cos ^3(x)+\frac{35}{128} a^4 \sin (x) \cos (x)","\frac{35 a^4 x}{128}+\frac{1}{8} a^4 \sin (x) \cos ^7(x)+\frac{7}{48} a^4 \sin (x) \cos ^5(x)+\frac{35}{192} a^4 \sin (x) \cos ^3(x)+\frac{35}{128} a^4 \sin (x) \cos (x)",1,"(35*a^4*x)/128 + (35*a^4*Cos[x]*Sin[x])/128 + (35*a^4*Cos[x]^3*Sin[x])/192 + (7*a^4*Cos[x]^5*Sin[x])/48 + (a^4*Cos[x]^7*Sin[x])/8","A",6,3,11,0.2727,1,"{3175, 2635, 8}"
35,1,62,0,0.0870735,"\int \frac{\sin ^7(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^7/(a - a*Sin[c + d*x]^2),x]","\frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos ^3(c+d x)}{a d}+\frac{3 \cos (c+d x)}{a d}+\frac{\sec (c+d x)}{a d}","\frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos ^3(c+d x)}{a d}+\frac{3 \cos (c+d x)}{a d}+\frac{\sec (c+d x)}{a d}",1,"(3*Cos[c + d*x])/(a*d) - Cos[c + d*x]^3/(a*d) + Cos[c + d*x]^5/(5*a*d) + Sec[c + d*x]/(a*d)","A",4,3,24,0.1250,1,"{3175, 2590, 270}"
36,1,46,0,0.0802002,"\int \frac{\sin ^5(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^5/(a - a*Sin[c + d*x]^2),x]","-\frac{\cos ^3(c+d x)}{3 a d}+\frac{2 \cos (c+d x)}{a d}+\frac{\sec (c+d x)}{a d}","-\frac{\cos ^3(c+d x)}{3 a d}+\frac{2 \cos (c+d x)}{a d}+\frac{\sec (c+d x)}{a d}",1,"(2*Cos[c + d*x])/(a*d) - Cos[c + d*x]^3/(3*a*d) + Sec[c + d*x]/(a*d)","A",4,3,24,0.1250,1,"{3175, 2590, 270}"
37,1,27,0,0.0654911,"\int \frac{\sin ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^3/(a - a*Sin[c + d*x]^2),x]","\frac{\cos (c+d x)}{a d}+\frac{\sec (c+d x)}{a d}","\frac{\cos (c+d x)}{a d}+\frac{\sec (c+d x)}{a d}",1,"Cos[c + d*x]/(a*d) + Sec[c + d*x]/(a*d)","A",4,3,24,0.1250,1,"{3175, 2590, 14}"
38,1,13,0,0.0338714,"\int \frac{\sin (c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]/(a - a*Sin[c + d*x]^2),x]","\frac{\sec (c+d x)}{a d}","\frac{\sec (c+d x)}{a d}",1,"Sec[c + d*x]/(a*d)","A",3,3,22,0.1364,1,"{3175, 2606, 8}"
39,1,29,0,0.0575448,"\int \frac{\csc (c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]/(a - a*Sin[c + d*x]^2),x]","\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}","\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"-(ArcTanh[Cos[c + d*x]]/(a*d)) + Sec[c + d*x]/(a*d)","A",4,4,22,0.1818,1,"{3175, 2622, 321, 207}"
40,1,58,0,0.0959269,"\int \frac{\csc ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]^3/(a - a*Sin[c + d*x]^2),x]","\frac{3 \sec (c+d x)}{2 a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\csc ^2(c+d x) \sec (c+d x)}{2 a d}","\frac{3 \sec (c+d x)}{2 a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\csc ^2(c+d x) \sec (c+d x)}{2 a d}",1,"(-3*ArcTanh[Cos[c + d*x]])/(2*a*d) + (3*Sec[c + d*x])/(2*a*d) - (Csc[c + d*x]^2*Sec[c + d*x])/(2*a*d)","A",5,5,24,0.2083,1,"{3175, 2622, 288, 321, 207}"
41,1,82,0,0.0945551,"\int \frac{\csc ^5(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]^5/(a - a*Sin[c + d*x]^2),x]","\frac{15 \sec (c+d x)}{8 a d}-\frac{15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\csc ^4(c+d x) \sec (c+d x)}{4 a d}-\frac{5 \csc ^2(c+d x) \sec (c+d x)}{8 a d}","\frac{15 \sec (c+d x)}{8 a d}-\frac{15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\csc ^4(c+d x) \sec (c+d x)}{4 a d}-\frac{5 \csc ^2(c+d x) \sec (c+d x)}{8 a d}",1,"(-15*ArcTanh[Cos[c + d*x]])/(8*a*d) + (15*Sec[c + d*x])/(8*a*d) - (5*Csc[c + d*x]^2*Sec[c + d*x])/(8*a*d) - (Csc[c + d*x]^4*Sec[c + d*x])/(4*a*d)","A",6,5,24,0.2083,1,"{3175, 2622, 288, 321, 207}"
42,1,73,0,0.0902474,"\int \frac{\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^6/(a - a*Sin[c + d*x]^2),x]","\frac{15 \tan (c+d x)}{8 a d}-\frac{\sin ^4(c+d x) \tan (c+d x)}{4 a d}-\frac{5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac{15 x}{8 a}","\frac{15 \tan (c+d x)}{8 a d}-\frac{\sin ^4(c+d x) \tan (c+d x)}{4 a d}-\frac{5 \sin ^2(c+d x) \tan (c+d x)}{8 a d}-\frac{15 x}{8 a}",1,"(-15*x)/(8*a) + (15*Tan[c + d*x])/(8*a*d) - (5*Sin[c + d*x]^2*Tan[c + d*x])/(8*a*d) - (Sin[c + d*x]^4*Tan[c + d*x])/(4*a*d)","A",6,5,24,0.2083,1,"{3175, 2591, 288, 321, 203}"
43,1,49,0,0.0815679,"\int \frac{\sin ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^4/(a - a*Sin[c + d*x]^2),x]","\frac{3 \tan (c+d x)}{2 a d}-\frac{\sin ^2(c+d x) \tan (c+d x)}{2 a d}-\frac{3 x}{2 a}","\frac{3 \tan (c+d x)}{2 a d}-\frac{\sin ^2(c+d x) \tan (c+d x)}{2 a d}-\frac{3 x}{2 a}",1,"(-3*x)/(2*a) + (3*Tan[c + d*x])/(2*a*d) - (Sin[c + d*x]^2*Tan[c + d*x])/(2*a*d)","A",5,5,24,0.2083,1,"{3175, 2591, 288, 321, 203}"
44,1,20,0,0.0631069,"\int \frac{\sin ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^2/(a - a*Sin[c + d*x]^2),x]","\frac{\tan (c+d x)}{a d}-\frac{x}{a}","\frac{\tan (c+d x)}{a d}-\frac{x}{a}",1,"-(x/a) + Tan[c + d*x]/(a*d)","A",4,4,24,0.1667,1,"{3171, 3175, 3767, 8}"
45,1,13,0,0.0224128,"\int \frac{1}{a-a \sin ^2(c+d x)} \, dx","Int[(a - a*Sin[c + d*x]^2)^(-1),x]","\frac{\tan (c+d x)}{a d}","\frac{\tan (c+d x)}{a d}",1,"Tan[c + d*x]/(a*d)","A",3,3,15,0.2000,1,"{3175, 3767, 8}"
46,1,28,0,0.07482,"\int \frac{\csc ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]^2/(a - a*Sin[c + d*x]^2),x]","\frac{\tan (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}","\frac{\tan (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}",1,"-(Cot[c + d*x]/(a*d)) + Tan[c + d*x]/(a*d)","A",4,3,24,0.1250,1,"{3175, 2620, 14}"
47,1,46,0,0.0801195,"\int \frac{\csc ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]^4/(a - a*Sin[c + d*x]^2),x]","\frac{\tan (c+d x)}{a d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{2 \cot (c+d x)}{a d}","\frac{\tan (c+d x)}{a d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{2 \cot (c+d x)}{a d}",1,"(-2*Cot[c + d*x])/(a*d) - Cot[c + d*x]^3/(3*a*d) + Tan[c + d*x]/(a*d)","A",4,3,24,0.1250,1,"{3175, 2620, 270}"
48,1,62,0,0.0844312,"\int \frac{\csc ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]^6/(a - a*Sin[c + d*x]^2),x]","\frac{\tan (c+d x)}{a d}-\frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{a d}-\frac{3 \cot (c+d x)}{a d}","\frac{\tan (c+d x)}{a d}-\frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{a d}-\frac{3 \cot (c+d x)}{a d}",1,"(-3*Cot[c + d*x])/(a*d) - Cot[c + d*x]^3/(a*d) - Cot[c + d*x]^5/(5*a*d) + Tan[c + d*x]/(a*d)","A",4,3,24,0.1250,1,"{3175, 2620, 270}"
49,1,65,0,0.0837538,"\int \frac{\sin ^7(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^7/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\cos ^3(c+d x)}{3 a^2 d}-\frac{3 \cos (c+d x)}{a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{3 \sec (c+d x)}{a^2 d}","\frac{\cos ^3(c+d x)}{3 a^2 d}-\frac{3 \cos (c+d x)}{a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{3 \sec (c+d x)}{a^2 d}",1,"(-3*Cos[c + d*x])/(a^2*d) + Cos[c + d*x]^3/(3*a^2*d) - (3*Sec[c + d*x])/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d)","A",4,3,24,0.1250,1,"{3175, 2590, 270}"
50,1,47,0,0.0694904,"\int \frac{\sin ^5(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^5/(a - a*Sin[c + d*x]^2)^2,x]","-\frac{\cos (c+d x)}{a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}","-\frac{\cos (c+d x)}{a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}",1,"-(Cos[c + d*x]/(a^2*d)) - (2*Sec[c + d*x])/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d)","A",4,3,24,0.1250,1,"{3175, 2590, 270}"
51,1,33,0,0.0615918,"\int \frac{\sin ^3(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^3/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{\sec (c+d x)}{a^2 d}","\frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{\sec (c+d x)}{a^2 d}",1,"-(Sec[c + d*x]/(a^2*d)) + Sec[c + d*x]^3/(3*a^2*d)","A",3,2,24,0.08333,1,"{3175, 2606}"
52,1,18,0,0.0421728,"\int \frac{\sin (c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\sec ^3(c+d x)}{3 a^2 d}","\frac{\sec ^3(c+d x)}{3 a^2 d}",1,"Sec[c + d*x]^3/(3*a^2*d)","A",3,3,22,0.1364,1,"{3175, 2606, 30}"
53,1,47,0,0.062176,"\int \frac{\csc (c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[Csc[c + d*x]/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\sec ^3(c+d x)}{3 a^2 d}+\frac{\sec (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}","\frac{\sec ^3(c+d x)}{3 a^2 d}+\frac{\sec (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}",1,"-(ArcTanh[Cos[c + d*x]]/(a^2*d)) + Sec[c + d*x]/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d)","A",5,4,22,0.1818,1,"{3175, 2622, 302, 207}"
54,1,78,0,0.0847661,"\int \frac{\csc ^3(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[Csc[c + d*x]^3/(a - a*Sin[c + d*x]^2)^2,x]","\frac{5 \sec ^3(c+d x)}{6 a^2 d}+\frac{5 \sec (c+d x)}{2 a^2 d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\csc ^2(c+d x) \sec ^3(c+d x)}{2 a^2 d}","\frac{5 \sec ^3(c+d x)}{6 a^2 d}+\frac{5 \sec (c+d x)}{2 a^2 d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\csc ^2(c+d x) \sec ^3(c+d x)}{2 a^2 d}",1,"(-5*ArcTanh[Cos[c + d*x]])/(2*a^2*d) + (5*Sec[c + d*x])/(2*a^2*d) + (5*Sec[c + d*x]^3)/(6*a^2*d) - (Csc[c + d*x]^2*Sec[c + d*x]^3)/(2*a^2*d)","A",6,5,24,0.2083,1,"{3175, 2622, 288, 302, 207}"
55,1,69,0,0.0818862,"\int \frac{\sin ^6(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^6/(a - a*Sin[c + d*x]^2)^2,x]","\frac{5 \tan ^3(c+d x)}{6 a^2 d}-\frac{5 \tan (c+d x)}{2 a^2 d}-\frac{\sin ^2(c+d x) \tan ^3(c+d x)}{2 a^2 d}+\frac{5 x}{2 a^2}","\frac{5 \tan ^3(c+d x)}{6 a^2 d}-\frac{5 \tan (c+d x)}{2 a^2 d}-\frac{\sin ^2(c+d x) \tan ^3(c+d x)}{2 a^2 d}+\frac{5 x}{2 a^2}",1,"(5*x)/(2*a^2) - (5*Tan[c + d*x])/(2*a^2*d) + (5*Tan[c + d*x]^3)/(6*a^2*d) - (Sin[c + d*x]^2*Tan[c + d*x]^3)/(2*a^2*d)","A",6,5,24,0.2083,1,"{3175, 2591, 288, 302, 203}"
56,1,38,0,0.0556989,"\int \frac{\sin ^4(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^4/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\tan ^3(c+d x)}{3 a^2 d}-\frac{\tan (c+d x)}{a^2 d}+\frac{x}{a^2}","\frac{\tan ^3(c+d x)}{3 a^2 d}-\frac{\tan (c+d x)}{a^2 d}+\frac{x}{a^2}",1,"x/a^2 - Tan[c + d*x]/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)","A",4,3,24,0.1250,1,"{3175, 3473, 8}"
57,1,18,0,0.0674126,"\int \frac{\sin ^2(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^2/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\tan ^3(c+d x)}{3 a^2 d}","\frac{\tan ^3(c+d x)}{3 a^2 d}",1,"Tan[c + d*x]^3/(3*a^2*d)","A",3,3,24,0.1250,1,"{3175, 2607, 30}"
58,1,32,0,0.0245773,"\int \frac{1}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[(a - a*Sin[c + d*x]^2)^(-2),x]","\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{\tan (c+d x)}{a^2 d}","\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{\tan (c+d x)}{a^2 d}",1,"Tan[c + d*x]/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)","A",3,2,15,0.1333,1,"{3175, 3767}"
59,1,47,0,0.0781224,"\int \frac{\csc ^2(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[Csc[c + d*x]^2/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{2 \tan (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}","\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{2 \tan (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}",1,"-(Cot[c + d*x]/(a^2*d)) + (2*Tan[c + d*x])/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)","A",4,3,24,0.1250,1,"{3175, 2620, 270}"
60,1,65,0,0.0808019,"\int \frac{\csc ^4(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Int[Csc[c + d*x]^4/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{3 \tan (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{3 \cot (c+d x)}{a^2 d}","\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{3 \tan (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{3 \cot (c+d x)}{a^2 d}",1,"(-3*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + (3*Tan[c + d*x])/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)","A",4,3,24,0.1250,1,"{3175, 2620, 270}"
61,1,29,0,0.0208021,"\int \frac{1}{\left(a-a \sin ^2(x)\right)^3} \, dx","Int[(a - a*Sin[x]^2)^(-3),x]","\frac{\tan ^5(x)}{5 a^3}+\frac{2 \tan ^3(x)}{3 a^3}+\frac{\tan (x)}{a^3}","\frac{\tan ^5(x)}{5 a^3}+\frac{2 \tan ^3(x)}{3 a^3}+\frac{\tan (x)}{a^3}",1,"Tan[x]/a^3 + (2*Tan[x]^3)/(3*a^3) + Tan[x]^5/(5*a^3)","A",3,2,11,0.1818,1,"{3175, 3767}"
62,1,37,0,0.02165,"\int \frac{1}{\left(a-a \sin ^2(x)\right)^4} \, dx","Int[(a - a*Sin[x]^2)^(-4),x]","\frac{\tan ^7(x)}{7 a^4}+\frac{3 \tan ^5(x)}{5 a^4}+\frac{\tan ^3(x)}{a^4}+\frac{\tan (x)}{a^4}","\frac{\tan ^7(x)}{7 a^4}+\frac{3 \tan ^5(x)}{5 a^4}+\frac{\tan ^3(x)}{a^4}+\frac{\tan (x)}{a^4}",1,"Tan[x]/a^4 + Tan[x]^3/a^4 + (3*Tan[x]^5)/(5*a^4) + Tan[x]^7/(7*a^4)","A",3,2,11,0.1818,1,"{3175, 3767}"
63,1,51,0,0.0263289,"\int \frac{1}{\left(a-a \sin ^2(x)\right)^5} \, dx","Int[(a - a*Sin[x]^2)^(-5),x]","\frac{\tan ^9(x)}{9 a^5}+\frac{4 \tan ^7(x)}{7 a^5}+\frac{6 \tan ^5(x)}{5 a^5}+\frac{4 \tan ^3(x)}{3 a^5}+\frac{\tan (x)}{a^5}","\frac{\tan ^9(x)}{9 a^5}+\frac{4 \tan ^7(x)}{7 a^5}+\frac{6 \tan ^5(x)}{5 a^5}+\frac{4 \tan ^3(x)}{3 a^5}+\frac{\tan (x)}{a^5}",1,"Tan[x]/a^5 + (4*Tan[x]^3)/(3*a^5) + (6*Tan[x]^5)/(5*a^5) + (4*Tan[x]^7)/(7*a^5) + Tan[x]^9/(9*a^5)","A",3,2,11,0.1818,1,"{3175, 3767}"
64,1,51,0,0.0451092,"\int \sin ^3(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Int[Sin[c + d*x]^3*(a + b*Sin[c + d*x]^2),x]","\frac{(a+2 b) \cos ^3(c+d x)}{3 d}-\frac{(a+b) \cos (c+d x)}{d}-\frac{b \cos ^5(c+d x)}{5 d}","\frac{(a+2 b) \cos ^3(c+d x)}{3 d}-\frac{(a+b) \cos (c+d x)}{d}-\frac{b \cos ^5(c+d x)}{5 d}",1,"-(((a + b)*Cos[c + d*x])/d) + ((a + 2*b)*Cos[c + d*x]^3)/(3*d) - (b*Cos[c + d*x]^5)/(5*d)","A",3,2,21,0.09524,1,"{3013, 373}"
65,1,31,0,0.0224598,"\int \sin (c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Int[Sin[c + d*x]*(a + b*Sin[c + d*x]^2),x]","\frac{b \cos ^3(c+d x)}{3 d}-\frac{(a+b) \cos (c+d x)}{d}","\frac{b \cos ^3(c+d x)}{3 d}-\frac{(a+b) \cos (c+d x)}{d}",1,"-(((a + b)*Cos[c + d*x])/d) + (b*Cos[c + d*x]^3)/(3*d)","A",2,1,19,0.05263,1,"{3013}"
66,1,26,0,0.0248953,"\int \csc (c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Int[Csc[c + d*x]*(a + b*Sin[c + d*x]^2),x]","-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b \cos (c+d x)}{d}","-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b \cos (c+d x)}{d}",1,"-((a*ArcTanh[Cos[c + d*x]])/d) - (b*Cos[c + d*x])/d","A",2,2,19,0.1053,1,"{3014, 3770}"
67,1,40,0,0.0308706,"\int \csc ^3(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Int[Csc[c + d*x]^3*(a + b*Sin[c + d*x]^2),x]","-\frac{(a+2 b) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}","-\frac{(a+2 b) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}",1,"-((a + 2*b)*ArcTanh[Cos[c + d*x]])/(2*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)","A",2,2,21,0.09524,1,"{3012, 3770}"
68,1,89,0,0.0528693,"\int \sin ^4(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Int[Sin[c + d*x]^4*(a + b*Sin[c + d*x]^2),x]","-\frac{(6 a+5 b) \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{(6 a+5 b) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x (6 a+5 b)-\frac{b \sin ^5(c+d x) \cos (c+d x)}{6 d}","-\frac{(6 a+5 b) \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{(6 a+5 b) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x (6 a+5 b)-\frac{b \sin ^5(c+d x) \cos (c+d x)}{6 d}",1,"((6*a + 5*b)*x)/16 - ((6*a + 5*b)*Cos[c + d*x]*Sin[c + d*x])/(16*d) - ((6*a + 5*b)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) - (b*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)","A",4,3,21,0.1429,1,"{3014, 2635, 8}"
69,1,61,0,0.0404287,"\int \sin ^2(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Int[Sin[c + d*x]^2*(a + b*Sin[c + d*x]^2),x]","-\frac{(4 a+3 b) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 a+3 b)-\frac{b \sin ^3(c+d x) \cos (c+d x)}{4 d}","-\frac{(4 a+3 b) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 a+3 b)-\frac{b \sin ^3(c+d x) \cos (c+d x)}{4 d}",1,"((4*a + 3*b)*x)/8 - ((4*a + 3*b)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",3,3,21,0.1429,1,"{3014, 2635, 8}"
70,1,30,0,0.0148417,"\int \left(a+b \sin ^2(c+d x)\right) \, dx","Int[a + b*Sin[c + d*x]^2,x]","a x-\frac{b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b x}{2}","a x-\frac{b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b x}{2}",1,"a*x + (b*x)/2 - (b*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",3,2,12,0.1667,1,"{2635, 8}"
71,1,16,0,0.0232825,"\int \csc ^2(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Int[Csc[c + d*x]^2*(a + b*Sin[c + d*x]^2),x]","b x-\frac{a \cot (c+d x)}{d}","b x-\frac{a \cot (c+d x)}{d}",1,"b*x - (a*Cot[c + d*x])/d","A",2,2,21,0.09524,1,"{3012, 8}"
72,1,43,0,0.0357116,"\int \csc ^4(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Int[Csc[c + d*x]^4*(a + b*Sin[c + d*x]^2),x]","-\frac{(2 a+3 b) \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}","-\frac{(2 a+3 b) \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}",1,"-((2*a + 3*b)*Cot[c + d*x])/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d)","A",3,3,21,0.1429,1,"{3012, 3767, 8}"
73,1,65,0,0.042355,"\int \csc ^6(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Int[Csc[c + d*x]^6*(a + b*Sin[c + d*x]^2),x]","-\frac{(4 a+5 b) \cot ^3(c+d x)}{15 d}-\frac{(4 a+5 b) \cot (c+d x)}{5 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}","-\frac{(4 a+5 b) \cot ^3(c+d x)}{15 d}-\frac{(4 a+5 b) \cot (c+d x)}{5 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}",1,"-((4*a + 5*b)*Cot[c + d*x])/(5*d) - ((4*a + 5*b)*Cot[c + d*x]^3)/(15*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d)","A",3,2,21,0.09524,1,"{3012, 3767}"
74,1,19,0,0.0092947,"\int \left(a+b \sin ^2(x)\right) \, dx","Int[a + b*Sin[x]^2,x]","a x+\frac{b x}{2}-\frac{1}{2} b \sin (x) \cos (x)","a x+\frac{b x}{2}-\frac{1}{2} b \sin (x) \cos (x)",1,"a*x + (b*x)/2 - (b*Cos[x]*Sin[x])/2","A",3,2,8,0.2500,1,"{2635, 8}"
75,1,50,0,0.0157437,"\int \left(a+b \sin ^2(x)\right)^2 \, dx","Int[(a + b*Sin[x]^2)^2,x]","\frac{1}{8} x \left(8 a^2+8 a b+3 b^2\right)-\frac{1}{8} b (8 a+3 b) \sin (x) \cos (x)-\frac{1}{4} b^2 \sin ^3(x) \cos (x)","\frac{1}{8} x \left(8 a^2+8 a b+3 b^2\right)-\frac{1}{8} b (8 a+3 b) \sin (x) \cos (x)-\frac{1}{4} b^2 \sin ^3(x) \cos (x)",1,"((8*a^2 + 8*a*b + 3*b^2)*x)/8 - (b*(8*a + 3*b)*Cos[x]*Sin[x])/8 - (b^2*Cos[x]*Sin[x]^3)/4","A",1,1,10,0.1000,1,"{3179}"
76,1,87,0,0.0830855,"\int \left(a+b \sin ^2(x)\right)^3 \, dx","Int[(a + b*Sin[x]^2)^3,x]","\frac{1}{16} x (2 a+b) \left(8 a^2+8 a b+5 b^2\right)-\frac{1}{48} b \left(64 a^2+54 a b+15 b^2\right) \sin (x) \cos (x)-\frac{5}{24} b^2 (2 a+b) \sin ^3(x) \cos (x)-\frac{1}{6} b \sin (x) \cos (x) \left(a+b \sin ^2(x)\right)^2","\frac{1}{16} x (2 a+b) \left(8 a^2+8 a b+5 b^2\right)-\frac{1}{48} b \left(64 a^2+54 a b+15 b^2\right) \sin (x) \cos (x)-\frac{5}{24} b^2 (2 a+b) \sin ^3(x) \cos (x)-\frac{1}{6} b \sin (x) \cos (x) \left(a+b \sin ^2(x)\right)^2",1,"((2*a + b)*(8*a^2 + 8*a*b + 5*b^2)*x)/16 - (b*(64*a^2 + 54*a*b + 15*b^2)*Cos[x]*Sin[x])/48 - (5*b^2*(2*a + b)*Cos[x]*Sin[x]^3)/24 - (b*Cos[x]*Sin[x]*(a + b*Sin[x]^2)^2)/6","A",2,2,10,0.2000,1,"{3180, 3169}"
77,1,140,0,0.1668942,"\int \left(a+b \sin ^2(x)\right)^4 \, dx","Int[(a + b*Sin[x]^2)^4,x]","\frac{1}{128} x \left(288 a^2 b^2+256 a^3 b+128 a^4+160 a b^3+35 b^4\right)-\frac{1}{192} b^2 \left(104 a^2+104 a b+35 b^2\right) \sin ^3(x) \cos (x)-\frac{1}{384} b \left(808 a^2 b+608 a^3+480 a b^2+105 b^3\right) \sin (x) \cos (x)-\frac{1}{8} b \sin (x) \cos (x) \left(a+b \sin ^2(x)\right)^3-\frac{7}{48} b (2 a+b) \sin (x) \cos (x) \left(a+b \sin ^2(x)\right)^2","\frac{1}{128} x \left(288 a^2 b^2+256 a^3 b+128 a^4+160 a b^3+35 b^4\right)-\frac{1}{192} b^2 \left(104 a^2+104 a b+35 b^2\right) \sin ^3(x) \cos (x)-\frac{1}{384} b \left(808 a^2 b+608 a^3+480 a b^2+105 b^3\right) \sin (x) \cos (x)-\frac{1}{8} b \sin (x) \cos (x) \left(a+b \sin ^2(x)\right)^3-\frac{7}{48} b (2 a+b) \sin (x) \cos (x) \left(a+b \sin ^2(x)\right)^2",1,"((128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*x)/128 - (b*(608*a^3 + 808*a^2*b + 480*a*b^2 + 105*b^3)*Cos[x]*Sin[x])/384 - (b^2*(104*a^2 + 104*a*b + 35*b^2)*Cos[x]*Sin[x]^3)/192 - (7*b*(2*a + b)*Cos[x]*Sin[x]*(a + b*Sin[x]^2)^2)/48 - (b*Cos[x]*Sin[x]*(a + b*Sin[x]^2)^3)/8","A",3,3,10,0.3000,1,"{3180, 3170, 3169}"
78,1,106,0,0.1126098,"\int \frac{\sin ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^7/(a + b*Sin[c + d*x]^2),x]","-\frac{\left(a^2-a b+b^2\right) \cos (c+d x)}{b^3 d}+\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{b^{7/2} d \sqrt{a+b}}-\frac{(a-2 b) \cos ^3(c+d x)}{3 b^2 d}-\frac{\cos ^5(c+d x)}{5 b d}","-\frac{\left(a^2-a b+b^2\right) \cos (c+d x)}{b^3 d}+\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{b^{7/2} d \sqrt{a+b}}-\frac{(a-2 b) \cos ^3(c+d x)}{3 b^2 d}-\frac{\cos ^5(c+d x)}{5 b d}",1,"(a^3*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(b^(7/2)*Sqrt[a + b]*d) - ((a^2 - a*b + b^2)*Cos[c + d*x])/(b^3*d) - ((a - 2*b)*Cos[c + d*x]^3)/(3*b^2*d) - Cos[c + d*x]^5/(5*b*d)","A",4,3,23,0.1304,1,"{3186, 390, 208}"
79,1,77,0,0.0916907,"\int \frac{\sin ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^5/(a + b*Sin[c + d*x]^2),x]","-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{b^{5/2} d \sqrt{a+b}}+\frac{(a-b) \cos (c+d x)}{b^2 d}+\frac{\cos ^3(c+d x)}{3 b d}","-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{b^{5/2} d \sqrt{a+b}}+\frac{(a-b) \cos (c+d x)}{b^2 d}+\frac{\cos ^3(c+d x)}{3 b d}",1,"-((a^2*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(b^(5/2)*Sqrt[a + b]*d)) + ((a - b)*Cos[c + d*x])/(b^2*d) + Cos[c + d*x]^3/(3*b*d)","A",4,3,23,0.1304,1,"{3186, 390, 208}"
80,1,52,0,0.0696408,"\int \frac{\sin ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^3/(a + b*Sin[c + d*x]^2),x]","\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{b^{3/2} d \sqrt{a+b}}-\frac{\cos (c+d x)}{b d}","\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{b^{3/2} d \sqrt{a+b}}-\frac{\cos (c+d x)}{b d}",1,"(a*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b]*d) - Cos[c + d*x]/(b*d)","A",3,3,23,0.1304,1,"{3186, 388, 208}"
81,1,37,0,0.0395803,"\int \frac{\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]/(a + b*Sin[c + d*x]^2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{b} d \sqrt{a+b}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{b} d \sqrt{a+b}}",1,"-(ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b]*d))","A",2,2,21,0.09524,1,"{3186, 208}"
82,1,55,0,0.0638404,"\int \frac{\csc (c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]/(a + b*Sin[c + d*x]^2),x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{a d \sqrt{a+b}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{a d \sqrt{a+b}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"-(ArcTanh[Cos[c + d*x]]/(a*d)) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]*d)","A",4,4,21,0.1905,1,"{3186, 391, 206, 208}"
83,1,85,0,0.1160977,"\int \frac{\csc ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]^3/(a + b*Sin[c + d*x]^2),x]","-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a+b}}-\frac{(a-2 b) \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}","-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a+b}}-\frac{(a-2 b) \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}",1,"-((a - 2*b)*ArcTanh[Cos[c + d*x]])/(2*a^2*d) - (b^(3/2)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(a^2*Sqrt[a + b]*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)","A",5,5,23,0.2174,1,"{3186, 414, 522, 206, 208}"
84,1,125,0,0.1880419,"\int \frac{\csc ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]^5/(a + b*Sin[c + d*x]^2),x]","-\frac{\left(3 a^2-4 a b+8 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a+b}}-\frac{(3 a-4 b) \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}","-\frac{\left(3 a^2-4 a b+8 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a+b}}-\frac{(3 a-4 b) \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}",1,"-((3*a^2 - 4*a*b + 8*b^2)*ArcTanh[Cos[c + d*x]])/(8*a^3*d) + (b^(5/2)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(a^3*Sqrt[a + b]*d) - ((3*a - 4*b)*Cot[c + d*x]*Csc[c + d*x])/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)","A",6,6,23,0.2609,1,"{3186, 414, 527, 522, 206, 208}"
85,1,163,0,0.3647345,"\int \frac{\sin ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^8/(a + b*Sin[c + d*x]^2),x]","\frac{a^{7/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{b^4 d \sqrt{a+b}}-\frac{\left(8 a^2-6 a b+5 b^2\right) \sin (c+d x) \cos (c+d x)}{16 b^3 d}-\frac{x \left(-8 a^2 b+16 a^3+6 a b^2-5 b^3\right)}{16 b^4}+\frac{(6 a-5 b) \sin ^3(c+d x) \cos (c+d x)}{24 b^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x)}{6 b d}","\frac{a^{7/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{b^4 d \sqrt{a+b}}-\frac{\left(8 a^2-6 a b+5 b^2\right) \sin (c+d x) \cos (c+d x)}{16 b^3 d}-\frac{x \left(-8 a^2 b+16 a^3+6 a b^2-5 b^3\right)}{16 b^4}+\frac{(6 a-5 b) \sin ^3(c+d x) \cos (c+d x)}{24 b^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x)}{6 b d}",1,"-((16*a^3 - 8*a^2*b + 6*a*b^2 - 5*b^3)*x)/(16*b^4) + (a^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(b^4*Sqrt[a + b]*d) - ((8*a^2 - 6*a*b + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*b^3*d) + ((6*a - 5*b)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^5)/(6*b*d)","A",7,6,23,0.2609,1,"{3187, 470, 578, 522, 203, 205}"
86,1,117,0,0.2236807,"\int \frac{\sin ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^6/(a + b*Sin[c + d*x]^2),x]","-\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{b^3 d \sqrt{a+b}}+\frac{x \left(8 a^2-4 a b+3 b^2\right)}{8 b^3}+\frac{(4 a-3 b) \sin (c+d x) \cos (c+d x)}{8 b^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d}","-\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{b^3 d \sqrt{a+b}}+\frac{x \left(8 a^2-4 a b+3 b^2\right)}{8 b^3}+\frac{(4 a-3 b) \sin (c+d x) \cos (c+d x)}{8 b^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d}",1,"((8*a^2 - 4*a*b + 3*b^2)*x)/(8*b^3) - (a^(5/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(b^3*Sqrt[a + b]*d) + ((4*a - 3*b)*Cos[c + d*x]*Sin[c + d*x])/(8*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*b*d)","A",6,6,23,0.2609,1,"{3187, 470, 578, 522, 203, 205}"
87,1,77,0,0.1140907,"\int \frac{\sin ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^4/(a + b*Sin[c + d*x]^2),x]","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{b^2 d \sqrt{a+b}}-\frac{x (2 a-b)}{2 b^2}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{b^2 d \sqrt{a+b}}-\frac{x (2 a-b)}{2 b^2}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}",1,"-((2*a - b)*x)/(2*b^2) + (a^(3/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(b^2*Sqrt[a + b]*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",5,5,23,0.2174,1,"{3187, 470, 522, 203, 205}"
88,1,46,0,0.0747304,"\int \frac{\sin ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Sin[c + d*x]^2/(a + b*Sin[c + d*x]^2),x]","\frac{x}{b}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{b d \sqrt{a+b}}","\frac{x}{b}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{b d \sqrt{a+b}}",1,"x/b - (Sqrt[a]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(b*Sqrt[a + b]*d)","A",3,3,23,0.1304,1,"{3171, 3181, 205}"
89,1,36,0,0.0239375,"\int \frac{1}{a+b \sin ^2(c+d x)} \, dx","Int[(a + b*Sin[c + d*x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d \sqrt{a+b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d \sqrt{a+b}}",1,"ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a + b]*d)","A",2,2,14,0.1429,1,"{3181, 205}"
90,1,53,0,0.073055,"\int \frac{\csc ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]^2/(a + b*Sin[c + d*x]^2),x]","-\frac{b \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{3/2} d \sqrt{a+b}}-\frac{\cot (c+d x)}{a d}","-\frac{b \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{3/2} d \sqrt{a+b}}-\frac{\cot (c+d x)}{a d}",1,"-((b*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(3/2)*Sqrt[a + b]*d)) - Cot[c + d*x]/(a*d)","A",3,3,23,0.1304,1,"{3187, 453, 205}"
91,1,77,0,0.1056931,"\int \frac{\csc ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]^4/(a + b*Sin[c + d*x]^2),x]","\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{5/2} d \sqrt{a+b}}-\frac{(a-b) \cot (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}","\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{5/2} d \sqrt{a+b}}-\frac{(a-b) \cot (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}",1,"(b^2*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(5/2)*Sqrt[a + b]*d) - ((a - b)*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a*d)","A",4,3,23,0.1304,1,"{3187, 461, 205}"
92,1,109,0,0.1247966,"\int \frac{\csc ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]^6/(a + b*Sin[c + d*x]^2),x]","-\frac{b^3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{7/2} d \sqrt{a+b}}-\frac{\left(a^2-a b+b^2\right) \cot (c+d x)}{a^3 d}-\frac{(2 a-b) \cot ^3(c+d x)}{3 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}","-\frac{b^3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{7/2} d \sqrt{a+b}}-\frac{\left(a^2-a b+b^2\right) \cot (c+d x)}{a^3 d}-\frac{(2 a-b) \cot ^3(c+d x)}{3 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}",1,"-((b^3*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(7/2)*Sqrt[a + b]*d)) - ((a^2 - a*b + b^2)*Cot[c + d*x])/(a^3*d) - ((2*a - b)*Cot[c + d*x]^3)/(3*a^2*d) - Cot[c + d*x]^5/(5*a*d)","A",4,3,23,0.1304,1,"{3187, 461, 205}"
93,1,140,0,0.1555282,"\int \frac{\csc ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Csc[c + d*x]^8/(a + b*Sin[c + d*x]^2),x]","\frac{b^4 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{9/2} d \sqrt{a+b}}-\frac{\left(3 a^2-2 a b+b^2\right) \cot ^3(c+d x)}{3 a^3 d}-\frac{(a-b) \left(a^2+b^2\right) \cot (c+d x)}{a^4 d}-\frac{(3 a-b) \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^7(c+d x)}{7 a d}","\frac{b^4 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{9/2} d \sqrt{a+b}}-\frac{\left(3 a^2-2 a b+b^2\right) \cot ^3(c+d x)}{3 a^3 d}-\frac{(a-b) \left(a^2+b^2\right) \cot (c+d x)}{a^4 d}-\frac{(3 a-b) \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^7(c+d x)}{7 a d}",1,"(b^4*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(9/2)*Sqrt[a + b]*d) - ((a - b)*(a^2 + b^2)*Cot[c + d*x])/(a^4*d) - ((3*a^2 - 2*a*b + b^2)*Cot[c + d*x]^3)/(3*a^3*d) - ((3*a - b)*Cot[c + d*x]^5)/(5*a^2*d) - Cot[c + d*x]^7/(7*a*d)","A",4,3,23,0.1304,1,"{3187, 461, 205}"
94,1,128,0,0.1861279,"\int \frac{\sin ^7(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^7/(a + b*Sin[c + d*x]^2)^2,x]","\frac{a^3 \cos (c+d x)}{2 b^3 d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}-\frac{a^2 (5 a+6 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 b^{7/2} d (a+b)^{3/2}}+\frac{(2 a-b) \cos (c+d x)}{b^3 d}+\frac{\cos ^3(c+d x)}{3 b^2 d}","\frac{a^3 \cos (c+d x)}{2 b^3 d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}-\frac{a^2 (5 a+6 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 b^{7/2} d (a+b)^{3/2}}+\frac{(2 a-b) \cos (c+d x)}{b^3 d}+\frac{\cos ^3(c+d x)}{3 b^2 d}",1,"-(a^2*(5*a + 6*b)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(2*b^(7/2)*(a + b)^(3/2)*d) + ((2*a - b)*Cos[c + d*x])/(b^3*d) + Cos[c + d*x]^3/(3*b^2*d) + (a^3*Cos[c + d*x])/(2*b^3*(a + b)*d*(a + b - b*Cos[c + d*x]^2))","A",5,4,23,0.1739,1,"{3186, 390, 385, 208}"
95,1,102,0,0.1456859,"\int \frac{\sin ^5(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^5/(a + b*Sin[c + d*x]^2)^2,x]","-\frac{a^2 \cos (c+d x)}{2 b^2 d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}+\frac{a (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 b^{5/2} d (a+b)^{3/2}}-\frac{\cos (c+d x)}{b^2 d}","-\frac{a^2 \cos (c+d x)}{2 b^2 d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}+\frac{a (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 b^{5/2} d (a+b)^{3/2}}-\frac{\cos (c+d x)}{b^2 d}",1,"(a*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(2*b^(5/2)*(a + b)^(3/2)*d) - Cos[c + d*x]/(b^2*d) - (a^2*Cos[c + d*x])/(2*b^2*(a + b)*d*(a + b - b*Cos[c + d*x]^2))","A",5,4,23,0.1739,1,"{3186, 390, 385, 208}"
96,1,83,0,0.0885921,"\int \frac{\sin ^3(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^3/(a + b*Sin[c + d*x]^2)^2,x]","\frac{a \cos (c+d x)}{2 b d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 b^{3/2} d (a+b)^{3/2}}","\frac{a \cos (c+d x)}{2 b d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 b^{3/2} d (a+b)^{3/2}}",1,"-((a + 2*b)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(2*b^(3/2)*(a + b)^(3/2)*d) + (a*Cos[c + d*x])/(2*b*(a + b)*d*(a + b - b*Cos[c + d*x]^2))","A",3,3,23,0.1304,1,"{3186, 385, 208}"
97,1,74,0,0.0539592,"\int \frac{\sin (c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]/(a + b*Sin[c + d*x]^2)^2,x]","-\frac{\cos (c+d x)}{2 d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 \sqrt{b} d (a+b)^{3/2}}","-\frac{\cos (c+d x)}{2 d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 \sqrt{b} d (a+b)^{3/2}}",1,"-ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]]/(2*Sqrt[b]*(a + b)^(3/2)*d) - Cos[c + d*x]/(2*(a + b)*d*(a + b - b*Cos[c + d*x]^2))","A",3,3,21,0.1429,1,"{3186, 199, 208}"
98,1,103,0,0.1291065,"\int \frac{\csc (c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[Csc[c + d*x]/(a + b*Sin[c + d*x]^2)^2,x]","\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 a^2 d (a+b)^{3/2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{b \cos (c+d x)}{2 a d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}","\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 a^2 d (a+b)^{3/2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{b \cos (c+d x)}{2 a d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}",1,"-(ArcTanh[Cos[c + d*x]]/(a^2*d)) + (Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(3/2)*d) + (b*Cos[c + d*x])/(2*a*(a + b)*d*(a + b - b*Cos[c + d*x]^2))","A",5,5,21,0.2381,1,"{3186, 414, 522, 206, 208}"
99,1,153,0,0.2433391,"\int \frac{\csc ^3(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[Csc[c + d*x]^3/(a + b*Sin[c + d*x]^2)^2,x]","-\frac{b^{3/2} (5 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 a^3 d (a+b)^{3/2}}-\frac{b (a+2 b) \cos (c+d x)}{2 a^2 d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}-\frac{(a-4 b) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d \left(a-b \cos ^2(c+d x)+b\right)}","-\frac{b^{3/2} (5 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right)}{2 a^3 d (a+b)^{3/2}}-\frac{b (a+2 b) \cos (c+d x)}{2 a^2 d (a+b) \left(a-b \cos ^2(c+d x)+b\right)}-\frac{(a-4 b) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d \left(a-b \cos ^2(c+d x)+b\right)}",1,"-((a - 4*b)*ArcTanh[Cos[c + d*x]])/(2*a^3*d) - (b^(3/2)*(5*a + 4*b)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(2*a^3*(a + b)^(3/2)*d) - (b*(a + 2*b)*Cos[c + d*x])/(2*a^2*(a + b)*d*(a + b - b*Cos[c + d*x]^2)) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b - b*Cos[c + d*x]^2))","A",6,6,23,0.2609,1,"{3186, 414, 527, 522, 206, 208}"
100,1,148,0,0.2916965,"\int \frac{\sin ^6(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^6/(a + b*Sin[c + d*x]^2)^2,x]","\frac{a^{3/2} (4 a+5 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 b^3 d (a+b)^{3/2}}-\frac{a (2 a+b) \tan (c+d x)}{2 b^2 d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)}-\frac{x (4 a-b)}{2 b^3}-\frac{\sin ^2(c+d x) \tan (c+d x)}{2 b d \left((a+b) \tan ^2(c+d x)+a\right)}","\frac{a^{3/2} (4 a+5 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 b^3 d (a+b)^{3/2}}-\frac{a (2 a+b) \tan (c+d x)}{2 b^2 d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)}-\frac{x (4 a-b)}{2 b^3}-\frac{\sin ^2(c+d x) \tan (c+d x)}{2 b d \left((a+b) \tan ^2(c+d x)+a\right)}",1,"-((4*a - b)*x)/(2*b^3) + (a^(3/2)*(4*a + 5*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*b^3*(a + b)^(3/2)*d) - (a*(2*a + b)*Tan[c + d*x])/(2*b^2*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2)) - (Sin[c + d*x]^2*Tan[c + d*x])/(2*b*d*(a + (a + b)*Tan[c + d*x]^2))","A",6,6,23,0.2609,1,"{3187, 470, 578, 522, 203, 205}"
101,1,93,0,0.1376444,"\int \frac{\sin ^4(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^4/(a + b*Sin[c + d*x]^2)^2,x]","-\frac{\sqrt{a} (2 a+3 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 b^2 d (a+b)^{3/2}}+\frac{a \tan (c+d x)}{2 b d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)}+\frac{x}{b^2}","-\frac{\sqrt{a} (2 a+3 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 b^2 d (a+b)^{3/2}}+\frac{a \tan (c+d x)}{2 b d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)}+\frac{x}{b^2}",1,"x/b^2 - (Sqrt[a]*(2*a + 3*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*b^2*(a + b)^(3/2)*d) + (a*Tan[c + d*x])/(2*b*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2))","A",5,5,23,0.2174,1,"{3187, 470, 522, 203, 205}"
102,1,78,0,0.0883624,"\int \frac{\sin ^2(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^2/(a + b*Sin[c + d*x]^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 \sqrt{a} d (a+b)^{3/2}}-\frac{\sin (c+d x) \cos (c+d x)}{2 d (a+b) \left(a+b \sin ^2(c+d x)\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 \sqrt{a} d (a+b)^{3/2}}-\frac{\sin (c+d x) \cos (c+d x)}{2 d (a+b) \left(a+b \sin ^2(c+d x)\right)}",1,"ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]]/(2*Sqrt[a]*(a + b)^(3/2)*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*(a + b)*d*(a + b*Sin[c + d*x]^2))","A",4,4,23,0.1739,1,"{3173, 12, 3181, 205}"
103,1,87,0,0.0625083,"\int \frac{1}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[(a + b*Sin[c + d*x]^2)^(-2),x]","\frac{(2 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a+b)^{3/2}}+\frac{b \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left(a+b \sin ^2(c+d x)\right)}","\frac{(2 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a+b)^{3/2}}+\frac{b \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left(a+b \sin ^2(c+d x)\right)}",1,"((2*a + b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(3/2)*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a + b)*d*(a + b*Sin[c + d*x]^2))","A",4,4,14,0.2857,1,"{3184, 12, 3181, 205}"
104,1,130,0,0.1464214,"\int \frac{\csc ^2(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[Csc[c + d*x]^2/(a + b*Sin[c + d*x]^2)^2,x]","-\frac{\left(2 a^2+4 a b+3 b^2\right) \tan (c+d x)}{2 a^2 d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)}-\frac{b (4 a+3 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{5/2} d (a+b)^{3/2}}-\frac{\cot (c+d x)}{a d \left((a+b) \tan ^2(c+d x)+a\right)}","-\frac{\left(2 a b+3 b^2\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d (a+b) \left(a+b \sin ^2(c+d x)\right)}-\frac{b (4 a+3 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{5/2} d (a+b)^{3/2}}-\frac{\cot (c+d x)}{a d \left(a+b \sin ^2(c+d x)\right)}",1,"-(b*(4*a + 3*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(5/2)*(a + b)^(3/2)*d) - Cot[c + d*x]/(a*d*(a + (a + b)*Tan[c + d*x]^2)) - ((2*a^2 + 4*a*b + 3*b^2)*Tan[c + d*x])/(2*a^2*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2))","A",4,4,23,0.1739,1,"{3187, 462, 385, 205}"
105,1,162,0,0.2028761,"\int \frac{\csc ^4(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Int[Csc[c + d*x]^4/(a + b*Sin[c + d*x]^2)^2,x]","\frac{b^2 (6 a+5 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{7/2} d (a+b)^{3/2}}-\frac{\left(2 a^2-a b-5 b^2\right) \cot (c+d x)}{2 a^3 d (a+b)}-\frac{(2 a+5 b) \cot ^3(c+d x)}{6 a^2 d (a+b)}+\frac{b \csc ^3(c+d x) \sec (c+d x)}{2 a d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)}","\frac{b^2 (6 a+5 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{7/2} d (a+b)^{3/2}}-\frac{\left(2 a^2-a b-5 b^2\right) \cot (c+d x)}{2 a^3 d (a+b)}-\frac{(2 a+5 b) \cot ^3(c+d x)}{6 a^2 d (a+b)}+\frac{b \csc ^3(c+d x) \sec (c+d x)}{2 a d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)}",1,"(b^2*(6*a + 5*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(7/2)*(a + b)^(3/2)*d) - ((2*a^2 - a*b - 5*b^2)*Cot[c + d*x])/(2*a^3*(a + b)*d) - ((2*a + 5*b)*Cot[c + d*x]^3)/(6*a^2*(a + b)*d) + (b*Csc[c + d*x]^3*Sec[c + d*x])/(2*a*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2))","A",5,4,23,0.1739,1,"{3187, 468, 570, 205}"
106,1,148,0,0.2788464,"\int \frac{\sin ^6(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]^6/(a + b*Sin[c + d*x]^2)^3,x]","-\frac{\sqrt{a} \left(8 a^2+20 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{8 b^3 d (a+b)^{5/2}}+\frac{a (4 a+7 b) \tan (c+d x)}{8 b^2 d (a+b)^2 \left((a+b) \tan ^2(c+d x)+a\right)}+\frac{a \tan ^3(c+d x)}{4 b d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)^2}+\frac{x}{b^3}","-\frac{\sqrt{a} \left(8 a^2+20 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{8 b^3 d (a+b)^{5/2}}+\frac{a (4 a+7 b) \tan (c+d x)}{8 b^2 d (a+b)^2 \left((a+b) \tan ^2(c+d x)+a\right)}+\frac{a \tan ^3(c+d x)}{4 b d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)^2}+\frac{x}{b^3}",1,"x/b^3 - (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(8*b^3*(a + b)^(5/2)*d) + (a*Tan[c + d*x]^3)/(4*b*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2)^2) + (a*(4*a + 7*b)*Tan[c + d*x])/(8*b^2*(a + b)^2*d*(a + (a + b)*Tan[c + d*x]^2))","A",6,6,23,0.2609,1,"{3187, 470, 578, 522, 203, 205}"
107,1,110,0,0.0956382,"\int \frac{\sin ^4(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]^4/(a + b*Sin[c + d*x]^2)^3,x]","-\frac{\tan ^3(c+d x)}{4 d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)^2}-\frac{3 \tan (c+d x)}{8 d (a+b)^2 \left((a+b) \tan ^2(c+d x)+a\right)}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{8 \sqrt{a} d (a+b)^{5/2}}","-\frac{\tan ^3(c+d x)}{4 d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)^2}-\frac{3 \tan (c+d x)}{8 d (a+b)^2 \left((a+b) \tan ^2(c+d x)+a\right)}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{8 \sqrt{a} d (a+b)^{5/2}}",1,"(3*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(8*Sqrt[a]*(a + b)^(5/2)*d) - Tan[c + d*x]^3/(4*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2)^2) - (3*Tan[c + d*x])/(8*(a + b)^2*d*(a + (a + b)*Tan[c + d*x]^2))","A",4,3,23,0.1304,1,"{3187, 288, 205}"
108,1,131,0,0.1489795,"\int \frac{\sin ^2(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]^2/(a + b*Sin[c + d*x]^2)^3,x]","\frac{(4 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{8 a^{3/2} d (a+b)^{5/2}}-\frac{(2 a-b) \sin (c+d x) \cos (c+d x)}{8 a d (a+b)^2 \left(a+b \sin ^2(c+d x)\right)}-\frac{\sin (c+d x) \cos (c+d x)}{4 d (a+b) \left(a+b \sin ^2(c+d x)\right)^2}","\frac{(4 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{8 a^{3/2} d (a+b)^{5/2}}-\frac{(2 a-b) \sin (c+d x) \cos (c+d x)}{8 a d (a+b)^2 \left(a+b \sin ^2(c+d x)\right)}-\frac{\sin (c+d x) \cos (c+d x)}{4 d (a+b) \left(a+b \sin ^2(c+d x)\right)^2}",1,"((4*a + b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(8*a^(3/2)*(a + b)^(5/2)*d) - (Cos[c + d*x]*Sin[c + d*x])/(4*(a + b)*d*(a + b*Sin[c + d*x]^2)^2) - ((2*a - b)*Cos[c + d*x]*Sin[c + d*x])/(8*a*(a + b)^2*d*(a + b*Sin[c + d*x]^2))","A",5,4,23,0.1739,1,"{3173, 12, 3181, 205}"
109,1,144,0,0.1464227,"\int \frac{1}{\left(a+b \sin ^2(c+d x)\right)^3} \, dx","Int[(a + b*Sin[c + d*x]^2)^(-3),x]","\frac{\left(8 a^2+8 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a+b)^{5/2}}+\frac{3 b (2 a+b) \sin (c+d x) \cos (c+d x)}{8 a^2 d (a+b)^2 \left(a+b \sin ^2(c+d x)\right)}+\frac{b \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left(a+b \sin ^2(c+d x)\right)^2}","\frac{\left(8 a^2+8 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a+b)^{5/2}}+\frac{3 b (2 a+b) \sin (c+d x) \cos (c+d x)}{8 a^2 d (a+b)^2 \left(a+b \sin ^2(c+d x)\right)}+\frac{b \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left(a+b \sin ^2(c+d x)\right)^2}",1,"((8*a^2 + 8*a*b + 3*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a + b)^(5/2)*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(4*a*(a + b)*d*(a + b*Sin[c + d*x]^2)^2) + (3*b*(2*a + b)*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*(a + b)^2*d*(a + b*Sin[c + d*x]^2))","A",5,5,14,0.3571,1,"{3184, 3173, 12, 3181, 205}"
110,1,196,0,0.254987,"\int \frac{\csc ^2(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^3} \, dx","Int[Csc[c + d*x]^2/(a + b*Sin[c + d*x]^2)^3,x]","-\frac{3 b \left(8 a^2+12 a b+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{8 a^{7/2} d (a+b)^{5/2}}-\frac{(2 a+3 b) (4 a+5 b) \cot (c+d x)}{8 a^3 d (a+b)^2}+\frac{b \cot (c+d x) \left((4 a+b) \tan ^2(c+d x)+4 a+5 b\right)}{8 a^2 d (a+b)^2 \left((a+b) \tan ^2(c+d x)+a\right)}+\frac{b \csc (c+d x) \sec ^3(c+d x)}{4 a d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)^2}","-\frac{3 b \left(8 a^2+12 a b+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{8 a^{7/2} d (a+b)^{5/2}}-\frac{(2 a+3 b) (4 a+5 b) \cot (c+d x)}{8 a^3 d (a+b)^2}+\frac{b \cot (c+d x) \left((4 a+b) \tan ^2(c+d x)+4 a+5 b\right)}{8 a^2 d (a+b)^2 \left((a+b) \tan ^2(c+d x)+a\right)}+\frac{b \csc (c+d x) \sec ^3(c+d x)}{4 a d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)^2}",1,"(-3*b*(8*a^2 + 12*a*b + 5*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(8*a^(7/2)*(a + b)^(5/2)*d) - ((2*a + 3*b)*(4*a + 5*b)*Cot[c + d*x])/(8*a^3*(a + b)^2*d) + (b*Csc[c + d*x]*Sec[c + d*x]^3)/(4*a*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2)^2) + (b*Cot[c + d*x]*(4*a + 5*b + (4*a + b)*Tan[c + d*x]^2))/(8*a^2*(a + b)^2*d*(a + (a + b)*Tan[c + d*x]^2))","A",5,5,23,0.2174,1,"{3187, 468, 577, 453, 205}"
111,1,206,0,0.2963163,"\int \frac{1}{\left(a+b \sin ^2(c+d x)\right)^4} \, dx","Int[(a + b*Sin[c + d*x]^2)^(-4),x]","\frac{(2 a+b) \left(8 a^2+8 a b+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{16 a^{7/2} d (a+b)^{7/2}}+\frac{b \left(44 a^2+44 a b+15 b^2\right) \sin (c+d x) \cos (c+d x)}{48 a^3 d (a+b)^3 \left(a+b \sin ^2(c+d x)\right)}+\frac{5 b (2 a+b) \sin (c+d x) \cos (c+d x)}{24 a^2 d (a+b)^2 \left(a+b \sin ^2(c+d x)\right)^2}+\frac{b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left(a+b \sin ^2(c+d x)\right)^3}","\frac{(2 a+b) \left(8 a^2+8 a b+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{16 a^{7/2} d (a+b)^{7/2}}+\frac{b \left(44 a^2+44 a b+15 b^2\right) \sin (c+d x) \cos (c+d x)}{48 a^3 d (a+b)^3 \left(a+b \sin ^2(c+d x)\right)}+\frac{5 b (2 a+b) \sin (c+d x) \cos (c+d x)}{24 a^2 d (a+b)^2 \left(a+b \sin ^2(c+d x)\right)^2}+\frac{b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left(a+b \sin ^2(c+d x)\right)^3}",1,"((2*a + b)*(8*a^2 + 8*a*b + 5*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(16*a^(7/2)*(a + b)^(7/2)*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(6*a*(a + b)*d*(a + b*Sin[c + d*x]^2)^3) + (5*b*(2*a + b)*Cos[c + d*x]*Sin[c + d*x])/(24*a^2*(a + b)^2*d*(a + b*Sin[c + d*x]^2)^2) + (b*(44*a^2 + 44*a*b + 15*b^2)*Cos[c + d*x]*Sin[c + d*x])/(48*a^3*(a + b)^3*d*(a + b*Sin[c + d*x]^2))","A",6,5,14,0.3571,1,"{3184, 3173, 12, 3181, 205}"
112,1,279,0,0.5308566,"\int \frac{1}{\left(a+b \sin ^2(c+d x)\right)^5} \, dx","Int[(a + b*Sin[c + d*x]^2)^(-5),x]","\frac{\left(288 a^2 b^2+256 a^3 b+128 a^4+160 a b^3+35 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{128 a^{9/2} d (a+b)^{9/2}}+\frac{5 b (2 a+b) \left(40 a^2+40 a b+21 b^2\right) \sin (c+d x) \cos (c+d x)}{384 a^4 d (a+b)^4 \left(a+b \sin ^2(c+d x)\right)}+\frac{b \left(104 a^2+104 a b+35 b^2\right) \sin (c+d x) \cos (c+d x)}{192 a^3 d (a+b)^3 \left(a+b \sin ^2(c+d x)\right)^2}+\frac{7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{48 a^2 d (a+b)^2 \left(a+b \sin ^2(c+d x)\right)^3}+\frac{b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left(a+b \sin ^2(c+d x)\right)^4}","\frac{\left(288 a^2 b^2+256 a^3 b+128 a^4+160 a b^3+35 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{128 a^{9/2} d (a+b)^{9/2}}+\frac{5 b (2 a+b) \left(40 a^2+40 a b+21 b^2\right) \sin (c+d x) \cos (c+d x)}{384 a^4 d (a+b)^4 \left(a+b \sin ^2(c+d x)\right)}+\frac{b \left(104 a^2+104 a b+35 b^2\right) \sin (c+d x) \cos (c+d x)}{192 a^3 d (a+b)^3 \left(a+b \sin ^2(c+d x)\right)^2}+\frac{7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{48 a^2 d (a+b)^2 \left(a+b \sin ^2(c+d x)\right)^3}+\frac{b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left(a+b \sin ^2(c+d x)\right)^4}",1,"((128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(128*a^(9/2)*(a + b)^(9/2)*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(8*a*(a + b)*d*(a + b*Sin[c + d*x]^2)^4) + (7*b*(2*a + b)*Cos[c + d*x]*Sin[c + d*x])/(48*a^2*(a + b)^2*d*(a + b*Sin[c + d*x]^2)^3) + (b*(104*a^2 + 104*a*b + 35*b^2)*Cos[c + d*x]*Sin[c + d*x])/(192*a^3*(a + b)^3*d*(a + b*Sin[c + d*x]^2)^2) + (5*b*(2*a + b)*(40*a^2 + 40*a*b + 21*b^2)*Cos[c + d*x]*Sin[c + d*x])/(384*a^4*(a + b)^4*d*(a + b*Sin[c + d*x]^2))","A",7,5,14,0.3571,1,"{3184, 3173, 12, 3181, 205}"
113,1,11,0,0.0263391,"\int \frac{\sin (x)}{\sqrt{1+\sin ^2(x)}} \, dx","Int[Sin[x]/Sqrt[1 + Sin[x]^2],x]","-\sin ^{-1}\left(\frac{\cos (x)}{\sqrt{2}}\right)","-\sin ^{-1}\left(\frac{\cos (x)}{\sqrt{2}}\right)",1,"-ArcSin[Cos[x]/Sqrt[2]]","A",2,2,13,0.1538,1,"{3186, 216}"
114,1,30,0,0.02998,"\int \sin (x) \sqrt{1+\sin ^2(x)} \, dx","Int[Sin[x]*Sqrt[1 + Sin[x]^2],x]","-\frac{1}{2} \cos (x) \sqrt{2-\cos ^2(x)}-\sin ^{-1}\left(\frac{\cos (x)}{\sqrt{2}}\right)","-\frac{1}{2} \cos (x) \sqrt{2-\cos ^2(x)}-\sin ^{-1}\left(\frac{\cos (x)}{\sqrt{2}}\right)",1,"-ArcSin[Cos[x]/Sqrt[2]] - (Cos[x]*Sqrt[2 - Cos[x]^2])/2","A",3,3,13,0.2308,1,"{3186, 195, 216}"
115,1,15,0,0.029909,"\int \frac{\sin (7+3 x)}{\sqrt{3+\sin ^2(7+3 x)}} \, dx","Int[Sin[7 + 3*x]/Sqrt[3 + Sin[7 + 3*x]^2],x]","-\frac{1}{3} \sin ^{-1}\left(\frac{1}{2} \cos (3 x+7)\right)","-\frac{1}{3} \sin ^{-1}\left(\frac{1}{2} \cos (3 x+7)\right)",1,"-ArcSin[Cos[7 + 3*x]/2]/3","A",2,2,21,0.09524,1,"{3186, 216}"
116,1,53,0,0.0540351,"\int \left(a-a \sin ^2(x)\right)^{5/2} \, dx","Int[(a - a*Sin[x]^2)^(5/2),x]","\frac{8}{15} a^2 \tan (x) \sqrt{a \cos ^2(x)}+\frac{1}{5} \tan (x) \left(a \cos ^2(x)\right)^{5/2}+\frac{4}{15} a \tan (x) \left(a \cos ^2(x)\right)^{3/2}","\frac{8}{15} a^2 \tan (x) \sqrt{a \cos ^2(x)}+\frac{1}{5} \tan (x) \left(a \cos ^2(x)\right)^{5/2}+\frac{4}{15} a \tan (x) \left(a \cos ^2(x)\right)^{3/2}",1,"(8*a^2*Sqrt[a*Cos[x]^2]*Tan[x])/15 + (4*a*(a*Cos[x]^2)^(3/2)*Tan[x])/15 + ((a*Cos[x]^2)^(5/2)*Tan[x])/5","A",5,4,13,0.3077,1,"{3176, 3203, 3207, 2637}"
117,1,34,0,0.0355551,"\int \left(a-a \sin ^2(x)\right)^{3/2} \, dx","Int[(a - a*Sin[x]^2)^(3/2),x]","\frac{1}{3} \tan (x) \left(a \cos ^2(x)\right)^{3/2}+\frac{2}{3} a \tan (x) \sqrt{a \cos ^2(x)}","\frac{1}{3} \tan (x) \left(a \cos ^2(x)\right)^{3/2}+\frac{2}{3} a \tan (x) \sqrt{a \cos ^2(x)}",1,"(2*a*Sqrt[a*Cos[x]^2]*Tan[x])/3 + ((a*Cos[x]^2)^(3/2)*Tan[x])/3","A",4,4,13,0.3077,1,"{3176, 3203, 3207, 2637}"
118,1,13,0,0.0259045,"\int \sqrt{a-a \sin ^2(x)} \, dx","Int[Sqrt[a - a*Sin[x]^2],x]","\tan (x) \sqrt{a \cos ^2(x)}","\tan (x) \sqrt{a \cos ^2(x)}",1,"Sqrt[a*Cos[x]^2]*Tan[x]","A",3,3,13,0.2308,1,"{3176, 3207, 2637}"
119,1,16,0,0.0306739,"\int \frac{1}{\sqrt{a-a \sin ^2(x)}} \, dx","Int[1/Sqrt[a - a*Sin[x]^2],x]","\frac{\cos (x) \tanh ^{-1}(\sin (x))}{\sqrt{a \cos ^2(x)}}","\frac{\cos (x) \tanh ^{-1}(\sin (x))}{\sqrt{a \cos ^2(x)}}",1,"(ArcTanh[Sin[x]]*Cos[x])/Sqrt[a*Cos[x]^2]","A",3,3,13,0.2308,1,"{3176, 3207, 3770}"
120,1,42,0,0.0374599,"\int \frac{1}{\left(a-a \sin ^2(x)\right)^{3/2}} \, dx","Int[(a - a*Sin[x]^2)^(-3/2),x]","\frac{\tan (x)}{2 a \sqrt{a \cos ^2(x)}}+\frac{\cos (x) \tanh ^{-1}(\sin (x))}{2 a \sqrt{a \cos ^2(x)}}","\frac{\tan (x)}{2 a \sqrt{a \cos ^2(x)}}+\frac{\cos (x) \tanh ^{-1}(\sin (x))}{2 a \sqrt{a \cos ^2(x)}}",1,"(ArcTanh[Sin[x]]*Cos[x])/(2*a*Sqrt[a*Cos[x]^2]) + Tan[x]/(2*a*Sqrt[a*Cos[x]^2])","A",4,4,13,0.3077,1,"{3176, 3204, 3207, 3770}"
121,1,61,0,0.0498615,"\int \frac{1}{\left(a-a \sin ^2(x)\right)^{5/2}} \, dx","Int[(a - a*Sin[x]^2)^(-5/2),x]","\frac{3 \tan (x)}{8 a^2 \sqrt{a \cos ^2(x)}}+\frac{3 \cos (x) \tanh ^{-1}(\sin (x))}{8 a^2 \sqrt{a \cos ^2(x)}}+\frac{\tan (x)}{4 a \left(a \cos ^2(x)\right)^{3/2}}","\frac{3 \tan (x)}{8 a^2 \sqrt{a \cos ^2(x)}}+\frac{3 \cos (x) \tanh ^{-1}(\sin (x))}{8 a^2 \sqrt{a \cos ^2(x)}}+\frac{\tan (x)}{4 a \left(a \cos ^2(x)\right)^{3/2}}",1,"(3*ArcTanh[Sin[x]]*Cos[x])/(8*a^2*Sqrt[a*Cos[x]^2]) + Tan[x]/(4*a*(a*Cos[x]^2)^(3/2)) + (3*Tan[x])/(8*a^2*Sqrt[a*Cos[x]^2])","A",5,4,13,0.3077,1,"{3176, 3204, 3207, 3770}"
122,1,125,0,0.1280165,"\int \sin ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sin[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{(a-3 b) (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{8 b^{3/2} f}-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{4 b f}+\frac{(a-3 b) \cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{8 b f}","\frac{(a-3 b) (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{8 b^{3/2} f}-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{4 b f}+\frac{(a-3 b) \cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{8 b f}",1,"((a - 3*b)*(a + b)*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(8*b^(3/2)*f) + ((a - 3*b)*Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(8*b*f) - (Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(3/2))/(4*b*f)","A",5,5,25,0.2000,1,"{3186, 388, 195, 217, 203}"
123,1,78,0,0.0609011,"\int \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 f}-\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 \sqrt{b} f}","-\frac{\cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 f}-\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 \sqrt{b} f}",1,"-((a + b)*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*Sqrt[b]*f) - (Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(2*f)","A",4,4,23,0.1739,1,"{3186, 195, 217, 203}"
124,1,83,0,0.0965129,"\int \csc (e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Csc[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{f}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{f}",1,"-((Sqrt[b]*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/f) - (Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/f","A",6,6,23,0.2609,1,"{3186, 402, 217, 203, 377, 206}"
125,1,84,0,0.100486,"\int \csc ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Csc[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{(a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 \sqrt{a} f}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 f}","-\frac{(a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 \sqrt{a} f}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 f}",1,"-((a + b)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*Sqrt[a]*f) - (Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(2*f)","A",4,4,25,0.1600,1,"{3186, 378, 377, 206}"
126,1,143,0,0.1323822,"\int \csc ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Csc[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{(3 a-b) (a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{8 a^{3/2} f}-\frac{\cot (e+f x) \csc ^3(e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{4 a f}-\frac{(3 a-b) \cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{8 a f}","-\frac{(3 a-b) (a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{8 a^{3/2} f}-\frac{\cot (e+f x) \csc ^3(e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{4 a f}-\frac{(3 a-b) \cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{8 a f}",1,"-((3*a - b)*(a + b)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(8*a^(3/2)*f) - ((3*a - b)*Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(8*a*f) - ((a + b - b*Cos[e + f*x]^2)^(3/2)*Cot[e + f*x]*Csc[e + f*x]^3)/(4*a*f)","A",5,5,25,0.2000,1,"{3186, 382, 378, 377, 206}"
127,1,259,0,0.3156684,"\int \sin ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sin[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\left(2 a^2-3 a b-8 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{15 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{2 a (a-2 b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{15 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}-\frac{(a+4 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 b f}","-\frac{\left(2 a^2-3 a b-8 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{15 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{2 a (a-2 b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{15 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}-\frac{(a+4 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 b f}",1,"-((a + 4*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b*f) - (Cos[e + f*x]*Sin[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2])/(5*f) - ((2*a^2 - 3*a*b - 8*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (2*a*(a - 2*b)*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(15*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3188, 478, 582, 524, 426, 424, 421, 419}"
128,1,159,0,0.1913246,"\int \sin ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sin[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+2 b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+2 b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-(Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) + ((a + 2*b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(a + b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,6,25,0.2400,1,"{3170, 3172, 3178, 3177, 3183, 3182}"
129,1,51,0,0.0351473,"\int \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])","A",2,2,16,0.1250,1,"{3178, 3177}"
130,1,174,0,0.1626315,"\int \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{(a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{\cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{(a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/f) - (Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3188, 475, 21, 423, 426, 424, 421, 419}"
131,1,234,0,0.2598762,"\int \csc ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Csc[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{(2 a+b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{2 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{(2 a+b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{2 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-((2*a + b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - ((2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (2*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3188, 475, 583, 524, 426, 424, 421, 419}"
132,1,169,0,0.148433,"\int \sin ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Sin[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{(a-5 b) (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{16 b^{3/2} f}-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{5/2}}{6 b f}+\frac{(a-5 b) \cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{24 b f}+\frac{(a-5 b) (a+b) \cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{16 b f}","\frac{(a-5 b) (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{16 b^{3/2} f}-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{5/2}}{6 b f}+\frac{(a-5 b) \cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{24 b f}+\frac{(a-5 b) (a+b) \cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{16 b f}",1,"((a - 5*b)*(a + b)^2*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(16*b^(3/2)*f) + ((a - 5*b)*(a + b)*Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(16*b*f) + ((a - 5*b)*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(3/2))/(24*b*f) - (Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(5/2))/(6*b*f)","A",6,5,25,0.2000,1,"{3186, 388, 195, 217, 203}"
133,1,114,0,0.075981,"\int \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{3 (a+b) \cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{8 f}-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{4 f}-\frac{3 (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{8 \sqrt{b} f}","-\frac{3 (a+b) \cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{8 f}-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{4 f}-\frac{3 (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{8 \sqrt{b} f}",1,"(-3*(a + b)^2*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(8*Sqrt[b]*f) - (3*(a + b)*Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(8*f) - (Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(3/2))/(4*f)","A",5,4,23,0.1739,1,"{3186, 195, 217, 203}"
134,1,122,0,0.144398,"\int \csc (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{f}-\frac{b \cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 f}-\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 f}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{f}-\frac{b \cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 f}-\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 f}",1,"-(Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*f) - (a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/f - (b*Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(2*f)","A",7,7,23,0.3043,1,"{3186, 416, 523, 217, 203, 377, 206}"
135,1,128,0,0.1512444,"\int \csc ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{f}-\frac{\sqrt{a} (a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 f}-\frac{a \cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 f}","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{f}-\frac{\sqrt{a} (a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 f}-\frac{a \cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 f}",1,"-((b^(3/2)*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/f) - (Sqrt[a]*(a + 3*b)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*f) - (a*Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(2*f)","A",7,7,25,0.2800,1,"{3186, 413, 523, 217, 203, 377, 206}"
136,1,128,0,0.125199,"\int \csc ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^5*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{3 (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{8 \sqrt{a} f}-\frac{\cot (e+f x) \csc ^3(e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{4 f}-\frac{3 (a+b) \cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{8 f}","-\frac{3 (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{8 \sqrt{a} f}-\frac{\cot (e+f x) \csc ^3(e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{4 f}-\frac{3 (a+b) \cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{8 f}",1,"(-3*(a + b)^2*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(8*Sqrt[a]*f) - (3*(a + b)*Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(8*f) - ((a + b - b*Cos[e + f*x]^2)^(3/2)*Cot[e + f*x]*Csc[e + f*x]^3)/(4*f)","A",5,4,25,0.1600,1,"{3186, 378, 377, 206}"
137,1,197,0,0.1749828,"\int \csc ^7(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^7*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{(5 a-b) (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{16 a^{3/2} f}-\frac{\cot (e+f x) \csc ^5(e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{5/2}}{6 a f}-\frac{(5 a-b) \cot (e+f x) \csc ^3(e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{24 a f}-\frac{(5 a-b) (a+b) \cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{16 a f}","-\frac{(5 a-b) (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{16 a^{3/2} f}-\frac{\cot (e+f x) \csc ^5(e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{5/2}}{6 a f}-\frac{(5 a-b) \cot (e+f x) \csc ^3(e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}{24 a f}-\frac{(5 a-b) (a+b) \cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{16 a f}",1,"-((5*a - b)*(a + b)^2*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(16*a^(3/2)*f) - ((5*a - b)*(a + b)*Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(16*a*f) - ((5*a - b)*(a + b - b*Cos[e + f*x]^2)^(3/2)*Cot[e + f*x]*Csc[e + f*x]^3)/(24*a*f) - ((a + b - b*Cos[e + f*x]^2)^(5/2)*Cot[e + f*x]*Csc[e + f*x]^5)/(6*a*f)","A",6,5,25,0.2000,1,"{3186, 382, 378, 377, 206}"
138,1,325,0,0.4823106,"\int \sin ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Sin[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{\left(a^2+11 a b+8 b^2\right) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 b f}+\frac{a (a+b) \left(2 a^2-5 a b-8 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{35 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \left(a^2-4 a b-4 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{35 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{b \sin ^5(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{7 f}-\frac{2 (4 a+3 b) \sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 f}","-\frac{\left(a^2+11 a b+8 b^2\right) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 b f}+\frac{a (a+b) \left(2 a^2-5 a b-8 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{35 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \left(a^2-4 a b-4 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{35 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{b \sin ^5(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{7 f}-\frac{2 (4 a+3 b) \sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 f}",1,"-((a^2 + 11*a*b + 8*b^2)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*b*f) - (2*(4*a + 3*b)*Cos[e + f*x]*Sin[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2])/(35*f) - (b*Cos[e + f*x]*Sin[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2])/(7*f) - (2*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(a + b)*(2*a^2 - 5*a*b - 8*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(35*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",9,8,25,0.3200,1,"{3188, 477, 582, 524, 426, 424, 421, 419}"
139,1,218,0,0.3039007,"\int \sin ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Sin[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\left(3 a^2+13 a b+8 b^2\right) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{15 b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{\sin (e+f x) \cos (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{5 f}-\frac{(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 f}-\frac{a (a+b) (3 a+4 b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{15 b f \sqrt{a+b \sin ^2(e+f x)}}","\frac{\left(3 a^2+13 a b+8 b^2\right) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{15 b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{\sin (e+f x) \cos (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{5 f}-\frac{(3 a+4 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 f}-\frac{a (a+b) (3 a+4 b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{15 b f \sqrt{a+b \sin ^2(e+f x)}}",1,"-((3*a + 4*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*f) - (Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(5*f) + ((3*a^2 + 13*a*b + 8*b^2)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(a + b)*(3*a + 4*b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(15*b*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,6,25,0.2400,1,"{3170, 3172, 3178, 3177, 3183, 3182}"
140,1,154,0,0.1635892,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-(b*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(a + b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,6,16,0.3750,1,"{3180, 3172, 3178, 3177, 3183, 3182}"
141,1,181,0,0.1920464,"\int \csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{a \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{a (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{a \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{a (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-((a*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/f) - ((a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,7,25,0.2800,1,"{3188, 474, 524, 426, 424, 421, 419}"
142,1,236,0,0.2929069,"\int \csc ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{2 (a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{(a+b) (2 a+3 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{2 (a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{(a+b) (2 a+3 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(-2*(a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - (a*Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - (2*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((a + b)*(2*a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3188, 474, 583, 524, 426, 424, 421, 419}"
143,1,210,0,0.2816863,"\int \left(a+b \sin ^2(c+d x)\right)^{5/2} \, dx","Int[(a + b*Sin[c + d*x]^2)^(5/2),x]","\frac{\left(23 a^2+23 a b+8 b^2\right) \sqrt{a+b \sin ^2(c+d x)} E\left(c+d x\left|-\frac{b}{a}\right.\right)}{15 d \sqrt{\frac{b \sin ^2(c+d x)}{a}+1}}-\frac{b \sin (c+d x) \cos (c+d x) \left(a+b \sin ^2(c+d x)\right)^{3/2}}{5 d}-\frac{4 b (2 a+b) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin ^2(c+d x)}}{15 d}-\frac{4 a (a+b) (2 a+b) \sqrt{\frac{b \sin ^2(c+d x)}{a}+1} F\left(c+d x\left|-\frac{b}{a}\right.\right)}{15 d \sqrt{a+b \sin ^2(c+d x)}}","\frac{\left(23 a^2+23 a b+8 b^2\right) \sqrt{a+b \sin ^2(c+d x)} E\left(c+d x\left|-\frac{b}{a}\right.\right)}{15 d \sqrt{\frac{b \sin ^2(c+d x)}{a}+1}}-\frac{b \sin (c+d x) \cos (c+d x) \left(a+b \sin ^2(c+d x)\right)^{3/2}}{5 d}-\frac{4 b (2 a+b) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin ^2(c+d x)}}{15 d}-\frac{4 a (a+b) (2 a+b) \sqrt{\frac{b \sin ^2(c+d x)}{a}+1} F\left(c+d x\left|-\frac{b}{a}\right.\right)}{15 d \sqrt{a+b \sin ^2(c+d x)}}",1,"(-4*b*(2*a + b)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]^2])/(15*d) - (b*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x]^2)^(3/2))/(5*d) + ((23*a^2 + 23*a*b + 8*b^2)*EllipticE[c + d*x, -(b/a)]*Sqrt[a + b*Sin[c + d*x]^2])/(15*d*Sqrt[1 + (b*Sin[c + d*x]^2)/a]) - (4*a*(a + b)*(2*a + b)*EllipticF[c + d*x, -(b/a)]*Sqrt[1 + (b*Sin[c + d*x]^2)/a])/(15*d*Sqrt[a + b*Sin[c + d*x]^2])","A",7,7,16,0.4375,1,"{3180, 3170, 3172, 3178, 3177, 3183, 3182}"
144,1,83,0,0.0955315,"\int \frac{\sin ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 b^{3/2} f}-\frac{\cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 b f}","\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 b^{3/2} f}-\frac{\cos (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 b f}",1,"((a - b)*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*b^(3/2)*f) - (Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(2*b*f)","A",4,4,25,0.1600,1,"{3186, 388, 217, 203}"
145,1,41,0,0.0472215,"\int \frac{\sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Sin[e + f*x]/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{\sqrt{b} f}","-\frac{\tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{\sqrt{b} f}",1,"-(ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(Sqrt[b]*f))","A",3,3,23,0.1304,1,"{3186, 217, 203}"
146,1,41,0,0.0759755,"\int \frac{\csc (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Csc[e + f*x]/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{\sqrt{a} f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{\sqrt{a} f}",1,"-(ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(Sqrt[a]*f))","A",3,3,23,0.1304,1,"{3186, 377, 206}"
147,1,89,0,0.1040877,"\int \frac{\csc ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{(a-b) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 a^{3/2} f}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 a f}","-\frac{(a-b) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{2 a^{3/2} f}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 a f}",1,"-((a - b)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*a^(3/2)*f) - (Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(2*a*f)","A",4,4,25,0.1600,1,"{3186, 382, 377, 206}"
148,1,206,0,0.198534,"\int \frac{\sin ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{a (2 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 b f}","\frac{a (2 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 b f}",1,"-(Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b*f) - (2*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(2*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,7,25,0.2800,1,"{3188, 479, 524, 426, 424, 421, 419}"
149,1,111,0,0.1260822,"\int \frac{\sin ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{a \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{b f \sqrt{a+b \sin ^2(e+f x)}}","\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{a \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{b f \sqrt{a+b \sin ^2(e+f x)}}",1,"(EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2])","A",5,5,25,0.2000,1,"{3172, 3178, 3177, 3183, 3182}"
150,1,51,0,0.0328417,"\int \frac{1}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[1/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{a+b \sin ^2(e+f x)}}","\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{a+b \sin ^2(e+f x)}}",1,"(EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])","A",2,2,16,0.1250,1,"{3183, 3182}"
151,1,177,0,0.1786869,"\int \frac{\csc ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{a f}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{\cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{a f}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*f)) - (Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3188, 480, 12, 493, 426, 424, 421, 419}"
152,1,244,0,0.2666941,"\int \frac{\csc ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{2 (a-b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac{2 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}+\frac{(2 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a f \sqrt{a+b \sin ^2(e+f x)}}","-\frac{2 (a-b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac{2 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}+\frac{(2 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a f \sqrt{a+b \sin ^2(e+f x)}}",1,"(-2*(a - b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*f) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f) - (2*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((2*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3188, 480, 583, 524, 426, 424, 421, 419}"
153,1,79,0,0.0988078,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{a \cos (e+f x)}{b f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{b^{3/2} f}","\frac{a \cos (e+f x)}{b f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{b^{3/2} f}",1,"-(ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(b^(3/2)*f)) + (a*Cos[e + f*x])/(b*(a + b)*f*Sqrt[a + b - b*Cos[e + f*x]^2])","A",4,4,25,0.1600,1,"{3186, 385, 217, 203}"
154,1,34,0,0.045294,"\int \frac{\sin (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{\cos (e+f x)}{f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}","-\frac{\cos (e+f x)}{f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}",1,"-(Cos[e + f*x]/((a + b)*f*Sqrt[a + b - b*Cos[e + f*x]^2]))","A",2,2,23,0.08696,1,"{3186, 191}"
155,1,79,0,0.0938341,"\int \frac{\csc (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{b \cos (e+f x)}{a f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{a^{3/2} f}","\frac{b \cos (e+f x)}{a f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{a^{3/2} f}",1,"-(ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(a^(3/2)*f)) + (b*Cos[e + f*x])/(a*(a + b)*f*Sqrt[a + b - b*Cos[e + f*x]^2])","A",4,4,23,0.1739,1,"{3186, 382, 377, 206}"
156,1,134,0,0.1644571,"\int \frac{\csc ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{b (a+3 b) \cos (e+f x)}{2 a^2 f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{(a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^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 \cos ^2(e+f x)+b}}","-\frac{b (a+3 b) \cos (e+f x)}{2 a^2 f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{(a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^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 \cos ^2(e+f x)+b}}",1,"-((a - 3*b)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*a^(5/2)*f) - (b*(a + 3*b)*Cos[e + f*x])/(2*a^2*(a + b)*f*Sqrt[a + b - b*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*Sqrt[a + b - b*Cos[e + f*x]^2])","A",6,6,25,0.2400,1,"{3186, 414, 527, 12, 377, 206}"
157,1,274,0,0.3142396,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{\left(8 a^2+3 a b-2 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^3 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{(4 a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 b^2 f (a+b)}+\frac{a (8 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{a \sin ^3(e+f x) \cos (e+f x)}{b f (a+b) \sqrt{a+b \sin ^2(e+f x)}}","-\frac{\left(8 a^2+3 a b-2 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^3 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{(4 a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 b^2 f (a+b)}+\frac{a (8 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{a \sin ^3(e+f x) \cos (e+f x)}{b f (a+b) \sqrt{a+b \sin ^2(e+f x)}}",1,"(a*Cos[e + f*x]*Sin[e + f*x]^3)/(b*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((4*a + b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^2*(a + b)*f) - ((8*a^2 + 3*a*b - 2*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^3*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(8*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3188, 470, 582, 524, 426, 424, 421, 419}"
158,1,202,0,0.1995856,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{2 a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{b^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{b^2 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{a \sin (e+f x) \cos (e+f x)}{b f (a+b) \sqrt{a+b \sin ^2(e+f x)}}","-\frac{2 a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{b^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{b^2 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{a \sin (e+f x) \cos (e+f x)}{b f (a+b) \sqrt{a+b \sin ^2(e+f x)}}",1,"(a*Cos[e + f*x]*Sin[e + f*x])/(b*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(b^2*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (2*a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,7,25,0.2800,1,"{3188, 470, 524, 426, 424, 421, 419}"
159,1,153,0,0.1916411,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{\sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{b f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{\sin (e+f x) \cos (e+f x)}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{b f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-((Cos[e + f*x]*Sin[e + f*x])/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])) - (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(b*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,6,25,0.2400,1,"{3173, 3172, 3178, 3177, 3183, 3182}"
160,1,101,0,0.0573837,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[(a + b*Sin[e + f*x]^2)^(-3/2),x]","\frac{b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{a f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{a f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(b*Cos[e + f*x]*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(a*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])","A",4,4,16,0.2500,1,"{3184, 21, 3178, 3177}"
161,1,235,0,0.268003,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{(a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{a^2 f (a+b)}-\frac{(a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a^2 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{b \cot (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a f \sqrt{a+b \sin ^2(e+f x)}}","-\frac{(a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{a^2 f (a+b)}-\frac{(a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a^2 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{b \cot (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a f \sqrt{a+b \sin ^2(e+f x)}}",1,"(b*Cot[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a^2*(a + b)*f) - ((a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a^2*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(a*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3188, 472, 583, 524, 426, 424, 421, 419}"
162,1,137,0,0.1368321,"\int \frac{\sin ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{a (3 a+5 b) \cos (e+f x)}{3 b^2 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{b^{5/2} f}+\frac{a \sin ^2(e+f x) \cos (e+f x)}{3 b f (a+b) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}","\frac{a (3 a+5 b) \cos (e+f x)}{3 b^2 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{b} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{b^{5/2} f}+\frac{a \sin ^2(e+f x) \cos (e+f x)}{3 b f (a+b) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}",1,"-(ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(b^(5/2)*f)) + (a*(3*a + 5*b)*Cos[e + f*x])/(3*b^2*(a + b)^2*f*Sqrt[a + b - b*Cos[e + f*x]^2]) + (a*Cos[e + f*x]*Sin[e + f*x]^2)/(3*b*(a + b)*f*(a + b - b*Cos[e + f*x]^2)^(3/2))","A",5,5,25,0.2000,1,"{3186, 413, 385, 217, 203}"
163,1,81,0,0.0951856,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{2 \cos (e+f x)}{3 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\sin ^2(e+f x) \cos (e+f x)}{3 f (a+b) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}","-\frac{2 \cos (e+f x)}{3 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\sin ^2(e+f x) \cos (e+f x)}{3 f (a+b) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}",1,"(-2*Cos[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b - b*Cos[e + f*x]^2]) - (Cos[e + f*x]*Sin[e + f*x]^2)/(3*(a + b)*f*(a + b - b*Cos[e + f*x]^2)^(3/2))","A",3,3,25,0.1200,1,"{3186, 378, 191}"
164,1,73,0,0.0580009,"\int \frac{\sin (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{2 \cos (e+f x)}{3 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\cos (e+f x)}{3 f (a+b) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}","-\frac{2 \cos (e+f x)}{3 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\cos (e+f x)}{3 f (a+b) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}",1,"-Cos[e + f*x]/(3*(a + b)*f*(a + b - b*Cos[e + f*x]^2)^(3/2)) - (2*Cos[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b - b*Cos[e + f*x]^2])","A",3,3,23,0.1304,1,"{3186, 192, 191}"
165,1,129,0,0.1525963,"\int \frac{\csc (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{b (5 a+3 b) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{a^{5/2} f}+\frac{b \cos (e+f x)}{3 a f (a+b) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}","\frac{b (5 a+3 b) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a-b \cos ^2(e+f x)+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right)}{a^{5/2} f}+\frac{b \cos (e+f x)}{3 a f (a+b) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}",1,"-(ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(a^(5/2)*f)) + (b*Cos[e + f*x])/(3*a*(a + b)*f*(a + b - b*Cos[e + f*x]^2)^(3/2)) + (b*(5*a + 3*b)*Cos[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b - b*Cos[e + f*x]^2])","A",6,6,23,0.2609,1,"{3186, 414, 527, 12, 377, 206}"
166,1,285,0,0.3323623,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\left(8 a^2+13 a b+3 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^3 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{2 a (2 a+3 b) \sin (e+f x) \cos (e+f x)}{3 b^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{a (8 a+9 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^3 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{a \sin ^3(e+f x) \cos (e+f x)}{3 b f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}","\frac{\left(8 a^2+13 a b+3 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^3 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{2 a (2 a+3 b) \sin (e+f x) \cos (e+f x)}{3 b^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{a (8 a+9 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^3 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{a \sin ^3(e+f x) \cos (e+f x)}{3 b f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"(a*Cos[e + f*x]*Sin[e + f*x]^3)/(3*b*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*a*(2*a + 3*b)*Cos[e + f*x]*Sin[e + f*x])/(3*b^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((8*a^2 + 13*a*b + 3*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^3*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(8*a + 9*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^3*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3188, 470, 578, 524, 426, 424, 421, 419}"
167,1,269,0,0.2756006,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{(2 a+3 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^2 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{2 (a+2 b) \sin (e+f x) \cos (e+f x)}{3 b f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{a \sin (e+f x) \cos (e+f x)}{3 b f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}","\frac{(2 a+3 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^2 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{2 (a+2 b) \sin (e+f x) \cos (e+f x)}{3 b f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{a \sin (e+f x) \cos (e+f x)}{3 b f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"(a*Cos[e + f*x]*Sin[e + f*x])/(3*b*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*(a + 2*b)*Cos[e + f*x]*Sin[e + f*x])/(3*b*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^2*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((2*a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3188, 470, 527, 524, 426, 424, 421, 419}"
168,1,221,0,0.2896656,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{(a-b) \sin (e+f x) \cos (e+f x)}{3 a f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 b f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{(a-b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a b f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{(a-b) \sin (e+f x) \cos (e+f x)}{3 a f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sin (e+f x) \cos (e+f x)}{3 f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 b f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{(a-b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a b f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-(Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) - ((a - b)*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((a - b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*b*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,6,25,0.2400,1,"{3173, 3172, 3178, 3177, 3183, 3182}"
169,1,223,0,0.255501,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[(a + b*Sin[e + f*x]^2)^(-5/2),x]","\frac{2 b (2 a+b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a^2 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{b \sin (e+f x) \cos (e+f x)}{3 a f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}","\frac{2 b (2 a+b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a^2 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{b \sin (e+f x) \cos (e+f x)}{3 a f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}",1,"(b*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*b*(2*a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,7,16,0.4375,1,"{3184, 3173, 3172, 3178, 3177, 3183, 3182}"
170,1,322,0,0.411473,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{\left(3 a^2+13 a b+8 b^2\right) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 f (a+b)^2}-\frac{\left(3 a^2+13 a b+8 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^3 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{2 b (3 a+2 b) \cot (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{(3 a+4 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{b \cot (e+f x)}{3 a f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}","-\frac{\left(3 a^2+13 a b+8 b^2\right) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 f (a+b)^2}-\frac{\left(3 a^2+13 a b+8 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^3 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{2 b (3 a+2 b) \cot (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{(3 a+4 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{b \cot (e+f x)}{3 a f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"(b*Cot[e + f*x])/(3*a*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*b*(3*a + 2*b)*Cot[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((3*a^2 + 13*a*b + 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*(a + b)^2*f) - ((3*a^2 + 13*a*b + 8*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((3*a + 4*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a^2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",9,9,25,0.3600,1,"{3188, 472, 579, 583, 524, 426, 424, 421, 419}"
171,1,122,0,0.1156657,"\int (d \sin (e+f x))^m \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x]^2)^p,x]","-\frac{d \cos (e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \sin (e+f x))^{m-1} \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};\frac{1-m}{2},-p;\frac{3}{2};\cos ^2(e+f x),\frac{b \cos ^2(e+f x)}{a+b}\right)}{f}","-\frac{d \cos (e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \sin (e+f x))^{m-1} \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};\frac{1-m}{2},-p;\frac{3}{2};\cos ^2(e+f x),\frac{b \cos ^2(e+f x)}{a+b}\right)}{f}",1,"-((d*AppellF1[1/2, (1 - m)/2, -p, 3/2, Cos[e + f*x]^2, (b*Cos[e + f*x]^2)/(a + b)]*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*(d*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(f*(1 - (b*Cos[e + f*x]^2)/(a + b))^p))","A",3,3,25,0.1200,1,"{3189, 430, 429}"
172,1,220,0,0.2248455,"\int \sin ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Sin[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p,x]","-\frac{\left(3 a^2-4 a b (p+1)+4 b^2 \left(p^2+3 p+2\right)\right) \cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{b \cos ^2(e+f x)}{a+b}\right)}{b^2 f (2 p+3) (2 p+5)}+\frac{(3 a-2 b (p+2)) \cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{p+1}}{b^2 f (2 p+3) (2 p+5)}-\frac{\sin ^2(e+f x) \cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{p+1}}{b f (2 p+5)}","-\frac{\left(3 a^2-4 a b (p+1)+4 b^2 \left(p^2+3 p+2\right)\right) \cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{b \cos ^2(e+f x)}{a+b}\right)}{b^2 f (2 p+3) (2 p+5)}+\frac{(3 a-2 b (p+2)) \cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{p+1}}{b^2 f (2 p+3) (2 p+5)}-\frac{\sin ^2(e+f x) \cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{p+1}}{b f (2 p+5)}",1,"((3*a - 2*b*(2 + p))*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(1 + p))/(b^2*f*(3 + 2*p)*(5 + 2*p)) - ((3*a^2 - 4*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (b*Cos[e + f*x]^2)/(a + b)])/(b^2*f*(3 + 2*p)*(5 + 2*p)*(1 - (b*Cos[e + f*x]^2)/(a + b))^p) - (Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(1 + p)*Sin[e + f*x]^2)/(b*f*(5 + 2*p))","A",5,5,23,0.2174,1,"{3186, 416, 388, 246, 245}"
173,1,131,0,0.1098824,"\int \sin ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Sin[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p,x]","\frac{(a-2 b (p+1)) \cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{b \cos ^2(e+f x)}{a+b}\right)}{b f (2 p+3)}-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{p+1}}{b f (2 p+3)}","\frac{(a-2 b (p+1)) \cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{b \cos ^2(e+f x)}{a+b}\right)}{b f (2 p+3)}-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^{p+1}}{b f (2 p+3)}",1,"-((Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(1 + p))/(b*f*(3 + 2*p))) + ((a - 2*b*(1 + p))*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (b*Cos[e + f*x]^2)/(a + b)])/(b*f*(3 + 2*p)*(1 - (b*Cos[e + f*x]^2)/(a + b))^p)","A",4,4,23,0.1739,1,"{3186, 388, 246, 245}"
174,1,74,0,0.0461089,"\int \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p,x]","-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{b \cos ^2(e+f x)}{a+b}\right)}{f}","-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};\frac{b \cos ^2(e+f x)}{a+b}\right)}{f}",1,"-((Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (b*Cos[e + f*x]^2)/(a + b)])/(f*(1 - (b*Cos[e + f*x]^2)/(a + b))^p))","A",3,3,21,0.1429,1,"{3186, 246, 245}"
175,1,83,0,0.0778559,"\int \csc (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]*(a + b*Sin[e + f*x]^2)^p,x]","-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\cos ^2(e+f x),\frac{b \cos ^2(e+f x)}{a+b}\right)}{f}","-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\cos ^2(e+f x),\frac{b \cos ^2(e+f x)}{a+b}\right)}{f}",1,"-((AppellF1[1/2, 1, -p, 3/2, Cos[e + f*x]^2, (b*Cos[e + f*x]^2)/(a + b)]*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p)/(f*(1 - (b*Cos[e + f*x]^2)/(a + b))^p))","A",3,3,21,0.1429,1,"{3186, 430, 429}"
176,1,83,0,0.0835968,"\int \csc ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p,x]","-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};2,-p;\frac{3}{2};\cos ^2(e+f x),\frac{b \cos ^2(e+f x)}{a+b}\right)}{f}","-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};2,-p;\frac{3}{2};\cos ^2(e+f x),\frac{b \cos ^2(e+f x)}{a+b}\right)}{f}",1,"-((AppellF1[1/2, 2, -p, 3/2, Cos[e + f*x]^2, (b*Cos[e + f*x]^2)/(a + b)]*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p)/(f*(1 - (b*Cos[e + f*x]^2)/(a + b))^p))","A",3,3,23,0.1304,1,"{3186, 430, 429}"
177,1,83,0,0.0852897,"\int \csc ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p,x]","-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};3,-p;\frac{3}{2};\cos ^2(e+f x),\frac{b \cos ^2(e+f x)}{a+b}\right)}{f}","-\frac{\cos (e+f x) \left(a-b \cos ^2(e+f x)+b\right)^p \left(1-\frac{b \cos ^2(e+f x)}{a+b}\right)^{-p} F_1\left(\frac{1}{2};3,-p;\frac{3}{2};\cos ^2(e+f x),\frac{b \cos ^2(e+f x)}{a+b}\right)}{f}",1,"-((AppellF1[1/2, 3, -p, 3/2, Cos[e + f*x]^2, (b*Cos[e + f*x]^2)/(a + b)]*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p)/(f*(1 - (b*Cos[e + f*x]^2)/(a + b))^p))","A",3,3,23,0.1304,1,"{3186, 430, 429}"
178,1,101,0,0.1030966,"\int \sin ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Sin[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sin ^4(e+f x) \sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{5}{2};\frac{1}{2},-p;\frac{7}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{5 f}","\frac{\sin ^4(e+f x) \sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{5}{2};\frac{1}{2},-p;\frac{7}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{5 f}",1,"(AppellF1[5/2, 1/2, -p, 7/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*Sin[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(5*f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3188, 511, 510}"
179,1,99,0,0.1658875,"\int \sin ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Sin[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\tan ^3(e+f x) \sec ^2(e+f x)^p \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{(a+b) \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};p+2,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{(a+b) \tan ^2(e+f x)}{a}\right)}{3 f}","\frac{\tan ^3(e+f x) \sec ^2(e+f x)^p \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{(a+b) \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};p+2,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{(a+b) \tan ^2(e+f x)}{a}\right)}{3 f}",1,"(AppellF1[3/2, 2 + p, -p, 5/2, -Tan[e + f*x]^2, -(((a + b)*Tan[e + f*x]^2)/a)]*(Sec[e + f*x]^2)^p*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x]^3)/(3*f*(1 + ((a + b)*Tan[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3174, 511, 510}"
180,1,97,0,0.1038253,"\int \csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p,x]","-\frac{\sqrt{\cos ^2(e+f x)} \csc (e+f x) \sec (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{1}{2};\frac{1}{2},-p;\frac{1}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}","-\frac{\sqrt{\cos ^2(e+f x)} \csc (e+f x) \sec (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{1}{2};\frac{1}{2},-p;\frac{1}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}",1,"-((AppellF1[-1/2, 1/2, -p, 1/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*Csc[e + f*x]*Sec[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(f*(1 + (b*Sin[e + f*x]^2)/a)^p))","A",3,3,23,0.1304,1,"{3188, 511, 510}"
181,1,101,0,0.0996099,"\int \csc ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p,x]","-\frac{\sqrt{\cos ^2(e+f x)} \csc ^3(e+f x) \sec (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{3}{2};\frac{1}{2},-p;-\frac{1}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{3 f}","-\frac{\sqrt{\cos ^2(e+f x)} \csc ^3(e+f x) \sec (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{3}{2};\frac{1}{2},-p;-\frac{1}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{3 f}",1,"-(AppellF1[-3/2, 1/2, -p, -1/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*Csc[e + f*x]^3*Sec[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(3*f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3188, 511, 510}"
182,1,335,0,0.699105,"\int \frac{\sin ^7(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Sin[c + d*x]^7/(a + b*Sin[c + d*x]^3),x]","\frac{2 (-1)^{2/3} a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^{7/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{7/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{-1} a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b^{7/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac{3 \sin (c+d x) \cos (c+d x)}{8 b d}+\frac{3 x}{8 b}","\frac{2 (-1)^{2/3} a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^{7/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{7/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{-1} a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b^{7/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac{3 \sin (c+d x) \cos (c+d x)}{8 b d}+\frac{3 x}{8 b}",1,"(3*x)/(8*b) + (2*(-1)^(2/3)*a^(5/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(7/3)*d) - (2*a^(5/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(7/3)*d) + (2*(-1)^(1/3)*a^(5/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b^(7/3)*d) + (a*Cos[c + d*x])/(b^2*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*b*d)","A",17,7,23,0.3043,1,"{3220, 2638, 2635, 8, 2660, 618, 204}"
183,1,273,0,0.5669714,"\int \frac{\sin ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Sin[c + d*x]^5/(a + b*Sin[c + d*x]^3),x]","-\frac{2 a \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{5/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 a \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 b^{5/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 a \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 b^{5/3} d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}+\frac{x}{2 b}","-\frac{2 a \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{5/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 a \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 b^{5/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 a \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 b^{5/3} d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}+\frac{x}{2 b}",1,"x/(2*b) - (2*a*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(5/3)*d) + (2*a*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]])/(3*Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]*b^(5/3)*d) + (2*a*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*b^(5/3)*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",15,7,23,0.3043,1,"{3220, 2635, 8, 2660, 618, 204, 206}"
184,1,259,0,0.4575086,"\int \frac{\sin ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Sin[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]","-\frac{2 \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{2 \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{x}{b}","-\frac{2 \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{2 \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{x}{b}",1,"x/b - (2*a^(1/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b*d) - (2*a^(1/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b*d) + (2*a^(1/3)*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b*d)","A",13,5,23,0.2174,1,"{3220, 3213, 2660, 618, 204}"
185,1,267,0,0.2622492,"\int \frac{\sin (c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Sin[c + d*x]/(a + b*Sin[c + d*x]^3),x]","\frac{2 (-1)^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{-1} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}","\frac{2 (-1)^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{-1} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}",1,"(2*(-1)^(2/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(1/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(1/3)*d) - (2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(1/3)*Sqrt[a^(2/3) - b^(2/3)]*b^(1/3)*d) + (2*(-1)^(1/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(1/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b^(1/3)*d)","A",11,4,21,0.1905,1,"{3220, 2660, 618, 204}"
186,1,264,0,0.364646,"\int \frac{\csc (c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Csc[c + d*x]/(a + b*Sin[c + d*x]^3),x]","-\frac{2 \sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{b} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 a d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 \sqrt[3]{b} \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 a d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}","-\frac{2 \sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{b} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 a d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 \sqrt[3]{b} \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 a d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"(-2*b^(1/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a*Sqrt[a^(2/3) - b^(2/3)]*d) - ArcTanh[Cos[c + d*x]]/(a*d) + (2*b^(1/3)*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]])/(3*a*Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]*d) + (2*b^(1/3)*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*a*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*d)","A",14,6,21,0.2857,1,"{3220, 3770, 2660, 618, 204, 206}"
187,1,287,0,0.4039046,"\int \frac{\csc ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Csc[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]","-\frac{2 b \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 b \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{2 b \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}","-\frac{2 b \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 b \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{2 b \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}",1,"(-2*b*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(5/3)*Sqrt[a^(2/3) - b^(2/3)]*d) - (2*b*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(5/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) + (2*b*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(5/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) - ArcTanh[Cos[c + d*x]]/(2*a*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)","A",15,7,23,0.3043,1,"{3220, 3768, 3770, 3213, 2660, 618, 204}"
188,1,344,0,0.477802,"\int \frac{\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Csc[c + d*x]^5/(a + b*Sin[c + d*x]^3),x]","\frac{2 (-1)^{2/3} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{7/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{7/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{-1} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{7/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{b \cot (c+d x)}{a^2 d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}","\frac{2 (-1)^{2/3} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{7/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{7/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{-1} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{7/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{b \cot (c+d x)}{a^2 d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}",1,"(2*(-1)^(2/3)*b^(5/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(7/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) - (2*b^(5/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(7/3)*Sqrt[a^(2/3) - b^(2/3)]*d) + (2*(-1)^(1/3)*b^(5/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(7/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - (3*ArcTanh[Cos[c + d*x]])/(8*a*d) + (b*Cot[c + d*x])/(a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)","A",18,8,23,0.3478,1,"{3220, 3767, 8, 3768, 3770, 2660, 618, 204}"
189,1,293,0,0.3939496,"\int \frac{\sin ^6(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Sin[c + d*x]^6/(a + b*Sin[c + d*x]^3),x]","\frac{2 a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^2 d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b^2 d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^2 d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{a x}{b^2}+\frac{\cos ^3(c+d x)}{3 b d}-\frac{\cos (c+d x)}{b d}","\frac{2 a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^2 d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b^2 d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^2 d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{a x}{b^2}+\frac{\cos ^3(c+d x)}{3 b d}-\frac{\cos (c+d x)}{b d}",1,"-((a*x)/b^2) + (2*a^(4/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^2*d) + (2*a^(4/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b^2*d) - (2*a^(4/3)*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^2*d) - Cos[c + d*x]/(b*d) + Cos[c + d*x]^3/(3*b*d)","A",15,6,23,0.2609,1,"{3220, 2633, 3213, 2660, 618, 204}"
190,1,281,0,0.4222127,"\int \frac{\sin ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Sin[c + d*x]^4/(a + b*Sin[c + d*x]^3),x]","-\frac{2 (-1)^{2/3} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \sqrt[3]{-1} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{\cos (c+d x)}{b d}","-\frac{2 (-1)^{2/3} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \sqrt[3]{-1} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{\cos (c+d x)}{b d}",1,"(-2*(-1)^(2/3)*a^(2/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(4/3)*d) + (2*a^(2/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(4/3)*d) - (2*(-1)^(1/3)*a^(2/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b^(4/3)*d) - Cos[c + d*x]/(b*d)","A",14,5,23,0.2174,1,"{3220, 2638, 2660, 618, 204}"
191,1,240,0,0.272057,"\int \frac{\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Sin[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 b^{2/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}-\frac{2 \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 b^{2/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}-\frac{2 \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}",1,"(2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(2/3)*d) - (2*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]])/(3*Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]*b^(2/3)*d) - (2*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*b^(2/3)*d)","A",11,5,23,0.2174,1,"{3220, 2660, 618, 204, 206}"
192,1,245,0,0.2607079,"\int \frac{1}{a+b \sin ^3(c+d x)} \, dx","Int[(a + b*Sin[c + d*x]^3)^(-1),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}",1,"(2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - b^(2/3)]*d) + (2*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - (2*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d)","A",11,4,14,0.2857,1,"{3213, 2660, 618, 204}"
193,1,281,0,0.4285113,"\int \frac{\csc ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Csc[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]","-\frac{2 (-1)^{2/3} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \sqrt[3]{-1} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{\cot (c+d x)}{a d}","-\frac{2 (-1)^{2/3} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \sqrt[3]{-1} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{\cot (c+d x)}{a d}",1,"(-2*(-1)^(2/3)*b^(2/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(4/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) + (2*b^(2/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(4/3)*Sqrt[a^(2/3) - b^(2/3)]*d) - (2*(-1)^(1/3)*b^(2/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(4/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - Cot[c + d*x]/(a*d)","A",15,6,23,0.2609,1,"{3220, 3767, 8, 2660, 618, 204}"
194,1,296,0,0.3933317,"\int \frac{\csc ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Csc[c + d*x]^4/(a + b*Sin[c + d*x]^3),x]","\frac{2 b^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^2 d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 b^{4/3} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 a^2 d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}-\frac{2 b^{4/3} \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 a^2 d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}","\frac{2 b^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^2 d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 b^{4/3} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 a^2 d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}-\frac{2 b^{4/3} \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 a^2 d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}",1,"(2*b^(4/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^2*Sqrt[a^(2/3) - b^(2/3)]*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - (2*b^(4/3)*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]])/(3*a^2*Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]*d) - (2*b^(4/3)*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*a^2*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*d) - Cot[c + d*x]/(a*d) - Cot[c + d*x]^3/(3*a*d)","A",16,7,23,0.3043,1,"{3220, 3770, 3767, 2660, 618, 204, 206}"
195,1,177,0,0.2519467,"\int \frac{\sin ^9(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4),x]","-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{9/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{9/4} d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{(a+b) \cos (c+d x)}{b^2 d}+\frac{\cos ^5(c+d x)}{5 b d}-\frac{2 \cos ^3(c+d x)}{3 b d}","-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{9/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{9/4} d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{(a+b) \cos (c+d x)}{b^2 d}+\frac{\cos ^5(c+d x)}{5 b d}-\frac{2 \cos ^3(c+d x)}{3 b d}",1,"-(a^(3/2)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(9/4)*d) - (a^(3/2)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(9/4)*d) + ((a + b)*Cos[c + d*x])/(b^2*d) - (2*Cos[c + d*x]^3)/(3*b*d) + Cos[c + d*x]^5/(5*b*d)","A",6,5,24,0.2083,1,"{3215, 1170, 1093, 205, 208}"
196,1,148,0,0.1755875,"\int \frac{\sin ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4),x]","-\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{7/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{7/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\cos ^3(c+d x)}{3 b d}+\frac{\cos (c+d x)}{b d}","-\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{7/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{7/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\cos ^3(c+d x)}{3 b d}+\frac{\cos (c+d x)}{b d}",1,"-(a*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(7/4)*d) + (a*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(7/4)*d) + Cos[c + d*x]/(b*d) - Cos[c + d*x]^3/(3*b*d)","A",6,5,24,0.2083,1,"{3215, 1170, 1166, 205, 208}"
197,1,138,0,0.1785274,"\int \frac{\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]","-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{5/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{5/4} d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\cos (c+d x)}{b d}","-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{5/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{5/4} d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\cos (c+d x)}{b d}",1,"-(Sqrt[a]*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(5/4)*d) - (Sqrt[a]*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(5/4)*d) + Cos[c + d*x]/(b*d)","A",6,5,24,0.2083,1,"{3215, 1170, 1093, 205, 208}"
198,1,115,0,0.1171663,"\int \frac{\sin ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{3/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{3/4} d \sqrt{\sqrt{a}-\sqrt{b}}}","\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{3/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{3/4} d \sqrt{\sqrt{a}-\sqrt{b}}}",1,"-ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/4)*d) + ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/4)*d)","A",4,4,24,0.1667,1,"{3215, 1166, 205, 208}"
199,1,125,0,0.1020766,"\int \frac{\sin (c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sin[c + d*x]/(a - b*Sin[c + d*x]^4),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 \sqrt{a} \sqrt[4]{b} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 \sqrt{a} \sqrt[4]{b} d \sqrt{\sqrt{a}+\sqrt{b}}}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 \sqrt{a} \sqrt[4]{b} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 \sqrt{a} \sqrt[4]{b} d \sqrt{\sqrt{a}+\sqrt{b}}}",1,"-ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*Sqrt[a]*Sqrt[Sqrt[a] - Sqrt[b]]*b^(1/4)*d) - ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(2*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(1/4)*d)","A",4,4,22,0.1818,1,"{3215, 1093, 205, 208}"
200,1,136,0,0.1784447,"\int \frac{\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Csc[c + d*x]/(a - b*Sin[c + d*x]^4),x]","-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}","-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"-(b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[Cos[c + d*x]]/(a*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] + Sqrt[b]]*d)","A",7,6,22,0.2727,1,"{3215, 1170, 207, 1166, 205, 208}"
201,1,184,0,0.2099861,"\int \frac{\csc ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Csc[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]","-\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^{3/2} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^{3/2} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{1}{4 a d (1-\cos (c+d x))}+\frac{1}{4 a d (\cos (c+d x)+1)}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}","-\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^{3/2} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^{3/2} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{1}{4 a d (1-\cos (c+d x))}+\frac{1}{4 a d (\cos (c+d x)+1)}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}",1,"-(b^(3/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^(3/2)*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[Cos[c + d*x]]/(2*a*d) - (b^(3/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(3/2)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - 1/(4*a*d*(1 - Cos[c + d*x])) + 1/(4*a*d*(1 + Cos[c + d*x]))","A",7,6,24,0.2500,1,"{3215, 1170, 207, 1093, 205, 208}"
202,1,229,0,0.2465672,"\int \frac{\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Csc[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]","-\frac{b^{5/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^2 d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b^{5/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^2 d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{(3 a+8 b) \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{3}{16 a d (1-\cos (c+d x))}+\frac{3}{16 a d (\cos (c+d x)+1)}-\frac{1}{16 a d (1-\cos (c+d x))^2}+\frac{1}{16 a d (\cos (c+d x)+1)^2}","-\frac{b^{5/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^2 d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b^{5/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^2 d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{(3 a+8 b) \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{3}{16 a d (1-\cos (c+d x))}+\frac{3}{16 a d (\cos (c+d x)+1)}-\frac{1}{16 a d (1-\cos (c+d x))^2}+\frac{1}{16 a d (\cos (c+d x)+1)^2}",1,"-(b^(5/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ((3*a + 8*b)*ArcTanh[Cos[c + d*x]])/(8*a^2*d) + (b^(5/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] + Sqrt[b]]*d) - 1/(16*a*d*(1 - Cos[c + d*x])^2) - 3/(16*a*d*(1 - Cos[c + d*x])) + 1/(16*a*d*(1 + Cos[c + d*x])^2) + 3/(16*a*d*(1 + Cos[c + d*x]))","A",7,6,24,0.2500,1,"{3215, 1170, 207, 1166, 205, 208}"
203,1,184,0,0.2926746,"\int \frac{\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]","\frac{a^{5/4} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^2 d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{a^{5/4} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^2 d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{x (a+b)}{b^2}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}+\frac{5 \sin (c+d x) \cos (c+d x)}{8 b d}+\frac{5 x}{8 b}","\frac{a^{5/4} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^2 d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{a^{5/4} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^2 d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{x (a+b)}{b^2}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}+\frac{5 \sin (c+d x) \cos (c+d x)}{8 b d}+\frac{5 x}{8 b}",1,"(5*x)/(8*b) - ((a + b)*x)/b^2 + (a^(5/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^2*d) + (a^(5/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^2*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)","A",12,6,24,0.2500,1,"{3217, 1287, 199, 203, 1166, 205}"
204,1,155,0,0.204312,"\int \frac{\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]","\frac{a^{3/4} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^{3/2} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{a^{3/4} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^{3/2} d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\sin (c+d x) \cos (c+d x)}{2 b d}-\frac{x}{2 b}","\frac{a^{3/4} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^{3/2} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{a^{3/4} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^{3/2} d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\sin (c+d x) \cos (c+d x)}{2 b d}-\frac{x}{2 b}",1,"-x/(2*b) + (a^(3/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/2)*d) - (a^(3/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/2)*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",9,6,24,0.2500,1,"{3217, 1287, 199, 203, 1130, 205}"
205,1,127,0,0.1851669,"\int \frac{\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sin[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]","\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{x}{b}","\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{x}{b}",1,"-(x/b) + (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b*d) + (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b*d)","A",7,5,24,0.2083,1,"{3217, 1287, 203, 1166, 205}"
206,1,125,0,0.1123757,"\int \frac{\sin ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt[4]{a} \sqrt{b} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt[4]{a} \sqrt{b} d \sqrt{\sqrt{a}+\sqrt{b}}}","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt[4]{a} \sqrt{b} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt[4]{a} \sqrt{b} d \sqrt{\sqrt{a}+\sqrt{b}}}",1,"ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(2*a^(1/4)*Sqrt[Sqrt[a] - Sqrt[b]]*Sqrt[b]*d) - ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(2*a^(1/4)*Sqrt[Sqrt[a] + Sqrt[b]]*Sqrt[b]*d)","A",4,3,24,0.1250,1,"{3217, 1130, 205}"
207,1,115,0,0.0904601,"\int \frac{1}{a-b \sin ^4(c+d x)} \, dx","Int[(a - b*Sin[c + d*x]^4)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \sqrt{\sqrt{a}+\sqrt{b}}}","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \sqrt{\sqrt{a}+\sqrt{b}}}",1,"ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d)","A",4,3,15,0.2000,1,"{3209, 1166, 205}"
208,1,139,0,0.1781174,"\int \frac{\csc ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{5/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{5/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\cot (c+d x)}{a d}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{5/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{5/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\cot (c+d x)}{a d}",1,"(Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(5/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) - (Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(5/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - Cot[c + d*x]/(a*d)","A",6,4,24,0.1667,1,"{3217, 1287, 1130, 205}"
209,1,149,0,0.1934414,"\int \frac{\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Csc[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]","\frac{b \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{7/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{7/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}","\frac{b \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{7/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{7/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}",1,"(b*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(7/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + (b*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(7/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - Cot[c + d*x]/(a*d) - Cot[c + d*x]^3/(3*a*d)","A",6,4,24,0.1667,1,"{3217, 1287, 1166, 205}"
210,1,178,0,0.2029673,"\int \frac{\csc ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Csc[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{9/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{9/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{(a+b) \cot (c+d x)}{a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}-\frac{2 \cot ^3(c+d x)}{3 a d}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{9/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{9/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{(a+b) \cot (c+d x)}{a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}-\frac{2 \cot ^3(c+d x)}{3 a d}",1,"(b^(3/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(9/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) - (b^(3/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(9/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - ((a + b)*Cot[c + d*x])/(a^2*d) - (2*Cot[c + d*x]^3)/(3*a*d) - Cot[c + d*x]^5/(5*a*d)","A",6,4,24,0.1667,1,"{3217, 1287, 1130, 205}"
211,1,197,0,0.2353563,"\int \frac{\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Csc[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]","\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{11/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{11/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac{(a+b) \cot (c+d x)}{a^2 d}-\frac{\cot ^7(c+d x)}{7 a d}-\frac{3 \cot ^5(c+d x)}{5 a d}","\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{11/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{11/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{(3 a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac{(a+b) \cot (c+d x)}{a^2 d}-\frac{\cot ^7(c+d x)}{7 a d}-\frac{3 \cot ^5(c+d x)}{5 a d}",1,"(b^2*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(11/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + (b^2*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(11/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - ((a + b)*Cot[c + d*x])/(a^2*d) - ((3*a + b)*Cot[c + d*x]^3)/(3*a^2*d) - (3*Cot[c + d*x]^5)/(5*a*d) - Cot[c + d*x]^7/(7*a*d)","A",6,4,24,0.1667,1,"{3217, 1287, 1166, 205}"
212,1,236,0,0.4797938,"\int \frac{\sin ^9(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^2,x]","-\frac{a \cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{4 b^2 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}+\frac{\sqrt{a} \left(5 \sqrt{a}-6 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 b^{9/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\sqrt{a} \left(5 \sqrt{a}+6 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 b^{9/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cos (c+d x)}{b^2 d}","-\frac{a \cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{4 b^2 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}+\frac{\sqrt{a} \left(5 \sqrt{a}-6 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 b^{9/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\sqrt{a} \left(5 \sqrt{a}+6 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 b^{9/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cos (c+d x)}{b^2 d}",1,"(Sqrt[a]*(5*Sqrt[a] - 6*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*(Sqrt[a] - Sqrt[b])^(3/2)*b^(9/4)*d) + (Sqrt[a]*(5*Sqrt[a] + 6*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(9/4)*d) - Cos[c + d*x]/(b^2*d) - (a*Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(4*(a - b)*b^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",7,6,24,0.2500,1,"{3215, 1205, 1676, 1166, 205, 208}"
213,1,210,0,0.3349374,"\int \frac{\sin ^7(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4)^2,x]","\frac{\left(3 \sqrt{a}-4 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 b^{7/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\left(3 \sqrt{a}+4 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 b^{7/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{a \cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{4 b d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}","\frac{\left(3 \sqrt{a}-4 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 b^{7/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\left(3 \sqrt{a}+4 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 b^{7/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{a \cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{4 b d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}",1,"((3*Sqrt[a] - 4*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*(Sqrt[a] - Sqrt[b])^(3/2)*b^(7/4)*d) - ((3*Sqrt[a] + 4*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(7/4)*d) - (a*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*(a - b)*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",5,5,24,0.2083,1,"{3215, 1205, 1166, 205, 208}"
214,1,217,0,0.2641572,"\int \frac{\sin ^5(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^2,x]","\frac{\left(\sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 \sqrt{a} b^{5/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\left(\sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 \sqrt{a} b^{5/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{4 b d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}","\frac{\left(\sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 \sqrt{a} b^{5/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\left(\sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 \sqrt{a} b^{5/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{4 b d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}",1,"((Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(3/2)*b^(5/4)*d) + ((Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(3/2)*b^(5/4)*d) - (Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(4*(a - b)*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",5,5,24,0.2083,1,"{3215, 1205, 1166, 205, 208}"
215,1,186,0,0.1847633,"\int \frac{\sin ^3(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4)^2,x]","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 \sqrt{a} b^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 \sqrt{a} b^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{4 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 \sqrt{a} b^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 \sqrt{a} b^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{4 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}",1,"-ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(8*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/4)*d) + ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(8*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/4)*d) - (Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",5,5,24,0.2083,1,"{3215, 1178, 1166, 205, 208}"
216,1,221,0,0.2652999,"\int \frac{\sin (c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]/(a - b*Sin[c + d*x]^4)^2,x]","-\frac{\left(3 \sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 a^{3/2} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\left(3 \sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 a^{3/2} \sqrt[4]{b} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{4 a d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}","-\frac{\left(3 \sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 a^{3/2} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\left(3 \sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 a^{3/2} \sqrt[4]{b} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{4 a d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}",1,"-((3*Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] - Sqrt[b])^(3/2)*b^(1/4)*d) - ((3*Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] + Sqrt[b])^(3/2)*b^(1/4)*d) - (Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(4*a*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",5,5,22,0.2273,1,"{3215, 1092, 1166, 205, 208}"
217,1,325,0,0.3355044,"\int \frac{\csc (c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[Csc[c + d*x]/(a - b*Sin[c + d*x]^4)^2,x]","-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^2 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 a^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^2 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 a^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{b \cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{4 a d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}","-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^2 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 a^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^2 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 a^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{b \cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{4 a d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}",1,"-(b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] - Sqrt[b])^(3/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] + Sqrt[b]]*d) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*a*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",11,7,22,0.3182,1,"{3215, 1238, 207, 1178, 1166, 205, 208}"
218,1,320,0,0.446449,"\int \frac{\sin ^8(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^2,x]","\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^2 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^2 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\tan ^5(c+d x)}{4 b d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}-\frac{\tan (c+d x)}{4 b d (a-b)}+\frac{x}{b^2}","\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^2 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\sqrt[4]{a} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 b^2 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\tan ^5(c+d x)}{4 b d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}-\frac{\tan (c+d x)}{4 b d (a-b)}+\frac{x}{b^2}",1,"x/b^2 - (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^2*d) + (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/2)*d) - (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^2*d) - (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/2)*d) - Tan[c + d*x]/(4*(a - b)*b*d) + Tan[c + d*x]^5/(4*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",14,9,24,0.3750,1,"{3217, 1313, 1275, 12, 1122, 1166, 205, 1287, 203}"
219,1,233,0,0.3511639,"\int \frac{\sin ^6(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^2,x]","-\frac{\left(2 \sqrt{a}-3 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 \sqrt[4]{a} b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\left(2 \sqrt{a}+3 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 \sqrt[4]{a} b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\tan (c+d x)}{4 b d (a-b)}+\frac{\tan ^3(c+d x) \sec ^2(c+d x)}{4 b d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}","-\frac{\left(2 \sqrt{a}-3 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 \sqrt[4]{a} b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\left(2 \sqrt{a}+3 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 \sqrt[4]{a} b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\tan (c+d x)}{4 b d (a-b)}+\frac{\tan ^3(c+d x) \sec ^2(c+d x)}{4 b d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}",1,"-((2*Sqrt[a] - 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(1/4)*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/2)*d) + ((2*Sqrt[a] + 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(1/4)*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/2)*d) - Tan[c + d*x]/(4*(a - b)*b*d) + (Sec[c + d*x]^2*Tan[c + d*x]^3)/(4*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",6,5,24,0.2083,1,"{3217, 1120, 1279, 1166, 205}"
220,1,195,0,0.2294343,"\int \frac{\sin ^4(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^4/(a - b*Sin[c + d*x]^4)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{3/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{3/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}+\frac{\tan ^5(c+d x)}{4 a d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}-\frac{\tan (c+d x)}{4 a d (a-b)}","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{3/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{3/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}+\frac{\tan ^5(c+d x)}{4 a d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}-\frac{\tan (c+d x)}{4 a d (a-b)}",1,"ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(8*a^(3/4)*(Sqrt[a] - Sqrt[b])^(3/2)*Sqrt[b]*d) - ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(8*a^(3/4)*(Sqrt[a] + Sqrt[b])^(3/2)*Sqrt[b]*d) - Tan[c + d*x]/(4*a*(a - b)*d) + Tan[c + d*x]^5/(4*a*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",7,6,24,0.2500,1,"{3217, 1275, 12, 1122, 1166, 205}"
221,1,219,0,0.2974498,"\int \frac{\sin ^2(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4)^2,x]","\frac{\left(2 \sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{5/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\left(2 \sqrt{a}+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{5/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\tan (c+d x) \left((a+b) \tan ^2(c+d x)+a\right)}{4 a d (a-b) \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}","\frac{\left(2 \sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{5/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\left(2 \sqrt{a}+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{5/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\tan (c+d x) \left((a+b) \tan ^2(c+d x)+a\right)}{4 a d (a-b) \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}",1,"((2*Sqrt[a] - Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(5/4)*(Sqrt[a] - Sqrt[b])^(3/2)*Sqrt[b]*d) - ((2*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(5/4)*(Sqrt[a] + Sqrt[b])^(3/2)*Sqrt[b]*d) - (Tan[c + d*x]*(a + (a + b)*Tan[c + d*x]^2))/(4*a*(a - b)*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",5,4,24,0.1667,1,"{3217, 1333, 1166, 205}"
222,1,210,0,0.2603983,"\int \frac{1}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[(a - b*Sin[c + d*x]^4)^(-2),x]","\frac{\left(4 \sqrt{a}-3 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{7/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\left(4 \sqrt{a}+3 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{7/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{b \tan (c+d x) \left(2 \tan ^2(c+d x)+1\right)}{4 a d (a-b) \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}","\frac{\left(4 \sqrt{a}-3 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{7/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\left(4 \sqrt{a}+3 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{7/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{b \tan (c+d x) \left(2 \tan ^2(c+d x)+1\right)}{4 a d (a-b) \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}",1,"((4*Sqrt[a] - 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt[a] - Sqrt[b])^(3/2)*d) + ((4*Sqrt[a] + 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) - (b*Tan[c + d*x]*(1 + 2*Tan[c + d*x]^2))/(4*a*(a - b)*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",5,4,15,0.2667,1,"{3209, 1205, 1166, 205}"
223,1,236,0,0.5341785,"\int \frac{\csc ^2(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Int[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4)^2,x]","\frac{\sqrt{b} \left(6 \sqrt{a}-5 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{9/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\sqrt{b} \left(6 \sqrt{a}+5 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{9/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{b \tan (c+d x) \left((a+b) \tan ^2(c+d x)+a\right)}{4 a^2 d (a-b) \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}-\frac{\cot (c+d x)}{a^2 d}","\frac{\sqrt{b} \left(6 \sqrt{a}-5 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{9/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\sqrt{b} \left(6 \sqrt{a}+5 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 a^{9/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{b \tan (c+d x) \left((a+b) \tan ^2(c+d x)+a\right)}{4 a^2 d (a-b) \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}-\frac{\cot (c+d x)}{a^2 d}",1,"((6*Sqrt[a] - 5*Sqrt[b])*Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(9/4)*(Sqrt[a] - Sqrt[b])^(3/2)*d) - ((6*Sqrt[a] + 5*Sqrt[b])*Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(9/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) - Cot[c + d*x]/(a^2*d) - (b*Tan[c + d*x]*(a + (a + b)*Tan[c + d*x]^2))/(4*a^2*(a - b)*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",7,5,24,0.2083,1,"{3217, 1334, 1664, 1166, 205}"
224,1,315,0,0.5747591,"\int \frac{\sin ^9(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^3,x]","\frac{\cos (c+d x) \left(9 a^2-2 b (2 a-5 b) \cos ^2(c+d x)-11 a b-10 b^2\right)}{32 b^2 d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{a \cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{8 b^2 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}-\frac{\left(-14 \sqrt{a} \sqrt{b}+5 a+12 b\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 \sqrt{a} b^{9/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{\left(14 \sqrt{a} \sqrt{b}+5 a+12 b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 \sqrt{a} b^{9/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}","\frac{\cos (c+d x) \left(9 a^2-2 b (2 a-5 b) \cos ^2(c+d x)-11 a b-10 b^2\right)}{32 b^2 d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{a \cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{8 b^2 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}-\frac{\left(-14 \sqrt{a} \sqrt{b}+5 a+12 b\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 \sqrt{a} b^{9/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{\left(14 \sqrt{a} \sqrt{b}+5 a+12 b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 \sqrt{a} b^{9/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}",1,"-((5*a - 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(5/2)*b^(9/4)*d) - ((5*a + 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(9/4)*d) - (a*Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(8*(a - b)*b^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) + (Cos[c + d*x]*(9*a^2 - 11*a*b - 10*b^2 - 2*(2*a - 5*b)*b*Cos[c + d*x]^2))/(32*(a - b)^2*b^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",6,6,24,0.2500,1,"{3215, 1205, 1678, 1166, 205, 208}"
225,1,290,0,0.4347513,"\int \frac{\sin ^7(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4)^3,x]","\frac{3 \left(\sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 \sqrt{a} b^{7/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{3 \left(\sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 \sqrt{a} b^{7/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\cos (c+d x) \left(-3 (a-3 b) \cos ^2(c+d x)+5 a-17 b\right)}{32 b d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{a \cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{8 b d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}","\frac{3 \left(\sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 \sqrt{a} b^{7/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{3 \left(\sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 \sqrt{a} b^{7/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\cos (c+d x) \left(-3 (a-3 b) \cos ^2(c+d x)+5 a-17 b\right)}{32 b d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{a \cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{8 b d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}",1,"(3*(Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(5/2)*b^(7/4)*d) - (3*(Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(7/4)*d) - (a*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*(a - b)*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) + (Cos[c + d*x]*(5*a - 17*b - 3*(a - 3*b)*Cos[c + d*x]^2))/(32*(a - b)^2*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",6,6,24,0.2500,1,"{3215, 1205, 1178, 1166, 205, 208}"
226,1,313,0,0.4720613,"\int \frac{\sin ^5(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^3,x]","\frac{\cos (c+d x) \left(a^2+2 b (2 a+b) \cos ^2(c+d x)-11 a b-2 b^2\right)}{32 a b d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}+\frac{\left(-10 \sqrt{a} \sqrt{b}+3 a+4 b\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{3/2} b^{5/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(10 \sqrt{a} \sqrt{b}+3 a+4 b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{3/2} b^{5/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{8 b d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}","\frac{\cos (c+d x) \left(a^2+2 b (2 a+b) \cos ^2(c+d x)-11 a b-2 b^2\right)}{32 a b d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}+\frac{\left(-10 \sqrt{a} \sqrt{b}+3 a+4 b\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{3/2} b^{5/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(10 \sqrt{a} \sqrt{b}+3 a+4 b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{3/2} b^{5/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{8 b d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}",1,"((3*a - 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(5/4)*d) + ((3*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(5/4)*d) - (Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(8*(a - b)*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) + (Cos[c + d*x]*(a^2 - 11*a*b - 2*b^2 + 2*b*(2*a + b)*Cos[c + d*x]^2))/(32*a*(a - b)^2*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",6,6,24,0.2500,1,"{3215, 1205, 1178, 1166, 205, 208}"
227,1,288,0,0.4975251,"\int \frac{\sin ^3(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4)^3,x]","-\frac{\left(5 \sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{3/2} b^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(5 \sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{3/2} b^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\cos (c+d x) \left(-(5 a+b) \cos ^2(c+d x)+11 a+b\right)}{32 a d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{\cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{8 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}","-\frac{\left(5 \sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{3/2} b^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(5 \sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{3/2} b^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\cos (c+d x) \left(-(5 a+b) \cos ^2(c+d x)+11 a+b\right)}{32 a d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{\cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{8 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}",1,"-((5*Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(3/4)*d) + ((5*Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/4)*d) - (Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^2))/(32*a*(a - b)^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",6,5,24,0.2083,1,"{3215, 1178, 1166, 205, 208}"
228,1,313,0,0.4585559,"\int \frac{\sin (c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]/(a - b*Sin[c + d*x]^4)^3,x]","-\frac{\cos (c+d x) \left((7 a-3 b) (a+2 b)-6 b (2 a-b) \cos ^2(c+d x)\right)}{32 a^2 d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{3 \left(-10 \sqrt{a} \sqrt{b}+7 a+4 b\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{5/2} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{3 \left(10 \sqrt{a} \sqrt{b}+7 a+4 b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{5/2} \sqrt[4]{b} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{8 a d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}","-\frac{\cos (c+d x) \left((7 a-3 b) (a+2 b)-6 b (2 a-b) \cos ^2(c+d x)\right)}{32 a^2 d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{3 \left(-10 \sqrt{a} \sqrt{b}+7 a+4 b\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{5/2} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{3 \left(10 \sqrt{a} \sqrt{b}+7 a+4 b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{5/2} \sqrt[4]{b} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{8 a d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}",1,"(-3*(7*a - 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(1/4)*d) - (3*(7*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(1/4)*d) - (Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(8*a*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (Cos[c + d*x]*((7*a - 3*b)*(a + 2*b) - 6*(2*a - b)*b*Cos[c + d*x]^2))/(32*a^2*(a - b)^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",6,6,22,0.2727,1,"{3215, 1092, 1178, 1166, 205, 208}"
229,1,617,0,0.8368832,"\int \frac{\csc (c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[Csc[c + d*x]/(a - b*Sin[c + d*x]^4)^3,x]","-\frac{b \cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{4 a^2 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{b \cos (c+d x) \left(-(5 a+b) \cos ^2(c+d x)+11 a+b\right)}{32 a^2 d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^3 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 a^{5/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\sqrt[4]{b} \left(5 \sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{5/2} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\sqrt[4]{b} \left(5 \sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{5/2} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^3 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 a^{5/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{b \cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{8 a d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}","-\frac{b \cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{4 a^2 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{b \cos (c+d x) \left(-(5 a+b) \cos ^2(c+d x)+11 a+b\right)}{32 a^2 d (a-b)^2 \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^3 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 a^{5/2} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\sqrt[4]{b} \left(5 \sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{64 a^{5/2} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\sqrt[4]{b} \left(5 \sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{64 a^{5/2} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^3 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 a^{5/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{b \cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{8 a d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)^2}",1,"-((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] - Sqrt[b])^(5/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a] - Sqrt[b])^(3/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] + Sqrt[b]]*d) + ((5*Sqrt[a] + 2*Sqrt[b])*b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*a*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*a^2*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)) - (b*Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^2))/(32*a^2*(a - b)^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))","A",16,7,22,0.3182,1,"{3215, 1238, 207, 1178, 1166, 205, 208}"
230,1,319,0,0.5301705,"\int \frac{\sin ^8(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^3,x]","-\frac{\left(2 \sqrt{a}-5 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{3/4} b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(2 \sqrt{a}+5 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{3/4} b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\tan ^9(c+d x)}{8 a d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}+\frac{\tan ^3(c+d x)}{32 a b d (a-b)}-\frac{(a+5 b) \tan (c+d x)}{32 a b d (a-b)^2}-\frac{\tan ^5(c+d x) \sec ^2(c+d x)}{32 a b d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}","-\frac{\left(2 \sqrt{a}-5 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{3/4} b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(2 \sqrt{a}+5 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{3/4} b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\tan ^9(c+d x)}{8 a d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}+\frac{\tan ^3(c+d x)}{32 a b d (a-b)}-\frac{(a+5 b) \tan (c+d x)}{32 a b d (a-b)^2}-\frac{\tan ^5(c+d x) \sec ^2(c+d x)}{32 a b d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}",1,"-((2*Sqrt[a] - 5*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(3/4)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(3/2)*d) + ((2*Sqrt[a] + 5*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(3/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) - ((a + 5*b)*Tan[c + d*x])/(32*a*(a - b)^2*b*d) + Tan[c + d*x]^3/(32*a*(a - b)*b*d) + Tan[c + d*x]^9/(8*a*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Sec[c + d*x]^2*Tan[c + d*x]^5)/(32*a*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",9,7,24,0.2917,1,"{3217, 1275, 12, 1120, 1279, 1166, 205}"
231,1,343,0,0.7652677,"\int \frac{\sin ^6(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^3,x]","-\frac{\left(-10 \sqrt{a} \sqrt{b}+4 a+3 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{5/4} b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(10 \sqrt{a} \sqrt{b}+4 a+3 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{5/4} b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\tan (c+d x) \left(\frac{\left(2 a^2+15 a b+3 b^2\right) \tan ^2(c+d x)}{(a-b)^2}+\frac{2 a \left(a^2-a b-8 b^2\right)}{(a-b)^3}\right)}{32 a b d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}-\frac{\tan (c+d x) \left(\left(a^2+6 a b+b^2\right) \tan ^2(c+d x)+a (a+3 b)\right)}{8 d (a-b)^3 \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}","-\frac{\left(-10 \sqrt{a} \sqrt{b}+4 a+3 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{5/4} b^{3/2} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(10 \sqrt{a} \sqrt{b}+4 a+3 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{5/4} b^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\tan (c+d x) \left(\frac{\left(2 a^2+15 a b+3 b^2\right) \tan ^2(c+d x)}{(a-b)^2}+\frac{2 a \left(a^2-a b-8 b^2\right)}{(a-b)^3}\right)}{32 a b d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}-\frac{\tan (c+d x) \left(\left(a^2+6 a b+b^2\right) \tan ^2(c+d x)+a (a+3 b)\right)}{8 d (a-b)^3 \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}",1,"-((4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(5/4)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(3/2)*d) + ((4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(5/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) - (Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b + b^2)*Tan[c + d*x]^2))/(8*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Tan[c + d*x]*((2*a*(a^2 - a*b - 8*b^2))/(a - b)^3 + ((2*a^2 + 15*a*b + 3*b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",6,5,24,0.2083,1,"{3217, 1333, 1678, 1166, 205}"
232,1,313,0,0.6950442,"\int \frac{\sin ^4(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]^4/(a - b*Sin[c + d*x]^4)^3,x]","-\frac{\tan (c+d x) \left(\frac{9 a^2-24 a b-b^2}{(a-b)^3}+\frac{(17 a+3 b) \tan ^2(c+d x)}{(a-b)^2}\right)}{32 a d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}+\frac{3 \left(2 \sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{7/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{3 \left(2 \sqrt{a}+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{7/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{b \tan (c+d x) \left(4 (a+b) \tan ^2(c+d x)+3 a+b\right)}{8 d (a-b)^3 \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}","-\frac{\tan (c+d x) \left(\frac{9 a^2-24 a b-b^2}{(a-b)^3}+\frac{(17 a+3 b) \tan ^2(c+d x)}{(a-b)^2}\right)}{32 a d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}+\frac{3 \left(2 \sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{7/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{3 \left(2 \sqrt{a}+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{7/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{b \tan (c+d x) \left(4 (a+b) \tan ^2(c+d x)+3 a+b\right)}{8 d (a-b)^3 \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}",1,"(3*(2*Sqrt[a] - Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(7/4)*(Sqrt[a] - Sqrt[b])^(5/2)*Sqrt[b]*d) - (3*(2*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(7/4)*(Sqrt[a] + Sqrt[b])^(5/2)*Sqrt[b]*d) - (b*Tan[c + d*x]*(3*a + b + 4*(a + b)*Tan[c + d*x]^2))/(8*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Tan[c + d*x]*((9*a^2 - 24*a*b - b^2)/(a - b)^3 + ((17*a + 3*b)*Tan[c + d*x]^2)/(a - b)^2))/(32*a*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",6,5,24,0.2083,1,"{3217, 1333, 1678, 1166, 205}"
233,1,347,0,0.724284,"\int \frac{\sin ^2(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]","-\frac{\tan (c+d x) \left(\frac{5 \left(2 a^2+3 a b-b^2\right) \tan ^2(c+d x)}{(a-b)^2}+\frac{2 a \left(5 a^2-9 a b-4 b^2\right)}{(a-b)^3}\right)}{32 a^2 d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}-\frac{b \tan (c+d x) \left(\left(a^2+6 a b+b^2\right) \tan ^2(c+d x)+a (a+3 b)\right)}{8 a d (a-b)^3 \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}+\frac{\left(-14 \sqrt{a} \sqrt{b}+12 a+5 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{9/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{\left(14 \sqrt{a} \sqrt{b}+12 a+5 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{9/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}","-\frac{\tan (c+d x) \left(\frac{5 \left(2 a^2+3 a b-b^2\right) \tan ^2(c+d x)}{(a-b)^2}+\frac{2 a \left(5 a^2-9 a b-4 b^2\right)}{(a-b)^3}\right)}{32 a^2 d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}-\frac{b \tan (c+d x) \left(\left(a^2+6 a b+b^2\right) \tan ^2(c+d x)+a (a+3 b)\right)}{8 a d (a-b)^3 \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}+\frac{\left(-14 \sqrt{a} \sqrt{b}+12 a+5 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{9/4} \sqrt{b} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{\left(14 \sqrt{a} \sqrt{b}+12 a+5 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{9/4} \sqrt{b} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}",1,"((12*a - 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a] - Sqrt[b])^(5/2)*Sqrt[b]*d) - ((12*a + 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a] + Sqrt[b])^(5/2)*Sqrt[b]*d) - (b*Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b + b^2)*Tan[c + d*x]^2))/(8*a*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Tan[c + d*x]*((2*a*(5*a^2 - 9*a*b - 4*b^2))/(a - b)^3 + (5*(2*a^2 + 3*a*b - b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a^2*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",6,5,24,0.2083,1,"{3217, 1333, 1678, 1166, 205}"
234,1,319,0,0.6518391,"\int \frac{1}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[(a - b*Sin[c + d*x]^4)^(-3),x]","-\frac{b \tan (c+d x) \left(\frac{17 a^2-40 a b+7 b^2}{(a-b)^3}+\frac{(33 a-13 b) \tan ^2(c+d x)}{(a-b)^2}\right)}{32 a^2 d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}+\frac{\left(-50 \sqrt{a} \sqrt{b}+32 a+21 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{11/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(50 \sqrt{a} \sqrt{b}+32 a+21 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{11/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{b^2 \tan (c+d x) \left(4 (a+b) \tan ^2(c+d x)+3 a+b\right)}{8 a d (a-b)^3 \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}","-\frac{b \tan (c+d x) \left(\frac{17 a^2-40 a b+7 b^2}{(a-b)^3}+\frac{(33 a-13 b) \tan ^2(c+d x)}{(a-b)^2}\right)}{32 a^2 d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}+\frac{\left(-50 \sqrt{a} \sqrt{b}+32 a+21 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{11/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{\left(50 \sqrt{a} \sqrt{b}+32 a+21 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{11/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{b^2 \tan (c+d x) \left(4 (a+b) \tan ^2(c+d x)+3 a+b\right)}{8 a d (a-b)^3 \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}",1,"((32*a - 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(11/4)*(Sqrt[a] - Sqrt[b])^(5/2)*d) + ((32*a + 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(11/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b^2*Tan[c + d*x]*(3*a + b + 4*(a + b)*Tan[c + d*x]^2))/(8*a*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (b*Tan[c + d*x]*((17*a^2 - 40*a*b + 7*b^2)/(a - b)^3 + ((33*a - 13*b)*Tan[c + d*x]^2)/(a - b)^2))/(32*a^2*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",6,5,15,0.3333,1,"{3209, 1205, 1678, 1166, 205}"
235,1,357,0,1.294251,"\int \frac{\csc ^2(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Int[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]","-\frac{b^2 \tan (c+d x) \left(\left(a^2+6 a b+b^2\right) \tan ^2(c+d x)+a (a+3 b)\right)}{8 a^2 d (a-b)^3 \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}-\frac{b \tan (c+d x) \left(\frac{\left(18 a^2+15 a b-13 b^2\right) \tan ^2(c+d x)}{(a-b)^2}+\frac{2 a^2 (9 a-17 b)}{(a-b)^3}\right)}{32 a^3 d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}+\frac{3 \sqrt{b} \left(-34 \sqrt{a} \sqrt{b}+20 a+15 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{13/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{3 \sqrt{b} \left(34 \sqrt{a} \sqrt{b}+20 a+15 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{13/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\cot (c+d x)}{a^3 d}","-\frac{b^2 \tan (c+d x) \left(\left(a^2+6 a b+b^2\right) \tan ^2(c+d x)+a (a+3 b)\right)}{8 a^2 d (a-b)^3 \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)^2}-\frac{b \tan (c+d x) \left(\frac{\left(18 a^2+15 a b-13 b^2\right) \tan ^2(c+d x)}{(a-b)^2}+\frac{2 a^2 (9 a-17 b)}{(a-b)^3}\right)}{32 a^3 d \left((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}+\frac{3 \sqrt{b} \left(-34 \sqrt{a} \sqrt{b}+20 a+15 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{13/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}-\frac{3 \sqrt{b} \left(34 \sqrt{a} \sqrt{b}+20 a+15 b\right) \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{64 a^{13/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}-\frac{\cot (c+d x)}{a^3 d}",1,"(3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(13/4)*(Sqrt[a] - Sqrt[b])^(5/2)*d) - (3*Sqrt[b]*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(13/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - Cot[c + d*x]/(a^3*d) - (b^2*Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b + b^2)*Tan[c + d*x]^2))/(8*a^2*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (b*Tan[c + d*x]*((2*a^2*(9*a - 17*b))/(a - b)^3 + ((18*a^2 + 15*a*b - 13*b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))","A",8,6,24,0.2500,1,"{3217, 1334, 1669, 1664, 1166, 205}"
236,1,45,0,0.0192821,"\int \frac{1}{1-\sin ^4(x)} \, dx","Int[(1 - Sin[x]^4)^(-1),x]","\frac{x}{2 \sqrt{2}}+\frac{\tan (x)}{2}+\frac{\tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)}{2 \sqrt{2}}","\frac{\tan ^{-1}\left(\sqrt{2} \tan (x)\right)}{2 \sqrt{2}}+\frac{\tan (x)}{2}",1,"x/(2*Sqrt[2]) + ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]/(2*Sqrt[2]) + Tan[x]/2","A",3,3,10,0.3000,1,"{3209, 388, 203}"
237,1,487,0,1.1373998,"\int \frac{1}{a+b \sin ^4(x)} \, dx","Int[(a + b*Sin[x]^4)^(-1),x]","-\frac{\left(\sqrt{a+b}+\sqrt{a}\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}-\sqrt{2} (a+b)^{3/4} \tan (x)}{\sqrt[4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left(\sqrt{a+b}+\sqrt{a}\right) \tan ^{-1}\left(\frac{\sqrt{2} (a+b)^{3/4} \tan (x)+\sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}{\sqrt[4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left(\sqrt{a}-\sqrt{a+b}\right) \log \left((a+b)^{3/4} \tan ^2(x)-\sqrt{2} \sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \tan (x)+\sqrt{a} \sqrt[4]{a+b}\right)}{4 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}-\frac{\left(\sqrt{a}-\sqrt{a+b}\right) \log \left((a+b)^{3/4} \tan ^2(x)+\sqrt{2} \sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \tan (x)+\sqrt{a} \sqrt[4]{a+b}\right)}{4 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}","-\frac{\left(\sqrt{a+b}+\sqrt{a}\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}-\sqrt{2} (a+b)^{3/4} \tan (x)}{\sqrt[4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left(\sqrt{a+b}+\sqrt{a}\right) \tan ^{-1}\left(\frac{\sqrt{2} (a+b)^{3/4} \tan (x)+\sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}{\sqrt[4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left(\sqrt{a}-\sqrt{a+b}\right) \log \left((a+b)^{3/4} \tan ^2(x)-\sqrt{2} \sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \tan (x)+\sqrt{a} \sqrt[4]{a+b}\right)}{4 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}-\frac{\left(\sqrt{a}-\sqrt{a+b}\right) \log \left((a+b)^{3/4} \tan ^2(x)+\sqrt{2} \sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \tan (x)+\sqrt{a} \sqrt[4]{a+b}\right)}{4 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}",1,"-((Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] - Sqrt[2]*(a + b)^(3/4)*Tan[x])/(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) + ((Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] + Sqrt[2]*(a + b)^(3/4)*Tan[x])/(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) + ((Sqrt[a] - Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) - Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*Tan[x] + (a + b)^(3/4)*Tan[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]) - ((Sqrt[a] - Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) + Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*Tan[x] + (a + b)^(3/4)*Tan[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])","A",10,6,10,0.6000,1,"{3209, 1169, 634, 618, 204, 628}"
238,1,309,0,0.202052,"\int \frac{1}{1+\sin ^4(x)} \, dx","Int[(1 + Sin[x]^4)^(-1),x]","\frac{x}{2 \sqrt{\sqrt{2}-1}}-\frac{1}{8} \sqrt{\sqrt{2}-1} \log \left(2 \tan ^2(x)-2 \sqrt{\sqrt{2}-1} \tan (x)+\sqrt{2}\right)+\frac{1}{8} \sqrt{\sqrt{2}-1} \log \left(\sqrt{2} \tan ^2(x)+\sqrt{2 \left(\sqrt{2}-1\right)} \tan (x)+1\right)+\frac{\tan ^{-1}\left(\frac{-2 \sqrt{\sqrt{2}-1} \cos ^2(x)-\left(\sqrt{2}-2\right) \sin (x) \cos (x)+\sqrt{\sqrt{2}-1}}{\left(\sqrt{2}-2\right) \cos ^2(x)-2 \sqrt{\sqrt{2}-1} \sin (x) \cos (x)+\sqrt{1+\sqrt{2}}+2}\right)}{4 \sqrt{\sqrt{2}-1}}-\frac{\tan ^{-1}\left(\frac{-2 \sqrt{\sqrt{2}-1} \cos ^2(x)+\left(\sqrt{2}-2\right) \sin (x) \cos (x)+\sqrt{\sqrt{2}-1}}{\left(\sqrt{2}-2\right) \cos ^2(x)+2 \sqrt{\sqrt{2}-1} \sin (x) \cos (x)+\sqrt{1+\sqrt{2}}+2}\right)}{4 \sqrt{\sqrt{2}-1}}","\frac{x}{2 \sqrt{\sqrt{2}-1}}-\frac{1}{8} \sqrt{\sqrt{2}-1} \log \left(2 \tan ^2(x)-2 \sqrt{\sqrt{2}-1} \tan (x)+\sqrt{2}\right)+\frac{1}{8} \sqrt{\sqrt{2}-1} \log \left(\sqrt{2} \tan ^2(x)+\sqrt{2 \left(\sqrt{2}-1\right)} \tan (x)+1\right)+\frac{\tan ^{-1}\left(\frac{-2 \sqrt{\sqrt{2}-1} \cos ^2(x)-\left(\sqrt{2}-2\right) \sin (x) \cos (x)+\sqrt{\sqrt{2}-1}}{\left(\sqrt{2}-2\right) \cos ^2(x)-2 \sqrt{\sqrt{2}-1} \sin (x) \cos (x)+\sqrt{1+\sqrt{2}}+2}\right)}{4 \sqrt{\sqrt{2}-1}}-\frac{\tan ^{-1}\left(\frac{-2 \sqrt{\sqrt{2}-1} \cos ^2(x)+\left(\sqrt{2}-2\right) \sin (x) \cos (x)+\sqrt{\sqrt{2}-1}}{\left(\sqrt{2}-2\right) \cos ^2(x)+2 \sqrt{\sqrt{2}-1} \sin (x) \cos (x)+\sqrt{1+\sqrt{2}}+2}\right)}{4 \sqrt{\sqrt{2}-1}}",1,"x/(2*Sqrt[-1 + Sqrt[2]]) + ArcTan[(Sqrt[-1 + Sqrt[2]] - 2*Sqrt[-1 + Sqrt[2]]*Cos[x]^2 - (-2 + Sqrt[2])*Cos[x]*Sin[x])/(2 + Sqrt[1 + Sqrt[2]] + (-2 + Sqrt[2])*Cos[x]^2 - 2*Sqrt[-1 + Sqrt[2]]*Cos[x]*Sin[x])]/(4*Sqrt[-1 + Sqrt[2]]) - ArcTan[(Sqrt[-1 + Sqrt[2]] - 2*Sqrt[-1 + Sqrt[2]]*Cos[x]^2 + (-2 + Sqrt[2])*Cos[x]*Sin[x])/(2 + Sqrt[1 + Sqrt[2]] + (-2 + Sqrt[2])*Cos[x]^2 + 2*Sqrt[-1 + Sqrt[2]]*Cos[x]*Sin[x])]/(4*Sqrt[-1 + Sqrt[2]]) - (Sqrt[-1 + Sqrt[2]]*Log[Sqrt[2] - 2*Sqrt[-1 + Sqrt[2]]*Tan[x] + 2*Tan[x]^2])/8 + (Sqrt[-1 + Sqrt[2]]*Log[1 + Sqrt[2*(-1 + Sqrt[2])]*Tan[x] + Sqrt[2]*Tan[x]^2])/8","A",10,6,8,0.7500,1,"{3209, 1169, 634, 618, 204, 628}"
239,1,477,0,0.3882566,"\int \sin (c+d x) \sqrt{a+b \sin ^4(c+d x)} \, dx","Int[Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{\cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{3 d}+\frac{2 \sqrt{b} \cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{3 d \sqrt{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)}+\frac{(a+b)^{3/4} \left(\sqrt{b}-\sqrt{a+b}\right) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{3 \sqrt[4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{2 \sqrt[4]{b} (a+b)^{3/4} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{3 d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}","-\frac{\cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{3 d}+\frac{2 \sqrt{b} \cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{3 d \sqrt{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)}+\frac{(a+b)^{3/4} \left(\sqrt{b}-\sqrt{a+b}\right) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{3 \sqrt[4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{2 \sqrt[4]{b} (a+b)^{3/4} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{3 d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}",1,"-(Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(3*d) + (2*Sqrt[b]*Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(3*Sqrt[a + b]*d*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])) - (2*b^(1/4)*(a + b)^(3/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(3*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) + ((a + b)^(3/4)*(Sqrt[b] - Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(3*b^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])","A",5,5,23,0.2174,1,"{3215, 1091, 1197, 1103, 1195}"
240,1,521,0,0.6781191,"\int \csc (c+d x) \sqrt{a+b \sin ^4(c+d x)} \, dx","Int[Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\sqrt{b} \cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{d \sqrt{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)}+\frac{\sqrt{-a} \tan ^{-1}\left(\frac{\sqrt{-a} \cos (c+d x)}{\sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right)}{2 d}-\frac{\sqrt[4]{b} (a+b)^{3/4} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{\sqrt[4]{a+b} \left(\sqrt{b}-\sqrt{a+b}\right)^2 \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} \Pi \left(\frac{\left(\sqrt{b}+\sqrt{a+b}\right)^2}{4 \sqrt{b} \sqrt{a+b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{4 \sqrt[4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}","\frac{\sqrt{b} \cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{d \sqrt{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)}+\frac{\sqrt{-a} \tan ^{-1}\left(\frac{\sqrt{-a} \cos (c+d x)}{\sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right)}{2 d}-\frac{\sqrt[4]{b} (a+b)^{3/4} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{\sqrt[4]{a+b} \left(\sqrt{b}-\sqrt{a+b}\right)^2 \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} \Pi \left(\frac{\left(\sqrt{b}+\sqrt{a+b}\right)^2}{4 \sqrt{b} \sqrt{a+b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{4 \sqrt[4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}",1,"(Sqrt[-a]*ArcTan[(Sqrt[-a]*Cos[c + d*x])/Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]])/(2*d) + (Sqrt[b]*Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(Sqrt[a + b]*d*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])) - (b^(1/4)*(a + b)^(3/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - ((a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])^2*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticPi[(Sqrt[b] + Sqrt[a + b])^2/(4*Sqrt[b]*Sqrt[a + b]), 2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(4*b^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])","A",8,7,23,0.3043,1,"{3215, 1208, 1197, 1103, 1195, 1216, 1706}"
241,1,484,0,0.4258026,"\int \frac{\sin ^5(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Sin[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\sqrt[4]{a+b} \left(2 \sqrt{b} \sqrt{a+b}+a-2 b\right) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{6 b^{5/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{2 (a+b)^{3/4} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{3 b^{3/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{\cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{3 b d}+\frac{2 \cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{3 \sqrt{b} d \sqrt{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)}","\frac{\sqrt[4]{a+b} \left(2 \sqrt{b} \sqrt{a+b}+a-2 b\right) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{6 b^{5/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{2 (a+b)^{3/4} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{3 b^{3/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{\cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{3 b d}+\frac{2 \cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{3 \sqrt{b} d \sqrt{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)}",1,"-(Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(3*b*d) + (2*Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(3*Sqrt[b]*Sqrt[a + b]*d*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])) - (2*(a + b)^(3/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(3*b^(3/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) + ((a + b)^(1/4)*(a - 2*b + 2*Sqrt[b]*Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(6*b^(5/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])","A",5,5,25,0.2000,1,"{3215, 1206, 1197, 1103, 1195}"
242,1,431,0,0.3024929,"\int \frac{\sin ^3(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Sin[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{\sqrt[4]{a+b} \left(\sqrt{b}-\sqrt{a+b}\right) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{2 b^{3/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{(a+b)^{3/4} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{b^{3/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac{\cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{\sqrt{b} d \sqrt{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)}","-\frac{\sqrt[4]{a+b} \left(\sqrt{b}-\sqrt{a+b}\right) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{2 b^{3/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{(a+b)^{3/4} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{b^{3/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac{\cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{\sqrt{b} d \sqrt{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)}",1,"(Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(Sqrt[b]*Sqrt[a + b]*d*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])) - ((a + b)^(3/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(b^(3/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - ((a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(2*b^(3/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])","A",4,4,25,0.1600,1,"{3215, 1197, 1103, 1195}"
243,1,171,0,0.1045037,"\int \frac{\sin (c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Sin[c + d*x]/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{\sqrt[4]{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{2 \sqrt[4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}","-\frac{\sqrt[4]{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{2 \sqrt[4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}",1,"-((a + b)^(1/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(2*b^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])","A",2,2,23,0.08696,1,"{3215, 1103}"
244,1,469,0,0.4343113,"\int \frac{\csc (c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Csc[c + d*x]/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{-a} \cos (c+d x)}{\sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right)}{2 \sqrt{-a} d}+\frac{\sqrt[4]{b} \sqrt[4]{a+b} \left(\sqrt{b}-\sqrt{a+b}\right) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{2 a d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{\sqrt[4]{a+b} \left(\sqrt{b}-\sqrt{a+b}\right)^2 \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} \Pi \left(\frac{\left(\sqrt{b}+\sqrt{a+b}\right)^2}{4 \sqrt{b} \sqrt{a+b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{4 a \sqrt[4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{-a} \cos (c+d x)}{\sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right)}{2 \sqrt{-a} d}+\frac{\sqrt[4]{b} \sqrt[4]{a+b} \left(\sqrt{b}-\sqrt{a+b}\right) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{2 a d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{\sqrt[4]{a+b} \left(\sqrt{b}-\sqrt{a+b}\right)^2 \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} \Pi \left(\frac{\left(\sqrt{b}+\sqrt{a+b}\right)^2}{4 \sqrt{b} \sqrt{a+b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{4 a \sqrt[4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}",1,"-ArcTan[(Sqrt[-a]*Cos[c + d*x])/Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]]/(2*Sqrt[-a]*d) + (b^(1/4)*(a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(2*a*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - ((a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])^2*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticPi[(Sqrt[b] + Sqrt[a + b])^2/(4*Sqrt[b]*Sqrt[a + b]), 2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(4*a*b^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])","A",4,4,23,0.1739,1,"{3215, 1216, 1103, 1706}"
245,1,776,0,1.0433905,"\int \frac{\csc ^3(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Csc[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{\sqrt{b} \cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d \sqrt{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{-a} \cos (c+d x)}{\sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right)}{4 \sqrt{-a} d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d}-\frac{\sqrt[4]{b} \left(-\sqrt{b} \sqrt{a+b}+a+b\right) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{2 a d \sqrt[4]{a+b} \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac{\sqrt[4]{b} (a+b)^{3/4} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{2 a d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{\sqrt[4]{a+b} \left(\sqrt{b}-\sqrt{a+b}\right)^2 \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} \Pi \left(\frac{\left(\sqrt{b}+\sqrt{a+b}\right)^2}{4 \sqrt{b} \sqrt{a+b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{8 a \sqrt[4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}","-\frac{\sqrt{b} \cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d \sqrt{a+b} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)}-\frac{\tan ^{-1}\left(\frac{\sqrt{-a} \cos (c+d x)}{\sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right)}{4 \sqrt{-a} d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d}-\frac{\sqrt[4]{b} \left(-\sqrt{b} \sqrt{a+b}+a+b\right) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{2 a d \sqrt[4]{a+b} \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac{\sqrt[4]{b} (a+b)^{3/4} \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{2 a d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{\sqrt[4]{a+b} \left(\sqrt{b}-\sqrt{a+b}\right)^2 \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left(\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right)^2}} \Pi \left(\frac{\left(\sqrt{b}+\sqrt{a+b}\right)^2}{4 \sqrt{b} \sqrt{a+b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt[4]{a+b}}\right)|\frac{1}{2} \left(\frac{\sqrt{b}}{\sqrt{a+b}}+1\right)\right)}{8 a \sqrt[4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}",1,"-ArcTan[(Sqrt[-a]*Cos[c + d*x])/Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]]/(4*Sqrt[-a]*d) - (Sqrt[b]*Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(2*a*Sqrt[a + b]*d*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])) - (Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]*Cot[c + d*x]*Csc[c + d*x])/(2*a*d) + (b^(1/4)*(a + b)^(3/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(2*a*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - (b^(1/4)*(a + b - Sqrt[b]*Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(2*a*(a + b)^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - ((a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])^2*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticPi[(Sqrt[b] + Sqrt[a + b])^2/(4*Sqrt[b]*Sqrt[a + b]), 2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(8*a*b^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])","A",7,7,25,0.2800,1,"{3215, 1223, 1714, 1195, 1708, 1103, 1706}"
246,1,499,0,0.6649414,"\int \frac{\sin ^2(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Sin[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{\cos ^2(c+d x) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}}\right) \sqrt{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}}{2 \sqrt{b} d \sqrt{a+b \sin ^4(c+d x)}}-\frac{\sqrt[4]{a} \left(\sqrt{a+b}+\sqrt{a}\right) \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 b d \sqrt[4]{a+b} \sqrt{a+b \sin ^4(c+d x)}}+\frac{\left(\sqrt{a+b}+\sqrt{a}\right)^2 \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{a+b}\right)^2}{4 \sqrt{a} \sqrt{a+b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{4 \sqrt[4]{a} b d \sqrt[4]{a+b} \sqrt{a+b \sin ^4(c+d x)}}","-\frac{\cos ^2(c+d x) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}}\right) \sqrt{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}}{2 \sqrt{b} d \sqrt{a+b \sin ^4(c+d x)}}-\frac{\sqrt[4]{a} \left(\sqrt{a+b}+\sqrt{a}\right) \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 b d \sqrt[4]{a+b} \sqrt{a+b \sin ^4(c+d x)}}+\frac{\left(\sqrt{a+b}+\sqrt{a}\right)^2 \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{a+b}\right)^2}{4 \sqrt{a} \sqrt{a+b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{4 \sqrt[4]{a} b d \sqrt[4]{a+b} \sqrt{a+b \sin ^4(c+d x)}}",1,"-(ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4]]*Cos[c + d*x]^2*Sqrt[a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4])/(2*Sqrt[b]*d*Sqrt[a + b*Sin[c + d*x]^4]) - (a^(1/4)*(Sqrt[a] + Sqrt[a + b])*Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1 - Sqrt[a]/Sqrt[a + b])/2]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*b*(a + b)^(1/4)*d*Sqrt[a + b*Sin[c + d*x]^4]) + ((Sqrt[a] + Sqrt[a + b])^2*Cos[c + d*x]^2*EllipticPi[-(Sqrt[a] - Sqrt[a + b])^2/(4*Sqrt[a]*Sqrt[a + b]), 2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1 - Sqrt[a]/Sqrt[a + b])/2]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(4*a^(1/4)*b*(a + b)^(1/4)*d*Sqrt[a + b*Sin[c + d*x]^4])","A",4,4,25,0.1600,1,"{3219, 1319, 1103, 1706}"
247,1,162,0,0.0793673,"\int \frac{1}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[1/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 \sqrt[4]{a} d \sqrt[4]{a+b} \sqrt{a+b \sin ^4(c+d x)}}","\frac{\cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 \sqrt[4]{a} d \sqrt[4]{a+b} \sqrt{a+b \sin ^4(c+d x)}}",1,"(Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1 - Sqrt[a]/Sqrt[a + b])/2]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*a^(1/4)*(a + b)^(1/4)*d*Sqrt[a + b*Sin[c + d*x]^4])","A",2,2,16,0.1250,1,"{3210, 1103}"
248,1,493,0,0.4193354,"\int \frac{\csc ^2(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Csc[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\left(\sqrt{a} \sqrt{a+b}+a+b\right) \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 a^{3/4} d (a+b)^{3/4} \sqrt{a+b \sin ^4(c+d x)}}-\frac{\sqrt[4]{a+b} \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{a^{3/4} d \sqrt{a+b \sin ^4(c+d x)}}+\frac{\sqrt{a+b} \sin (c+d x) \cos (c+d x) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}{a d \sqrt{a+b \sin ^4(c+d x)} \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)}-\frac{\cos ^2(c+d x) \cot (c+d x) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}{a d \sqrt{a+b \sin ^4(c+d x)}}","\frac{\left(\sqrt{a} \sqrt{a+b}+a+b\right) \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 a^{3/4} d (a+b)^{3/4} \sqrt{a+b \sin ^4(c+d x)}}-\frac{\sqrt[4]{a+b} \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{a^{3/4} d \sqrt{a+b \sin ^4(c+d x)}}+\frac{\sqrt{a+b} \sin (c+d x) \cos (c+d x) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}{a d \sqrt{a+b \sin ^4(c+d x)} \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)}-\frac{\cos ^2(c+d x) \cot (c+d x) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}{a d \sqrt{a+b \sin ^4(c+d x)}}",1,"-((Cos[c + d*x]^2*Cot[c + d*x]*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4))/(a*d*Sqrt[a + b*Sin[c + d*x]^4])) + (Sqrt[a + b]*Cos[c + d*x]*Sin[c + d*x]*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4))/(a*d*Sqrt[a + b*Sin[c + d*x]^4]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)) - ((a + b)^(1/4)*Cos[c + d*x]^2*EllipticE[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1 - Sqrt[a]/Sqrt[a + b])/2]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(a^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4]) + ((a + b + Sqrt[a]*Sqrt[a + b])*Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1 - Sqrt[a]/Sqrt[a + b])/2]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*a^(3/4)*(a + b)^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4])","A",5,5,25,0.2000,1,"{3219, 1281, 1197, 1103, 1195}"
249,1,384,0,0.7143236,"\int \frac{1}{a+b \sin ^5(x)} \, dx","Int[(a + b*Sin[x]^5)^(-1),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{b}}{\sqrt{a^{2/5}-b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+(-1)^{2/5} \sqrt[5]{b}}{\sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+(-1)^{4/5} \sqrt[5]{b}}{\sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{3/5} \left((-1)^{2/5} \sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{b}\right)}{\sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{-1} \left((-1)^{4/5} \sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{b}\right)}{\sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{b}}{\sqrt{a^{2/5}-b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+(-1)^{2/5} \sqrt[5]{b}}{\sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+(-1)^{4/5} \sqrt[5]{b}}{\sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{3/5} \left((-1)^{2/5} \sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{b}\right)}{\sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{-1} \left((-1)^{4/5} \sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{b}\right)}{\sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}",1,"(2*ArcTan[(b^(1/5) + a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - b^(2/5)]) + (2*ArcTan[((-1)^(2/5)*b^(1/5) + a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - (-1)^(4/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - (-1)^(4/5)*b^(2/5)]) + (2*ArcTan[((-1)^(4/5)*b^(1/5) + a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) + (-1)^(3/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + (-1)^(3/5)*b^(2/5)]) - (2*ArcTan[((-1)^(3/5)*(b^(1/5) + (-1)^(2/5)*a^(1/5)*Tan[x/2]))/Sqrt[a^(2/5) + (-1)^(1/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + (-1)^(1/5)*b^(2/5)]) - (2*ArcTan[((-1)^(1/5)*(b^(1/5) + (-1)^(4/5)*a^(1/5)*Tan[x/2]))/Sqrt[a^(2/5) - (-1)^(2/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - (-1)^(2/5)*b^(2/5)])","A",17,4,10,0.4000,1,"{3213, 2660, 618, 204}"
250,1,171,0,0.2565203,"\int \frac{1}{a+b \sin ^6(x)} \, dx","Int[(a + b*Sin[x]^6)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+\sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+\sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-\sqrt[3]{-1} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-\sqrt[3]{-1} \sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b}}}","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+\sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+\sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-\sqrt[3]{-1} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-\sqrt[3]{-1} \sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b}}}",1,"ArcTan[(Sqrt[a^(1/3) + b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])","A",7,3,10,0.3000,1,"{3211, 3181, 203}"
251,1,245,0,0.5277576,"\int \frac{1}{a+b \sin ^8(x)} \, dx","Int[(a + b*Sin[x]^8)^(-1),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}-i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}-i \sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}+i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}+i \sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}+\sqrt[4]{b}} \tan (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}+\sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a \sqrt[4]{b}+(-a)^{5/4}} \tan (x)}{(-a)^{5/8}}\right)}{4 (-a)^{3/8} \sqrt{a \sqrt[4]{b}+(-a)^{5/4}}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}-\sqrt[4]{b}} \tan (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}-\sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}-i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}-i \sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}+i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}+i \sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}+\sqrt[4]{b}} \tan (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}+\sqrt[4]{b}}}",1,"-ArcTan[(Sqrt[(-a)^(1/4) - I*b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - I*b^(1/4)]) - ArcTan[(Sqrt[(-a)^(1/4) + I*b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) - ArcTan[(Sqrt[(-a)^(1/4) + b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)]) - ArcTan[(Sqrt[(-a)^(5/4) + a*b^(1/4)]*Tan[x])/(-a)^(5/8)]/(4*(-a)^(3/8)*Sqrt[(-a)^(5/4) + a*b^(1/4)])","A",9,3,10,0.3000,1,"{3211, 3181, 203}"
252,1,379,0,0.4767197,"\int \frac{1}{a-b \sin ^5(x)} \, dx","Int[(a - b*Sin[x]^5)^(-1),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{b}-\sqrt[5]{a} \tan \left(\frac{x}{2}\right)}{\sqrt{a^{2/5}-b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{2/5} \sqrt[5]{b}-\sqrt[5]{a} \tan \left(\frac{x}{2}\right)}{\sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{4/5} \sqrt[5]{b}-\sqrt[5]{a} \tan \left(\frac{x}{2}\right)}{\sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{-1} \sqrt[5]{b}}{\sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+(-1)^{3/5} \sqrt[5]{b}}{\sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{b}-\sqrt[5]{a} \tan \left(\frac{x}{2}\right)}{\sqrt{a^{2/5}-b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{2/5} \sqrt[5]{b}-\sqrt[5]{a} \tan \left(\frac{x}{2}\right)}{\sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{4/5} \sqrt[5]{b}-\sqrt[5]{a} \tan \left(\frac{x}{2}\right)}{\sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{-1} \sqrt[5]{b}}{\sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+(-1)^{3/5} \sqrt[5]{b}}{\sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}",1,"(-2*ArcTan[(b^(1/5) - a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - b^(2/5)]) - (2*ArcTan[((-1)^(2/5)*b^(1/5) - a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - (-1)^(4/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - (-1)^(4/5)*b^(2/5)]) - (2*ArcTan[((-1)^(4/5)*b^(1/5) - a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) + (-1)^(3/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + (-1)^(3/5)*b^(2/5)]) + (2*ArcTan[((-1)^(1/5)*b^(1/5) + a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - (-1)^(2/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - (-1)^(2/5)*b^(2/5)]) + (2*ArcTan[((-1)^(3/5)*b^(1/5) + a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) + (-1)^(1/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + (-1)^(1/5)*b^(2/5)])","A",17,4,11,0.3636,1,"{3213, 2660, 618, 204}"
253,1,175,0,0.2604252,"\int \frac{1}{a-b \sin ^6(x)} \, dx","Int[(a - b*Sin[x]^6)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-\sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-\sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b}}}","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-\sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-\sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b}}}",1,"ArcTan[(Sqrt[a^(1/3) - b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)])","A",7,3,11,0.2727,1,"{3211, 3181, 203}"
254,1,213,0,0.2178176,"\int \frac{1}{a-b \sin ^8(x)} \, dx","Int[(a - b*Sin[x]^8)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}-\sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}-\sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}-i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}-i \sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}+i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}+i \sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}+\sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}+\sqrt[4]{b}}}","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}-\sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}-\sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}-i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}-i \sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}+i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}+i \sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}+\sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}+\sqrt[4]{b}}}",1,"ArcTan[(Sqrt[a^(1/4) - b^(1/4)]*Tan[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) - b^(1/4)]) + ArcTan[(Sqrt[a^(1/4) - I*b^(1/4)]*Tan[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) - I*b^(1/4)]) + ArcTan[(Sqrt[a^(1/4) + I*b^(1/4)]*Tan[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) + I*b^(1/4)]) + ArcTan[(Sqrt[a^(1/4) + b^(1/4)]*Tan[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) + b^(1/4)])","A",9,3,11,0.2727,1,"{3211, 3181, 203}"
255,1,195,0,0.3826044,"\int \frac{1}{1+\sin ^5(x)} \, dx","Int[(1 + Sin[x]^5)^(-1),x]","\frac{2 \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+(-1)^{2/5}}{\sqrt{1-(-1)^{4/5}}}\right)}{5 \sqrt{1-(-1)^{4/5}}}+\frac{2 \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+(-1)^{4/5}}{\sqrt{1+(-1)^{3/5}}}\right)}{5 \sqrt{1+(-1)^{3/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{3/5} \left((-1)^{2/5} \tan \left(\frac{x}{2}\right)+1\right)}{\sqrt{1+\sqrt[5]{-1}}}\right)}{5 \sqrt{1+\sqrt[5]{-1}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{-1} \left((-1)^{4/5} \tan \left(\frac{x}{2}\right)+1\right)}{\sqrt{1-(-1)^{2/5}}}\right)}{5 \sqrt{1-(-1)^{2/5}}}-\frac{\cos (x)}{5 (\sin (x)+1)}","\frac{2 \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+(-1)^{2/5}}{\sqrt{1-(-1)^{4/5}}}\right)}{5 \sqrt{1-(-1)^{4/5}}}+\frac{2 \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+(-1)^{4/5}}{\sqrt{1+(-1)^{3/5}}}\right)}{5 \sqrt{1+(-1)^{3/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{3/5} \left((-1)^{2/5} \tan \left(\frac{x}{2}\right)+1\right)}{\sqrt{1+\sqrt[5]{-1}}}\right)}{5 \sqrt{1+\sqrt[5]{-1}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{-1} \left((-1)^{4/5} \tan \left(\frac{x}{2}\right)+1\right)}{\sqrt{1-(-1)^{2/5}}}\right)}{5 \sqrt{1-(-1)^{2/5}}}-\frac{\cos (x)}{5 (\sin (x)+1)}",1,"(2*ArcTan[((-1)^(2/5) + Tan[x/2])/Sqrt[1 - (-1)^(4/5)]])/(5*Sqrt[1 - (-1)^(4/5)]) + (2*ArcTan[((-1)^(4/5) + Tan[x/2])/Sqrt[1 + (-1)^(3/5)]])/(5*Sqrt[1 + (-1)^(3/5)]) - (2*ArcTan[((-1)^(3/5)*(1 + (-1)^(2/5)*Tan[x/2]))/Sqrt[1 + (-1)^(1/5)]])/(5*Sqrt[1 + (-1)^(1/5)]) - (2*ArcTan[((-1)^(1/5)*(1 + (-1)^(4/5)*Tan[x/2]))/Sqrt[1 - (-1)^(2/5)]])/(5*Sqrt[1 - (-1)^(2/5)]) - Cos[x]/(5*(1 + Sin[x]))","A",15,5,8,0.6250,1,"{3213, 2648, 2660, 618, 204}"
256,1,103,0,0.103638,"\int \frac{1}{1+\sin ^6(x)} \, dx","Int[(1 + Sin[x]^6)^(-1),x]","\frac{x}{3 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{1-\sqrt[3]{-1}} \tan (x)\right)}{3 \sqrt{1-\sqrt[3]{-1}}}+\frac{\tan ^{-1}\left(\sqrt{1+(-1)^{2/3}} \tan (x)\right)}{3 \sqrt{1+(-1)^{2/3}}}+\frac{\tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)}{3 \sqrt{2}}","\frac{x}{3 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{1-\sqrt[3]{-1}} \tan (x)\right)}{3 \sqrt{1-\sqrt[3]{-1}}}+\frac{\tan ^{-1}\left(\sqrt{1+(-1)^{2/3}} \tan (x)\right)}{3 \sqrt{1+(-1)^{2/3}}}+\frac{\tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)}{3 \sqrt{2}}",1,"x/(3*Sqrt[2]) + ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]/(3*Sqrt[2]) + ArcTan[Sqrt[1 - (-1)^(1/3)]*Tan[x]]/(3*Sqrt[1 - (-1)^(1/3)]) + ArcTan[Sqrt[1 + (-1)^(2/3)]*Tan[x]]/(3*Sqrt[1 + (-1)^(2/3)])","A",7,3,8,0.3750,1,"{3211, 3181, 203}"
257,1,129,0,0.2001968,"\int \frac{1}{1+\sin ^8(x)} \, dx","Int[(1 + Sin[x]^8)^(-1),x]","\frac{\tan ^{-1}\left(\sqrt{1-\sqrt[4]{-1}} \tan (x)\right)}{4 \sqrt{1-\sqrt[4]{-1}}}+\frac{\tan ^{-1}\left(\sqrt{1+\sqrt[4]{-1}} \tan (x)\right)}{4 \sqrt{1+\sqrt[4]{-1}}}+\frac{\tan ^{-1}\left(\sqrt{1-(-1)^{3/4}} \tan (x)\right)}{4 \sqrt{1-(-1)^{3/4}}}+\frac{\tan ^{-1}\left(\sqrt{1+(-1)^{3/4}} \tan (x)\right)}{4 \sqrt{1+(-1)^{3/4}}}","\frac{1}{8} \left(\sqrt{1+\sqrt{4-2 \sqrt{2}}}+\sqrt{2+2 \sqrt[4]{2}+2 \sqrt{1+\sqrt{2}}+2 \sqrt{2+\sqrt{2}}}+\sqrt{1+\sqrt{4+2 \sqrt{2}}}\right) \left(x-\tan ^{-1}(\tan (x))\right)+\frac{\tan ^{-1}\left(\sqrt{1-\sqrt[4]{-1}} \tan (x)\right)}{4 \sqrt{1-\sqrt[4]{-1}}}+\frac{\tan ^{-1}\left(\sqrt{1+\sqrt[4]{-1}} \tan (x)\right)}{4 \sqrt{1+\sqrt[4]{-1}}}+\frac{\tan ^{-1}\left(\sqrt{1-(-1)^{3/4}} \tan (x)\right)}{4 \sqrt{1-(-1)^{3/4}}}+\frac{\tan ^{-1}\left(\sqrt{1+(-1)^{3/4}} \tan (x)\right)}{4 \sqrt{1+(-1)^{3/4}}}",1,"ArcTan[Sqrt[1 - (-1)^(1/4)]*Tan[x]]/(4*Sqrt[1 - (-1)^(1/4)]) + ArcTan[Sqrt[1 + (-1)^(1/4)]*Tan[x]]/(4*Sqrt[1 + (-1)^(1/4)]) + ArcTan[Sqrt[1 - (-1)^(3/4)]*Tan[x]]/(4*Sqrt[1 - (-1)^(3/4)]) + ArcTan[Sqrt[1 + (-1)^(3/4)]*Tan[x]]/(4*Sqrt[1 + (-1)^(3/4)])","A",9,3,8,0.3750,1,"{3211, 3181, 203}"
258,1,187,0,0.2685139,"\int \frac{1}{1-\sin ^5(x)} \, dx","Int[(1 - Sin[x]^5)^(-1),x]","-\frac{2 \tan ^{-1}\left(\frac{(-1)^{2/5}-\tan \left(\frac{x}{2}\right)}{\sqrt{1-(-1)^{4/5}}}\right)}{5 \sqrt{1-(-1)^{4/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{4/5}-\tan \left(\frac{x}{2}\right)}{\sqrt{1+(-1)^{3/5}}}\right)}{5 \sqrt{1+(-1)^{3/5}}}+\frac{2 \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+\sqrt[5]{-1}}{\sqrt{1-(-1)^{2/5}}}\right)}{5 \sqrt{1-(-1)^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+(-1)^{3/5}}{\sqrt{1+\sqrt[5]{-1}}}\right)}{5 \sqrt{1+\sqrt[5]{-1}}}+\frac{\cos (x)}{5 (1-\sin (x))}","-\frac{2 \tan ^{-1}\left(\frac{(-1)^{2/5}-\tan \left(\frac{x}{2}\right)}{\sqrt{1-(-1)^{4/5}}}\right)}{5 \sqrt{1-(-1)^{4/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{4/5}-\tan \left(\frac{x}{2}\right)}{\sqrt{1+(-1)^{3/5}}}\right)}{5 \sqrt{1+(-1)^{3/5}}}+\frac{2 \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+\sqrt[5]{-1}}{\sqrt{1-(-1)^{2/5}}}\right)}{5 \sqrt{1-(-1)^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+(-1)^{3/5}}{\sqrt{1+\sqrt[5]{-1}}}\right)}{5 \sqrt{1+\sqrt[5]{-1}}}+\frac{\cos (x)}{5 (1-\sin (x))}",1,"(-2*ArcTan[((-1)^(2/5) - Tan[x/2])/Sqrt[1 - (-1)^(4/5)]])/(5*Sqrt[1 - (-1)^(4/5)]) - (2*ArcTan[((-1)^(4/5) - Tan[x/2])/Sqrt[1 + (-1)^(3/5)]])/(5*Sqrt[1 + (-1)^(3/5)]) + (2*ArcTan[((-1)^(1/5) + Tan[x/2])/Sqrt[1 - (-1)^(2/5)]])/(5*Sqrt[1 - (-1)^(2/5)]) + (2*ArcTan[((-1)^(3/5) + Tan[x/2])/Sqrt[1 + (-1)^(1/5)]])/(5*Sqrt[1 + (-1)^(1/5)]) + Cos[x]/(5*(1 - Sin[x]))","A",15,5,10,0.5000,1,"{3213, 2648, 2660, 618, 204}"
259,1,71,0,0.135382,"\int \frac{1}{1-\sin ^6(x)} \, dx","Int[(1 - Sin[x]^6)^(-1),x]","\frac{\tan ^{-1}\left(\sqrt{1+\sqrt[3]{-1}} \tan (x)\right)}{3 \sqrt{1+\sqrt[3]{-1}}}+\frac{\tan ^{-1}\left(\sqrt{1-(-1)^{2/3}} \tan (x)\right)}{3 \sqrt{1-(-1)^{2/3}}}+\frac{\tan (x)}{3}","\frac{\tan ^{-1}\left(\sqrt{1+\sqrt[3]{-1}} \tan (x)\right)}{3 \sqrt{1+\sqrt[3]{-1}}}+\frac{\tan ^{-1}\left(\sqrt{1-(-1)^{2/3}} \tan (x)\right)}{3 \sqrt{1-(-1)^{2/3}}}+\frac{\tan (x)}{3}",1,"ArcTan[Sqrt[1 + (-1)^(1/3)]*Tan[x]]/(3*Sqrt[1 + (-1)^(1/3)]) + ArcTan[Sqrt[1 - (-1)^(2/3)]*Tan[x]]/(3*Sqrt[1 - (-1)^(2/3)]) + Tan[x]/3","A",8,6,10,0.6000,1,"{3211, 3181, 203, 3175, 3767, 8}"
260,1,89,0,0.0770086,"\int \frac{1}{1-\sin ^8(x)} \, dx","Int[(1 - Sin[x]^8)^(-1),x]","\frac{x}{4 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{1-i} \tan (x)\right)}{4 \sqrt{1-i}}+\frac{\tan ^{-1}\left(\sqrt{1+i} \tan (x)\right)}{4 \sqrt{1+i}}+\frac{\tan (x)}{4}+\frac{\tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)}{4 \sqrt{2}}","\frac{x}{4 \sqrt{2}}+\frac{\tan ^{-1}\left(\sqrt{1-i} \tan (x)\right)}{4 \sqrt{1-i}}+\frac{\tan ^{-1}\left(\sqrt{1+i} \tan (x)\right)}{4 \sqrt{1+i}}+\frac{\tan (x)}{4}+\frac{\tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)}{4 \sqrt{2}}",1,"x/(4*Sqrt[2]) + ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]/(4*Sqrt[2]) + ArcTan[Sqrt[1 - I]*Tan[x]]/(4*Sqrt[1 - I]) + ArcTan[Sqrt[1 + I]*Tan[x]]/(4*Sqrt[1 + I]) + Tan[x]/4","A",10,6,10,0.6000,1,"{3211, 3181, 203, 3175, 3767, 8}"
261,1,38,0,0.0538224,"\int \frac{\cos ^9(x)}{a-a \sin ^2(x)} \, dx","Int[Cos[x]^9/(a - a*Sin[x]^2),x]","-\frac{\sin ^7(x)}{7 a}+\frac{3 \sin ^5(x)}{5 a}-\frac{\sin ^3(x)}{a}+\frac{\sin (x)}{a}","-\frac{\sin ^7(x)}{7 a}+\frac{3 \sin ^5(x)}{5 a}-\frac{\sin ^3(x)}{a}+\frac{\sin (x)}{a}",1,"Sin[x]/a - Sin[x]^3/a + (3*Sin[x]^5)/(5*a) - Sin[x]^7/(7*a)","A",3,2,16,0.1250,1,"{3175, 2633}"
262,1,29,0,0.0523097,"\int \frac{\cos ^7(x)}{a-a \sin ^2(x)} \, dx","Int[Cos[x]^7/(a - a*Sin[x]^2),x]","\frac{\sin ^5(x)}{5 a}-\frac{2 \sin ^3(x)}{3 a}+\frac{\sin (x)}{a}","\frac{\sin ^5(x)}{5 a}-\frac{2 \sin ^3(x)}{3 a}+\frac{\sin (x)}{a}",1,"Sin[x]/a - (2*Sin[x]^3)/(3*a) + Sin[x]^5/(5*a)","A",3,2,16,0.1250,1,"{3175, 2633}"
263,1,18,0,0.0496366,"\int \frac{\cos ^5(x)}{a-a \sin ^2(x)} \, dx","Int[Cos[x]^5/(a - a*Sin[x]^2),x]","\frac{\sin (x)}{a}-\frac{\sin ^3(x)}{3 a}","\frac{\sin (x)}{a}-\frac{\sin ^3(x)}{3 a}",1,"Sin[x]/a - Sin[x]^3/(3*a)","A",3,2,16,0.1250,1,"{3175, 2633}"
264,1,6,0,0.0430568,"\int \frac{\cos ^3(x)}{a-a \sin ^2(x)} \, dx","Int[Cos[x]^3/(a - a*Sin[x]^2),x]","\frac{\sin (x)}{a}","\frac{\sin (x)}{a}",1,"Sin[x]/a","A",2,2,16,0.1250,1,"{3175, 2637}"
265,1,7,0,0.0260759,"\int \frac{\cos (x)}{a-a \sin ^2(x)} \, dx","Int[Cos[x]/(a - a*Sin[x]^2),x]","\frac{\tanh ^{-1}(\sin (x))}{a}","\frac{\tanh ^{-1}(\sin (x))}{a}",1,"ArcTanh[Sin[x]]/a","A",2,2,14,0.1429,1,"{3175, 3770}"
266,1,35,0,0.0583,"\int \frac{\sec ^3(x)}{a-a \sin ^2(x)} \, dx","Int[Sec[x]^3/(a - a*Sin[x]^2),x]","\frac{3 \tanh ^{-1}(\sin (x))}{8 a}+\frac{\tan (x) \sec ^3(x)}{4 a}+\frac{3 \tan (x) \sec (x)}{8 a}","\frac{3 \tanh ^{-1}(\sin (x))}{8 a}+\frac{\tan (x) \sec ^3(x)}{4 a}+\frac{3 \tan (x) \sec (x)}{8 a}",1,"(3*ArcTanh[Sin[x]])/(8*a) + (3*Sec[x]*Tan[x])/(8*a) + (Sec[x]^3*Tan[x])/(4*a)","A",4,3,16,0.1875,1,"{3175, 3768, 3770}"
267,1,33,0,0.055304,"\int \frac{\cos ^6(x)}{a-a \sin ^2(x)} \, dx","Int[Cos[x]^6/(a - a*Sin[x]^2),x]","\frac{3 x}{8 a}+\frac{\sin (x) \cos ^3(x)}{4 a}+\frac{3 \sin (x) \cos (x)}{8 a}","\frac{3 x}{8 a}+\frac{\sin (x) \cos ^3(x)}{4 a}+\frac{3 \sin (x) \cos (x)}{8 a}",1,"(3*x)/(8*a) + (3*Cos[x]*Sin[x])/(8*a) + (Cos[x]^3*Sin[x])/(4*a)","A",4,3,16,0.1875,1,"{3175, 2635, 8}"
268,1,20,0,0.0479074,"\int \frac{\cos ^4(x)}{a-a \sin ^2(x)} \, dx","Int[Cos[x]^4/(a - a*Sin[x]^2),x]","\frac{x}{2 a}+\frac{\sin (x) \cos (x)}{2 a}","\frac{x}{2 a}+\frac{\sin (x) \cos (x)}{2 a}",1,"x/(2*a) + (Cos[x]*Sin[x])/(2*a)","A",3,3,16,0.1875,1,"{3175, 2635, 8}"
269,1,5,0,0.040586,"\int \frac{\cos ^2(x)}{a-a \sin ^2(x)} \, dx","Int[Cos[x]^2/(a - a*Sin[x]^2),x]","\frac{x}{a}","\frac{x}{a}",1,"x/a","A",2,2,16,0.1250,1,"{3175, 8}"
270,1,22,0,0.0434152,"\int \frac{\sec (x)}{a-a \sin ^2(x)} \, dx","Int[Sec[x]/(a - a*Sin[x]^2),x]","\frac{\tanh ^{-1}(\sin (x))}{2 a}+\frac{\tan (x) \sec (x)}{2 a}","\frac{\tanh ^{-1}(\sin (x))}{2 a}+\frac{\tan (x) \sec (x)}{2 a}",1,"ArcTanh[Sin[x]]/(2*a) + (Sec[x]*Tan[x])/(2*a)","A",3,3,14,0.2143,1,"{3175, 3768, 3770}"
271,1,18,0,0.0494037,"\int \frac{\sec ^2(x)}{a-a \sin ^2(x)} \, dx","Int[Sec[x]^2/(a - a*Sin[x]^2),x]","\frac{\tan ^3(x)}{3 a}+\frac{\tan (x)}{a}","\frac{\tan ^3(x)}{3 a}+\frac{\tan (x)}{a}",1,"Tan[x]/a + Tan[x]^3/(3*a)","A",3,2,16,0.1250,1,"{3175, 3767}"
272,1,29,0,0.0516302,"\int \frac{\sec ^4(x)}{a-a \sin ^2(x)} \, dx","Int[Sec[x]^4/(a - a*Sin[x]^2),x]","\frac{\tan ^5(x)}{5 a}+\frac{2 \tan ^3(x)}{3 a}+\frac{\tan (x)}{a}","\frac{\tan ^5(x)}{5 a}+\frac{2 \tan ^3(x)}{3 a}+\frac{\tan (x)}{a}",1,"Tan[x]/a + (2*Tan[x]^3)/(3*a) + Tan[x]^5/(5*a)","A",3,2,16,0.1250,1,"{3175, 3767}"
273,1,29,0,0.0461488,"\int \frac{\cos ^9(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^9/(a - a*Sin[x]^2)^2,x]","\frac{\sin ^5(x)}{5 a^2}-\frac{2 \sin ^3(x)}{3 a^2}+\frac{\sin (x)}{a^2}","\frac{\sin ^5(x)}{5 a^2}-\frac{2 \sin ^3(x)}{3 a^2}+\frac{\sin (x)}{a^2}",1,"Sin[x]/a^2 - (2*Sin[x]^3)/(3*a^2) + Sin[x]^5/(5*a^2)","A",3,2,16,0.1250,1,"{3175, 2633}"
274,1,18,0,0.044215,"\int \frac{\cos ^7(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^7/(a - a*Sin[x]^2)^2,x]","\frac{\sin (x)}{a^2}-\frac{\sin ^3(x)}{3 a^2}","\frac{\sin (x)}{a^2}-\frac{\sin ^3(x)}{3 a^2}",1,"Sin[x]/a^2 - Sin[x]^3/(3*a^2)","A",3,2,16,0.1250,1,"{3175, 2633}"
275,1,6,0,0.0385285,"\int \frac{\cos ^5(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^5/(a - a*Sin[x]^2)^2,x]","\frac{\sin (x)}{a^2}","\frac{\sin (x)}{a^2}",1,"Sin[x]/a^2","A",2,2,16,0.1250,1,"{3175, 2637}"
276,1,7,0,0.0393184,"\int \frac{\cos ^3(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^3/(a - a*Sin[x]^2)^2,x]","\frac{\tanh ^{-1}(\sin (x))}{a^2}","\frac{\tanh ^{-1}(\sin (x))}{a^2}",1,"ArcTanh[Sin[x]]/a^2","A",2,2,16,0.1250,1,"{3175, 3770}"
277,1,22,0,0.0316159,"\int \frac{\cos (x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Cos[x]/(a - a*Sin[x]^2)^2,x]","\frac{\tanh ^{-1}(\sin (x))}{2 a^2}+\frac{\tan (x) \sec (x)}{2 a^2}","\frac{\tanh ^{-1}(\sin (x))}{2 a^2}+\frac{\tan (x) \sec (x)}{2 a^2}",1,"ArcTanh[Sin[x]]/(2*a^2) + (Sec[x]*Tan[x])/(2*a^2)","A",3,3,14,0.2143,1,"{3175, 3768, 3770}"
278,1,35,0,0.0480322,"\int \frac{\sec (x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Sec[x]/(a - a*Sin[x]^2)^2,x]","\frac{3 \tanh ^{-1}(\sin (x))}{8 a^2}+\frac{\tan (x) \sec ^3(x)}{4 a^2}+\frac{3 \tan (x) \sec (x)}{8 a^2}","\frac{3 \tanh ^{-1}(\sin (x))}{8 a^2}+\frac{\tan (x) \sec ^3(x)}{4 a^2}+\frac{3 \tan (x) \sec (x)}{8 a^2}",1,"(3*ArcTanh[Sin[x]])/(8*a^2) + (3*Sec[x]*Tan[x])/(8*a^2) + (Sec[x]^3*Tan[x])/(4*a^2)","A",4,3,14,0.2143,1,"{3175, 3768, 3770}"
279,1,33,0,0.0501656,"\int \frac{\cos ^8(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^8/(a - a*Sin[x]^2)^2,x]","\frac{3 x}{8 a^2}+\frac{\sin (x) \cos ^3(x)}{4 a^2}+\frac{3 \sin (x) \cos (x)}{8 a^2}","\frac{3 x}{8 a^2}+\frac{\sin (x) \cos ^3(x)}{4 a^2}+\frac{3 \sin (x) \cos (x)}{8 a^2}",1,"(3*x)/(8*a^2) + (3*Cos[x]*Sin[x])/(8*a^2) + (Cos[x]^3*Sin[x])/(4*a^2)","A",4,3,16,0.1875,1,"{3175, 2635, 8}"
280,1,20,0,0.0434516,"\int \frac{\cos ^6(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^6/(a - a*Sin[x]^2)^2,x]","\frac{x}{2 a^2}+\frac{\sin (x) \cos (x)}{2 a^2}","\frac{x}{2 a^2}+\frac{\sin (x) \cos (x)}{2 a^2}",1,"x/(2*a^2) + (Cos[x]*Sin[x])/(2*a^2)","A",3,3,16,0.1875,1,"{3175, 2635, 8}"
281,1,5,0,0.037163,"\int \frac{\cos ^4(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^4/(a - a*Sin[x]^2)^2,x]","\frac{x}{a^2}","\frac{x}{a^2}",1,"x/a^2","A",2,2,16,0.1250,1,"{3175, 8}"
282,1,6,0,0.0430509,"\int \frac{\cos ^2(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^2/(a - a*Sin[x]^2)^2,x]","\frac{\tan (x)}{a^2}","\frac{\tan (x)}{a^2}",1,"Tan[x]/a^2","A",3,3,16,0.1875,1,"{3175, 3767, 8}"
283,1,29,0,0.046838,"\int \frac{\sec ^2(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Sec[x]^2/(a - a*Sin[x]^2)^2,x]","\frac{\tan ^5(x)}{5 a^2}+\frac{2 \tan ^3(x)}{3 a^2}+\frac{\tan (x)}{a^2}","\frac{\tan ^5(x)}{5 a^2}+\frac{2 \tan ^3(x)}{3 a^2}+\frac{\tan (x)}{a^2}",1,"Tan[x]/a^2 + (2*Tan[x]^3)/(3*a^2) + Tan[x]^5/(5*a^2)","A",3,2,16,0.1250,1,"{3175, 3767}"
284,1,37,0,0.0481523,"\int \frac{\sec ^4(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Int[Sec[x]^4/(a - a*Sin[x]^2)^2,x]","\frac{\tan ^7(x)}{7 a^2}+\frac{3 \tan ^5(x)}{5 a^2}+\frac{\tan ^3(x)}{a^2}+\frac{\tan (x)}{a^2}","\frac{\tan ^7(x)}{7 a^2}+\frac{3 \tan ^5(x)}{5 a^2}+\frac{\tan ^3(x)}{a^2}+\frac{\tan (x)}{a^2}",1,"Tan[x]/a^2 + Tan[x]^3/a^2 + (3*Tan[x]^5)/(5*a^2) + Tan[x]^7/(7*a^2)","A",3,2,16,0.1250,1,"{3175, 3767}"
285,1,109,0,0.0645058,"\int \cos ^6(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Int[Cos[e + f*x]^6*(a + b*Sin[e + f*x]^2),x]","\frac{(8 a+b) \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{5 (8 a+b) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{5 (8 a+b) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{5}{128} x (8 a+b)-\frac{b \sin (e+f x) \cos ^7(e+f x)}{8 f}","\frac{(8 a+b) \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{5 (8 a+b) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{5 (8 a+b) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{5}{128} x (8 a+b)-\frac{b \sin (e+f x) \cos ^7(e+f x)}{8 f}",1,"(5*(8*a + b)*x)/128 + (5*(8*a + b)*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (5*(8*a + b)*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + ((8*a + b)*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) - (b*Cos[e + f*x]^7*Sin[e + f*x])/(8*f)","A",6,4,21,0.1905,1,"{3191, 385, 199, 203}"
286,1,83,0,0.0511575,"\int \cos ^4(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Int[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2),x]","\frac{(6 a+b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{(6 a+b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x (6 a+b)-\frac{b \sin (e+f x) \cos ^5(e+f x)}{6 f}","\frac{(6 a+b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{(6 a+b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x (6 a+b)-\frac{b \sin (e+f x) \cos ^5(e+f x)}{6 f}",1,"((6*a + b)*x)/16 + ((6*a + b)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + ((6*a + b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - (b*Cos[e + f*x]^5*Sin[e + f*x])/(6*f)","A",5,4,21,0.1905,1,"{3191, 385, 199, 203}"
287,1,57,0,0.0444805,"\int \cos ^2(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2),x]","\frac{(4 a+b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} x (4 a+b)-\frac{b \sin (e+f x) \cos ^3(e+f x)}{4 f}","\frac{(4 a+b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} x (4 a+b)-\frac{b \sin (e+f x) \cos ^3(e+f x)}{4 f}",1,"((4*a + b)*x)/8 + ((4*a + b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (b*Cos[e + f*x]^3*Sin[e + f*x])/(4*f)","A",4,4,21,0.1905,1,"{3191, 385, 199, 203}"
288,1,30,0,0.0153733,"\int \left(a+b \sin ^2(e+f x)\right) \, dx","Int[a + b*Sin[e + f*x]^2,x]","a x-\frac{b \sin (e+f x) \cos (e+f x)}{2 f}+\frac{b x}{2}","a x-\frac{b \sin (e+f x) \cos (e+f x)}{2 f}+\frac{b x}{2}",1,"a*x + (b*x)/2 - (b*Cos[e + f*x]*Sin[e + f*x])/(2*f)","A",3,2,12,0.1667,1,"{2635, 8}"
289,1,18,0,0.0325623,"\int \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Int[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2),x]","\frac{(a+b) \tan (e+f x)}{f}-b x","\frac{(a+b) \tan (e+f x)}{f}-b x",1,"-(b*x) + ((a + b)*Tan[e + f*x])/f","A",3,3,21,0.1429,1,"{3191, 388, 203}"
290,1,30,0,0.032005,"\int \sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Int[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2),x]","\frac{(a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f}","\frac{(a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f}",1,"(a*Tan[e + f*x])/f + ((a + b)*Tan[e + f*x]^3)/(3*f)","A",2,1,21,0.04762,1,"{3191}"
291,1,50,0,0.0444607,"\int \sec ^6(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Int[Sec[e + f*x]^6*(a + b*Sin[e + f*x]^2),x]","\frac{(a+b) \tan ^5(e+f x)}{5 f}+\frac{(2 a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f}","\frac{(a+b) \tan ^5(e+f x)}{5 f}+\frac{(2 a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f}",1,"(a*Tan[e + f*x])/f + ((2*a + b)*Tan[e + f*x]^3)/(3*f) + ((a + b)*Tan[e + f*x]^5)/(5*f)","A",3,2,21,0.09524,1,"{3191, 373}"
292,1,72,0,0.0552121,"\int \sec ^8(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Int[Sec[e + f*x]^8*(a + b*Sin[e + f*x]^2),x]","\frac{(a+b) \tan ^7(e+f x)}{7 f}+\frac{(3 a+2 b) \tan ^5(e+f x)}{5 f}+\frac{(3 a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f}","\frac{(a+b) \tan ^7(e+f x)}{7 f}+\frac{(3 a+2 b) \tan ^5(e+f x)}{5 f}+\frac{(3 a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f}",1,"(a*Tan[e + f*x])/f + ((3*a + b)*Tan[e + f*x]^3)/(3*f) + ((3*a + 2*b)*Tan[e + f*x]^5)/(5*f) + ((a + b)*Tan[e + f*x]^7)/(7*f)","A",3,2,21,0.09524,1,"{3191, 373}"
293,1,156,0,0.1497023,"\int \cos ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Int[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2)^2,x]","\frac{\left(48 a^2+16 a b+3 b^2\right) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{\left(48 a^2+16 a b+3 b^2\right) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{1}{128} x \left(48 a^2+16 a b+3 b^2\right)-\frac{b (10 a+3 b) \sin (e+f x) \cos ^5(e+f x)}{48 f}-\frac{b \sin (e+f x) \cos ^7(e+f x) \left((a+b) \tan ^2(e+f x)+a\right)}{8 f}","\frac{\left(48 a^2+16 a b+3 b^2\right) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{\left(48 a^2+16 a b+3 b^2\right) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{1}{128} x \left(48 a^2+16 a b+3 b^2\right)-\frac{b (10 a+3 b) \sin (e+f x) \cos ^5(e+f x)}{48 f}-\frac{b \sin (e+f x) \cos ^7(e+f x) \left((a+b) \tan ^2(e+f x)+a\right)}{8 f}",1,"((48*a^2 + 16*a*b + 3*b^2)*x)/128 + ((48*a^2 + 16*a*b + 3*b^2)*Cos[e + f*x]*Sin[e + f*x])/(128*f) + ((48*a^2 + 16*a*b + 3*b^2)*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) - (b*(10*a + 3*b)*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) - (b*Cos[e + f*x]^7*Sin[e + f*x]*(a + (a + b)*Tan[e + f*x]^2))/(8*f)","A",6,5,23,0.2174,1,"{3191, 413, 385, 199, 203}"
294,1,116,0,0.1455552,"\int \cos ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2)^2,x]","\frac{\left(8 a^2+4 a b+b^2\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x \left(8 a^2+4 a b+b^2\right)-\frac{b (8 a+3 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}-\frac{b \sin (e+f x) \cos ^5(e+f x) \left((a+b) \tan ^2(e+f x)+a\right)}{6 f}","\frac{\left(8 a^2+4 a b+b^2\right) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x \left(8 a^2+4 a b+b^2\right)-\frac{b (8 a+3 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}-\frac{b \sin (e+f x) \cos ^5(e+f x) \left((a+b) \tan ^2(e+f x)+a\right)}{6 f}",1,"((8*a^2 + 4*a*b + b^2)*x)/16 + ((8*a^2 + 4*a*b + b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (b*(8*a + 3*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - (b*Cos[e + f*x]^5*Sin[e + f*x]*(a + (a + b)*Tan[e + f*x]^2))/(6*f)","A",5,5,23,0.2174,1,"{3191, 413, 385, 199, 203}"
295,1,72,0,0.020182,"\int \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Int[(a + b*Sin[e + f*x]^2)^2,x]","\frac{1}{8} x \left(8 a^2+8 a b+3 b^2\right)-\frac{b (8 a+3 b) \sin (e+f x) \cos (e+f x)}{8 f}-\frac{b^2 \sin ^3(e+f x) \cos (e+f x)}{4 f}","\frac{1}{8} x \left(8 a^2+8 a b+3 b^2\right)-\frac{b (8 a+3 b) \sin (e+f x) \cos (e+f x)}{8 f}-\frac{b^2 \sin ^3(e+f x) \cos (e+f x)}{4 f}",1,"((8*a^2 + 8*a*b + 3*b^2)*x)/8 - (b*(8*a + 3*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (b^2*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f)","A",1,1,14,0.07143,1,"{3179}"
296,1,51,0,0.0892798,"\int \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Int[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^2,x]","\frac{(a+b)^2 \tan (e+f x)}{f}-\frac{1}{2} b x (4 a+3 b)+\frac{b^2 \sin (e+f x) \cos (e+f x)}{2 f}","\frac{(a+b)^2 \tan (e+f x)}{f}-\frac{1}{2} b x (4 a+3 b)+\frac{b^2 \sin (e+f x) \cos (e+f x)}{2 f}",1,"-(b*(4*a + 3*b)*x)/2 + (b^2*Cos[e + f*x]*Sin[e + f*x])/(2*f) + ((a + b)^2*Tan[e + f*x])/f","A",5,4,23,0.1739,1,"{3191, 390, 385, 203}"
297,1,45,0,0.062085,"\int \sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Int[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^2,x]","\frac{\left(a^2-b^2\right) \tan (e+f x)}{f}+\frac{(a+b)^2 \tan ^3(e+f x)}{3 f}+b^2 x","\frac{\left(a^2-b^2\right) \tan (e+f x)}{f}+\frac{(a+b)^2 \tan ^3(e+f x)}{3 f}+b^2 x",1,"b^2*x + ((a^2 - b^2)*Tan[e + f*x])/f + ((a + b)^2*Tan[e + f*x]^3)/(3*f)","A",4,3,23,0.1304,1,"{3191, 390, 203}"
298,1,53,0,0.0572108,"\int \sec ^6(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Int[Sec[e + f*x]^6*(a + b*Sin[e + f*x]^2)^2,x]","\frac{a^2 \tan (e+f x)}{f}+\frac{(a+b)^2 \tan ^5(e+f x)}{5 f}+\frac{2 a (a+b) \tan ^3(e+f x)}{3 f}","\frac{a^2 \tan (e+f x)}{f}+\frac{(a+b)^2 \tan ^5(e+f x)}{5 f}+\frac{2 a (a+b) \tan ^3(e+f x)}{3 f}",1,"(a^2*Tan[e + f*x])/f + (2*a*(a + b)*Tan[e + f*x]^3)/(3*f) + ((a + b)^2*Tan[e + f*x]^5)/(5*f)","A",3,2,23,0.08696,1,"{3191, 194}"
299,1,80,0,0.0757315,"\int \sec ^8(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Int[Sec[e + f*x]^8*(a + b*Sin[e + f*x]^2)^2,x]","\frac{a^2 \tan (e+f x)}{f}+\frac{(a+b)^2 \tan ^7(e+f x)}{7 f}+\frac{(a+b) (3 a+b) \tan ^5(e+f x)}{5 f}+\frac{a (3 a+2 b) \tan ^3(e+f x)}{3 f}","\frac{a^2 \tan (e+f x)}{f}+\frac{(a+b)^2 \tan ^7(e+f x)}{7 f}+\frac{(a+b) (3 a+b) \tan ^5(e+f x)}{5 f}+\frac{a (3 a+2 b) \tan ^3(e+f x)}{3 f}",1,"(a^2*Tan[e + f*x])/f + (a*(3*a + 2*b)*Tan[e + f*x]^3)/(3*f) + ((a + b)*(3*a + b)*Tan[e + f*x]^5)/(5*f) + ((a + b)^2*Tan[e + f*x]^7)/(7*f)","A",3,2,23,0.08696,1,"{3191, 373}"
300,1,106,0,0.0927193,"\int \sec ^{10}(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Int[Sec[e + f*x]^10*(a + b*Sin[e + f*x]^2)^2,x]","\frac{\left(6 a^2+6 a b+b^2\right) \tan ^5(e+f x)}{5 f}+\frac{a^2 \tan (e+f x)}{f}+\frac{(a+b)^2 \tan ^9(e+f x)}{9 f}+\frac{2 (a+b) (2 a+b) \tan ^7(e+f x)}{7 f}+\frac{2 a (2 a+b) \tan ^3(e+f x)}{3 f}","\frac{\left(6 a^2+6 a b+b^2\right) \tan ^5(e+f x)}{5 f}+\frac{a^2 \tan (e+f x)}{f}+\frac{(a+b)^2 \tan ^9(e+f x)}{9 f}+\frac{2 (a+b) (2 a+b) \tan ^7(e+f x)}{7 f}+\frac{2 a (2 a+b) \tan ^3(e+f x)}{3 f}",1,"(a^2*Tan[e + f*x])/f + (2*a*(2*a + b)*Tan[e + f*x]^3)/(3*f) + ((6*a^2 + 6*a*b + b^2)*Tan[e + f*x]^5)/(5*f) + (2*(a + b)*(2*a + b)*Tan[e + f*x]^7)/(7*f) + ((a + b)^2*Tan[e + f*x]^9)/(9*f)","A",3,2,23,0.08696,1,"{3191, 373}"
301,1,78,0,0.0895978,"\int \frac{\cos ^7(x)}{a+b \sin ^2(x)} \, dx","Int[Cos[x]^7/(a + b*Sin[x]^2),x]","-\frac{\left(a^2+3 a b+3 b^2\right) \sin (x)}{b^3}+\frac{(a+3 b) \sin ^3(x)}{3 b^2}+\frac{(a+b)^3 \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}-\frac{\sin ^5(x)}{5 b}","-\frac{\left(a^2+3 a b+3 b^2\right) \sin (x)}{b^3}+\frac{(a+3 b) \sin ^3(x)}{3 b^2}+\frac{(a+b)^3 \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}-\frac{\sin ^5(x)}{5 b}",1,"((a + b)^3*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*b^(7/2)) - ((a^2 + 3*a*b + 3*b^2)*Sin[x])/b^3 + ((a + 3*b)*Sin[x]^3)/(3*b^2) - Sin[x]^5/(5*b)","A",4,3,15,0.2000,1,"{3190, 390, 205}"
302,1,87,0,0.1869,"\int \frac{\cos ^6(x)}{a+b \sin ^2(x)} \, dx","Int[Cos[x]^6/(a + b*Sin[x]^2),x]","-\frac{x \left(8 a^2+20 a b+15 b^2\right)}{8 b^3}+\frac{(a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} b^3}-\frac{(4 a+7 b) \sin (x) \cos (x)}{8 b^2}-\frac{\sin (x) \cos ^3(x)}{4 b}","-\frac{x \left(8 a^2+20 a b+15 b^2\right)}{8 b^3}+\frac{(a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} b^3}-\frac{(4 a+7 b) \sin (x) \cos (x)}{8 b^2}-\frac{\sin (x) \cos ^3(x)}{4 b}",1,"-((8*a^2 + 20*a*b + 15*b^2)*x)/(8*b^3) + ((a + b)^(5/2)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*b^3) - ((4*a + 7*b)*Cos[x]*Sin[x])/(8*b^2) - (Cos[x]^3*Sin[x])/(4*b)","A",6,6,15,0.4000,1,"{3191, 414, 527, 522, 203, 205}"
303,1,54,0,0.0732421,"\int \frac{\cos ^5(x)}{a+b \sin ^2(x)} \, dx","Int[Cos[x]^5/(a + b*Sin[x]^2),x]","-\frac{(a+2 b) \sin (x)}{b^2}+\frac{(a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2}}+\frac{\sin ^3(x)}{3 b}","-\frac{(a+2 b) \sin (x)}{b^2}+\frac{(a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2}}+\frac{\sin ^3(x)}{3 b}",1,"((a + b)^2*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)) - ((a + 2*b)*Sin[x])/b^2 + Sin[x]^3/(3*b)","A",4,3,15,0.2000,1,"{3190, 390, 205}"
304,1,59,0,0.1084882,"\int \frac{\cos ^4(x)}{a+b \sin ^2(x)} \, dx","Int[Cos[x]^4/(a + b*Sin[x]^2),x]","-\frac{x (2 a+3 b)}{2 b^2}+\frac{(a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} b^2}-\frac{\sin (x) \cos (x)}{2 b}","-\frac{x (2 a+3 b)}{2 b^2}+\frac{(a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} b^2}-\frac{\sin (x) \cos (x)}{2 b}",1,"-((2*a + 3*b)*x)/(2*b^2) + ((a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*b^2) - (Cos[x]*Sin[x])/(2*b)","A",5,5,15,0.3333,1,"{3191, 414, 522, 203, 205}"
305,1,36,0,0.0563766,"\int \frac{\cos ^3(x)}{a+b \sin ^2(x)} \, dx","Int[Cos[x]^3/(a + b*Sin[x]^2),x]","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}-\frac{\sin (x)}{b}","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}-\frac{\sin (x)}{b}",1,"((a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)) - Sin[x]/b","A",3,3,15,0.2000,1,"{3190, 388, 205}"
306,1,39,0,0.059619,"\int \frac{\cos ^2(x)}{a+b \sin ^2(x)} \, dx","Int[Cos[x]^2/(a + b*Sin[x]^2),x]","\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} b}-\frac{x}{b}","\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} b}-\frac{x}{b}",1,"-(x/b) + (Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*b)","A",4,4,15,0.2667,1,"{3191, 391, 203, 205}"
307,1,25,0,0.0288314,"\int \frac{\cos (x)}{a+b \sin ^2(x)} \, dx","Int[Cos[x]/(a + b*Sin[x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,13,0.1538,1,"{3190, 205}"
308,1,40,0,0.0509571,"\int \frac{\sec (x)}{a+b \sin ^2(x)} \, dx","Int[Sec[x]/(a + b*Sin[x]^2),x]","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)}+\frac{\tanh ^{-1}(\sin (x))}{a+b}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)}+\frac{\tanh ^{-1}(\sin (x))}{a+b}",1,"(Sqrt[b]*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*(a + b)) + ArcTanh[Sin[x]]/(a + b)","A",4,4,13,0.3077,1,"{3190, 391, 206, 205}"
309,1,39,0,0.0606788,"\int \frac{\sec ^2(x)}{a+b \sin ^2(x)} \, dx","Int[Sec[x]^2/(a + b*Sin[x]^2),x]","\frac{b \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^{3/2}}+\frac{\tan (x)}{a+b}","\frac{b \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^{3/2}}+\frac{\tan (x)}{a+b}",1,"(b*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(3/2)) + Tan[x]/(a + b)","A",3,3,15,0.2000,1,"{3191, 388, 205}"
310,1,61,0,0.0867202,"\int \frac{\sec ^3(x)}{a+b \sin ^2(x)} \, dx","Int[Sec[x]^3/(a + b*Sin[x]^2),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^2}+\frac{(a+3 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^2}+\frac{\tan (x) \sec (x)}{2 (a+b)}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^2}+\frac{(a+3 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^2}+\frac{\tan (x) \sec (x)}{2 (a+b)}",1,"(b^(3/2)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^2) + ((a + 3*b)*ArcTanh[Sin[x]])/(2*(a + b)^2) + (Sec[x]*Tan[x])/(2*(a + b))","A",5,5,15,0.3333,1,"{3190, 414, 522, 206, 205}"
311,1,59,0,0.0813427,"\int \frac{\sec ^4(x)}{a+b \sin ^2(x)} \, dx","Int[Sec[x]^4/(a + b*Sin[x]^2),x]","\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^{5/2}}+\frac{\tan ^3(x)}{3 (a+b)}+\frac{(a+2 b) \tan (x)}{(a+b)^2}","\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^{5/2}}+\frac{\tan ^3(x)}{3 (a+b)}+\frac{(a+2 b) \tan (x)}{(a+b)^2}",1,"(b^2*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(5/2)) + ((a + 2*b)*Tan[x])/(a + b)^2 + Tan[x]^3/(3*(a + b))","A",4,3,15,0.2000,1,"{3191, 390, 205}"
312,1,93,0,0.145,"\int \frac{\sec ^5(x)}{a+b \sin ^2(x)} \, dx","Int[Sec[x]^5/(a + b*Sin[x]^2),x]","\frac{\left(3 a^2+10 a b+15 b^2\right) \tanh ^{-1}(\sin (x))}{8 (a+b)^3}+\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^3}+\frac{\tan (x) \sec ^3(x)}{4 (a+b)}+\frac{(3 a+7 b) \tan (x) \sec (x)}{8 (a+b)^2}","\frac{\left(3 a^2+10 a b+15 b^2\right) \tanh ^{-1}(\sin (x))}{8 (a+b)^3}+\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^3}+\frac{\tan (x) \sec ^3(x)}{4 (a+b)}+\frac{(3 a+7 b) \tan (x) \sec (x)}{8 (a+b)^2}",1,"(b^(5/2)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^3) + ((3*a^2 + 10*a*b + 15*b^2)*ArcTanh[Sin[x]])/(8*(a + b)^3) + ((3*a + 7*b)*Sec[x]*Tan[x])/(8*(a + b)^2) + (Sec[x]^3*Tan[x])/(4*(a + b))","A",6,6,15,0.4000,1,"{3190, 414, 527, 522, 206, 205}"
313,1,87,0,0.1134055,"\int \frac{\sec ^6(x)}{a+b \sin ^2(x)} \, dx","Int[Sec[x]^6/(a + b*Sin[x]^2),x]","\frac{\left(a^2+3 a b+3 b^2\right) \tan (x)}{(a+b)^3}+\frac{b^3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^{7/2}}+\frac{\tan ^5(x)}{5 (a+b)}+\frac{(2 a+3 b) \tan ^3(x)}{3 (a+b)^2}","\frac{\left(a^2+3 a b+3 b^2\right) \tan (x)}{(a+b)^3}+\frac{b^3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^{7/2}}+\frac{\tan ^5(x)}{5 (a+b)}+\frac{(2 a+3 b) \tan ^3(x)}{3 (a+b)^2}",1,"(b^3*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(7/2)) + ((a^2 + 3*a*b + 3*b^2)*Tan[x])/(a + b)^3 + ((2*a + 3*b)*Tan[x]^3)/(3*(a + b)^2) + Tan[x]^5/(5*(a + b))","A",4,3,15,0.2000,1,"{3191, 390, 205}"
314,1,113,0,0.2231407,"\int \frac{\cos ^6(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^6/(a + b*Sin[x]^2)^2,x]","-\frac{(4 a-b) (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{2 a^{3/2} b^3}+\frac{x (4 a+5 b)}{2 b^3}+\frac{(2 a+b) (a+b) \tan (x)}{2 a b^2 \left((a+b) \tan ^2(x)+a\right)}-\frac{\sin (x) \cos (x)}{2 b \left((a+b) \tan ^2(x)+a\right)}","-\frac{(4 a-b) (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{2 a^{3/2} b^3}+\frac{x (4 a+5 b)}{2 b^3}+\frac{(2 a+b) (a+b) \tan (x)}{2 a b^2 \left((a+b) \tan ^2(x)+a\right)}-\frac{\sin (x) \cos (x)}{2 b \left((a+b) \tan ^2(x)+a\right)}",1,"((4*a + 5*b)*x)/(2*b^3) - ((4*a - b)*(a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(2*a^(3/2)*b^3) - (Cos[x]*Sin[x])/(2*b*(a + (a + b)*Tan[x]^2)) + ((a + b)*(2*a + b)*Tan[x])/(2*a*b^2*(a + (a + b)*Tan[x]^2))","A",6,6,15,0.4000,1,"{3191, 414, 527, 522, 203, 205}"
315,1,72,0,0.1186327,"\int \frac{\cos ^5(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^5/(a + b*Sin[x]^2)^2,x]","-\frac{(3 a-b) (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2}}+\frac{(a+b)^2 \sin (x)}{2 a b^2 \left(a+b \sin ^2(x)\right)}+\frac{\sin (x)}{b^2}","-\frac{(3 a-b) (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2}}+\frac{(a+b)^2 \sin (x)}{2 a b^2 \left(a+b \sin ^2(x)\right)}+\frac{\sin (x)}{b^2}",1,"-((3*a - b)*(a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(2*a^(3/2)*b^(5/2)) + Sin[x]/b^2 + ((a + b)^2*Sin[x])/(2*a*b^2*(a + b*Sin[x]^2))","A",5,4,15,0.2667,1,"{3190, 390, 385, 205}"
316,1,75,0,0.1055462,"\int \frac{\cos ^4(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^4/(a + b*Sin[x]^2)^2,x]","-\frac{(2 a-b) \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{2 a^{3/2} b^2}+\frac{(a+b) \tan (x)}{2 a b \left((a+b) \tan ^2(x)+a\right)}+\frac{x}{b^2}","-\frac{(2 a-b) \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{2 a^{3/2} b^2}+\frac{(a+b) \tan (x)}{2 a b \left((a+b) \tan ^2(x)+a\right)}+\frac{x}{b^2}",1,"x/b^2 - ((2*a - b)*Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(2*a^(3/2)*b^2) + ((a + b)*Tan[x])/(2*a*b*(a + (a + b)*Tan[x]^2))","A",5,5,15,0.3333,1,"{3191, 414, 522, 203, 205}"
317,1,59,0,0.0581525,"\int \frac{\cos ^3(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^3/(a + b*Sin[x]^2)^2,x]","\frac{(a+b) \sin (x)}{2 a b \left(a+b \sin ^2(x)\right)}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}","\frac{(a+b) \sin (x)}{2 a b \left(a+b \sin ^2(x)\right)}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2}}",1,"-((a - b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(2*a^(3/2)*b^(3/2)) + ((a + b)*Sin[x])/(2*a*b*(a + b*Sin[x]^2))","A",3,3,15,0.2000,1,"{3190, 385, 205}"
318,1,54,0,0.0559371,"\int \frac{\cos ^2(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Int[Cos[x]^2/(a + b*Sin[x]^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{a+b}}+\frac{\tan (x)}{2 a \left((a+b) \tan ^2(x)+a\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{a+b}}+\frac{\tan (x)}{2 a \left((a+b) \tan ^2(x)+a\right)}",1,"ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[a + b]) + Tan[x]/(2*a*(a + (a + b)*Tan[x]^2))","A",3,3,15,0.2000,1,"{3191, 199, 205}"
319,1,48,0,0.0348803,"\int \frac{\cos (x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Int[Cos[x]/(a + b*Sin[x]^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b}}+\frac{\sin (x)}{2 a \left(a+b \sin ^2(x)\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b}}+\frac{\sin (x)}{2 a \left(a+b \sin ^2(x)\right)}",1,"ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]) + Sin[x]/(2*a*(a + b*Sin[x]^2))","A",3,3,13,0.2308,1,"{3190, 199, 205}"
320,1,73,0,0.0903967,"\int \frac{\sec (x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Int[Sec[x]/(a + b*Sin[x]^2)^2,x]","\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{2 a^{3/2} (a+b)^2}+\frac{b \sin (x)}{2 a (a+b) \left(a+b \sin ^2(x)\right)}+\frac{\tanh ^{-1}(\sin (x))}{(a+b)^2}","\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{2 a^{3/2} (a+b)^2}+\frac{b \sin (x)}{2 a (a+b) \left(a+b \sin ^2(x)\right)}+\frac{\tanh ^{-1}(\sin (x))}{(a+b)^2}",1,"(Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^2) + ArcTanh[Sin[x]]/(a + b)^2 + (b*Sin[x])/(2*a*(a + b)*(a + b*Sin[x]^2))","A",5,5,13,0.3846,1,"{3190, 414, 522, 206, 205}"
321,1,76,0,0.1237711,"\int \frac{\sec ^2(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Int[Sec[x]^2/(a + b*Sin[x]^2)^2,x]","\frac{b (4 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{2 a^{3/2} (a+b)^{5/2}}+\frac{b^2 \tan (x)}{2 a (a+b)^2 \left((a+b) \tan ^2(x)+a\right)}+\frac{\tan (x)}{(a+b)^2}","\frac{b (4 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{2 a^{3/2} (a+b)^{5/2}}+\frac{b^2 \tan (x)}{2 a (a+b)^2 \left((a+b) \tan ^2(x)+a\right)}+\frac{\tan (x)}{(a+b)^2}",1,"(b*(4*a + b)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(5/2)) + Tan[x]/(a + b)^2 + (b^2*Tan[x])/(2*a*(a + b)^2*(a + (a + b)*Tan[x]^2))","A",5,4,15,0.2667,1,"{3191, 390, 385, 205}"
322,1,109,0,0.1631646,"\int \frac{\sec ^3(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Int[Sec[x]^3/(a + b*Sin[x]^2)^2,x]","\frac{b^{3/2} (5 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{2 a^{3/2} (a+b)^3}-\frac{b (a-b) \sin (x)}{2 a (a+b)^2 \left(a+b \sin ^2(x)\right)}+\frac{(a+5 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^3}+\frac{\tan (x) \sec (x)}{2 (a+b) \left(a+b \sin ^2(x)\right)}","\frac{b^{3/2} (5 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{2 a^{3/2} (a+b)^3}-\frac{b (a-b) \sin (x)}{2 a (a+b)^2 \left(a+b \sin ^2(x)\right)}+\frac{(a+5 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^3}+\frac{\tan (x) \sec (x)}{2 (a+b) \left(a+b \sin ^2(x)\right)}",1,"(b^(3/2)*(5*a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^3) + ((a + 5*b)*ArcTanh[Sin[x]])/(2*(a + b)^3) - ((a - b)*b*Sin[x])/(2*a*(a + b)^2*(a + b*Sin[x]^2)) + (Sec[x]*Tan[x])/(2*(a + b)*(a + b*Sin[x]^2))","A",6,6,15,0.4000,1,"{3190, 414, 527, 522, 206, 205}"
323,1,96,0,0.1476733,"\int \frac{\sec ^4(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Int[Sec[x]^4/(a + b*Sin[x]^2)^2,x]","\frac{b^2 (6 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{2 a^{3/2} (a+b)^{7/2}}+\frac{b^3 \tan (x)}{2 a (a+b)^3 \left((a+b) \tan ^2(x)+a\right)}+\frac{\tan ^3(x)}{3 (a+b)^2}+\frac{(a+3 b) \tan (x)}{(a+b)^3}","\frac{b^2 (6 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{2 a^{3/2} (a+b)^{7/2}}+\frac{b^3 \tan (x)}{2 a (a+b)^3 \left((a+b) \tan ^2(x)+a\right)}+\frac{\tan ^3(x)}{3 (a+b)^2}+\frac{(a+3 b) \tan (x)}{(a+b)^3}",1,"(b^2*(6*a + b)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(7/2)) + ((a + 3*b)*Tan[x])/(a + b)^3 + Tan[x]^3/(3*(a + b)^2) + (b^3*Tan[x])/(2*a*(a + b)^3*(a + (a + b)*Tan[x]^2))","A",5,4,15,0.2667,1,"{3191, 390, 385, 205}"
324,1,117,0,0.1182361,"\int \cos ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Cos[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{a (a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{8 b^{3/2} f}-\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 b f}+\frac{(a+4 b) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 b f}","\frac{a (a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{8 b^{3/2} f}-\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 b f}+\frac{(a+4 b) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 b f}",1,"(a*(a + 4*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(8*b^(3/2)*f) + ((a + 4*b)*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(8*b*f) - (Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(4*b*f)","A",5,5,25,0.2000,1,"{3190, 388, 195, 217, 206}"
325,1,72,0,0.0547196,"\int \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 \sqrt{b} f}","\frac{\sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 \sqrt{b} f}",1,"(a*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*Sqrt[b]*f) + (Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(2*f)","A",4,4,23,0.1739,1,"{3190, 195, 217, 206}"
326,1,82,0,0.0923954,"\int \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f}","\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f}",1,"-((Sqrt[b]*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/f) + (Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/f","A",6,5,23,0.2174,1,"{3190, 402, 217, 206, 377}"
327,1,82,0,0.0941853,"\int \sec ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sec[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{a \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 f \sqrt{a+b}}+\frac{\tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f}","\frac{a \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 f \sqrt{a+b}}+\frac{\tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f}",1,"(a*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*Sqrt[a + b]*f) + (Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(2*f)","A",4,4,25,0.1600,1,"{3190, 378, 377, 206}"
328,1,143,0,0.1393332,"\int \sec ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sec[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{a (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{8 f (a+b)^{3/2}}+\frac{\tan (e+f x) \sec ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 f (a+b)}+\frac{(3 a+4 b) \tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 f (a+b)}","\frac{a (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{8 f (a+b)^{3/2}}+\frac{\tan (e+f x) \sec ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 f (a+b)}+\frac{(3 a+4 b) \tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 f (a+b)}",1,"(a*(3*a + 4*b)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(8*(a + b)^(3/2)*f) + ((3*a + 4*b)*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(8*(a + b)*f) + (Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x])/(4*(a + b)*f)","A",5,5,25,0.2000,1,"{3190, 382, 378, 377, 206}"
329,1,260,0,0.2751731,"\int \cos ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Cos[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\left(2 a^2+7 a b-3 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{15 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{2 a (a+b) (a+3 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{15 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sin (e+f x) \cos (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{5 b f}+\frac{2 (a+3 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 b f}","-\frac{\left(2 a^2+7 a b-3 b^2\right) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{15 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{2 a (a+b) (a+3 b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{15 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sin (e+f x) \cos (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{5 b f}+\frac{2 (a+3 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 b f}",1,"(2*(a + 3*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b*f) - (Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(5*b*f) - ((2*a^2 + 7*a*b - 3*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (2*a*(a + b)*(a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(15*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3192, 416, 528, 524, 426, 424, 421, 419}"
330,1,199,0,0.1837633,"\int \cos ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Cos[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{a (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{a (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(a-b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - ((a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,7,25,0.2800,1,"{3192, 417, 524, 426, 424, 421, 419}"
331,1,51,0,0.0352502,"\int \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])","A",2,2,16,0.1250,1,"{3178, 3177}"
332,1,171,0,0.1652781,"\int \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{a \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-((Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/f","A",8,8,25,0.3200,1,"{3192, 412, 12, 493, 426, 424, 421, 419}"
333,1,236,0,0.2231344,"\int \sec ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sec[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{(2 a+b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)}+\frac{\tan (e+f x) \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{2 a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{(2 a+b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)}+\frac{2 a \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{\tan (e+f x) \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}",1,"-((2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (2*a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((2*a + b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)*f) + (Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*f)","A",8,8,25,0.3200,1,"{3192, 412, 527, 524, 426, 424, 421, 419}"
334,1,157,0,0.1376055,"\int \cos ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Cos[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{a^2 (a+6 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{16 b^{3/2} f}-\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{6 b f}+\frac{(a+6 b) \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{24 b f}+\frac{a (a+6 b) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{16 b f}","\frac{a^2 (a+6 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{16 b^{3/2} f}-\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{6 b f}+\frac{(a+6 b) \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{24 b f}+\frac{a (a+6 b) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{16 b f}",1,"(a^2*(a + 6*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(16*b^(3/2)*f) + (a*(a + 6*b)*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(16*b*f) + ((a + 6*b)*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(24*b*f) - (Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(5/2))/(6*b*f)","A",6,5,25,0.2000,1,"{3190, 388, 195, 217, 206}"
335,1,104,0,0.0692238,"\int \cos (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Cos[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{8 \sqrt{b} f}+\frac{3 a \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 f}+\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 f}","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{8 \sqrt{b} f}+\frac{3 a \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 f}+\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 f}",1,"(3*a^2*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(8*Sqrt[b]*f) + (3*a*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(8*f) + (Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(4*f)","A",5,4,23,0.1739,1,"{3190, 195, 217, 206}"
336,1,121,0,0.1439205,"\int \sec (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{b \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f}+\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f}-\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 f}","-\frac{b \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f}+\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f}-\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 f}",1,"-(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*f) + ((a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/f - (b*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(2*f)","A",7,6,23,0.2609,1,"{3190, 416, 523, 217, 206, 377}"
337,1,127,0,0.1458344,"\int \sec ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f}+\frac{(a-2 b) \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 f}+\frac{(a+b) \tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f}","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f}+\frac{(a-2 b) \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 f}+\frac{(a+b) \tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f}",1,"(b^(3/2)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/f + ((a - 2*b)*Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*f) + ((a + b)*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(2*f)","A",7,6,25,0.2400,1,"{3190, 413, 523, 217, 206, 377}"
338,1,122,0,0.1207784,"\int \sec ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]^5*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{8 f \sqrt{a+b}}+\frac{\tan (e+f x) \sec ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 f}+\frac{3 a \tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 f}","\frac{3 a^2 \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{8 f \sqrt{a+b}}+\frac{\tan (e+f x) \sec ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 f}+\frac{3 a \tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 f}",1,"(3*a^2*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(8*Sqrt[a + b]*f) + (3*a*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(8*f) + (Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x])/(4*f)","A",5,4,25,0.1600,1,"{3190, 378, 377, 206}"
339,1,195,0,0.176011,"\int \sec ^7(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]^7*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{a^2 (5 a+6 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{16 f (a+b)^{3/2}}+\frac{\tan (e+f x) \sec ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{6 f (a+b)}+\frac{(5 a+6 b) \tan (e+f x) \sec ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{24 f (a+b)}+\frac{a (5 a+6 b) \tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{16 f (a+b)}","\frac{a^2 (5 a+6 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{16 f (a+b)^{3/2}}+\frac{\tan (e+f x) \sec ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{6 f (a+b)}+\frac{(5 a+6 b) \tan (e+f x) \sec ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{24 f (a+b)}+\frac{a (5 a+6 b) \tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{16 f (a+b)}",1,"(a^2*(5*a + 6*b)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(16*(a + b)^(3/2)*f) + (a*(5*a + 6*b)*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(16*(a + b)*f) + ((5*a + 6*b)*Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x])/(24*(a + b)*f) + (Sec[e + f*x]^5*(a + b*Sin[e + f*x]^2)^(5/2)*Tan[e + f*x])/(6*(a + b)*f)","A",6,5,25,0.2000,1,"{3190, 382, 378, 377, 206}"
340,1,321,0,0.3891748,"\int \cos ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{\left(a^2-9 a b-2 b^2\right) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 b f}+\frac{a (a+b) \left(2 a^2+9 a b-b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{35 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a-b) \left(a^2+6 a b+b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{35 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{b \sin (e+f x) \cos ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{7 f}+\frac{2 (4 a+b) \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 f}","-\frac{\left(a^2-9 a b-2 b^2\right) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 b f}+\frac{a (a+b) \left(2 a^2+9 a b-b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{35 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a-b) \left(a^2+6 a b+b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{35 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{b \sin (e+f x) \cos ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{7 f}+\frac{2 (4 a+b) \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 f}",1,"-((a^2 - 9*a*b - 2*b^2)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*b*f) + (2*(4*a + b)*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*f) - (b*Cos[e + f*x]^5*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(7*f) - (2*(a - b)*(a^2 + 6*a*b + b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(a + b)*(2*a^2 + 9*a*b - b^2)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(35*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",9,8,25,0.3200,1,"{3192, 416, 528, 524, 426, 424, 421, 419}"
341,1,259,0,0.2849608,"\int \cos ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{\left(3 a^2-7 a b-2 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{15 b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{b \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}+\frac{2 (3 a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 f}+\frac{a (3 a-b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{15 b f \sqrt{a+b \sin ^2(e+f x)}}","-\frac{\left(3 a^2-7 a b-2 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{15 b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{b \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}+\frac{2 (3 a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 f}+\frac{a (3 a-b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{15 b f \sqrt{a+b \sin ^2(e+f x)}}",1,"(2*(3*a + b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*f) - (b*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(5*f) - ((3*a^2 - 7*a*b - 2*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(3*a - b)*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(15*b*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3192, 416, 528, 524, 426, 424, 421, 419}"
342,1,154,0,0.1650997,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-(b*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(a + b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,6,16,0.3750,1,"{3180, 3172, 3178, 3177, 3183, 3182}"
343,1,182,0,0.1800637,"\int \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{(a+b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{a (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{(a+b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{a (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-(((a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (a*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2]) + ((a + b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/f","A",7,7,25,0.2800,1,"{3192, 413, 524, 426, 424, 421, 419}"
344,1,236,0,0.2564086,"\int \sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{2 (a-b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{(a+b) \tan (e+f x) \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{a (2 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{2 (a-b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{(a+b) \tan (e+f x) \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{a (2 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(-2*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(2*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2]) + (2*(a - b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*f) + ((a + b)*Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*f)","A",8,8,25,0.3200,1,"{3192, 413, 527, 524, 426, 424, 421, 419}"
345,1,79,0,0.0931914,"\int \frac{\cos ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Cos[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 b^{3/2} f}-\frac{\sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 b f}","\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 b^{3/2} f}-\frac{\sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 b f}",1,"((a + 2*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*b^(3/2)*f) - (Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(2*b*f)","A",4,4,25,0.1600,1,"{3190, 388, 217, 206}"
346,1,38,0,0.046163,"\int \frac{\cos (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{\sqrt{b} f}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{\sqrt{b} f}",1,"ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/(Sqrt[b]*f)","A",3,3,23,0.1304,1,"{3190, 217, 206}"
347,1,42,0,0.0720426,"\int \frac{\sec (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Sec[e + f*x]/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f \sqrt{a+b}}",1,"ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/(Sqrt[a + b]*f)","A",3,3,23,0.1304,1,"{3190, 377, 206}"
348,1,91,0,0.1113147,"\int \frac{\sec ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Sec[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 f (a+b)^{3/2}}+\frac{\tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f (a+b)}","\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 f (a+b)^{3/2}}+\frac{\tan (e+f x) \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f (a+b)}",1,"((a + 2*b)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*(a + b)^(3/2)*f) + (Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(2*(a + b)*f)","A",4,4,25,0.1600,1,"{3190, 382, 377, 206}"
349,1,208,0,0.1941835,"\int \frac{\cos ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Cos[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{(a+b) (2 a+3 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 b f}","\frac{(a+b) (2 a+3 b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 b f}",1,"-(Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b*f) - (2*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((a + b)*(2*a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,7,25,0.2800,1,"{3192, 416, 524, 426, 424, 421, 419}"
350,1,153,0,0.1406544,"\int \frac{\cos ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Cos[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{(a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{(a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{a \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{b f \sqrt{a+b \sin ^2(e+f x)}}",1,"-((Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + ((a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,6,25,0.2400,1,"{3192, 423, 426, 424, 421, 419}"
351,1,51,0,0.0347665,"\int \frac{1}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[1/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{a+b \sin ^2(e+f x)}}","\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{a+b \sin ^2(e+f x)}}",1,"(EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])","A",2,2,16,0.1250,1,"{3183, 3182}"
352,1,180,0,0.1689162,"\int \frac{\sec ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Sec[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f (a+b)}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f (a+b)}+\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-((Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/((a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/((a + b)*f)","A",8,8,25,0.3200,1,"{3192, 414, 21, 423, 426, 424, 421, 419}"
353,1,252,0,0.2481343,"\int \frac{\sec ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Sec[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{2 (a+2 b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)^2}+\frac{\tan (e+f x) \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)}+\frac{(2 a+3 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{2 (a+2 b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)^2}+\frac{(2 a+3 b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{\tan (e+f x) \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)}",1,"(-2*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((2*a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + (2*(a + 2*b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)^2*f) + (Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)*f)","A",8,8,25,0.3200,1,"{3192, 414, 527, 524, 426, 424, 421, 419}"
354,1,75,0,0.1004898,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cos[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{(a+b) \sin (e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{b^{3/2} f}","\frac{(a+b) \sin (e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{b^{3/2} f}",1,"-(ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/(b^(3/2)*f)) + ((a + b)*Sin[e + f*x])/(a*b*f*Sqrt[a + b*Sin[e + f*x]^2])","A",4,4,25,0.1600,1,"{3190, 385, 217, 206}"
355,1,29,0,0.0439352,"\int \frac{\cos (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cos[e + f*x]/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sin (e+f x)}{a f \sqrt{a+b \sin ^2(e+f x)}}","\frac{\sin (e+f x)}{a f \sqrt{a+b \sin ^2(e+f x)}}",1,"Sin[e + f*x]/(a*f*Sqrt[a + b*Sin[e + f*x]^2])","A",2,2,23,0.08696,1,"{3190, 191}"
356,1,78,0,0.1012234,"\int \frac{\sec (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Sec[e + f*x]/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{b \sin (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f (a+b)^{3/2}}","\frac{b \sin (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f (a+b)^{3/2}}",1,"ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/((a + b)^(3/2)*f) + (b*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",4,4,23,0.1739,1,"{3190, 382, 377, 206}"
357,1,134,0,0.1759526,"\int \frac{\sec ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Sec[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{b (a-2 b) \sin (e+f x)}{2 a f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+4 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 f (a+b)^{5/2}}+\frac{\tan (e+f x) \sec (e+f x)}{2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}","-\frac{b (a-2 b) \sin (e+f x)}{2 a f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+4 b) \tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{2 f (a+b)^{5/2}}+\frac{\tan (e+f x) \sec (e+f x)}{2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}",1,"((a + 4*b)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*(a + b)^(5/2)*f) - ((a - 2*b)*b*Sin[e + f*x])/(2*a*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sec[e + f*x]*Tan[e + f*x])/(2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,6,25,0.2400,1,"{3190, 414, 527, 12, 377, 206}"
358,1,274,0,0.2899348,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cos[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\left(8 a^2+13 a b+3 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a b^3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(4 a+3 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a b^2 f}-\frac{(a+b) (8 a+9 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+b) \sin (e+f x) \cos ^3(e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}","\frac{\left(8 a^2+13 a b+3 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a b^3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(4 a+3 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a b^2 f}-\frac{(a+b) (8 a+9 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 b^3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+b) \sin (e+f x) \cos ^3(e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}",1,"((a + b)*Cos[e + f*x]^3*Sin[e + f*x])/(a*b*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((4*a + 3*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*b^2*f) + ((8*a^2 + 13*a*b + 3*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*b^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - ((a + b)*(8*a + 9*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3192, 413, 528, 524, 426, 424, 421, 419}"
359,1,202,0,0.2059151,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cos[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{2 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{b^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(a+b) \sin (e+f x) \cos (e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}","-\frac{2 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{b^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(a+b) \sin (e+f x) \cos (e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}",1,"((a + b)*Cos[e + f*x]*Sin[e + f*x])/(a*b*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (2*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,7,25,0.2800,1,"{3192, 413, 524, 426, 424, 421, 419}"
360,1,188,0,0.1838018,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cos[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sin (e+f x) \cos (e+f x)}{a f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\sin (e+f x) \cos (e+f x)}{a f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(Cos[e + f*x]*Sin[e + f*x])/(a*f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,7,25,0.2800,1,"{3192, 412, 493, 426, 424, 421, 419}"
361,1,101,0,0.0599659,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[(a + b*Sin[e + f*x]^2)^(-3/2),x]","\frac{b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{a f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{a f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(b*Cos[e + f*x]*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(a*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])","A",4,4,16,0.2500,1,"{3184, 21, 3178, 3177}"
362,1,240,0,0.239238,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Sec[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x)}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{b (a-b) \sin (e+f x) \cos (e+f x)}{a f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{(a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\tan (e+f x)}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{b (a-b) \sin (e+f x) \cos (e+f x)}{a f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{(a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-(((a - b)*b*Cos[e + f*x]*Sin[e + f*x])/(a*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2])) - ((a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + Tan[e + f*x]/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3192, 414, 527, 524, 426, 424, 421, 419}"
363,1,130,0,0.134544,"\int \frac{\cos ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{(3 a-2 b) (a+b) \sin (e+f x)}{3 a^2 b^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{b^{5/2} f}+\frac{(a+b) \sin (e+f x) \cos ^2(e+f x)}{3 a b f \left(a+b \sin ^2(e+f x)\right)^{3/2}}","-\frac{(3 a-2 b) (a+b) \sin (e+f x)}{3 a^2 b^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{b^{5/2} f}+\frac{(a+b) \sin (e+f x) \cos ^2(e+f x)}{3 a b f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/(b^(5/2)*f) + ((a + b)*Cos[e + f*x]^2*Sin[e + f*x])/(3*a*b*f*(a + b*Sin[e + f*x]^2)^(3/2)) - ((3*a - 2*b)*(a + b)*Sin[e + f*x])/(3*a^2*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",5,5,25,0.2000,1,"{3190, 413, 385, 217, 206}"
364,1,73,0,0.0922958,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{2 \sin (e+f x)}{3 a^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sin (e+f x) \cos ^2(e+f x)}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}","\frac{2 \sin (e+f x)}{3 a^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sin (e+f x) \cos ^2(e+f x)}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"(Cos[e + f*x]^2*Sin[e + f*x])/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*Sin[e + f*x])/(3*a^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",3,3,25,0.1200,1,"{3190, 378, 191}"
365,1,65,0,0.0552965,"\int \frac{\cos (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{2 \sin (e+f x)}{3 a^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sin (e+f x)}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}","\frac{2 \sin (e+f x)}{3 a^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sin (e+f x)}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"Sin[e + f*x]/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*Sin[e + f*x])/(3*a^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",3,3,23,0.1304,1,"{3190, 192, 191}"
366,1,126,0,0.1589756,"\int \frac{\sec (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Sec[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{b (5 a+2 b) \sin (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{b \sin (e+f x)}{3 a f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f (a+b)^{5/2}}","\frac{b (5 a+2 b) \sin (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{b \sin (e+f x)}{3 a f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b} \sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}}\right)}{f (a+b)^{5/2}}",1,"ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/((a + b)^(5/2)*f) + (b*Sin[e + f*x])/(3*a*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (b*(5*a + 2*b)*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,6,23,0.2609,1,"{3190, 414, 527, 12, 377, 206}"
367,1,283,0,0.3096952,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{2 (2 a-b) (a+b) \sin (e+f x) \cos (e+f x)}{3 a^2 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\left(8 a^2+3 a b-2 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 b^3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(8 a-b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a b^3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+b) \sin (e+f x) \cos ^3(e+f x)}{3 a b f \left(a+b \sin ^2(e+f x)\right)^{3/2}}","-\frac{2 (2 a-b) (a+b) \sin (e+f x) \cos (e+f x)}{3 a^2 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\left(8 a^2+3 a b-2 b^2\right) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a^2 b^3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(8 a-b) (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a b^3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+b) \sin (e+f x) \cos ^3(e+f x)}{3 a b f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"((a + b)*Cos[e + f*x]^3*Sin[e + f*x])/(3*a*b*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*(2*a - b)*(a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*b^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((8*a^2 + 3*a*b - 2*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*b^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((8*a - b)*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*b^3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3192, 413, 526, 524, 426, 424, 421, 419}"
368,1,263,0,0.2833162,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{2 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{2 (a-b) \sin (e+f x) \cos (e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(2 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a b^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+b) \sin (e+f x) \cos (e+f x)}{3 a b f \left(a+b \sin ^2(e+f x)\right)^{3/2}}","-\frac{2 (a-b) \sin (e+f x) \cos (e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(2 a-b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a-b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a b^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+b) \sin (e+f x) \cos (e+f x)}{3 a b f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"((a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a*b*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*(a - b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*b*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((2*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3192, 413, 527, 524, 426, 424, 421, 419}"
369,1,257,0,0.2417581,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Cos[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{(a+2 b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 b f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{\sin (e+f x) \cos (e+f x)}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a b f \sqrt{a+b \sin ^2(e+f x)}}","\frac{(a+2 b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sin (e+f x) \cos (e+f x)}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(a+2 b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a b f (a+b) \sqrt{a+b \sin ^2(e+f x)}}",1,"(Cos[e + f*x]*Sin[e + f*x])/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + ((a + 2*b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*b*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*b*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3192, 412, 527, 524, 426, 424, 421, 419}"
370,1,223,0,0.2655122,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[(a + b*Sin[e + f*x]^2)^(-5/2),x]","\frac{2 b (2 a+b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a^2 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{b \sin (e+f x) \cos (e+f x)}{3 a f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}","\frac{2 b (2 a+b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a^2 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{b \sin (e+f x) \cos (e+f x)}{3 a f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}",1,"(b*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*b*(2*a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,7,16,0.4375,1,"{3184, 3173, 3172, 3178, 3177, 3183, 3182}"
371,1,328,0,0.3565588,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Sec[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{b \left(3 a^2-7 a b-2 b^2\right) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}-\frac{\left(3 a^2-7 a b-2 b^2\right) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 f (a+b)^3 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{\tan (e+f x)}{f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{b (3 a-b) \sin (e+f x) \cos (e+f x)}{3 a f (a+b)^2 \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{(3 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}","-\frac{b \left(3 a^2-7 a b-2 b^2\right) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}-\frac{\left(3 a^2-7 a b-2 b^2\right) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}+\frac{\tan (e+f x)}{f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{b (3 a-b) \sin (e+f x) \cos (e+f x)}{3 a f (a+b)^2 \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{(3 a-b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}",1,"-((3*a - b)*b*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (b*(3*a^2 - 7*a*b - 2*b^2)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((3*a^2 - 7*a*b - 2*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*(a + b)^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((3*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + Tan[e + f*x]/((a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2))","A",9,8,25,0.3200,1,"{3192, 414, 527, 524, 426, 424, 421, 419}"
372,1,115,0,0.0900169,"\int (d \cos (e+f x))^m \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[(d*Cos[e + f*x])^m*(a + b*Sin[e + f*x]^2)^p,x]","\frac{d \sin (e+f x) \cos ^2(e+f x)^{\frac{1-m}{2}} (d \cos (e+f x))^{m-1} \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{1-m}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}","\frac{d \sin (e+f x) \cos ^2(e+f x)^{\frac{1-m}{2}} (d \cos (e+f x))^{m-1} \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{1-m}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}",1,"(d*AppellF1[1/2, (1 - m)/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*(d*Cos[e + f*x])^(-1 + m)*(Cos[e + f*x]^2)^((1 - m)/2)*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,25,0.1200,1,"{3193, 430, 429}"
373,1,214,0,0.2090682,"\int \cos ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\left(3 a^2+2 a b (2 p+5)+b^2 \left(4 p^2+16 p+15\right)\right) \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^2(e+f x)}{a}\right)}{b^2 f (2 p+3) (2 p+5)}-\frac{(3 a+b (2 p+7)) \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{p+1}}{b^2 f (2 p+3) (2 p+5)}-\frac{\sin (e+f x) \cos ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{p+1}}{b f (2 p+5)}","\frac{\left(3 a^2+2 a b (2 p+5)+b^2 \left(4 p^2+16 p+15\right)\right) \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^2(e+f x)}{a}\right)}{b^2 f (2 p+3) (2 p+5)}-\frac{(3 a+b (2 p+7)) \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{p+1}}{b^2 f (2 p+3) (2 p+5)}-\frac{\sin (e+f x) \cos ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{p+1}}{b f (2 p+5)}",1,"-(((3*a + b*(7 + 2*p))*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(1 + p))/(b^2*f*(3 + 2*p)*(5 + 2*p))) - (Cos[e + f*x]^2*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(1 + p))/(b*f*(5 + 2*p)) + ((3*a^2 + 2*a*b*(5 + 2*p) + b^2*(15 + 16*p + 4*p^2))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sin[e + f*x]^2)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(b^2*f*(3 + 2*p)*(5 + 2*p)*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",5,5,23,0.2174,1,"{3190, 416, 388, 246, 245}"
374,1,119,0,0.1040947,"\int \cos ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\left(\frac{a}{2 b p+3 b}+1\right) \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^2(e+f x)}{a}\right)}{f}-\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{p+1}}{b f (2 p+3)}","\frac{(a+b (2 p+3)) \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^2(e+f x)}{a}\right)}{b f (2 p+3)}-\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{p+1}}{b f (2 p+3)}",1,"-((Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(1 + p))/(b*f*(3 + 2*p))) + ((1 + a/(3*b + 2*b*p))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sin[e + f*x]^2)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",4,4,23,0.1739,1,"{3190, 388, 246, 245}"
375,1,67,0,0.0440752,"\int \cos (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Cos[e + f*x]*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^2(e+f x)}{a}\right)}{f}","\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^2(e+f x)}{a}\right)}{f}",1,"(Hypergeometric2F1[1/2, -p, 3/2, -((b*Sin[e + f*x]^2)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,21,0.1429,1,"{3190, 246, 245}"
376,1,76,0,0.0741071,"\int \sec (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Sec[e + f*x]*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}","\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/2, 1, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,21,0.1429,1,"{3190, 430, 429}"
377,1,76,0,0.0820807,"\int \sec ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};2,-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}","\frac{\sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};2,-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/2, 2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3190, 430, 429}"
378,1,90,0,0.082417,"\int \cos ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-\frac{3}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-\frac{3}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/2, -3/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3192, 430, 429}"
379,1,90,0,0.0815529,"\int \cos ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};-\frac{1}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/2, -1/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3192, 430, 429}"
380,1,90,0,0.052151,"\int \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{1}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/2, 1/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,14,0.2143,1,"{3185, 430, 429}"
381,1,90,0,0.080179,"\int \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{3}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/2, 3/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3192, 430, 429}"
382,1,90,0,0.0812078,"\int \sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{5}{2},-p;\frac{3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/2, 5/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(f*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3192, 430, 429}"
383,1,219,0,0.2895023,"\int \frac{\cos ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Cos[c + d*x]^5/(a + b*Sin[c + d*x]^3),x]","-\frac{\left(a^{4/3}+b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 a^{2/3} b^{5/3} d}+\frac{\left(a^{4/3}+b^{4/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{3 a^{2/3} b^{5/3} d}+\frac{\left(a^{4/3}-b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} b^{5/3} d}-\frac{2 \log \left(a+b \sin ^3(c+d x)\right)}{3 b d}+\frac{\sin ^2(c+d x)}{2 b d}","-\frac{\left(a^{4/3}+b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 a^{2/3} b^{5/3} d}+\frac{\left(a^{4/3}+b^{4/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{3 a^{2/3} b^{5/3} d}+\frac{\left(a^{4/3}-b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} b^{5/3} d}-\frac{2 \log \left(a+b \sin ^3(c+d x)\right)}{3 b d}+\frac{\sin ^2(c+d x)}{2 b d}",1,"((a^(4/3) - b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(5/3)*d) + ((a^(4/3) + b^(4/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(2/3)*b^(5/3)*d) - ((a^(4/3) + b^(4/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(2/3)*b^(5/3)*d) - (2*Log[a + b*Sin[c + d*x]^3])/(3*b*d) + Sin[c + d*x]^2/(2*b*d)","A",11,10,23,0.4348,1,"{3223, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260}"
384,1,167,0,0.1483564,"\int \frac{\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Cos[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 a^{2/3} \sqrt[3]{b} d}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{3 a^{2/3} \sqrt[3]{b} d}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} \sqrt[3]{b} d}-\frac{\log \left(a+b \sin ^3(c+d x)\right)}{3 b d}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 a^{2/3} \sqrt[3]{b} d}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{3 a^{2/3} \sqrt[3]{b} d}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} \sqrt[3]{b} d}-\frac{\log \left(a+b \sin ^3(c+d x)\right)}{3 b d}",1,"-(ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(1/3)*d)) + Log[a^(1/3) + b^(1/3)*Sin[c + d*x]]/(3*a^(2/3)*b^(1/3)*d) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]/(6*a^(2/3)*b^(1/3)*d) - Log[a + b*Sin[c + d*x]^3]/(3*b*d)","A",9,9,23,0.3913,1,"{3223, 1871, 200, 31, 634, 617, 204, 628, 260}"
385,1,144,0,0.0999976,"\int \frac{\cos (c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Cos[c + d*x]/(a + b*Sin[c + d*x]^3),x]","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 a^{2/3} \sqrt[3]{b} d}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{3 a^{2/3} \sqrt[3]{b} d}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} \sqrt[3]{b} d}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 a^{2/3} \sqrt[3]{b} d}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{3 a^{2/3} \sqrt[3]{b} d}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} \sqrt[3]{b} d}",1,"-(ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(1/3)*d)) + Log[a^(1/3) + b^(1/3)*Sin[c + d*x]]/(3*a^(2/3)*b^(1/3)*d) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]/(6*a^(2/3)*b^(1/3)*d)","A",7,7,21,0.3333,1,"{3223, 200, 31, 634, 617, 204, 628}"
386,1,290,0,0.3293309,"\int \frac{\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Sec[c + d*x]/(a + b*Sin[c + d*x]^3),x]","\frac{\sqrt[3]{b} \left(a^{4/3}+b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 a^{2/3} d \left(a^2-b^2\right)}-\frac{b \log \left(a+b \sin ^3(c+d x)\right)}{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} \sin (c+d x)\right)}{3 a^{2/3} d \left(a^2-b^2\right)}-\frac{\sqrt[3]{b} \left(a^{4/3}-b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}","\frac{\sqrt[3]{b} \left(a^{4/3}+b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 a^{2/3} d \left(a^2-b^2\right)}-\frac{b \log \left(a+b \sin ^3(c+d x)\right)}{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} \sin (c+d x)\right)}{3 a^{2/3} d \left(a^2-b^2\right)}-\frac{\sqrt[3]{b} \left(a^{4/3}-b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}",1,"-((b^(1/3)*(a^(4/3) - b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(a^2 - b^2)*d)) - Log[1 - Sin[c + d*x]]/(2*(a + b)*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)*d) - (b^(1/3)*(a^(4/3) + b^(4/3))*Log[a^(1/3) + b^(1/3)*Sin[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)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(2/3)*(a^2 - b^2)*d) - (b*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)*d)","A",11,10,21,0.4762,1,"{3223, 2074, 1871, 1860, 31, 634, 617, 204, 628, 260}"
387,1,385,0,0.5043306,"\int \frac{\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Sec[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]","-\frac{b^{5/3} \left(3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 a^{2/3} d \left(a^2-b^2\right)^2}+\frac{b \left(a^2+2 b^2\right) \log \left(a+b \sin ^3(c+d x)\right)}{3 d \left(a^2-b^2\right)^2}+\frac{b^{5/3} \left(3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{3 a^{2/3} d \left(a^2-b^2\right)^2}-\frac{b^{5/3} \left(-3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} d \left(a^2-b^2\right)^2}+\frac{1}{4 d (a+b) (1-\sin (c+d x))}-\frac{1}{4 d (a-b) (\sin (c+d x)+1)}-\frac{(a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}","-\frac{b^{5/3} \left(3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 a^{2/3} d \left(a^2-b^2\right)^2}+\frac{b \left(a^2+2 b^2\right) \log \left(a+b \sin ^3(c+d x)\right)}{3 d \left(a^2-b^2\right)^2}+\frac{b^{5/3} \left(3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{3 a^{2/3} d \left(a^2-b^2\right)^2}-\frac{b^{5/3} \left(-3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} d \left(a^2-b^2\right)^2}+\frac{1}{4 d (a+b) (1-\sin (c+d x))}-\frac{1}{4 d (a-b) (\sin (c+d x)+1)}-\frac{(a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}",1,"-((b^(5/3)*(2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(a^2 - b^2)^2*d)) - ((a + 4*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((a - 4*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) + (b^(5/3)*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(2/3)*(a^2 - b^2)^2*d) - (b^(5/3)*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(2/3)*(a^2 - b^2)^2*d) + (b*(a^2 + 2*b^2)*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Sin[c + d*x])) - 1/(4*(a - b)*d*(1 + Sin[c + d*x]))","A",11,10,23,0.4348,1,"{3223, 2074, 1871, 1860, 31, 634, 617, 204, 628, 260}"
388,1,764,0,1.530423,"\int \frac{\cos ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3),x]","-\frac{2 (-1)^{2/3} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{4 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \sqrt[3]{-1} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{4 \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 b^{2/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{4 \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}-\frac{\cos (c+d x)}{b d}","-\frac{2 (-1)^{2/3} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{4 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \sqrt[3]{-1} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{4 \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 b^{2/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{4 \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}-\frac{\cos (c+d x)}{b d}",1,"(-2*(-1)^(2/3)*a^(2/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(4/3)*d) + (2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - b^(2/3)]*d) + (2*a^(2/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(4/3)*d) - (4*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(2/3)*d) + (2*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - (2*(-1)^(1/3)*a^(2/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b^(4/3)*d) - (2*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) + (4*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]])/(3*Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]*b^(2/3)*d) + (4*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*b^(2/3)*d) - Cos[c + d*x]/(b*d)","A",38,8,23,0.3478,1,"{3226, 3213, 2660, 618, 204, 3220, 206, 2638}"
389,1,484,0,0.6344572,"\int \frac{\cos ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 b^{2/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 b^{2/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}",1,"(2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - b^(2/3)]*d) - (2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(2/3)*d) + (2*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - (2*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) + (2*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]])/(3*Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]*b^(2/3)*d) + (2*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*b^(2/3)*d)","A",24,7,23,0.3043,1,"{3226, 3213, 2660, 618, 204, 3220, 206}"
390,1,245,0,0.2514389,"\int \frac{1}{a+b \sin ^3(c+d x)} \, dx","Int[(a + b*Sin[c + d*x]^3)^(-1),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}",1,"(2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - b^(2/3)]*d) + (2*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - (2*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d)","A",11,4,14,0.2857,1,"{3213, 2660, 618, 204}"
391,0,0,0,0.0453848,"\int \frac{\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]","\int \frac{\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx","\frac{2 (-1)^{2/3} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \left(a^{2/3}-(-1)^{2/3} b^{2/3}\right)^{3/2}}-\frac{2 b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \left(a^{2/3}-b^{2/3}\right)^{3/2}}+\frac{2 \sqrt[3]{-1} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \left(a^{2/3}+\sqrt[3]{-1} b^{2/3}\right)^{3/2}}+\frac{\sec (c+d x) (b-a \sin (c+d x))}{d \left(b^2-a^2\right)}",1,"Defer[Int][Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3), x]","F",0,0,0,0,-1,"{}"
392,0,0,0,0.0462909,"\int \frac{\sec ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3),x]","\int \frac{\sec ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx","-\frac{2 (-1)^{2/3} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right) b^{8/3}}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}} \left(a^2-b^2\right)^2 d}+\frac{2 a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right) b^{8/3}}{\sqrt{a^{2/3}-b^{2/3}} \left(a^2-b^2\right)^2 d}-\frac{2 \sqrt[3]{-1} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right) b^{8/3}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}} \left(a^2-b^2\right)^2 d}-\frac{2 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right) b^2}{3 a^{2/3} \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}} \left(a^2-b^2\right)^2 d}+\frac{2 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right) b^2}{3 a^{2/3} \sqrt{a^{2/3}-b^{2/3}} \left(a^2-b^2\right)^2 d}+\frac{2 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right) b^2}{3 a^{2/3} \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}} \left(a^2-b^2\right)^2 d}+\frac{2 \left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right) b^{4/3}}{3 \sqrt{a^{2/3}-b^{2/3}} \left(a^2-b^2\right)^2 d}-\frac{2 \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right) b^{4/3}}{3 \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}} \left(a^2-b^2\right)^2 d}-\frac{2 \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right) b^{4/3}}{3 \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}} \left(a^2-b^2\right)^2 d}+\frac{(a+4 b) \cos (c+d x)}{4 (a+b)^2 d (1-\sin (c+d x))}+\frac{\cos (c+d x)}{12 (a+b) d (1-\sin (c+d x))}-\frac{\cos (c+d x)}{12 (a-b) d (\sin (c+d x)+1)}-\frac{(a-4 b) \cos (c+d x)}{4 (a-b)^2 d (\sin (c+d x)+1)}+\frac{\cos (c+d x)}{12 (a+b) d (1-\sin (c+d x))^2}-\frac{\cos (c+d x)}{12 (a-b) d (\sin (c+d x)+1)^2}",1,"Defer[Int][Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3), x]","F",0,0,0,0,-1,"{}"
393,1,288,0,0.3350407,"\int \frac{\cos ^7(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Int[Cos[c + d*x]^7/(a + b*Sin[c + d*x]^3)^2,x]","-\frac{\sin (c+d x) \left(a^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)-b^2\right)}{3 a b^2 d \left(a+b \sin ^3(c+d x)\right)}-\frac{\left(-3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{9 a^{5/3} b^{7/3} d}+\frac{2 \left(-3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{9 a^{5/3} b^{7/3} d}-\frac{2 \left(3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} b^{7/3} d}-\frac{\sin (c+d x)}{b^2 d}","-\frac{\sin (c+d x) \left(a^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)-b^2\right)}{3 a b^2 d \left(a+b \sin ^3(c+d x)\right)}-\frac{\left(-3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{9 a^{5/3} b^{7/3} d}+\frac{2 \left(-3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{9 a^{5/3} b^{7/3} d}-\frac{2 \left(3 a^{4/3} b^{2/3}+2 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} b^{7/3} d}-\frac{\sin (c+d x)}{b^2 d}",1,"(-2*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(7/3)*d) + (2*(2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*b^(7/3)*d) - ((2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(9*a^(5/3)*b^(7/3)*d) - Sin[c + d*x]/(b^2*d) - (Sin[c + d*x]*(a^2 - b^2 + 3*a*b*Sin[c + d*x] + 3*b^2*Sin[c + d*x]^2))/(3*a*b^2*d*(a + b*Sin[c + d*x]^3))","A",10,9,23,0.3913,1,"{3223, 1858, 1887, 1860, 31, 634, 617, 204, 628}"
394,1,238,0,0.2245609,"\int \frac{\cos ^5(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Int[Cos[c + d*x]^5/(a + b*Sin[c + d*x]^3)^2,x]","\frac{\left(a^{4/3}-b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{9 a^{5/3} b^{5/3} d}-\frac{2 \left(a^{4/3}-b^{4/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{9 a^{5/3} b^{5/3} d}-\frac{2 \left(a^{4/3}+b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} b^{5/3} d}+\frac{\sin (c+d x) \left(-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right)}{3 a b d \left(a+b \sin ^3(c+d x)\right)}","\frac{\left(a^{4/3}-b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{9 a^{5/3} b^{5/3} d}-\frac{2 \left(a^{4/3}-b^{4/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{9 a^{5/3} b^{5/3} d}-\frac{2 \left(a^{4/3}+b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} b^{5/3} d}+\frac{\sin (c+d x) \left(-a \sin (c+d x)-2 b \sin ^2(c+d x)+b\right)}{3 a b d \left(a+b \sin ^3(c+d x)\right)}",1,"(-2*(a^(4/3) + b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(5/3)*d) - (2*(a^(4/3) - b^(4/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*b^(5/3)*d) + ((a^(4/3) - b^(4/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(9*a^(5/3)*b^(5/3)*d) + (Sin[c + d*x]*(b - a*Sin[c + d*x] - 2*b*Sin[c + d*x]^2))/(3*a*b*d*(a + b*Sin[c + d*x]^3))","A",8,8,23,0.3478,1,"{3223, 1858, 1860, 31, 634, 617, 204, 628}"
395,1,183,0,0.1624158,"\int \frac{\cos ^3(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Int[Cos[c + d*x]^3/(a + b*Sin[c + d*x]^3)^2,x]","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{9 a^{5/3} \sqrt[3]{b} d}+\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{9 a^{5/3} \sqrt[3]{b} d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} \sqrt[3]{b} d}+\frac{a+b \sin (c+d x)}{3 a b d \left(a+b \sin ^3(c+d x)\right)}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{9 a^{5/3} \sqrt[3]{b} d}+\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{9 a^{5/3} \sqrt[3]{b} d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} \sqrt[3]{b} d}+\frac{a+b \sin (c+d x)}{3 a b d \left(a+b \sin ^3(c+d x)\right)}",1,"(-2*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(1/3)*d) + (2*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*b^(1/3)*d) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]/(9*a^(5/3)*b^(1/3)*d) + (a + b*Sin[c + d*x])/(3*a*b*d*(a + b*Sin[c + d*x]^3))","A",9,9,23,0.3913,1,"{3223, 1854, 12, 200, 31, 634, 617, 204, 628}"
396,1,176,0,0.1151854,"\int \frac{\cos (c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Int[Cos[c + d*x]/(a + b*Sin[c + d*x]^3)^2,x]","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{9 a^{5/3} \sqrt[3]{b} d}+\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{9 a^{5/3} \sqrt[3]{b} d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} \sqrt[3]{b} d}+\frac{\sin (c+d x)}{3 a d \left(a+b \sin ^3(c+d x)\right)}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{9 a^{5/3} \sqrt[3]{b} d}+\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{9 a^{5/3} \sqrt[3]{b} d}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} \sqrt[3]{b} d}+\frac{\sin (c+d x)}{3 a d \left(a+b \sin ^3(c+d x)\right)}",1,"(-2*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(1/3)*d) + (2*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*b^(1/3)*d) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]/(9*a^(5/3)*b^(1/3)*d) + Sin[c + d*x]/(3*a*d*(a + b*Sin[c + d*x]^3))","A",8,8,21,0.3810,1,"{3223, 199, 200, 31, 634, 617, 204, 628}"
397,1,587,0,0.6875266,"\int \frac{\sec (c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]/(a + b*Sin[c + d*x]^3)^2,x]","\frac{b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a d \left(a^2-b^2\right) \left(a+b \sin ^3(c+d x)\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} \sin (c+d x)+b^{2/3} \sin ^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(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{18 a^{5/3} d \left(a^2-b^2\right)}-\frac{2 a b \log \left(a+b \sin ^3(c+d x)\right)}{3 d \left(a^2-b^2\right)^2}-\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} \sin (c+d x)\right)}{3 \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} \sin (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) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} d \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} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^2}","\frac{b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a d \left(a^2-b^2\right) \left(a+b \sin ^3(c+d x)\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} \sin (c+d x)+b^{2/3} \sin ^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(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{18 a^{5/3} d \left(a^2-b^2\right)}-\frac{2 a b \log \left(a+b \sin ^3(c+d x)\right)}{3 d \left(a^2-b^2\right)^2}-\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} \sin (c+d x)\right)}{3 \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} \sin (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) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} d \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} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^2}",1,"-(b^(1/3)*(a^(4/3) - 2*b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(a^2 - b^2)*d) - (b^(1/3)*(a^2 - 2*a^(2/3)*b^(4/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 - b^2)^2*d) - Log[1 - Sin[c + d*x]]/(2*(a + b)^2*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*d) - (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(1/3) + b^(1/3)*Sin[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^(1/3) + b^(1/3)*Sin[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^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(18*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)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(1/3)*(a^2 - b^2)^2*d) - (2*a*b*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^2*d) + (b*(a - Sin[c + d*x]*(b - a*Sin[c + d*x])))/(3*a*(a^2 - b^2)*d*(a + b*Sin[c + d*x]^3))","A",18,11,21,0.5238,1,"{3223, 2074, 1854, 1860, 31, 634, 617, 204, 628, 1871, 260}"
398,1,747,0,1.0215177,"\int \frac{\sec ^3(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]^3/(a + b*Sin[c + d*x]^3)^2,x]","-\frac{b \left(a \left(a^2+2 b^2\right)-b \sin (c+d x) \left(2 a^2-3 a b \sin (c+d x)+b^2\right)\right)}{3 a d \left(a^2-b^2\right)^2 \left(a+b \sin ^3(c+d x)\right)}-\frac{b^{5/3} \left(3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{18 a^{5/3} d \left(a^2-b^2\right)^2}-\frac{b^{5/3} \left(3 b^{2/3} \left(3 a^2+b^2\right)+4 a^{2/3} \left(a^2+2 b^2\right)\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 \sqrt[3]{a} d \left(a^2-b^2\right)^3}+\frac{2 a b \left(a^2+5 b^2\right) \log \left(a+b \sin ^3(c+d x)\right)}{3 d \left(a^2-b^2\right)^3}+\frac{b^{5/3} \left(3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{9 a^{5/3} d \left(a^2-b^2\right)^2}+\frac{b^{5/3} \left(3 b^{2/3} \left(3 a^2+b^2\right)+4 a^{2/3} \left(a^2+2 b^2\right)\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{3 \sqrt[3]{a} d \left(a^2-b^2\right)^3}-\frac{b^{5/3} \left(-3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} d \left(a^2-b^2\right)^2}-\frac{b^{5/3} \left(-9 a^2 b^{2/3}+8 a^{2/3} b^2+4 a^{8/3}-3 b^{8/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} d \left(a^2-b^2\right)^3}+\frac{1}{4 d (a+b)^2 (1-\sin (c+d x))}-\frac{1}{4 d (a-b)^2 (\sin (c+d x)+1)}-\frac{(a+7 b) \log (1-\sin (c+d x))}{4 d (a+b)^3}+\frac{(a-7 b) \log (\sin (c+d x)+1)}{4 d (a-b)^3}","-\frac{b \left(a \left(a^2+2 b^2\right)-b \sin (c+d x) \left(2 a^2-3 a b \sin (c+d x)+b^2\right)\right)}{3 a d \left(a^2-b^2\right)^2 \left(a+b \sin ^3(c+d x)\right)}-\frac{b^{5/3} \left(3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{18 a^{5/3} d \left(a^2-b^2\right)^2}-\frac{b^{5/3} \left(3 b^{2/3} \left(3 a^2+b^2\right)+4 a^{2/3} \left(a^2+2 b^2\right)\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{6 \sqrt[3]{a} d \left(a^2-b^2\right)^3}+\frac{2 a b \left(a^2+5 b^2\right) \log \left(a+b \sin ^3(c+d x)\right)}{3 d \left(a^2-b^2\right)^3}+\frac{b^{5/3} \left(3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{9 a^{5/3} d \left(a^2-b^2\right)^2}+\frac{b^{5/3} \left(3 b^{2/3} \left(3 a^2+b^2\right)+4 a^{2/3} \left(a^2+2 b^2\right)\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{3 \sqrt[3]{a} d \left(a^2-b^2\right)^3}-\frac{b^{5/3} \left(-3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} d \left(a^2-b^2\right)^2}-\frac{b^{5/3} \left(-9 a^2 b^{2/3}+8 a^{2/3} b^2+4 a^{8/3}-3 b^{8/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} d \left(a^2-b^2\right)^3}+\frac{1}{4 d (a+b)^2 (1-\sin (c+d x))}-\frac{1}{4 d (a-b)^2 (\sin (c+d x)+1)}-\frac{(a+7 b) \log (1-\sin (c+d x))}{4 d (a+b)^3}+\frac{(a-7 b) \log (\sin (c+d x)+1)}{4 d (a-b)^3}",1,"-(b^(5/3)*(4*a^2 - 3*a^(4/3)*b^(2/3) + 2*b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(a^2 - b^2)^2*d) - (b^(5/3)*(4*a^(8/3) - 9*a^2*b^(2/3) + 8*a^(2/3)*b^2 - 3*b^(8/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 - b^2)^3*d) - ((a + 7*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^3*d) + ((a - 7*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^3*d) + (b^(5/3)*(4*a^2 + 3*a^(4/3)*b^(2/3) + 2*b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*(a^2 - b^2)^2*d) + (b^(5/3)*(3*b^(2/3)*(3*a^2 + b^2) + 4*a^(2/3)*(a^2 + 2*b^2))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(1/3)*(a^2 - b^2)^3*d) - (b^(5/3)*(4*a^2 + 3*a^(4/3)*b^(2/3) + 2*b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(18*a^(5/3)*(a^2 - b^2)^2*d) - (b^(5/3)*(3*b^(2/3)*(3*a^2 + b^2) + 4*a^(2/3)*(a^2 + 2*b^2))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(1/3)*(a^2 - b^2)^3*d) + (2*a*b*(a^2 + 5*b^2)*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^3*d) + 1/(4*(a + b)^2*d*(1 - Sin[c + d*x])) - 1/(4*(a - b)^2*d*(1 + Sin[c + d*x])) - (b*(a*(a^2 + 2*b^2) - b*Sin[c + d*x]*(2*a^2 + b^2 - 3*a*b*Sin[c + d*x])))/(3*a*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]^3))","A",18,11,23,0.4783,1,"{3223, 2074, 1854, 1860, 31, 634, 617, 204, 628, 1871, 260}"
399,0,0,0,0.0437858,"\int \frac{\cos ^4(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Int[Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]","\int \frac{\cos ^4(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","\text{Int}\left(\frac{\cos ^4(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2},x\right)",0,"Defer[Int][Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2, x]","A",0,0,0,0,-1,"{}"
400,0,0,0,0.0425309,"\int \frac{\cos ^2(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Int[Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]","\int \frac{\cos ^2(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","\text{Int}\left(\frac{\cos ^2(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2},x\right)",0,"Defer[Int][Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2, x]","A",0,0,0,0,-1,"{}"
401,0,0,0,0.0118343,"\int \frac{1}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Int[(a + b*Sin[c + d*x]^3)^(-2),x]","\int \frac{1}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(a+b \sin ^3(c+d x)\right)^2},x\right)",0,"Defer[Int][(a + b*Sin[c + d*x]^3)^(-2), x]","A",0,0,0,0,-1,"{}"
402,0,0,0,0.0432045,"\int \frac{\sec ^2(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]","\int \frac{\sec ^2(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","\text{Int}\left(\frac{\sec ^2(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2},x\right)",0,"Defer[Int][Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2, x]","A",0,0,0,0,-1,"{}"
403,0,0,0,0.0430803,"\int \frac{\sec ^4(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]","\int \frac{\sec ^4(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","\text{Int}\left(\frac{\sec ^4(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2},x\right)",0,"Defer[Int][Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2, x]","A",0,0,0,0,-1,"{}"
404,1,131,0,0.1899837,"\int \frac{\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Cos[c + d*x]^7/(a - b*Sin[c + d*x]^4),x]","\frac{\left(\sqrt{a}+\sqrt{b}\right)^3 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{7/4} d}-\frac{\left(\sqrt{a}-\sqrt{b}\right)^3 \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{7/4} d}+\frac{\sin ^3(c+d x)}{3 b d}-\frac{3 \sin (c+d x)}{b d}","\frac{\left(\sqrt{a}+\sqrt{b}\right)^3 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{7/4} d}-\frac{\left(\sqrt{a}-\sqrt{b}\right)^3 \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{7/4} d}+\frac{\sin ^3(c+d x)}{3 b d}-\frac{3 \sin (c+d x)}{b d}",1,"((Sqrt[a] + Sqrt[b])^3*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(7/4)*d) - ((Sqrt[a] - Sqrt[b])^3*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(7/4)*d) - (3*Sin[c + d*x])/(b*d) + Sin[c + d*x]^3/(3*b*d)","A",6,5,24,0.2083,1,"{3223, 1171, 1167, 205, 208}"
405,1,113,0,0.1616787,"\int \frac{\cos ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Cos[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]","\frac{\left(\sqrt{a}+\sqrt{b}\right)^2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{5/4} d}+\frac{\left(-2 \sqrt{a} \sqrt{b}+a+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{5/4} d}-\frac{\sin (c+d x)}{b d}","\frac{\left(\sqrt{a}+\sqrt{b}\right)^2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{5/4} d}+\frac{\left(-2 \sqrt{a} \sqrt{b}+a+b\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{5/4} d}-\frac{\sin (c+d x)}{b d}",1,"((Sqrt[a] + Sqrt[b])^2*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(5/4)*d) + ((a - 2*Sqrt[a]*Sqrt[b] + b)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(5/4)*d) - Sin[c + d*x]/(b*d)","A",6,5,24,0.2083,1,"{3223, 1171, 1167, 205, 208}"
406,1,95,0,0.1017723,"\int \frac{\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Cos[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]","\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4} d}-\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4} d}","\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4} d}-\frac{\left(\sqrt{a}-\sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/4} d}",1,"((Sqrt[a] + Sqrt[b])*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(3/4)*d) - ((Sqrt[a] - Sqrt[b])*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(3/4)*d)","A",4,4,24,0.1667,1,"{3223, 1167, 205, 208}"
407,1,71,0,0.0673509,"\int \frac{\cos (c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Cos[c + d*x]/(a - b*Sin[c + d*x]^4),x]","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt[4]{b} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt[4]{b} d}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt[4]{b} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt[4]{b} d}",1,"ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)]/(2*a^(3/4)*b^(1/4)*d) + ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)]/(2*a^(3/4)*b^(1/4)*d)","A",4,4,22,0.1818,1,"{3223, 212, 208, 205}"
408,1,117,0,0.1534835,"\int \frac{\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sec[c + d*x]/(a - b*Sin[c + d*x]^4),x]","\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)}-\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{d (a-b)}","\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)}-\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{d (a-b)}",1,"(b^(1/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])*d) + ArcTanh[Sin[c + d*x]]/((a - b)*d) - (b^(1/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])*d)","A",7,6,22,0.2727,1,"{3223, 1171, 207, 1167, 205, 208}"
409,1,175,0,0.2137217,"\int \frac{\sec ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sec[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]","\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^2}+\frac{b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^2}+\frac{1}{4 d (a-b) (1-\sin (c+d x))}-\frac{1}{4 d (a-b) (\sin (c+d x)+1)}+\frac{(a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 d (a-b)^2}","\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^2}+\frac{b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^2}+\frac{1}{4 d (a-b) (1-\sin (c+d x))}-\frac{1}{4 d (a-b) (\sin (c+d x)+1)}+\frac{(a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 d (a-b)^2}",1,"(b^(3/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^2*d) + ((a - 5*b)*ArcTanh[Sin[c + d*x]])/(2*(a - b)^2*d) + (b^(3/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^2*d) + 1/(4*(a - b)*d*(1 - Sin[c + d*x])) - 1/(4*(a - b)*d*(1 + Sin[c + d*x]))","A",7,6,24,0.2500,1,"{3223, 1171, 207, 1167, 205, 208}"
410,1,249,0,0.2967951,"\int \frac{\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sec[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]","\frac{b^{5/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^3}-\frac{b^{5/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^3}+\frac{\left(3 a^2-6 a b+35 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d (a-b)^3}+\frac{3 a-11 b}{16 d (a-b)^2 (1-\sin (c+d x))}-\frac{3 a-11 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a-b) (1-\sin (c+d x))^2}-\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2}","\frac{b^{5/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^3}-\frac{b^{5/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \sin (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^3}+\frac{\left(3 a^2-6 a b+35 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d (a-b)^3}+\frac{3 a-11 b}{16 d (a-b)^2 (1-\sin (c+d x))}-\frac{3 a-11 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a-b) (1-\sin (c+d x))^2}-\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2}",1,"(b^(5/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^3*d) + ((3*a^2 - 6*a*b + 35*b^2)*ArcTanh[Sin[c + d*x]])/(8*(a - b)^3*d) - (b^(5/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^3*d) + 1/(16*(a - b)*d*(1 - Sin[c + d*x])^2) + (3*a - 11*b)/(16*(a - b)^2*d*(1 - Sin[c + d*x])) - 1/(16*(a - b)*d*(1 + Sin[c + d*x])^2) - (3*a - 11*b)/(16*(a - b)^2*d*(1 + Sin[c + d*x]))","A",7,6,24,0.2500,1,"{3223, 1171, 207, 1167, 205, 208}"
411,1,252,0,0.4395217,"\int \frac{\cos ^{10}(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Cos[c + d*x]^10/(a - b*Sin[c + d*x]^4),x]","-\frac{\left(\sqrt{a}-\sqrt{b}\right)^{9/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{5/2} d}+\frac{\left(\sqrt{a}+\sqrt{b}\right)^{9/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{5/2} d}-\frac{(a+3 b) \sin (c+d x) \cos (c+d x)}{2 b^2 d}-\frac{4 x (a+b)}{b^2}-\frac{x (a+3 b)}{2 b^2}-\frac{\sin (c+d x) \cos ^5(c+d x)}{6 b d}-\frac{17 \sin (c+d x) \cos ^3(c+d x)}{24 b d}-\frac{17 \sin (c+d x) \cos (c+d x)}{16 b d}-\frac{17 x}{16 b}","-\frac{\left(\sqrt{a}-\sqrt{b}\right)^{9/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{5/2} d}+\frac{\left(\sqrt{a}+\sqrt{b}\right)^{9/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{5/2} d}-\frac{(a+3 b) \sin (c+d x) \cos (c+d x)}{2 b^2 d}-\frac{4 x (a+b)}{b^2}-\frac{x (a+3 b)}{2 b^2}-\frac{\sin (c+d x) \cos ^5(c+d x)}{6 b d}-\frac{17 \sin (c+d x) \cos ^3(c+d x)}{24 b d}-\frac{17 \sin (c+d x) \cos (c+d x)}{16 b d}-\frac{17 x}{16 b}",1,"(-17*x)/(16*b) - (4*(a + b)*x)/b^2 - ((a + 3*b)*x)/(2*b^2) - ((Sqrt[a] - Sqrt[b])^(9/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(5/2)*d) + ((Sqrt[a] + Sqrt[b])^(9/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(5/2)*d) - (17*Cos[c + d*x]*Sin[c + d*x])/(16*b*d) - ((a + 3*b)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) - (17*Cos[c + d*x]^3*Sin[c + d*x])/(24*b*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(6*b*d)","A",16,6,24,0.2500,1,"{3224, 1170, 199, 203, 1166, 205}"
412,1,186,0,0.3261018,"\int \frac{\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Cos[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]","\frac{\left(\sqrt{a}-\sqrt{b}\right)^{7/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^2 d}+\frac{\left(\sqrt{a}+\sqrt{b}\right)^{7/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^2 d}-\frac{x (a+3 b)}{b^2}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}-\frac{11 \sin (c+d x) \cos (c+d x)}{8 b d}-\frac{11 x}{8 b}","\frac{\left(\sqrt{a}-\sqrt{b}\right)^{7/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^2 d}+\frac{\left(\sqrt{a}+\sqrt{b}\right)^{7/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^2 d}-\frac{x (a+3 b)}{b^2}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}-\frac{11 \sin (c+d x) \cos (c+d x)}{8 b d}-\frac{11 x}{8 b}",1,"(-11*x)/(8*b) - ((a + 3*b)*x)/b^2 + ((Sqrt[a] - Sqrt[b])^(7/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^2*d) + ((Sqrt[a] + Sqrt[b])^(7/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^2*d) - (11*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)","A",12,6,24,0.2500,1,"{3224, 1170, 199, 203, 1166, 205}"
413,1,155,0,0.2856366,"\int \frac{\cos ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Cos[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]","-\frac{\left(\sqrt{a}-\sqrt{b}\right)^{5/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/2} d}+\frac{\left(\sqrt{a}+\sqrt{b}\right)^{5/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/2} d}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}-\frac{5 x}{2 b}","-\frac{\left(\sqrt{a}-\sqrt{b}\right)^{5/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/2} d}+\frac{\left(\sqrt{a}+\sqrt{b}\right)^{5/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b^{3/2} d}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}-\frac{5 x}{2 b}",1,"(-5*x)/(2*b) - ((Sqrt[a] - Sqrt[b])^(5/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(3/2)*d) + ((Sqrt[a] + Sqrt[b])^(5/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(3/2)*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",9,6,24,0.2500,1,"{3224, 1170, 199, 203, 1166, 205}"
414,1,127,0,0.2354601,"\int \frac{\cos ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Cos[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]","\frac{\left(\sqrt{a}-\sqrt{b}\right)^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b d}+\frac{\left(\sqrt{a}+\sqrt{b}\right)^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b d}-\frac{x}{b}","\frac{\left(\sqrt{a}-\sqrt{b}\right)^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b d}+\frac{\left(\sqrt{a}+\sqrt{b}\right)^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} b d}-\frac{x}{b}",1,"-(x/b) + ((Sqrt[a] - Sqrt[b])^(3/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b*d) + ((Sqrt[a] + Sqrt[b])^(3/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b*d)","A",7,5,24,0.2083,1,"{3224, 1170, 203, 1166, 205}"
415,1,125,0,0.1093137,"\int \frac{\cos ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Cos[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]","\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt{b} d}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt{b} d}","\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt{b} d}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt{b} d}",1,"-(Sqrt[Sqrt[a] - Sqrt[b]]*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*Sqrt[b]*d) + (Sqrt[Sqrt[a] + Sqrt[b]]*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*Sqrt[b]*d)","A",4,3,24,0.1250,1,"{3224, 1093, 205}"
416,1,142,0,0.2320295,"\int \frac{\sec ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sec[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}+\frac{\tan (c+d x)}{d (a-b)}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}+\frac{\tan (c+d x)}{d (a-b)}",1,"-(Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^(3/2)*d) + (Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + Tan[c + d*x]/((a - b)*d)","A",6,4,24,0.1667,1,"{3224, 1170, 1166, 205}"
417,1,161,0,0.3476035,"\int \frac{\sec ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sec[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]","\frac{b \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\tan ^3(c+d x)}{3 d (a-b)}+\frac{(a-3 b) \tan (c+d x)}{d (a-b)^2}","\frac{b \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{5/2}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{5/2}}+\frac{\tan ^3(c+d x)}{3 d (a-b)}+\frac{(a-3 b) \tan (c+d x)}{d (a-b)^2}",1,"(b*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^(5/2)*d) + (b*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) + ((a - 3*b)*Tan[c + d*x])/((a - b)^2*d) + Tan[c + d*x]^3/(3*(a - b)*d)","A",6,4,24,0.1667,1,"{3224, 1170, 1166, 205}"
418,1,204,0,0.3779655,"\int \frac{\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Int[Sec[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{7/2}}+\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{7/2}}+\frac{\left(a^2-3 a b+6 b^2\right) \tan (c+d x)}{d (a-b)^3}+\frac{\tan ^5(c+d x)}{5 d (a-b)}+\frac{2 (a-2 b) \tan ^3(c+d x)}{3 d (a-b)^2}","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{7/2}}+\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{7/2}}+\frac{\left(a^2-3 a b+6 b^2\right) \tan (c+d x)}{d (a-b)^3}+\frac{\tan ^5(c+d x)}{5 d (a-b)}+\frac{2 (a-2 b) \tan ^3(c+d x)}{3 d (a-b)^2}",1,"-(b^(3/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^(7/2)*d) + (b^(3/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^(7/2)*d) + ((a^2 - 3*a*b + 6*b^2)*Tan[c + d*x])/((a - b)^3*d) + (2*(a - 2*b)*Tan[c + d*x]^3)/(3*(a - b)^2*d) + Tan[c + d*x]^5/(5*(a - b)*d)","A",6,4,24,0.1667,1,"{3224, 1170, 1166, 205}"
419,0,0,0,0.0439154,"\int \cos ^m(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^m*(a + b*Sin[e + f*x]^4)^p,x]","\int \cos ^m(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","\text{Int}\left(\cos ^m(e+f x) \left(a+b \sin ^4(e+f x)\right)^p,x\right)",0,"Defer[Int][Cos[e + f*x]^m*(a + b*Sin[e + f*x]^4)^p, x]","A",0,0,0,0,-1,"{}"
420,1,191,0,0.2219719,"\int \cos ^5(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^5*(a + b*Sin[e + f*x]^4)^p,x]","-\frac{2 \sin ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b \sin ^4(e+f x)}{a}\right)}{3 f}+\frac{\left(1-\frac{a}{4 b p+5 b}\right) \sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b \sin ^4(e+f x)}{a}\right)}{f}+\frac{\sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^{p+1}}{b f (4 p+5)}","-\frac{2 \sin ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b \sin ^4(e+f x)}{a}\right)}{3 f}-\frac{(a-b (4 p+5)) \sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b \sin ^4(e+f x)}{a}\right)}{b f (4 p+5)}+\frac{\sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^{p+1}}{b f (4 p+5)}",1,"(Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^(1 + p))/(b*f*(5 + 4*p)) + ((1 - a/(5*b + 4*b*p))*Hypergeometric2F1[1/4, -p, 5/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^p)/(f*(1 + (b*Sin[e + f*x]^4)/a)^p) - (2*Hypergeometric2F1[3/4, -p, 7/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p)/(3*f*(1 + (b*Sin[e + f*x]^4)/a)^p)","A",8,7,23,0.3043,1,"{3223, 1207, 1204, 246, 245, 365, 364}"
421,1,140,0,0.0980214,"\int \cos ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p,x]","\frac{\sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b \sin ^4(e+f x)}{a}\right)}{f}-\frac{\sin ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b \sin ^4(e+f x)}{a}\right)}{3 f}","\frac{\sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b \sin ^4(e+f x)}{a}\right)}{f}-\frac{\sin ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b \sin ^4(e+f x)}{a}\right)}{3 f}",1,"(Hypergeometric2F1[1/4, -p, 5/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^p)/(f*(1 + (b*Sin[e + f*x]^4)/a)^p) - (Hypergeometric2F1[3/4, -p, 7/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p)/(3*f*(1 + (b*Sin[e + f*x]^4)/a)^p)","A",7,6,23,0.2609,1,"{3223, 1204, 246, 245, 365, 364}"
422,1,67,0,0.0420269,"\int \cos (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Int[Cos[e + f*x]*(a + b*Sin[e + f*x]^4)^p,x]","\frac{\sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b \sin ^4(e+f x)}{a}\right)}{f}","\frac{\sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b \sin ^4(e+f x)}{a}\right)}{f}",1,"(Hypergeometric2F1[1/4, -p, 5/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^p)/(f*(1 + (b*Sin[e + f*x]^4)/a)^p)","A",3,3,21,0.1429,1,"{3223, 246, 245}"
423,1,158,0,0.1502377,"\int \sec (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Int[Sec[e + f*x]*(a + b*Sin[e + f*x]^4)^p,x]","\frac{\sin ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{4};1,-p;\frac{7}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right)}{3 f}+\frac{\sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{4};1,-p;\frac{5}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right)}{f}","\frac{\sin ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{4};1,-p;\frac{7}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right)}{3 f}+\frac{\sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{4};1,-p;\frac{5}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/4, 1, -p, 5/4, Sin[e + f*x]^4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^p)/(f*(1 + (b*Sin[e + f*x]^4)/a)^p) + (AppellF1[3/4, 1, -p, 7/4, Sin[e + f*x]^4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p)/(3*f*(1 + (b*Sin[e + f*x]^4)/a)^p)","A",7,6,21,0.2857,1,"{3223, 1240, 430, 429, 511, 510}"
424,1,239,0,0.2166923,"\int \sec ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p,x]","\frac{\sin ^5(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{5}{4};2,-p;\frac{9}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right)}{5 f}+\frac{2 \sin ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{4};2,-p;\frac{7}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right)}{3 f}+\frac{\sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{4};2,-p;\frac{5}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right)}{f}","\frac{\sin ^5(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{5}{4};2,-p;\frac{9}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right)}{5 f}+\frac{2 \sin ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{4};2,-p;\frac{7}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right)}{3 f}+\frac{\sin (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \left(\frac{b \sin ^4(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{4};2,-p;\frac{5}{4};\sin ^4(e+f x),-\frac{b \sin ^4(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/4, 2, -p, 5/4, Sin[e + f*x]^4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^p)/(f*(1 + (b*Sin[e + f*x]^4)/a)^p) + (2*AppellF1[3/4, 2, -p, 7/4, Sin[e + f*x]^4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p)/(3*f*(1 + (b*Sin[e + f*x]^4)/a)^p) + (AppellF1[5/4, 2, -p, 9/4, Sin[e + f*x]^4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^5*(a + b*Sin[e + f*x]^4)^p)/(5*f*(1 + (b*Sin[e + f*x]^4)/a)^p)","A",9,6,23,0.2609,1,"{3223, 1240, 430, 429, 511, 510}"
425,0,0,0,0.0399659,"\int \cos ^4(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p,x]","\int \cos ^4(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","\text{Int}\left(\cos ^4(e+f x) \left(a+b \sin ^4(e+f x)\right)^p,x\right)",0,"Defer[Int][Cos[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p, x]","A",0,0,0,0,-1,"{}"
426,0,0,0,0.0406979,"\int \cos ^2(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^4)^p,x]","\int \cos ^2(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","\text{Int}\left(\cos ^2(e+f x) \left(a+b \sin ^4(e+f x)\right)^p,x\right)",0,"Defer[Int][Cos[e + f*x]^2*(a + b*Sin[e + f*x]^4)^p, x]","A",0,0,0,0,-1,"{}"
427,0,0,0,0.010978,"\int \left(a+b \sin ^4(e+f x)\right)^p \, dx","Int[(a + b*Sin[e + f*x]^4)^p,x]","\int \left(a+b \sin ^4(e+f x)\right)^p \, dx","\text{Int}\left(\left(a+b \sin ^4(e+f x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[e + f*x]^4)^p, x]","A",0,0,0,0,-1,"{}"
428,0,0,0,0.043307,"\int \sec ^2(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^4)^p,x]","\int \sec ^2(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","\text{Int}\left(\sec ^2(e+f x) \left(a+b \sin ^4(e+f x)\right)^p,x\right)",0,"Defer[Int][Sec[e + f*x]^2*(a + b*Sin[e + f*x]^4)^p, x]","A",0,0,0,0,-1,"{}"
429,0,0,0,0.0414706,"\int \sec ^4(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p,x]","\int \sec ^4(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","\text{Int}\left(\sec ^4(e+f x) \left(a+b \sin ^4(e+f x)\right)^p,x\right)",0,"Defer[Int][Sec[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p, x]","A",0,0,0,0,-1,"{}"
430,0,0,0,0.0490627,"\int \cos ^m(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^m*(a + b*Sin[e + f*x]^n)^p,x]","\int \cos ^m(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","\text{Int}\left(\cos ^m(e+f x) \left(a+b \sin ^n(e+f x)\right)^p,x\right)",0,"Defer[Int][Cos[e + f*x]^m*(a + b*Sin[e + f*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
431,1,226,0,0.1743414,"\int \cos ^5(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^5*(a + b*Sin[e + f*x]^n)^p,x]","\frac{\sin ^5(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{n},-p;\frac{n+5}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{5 f}-\frac{2 \sin ^3(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{3 f}+\frac{\sin (e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{f}","\frac{\sin ^5(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{n},-p;\frac{n+5}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{5 f}-\frac{2 \sin ^3(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{3 f}+\frac{\sin (e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{f}",1,"(Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^n)^p)/(f*(1 + (b*Sin[e + f*x]^n)/a)^p) - (2*Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^n)^p)/(3*f*(1 + (b*Sin[e + f*x]^n)/a)^p) + (Hypergeometric2F1[5/n, -p, (5 + n)/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]^5*(a + b*Sin[e + f*x]^n)^p)/(5*f*(1 + (b*Sin[e + f*x]^n)/a)^p)","A",9,6,23,0.2609,1,"{3223, 1893, 246, 245, 365, 364}"
432,1,148,0,0.1180103,"\int \cos ^3(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^3*(a + b*Sin[e + f*x]^n)^p,x]","\frac{\sin (e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{f}-\frac{\sin ^3(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{3 f}","\frac{\sin (e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{f}-\frac{\sin ^3(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{3 f}",1,"(Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^n)^p)/(f*(1 + (b*Sin[e + f*x]^n)/a)^p) - (Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^n)^p)/(3*f*(1 + (b*Sin[e + f*x]^n)/a)^p)","A",7,6,23,0.2609,1,"{3223, 1893, 246, 245, 365, 364}"
433,1,69,0,0.0493514,"\int \cos (e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Int[Cos[e + f*x]*(a + b*Sin[e + f*x]^n)^p,x]","\frac{\sin (e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{f}","\frac{\sin (e+f x) \left(a+b \sin ^n(e+f x)\right)^p \left(\frac{b \sin ^n(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right)}{f}",1,"(Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^n)^p)/(f*(1 + (b*Sin[e + f*x]^n)/a)^p)","A",3,3,21,0.1429,1,"{3223, 246, 245}"
434,0,0,0,0.039806,"\int \sec (e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Int[Sec[e + f*x]*(a + b*Sin[e + f*x]^n)^p,x]","\int \sec (e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","\text{Int}\left(\sec (e+f x) \left(a+b \sin ^n(e+f x)\right)^p,x\right)",0,"Defer[Int][Sec[e + f*x]*(a + b*Sin[e + f*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
435,0,0,0,0.0511064,"\int \sec ^3(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^3*(a + b*Sin[e + f*x]^n)^p,x]","\int \sec ^3(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","\text{Int}\left(\sec ^3(e+f x) \left(a+b \sin ^n(e+f x)\right)^p,x\right)",0,"Defer[Int][Sec[e + f*x]^3*(a + b*Sin[e + f*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
436,0,0,0,0.0497227,"\int \cos ^4(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p,x]","\int \cos ^4(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","\text{Int}\left(\cos ^4(e+f x) \left(a+b \sin ^n(e+f x)\right)^p,x\right)",0,"Defer[Int][Cos[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
437,0,0,0,0.048352,"\int \cos ^2(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^n)^p,x]","\int \cos ^2(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","\text{Int}\left(\cos ^2(e+f x) \left(a+b \sin ^n(e+f x)\right)^p,x\right)",0,"Defer[Int][Cos[e + f*x]^2*(a + b*Sin[e + f*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
438,0,0,0,0.0129351,"\int \left(a+b \sin ^n(e+f x)\right)^p \, dx","Int[(a + b*Sin[e + f*x]^n)^p,x]","\int \left(a+b \sin ^n(e+f x)\right)^p \, dx","\text{Int}\left(\left(a+b \sin ^n(e+f x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[e + f*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
439,0,0,0,0.0484019,"\int \sec ^2(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^n)^p,x]","\int \sec ^2(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","\text{Int}\left(\sec ^2(e+f x) \left(a+b \sin ^n(e+f x)\right)^p,x\right)",0,"Defer[Int][Sec[e + f*x]^2*(a + b*Sin[e + f*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
440,0,0,0,0.0488473,"\int \sec ^4(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Int[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p,x]","\int \sec ^4(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","\text{Int}\left(\sec ^4(e+f x) \left(a+b \sin ^n(e+f x)\right)^p,x\right)",0,"Defer[Int][Sec[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
441,1,128,0,0.1316902,"\int \frac{\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Tan[c + d*x]^7/(a + b*Sin[c + d*x]^2),x]","\frac{\left(3 a^2+3 a b+b^2\right) \sec ^2(c+d x)}{2 d (a+b)^3}-\frac{a^3 \log \left(a+b \sin ^2(c+d x)\right)}{2 d (a+b)^4}+\frac{a^3 \log (\cos (c+d x))}{d (a+b)^4}+\frac{\sec ^6(c+d x)}{6 d (a+b)}-\frac{(3 a+2 b) \sec ^4(c+d x)}{4 d (a+b)^2}","\frac{\left(3 a^2+3 a b+b^2\right) \sec ^2(c+d x)}{2 d (a+b)^3}-\frac{a^3 \log \left(a+b \sin ^2(c+d x)\right)}{2 d (a+b)^4}+\frac{a^3 \log (\cos (c+d x))}{d (a+b)^4}+\frac{\sec ^6(c+d x)}{6 d (a+b)}-\frac{(3 a+2 b) \sec ^4(c+d x)}{4 d (a+b)^2}",1,"(a^3*Log[Cos[c + d*x]])/((a + b)^4*d) - (a^3*Log[a + b*Sin[c + d*x]^2])/(2*(a + b)^4*d) + ((3*a^2 + 3*a*b + b^2)*Sec[c + d*x]^2)/(2*(a + b)^3*d) - ((3*a + 2*b)*Sec[c + d*x]^4)/(4*(a + b)^2*d) + Sec[c + d*x]^6/(6*(a + b)*d)","A",3,2,23,0.08696,1,"{3194, 88}"
442,1,94,0,0.1024057,"\int \frac{\tan ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Tan[c + d*x]^5/(a + b*Sin[c + d*x]^2),x]","\frac{a^2 \log \left(a+b \sin ^2(c+d x)\right)}{2 d (a+b)^3}-\frac{a^2 \log (\cos (c+d x))}{d (a+b)^3}+\frac{\sec ^4(c+d x)}{4 d (a+b)}-\frac{(2 a+b) \sec ^2(c+d x)}{2 d (a+b)^2}","\frac{a^2 \log \left(a+b \sin ^2(c+d x)\right)}{2 d (a+b)^3}-\frac{a^2 \log (\cos (c+d x))}{d (a+b)^3}+\frac{\sec ^4(c+d x)}{4 d (a+b)}-\frac{(2 a+b) \sec ^2(c+d x)}{2 d (a+b)^2}",1,"-((a^2*Log[Cos[c + d*x]])/((a + b)^3*d)) + (a^2*Log[a + b*Sin[c + d*x]^2])/(2*(a + b)^3*d) - ((2*a + b)*Sec[c + d*x]^2)/(2*(a + b)^2*d) + Sec[c + d*x]^4/(4*(a + b)*d)","A",3,2,23,0.08696,1,"{3194, 88}"
443,1,64,0,0.0757577,"\int \frac{\tan ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Tan[c + d*x]^3/(a + b*Sin[c + d*x]^2),x]","\frac{\sec ^2(c+d x)}{2 d (a+b)}-\frac{a \log \left(a+b \sin ^2(c+d x)\right)}{2 d (a+b)^2}+\frac{a \log (\cos (c+d x))}{d (a+b)^2}","\frac{\sec ^2(c+d x)}{2 d (a+b)}-\frac{a \log \left(a+b \sin ^2(c+d x)\right)}{2 d (a+b)^2}+\frac{a \log (\cos (c+d x))}{d (a+b)^2}",1,"(a*Log[Cos[c + d*x]])/((a + b)^2*d) - (a*Log[a + b*Sin[c + d*x]^2])/(2*(a + b)^2*d) + Sec[c + d*x]^2/(2*(a + b)*d)","A",3,2,23,0.08696,1,"{3194, 77}"
444,1,43,0,0.0385448,"\int \frac{\tan (c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Tan[c + d*x]/(a + b*Sin[c + d*x]^2),x]","\frac{\log \left(a+b \sin ^2(c+d x)\right)}{2 d (a+b)}-\frac{\log (\cos (c+d x))}{d (a+b)}","\frac{\log \left(a+b \sin ^2(c+d x)\right)}{2 d (a+b)}-\frac{\log (\cos (c+d x))}{d (a+b)}",1,"-(Log[Cos[c + d*x]]/((a + b)*d)) + Log[a + b*Sin[c + d*x]^2]/(2*(a + b)*d)","A",4,3,21,0.1429,1,"{3194, 36, 31}"
445,1,38,0,0.0431496,"\int \frac{\cot (c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Cot[c + d*x]/(a + b*Sin[c + d*x]^2),x]","\frac{\log (\sin (c+d x))}{a d}-\frac{\log \left(a+b \sin ^2(c+d x)\right)}{2 a d}","\frac{\log (\sin (c+d x))}{a d}-\frac{\log \left(a+b \sin ^2(c+d x)\right)}{2 a d}",1,"Log[Sin[c + d*x]]/(a*d) - Log[a + b*Sin[c + d*x]^2]/(2*a*d)","A",4,4,21,0.1905,1,"{3194, 36, 29, 31}"
446,1,63,0,0.0749722,"\int \frac{\cot ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Cot[c + d*x]^3/(a + b*Sin[c + d*x]^2),x]","\frac{(a+b) \log \left(a+b \sin ^2(c+d x)\right)}{2 a^2 d}-\frac{(a+b) \log (\sin (c+d x))}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}","\frac{(a+b) \log \left(a+b \sin ^2(c+d x)\right)}{2 a^2 d}-\frac{(a+b) \log (\sin (c+d x))}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}",1,"-Csc[c + d*x]^2/(2*a*d) - ((a + b)*Log[Sin[c + d*x]])/(a^2*d) + ((a + b)*Log[a + b*Sin[c + d*x]^2])/(2*a^2*d)","A",3,2,23,0.08696,1,"{3194, 77}"
447,1,89,0,0.0889625,"\int \frac{\cot ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Cot[c + d*x]^5/(a + b*Sin[c + d*x]^2),x]","\frac{(2 a+b) \csc ^2(c+d x)}{2 a^2 d}-\frac{(a+b)^2 \log \left(a+b \sin ^2(c+d x)\right)}{2 a^3 d}+\frac{(a+b)^2 \log (\sin (c+d x))}{a^3 d}-\frac{\csc ^4(c+d x)}{4 a d}","\frac{(2 a+b) \csc ^2(c+d x)}{2 a^2 d}-\frac{(a+b)^2 \log \left(a+b \sin ^2(c+d x)\right)}{2 a^3 d}+\frac{(a+b)^2 \log (\sin (c+d x))}{a^3 d}-\frac{\csc ^4(c+d x)}{4 a d}",1,"((2*a + b)*Csc[c + d*x]^2)/(2*a^2*d) - Csc[c + d*x]^4/(4*a*d) + ((a + b)^2*Log[Sin[c + d*x]])/(a^3*d) - ((a + b)^2*Log[a + b*Sin[c + d*x]^2])/(2*a^3*d)","A",3,2,23,0.08696,1,"{3194, 88}"
448,1,121,0,0.1119729,"\int \frac{\cot ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Cot[c + d*x]^7/(a + b*Sin[c + d*x]^2),x]","-\frac{\left(3 a^2+3 a b+b^2\right) \csc ^2(c+d x)}{2 a^3 d}+\frac{(3 a+b) \csc ^4(c+d x)}{4 a^2 d}+\frac{(a+b)^3 \log \left(a+b \sin ^2(c+d x)\right)}{2 a^4 d}-\frac{(a+b)^3 \log (\sin (c+d x))}{a^4 d}-\frac{\csc ^6(c+d x)}{6 a d}","-\frac{\left(3 a^2+3 a b+b^2\right) \csc ^2(c+d x)}{2 a^3 d}+\frac{(3 a+b) \csc ^4(c+d x)}{4 a^2 d}+\frac{(a+b)^3 \log \left(a+b \sin ^2(c+d x)\right)}{2 a^4 d}-\frac{(a+b)^3 \log (\sin (c+d x))}{a^4 d}-\frac{\csc ^6(c+d x)}{6 a d}",1,"-((3*a^2 + 3*a*b + b^2)*Csc[c + d*x]^2)/(2*a^3*d) + ((3*a + b)*Csc[c + d*x]^4)/(4*a^2*d) - Csc[c + d*x]^6/(6*a*d) - ((a + b)^3*Log[Sin[c + d*x]])/(a^4*d) + ((a + b)^3*Log[a + b*Sin[c + d*x]^2])/(2*a^4*d)","A",3,2,23,0.08696,1,"{3194, 88}"
449,1,120,0,0.1254296,"\int \frac{\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Tan[c + d*x]^8/(a + b*Sin[c + d*x]^2),x]","\frac{a^2 \tan ^3(c+d x)}{3 d (a+b)^3}+\frac{a^{7/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{d (a+b)^{9/2}}-\frac{a^3 \tan (c+d x)}{d (a+b)^4}+\frac{\tan ^7(c+d x)}{7 d (a+b)}-\frac{a \tan ^5(c+d x)}{5 d (a+b)^2}","\frac{a^2 \tan ^3(c+d x)}{3 d (a+b)^3}+\frac{a^{7/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{d (a+b)^{9/2}}-\frac{a^3 \tan (c+d x)}{d (a+b)^4}+\frac{\tan ^7(c+d x)}{7 d (a+b)}-\frac{a \tan ^5(c+d x)}{5 d (a+b)^2}",1,"(a^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/((a + b)^(9/2)*d) - (a^3*Tan[c + d*x])/((a + b)^4*d) + (a^2*Tan[c + d*x]^3)/(3*(a + b)^3*d) - (a*Tan[c + d*x]^5)/(5*(a + b)^2*d) + Tan[c + d*x]^7/(7*(a + b)*d)","A",4,3,23,0.1304,1,"{3195, 302, 205}"
450,1,97,0,0.1068329,"\int \frac{\tan ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Tan[c + d*x]^6/(a + b*Sin[c + d*x]^2),x]","-\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{d (a+b)^{7/2}}+\frac{a^2 \tan (c+d x)}{d (a+b)^3}+\frac{\tan ^5(c+d x)}{5 d (a+b)}-\frac{a \tan ^3(c+d x)}{3 d (a+b)^2}","-\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{d (a+b)^{7/2}}+\frac{a^2 \tan (c+d x)}{d (a+b)^3}+\frac{\tan ^5(c+d x)}{5 d (a+b)}-\frac{a \tan ^3(c+d x)}{3 d (a+b)^2}",1,"-((a^(5/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/((a + b)^(7/2)*d)) + (a^2*Tan[c + d*x])/((a + b)^3*d) - (a*Tan[c + d*x]^3)/(3*(a + b)^2*d) + Tan[c + d*x]^5/(5*(a + b)*d)","A",4,3,23,0.1304,1,"{3195, 302, 205}"
451,1,74,0,0.0976003,"\int \frac{\tan ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Tan[c + d*x]^4/(a + b*Sin[c + d*x]^2),x]","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{d (a+b)^{5/2}}+\frac{\tan ^3(c+d x)}{3 d (a+b)}-\frac{a \tan (c+d x)}{d (a+b)^2}","\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{d (a+b)^{5/2}}+\frac{\tan ^3(c+d x)}{3 d (a+b)}-\frac{a \tan (c+d x)}{d (a+b)^2}",1,"(a^(3/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/((a + b)^(5/2)*d) - (a*Tan[c + d*x])/((a + b)^2*d) + Tan[c + d*x]^3/(3*(a + b)*d)","A",4,3,23,0.1304,1,"{3195, 302, 205}"
452,1,53,0,0.0758715,"\int \frac{\tan ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Tan[c + d*x]^2/(a + b*Sin[c + d*x]^2),x]","\frac{\tan (c+d x)}{d (a+b)}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{d (a+b)^{3/2}}","\frac{\tan (c+d x)}{d (a+b)}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{d (a+b)^{3/2}}",1,"-((Sqrt[a]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/((a + b)^(3/2)*d)) + Tan[c + d*x]/((a + b)*d)","A",3,3,23,0.1304,1,"{3195, 321, 205}"
453,1,52,0,0.0675979,"\int \frac{\cot ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Cot[c + d*x]^2/(a + b*Sin[c + d*x]^2),x]","-\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{\cot (c+d x)}{a d}","-\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{3/2} d}-\frac{\cot (c+d x)}{a d}",1,"-((Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(3/2)*d)) - Cot[c + d*x]/(a*d)","A",3,3,23,0.1304,1,"{3195, 325, 205}"
454,1,71,0,0.0793176,"\int \frac{\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x]^2),x]","\frac{(a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{(a+b) \cot (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}","\frac{(a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{(a+b) \cot (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}",1,"((a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(5/2)*d) + ((a + b)*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a*d)","A",4,3,23,0.1304,1,"{3195, 325, 205}"
455,1,96,0,0.0961273,"\int \frac{\cot ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Cot[c + d*x]^6/(a + b*Sin[c + d*x]^2),x]","-\frac{(a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{7/2} d}+\frac{(a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac{(a+b)^2 \cot (c+d x)}{a^3 d}-\frac{\cot ^5(c+d x)}{5 a d}","-\frac{(a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{7/2} d}+\frac{(a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac{(a+b)^2 \cot (c+d x)}{a^3 d}-\frac{\cot ^5(c+d x)}{5 a d}",1,"-(((a + b)^(5/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(7/2)*d)) - ((a + b)^2*Cot[c + d*x])/(a^3*d) + ((a + b)*Cot[c + d*x]^3)/(3*a^2*d) - Cot[c + d*x]^5/(5*a*d)","A",5,3,23,0.1304,1,"{3195, 325, 205}"
456,1,117,0,0.1120051,"\int \frac{\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Int[Cot[c + d*x]^8/(a + b*Sin[c + d*x]^2),x]","\frac{(a+b)^{7/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{9/2} d}+\frac{(a+b) \cot ^5(c+d x)}{5 a^2 d}-\frac{(a+b)^2 \cot ^3(c+d x)}{3 a^3 d}+\frac{(a+b)^3 \cot (c+d x)}{a^4 d}-\frac{\cot ^7(c+d x)}{7 a d}","\frac{(a+b)^{7/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{9/2} d}+\frac{(a+b) \cot ^5(c+d x)}{5 a^2 d}-\frac{(a+b)^2 \cot ^3(c+d x)}{3 a^3 d}+\frac{(a+b)^3 \cot (c+d x)}{a^4 d}-\frac{\cot ^7(c+d x)}{7 a d}",1,"((a + b)^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(9/2)*d) + ((a + b)^3*Cot[c + d*x])/(a^4*d) - ((a + b)^2*Cot[c + d*x]^3)/(3*a^3*d) + ((a + b)*Cot[c + d*x]^5)/(5*a^2*d) - Cot[c + d*x]^7/(7*a*d)","A",6,3,23,0.1304,1,"{3195, 325, 205}"
457,1,64,0,0.1096091,"\int \sqrt{a-a \sin ^2(e+f x)} \tan ^5(e+f x) \, dx","Int[Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^5,x]","\frac{a^2}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}-\frac{2 a}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\sqrt{a \cos ^2(e+f x)}}{f}","\frac{a^2}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}-\frac{2 a}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\sqrt{a \cos ^2(e+f x)}}{f}",1,"a^2/(3*f*(a*Cos[e + f*x]^2)^(3/2)) - (2*a)/(f*Sqrt[a*Cos[e + f*x]^2]) - Sqrt[a*Cos[e + f*x]^2]/f","A",5,4,26,0.1538,1,"{3176, 3205, 16, 43}"
458,1,38,0,0.1046876,"\int \sqrt{a-a \sin ^2(e+f x)} \tan ^3(e+f x) \, dx","Int[Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^3,x]","\frac{a}{f \sqrt{a \cos ^2(e+f x)}}+\frac{\sqrt{a \cos ^2(e+f x)}}{f}","\frac{a}{f \sqrt{a \cos ^2(e+f x)}}+\frac{\sqrt{a \cos ^2(e+f x)}}{f}",1,"a/(f*Sqrt[a*Cos[e + f*x]^2]) + Sqrt[a*Cos[e + f*x]^2]/f","A",5,4,26,0.1538,1,"{3176, 3205, 16, 43}"
459,1,19,0,0.0629348,"\int \sqrt{a-a \sin ^2(e+f x)} \tan (e+f x) \, dx","Int[Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x],x]","-\frac{\sqrt{a \cos ^2(e+f x)}}{f}","-\frac{\sqrt{a \cos ^2(e+f x)}}{f}",1,"-(Sqrt[a*Cos[e + f*x]^2]/f)","A",4,4,24,0.1667,1,"{3176, 3205, 16, 32}"
460,1,50,0,0.0790169,"\int \cot (e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx","Int[Cot[e + f*x]*Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\sqrt{a \cos ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{f}","\frac{\sqrt{a \cos ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{f}",1,"-((Sqrt[a]*ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]])/f) + Sqrt[a*Cos[e + f*x]^2]/f","A",5,5,24,0.2083,1,"{3176, 3205, 50, 63, 206}"
461,1,87,0,0.1182426,"\int \cot ^3(e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx","Int[Cot[e + f*x]^3*Sqrt[a - a*Sin[e + f*x]^2],x]","-\frac{3 \sqrt{a \cos ^2(e+f x)}}{2 f}-\frac{\csc ^2(e+f x) \left(a \cos ^2(e+f x)\right)^{3/2}}{2 a f}+\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{2 f}","-\frac{3 \sqrt{a \cos ^2(e+f x)}}{2 f}-\frac{\csc ^2(e+f x) \left(a \cos ^2(e+f x)\right)^{3/2}}{2 a f}+\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{2 f}",1,"(3*Sqrt[a]*ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]])/(2*f) - (3*Sqrt[a*Cos[e + f*x]^2])/(2*f) - ((a*Cos[e + f*x]^2)^(3/2)*Csc[e + f*x]^2)/(2*a*f)","A",7,7,26,0.2692,1,"{3176, 3205, 16, 47, 50, 63, 206}"
462,1,120,0,0.1286483,"\int \sqrt{a-a \sin ^2(e+f x)} \tan ^6(e+f x) \, dx","Int[Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^6,x]","\frac{\tan ^5(e+f x) \sqrt{a \cos ^2(e+f x)}}{4 f}-\frac{5 \tan ^3(e+f x) \sqrt{a \cos ^2(e+f x)}}{8 f}-\frac{15 \tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{8 f}+\frac{15 \sec (e+f x) \sqrt{a \cos ^2(e+f x)} \tanh ^{-1}(\sin (e+f x))}{8 f}","\frac{\tan ^5(e+f x) \sqrt{a \cos ^2(e+f x)}}{4 f}-\frac{5 \tan ^3(e+f x) \sqrt{a \cos ^2(e+f x)}}{8 f}-\frac{15 \tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{8 f}+\frac{15 \sec (e+f x) \sqrt{a \cos ^2(e+f x)} \tanh ^{-1}(\sin (e+f x))}{8 f}",1,"(15*ArcTanh[Sin[e + f*x]]*Sqrt[a*Cos[e + f*x]^2]*Sec[e + f*x])/(8*f) - (15*Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/(8*f) - (5*Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x]^3)/(8*f) + (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x]^5)/(4*f)","A",7,6,26,0.2308,1,"{3176, 3207, 2592, 288, 321, 206}"
463,1,91,0,0.1237612,"\int \sqrt{a-a \sin ^2(e+f x)} \tan ^4(e+f x) \, dx","Int[Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^4,x]","\frac{\tan ^3(e+f x) \sqrt{a \cos ^2(e+f x)}}{2 f}+\frac{3 \tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{2 f}-\frac{3 \sec (e+f x) \sqrt{a \cos ^2(e+f x)} \tanh ^{-1}(\sin (e+f x))}{2 f}","\frac{\tan ^3(e+f x) \sqrt{a \cos ^2(e+f x)}}{2 f}+\frac{3 \tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{2 f}-\frac{3 \sec (e+f x) \sqrt{a \cos ^2(e+f x)} \tanh ^{-1}(\sin (e+f x))}{2 f}",1,"(-3*ArcTanh[Sin[e + f*x]]*Sqrt[a*Cos[e + f*x]^2]*Sec[e + f*x])/(2*f) + (3*Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/(2*f) + (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x]^3)/(2*f)","A",6,6,26,0.2308,1,"{3176, 3207, 2592, 288, 321, 206}"
464,1,57,0,0.103233,"\int \sqrt{a-a \sin ^2(e+f x)} \tan ^2(e+f x) \, dx","Int[Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^2,x]","\frac{\sec (e+f x) \sqrt{a \cos ^2(e+f x)} \tanh ^{-1}(\sin (e+f x))}{f}-\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}","\frac{\sec (e+f x) \sqrt{a \cos ^2(e+f x)} \tanh ^{-1}(\sin (e+f x))}{f}-\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}",1,"(ArcTanh[Sin[e + f*x]]*Sqrt[a*Cos[e + f*x]^2]*Sec[e + f*x])/f - (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/f","A",5,5,26,0.1923,1,"{3176, 3207, 2592, 321, 206}"
465,1,57,0,0.1116691,"\int \cot ^2(e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx","Int[Cot[e + f*x]^2*Sqrt[a - a*Sin[e + f*x]^2],x]","-\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{\csc (e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}","-\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{\csc (e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}",1,"-((Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]*Sec[e + f*x])/f) - (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/f","A",5,4,26,0.1538,1,"{3176, 3207, 2590, 14}"
466,1,91,0,0.1151269,"\int \cot ^4(e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx","Int[Cot[e + f*x]^4*Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{\csc ^3(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{3 f}+\frac{2 \csc (e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}","\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{\csc ^3(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{3 f}+\frac{2 \csc (e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}",1,"(2*Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]*Sec[e + f*x])/f - (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^3*Sec[e + f*x])/(3*f) + (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/f","A",5,4,26,0.1538,1,"{3176, 3207, 2590, 270}"
467,1,124,0,0.1215114,"\int \cot ^6(e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx","Int[Cot[e + f*x]^6*Sqrt[a - a*Sin[e + f*x]^2],x]","-\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{\csc ^5(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{5 f}+\frac{\csc ^3(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{3 \csc (e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}","-\frac{\tan (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{\csc ^5(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{5 f}+\frac{\csc ^3(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}-\frac{3 \csc (e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{f}",1,"(-3*Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]*Sec[e + f*x])/f + (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^3*Sec[e + f*x])/f - (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^5*Sec[e + f*x])/(5*f) - (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/f","A",5,4,26,0.1538,1,"{3176, 3207, 2590, 270}"
468,1,65,0,0.1151203,"\int \frac{\tan ^5(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^5/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{a^2}{5 f \left(a \cos ^2(e+f x)\right)^{5/2}}-\frac{2 a}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}+\frac{1}{f \sqrt{a \cos ^2(e+f x)}}","\frac{a^2}{5 f \left(a \cos ^2(e+f x)\right)^{5/2}}-\frac{2 a}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}+\frac{1}{f \sqrt{a \cos ^2(e+f x)}}",1,"a^2/(5*f*(a*Cos[e + f*x]^2)^(5/2)) - (2*a)/(3*f*(a*Cos[e + f*x]^2)^(3/2)) + 1/(f*Sqrt[a*Cos[e + f*x]^2])","A",5,4,26,0.1538,1,"{3176, 3205, 16, 43}"
469,1,42,0,0.1070253,"\int \frac{\tan ^3(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^3/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{a}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}-\frac{1}{f \sqrt{a \cos ^2(e+f x)}}","\frac{a}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}-\frac{1}{f \sqrt{a \cos ^2(e+f x)}}",1,"a/(3*f*(a*Cos[e + f*x]^2)^(3/2)) - 1/(f*Sqrt[a*Cos[e + f*x]^2])","A",5,4,26,0.1538,1,"{3176, 3205, 16, 43}"
470,1,18,0,0.0645136,"\int \frac{\tan (e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Int[Tan[e + f*x]/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{1}{f \sqrt{a \cos ^2(e+f x)}}","\frac{1}{f \sqrt{a \cos ^2(e+f x)}}",1,"1/(f*Sqrt[a*Cos[e + f*x]^2])","A",4,4,24,0.1667,1,"{3176, 3205, 16, 32}"
471,1,31,0,0.0758756,"\int \frac{\cot (e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Int[Cot[e + f*x]/Sqrt[a - a*Sin[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"-(ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f))","A",4,4,24,0.1667,1,"{3176, 3205, 63, 206}"
472,1,66,0,0.1137198,"\int \frac{\cot ^3(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^3/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{2 \sqrt{a} f}-\frac{\csc ^2(e+f x) \sqrt{a \cos ^2(e+f x)}}{2 a f}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{2 \sqrt{a} f}-\frac{\csc ^2(e+f x) \sqrt{a \cos ^2(e+f x)}}{2 a f}",1,"ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]]/(2*Sqrt[a]*f) - (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^2)/(2*a*f)","A",6,6,26,0.2308,1,"{3176, 3205, 16, 47, 63, 206}"
473,1,91,0,0.1385092,"\int \frac{\tan ^4(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^4/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\tan ^3(e+f x)}{4 f \sqrt{a \cos ^2(e+f x)}}-\frac{3 \tan (e+f x)}{8 f \sqrt{a \cos ^2(e+f x)}}+\frac{3 \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 f \sqrt{a \cos ^2(e+f x)}}","\frac{\tan ^3(e+f x)}{4 f \sqrt{a \cos ^2(e+f x)}}-\frac{3 \tan (e+f x)}{8 f \sqrt{a \cos ^2(e+f x)}}+\frac{3 \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 f \sqrt{a \cos ^2(e+f x)}}",1,"(3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*f*Sqrt[a*Cos[e + f*x]^2]) - (3*Tan[e + f*x])/(8*f*Sqrt[a*Cos[e + f*x]^2]) + Tan[e + f*x]^3/(4*f*Sqrt[a*Cos[e + f*x]^2])","A",5,4,26,0.1538,1,"{3176, 3207, 2611, 3770}"
474,1,62,0,0.1193547,"\int \frac{\tan ^2(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^2/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\tan (e+f x)}{2 f \sqrt{a \cos ^2(e+f x)}}-\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 f \sqrt{a \cos ^2(e+f x)}}","\frac{\tan (e+f x)}{2 f \sqrt{a \cos ^2(e+f x)}}-\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 f \sqrt{a \cos ^2(e+f x)}}",1,"-(ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*f*Sqrt[a*Cos[e + f*x]^2]) + Tan[e + f*x]/(2*f*Sqrt[a*Cos[e + f*x]^2])","A",4,4,26,0.1538,1,"{3176, 3207, 2611, 3770}"
475,1,25,0,0.1018294,"\int \frac{\cot ^2(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^2/Sqrt[a - a*Sin[e + f*x]^2],x]","-\frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}}","-\frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}}",1,"-(Cot[e + f*x]/(f*Sqrt[a*Cos[e + f*x]^2]))","A",4,4,26,0.1538,1,"{3176, 3207, 2606, 8}"
476,1,60,0,0.1172423,"\int \frac{\cot ^4(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^4/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt{a \cos ^2(e+f x)}}","\frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt{a \cos ^2(e+f x)}}",1,"Cot[e + f*x]/(f*Sqrt[a*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^2)/(3*f*Sqrt[a*Cos[e + f*x]^2])","A",4,3,26,0.1154,1,"{3176, 3207, 2606}"
477,1,96,0,0.1213203,"\int \frac{\cot ^6(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^6/Sqrt[a - a*Sin[e + f*x]^2],x]","-\frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^4(e+f x)}{5 f \sqrt{a \cos ^2(e+f x)}}+\frac{2 \cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt{a \cos ^2(e+f x)}}","-\frac{\cot (e+f x)}{f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^4(e+f x)}{5 f \sqrt{a \cos ^2(e+f x)}}+\frac{2 \cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt{a \cos ^2(e+f x)}}",1,"-(Cot[e + f*x]/(f*Sqrt[a*Cos[e + f*x]^2])) + (2*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f*Sqrt[a*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^4)/(5*f*Sqrt[a*Cos[e + f*x]^2])","A",5,4,26,0.1538,1,"{3176, 3207, 2606, 194}"
478,1,68,0,0.1269104,"\int \frac{\tan ^5(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^5/(a - a*Sin[e + f*x]^2)^(3/2),x]","\frac{a^2}{7 f \left(a \cos ^2(e+f x)\right)^{7/2}}-\frac{2 a}{5 f \left(a \cos ^2(e+f x)\right)^{5/2}}+\frac{1}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}","\frac{a^2}{7 f \left(a \cos ^2(e+f x)\right)^{7/2}}-\frac{2 a}{5 f \left(a \cos ^2(e+f x)\right)^{5/2}}+\frac{1}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}",1,"a^2/(7*f*(a*Cos[e + f*x]^2)^(7/2)) - (2*a)/(5*f*(a*Cos[e + f*x]^2)^(5/2)) + 1/(3*f*(a*Cos[e + f*x]^2)^(3/2))","A",5,4,26,0.1538,1,"{3176, 3205, 16, 43}"
479,1,44,0,0.1194883,"\int \frac{\tan ^3(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^3/(a - a*Sin[e + f*x]^2)^(3/2),x]","\frac{a}{5 f \left(a \cos ^2(e+f x)\right)^{5/2}}-\frac{1}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}","\frac{a}{5 f \left(a \cos ^2(e+f x)\right)^{5/2}}-\frac{1}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}",1,"a/(5*f*(a*Cos[e + f*x]^2)^(5/2)) - 1/(3*f*(a*Cos[e + f*x]^2)^(3/2))","A",5,4,26,0.1538,1,"{3176, 3205, 16, 43}"
480,1,21,0,0.0744949,"\int \frac{\tan (e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]/(a - a*Sin[e + f*x]^2)^(3/2),x]","\frac{1}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}","\frac{1}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}",1,"1/(3*f*(a*Cos[e + f*x]^2)^(3/2))","A",4,4,24,0.1667,1,"{3176, 3205, 16, 32}"
481,1,53,0,0.0922653,"\int \frac{\cot (e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]/(a - a*Sin[e + f*x]^2)^(3/2),x]","\frac{1}{a f \sqrt{a \cos ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}","\frac{1}{a f \sqrt{a \cos ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}",1,"-(ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + 1/(a*f*Sqrt[a*Cos[e + f*x]^2])","A",5,5,24,0.2083,1,"{3176, 3205, 51, 63, 206}"
482,1,66,0,0.1266031,"\int \frac{\cot ^3(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^3/(a - a*Sin[e + f*x]^2)^(3/2),x]","-\frac{\csc ^2(e+f x) \sqrt{a \cos ^2(e+f x)}}{2 a^2 f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{3/2} f}","-\frac{\csc ^2(e+f x) \sqrt{a \cos ^2(e+f x)}}{2 a^2 f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{3/2} f}",1,"-ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]]/(2*a^(3/2)*f) - (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^2)/(2*a^2*f)","A",6,6,26,0.2308,1,"{3176, 3205, 16, 51, 63, 206}"
483,1,106,0,0.1572679,"\int \frac{\tan ^2(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^2/(a - a*Sin[e + f*x]^2)^(3/2),x]","-\frac{\tan (e+f x)}{8 a f \sqrt{a \cos ^2(e+f x)}}-\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 a f \sqrt{a \cos ^2(e+f x)}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{4 a f \sqrt{a \cos ^2(e+f x)}}","-\frac{\tan (e+f x)}{8 a f \sqrt{a \cos ^2(e+f x)}}-\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{8 a f \sqrt{a \cos ^2(e+f x)}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{4 a f \sqrt{a \cos ^2(e+f x)}}",1,"-(ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*a*f*Sqrt[a*Cos[e + f*x]^2]) - Tan[e + f*x]/(8*a*f*Sqrt[a*Cos[e + f*x]^2]) + (Sec[e + f*x]^2*Tan[e + f*x])/(4*a*f*Sqrt[a*Cos[e + f*x]^2])","A",5,5,26,0.1923,1,"{3176, 3207, 2611, 3768, 3770}"
484,1,63,0,0.1324674,"\int \frac{\cot ^2(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^2/(a - a*Sin[e + f*x]^2)^(3/2),x]","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{a f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x)}{a f \sqrt{a \cos ^2(e+f x)}}","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{a f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x)}{a f \sqrt{a \cos ^2(e+f x)}}",1,"(ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(a*f*Sqrt[a*Cos[e + f*x]^2]) - Cot[e + f*x]/(a*f*Sqrt[a*Cos[e + f*x]^2])","A",5,5,26,0.1923,1,"{3176, 3207, 2621, 321, 207}"
485,1,38,0,0.1262793,"\int \frac{\cot ^4(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^4/(a - a*Sin[e + f*x]^2)^(3/2),x]","-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt{a \cos ^2(e+f x)}}","-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt{a \cos ^2(e+f x)}}",1,"-(Cot[e + f*x]*Csc[e + f*x]^2)/(3*a*f*Sqrt[a*Cos[e + f*x]^2])","A",4,4,26,0.1538,1,"{3176, 3207, 2606, 30}"
486,1,77,0,0.1440773,"\int \frac{\cot ^6(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^6/(a - a*Sin[e + f*x]^2)^(3/2),x]","\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt{a \cos ^2(e+f x)}}","\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt{a \cos ^2(e+f x)}}",1,"(Cot[e + f*x]*Csc[e + f*x]^2)/(3*a*f*Sqrt[a*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^4)/(5*a*f*Sqrt[a*Cos[e + f*x]^2])","A",5,4,26,0.1538,1,"{3176, 3207, 2606, 14}"
487,1,115,0,0.1518664,"\int \frac{\cot ^8(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^8/(a - a*Sin[e + f*x]^2)^(3/2),x]","-\frac{\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt{a \cos ^2(e+f x)}}+\frac{2 \cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt{a \cos ^2(e+f x)}}","-\frac{\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt{a \cos ^2(e+f x)}}+\frac{2 \cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt{a \cos ^2(e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt{a \cos ^2(e+f x)}}",1,"-(Cot[e + f*x]*Csc[e + f*x]^2)/(3*a*f*Sqrt[a*Cos[e + f*x]^2]) + (2*Cot[e + f*x]*Csc[e + f*x]^4)/(5*a*f*Sqrt[a*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^6)/(7*a*f*Sqrt[a*Cos[e + f*x]^2])","A",5,4,26,0.1538,1,"{3176, 3207, 2606, 270}"
488,1,177,0,0.2107428,"\int \sqrt{a+b \sin ^2(e+f x)} \tan ^5(e+f x) \, dx","Int[Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^5,x]","-\frac{\left(8 a^2+24 a b+15 b^2\right) \sqrt{a+b \sin ^2(e+f x)}}{8 f (a+b)^2}+\frac{\left(8 a^2+24 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{3/2}}+\frac{\sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 f (a+b)}-\frac{(8 a+7 b) \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{8 f (a+b)^2}","-\frac{\left(8 a^2+24 a b+15 b^2\right) \sqrt{a+b \sin ^2(e+f x)}}{8 f (a+b)^2}+\frac{\left(8 a^2+24 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{3/2}}+\frac{\sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 f (a+b)}-\frac{(8 a+7 b) \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{8 f (a+b)^2}",1,"((8*a^2 + 24*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(3/2)*f) - ((8*a^2 + 24*a*b + 15*b^2)*Sqrt[a + b*Sin[e + f*x]^2])/(8*(a + b)^2*f) - ((8*a + 7*b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2))/(8*(a + b)^2*f) + (Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2))/(4*(a + b)*f)","A",6,6,25,0.2400,1,"{3194, 89, 78, 50, 63, 208}"
489,1,118,0,0.1110781,"\int \sqrt{a+b \sin ^2(e+f x)} \tan ^3(e+f x) \, dx","Int[Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^3,x]","\frac{(2 a+3 b) \sqrt{a+b \sin ^2(e+f x)}}{2 f (a+b)}-\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f \sqrt{a+b}}+\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{2 f (a+b)}","\frac{(2 a+3 b) \sqrt{a+b \sin ^2(e+f x)}}{2 f (a+b)}-\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f \sqrt{a+b}}+\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{2 f (a+b)}",1,"-((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(2*Sqrt[a + b]*f) + ((2*a + 3*b)*Sqrt[a + b*Sin[e + f*x]^2])/(2*(a + b)*f) + (Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2))/(2*(a + b)*f)","A",5,5,25,0.2000,1,"{3194, 78, 50, 63, 208}"
490,1,58,0,0.0559427,"\int \sqrt{a+b \sin ^2(e+f x)} \tan (e+f x) \, dx","Int[Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x],x]","\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{f}-\frac{\sqrt{a+b \sin ^2(e+f x)}}{f}","\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{f}-\frac{\sqrt{a+b \sin ^2(e+f x)}}{f}",1,"(Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/f - Sqrt[a + b*Sin[e + f*x]^2]/f","A",4,4,23,0.1739,1,"{3194, 50, 63, 208}"
491,1,54,0,0.0649755,"\int \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{a+b \sin ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{f}","\frac{\sqrt{a+b \sin ^2(e+f x)}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{f}",1,"-((Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/f) + Sqrt[a + b*Sin[e + f*x]^2]/f","A",4,4,23,0.1739,1,"{3194, 50, 63, 208}"
492,1,110,0,0.102533,"\int \cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Cot[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{(2 a-b) \sqrt{a+b \sin ^2(e+f x)}}{2 a f}+\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{2 \sqrt{a} f}-\frac{\csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{2 a f}","-\frac{(2 a-b) \sqrt{a+b \sin ^2(e+f x)}}{2 a f}+\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{2 \sqrt{a} f}-\frac{\csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{2 a f}",1,"((2*a - b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(2*Sqrt[a]*f) - ((2*a - b)*Sqrt[a + b*Sin[e + f*x]^2])/(2*a*f) - (Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2))/(2*a*f)","A",5,5,25,0.2000,1,"{3194, 78, 50, 63, 208}"
493,1,165,0,0.1548059,"\int \cot ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Cot[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\left(8 a^2-8 a b-b^2\right) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac{\left(8 a^2-8 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{3/2} f}+\frac{(8 a+b) \csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{8 a^2 f}-\frac{\csc ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 a f}","\frac{\left(8 a^2-8 a b-b^2\right) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac{\left(8 a^2-8 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{3/2} f}+\frac{(8 a+b) \csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{8 a^2 f}-\frac{\csc ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{4 a f}",1,"-((8*a^2 - 8*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(8*a^(3/2)*f) + ((8*a^2 - 8*a*b - b^2)*Sqrt[a + b*Sin[e + f*x]^2])/(8*a^2*f) + ((8*a + b)*Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2))/(8*a^2*f) - (Csc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2))/(4*a*f)","A",6,6,25,0.2400,1,"{3194, 89, 78, 50, 63, 208}"
494,1,234,0,0.2768236,"\int \sqrt{a+b \sin ^2(e+f x)} \tan ^4(e+f x) \, dx","Int[Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^4,x]","\frac{\tan ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{(3 a+4 b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)}-\frac{4 a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\tan ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{(3 a+4 b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)}-\frac{4 a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"((7*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (4*a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((3*a + 4*b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)*f) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^3)/(3*f)","A",8,8,25,0.3200,1,"{3196, 467, 578, 524, 426, 424, 421, 419}"
495,1,171,0,0.1586626,"\int \sqrt{a+b \sin ^2(e+f x)} \tan ^2(e+f x) \, dx","Int[Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^2,x]","\frac{\tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(-2*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/f","A",7,7,25,0.2800,1,"{3196, 467, 524, 426, 424, 421, 419}"
496,1,51,0,0.0330833,"\int \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])","A",2,2,16,0.1250,1,"{3178, 3177}"
497,1,174,0,0.1716765,"\int \cot ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Cot[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{(a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{\cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{(a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/f) - (2*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,7,25,0.2800,1,"{3196, 473, 524, 426, 424, 421, 419}"
498,1,232,0,0.2679559,"\int \cot ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Int[Cot[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{4 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{\cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{4 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"((3*a - b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f) - (Cot[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) + ((7*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (4*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3196, 473, 580, 524, 426, 424, 421, 419}"
499,1,220,0,0.2578007,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \tan ^5(e+f x) \, dx","Int[(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^5,x]","-\frac{\left(8 a^2+40 a b+35 b^2\right) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{24 f (a+b)^2}-\frac{\left(8 a^2+40 a b+35 b^2\right) \sqrt{a+b \sin ^2(e+f x)}}{8 f (a+b)}+\frac{\left(8 a^2+40 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f \sqrt{a+b}}+\frac{\sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{4 f (a+b)}-\frac{(8 a+9 b) \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{8 f (a+b)^2}","-\frac{\left(8 a^2+40 a b+35 b^2\right) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{24 f (a+b)^2}-\frac{\left(8 a^2+40 a b+35 b^2\right) \sqrt{a+b \sin ^2(e+f x)}}{8 f (a+b)}+\frac{\left(8 a^2+40 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f \sqrt{a+b}}+\frac{\sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{4 f (a+b)}-\frac{(8 a+9 b) \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{8 f (a+b)^2}",1,"((8*a^2 + 40*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(8*Sqrt[a + b]*f) - ((8*a^2 + 40*a*b + 35*b^2)*Sqrt[a + b*Sin[e + f*x]^2])/(8*(a + b)*f) - ((8*a^2 + 40*a*b + 35*b^2)*(a + b*Sin[e + f*x]^2)^(3/2))/(24*(a + b)^2*f) - ((8*a + 9*b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(5/2))/(8*(a + b)^2*f) + (Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(5/2))/(4*(a + b)*f)","A",7,6,25,0.2400,1,"{3194, 89, 78, 50, 63, 208}"
500,1,148,0,0.1361331,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \tan ^3(e+f x) \, dx","Int[(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^3,x]","\frac{(2 a+5 b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{6 f (a+b)}+\frac{(2 a+5 b) \sqrt{a+b \sin ^2(e+f x)}}{2 f}-\frac{\sqrt{a+b} (2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f}+\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{2 f (a+b)}","\frac{(2 a+5 b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{6 f (a+b)}+\frac{(2 a+5 b) \sqrt{a+b \sin ^2(e+f x)}}{2 f}-\frac{\sqrt{a+b} (2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f}+\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{2 f (a+b)}",1,"-(Sqrt[a + b]*(2*a + 5*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(2*f) + ((2*a + 5*b)*Sqrt[a + b*Sin[e + f*x]^2])/(2*f) + ((2*a + 5*b)*(a + b*Sin[e + f*x]^2)^(3/2))/(6*(a + b)*f) + (Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(5/2))/(2*(a + b)*f)","A",6,5,25,0.2000,1,"{3194, 78, 50, 63, 208}"
501,1,84,0,0.0775713,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \tan (e+f x) \, dx","Int[(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x],x]","-\frac{(a+b) \sqrt{a+b \sin ^2(e+f x)}}{f}-\frac{\left(a+b \sin ^2(e+f x)\right)^{3/2}}{3 f}+\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{f}","-\frac{(a+b) \sqrt{a+b \sin ^2(e+f x)}}{f}-\frac{\left(a+b \sin ^2(e+f x)\right)^{3/2}}{3 f}+\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{f}",1,"((a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/f - ((a + b)*Sqrt[a + b*Sin[e + f*x]^2])/f - (a + b*Sin[e + f*x]^2)^(3/2)/(3*f)","A",5,4,23,0.1739,1,"{3194, 50, 63, 208}"
502,1,78,0,0.0791743,"\int \cot (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{a \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{\left(a+b \sin ^2(e+f x)\right)^{3/2}}{3 f}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{a \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{\left(a+b \sin ^2(e+f x)\right)^{3/2}}{3 f}",1,"-((a^(3/2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/f) + (a*Sqrt[a + b*Sin[e + f*x]^2])/f + (a + b*Sin[e + f*x]^2)^(3/2)/(3*f)","A",5,4,23,0.1739,1,"{3194, 50, 63, 208}"
503,1,140,0,0.1270714,"\int \cot ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{(2 a-3 b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{6 a f}-\frac{(2 a-3 b) \sqrt{a+b \sin ^2(e+f x)}}{2 f}+\frac{\sqrt{a} (2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{2 f}-\frac{\csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{2 a f}","-\frac{(2 a-3 b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{6 a f}-\frac{(2 a-3 b) \sqrt{a+b \sin ^2(e+f x)}}{2 f}+\frac{\sqrt{a} (2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{2 f}-\frac{\csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{2 a f}",1,"(Sqrt[a]*(2*a - 3*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(2*f) - ((2*a - 3*b)*Sqrt[a + b*Sin[e + f*x]^2])/(2*f) - ((2*a - 3*b)*(a + b*Sin[e + f*x]^2)^(3/2))/(6*a*f) - (Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(5/2))/(2*a*f)","A",6,5,25,0.2000,1,"{3194, 78, 50, 63, 208}"
504,1,208,0,0.1919956,"\int \cot ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^5*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\left(8 a^2-24 a b+3 b^2\right) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{24 a^2 f}+\frac{\left(8 a^2-24 a b+3 b^2\right) \sqrt{a+b \sin ^2(e+f x)}}{8 a f}-\frac{\left(8 a^2-24 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{8 \sqrt{a} f}+\frac{(8 a-b) \csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{8 a^2 f}-\frac{\csc ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{4 a f}","\frac{\left(8 a^2-24 a b+3 b^2\right) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{24 a^2 f}+\frac{\left(8 a^2-24 a b+3 b^2\right) \sqrt{a+b \sin ^2(e+f x)}}{8 a f}-\frac{\left(8 a^2-24 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{8 \sqrt{a} f}+\frac{(8 a-b) \csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{8 a^2 f}-\frac{\csc ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{4 a f}",1,"-((8*a^2 - 24*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(8*Sqrt[a]*f) + ((8*a^2 - 24*a*b + 3*b^2)*Sqrt[a + b*Sin[e + f*x]^2])/(8*a*f) + ((8*a^2 - 24*a*b + 3*b^2)*(a + b*Sin[e + f*x]^2)^(3/2))/(24*a^2*f) + ((8*a - b)*Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(5/2))/(8*a^2*f) - (Csc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(5/2))/(4*a*f)","A",7,6,25,0.2400,1,"{3194, 89, 78, 50, 63, 208}"
505,1,275,0,0.3688097,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \tan ^4(e+f x) \, dx","Int[(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^4,x]","\frac{\tan ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{3 f}-\frac{(a+2 b) \sin ^2(e+f x) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}-\frac{(3 a+8 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (5 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\tan ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{3 f}-\frac{(a+2 b) \sin ^2(e+f x) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}-\frac{(3 a+8 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (5 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-((3*a + 8*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) + (8*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(5*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((a + 2*b)*Sin[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/f + ((a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^3)/(3*f)","A",9,9,25,0.3600,1,"{3196, 467, 577, 582, 524, 426, 424, 421, 419}"
506,1,222,0,0.2308663,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \tan ^2(e+f x) \, dx","Int[(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^2,x]","\frac{\tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{f}+\frac{4 b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{4 a (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(7 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\tan (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{f}+\frac{4 b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{4 a (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(7 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(4*b*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - ((7*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (4*a*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x])/f","A",8,8,25,0.3200,1,"{3196, 467, 528, 524, 426, 424, 421, 419}"
507,1,154,0,0.1671019,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-(b*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(a + b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,6,16,0.3750,1,"{3180, 3172, 3178, 3177, 3183, 3182}"
508,1,223,0,0.256591,"\int \cot ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{4 b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\cot (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{f}+\frac{4 a (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{4 b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\cot (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{f}+\frac{4 a (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(4*b*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - (Cot[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/f - ((7*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (4*a*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3196, 473, 528, 524, 426, 424, 421, 419}"
509,1,276,0,0.3508877,"\int \cot ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{(3 a-5 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\cot ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{3 f}+\frac{(a-b) \cos ^2(e+f x) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}-\frac{(5 a-3 b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{(3 a-5 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\cot ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{3 f}+\frac{(a-b) \cos ^2(e+f x) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}-\frac{(5 a-3 b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"((a - b)*Cos[e + f*x]^2*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/f + ((3*a - 5*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - (Cot[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2))/(3*f) + (8*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - ((5*a - 3*b)*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",9,9,25,0.3600,1,"{3196, 473, 580, 528, 524, 426, 424, 421, 419}"
510,1,134,0,0.1738415,"\int \frac{\tan ^5(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^5/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\left(8 a^2+8 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{5/2}}+\frac{\sec ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{4 f (a+b)}-\frac{(8 a+5 b) \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 f (a+b)^2}","\frac{\left(8 a^2+8 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{5/2}}+\frac{\sec ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{4 f (a+b)}-\frac{(8 a+5 b) \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 f (a+b)^2}",1,"((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(5/2)*f) - ((8*a + 5*b)*Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(8*(a + b)^2*f) + (Sec[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2])/(4*(a + b)*f)","A",5,5,25,0.2000,1,"{3194, 89, 78, 63, 208}"
511,1,81,0,0.0977817,"\int \frac{\tan ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f (a+b)}-\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{3/2}}","\frac{\sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 f (a+b)}-\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{3/2}}",1,"-((2*a + b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(3/2)*f) + (Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(2*(a + b)*f)","A",4,4,25,0.1600,1,"{3194, 78, 63, 208}"
512,1,36,0,0.0506633,"\int \frac{\tan (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Tan[e + f*x]/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{f \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{f \sqrt{a+b}}",1,"ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]]/(Sqrt[a + b]*f)","A",3,3,23,0.1304,1,"{3194, 63, 208}"
513,1,33,0,0.0638331,"\int \frac{\cot (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Cot[e + f*x]/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"-(ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f))","A",3,3,23,0.1304,1,"{3194, 63, 208}"
514,1,75,0,0.0898619,"\int \frac{\cot ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{3/2} f}-\frac{\csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 a f}","\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{3/2} f}-\frac{\csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{2 a f}",1,"((2*a + b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(2*a^(3/2)*f) - (Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(2*a*f)","A",4,4,25,0.1600,1,"{3194, 78, 63, 208}"
515,1,126,0,0.1285165,"\int \frac{\cot ^5(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^5/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\left(8 a^2+8 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{5/2} f}+\frac{(8 a+3 b) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac{\csc ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{4 a f}","-\frac{\left(8 a^2+8 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{5/2} f}+\frac{(8 a+3 b) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac{\csc ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{4 a f}",1,"-((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(8*a^(5/2)*f) + ((8*a + 3*b)*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(8*a^2*f) - (Csc[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2])/(4*a*f)","A",5,5,25,0.2000,1,"{3194, 89, 78, 63, 208}"
516,1,246,0,0.2375283,"\int \frac{\tan ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{2 (2 a+b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)^2}+\frac{\tan (e+f x) \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)}-\frac{a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{2 (2 a+b) \tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)^2}+\frac{\tan (e+f x) \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f (a+b)}-\frac{a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(2*(2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*(2*a + b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)^2*f) + (Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)*f)","A",8,8,25,0.3200,1,"{3196, 470, 527, 524, 426, 424, 421, 419}"
517,1,109,0,0.1177748,"\int \frac{\tan ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f (a+b)}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\tan (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f (a+b)}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-((Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/((a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/((a + b)*f)","A",4,4,25,0.1600,1,"{3196, 471, 426, 424}"
518,1,51,0,0.033892,"\int \frac{1}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[1/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{a+b \sin ^2(e+f x)}}","\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{a+b \sin ^2(e+f x)}}",1,"(EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])","A",2,2,16,0.1250,1,"{3183, 3182}"
519,1,106,0,0.115462,"\int \frac{\cot ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{a f}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{\cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{a f}-\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"-((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*f)) - (Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])","A",5,5,25,0.2000,1,"{3196, 475, 21, 426, 424}"
520,1,240,0,0.267876,"\int \frac{\cot ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{2 (2 a+b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^2 f}+\frac{2 (2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{(a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a f \sqrt{a+b \sin ^2(e+f x)}}","\frac{2 (2 a+b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^2 f}+\frac{2 (2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{(a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a f \sqrt{a+b \sin ^2(e+f x)}}",1,"(2*(2*a + b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*f) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f) + (2*(2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - ((a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3196, 474, 583, 524, 426, 424, 421, 419}"
521,1,177,0,0.2293944,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{8 a^2-8 a b-b^2}{8 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}+\frac{\left(8 a^2-8 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{7/2}}+\frac{\sec ^4(e+f x)}{4 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{(8 a+3 b) \sec ^2(e+f x)}{8 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}","-\frac{8 a^2-8 a b-b^2}{8 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}+\frac{\left(8 a^2-8 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{7/2}}+\frac{\sec ^4(e+f x)}{4 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{(8 a+3 b) \sec ^2(e+f x)}{8 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}",1,"((8*a^2 - 8*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(7/2)*f) - (8*a^2 - 8*a*b - b^2)/(8*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((8*a + 3*b)*Sec[e + f*x]^2)/(8*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + Sec[e + f*x]^4/(4*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,6,25,0.2400,1,"{3194, 89, 78, 51, 63, 208}"
522,1,118,0,0.122448,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{2 a-b}{2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{5/2}}+\frac{\sec ^2(e+f x)}{2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}","\frac{2 a-b}{2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{5/2}}+\frac{\sec ^2(e+f x)}{2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}",1,"-((2*a - b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(5/2)*f) + (2*a - b)/(2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + Sec[e + f*x]^2/(2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",5,5,25,0.2000,1,"{3194, 78, 51, 63, 208}"
523,1,63,0,0.0677903,"\int \frac{\tan (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{f (a+b)^{3/2}}-\frac{1}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{f (a+b)^{3/2}}-\frac{1}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}",1,"ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]]/((a + b)^(3/2)*f) - 1/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",4,4,23,0.1739,1,"{3194, 51, 63, 208}"
524,1,57,0,0.0756698,"\int \frac{\cot (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{1}{a f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}","\frac{1}{a f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}",1,"-(ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + 1/(a*f*Sqrt[a + b*Sin[e + f*x]^2])","A",4,4,23,0.1739,1,"{3194, 51, 63, 208}"
525,1,110,0,0.1226422,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{2 a+3 b}{2 a^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{5/2} f}-\frac{\csc ^2(e+f x)}{2 a f \sqrt{a+b \sin ^2(e+f x)}}","-\frac{2 a+3 b}{2 a^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{5/2} f}-\frac{\csc ^2(e+f x)}{2 a f \sqrt{a+b \sin ^2(e+f x)}}",1,"((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(2*a^(5/2)*f) - (2*a + 3*b)/(2*a^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - Csc[e + f*x]^2/(2*a*f*Sqrt[a + b*Sin[e + f*x]^2])","A",5,5,25,0.2000,1,"{3194, 78, 51, 63, 208}"
526,1,167,0,0.162834,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{8 a^2+24 a b+15 b^2}{8 a^3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\left(8 a^2+24 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{7/2} f}+\frac{(8 a+5 b) \csc ^2(e+f x)}{8 a^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\csc ^4(e+f x)}{4 a f \sqrt{a+b \sin ^2(e+f x)}}","\frac{8 a^2+24 a b+15 b^2}{8 a^3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\left(8 a^2+24 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{7/2} f}+\frac{(8 a+5 b) \csc ^2(e+f x)}{8 a^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\csc ^4(e+f x)}{4 a f \sqrt{a+b \sin ^2(e+f x)}}",1,"-((8*a^2 + 24*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(8*a^(7/2)*f) + (8*a^2 + 24*a*b + 15*b^2)/(8*a^3*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((8*a + 5*b)*Csc[e + f*x]^2)/(8*a^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - Csc[e + f*x]^4/(4*a*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,6,25,0.2400,1,"{3194, 89, 78, 51, 63, 208}"
527,1,292,0,0.3148683,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{4 a \tan (e+f x)}{3 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{b (7 a-b) \sin (e+f x) \cos (e+f x)}{3 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{3 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{4 a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^3 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{4 a \tan (e+f x)}{3 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{b (7 a-b) \sin (e+f x) \cos (e+f x)}{3 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{3 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{4 a \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^3 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"((7*a - b)*b*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((7*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (4*a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - (4*a*Tan[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",9,8,25,0.3200,1,"{3196, 470, 527, 524, 426, 424, 421, 419}"
528,1,224,0,0.2259155,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x)}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 b \sin (e+f x) \cos (e+f x)}{f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\tan (e+f x)}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 b \sin (e+f x) \cos (e+f x)}{f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f (a+b) \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(-2*b*Cos[e + f*x]*Sin[e + f*x])/((a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/((a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + Tan[e + f*x]/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3196, 471, 527, 524, 426, 424, 421, 419}"
529,1,101,0,0.0621282,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[(a + b*Sin[e + f*x]^2)^(-3/2),x]","\frac{b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{a f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{b \sin (e+f x) \cos (e+f x)}{a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{a f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(b*Cos[e + f*x]*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(a*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])","A",4,4,16,0.2500,1,"{3184, 21, 3178, 3177}"
530,1,209,0,0.2416046,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{2 \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{a^2 f}-\frac{2 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{\cot (e+f x)}{a f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a f \sqrt{a+b \sin ^2(e+f x)}}","-\frac{2 \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{a^2 f}-\frac{2 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{\cot (e+f x)}{a f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{a f \sqrt{a+b \sin ^2(e+f x)}}",1,"Cot[e + f*x]/(a*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a^2*f) - (2*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(a*f*Sqrt[a + b*Sin[e + f*x]^2])","A",8,8,25,0.3200,1,"{3196, 469, 583, 524, 426, 424, 421, 419}"
531,1,297,0,0.3741231,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{(7 a+8 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 f}-\frac{(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^2 b f}-\frac{4 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}","\frac{(7 a+8 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 f}-\frac{(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^2 b f}-\frac{4 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}",1,"((a + b)*Cot[e + f*x]*Csc[e + f*x]^2)/(a*b*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((7*a + 8*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*f) - ((3*a + 4*b)*Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*b*f) + ((7*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (4*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",9,8,25,0.3200,1,"{3196, 468, 583, 524, 426, 424, 421, 419}"
532,1,218,0,0.2772095,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{8 a^2-24 a b+3 b^2}{8 f (a+b)^4 \sqrt{a+b \sin ^2(e+f x)}}-\frac{8 a^2-24 a b+3 b^2}{24 f (a+b)^3 \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{\left(8 a^2-24 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{9/2}}+\frac{\sec ^4(e+f x)}{4 f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{(8 a+b) \sec ^2(e+f x)}{8 f (a+b)^2 \left(a+b \sin ^2(e+f x)\right)^{3/2}}","-\frac{8 a^2-24 a b+3 b^2}{8 f (a+b)^4 \sqrt{a+b \sin ^2(e+f x)}}-\frac{8 a^2-24 a b+3 b^2}{24 f (a+b)^3 \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{\left(8 a^2-24 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{8 f (a+b)^{9/2}}+\frac{\sec ^4(e+f x)}{4 f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{(8 a+b) \sec ^2(e+f x)}{8 f (a+b)^2 \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"((8*a^2 - 24*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(9/2)*f) - (8*a^2 - 24*a*b + 3*b^2)/(24*(a + b)^3*f*(a + b*Sin[e + f*x]^2)^(3/2)) - ((8*a + b)*Sec[e + f*x]^2)/(8*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) + Sec[e + f*x]^4/(4*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (8*a^2 - 24*a*b + 3*b^2)/(8*(a + b)^4*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,6,25,0.2400,1,"{3194, 89, 78, 51, 63, 208}"
533,1,153,0,0.1543892,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{2 a-3 b}{2 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 a-3 b}{6 f (a+b)^2 \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{(2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{7/2}}+\frac{\sec ^2(e+f x)}{2 f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}","\frac{2 a-3 b}{2 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 a-3 b}{6 f (a+b)^2 \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{(2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{2 f (a+b)^{7/2}}+\frac{\sec ^2(e+f x)}{2 f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"-((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(7/2)*f) + (2*a - 3*b)/(6*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) + Sec[e + f*x]^2/(2*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*a - 3*b)/(2*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,5,25,0.2000,1,"{3194, 78, 51, 63, 208}"
534,1,91,0,0.0845923,"\int \frac{\tan (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{1}{f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{1}{3 f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{f (a+b)^{5/2}}","-\frac{1}{f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{1}{3 f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{f (a+b)^{5/2}}",1,"ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]]/((a + b)^(5/2)*f) - 1/(3*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) - 1/((a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",5,4,23,0.1739,1,"{3194, 51, 63, 208}"
535,1,83,0,0.0876517,"\int \frac{\cot (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{1}{a^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}+\frac{1}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}","\frac{1}{a^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}+\frac{1}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"-(ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f)) + 1/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + 1/(a^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",5,4,23,0.1739,1,"{3194, 51, 63, 208}"
536,1,143,0,0.1372295,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{2 a+5 b}{2 a^3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 a+5 b}{6 a^2 f \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{(2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{7/2} f}-\frac{\csc ^2(e+f x)}{2 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}","-\frac{2 a+5 b}{2 a^3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 a+5 b}{6 a^2 f \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{(2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{7/2} f}-\frac{\csc ^2(e+f x)}{2 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"((2*a + 5*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(2*a^(7/2)*f) - (2*a + 5*b)/(6*a^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) - Csc[e + f*x]^2/(2*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*a + 5*b)/(2*a^3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",6,5,25,0.2000,1,"{3194, 78, 51, 63, 208}"
537,1,208,0,0.1959774,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{8 a^2+40 a b+35 b^2}{8 a^4 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 a^2+40 a b+35 b^2}{24 a^3 f \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\left(8 a^2+40 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{9/2} f}+\frac{(8 a+7 b) \csc ^2(e+f x)}{8 a^2 f \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\csc ^4(e+f x)}{4 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}","\frac{8 a^2+40 a b+35 b^2}{8 a^4 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 a^2+40 a b+35 b^2}{24 a^3 f \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\left(8 a^2+40 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{9/2} f}+\frac{(8 a+7 b) \csc ^2(e+f x)}{8 a^2 f \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\csc ^4(e+f x)}{4 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"-((8*a^2 + 40*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(8*a^(9/2)*f) + (8*a^2 + 40*a*b + 35*b^2)/(24*a^3*f*(a + b*Sin[e + f*x]^2)^(3/2)) + ((8*a + 7*b)*Csc[e + f*x]^2)/(8*a^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) - Csc[e + f*x]^4/(4*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (8*a^2 + 40*a*b + 35*b^2)/(8*a^4*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,6,25,0.2400,1,"{3194, 89, 78, 51, 63, 208}"
538,1,348,0,0.4264389,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{2 (2 a-b) \tan (e+f x)}{3 f (a+b)^2 \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{8 b (a-b) \sin (e+f x) \cos (e+f x)}{3 f (a+b)^4 \sqrt{a+b \sin ^2(e+f x)}}+\frac{b (5 a-3 b) \sin (e+f x) \cos (e+f x)}{3 f (a+b)^3 \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{3 f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{(5 a-3 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^4 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","-\frac{2 (2 a-b) \tan (e+f x)}{3 f (a+b)^2 \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{8 b (a-b) \sin (e+f x) \cos (e+f x)}{3 f (a+b)^4 \sqrt{a+b \sin ^2(e+f x)}}+\frac{b (5 a-3 b) \sin (e+f x) \cos (e+f x)}{3 f (a+b)^3 \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{\tan (e+f x) \sec ^2(e+f x)}{3 f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{(5 a-3 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^4 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"((5*a - 3*b)*b*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)^3*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (8*(a - b)*b*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)^4*f*Sqrt[a + b*Sin[e + f*x]^2]) + (8*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)^4*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - ((5*a - 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*(2*a - b)*Tan[e + f*x])/(3*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2))","A",10,8,25,0.3200,1,"{3196, 470, 527, 524, 426, 424, 421, 419}"
539,1,292,0,0.3093447,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\tan (e+f x)}{f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{b (7 a-b) \sin (e+f x) \cos (e+f x)}{3 a f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}-\frac{4 b \sin (e+f x) \cos (e+f x)}{3 f (a+b)^2 \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{4 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a f (a+b)^3 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\frac{\tan (e+f x)}{f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{b (7 a-b) \sin (e+f x) \cos (e+f x)}{3 a f (a+b)^3 \sqrt{a+b \sin ^2(e+f x)}}-\frac{4 b \sin (e+f x) \cos (e+f x)}{3 f (a+b)^2 \left(a+b \sin ^2(e+f x)\right)^{3/2}}+\frac{4 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}-\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a f (a+b)^3 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}",1,"(-4*b*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) - ((7*a - b)*b*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((7*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*(a + b)^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (4*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + Tan[e + f*x]/((a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2))","A",9,8,25,0.3200,1,"{3196, 471, 527, 524, 426, 424, 421, 419}"
540,1,223,0,0.2588043,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[(a + b*Sin[e + f*x]^2)^(-5/2),x]","\frac{2 b (2 a+b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a^2 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{b \sin (e+f x) \cos (e+f x)}{3 a f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}","\frac{2 b (2 a+b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a^2 f (a+b)^2 \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{b \sin (e+f x) \cos (e+f x)}{3 a f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}-\frac{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a f (a+b) \sqrt{a+b \sin ^2(e+f x)}}",1,"(b*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*b*(2*a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",7,7,16,0.4375,1,"{3184, 3173, 3172, 3178, 3177, 3183, 3182}"
541,1,287,0,0.3659446,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{(7 a+8 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 f (a+b)}+\frac{(3 a+4 b) \cot (e+f x)}{3 a^2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{4 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(7 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^3 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{\cot (e+f x)}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}","-\frac{(7 a+8 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 f (a+b)}+\frac{(3 a+4 b) \cot (e+f x)}{3 a^2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{4 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(7 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^3 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{\cot (e+f x)}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"Cot[e + f*x]/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + ((3*a + 4*b)*Cot[e + f*x])/(3*a^2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((7*a + 8*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*(a + b)*f) - ((7*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (4*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a^2*f*Sqrt[a + b*Sin[e + f*x]^2])","A",9,9,25,0.3600,1,"{3196, 469, 579, 583, 524, 426, 424, 421, 419}"
542,1,348,0,0.5226077,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{8 (a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac{(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac{2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(5 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^4 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left(a+b \sin ^2(e+f x)\right)^{3/2}}","\frac{8 (a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac{(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac{2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(5 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left(\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right)}{3 a^4 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"((a + b)*Cot[e + f*x]*Csc[e + f*x]^2)/(3*a*b*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*(a + 3*b)*Cot[e + f*x]*Csc[e + f*x]^2)/(3*a^2*b*f*Sqrt[a + b*Sin[e + f*x]^2]) + (8*(a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^4*f) - ((3*a + 8*b)*Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*b*f) + (8*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^4*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - ((5*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a^3*f*Sqrt[a + b*Sin[e + f*x]^2])","A",10,9,25,0.3600,1,"{3196, 468, 579, 583, 524, 426, 424, 421, 419}"
543,1,120,0,0.1218114,"\int \left(a+b \sin ^2(e+f x)\right)^p (d \tan (e+f x))^m \, dx","Int[(a + b*Sin[e + f*x]^2)^p*(d*Tan[e + f*x])^m,x]","\frac{\cos ^2(e+f x)^{\frac{m+1}{2}} (d \tan (e+f x))^{m+1} \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};\frac{m+1}{2},-p;\frac{m+3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{d f (m+1)}","\frac{\cos ^2(e+f x)^{\frac{m+1}{2}} (d \tan (e+f x))^{m+1} \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};\frac{m+1}{2},-p;\frac{m+3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{d f (m+1)}",1,"(AppellF1[(1 + m)/2, (1 + m)/2, -p, (3 + m)/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*(Cos[e + f*x]^2)^((1 + m)/2)*(a + b*Sin[e + f*x]^2)^p*(d*Tan[e + f*x])^(1 + m))/(d*f*(1 + m)*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,25,0.1200,1,"{3197, 511, 510}"
544,1,102,0,0.0936865,"\int \left(a+b \sin ^2(c+d x)\right)^p \tan ^3(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x]^3,x]","\frac{\sec ^2(c+d x) \left(a+b \sin ^2(c+d x)\right)^{p+1}}{2 d (a+b)}-\frac{(a+b p+b) \left(a+b \sin ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^2(c+d x)+a}{a+b}\right)}{2 d (p+1) (a+b)^2}","\frac{\sec ^2(c+d x) \left(a+b \sin ^2(c+d x)\right)^{p+1}}{2 d (a+b)}-\frac{(a+b p+b) \left(a+b \sin ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^2(c+d x)+a}{a+b}\right)}{2 d (p+1) (a+b)^2}",1,"-((a + b + b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sin[c + d*x]^2)/(a + b)]*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*(a + b)^2*d*(1 + p)) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*(a + b)*d)","A",3,3,23,0.1304,1,"{3194, 78, 68}"
545,1,59,0,0.0434174,"\int \left(a+b \sin ^2(c+d x)\right)^p \tan (c+d x) \, dx","Int[(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x],x]","\frac{\left(a+b \sin ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^2(c+d x)+a}{a+b}\right)}{2 d (p+1) (a+b)}","\frac{\left(a+b \sin ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^2(c+d x)+a}{a+b}\right)}{2 d (p+1) (a+b)}",1,"(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sin[c + d*x]^2)/(a + b)]*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*(a + b)*d*(1 + p))","A",2,2,21,0.09524,1,"{3194, 68}"
546,1,54,0,0.0557917,"\int \cot (c+d x) \left(a+b \sin ^2(c+d x)\right)^p \, dx","Int[Cot[c + d*x]*(a + b*Sin[c + d*x]^2)^p,x]","-\frac{\left(a+b \sin ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^2(c+d x)}{a}+1\right)}{2 a d (p+1)}","-\frac{\left(a+b \sin ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^2(c+d x)}{a}+1\right)}{2 a d (p+1)}",1,"-(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^2)/a]*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*a*d*(1 + p))","A",2,2,21,0.09524,1,"{3194, 65}"
547,1,95,0,0.075998,"\int \cot ^3(c+d x) \left(a+b \sin ^2(c+d x)\right)^p \, dx","Int[Cot[c + d*x]^3*(a + b*Sin[c + d*x]^2)^p,x]","\frac{(a-b p) \left(a+b \sin ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^2(c+d x)}{a}+1\right)}{2 a^2 d (p+1)}-\frac{\csc ^2(c+d x) \left(a+b \sin ^2(c+d x)\right)^{p+1}}{2 a d}","\frac{(a-b p) \left(a+b \sin ^2(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^2(c+d x)}{a}+1\right)}{2 a^2 d (p+1)}-\frac{\csc ^2(c+d x) \left(a+b \sin ^2(c+d x)\right)^{p+1}}{2 a d}",1,"-(Csc[c + d*x]^2*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*a*d) + ((a - b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^2)/a]*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*a^2*d*(1 + p))","A",3,3,23,0.1304,1,"{3194, 78, 65}"
548,1,101,0,0.1274908,"\int \left(a+b \sin ^2(c+d x)\right)^p \tan ^4(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x]^4,x]","\frac{\sin ^4(c+d x) \sqrt{\cos ^2(c+d x)} \tan (c+d x) \left(a+b \sin ^2(c+d x)\right)^p \left(\frac{b \sin ^2(c+d x)}{a}+1\right)^{-p} F_1\left(\frac{5}{2};\frac{5}{2},-p;\frac{7}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right)}{5 d}","\frac{\sin ^4(c+d x) \sqrt{\cos ^2(c+d x)} \tan (c+d x) \left(a+b \sin ^2(c+d x)\right)^p \left(\frac{b \sin ^2(c+d x)}{a}+1\right)^{-p} F_1\left(\frac{5}{2};\frac{5}{2},-p;\frac{7}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right)}{5 d}",1,"(AppellF1[5/2, 5/2, -p, 7/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt[Cos[c + d*x]^2]*Sin[c + d*x]^4*(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x])/(5*d*(1 + (b*Sin[c + d*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3196, 511, 510}"
549,1,101,0,0.1049741,"\int \left(a+b \sin ^2(c+d x)\right)^p \tan ^2(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x]^2,x]","\frac{\sin ^2(c+d x) \sqrt{\cos ^2(c+d x)} \tan (c+d x) \left(a+b \sin ^2(c+d x)\right)^p \left(\frac{b \sin ^2(c+d x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right)}{3 d}","\frac{\sin ^2(c+d x) \sqrt{\cos ^2(c+d x)} \tan (c+d x) \left(a+b \sin ^2(c+d x)\right)^p \left(\frac{b \sin ^2(c+d x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};\frac{3}{2},-p;\frac{5}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right)}{3 d}",1,"(AppellF1[3/2, 3/2, -p, 5/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt[Cos[c + d*x]^2]*Sin[c + d*x]^2*(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x])/(3*d*(1 + (b*Sin[c + d*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3196, 511, 510}"
550,1,97,0,0.1004705,"\int \cot ^2(c+d x) \left(a+b \sin ^2(c+d x)\right)^p \, dx","Int[Cot[c + d*x]^2*(a + b*Sin[c + d*x]^2)^p,x]","-\frac{\sqrt{\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left(a+b \sin ^2(c+d x)\right)^p \left(\frac{b \sin ^2(c+d x)}{a}+1\right)^{-p} F_1\left(-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right)}{d}","-\frac{\sqrt{\cos ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left(a+b \sin ^2(c+d x)\right)^p \left(\frac{b \sin ^2(c+d x)}{a}+1\right)^{-p} F_1\left(-\frac{1}{2};-\frac{1}{2},-p;\frac{1}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right)}{d}",1,"-((AppellF1[-1/2, -1/2, -p, 1/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]*Sec[c + d*x]*(a + b*Sin[c + d*x]^2)^p)/(d*(1 + (b*Sin[c + d*x]^2)/a)^p))","A",3,3,23,0.1304,1,"{3196, 511, 510}"
551,1,101,0,0.1023053,"\int \cot ^4(c+d x) \left(a+b \sin ^2(c+d x)\right)^p \, dx","Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^2)^p,x]","-\frac{\sqrt{\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left(a+b \sin ^2(c+d x)\right)^p \left(\frac{b \sin ^2(c+d x)}{a}+1\right)^{-p} F_1\left(-\frac{3}{2};-\frac{3}{2},-p;-\frac{1}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right)}{3 d}","-\frac{\sqrt{\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left(a+b \sin ^2(c+d x)\right)^p \left(\frac{b \sin ^2(c+d x)}{a}+1\right)^{-p} F_1\left(-\frac{3}{2};-\frac{3}{2},-p;-\frac{1}{2};\sin ^2(c+d x),-\frac{b \sin ^2(c+d x)}{a}\right)}{3 d}",1,"-(AppellF1[-3/2, -3/2, -p, -1/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]^3*Sec[c + d*x]*(a + b*Sin[c + d*x]^2)^p)/(3*d*(1 + (b*Sin[c + d*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3196, 511, 510}"
552,1,153,0,0.1908707,"\int \frac{\cot ^3(x)}{a+b \sin ^3(x)} \, dx","Int[Cot[x]^3/(a + b*Sin[x]^3),x]","\frac{b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (x)+b^{2/3} \sin ^2(x)\right)}{6 a^{5/3}}-\frac{b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (x)\right)}{3 a^{5/3}}+\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{5/3}}+\frac{\log \left(a+b \sin ^3(x)\right)}{3 a}-\frac{\csc ^2(x)}{2 a}-\frac{\log (\sin (x))}{a}","\frac{b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (x)+b^{2/3} \sin ^2(x)\right)}{6 a^{5/3}}-\frac{b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (x)\right)}{3 a^{5/3}}+\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{5/3}}+\frac{\log \left(a+b \sin ^3(x)\right)}{3 a}-\frac{\csc ^2(x)}{2 a}-\frac{\log (\sin (x))}{a}",1,"(b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)) - Csc[x]^2/(2*a) - Log[Sin[x]]/a - (b^(2/3)*Log[a^(1/3) + b^(1/3)*Sin[x]])/(3*a^(5/3)) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[x] + b^(2/3)*Sin[x]^2])/(6*a^(5/3)) + Log[a + b*Sin[x]^3]/(3*a)","A",11,10,15,0.6667,1,"{3230, 1834, 1871, 200, 31, 634, 617, 204, 628, 260}"
553,1,45,0,0.0740431,"\int \cot (x) \sqrt{a+b \sin ^3(x)} \, dx","Int[Cot[x]*Sqrt[a + b*Sin[x]^3],x]","\frac{2}{3} \sqrt{a+b \sin ^3(x)}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^3(x)}}{\sqrt{a}}\right)","\frac{2}{3} \sqrt{a+b \sin ^3(x)}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^3(x)}}{\sqrt{a}}\right)",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[x]^3]/Sqrt[a]])/3 + (2*Sqrt[a + b*Sin[x]^3])/3","A",5,5,15,0.3333,1,"{3230, 266, 50, 63, 208}"
554,1,28,0,0.0715143,"\int \frac{\cot (x)}{\sqrt{a+b \sin ^3(x)}} \, dx","Int[Cot[x]/Sqrt[a + b*Sin[x]^3],x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^3(x)}}{\sqrt{a}}\right)}{3 \sqrt{a}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^3(x)}}{\sqrt{a}}\right)}{3 \sqrt{a}}",1,"(-2*ArcTanh[Sqrt[a + b*Sin[x]^3]/Sqrt[a]])/(3*Sqrt[a])","A",4,4,15,0.2667,1,"{3230, 266, 63, 208}"
555,1,59,0,0.0889988,"\int \cot (c+d x) \sqrt{a+b \sin ^4(c+d x)} \, dx","Int[Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\sqrt{a+b \sin ^4(c+d x)}}{2 d}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^4(c+d x)}}{\sqrt{a}}\right)}{2 d}","\frac{\sqrt{a+b \sin ^4(c+d x)}}{2 d}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^4(c+d x)}}{\sqrt{a}}\right)}{2 d}",1,"-(Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]])/(2*d) + Sqrt[a + b*Sin[c + d*x]^4]/(2*d)","A",5,5,23,0.2174,1,"{3229, 266, 50, 63, 208}"
556,1,89,0,0.116921,"\int \frac{\tan ^3(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Tan[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\sec ^2(c+d x) \sqrt{a+b \sin ^4(c+d x)}}{2 d (a+b)}-\frac{a \tanh ^{-1}\left(\frac{a+b \sin ^2(c+d x)}{\sqrt{a+b} \sqrt{a+b \sin ^4(c+d x)}}\right)}{2 d (a+b)^{3/2}}","\frac{\sec ^2(c+d x) \sqrt{a+b \sin ^4(c+d x)}}{2 d (a+b)}-\frac{a \tanh ^{-1}\left(\frac{a+b \sin ^2(c+d x)}{\sqrt{a+b} \sqrt{a+b \sin ^4(c+d x)}}\right)}{2 d (a+b)^{3/2}}",1,"-(a*ArcTanh[(a + b*Sin[c + d*x]^2)/(Sqrt[a + b]*Sqrt[a + b*Sin[c + d*x]^4])])/(2*(a + b)^(3/2)*d) + (Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]^4])/(2*(a + b)*d)","A",4,4,25,0.1600,1,"{3229, 807, 725, 206}"
557,1,51,0,0.0555082,"\int \frac{\tan (c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Tan[c + d*x]/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\tanh ^{-1}\left(\frac{a+b \sin ^2(c+d x)}{\sqrt{a+b} \sqrt{a+b \sin ^4(c+d x)}}\right)}{2 d \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{a+b \sin ^2(c+d x)}{\sqrt{a+b} \sqrt{a+b \sin ^4(c+d x)}}\right)}{2 d \sqrt{a+b}}",1,"ArcTanh[(a + b*Sin[c + d*x]^2)/(Sqrt[a + b]*Sqrt[a + b*Sin[c + d*x]^4])]/(2*Sqrt[a + b]*d)","A",3,3,23,0.1304,1,"{3229, 725, 206}"
558,1,35,0,0.0711073,"\int \frac{\cot (c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Cot[c + d*x]/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^4(c+d x)}}{\sqrt{a}}\right)}{2 \sqrt{a} d}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^4(c+d x)}}{\sqrt{a}}\right)}{2 \sqrt{a} d}",1,"-ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]]/(2*Sqrt[a]*d)","A",4,4,23,0.1739,1,"{3229, 266, 63, 208}"
559,1,70,0,0.0989119,"\int \frac{\cot ^3(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Cot[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^4(c+d x)}}{\sqrt{a}}\right)}{2 \sqrt{a} d}-\frac{\csc ^2(c+d x) \sqrt{a+b \sin ^4(c+d x)}}{2 a d}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^4(c+d x)}}{\sqrt{a}}\right)}{2 \sqrt{a} d}-\frac{\csc ^2(c+d x) \sqrt{a+b \sin ^4(c+d x)}}{2 a d}",1,"ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]]/(2*Sqrt[a]*d) - (Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]^4])/(2*a*d)","A",5,5,25,0.2000,1,"{3229, 807, 266, 63, 208}"
560,1,108,0,0.1755691,"\int \frac{\cot ^5(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Cot[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^4(c+d x)}}{\sqrt{a}}\right)}{4 a^{3/2} d}-\frac{\csc ^4(c+d x) \sqrt{a+b \sin ^4(c+d x)}}{4 a d}+\frac{\csc ^2(c+d x) \sqrt{a+b \sin ^4(c+d x)}}{a d}","-\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^4(c+d x)}}{\sqrt{a}}\right)}{4 a^{3/2} d}-\frac{\csc ^4(c+d x) \sqrt{a+b \sin ^4(c+d x)}}{4 a d}+\frac{\csc ^2(c+d x) \sqrt{a+b \sin ^4(c+d x)}}{a d}",1,"-((2*a - b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]])/(4*a^(3/2)*d) + (Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]^4])/(a*d) - (Csc[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]^4])/(4*a*d)","A",6,6,25,0.2400,1,"{3229, 1807, 807, 266, 63, 208}"
561,1,411,0,0.379546,"\int \frac{\tan ^2(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Tan[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\sin (c+d x) \cos (c+d x) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}{d \sqrt{a+b} \sqrt{a+b \sin ^4(c+d x)} \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)}+\frac{\sqrt[4]{a} \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 d (a+b)^{3/4} \sqrt{a+b \sin ^4(c+d x)}}-\frac{\sqrt[4]{a} \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{d (a+b)^{3/4} \sqrt{a+b \sin ^4(c+d x)}}","\frac{\sin (c+d x) \cos (c+d x) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}{d \sqrt{a+b} \sqrt{a+b \sin ^4(c+d x)} \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)}+\frac{\sqrt[4]{a} \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 d (a+b)^{3/4} \sqrt{a+b \sin ^4(c+d x)}}-\frac{\sqrt[4]{a} \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{d (a+b)^{3/4} \sqrt{a+b \sin ^4(c+d x)}}",1,"(Cos[c + d*x]*Sin[c + d*x]*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4))/(Sqrt[a + b]*d*Sqrt[a + b*Sin[c + d*x]^4]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)) - (a^(1/4)*Cos[c + d*x]^2*EllipticE[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1 - Sqrt[a]/Sqrt[a + b])/2]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/((a + b)^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4]) + (a^(1/4)*Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1 - Sqrt[a]/Sqrt[a + b])/2]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*(a + b)^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4])","A",4,4,25,0.1600,1,"{3232, 1139, 1103, 1195}"
562,1,162,0,0.08279,"\int \frac{1}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[1/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 \sqrt[4]{a} d \sqrt[4]{a+b} \sqrt{a+b \sin ^4(c+d x)}}","\frac{\cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 \sqrt[4]{a} d \sqrt[4]{a+b} \sqrt{a+b \sin ^4(c+d x)}}",1,"(Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1 - Sqrt[a]/Sqrt[a + b])/2]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*a^(1/4)*(a + b)^(1/4)*d*Sqrt[a + b*Sin[c + d*x]^4])","A",2,2,16,0.1250,1,"{3210, 1103}"
563,1,477,0,0.3655761,"\int \frac{\cot ^2(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Int[Cot[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\sqrt[4]{a+b} \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 a^{3/4} d \sqrt{a+b \sin ^4(c+d x)}}-\frac{\sqrt[4]{a+b} \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{a^{3/4} d \sqrt{a+b \sin ^4(c+d x)}}+\frac{\sqrt{a+b} \sin (c+d x) \cos (c+d x) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}{a d \sqrt{a+b \sin ^4(c+d x)} \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)}-\frac{\cos ^2(c+d x) \cot (c+d x) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}{a d \sqrt{a+b \sin ^4(c+d x)}}","\frac{\sqrt[4]{a+b} \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{2 a^{3/4} d \sqrt{a+b \sin ^4(c+d x)}}-\frac{\sqrt[4]{a+b} \cos ^2(c+d x) \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right) \sqrt{\frac{(a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}{\left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2} \left(1-\frac{\sqrt{a}}{\sqrt{a+b}}\right)\right)}{a^{3/4} d \sqrt{a+b \sin ^4(c+d x)}}+\frac{\sqrt{a+b} \sin (c+d x) \cos (c+d x) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}{a d \sqrt{a+b \sin ^4(c+d x)} \left(\sqrt{a+b} \tan ^2(c+d x)+\sqrt{a}\right)}-\frac{\cos ^2(c+d x) \cot (c+d x) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}{a d \sqrt{a+b \sin ^4(c+d x)}}",1,"-((Cos[c + d*x]^2*Cot[c + d*x]*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4))/(a*d*Sqrt[a + b*Sin[c + d*x]^4])) + (Sqrt[a + b]*Cos[c + d*x]*Sin[c + d*x]*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4))/(a*d*Sqrt[a + b*Sin[c + d*x]^4]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)) - ((a + b)^(1/4)*Cos[c + d*x]^2*EllipticE[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1 - Sqrt[a]/Sqrt[a + b])/2]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(a^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4]) + ((a + b)^(1/4)*Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1 - Sqrt[a]/Sqrt[a + b])/2]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*a^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4])","A",6,6,25,0.2400,1,"{3232, 1123, 12, 1139, 1103, 1195}"
564,0,0,0,0.0419558,"\int \left(a+b \sin ^4(c+d x)\right)^p \tan ^m(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^m,x]","\int \left(a+b \sin ^4(c+d x)\right)^p \tan ^m(c+d x) \, dx","\text{Int}\left(\tan ^m(c+d x) \left(a+b \sin ^4(c+d x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^m, x]","A",0,0,0,0,-1,"{}"
565,1,279,0,0.2937756,"\int \left(a+b \sin ^4(c+d x)\right)^p \tan ^3(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^3,x]","-\frac{(a+2 b p+b) \sin ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \left(\frac{b \sin ^4(c+d x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\sin ^4(c+d x),-\frac{b \sin ^4(c+d x)}{a}\right)}{2 d (a+b)}+\frac{b (2 p+1) \sin ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \left(\frac{b \sin ^4(c+d x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^4(c+d x)}{a}\right)}{2 d (a+b)}-\frac{(a+2 b p+b) \left(a+b \sin ^4(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^4(c+d x)+a}{a+b}\right)}{4 d (p+1) (a+b)^2}+\frac{\sec ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^{p+1}}{2 d (a+b)}","-\frac{(a+2 b p+b) \sin ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \left(\frac{b \sin ^4(c+d x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\sin ^4(c+d x),-\frac{b \sin ^4(c+d x)}{a}\right)}{2 d (a+b)}+\frac{b (2 p+1) \sin ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \left(\frac{b \sin ^4(c+d x)}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^4(c+d x)}{a}\right)}{2 d (a+b)}-\frac{(a+2 b p+b) \left(a+b \sin ^4(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^4(c+d x)+a}{a+b}\right)}{4 d (p+1) (a+b)^2}+\frac{\sec ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^{p+1}}{2 d (a+b)}",1,"-((a + b + 2*b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sin[c + d*x]^4)/(a + b)]*(a + b*Sin[c + d*x]^4)^(1 + p))/(4*(a + b)^2*d*(1 + p)) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x]^4)^(1 + p))/(2*(a + b)*d) - ((a + b + 2*b*p)*AppellF1[1/2, 1, -p, 3/2, Sin[c + d*x]^4, -((b*Sin[c + d*x]^4)/a)]*Sin[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p)/(2*(a + b)*d*(1 + (b*Sin[c + d*x]^4)/a)^p) + (b*(1 + 2*p)*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sin[c + d*x]^4)/a)]*Sin[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p)/(2*(a + b)*d*(1 + (b*Sin[c + d*x]^4)/a)^p)","A",11,10,23,0.4348,1,"{3229, 835, 844, 246, 245, 757, 430, 429, 444, 68}"
566,1,141,0,0.1192689,"\int \left(a+b \sin ^4(c+d x)\right)^p \tan (c+d x) \, dx","Int[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x],x]","\frac{\sin ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \left(\frac{b \sin ^4(c+d x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\sin ^4(c+d x),-\frac{b \sin ^4(c+d x)}{a}\right)}{2 d}+\frac{\left(a+b \sin ^4(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^4(c+d x)+a}{a+b}\right)}{4 d (p+1) (a+b)}","\frac{\sin ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \left(\frac{b \sin ^4(c+d x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\sin ^4(c+d x),-\frac{b \sin ^4(c+d x)}{a}\right)}{2 d}+\frac{\left(a+b \sin ^4(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^4(c+d x)+a}{a+b}\right)}{4 d (p+1) (a+b)}",1,"(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sin[c + d*x]^4)/(a + b)]*(a + b*Sin[c + d*x]^4)^(1 + p))/(4*(a + b)*d*(1 + p)) + (AppellF1[1/2, 1, -p, 3/2, Sin[c + d*x]^4, -((b*Sin[c + d*x]^4)/a)]*Sin[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p)/(2*d*(1 + (b*Sin[c + d*x]^4)/a)^p)","A",7,6,21,0.2857,1,"{3229, 757, 430, 429, 444, 68}"
567,1,54,0,0.0649503,"\int \cot (c+d x) \left(a+b \sin ^4(c+d x)\right)^p \, dx","Int[Cot[c + d*x]*(a + b*Sin[c + d*x]^4)^p,x]","-\frac{\left(a+b \sin ^4(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^4(c+d x)}{a}+1\right)}{4 a d (p+1)}","-\frac{\left(a+b \sin ^4(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^4(c+d x)}{a}+1\right)}{4 a d (p+1)}",1,"-(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^4)/a]*(a + b*Sin[c + d*x]^4)^(1 + p))/(4*a*d*(1 + p))","A",3,3,21,0.1429,1,"{3229, 266, 65}"
568,1,127,0,0.1017917,"\int \cot ^3(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \, dx","Int[Cot[c + d*x]^3*(a + b*Sin[c + d*x]^4)^p,x]","\frac{\left(a+b \sin ^4(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^4(c+d x)}{a}+1\right)}{4 a d (p+1)}-\frac{\csc ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \left(\frac{b \sin ^4(c+d x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sin ^4(c+d x)}{a}\right)}{2 d}","\frac{\left(a+b \sin ^4(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^4(c+d x)}{a}+1\right)}{4 a d (p+1)}-\frac{\csc ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \left(\frac{b \sin ^4(c+d x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sin ^4(c+d x)}{a}\right)}{2 d}",1,"(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^4)/a]*(a + b*Sin[c + d*x]^4)^(1 + p))/(4*a*d*(1 + p)) - (Csc[c + d*x]^2*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sin[c + d*x]^4)/a)]*(a + b*Sin[c + d*x]^4)^p)/(2*d*(1 + (b*Sin[c + d*x]^4)/a)^p)","A",6,6,23,0.2609,1,"{3229, 764, 365, 364, 266, 65}"
569,0,0,0,0.0411529,"\int \left(a+b \sin ^4(c+d x)\right)^p \tan ^4(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^4,x]","\int \left(a+b \sin ^4(c+d x)\right)^p \tan ^4(c+d x) \, dx","\text{Int}\left(\tan ^4(c+d x) \left(a+b \sin ^4(c+d x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^4, x]","A",0,0,0,0,-1,"{}"
570,0,0,0,0.0423265,"\int \left(a+b \sin ^4(c+d x)\right)^p \tan ^2(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^2,x]","\int \left(a+b \sin ^4(c+d x)\right)^p \tan ^2(c+d x) \, dx","\text{Int}\left(\tan ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^2, x]","A",0,0,0,0,-1,"{}"
571,0,0,0,0.0110318,"\int \left(a+b \sin ^4(c+d x)\right)^p \, dx","Int[(a + b*Sin[c + d*x]^4)^p,x]","\int \left(a+b \sin ^4(c+d x)\right)^p \, dx","\text{Int}\left(\left(a+b \sin ^4(c+d x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[c + d*x]^4)^p, x]","A",0,0,0,0,-1,"{}"
572,0,0,0,0.0407652,"\int \cot ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \, dx","Int[Cot[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p,x]","\int \cot ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \, dx","\text{Int}\left(\cot ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p,x\right)",0,"Defer[Int][Cot[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p, x]","A",0,0,0,0,-1,"{}"
573,0,0,0,0.0401924,"\int \cot ^4(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \, dx","Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^4)^p,x]","\int \cot ^4(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \, dx","\text{Int}\left(\cot ^4(c+d x) \left(a+b \sin ^4(c+d x)\right)^p,x\right)",0,"Defer[Int][Cot[c + d*x]^4*(a + b*Sin[c + d*x]^4)^p, x]","A",0,0,0,0,-1,"{}"
574,1,306,0,0.4293908,"\int \left(a+b \sin ^n(c+d x)\right)^3 \tan ^m(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^n)^3*Tan[c + d*x]^m,x]","\frac{3 a^2 b \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^n(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\sin ^2(c+d x)\right)}{d (m+n+1)}+\frac{a^3 \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1)}+\frac{3 a b^2 \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^{2 n}(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+2 n+1);\frac{1}{2} (m+2 n+3);\sin ^2(c+d x)\right)}{d (m+2 n+1)}+\frac{b^3 \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^{3 n}(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+3 n+1);\frac{1}{2} (m+3 n+3);\sin ^2(c+d x)\right)}{d (m+3 n+1)}","\frac{3 a^2 b \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^n(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\sin ^2(c+d x)\right)}{d (m+n+1)}+\frac{a^3 \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1)}+\frac{3 a b^2 \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^{2 n}(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+2 n+1);\frac{1}{2} (m+2 n+3);\sin ^2(c+d x)\right)}{d (m+2 n+1)}+\frac{b^3 \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^{3 n}(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+3 n+1);\frac{1}{2} (m+3 n+3);\sin ^2(c+d x)\right)}{d (m+3 n+1)}",1,"(a^3*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (3*a^2*b*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1 + m + n)/2, (3 + m + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^n*Tan[c + d*x]^(1 + m))/(d*(1 + m + n)) + (3*a*b^2*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1 + m + 2*n)/2, (3 + m + 2*n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2*n)*Tan[c + d*x]^(1 + m))/(d*(1 + m + 2*n)) + (b^3*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1 + m + 3*n)/2, (3 + m + 3*n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(3*n)*Tan[c + d*x]^(1 + m))/(d*(1 + m + 3*n))","A",10,5,23,0.2174,1,"{3234, 3476, 364, 2602, 2577}"
575,1,215,0,0.2953588,"\int \left(a+b \sin ^n(c+d x)\right)^2 \tan ^m(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^n)^2*Tan[c + d*x]^m,x]","\frac{a^2 \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1)}+\frac{2 a b \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^n(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\sin ^2(c+d x)\right)}{d (m+n+1)}+\frac{b^2 \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^{2 n}(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+2 n+1);\frac{1}{2} (m+2 n+3);\sin ^2(c+d x)\right)}{d (m+2 n+1)}","\frac{a^2 \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1)}+\frac{2 a b \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^n(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\sin ^2(c+d x)\right)}{d (m+n+1)}+\frac{b^2 \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^{2 n}(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+2 n+1);\frac{1}{2} (m+2 n+3);\sin ^2(c+d x)\right)}{d (m+2 n+1)}",1,"(a^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (2*a*b*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1 + m + n)/2, (3 + m + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^n*Tan[c + d*x]^(1 + m))/(d*(1 + m + n)) + (b^2*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1 + m + 2*n)/2, (3 + m + 2*n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2*n)*Tan[c + d*x]^(1 + m))/(d*(1 + m + 2*n))","A",8,5,23,0.2174,1,"{3234, 3476, 364, 2602, 2577}"
576,1,124,0,0.1613954,"\int \left(a+b \sin ^n(c+d x)\right) \tan ^m(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^n)*Tan[c + d*x]^m,x]","\frac{a \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1)}+\frac{b \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^n(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\sin ^2(c+d x)\right)}{d (m+n+1)}","\frac{a \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(c+d x)\right)}{d (m+1)}+\frac{b \cos ^2(c+d x)^{\frac{m+1}{2}} \tan ^{m+1}(c+d x) \sin ^n(c+d x) \, _2F_1\left(\frac{m+1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\sin ^2(c+d x)\right)}{d (m+n+1)}",1,"(a*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (b*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1 + m + n)/2, (3 + m + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^n*Tan[c + d*x]^(1 + m))/(d*(1 + m + n))","A",6,5,21,0.2381,1,"{3234, 3476, 364, 2602, 2577}"
577,0,0,0,0.0577452,"\int \frac{\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx","Int[Tan[c + d*x]^m/(a + b*Sin[c + d*x]^n),x]","\int \frac{\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx","\text{Int}\left(\frac{\tan ^m(c+d x)}{a+b \sin ^n(c+d x)},x\right)",0,"Defer[Int][Tan[c + d*x]^m/(a + b*Sin[c + d*x]^n), x]","A",0,0,0,0,-1,"{}"
578,0,0,0,0.0580189,"\int \frac{\tan ^m(c+d x)}{\left(a+b \sin ^n(c+d x)\right)^2} \, dx","Int[Tan[c + d*x]^m/(a + b*Sin[c + d*x]^n)^2,x]","\int \frac{\tan ^m(c+d x)}{\left(a+b \sin ^n(c+d x)\right)^2} \, dx","\text{Int}\left(\frac{\tan ^m(c+d x)}{\left(a+b \sin ^n(c+d x)\right)^2},x\right)",0,"Defer[Int][Tan[c + d*x]^m/(a + b*Sin[c + d*x]^n)^2, x]","A",0,0,0,0,-1,"{}"
579,1,47,0,0.0811218,"\int \cot (x) \sqrt{a+b \sin ^n(x)} \, dx","Int[Cot[x]*Sqrt[a + b*Sin[x]^n],x]","\frac{2 \sqrt{a+b \sin ^n(x)}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^n(x)}}{\sqrt{a}}\right)}{n}","\frac{2 \sqrt{a+b \sin ^n(x)}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^n(x)}}{\sqrt{a}}\right)}{n}",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[x]^n]/Sqrt[a]])/n + (2*Sqrt[a + b*Sin[x]^n])/n","A",5,5,15,0.3333,1,"{3230, 266, 50, 63, 208}"
580,1,29,0,0.0786557,"\int \frac{\cot (x)}{\sqrt{a+b \sin ^n(x)}} \, dx","Int[Cot[x]/Sqrt[a + b*Sin[x]^n],x]","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^n(x)}}{\sqrt{a}}\right)}{\sqrt{a} n}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^n(x)}}{\sqrt{a}}\right)}{\sqrt{a} n}",1,"(-2*ArcTanh[Sqrt[a + b*Sin[x]^n]/Sqrt[a]])/(Sqrt[a]*n)","A",4,4,15,0.2667,1,"{3230, 266, 63, 208}"
581,0,0,0,0.0521793,"\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^m(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^m,x]","\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^m(c+d x) \, dx","\text{Int}\left(\tan ^m(c+d x) \left(a+b \sin ^n(c+d x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^m, x]","A",0,0,0,0,-1,"{}"
582,0,0,0,0.0511328,"\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^3(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^3,x]","\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^3(c+d x) \, dx","\text{Int}\left(\tan ^3(c+d x) \left(a+b \sin ^n(c+d x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^3, x]","A",0,0,0,0,-1,"{}"
583,0,0,0,0.026456,"\int \left(a+b \sin ^n(c+d x)\right)^p \tan (c+d x) \, dx","Int[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x],x]","\int \left(a+b \sin ^n(c+d x)\right)^p \tan (c+d x) \, dx","\text{Int}\left(\tan (c+d x) \left(a+b \sin ^n(c+d x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x], x]","A",0,0,0,0,-1,"{}"
584,1,55,0,0.0765276,"\int \cot (c+d x) \left(a+b \sin ^n(c+d x)\right)^p \, dx","Int[Cot[c + d*x]*(a + b*Sin[c + d*x]^n)^p,x]","-\frac{\left(a+b \sin ^n(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^n(c+d x)}{a}+1\right)}{a d n (p+1)}","-\frac{\left(a+b \sin ^n(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^n(c+d x)}{a}+1\right)}{a d n (p+1)}",1,"-((Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^n)/a]*(a + b*Sin[c + d*x]^n)^(1 + p))/(a*d*n*(1 + p)))","A",3,3,21,0.1429,1,"{3230, 266, 65}"
585,1,136,0,0.151623,"\int \cot ^3(c+d x) \left(a+b \sin ^n(c+d x)\right)^p \, dx","Int[Cot[c + d*x]^3*(a + b*Sin[c + d*x]^n)^p,x]","\frac{\left(a+b \sin ^n(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^n(c+d x)}{a}+1\right)}{a d n (p+1)}-\frac{\csc ^2(c+d x) \left(a+b \sin ^n(c+d x)\right)^p \left(\frac{b \sin ^n(c+d x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{2}{n},-p;-\frac{2-n}{n};-\frac{b \sin ^n(c+d x)}{a}\right)}{2 d}","\frac{\left(a+b \sin ^n(c+d x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^n(c+d x)}{a}+1\right)}{a d n (p+1)}-\frac{\csc ^2(c+d x) \left(a+b \sin ^n(c+d x)\right)^p \left(\frac{b \sin ^n(c+d x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{2}{n},-p;-\frac{2-n}{n};-\frac{b \sin ^n(c+d x)}{a}\right)}{2 d}",1,"(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^n)/a]*(a + b*Sin[c + d*x]^n)^(1 + p))/(a*d*n*(1 + p)) - (Csc[c + d*x]^2*Hypergeometric2F1[-2/n, -p, -((2 - n)/n), -((b*Sin[c + d*x]^n)/a)]*(a + b*Sin[c + d*x]^n)^p)/(2*d*(1 + (b*Sin[c + d*x]^n)/a)^p)","A",7,6,23,0.2609,1,"{3230, 1844, 365, 364, 266, 65}"
586,0,0,0,0.0488318,"\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^4(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^4,x]","\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^4(c+d x) \, dx","\text{Int}\left(\tan ^4(c+d x) \left(a+b \sin ^n(c+d x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^4, x]","A",0,0,0,0,-1,"{}"
587,0,0,0,0.0511503,"\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^2(c+d x) \, dx","Int[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^2,x]","\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^2(c+d x) \, dx","\text{Int}\left(\tan ^2(c+d x) \left(a+b \sin ^n(c+d x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^2, x]","A",0,0,0,0,-1,"{}"
588,0,0,0,0.0132638,"\int \left(a+b \sin ^n(c+d x)\right)^p \, dx","Int[(a + b*Sin[c + d*x]^n)^p,x]","\int \left(a+b \sin ^n(c+d x)\right)^p \, dx","\text{Int}\left(\left(a+b \sin ^n(c+d x)\right)^p,x\right)",0,"Defer[Int][(a + b*Sin[c + d*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
589,0,0,0,0.0491673,"\int \cot ^2(c+d x) \left(a+b \sin ^n(c+d x)\right)^p \, dx","Int[Cot[c + d*x]^2*(a + b*Sin[c + d*x]^n)^p,x]","\int \cot ^2(c+d x) \left(a+b \sin ^n(c+d x)\right)^p \, dx","\text{Int}\left(\cot ^2(c+d x) \left(a+b \sin ^n(c+d x)\right)^p,x\right)",0,"Defer[Int][Cot[c + d*x]^2*(a + b*Sin[c + d*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
590,0,0,0,0.0501024,"\int \cot ^4(c+d x) \left(a+b \sin ^n(c+d x)\right)^p \, dx","Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^n)^p,x]","\int \cot ^4(c+d x) \left(a+b \sin ^n(c+d x)\right)^p \, dx","\text{Int}\left(\cot ^4(c+d x) \left(a+b \sin ^n(c+d x)\right)^p,x\right)",0,"Defer[Int][Cot[c + d*x]^4*(a + b*Sin[c + d*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
591,1,114,0,0.1814461,"\int \frac{a+b \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt{d \sin (e+f x)}} \, dx","Int[(a + b*Sin[e + f*x]^2)/((g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]]),x]","\frac{2 (a+b) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}-\frac{2 (2 a-b) \left(1-\csc ^2(e+f x)\right)^{3/4} (d \sin (e+f x))^{3/2} F\left(\left.\frac{1}{2} \csc ^{-1}(\sin (e+f x))\right|2\right)}{3 d^2 f g (g \cos (e+f x))^{3/2}}","\frac{(2 a-b) \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{3 f g^2 \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 (a+b) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}",1,"(2*(a + b)*Sqrt[d*Sin[e + f*x]])/(3*d*f*g*(g*Cos[e + f*x])^(3/2)) - (2*(2*a - b)*(1 - Csc[e + f*x]^2)^(3/4)*EllipticF[ArcCsc[Sin[e + f*x]]/2, 2]*(d*Sin[e + f*x])^(3/2))/(3*d^2*f*g*(g*Cos[e + f*x])^(3/2))","A",7,7,37,0.1892,1,"{3202, 457, 329, 237, 335, 275, 232}"
592,1,137,0,0.2106493,"\int (c \cos (e+f x))^m (d \sin (e+f x))^n \left(a+b \sin ^2(e+f x)\right)^p \, dx","Int[(c*Cos[e + f*x])^m*(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x]^2)^p,x]","\frac{c \cos ^2(e+f x)^{\frac{1-m}{2}} (c \cos (e+f x))^{m-1} (d \sin (e+f x))^{n+1} \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{n+1}{2};\frac{1-m}{2},-p;\frac{n+3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{d f (n+1)}","\frac{c \cos ^2(e+f x)^{\frac{1-m}{2}} (c \cos (e+f x))^{m-1} (d \sin (e+f x))^{n+1} \left(a+b \sin ^2(e+f x)\right)^p \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{n+1}{2};\frac{1-m}{2},-p;\frac{n+3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{d f (n+1)}",1,"(c*AppellF1[(1 + n)/2, (1 - m)/2, -p, (3 + n)/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*(c*Cos[e + f*x])^(-1 + m)*(Cos[e + f*x]^2)^((1 - m)/2)*(d*Sin[e + f*x])^(1 + n)*(a + b*Sin[e + f*x]^2)^p)/(d*f*(1 + n)*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",3,3,35,0.08571,1,"{3202, 511, 510}"
593,0,0,0,0.6985816,"\int \sqrt{a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx","Int[Sqrt[a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2],x]","\int \sqrt{a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx","\frac{\sqrt{a+(b \sin (e+f x)+c \cos (e+f x))^2} E\left(e+f x+\tan ^{-1}(b,c)|-\frac{b^2+c^2}{a}\right)}{f \sqrt{\frac{(b \sin (e+f x)+c \cos (e+f x))^2}{a}+1}}",1,"((I/2)*Defer[Subst][Defer[Int][Sqrt[a + (c + b*x)^2/(1 + x^2)]/(I - x), x], x, Tan[e + f*x]])/f + ((I/2)*Defer[Subst][Defer[Int][Sqrt[a + (c + b*x)^2/(1 + x^2)]/(I + x), x], x, Tan[e + f*x]])/f","F",0,0,0,0,-1,"{}"
594,0,0,0,0.6971653,"\int \frac{1}{\sqrt{a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx","Int[1/Sqrt[a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2],x]","\int \frac{1}{\sqrt{a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx","\frac{\sqrt{\frac{(b \sin (e+f x)+c \cos (e+f x))^2}{a}+1} F\left(e+f x+\tan ^{-1}(b,c)|-\frac{b^2+c^2}{a}\right)}{f \sqrt{a+(b \sin (e+f x)+c \cos (e+f x))^2}}",1,"((I/2)*Defer[Subst][Defer[Int][1/((I - x)*Sqrt[a + (c + b*x)^2/(1 + x^2)]), x], x, Tan[e + f*x]])/f + ((I/2)*Defer[Subst][Defer[Int][1/((I + x)*Sqrt[a + (c + b*x)^2/(1 + x^2)]), x], x, Tan[e + f*x]])/f","F",0,0,0,0,-1,"{}"