1,1,36,53,0.0325362,"\int \left(a \sin ^2(x)\right)^{5/2} \, dx","Integrate[(a*Sin[x]^2)^(5/2),x]","-\frac{1}{240} a^2 (150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \csc (x) \sqrt{a \sin ^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}",1,"-1/240*(a^2*(150*Cos[x] - 25*Cos[3*x] + 3*Cos[5*x])*Csc[x]*Sqrt[a*Sin[x]^2])","A",1
2,1,26,34,0.0389432,"\int \left(a \sin ^2(x)\right)^{3/2} \, dx","Integrate[(a*Sin[x]^2)^(3/2),x]","\frac{1}{12} a (\cos (3 x)-9 \cos (x)) \csc (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,"(a*(-9*Cos[x] + Cos[3*x])*Csc[x]*Sqrt[a*Sin[x]^2])/12","A",1
3,1,14,14,0.0044447,"\int \sqrt{a \sin ^2(x)} \, dx","Integrate[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",1
4,1,30,17,0.0146115,"\int \frac{1}{\sqrt{a \sin ^2(x)}} \, dx","Integrate[1/Sqrt[a*Sin[x]^2],x]","\frac{\sin (x) \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)}{\sqrt{a \sin ^2(x)}}","-\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{a \sin ^2(x)}}",1,"((-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x])/Sqrt[a*Sin[x]^2]","A",1
5,1,55,42,0.0619813,"\int \frac{1}{\left(a \sin ^2(x)\right)^{3/2}} \, dx","Integrate[(a*Sin[x]^2)^(-3/2),x]","-\frac{\sin ^3(x) \left(\csc ^2\left(\frac{x}{2}\right)-\sec ^2\left(\frac{x}{2}\right)-4 \log \left(\sin \left(\frac{x}{2}\right)\right)+4 \log \left(\cos \left(\frac{x}{2}\right)\right)\right)}{8 \left(a \sin ^2(x)\right)^{3/2}}","-\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,"-1/8*((Csc[x/2]^2 + 4*Log[Cos[x/2]] - 4*Log[Sin[x/2]] - Sec[x/2]^2)*Sin[x]^3)/(a*Sin[x]^2)^(3/2)","A",1
6,1,77,61,0.2074818,"\int \frac{1}{\left(a \sin ^2(x)\right)^{5/2}} \, dx","Integrate[(a*Sin[x]^2)^(-5/2),x]","-\frac{\csc (x) \sqrt{a \sin ^2(x)} \left(\csc ^4\left(\frac{x}{2}\right)+6 \csc ^2\left(\frac{x}{2}\right)-\sec ^4\left(\frac{x}{2}\right)-6 \sec ^2\left(\frac{x}{2}\right)+24 \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)\right)}{64 a^3}","-\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,"-1/64*(Csc[x]*(6*Csc[x/2]^2 + Csc[x/2]^4 + 24*(Log[Cos[x/2]] - Log[Sin[x/2]]) - 6*Sec[x/2]^2 - Sec[x/2]^4)*Sqrt[a*Sin[x]^2])/a^3","A",1
7,1,65,123,0.1748458,"\int \left(a \sin ^3(x)\right)^{5/2} \, dx","Integrate[(a*Sin[x]^3)^(5/2),x]","\frac{a \left(a \sin ^3(x)\right)^{3/2} \left(\sqrt{\sin (x)} (-15465 \cos (x)+3657 \cos (3 x)-749 \cos (5 x)+77 \cos (7 x))-12480 F\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)\right)}{36960 \sin ^{\frac{9}{2}}(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{2}{15} a^2 \sin ^5(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,"(a*(-12480*EllipticF[(Pi - 2*x)/4, 2] + (-15465*Cos[x] + 3657*Cos[3*x] - 749*Cos[5*x] + 77*Cos[7*x])*Sqrt[Sin[x]])*(a*Sin[x]^3)^(3/2))/(36960*Sin[x]^(9/2))","A",1
8,1,54,73,0.1010747,"\int \left(a \sin ^3(x)\right)^{3/2} \, dx","Integrate[(a*Sin[x]^3)^(3/2),x]","\frac{\left(a \sin ^3(x)\right)^{3/2} \left(\sqrt{\sin (x)} (5 \sin (4 x)-38 \sin (2 x))-168 E\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)\right)}{180 \sin ^{\frac{9}{2}}(x)}","-\frac{14}{45} a \cos (x) \sqrt{a \sin ^3(x)}-\frac{2}{9} a \sin ^2(x) \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,"((a*Sin[x]^3)^(3/2)*(-168*EllipticE[(Pi - 2*x)/4, 2] + Sqrt[Sin[x]]*(-38*Sin[2*x] + 5*Sin[4*x])))/(180*Sin[x]^(9/2))","A",1
9,1,41,50,0.0308453,"\int \sqrt{a \sin ^3(x)} \, dx","Integrate[Sqrt[a*Sin[x]^3],x]","-\frac{2 \sqrt{a \sin ^3(x)} \left(F\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)+\sqrt{\sin (x)} \cos (x)\right)}{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*(EllipticF[(Pi - 2*x)/4, 2] + Cos[x]*Sqrt[Sin[x]])*Sqrt[a*Sin[x]^3])/(3*Sin[x]^(3/2))","A",1
10,1,37,48,0.0241771,"\int \frac{1}{\sqrt{a \sin ^3(x)}} \, dx","Integrate[1/Sqrt[a*Sin[x]^3],x]","\frac{2 \sin ^{\frac{3}{2}}(x) E\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)-\sin (2 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*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(3/2) - Sin[2*x])/Sqrt[a*Sin[x]^3]","A",1
11,1,48,77,0.06841,"\int \frac{1}{\left(a \sin ^3(x)\right)^{3/2}} \, dx","Integrate[(a*Sin[x]^3)^(-3/2),x]","-\frac{2 \sin ^2(x) \left(3 \cot (x)+5 \sin (x) \cos (x)+5 \sin ^{\frac{5}{2}}(x) F\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)\right)}{21 \left(a \sin ^3(x)\right)^{3/2}}","-\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,"(-2*Sin[x]^2*(3*Cot[x] + 5*Cos[x]*Sin[x] + 5*EllipticF[(Pi - 2*x)/4, 2]*Sin[x]^(5/2)))/(21*(a*Sin[x]^3)^(3/2))","A",1
12,1,60,123,0.216076,"\int \frac{1}{\left(a \sin ^3(x)\right)^{5/2}} \, dx","Integrate[(a*Sin[x]^3)^(-5/2),x]","-\frac{2 \left(231 \sin (x) \cos (x)+\cot (x) \left(45 \csc ^4(x)+55 \csc ^2(x)+77\right)-231 \sin ^{\frac{3}{2}}(x) E\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)\right)}{585 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,"(-2*(Cot[x]*(77 + 55*Csc[x]^2 + 45*Csc[x]^4) + 231*Cos[x]*Sin[x] - 231*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(3/2)))/(585*a^2*Sqrt[a*Sin[x]^3])","A",1
13,1,53,132,0.1751146,"\int \left(a \sin ^4(x)\right)^{5/2} \, dx","Integrate[(a*Sin[x]^4)^(5/2),x]","\frac{a (2520 x-2100 \sin (2 x)+600 \sin (4 x)-150 \sin (6 x)+25 \sin (8 x)-2 \sin (10 x)) \csc ^6(x) \left(a \sin ^4(x)\right)^{3/2}}{10240}","-\frac{21}{128} a^2 \sin (x) \cos (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{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,"(a*Csc[x]^6*(a*Sin[x]^4)^(3/2)*(2520*x - 2100*Sin[2*x] + 600*Sin[4*x] - 150*Sin[6*x] + 25*Sin[8*x] - 2*Sin[10*x]))/10240","A",1
14,1,38,78,0.0985658,"\int \left(a \sin ^4(x)\right)^{3/2} \, dx","Integrate[(a*Sin[x]^4)^(3/2),x]","-\frac{1}{192} (-60 x+45 \sin (2 x)-9 \sin (4 x)+\sin (6 x)) \csc ^6(x) \left(a \sin ^4(x)\right)^{3/2}","-\frac{5}{24} a \sin (x) \cos (x) \sqrt{a \sin ^4(x)}-\frac{1}{6} a \sin ^3(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,"-1/192*(Csc[x]^6*(a*Sin[x]^4)^(3/2)*(-60*x + 45*Sin[2*x] - 9*Sin[4*x] + Sin[6*x]))","A",1
15,1,25,36,0.0168173,"\int \sqrt{a \sin ^4(x)} \, dx","Integrate[Sqrt[a*Sin[x]^4],x]","\frac{1}{2} \csc (x) \sqrt{a \sin ^4(x)} (x \csc (x)-\cos (x))","\frac{1}{2} x \csc ^2(x) \sqrt{a \sin ^4(x)}-\frac{1}{2} \cot (x) \sqrt{a \sin ^4(x)}",1,"(Csc[x]*(-Cos[x] + x*Csc[x])*Sqrt[a*Sin[x]^4])/2","A",1
16,1,16,16,0.0058844,"\int \frac{1}{\sqrt{a \sin ^4(x)}} \, dx","Integrate[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",1
17,1,34,68,0.0331234,"\int \frac{1}{\left(a \sin ^4(x)\right)^{3/2}} \, dx","Integrate[(a*Sin[x]^4)^(-3/2),x]","-\frac{\sin ^5(x) \cos (x) \left(3 \csc ^4(x)+4 \csc ^2(x)+8\right)}{15 \left(a \sin ^4(x)\right)^{3/2}}","-\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,"-1/15*(Cos[x]*(8 + 4*Csc[x]^2 + 3*Csc[x]^4)*Sin[x]^5)/(a*Sin[x]^4)^(3/2)","A",1
18,1,47,118,0.0524927,"\int \frac{1}{\left(a \sin ^4(x)\right)^{5/2}} \, dx","Integrate[(a*Sin[x]^4)^(-5/2),x]","-\frac{\sin (x) \cos (x) \left(35 \csc ^8(x)+40 \csc ^6(x)+48 \csc ^4(x)+64 \csc ^2(x)+128\right)}{315 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,"-1/315*(Cos[x]*(128 + 64*Csc[x]^2 + 48*Csc[x]^4 + 40*Csc[x]^6 + 35*Csc[x]^8)*Sin[x])/(a^2*Sqrt[a*Sin[x]^4])","A",1
19,1,74,89,0.1695678,"\int \left(c \sin ^m(a+b x)\right)^{5/2} \, dx","Integrate[(c*Sin[a + b*x]^m)^(5/2),x]","\frac{2 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \left(c \sin ^m(a+b x)\right)^{5/2} \, _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)}","\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*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (2 + 5*m)/4, (6 + 5*m)/4, Sin[a + b*x]^2]*(c*Sin[a + b*x]^m)^(5/2)*Tan[a + b*x])/(b*(2 + 5*m))","A",1
20,1,72,83,0.1143127,"\int \left(c \sin ^m(a+b x)\right)^{3/2} \, dx","Integrate[(c*Sin[a + b*x]^m)^(3/2),x]","\frac{2 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \left(c \sin ^m(a+b x)\right)^{3/2} \, _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)}","\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*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (2 + 3*m)/4, (3*(2 + m))/4, Sin[a + b*x]^2]*(c*Sin[a + b*x]^m)^(3/2)*Tan[a + b*x])/(b*(2 + 3*m))","A",1
21,1,68,74,0.0681474,"\int \sqrt{c \sin ^m(a+b x)} \, dx","Integrate[Sqrt[c*Sin[a + b*x]^m],x]","\frac{2 \sqrt{\cos ^2(a+b x)} \tan (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)}","\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*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (2 + m)/4, (6 + m)/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]^m]*Tan[a + b*x])/(b*(2 + m))","A",1
22,1,72,80,0.0759878,"\int \frac{1}{\sqrt{c \sin ^m(a+b x)}} \, dx","Integrate[1/Sqrt[c*Sin[a + b*x]^m],x]","-\frac{2 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left(\frac{1}{2},\frac{2-m}{4};\frac{6-m}{4};\sin ^2(a+b x)\right)}{b (m-2) \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*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (2 - m)/4, (6 - m)/4, Sin[a + b*x]^2]*Tan[a + b*x])/(b*(-2 + m)*Sqrt[c*Sin[a + b*x]^m])","A",1
23,1,71,89,0.1079444,"\int \frac{1}{\left(c \sin ^m(a+b x)\right)^{3/2}} \, dx","Integrate[(c*Sin[a + b*x]^m)^(-3/2),x]","\frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2-3 m);-\frac{3}{4} (m-2);\sin ^2(a+b x)\right)}{\left(b-\frac{3 b m}{2}\right) \left(c \sin ^m(a+b x)\right)^{3/2}}","\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,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (2 - 3*m)/4, (-3*(-2 + m))/4, Sin[a + b*x]^2]*Tan[a + b*x])/((b - (3*b*m)/2)*(c*Sin[a + b*x]^m)^(3/2))","A",1
24,1,73,89,0.1065135,"\int \frac{1}{\left(c \sin ^m(a+b x)\right)^{5/2}} \, dx","Integrate[(c*Sin[a + b*x]^m)^(-5/2),x]","\frac{\sqrt{\cos ^2(a+b x)} \tan (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)}{\left(b-\frac{5 b m}{2}\right) \left(c \sin ^m(a+b x)\right)^{5/2}}","\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,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (2 - 5*m)/4, (6 - 5*m)/4, Sin[a + b*x]^2]*Tan[a + b*x])/((b - (5*b*m)/2)*(c*Sin[a + b*x]^m)^(5/2))","A",1
25,1,71,77,0.0580943,"\int \left(b \sin ^n(c+d x)\right)^p \, dx","Integrate[(b*Sin[c + d*x]^n)^p,x]","\frac{\sqrt{\cos ^2(c+d x)} \tan (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)}","\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,"(Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[c + d*x]^2]*(b*Sin[c + d*x]^n)^p*Tan[c + d*x])/(d*(1 + n*p))","A",1
26,1,61,77,0.0718335,"\int \left(c \sin ^2(a+b x)\right)^p \, dx","Integrate[(c*Sin[a + b*x]^2)^p,x]","\frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) \left(c \sin ^2(a+b x)\right)^p \, _2F_1\left(\frac{1}{2},p+\frac{1}{2};p+\frac{3}{2};\sin ^2(a+b x)\right)}{2 b p+b}","\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,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, 1/2 + p, 3/2 + p, Sin[a + b*x]^2]*(c*Sin[a + b*x]^2)^p*Tan[a + b*x])/(b + 2*b*p)","A",1
27,1,67,75,0.0735286,"\int \left(c \sin ^3(a+b x)\right)^p \, dx","Integrate[(c*Sin[a + b*x]^3)^p,x]","\frac{\sqrt{\cos ^2(a+b x)} \tan (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)}{3 b p+b}","\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,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (1 + 3*p)/2, (3*(1 + p))/2, Sin[a + b*x]^2]*(c*Sin[a + b*x]^3)^p*Tan[a + b*x])/(b + 3*b*p)","A",1
28,1,65,77,0.0835755,"\int \left(c \sin ^4(a+b x)\right)^p \, dx","Integrate[(c*Sin[a + b*x]^4)^p,x]","\frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) \left(c \sin ^4(a+b x)\right)^p \, _2F_1\left(\frac{1}{2},2 p+\frac{1}{2};2 p+\frac{3}{2};\sin ^2(a+b x)\right)}{4 b p+b}","\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,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, 1/2 + 2*p, 3/2 + 2*p, Sin[a + b*x]^2]*(c*Sin[a + b*x]^4)^p*Tan[a + b*x])/(b + 4*b*p)","A",1
29,1,25,25,0.0325016,"\int \left(c \sin ^n(a+b x)\right)^{\frac{1}{n}} \, dx","Integrate[(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",1
30,1,73,79,0.0513625,"\int \left(a (b \sin (c+d x))^p\right)^n \, dx","Integrate[(a*(b*Sin[c + d*x])^p)^n,x]","\frac{\sqrt{\cos ^2(c+d x)} \tan (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)}","\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,"(Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[c + d*x]^2]*(a*(b*Sin[c + d*x])^p)^n*Tan[c + d*x])/(d*(1 + n*p))","A",1
31,1,16,16,0.0029621,"\int \left(a-a \sin ^2(x)\right) \, dx","Integrate[a - a*Sin[x]^2,x]","a \left(\frac{x}{2}+\frac{1}{4} \sin (2 x)\right)","\frac{a x}{2}+\frac{1}{2} a \sin (x) \cos (x)",1,"a*(x/2 + Sin[2*x]/4)","A",1
32,1,26,33,0.0030345,"\int \left(a-a \sin ^2(x)\right)^2 \, dx","Integrate[(a - a*Sin[x]^2)^2,x]","a^2 \left(\frac{3 x}{8}+\frac{1}{4} \sin (2 x)+\frac{1}{32} \sin (4 x)\right)","\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,"a^2*((3*x)/8 + Sin[2*x]/4 + Sin[4*x]/32)","A",1
33,1,34,46,0.0030201,"\int \left(a-a \sin ^2(x)\right)^3 \, dx","Integrate[(a - a*Sin[x]^2)^3,x]","a^3 \left(\frac{5 x}{16}+\frac{15}{64} \sin (2 x)+\frac{3}{64} \sin (4 x)+\frac{1}{192} \sin (6 x)\right)","\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,"a^3*((5*x)/16 + (15*Sin[2*x])/64 + (3*Sin[4*x])/64 + Sin[6*x]/192)","A",1
34,1,42,59,0.0031666,"\int \left(a-a \sin ^2(x)\right)^4 \, dx","Integrate[(a - a*Sin[x]^2)^4,x]","a^4 \left(\frac{35 x}{128}+\frac{7}{32} \sin (2 x)+\frac{7}{128} \sin (4 x)+\frac{1}{96} \sin (6 x)+\frac{\sin (8 x)}{1024}\right)","\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,"a^4*((35*x)/128 + (7*Sin[2*x])/32 + (7*Sin[4*x])/128 + Sin[6*x]/96 + Sin[8*x]/1024)","A",1
35,1,58,62,0.0583462,"\int \frac{\sin ^7(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^7/(a - a*Sin[c + d*x]^2),x]","\frac{\frac{19 \cos (c+d x)}{8 d}-\frac{3 \cos (3 (c+d x))}{16 d}+\frac{\cos (5 (c+d x))}{80 d}+\frac{\sec (c+d x)}{d}}{a}","\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,"((19*Cos[c + d*x])/(8*d) - (3*Cos[3*(c + d*x)])/(16*d) + Cos[5*(c + d*x)]/(80*d) + Sec[c + d*x]/d)/a","A",1
36,1,43,46,0.0369415,"\int \frac{\sin ^5(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^5/(a - a*Sin[c + d*x]^2),x]","\frac{\frac{7 \cos (c+d x)}{4 d}-\frac{\cos (3 (c+d x))}{12 d}+\frac{\sec (c+d x)}{d}}{a}","-\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,"((7*Cos[c + d*x])/(4*d) - Cos[3*(c + d*x)]/(12*d) + Sec[c + d*x]/d)/a","A",1
37,1,25,27,0.031625,"\int \frac{\sin ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^3/(a - a*Sin[c + d*x]^2),x]","\frac{\frac{\cos (c+d x)}{d}+\frac{\sec (c+d x)}{d}}{a}","\frac{\cos (c+d x)}{a d}+\frac{\sec (c+d x)}{a d}",1,"(Cos[c + d*x]/d + Sec[c + d*x]/d)/a","A",1
38,1,13,13,0.0123811,"\int \frac{\sin (c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[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",1
39,1,46,29,0.038921,"\int \frac{\csc (c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]/(a - a*Sin[c + d*x]^2),x]","\frac{\frac{\sec (c+d x)}{d}+\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}}{a}","\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"(-(Log[Cos[(c + d*x)/2]]/d) + Log[Sin[(c + d*x)/2]]/d + Sec[c + d*x]/d)/a","A",1
40,1,146,58,0.2622505,"\int \frac{\csc ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]^3/(a - a*Sin[c + d*x]^2),x]","\frac{\csc ^4(c+d x) \left(-6 \cos (2 (c+d x))+2 \cos (3 (c+d x))+3 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-3 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\cos (c+d x) \left(3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2\right)+2\right)}{2 a d \left(\csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^2\left(\frac{1}{2} (c+d x)\right)\right)}","\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,"(Csc[c + d*x]^4*(2 - 6*Cos[2*(c + d*x)] + 2*Cos[3*(c + d*x)] + 3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 3*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] + Cos[c + d*x]*(-2 - 3*Log[Cos[(c + d*x)/2]] + 3*Log[Sin[(c + d*x)/2]])))/(2*a*d*(Csc[(c + d*x)/2]^2 - Sec[(c + d*x)/2]^2))","B",1
41,1,132,82,4.2621668,"\int \frac{\csc ^5(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]^5/(a - a*Sin[c + d*x]^2),x]","-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right)+14 \csc ^2\left(\frac{1}{2} (c+d x)\right)+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-14 \tan ^2\left(\frac{1}{2} (c+d x)\right)+\cos (c+d x) \left(\sec ^4\left(\frac{1}{2} (c+d x)\right)-8 \left(-15 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+8\right)\right)+78\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)-1}}{64 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,"-1/64*(14*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^4 + (Sec[(c + d*x)/2]^2*(78 + Cos[c + d*x]*(-8*(8 + 15*Log[Cos[(c + d*x)/2]] - 15*Log[Sin[(c + d*x)/2]]) + Sec[(c + d*x)/2]^4) - 14*Tan[(c + d*x)/2]^2))/(-1 + Tan[(c + d*x)/2]^2))/(a*d)","A",1
42,1,44,73,0.1864327,"\int \frac{\sin ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^6/(a - a*Sin[c + d*x]^2),x]","-\frac{-16 \sin (2 (c+d x))+\sin (4 (c+d x))-32 \tan (c+d x)+60 c+60 d x}{32 a d}","\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,"-1/32*(60*c + 60*d*x - 16*Sin[2*(c + d*x)] + Sin[4*(c + d*x)] - 32*Tan[c + d*x])/(a*d)","A",1
43,1,34,49,0.1264552,"\int \frac{\sin ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^4/(a - a*Sin[c + d*x]^2),x]","\frac{-6 (c+d x)+\sin (2 (c+d x))+4 \tan (c+d x)}{4 a d}","\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,"(-6*(c + d*x) + Sin[2*(c + d*x)] + 4*Tan[c + d*x])/(4*a*d)","A",1
44,1,27,20,0.0120577,"\int \frac{\sin ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^2/(a - a*Sin[c + d*x]^2),x]","\frac{\frac{\tan (c+d x)}{d}-\frac{\tan ^{-1}(\tan (c+d x))}{d}}{a}","\frac{\tan (c+d x)}{a d}-\frac{x}{a}",1,"(-(ArcTan[Tan[c + d*x]]/d) + Tan[c + d*x]/d)/a","A",1
45,1,13,13,0.005823,"\int \frac{1}{a-a \sin ^2(c+d x)} \, dx","Integrate[(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",1
46,1,16,28,0.0272572,"\int \frac{\csc ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]^2/(a - a*Sin[c + d*x]^2),x]","-\frac{2 \cot (2 (c+d x))}{a d}","\frac{\tan (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}",1,"(-2*Cot[2*(c + d*x)])/(a*d)","A",1
47,1,49,46,0.043784,"\int \frac{\csc ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]^4/(a - a*Sin[c + d*x]^2),x]","\frac{\frac{\tan (c+d x)}{d}-\frac{5 \cot (c+d x)}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 d}}{a}","\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,"((-5*Cot[c + d*x])/(3*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) + Tan[c + d*x]/d)/a","A",1
48,1,70,62,0.0372085,"\int \frac{\csc ^6(c+d x)}{a-a \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]^6/(a - a*Sin[c + d*x]^2),x]","\frac{\frac{\tan (c+d x)}{d}-\frac{11 \cot (c+d x)}{5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{3 \cot (c+d x) \csc ^2(c+d x)}{5 d}}{a}","\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,"((-11*Cot[c + d*x])/(5*d) - (3*Cot[c + d*x]*Csc[c + d*x]^2)/(5*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*d) + Tan[c + d*x]/d)/a","A",1
49,1,59,65,0.0497528,"\int \frac{\sin ^7(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^7/(a - a*Sin[c + d*x]^2)^2,x]","\frac{-\frac{11 \cos (c+d x)}{4 d}+\frac{\cos (3 (c+d x))}{12 d}+\frac{\sec ^3(c+d x)}{3 d}-\frac{3 \sec (c+d x)}{d}}{a^2}","\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,"((-11*Cos[c + d*x])/(4*d) + Cos[3*(c + d*x)]/(12*d) - (3*Sec[c + d*x])/d + Sec[c + d*x]^3/(3*d))/a^2","A",1
50,1,42,47,0.0326383,"\int \frac{\sin ^5(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^5/(a - a*Sin[c + d*x]^2)^2,x]","\frac{-\frac{\cos (c+d x)}{d}+\frac{\sec ^3(c+d x)}{3 d}-\frac{2 \sec (c+d x)}{d}}{a^2}","-\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]/d) - (2*Sec[c + d*x])/d + Sec[c + d*x]^3/(3*d))/a^2","A",1
51,1,31,33,0.0314309,"\int \frac{\sin ^3(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^3/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\frac{\sec ^3(c+d x)}{3 d}-\frac{\sec (c+d x)}{d}}{a^2}","\frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{\sec (c+d x)}{a^2 d}",1,"(-(Sec[c + d*x]/d) + Sec[c + d*x]^3/(3*d))/a^2","A",1
52,1,18,18,0.0125466,"\int \frac{\sin (c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[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",1
53,1,61,47,0.0392494,"\int \frac{\csc (c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[Csc[c + d*x]/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\frac{\sec ^3(c+d x)}{3 d}+\frac{\sec (c+d x)}{d}+\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}}{a^2}","\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,"(-(Log[Cos[(c + d*x)/2]]/d) + Log[Sin[(c + d*x)/2]]/d + Sec[c + d*x]/d + Sec[c + d*x]^3/(3*d))/a^2","A",1
54,1,208,78,0.4407001,"\int \frac{\csc ^3(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[Csc[c + d*x]^3/(a - a*Sin[c + d*x]^2)^2,x]","\frac{2 \csc ^8(c+d x) \left(-40 \cos (2 (c+d x))+13 \cos (3 (c+d x))-30 \cos (4 (c+d x))+13 \cos (5 (c+d x))+15 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+15 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-15 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-15 \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\cos (c+d x) \left(30 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-30 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-26\right)+22\right)}{3 a^2 d \left(\csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^3}","\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,"(2*Csc[c + d*x]^8*(22 - 40*Cos[2*(c + d*x)] + 13*Cos[3*(c + d*x)] - 30*Cos[4*(c + d*x)] + 13*Cos[5*(c + d*x)] + 15*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 15*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 15*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 15*Cos[5*(c + d*x)]*Log[Sin[(c + d*x)/2]] + Cos[c + d*x]*(-26 - 30*Log[Cos[(c + d*x)/2]] + 30*Log[Sin[(c + d*x)/2]])))/(3*a^2*d*(Csc[(c + d*x)/2]^2 - Sec[(c + d*x)/2]^2)^3)","B",1
55,1,46,69,0.2160128,"\int \frac{\sin ^6(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^6/(a - a*Sin[c + d*x]^2)^2,x]","\frac{30 (c+d x)-3 \sin (2 (c+d x))+4 \tan (c+d x) \left(\sec ^2(c+d x)-7\right)}{12 a^2 d}","\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,"(30*(c + d*x) - 3*Sin[2*(c + d*x)] + 4*(-7 + Sec[c + d*x]^2)*Tan[c + d*x])/(12*a^2*d)","A",1
56,1,42,38,0.0149562,"\int \frac{\sin ^4(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^4/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\frac{\tan ^{-1}(\tan (c+d x))}{d}+\frac{\tan ^3(c+d x)}{3 d}-\frac{\tan (c+d x)}{d}}{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,"(ArcTan[Tan[c + d*x]]/d - Tan[c + d*x]/d + Tan[c + d*x]^3/(3*d))/a^2","A",1
57,1,18,18,0.0163244,"\int \frac{\sin ^2(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[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",1
58,1,26,32,0.039122,"\int \frac{1}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[(a - a*Sin[c + d*x]^2)^(-2),x]","\frac{\frac{1}{3} \tan ^3(c+d x)+\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] + Tan[c + d*x]^3/3)/(a^2*d)","A",1
59,1,50,47,0.0422044,"\int \frac{\csc ^2(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[Csc[c + d*x]^2/(a - a*Sin[c + d*x]^2)^2,x]","\frac{\frac{5 \tan (c+d x)}{3 d}-\frac{\cot (c+d x)}{d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 d}}{a^2}","\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]/d) + (5*Tan[c + d*x])/(3*d) + (Sec[c + d*x]^2*Tan[c + d*x])/(3*d))/a^2","A",1
60,1,46,65,0.0266701,"\int \frac{\csc ^4(c+d x)}{\left(a-a \sin ^2(c+d x)\right)^2} \, dx","Integrate[Csc[c + d*x]^4/(a - a*Sin[c + d*x]^2)^2,x]","\frac{16 \left(-\frac{\cot (2 (c+d x))}{3 d}-\frac{\cot (2 (c+d x)) \csc ^2(2 (c+d x))}{6 d}\right)}{a^2}","\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,"(16*(-1/3*Cot[2*(c + d*x)]/d - (Cot[2*(c + d*x)]*Csc[2*(c + d*x)]^2)/(6*d)))/a^2","A",1
61,1,31,29,0.0051622,"\int \frac{1}{\left(a-a \sin ^2(x)\right)^3} \, dx","Integrate[(a - a*Sin[x]^2)^(-3),x]","\frac{\frac{8 \tan (x)}{15}+\frac{1}{5} \tan (x) \sec ^4(x)+\frac{4}{15} \tan (x) \sec ^2(x)}{a^3}","\frac{\tan ^5(x)}{5 a^3}+\frac{2 \tan ^3(x)}{3 a^3}+\frac{\tan (x)}{a^3}",1,"((8*Tan[x])/15 + (4*Sec[x]^2*Tan[x])/15 + (Sec[x]^4*Tan[x])/5)/a^3","A",1
62,1,41,37,0.0055182,"\int \frac{1}{\left(a-a \sin ^2(x)\right)^4} \, dx","Integrate[(a - a*Sin[x]^2)^(-4),x]","\frac{\frac{16 \tan (x)}{35}+\frac{1}{7} \tan (x) \sec ^6(x)+\frac{6}{35} \tan (x) \sec ^4(x)+\frac{8}{35} \tan (x) \sec ^2(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,"((16*Tan[x])/35 + (8*Sec[x]^2*Tan[x])/35 + (6*Sec[x]^4*Tan[x])/35 + (Sec[x]^6*Tan[x])/7)/a^4","A",1
63,1,51,51,0.0059714,"\int \frac{1}{\left(a-a \sin ^2(x)\right)^5} \, dx","Integrate[(a - a*Sin[x]^2)^(-5),x]","\frac{\frac{128 \tan (x)}{315}+\frac{1}{9} \tan (x) \sec ^8(x)+\frac{8}{63} \tan (x) \sec ^6(x)+\frac{16}{105} \tan (x) \sec ^4(x)+\frac{64}{315} \tan (x) \sec ^2(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,"((128*Tan[x])/315 + (64*Sec[x]^2*Tan[x])/315 + (16*Sec[x]^4*Tan[x])/105 + (8*Sec[x]^6*Tan[x])/63 + (Sec[x]^8*Tan[x])/9)/a^5","A",1
64,1,77,51,0.0295024,"\int \sin ^3(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Integrate[Sin[c + d*x]^3*(a + b*Sin[c + d*x]^2),x]","-\frac{3 a \cos (c+d x)}{4 d}+\frac{a \cos (3 (c+d x))}{12 d}-\frac{5 b \cos (c+d x)}{8 d}+\frac{5 b \cos (3 (c+d x))}{48 d}-\frac{b \cos (5 (c+d x))}{80 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,"(-3*a*Cos[c + d*x])/(4*d) - (5*b*Cos[c + d*x])/(8*d) + (a*Cos[3*(c + d*x)])/(12*d) + (5*b*Cos[3*(c + d*x)])/(48*d) - (b*Cos[5*(c + d*x)])/(80*d)","A",1
65,1,54,31,0.0212136,"\int \sin (c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Integrate[Sin[c + d*x]*(a + b*Sin[c + d*x]^2),x]","\frac{a \sin (c) \sin (d x)}{d}-\frac{a \cos (c) \cos (d x)}{d}-\frac{3 b \cos (c+d x)}{4 d}+\frac{b \cos (3 (c+d x))}{12 d}","\frac{b \cos ^3(c+d x)}{3 d}-\frac{(a+b) \cos (c+d x)}{d}",1,"-((a*Cos[c]*Cos[d*x])/d) - (3*b*Cos[c + d*x])/(4*d) + (b*Cos[3*(c + d*x)])/(12*d) + (a*Sin[c]*Sin[d*x])/d","A",1
66,1,63,26,0.0241548,"\int \csc (c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Integrate[Csc[c + d*x]*(a + b*Sin[c + d*x]^2),x]","\frac{a \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}+\frac{b \sin (c) \sin (d x)}{d}-\frac{b \cos (c) \cos (d x)}{d}","-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b \cos (c+d x)}{d}",1,"-((b*Cos[c]*Cos[d*x])/d) - (a*Log[Cos[c/2 + (d*x)/2]])/d + (a*Log[Sin[c/2 + (d*x)/2]])/d + (b*Sin[c]*Sin[d*x])/d","B",1
67,1,118,40,0.0419614,"\int \csc ^3(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Integrate[Csc[c + d*x]^3*(a + b*Sin[c + d*x]^2),x]","-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{b \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{b \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{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,"-1/8*(a*Csc[(c + d*x)/2]^2)/d - (b*Log[Cos[c/2 + (d*x)/2]])/d - (a*Log[Cos[(c + d*x)/2]])/(2*d) + (b*Log[Sin[c/2 + (d*x)/2]])/d + (a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)","B",1
68,1,70,89,0.1079759,"\int \sin ^4(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Integrate[Sin[c + d*x]^4*(a + b*Sin[c + d*x]^2),x]","\frac{-3 (16 a+15 b) \sin (2 (c+d x))+(6 a+9 b) \sin (4 (c+d x))+72 a c+72 a d x-b \sin (6 (c+d x))+60 b c+60 b d x}{192 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,"(72*a*c + 60*b*c + 72*a*d*x + 60*b*d*x - 3*(16*a + 15*b)*Sin[2*(c + d*x)] + (6*a + 9*b)*Sin[4*(c + d*x)] - b*Sin[6*(c + d*x)])/(192*d)","A",1
69,1,45,61,0.0938799,"\int \sin ^2(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Integrate[Sin[c + d*x]^2*(a + b*Sin[c + d*x]^2),x]","\frac{4 (4 a+3 b) (c+d x)-8 (a+b) \sin (2 (c+d x))+b \sin (4 (c+d x))}{32 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*(4*a + 3*b)*(c + d*x) - 8*(a + b)*Sin[2*(c + d*x)] + b*Sin[4*(c + d*x)])/(32*d)","A",1
70,1,33,30,0.030746,"\int \left(a+b \sin ^2(c+d x)\right) \, dx","Integrate[a + b*Sin[c + d*x]^2,x]","a x+\frac{b (c+d x)}{2 d}-\frac{b \sin (2 (c+d x))}{4 d}","a x-\frac{b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b x}{2}",1,"a*x + (b*(c + d*x))/(2*d) - (b*Sin[2*(c + d*x)])/(4*d)","A",1
71,1,16,16,0.0187517,"\int \csc ^2(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Integrate[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",1
72,1,49,43,0.02774,"\int \csc ^4(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Integrate[Csc[c + d*x]^4*(a + b*Sin[c + d*x]^2),x]","-\frac{2 a \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac{b \cot (c+d x)}{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*Cot[c + d*x])/(3*d) - (b*Cot[c + d*x])/d - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d)","A",1
73,1,95,65,0.0293324,"\int \csc ^6(c+d x) \left(a+b \sin ^2(c+d x)\right) \, dx","Integrate[Csc[c + d*x]^6*(a + b*Sin[c + d*x]^2),x]","-\frac{8 a \cot (c+d x)}{15 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{4 a \cot (c+d x) \csc ^2(c+d x)}{15 d}-\frac{2 b \cot (c+d x)}{3 d}-\frac{b \cot (c+d x) \csc ^2(c+d x)}{3 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,"(-8*a*Cot[c + d*x])/(15*d) - (2*b*Cot[c + d*x])/(3*d) - (4*a*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) - (b*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d)","A",1
74,1,19,19,0.0042923,"\int \left(a+b \sin ^2(x)\right) \, dx","Integrate[a + b*Sin[x]^2,x]","a x+\frac{b x}{2}-\frac{1}{4} b \sin (2 x)","a x+\frac{b x}{2}-\frac{1}{2} b \sin (x) \cos (x)",1,"a*x + (b*x)/2 - (b*Sin[2*x])/4","A",1
75,1,43,50,0.0613171,"\int \left(a+b \sin ^2(x)\right)^2 \, dx","Integrate[(a + b*Sin[x]^2)^2,x]","\frac{1}{32} \left(4 x \left(8 a^2+8 a b+3 b^2\right)-8 b (2 a+b) \sin (2 x)+b^2 \sin (4 x)\right)","\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,"(4*(8*a^2 + 8*a*b + 3*b^2)*x - 8*b*(2*a + b)*Sin[2*x] + b^2*Sin[4*x])/32","A",1
76,1,80,87,0.1024187,"\int \left(a+b \sin ^2(x)\right)^3 \, dx","Integrate[(a + b*Sin[x]^2)^3,x]","\frac{1}{192} \left(12 x (2 a+b) \left(8 a^2+8 a b+5 b^2\right)+9 b^2 (2 a+b) \sin (4 x)+9 i b (4 i a+(1+2 i) b) (4 a+(2+i) b) \sin (2 x)+b^3 (-\sin (6 x))\right)","\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,"(12*(2*a + b)*(8*a^2 + 8*a*b + 5*b^2)*x + (9*I)*b*((4*I)*a + (1 + 2*I)*b)*(4*a + (2 + I)*b)*Sin[2*x] + 9*b^2*(2*a + b)*Sin[4*x] - b^3*Sin[6*x])/192","C",1
77,1,113,140,0.1554438,"\int \left(a+b \sin ^2(x)\right)^4 \, dx","Integrate[(a + b*Sin[x]^2)^4,x]","\frac{24 b^2 \left(24 a^2+24 a b+7 b^2\right) \sin (4 x)-96 b (2 a+b) \left(16 a^2+16 a b+7 b^2\right) \sin (2 x)+24 x \left(128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right)-32 b^3 (2 a+b) \sin (6 x)+3 b^4 \sin (8 x)}{3072}","-\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(608 a^3+808 a^2 b+480 a b^2+105 b^3\right) \sin (x) \cos (x)+\frac{1}{128} x \left(128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right)-\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,"(24*(128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*x - 96*b*(2*a + b)*(16*a^2 + 16*a*b + 7*b^2)*Sin[2*x] + 24*b^2*(24*a^2 + 24*a*b + 7*b^2)*Sin[4*x] - 32*b^3*(2*a + b)*Sin[6*x] + 3*b^4*Sin[8*x])/3072","A",1
78,1,180,106,1.4405241,"\int \frac{\sin ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^7/(a + b*Sin[c + d*x]^2),x]","\frac{-240 a^3 \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)-240 a^3 \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)-2 \sqrt{b} \sqrt{-a-b} \cos (c+d x) \left(120 a^2+4 b (5 a-7 b) \cos (2 (c+d x))-100 a b+3 b^2 \cos (4 (c+d x))+89 b^2\right)}{240 b^{7/2} d \sqrt{-a-b}}","\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{\left(a^2-a b+b^2\right) \cos (c+d x)}{b^3 d}-\frac{(a-2 b) \cos ^3(c+d x)}{3 b^2 d}-\frac{\cos ^5(c+d x)}{5 b d}",1,"(-240*a^3*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] - 240*a^3*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] - 2*Sqrt[-a - b]*Sqrt[b]*Cos[c + d*x]*(120*a^2 - 100*a*b + 89*b^2 + 4*(5*a - 7*b)*b*Cos[2*(c + d*x)] + 3*b^2*Cos[4*(c + d*x)]))/(240*Sqrt[-a - b]*b^(7/2)*d)","C",1
79,1,150,77,0.5055178,"\int \frac{\sin ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^5/(a + b*Sin[c + d*x]^2),x]","\frac{6 a^2 \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)+6 a^2 \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)+\sqrt{b} \sqrt{-a-b} \cos (c+d x) (6 a+b \cos (2 (c+d x))-5 b)}{6 b^{5/2} d \sqrt{-a-b}}","-\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,"(6*a^2*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] + 6*a^2*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] + Sqrt[-a - b]*Sqrt[b]*Cos[c + d*x]*(6*a - 5*b + b*Cos[2*(c + d*x)]))/(6*Sqrt[-a - b]*b^(5/2)*d)","C",1
80,1,125,52,0.2426459,"\int \frac{\sin ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^3/(a + b*Sin[c + d*x]^2),x]","-\frac{\sqrt{b} \sqrt{-a-b} \cos (c+d x)+a \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)+a \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{b^{3/2} d \sqrt{-a-b}}","\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*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] + a*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] + Sqrt[-a - b]*Sqrt[b]*Cos[c + d*x])/(Sqrt[-a - b]*b^(3/2)*d))","C",1
81,1,97,37,0.136511,"\int \frac{\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]/(a + b*Sin[c + d*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)+\tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\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,"(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/(Sqrt[-a - b]*Sqrt[b]*d)","C",1
82,1,143,55,0.2910734,"\int \frac{\csc (c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]/(a + b*Sin[c + d*x]^2),x]","-\frac{\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{\sqrt{-a-b}}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{\sqrt{-a-b}}-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a d}","\frac{\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,"-(((Sqrt[b]*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/Sqrt[-a - b] + (Sqrt[b]*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/Sqrt[-a - b] + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])/(a*d))","C",1
83,1,224,85,2.2098236,"\int \frac{\csc ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]^3/(a + b*Sin[c + d*x]^2),x]","-\frac{\csc ^2(c+d x) (2 a-b \cos (2 (c+d x))+b) \left(-8 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)-8 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)+\sqrt{-a-b} \left(4 (a-2 b) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+a \csc ^2\left(\frac{1}{2} (c+d x)\right)-a \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{16 a^2 d \sqrt{-a-b} \left(a \csc ^2(c+d x)+b\right)}","-\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,"-1/16*((2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x]^2*(-8*b^(3/2)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] - 8*b^(3/2)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]] + Sqrt[-a - b]*(a*Csc[(c + d*x)/2]^2 + 4*(a - 2*b)*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - a*Sec[(c + d*x)/2]^2)))/(a^2*Sqrt[-a - b]*d*(b + a*Csc[c + d*x]^2))","C",1
84,1,657,125,6.3040055,"\int \frac{\csc ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]^5/(a + b*Sin[c + d*x]^2),x]","\frac{b^{5/2} \csc ^2(c+d x) (-2 a+b \cos (2 (c+d x))-b) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(\sqrt{b} \cos \left(\frac{1}{2} (c+d x)\right)-i \sqrt{a} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{-a-b}}\right)}{2 a^3 d \sqrt{-a-b} \left(a \csc ^2(c+d x)+b\right)}+\frac{b^{5/2} \csc ^2(c+d x) (-2 a+b \cos (2 (c+d x))-b) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(\sqrt{b} \cos \left(\frac{1}{2} (c+d x)\right)+i \sqrt{a} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{-a-b}}\right)}{2 a^3 d \sqrt{-a-b} \left(a \csc ^2(c+d x)+b\right)}+\frac{(3 a-4 b) \csc ^2(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (-2 a+b \cos (2 (c+d x))-b)}{64 a^2 d \left(a \csc ^2(c+d x)+b\right)}+\frac{(4 b-3 a) \csc ^2(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (-2 a+b \cos (2 (c+d x))-b)}{64 a^2 d \left(a \csc ^2(c+d x)+b\right)}+\frac{\left(3 a^2-4 a b+8 b^2\right) \csc ^2(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (-2 a+b \cos (2 (c+d x))-b)}{16 a^3 d \left(a \csc ^2(c+d x)+b\right)}+\frac{\left(-3 a^2+4 a b-8 b^2\right) \csc ^2(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (-2 a+b \cos (2 (c+d x))-b)}{16 a^3 d \left(a \csc ^2(c+d x)+b\right)}+\frac{\csc ^2(c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right) (-2 a+b \cos (2 (c+d x))-b)}{128 a d \left(a \csc ^2(c+d x)+b\right)}-\frac{\csc ^2(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (-2 a+b \cos (2 (c+d x))-b)}{128 a d \left(a \csc ^2(c+d x)+b\right)}","\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{\left(3 a^2-4 a b+8 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}",1,"(b^(5/2)*ArcTan[(Sec[(c + d*x)/2]*(Sqrt[b]*Cos[(c + d*x)/2] - I*Sqrt[a]*Sin[(c + d*x)/2]))/Sqrt[-a - b]]*(-2*a - b + b*Cos[2*(c + d*x)])*Csc[c + d*x]^2)/(2*a^3*Sqrt[-a - b]*d*(b + a*Csc[c + d*x]^2)) + (b^(5/2)*ArcTan[(Sec[(c + d*x)/2]*(Sqrt[b]*Cos[(c + d*x)/2] + I*Sqrt[a]*Sin[(c + d*x)/2]))/Sqrt[-a - b]]*(-2*a - b + b*Cos[2*(c + d*x)])*Csc[c + d*x]^2)/(2*a^3*Sqrt[-a - b]*d*(b + a*Csc[c + d*x]^2)) + ((3*a - 4*b)*(-2*a - b + b*Cos[2*(c + d*x)])*Csc[(c + d*x)/2]^2*Csc[c + d*x]^2)/(64*a^2*d*(b + a*Csc[c + d*x]^2)) + ((-2*a - b + b*Cos[2*(c + d*x)])*Csc[(c + d*x)/2]^4*Csc[c + d*x]^2)/(128*a*d*(b + a*Csc[c + d*x]^2)) + ((3*a^2 - 4*a*b + 8*b^2)*(-2*a - b + b*Cos[2*(c + d*x)])*Csc[c + d*x]^2*Log[Cos[(c + d*x)/2]])/(16*a^3*d*(b + a*Csc[c + d*x]^2)) + ((-3*a^2 + 4*a*b - 8*b^2)*(-2*a - b + b*Cos[2*(c + d*x)])*Csc[c + d*x]^2*Log[Sin[(c + d*x)/2]])/(16*a^3*d*(b + a*Csc[c + d*x]^2)) + ((-3*a + 4*b)*(-2*a - b + b*Cos[2*(c + d*x)])*Csc[c + d*x]^2*Sec[(c + d*x)/2]^2)/(64*a^2*d*(b + a*Csc[c + d*x]^2)) - ((-2*a - b + b*Cos[2*(c + d*x)])*Csc[c + d*x]^2*Sec[(c + d*x)/2]^4)/(128*a*d*(b + a*Csc[c + d*x]^2))","C",1
85,1,133,163,1.18375,"\int \frac{\sin ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^8/(a + b*Sin[c + d*x]^2),x]","-\frac{-\frac{192 a^{7/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a+b}}+3 b \left(16 a^2-16 a b+15 b^2\right) \sin (2 (c+d x))+12 \left(16 a^3-8 a^2 b+6 a b^2-5 b^3\right) (c+d x)+3 b^2 (2 a-3 b) \sin (4 (c+d x))+b^3 \sin (6 (c+d x))}{192 b^4 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(16 a^3-8 a^2 b+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,"-1/192*(12*(16*a^3 - 8*a^2*b + 6*a*b^2 - 5*b^3)*(c + d*x) - (192*a^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/Sqrt[a + b] + 3*b*(16*a^2 - 16*a*b + 15*b^2)*Sin[2*(c + d*x)] + 3*(2*a - 3*b)*b^2*Sin[4*(c + d*x)] + b^3*Sin[6*(c + d*x)])/(b^4*d)","A",1
86,1,95,117,0.4598545,"\int \frac{\sin ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^6/(a + b*Sin[c + d*x]^2),x]","\frac{-\frac{32 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a+b}}+4 \left(8 a^2-4 a b+3 b^2\right) (c+d x)+8 b (a-b) \sin (2 (c+d x))+b^2 \sin (4 (c+d x))}{32 b^3 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,"(4*(8*a^2 - 4*a*b + 3*b^2)*(c + d*x) - (32*a^(5/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/Sqrt[a + b] + 8*(a - b)*b*Sin[2*(c + d*x)] + b^2*Sin[4*(c + d*x)])/(32*b^3*d)","A",1
87,1,69,77,0.3207568,"\int \frac{\sin ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^4/(a + b*Sin[c + d*x]^2),x]","-\frac{-\frac{4 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a+b}}+2 (2 a-b) (c+d x)+b \sin (2 (c+d x))}{4 b^2 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,"-1/4*(2*(2*a - b)*(c + d*x) - (4*a^(3/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/Sqrt[a + b] + b*Sin[2*(c + d*x)])/(b^2*d)","A",1
88,1,46,46,0.150283,"\int \frac{\sin ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Sin[c + d*x]^2),x]","\frac{-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a+b}}+c+d x}{b d}","\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,"(c + d*x - (Sqrt[a]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/Sqrt[a + b])/(b*d)","A",1
89,1,36,36,0.081429,"\int \frac{1}{a+b \sin ^2(c+d x)} \, dx","Integrate[(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",1
90,1,53,53,0.3077333,"\int \frac{\csc ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Sin[c + d*x]^2),x]","\frac{-\frac{b \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a+b}}-\sqrt{a} \cot (c+d x)}{a^{3/2} 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]])/Sqrt[a + b]) - Sqrt[a]*Cot[c + d*x])/(a^(3/2)*d)","A",1
91,1,119,77,0.6691933,"\int \frac{\csc ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]^4/(a + b*Sin[c + d*x]^2),x]","-\frac{\csc ^2(c+d x) (2 a-b \cos (2 (c+d x))+b) \left(\sqrt{a} \sqrt{a+b} \cot (c+d x) \left(a \csc ^2(c+d x)+2 a-3 b\right)-3 b^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)\right)}{6 a^{5/2} d \sqrt{a+b} \left(a \csc ^2(c+d x)+b\right)}","\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,"-1/6*((2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x]^2*(-3*b^2*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]] + Sqrt[a]*Sqrt[a + b]*Cot[c + d*x]*(2*a - 3*b + a*Csc[c + d*x]^2)))/(a^(5/2)*Sqrt[a + b]*d*(b + a*Csc[c + d*x]^2))","A",1
92,1,147,109,1.5157015,"\int \frac{\csc ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Csc[c + d*x]^6/(a + b*Sin[c + d*x]^2),x]","-\frac{\csc ^2(c+d x) (2 a-b \cos (2 (c+d x))+b) \left(\sqrt{a} \sqrt{a+b} \cot (c+d x) \left(3 a^2 \csc ^4(c+d x)+8 a^2+a (4 a-5 b) \csc ^2(c+d x)-10 a b+15 b^2\right)+15 b^3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)\right)}{30 a^{7/2} d \sqrt{a+b} \left(a \csc ^2(c+d x)+b\right)}","-\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{(2 a-b) \cot ^3(c+d x)}{3 a^2 d}-\frac{\left(a^2-a b+b^2\right) \cot (c+d x)}{a^3 d}-\frac{\cot ^5(c+d x)}{5 a d}",1,"-1/30*((2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x]^2*(15*b^3*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]] + Sqrt[a]*Sqrt[a + b]*Cot[c + d*x]*(8*a^2 - 10*a*b + 15*b^2 + a*(4*a - 5*b)*Csc[c + d*x]^2 + 3*a^2*Csc[c + d*x]^4)))/(a^(7/2)*Sqrt[a + b]*d*(b + a*Csc[c + d*x]^2))","A",1
93,1,137,140,1.7075542,"\int \frac{\csc ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[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{\cot (c+d x) \left(15 a^3 \csc ^6(c+d x)+48 a^3+a \left(24 a^2-28 a b+35 b^2\right) \csc ^2(c+d x)+3 a^2 (6 a-7 b) \csc ^4(c+d x)-56 a^2 b+70 a b^2-105 b^3\right)}{105 a^4 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{(3 a-b) \cot ^5(c+d x)}{5 a^2 d}-\frac{(a-b) \left(a^2+b^2\right) \cot (c+d x)}{a^4 d}-\frac{\left(3 a^2-2 a b+b^2\right) \cot ^3(c+d x)}{3 a^3 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) - (Cot[c + d*x]*(48*a^3 - 56*a^2*b + 70*a*b^2 - 105*b^3 + a*(24*a^2 - 28*a*b + 35*b^2)*Csc[c + d*x]^2 + 3*a^2*(6*a - 7*b)*Csc[c + d*x]^4 + 15*a^3*Csc[c + d*x]^6))/(105*a^4*d)","A",1
94,1,194,128,1.5688206,"\int \frac{\sin ^7(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^7/(a + b*Sin[c + d*x]^2)^2,x]","\frac{\sqrt{b} \left(\cos (c+d x) \left(\frac{12 a^3}{(a+b) (2 a-b \cos (2 (c+d x))+b)}+24 a-9 b\right)+b \cos (3 (c+d x))\right)-\frac{6 a^2 (5 a+6 b) \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{(-a-b)^{3/2}}-\frac{6 a^2 (5 a+6 b) \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{(-a-b)^{3/2}}}{12 b^{7/2} 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,"((-6*a^2*(5*a + 6*b)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/(-a - b)^(3/2) - (6*a^2*(5*a + 6*b)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/(-a - b)^(3/2) + Sqrt[b]*(Cos[c + d*x]*(24*a - 9*b + (12*a^3)/((a + b)*(2*a + b - b*Cos[2*(c + d*x)]))) + b*Cos[3*(c + d*x)]))/(12*b^(7/2)*d)","C",1
95,1,172,102,0.9150176,"\int \frac{\sin ^5(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^5/(a + b*Sin[c + d*x]^2)^2,x]","\frac{2 \sqrt{b} \cos (c+d x) \left(-\frac{a^2}{(a+b) (2 a-b \cos (2 (c+d x))+b)}-1\right)+\frac{a (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{(-a-b)^{3/2}}+\frac{a (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{(-a-b)^{3/2}}}{2 b^{5/2} 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)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/(-a - b)^(3/2) + (a*(3*a + 4*b)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/(-a - b)^(3/2) + 2*Sqrt[b]*Cos[c + d*x]*(-1 - a^2/((a + b)*(2*a + b - b*Cos[2*(c + d*x)]))))/(2*b^(5/2)*d)","C",1
96,1,160,83,0.4849914,"\int \frac{\sin ^3(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^3/(a + b*Sin[c + d*x]^2)^2,x]","\frac{\frac{2 a \sqrt{b} \cos (c+d x)}{2 a-b \cos (2 (c+d x))+b}+\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{\sqrt{-a-b}}+\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{\sqrt{-a-b}}}{2 b^{3/2} d (a+b)}","\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)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/Sqrt[-a - b] + ((a + 2*b)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/Sqrt[-a - b] + (2*a*Sqrt[b]*Cos[c + d*x])/(2*a + b - b*Cos[2*(c + d*x)]))/(2*b^(3/2)*(a + b)*d)","C",1
97,1,149,74,0.2948835,"\int \frac{\sin (c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]/(a + b*Sin[c + d*x]^2)^2,x]","\frac{-\frac{2 \cos (c+d x)}{2 a-b \cos (2 (c+d x))+b}+\frac{\tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{\sqrt{b} \sqrt{-a-b}}+\frac{\tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{\sqrt{b} \sqrt{-a-b}}}{2 d (a+b)}","-\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,"(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]]/(Sqrt[-a - b]*Sqrt[b]) + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]]/(Sqrt[-a - b]*Sqrt[b]) - (2*Cos[c + d*x])/(2*a + b - b*Cos[2*(c + d*x)]))/(2*(a + b)*d)","C",1
98,1,194,103,0.803576,"\int \frac{\csc (c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[Csc[c + d*x]/(a + b*Sin[c + d*x]^2)^2,x]","\frac{\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{(-a-b)^{3/2}}+\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{(-a-b)^{3/2}}+2 \left(\frac{a b \cos (c+d x)}{(a+b) (2 a-b \cos (2 (c+d x))+b)}+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 a^2 d}","\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,"((Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/(-a - b)^(3/2) + (Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]])/(-a - b)^(3/2) + 2*((a*b*Cos[c + d*x])/((a + b)*(2*a + b - b*Cos[2*(c + d*x)])) - Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]))/(2*a^2*d)","C",1
99,1,390,153,1.5705647,"\int \frac{\csc ^3(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[Csc[c + d*x]^3/(a + b*Sin[c + d*x]^2)^2,x]","\frac{\csc ^3(c+d x) (-2 a+b \cos (2 (c+d x))-b) \left(\frac{4 b^{3/2} (5 a+4 b) \csc (c+d x) (2 a-b \cos (2 (c+d x))+b) \tan ^{-1}\left(\frac{\sqrt{b}-i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{(-a-b)^{3/2}}+\frac{4 b^{3/2} (5 a+4 b) \csc (c+d x) (2 a-b \cos (2 (c+d x))+b) \tan ^{-1}\left(\frac{\sqrt{b}+i \sqrt{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{-a-b}}\right)}{(-a-b)^{3/2}}+\frac{8 a b^2 \cot (c+d x)}{a+b}+a \csc ^2\left(\frac{1}{2} (c+d x)\right) \csc (c+d x) (2 a-b \cos (2 (c+d x))+b)+4 (a-4 b) \csc (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (2 a-b \cos (2 (c+d x))+b)-a \csc (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (2 a-b \cos (2 (c+d x))+b)-4 (a-4 b) \csc (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (2 a-b \cos (2 (c+d x))+b)\right)}{32 a^3 d \left(a \csc ^2(c+d x)+b\right)^2}","-\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{(a-4 b) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\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{\cot (c+d x) \csc (c+d x)}{2 a d \left(a-b \cos ^2(c+d x)+b\right)}",1,"((-2*a - b + b*Cos[2*(c + d*x)])*Csc[c + d*x]^3*((8*a*b^2*Cot[c + d*x])/(a + b) + (4*b^(3/2)*(5*a + 4*b)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]]*(2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x])/(-a - b)^(3/2) + (4*b^(3/2)*(5*a + 4*b)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tan[(c + d*x)/2])/Sqrt[-a - b]]*(2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x])/(-a - b)^(3/2) + a*(2*a + b - b*Cos[2*(c + d*x)])*Csc[(c + d*x)/2]^2*Csc[c + d*x] + 4*(a - 4*b)*(2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x]*Log[Cos[(c + d*x)/2]] - 4*(a - 4*b)*(2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x]*Log[Sin[(c + d*x)/2]] - a*(2*a + b - b*Cos[2*(c + d*x)])*Csc[c + d*x]*Sec[(c + d*x)/2]^2))/(32*a^3*d*(b + a*Csc[c + d*x]^2)^2)","C",1
100,1,106,148,1.5322612,"\int \frac{\sin ^6(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^6/(a + b*Sin[c + d*x]^2)^2,x]","\frac{\frac{2 a^{3/2} (4 a+5 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{(a+b)^{3/2}}+b \sin (2 (c+d x)) \left(-\frac{2 a^2}{(a+b) (2 a-b \cos (2 (c+d x))+b)}-1\right)-2 (4 a-b) (c+d x)}{4 b^3 d}","\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{x (4 a-b)}{2 b^3}-\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{\sin ^2(c+d x) \tan (c+d x)}{2 b d \left((a+b) \tan ^2(c+d x)+a\right)}",1,"(-2*(4*a - b)*(c + d*x) + (2*a^(3/2)*(4*a + 5*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a + b)^(3/2) + b*(-1 - (2*a^2)/((a + b)*(2*a + b - b*Cos[2*(c + d*x)])))*Sin[2*(c + d*x)])/(4*b^3*d)","A",1
101,1,93,93,0.855872,"\int \frac{\sin ^4(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^4/(a + b*Sin[c + d*x]^2)^2,x]","\frac{-\frac{\sqrt{a} (2 a+3 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{(a+b)^{3/2}}+\frac{a b \sin (2 (c+d x))}{(a+b) (2 a-b \cos (2 (c+d x))+b)}+2 (c+d x)}{2 b^2 d}","-\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,"(2*(c + d*x) - (Sqrt[a]*(2*a + 3*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a + b)^(3/2) + (a*b*Sin[2*(c + d*x)])/((a + b)*(2*a + b - b*Cos[2*(c + d*x)])))/(2*b^2*d)","A",1
102,1,74,78,0.5341368,"\int \frac{\sin ^2(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Sin[c + d*x]^2)^2,x]","\frac{\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^{3/2}}-\frac{\sin (2 (c+d x))}{(a+b) (2 a-b \cos (2 (c+d x))+b)}}{2 d}","\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]]/(Sqrt[a]*(a + b)^(3/2)) - Sin[2*(c + d*x)]/((a + b)*(2*a + b - b*Cos[2*(c + d*x)])))/(2*d)","A",1
103,1,84,87,0.4134025,"\int \frac{1}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[(a + b*Sin[c + d*x]^2)^(-2),x]","\frac{\frac{(2 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{(a+b)^{3/2}}+\frac{\sqrt{a} b \sin (2 (c+d x))}{(a+b) (2 a-b \cos (2 (c+d x))+b)}}{2 a^{3/2} d}","\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]])/(a + b)^(3/2) + (Sqrt[a]*b*Sin[2*(c + d*x)])/((a + b)*(2*a + b - b*Cos[2*(c + d*x)])))/(2*a^(3/2)*d)","A",1
104,1,155,127,1.1852306,"\int \frac{\csc ^2(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Sin[c + d*x]^2)^2,x]","-\frac{\csc ^4(c+d x) (2 a-b \cos (2 (c+d x))+b) \left(\sqrt{a} \sqrt{a+b} \cot (c+d x) \left(4 a^2-b (2 a+3 b) \cos (2 (c+d x))+6 a b+3 b^2\right)+b (4 a+3 b) (2 a-b \cos (2 (c+d x))+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)\right)}{8 a^{5/2} d (a+b)^{3/2} \left(a \csc ^2(c+d x)+b\right)^2}","-\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{\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{\cot (c+d x)}{a d \left(a+b \sin ^2(c+d x)\right)}",1,"-1/8*((2*a + b - b*Cos[2*(c + d*x)])*(b*(4*a + 3*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]]*(2*a + b - b*Cos[2*(c + d*x)]) + Sqrt[a]*Sqrt[a + b]*(4*a^2 + 6*a*b + 3*b^2 - b*(2*a + 3*b)*Cos[2*(c + d*x)])*Cot[c + d*x])*Csc[c + d*x]^4)/(a^(5/2)*(a + b)^(3/2)*d*(b + a*Csc[c + d*x]^2)^2)","A",1
105,1,202,162,1.2999377,"\int \frac{\csc ^4(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[Csc[c + d*x]^4/(a + b*Sin[c + d*x]^2)^2,x]","\frac{\csc ^4(c+d x) (-2 a+b \cos (2 (c+d x))-b) \left(2 a^{3/2} \cot (c+d x) \csc ^2(c+d x) (2 a-b \cos (2 (c+d x))+b)-\frac{3 \sqrt{a} b^3 \sin (2 (c+d x))}{a+b}+\frac{3 b^2 (6 a+5 b) (-2 a+b \cos (2 (c+d x))-b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{(a+b)^{3/2}}+4 \sqrt{a} (a-3 b) \cot (c+d x) (2 a-b \cos (2 (c+d x))+b)\right)}{24 a^{7/2} d \left(a \csc ^2(c+d x)+b\right)^2}","\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{(2 a+5 b) \cot ^3(c+d x)}{6 a^2 d (a+b)}-\frac{\left(2 a^2-a b-5 b^2\right) \cot (c+d x)}{2 a^3 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,"((-2*a - b + b*Cos[2*(c + d*x)])*Csc[c + d*x]^4*((3*b^2*(6*a + 5*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]]*(-2*a - b + b*Cos[2*(c + d*x)]))/(a + b)^(3/2) + 4*Sqrt[a]*(a - 3*b)*(2*a + b - b*Cos[2*(c + d*x)])*Cot[c + d*x] + 2*a^(3/2)*(2*a + b - b*Cos[2*(c + d*x)])*Cot[c + d*x]*Csc[c + d*x]^2 - (3*Sqrt[a]*b^3*Sin[2*(c + d*x)])/(a + b)))/(24*a^(7/2)*d*(b + a*Csc[c + d*x]^2)^2)","A",1
106,1,134,148,2.6977972,"\int \frac{\sin ^6(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]^6/(a + b*Sin[c + d*x]^2)^3,x]","\frac{-\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)}{(a+b)^{5/2}}+\frac{a b \sin (2 (c+d x)) \left(8 a^2-3 b (2 a+3 b) \cos (2 (c+d x))+20 a b+9 b^2\right)}{(a+b)^2 (2 a-b \cos (2 (c+d x))+b)^2}+8 (c+d x)}{8 b^3 d}","-\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,"(8*(c + d*x) - (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a + b)^(5/2) + (a*b*(8*a^2 + 20*a*b + 9*b^2 - 3*b*(2*a + 3*b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/((a + b)^2*(2*a + b - b*Cos[2*(c + d*x)])^2))/(8*b^3*d)","A",1
107,1,97,110,1.3170273,"\int \frac{\sin ^4(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]^4/(a + b*Sin[c + d*x]^2)^3,x]","\frac{\frac{3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^{5/2}}+\frac{\sin (2 (c+d x)) ((2 a+5 b) \cos (2 (c+d x))-8 a-5 b)}{(a+b)^2 (2 a-b \cos (2 (c+d x))+b)^2}}{8 d}","\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{3 \tan (c+d x)}{8 d (a+b)^2 \left((a+b) \tan ^2(c+d x)+a\right)}-\frac{\tan ^3(c+d x)}{4 d (a+b) \left((a+b) \tan ^2(c+d x)+a\right)^2}",1,"((3*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(5/2)) + ((-8*a - 5*b + (2*a + 5*b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/((a + b)^2*(2*a + b - b*Cos[2*(c + d*x)])^2))/(8*d)","A",1
108,1,112,131,1.3886547,"\int \frac{\sin ^2(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Sin[c + d*x]^2)^3,x]","\frac{\frac{(4 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{a^{3/2} (a+b)^{5/2}}-\frac{\sin (2 (c+d x)) \left(8 a^2+b (b-2 a) \cos (2 (c+d x))+4 a b-b^2\right)}{a (a+b)^2 (2 a-b \cos (2 (c+d x))+b)^2}}{8 d}","\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]])/(a^(3/2)*(a + b)^(5/2)) - ((8*a^2 + 4*a*b - b^2 + b*(-2*a + b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(a*(a + b)^2*(2*a + b - b*Cos[2*(c + d*x)])^2))/(8*d)","A",1
109,1,125,144,1.2702368,"\int \frac{1}{\left(a+b \sin ^2(c+d x)\right)^3} \, dx","Integrate[(a + b*Sin[c + d*x]^2)^(-3),x]","\frac{\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)}{(a+b)^{5/2}}+\frac{\sqrt{a} b \sin (2 (c+d x)) \left(16 a^2-3 b (2 a+b) \cos (2 (c+d x))+16 a b+3 b^2\right)}{(a+b)^2 (2 a-b \cos (2 (c+d x))+b)^2}}{8 a^{5/2} d}","\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{\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{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]])/(a + b)^(5/2) + (Sqrt[a]*b*(16*a^2 + 16*a*b + 3*b^2 - 3*b*(2*a + b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/((a + b)^2*(2*a + b - b*Cos[2*(c + d*x)])^2))/(8*a^(5/2)*d)","A",1
110,1,214,196,1.7158277,"\int \frac{\csc ^2(c+d x)}{\left(a+b \sin ^2(c+d x)\right)^3} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Sin[c + d*x]^2)^3,x]","\frac{\csc ^6(c+d x) (-2 a+b \cos (2 (c+d x))-b) \left(\frac{4 a^{3/2} b^2 \sin (2 (c+d x))}{a+b}+\frac{3 b \left(8 a^2+12 a b+5 b^2\right) (2 a-b \cos (2 (c+d x))+b)^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{(a+b)^{5/2}}+\frac{\sqrt{a} b^2 (10 a+7 b) \sin (2 (c+d x)) (2 a-b \cos (2 (c+d x))+b)}{(a+b)^2}+8 \sqrt{a} \cot (c+d x) (2 a-b \cos (2 (c+d x))+b)^2\right)}{64 a^{7/2} d \left(a \csc ^2(c+d x)+b\right)^3}","-\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{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{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,"((-2*a - b + b*Cos[2*(c + d*x)])*Csc[c + d*x]^6*((3*b*(8*a^2 + 12*a*b + 5*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]]*(2*a + b - b*Cos[2*(c + d*x)])^2)/(a + b)^(5/2) + 8*Sqrt[a]*(2*a + b - b*Cos[2*(c + d*x)])^2*Cot[c + d*x] + (4*a^(3/2)*b^2*Sin[2*(c + d*x)])/(a + b) + (Sqrt[a]*b^2*(10*a + 7*b)*(2*a + b - b*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(a + b)^2))/(64*a^(7/2)*d*(b + a*Csc[c + d*x]^2)^3)","A",1
111,1,201,206,1.4726282,"\int \frac{1}{\left(a+b \sin ^2(c+d x)\right)^4} \, dx","Integrate[(a + b*Sin[c + d*x]^2)^(-4),x]","\frac{\frac{32 a^{5/2} b \sin (2 (c+d x))}{(a+b) (2 a-b \cos (2 (c+d x))+b)^3}+\frac{20 a^{3/2} b (2 a+b) \sin (2 (c+d x))}{(a+b)^2 (2 a-b \cos (2 (c+d x))+b)^2}+\frac{\sqrt{a} b \left(44 a^2+44 a b+15 b^2\right) \sin (2 (c+d x))}{(a+b)^3 (2 a-b \cos (2 (c+d x))+b)}+\frac{3 \left(16 a^3+24 a^2 b+18 a b^2+5 b^3\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{(a+b)^{7/2}}}{48 a^{7/2} d}","\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{(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{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,"((3*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a + b)^(7/2) + (32*a^(5/2)*b*Sin[2*(c + d*x)])/((a + b)*(2*a + b - b*Cos[2*(c + d*x)])^3) + (20*a^(3/2)*b*(2*a + b)*Sin[2*(c + d*x)])/((a + b)^2*(2*a + b - b*Cos[2*(c + d*x)])^2) + (Sqrt[a]*b*(44*a^2 + 44*a*b + 15*b^2)*Sin[2*(c + d*x)])/((a + b)^3*(2*a + b - b*Cos[2*(c + d*x)])))/(48*a^(7/2)*d)","A",1
112,1,312,279,1.92737,"\int \frac{1}{\left(a+b \sin ^2(c+d x)\right)^5} \, dx","Integrate[(a + b*Sin[c + d*x]^2)^(-5),x]","\frac{\frac{24 \left(128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{(a+b)^{9/2}}+\frac{2 \sqrt{a} b \sin (2 (c+d x)) \left(24576 a^6+73728 a^5 b+97280 a^4 b^2-400 a^3 b^3 \cos (6 (c+d x))+71680 a^3 b^3-600 a^2 b^4 \cos (6 (c+d x))+32272 a^2 b^4+2 b^2 \left(2816 a^4+5632 a^3 b+4816 a^2 b^2+2000 a b^3+315 b^4\right) \cos (4 (c+d x))-b \left(27648 a^5+69120 a^4 b+73616 a^3 b^2+41304 a^2 b^3+12310 a b^4+1575 b^5\right) \cos (2 (c+d x))-410 a b^5 \cos (6 (c+d x))+8720 a b^5-105 b^6 \cos (6 (c+d x))+1050 b^6\right)}{(a+b)^4 (2 a-b \cos (2 (c+d x))+b)^4}}{3072 a^{9/2} d}","\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{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{\left(128 a^4+256 a^3 b+288 a^2 b^2+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{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,"((24*(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]])/(a + b)^(9/2) + (2*Sqrt[a]*b*(24576*a^6 + 73728*a^5*b + 97280*a^4*b^2 + 71680*a^3*b^3 + 32272*a^2*b^4 + 8720*a*b^5 + 1050*b^6 - b*(27648*a^5 + 69120*a^4*b + 73616*a^3*b^2 + 41304*a^2*b^3 + 12310*a*b^4 + 1575*b^5)*Cos[2*(c + d*x)] + 2*b^2*(2816*a^4 + 5632*a^3*b + 4816*a^2*b^2 + 2000*a*b^3 + 315*b^4)*Cos[4*(c + d*x)] - 400*a^3*b^3*Cos[6*(c + d*x)] - 600*a^2*b^4*Cos[6*(c + d*x)] - 410*a*b^5*Cos[6*(c + d*x)] - 105*b^6*Cos[6*(c + d*x)])*Sin[2*(c + d*x)])/((a + b)^4*(2*a + b - b*Cos[2*(c + d*x)])^4))/(3072*a^(9/2)*d)","A",1
113,1,29,11,0.0643183,"\int \frac{\sin (x)}{\sqrt{1+\sin ^2(x)}} \, dx","Integrate[Sin[x]/Sqrt[1 + Sin[x]^2],x]","i \log \left(\sqrt{3-\cos (2 x)}+i \sqrt{2} \cos (x)\right)","-\sin ^{-1}\left(\frac{\cos (x)}{\sqrt{2}}\right)",1,"I*Log[I*Sqrt[2]*Cos[x] + Sqrt[3 - Cos[2*x]]]","C",1
114,1,53,30,0.0525174,"\int \sin (x) \sqrt{1+\sin ^2(x)} \, dx","Integrate[Sin[x]*Sqrt[1 + Sin[x]^2],x]","-\frac{\cos (x) \sqrt{3-\cos (2 x)}}{2 \sqrt{2}}+i \log \left(\sqrt{3-\cos (2 x)}+i \sqrt{2} \cos (x)\right)","-\frac{1}{2} \cos (x) \sqrt{2-\cos ^2(x)}-\sin ^{-1}\left(\frac{\cos (x)}{\sqrt{2}}\right)",1,"-1/2*(Cos[x]*Sqrt[3 - Cos[2*x]])/Sqrt[2] + I*Log[I*Sqrt[2]*Cos[x] + Sqrt[3 - Cos[2*x]]]","C",1
115,1,39,15,0.093428,"\int \frac{\sin (7+3 x)}{\sqrt{3+\sin ^2(7+3 x)}} \, dx","Integrate[Sin[7 + 3*x]/Sqrt[3 + Sin[7 + 3*x]^2],x]","\frac{1}{3} i \log \left(\sqrt{7-\cos (2 (3 x+7))}+i \sqrt{2} \cos (3 x+7)\right)","-\frac{1}{3} \sin ^{-1}\left(\frac{1}{2} \cos (3 x+7)\right)",1,"(I/3)*Log[I*Sqrt[2]*Cos[7 + 3*x] + Sqrt[7 - Cos[2*(7 + 3*x)]]]","C",1
116,1,36,53,0.0215362,"\int \left(a-a \sin ^2(x)\right)^{5/2} \, dx","Integrate[(a - a*Sin[x]^2)^(5/2),x]","\frac{1}{240} a^2 (150 \sin (x)+25 \sin (3 x)+3 \sin (5 x)) \sec (x) \sqrt{a \cos ^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}",1,"(a^2*Sqrt[a*Cos[x]^2]*Sec[x]*(150*Sin[x] + 25*Sin[3*x] + 3*Sin[5*x]))/240","A",1
117,1,26,34,0.0098692,"\int \left(a-a \sin ^2(x)\right)^{3/2} \, dx","Integrate[(a - a*Sin[x]^2)^(3/2),x]","\frac{1}{12} a (9 \sin (x)+\sin (3 x)) \sec (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,"(a*Sqrt[a*Cos[x]^2]*Sec[x]*(9*Sin[x] + Sin[3*x]))/12","A",1
118,1,13,13,0.004751,"\int \sqrt{a-a \sin ^2(x)} \, dx","Integrate[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",1
119,1,46,16,0.0213483,"\int \frac{1}{\sqrt{a-a \sin ^2(x)}} \, dx","Integrate[1/Sqrt[a - a*Sin[x]^2],x]","\frac{\cos (x) \left(\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)\right)}{\sqrt{a \cos ^2(x)}}","\frac{\cos (x) \tanh ^{-1}(\sin (x))}{\sqrt{a \cos ^2(x)}}",1,"(Cos[x]*(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]]))/Sqrt[a*Cos[x]^2]","B",1
120,1,91,42,0.0666804,"\int \frac{1}{\left(a-a \sin ^2(x)\right)^{3/2}} \, dx","Integrate[(a - a*Sin[x]^2)^(-3/2),x]","-\frac{\cos (x) \left(-2 \sin (x)+\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\cos (2 x) \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{4 \left(a \cos ^2(x)\right)^{3/2}}","\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,"-1/4*(Cos[x]*(Log[Cos[x/2] - Sin[x/2]] + Cos[2*x]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) - Log[Cos[x/2] + Sin[x/2]] - 2*Sin[x]))/(a*Cos[x]^2)^(3/2)","B",1
121,1,72,61,0.1592563,"\int \frac{1}{\left(a-a \sin ^2(x)\right)^{5/2}} \, dx","Integrate[(a - a*Sin[x]^2)^(-5/2),x]","\frac{\cos ^5(x) \left(\frac{1}{2} (11 \sin (x)+3 \sin (3 x)) \sec ^4(x)-6 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+6 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{16 \left(a \cos ^2(x)\right)^{5/2}}","\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,"(Cos[x]^5*(-6*Log[Cos[x/2] - Sin[x/2]] + 6*Log[Cos[x/2] + Sin[x/2]] + (Sec[x]^4*(11*Sin[x] + 3*Sin[3*x]))/2))/(16*(a*Cos[x]^2)^(5/2))","A",1
122,1,119,125,0.4298057,"\int \sin ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\frac{\cos (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b} (-a+b \cos (2 (e+f x))-4 b)}{\sqrt{2} b}+\frac{(a+b) (3 b-a) \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{-b} \cos (e+f x)\right)}{(-b)^{3/2}}}{8 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,"((Cos[e + f*x]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*(-a - 4*b + b*Cos[2*(e + f*x)]))/(Sqrt[2]*b) + ((a + b)*(-a + 3*b)*Log[Sqrt[2]*Sqrt[-b]*Cos[e + f*x] + Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/(-b)^(3/2))/(8*f)","A",1
123,1,93,78,0.264524,"\int \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\sqrt{2} \cos (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b}+\frac{2 (a+b) \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{-b} \cos (e+f x)\right)}{\sqrt{-b}}}{4 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,"-1/4*(Sqrt[2]*Cos[e + f*x]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]] + (2*(a + b)*Log[Sqrt[2]*Sqrt[-b]*Cos[e + f*x] + Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/Sqrt[-b])/f","A",1
124,1,99,83,0.1164463,"\int \csc (e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{-b} \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{-b} \cos (e+f x)\right)-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 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[a]*ArcTanh[(Sqrt[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]]) + Sqrt[-b]*Log[Sqrt[2]*Sqrt[-b]*Cos[e + f*x] + Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/f","A",1
125,1,100,84,0.2888906,"\int \csc ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{-2 (a+b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)-\sqrt{2} \sqrt{a} \cot (e+f x) \csc (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b}}{4 \sqrt{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 \sqrt{a} f}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{2 f}",1,"(-2*(a + b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] - Sqrt[2]*Sqrt[a]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*Cot[e + f*x]*Csc[e + f*x])/(4*Sqrt[a]*f)","A",1
126,1,127,143,0.5079673,"\int \csc ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\left(-6 a^2-4 a b+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)-\sqrt{2} \sqrt{a} \cot (e+f x) \csc (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b} \left(2 a \csc ^2(e+f x)+3 a+b\right)}{16 a^{3/2} 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,"((-6*a^2 - 4*a*b + 2*b^2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] - Sqrt[2]*Sqrt[a]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*Cot[e + f*x]*Csc[e + f*x]*(3*a + b + 2*a*Csc[e + f*x]^2))/(16*a^(3/2)*f)","A",1
127,1,199,259,1.4277368,"\int \sin ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{-\sqrt{2} b \sin (2 (e+f x)) \left(8 a^2-4 b (4 a+7 b) \cos (2 (e+f x))+48 a b+3 b^2 \cos (4 (e+f x))+25 b^2\right)+32 a \left(a^2-a b-2 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-16 a \left(2 a^2-3 a b-8 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{240 b^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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{(a+4 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 b f}-\frac{\sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}",1,"(-16*a*(2*a^2 - 3*a*b - 8*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 32*a*(a^2 - a*b - 2*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(8*a^2 + 48*a*b + 25*b^2 - 4*b*(4*a + 7*b)*Cos[2*(e + f*x)] + 3*b^2*Cos[4*(e + f*x)])*Sin[2*(e + f*x)])/(240*b^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
128,1,159,159,0.8280284,"\int \sin ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sin[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{b \sin (2 (e+f x)) (-2 a+b \cos (2 (e+f x))-b)-2 \sqrt{2} a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 \sqrt{2} a (a+2 b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 \sqrt{2} b f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(2*Sqrt[2]*a*(a + 2*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 2*Sqrt[2]*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + b*(-2*a - b + b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*Sqrt[2]*b*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
129,1,61,51,0.0870618,"\int \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)])/(f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
130,1,137,174,0.6055509,"\int \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{-\sqrt{2} \cot (e+f x) (2 a-b \cos (2 (e+f x))+b)+2 (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(-(Sqrt[2]*(2*a + b - b*Cos[2*(e + f*x)])*Cot[e + f*x]) - 2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
131,1,188,234,3.1935096,"\int \csc ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Csc[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\frac{\cot (e+f x) \csc ^2(e+f x) \left(4 \left(2 a^2+4 a b+b^2\right) \cos (2 (e+f x))-(2 a+b) (8 a+b \cos (4 (e+f x))+3 b)\right)}{2 \sqrt{2}}+4 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a (2 a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 a f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(((4*(2*a^2 + 4*a*b + b^2)*Cos[2*(e + f*x)] - (2*a + b)*(8*a + 3*b + b*Cos[4*(e + f*x)]))*Cot[e + f*x]*Csc[e + f*x]^2)/(2*Sqrt[2]) - 2*a*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 4*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(6*a*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
132,1,152,169,0.7812628,"\int \sin ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sin[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\frac{(a+b)^2 (5 b-a) \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{-b} \cos (e+f x)\right)}{(-b)^{3/2}}-\frac{\cos (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b} \left(3 a^2-b (7 a+9 b) \cos (2 (e+f x))+29 a b+b^2 \cos (4 (e+f x))+23 b^2\right)}{3 \sqrt{2} b}}{16 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,"(-1/3*(Cos[e + f*x]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*(3*a^2 + 29*a*b + 23*b^2 - b*(7*a + 9*b)*Cos[2*(e + f*x)] + b^2*Cos[4*(e + f*x)]))/(Sqrt[2]*b) + ((a + b)^2*(-a + 5*b)*Log[Sqrt[2]*Sqrt[-b]*Cos[e + f*x] + Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/(-b)^(3/2))/(16*f)","A",0
133,1,113,114,0.4150886,"\int \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{\frac{\cos (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b} (5 a-b \cos (2 (e+f x))+4 b)}{\sqrt{2}}+\frac{3 (a+b)^2 \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{-b} \cos (e+f x)\right)}{\sqrt{-b}}}{8 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,"-1/8*((Cos[e + f*x]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*(5*a + 4*b - b*Cos[2*(e + f*x)]))/Sqrt[2] + (3*(a + b)^2*Log[Sqrt[2]*Sqrt[-b]*Cos[e + f*x] + Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/Sqrt[-b])/f","A",1
134,1,141,122,0.9196709,"\int \csc (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{4 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)+\sqrt{2} b \cos (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b}-2 \sqrt{-b} (3 a+b) \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{-b} \cos (e+f x)\right)}{4 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,"-1/4*(4*a^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] + Sqrt[2]*b*Cos[e + f*x]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]] - 2*Sqrt[-b]*(3*a + b)*Log[Sqrt[2]*Sqrt[-b]*Cos[e + f*x] + Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/f","A",1
135,1,147,128,1.221146,"\int \csc ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{4 (-b)^{3/2} \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{-b} \cos (e+f x)\right)+2 \sqrt{a} (a+3 b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)+\sqrt{2} a \cot (e+f x) \csc (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b}}{4 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,"-1/4*(2*Sqrt[a]*(a + 3*b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] + Sqrt[2]*a*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*Cot[e + f*x]*Csc[e + f*x] + 4*(-b)^(3/2)*Log[Sqrt[2]*Sqrt[-b]*Cos[e + f*x] + Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/f","A",1
136,1,114,128,0.7193408,"\int \csc ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^5*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{\frac{6 (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)}{\sqrt{a}}+\sqrt{2} \cot (e+f x) \csc (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b} \left(2 a \csc ^2(e+f x)+3 a+5 b\right)}{16 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,"-1/16*((6*(a + b)^2*ArcTanh[(Sqrt[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/Sqrt[a] + Sqrt[2]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*Cot[e + f*x]*Csc[e + f*x]*(3*a + 5*b + 2*a*Csc[e + f*x]^2))/f","A",1
137,1,161,197,1.1766571,"\int \csc ^7(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^7*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{-\sqrt{2} \sqrt{a} \csc ^2(e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b} \left(\left(15 a^2+22 a b+3 b^2\right) \cos (e+f x)+2 a \cot (e+f x) \csc (e+f x) \left(4 a \csc ^2(e+f x)+5 a+7 b\right)\right)-6 (5 a-b) (a+b)^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)}{96 a^{3/2} 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,"(-6*(5*a - b)*(a + b)^2*ArcTanh[(Sqrt[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] - Sqrt[2]*Sqrt[a]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*Csc[e + f*x]^2*((15*a^2 + 22*a*b + 3*b^2)*Cos[e + f*x] + 2*a*Cot[e + f*x]*Csc[e + f*x]*(5*a + 7*b + 4*a*Csc[e + f*x]^2)))/(96*a^(3/2)*f)","A",1
138,1,249,325,2.6540595,"\int \sin ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sin[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} b \sin (2 (e+f x)) \left(-32 a^3+b \left(144 a^2+480 a b+299 b^2\right) \cos (2 (e+f x))-496 a^2 b-2 b^2 (26 a+27 b) \cos (4 (e+f x))-684 a b^2+5 b^3 \cos (6 (e+f x))-250 b^3\right)+64 a \left(2 a^3-3 a^2 b-13 a b^2-8 b^3\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-128 a \left(a^3-2 a^2 b-12 a b^2-8 b^3\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{2240 b^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(-128*a*(a^3 - 2*a^2*b - 12*a*b^2 - 8*b^3)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 64*a*(2*a^3 - 3*a^2*b - 13*a*b^2 - 8*b^3)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + Sqrt[2]*b*(-32*a^3 - 496*a^2*b - 684*a*b^2 - 250*b^3 + b*(144*a^2 + 480*a*b + 299*b^2)*Cos[2*(e + f*x)] - 2*b^2*(26*a + 27*b)*Cos[4*(e + f*x)] + 5*b^3*Cos[6*(e + f*x)])*Sin[2*(e + f*x)])/(2240*b^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
139,1,201,218,1.382471,"\int \sin ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sin[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{-\sqrt{2} b \sin (2 (e+f x)) \left(48 a^2-4 b (9 a+7 b) \cos (2 (e+f x))+68 a b+3 b^2 \cos (4 (e+f x))+25 b^2\right)-16 a \left(3 a^2+7 a b+4 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+16 a \left(3 a^2+13 a b+8 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{240 b f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(16*a*(3*a^2 + 13*a*b + 8*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 16*a*(3*a^2 + 7*a*b + 4*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(48*a^2 + 68*a*b + 25*b^2 - 4*b*(9*a + 7*b)*Cos[2*(e + f*x)] + 3*b^2*Cos[4*(e + f*x)])*Sin[2*(e + f*x)])/(240*b*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
140,1,156,154,0.8038914,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{b \sin (2 (e+f x)) (-2 a+b \cos (2 (e+f x))-b)-2 \sqrt{2} a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+4 \sqrt{2} a (2 a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 \sqrt{2} f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(4*Sqrt[2]*a*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 2*Sqrt[2]*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + b*(-2*a - b + b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*Sqrt[2]*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
141,1,141,181,1.3663684,"\int \csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{a \left(\sqrt{2} \cot (e+f x) (2 a-b \cos (2 (e+f x))+b)-2 (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 (a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)\right)}{2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"-1/2*(a*(Sqrt[2]*(2*a + b - b*Cos[2*(e + f*x)])*Cot[e + f*x] + 2*(a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 2*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)]))/(f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
142,1,201,236,4.5107075,"\int \csc ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Csc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{2 \left(2 a^2+5 a b+3 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+\frac{\cot (e+f x) \csc ^2(e+f x) \left(2 \left(2 a^2+7 a b+4 b^2\right) \cos (2 (e+f x))-8 a^2-b (a+2 b) \cos (4 (e+f x))-13 a b-6 b^2\right)}{\sqrt{2}}-4 a (a+2 b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(((-8*a^2 - 13*a*b - 6*b^2 + 2*(2*a^2 + 7*a*b + 4*b^2)*Cos[2*(e + f*x)] - b*(a + 2*b)*Cos[4*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^2)/Sqrt[2] - 4*a*(a + 2*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*(2*a^2 + 5*a*b + 3*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(6*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
143,1,194,210,1.4192866,"\int \left(a+b \sin ^2(c+d x)\right)^{5/2} \, dx","Integrate[(a + b*Sin[c + d*x]^2)^(5/2),x]","\frac{-\sqrt{2} b \sin (2 (c+d x)) \left(88 a^2-28 b (2 a+b) \cos (2 (c+d x))+88 a b+3 b^2 \cos (4 (c+d x))+25 b^2\right)-64 a \left(2 a^2+3 a b+b^2\right) \sqrt{\frac{2 a-b \cos (2 (c+d x))+b}{a}} F\left(c+d x\left|-\frac{b}{a}\right.\right)+16 a \left(23 a^2+23 a b+8 b^2\right) \sqrt{\frac{2 a-b \cos (2 (c+d x))+b}{a}} E\left(c+d x\left|-\frac{b}{a}\right.\right)}{240 d \sqrt{2 a-b \cos (2 (c+d x))+b}}","\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,"(16*a*(23*a^2 + 23*a*b + 8*b^2)*Sqrt[(2*a + b - b*Cos[2*(c + d*x)])/a]*EllipticE[c + d*x, -(b/a)] - 64*a*(2*a^2 + 3*a*b + b^2)*Sqrt[(2*a + b - b*Cos[2*(c + d*x)])/a]*EllipticF[c + d*x, -(b/a)] - Sqrt[2]*b*(88*a^2 + 88*a*b + 25*b^2 - 28*b*(2*a + b)*Cos[2*(c + d*x)] + 3*b^2*Cos[4*(c + d*x)])*Sin[2*(c + d*x)])/(240*d*Sqrt[2*a + b - b*Cos[2*(c + d*x)]])","A",1
144,1,105,83,0.2789081,"\int \frac{\sin ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{(a-b) \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{-b} \cos (e+f x)\right)}{2 \sqrt{-b} b f}-\frac{\cos (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b}}{2 \sqrt{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,"-1/2*(Cos[e + f*x]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])/(Sqrt[2]*b*f) + ((a - b)*Log[Sqrt[2]*Sqrt[-b]*Cos[e + f*x] + Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/(2*Sqrt[-b]*b*f)","A",1
145,1,53,41,0.1104776,"\int \frac{\sin (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{-b} \cos (e+f x)\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,"-(Log[Sqrt[2]*Sqrt[-b]*Cos[e + f*x] + Sqrt[2*a + b - b*Cos[2*(e + f*x)]]]/(Sqrt[-b]*f))","A",1
146,1,48,41,0.1651589,"\int \frac{\csc (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 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[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]]/(Sqrt[a]*f))","A",1
147,1,102,89,0.313881,"\int \frac{\csc ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{-2 (a-b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)-\sqrt{2} \sqrt{a} \cot (e+f x) \csc (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b}}{4 a^{3/2} 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 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,"(-2*(a - b)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] - Sqrt[2]*Sqrt[a]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*Cot[e + f*x]*Csc[e + f*x])/(4*a^(3/2)*f)","A",1
148,1,163,206,0.9013682,"\int \frac{\sin ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{b \sin (2 (e+f x)) (-2 a+b \cos (2 (e+f x))-b)+2 \sqrt{2} a (2 a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-4 \sqrt{2} a (a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 \sqrt{2} b^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-4*Sqrt[2]*a*(a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*Sqrt[2]*a*(2*a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + b*(-2*a - b + b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*Sqrt[2]*b^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
149,1,78,111,0.2242888,"\int \frac{\sin ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Sin[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{2 a-b \cos (2 (e+f x))+b} \left(E\left(e+f x\left|-\frac{b}{a}\right.\right)-F\left(e+f x\left|-\frac{b}{a}\right.\right)\right)}{b f \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}}}","\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,"(Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*(EllipticE[e + f*x, -(b/a)] - EllipticF[e + f*x, -(b/a)]))/(b*f*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a])","A",1
150,1,60,51,0.0880121,"\int \frac{1}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{2 a-b \cos (2 (e+f x))+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)}}",1,"(Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
151,1,138,177,0.6358915,"\int \frac{\csc ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{-\sqrt{2} \cot (e+f x) (2 a-b \cos (2 (e+f x))+b)+2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{2 a f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(-(Sqrt[2]*(2*a + b - b*Cos[2*(e + f*x)])*Cot[e + f*x]) - 2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(2*a*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
152,1,195,244,3.9801813,"\int \frac{\csc ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Csc[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\frac{\cot (e+f x) \csc ^2(e+f x) \left(2 \left(2 a^2+a b-2 b^2\right) \cos (2 (e+f x))-8 a^2+b (b-a) \cos (4 (e+f x))-a b+3 b^2\right)}{\sqrt{2}}+2 a (2 a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-4 a (a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 a^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(((-8*a^2 - a*b + 3*b^2 + 2*(2*a^2 + a*b - 2*b^2)*Cos[2*(e + f*x)] + b*(-a + b)*Cos[4*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^2)/Sqrt[2] - 4*a*(a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*a*(2*a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(6*a^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
153,1,96,79,0.4229043,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\frac{\sqrt{2} a b \cos (e+f x)}{(a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}+\sqrt{-b} \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{-b} \cos (e+f x)\right)}{b^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,"((Sqrt[2]*a*b*Cos[e + f*x])/((a + b)*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]) + Sqrt[-b]*Log[Sqrt[2]*Sqrt[-b]*Cos[e + f*x] + Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/(b^2*f)","A",1
154,1,41,34,0.1173038,"\int \frac{\sin (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{\sqrt{2} \cos (e+f x)}{f (a+b) \sqrt{2 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,"-((Sqrt[2]*Cos[e + f*x])/((a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]))","A",1
155,1,93,79,0.3527416,"\int \frac{\csc (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\frac{\sqrt{2} \sqrt{a} b \cos (e+f x)}{(a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}-\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 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[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] + (Sqrt[2]*Sqrt[a]*b*Cos[e + f*x])/((a + b)*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]))/(a^(3/2)*f)","A",1
156,1,134,134,0.703437,"\int \frac{\csc ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\frac{\cot (e+f x) \csc (e+f x) \left(-2 a^2+b (a+3 b) \cos (2 (e+f x))-3 a b-3 b^2\right)}{\sqrt{2} a^2 (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}-\frac{(a-3 b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)}{a^{5/2}}}{2 f}","-\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{b (a+3 b) \cos (e+f x)}{2 a^2 f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}-\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[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/a^(5/2)) + ((-2*a^2 - 3*a*b - 3*b^2 + b*(a + 3*b)*Cos[2*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x])/(Sqrt[2]*a^2*(a + b)*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]))/(2*f)","A",1
157,1,197,274,1.3132729,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{b \sin (2 (e+f x)) \left(-8 a^2+b (a+b) \cos (2 (e+f x))-3 a b-b^2\right)+2 \sqrt{2} a \left(8 a^2+7 a b-b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 \sqrt{2} a \left(8 a^2+3 a b-2 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 \sqrt{2} b^3 f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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{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{(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 \sin ^3(e+f x) \cos (e+f x)}{b f (a+b) \sqrt{a+b \sin ^2(e+f x)}}",1,"(-2*Sqrt[2]*a*(8*a^2 + 3*a*b - 2*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*Sqrt[2]*a*(8*a^2 + 7*a*b - b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + b*(-8*a^2 - 3*a*b - b^2 + b*(a + b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*Sqrt[2]*b^3*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
158,1,136,202,0.6939888,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{a \left(-4 (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 (2 a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)+\sqrt{2} b \sin (2 (e+f x))\right)}{2 b^2 f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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*(2*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 4*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + Sqrt[2]*b*Sin[2*(e + f*x)]))/(2*b^2*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
159,1,138,153,0.4523606,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-\sqrt{2} a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)-b \sin (2 (e+f x))}{\sqrt{2} b f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(-(Sqrt[2]*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)]) + Sqrt[2]*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] - b*Sin[2*(e + f*x)])/(Sqrt[2]*b*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
160,1,90,101,0.1498687,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(-3/2),x]","\frac{2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)+\sqrt{2} b \sin (2 (e+f x))}{2 a f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + Sqrt[2]*b*Sin[2*(e + f*x)])/(2*a*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
161,1,170,235,1.2838456,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\cot (e+f x) \left(-2 a^2+b (a+2 b) \cos (2 (e+f x))-3 a b-2 b^2\right)+\sqrt{2} a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-\sqrt{2} a (a+2 b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{\sqrt{2} a^2 f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"((-2*a^2 - 3*a*b - 2*b^2 + b*(a + 2*b)*Cos[2*(e + f*x)])*Cot[e + f*x] - Sqrt[2]*a*(a + 2*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + Sqrt[2]*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(Sqrt[2]*a^2*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
162,1,133,137,0.7798248,"\int \frac{\sin ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\frac{2 \sqrt{2} a \cos (e+f x) \left(3 a^2-b (2 a+3 b) \cos (2 (e+f x))+7 a b+3 b^2\right)}{(a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}}-\frac{3 \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{-b} \cos (e+f x)\right)}{\sqrt{-b}}}{3 b^2 f}","-\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 (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{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,"((2*Sqrt[2]*a*Cos[e + f*x]*(3*a^2 + 7*a*b + 3*b^2 - b*(2*a + 3*b)*Cos[2*(e + f*x)]))/((a + b)^2*(2*a + b - b*Cos[2*(e + f*x)])^(3/2)) - (3*Log[Sqrt[2]*Sqrt[-b]*Cos[e + f*x] + Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/Sqrt[-b])/(3*b^2*f)","A",0
163,1,64,81,0.3237377,"\int \frac{\sin ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\sqrt{2} \cos (e+f x) ((a+3 b) \cos (2 (e+f x))-5 a-3 b)}{3 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{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,"(Sqrt[2]*Cos[e + f*x]*(-5*a - 3*b + (a + 3*b)*Cos[2*(e + f*x)]))/(3*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
164,1,60,73,0.1793304,"\int \frac{\sin (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{2 \sqrt{2} \cos (e+f x) (-3 a+b \cos (2 (e+f x))-2 b)}{3 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{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,"(2*Sqrt[2]*Cos[e + f*x]*(-3*a - 2*b + b*Cos[2*(e + f*x)]))/(3*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
165,1,127,129,0.6300212,"\int \frac{\csc (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\frac{\sqrt{2} b \cos (e+f x) \left(12 a^2-b (5 a+3 b) \cos (2 (e+f x))+13 a b+3 b^2\right)}{3 a^2 (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)}{a^{5/2}}}{f}","-\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 (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{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[2]*Sqrt[a]*Cos[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]]/a^(5/2)) + (Sqrt[2]*b*Cos[e + f*x]*(12*a^2 + 13*a*b + 3*b^2 - b*(5*a + 3*b)*Cos[2*(e + f*x)]))/(3*a^2*(a + b)^2*(2*a + b - b*Cos[2*(e + f*x)])^(3/2)))/f","A",1
166,1,192,285,2.1413344,"\int \frac{\sin ^6(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{a \left(\sqrt{2} b \sin (2 (e+f x)) \left(-8 a^2+b (5 a+7 b) \cos (2 (e+f x))-17 a b-7 b^2\right)+2 a \left(8 a^2+17 a b+9 b^2\right) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a \left(8 a^2+13 a b+3 b^2\right) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)\right)}{6 b^3 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{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{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{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 \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,"-1/6*(a*(-2*a*(8*a^2 + 13*a*b + 3*b^2)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] + 2*a*(8*a^2 + 17*a*b + 9*b^2)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] + Sqrt[2]*b*(-8*a^2 - 17*a*b - 7*b^2 + b*(5*a + 7*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))/(b^3*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
167,1,182,269,1.705858,"\int \frac{\sin ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{-\sqrt{2} b \sin (2 (e+f x)) \left(-a^2+b (a+2 b) \cos (2 (e+f x))-4 a b-2 b^2\right)-a \left(2 a^2+5 a b+3 b^2\right) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 a^2 (a+2 b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 b^2 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{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,"-1/3*(2*a^2*(a + 2*b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] - a*(2*a^2 + 5*a*b + 3*b^2)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(-a^2 - 4*a*b - 2*b^2 + b*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(b^2*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
168,1,174,221,1.5255102,"\int \frac{\sin ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sin[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{-\sqrt{2} b \sin (2 (e+f x)) \left(4 a^2+b (b-a) \cos (2 (e+f x))+a b-b^2\right)+2 a^2 (a+b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a^2 (a-b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 a b f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}}","-\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,"(-2*a^2*(a - b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] + 2*a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(4*a^2 + a*b - b^2 + b*(-a + b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*a*b*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
169,1,172,223,1.379624,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(-5/2),x]","\frac{-\sqrt{2} b \sin (2 (e+f x)) \left(-5 a^2+b (2 a+b) \cos (2 (e+f x))-5 a b-b^2\right)-a^2 (a+b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 a^2 (2 a+b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a^2 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}}","\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,"(2*a^2*(2*a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] - a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(-5*a^2 - 5*a*b - b^2 + b*(2*a + b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(3*a^2*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
170,1,214,322,2.3577363,"\int \frac{\csc ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{4 a^2 \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} \left(\left(3 a^2+7 a b+4 b^2\right) F\left(e+f x\left|-\frac{b}{a}\right.\right)-\left(3 a^2+13 a b+8 b^2\right) E\left(e+f x\left|-\frac{b}{a}\right.\right)\right)-2 \sqrt{2} \left(2 a b^2 (a+b) \sin (2 (e+f x))+b^2 (7 a+5 b) \sin (2 (e+f x)) (2 a-b \cos (2 (e+f x))+b)+3 (a+b)^2 \cot (e+f x) (2 a-b \cos (2 (e+f x))+b)^2\right)}{12 a^3 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}}","\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{\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{b \cot (e+f x)}{3 a f (a+b) \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"(4*a^2*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*(-((3*a^2 + 13*a*b + 8*b^2)*EllipticE[e + f*x, -(b/a)]) + (3*a^2 + 7*a*b + 4*b^2)*EllipticF[e + f*x, -(b/a)]) - 2*Sqrt[2]*(3*(a + b)^2*(2*a + b - b*Cos[2*(e + f*x)])^2*Cot[e + f*x] + 2*a*b^2*(a + b)*Sin[2*(e + f*x)] + b^2*(7*a + 5*b)*(2*a + b - b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))/(12*a^3*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
171,1,113,122,0.4693639,"\int (d \sin (e+f x))^m \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) (d \sin (e+f x))^m \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{1}{2},-p;\frac{m+3}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{f (m+1)}","-\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,"(AppellF1[(1 + m)/2, 1/2, -p, (3 + m)/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(f*(1 + m)*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",0
172,1,98,220,0.5497511,"\int \sin ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Sin[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sin ^5(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{a+b \sin ^2(e+f x)}{a}\right)^{-p} F_1\left(3;\frac{1}{2},-p;4;\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{6 f}","-\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,"(AppellF1[3, 1/2, -p, 4, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*Sin[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(6*f*((a + b*Sin[e + f*x]^2)/a)^p)","C",0
173,1,98,131,0.3525223,"\int \sin ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Sin[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\sin ^3(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{a+b \sin ^2(e+f x)}{a}\right)^{-p} F_1\left(2;\frac{1}{2},-p;3;\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)}{4 f}","\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,"(AppellF1[2, 1/2, -p, 3, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(4*f*((a + b*Sin[e + f*x]^2)/a)^p)","C",0
174,1,74,74,0.203363,"\int \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[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",1
175,0,0,83,4.8420677,"\int \csc (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]*(a + b*Sin[e + f*x]^2)^p,x]","\int \csc (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","-\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,"Integrate[Csc[e + f*x]*(a + b*Sin[e + f*x]^2)^p, x]","F",-1
176,0,0,83,92.3814567,"\int \csc ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p,x]","\int \csc ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","-\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,"Integrate[Csc[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p, x]","F",-1
177,0,0,83,100.930672,"\int \csc ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p,x]","\int \csc ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","-\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,"Integrate[Csc[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p, x]","F",-1
178,1,102,101,0.5436446,"\int \sin ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[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{a+b \sin ^2(e+f x)}{a}\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*((a + b*Sin[e + f*x]^2)/a)^p)","A",1
179,1,240,99,0.6910049,"\int \sin ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Sin[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p,x]","-\frac{2^{-p-2} \csc (2 (e+f x)) \sqrt{-\frac{b \sin ^2(e+f x)}{a}} \sqrt{\frac{b \cos ^2(e+f x)}{a+b}} (2 a-b \cos (2 (e+f x))+b)^{p+1} \left(2 a (p+2) F_1\left(p+1;\frac{1}{2},\frac{1}{2};p+2;\frac{2 a+b-b \cos (2 (e+f x))}{2 (a+b)},\frac{2 a+b-b \cos (2 (e+f x))}{2 a}\right)-(p+1) (2 a-b \cos (2 (e+f x))+b) F_1\left(p+2;\frac{1}{2},\frac{1}{2};p+3;\frac{2 a+b-b \cos (2 (e+f x))}{2 (a+b)},\frac{2 a+b-b \cos (2 (e+f x))}{2 a}\right)\right)}{b^2 f (p+1) (p+2)}","\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,"-((2^(-2 - p)*Sqrt[(b*Cos[e + f*x]^2)/(a + b)]*(2*a + b - b*Cos[2*(e + f*x)])^(1 + p)*(2*a*(2 + p)*AppellF1[1 + p, 1/2, 1/2, 2 + p, (2*a + b - b*Cos[2*(e + f*x)])/(2*(a + b)), (2*a + b - b*Cos[2*(e + f*x)])/(2*a)] - (1 + p)*AppellF1[2 + p, 1/2, 1/2, 3 + p, (2*a + b - b*Cos[2*(e + f*x)])/(2*(a + b)), (2*a + b - b*Cos[2*(e + f*x)])/(2*a)]*(2*a + b - b*Cos[2*(e + f*x)]))*Csc[2*(e + f*x)]*Sqrt[-((b*Sin[e + f*x]^2)/a)])/(b^2*f*(1 + p)*(2 + p)))","B",0
180,0,0,97,4.8556732,"\int \csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p,x]","\int \csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","-\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,"Integrate[Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p, x]","F",-1
181,0,0,101,6.5953005,"\int \csc ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Csc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p,x]","\int \csc ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","-\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,"Integrate[Csc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p, x]","F",-1
182,1,219,335,0.5150811,"\int \frac{\sin ^7(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Sin[c + d*x]^7/(a + b*Sin[c + d*x]^3),x]","\frac{-32 a^2 \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{-i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b-4 i \text{$\#$1} a+b}\&\right]+96 a \cos (c+d x)+3 b (12 (c+d x)-8 \sin (2 (c+d x))+\sin (4 (c+d x)))}{96 b^2 d}","\frac{2 (-1)^{2/3} a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^{7/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{7/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{-1} a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b^{7/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac{3 \sin (c+d x) \cos (c+d x)}{8 b d}+\frac{3 x}{8 b}",1,"(96*a*Cos[c + d*x] - 32*a^2*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ] + 3*b*(12*(c + d*x) - 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(96*b^2*d)","C",1
183,1,255,273,0.2798954,"\int \frac{\sin ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Sin[c + d*x]^5/(a + b*Sin[c + d*x]^3),x]","\frac{-2 i a \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-4 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]+6 (c+d x)-3 \sin (2 (c+d x))}{12 b d}","-\frac{2 a \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{5/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 a \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 b^{5/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 a \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 b^{5/3} d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}+\frac{x}{2 b}",1,"(6*(c + d*x) - (2*I)*a*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (2*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] - 3*Sin[2*(c + d*x)])/(12*b*d)","C",1
184,1,140,259,0.1805351,"\int \frac{\sin ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Sin[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]","\frac{2 i a \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1} \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b-4 i \text{$\#$1} a+b}\&\right]+3 c+3 d x}{3 b d}","-\frac{2 \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{2 \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{x}{b}",1,"(3*c + 3*d*x + (2*I)*a*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ])/(3*b*d)","C",1
185,1,172,267,0.1835318,"\int \frac{\sin (c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Sin[c + d*x]/(a + b*Sin[c + d*x]^3),x]","-\frac{\text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{-i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b-4 i \text{$\#$1} a+b}\&\right]}{3 d}","\frac{2 (-1)^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{-1} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 \sqrt[3]{a} \sqrt[3]{b} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}",1,"-1/3*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ]/d","C",1
186,1,264,264,0.2738213,"\int \frac{\csc (c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Csc[c + d*x]/(a + b*Sin[c + d*x]^3),x]","-\frac{i b \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-4 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]-6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a d}","-\frac{2 \sqrt[3]{b} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{b} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 a d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 \sqrt[3]{b} \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 a d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"-1/6*(6*Log[Cos[(c + d*x)/2]] - 6*Log[Sin[(c + d*x)/2]] + I*b*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (2*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(a*d)","C",1
187,1,181,287,0.4045753,"\int \frac{\csc ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Csc[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]","\frac{-3 \left(\csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^2\left(\frac{1}{2} (c+d x)\right)-4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+16 i b \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b-8 i \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1} \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b-4 i \text{$\#$1} a+b}\&\right]}{24 a d}","-\frac{2 b \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 b \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{2 b \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{5/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}",1,"((16*I)*b*RootSum[-b + 3*b*#1^2 - (8*I)*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ] - 3*(Csc[(c + d*x)/2]^2 + 4*Log[Cos[(c + d*x)/2]] - 4*Log[Sin[(c + d*x)/2]] - Sec[(c + d*x)/2]^2))/(24*a*d)","C",1
188,1,290,344,2.0231533,"\int \frac{\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Csc[c + d*x]^5/(a + b*Sin[c + d*x]^3),x]","\frac{3 \left(-a \csc ^4\left(\frac{1}{2} (c+d x)\right)-6 a \csc ^2\left(\frac{1}{2} (c+d x)\right)+a \sec ^4\left(\frac{1}{2} (c+d x)\right)+6 a \sec ^2\left(\frac{1}{2} (c+d x)\right)+24 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-24 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-32 b \tan \left(\frac{1}{2} (c+d x)\right)+32 b \cot \left(\frac{1}{2} (c+d x)\right)\right)-64 b^2 \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b-8 i \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{-i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b-4 i \text{$\#$1} a+b}\&\right]}{192 a^2 d}","\frac{2 (-1)^{2/3} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{7/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{2 b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{7/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \sqrt[3]{-1} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{7/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}+\frac{b \cot (c+d x)}{a^2 d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}",1,"(-64*b^2*RootSum[-b + 3*b*#1^2 - (8*I)*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ] + 3*(32*b*Cot[(c + d*x)/2] - 6*a*Csc[(c + d*x)/2]^2 - a*Csc[(c + d*x)/2]^4 - 24*a*Log[Cos[(c + d*x)/2]] + 24*a*Log[Sin[(c + d*x)/2]] + 6*a*Sec[(c + d*x)/2]^2 + a*Sec[(c + d*x)/2]^4 - 32*b*Tan[(c + d*x)/2]))/(192*a^2*d)","C",0
189,1,164,293,0.2826656,"\int \frac{\sin ^6(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Sin[c + d*x]^6/(a + b*Sin[c + d*x]^3),x]","-\frac{8 i a^2 \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1} \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b-4 i \text{$\#$1} a+b}\&\right]+12 a c+12 a d x+9 b \cos (c+d x)-b \cos (3 (c+d x))}{12 b^2 d}","\frac{2 a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^2 d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b^2 d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^2 d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac{a x}{b^2}+\frac{\cos ^3(c+d x)}{3 b d}-\frac{\cos (c+d x)}{b d}",1,"-1/12*(12*a*c + 12*a*d*x + 9*b*Cos[c + d*x] - b*Cos[3*(c + d*x)] + (8*I)*a^2*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ])/(b^2*d)","C",1
190,1,186,281,0.2608553,"\int \frac{\sin ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Sin[c + d*x]^4/(a + b*Sin[c + d*x]^3),x]","\frac{-3 \cos (c+d x)+a \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{-i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b-4 i \text{$\#$1} a+b}\&\right]}{3 b d}","-\frac{2 (-1)^{2/3} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \sqrt[3]{-1} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 b^{4/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{\cos (c+d x)}{b d}",1,"(-3*Cos[c + d*x] + a*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ])/(3*b*d)","C",1
191,1,231,240,0.1861962,"\int \frac{\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]","\frac{i \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-4 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]}{6 d}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 b^{2/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}-\frac{2 \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}",1,"((I/6)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (2*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/d","C",1
192,1,126,245,0.1270739,"\int \frac{1}{a+b \sin ^3(c+d x)} \, dx","Integrate[(a + b*Sin[c + d*x]^3)^(-1),x]","-\frac{2 i \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1} \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b-4 i \text{$\#$1} a+b}\&\right]}{3 d}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}",1,"(((-2*I)/3)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ])/d","C",1
193,1,196,281,0.3119081,"\int \frac{\csc ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]","\frac{2 b \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b-8 i \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{-i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b-4 i \text{$\#$1} a+b}\&\right]+3 \tan \left(\frac{1}{2} (c+d x)\right)-3 \cot \left(\frac{1}{2} (c+d x)\right)}{6 a d}","-\frac{2 (-1)^{2/3} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{-1} \sqrt[3]{b}-\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \sqrt[3]{-1} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{4/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{\cot (c+d x)}{a d}",1,"(-3*Cot[(c + d*x)/2] + 2*b*RootSum[-b + 3*b*#1^2 - (8*I)*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ] + 3*Tan[(c + d*x)/2])/(6*a*d)","C",1
194,1,333,296,2.1497816,"\int \frac{\csc ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Csc[c + d*x]^4/(a + b*Sin[c + d*x]^3),x]","\frac{4 i b^2 \text{RootSum}\left[\text{$\#$1}^6 b-3 \text{$\#$1}^4 b-8 i \text{$\#$1}^3 a+3 \text{$\#$1}^2 b-b\&,\frac{2 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-4 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]+8 a \tan \left(\frac{1}{2} (c+d x)\right)-8 a \cot \left(\frac{1}{2} (c+d x)\right)-\frac{1}{2} a \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+8 a \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-24 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+24 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{24 a^2 d}","\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{2 b^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^2 d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 b^{4/3} \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 a^2 d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}-\frac{2 b^{4/3} \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 a^2 d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}",1,"(-8*a*Cot[(c + d*x)/2] + 24*b*Log[Cos[(c + d*x)/2]] - 24*b*Log[Sin[(c + d*x)/2]] + (4*I)*b^2*RootSum[-b + 3*b*#1^2 - (8*I)*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (2*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + 8*a*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - (a*Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 + 8*a*Tan[(c + d*x)/2])/(24*a^2*d)","C",1
195,1,228,177,0.477921,"\int \frac{\sin ^9(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4),x]","\frac{\cos (c+d x) (120 a-28 b \cos (2 (c+d x))+3 b \cos (4 (c+d x))+89 b)+60 i a^2 \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+i \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \text{$\#$1} \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^6 b-3 \text{$\#$1}^4 b-8 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b-b}\&\right]}{120 b^2 d}","-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{9/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{9/4} d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{(a+b) \cos (c+d x)}{b^2 d}+\frac{\cos ^5(c+d x)}{5 b d}-\frac{2 \cos ^3(c+d x)}{3 b d}",1,"(Cos[c + d*x]*(120*a + 89*b - 28*b*Cos[2*(c + d*x)] + 3*b*Cos[4*(c + d*x)]) + (60*I)*a^2*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ])/(120*b^2*d)","C",1
196,1,310,148,0.286865,"\int \frac{\sin ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4),x]","\frac{-3 i a \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-6 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-3 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+6 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+3 i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]+18 \cos (c+d x)-2 \cos (3 (c+d x))}{24 b d}","-\frac{a \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{7/4} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{7/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\cos ^3(c+d x)}{3 b d}+\frac{\cos (c+d x)}{b d}",1,"(18*Cos[c + d*x] - 2*Cos[3*(c + d*x)] - (3*I)*a*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 6*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 6*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(24*b*d)","C",0
197,1,198,138,0.2505166,"\int \frac{\sin ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]","\frac{2 \cos (c+d x)+i a \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+i \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \text{$\#$1} \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^6 b-3 \text{$\#$1}^4 b-8 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b-b}\&\right]}{2 b d}","-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{5/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{5/4} d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\cos (c+d x)}{b d}",1,"(2*Cos[c + d*x] + I*a*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ])/(2*b*d)","C",1
198,1,285,115,0.1705056,"\int \frac{\sin ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]","-\frac{i \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-6 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-3 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+6 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+3 i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{8 d}","\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 b^{3/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 b^{3/4} d \sqrt{\sqrt{a}-\sqrt{b}}}",1,"((-1/8*I)*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 6*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 6*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/d","C",0
199,1,183,125,0.1596456,"\int \frac{\sin (c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sin[c + d*x]/(a - b*Sin[c + d*x]^4),x]","\frac{i \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+i \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \text{$\#$1} \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^6 b-3 \text{$\#$1}^4 b-8 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b-b}\&\right]}{2 d}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 \sqrt{a} \sqrt[4]{b} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 \sqrt{a} \sqrt[4]{b} d \sqrt{\sqrt{a}+\sqrt{b}}}",1,"((I/2)*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ])/d","C",1
200,1,318,136,0.2562991,"\int \frac{\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Csc[c + d*x]/(a - b*Sin[c + d*x]^4),x]","-\frac{i b \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-6 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-3 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+6 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+3 i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]-8 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a d}","-\frac{\sqrt[4]{b} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt[4]{b} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"-1/8*(8*Log[Cos[(c + d*x)/2]] - 8*Log[Sin[(c + d*x)/2]] + I*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 6*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 6*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a*d)","C",0
201,1,242,184,0.3398069,"\int \frac{\csc ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Csc[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]","\frac{4 i b \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+i \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \text{$\#$1} \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^6 b-3 \text{$\#$1}^4 b-8 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b-b}\&\right]-\csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^2\left(\frac{1}{2} (c+d x)\right)+4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a d}","-\frac{b^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{2 a^{3/2} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{b^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{2 a^{3/2} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{1}{4 a d (1-\cos (c+d x))}+\frac{1}{4 a d (\cos (c+d x)+1)}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}",1,"(-Csc[(c + d*x)/2]^2 - 4*Log[Cos[(c + d*x)/2]] + 4*Log[Sin[(c + d*x)/2]] + (4*I)*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ] + Sec[(c + d*x)/2]^2)/(8*a*d)","C",1
202,1,409,229,1.1742045,"\int \frac{\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Csc[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]","\frac{-8 i b^2 \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-6 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-3 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+6 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+3 i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]-a \csc ^4\left(\frac{1}{2} (c+d x)\right)-6 a \csc ^2\left(\frac{1}{2} (c+d x)\right)+a \sec ^4\left(\frac{1}{2} (c+d x)\right)+6 a \sec ^2\left(\frac{1}{2} (c+d x)\right)+24 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-24 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+64 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-64 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 a^2 d}","-\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,"(-6*a*Csc[(c + d*x)/2]^2 - a*Csc[(c + d*x)/2]^4 - 24*a*Log[Cos[(c + d*x)/2]] - 64*b*Log[Cos[(c + d*x)/2]] + 24*a*Log[Sin[(c + d*x)/2]] + 64*b*Log[Sin[(c + d*x)/2]] - (8*I)*b^2*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 6*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 6*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ] + 6*a*Sec[(c + d*x)/2]^2 + a*Sec[(c + d*x)/2]^4)/(64*a^2*d)","C",0
203,1,172,184,0.920019,"\int \frac{\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]","-\frac{-\frac{16 a^{3/2} \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{16 a^{3/2} \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}+4 (8 a+3 b) (c+d x)-8 b \sin (2 (c+d x))+b \sin (4 (c+d x))}{32 b^2 d}","\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,"-1/32*(4*(8*a + 3*b)*(c + d*x) - (16*a^(3/2)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] + (16*a^(3/2)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] - 8*b*Sin[2*(c + d*x)] + b*Sin[4*(c + d*x)])/(b^2*d)","A",1
204,1,157,155,0.8321818,"\int \frac{\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]","\frac{-\frac{2 a \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{2 a \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}-2 \sqrt{b} (c+d x)+\sqrt{b} \sin (2 (c+d x))}{4 b^{3/2} d}","\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,"(-2*Sqrt[b]*(c + d*x) - (2*a*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - (2*a*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + Sqrt[b]*Sin[2*(c + d*x)])/(4*b^(3/2)*d)","A",1
205,1,143,127,0.4168973,"\int \frac{\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sin[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]","\frac{\frac{\sqrt{a} \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}-2 (c+d x)}{2 b d}","\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,"(-2*(c + d*x) + (Sqrt[a]*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - (Sqrt[a]*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]])/(2*b*d)","A",1
206,1,137,125,0.3569497,"\int \frac{\sin ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]","-\frac{\tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{2 \sqrt{b} d \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{2 \sqrt{b} d \sqrt{\sqrt{a} \sqrt{b}-a}}","\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,"-1/2*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]]/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]*d) - ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]]/(2*Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]*d)","A",1
207,1,128,115,0.2639885,"\int \frac{1}{a-b \sin ^4(c+d x)} \, dx","Integrate[(a - b*Sin[c + d*x]^4)^(-1),x]","\frac{\frac{\tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}}{2 \sqrt{a} d}","\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[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]]/Sqrt[a + Sqrt[a]*Sqrt[b]] - ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]]/Sqrt[-a + Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*d)","A",1
208,1,143,139,1.0830126,"\int \frac{\csc ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]","-\frac{\frac{\sqrt{b} \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}+2 \cot (c+d x)}{2 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,"-1/2*((Sqrt[b]*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] + (Sqrt[b]*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + 2*Cot[c + d*x])/(a*d)","A",1
209,1,165,149,1.5583583,"\int \frac{\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Csc[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]","\frac{\frac{3 b \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{3 b \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}-4 \sqrt{a} \cot (c+d x)-2 \sqrt{a} \cot (c+d x) \csc ^2(c+d x)}{6 a^{3/2} 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,"((3*b*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - (3*b*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] - 4*Sqrt[a]*Cot[c + d*x] - 2*Sqrt[a]*Cot[c + d*x]*Csc[c + d*x]^2)/(6*a^(3/2)*d)","A",1
210,1,174,178,4.6736621,"\int \frac{\csc ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Csc[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]","-\frac{\frac{15 b^{3/2} \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{15 b^{3/2} \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}+2 \cot (c+d x) \left(3 a \csc ^4(c+d x)+4 a \csc ^2(c+d x)+8 a+15 b\right)}{30 a^2 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,"-1/30*((15*b^(3/2)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] + (15*b^(3/2)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + 2*Cot[c + d*x]*(8*a + 15*b + 4*a*Csc[c + d*x]^2 + 3*a*Csc[c + d*x]^4))/(a^2*d)","A",1
211,1,277,197,6.3204134,"\int \frac{\csc ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Csc[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]","\frac{b^2 \tan ^{-1}\left(\frac{\left(\sqrt{a} \sqrt{b}+b\right) \tan (c+d x)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{2 a^{5/2} d \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{b^2 \tanh ^{-1}\left(\frac{\left(\sqrt{a} \sqrt{b}-b\right) \tan (c+d x)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{2 a^{5/2} d \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{\csc ^3(c+d x) (-24 a \cos (c+d x)-35 b \cos (c+d x))}{105 a^2 d}-\frac{2 \csc (c+d x) (24 a \cos (c+d x)+35 b \cos (c+d x))}{105 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac{6 \cot (c+d x) \csc ^4(c+d x)}{35 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[a]*Sqrt[b] + b)*Tan[c + d*x])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b])])/(2*a^(5/2)*Sqrt[a + Sqrt[a]*Sqrt[b]]*d) - (b^2*ArcTanh[((Sqrt[a]*Sqrt[b] - b)*Tan[c + d*x])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b])])/(2*a^(5/2)*Sqrt[-a + Sqrt[a]*Sqrt[b]]*d) - (2*(24*a*Cos[c + d*x] + 35*b*Cos[c + d*x])*Csc[c + d*x])/(105*a^2*d) + ((-24*a*Cos[c + d*x] - 35*b*Cos[c + d*x])*Csc[c + d*x]^3)/(105*a^2*d) - (6*Cot[c + d*x]*Csc[c + d*x]^4)/(35*a*d) - (Cot[c + d*x]*Csc[c + d*x]^6)/(7*a*d)","A",1
212,1,486,236,1.1919317,"\int \frac{\sin ^9(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^2,x]","-\frac{\frac{i a \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+40 \text{$\#$1}^4 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-54 \text{$\#$1}^4 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+20 i \text{$\#$1}^2 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-40 \text{$\#$1}^2 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-27 i \text{$\#$1}^2 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+54 \text{$\#$1}^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-20 i \text{$\#$1}^4 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+27 i \text{$\#$1}^4 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{a-b}+\frac{32 a \cos (c+d x) (2 a-b \cos (2 (c+d x))+b)}{(a-b) (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+32 \cos (c+d x)}{32 b^2 d}","\frac{\sqrt{a} \left(5 \sqrt{a}-6 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 b^{9/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\sqrt{a} \left(5 \sqrt{a}+6 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 b^{9/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{a \cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{4 b^2 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}-\frac{\cos (c+d x)}{b^2 d}",1,"-1/32*(32*Cos[c + d*x] + (32*a*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) + (I*a*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 40*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 54*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (20*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (27*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 40*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 54*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (20*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (27*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b))/(b^2*d)","C",0
213,1,565,210,0.6056784,"\int \frac{\sin ^7(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4)^2,x]","\frac{\frac{16 a (\cos (3 (c+d x))-5 \cos (c+d x))}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-i \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{-6 \text{$\#$1}^6 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+8 \text{$\#$1}^6 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+10 \text{$\#$1}^4 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-24 \text{$\#$1}^4 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+5 i \text{$\#$1}^2 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-3 i a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-10 \text{$\#$1}^2 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-12 i \text{$\#$1}^2 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+4 i b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+24 \text{$\#$1}^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+3 i \text{$\#$1}^6 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-4 i \text{$\#$1}^6 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-5 i \text{$\#$1}^4 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+12 i \text{$\#$1}^4 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+6 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-8 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{32 b d (a-b)}","\frac{\left(3 \sqrt{a}-4 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 b^{7/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\left(3 \sqrt{a}+4 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 b^{7/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{a \cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{4 b d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}",1,"((16*a*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) - I*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (6*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - 8*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (3*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (4*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 24*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (12*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 24*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (12*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - 6*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + 8*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + (3*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 - (4*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(32*(a - b)*b*d)","C",0
214,1,469,217,0.6640303,"\int \frac{\sin ^5(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^2,x]","-\frac{\frac{32 \cos (c+d x) (2 a-b \cos (2 (c+d x))+b)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}+i \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+8 \text{$\#$1}^4 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-22 \text{$\#$1}^4 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+4 i \text{$\#$1}^2 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-8 \text{$\#$1}^2 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-11 i \text{$\#$1}^2 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+22 \text{$\#$1}^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-4 i \text{$\#$1}^4 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+11 i \text{$\#$1}^4 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{32 b d (a-b)}","\frac{\left(\sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 \sqrt{a} b^{5/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\left(\sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 \sqrt{a} b^{5/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{4 b d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}",1,"-1/32*((32*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) + I*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 8*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 22*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (4*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (11*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 8*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 22*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (4*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (11*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/((a - b)*b*d)","C",0
215,1,345,186,0.3410199,"\int \frac{\sin ^3(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4)^2,x]","\frac{\frac{16 (\cos (3 (c+d x))-5 \cos (c+d x))}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-i \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-14 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-7 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+14 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+7 i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{32 d (a-b)}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 \sqrt{a} b^{3/4} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 \sqrt{a} b^{3/4} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cos (c+d x) \left(2-\cos ^2(c+d x)\right)}{4 d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}",1,"((16*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) - I*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 14*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (7*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 14*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (7*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(32*(a - b)*d)","C",0
216,1,469,221,0.4542316,"\int \frac{\sin (c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]/(a - b*Sin[c + d*x]^4)^2,x]","-\frac{\frac{32 \cos (c+d x) (2 a-b \cos (2 (c+d x))+b)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}+i \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^6 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-24 \text{$\#$1}^4 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+10 \text{$\#$1}^4 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-12 i \text{$\#$1}^2 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+24 \text{$\#$1}^2 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+5 i \text{$\#$1}^2 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-10 \text{$\#$1}^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+12 i \text{$\#$1}^4 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-5 i \text{$\#$1}^4 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{32 a d (a-b)}","-\frac{\left(3 \sqrt{a}-2 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right)}{8 a^{3/2} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right)^{3/2}}-\frac{\left(3 \sqrt{a}+2 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\sqrt[4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right)}{8 a^{3/2} \sqrt[4]{b} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\frac{\cos (c+d x) \left(a-b \cos ^2(c+d x)+b\right)}{4 a d (a-b) \left(a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right)}",1,"-1/32*((32*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) + I*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 24*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 10*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (12*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (5*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 24*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 10*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (12*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (5*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a*(a - b)*d)","C",0
217,1,600,325,0.8628436,"\int \frac{\csc (c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[Csc[c + d*x]/(a - b*Sin[c + d*x]^4)^2,x]","\frac{-\frac{i b \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{10 \text{$\#$1}^6 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-8 \text{$\#$1}^6 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-38 \text{$\#$1}^4 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+24 \text{$\#$1}^4 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-19 i \text{$\#$1}^2 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+5 i a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+38 \text{$\#$1}^2 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+12 i \text{$\#$1}^2 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-4 i b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-24 \text{$\#$1}^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-5 i \text{$\#$1}^6 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+4 i \text{$\#$1}^6 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+19 i \text{$\#$1}^4 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-12 i \text{$\#$1}^4 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-10 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+8 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{a-b}+\frac{16 a b (\cos (3 (c+d x))-5 \cos (c+d x))}{(a-b) (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+32 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-32 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{32 a^2 d}","-\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)}{8 a^{3/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\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} \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{\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,"((16*a*b*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) - 32*Log[Cos[(c + d*x)/2]] + 32*Log[Sin[(c + d*x)/2]] - (I*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 8*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (4*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 38*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 24*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (19*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (12*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 38*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 24*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (19*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (12*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 8*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 + (4*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b))/(32*a^2*d)","C",0
218,1,262,320,5.0223018,"\int \frac{\sin ^8(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^2,x]","\frac{-\frac{\sqrt{a} \left(4 \sqrt{a}+5 \sqrt{b}\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\left(\sqrt{a}+\sqrt{b}\right) \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{2 a b (\sin (4 (c+d x))-6 \sin (2 (c+d x)))}{(a-b) (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+\frac{\sqrt{a} \left(4 \sqrt{a}-5 \sqrt{b}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\left(\sqrt{a}-\sqrt{b}\right) \sqrt{\sqrt{a} \sqrt{b}-a}}+8 (c+d x)}{8 b^2 d}","\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)}{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)}{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,"(8*(c + d*x) - (Sqrt[a]*(4*Sqrt[a] + 5*Sqrt[b])*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (Sqrt[a]*(4*Sqrt[a] - 5*Sqrt[b])*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + (2*a*b*(-6*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])))/(8*b^2*d)","A",1
219,1,238,233,2.660127,"\int \frac{\sin ^6(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^2,x]","\frac{\frac{\sqrt{b} \left(\sqrt{a} \sqrt{b}+2 a-3 b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{4 b \sin (2 (c+d x)) (-2 a+b \cos (2 (c+d x))-b)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-\frac{\sqrt{b} \left(\sqrt{a} \sqrt{b}-2 a+3 b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}}{8 b^2 d (a-b)}","-\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*a + Sqrt[a]*Sqrt[b] - 3*b)*Sqrt[b]*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - (Sqrt[b]*(-2*a + Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (4*b*(-2*a - b + b*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]))/(8*(a - b)*b^2*d)","A",1
220,1,225,195,4.4069448,"\int \frac{\sin ^4(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^4/(a - b*Sin[c + d*x]^4)^2,x]","-\frac{\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{2 (\sin (4 (c+d x))-6 \sin (2 (c+d x)))}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}+\frac{\left(\sqrt{a}+\sqrt{b}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}-a}}}{8 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,"-1/8*(((Sqrt[a] - Sqrt[b])*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + ((Sqrt[a] + Sqrt[b])*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) - (2*(-6*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]))/((a - b)*d)","A",1
221,1,255,219,2.1415213,"\int \frac{\sin ^2(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4)^2,x]","\frac{-\frac{\sqrt{a} \left(-\sqrt{a} \sqrt{b}+2 a-b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{4 \sqrt{a} \sin (2 (c+d x)) (2 a-b \cos (2 (c+d x))+b)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-\frac{\sqrt{a} \left(\sqrt{a} \sqrt{b}+2 a-b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}-a}}}{8 a^{3/2} d (a-b)}","\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,"(-((Sqrt[a]*(2*a - Sqrt[a]*Sqrt[b] - b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b])) - (Sqrt[a]*(2*a + Sqrt[a]*Sqrt[b] - b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) - (4*Sqrt[a]*(2*a + b - b*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]))/(8*a^(3/2)*(a - b)*d)","A",1
222,1,230,210,2.870745,"\int \frac{1}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[(a - b*Sin[c + d*x]^4)^(-2),x]","\frac{\frac{\left(-\sqrt{a} \sqrt{b}+4 a-3 b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{2 \sqrt{a} b (\sin (4 (c+d x))-6 \sin (2 (c+d x)))}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-\frac{\left(\sqrt{a} \sqrt{b}+4 a-3 b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}}{8 a^{3/2} d (a-b)}","\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*a - Sqrt[a]*Sqrt[b] - 3*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - ((4*a + Sqrt[a]*Sqrt[b] - 3*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (2*Sqrt[a]*b*(-6*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]))/(8*a^(3/2)*(a - b)*d)","A",1
223,1,274,236,2.2770033,"\int \frac{\csc ^2(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^2} \, dx","Integrate[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4)^2,x]","\frac{-\frac{\left(6 a \sqrt{b}+5 \sqrt{a} b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\left(\sqrt{a}+\sqrt{b}\right) \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{4 \sqrt{a} b \sin (2 (c+d x)) (2 a-b \cos (2 (c+d x))+b)}{(a-b) (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}-\frac{\left(6 a \sqrt{b}-5 \sqrt{a} b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\left(\sqrt{a}-\sqrt{b}\right) \sqrt{\sqrt{a} \sqrt{b}-a}}-8 \sqrt{a} \cot (c+d x)}{8 a^{5/2} 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*a*Sqrt[b] + 5*Sqrt[a]*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])*Sqrt[a + Sqrt[a]*Sqrt[b]])) - ((6*a*Sqrt[b] - 5*Sqrt[a]*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])*Sqrt[-a + Sqrt[a]*Sqrt[b]]) - 8*Sqrt[a]*Cot[c + d*x] - (4*Sqrt[a]*b*(2*a + b - b*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/((a - b)*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])))/(8*a^(5/2)*d)","A",1
224,1,785,315,1.59044,"\int \frac{\sin ^9(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^3,x]","\frac{i \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{4 \text{$\#$1}^6 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-10 \text{$\#$1}^6 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+20 \text{$\#$1}^4 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-56 \text{$\#$1}^4 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+78 \text{$\#$1}^4 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+10 i \text{$\#$1}^2 a^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-20 \text{$\#$1}^2 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-28 i \text{$\#$1}^2 a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 i a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+56 \text{$\#$1}^2 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+39 i \text{$\#$1}^2 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-5 i b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-78 \text{$\#$1}^2 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-2 i \text{$\#$1}^6 a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+5 i \text{$\#$1}^6 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-10 i \text{$\#$1}^4 a^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+28 i \text{$\#$1}^4 a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-39 i \text{$\#$1}^4 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-4 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+10 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]-\frac{32 \cos (c+d x) \left(-9 a^2+b (2 a-5 b) \cos (2 (c+d x))+13 a b+5 b^2\right)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-\frac{512 a (a-b) \cos (c+d x) (2 a-b \cos (2 (c+d x))+b)}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}}{128 b^2 d (a-b)^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{\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{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}",1,"((-32*Cos[c + d*x]*(-9*a^2 + 13*a*b + 5*b^2 + (2*a - 5*b)*b*Cos[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) - (512*a*(a - b)*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2 + I*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-4*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + (2*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 20*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 56*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 78*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (10*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (28*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (39*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 20*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 56*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 78*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (10*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (28*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (39*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 4*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - (2*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 + (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(128*(a - b)^2*b^2*d)","C",0
225,1,630,290,1.1476734,"\int \frac{\sin ^7(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4)^3,x]","\frac{-3 i \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{-2 \text{$\#$1}^6 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+6 \text{$\#$1}^6 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+6 \text{$\#$1}^4 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-34 \text{$\#$1}^4 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+3 i \text{$\#$1}^2 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-6 \text{$\#$1}^2 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-17 i \text{$\#$1}^2 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+3 i b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+34 \text{$\#$1}^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+i \text{$\#$1}^6 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-3 i \text{$\#$1}^6 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-3 i \text{$\#$1}^4 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+17 i \text{$\#$1}^4 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-6 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]-\frac{32 \cos (c+d x) (3 (a-3 b) \cos (2 (c+d x))-7 a+25 b)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}+\frac{512 a (a-b) (\cos (3 (c+d x))-5 \cos (c+d x))}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}}{256 b d (a-b)^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,"((-32*Cos[c + d*x]*(-7*a + 25*b + 3*(a - 3*b)*Cos[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) + (512*a*(a - b)*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2 - (3*I)*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (2*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - 6*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (3*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 6*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 34*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (3*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (17*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 6*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 34*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (3*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (17*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - 2*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + 6*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + I*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 - (3*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(256*(a - b)^2*b*d)","C",0
226,1,786,313,1.3982529,"\int \frac{\sin ^5(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^3,x]","\frac{\frac{i \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{-4 \text{$\#$1}^6 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-2 \text{$\#$1}^6 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-12 \text{$\#$1}^4 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+64 \text{$\#$1}^4 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-10 \text{$\#$1}^4 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-6 i \text{$\#$1}^2 a^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+12 \text{$\#$1}^2 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+32 i \text{$\#$1}^2 a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 i a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-64 \text{$\#$1}^2 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-5 i \text{$\#$1}^2 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+10 \text{$\#$1}^2 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 i \text{$\#$1}^6 a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \text{$\#$1}^6 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+6 i \text{$\#$1}^4 a^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-32 i \text{$\#$1}^4 a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+5 i \text{$\#$1}^4 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+4 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{a}+\frac{32 \cos (c+d x) \left(a^2+b (2 a+b) \cos (2 (c+d x))-9 a b-b^2\right)}{a (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}-\frac{512 (a-b) \cos (c+d x) (2 a-b \cos (2 (c+d x))+b)}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}}{128 b d (a-b)^2}","\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^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{\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,"((32*Cos[c + d*x]*(a^2 - 9*a*b - b^2 + b*(2*a + b)*Cos[2*(c + d*x)]))/(a*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) - (512*(a - b)*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2 + (I*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (4*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (2*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 12*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 64*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (6*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (32*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 12*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 64*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (6*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (32*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - 4*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + (2*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 + I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/a)/(128*(a - b)^2*b*d)","C",0
227,1,631,288,1.1223301,"\int \frac{\sin ^3(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4)^3,x]","\frac{\frac{i \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{-10 \text{$\#$1}^6 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-2 \text{$\#$1}^6 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+94 \text{$\#$1}^4 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-10 \text{$\#$1}^4 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+47 i \text{$\#$1}^2 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-5 i a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-94 \text{$\#$1}^2 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-5 i \text{$\#$1}^2 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+10 \text{$\#$1}^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+5 i \text{$\#$1}^6 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i \text{$\#$1}^6 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-47 i \text{$\#$1}^4 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+5 i \text{$\#$1}^4 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+10 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]}{a}+\frac{32 \cos (c+d x) ((5 a+b) \cos (2 (c+d x))-17 a-b)}{a (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+\frac{512 (a-b) (\cos (3 (c+d x))-5 \cos (c+d x))}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}}{256 d (a-b)^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(-\left((5 a+b) \cos ^2(c+d x)\right)+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,"((32*Cos[c + d*x]*(-17*a - b + (5*a + b)*Cos[2*(c + d*x)]))/(a*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) + (512*(a - b)*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2 + (I*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 94*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 10*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (47*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (5*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 94*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 10*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (47*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (5*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - 10*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + (5*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 + I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/a)/(256*(a - b)^2*d)","C",0
228,1,784,313,1.2798777,"\int \frac{\sin (c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]/(a - b*Sin[c + d*x]^4)^3,x]","\frac{3 i \text{RootSum}\left[\text{$\#$1}^8 b-4 \text{$\#$1}^6 b-16 \text{$\#$1}^4 a+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\&,\frac{-4 \text{$\#$1}^6 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 \text{$\#$1}^6 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+28 \text{$\#$1}^4 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-24 \text{$\#$1}^4 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+10 \text{$\#$1}^4 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+14 i \text{$\#$1}^2 a^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-28 \text{$\#$1}^2 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-12 i \text{$\#$1}^2 a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 i a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+24 \text{$\#$1}^2 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+5 i \text{$\#$1}^2 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+i b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-10 \text{$\#$1}^2 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 i \text{$\#$1}^6 a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \text{$\#$1}^6 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-14 i \text{$\#$1}^4 a^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+12 i \text{$\#$1}^4 a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-5 i \text{$\#$1}^4 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+4 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-2 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^7 b-3 \text{$\#$1}^5 b-8 \text{$\#$1}^3 a+3 \text{$\#$1}^3 b-\text{$\#$1} b}\&\right]-\frac{32 \cos (c+d x) \left(7 a^2+3 b (b-2 a) \cos (2 (c+d x))+5 a b-3 b^2\right)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-\frac{512 a (a-b) \cos (c+d x) (2 a-b \cos (2 (c+d x))+b)}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}}{128 a^2 d (a-b)^2}","-\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((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{\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,"((-32*Cos[c + d*x]*(7*a^2 + 5*a*b - 3*b^2 + 3*b*(-2*a + b)*Cos[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) - (512*a*(a - b)*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2 + (3*I)*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (4*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (2*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 28*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 24*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (14*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (12*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 28*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 24*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 10*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (14*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (12*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (5*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - 4*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + (2*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(128*a^2*(a - b)^2*d)","C",0
229,1,920,617,4.2846768,"\int \frac{\csc (c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[Csc[c + d*x]/(a - b*Sin[c + d*x]^4)^3,x]","\frac{\frac{512 b (\cos (3 (c+d x))-5 \cos (c+d x)) a^2}{(a-b) (-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}+\frac{32 b \cos (c+d x) (-41 a+23 b+(13 a-7 b) \cos (2 (c+d x))) a}{(a-b)^2 (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}-256 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+256 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{i b \text{RootSum}\left[b \text{$\#$1}^8-4 b \text{$\#$1}^6-16 a \text{$\#$1}^4+6 b \text{$\#$1}^4-4 b \text{$\#$1}^2+b\&,\frac{90 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^6+64 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^6-142 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^6-45 i a^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^6-32 i b^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^6+71 i a b \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^6-398 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^4-192 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^4+506 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^4+199 i a^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^4+96 i b^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^4-253 i a b \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^4+398 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^2+192 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^2-506 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^2-199 i a^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^2-96 i b^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^2+253 i a b \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^2-90 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-64 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+142 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+45 i a^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right)+32 i b^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right)-71 i a b \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right)}{b \text{$\#$1}^7-3 b \text{$\#$1}^5-8 a \text{$\#$1}^3+3 b \text{$\#$1}^3-b \text{$\#$1}}\&\right]}{(a-b)^2}}{256 a^3 d}","-\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)}{8 a^{5/2} d \left(\sqrt{a}+\sqrt{b}\right)^{3/2}}-\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} \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{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\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(-\left((5 a+b) \cos ^2(c+d x)\right)+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{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,"((32*a*b*Cos[c + d*x]*(-41*a + 23*b + (13*a - 7*b)*Cos[2*(c + d*x)]))/((a - b)^2*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) + (512*a^2*b*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/((a - b)*(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2) - 256*Log[Cos[(c + d*x)/2]] + 256*Log[Sin[(c + d*x)/2]] - (I*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-90*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 142*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - 64*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + (45*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (71*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (32*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 398*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 506*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 192*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (199*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (253*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (96*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 398*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 506*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 192*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (199*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (253*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (96*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 90*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 142*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 + 64*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - (45*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 + (71*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 - (32*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b)^2)/(256*a^3*d)","C",0
230,1,331,319,3.9795751,"\int \frac{\sin ^8(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^3,x]","\frac{\frac{\left(2 a^{3/2} \sqrt{b}-8 \sqrt{a} b^{3/2}+a b+5 b^2\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{8 b \sin (2 (c+d x)) ((5 b-2 a) \cos (2 (c+d x))+5 a-14 b)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}+\frac{64 a b (a-b) (\sin (4 (c+d x))-6 \sin (2 (c+d x)))}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}+\frac{\sqrt{b} \left(2 \sqrt{a}-5 \sqrt{b}\right) \left(\sqrt{a}+\sqrt{b}\right)^2 \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}}{64 b^2 d (a-b)^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{\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 ^3(c+d x)}{32 a b d (a-b)}+\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{(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*a^(3/2)*Sqrt[b] + a*b - 8*Sqrt[a]*b^(3/2) + 5*b^2)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) + ((2*Sqrt[a] - 5*Sqrt[b])*(Sqrt[a] + Sqrt[b])^2*Sqrt[b]*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + (8*b*(5*a - 14*b + (-2*a + 5*b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) + (64*a*(a - b)*b*(-6*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2)/(64*(a - b)^2*b^2*d)","A",1
231,1,350,343,3.7632331,"\int \frac{\sin ^6(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^3,x]","\frac{\frac{4 b \sin (2 (c+d x)) \left(4 a^2+3 b (a+b) \cos (2 (c+d x))-19 a b-3 b^2\right)}{a (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+\frac{\sqrt{b} \left(10 \sqrt{a} \sqrt{b}+4 a+3 b\right) \left(\sqrt{a}-\sqrt{b}\right)^2 \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{a \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{128 b (a-b) \sin (2 (c+d x)) (2 a-b \cos (2 (c+d x))+b)}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}+\frac{\sqrt{b} \left(\sqrt{a}+\sqrt{b}\right)^2 \left(-10 \sqrt{a} \sqrt{b}+4 a+3 b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{a \sqrt{\sqrt{a} \sqrt{b}-a}}}{64 b^2 d (a-b)^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,"(((Sqrt[a] - Sqrt[b])^2*Sqrt[b]*(4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(a*Sqrt[a + Sqrt[a]*Sqrt[b]]) + ((Sqrt[a] + Sqrt[b])^2*Sqrt[b]*(4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(a*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + (4*b*(4*a^2 - 19*a*b - 3*b^2 + 3*b*(a + b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(a*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) - (128*(a - b)*b*(2*a + b - b*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2)/(64*(a - b)^2*b^2*d)","A",1
232,1,316,313,4.8370815,"\int \frac{\sin ^4(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]^4/(a - b*Sin[c + d*x]^4)^3,x]","\frac{-\frac{3 \left(2 a^{3/2}-3 a \sqrt{b}+b^{3/2}\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{a^{3/2} \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{3 \left(2 a^{3/2}+3 a \sqrt{b}-b^{3/2}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{a^{3/2} \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{8 \sin (2 (c+d x)) ((2 a+b) \cos (2 (c+d x))-7 a-2 b)}{a (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+\frac{64 (a-b) (\sin (4 (c+d x))-6 \sin (2 (c+d x)))}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}}{64 d (a-b)^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{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{\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{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*a^(3/2) - 3*a*Sqrt[b] + b^(3/2))*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(a^(3/2)*Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) - (3*(2*a^(3/2) + 3*a*Sqrt[b] - b^(3/2))*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(a^(3/2)*Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + (8*(-7*a - 2*b + (2*a + b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(a*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) + (64*(a - b)*(-6*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2)/(64*(a - b)^2*d)","A",1
233,1,457,347,6.4298931,"\int \frac{\sin ^2(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]","\frac{24 a^2 \sin (2 (c+d x))+22 a b \sin (2 (c+d x))-11 a b \sin (4 (c+d x))-10 b^2 \sin (2 (c+d x))+5 b^2 \sin (4 (c+d x))}{32 a^2 d (a-b)^2 (-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)}+\frac{\left(11 a^{3/2} b^{3/2}-12 a^{5/2} \sqrt{b}+10 a^2 b-5 \sqrt{a} b^{5/2}-4 a b^2\right) \tan ^{-1}\left(\frac{\left(\sqrt{a} \sqrt{b}+b\right) \tan (c+d x)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{64 a^{5/2} b d \sqrt{\sqrt{a} \sqrt{b}+a} (a-b)^2}-\frac{\left(-11 a^{3/2} b^{3/2}+12 a^{5/2} \sqrt{b}+10 a^2 b+5 \sqrt{a} b^{5/2}-4 a b^2\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a} \sqrt{b}-b\right) \tan (c+d x)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{64 a^{5/2} b d \sqrt{\sqrt{a} \sqrt{b}-a} (a-b)^2}+\frac{-4 a \sin (2 (c+d x))-2 b \sin (2 (c+d x))+b \sin (4 (c+d x))}{a d (a-b) (-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^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}",1,"((-12*a^(5/2)*Sqrt[b] + 10*a^2*b + 11*a^(3/2)*b^(3/2) - 4*a*b^2 - 5*Sqrt[a]*b^(5/2))*ArcTan[((Sqrt[a]*Sqrt[b] + b)*Tan[c + d*x])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b])])/(64*a^(5/2)*Sqrt[a + Sqrt[a]*Sqrt[b]]*(a - b)^2*b*d) - ((12*a^(5/2)*Sqrt[b] + 10*a^2*b - 11*a^(3/2)*b^(3/2) - 4*a*b^2 + 5*Sqrt[a]*b^(5/2))*ArcTanh[((Sqrt[a]*Sqrt[b] - b)*Tan[c + d*x])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b])])/(64*a^(5/2)*Sqrt[-a + Sqrt[a]*Sqrt[b]]*(a - b)^2*b*d) + (-4*a*Sin[2*(c + d*x)] - 2*b*Sin[2*(c + d*x)] + b*Sin[4*(c + d*x)])/(a*(a - b)*d*(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2) + (24*a^2*Sin[2*(c + d*x)] + 22*a*b*Sin[2*(c + d*x)] - 10*b^2*Sin[2*(c + d*x)] - 11*a*b*Sin[4*(c + d*x)] + 5*b^2*Sin[4*(c + d*x)])/(32*a^2*(a - b)^2*d*(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)]))","A",1
234,1,333,319,2.9969049,"\int \frac{1}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[(a - b*Sin[c + d*x]^4)^(-3),x]","\frac{\frac{64 a^{3/2} b (a-b) (\sin (4 (c+d x))-6 \sin (2 (c+d x)))}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}+\frac{\left(50 \sqrt{a} \sqrt{b}+32 a+21 b\right) \left(\sqrt{a}-\sqrt{b}\right)^2 \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{8 \sqrt{a} b \sin (2 (c+d x)) ((6 a-3 b) \cos (2 (c+d x))-19 a+10 b)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-\frac{\left(\sqrt{a}+\sqrt{b}\right)^2 \left(-50 \sqrt{a} \sqrt{b}+32 a+21 b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}}{64 a^{5/2} d (a-b)^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{\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 \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{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,"(((Sqrt[a] - Sqrt[b])^2*(32*a + 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - ((Sqrt[a] + Sqrt[b])^2*(32*a - 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (8*Sqrt[a]*b*(-19*a + 10*b + (6*a - 3*b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) + (64*a^(3/2)*(a - b)*b*(-6*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2)/(64*a^(5/2)*(a - b)^2*d)","A",1
235,1,357,357,5.0308791,"\int \frac{\csc ^2(c+d x)}{\left(a-b \sin ^4(c+d x)\right)^3} \, dx","Integrate[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]","-\frac{\frac{4 b \sin (2 (c+d x)) \left(28 a^2+b (13 b-19 a) \cos (2 (c+d x))+3 a b-13 b^2\right)}{(a-b)^2 (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+\frac{3 \sqrt{b} \left(34 \sqrt{a} \sqrt{b}+20 a+15 b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\left(\sqrt{a}+\sqrt{b}\right)^2 \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{128 a b \sin (2 (c+d x)) (2 a-b \cos (2 (c+d x))+b)}{(a-b) (-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}+\frac{3 \sqrt{b} \left(-34 \sqrt{a} \sqrt{b}+20 a+15 b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\left(\sqrt{a}-\sqrt{b}\right)^2 \sqrt{\sqrt{a} \sqrt{b}-a}}+64 \cot (c+d x)}{64 a^3 d}","\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)}",1,"-1/64*((3*Sqrt[b]*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])^2*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])^2*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 64*Cot[c + d*x] + (4*b*(28*a^2 + 3*a*b - 13*b^2 + b*(-19*a + 13*b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/((a - b)^2*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) + (128*a*b*(2*a + b - b*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/((a - b)*(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2))/(a^3*d)","A",1
236,1,24,25,0.0530534,"\int \frac{1}{1-\sin ^4(x)} \, dx","Integrate[(1 - Sin[x]^4)^(-1),x]","\frac{1}{4} \left(\sqrt{2} \tan ^{-1}\left(\sqrt{2} \tan (x)\right)+2 \tan (x)\right)","\frac{\tan ^{-1}\left(\sqrt{2} \tan (x)\right)}{2 \sqrt{2}}+\frac{\tan (x)}{2}",1,"(Sqrt[2]*ArcTan[Sqrt[2]*Tan[x]] + 2*Tan[x])/4","A",1
237,1,148,487,0.3100652,"\int \frac{1}{a+b \sin ^4(x)} \, dx","Integrate[(a + b*Sin[x]^4)^(-1),x]","\frac{\left(\sqrt{a}-i \sqrt{b}\right) \sqrt{a+i \sqrt{a} \sqrt{b}} \tan ^{-1}\left(\frac{\sqrt{a+i \sqrt{a} \sqrt{b}} \tan (x)}{\sqrt{a}}\right)-\left(\sqrt{a}+i \sqrt{b}\right) \sqrt{-a+i \sqrt{a} \sqrt{b}} \tanh ^{-1}\left(\frac{\sqrt{-a+i \sqrt{a} \sqrt{b}} \tan (x)}{\sqrt{a}}\right)}{2 a (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] - I*Sqrt[b])*Sqrt[a + I*Sqrt[a]*Sqrt[b]]*ArcTan[(Sqrt[a + I*Sqrt[a]*Sqrt[b]]*Tan[x])/Sqrt[a]] - (Sqrt[a] + I*Sqrt[b])*Sqrt[-a + I*Sqrt[a]*Sqrt[b]]*ArcTanh[(Sqrt[-a + I*Sqrt[a]*Sqrt[b]]*Tan[x])/Sqrt[a]])/(2*a*(a + b))","C",1
238,1,45,309,0.074277,"\int \frac{1}{1+\sin ^4(x)} \, dx","Integrate[(1 + Sin[x]^4)^(-1),x]","\frac{\tan ^{-1}\left(\sqrt{1-i} \tan (x)\right)}{2 \sqrt{1-i}}+\frac{\tan ^{-1}\left(\sqrt{1+i} \tan (x)\right)}{2 \sqrt{1+i}}","\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,"ArcTan[Sqrt[1 - I]*Tan[x]]/(2*Sqrt[1 - I]) + ArcTan[Sqrt[1 + I]*Tan[x]]/(2*Sqrt[1 + I])","C",1
239,1,47242,477,31.5947552,"\int \sin (c+d x) \sqrt{a+b \sin ^4(c+d x)} \, dx","Integrate[Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4],x]","\text{Result too large to show}","-\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,"Result too large to show","C",0
240,1,118912,521,31.9983086,"\int \csc (c+d x) \sqrt{a+b \sin ^4(c+d x)} \, dx","Integrate[Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4],x]","\text{Result too large to show}","\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,"Result too large to show","C",0
241,1,47246,484,31.7237659,"\int \frac{\sin ^5(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Sin[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]^4],x]","\text{Result too large to show}","\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,"Result too large to show","C",0
242,1,89374,431,31.827758,"\int \frac{\sin ^3(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Sin[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]","\text{Result too large to show}","-\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,"Result too large to show","C",0
243,1,13300,171,25.3590742,"\int \frac{\sin (c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Sin[c + d*x]/Sqrt[a + b*Sin[c + d*x]^4],x]","\text{Result too large to show}","-\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,"Result too large to show","C",0
244,1,63281,469,31.4712977,"\int \frac{\csc (c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Csc[c + d*x]/Sqrt[a + b*Sin[c + d*x]^4],x]","\text{Result too large to show}","-\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,"Result too large to show","C",0
245,1,119171,776,32.6277478,"\int \frac{\csc ^3(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Csc[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]","\text{Result too large to show}","-\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,"Result too large to show","C",0
246,1,287,499,2.8759174,"\int \frac{\sin ^2(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Sin[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{2 i \cos ^2(c+d x) \sqrt{1+\left(1+\frac{i \sqrt{b}}{\sqrt{a}}\right) \tan ^2(c+d x)} \sqrt{2+\left(2-\frac{2 i \sqrt{b}}{\sqrt{a}}\right) \tan ^2(c+d x)} \left(F\left(i \sinh ^{-1}\left(\sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)|\frac{\sqrt{a}+i \sqrt{b}}{\sqrt{a}-i \sqrt{b}}\right)-\Pi \left(\frac{\sqrt{a}}{\sqrt{a}-i \sqrt{b}};i \sinh ^{-1}\left(\sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)|\frac{\sqrt{a}+i \sqrt{b}}{\sqrt{a}-i \sqrt{b}}\right)\right)}{d \sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b}}","-\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,"((-2*I)*Cos[c + d*x]^2*(EllipticF[I*ArcSinh[Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], (Sqrt[a] + I*Sqrt[b])/(Sqrt[a] - I*Sqrt[b])] - EllipticPi[Sqrt[a]/(Sqrt[a] - I*Sqrt[b]), I*ArcSinh[Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], (Sqrt[a] + I*Sqrt[b])/(Sqrt[a] - I*Sqrt[b])])*Sqrt[1 + (1 + (I*Sqrt[b])/Sqrt[a])*Tan[c + d*x]^2]*Sqrt[2 + (2 - ((2*I)*Sqrt[b])/Sqrt[a])*Tan[c + d*x]^2])/(Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*d*Sqrt[8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)]])","C",1
247,1,304,162,8.7011633,"\int \frac{1}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{2 \sqrt{2} \left(\sqrt{b}+i \sqrt{a}\right) \sin ^2(c+d x) \tan (c+d x) \left(2 \sqrt{a}+i \sqrt{b} \cos (2 (c+d x))-i \sqrt{b}\right) \left(2 i \sqrt{a}+\sqrt{b} \cos (2 (c+d x))-\sqrt{b}\right) \sqrt{\csc ^2(c+d x) \left(-\frac{2 i \sqrt{a}}{\sqrt{b}}-\cos (2 (c+d x))+1\right)} \sqrt{\frac{\cot ^2(c+d x) \left(-a \csc ^2(c+d x)+i \sqrt{a} \sqrt{b}\right)}{\left(\sqrt{a}-i \sqrt{b}\right)^2}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt{a} \csc ^2(c+d x)-i \sqrt{b}}{\sqrt{a}-i \sqrt{b}}}\right)|\frac{i \sqrt{a}}{2 \sqrt{b}}+\frac{1}{2}\right)}{\sqrt{a} d (8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^{3/2}}","\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,"(2*Sqrt[2]*(I*Sqrt[a] + Sqrt[b])*(2*Sqrt[a] - I*Sqrt[b] + I*Sqrt[b]*Cos[2*(c + d*x)])*((2*I)*Sqrt[a] - Sqrt[b] + Sqrt[b]*Cos[2*(c + d*x)])*Sqrt[(1 - ((2*I)*Sqrt[a])/Sqrt[b] - Cos[2*(c + d*x)])*Csc[c + d*x]^2]*Sqrt[(Cot[c + d*x]^2*(I*Sqrt[a]*Sqrt[b] - a*Csc[c + d*x]^2))/(Sqrt[a] - I*Sqrt[b])^2]*EllipticF[ArcSin[Sqrt[((-I)*Sqrt[b] + Sqrt[a]*Csc[c + d*x]^2)/(Sqrt[a] - I*Sqrt[b])]], 1/2 + ((I/2)*Sqrt[a])/Sqrt[b]]*Sin[c + d*x]^2*Tan[c + d*x])/(Sqrt[a]*d*(8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^(3/2))","C",1
248,1,498,493,16.2013225,"\int \frac{\csc ^2(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Csc[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{\cot (c+d x) \sqrt{8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b}}{2 \sqrt{2} a d}-\frac{\sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x) \left(a \left(\tan ^2(c+d x)+1\right)^2+b \tan ^4(c+d x)\right)-\sqrt{a} \sqrt{b} \left(\tan ^2(c+d x)+1\right) \sqrt{1+\left(1-\frac{i \sqrt{b}}{\sqrt{a}}\right) \tan ^2(c+d x)} \sqrt{1+\left(1+\frac{i \sqrt{b}}{\sqrt{a}}\right) \tan ^2(c+d x)} F\left(i \sinh ^{-1}\left(\sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)|\frac{\sqrt{a}+i \sqrt{b}}{\sqrt{a}-i \sqrt{b}}\right)+\sqrt{a} \left(\sqrt{b}+i \sqrt{a}\right) \left(\tan ^2(c+d x)+1\right) \sqrt{1+\left(1-\frac{i \sqrt{b}}{\sqrt{a}}\right) \tan ^2(c+d x)} \sqrt{1+\left(1+\frac{i \sqrt{b}}{\sqrt{a}}\right) \tan ^2(c+d x)} E\left(i \sinh ^{-1}\left(\sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)|\frac{\sqrt{a}+i \sqrt{b}}{\sqrt{a}-i \sqrt{b}}\right)}{a d \sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \left(\tan ^2(c+d x)+1\right)^2 \sqrt{\frac{a \left(\tan ^2(c+d x)+1\right)^2+b \tan ^4(c+d x)}{\left(\tan ^2(c+d x)+1\right)^2}}}","\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,"-1/2*(Sqrt[8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)]]*Cot[c + d*x])/(Sqrt[2]*a*d) - (Sqrt[a]*(I*Sqrt[a] + Sqrt[b])*EllipticE[I*ArcSinh[Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], (Sqrt[a] + I*Sqrt[b])/(Sqrt[a] - I*Sqrt[b])]*(1 + Tan[c + d*x]^2)*Sqrt[1 + (1 - (I*Sqrt[b])/Sqrt[a])*Tan[c + d*x]^2]*Sqrt[1 + (1 + (I*Sqrt[b])/Sqrt[a])*Tan[c + d*x]^2] - Sqrt[a]*Sqrt[b]*EllipticF[I*ArcSinh[Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], (Sqrt[a] + I*Sqrt[b])/(Sqrt[a] - I*Sqrt[b])]*(1 + Tan[c + d*x]^2)*Sqrt[1 + (1 - (I*Sqrt[b])/Sqrt[a])*Tan[c + d*x]^2]*Sqrt[1 + (1 + (I*Sqrt[b])/Sqrt[a])*Tan[c + d*x]^2] + Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]*(b*Tan[c + d*x]^4 + a*(1 + Tan[c + d*x]^2)^2))/(a*Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*d*(1 + Tan[c + d*x]^2)^2*Sqrt[(b*Tan[c + d*x]^4 + a*(1 + Tan[c + d*x]^2)^2)/(1 + Tan[c + d*x]^2)^2])","C",1
249,1,149,384,0.2131082,"\int \frac{1}{a+b \sin ^5(x)} \, dx","Integrate[(a + b*Sin[x]^5)^(-1),x]","\frac{8}{5} i \text{RootSum}\left[-i \text{$\#$1}^{10} b+5 i \text{$\#$1}^8 b-10 i \text{$\#$1}^6 b+32 \text{$\#$1}^5 a+10 i \text{$\#$1}^4 b-5 i \text{$\#$1}^2 b+i b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)}{\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b+16 i \text{$\#$1}^3 a-4 \text{$\#$1}^2 b+b}\&\right]","\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{b}}{\sqrt{a^{2/5}-b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+(-1)^{2/5} \sqrt[5]{b}}{\sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+(-1)^{4/5} \sqrt[5]{b}}{\sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{3/5} \left((-1)^{2/5} \sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{b}\right)}{\sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{-1} \left((-1)^{4/5} \sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{b}\right)}{\sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}",1,"((8*I)/5)*RootSum[I*b - (5*I)*b*#1^2 + (10*I)*b*#1^4 + 32*a*#1^5 - (10*I)*b*#1^6 + (5*I)*b*#1^8 - I*b*#1^10 & , (2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3)/(b - 4*b*#1^2 + (16*I)*a*#1^3 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8) & ]","C",0
250,1,148,171,0.221006,"\int \frac{1}{a+b \sin ^6(x)} \, dx","Integrate[(a + b*Sin[x]^6)^(-1),x]","-\frac{8}{3} \text{RootSum}\left[\text{$\#$1}^6 b-6 \text{$\#$1}^5 b+15 \text{$\#$1}^4 b-64 \text{$\#$1}^3 a-20 \text{$\#$1}^3 b+15 \text{$\#$1}^2 b-6 \text{$\#$1} b+b\&,\frac{2 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right)-i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right)}{\text{$\#$1}^5 b-5 \text{$\#$1}^4 b+10 \text{$\#$1}^3 b-32 \text{$\#$1}^2 a-10 \text{$\#$1}^2 b+5 \text{$\#$1} b-b}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+\sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+\sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-\sqrt[3]{-1} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-\sqrt[3]{-1} \sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b}}}",1,"(-8*RootSum[b - 6*b*#1 + 15*b*#1^2 - 64*a*#1^3 - 20*b*#1^3 + 15*b*#1^4 - 6*b*#1^5 + b*#1^6 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^2)/(-b + 5*b*#1 - 32*a*#1^2 - 10*b*#1^2 + 10*b*#1^3 - 5*b*#1^4 + b*#1^5) & ])/3","C",1
251,1,174,245,0.2608209,"\int \frac{1}{a+b \sin ^8(x)} \, dx","Integrate[(a + b*Sin[x]^8)^(-1),x]","8 \text{RootSum}\left[\text{$\#$1}^8 b-8 \text{$\#$1}^7 b+28 \text{$\#$1}^6 b-56 \text{$\#$1}^5 b+256 \text{$\#$1}^4 a+70 \text{$\#$1}^4 b-56 \text{$\#$1}^3 b+28 \text{$\#$1}^2 b-8 \text{$\#$1} b+b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right)}{\text{$\#$1}^7 b-7 \text{$\#$1}^6 b+21 \text{$\#$1}^5 b-35 \text{$\#$1}^4 b+128 \text{$\#$1}^3 a+35 \text{$\#$1}^3 b-21 \text{$\#$1}^2 b+7 \text{$\#$1} b-b}\&\right]","-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}-\sqrt[4]{b}} \tan (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}-\sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}-i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}-i \sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}+i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}+i \sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}+\sqrt[4]{b}} \tan (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}+\sqrt[4]{b}}}",1,"8*RootSum[b - 8*b*#1 + 28*b*#1^2 - 56*b*#1^3 + 256*a*#1^4 + 70*b*#1^4 - 56*b*#1^5 + 28*b*#1^6 - 8*b*#1^7 + b*#1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3)/(-b + 7*b*#1 - 21*b*#1^2 + 128*a*#1^3 + 35*b*#1^3 - 35*b*#1^4 + 21*b*#1^5 - 7*b*#1^6 + b*#1^7) & ]","C",0
252,1,149,379,0.1909339,"\int \frac{1}{a-b \sin ^5(x)} \, dx","Integrate[(a - b*Sin[x]^5)^(-1),x]","-\frac{8}{5} i \text{RootSum}\left[i \text{$\#$1}^{10} b-5 i \text{$\#$1}^8 b+10 i \text{$\#$1}^6 b+32 \text{$\#$1}^5 a-10 i \text{$\#$1}^4 b+5 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)}{\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-16 i \text{$\#$1}^3 a-4 \text{$\#$1}^2 b+b}\&\right]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{b}-\sqrt[5]{a} \tan \left(\frac{x}{2}\right)}{\sqrt{a^{2/5}-b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{2/5} \sqrt[5]{b}-\sqrt[5]{a} \tan \left(\frac{x}{2}\right)}{\sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{4/5} b^{2/5}}}-\frac{2 \tan ^{-1}\left(\frac{(-1)^{4/5} \sqrt[5]{b}-\sqrt[5]{a} \tan \left(\frac{x}{2}\right)}{\sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+(-1)^{3/5} b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+\sqrt[5]{-1} \sqrt[5]{b}}{\sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}-(-1)^{2/5} b^{2/5}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[5]{a} \tan \left(\frac{x}{2}\right)+(-1)^{3/5} \sqrt[5]{b}}{\sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}\right)}{5 a^{4/5} \sqrt{a^{2/5}+\sqrt[5]{-1} b^{2/5}}}",1,"((-8*I)/5)*RootSum[(-I)*b + (5*I)*b*#1^2 - (10*I)*b*#1^4 + 32*a*#1^5 + (10*I)*b*#1^6 - (5*I)*b*#1^8 + I*b*#1^10 & , (2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3)/(b - 4*b*#1^2 - (16*I)*a*#1^3 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8) & ]","C",0
253,1,148,175,0.1750342,"\int \frac{1}{a-b \sin ^6(x)} \, dx","Integrate[(a - b*Sin[x]^6)^(-1),x]","\frac{8}{3} \text{RootSum}\left[\text{$\#$1}^6 b-6 \text{$\#$1}^5 b+15 \text{$\#$1}^4 b+64 \text{$\#$1}^3 a-20 \text{$\#$1}^3 b+15 \text{$\#$1}^2 b-6 \text{$\#$1} b+b\&,\frac{2 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right)-i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right)}{\text{$\#$1}^5 b-5 \text{$\#$1}^4 b+10 \text{$\#$1}^3 b+32 \text{$\#$1}^2 a-10 \text{$\#$1}^2 b+5 \text{$\#$1} b-b}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-\sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-\sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b}} \tan (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b}}}",1,"(8*RootSum[b - 6*b*#1 + 15*b*#1^2 + 64*a*#1^3 - 20*b*#1^3 + 15*b*#1^4 - 6*b*#1^5 + b*#1^6 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^2)/(-b + 5*b*#1 + 32*a*#1^2 - 10*b*#1^2 + 10*b*#1^3 - 5*b*#1^4 + b*#1^5) & ])/3","C",1
254,1,174,213,0.2125258,"\int \frac{1}{a-b \sin ^8(x)} \, dx","Integrate[(a - b*Sin[x]^8)^(-1),x]","-8 \text{RootSum}\left[\text{$\#$1}^8 b-8 \text{$\#$1}^7 b+28 \text{$\#$1}^6 b-56 \text{$\#$1}^5 b-256 \text{$\#$1}^4 a+70 \text{$\#$1}^4 b-56 \text{$\#$1}^3 b+28 \text{$\#$1}^2 b-8 \text{$\#$1} b+b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right)}{\text{$\#$1}^7 b-7 \text{$\#$1}^6 b+21 \text{$\#$1}^5 b-35 \text{$\#$1}^4 b-128 \text{$\#$1}^3 a+35 \text{$\#$1}^3 b-21 \text{$\#$1}^2 b+7 \text{$\#$1} b-b}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}-\sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}-\sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}-i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}-i \sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}+i \sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}+i \sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}+\sqrt[4]{b}} \tan (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}+\sqrt[4]{b}}}",1,"-8*RootSum[b - 8*b*#1 + 28*b*#1^2 - 56*b*#1^3 - 256*a*#1^4 + 70*b*#1^4 - 56*b*#1^5 + 28*b*#1^6 - 8*b*#1^7 + b*#1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3)/(-b + 7*b*#1 - 21*b*#1^2 - 128*a*#1^3 + 35*b*#1^3 - 35*b*#1^4 + 21*b*#1^5 - 7*b*#1^6 + b*#1^7) & ]","C",0
255,1,411,195,0.1460204,"\int \frac{1}{1+\sin ^5(x)} \, dx","Integrate[(1 + Sin[x]^5)^(-1),x]","\frac{2 \sin \left(\frac{x}{2}\right)}{5 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}-\frac{1}{10} i \text{RootSum}\left[\text{$\#$1}^8-2 i \text{$\#$1}^7-8 \text{$\#$1}^6+14 i \text{$\#$1}^5+30 \text{$\#$1}^4-14 i \text{$\#$1}^3-8 \text{$\#$1}^2+2 i \text{$\#$1}+1\&,\frac{2 \text{$\#$1}^6 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-8 i \text{$\#$1}^5 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-30 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)+80 i \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-15 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-4 \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+30 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-4 \text{$\#$1}^5 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+15 i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+40 \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-8 i \text{$\#$1} \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-2 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)}{4 \text{$\#$1}^7-7 i \text{$\#$1}^6-24 \text{$\#$1}^5+35 i \text{$\#$1}^4+60 \text{$\#$1}^3-21 i \text{$\#$1}^2-8 \text{$\#$1}+i}\&\right]","\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,"(-1/10*I)*RootSum[1 + (2*I)*#1 - 8*#1^2 - (14*I)*#1^3 + 30*#1^4 + (14*I)*#1^5 - 8*#1^6 - (2*I)*#1^7 + #1^8 & , (-2*ArcTan[Sin[x]/(Cos[x] - #1)] + I*Log[1 - 2*Cos[x]*#1 + #1^2] - (8*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 - 4*Log[1 - 2*Cos[x]*#1 + #1^2]*#1 + 30*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^2 - (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^2 + (80*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 + 40*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3 - 30*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^4 + (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^4 - (8*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^5 - 4*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^5 + 2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^6 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^6)/(I - 8*#1 - (21*I)*#1^2 + 60*#1^3 + (35*I)*#1^4 - 24*#1^5 - (7*I)*#1^6 + 4*#1^7) & ] + (2*Sin[x/2])/(5*(Cos[x/2] + Sin[x/2]))","C",1
256,1,79,103,0.1649656,"\int \frac{1}{1+\sin ^6(x)} \, dx","Integrate[(1 + Sin[x]^6)^(-1),x]","\frac{1}{12} \left(-2 \sqrt{3} \tan ^{-1}\left(\frac{1-2 \tan (x)}{\sqrt{3}}\right)+2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \tan (x)\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 \tan (x)+1}{\sqrt{3}}\right)-\log (2-\sin (2 x))+\log (\sin (2 x)+2)\right)","\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,"(-2*Sqrt[3]*ArcTan[(1 - 2*Tan[x])/Sqrt[3]] + 2*Sqrt[2]*ArcTan[Sqrt[2]*Tan[x]] + 2*Sqrt[3]*ArcTan[(1 + 2*Tan[x])/Sqrt[3]] - Log[2 - Sin[2*x]] + Log[2 + Sin[2*x]])/12","A",1
257,1,141,218,0.147024,"\int \frac{1}{1+\sin ^8(x)} \, dx","Integrate[(1 + Sin[x]^8)^(-1),x]","8 \text{RootSum}\left[\text{$\#$1}^8-8 \text{$\#$1}^7+28 \text{$\#$1}^6-56 \text{$\#$1}^5+326 \text{$\#$1}^4-56 \text{$\#$1}^3+28 \text{$\#$1}^2-8 \text{$\#$1}+1\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right)}{\text{$\#$1}^7-7 \text{$\#$1}^6+21 \text{$\#$1}^5-35 \text{$\#$1}^4+163 \text{$\#$1}^3-21 \text{$\#$1}^2+7 \text{$\#$1}-1}\&\right]","\frac{1}{8} \left(\sqrt{1+\sqrt{4-2 \sqrt{2}}}+\sqrt{2+2 \sqrt[4]{2}+2 \sqrt{1+\sqrt{2}}+2 \sqrt{2+\sqrt{2}}}+\sqrt{1+\sqrt{4+2 \sqrt{2}}}\right) \left(x-\tan ^{-1}(\tan (x))\right)+\frac{\tan ^{-1}\left(\sqrt{1-\sqrt[4]{-1}} \tan (x)\right)}{4 \sqrt{1-\sqrt[4]{-1}}}+\frac{\tan ^{-1}\left(\sqrt{1+\sqrt[4]{-1}} \tan (x)\right)}{4 \sqrt{1+\sqrt[4]{-1}}}+\frac{\tan ^{-1}\left(\sqrt{1-(-1)^{3/4}} \tan (x)\right)}{4 \sqrt{1-(-1)^{3/4}}}+\frac{\tan ^{-1}\left(\sqrt{1+(-1)^{3/4}} \tan (x)\right)}{4 \sqrt{1+(-1)^{3/4}}}",1,"8*RootSum[1 - 8*#1 + 28*#1^2 - 56*#1^3 + 326*#1^4 - 56*#1^5 + 28*#1^6 - 8*#1^7 + #1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3)/(-1 + 7*#1 - 21*#1^2 + 163*#1^3 - 35*#1^4 + 21*#1^5 - 7*#1^6 + #1^7) & ]","C",1
258,1,413,187,0.1374237,"\int \frac{1}{1-\sin ^5(x)} \, dx","Integrate[(1 - Sin[x]^5)^(-1),x]","\frac{2 \sin \left(\frac{x}{2}\right)}{5 \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)}+\frac{1}{10} i \text{RootSum}\left[\text{$\#$1}^8+2 i \text{$\#$1}^7-8 \text{$\#$1}^6-14 i \text{$\#$1}^5+30 \text{$\#$1}^4+14 i \text{$\#$1}^3-8 \text{$\#$1}^2-2 i \text{$\#$1}+1\&,\frac{2 \text{$\#$1}^6 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)+8 i \text{$\#$1}^5 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-30 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-80 i \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-15 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+4 \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+30 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+4 \text{$\#$1}^5 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+15 i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-40 \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+8 i \text{$\#$1} \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-2 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)}{4 \text{$\#$1}^7+7 i \text{$\#$1}^6-24 \text{$\#$1}^5-35 i \text{$\#$1}^4+60 \text{$\#$1}^3+21 i \text{$\#$1}^2-8 \text{$\#$1}-i}\&\right]","-\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,"(I/10)*RootSum[1 - (2*I)*#1 - 8*#1^2 + (14*I)*#1^3 + 30*#1^4 - (14*I)*#1^5 - 8*#1^6 + (2*I)*#1^7 + #1^8 & , (-2*ArcTan[Sin[x]/(Cos[x] - #1)] + I*Log[1 - 2*Cos[x]*#1 + #1^2] + (8*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 + 4*Log[1 - 2*Cos[x]*#1 + #1^2]*#1 + 30*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^2 - (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^2 - (80*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 - 40*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3 - 30*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^4 + (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^4 + (8*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^5 + 4*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^5 + 2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^6 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^6)/(-I - 8*#1 + (21*I)*#1^2 + 60*#1^3 - (35*I)*#1^4 - 24*#1^5 + (7*I)*#1^6 + 4*#1^7) & ] + (2*Sin[x/2])/(5*(Cos[x/2] - Sin[x/2]))","C",1
259,1,117,71,0.283735,"\int \frac{1}{1-\sin ^6(x)} \, dx","Integrate[(1 - Sin[x]^6)^(-1),x]","\frac{\cos (x) (-8 \cos (2 x)+\cos (4 x)+15) \left(-6 \sin (x)+i \sqrt[4]{-3} \left(\sqrt{3}+3 i\right) \cos (x) \tan ^{-1}\left(\frac{1}{2} \sqrt[4]{-\frac{1}{3}} \left(\sqrt{3}-3 i\right) \tan (x)\right)+\sqrt[4]{-3} \left(\sqrt{3}-3 i\right) \cos (x) \tan ^{-1}\left(\frac{(-1)^{3/4} \left(\sqrt{3}+3 i\right) \tan (x)}{2 \sqrt[4]{3}}\right)\right)}{144 \left(\sin ^6(x)-1\right)}","\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,"(Cos[x]*(15 - 8*Cos[2*x] + Cos[4*x])*(I*(-3)^(1/4)*(3*I + Sqrt[3])*ArcTan[((-1/3)^(1/4)*(-3*I + Sqrt[3])*Tan[x])/2]*Cos[x] + (-3)^(1/4)*(-3*I + Sqrt[3])*ArcTan[((-1)^(3/4)*(3*I + Sqrt[3])*Tan[x])/(2*3^(1/4))]*Cos[x] - 6*Sin[x]))/(144*(-1 + Sin[x]^6))","C",1
260,1,64,89,0.1695869,"\int \frac{1}{1-\sin ^8(x)} \, dx","Integrate[(1 - Sin[x]^8)^(-1),x]","\frac{1}{8} \left(\frac{2 \tan ^{-1}\left(\sqrt{1-i} \tan (x)\right)}{\sqrt{1-i}}+\frac{2 \tan ^{-1}\left(\sqrt{1+i} \tan (x)\right)}{\sqrt{1+i}}+\sqrt{2} \tan ^{-1}\left(\sqrt{2} \tan (x)\right)+2 \tan (x)\right)","\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,"((2*ArcTan[Sqrt[1 - I]*Tan[x]])/Sqrt[1 - I] + (2*ArcTan[Sqrt[1 + I]*Tan[x]])/Sqrt[1 + I] + Sqrt[2]*ArcTan[Sqrt[2]*Tan[x]] + 2*Tan[x])/8","A",1
261,1,35,38,0.0046335,"\int \frac{\cos ^9(x)}{a-a \sin ^2(x)} \, dx","Integrate[Cos[x]^9/(a - a*Sin[x]^2),x]","\frac{\frac{35 \sin (x)}{64}+\frac{7}{64} \sin (3 x)+\frac{7}{320} \sin (5 x)+\frac{1}{448} \sin (7 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,"((35*Sin[x])/64 + (7*Sin[3*x])/64 + (7*Sin[5*x])/320 + Sin[7*x]/448)/a","A",1
262,1,27,29,0.0032059,"\int \frac{\cos ^7(x)}{a-a \sin ^2(x)} \, dx","Integrate[Cos[x]^7/(a - a*Sin[x]^2),x]","\frac{\frac{5 \sin (x)}{8}+\frac{5}{48} \sin (3 x)+\frac{1}{80} \sin (5 x)}{a}","\frac{\sin ^5(x)}{5 a}-\frac{2 \sin ^3(x)}{3 a}+\frac{\sin (x)}{a}",1,"((5*Sin[x])/8 + (5*Sin[3*x])/48 + Sin[5*x]/80)/a","A",1
263,1,19,18,0.0032265,"\int \frac{\cos ^5(x)}{a-a \sin ^2(x)} \, dx","Integrate[Cos[x]^5/(a - a*Sin[x]^2),x]","\frac{\frac{3 \sin (x)}{4}+\frac{1}{12} \sin (3 x)}{a}","\frac{\sin (x)}{a}-\frac{\sin ^3(x)}{3 a}",1,"((3*Sin[x])/4 + Sin[3*x]/12)/a","A",1
264,1,6,6,0.0023046,"\int \frac{\cos ^3(x)}{a-a \sin ^2(x)} \, dx","Integrate[Cos[x]^3/(a - a*Sin[x]^2),x]","\frac{\sin (x)}{a}","\frac{\sin (x)}{a}",1,"Sin[x]/a","A",1
265,1,37,7,0.0043985,"\int \frac{\cos (x)}{a-a \sin ^2(x)} \, dx","Integrate[Cos[x]/(a - a*Sin[x]^2),x]","\frac{\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)}{a}","\frac{\tanh ^{-1}(\sin (x))}{a}",1,"(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])/a","B",1
266,1,61,35,0.125493,"\int \frac{\sec ^3(x)}{a-a \sin ^2(x)} \, dx","Integrate[Sec[x]^3/(a - a*Sin[x]^2),x]","\frac{\frac{1}{2} (11 \sin (x)+3 \sin (3 x)) \sec ^4(x)-6 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+6 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{16 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,"(-6*Log[Cos[x/2] - Sin[x/2]] + 6*Log[Cos[x/2] + Sin[x/2]] + (Sec[x]^4*(11*Sin[x] + 3*Sin[3*x]))/2)/(16*a)","A",1
267,1,26,33,0.0032878,"\int \frac{\cos ^6(x)}{a-a \sin ^2(x)} \, dx","Integrate[Cos[x]^6/(a - a*Sin[x]^2),x]","\frac{\frac{3 x}{8}+\frac{1}{4} \sin (2 x)+\frac{1}{32} \sin (4 x)}{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 + Sin[2*x]/4 + Sin[4*x]/32)/a","A",1
268,1,18,20,0.0029656,"\int \frac{\cos ^4(x)}{a-a \sin ^2(x)} \, dx","Integrate[Cos[x]^4/(a - a*Sin[x]^2),x]","\frac{\frac{x}{2}+\frac{1}{4} \sin (2 x)}{a}","\frac{x}{2 a}+\frac{\sin (x) \cos (x)}{2 a}",1,"(x/2 + Sin[2*x]/4)/a","A",1
269,1,5,5,0.0008706,"\int \frac{\cos ^2(x)}{a-a \sin ^2(x)} \, dx","Integrate[Cos[x]^2/(a - a*Sin[x]^2),x]","\frac{x}{a}","\frac{x}{a}",1,"x/a","A",1
270,1,45,22,0.0413473,"\int \frac{\sec (x)}{a-a \sin ^2(x)} \, dx","Integrate[Sec[x]/(a - a*Sin[x]^2),x]","\frac{\tan (x) \sec (x)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{2 a}","\frac{\tanh ^{-1}(\sin (x))}{2 a}+\frac{\tan (x) \sec (x)}{2 a}",1,"(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]] + Sec[x]*Tan[x])/(2*a)","B",1
271,1,21,18,0.003712,"\int \frac{\sec ^2(x)}{a-a \sin ^2(x)} \, dx","Integrate[Sec[x]^2/(a - a*Sin[x]^2),x]","\frac{\frac{2 \tan (x)}{3}+\frac{1}{3} \tan (x) \sec ^2(x)}{a}","\frac{\tan ^3(x)}{3 a}+\frac{\tan (x)}{a}",1,"((2*Tan[x])/3 + (Sec[x]^2*Tan[x])/3)/a","A",1
272,1,31,29,0.0041615,"\int \frac{\sec ^4(x)}{a-a \sin ^2(x)} \, dx","Integrate[Sec[x]^4/(a - a*Sin[x]^2),x]","\frac{\frac{8 \tan (x)}{15}+\frac{1}{5} \tan (x) \sec ^4(x)+\frac{4}{15} \tan (x) \sec ^2(x)}{a}","\frac{\tan ^5(x)}{5 a}+\frac{2 \tan ^3(x)}{3 a}+\frac{\tan (x)}{a}",1,"((8*Tan[x])/15 + (4*Sec[x]^2*Tan[x])/15 + (Sec[x]^4*Tan[x])/5)/a","A",1
273,1,27,29,0.0037923,"\int \frac{\cos ^9(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[Cos[x]^9/(a - a*Sin[x]^2)^2,x]","\frac{\frac{5 \sin (x)}{8}+\frac{5}{48} \sin (3 x)+\frac{1}{80} \sin (5 x)}{a^2}","\frac{\sin ^5(x)}{5 a^2}-\frac{2 \sin ^3(x)}{3 a^2}+\frac{\sin (x)}{a^2}",1,"((5*Sin[x])/8 + (5*Sin[3*x])/48 + Sin[5*x]/80)/a^2","A",1
274,1,19,18,0.0027506,"\int \frac{\cos ^7(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[Cos[x]^7/(a - a*Sin[x]^2)^2,x]","\frac{\frac{3 \sin (x)}{4}+\frac{1}{12} \sin (3 x)}{a^2}","\frac{\sin (x)}{a^2}-\frac{\sin ^3(x)}{3 a^2}",1,"((3*Sin[x])/4 + Sin[3*x]/12)/a^2","A",1
275,1,6,6,0.0018889,"\int \frac{\cos ^5(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[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",1
276,1,37,7,0.0037939,"\int \frac{\cos ^3(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[Cos[x]^3/(a - a*Sin[x]^2)^2,x]","\frac{\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)}{a^2}","\frac{\tanh ^{-1}(\sin (x))}{a^2}",1,"(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])/a^2","B",1
277,1,45,22,0.005948,"\int \frac{\cos (x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[Cos[x]/(a - a*Sin[x]^2)^2,x]","\frac{\tan (x) \sec (x)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{2 a^2}","\frac{\tanh ^{-1}(\sin (x))}{2 a^2}+\frac{\tan (x) \sec (x)}{2 a^2}",1,"(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]] + Sec[x]*Tan[x])/(2*a^2)","B",1
278,1,61,35,0.0078552,"\int \frac{\sec (x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[Sec[x]/(a - a*Sin[x]^2)^2,x]","\frac{\frac{1}{2} (11 \sin (x)+3 \sin (3 x)) \sec ^4(x)-6 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+6 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{16 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,"(-6*Log[Cos[x/2] - Sin[x/2]] + 6*Log[Cos[x/2] + Sin[x/2]] + (Sec[x]^4*(11*Sin[x] + 3*Sin[3*x]))/2)/(16*a^2)","A",1
279,1,26,33,0.0028343,"\int \frac{\cos ^8(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[Cos[x]^8/(a - a*Sin[x]^2)^2,x]","\frac{\frac{3 x}{8}+\frac{1}{4} \sin (2 x)+\frac{1}{32} \sin (4 x)}{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 + Sin[2*x]/4 + Sin[4*x]/32)/a^2","A",1
280,1,18,20,0.002627,"\int \frac{\cos ^6(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[Cos[x]^6/(a - a*Sin[x]^2)^2,x]","\frac{\frac{x}{2}+\frac{1}{4} \sin (2 x)}{a^2}","\frac{x}{2 a^2}+\frac{\sin (x) \cos (x)}{2 a^2}",1,"(x/2 + Sin[2*x]/4)/a^2","A",1
281,1,5,5,0.0004607,"\int \frac{\cos ^4(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[Cos[x]^4/(a - a*Sin[x]^2)^2,x]","\frac{x}{a^2}","\frac{x}{a^2}",1,"x/a^2","A",1
282,1,6,6,0.0023697,"\int \frac{\cos ^2(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[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",1
283,1,31,29,0.0035578,"\int \frac{\sec ^2(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[Sec[x]^2/(a - a*Sin[x]^2)^2,x]","\frac{\frac{8 \tan (x)}{15}+\frac{1}{5} \tan (x) \sec ^4(x)+\frac{4}{15} \tan (x) \sec ^2(x)}{a^2}","\frac{\tan ^5(x)}{5 a^2}+\frac{2 \tan ^3(x)}{3 a^2}+\frac{\tan (x)}{a^2}",1,"((8*Tan[x])/15 + (4*Sec[x]^2*Tan[x])/15 + (Sec[x]^4*Tan[x])/5)/a^2","A",1
284,1,41,37,0.0040806,"\int \frac{\sec ^4(x)}{\left(a-a \sin ^2(x)\right)^2} \, dx","Integrate[Sec[x]^4/(a - a*Sin[x]^2)^2,x]","\frac{\frac{16 \tan (x)}{35}+\frac{1}{7} \tan (x) \sec ^6(x)+\frac{6}{35} \tan (x) \sec ^4(x)+\frac{8}{35} \tan (x) \sec ^2(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,"((16*Tan[x])/35 + (8*Sec[x]^2*Tan[x])/35 + (6*Sec[x]^4*Tan[x])/35 + (Sec[x]^6*Tan[x])/7)/a^2","A",1
285,1,87,109,0.2745305,"\int \cos ^6(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^6*(a + b*Sin[e + f*x]^2),x]","\frac{48 (15 a+b) \sin (2 (e+f x))+24 (6 a-b) \sin (4 (e+f x))+16 a \sin (6 (e+f x))+960 a e+960 a f x-16 b \sin (6 (e+f x))-3 b \sin (8 (e+f x))+120 b f x}{3072 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,"(960*a*e + 960*a*f*x + 120*b*f*x + 48*(15*a + b)*Sin[2*(e + f*x)] + 24*(6*a - b)*Sin[4*(e + f*x)] + 16*a*Sin[6*(e + f*x)] - 16*b*Sin[6*(e + f*x)] - 3*b*Sin[8*(e + f*x)])/(3072*f)","A",1
286,1,64,83,0.1514144,"\int \cos ^4(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2),x]","\frac{3 (16 a+b) \sin (2 (e+f x))+(6 a-3 b) \sin (4 (e+f x))+72 a e+72 a f x-b \sin (6 (e+f x))+12 b f x}{192 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,"(72*a*e + 72*a*f*x + 12*b*f*x + 3*(16*a + b)*Sin[2*(e + f*x)] + (6*a - 3*b)*Sin[4*(e + f*x)] - b*Sin[6*(e + f*x)])/(192*f)","A",1
287,1,46,57,0.0802047,"\int \cos ^2(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2),x]","\frac{4 (4 a e+4 a f x+b f x)+8 a \sin (2 (e+f x))-b \sin (4 (e+f x))}{32 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*(4*a*e + 4*a*f*x + b*f*x) + 8*a*Sin[2*(e + f*x)] - b*Sin[4*(e + f*x)])/(32*f)","A",1
288,1,33,30,0.0294239,"\int \left(a+b \sin ^2(e+f x)\right) \, dx","Integrate[a + b*Sin[e + f*x]^2,x]","a x+\frac{b (e+f x)}{2 f}-\frac{b \sin (2 (e+f x))}{4 f}","a x-\frac{b \sin (e+f x) \cos (e+f x)}{2 f}+\frac{b x}{2}",1,"a*x + (b*(e + f*x))/(2*f) - (b*Sin[2*(e + f*x)])/(4*f)","A",1
289,1,36,18,0.0140131,"\int \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2),x]","\frac{a \tan (e+f x)}{f}-\frac{b \tan ^{-1}(\tan (e+f x))}{f}+\frac{b \tan (e+f x)}{f}","\frac{(a+b) \tan (e+f x)}{f}-b x",1,"-((b*ArcTan[Tan[e + f*x]])/f) + (a*Tan[e + f*x])/f + (b*Tan[e + f*x])/f","A",1
290,1,41,30,0.0728151,"\int \sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2),x]","\frac{a \left(\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)\right)}{f}+\frac{b \tan ^3(e+f x)}{3 f}","\frac{(a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f}",1,"(b*Tan[e + f*x]^3)/(3*f) + (a*(Tan[e + f*x] + Tan[e + f*x]^3/3))/f","A",1
291,1,64,50,0.1824995,"\int \sec ^6(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^6*(a + b*Sin[e + f*x]^2),x]","\frac{\tan (e+f x) \left(3 a \tan ^4(e+f x)+10 a \tan ^2(e+f x)+15 a+3 b \sec ^4(e+f x)-b \sec ^2(e+f x)-2 b\right)}{15 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,"(Tan[e + f*x]*(15*a - 2*b - b*Sec[e + f*x]^2 + 3*b*Sec[e + f*x]^4 + 10*a*Tan[e + f*x]^2 + 3*a*Tan[e + f*x]^4))/(15*f)","A",1
292,1,86,72,0.3131529,"\int \sec ^8(e+f x) \left(a+b \sin ^2(e+f x)\right) \, dx","Integrate[Sec[e + f*x]^8*(a + b*Sin[e + f*x]^2),x]","\frac{\tan (e+f x) \left(15 a \tan ^6(e+f x)+63 a \tan ^4(e+f x)+105 a \tan ^2(e+f x)+105 a+15 b \sec ^6(e+f x)-3 b \sec ^4(e+f x)-4 b \sec ^2(e+f x)-8 b\right)}{105 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,"(Tan[e + f*x]*(105*a - 8*b - 4*b*Sec[e + f*x]^2 - 3*b*Sec[e + f*x]^4 + 15*b*Sec[e + f*x]^6 + 105*a*Tan[e + f*x]^2 + 63*a*Tan[e + f*x]^4 + 15*a*Tan[e + f*x]^6))/(105*f)","A",1
293,1,96,156,0.3226507,"\int \cos ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Integrate[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2)^2,x]","\frac{24 \left(48 a^2+16 a b+3 b^2\right) (e+f x)+24 \left(4 a^2-4 a b-b^2\right) \sin (4 (e+f x))-32 a b \sin (6 (e+f x))+96 a (8 a+b) \sin (2 (e+f x))+3 b^2 \sin (8 (e+f x))}{3072 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,"(24*(48*a^2 + 16*a*b + 3*b^2)*(e + f*x) + 96*a*(8*a + b)*Sin[2*(e + f*x)] + 24*(4*a^2 - 4*a*b - b^2)*Sin[4*(e + f*x)] - 32*a*b*Sin[6*(e + f*x)] + 3*b^2*Sin[8*(e + f*x)])/(3072*f)","A",1
294,1,79,116,0.2681776,"\int \cos ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2)^2,x]","\frac{12 (b+(2-2 i) a) (b+(2+2 i) a) (e+f x)-3 b (4 a+b) \sin (4 (e+f x))+3 (4 a-b) (4 a+b) \sin (2 (e+f x))+b^2 \sin (6 (e+f x))}{192 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,"(12*((2 - 2*I)*a + b)*((2 + 2*I)*a + b)*(e + f*x) + 3*(4*a - b)*(4*a + b)*Sin[2*(e + f*x)] - 3*b*(4*a + b)*Sin[4*(e + f*x)] + b^2*Sin[6*(e + f*x)])/(192*f)","C",1
295,1,58,72,0.1188743,"\int \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Integrate[(a + b*Sin[e + f*x]^2)^2,x]","\frac{4 \left(8 a^2+8 a b+3 b^2\right) (e+f x)-8 b (2 a+b) \sin (2 (e+f x))+b^2 \sin (4 (e+f x))}{32 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,"(4*(8*a^2 + 8*a*b + 3*b^2)*(e + f*x) - 8*b*(2*a + b)*Sin[2*(e + f*x)] + b^2*Sin[4*(e + f*x)])/(32*f)","A",1
296,1,48,51,0.3131936,"\int \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Integrate[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^2,x]","\frac{-2 b (4 a+3 b) (e+f x)+4 (a+b)^2 \tan (e+f x)+b^2 \sin (2 (e+f x))}{4 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,"(-2*b*(4*a + 3*b)*(e + f*x) + b^2*Sin[2*(e + f*x)] + 4*(a + b)^2*Tan[e + f*x])/(4*f)","A",1
297,1,57,45,0.3251639,"\int \sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Integrate[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^2,x]","\frac{(a+b) \tan (e+f x) \sec ^2(e+f x) ((a-2 b) \cos (2 (e+f x))+2 a-b)+3 b^2 (e+f x)}{3 f}","\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,"(3*b^2*(e + f*x) + (a + b)*(2*a - b + (a - 2*b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2*Tan[e + f*x])/(3*f)","A",1
298,1,67,53,0.355055,"\int \sec ^6(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Integrate[Sec[e + f*x]^6*(a + b*Sin[e + f*x]^2)^2,x]","\frac{\tan (e+f x) \left(\left(4 a^2-2 a b-6 b^2\right) \sec ^2(e+f x)+8 a^2+3 (a+b)^2 \sec ^4(e+f x)-4 a b+3 b^2\right)}{15 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,"((8*a^2 - 4*a*b + 3*b^2 + (4*a^2 - 2*a*b - 6*b^2)*Sec[e + f*x]^2 + 3*(a + b)^2*Sec[e + f*x]^4)*Tan[e + f*x])/(15*f)","A",1
299,1,92,80,0.4793048,"\int \sec ^8(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Integrate[Sec[e + f*x]^8*(a + b*Sin[e + f*x]^2)^2,x]","\frac{\tan (e+f x) \left(6 \left(3 a^2-a b-4 b^2\right) \sec ^4(e+f x)+\left(24 a^2-8 a b+3 b^2\right) \sec ^2(e+f x)+48 a^2+15 (a+b)^2 \sec ^6(e+f x)-16 a b+6 b^2\right)}{105 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,"((48*a^2 - 16*a*b + 6*b^2 + (24*a^2 - 8*a*b + 3*b^2)*Sec[e + f*x]^2 + 6*(3*a^2 - a*b - 4*b^2)*Sec[e + f*x]^4 + 15*(a + b)^2*Sec[e + f*x]^6)*Tan[e + f*x])/(105*f)","A",1
300,1,107,106,0.468559,"\int \sec ^{10}(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, dx","Integrate[Sec[e + f*x]^10*(a + b*Sin[e + f*x]^2)^2,x]","\frac{\sec ^9(e+f x) \left(252 \left(8 a^2+8 a b+3 b^2\right) \sin (e+f x)+336 \left(4 a^2-a b-b^2\right) \sin (3 (e+f x))+\left(16 a^2-4 a b+b^2\right) (36 \sin (5 (e+f x))+9 \sin (7 (e+f x))+\sin (9 (e+f x)))\right)}{10080 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,"(Sec[e + f*x]^9*(252*(8*a^2 + 8*a*b + 3*b^2)*Sin[e + f*x] + 336*(4*a^2 - a*b - b^2)*Sin[3*(e + f*x)] + (16*a^2 - 4*a*b + b^2)*(36*Sin[5*(e + f*x)] + 9*Sin[7*(e + f*x)] + Sin[9*(e + f*x)])))/(10080*f)","A",1
301,1,109,78,0.275793,"\int \frac{\cos ^7(x)}{a+b \sin ^2(x)} \, dx","Integrate[Cos[x]^7/(a + b*Sin[x]^2),x]","\frac{-2 \sqrt{a} \sqrt{b} \sin (x) \left(120 a^2+4 b (5 a+12 b) \cos (2 x)+340 a b+3 b^2 \cos (4 x)+309 b^2\right)+120 (a+b)^3 \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)-120 (a+b)^3 \tan ^{-1}\left(\frac{\sqrt{a} \csc (x)}{\sqrt{b}}\right)}{240 \sqrt{a} b^{7/2}}","-\frac{\left(a^2+3 a b+3 b^2\right) \sin (x)}{b^3}+\frac{(a+b)^3 \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}+\frac{(a+3 b) \sin ^3(x)}{3 b^2}-\frac{\sin ^5(x)}{5 b}",1,"(-120*(a + b)^3*ArcTan[(Sqrt[a]*Csc[x])/Sqrt[b]] + 120*(a + b)^3*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]] - 2*Sqrt[a]*Sqrt[b]*(120*a^2 + 340*a*b + 309*b^2 + 4*b*(5*a + 12*b)*Cos[2*x] + 3*b^2*Cos[4*x])*Sin[x])/(240*Sqrt[a]*b^(7/2))","A",1
302,1,79,87,0.1873487,"\int \frac{\cos ^6(x)}{a+b \sin ^2(x)} \, dx","Integrate[Cos[x]^6/(a + b*Sin[x]^2),x]","\frac{(a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} b^3}-\frac{4 x \left(8 a^2+20 a b+15 b^2\right)+8 b (a+2 b) \sin (2 x)+b^2 \sin (4 x)}{32 b^3}","-\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,"((a + b)^(5/2)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*b^3) - (4*(8*a^2 + 20*a*b + 15*b^2)*x + 8*b*(a + 2*b)*Sin[2*x] + b^2*Sin[4*x])/(32*b^3)","A",1
303,1,84,54,0.1733757,"\int \frac{\cos ^5(x)}{a+b \sin ^2(x)} \, dx","Integrate[Cos[x]^5/(a + b*Sin[x]^2),x]","\frac{6 (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)-2 \sqrt{a} \sqrt{b} \sin (x) (6 a+b \cos (2 x)+11 b)-6 (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a} \csc (x)}{\sqrt{b}}\right)}{12 \sqrt{a} b^{5/2}}","\frac{(a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2}}-\frac{(a+2 b) \sin (x)}{b^2}+\frac{\sin ^3(x)}{3 b}",1,"(-6*(a + b)^2*ArcTan[(Sqrt[a]*Csc[x])/Sqrt[b]] + 6*(a + b)^2*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]] - 2*Sqrt[a]*Sqrt[b]*(6*a + 11*b + b*Cos[2*x])*Sin[x])/(12*Sqrt[a]*b^(5/2))","A",1
304,1,55,59,0.1506899,"\int \frac{\cos ^4(x)}{a+b \sin ^2(x)} \, dx","Integrate[Cos[x]^4/(a + b*Sin[x]^2),x]","\frac{\frac{4 (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a}}-2 (2 a x+3 b x+b \sin (x) \cos (x))}{4 b^2}","-\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,"((4*(a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/Sqrt[a] - 2*(2*a*x + 3*b*x + b*Cos[x]*Sin[x]))/(4*b^2)","A",1
305,1,36,36,0.0232929,"\int \frac{\cos ^3(x)}{a+b \sin ^2(x)} \, dx","Integrate[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",1
306,1,39,39,0.0560955,"\int \frac{\cos ^2(x)}{a+b \sin ^2(x)} \, dx","Integrate[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",1
307,1,25,25,0.0073813,"\int \frac{\cos (x)}{a+b \sin ^2(x)} \, dx","Integrate[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",1
308,1,96,40,0.1261201,"\int \frac{\sec (x)}{a+b \sin ^2(x)} \, dx","Integrate[Sec[x]/(a + b*Sin[x]^2),x]","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)-\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \csc (x)}{\sqrt{b}}\right)+2 \sqrt{a} \left(\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)\right)}{2 \sqrt{a} (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[a]*Csc[x])/Sqrt[b]]) + Sqrt[b]*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]] + 2*Sqrt[a]*(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]]))/(2*Sqrt[a]*(a + b))","B",1
309,1,39,39,0.0830449,"\int \frac{\sec ^2(x)}{a+b \sin ^2(x)} \, dx","Integrate[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",1
310,1,147,61,0.3199895,"\int \frac{\sec ^3(x)}{a+b \sin ^2(x)} \, dx","Integrate[Sec[x]^3/(a + b*Sin[x]^2),x]","\frac{\frac{2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \csc (x)}{\sqrt{b}}\right)}{\sqrt{a}}+\frac{a+b}{\left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)^2}-\frac{a+b}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}-2 (a+3 b) \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+2 (a+3 b) \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{4 (a+b)^2}","\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,"((-2*b^(3/2)*ArcTan[(Sqrt[a]*Csc[x])/Sqrt[b]])/Sqrt[a] + (2*b^(3/2)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/Sqrt[a] - 2*(a + 3*b)*Log[Cos[x/2] - Sin[x/2]] + 2*(a + 3*b)*Log[Cos[x/2] + Sin[x/2]] + (a + b)/(Cos[x/2] - Sin[x/2])^2 - (a + b)/(Cos[x/2] + Sin[x/2])^2)/(4*(a + b)^2)","B",1
311,1,59,59,0.2192113,"\int \frac{\sec ^4(x)}{a+b \sin ^2(x)} \, dx","Integrate[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 (x) \left((a+b) \sec ^2(x)+2 a+5 b\right)}{3 (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)) + ((2*a + 5*b + (a + b)*Sec[x]^2)*Tan[x])/(3*(a + b)^2)","A",1
312,1,214,93,1.2453094,"\int \frac{\sec ^5(x)}{a+b \sin ^2(x)} \, dx","Integrate[Sec[x]^5/(a + b*Sin[x]^2),x]","-\frac{2 \left(3 a^2+10 a b+15 b^2\right) \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-2 \left(3 a^2+10 a b+15 b^2\right) \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-\frac{8 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{8 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \csc (x)}{\sqrt{b}}\right)}{\sqrt{a}}+\frac{(a+b) (3 a+7 b)}{\sin (x)-1}+\frac{(a+b) (3 a+7 b)}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}-\frac{(a+b)^2}{\left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)^4}+\frac{(a+b)^2}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4}}{16 (a+b)^3}","\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,"-1/16*((8*b^(5/2)*ArcTan[(Sqrt[a]*Csc[x])/Sqrt[b]])/Sqrt[a] - (8*b^(5/2)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/Sqrt[a] + 2*(3*a^2 + 10*a*b + 15*b^2)*Log[Cos[x/2] - Sin[x/2]] - 2*(3*a^2 + 10*a*b + 15*b^2)*Log[Cos[x/2] + Sin[x/2]] - (a + b)^2/(Cos[x/2] - Sin[x/2])^4 + (a + b)^2/(Cos[x/2] + Sin[x/2])^4 + ((a + b)*(3*a + 7*b))/(Cos[x/2] + Sin[x/2])^2 + ((a + b)*(3*a + 7*b))/(-1 + Sin[x]))/(a + b)^3","B",1
313,1,90,87,0.3750558,"\int \frac{\sec ^6(x)}{a+b \sin ^2(x)} \, dx","Integrate[Sec[x]^6/(a + b*Sin[x]^2),x]","\frac{\tan (x) \left(\left(4 a^2+13 a b+9 b^2\right) \sec ^2(x)+8 a^2+3 (a+b)^2 \sec ^4(x)+26 a b+33 b^2\right)}{15 (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{\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)) + ((8*a^2 + 26*a*b + 33*b^2 + (4*a^2 + 13*a*b + 9*b^2)*Sec[x]^2 + 3*(a + b)^2*Sec[x]^4)*Tan[x])/(15*(a + b)^3)","A",1
314,1,90,113,0.2972427,"\int \frac{\cos ^6(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Integrate[Cos[x]^6/(a + b*Sin[x]^2)^2,x]","\frac{-\frac{2 (4 a-b) (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{a^{3/2}}+2 x (4 a+5 b)+\frac{2 b (a+b)^2 \sin (2 x)}{a (2 a-b \cos (2 x)+b)}+b \sin (2 x)}{4 b^3}","-\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,"(2*(4*a + 5*b)*x - (2*(4*a - b)*(a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/a^(3/2) + b*Sin[2*x] + (2*b*(a + b)^2*Sin[2*x])/(a*(2*a + b - b*Cos[2*x])))/(4*b^3)","A",1
315,1,118,72,0.3398361,"\int \frac{\cos ^5(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Integrate[Cos[x]^5/(a + b*Sin[x]^2)^2,x]","\frac{\frac{\left(-3 a^2-2 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{\left(3 a^2+2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \csc (x)}{\sqrt{b}}\right)}{a^{3/2}}+\frac{4 \sqrt{b} (a+b)^2 \sin (x)}{a (2 a-b \cos (2 x)+b)}+4 \sqrt{b} \sin (x)}{4 b^{5/2}}","-\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^2 + 2*a*b - b^2)*ArcTan[(Sqrt[a]*Csc[x])/Sqrt[b]])/a^(3/2) + ((-3*a^2 - 2*a*b + b^2)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/a^(3/2) + 4*Sqrt[b]*Sin[x] + (4*Sqrt[b]*(a + b)^2*Sin[x])/(a*(2*a + b - b*Cos[2*x])))/(4*b^(5/2))","A",1
316,1,79,75,0.3187128,"\int \frac{\cos ^4(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Integrate[Cos[x]^4/(a + b*Sin[x]^2)^2,x]","\frac{\frac{\left(-2 a^2-a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{a^{3/2} \sqrt{a+b}}+\frac{b (a+b) \sin (2 x)}{a (2 a-b \cos (2 x)+b)}+2 x}{2 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,"(2*x + ((-2*a^2 - a*b + b^2)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(a^(3/2)*Sqrt[a + b]) + (b*(a + b)*Sin[2*x])/(a*(2*a + b - b*Cos[2*x])))/(2*b^2)","A",1
317,1,59,59,0.0667609,"\int \frac{\cos ^3(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Integrate[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,"-1/2*((a - b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(a^(3/2)*b^(3/2)) + ((a + b)*Sin[x])/(2*a*b*(a + b*Sin[x]^2))","A",1
318,1,59,54,0.1326256,"\int \frac{\cos ^2(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Integrate[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{\sin (2 x)}{2 a (-2 a+b \cos (2 x)-b)}","\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]) - Sin[2*x]/(2*a*(-2*a - b + b*Cos[2*x]))","A",1
319,1,48,48,0.0550236,"\int \frac{\cos (x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Integrate[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",1
320,1,130,73,0.5286372,"\int \frac{\sec (x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Integrate[Sec[x]/(a + b*Sin[x]^2)^2,x]","\frac{\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{a^{3/2}}-\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{a} \csc (x)}{\sqrt{b}}\right)}{a^{3/2}}+4 \left(\frac{b (a+b) \sin (x)}{a (2 a-b \cos (2 x)+b)}-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{4 (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[a]*Csc[x])/Sqrt[b]])/a^(3/2)) + (Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/a^(3/2) + 4*(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]] + (b*(a + b)*Sin[x])/(a*(2*a + b - b*Cos[2*x]))))/(4*(a + b)^2)","A",1
321,1,76,76,0.5102791,"\int \frac{\sec ^2(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Integrate[Sec[x]^2/(a + b*Sin[x]^2)^2,x]","\frac{1}{2} \left(\frac{b (4 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{a^{3/2} (a+b)^{5/2}}+\frac{\frac{b^2 \sin (2 x)}{a (2 a-b \cos (2 x)+b)}+2 \tan (x)}{(a+b)^2}\right)","\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]])/(a^(3/2)*(a + b)^(5/2)) + ((b^2*Sin[2*x])/(a*(2*a + b - b*Cos[2*x])) + 2*Tan[x])/(a + b)^2)/2","A",1
322,1,183,109,1.0426558,"\int \frac{\sec ^3(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Integrate[Sec[x]^3/(a + b*Sin[x]^2)^2,x]","\frac{\frac{b^{3/2} (5 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right)}{a^{3/2}}-\frac{b^{3/2} (5 a+b) \tan ^{-1}\left(\frac{\sqrt{a} \csc (x)}{\sqrt{b}}\right)}{a^{3/2}}+\frac{4 b^2 (a+b) \sin (x)}{a (2 a-b \cos (2 x)+b)}+\frac{a+b}{\left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)^2}-\frac{a+b}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}-2 (a+5 b) \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+2 (a+5 b) \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{4 (a+b)^3}","\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[a]*Csc[x])/Sqrt[b]])/a^(3/2)) + (b^(3/2)*(5*a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/a^(3/2) - 2*(a + 5*b)*Log[Cos[x/2] - Sin[x/2]] + 2*(a + 5*b)*Log[Cos[x/2] + Sin[x/2]] + (a + b)/(Cos[x/2] - Sin[x/2])^2 - (a + b)/(Cos[x/2] + Sin[x/2])^2 + (4*b^2*(a + b)*Sin[x])/(a*(2*a + b - b*Cos[2*x])))/(4*(a + b)^3)","A",1
323,1,97,96,0.9577238,"\int \frac{\sec ^4(x)}{\left(a+b \sin ^2(x)\right)^2} \, dx","Integrate[Sec[x]^4/(a + b*Sin[x]^2)^2,x]","\frac{1}{6} \left(\frac{3 b^2 (6 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right)}{a^{3/2} (a+b)^{7/2}}+\frac{\frac{3 b^3 \sin (2 x)}{a (2 a-b \cos (2 x)+b)}+2 (a+b) \tan (x) \sec ^2(x)+4 a \tan (x)+16 b \tan (x)}{(a+b)^3}\right)","\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,"((3*b^2*(6*a + b)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(a^(3/2)*(a + b)^(7/2)) + ((3*b^3*Sin[2*x])/(a*(2*a + b - b*Cos[2*x])) + 4*a*Tan[x] + 16*b*Tan[x] + 2*(a + b)*Sec[x]^2*Tan[x])/(a + b)^3)/6","A",1
324,1,125,117,0.4363308,"\int \cos ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{a+b \sin ^2(e+f x)} \left(\sqrt{a} (a+4 b) \sinh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a}}\right)-\sqrt{b} \sin (e+f x) \left(a+2 b \sin ^2(e+f x)-4 b\right) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}\right)}{8 b^{3/2} f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\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,"(Sqrt[a + b*Sin[e + f*x]^2]*(Sqrt[a]*(a + 4*b)*ArcSinh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a]] - Sqrt[b]*Sin[e + f*x]*(a - 4*b + 2*b*Sin[e + f*x]^2)*Sqrt[1 + (b*Sin[e + f*x]^2)/a]))/(8*b^(3/2)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])","A",1
325,1,96,72,0.2611754,"\int \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{a^{3/2} \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a}}\right)+\sqrt{b} \sin (e+f x) \left(a+b \sin ^2(e+f x)\right)}{2 \sqrt{b} f \sqrt{a+b \sin ^2(e+f 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}",1,"(Sqrt[b]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2) + a^(3/2)*ArcSinh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a]]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(2*Sqrt[b]*f*Sqrt[a + b*Sin[e + f*x]^2])","A",1
326,1,129,82,0.2718868,"\int \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\frac{\sqrt{a} \sqrt{-b} \sin ^{-1}\left(\frac{\sqrt{-b} \sin (e+f x)}{\sqrt{a}}\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}}}{\sqrt{2 a-b \cos (2 (e+f x))+b}}+\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{2 a+2 b} \sin (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\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[a + b]*ArcTanh[(Sqrt[2*a + 2*b]*Sin[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] + (Sqrt[a]*Sqrt[-b]*ArcSin[(Sqrt[-b]*Sin[e + f*x])/Sqrt[a]]*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]])/f","A",1
327,1,164,82,2.1650633,"\int \sec ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sin (e+f x) \left(\sqrt{2} a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} \tanh ^{-1}\left(\frac{\sqrt{\frac{(a+b) \sin ^2(e+f x)}{a}}}{\sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}\right)+\sec ^2(e+f x) \sqrt{\frac{(a+b) \sin ^2(e+f x)}{a}} (2 a-b \cos (2 (e+f x))+b)\right)}{4 f \sqrt{\frac{(a+b) \sin ^2(e+f x)}{a}} \sqrt{a+b \sin ^2(e+f 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}",1,"(Sin[e + f*x]*(Sqrt[2]*a*ArcTanh[Sqrt[((a + b)*Sin[e + f*x]^2)/a]/Sqrt[1 + (b*Sin[e + f*x]^2)/a]]*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a] + (2*a + b - b*Cos[2*(e + f*x)])*Sec[e + f*x]^2*Sqrt[((a + b)*Sin[e + f*x]^2)/a]))/(4*f*Sqrt[((a + b)*Sin[e + f*x]^2)/a]*Sqrt[a + b*Sin[e + f*x]^2])","A",1
328,1,669,143,13.3015214,"\int \sec ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\tan (e+f x) \sec ^3(e+f x) \left(\frac{b \sin ^2(e+f x)}{a}+1\right) \left(10 b \sin ^2(e+f x) \sqrt{-\frac{(a+b) \tan ^2(e+f x) \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a^2}}+15 a \sqrt{-\frac{(a+b) \tan ^2(e+f x) \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a^2}}+32 b \sin ^2(e+f x) \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{7/2} \, _2F_1\left(2,4;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}+32 a \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{7/2} \, _2F_1\left(2,4;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}-32 b \sin ^2(e+f x) \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{5/2} \, _2F_1\left(2,4;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}-32 a \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{5/2} \, _2F_1\left(2,4;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}-15 a \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right)-10 b \sin ^2(e+f x) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right)-20 b \sin ^2(e+f x) \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}-30 a \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}\right)}{40 f \sqrt{a+b \sin ^2(e+f x)} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}}","\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,"-1/40*(Sec[e + f*x]^3*(1 + (b*Sin[e + f*x]^2)/a)*Tan[e + f*x]*(-15*a*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]] - 10*b*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^2 - 30*a*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2) - 20*b*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2) - 32*a*Hypergeometric2F1[2, 4, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2) - 32*b*Hypergeometric2F1[2, 4, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2) + 32*a*Hypergeometric2F1[2, 4, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(7/2) + 32*b*Hypergeometric2F1[2, 4, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(7/2) + 15*a*Sqrt[-(((a + b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)*Tan[e + f*x]^2)/a^2)] + 10*b*Sin[e + f*x]^2*Sqrt[-(((a + b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)*Tan[e + f*x]^2)/a^2)]))/(f*Sqrt[a + b*Sin[e + f*x]^2]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2))","C",0
329,1,199,220,1.3798315,"\int \cos ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{-\sqrt{2} b \sin (2 (e+f x)) \left(8 a^2-4 b (4 a-3 b) \cos (2 (e+f x))-32 a b+3 b^2 \cos (4 (e+f x))-15 b^2\right)+32 a \left(a^2+4 a b+3 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-16 a \left(2 a^2+7 a b-3 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{240 b^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(-16*a*(2*a^2 + 7*a*b - 3*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 32*a*(a^2 + 4*a*b + 3*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(8*a^2 - 32*a*b - 15*b^2 - 4*(4*a - 3*b)*b*Cos[2*(e + f*x)] + 3*b^2*Cos[4*(e + f*x)])*Sin[2*(e + f*x)])/(240*b^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
330,1,158,159,0.8626322,"\int \cos ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Cos[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{b \sin (2 (e+f x)) (2 a-b \cos (2 (e+f x))+b)+2 \sqrt{2} a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 \sqrt{2} a (a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 \sqrt{2} b f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-2*Sqrt[2]*a*(a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*Sqrt[2]*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + b*(2*a + b - b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*Sqrt[2]*b*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
331,1,61,51,0.0782933,"\int \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)])/(f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
332,1,134,131,0.472461,"\int \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{2} \tan (e+f x) (2 a-b \cos (2 (e+f x))+b)+2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + Sqrt[2]*(2*a + b - b*Cos[2*(e + f*x)])*Tan[e + f*x])/(2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
333,1,187,196,1.8298792,"\int \sec ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sec[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\frac{\tan (e+f x) \sec ^2(e+f x) \left(\left(8 a^2-4 b^2\right) \cos (2 (e+f x))+(2 a+b) (8 a-b \cos (4 (e+f x))+5 b)\right)}{2 \sqrt{2}}+4 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a (2 a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 4*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + (((8*a^2 - 4*b^2)*Cos[2*(e + f*x)] + (2*a + b)*(8*a + 5*b - b*Cos[4*(e + f*x)]))*Sec[e + f*x]^2*Tan[e + f*x])/(2*Sqrt[2]))/(6*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
334,1,149,157,0.8106818,"\int \cos ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cos[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a+b \sin ^2(e+f x)} \left(3 a^{3/2} (a+6 b) \sinh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a}}\right)+\sqrt{b} \sin (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} \left(-2 b (7 a-6 b) \sin ^2(e+f x)-3 a (a-10 b)-8 b^2 \sin ^4(e+f x)\right)\right)}{48 b^{3/2} f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}","\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,"(Sqrt[a + b*Sin[e + f*x]^2]*(3*a^(3/2)*(a + 6*b)*ArcSinh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a]] + Sqrt[b]*Sin[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a]*(-3*a*(a - 10*b) - 2*(7*a - 6*b)*b*Sin[e + f*x]^2 - 8*b^2*Sin[e + f*x]^4)))/(48*b^(3/2)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])","A",1
335,1,93,104,0.4910629,"\int \cos (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cos[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a+b \sin ^2(e+f x)} \left(\frac{3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+5 a \sin (e+f x)+2 b \sin ^3(e+f x)\right)}{8 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,"(Sqrt[a + b*Sin[e + f*x]^2]*(5*a*Sin[e + f*x] + 2*b*Sin[e + f*x]^3 + (3*a^(3/2)*ArcSinh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a]])/(Sqrt[b]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])))/(8*f)","A",1
336,1,233,121,0.6654165,"\int \sec (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} \left(4 a^2+5 a b+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{2 a+2 b} \sin (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)-2 \sqrt{b} \sqrt{a+b} \left(\sqrt{b} \sin (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} (3 a+2 b) \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{b} \sin (e+f x)\right)\right)+\sqrt{2} b (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \sin (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)}{4 \sqrt{2} f \sqrt{a+b}}","-\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[2]*b*(3*a + 2*b)*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Sin[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] + Sqrt[2]*(4*a^2 + 5*a*b + 2*b^2)*ArcTanh[(Sqrt[2*a + 2*b]*Sin[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] - 2*Sqrt[b]*Sqrt[a + b]*(Sqrt[2]*(3*a + 2*b)*Log[Sqrt[2*a + b - b*Cos[2*(e + f*x)]] + Sqrt[2]*Sqrt[b]*Sin[e + f*x]] + Sqrt[b]*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*Sin[e + f*x]))/(4*Sqrt[2]*Sqrt[a + b]*f)","A",1
337,1,210,127,0.9495258,"\int \sec ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{2 \left(a^2-a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{2 a+2 b} \sin (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)+\sqrt{a+b} \left(4 b^{3/2} \log \left(\sqrt{2 a-b \cos (2 (e+f x))+b}+\sqrt{2} \sqrt{b} \sin (e+f x)\right)+\sqrt{2} (a+b) \tan (e+f x) \sec (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b}\right)-2 b^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a+b} \sin (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)}{4 f \sqrt{a+b}}","\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,"(-2*b^2*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Sin[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] + 2*(a^2 - a*b - b^2)*ArcTanh[(Sqrt[2*a + 2*b]*Sin[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]] + Sqrt[a + b]*(4*b^(3/2)*Log[Sqrt[2*a + b - b*Cos[2*(e + f*x)]] + Sqrt[2]*Sqrt[b]*Sin[e + f*x]] + Sqrt[2]*(a + b)*Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*Sec[e + f*x]*Tan[e + f*x]))/(4*Sqrt[a + b]*f)","A",1
338,1,63,122,0.1271841,"\int \sec ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]^5*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{a^2 \sin (e+f x) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{(a+b) \sin ^2(e+f x)}{b \sin ^2(e+f x)+a}\right)}{f \sqrt{a+b \sin ^2(e+f 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}",1,"(a^2*Hypergeometric2F1[1/2, 3, 3/2, ((a + b)*Sin[e + f*x]^2)/(a + b*Sin[e + f*x]^2)]*Sin[e + f*x])/(f*Sqrt[a + b*Sin[e + f*x]^2])","C",1
339,1,938,195,15.2166409,"\int \sec ^7(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]^7*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{a^2 \sec ^3(e+f x) \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^2 \tan (e+f x) \left(256 b \, _2F_1\left(2,5;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sin ^2(e+f x) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{9/2}+256 a \, _2F_1\left(2,5;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{9/2}-512 b \, _2F_1\left(2,5;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sin ^2(e+f x) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{7/2}-512 a \, _2F_1\left(2,5;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{7/2}-80 b \sin ^2(e+f x) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{5/2}+256 b \, _2F_1\left(2,5;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sin ^2(e+f x) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{5/2}-120 a \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{5/2}+256 a \, _2F_1\left(2,5;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{5/2}+140 b \sin ^2(e+f x) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{3/2}+210 a \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{3/2}+30 b \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right) \sin ^2(e+f x)+45 a \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right)-30 b \sin ^2(e+f x) \sqrt{-\frac{(a+b) \sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right) \tan ^2(e+f x)}{a^2}}-45 a \sqrt{-\frac{(a+b) \sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right) \tan ^2(e+f x)}{a^2}}\right)}{240 f \left(b \sin ^2(e+f x)+a\right)^{3/2} \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{3/2}}","\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*Sec[e + f*x]^3*(1 + (b*Sin[e + f*x]^2)/a)^2*Tan[e + f*x]*(45*a*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]] + 30*b*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^2 + 210*a*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2) + 140*b*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2) - 120*a*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2) + 256*a*Hypergeometric2F1[2, 5, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2) - 80*b*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2) + 256*b*Hypergeometric2F1[2, 5, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2) - 512*a*Hypergeometric2F1[2, 5, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(7/2) - 512*b*Hypergeometric2F1[2, 5, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(7/2) + 256*a*Hypergeometric2F1[2, 5, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(9/2) + 256*b*Hypergeometric2F1[2, 5, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(9/2) - 45*a*Sqrt[-(((a + b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)*Tan[e + f*x]^2)/a^2)] - 30*b*Sin[e + f*x]^2*Sqrt[-(((a + b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)*Tan[e + f*x]^2)/a^2)]))/(240*f*(a + b*Sin[e + f*x]^2)^(3/2)*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2))","C",0
340,1,247,321,2.5681058,"\int \cos ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} b \sin (2 (e+f x)) \left(-32 a^3+b \left(144 a^2-192 a b-37 b^2\right) \cos (2 (e+f x))+400 a^2 b+2 b^2 (b-26 a) \cos (4 (e+f x))+212 a b^2+5 b^3 \cos (6 (e+f x))+30 b^3\right)+64 a \left(2 a^3+11 a^2 b+8 a b^2-b^3\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-128 a \left(a^3+5 a^2 b-5 a b^2-b^3\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{2240 b^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(-128*a*(a^3 + 5*a^2*b - 5*a*b^2 - b^3)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 64*a*(2*a^3 + 11*a^2*b + 8*a*b^2 - b^3)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + Sqrt[2]*b*(-32*a^3 + 400*a^2*b + 212*a*b^2 + 30*b^3 + b*(144*a^2 - 192*a*b - 37*b^2)*Cos[2*(e + f*x)] + 2*b^2*(-26*a + b)*Cos[4*(e + f*x)] + 5*b^3*Cos[6*(e + f*x)])*Sin[2*(e + f*x)])/(2240*b^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
341,1,200,259,1.3415777,"\int \cos ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} b \sin (2 (e+f x)) \left(48 a^2-4 b (9 a+2 b) \cos (2 (e+f x))+28 a b+3 b^2 \cos (4 (e+f x))+5 b^2\right)+16 a \left(3 a^2+2 a b-b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-16 a \left(3 a^2-7 a b-2 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{240 b f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(-16*a*(3*a^2 - 7*a*b - 2*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 16*a*(3*a^2 + 2*a*b - b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + Sqrt[2]*b*(48*a^2 + 28*a*b + 5*b^2 - 4*b*(9*a + 2*b)*Cos[2*(e + f*x)] + 3*b^2*Cos[4*(e + f*x)])*Sin[2*(e + f*x)])/(240*b*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
342,1,156,154,0.75773,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{b \sin (2 (e+f x)) (-2 a+b \cos (2 (e+f x))-b)-2 \sqrt{2} a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+4 \sqrt{2} a (2 a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 \sqrt{2} f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(4*Sqrt[2]*a*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 2*Sqrt[2]*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + b*(-2*a - b + b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*Sqrt[2]*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
343,1,144,182,0.8553761,"\int \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{(a+b) \left(\sqrt{2} \tan (e+f x) (2 a-b \cos (2 (e+f x))+b)+2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)\right)-2 a (a+2 b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-2*a*(a + 2*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + (a + b)*(2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + Sqrt[2]*(2*a + b - b*Cos[2*(e + f*x)])*Tan[e + f*x]))/(2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
344,1,190,236,2.0263282,"\int \sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\frac{\tan (e+f x) \sec ^2(e+f x) \left(\left(4 a^2-6 a b-2 b^2\right) \cos (2 (e+f x))+8 a^2+b (b-a) \cos (4 (e+f x))+3 a b+b^2\right)}{\sqrt{2}}+2 a (2 a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-4 a (a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-4*a*(a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*a*(2*a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + ((8*a^2 + 3*a*b + b^2 + (4*a^2 - 6*a*b - 2*b^2)*Cos[2*(e + f*x)] + b*(-a + b)*Cos[4*(e + f*x)])*Sec[e + f*x]^2*Tan[e + f*x])/Sqrt[2])/(6*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
345,1,79,79,0.1129728,"\int \frac{\cos ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Cos[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{-\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}}-\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,"(-1/2*((-a - 2*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/b^(3/2) - (Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(2*b))/f","A",1
346,1,38,38,0.015472,"\int \frac{\cos (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[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",1
347,1,42,42,0.0288508,"\int \frac{\sec (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[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",1
348,1,436,91,9.7489794,"\int \frac{\sec ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Sec[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\tan (e+f x) \sec ^3(e+f x) \left(\frac{b \sin ^2(e+f x)}{a}+1\right) \left(-30 b \sin ^2(e+f x) \sqrt{-\frac{\tan ^2(e+f x) \sec ^2(e+f x) \left(a^2+a b \left(\sin ^2(e+f x)+1\right)+b^2 \sin ^2(e+f x)\right)}{a^2}}-45 a \sqrt{-\frac{\tan ^2(e+f x) \sec ^2(e+f x) \left(a^2+a b \left(\sin ^2(e+f x)+1\right)+b^2 \sin ^2(e+f x)\right)}{a^2}}+16 b \sin ^2(e+f x) \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{5/2} \, _2F_1\left(2,3;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}+16 a \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{5/2} \, _2F_1\left(2,3;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}+45 a \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right)+30 b \sin ^2(e+f x) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right)\right)}{30 a f \sqrt{a+b \sin ^2(e+f x)} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}}","\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,"(Sec[e + f*x]^3*(1 + (b*Sin[e + f*x]^2)/a)*Tan[e + f*x]*(45*a*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]] + 30*b*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^2 + 16*a*Hypergeometric2F1[2, 3, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2) + 16*b*Hypergeometric2F1[2, 3, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2) - 45*a*Sqrt[-((Sec[e + f*x]^2*(a^2 + b^2*Sin[e + f*x]^2 + a*b*(1 + Sin[e + f*x]^2))*Tan[e + f*x]^2)/a^2)] - 30*b*Sin[e + f*x]^2*Sqrt[-((Sec[e + f*x]^2*(a^2 + b^2*Sin[e + f*x]^2 + a*b*(1 + Sin[e + f*x]^2))*Tan[e + f*x]^2)/a^2)]))/(30*a*f*Sqrt[a + b*Sin[e + f*x]^2]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2))","C",0
349,1,170,168,0.8527133,"\int \frac{\cos ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Cos[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{2 \sqrt{2} \left(2 a^2+5 a b+3 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+b \sin (2 (e+f x)) (-2 a+b \cos (2 (e+f x))-b)-4 \sqrt{2} a (a+2 b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 \sqrt{2} b^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-4*Sqrt[2]*a*(a + 2*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*Sqrt[2]*(2*a^2 + 5*a*b + 3*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + b*(-2*a - b + b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*Sqrt[2]*b^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
350,1,83,114,0.196405,"\int \frac{\cos ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Cos[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} \left((a+b) F\left(e+f x\left|-\frac{b}{a}\right.\right)-a E\left(e+f x\left|-\frac{b}{a}\right.\right)\right)}{b f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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[(2*a + b - b*Cos[2*(e + f*x)])/a]*(-(a*EllipticE[e + f*x, -(b/a)]) + (a + b)*EllipticF[e + f*x, -(b/a)]))/(b*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
351,1,60,51,0.0744193,"\int \frac{1}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{2 a-b \cos (2 (e+f x))+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)}}",1,"(Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
352,1,141,140,0.6232371,"\int \frac{\sec ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Sec[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{2} \tan (e+f x) (2 a-b \cos (2 (e+f x))+b)+2 (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{2 f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + Sqrt[2]*(2*a + b - b*Cos[2*(e + f*x)])*Tan[e + f*x])/(2*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
353,1,205,212,2.1874701,"\int \frac{\sec ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Sec[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{2 \left(2 a^2+5 a b+3 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+\frac{\tan (e+f x) \sec ^2(e+f x) \left(\left(4 a^2+6 a b-2 b^2\right) \cos (2 (e+f x))+8 a^2-b (a+2 b) \cos (4 (e+f x))+15 a b+4 b^2\right)}{\sqrt{2}}-4 a (a+2 b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 f (a+b)^2 \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-4*a*(a + 2*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 2*(2*a^2 + 5*a*b + 3*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + ((8*a^2 + 15*a*b + 4*b^2 + (4*a^2 + 6*a*b - 2*b^2)*Cos[2*(e + f*x)] - b*(a + 2*b)*Cos[4*(e + f*x)])*Sec[e + f*x]^2*Tan[e + f*x])/Sqrt[2])/(6*(a + b)^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
354,1,88,75,0.1718316,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cos[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{b} (a+b) \sin (e+f x)-a^{3/2} \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sin (e+f x)}{\sqrt{a}}\right)}{a b^{3/2} f \sqrt{a+b \sin ^2(e+f 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}",1,"(Sqrt[b]*(a + b)*Sin[e + f*x] - a^(3/2)*ArcSinh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a]]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(a*b^(3/2)*f*Sqrt[a + b*Sin[e + f*x]^2])","A",1
355,1,29,29,0.0270701,"\int \frac{\cos (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[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",1
356,1,480,78,7.4092699,"\int \frac{\sec (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sec[e + f*x]/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x) \sec (e+f x) \left(-\frac{30 b (a+b) \sin ^2(e+f x) \tan ^2(e+f x) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right)}{a^2}+\frac{30 b \sin ^2(e+f x) \sqrt{-\frac{(a+b) \tan ^2(e+f x) \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a^2}}}{a}+45 \sqrt{-\frac{(a+b) \tan ^2(e+f x) \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a^2}}+\frac{4 b \sin ^2(e+f x) \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{5/2} \, _2F_1\left(2,2;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}}{a}+4 \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{5/2} \, _2F_1\left(2,2;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}-\frac{45 (a+b) \tan ^2(e+f x) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right)}{a}-45 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right)-\frac{30 b \sin ^2(e+f x) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right)}{a}\right)}{15 a f \sqrt{a+b \sin ^2(e+f x)} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{3/2} \sqrt{\frac{\sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)}{a}}}","\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,"(Sec[e + f*x]*Tan[e + f*x]*(-45*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]] - (30*b*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^2)/a - (45*(a + b)*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Tan[e + f*x]^2)/a - (30*b*(a + b)*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^2*Tan[e + f*x]^2)/a^2 + 4*Hypergeometric2F1[2, 2, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2) + (4*b*Hypergeometric2F1[2, 2, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2))/a + 45*Sqrt[-(((a + b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)*Tan[e + f*x]^2)/a^2)] + (30*b*Sin[e + f*x]^2*Sqrt[-(((a + b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)*Tan[e + f*x]^2)/a^2)])/a))/(15*a*f*Sqrt[a + b*Sin[e + f*x]^2]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2))","C",0
357,1,224,134,5.6150786,"\int \frac{\sec ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sec[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{\tan (e+f x) \sec ^5(e+f x) \left(16 (a+b) \sin ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^2 \, _3F_2\left(2,2,3;1,\frac{9}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right)+16 (a+b) \sin ^2(e+f x) \left(4 a^2+7 a b \sin ^2(e+f x)+3 b^2 \sin ^4(e+f x)\right) \, _2F_1\left(2,3;\frac{9}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right)-7 a \cos ^2(e+f x) \left(15 a^2+20 a b \sin ^2(e+f x)+8 b^2 \sin ^4(e+f x)\right) \, _2F_1\left(1,2;\frac{7}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right)\right)}{105 a^4 f \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,"-1/105*(Sec[e + f*x]^5*(16*(a + b)*HypergeometricPFQ[{2, 2, 3}, {1, 9/2}, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*(a + b*Sin[e + f*x]^2)^2 + 16*(a + b)*Hypergeometric2F1[2, 3, 9/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*(4*a^2 + 7*a*b*Sin[e + f*x]^2 + 3*b^2*Sin[e + f*x]^4) - 7*a*Cos[e + f*x]^2*Hypergeometric2F1[1, 2, 7/2, -(((a + b)*Tan[e + f*x]^2)/a)]*(15*a^2 + 20*a*b*Sin[e + f*x]^2 + 8*b^2*Sin[e + f*x]^4))*Tan[e + f*x])/(a^4*f*Sqrt[a + b*Sin[e + f*x]^2])","C",0
358,1,184,274,1.121139,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cos[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} b \sin (2 (e+f x)) \left(8 a^2-a b \cos (2 (e+f x))+13 a b+6 b^2\right)-4 a \left(8 a^2+17 a b+9 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+4 a \left(8 a^2+13 a b+3 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{12 a b^3 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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{(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{(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) \sin (e+f x) \cos ^3(e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}",1,"(4*a*(8*a^2 + 13*a*b + 3*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 4*a*(8*a^2 + 17*a*b + 9*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + Sqrt[2]*b*(8*a^2 + 13*a*b + 6*b^2 - a*b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(12*a*b^3*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
359,1,139,202,0.5575021,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cos[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{2 a (2 a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)-(a+b) \left(4 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-\sqrt{2} b \sin (2 (e+f x))\right)}{2 a b^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(2*a*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - (a + b)*(4*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*Sin[2*(e + f*x)]))/(2*a*b^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
360,1,133,188,0.302308,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cos[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{-\sqrt{2} a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+\sqrt{2} a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)+b \sin (2 (e+f x))}{\sqrt{2} a b f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(Sqrt[2]*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - Sqrt[2]*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + b*Sin[2*(e + f*x)])/(Sqrt[2]*a*b*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
361,1,90,101,0.1465195,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(-3/2),x]","\frac{2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)+\sqrt{2} b \sin (2 (e+f x))}{2 a f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + Sqrt[2]*b*Sin[2*(e + f*x)])/(2*a*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
362,1,167,240,1.2254707,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Sec[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x) \left(2 a^2+b (b-a) \cos (2 (e+f x))+a b+b^2\right)+\sqrt{2} a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-\sqrt{2} a (a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{\sqrt{2} a f (a+b)^2 \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-(Sqrt[2]*a*(a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)]) + Sqrt[2]*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + (2*a^2 + a*b + b^2 + b*(-a + b)*Cos[2*(e + f*x)])*Tan[e + f*x])/(Sqrt[2]*a*(a + b)^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
363,1,128,130,0.810657,"\int \frac{\cos ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\frac{2 \sqrt{2} (a+b) \sin (e+f x) \left(-3 a^2+b (2 a-b) \cos (2 (e+f x))+a b+b^2\right)}{a^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-b} \sin (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right)}{\sqrt{-b}}}{3 b^2 f}","-\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,"((3*ArcTan[(Sqrt[2]*Sqrt[-b]*Sin[e + f*x])/Sqrt[2*a + b - b*Cos[2*(e + f*x)]]])/Sqrt[-b] + (2*Sqrt[2]*(a + b)*(-3*a^2 + a*b + b^2 + (2*a - b)*b*Cos[2*(e + f*x)])*Sin[e + f*x])/(a^2*(2*a + b - b*Cos[2*(e + f*x)])^(3/2)))/(3*b^2*f)","A",1
364,1,51,73,0.1046732,"\int \frac{\cos ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{3 a \sin (e+f x)-(a-2 b) \sin ^3(e+f x)}{3 a^2 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,"(3*a*Sin[e + f*x] - (a - 2*b)*Sin[e + f*x]^3)/(3*a^2*f*(a + b*Sin[e + f*x]^2)^(3/2))","A",1
365,1,47,65,0.0448524,"\int \frac{\cos (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\sin (e+f x) \left(3 a+2 b \sin ^2(e+f x)\right)}{3 a^2 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 + 2*b*Sin[e + f*x]^2))/(3*a^2*f*(a + b*Sin[e + f*x]^2)^(3/2))","A",1
366,1,1291,126,9.3650229,"\int \frac{\sec (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sec[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\sec (e+f x) \tan (e+f x) \left(\frac{840 b^2 (a+b)^2 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right) \tan ^4(e+f x) \sin ^4(e+f x)}{a^4}+\frac{72 b^2 \, _2F_1\left(2,2;\frac{9}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{7/2} \sin ^4(e+f x)}{a^2}+\frac{24 b^2 \, _3F_2\left(2,2,2;1,\frac{9}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{7/2} \sin ^4(e+f x)}{a^2}+\frac{1680 b^2 (a+b) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right) \tan ^2(e+f x) \sin ^4(e+f x)}{a^3}+\frac{1120 b^2 \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{3/2} \sin ^4(e+f x)}{a^2}+\frac{840 b^2 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right) \sin ^4(e+f x)}{a^2}-\frac{840 b^2 \sqrt{-\frac{(a+b) \sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right) \tan ^2(e+f x)}{a^2}} \sin ^4(e+f x)}{a^2}+\frac{2100 b (a+b)^2 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right) \tan ^4(e+f x) \sin ^2(e+f x)}{a^3}+\frac{168 b \, _2F_1\left(2,2;\frac{9}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{7/2} \sin ^2(e+f x)}{a}+\frac{48 b \, _3F_2\left(2,2,2;1,\frac{9}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{7/2} \sin ^2(e+f x)}{a}+\frac{4200 b (a+b) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right) \tan ^2(e+f x) \sin ^2(e+f x)}{a^2}+\frac{2800 b \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{3/2} \sin ^2(e+f x)}{a}+\frac{2100 b \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right) \sin ^2(e+f x)}{a}-\frac{2100 b \sqrt{-\frac{(a+b) \sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right) \tan ^2(e+f x)}{a^2}} \sin ^2(e+f x)}{a}+\frac{1575 (a+b)^2 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right) \tan ^4(e+f x)}{a^2}+96 \, _2F_1\left(2,2;\frac{9}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{7/2}+24 \, _3F_2\left(2,2,2;1,\frac{9}{2};-\frac{(a+b) \tan ^2(e+f x)}{a}\right) \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{7/2}+\frac{3150 (a+b) \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right) \tan ^2(e+f x)}{a}+2100 \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{3/2}+1575 \sin ^{-1}\left(\sqrt{-\frac{(a+b) \tan ^2(e+f x)}{a}}\right)-1575 \sqrt{-\frac{(a+b) \sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right) \tan ^2(e+f x)}{a^2}}\right)}{315 a^2 f \sqrt{b \sin ^2(e+f x)+a} \sqrt{\frac{\sec ^2(e+f x) \left(b \sin ^2(e+f x)+a\right)}{a}} \left(\frac{b \sin ^2(e+f x)}{a}+1\right) \left(-\frac{(a+b) \tan ^2(e+f x)}{a}\right)^{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,"(Sec[e + f*x]*Tan[e + f*x]*(1575*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]] + (2100*b*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^2)/a + (840*b^2*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^4)/a^2 + (3150*(a + b)*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Tan[e + f*x]^2)/a + (4200*b*(a + b)*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^2*Tan[e + f*x]^2)/a^2 + (1680*b^2*(a + b)*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^4*Tan[e + f*x]^2)/a^3 + (1575*(a + b)^2*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Tan[e + f*x]^4)/a^2 + (2100*b*(a + b)^2*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^2*Tan[e + f*x]^4)/a^3 + (840*b^2*(a + b)^2*ArcSin[Sqrt[-(((a + b)*Tan[e + f*x]^2)/a)]]*Sin[e + f*x]^4*Tan[e + f*x]^4)/a^4 + 2100*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2) + (2800*b*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2))/a + (1120*b^2*Sin[e + f*x]^4*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(3/2))/a^2 + 96*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(7/2) + 24*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Tan[e + f*x]^2)/a)]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(7/2) + (168*b*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(7/2))/a + (48*b*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^2*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(7/2))/a + (72*b^2*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^4*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(7/2))/a^2 + (24*b^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Tan[e + f*x]^2)/a)]*Sin[e + f*x]^4*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(-(((a + b)*Tan[e + f*x]^2)/a))^(7/2))/a^2 - 1575*Sqrt[-(((a + b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)*Tan[e + f*x]^2)/a^2)] - (2100*b*Sin[e + f*x]^2*Sqrt[-(((a + b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)*Tan[e + f*x]^2)/a^2)])/a - (840*b^2*Sin[e + f*x]^4*Sqrt[-(((a + b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)*Tan[e + f*x]^2)/a^2)])/a^2))/(315*a^2*f*Sqrt[a + b*Sin[e + f*x]^2]*Sqrt[(Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2))/a]*(1 + (b*Sin[e + f*x]^2)/a)*(-(((a + b)*Tan[e + f*x]^2)/a))^(5/2))","C",0
367,1,194,243,2.1161389,"\int \frac{\cos ^6(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\frac{1}{2} (a+b) \left(4 a^2 (8 a-b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 \sqrt{2} b \sin (2 (e+f x)) \left(8 a^2+b (2 b-5 a) \cos (2 (e+f x))-a b-2 b^2\right)\right)-2 a^2 \left(8 a^2+3 a b-2 b^2\right) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 a^2 b^3 f (2 a-b \cos (2 (e+f x))+b)^{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,"(-2*a^2*(8*a^2 + 3*a*b - 2*b^2)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] + ((a + b)*(4*a^2*(8*a - b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] - 2*Sqrt[2]*b*(8*a^2 - a*b - 2*b^2 + b*(-5*a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))/2)/(6*a^2*b^3*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
368,1,171,223,1.4127752,"\int \frac{\cos ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{-\sqrt{2} b \sin (2 (e+f x)) \left(a^2+b (b-a) \cos (2 (e+f x))-2 a b-b^2\right)+a^2 (2 a-b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a^2 (a-b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a^2 b^2 f (2 a-b \cos (2 (e+f x))+b)^{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,"(-2*a^2*(a - b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] + a^2*(2*a - b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(a^2 - 2*a*b - b^2 + b*(-a + b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(3*a^2*b^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
369,1,175,217,1.4305662,"\int \frac{\cos ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cos[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{-\sqrt{2} b \sin (2 (e+f x)) \left(-4 a^2+b (a+2 b) \cos (2 (e+f x))-7 a b-2 b^2\right)-2 a^2 (a+b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 a^2 (a+2 b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 a^2 b f (a+b) (2 a-b \cos (2 (e+f x))+b)^{3/2}}","\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,"(2*a^2*(a + 2*b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] - 2*a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(-4*a^2 - 7*a*b - 2*b^2 + b*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*a^2*b*(a + b)*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
370,1,172,223,1.2319842,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(-5/2),x]","\frac{-\sqrt{2} b \sin (2 (e+f x)) \left(-5 a^2+b (2 a+b) \cos (2 (e+f x))-5 a b-b^2\right)-a^2 (a+b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 a^2 (2 a+b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a^2 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}}","\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,"(2*a^2*(2*a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] - a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(-5*a^2 - 5*a*b - b^2 + b*(2*a + b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(3*a^2*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
371,1,245,288,3.3484114,"\int \frac{\sec ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Sec[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{2 a^2 \left(3 a^2+2 a b-b^2\right) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a^2 \left(3 a^2-7 a b-2 b^2\right) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)+\frac{\tan (e+f x) \left(24 a^4+24 a^3 b-4 a b \left(6 a^2-5 a b-3 b^2\right) \cos (2 (e+f x))+b^2 \left(3 a^2-7 a b-2 b^2\right) \cos (4 (e+f x))+41 a^2 b^2+19 a b^3+2 b^4\right)}{\sqrt{2}}}{6 a^2 f (a+b)^3 (2 a-b \cos (2 (e+f x))+b)^{3/2}}","-\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,"(-2*a^2*(3*a^2 - 7*a*b - 2*b^2)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] + 2*a^2*(3*a^2 + 2*a*b - b^2)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] + ((24*a^4 + 24*a^3*b + 41*a^2*b^2 + 19*a*b^3 + 2*b^4 - 4*a*b*(6*a^2 - 5*a*b - 3*b^2)*Cos[2*(e + f*x)] + b^2*(3*a^2 - 7*a*b - 2*b^2)*Cos[4*(e + f*x)])*Tan[e + f*x])/Sqrt[2])/(6*a^2*(a + b)^3*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
372,1,228,115,0.8995024,"\int (d \cos (e+f x))^m \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[(d*Cos[e + f*x])^m*(a + b*Sin[e + f*x]^2)^p,x]","\frac{3 a \tan (e+f x) (d \cos (e+f x))^m \left(a+b \sin ^2(e+f x)\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 \left(\sin ^2(e+f x) \left(2 b p F_1\left(\frac{3}{2};\frac{1-m}{2},1-p;\frac{5}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)-a (m-1) F_1\left(\frac{3}{2};\frac{3-m}{2},-p;\frac{5}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)\right)+3 a 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)\right)}","\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,"(3*a*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])^m*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(f*(3*a*AppellF1[1/2, (1 - m)/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, (1 - m)/2, 1 - p, 5/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)] - a*(-1 + m)*AppellF1[3/2, (3 - m)/2, -p, 5/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)])*Sin[e + f*x]^2))","A",0
373,1,191,214,0.3965142,"\int \cos ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Cos[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p,x]","\frac{3 a \sin (e+f x) \cos ^4(e+f x) \left(a+b \sin ^2(e+f x)\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 \left(2 \sin ^2(e+f x) \left(b p F_1\left(\frac{3}{2};-2,1-p;\frac{5}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)-2 a F_1\left(\frac{3}{2};-1,-p;\frac{5}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)\right)+3 a 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)\right)}","\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*AppellF1[1/2, -2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Cos[e + f*x]^4*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(f*(3*a*AppellF1[1/2, -2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)] + 2*(b*p*AppellF1[3/2, -2, 1 - p, 5/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)] - 2*a*AppellF1[3/2, -1, -p, 5/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)])*Sin[e + f*x]^2))","C",0
374,1,120,124,0.1907239,"\int \cos ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Cos[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} \left(\left(a+b \sin ^2(e+f x)\right) \left(\frac{b \sin ^2(e+f x)}{a}+1\right)^p-(a+b (2 p+3)) \, _2F_1\left(\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^2(e+f x)}{a}\right)\right)}{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)^p*(-((a + b*(3 + 2*p))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sin[e + f*x]^2)/a)]) + (a + b*Sin[e + f*x]^2)*(1 + (b*Sin[e + f*x]^2)/a)^p))/(b*f*(3 + 2*p)*(1 + (b*Sin[e + f*x]^2)/a)^p))","A",1
375,1,67,67,0.0249659,"\int \cos (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[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",1
376,0,0,76,4.4461796,"\int \sec (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Sec[e + f*x]*(a + b*Sin[e + f*x]^2)^p,x]","\int \sec (e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","\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,"Integrate[Sec[e + f*x]*(a + b*Sin[e + f*x]^2)^p, x]","F",-1
377,0,0,76,6.5633187,"\int \sec ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p,x]","\int \sec ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","\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,"Integrate[Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p, x]","F",-1
378,1,199,90,0.5780398,"\int \cos ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p,x]","\frac{3 a \sin (e+f x) \cos ^3(e+f x) \left(a+b \sin ^2(e+f x)\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 \left(\sin ^2(e+f x) \left(2 b p F_1\left(\frac{3}{2};-\frac{3}{2},1-p;\frac{5}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)-3 a F_1\left(\frac{3}{2};-\frac{1}{2},-p;\frac{5}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)\right)+3 a 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)\right)}","\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,"(3*a*AppellF1[1/2, -3/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Cos[e + f*x]^3*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(f*(3*a*AppellF1[1/2, -3/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, -3/2, 1 - p, 5/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)] - 3*a*AppellF1[3/2, -1/2, -p, 5/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)])*Sin[e + f*x]^2))","B",0
379,1,195,90,0.4677851,"\int \cos ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p,x]","\frac{3 a \sin (2 (e+f x)) \left(a+b \sin ^2(e+f x)\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)}{2 f \left(\sin ^2(e+f x) \left(2 b p F_1\left(\frac{3}{2};-\frac{1}{2},1-p;\frac{5}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)-a F_1\left(\frac{3}{2};\frac{1}{2},-p;\frac{5}{2};\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right)\right)+3 a 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)\right)}","\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,"(3*a*AppellF1[1/2, -1/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*(a + b*Sin[e + f*x]^2)^p*Sin[2*(e + f*x)])/(2*f*(3*a*AppellF1[1/2, -1/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)] + (2*b*p*AppellF1[3/2, -1/2, 1 - p, 5/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)] - a*AppellF1[3/2, 1/2, -p, 5/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)])*Sin[e + f*x]^2))","B",0
380,1,145,90,0.5025926,"\int \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[(a + b*Sin[e + f*x]^2)^p,x]","\frac{2^{-p-1} \csc (2 (e+f x)) \sqrt{-\frac{b \sin ^2(e+f x)}{a}} \sqrt{\frac{b \cos ^2(e+f x)}{a+b}} (2 a-b \cos (2 (e+f x))+b)^{p+1} F_1\left(p+1;\frac{1}{2},\frac{1}{2};p+2;\frac{2 a+b-b \cos (2 (e+f x))}{2 (a+b)},\frac{2 a+b-b \cos (2 (e+f x))}{2 a}\right)}{b f (p+1)}","\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,"(2^(-1 - p)*AppellF1[1 + p, 1/2, 1/2, 2 + p, (2*a + b - b*Cos[2*(e + f*x)])/(2*(a + b)), (2*a + b - b*Cos[2*(e + f*x)])/(2*a)]*Sqrt[(b*Cos[e + f*x]^2)/(a + b)]*(2*a + b - b*Cos[2*(e + f*x)])^(1 + p)*Csc[2*(e + f*x)]*Sqrt[-((b*Sin[e + f*x]^2)/a)])/(b*f*(1 + p))","A",0
381,0,0,90,4.5546522,"\int \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p,x]","\int \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","\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,"Integrate[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p, x]","F",-1
382,0,0,90,6.7817669,"\int \sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p,x]","\int \sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^p \, dx","\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,"Integrate[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p, x]","F",-1
383,1,203,219,0.2361774,"\int \frac{\cos ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Sin[c + d*x]^3),x]","\frac{-b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)-3 a^{2/3} \sin ^2(c+d x) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right)-4 a^{2/3} \log \left(a+b \sin ^3(c+d x)\right)+3 a^{2/3} \sin ^2(c+d x)+2 b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)-2 \sqrt{3} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{6 a^{2/3} 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,"(-2*Sqrt[3]*b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] + 2*b^(2/3)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]] - b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2] - 4*a^(2/3)*Log[a + b*Sin[c + d*x]^3] + 3*a^(2/3)*Sin[c + d*x]^2 - 3*a^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(6*a^(2/3)*b*d)","C",1
384,1,139,167,0.1435036,"\int \frac{\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]","\frac{\left((-1)^{2/3} b^{2/3}-a^{2/3}\right) \log \left(-(-1)^{2/3} \sqrt[3]{a}-\sqrt[3]{b} \sin (c+d x)\right)+\left(b^{2/3}-a^{2/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)-\left(a^{2/3}+\sqrt[3]{-1} b^{2/3}\right) \log \left(\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} \sin (c+d x)\right)}{3 a^{2/3} 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,"((-a^(2/3) + (-1)^(2/3)*b^(2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sin[c + d*x]] + (-a^(2/3) + b^(2/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]] - (a^(2/3) + (-1)^(1/3)*b^(2/3))*Log[a^(1/3) + (-1)^(2/3)*b^(1/3)*Sin[c + d*x]])/(3*a^(2/3)*b*d)","A",1
385,1,116,144,0.0513038,"\int \frac{\cos (c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[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)-2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{6 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,"-1/6*(2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] - 2*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]] + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(a^(2/3)*b^(1/3)*d)","A",1
386,1,268,290,0.2048643,"\int \frac{\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x]^3),x]","\frac{b^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)+3 a^{2/3} b \sin ^2(c+d x) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right)-2 a^{2/3} b \log \left(a+b \sin ^3(c+d x)\right)+3 a^{2/3} b \log (1-\sin (c+d x))+3 a^{2/3} b \log (\sin (c+d x)+1)-3 a^{5/3} \log (1-\sin (c+d x))+3 a^{5/3} \log (\sin (c+d x)+1)-2 b^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)+2 \sqrt{3} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{6 a^{2/3} d (a-b) (a+b)}","-\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(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{\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,"(2*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] - 3*a^(5/3)*Log[1 - Sin[c + d*x]] + 3*a^(2/3)*b*Log[1 - Sin[c + d*x]] + 3*a^(5/3)*Log[1 + Sin[c + d*x]] + 3*a^(2/3)*b*Log[1 + Sin[c + d*x]] - 2*b^(5/3)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]] + b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2] - 2*a^(2/3)*b*Log[a + b*Sin[c + d*x]^3] + 3*a^(2/3)*b*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(6*a^(2/3)*(a - b)*(a + b)*d)","C",1
387,1,333,385,2.1879246,"\int \frac{\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]","-\frac{-\frac{4 b \left(a^2+2 b^2\right) \log \left(a+b \sin ^3(c+d x)\right)}{\left(a^2-b^2\right)^2}+\frac{18 b^3 \sin ^2(c+d x) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right)}{\left(a^2-b^2\right)^2}-\frac{4 b^{5/3} \left(2 a^2+b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{a^{2/3} \left(a^2-b^2\right)^2}+\frac{2 b^{5/3} \left(2 a^2+b^2\right) \left(\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)\right)}{a^{2/3} \left(a^2-b^2\right)^2}+\frac{3}{(a+b) (\sin (c+d x)-1)}+\frac{3}{(a-b) (\sin (c+d x)+1)}+\frac{3 (a+4 b) \log (1-\sin (c+d x))}{(a+b)^2}-\frac{3 (a-4 b) \log (\sin (c+d x)+1)}{(a-b)^2}}{12 d}","\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(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^{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,"-1/12*((3*(a + 4*b)*Log[1 - Sin[c + d*x]])/(a + b)^2 - (3*(a - 4*b)*Log[1 + Sin[c + d*x]])/(a - b)^2 - (4*b^(5/3)*(2*a^2 + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(2/3)*(a^2 - b^2)^2) + (2*b^(5/3)*(2*a^2 + b^2)*(2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]))/(a^(2/3)*(a^2 - b^2)^2) - (4*b*(a^2 + 2*b^2)*Log[a + b*Sin[c + d*x]^3])/(a^2 - b^2)^2 + 3/((a + b)*(-1 + Sin[c + d*x])) + (18*b^3*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a^2 - b^2)^2 + 3/((a - b)*(1 + Sin[c + d*x])))/d","C",1
388,1,300,764,0.1479746,"\int \frac{\cos ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3),x]","-\frac{3 \cos (c+d x)+i \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 i \text{$\#$1}^3 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-\text{$\#$1} a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \text{$\#$1}^4 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+\text{$\#$1}^3 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 i \text{$\#$1} a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]}{3 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,"-1/3*(3*Cos[c + d*x] + I*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (2*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 - a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + (2*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 + a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(b*d)","C",1
389,1,231,484,0.0991644,"\int \frac{\cos ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]","-\frac{i \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-2 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+4 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]}{6 d}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt[3]{b}-\sqrt[3]{-1} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}\right)}{3 b^{2/3} d \sqrt{b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac{2 \tanh ^{-1}\left(\frac{(-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}\right)}{3 b^{2/3} d \sqrt{\sqrt[3]{-1} a^{2/3}+b^{2/3}}}",1,"((-1/6*I)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (2*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/d","C",1
390,1,126,245,0.1552682,"\int \frac{1}{a+b \sin ^3(c+d x)} \, dx","Integrate[(a + b*Sin[c + d*x]^3)^(-1),x]","-\frac{2 i \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1} \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b-4 i \text{$\#$1} a+b}\&\right]}{3 d}","\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}}{\sqrt{a^{2/3}-b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-b^{2/3}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+(-1)^{2/3} \sqrt[3]{b}}{\sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}+\sqrt[3]{-1} b^{2/3}}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{-1} \left((-1)^{2/3} \sqrt[3]{a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt[3]{b}\right)}{\sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right)}{3 a^{2/3} d \sqrt{a^{2/3}-(-1)^{2/3} b^{2/3}}}",1,"(((-2*I)/3)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ])/d","C",1
391,1,432,299,0.2568134,"\int \frac{\sec ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]","\frac{-i b \cos (c+d x) \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-4 i \text{$\#$1}^3 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 \text{$\#$1} a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+6 i \text{$\#$1}^2 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-12 \text{$\#$1}^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^4 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \text{$\#$1}^3 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+4 i \text{$\#$1} a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]+6 a \sin (c+d x)+6 b \cos (c+d x)-6 b}{6 d (a-b) (a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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,"(-6*b + 6*b*Cos[c + d*x] - I*b*Cos[c + d*x]*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (4*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 2*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 12*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (6*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (4*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + 6*a*Sin[c + d*x])/(6*(a - b)*(a + b)*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
392,1,679,1093,1.7049633,"\int \frac{\sec ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3),x]","\frac{\sec ^3(c+d x) \left(12 a^3 \sin (c+d x)+4 a^3 \sin (3 (c+d x))-3 b \left(5 a^2+13 b^2\right) \cos (c+d x)+12 b \left(a^2+2 b^2\right) \cos (2 (c+d x))-5 a^2 b \cos (3 (c+d x))+4 a^2 b-30 a b^2 \sin (c+d x)-22 a b^2 \sin (3 (c+d x))-13 b^3 \cos (3 (c+d x))+32 b^3\right)+4 i b^2 \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+4 \text{$\#$1}^4 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-12 i \text{$\#$1}^3 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+10 i \text{$\#$1}^2 a^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i a^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-20 \text{$\#$1}^2 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+6 \text{$\#$1} a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+8 i \text{$\#$1}^2 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 i b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-16 \text{$\#$1}^2 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^4 a^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 i \text{$\#$1}^4 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-6 \text{$\#$1}^3 a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+12 i \text{$\#$1} a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+4 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]}{24 d (a-b)^2 (a+b)^2}","-\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,"((4*I)*b^2*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 4*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (2*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (12*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 6*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 20*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 16*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (10*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (8*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (12*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 6*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 4*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (2*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + Sec[c + d*x]^3*(4*a^2*b + 32*b^3 - 3*b*(5*a^2 + 13*b^2)*Cos[c + d*x] + 12*b*(a^2 + 2*b^2)*Cos[2*(c + d*x)] - 5*a^2*b*Cos[3*(c + d*x)] - 13*b^3*Cos[3*(c + d*x)] + 12*a^3*Sin[c + d*x] - 30*a*b^2*Sin[c + d*x] + 4*a^3*Sin[3*(c + d*x)] - 22*a*b^2*Sin[3*(c + d*x)]))/(24*(a - b)^2*(a + b)^2*d)","C",1
393,1,402,288,3.4977169,"\int \frac{\cos ^7(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Integrate[Cos[c + d*x]^7/(a + b*Sin[c + d*x]^3)^2,x]","\frac{\frac{6 \sqrt[3]{-1} \left(2 \sqrt[3]{-1} a^{2/3}+3 b^{2/3}\right) \log \left(-(-1)^{2/3} \sqrt[3]{a}-\sqrt[3]{b} \sin (c+d x)\right)}{\sqrt[3]{a} b^{7/3}}+\frac{6 \left(2 a^{2/3}-3 b^{2/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{\sqrt[3]{a} b^{7/3}}-\frac{6 \sqrt[3]{-1} \left(2 a^{2/3}+3 \sqrt[3]{-1} b^{2/3}\right) \log \left(\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} \sin (c+d x)\right)}{\sqrt[3]{a} b^{7/3}}+\frac{6 \left(1-\frac{a^2}{b^2}\right) \sin (c+d x)}{a \left(a+b \sin ^3(c+d x)\right)}+\frac{2 \left(a^2-b^2\right) \left(\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)-2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)\right)}{a^{5/3} b^{7/3}}-\frac{27 \sin ^2(c+d x) \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right)}{a b}+\frac{18}{b \left(a+b \sin ^3(c+d x)\right)}-\frac{18 \sin (c+d x)}{b^2}}{18 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,"((6*(-1)^(1/3)*(2*(-1)^(1/3)*a^(2/3) + 3*b^(2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sin[c + d*x]])/(a^(1/3)*b^(7/3)) + (6*(2*a^(2/3) - 3*b^(2/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(1/3)*b^(7/3)) - (6*(-1)^(1/3)*(2*a^(2/3) + 3*(-1)^(1/3)*b^(2/3))*Log[a^(1/3) + (-1)^(2/3)*b^(1/3)*Sin[c + d*x]])/(a^(1/3)*b^(7/3)) + (2*(a^2 - b^2)*(2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] - 2*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]] + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]))/(a^(5/3)*b^(7/3)) - (18*Sin[c + d*x])/b^2 - (27*Hypergeometric2F1[2/3, 2, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a*b) + 18/(b*(a + b*Sin[c + d*x]^3)) + (6*(1 - a^2/b^2)*Sin[c + d*x])/(a*(a + b*Sin[c + d*x]^3)))/(18*d)","C",1
394,1,258,238,1.0872748,"\int \frac{\cos ^5(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Sin[c + d*x]^3)^2,x]","\frac{-\frac{2 \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{a^{5/3} \sqrt[3]{b}}+\frac{4 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{a^{5/3} \sqrt[3]{b}}-\frac{4 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{a^{5/3} \sqrt[3]{b}}+\frac{9 \sin ^2(c+d x) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right)}{a b}-\frac{9 \sin ^2(c+d x) \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right)}{a b}+\frac{6 \sin (c+d x)}{a \left(a+b \sin ^3(c+d x)\right)}+\frac{12}{b \left(a+b \sin ^3(c+d x)\right)}}{18 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)}{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,"((-4*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(a^(5/3)*b^(1/3)) + (4*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(5/3)*b^(1/3)) - (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(a^(5/3)*b^(1/3)) + (9*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a*b) - (9*Hypergeometric2F1[2/3, 2, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a*b) + 12/(b*(a + b*Sin[c + d*x]^3)) + (6*Sin[c + d*x])/(a*(a + b*Sin[c + d*x]^3)))/(18*d)","C",1
395,1,184,183,0.785297,"\int \frac{\cos ^3(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Sin[c + d*x]^3)^2,x]","\frac{\frac{-\frac{b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)}{a^{5/3}}+\frac{2 b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{a^{5/3}}+\frac{3}{a+b \sin ^3(c+d x)}}{b}-\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{a^{5/3} \sqrt[3]{b}}+\frac{3 \sin (c+d x)}{a \left(a+b \sin ^3(c+d x)\right)}}{9 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)}{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*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(a^(5/3)*b^(1/3)) + (3*Sin[c + d*x])/(a*(a + b*Sin[c + d*x]^3)) + ((2*b^(2/3)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/a^(5/3) - (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/a^(5/3) + 3/(a + b*Sin[c + d*x]^3))/b)/(9*d)","A",1
396,1,152,176,0.4666025,"\int \frac{\cos (c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Integrate[Cos[c + d*x]/(a + b*Sin[c + d*x]^3)^2,x]","\frac{\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\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)}{\sqrt[3]{b}}+\frac{3 a^{2/3} \sin (c+d x)}{a+b \sin ^3(c+d x)}-\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt[3]{b}}}{9 a^{5/3} 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)}{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*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/b^(1/3) + (2*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]] - Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/b^(1/3) + (3*a^(2/3)*Sin[c + d*x])/(a + b*Sin[c + d*x]^3))/(9*a^(5/3)*d)","A",1
397,1,503,587,4.3027892,"\int \frac{\sec (c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x]^3)^2,x]","\frac{\frac{9 b \sin ^2(c+d x) \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right)}{a^3-a b^2}+\frac{9 b \left(a^2+b^2\right) \sin ^2(c+d x) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right)}{a \left(a^2-b^2\right)^2}-\frac{6 b^2 \sin (c+d x)}{a \left(a^2-b^2\right) \left(a+b \sin ^3(c+d x)\right)}+\frac{6 b}{\left(a^2-b^2\right) \left(a+b \sin ^3(c+d x)\right)}-\frac{12 a b \log \left(a+b \sin ^3(c+d x)\right)}{\left(a^2-b^2\right)^2}-\frac{12 \sqrt[3]{a} b^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{\left(a^2-b^2\right)^2}+\frac{6 \sqrt[3]{a} b^{5/3} \left(\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)\right)}{\left(a^2-b^2\right)^2}+\frac{2 b^{5/3} \left(\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)-2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)\right)}{a^{5/3} \left(a^2-b^2\right)}-\frac{9 \log (1-\sin (c+d x))}{(a+b)^2}+\frac{9 \log (\sin (c+d x)+1)}{(a-b)^2}}{18 d}","\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{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(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{\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,"((-9*Log[1 - Sin[c + d*x]])/(a + b)^2 + (9*Log[1 + Sin[c + d*x]])/(a - b)^2 - (12*a^(1/3)*b^(5/3)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^2 - b^2)^2 + (6*a^(1/3)*b^(5/3)*(2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]))/(a^2 - b^2)^2 + (2*b^(5/3)*(2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] - 2*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]] + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]))/(a^(5/3)*(a^2 - b^2)) - (12*a*b*Log[a + b*Sin[c + d*x]^3])/(a^2 - b^2)^2 + (9*b*(a^2 + b^2)*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a*(a^2 - b^2)^2) + (9*b*Hypergeometric2F1[2/3, 2, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a^3 - a*b^2) + (6*b)/((a^2 - b^2)*(a + b*Sin[c + d*x]^3)) - (6*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*(a + b*Sin[c + d*x]^3)))/(18*d)","C",1
398,1,657,747,6.3831737,"\int \frac{\sec ^3(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sin[c + d*x]^3)^2,x]","\frac{\frac{a b^2 \left(\frac{b^2}{a^2}+2\right) \sin (c+d x)}{3 \left(a^2-b^2\right)^2 \left(a+b \sin ^3(c+d x)\right)}-\frac{b \left(a^2+2 b^2\right)}{3 \left(a^2-b^2\right)^2 \left(a+b \sin ^3(c+d x)\right)}+\frac{2 a b \left(a^2+5 b^2\right) \log \left(a+b \sin ^3(c+d x)\right)}{3 \left(a^2-b^2\right)^3}+\frac{4 \sqrt[3]{a} b^{5/3} \left(a^2+2 b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)}{3 \left(a^2-b^2\right)^3}-\frac{3 b^3 \left(3 a^2+b^2\right) \sin ^2(c+d x) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right)}{2 a \left(a^2-b^2\right)^3}-\frac{3 b^3 \sin ^2(c+d x) \, _2F_1\left(\frac{2}{3},2;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right)}{2 a \left(a^2-b^2\right)^2}-\frac{2 \sqrt[3]{a} \left(a^2+2 b^2\right) \left(b^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)+2 \sqrt{3} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)\right)}{3 \left(a^2-b^2\right)^3}+\frac{\left(\frac{b^2}{a^2}+2\right) \left(2 \sqrt[3]{a} b^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (c+d x)\right)-\sqrt[3]{a} \left(b^{5/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right)+2 \sqrt{3} b^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sin (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)\right)\right)}{9 \left(a^2-b^2\right)^2}+\frac{1}{4 (a+b)^2 (1-\sin (c+d x))}-\frac{1}{4 (a-b)^2 (\sin (c+d x)+1)}-\frac{(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3}+\frac{(a-7 b) \log (\sin (c+d x)+1)}{4 (a-b)^3}}{d}","-\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{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(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{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(8 a^{2/3} b^2+4 a^{8/3}-9 a^2 b^{2/3}-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,"(-1/4*((a + 7*b)*Log[1 - Sin[c + d*x]])/(a + b)^3 + ((a - 7*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^3) + (4*a^(1/3)*b^(5/3)*(a^2 + 2*b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*(a^2 - b^2)^3) - (2*a^(1/3)*(a^2 + 2*b^2)*(2*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] + b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]))/(3*(a^2 - b^2)^3) + ((2 + b^2/a^2)*(2*a^(1/3)*b^(5/3)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]] - a^(1/3)*(2*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] + b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])))/(9*(a^2 - b^2)^2) + (2*a*b*(a^2 + 5*b^2)*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^3) + 1/(4*(a + b)^2*(1 - Sin[c + d*x])) - (3*b^3*(3*a^2 + b^2)*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(2*a*(a^2 - b^2)^3) - (3*b^3*Hypergeometric2F1[2/3, 2, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(2*a*(a^2 - b^2)^2) - 1/(4*(a - b)^2*(1 + Sin[c + d*x])) - (b*(a^2 + 2*b^2))/(3*(a^2 - b^2)^2*(a + b*Sin[c + d*x]^3)) + (a*b^2*(2 + b^2/a^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*(a + b*Sin[c + d*x]^3)))/d","C",1
399,1,394,26,0.3573461,"\int \frac{\cos ^4(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]","\frac{\frac{24 \cos (c+d x) (a+b \sin (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}-i \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-4 i \text{$\#$1}^3 a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 \text{$\#$1} a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-6 i \text{$\#$1}^2 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+12 \text{$\#$1}^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^4 b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \text{$\#$1}^3 a \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+4 i \text{$\#$1} a \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]}{18 a b d}","\text{Int}\left(\frac{\cos ^4(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2},x\right)",0,"((-I)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (4*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 2*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 12*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (6*I)*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (4*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + (24*Cos[c + d*x]*(a + b*Sin[c + d*x]))/(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)]))/(18*a*b*d)","C",1
400,1,273,26,0.2369779,"\int \frac{\cos ^2(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]","\frac{\frac{12 \sin (2 (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}-i \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-6 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+12 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]}{18 a d}","\text{Int}\left(\frac{\cos ^2(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2},x\right)",0,"((-I)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 12*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (6*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + (12*Sin[2*(c + d*x)])/(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)]))/(18*a*d)","C",1
401,1,502,17,0.4764765,"\int \frac{1}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Integrate[(a + b*Sin[c + d*x]^3)^(-2),x]","\frac{-\frac{12 b \cos (c+d x) (a \cos (2 (c+d x))-3 a+2 b \sin (c+d x))}{(a-b) (a+b) (4 a+3 b \sin (c+d x)-b \sin (3 (c+d x)))}+\frac{i \text{RootSum}\left[i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+8 \text{$\#$1}^3 a+3 i \text{$\#$1}^2 b-i b\&,\frac{2 \text{$\#$1}^4 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-4 i \text{$\#$1}^3 a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+12 i \text{$\#$1}^2 a^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-24 \text{$\#$1}^2 a^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 \text{$\#$1} a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-6 i \text{$\#$1}^2 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-i b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+12 \text{$\#$1}^2 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i \text{$\#$1}^4 b^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)-2 \text{$\#$1}^3 a b \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right)+4 i \text{$\#$1} a b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+2 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)}{\text{$\#$1}^5 b-2 \text{$\#$1}^3 b-4 i \text{$\#$1}^2 a+\text{$\#$1} b}\&\right]}{a^2-b^2}}{18 a d}","\text{Int}\left(\frac{1}{\left(a+b \sin ^3(c+d x)\right)^2},x\right)",0,"((I*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 2*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 24*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (12*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(a^2 - b^2) - (12*b*Cos[c + d*x]*(-3*a + a*Cos[2*(c + d*x)] + 2*b*Sin[c + d*x]))/((a - b)*(a + b)*(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)])))/(18*a*d)","C",1
402,1,845,26,1.604122,"\int \frac{\sec ^2(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]","\frac{-\frac{i b \text{RootSum}\left[i b \text{$\#$1}^6-3 i b \text{$\#$1}^4+8 a \text{$\#$1}^3+3 i b \text{$\#$1}^2-i b\&,\frac{2 b^3 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^4+16 a^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^4-i b^3 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^4-8 i a^2 b \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^4-20 i a^3 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^3-16 i a b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^3-10 a^3 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^3-8 a b^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^3+12 b^3 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^2-120 a^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^2-6 i b^3 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^2+60 i a^2 b \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^2+20 i a^3 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}+16 i a b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}+10 a^3 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}+8 a b^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}+2 b^3 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)+16 a^2 b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i b^3 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right)-8 i a^2 b \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right)}{b \text{$\#$1}^5-2 b \text{$\#$1}^3-4 i a \text{$\#$1}^2+b \text{$\#$1}}\&\right]}{a \left(a^2-b^2\right)^2}+\frac{18 \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{18 \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{12 b \cos (c+d x) \left(-2 a^3-7 b^2 a+3 b^2 \cos (2 (c+d x)) a+2 b \left(2 a^2+b^2\right) \sin (c+d x)\right)}{a (a-b)^2 (a+b)^2 (4 a+3 b \sin (c+d x)-b \sin (3 (c+d x)))}}{18 d}","\text{Int}\left(\frac{\sec ^2(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2},x\right)",0,"(((-I)*b*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (16*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 2*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - I*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (20*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + (16*I)*a*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 10*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 8*a*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 120*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (60*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (20*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (16*I)*a*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 10*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 - 8*a*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 16*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 2*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - I*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(a*(a^2 - b^2)^2) + (18*Sin[(c + d*x)/2])/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (18*Sin[(c + d*x)/2])/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (12*b*Cos[c + d*x]*(-2*a^3 - 7*a*b^2 + 3*a*b^2*Cos[2*(c + d*x)] + 2*b*(2*a^2 + b^2)*Sin[c + d*x]))/(a*(a - b)^2*(a + b)^2*(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)])))/(18*d)","C",1
403,1,1158,26,1.7438486,"\int \frac{\sec ^4(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]","\frac{\frac{3 \left(48 \sin (c+d x) a^6+16 \sin (3 (c+d x)) a^6+48 b a^5+2 b \cos (6 (c+d x)) a^5-244 b^2 \sin (c+d x) a^4-194 b^2 \sin (3 (c+d x)) a^4-14 b^2 \sin (5 (c+d x)) a^4+568 b^3 a^3-30 b^3 \cos (6 (c+d x)) a^3+20 b^4 \sin (c+d x) a^2-86 b^4 \sin (3 (c+d x)) a^2-74 b^4 \sin (5 (c+d x)) a^2+14 b^5 a+18 b^3 \left(4 a^2+b^2\right) \cos (4 (c+d x)) a-17 b^5 \cos (6 (c+d x)) a+\left(78 b a^5+606 b^3 a^3+81 b^5 a\right) \cos (2 (c+d x))-4 b^6 \sin (c+d x)-6 b^6 \sin (3 (c+d x))-2 b^6 \sin (5 (c+d x))\right) \sec ^3(c+d x)}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}+4 i b^2 \text{RootSum}\left[i b \text{$\#$1}^6-3 i b \text{$\#$1}^4+8 a \text{$\#$1}^3+3 i b \text{$\#$1}^2-i b\&,\frac{14 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^4 a^4-7 i \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^4 a^4-180 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^2 a^4+90 i \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^2 a^4+14 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) a^4-7 i \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) a^4-144 i b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^3 a^3-72 b \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^3 a^3+144 i b \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1} a^3+72 b \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1} a^3+74 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^4 a^2-37 i b^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^4 a^2-372 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^2 a^2+186 i b^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^2 a^2+74 b^2 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) a^2-37 i b^2 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) a^2-36 i b^3 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^3 a-18 b^3 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^3 a+36 i b^3 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1} a+18 b^3 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1} a+2 b^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^4-i b^4 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^4+12 b^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right) \text{$\#$1}^2-6 i b^4 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right) \text{$\#$1}^2+2 b^4 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right)-i b^4 \log \left(\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right)}{b \text{$\#$1}^5-2 b \text{$\#$1}^3-4 i a \text{$\#$1}^2+b \text{$\#$1}}\&\right]}{72 a \left(a^2-b^2\right)^3 d}","\text{Int}\left(\frac{\sec ^4(c+d x)}{\left(a+b \sin ^3(c+d x)\right)^2},x\right)",0,"((4*I)*b^2*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (14*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 74*a^2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (7*I)*a^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (37*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - I*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (144*I)*a^3*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + (36*I)*a*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 18*a*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 180*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 372*a^2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (90*I)*a^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (186*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (144*I)*a^3*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (36*I)*a*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 - 18*a*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 14*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 74*a^2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (7*I)*a^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (37*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - I*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + (3*Sec[c + d*x]^3*(48*a^5*b + 568*a^3*b^3 + 14*a*b^5 + (78*a^5*b + 606*a^3*b^3 + 81*a*b^5)*Cos[2*(c + d*x)] + 18*a*b^3*(4*a^2 + b^2)*Cos[4*(c + d*x)] + 2*a^5*b*Cos[6*(c + d*x)] - 30*a^3*b^3*Cos[6*(c + d*x)] - 17*a*b^5*Cos[6*(c + d*x)] + 48*a^6*Sin[c + d*x] - 244*a^4*b^2*Sin[c + d*x] + 20*a^2*b^4*Sin[c + d*x] - 4*b^6*Sin[c + d*x] + 16*a^6*Sin[3*(c + d*x)] - 194*a^4*b^2*Sin[3*(c + d*x)] - 86*a^2*b^4*Sin[3*(c + d*x)] - 6*b^6*Sin[3*(c + d*x)] - 14*a^4*b^2*Sin[5*(c + d*x)] - 74*a^2*b^4*Sin[5*(c + d*x)] - 2*b^6*Sin[5*(c + d*x)]))/(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)]))/(72*a*(a^2 - b^2)^3*d)","C",0
404,1,207,131,0.2627628,"\int \frac{\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Cos[c + d*x]^7/(a - b*Sin[c + d*x]^4),x]","\frac{4 a^{3/4} b^{3/4} \sin ^3(c+d x)-36 a^{3/4} b^{3/4} \sin (c+d x)+3 \left(\sqrt{a}-\sqrt{b}\right)^3 \log \left(\sqrt[4]{a}-\sqrt[4]{b} \sin (c+d x)\right)-3 \left(\sqrt{a}-\sqrt{b}\right)^3 \log \left(\sqrt[4]{a}+\sqrt[4]{b} \sin (c+d x)\right)+3 i \left(\sqrt{a}+\sqrt{b}\right)^3 \log \left(\sqrt[4]{a}-i \sqrt[4]{b} \sin (c+d x)\right)-3 i \left(\sqrt{a}+\sqrt{b}\right)^3 \log \left(\sqrt[4]{a}+i \sqrt[4]{b} \sin (c+d x)\right)}{12 a^{3/4} b^{7/4} 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,"(3*(Sqrt[a] - Sqrt[b])^3*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]] + (3*I)*(Sqrt[a] + Sqrt[b])^3*Log[a^(1/4) - I*b^(1/4)*Sin[c + d*x]] - (3*I)*(Sqrt[a] + Sqrt[b])^3*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x]] - 3*(Sqrt[a] - Sqrt[b])^3*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]] - 36*a^(3/4)*b^(3/4)*Sin[c + d*x] + 4*a^(3/4)*b^(3/4)*Sin[c + d*x]^3)/(12*a^(3/4)*b^(7/4)*d)","C",1
405,1,189,113,0.2002168,"\int \frac{\cos ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Cos[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]","\frac{-4 a^{3/4} \sqrt[4]{b} \sin (c+d x)+\left(\sqrt{a}-\sqrt{b}\right)^2 \left(-\log \left(\sqrt[4]{a}-\sqrt[4]{b} \sin (c+d x)\right)\right)+i \left(-i \left(\sqrt{a}-\sqrt{b}\right)^2 \log \left(\sqrt[4]{a}+\sqrt[4]{b} \sin (c+d x)\right)+\left(\sqrt{a}+\sqrt{b}\right)^2 \log \left(\sqrt[4]{a}-i \sqrt[4]{b} \sin (c+d x)\right)-\left(\sqrt{a}+\sqrt{b}\right)^2 \log \left(\sqrt[4]{a}+i \sqrt[4]{b} \sin (c+d x)\right)\right)}{4 a^{3/4} b^{5/4} 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*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]]) + I*((Sqrt[a] + Sqrt[b])^2*Log[a^(1/4) - I*b^(1/4)*Sin[c + d*x]] - (Sqrt[a] + Sqrt[b])^2*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x]] - I*(Sqrt[a] - Sqrt[b])^2*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]]) - 4*a^(3/4)*b^(1/4)*Sin[c + d*x])/(4*a^(3/4)*b^(5/4)*d)","C",1
406,1,160,95,0.083906,"\int \frac{\cos ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]","\frac{\left(\sqrt{a}-\sqrt{b}\right) \log \left(\sqrt[4]{a}-\sqrt[4]{b} \sin (c+d x)\right)+i \left(\sqrt{a}+\sqrt{b}\right) \log \left(\sqrt[4]{a}-i \sqrt[4]{b} \sin (c+d x)\right)-i \left(\sqrt{a}+\sqrt{b}\right) \log \left(\sqrt[4]{a}+i \sqrt[4]{b} \sin (c+d x)\right)-\left(\sqrt{a}-\sqrt{b}\right) \log \left(\sqrt[4]{a}+\sqrt[4]{b} \sin (c+d x)\right)}{4 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])*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]] + I*(Sqrt[a] + Sqrt[b])*Log[a^(1/4) - I*b^(1/4)*Sin[c + d*x]] - I*(Sqrt[a] + Sqrt[b])*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x]] - (Sqrt[a] - Sqrt[b])*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]])/(4*a^(3/4)*b^(3/4)*d)","C",1
407,1,54,71,0.0225941,"\int \frac{\cos (c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[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)+\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)] + ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(1/4)*d)","A",1
408,1,184,117,0.1798026,"\int \frac{\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sec[c + d*x]/(a - b*Sin[c + d*x]^4),x]","\frac{4 a^{3/4} \tanh ^{-1}(\sin (c+d x))+\sqrt[4]{b} \left(\left(\sqrt{a}+\sqrt{b}\right) \log \left(\sqrt[4]{a}-\sqrt[4]{b} \sin (c+d x)\right)+i \left(\left(\sqrt{a}-\sqrt{b}\right) \log \left(\sqrt[4]{a}-i \sqrt[4]{b} \sin (c+d x)\right)+\left(\sqrt{b}-\sqrt{a}\right) \log \left(\sqrt[4]{a}+i \sqrt[4]{b} \sin (c+d x)\right)+i \left(\sqrt{a}+\sqrt{b}\right) \log \left(\sqrt[4]{a}+\sqrt[4]{b} \sin (c+d x)\right)\right)\right)}{4 a^{3/4} 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,"(4*a^(3/4)*ArcTanh[Sin[c + d*x]] + b^(1/4)*((Sqrt[a] + Sqrt[b])*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]] + I*((Sqrt[a] - Sqrt[b])*Log[a^(1/4) - I*b^(1/4)*Sin[c + d*x]] + (-Sqrt[a] + Sqrt[b])*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x]] + I*(Sqrt[a] + Sqrt[b])*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]])))/(4*a^(3/4)*(a - b)*d)","C",1
409,1,255,175,1.0180234,"\int \frac{\sec ^3(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a - b*Sin[c + d*x]^4),x]","-\frac{\frac{b^{3/4} \log \left(\sqrt[4]{a}-\sqrt[4]{b} \sin (c+d x)\right)}{a^{3/4} \left(\sqrt{a}-\sqrt{b}\right)^2}-\frac{i b^{3/4} \log \left(\sqrt[4]{a}-i \sqrt[4]{b} \sin (c+d x)\right)}{a^{3/4} \left(\sqrt{a}+\sqrt{b}\right)^2}+\frac{i b^{3/4} \log \left(\sqrt[4]{a}+i \sqrt[4]{b} \sin (c+d x)\right)}{a^{3/4} \left(\sqrt{a}+\sqrt{b}\right)^2}-\frac{b^{3/4} \log \left(\sqrt[4]{a}+\sqrt[4]{b} \sin (c+d x)\right)}{a^{3/4} \left(\sqrt{a}-\sqrt{b}\right)^2}+\frac{1}{(a-b) (\sin (c+d x)-1)}+\frac{1}{(a-b) (\sin (c+d x)+1)}-\frac{2 (a-5 b) \tanh ^{-1}(\sin (c+d x))}{(a-b)^2}}{4 d}","\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,"-1/4*((-2*(a - 5*b)*ArcTanh[Sin[c + d*x]])/(a - b)^2 + (b^(3/4)*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] - Sqrt[b])^2) - (I*b^(3/4)*Log[a^(1/4) - I*b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] + Sqrt[b])^2) + (I*b^(3/4)*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] + Sqrt[b])^2) - (b^(3/4)*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] - Sqrt[b])^2) + 1/((a - b)*(-1 + Sin[c + d*x])) + 1/((a - b)*(1 + Sin[c + d*x])))/d","C",1
410,1,317,249,5.6021381,"\int \frac{\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sec[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]","\frac{\frac{4 b^{5/4} \log \left(\sqrt[4]{a}-\sqrt[4]{b} \sin (c+d x)\right)}{a^{3/4} \left(\sqrt{a}-\sqrt{b}\right)^3}+\frac{4 i b^{5/4} \log \left(\sqrt[4]{a}-i \sqrt[4]{b} \sin (c+d x)\right)}{a^{3/4} \left(\sqrt{a}+\sqrt{b}\right)^3}-\frac{4 i b^{5/4} \log \left(\sqrt[4]{a}+i \sqrt[4]{b} \sin (c+d x)\right)}{a^{3/4} \left(\sqrt{a}+\sqrt{b}\right)^3}-\frac{4 b^{5/4} \log \left(\sqrt[4]{a}+\sqrt[4]{b} \sin (c+d x)\right)}{a^{3/4} \left(\sqrt{a}-\sqrt{b}\right)^3}+\frac{2 \left(3 a^2-6 a b+35 b^2\right) \tanh ^{-1}(\sin (c+d x))}{(a-b)^3}+\frac{11 b-3 a}{(a-b)^2 (\sin (c+d x)-1)}+\frac{11 b-3 a}{(a-b)^2 (\sin (c+d x)+1)}+\frac{1}{(a-b) (\sin (c+d x)-1)^2}-\frac{1}{(a-b) (\sin (c+d x)+1)^2}}{16 d}","\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,"((2*(3*a^2 - 6*a*b + 35*b^2)*ArcTanh[Sin[c + d*x]])/(a - b)^3 + (4*b^(5/4)*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] - Sqrt[b])^3) + ((4*I)*b^(5/4)*Log[a^(1/4) - I*b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] + Sqrt[b])^3) - ((4*I)*b^(5/4)*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] + Sqrt[b])^3) - (4*b^(5/4)*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] - Sqrt[b])^3) + 1/((a - b)*(-1 + Sin[c + d*x])^2) + (-3*a + 11*b)/((a - b)^2*(-1 + Sin[c + d*x])) - 1/((a - b)*(1 + Sin[c + d*x])^2) + (-3*a + 11*b)/((a - b)^2*(1 + Sin[c + d*x])))/(16*d)","C",1
411,1,233,252,0.9066543,"\int \frac{\cos ^{10}(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Cos[c + d*x]^10/(a - b*Sin[c + d*x]^4),x]","-\frac{36 b (24 a+35 b) (c+d x)+3 b (16 a+95 b) \sin (2 (c+d x))-\frac{96 \sqrt{b} \left(\sqrt{a}+\sqrt{b}\right)^5 \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{96 \sqrt{b} \left(\sqrt{a}-\sqrt{b}\right)^5 \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}+21 b^2 \sin (4 (c+d x))+b^2 \sin (6 (c+d x))}{192 b^3 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{\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,"-1/192*(36*b*(24*a + 35*b)*(c + d*x) - (96*(Sqrt[a] + Sqrt[b])^5*Sqrt[b]*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) - (96*(Sqrt[a] - Sqrt[b])^5*Sqrt[b]*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 3*b*(16*a + 95*b)*Sin[2*(c + d*x)] + 21*b^2*Sin[4*(c + d*x)] + b^2*Sin[6*(c + d*x)])/(b^3*d)","A",1
412,1,200,186,0.676913,"\int \frac{\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Cos[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]","-\frac{4 (8 a+35 b) (c+d x)-\frac{16 \left(\sqrt{a}+\sqrt{b}\right)^4 \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{16 \left(\sqrt{a}-\sqrt{b}\right)^4 \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}+24 b \sin (2 (c+d x))+b \sin (4 (c+d x))}{32 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{\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,"-1/32*(4*(8*a + 35*b)*(c + d*x) - (16*(Sqrt[a] + Sqrt[b])^4*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (16*(Sqrt[a] - Sqrt[b])^4*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 24*b*Sin[2*(c + d*x)] + b*Sin[4*(c + d*x)])/(b^2*d)","A",1
413,1,194,155,0.4814308,"\int \frac{\cos ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Cos[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]","\frac{\frac{2 \sqrt{b} \left(\sqrt{a}+\sqrt{b}\right)^3 \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{2 \sqrt{b} \left(\sqrt{a}-\sqrt{b}\right)^3 \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}-10 b (c+d x)-b \sin (2 (c+d x))}{4 b^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{\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,"(-10*b*(c + d*x) + (2*(Sqrt[a] + Sqrt[b])^3*Sqrt[b]*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (2*(Sqrt[a] - Sqrt[b])^3*Sqrt[b]*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) - b*Sin[2*(c + d*x)])/(4*b^2*d)","A",1
414,1,171,127,0.2533309,"\int \frac{\cos ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]","\frac{\frac{\left(\sqrt{a}+\sqrt{b}\right)^2 \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2 \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}-2 (c+d x)}{2 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{\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,"(-2*(c + d*x) + ((Sqrt[a] + Sqrt[b])^2*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) - ((Sqrt[a] - Sqrt[b])^2*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]))/(2*b*d)","A",1
415,1,158,125,0.2828462,"\int \frac{\cos ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]","\frac{\frac{\left(\sqrt{a} \sqrt{b}+b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\left(\sqrt{a} \sqrt{b}-b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{\sqrt{a} \sqrt{b}-a}}}{2 \sqrt{a} 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[a]*Sqrt[b] + b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] + ((Sqrt[a]*Sqrt[b] - b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*b*d)","A",1
416,1,175,142,0.4874335,"\int \frac{\sec ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]","\frac{\frac{\left(\sqrt{a} \sqrt{b}-b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\left(\sqrt{a} \sqrt{b}+b\right) \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}+2 \tan (c+d x)}{2 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[a]*Sqrt[b] - b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) + ((Sqrt[a]*Sqrt[b] + b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 2*Tan[c + d*x])/(2*(a - b)*d)","A",1
417,1,205,161,1.0059971,"\int \frac{\sec ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]","\frac{\frac{3 b \left(-2 \sqrt{a} \sqrt{b}+a+b\right) \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}+4 (a-4 b) \tan (c+d x)-\frac{3 b \left(\sqrt{a}+\sqrt{b}\right)^2 \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}+2 (a-b) \tan (c+d x) \sec ^2(c+d x)}{6 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,"((3*b*(a - 2*Sqrt[a]*Sqrt[b] + b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) - (3*(Sqrt[a] + Sqrt[b])^2*b*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 4*(a - 4*b)*Tan[c + d*x] + 2*(a - b)*Sec[c + d*x]^2*Tan[c + d*x])/(6*(a - b)^2*d)","A",1
418,1,253,204,1.3196938,"\int \frac{\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx","Integrate[Sec[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]","\frac{2 \left(8 a^2-21 a b+73 b^2\right) \tan (c+d x)+\frac{15 b^{3/2} \left(\sqrt{a}-\sqrt{b}\right)^3 \tan ^{-1}\left(\frac{\left(\sqrt{a}+\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{15 b^{3/2} \left(\sqrt{a}+\sqrt{b}\right)^3 \tanh ^{-1}\left(\frac{\left(\sqrt{a}-\sqrt{b}\right) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}+6 (a-b)^2 \tan (c+d x) \sec ^4(c+d x)+4 (2 a-7 b) (a-b) \tan (c+d x) \sec ^2(c+d x)}{30 d (a-b)^3}","-\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,"((15*(Sqrt[a] - Sqrt[b])^3*b^(3/2)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (15*(Sqrt[a] + Sqrt[b])^3*b^(3/2)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 2*(8*a^2 - 21*a*b + 73*b^2)*Tan[c + d*x] + 4*(2*a - 7*b)*(a - b)*Sec[c + d*x]^2*Tan[c + d*x] + 6*(a - b)^2*Sec[c + d*x]^4*Tan[c + d*x])/(30*(a - b)^3*d)","A",1
419,0,0,26,8.5353832,"\int \cos ^m(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Cos[e + f*x]^m*(a + b*Sin[e + f*x]^4)^p, x]","A",-1
420,1,141,197,0.139411,"\int \cos ^5(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Integrate[Cos[e + f*x]^5*(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} \left(3 \sin ^4(e+f x) \, _2F_1\left(\frac{5}{4},-p;\frac{9}{4};-\frac{b \sin ^4(e+f x)}{a}\right)+15 \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b \sin ^4(e+f x)}{a}\right)-10 \sin ^2(e+f x) \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b \sin ^4(e+f x)}{a}\right)\right)}{15 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{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{\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)^p*(15*Hypergeometric2F1[1/4, -p, 5/4, -((b*Sin[e + f*x]^4)/a)] - 10*Hypergeometric2F1[3/4, -p, 7/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^2 + 3*Hypergeometric2F1[5/4, -p, 9/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^4))/(15*f*(1 + (b*Sin[e + f*x]^4)/a)^p)","A",1
421,1,106,140,0.0505944,"\int \cos ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Integrate[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} \left(\sin ^2(e+f x) \, _2F_1\left(\frac{3}{4},-p;\frac{7}{4};-\frac{b \sin ^4(e+f x)}{a}\right)-3 \, _2F_1\left(\frac{1}{4},-p;\frac{5}{4};-\frac{b \sin ^4(e+f x)}{a}\right)\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,"-1/3*(Sin[e + f*x]*(-3*Hypergeometric2F1[1/4, -p, 5/4, -((b*Sin[e + f*x]^4)/a)] + Hypergeometric2F1[3/4, -p, 7/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^2)*(a + b*Sin[e + f*x]^4)^p)/(f*(1 + (b*Sin[e + f*x]^4)/a)^p)","A",1
422,1,67,67,0.0158368,"\int \cos (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Integrate[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",1
423,0,0,158,5.9275189,"\int \sec (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Integrate[Sec[e + f*x]*(a + b*Sin[e + f*x]^4)^p,x]","\int \sec (e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","\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}",1,"Integrate[Sec[e + f*x]*(a + b*Sin[e + f*x]^4)^p, x]","F",-1
424,0,0,239,9.2389308,"\int \sec ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Integrate[Sec[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p,x]","\int \sec ^3(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","\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}",1,"Integrate[Sec[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p, x]","F",-1
425,0,0,26,5.3100877,"\int \cos ^4(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p, x]","A",-1
426,0,0,26,17.2415573,"\int \cos ^2(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^4)^p, x]","A",-1
427,0,0,17,1.1342076,"\int \left(a+b \sin ^4(e+f x)\right)^p \, dx","Integrate[(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,"Integrate[(a + b*Sin[e + f*x]^4)^p, x]","A",-1
428,0,0,26,6.0031922,"\int \sec ^2(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^4)^p, x]","A",-1
429,0,0,26,9.0792476,"\int \sec ^4(e+f x) \left(a+b \sin ^4(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p, x]","A",-1
430,0,0,26,4.5927155,"\int \cos ^m(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Cos[e + f*x]^m*(a + b*Sin[e + f*x]^n)^p, x]","A",-1
431,1,155,226,0.2160804,"\int \cos ^5(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Integrate[Cos[e + f*x]^5*(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} \left(15 \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right)+3 \sin ^4(e+f x) \, _2F_1\left(\frac{5}{n},-p;\frac{n+5}{n};-\frac{b \sin ^n(e+f x)}{a}\right)-10 \sin ^2(e+f x) \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b \sin ^n(e+f x)}{a}\right)\right)}{15 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}",1,"(Sin[e + f*x]*(15*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*Sin[e + f*x]^n)/a)] - 10*Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]^2 + 3*Hypergeometric2F1[5/n, -p, (5 + n)/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]^4)*(a + b*Sin[e + f*x]^n)^p)/(15*f*(1 + (b*Sin[e + f*x]^n)/a)^p)","A",1
432,1,114,148,0.0871048,"\int \cos ^3(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Integrate[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} \left(\sin ^2(e+f x) \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b \sin ^n(e+f x)}{a}\right)-3 \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b \sin ^n(e+f x)}{a}\right)\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,"-1/3*(Sin[e + f*x]*(-3*Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*Sin[e + f*x]^n)/a)] + Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]^2)*(a + b*Sin[e + f*x]^n)^p)/(f*(1 + (b*Sin[e + f*x]^n)/a)^p)","A",1
433,1,69,69,0.0215752,"\int \cos (e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Integrate[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",1
434,0,0,24,2.85434,"\int \sec (e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Sec[e + f*x]*(a + b*Sin[e + f*x]^n)^p, x]","A",-1
435,0,0,26,8.1474998,"\int \sec ^3(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Sec[e + f*x]^3*(a + b*Sin[e + f*x]^n)^p, x]","A",-1
436,0,0,26,19.6583332,"\int \cos ^4(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p, x]","A",-1
437,0,0,26,11.7106994,"\int \cos ^2(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^n)^p, x]","A",-1
438,0,0,17,1.5725126,"\int \left(a+b \sin ^n(e+f x)\right)^p \, dx","Integrate[(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,"Integrate[(a + b*Sin[e + f*x]^n)^p, x]","A",-1
439,0,0,26,4.3839625,"\int \sec ^2(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^n)^p, x]","A",-1
440,0,0,26,8.0566964,"\int \sec ^4(e+f x) \left(a+b \sin ^n(e+f x)\right)^p \, dx","Integrate[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,"Integrate[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p, x]","A",-1
441,1,113,128,0.2842665,"\int \frac{\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Tan[c + d*x]^7/(a + b*Sin[c + d*x]^2),x]","\frac{-\frac{6 a^3 \log \left(a+b \sin ^2(c+d x)\right)}{(a+b)^4}+\frac{12 a^3 \log (\cos (c+d x))}{(a+b)^4}+\frac{6 \left(3 a^2+3 a b+b^2\right) \sec ^2(c+d x)}{(a+b)^3}+\frac{2 \sec ^6(c+d x)}{a+b}-\frac{3 (3 a+2 b) \sec ^4(c+d x)}{(a+b)^2}}{12 d}","-\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{\left(3 a^2+3 a b+b^2\right) \sec ^2(c+d x)}{2 d (a+b)^3}+\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,"((12*a^3*Log[Cos[c + d*x]])/(a + b)^4 - (6*a^3*Log[a + b*Sin[c + d*x]^2])/(a + b)^4 + (6*(3*a^2 + 3*a*b + b^2)*Sec[c + d*x]^2)/(a + b)^3 - (3*(3*a + 2*b)*Sec[c + d*x]^4)/(a + b)^2 + (2*Sec[c + d*x]^6)/(a + b))/(12*d)","A",1
442,1,78,94,0.2601375,"\int \frac{\tan ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Sin[c + d*x]^2),x]","\frac{-2 \left(2 a^2+3 a b+b^2\right) \sec ^2(c+d x)+2 a^2 \left(\log \left(a+b \sin ^2(c+d x)\right)-2 \log (\cos (c+d x))\right)+(a+b)^2 \sec ^4(c+d x)}{4 d (a+b)^3}","\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,"(2*a^2*(-2*Log[Cos[c + d*x]] + Log[a + b*Sin[c + d*x]^2]) - 2*(2*a^2 + 3*a*b + b^2)*Sec[c + d*x]^2 + (a + b)^2*Sec[c + d*x]^4)/(4*(a + b)^3*d)","A",1
443,1,52,64,0.0939431,"\int \frac{\tan ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Sin[c + d*x]^2),x]","\frac{(a+b) \sec ^2(c+d x)+a \left(2 \log (\cos (c+d x))-\log \left(a+b \sin ^2(c+d x)\right)\right)}{2 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*(2*Log[Cos[c + d*x]] - Log[a + b*Sin[c + d*x]^2]) + (a + b)*Sec[c + d*x]^2)/(2*(a + b)^2*d)","A",1
444,1,37,43,0.0333729,"\int \frac{\tan (c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Tan[c + d*x]/(a + b*Sin[c + d*x]^2),x]","\frac{\log \left(a-b \cos ^2(c+d x)+b\right)-2 \log (\cos (c+d x))}{2 a d+2 b d}","\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,"(-2*Log[Cos[c + d*x]] + Log[a + b - b*Cos[c + d*x]^2])/(2*a*d + 2*b*d)","A",1
445,1,38,38,0.0206591,"\int \frac{\cot (c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[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",1
446,1,50,63,0.147905,"\int \frac{\cot ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Cot[c + d*x]^3/(a + b*Sin[c + d*x]^2),x]","-\frac{(a+b) \left(2 \log (\sin (c+d x))-\log \left(a+b \sin ^2(c+d x)\right)\right)+a \csc ^2(c+d x)}{2 a^2 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,"-1/2*(a*Csc[c + d*x]^2 + (a + b)*(2*Log[Sin[c + d*x]] - Log[a + b*Sin[c + d*x]^2]))/(a^2*d)","A",1
447,1,72,89,0.5202263,"\int \frac{\cot ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Cot[c + d*x]^5/(a + b*Sin[c + d*x]^2),x]","\frac{-a^2 \csc ^4(c+d x)+2 a (2 a+b) \csc ^2(c+d x)+2 (a+b)^2 \left(2 \log (\sin (c+d x))-\log \left(a+b \sin ^2(c+d x)\right)\right)}{4 a^3 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{(2 a+b) \csc ^2(c+d x)}{2 a^2 d}-\frac{\csc ^4(c+d x)}{4 a d}",1,"(2*a*(2*a + b)*Csc[c + d*x]^2 - a^2*Csc[c + d*x]^4 + 2*(a + b)^2*(2*Log[Sin[c + d*x]] - Log[a + b*Sin[c + d*x]^2]))/(4*a^3*d)","A",1
448,1,100,121,0.2638648,"\int \frac{\cot ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Cot[c + d*x]^7/(a + b*Sin[c + d*x]^2),x]","-\frac{2 a^3 \csc ^6(c+d x)+6 a \left(3 a^2+3 a b+b^2\right) \csc ^2(c+d x)-3 a^2 (3 a+b) \csc ^4(c+d x)-6 (a+b)^3 \log \left(a+b \sin ^2(c+d x)\right)+12 (a+b)^3 \log (\sin (c+d x))}{12 a^4 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{(3 a+b) \csc ^4(c+d x)}{4 a^2 d}-\frac{\left(3 a^2+3 a b+b^2\right) \csc ^2(c+d x)}{2 a^3 d}-\frac{\csc ^6(c+d x)}{6 a d}",1,"-1/12*(6*a*(3*a^2 + 3*a*b + b^2)*Csc[c + d*x]^2 - 3*a^2*(3*a + b)*Csc[c + d*x]^4 + 2*a^3*Csc[c + d*x]^6 + 12*(a + b)^3*Log[Sin[c + d*x]] - 6*(a + b)^3*Log[a + b*Sin[c + d*x]^2])/(a^4*d)","A",1
449,1,147,120,2.3866192,"\int \frac{\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Tan[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)}{d (a+b)^{9/2}}+\frac{\tan (c+d x) \left(-176 a^3-122 a^2 b+\left(122 a^3+254 a^2 b+177 a b^2+45 b^3\right) \sec ^2(c+d x)-66 a b^2+15 (a+b)^3 \sec ^6(c+d x)-3 (a+b)^2 (22 a+15 b) \sec ^4(c+d x)-15 b^3\right)}{105 d (a+b)^4}","\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{a^2 \tan ^3(c+d x)}{3 d (a+b)^3}+\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) + ((-176*a^3 - 122*a^2*b - 66*a*b^2 - 15*b^3 + (122*a^3 + 254*a^2*b + 177*a*b^2 + 45*b^3)*Sec[c + d*x]^2 - 3*(a + b)^2*(22*a + 15*b)*Sec[c + d*x]^4 + 15*(a + b)^3*Sec[c + d*x]^6)*Tan[c + d*x])/(105*(a + b)^4*d)","A",1
450,1,111,97,0.7544713,"\int \frac{\tan ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Tan[c + d*x]^6/(a + b*Sin[c + d*x]^2),x]","\frac{\sqrt{a+b} \tan (c+d x) \left(-\left(11 a^2+17 a b+6 b^2\right) \sec ^2(c+d x)+23 a^2+3 (a+b)^2 \sec ^4(c+d x)+11 a b+3 b^2\right)-15 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{15 d (a+b)^{7/2}}","-\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,"(-15*a^(5/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]] + Sqrt[a + b]*(23*a^2 + 11*a*b + 3*b^2 - (11*a^2 + 17*a*b + 6*b^2)*Sec[c + d*x]^2 + 3*(a + b)^2*Sec[c + d*x]^4)*Tan[c + d*x])/(15*(a + b)^(7/2)*d)","A",1
451,1,75,74,0.3043869,"\int \frac{\tan ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Sin[c + d*x]^2),x]","\frac{3 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)+\sqrt{a+b} \tan (c+d x) \left((a+b) \sec ^2(c+d x)-4 a-b\right)}{3 d (a+b)^{5/2}}","\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,"(3*a^(3/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]] + Sqrt[a + b]*(-4*a - b + (a + b)*Sec[c + d*x]^2)*Tan[c + d*x])/(3*(a + b)^(5/2)*d)","A",1
452,1,53,53,0.1281187,"\int \frac{\tan ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[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",1
453,1,52,52,0.1775515,"\int \frac{\cot ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[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)-\sqrt{a} \cot (c+d x)}{a^{3/2} 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]]) - Sqrt[a]*Cot[c + d*x])/(a^(3/2)*d)","A",1
454,1,72,71,0.2923144,"\int \frac{\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x]^2),x]","\frac{3 (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)+\sqrt{a} \cot (c+d x) \left(-a \csc ^2(c+d x)+4 a+3 b\right)}{3 a^{5/2} 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,"(3*(a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]] + Sqrt[a]*Cot[c + d*x]*(4*a + 3*b - a*Csc[c + d*x]^2))/(3*a^(5/2)*d)","A",1
455,1,101,96,0.8614651,"\int \frac{\cot ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[Cot[c + d*x]^6/(a + b*Sin[c + d*x]^2),x]","\frac{-\sqrt{a} \cot (c+d x) \left(3 a^2 \csc ^4(c+d x)+23 a^2-a (11 a+5 b) \csc ^2(c+d x)+35 a b+15 b^2\right)-15 (a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{15 a^{7/2} 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)^2 \cot (c+d x)}{a^3 d}+\frac{(a+b) \cot ^3(c+d x)}{3 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}",1,"(-15*(a + b)^(5/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]] - Sqrt[a]*Cot[c + d*x]*(23*a^2 + 35*a*b + 15*b^2 - a*(11*a + 5*b)*Csc[c + d*x]^2 + 3*a^2*Csc[c + d*x]^4))/(15*a^(7/2)*d)","A",1
456,1,135,117,1.0771229,"\int \frac{\cot ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx","Integrate[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{\cot (c+d x) \left(-15 a^3 \csc ^6(c+d x)+176 a^3-a \left(122 a^2+112 a b+35 b^2\right) \csc ^2(c+d x)+3 a^2 (22 a+7 b) \csc ^4(c+d x)+406 a^2 b+350 a b^2+105 b^3\right)}{105 a^4 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)^3 \cot (c+d x)}{a^4 d}-\frac{(a+b)^2 \cot ^3(c+d x)}{3 a^3 d}+\frac{(a+b) \cot ^5(c+d x)}{5 a^2 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) + (Cot[c + d*x]*(176*a^3 + 406*a^2*b + 350*a*b^2 + 105*b^3 - a*(122*a^2 + 112*a*b + 35*b^2)*Csc[c + d*x]^2 + 3*a^2*(22*a + 7*b)*Csc[c + d*x]^4 - 15*a^3*Csc[c + d*x]^6))/(105*a^4*d)","A",1
457,1,51,64,0.0866124,"\int \sqrt{a-a \sin ^2(e+f x)} \tan ^5(e+f x) \, dx","Integrate[Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^5,x]","-\frac{\left(3 \cos ^4(e+f x)+6 \cos ^2(e+f x)-1\right) \sec ^4(e+f x) \sqrt{a \cos ^2(e+f x)}}{3 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,"-1/3*(Sqrt[a*Cos[e + f*x]^2]*(-1 + 6*Cos[e + f*x]^2 + 3*Cos[e + f*x]^4)*Sec[e + f*x]^4)/f","A",1
458,1,29,38,0.0771973,"\int \sqrt{a-a \sin ^2(e+f x)} \tan ^3(e+f x) \, dx","Integrate[Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^3,x]","\frac{a \left(\cos ^2(e+f x)+1\right)}{f \sqrt{a \cos ^2(e+f x)}}","\frac{a}{f \sqrt{a \cos ^2(e+f x)}}+\frac{\sqrt{a \cos ^2(e+f x)}}{f}",1,"(a*(1 + Cos[e + f*x]^2))/(f*Sqrt[a*Cos[e + f*x]^2])","A",1
459,1,19,19,0.0403015,"\int \sqrt{a-a \sin ^2(e+f x)} \tan (e+f x) \, dx","Integrate[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",1
460,1,55,50,0.0673195,"\int \cot (e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]*Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\sec (e+f x) \sqrt{a \cos ^2(e+f x)} \left(\cos (e+f x)+\log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\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*Cos[e + f*x]^2]*(Cos[e + f*x] - Log[Cos[(e + f*x)/2]] + Log[Sin[(e + f*x)/2]])*Sec[e + f*x])/f","A",1
461,1,88,87,0.4163128,"\int \cot ^3(e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^3*Sqrt[a - a*Sin[e + f*x]^2],x]","-\frac{\sec (e+f x) \sqrt{a \cos ^2(e+f x)} \left(8 \cos (e+f x)+\csc ^2\left(\frac{1}{2} (e+f x)\right)-\sec ^2\left(\frac{1}{2} (e+f x)\right)+12 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-12 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{8 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,"-1/8*(Sqrt[a*Cos[e + f*x]^2]*(8*Cos[e + f*x] + Csc[(e + f*x)/2]^2 - 12*Log[Cos[(e + f*x)/2]] + 12*Log[Sin[(e + f*x)/2]] - Sec[(e + f*x)/2]^2)*Sec[e + f*x])/f","A",1
462,1,75,120,0.327636,"\int \sqrt{a-a \sin ^2(e+f x)} \tan ^6(e+f x) \, dx","Integrate[Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^6,x]","\frac{\sec ^5(e+f x) \sqrt{a \cos ^2(e+f x)} \left(-5 \sin (e+f x)-15 \sin (3 (e+f x))-2 \sin (5 (e+f x))+60 \cos ^4(e+f x) \tanh ^{-1}(\sin (e+f x))\right)}{32 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,"(Sqrt[a*Cos[e + f*x]^2]*Sec[e + f*x]^5*(60*ArcTanh[Sin[e + f*x]]*Cos[e + f*x]^4 - 5*Sin[e + f*x] - 15*Sin[3*(e + f*x)] - 2*Sin[5*(e + f*x)]))/(32*f)","A",1
463,1,55,91,0.1907171,"\int \sqrt{a-a \sin ^2(e+f x)} \tan ^4(e+f x) \, dx","Integrate[Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^4,x]","\frac{a \left((\cos (2 (e+f x))+2) \tan (e+f x)-3 \cos (e+f x) \tanh ^{-1}(\sin (e+f x))\right)}{2 f \sqrt{a \cos ^2(e+f 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}",1,"(a*(-3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x] + (2 + Cos[2*(e + f*x)])*Tan[e + f*x]))/(2*f*Sqrt[a*Cos[e + f*x]^2])","A",1
464,1,40,57,0.047181,"\int \sqrt{a-a \sin ^2(e+f x)} \tan ^2(e+f x) \, dx","Integrate[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)} \left(\tanh ^{-1}(\sin (e+f x))-\sin (e+f x)\right)}{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,"(Sqrt[a*Cos[e + f*x]^2]*Sec[e + f*x]*(ArcTanh[Sin[e + f*x]] - Sin[e + f*x]))/f","A",1
465,1,35,57,0.0714974,"\int \cot ^2(e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^2*Sqrt[a - a*Sin[e + f*x]^2],x]","-\frac{\tan (e+f x) \left(\csc ^2(e+f x)+1\right) \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]*(1 + Csc[e + f*x]^2)*Tan[e + f*x])/f)","A",1
466,1,47,91,0.0736339,"\int \cot ^4(e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^4*Sqrt[a - a*Sin[e + f*x]^2],x]","-\frac{\tan (e+f x) \left(\csc ^4(e+f x)-6 \csc ^2(e+f x)-3\right) \sqrt{a \cos ^2(e+f x)}}{3 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,"-1/3*(Sqrt[a*Cos[e + f*x]^2]*(-3 - 6*Csc[e + f*x]^2 + Csc[e + f*x]^4)*Tan[e + f*x])/f","A",1
467,1,67,124,0.1817474,"\int \cot ^6(e+f x) \sqrt{a-a \sin ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^6*Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{(235 \cos (2 (e+f x))-90 \cos (4 (e+f x))+5 \cos (6 (e+f x))-182) \csc ^5(e+f x) \sec (e+f x) \sqrt{a \cos ^2(e+f x)}}{160 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,"(Sqrt[a*Cos[e + f*x]^2]*(-182 + 235*Cos[2*(e + f*x)] - 90*Cos[4*(e + f*x)] + 5*Cos[6*(e + f*x)])*Csc[e + f*x]^5*Sec[e + f*x])/(160*f)","A",1
468,1,43,65,0.0778721,"\int \frac{\tan ^5(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^5/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{3 \sec ^4(e+f x)-10 \sec ^2(e+f x)+15}{15 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,"(15 - 10*Sec[e + f*x]^2 + 3*Sec[e + f*x]^4)/(15*f*Sqrt[a*Cos[e + f*x]^2])","A",1
469,1,31,42,0.0636195,"\int \frac{\tan ^3(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^3/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\sec ^2(e+f x)-3}{3 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,"(-3 + Sec[e + f*x]^2)/(3*f*Sqrt[a*Cos[e + f*x]^2])","A",1
470,1,18,18,0.0254755,"\int \frac{\tan (e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Integrate[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",1
471,1,49,31,0.0421036,"\int \frac{\cot (e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\cos (e+f x) \left(\log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{f \sqrt{a \cos ^2(e+f x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"(Cos[e + f*x]*(-Log[Cos[(e + f*x)/2]] + Log[Sin[(e + f*x)/2]]))/(f*Sqrt[a*Cos[e + f*x]^2])","A",1
472,1,80,66,0.1726724,"\int \frac{\cot ^3(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^3/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\cos (e+f x) \left(-\csc ^2\left(\frac{1}{2} (e+f x)\right)+\sec ^2\left(\frac{1}{2} (e+f x)\right)-4 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{8 f \sqrt{a \cos ^2(e+f 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}",1,"(Cos[e + f*x]*(-Csc[(e + f*x)/2]^2 + 4*Log[Cos[(e + f*x)/2]] - 4*Log[Sin[(e + f*x)/2]] + Sec[(e + f*x)/2]^2))/(8*f*Sqrt[a*Cos[e + f*x]^2])","A",1
473,1,66,91,0.1327449,"\int \frac{\tan ^4(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^4/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\tan (e+f x) \left(8 \tan ^2(e+f x)-6 \sec ^2(e+f x)+3\right)+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] + Tan[e + f*x]*(3 - 6*Sec[e + f*x]^2 + 8*Tan[e + f*x]^2))/(8*f*Sqrt[a*Cos[e + f*x]^2])","A",1
474,1,43,62,0.0450165,"\int \frac{\tan ^2(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^2/Sqrt[a - a*Sin[e + f*x]^2],x]","\frac{\tan (e+f x)-\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]) + Tan[e + f*x])/(2*f*Sqrt[a*Cos[e + f*x]^2])","A",1
475,1,25,25,0.028297,"\int \frac{\cot ^2(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Integrate[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",1
476,1,37,60,0.0613834,"\int \frac{\cot ^4(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^4/Sqrt[a - a*Sin[e + f*x]^2],x]","-\frac{\cot (e+f x) \left(\csc ^2(e+f x)-3\right)}{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,"-1/3*(Cot[e + f*x]*(-3 + Csc[e + f*x]^2))/(f*Sqrt[a*Cos[e + f*x]^2])","A",1
477,1,49,96,0.0688766,"\int \frac{\cot ^6(e+f x)}{\sqrt{a-a \sin ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^6/Sqrt[a - a*Sin[e + f*x]^2],x]","-\frac{\cot (e+f x) \left(3 \csc ^4(e+f x)-10 \csc ^2(e+f x)+15\right)}{15 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,"-1/15*(Cot[e + f*x]*(15 - 10*Csc[e + f*x]^2 + 3*Csc[e + f*x]^4))/(f*Sqrt[a*Cos[e + f*x]^2])","A",1
478,1,51,68,0.1006495,"\int \frac{\tan ^5(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^5/(a - a*Sin[e + f*x]^2)^(3/2),x]","\frac{\left(35 \cos ^4(e+f x)-42 \cos ^2(e+f x)+15\right) \sec ^4(e+f x)}{105 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,"((15 - 42*Cos[e + f*x]^2 + 35*Cos[e + f*x]^4)*Sec[e + f*x]^4)/(105*f*(a*Cos[e + f*x]^2)^(3/2))","A",1
479,1,34,44,0.109787,"\int \frac{\tan ^3(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^3/(a - a*Sin[e + f*x]^2)^(3/2),x]","\frac{a \left(3-5 \cos ^2(e+f x)\right)}{15 f \left(a \cos ^2(e+f x)\right)^{5/2}}","\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*(3 - 5*Cos[e + f*x]^2))/(15*f*(a*Cos[e + f*x]^2)^(5/2))","A",1
480,1,21,21,0.0289525,"\int \frac{\tan (e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[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",1
481,1,55,53,0.0675955,"\int \frac{\cot (e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]/(a - a*Sin[e + f*x]^2)^(3/2),x]","\frac{\cos (e+f x) \left(\log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+1}{a f \sqrt{a \cos ^2(e+f 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}",1,"(1 + Cos[e + f*x]*(-Log[Cos[(e + f*x)/2]] + Log[Sin[(e + f*x)/2]]))/(a*f*Sqrt[a*Cos[e + f*x]^2])","A",1
482,1,82,66,0.1725497,"\int \frac{\cot ^3(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^3/(a - a*Sin[e + f*x]^2)^(3/2),x]","-\frac{\cos ^3(e+f x) \left(\csc ^2\left(\frac{1}{2} (e+f x)\right)-\sec ^2\left(\frac{1}{2} (e+f x)\right)-4 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{8 f \left(a \cos ^2(e+f x)\right)^{3/2}}","-\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}",1,"-1/8*(Cos[e + f*x]^3*(Csc[(e + f*x)/2]^2 + 4*Log[Cos[(e + f*x)/2]] - 4*Log[Sin[(e + f*x)/2]] - Sec[(e + f*x)/2]^2))/(f*(a*Cos[e + f*x]^2)^(3/2))","A",1
483,1,59,106,0.0940225,"\int \frac{\tan ^2(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^2/(a - a*Sin[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x) \left(2 \sec ^2(e+f x)-1\right)-\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)}{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]) + (-1 + 2*Sec[e + f*x]^2)*Tan[e + f*x])/(8*a*f*Sqrt[a*Cos[e + f*x]^2])","A",1
484,1,44,63,0.0725306,"\int \frac{\cot ^2(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^2/(a - a*Sin[e + f*x]^2)^(3/2),x]","-\frac{\cot (e+f x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sin ^2(e+f x)\right)}{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,"-((Cot[e + f*x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[e + f*x]^2])/(a*f*Sqrt[a*Cos[e + f*x]^2]))","C",1
485,1,29,38,0.0338035,"\int \frac{\cot ^4(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^4/(a - a*Sin[e + f*x]^2)^(3/2),x]","-\frac{\cot ^3(e+f x)}{3 f \left(a \cos ^2(e+f x)\right)^{3/2}}","-\frac{\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt{a \cos ^2(e+f x)}}",1,"-1/3*Cot[e + f*x]^3/(f*(a*Cos[e + f*x]^2)^(3/2))","A",1
486,1,41,77,0.0911864,"\int \frac{\cot ^6(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^6/(a - a*Sin[e + f*x]^2)^(3/2),x]","-\frac{\cot ^3(e+f x) \left(3 \csc ^2(e+f x)-5\right)}{15 f \left(a \cos ^2(e+f x)\right)^{3/2}}","\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,"-1/15*(Cot[e + f*x]^3*(-5 + 3*Csc[e + f*x]^2))/(f*(a*Cos[e + f*x]^2)^(3/2))","A",1
487,1,51,115,0.129843,"\int \frac{\cot ^8(e+f x)}{\left(a-a \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^8/(a - a*Sin[e + f*x]^2)^(3/2),x]","-\frac{\cot ^3(e+f x) \left(15 \csc ^4(e+f x)-42 \csc ^2(e+f x)+35\right)}{105 f \left(a \cos ^2(e+f x)\right)^{3/2}}","-\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,"-1/105*(Cot[e + f*x]^3*(35 - 42*Csc[e + f*x]^2 + 15*Csc[e + f*x]^4))/(f*(a*Cos[e + f*x]^2)^(3/2))","A",1
488,1,143,177,0.5564602,"\int \sqrt{a+b \sin ^2(e+f x)} \tan ^5(e+f x) \, dx","Integrate[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) \left(\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)-\sqrt{a+b \sin ^2(e+f x)}\right)+2 (a+b) \sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}-(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 + 7*b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2)) + 2*(a + b)*Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2) + (8*a^2 + 24*a*b + 15*b^2)*(Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]] - Sqrt[a + b*Sin[e + f*x]^2]))/(8*(a + b)^2*f)","A",1
489,1,84,118,0.3878205,"\int \sqrt{a+b \sin ^2(e+f x)} \tan ^3(e+f x) \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^3,x]","\frac{(\cos (2 (e+f x))+2) \sec ^2(e+f x) \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+b}}\right)}{\sqrt{a+b}}}{2 f}","\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]])/Sqrt[a + b]) + (2 + Cos[2*(e + f*x)])*Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(2*f)","A",1
490,1,60,58,0.0566346,"\int \sqrt{a+b \sin ^2(e+f x)} \tan (e+f x) \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x],x]","\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a-b \cos ^2(e+f x)+b}}{\sqrt{a+b}}\right)-\sqrt{a-b \cos ^2(e+f x)+b}}{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 - b*Cos[e + f*x]^2]/Sqrt[a + b]] - Sqrt[a + b - b*Cos[e + f*x]^2])/f","A",1
491,1,53,54,0.0467085,"\int \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)-\sqrt{a+b \sin ^2(e+f x)}}{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]] - Sqrt[a + b*Sin[e + f*x]^2])/f)","A",1
492,1,77,110,0.1997797,"\int \cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[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)-\sqrt{a} \left(\csc ^2(e+f x)+2\right) \sqrt{a+b \sin ^2(e+f x)}}{2 \sqrt{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]] - Sqrt[a]*(2 + Csc[e + f*x]^2)*Sqrt[a + b*Sin[e + f*x]^2])/(2*Sqrt[a]*f)","A",1
493,1,103,165,0.5685892,"\int \cot ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[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) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)+\sqrt{a} \sqrt{a+b \sin ^2(e+f x)} \left((8 a-b) \csc ^2(e+f x)-2 a \csc ^4(e+f x)+8 a\right)}{8 a^{3/2} 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{(8 a+b) \csc ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2}}{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{\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]] + Sqrt[a]*(8*a + (8*a - b)*Csc[e + f*x]^2 - 2*a*Csc[e + f*x]^4)*Sqrt[a + b*Sin[e + f*x]^2])/(8*a^(3/2)*f)","A",1
494,1,198,234,1.9927705,"\int \sqrt{a+b \sin ^2(e+f x)} \tan ^4(e+f x) \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^4,x]","\frac{-\frac{\tan (e+f x) \sec ^2(e+f x) \left(4 \left(4 a^2+6 a b+b^2\right) \cos (2 (e+f x))+8 a^2-b (4 a+5 b) \cos (4 (e+f x))+12 a b+b^2\right)}{2 \sqrt{2}}-8 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 a (7 a+8 b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(2*a*(7*a + 8*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 8*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] - ((8*a^2 + 12*a*b + b^2 + 4*(4*a^2 + 6*a*b + b^2)*Cos[2*(e + f*x)] - b*(4*a + 5*b)*Cos[4*(e + f*x)])*Sec[e + f*x]^2*Tan[e + f*x])/(2*Sqrt[2]))/(6*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
495,1,140,171,0.4731722,"\int \sqrt{a+b \sin ^2(e+f x)} \tan ^2(e+f x) \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^2,x]","\frac{\tan (e+f x) (2 a-b \cos (2 (e+f x))+b)+\sqrt{2} a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 \sqrt{2} a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{\sqrt{2} f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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[2]*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + Sqrt[2]*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + (2*a + b - b*Cos[2*(e + f*x)])*Tan[e + f*x])/(Sqrt[2]*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
496,1,61,51,0.0822264,"\int \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)])/(f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
497,1,143,174,0.5483538,"\int \cot ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{-(\cot (e+f x) (2 a-b \cos (2 (e+f x))+b))+\sqrt{2} (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 \sqrt{2} a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{\sqrt{2} f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(-((2*a + b - b*Cos[2*(e + f*x)])*Cot[e + f*x]) - 2*Sqrt[2]*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + Sqrt[2]*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(Sqrt[2]*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
498,1,197,232,3.1636811,"\int \cot ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx","Integrate[Cot[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{-\frac{\cot (e+f x) \csc ^2(e+f x) \left(4 \left(4 a^2+2 a b-b^2\right) \cos (2 (e+f x))-8 a^2+b (b-4 a) \cos (4 (e+f x))-4 a b+3 b^2\right)}{2 \sqrt{2}}-8 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 a (7 a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 a f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(-1/2*((-8*a^2 - 4*a*b + 3*b^2 + 4*(4*a^2 + 2*a*b - b^2)*Cos[2*(e + f*x)] + b*(-4*a + b)*Cos[4*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^2)/Sqrt[2] + 2*a*(7*a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 8*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(6*a*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
499,1,160,220,2.0110337,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \tan ^5(e+f x) \, dx","Integrate[(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(\sqrt{a+b \sin ^2(e+f x)} \left(4 a+b \sin ^2(e+f x)+3 b\right)-3 (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)\right)-6 (a+b) \sec ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}+3 (8 a+9 b) \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{24 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,"-1/24*(3*(8*a + 9*b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(5/2) - 6*(a + b)*Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(5/2) + (8*a^2 + 40*a*b + 35*b^2)*(-3*(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]] + Sqrt[a + b*Sin[e + f*x]^2]*(4*a + 3*b + b*Sin[e + f*x]^2)))/((a + b)^2*f)","A",1
500,1,116,148,0.5073659,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \tan ^3(e+f x) \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^3,x]","\frac{(2 a+5 b) \left(\sqrt{a+b \sin ^2(e+f x)} \left(4 a+b \sin ^2(e+f x)+3 b\right)-3 (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)\right)+3 \sec ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{5/2}}{6 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,"(3*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(5/2) + (2*a + 5*b)*(-3*(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]] + Sqrt[a + b*Sin[e + f*x]^2]*(4*a + 3*b + b*Sin[e + f*x]^2)))/(6*(a + b)*f)","A",1
501,1,79,84,0.1503935,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \tan (e+f x) \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x],x]","\frac{\sqrt{a-b \cos ^2(e+f x)+b} \left(b \cos ^2(e+f x)-4 (a+b)\right)+3 (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b \cos ^2(e+f x)+b}}{\sqrt{a+b}}\right)}{3 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,"(3*(a + b)^(3/2)*ArcTanh[Sqrt[a + b - b*Cos[e + f*x]^2]/Sqrt[a + b]] + Sqrt[a + b - b*Cos[e + f*x]^2]*(-4*(a + b) + b*Cos[e + f*x]^2))/(3*f)","A",1
502,1,69,78,0.1254072,"\int \cot (e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a+b \sin ^2(e+f x)} \left(4 a+b \sin ^2(e+f x)\right)-3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{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,"(-3*a^(3/2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]] + Sqrt[a + b*Sin[e + f*x]^2]*(4*a + b*Sin[e + f*x]^2))/(3*f)","A",1
503,1,90,140,0.4567492,"\int \cot ^3(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{3 \sqrt{a} (2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)+\sqrt{a+b \sin ^2(e+f x)} \left(-3 a \csc ^2(e+f x)-8 a+b \cos (2 (e+f x))+5 b\right)}{6 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,"(3*Sqrt[a]*(2*a - 3*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]] + (-8*a + 5*b + b*Cos[2*(e + f*x)] - 3*a*Csc[e + f*x]^2)*Sqrt[a + b*Sin[e + f*x]^2])/(6*f)","A",1
504,1,123,208,0.7983628,"\int \cot ^5(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^5*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a} \sqrt{a+b \sin ^2(e+f x)} \left(8 \left(4 a+b \sin ^2(e+f x)-6 b\right)+3 (8 a-5 b) \csc ^2(e+f x)-6 a \csc ^4(e+f x)\right)-3 \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)}{24 \sqrt{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,"(-3*(8*a^2 - 24*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]] + Sqrt[a]*Sqrt[a + b*Sin[e + f*x]^2]*(3*(8*a - 5*b)*Csc[e + f*x]^2 - 6*a*Csc[e + f*x]^4 + 8*(4*a - 6*b + b*Sin[e + f*x]^2)))/(24*Sqrt[a]*f)","A",1
505,1,211,275,2.7392536,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \tan ^4(e+f x) \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^4,x]","\frac{-\frac{\tan (e+f x) \sec ^2(e+f x) \left(\left(64 a^2+160 a b+17 b^2\right) \cos (2 (e+f x))+32 a^2-2 b (6 a+17 b) \cos (4 (e+f x))+108 a b-b^2 \cos (6 (e+f x))+18 b^2\right)}{4 \sqrt{2}}-4 a (5 a+8 b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+32 a (a+2 b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{12 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(32*a*(a + 2*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 4*a*(5*a + 8*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] - ((32*a^2 + 108*a*b + 18*b^2 + (64*a^2 + 160*a*b + 17*b^2)*Cos[2*(e + f*x)] - 2*b*(6*a + 17*b)*Cos[4*(e + f*x)] - b^2*Cos[6*(e + f*x)])*Sec[e + f*x]^2*Tan[e + f*x])/(4*Sqrt[2]))/(12*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
506,1,174,222,2.8169316,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \tan ^2(e+f x) \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^2,x]","\frac{\sqrt{2} \tan (e+f x) \left(24 a^2-4 b (2 a+3 b) \cos (2 (e+f x))+40 a b-b^2 \cos (4 (e+f x))+13 b^2\right)+32 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-8 a (7 a+8 b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{24 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-8*a*(7*a + 8*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 32*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + Sqrt[2]*(24*a^2 + 40*a*b + 13*b^2 - 4*b*(2*a + 3*b)*Cos[2*(e + f*x)] - b^2*Cos[4*(e + f*x)])*Tan[e + f*x])/(24*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
507,1,156,154,0.7676077,"\int \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{b \sin (2 (e+f x)) (-2 a+b \cos (2 (e+f x))-b)-2 \sqrt{2} a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+4 \sqrt{2} a (2 a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 \sqrt{2} f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(4*Sqrt[2]*a*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 2*Sqrt[2]*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + b*(-2*a - b + b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(6*Sqrt[2]*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
508,1,173,223,2.2337023,"\int \cot ^2(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\sqrt{2} \cot (e+f x) \left(-24 a^2+4 b (2 a-b) \cos (2 (e+f x))-8 a b+b^2 \cos (4 (e+f x))+3 b^2\right)+32 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-8 a (7 a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{24 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(Sqrt[2]*(-24*a^2 - 8*a*b + 3*b^2 + 4*(2*a - b)*b*Cos[2*(e + f*x)] + b^2*Cos[4*(e + f*x)])*Cot[e + f*x] - 8*a*(7*a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + 32*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(24*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
509,1,218,276,4.7880324,"\int \cot ^4(e+f x) \left(a+b \sin ^2(e+f x)\right)^{3/2} \, dx","Integrate[Cot[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{-4 \left(5 a^2+2 a b-3 b^2\right) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-\frac{\cot (e+f x) \csc ^2(e+f x) \left(\left(64 a^2-32 a b-79 b^2\right) \cos (2 (e+f x))-32 a^2-2 b (6 a-11 b) \cos (4 (e+f x))+44 a b-b^2 \cos (6 (e+f x))+58 b^2\right)}{4 \sqrt{2}}+32 a (a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{12 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-1/4*((-32*a^2 + 44*a*b + 58*b^2 + (64*a^2 - 32*a*b - 79*b^2)*Cos[2*(e + f*x)] - 2*(6*a - 11*b)*b*Cos[4*(e + f*x)] - b^2*Cos[6*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^2)/Sqrt[2] + 32*a*(a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 4*(5*a^2 + 2*a*b - 3*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(12*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
510,1,108,134,0.4737807,"\int \frac{\tan ^5(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[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)+\sqrt{a+b} \sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \left(2 (a+b) \sec ^2(e+f x)-8 a-5 b\right)}{8 f (a+b)^{5/2}}","\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]] + Sqrt[a + b]*Sec[e + f*x]^2*(-8*a - 5*b + 2*(a + b)*Sec[e + f*x]^2)*Sqrt[a + b*Sin[e + f*x]^2])/(8*(a + b)^(5/2)*f)","A",1
511,1,77,81,0.218962,"\int \frac{\tan ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a+b}}\right)}{(a+b)^{3/2}}-\frac{\sec ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{a+b}}{2 f}","\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,"-1/2*(((2*a + b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(a + b)^(3/2) - (Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(a + b))/f","A",1
512,1,38,36,0.0381903,"\int \frac{\tan (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b \cos ^2(e+f x)+b}}{\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 - b*Cos[e + f*x]^2]/Sqrt[a + b]]/(Sqrt[a + b]*f)","A",1
513,1,33,33,0.0339374,"\int \frac{\cot (e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[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",1
514,1,71,75,0.1665084,"\int \frac{\cot ^3(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{\csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{a}}{2 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]])/a^(3/2) - (Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/a)/(2*f)","A",1
515,1,101,126,0.3609077,"\int \frac{\cot ^5(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^5/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{a} \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)} \left(-2 a \csc ^2(e+f x)+8 a+3 b\right)-\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{\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{\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]]) + Sqrt[a]*Csc[e + f*x]^2*(8*a + 3*b - 2*a*Csc[e + f*x]^2)*Sqrt[a + b*Sin[e + f*x]^2])/(8*a^(5/2)*f)","A",1
516,1,188,246,2.1354897,"\int \frac{\tan ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{-\frac{\tan (e+f x) \sec ^2(e+f x) \left(2 \left(4 a^2+3 a b+b^2\right) \cos (2 (e+f x))+(2 a+b) (2 a-b \cos (4 (e+f x))-b)\right)}{\sqrt{2}}-2 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+4 a (2 a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 f (a+b)^2 \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(4*a*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 2*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] - ((2*(4*a^2 + 3*a*b + b^2)*Cos[2*(e + f*x)] + (2*a + b)*(2*a - b - b*Cos[4*(e + f*x)]))*Sec[e + f*x]^2*Tan[e + f*x])/Sqrt[2])/(6*(a + b)^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
517,1,100,109,0.3841032,"\int \frac{\tan ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Tan[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{2} \tan (e+f x) (2 a-b \cos (2 (e+f x))+b)-2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{2 f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(-2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + Sqrt[2]*(2*a + b - b*Cos[2*(e + f*x)])*Tan[e + f*x])/(2*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
518,1,60,51,0.0763956,"\int \frac{1}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{2 a-b \cos (2 (e+f x))+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)}}",1,"(Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
519,1,101,106,0.364057,"\int \frac{\cot ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b}}{\sqrt{2} a f}-\frac{\sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"-((Sqrt[2*a + b - b*Cos[2*(e + f*x)]]*Cot[e + f*x])/(Sqrt[2]*a*f)) - (Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)])/(f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
520,1,186,240,3.9685075,"\int \frac{\cot ^4(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx","Integrate[Cot[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]","\frac{\frac{\cot (e+f x) \csc ^2(e+f x) \left((2 a+b) (2 a+b \cos (4 (e+f x))+3 b)-2 \left(4 a^2+5 a b+2 b^2\right) \cos (2 (e+f x))\right)}{\sqrt{2}}-2 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+4 a (2 a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 a^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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*(4*a^2 + 5*a*b + 2*b^2)*Cos[2*(e + f*x)] + (2*a + b)*(2*a + 3*b + b*Cos[4*(e + f*x)]))*Cot[e + f*x]*Csc[e + f*x]^2)/Sqrt[2] + 4*a*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 2*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(6*a^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
521,1,107,177,0.4689054,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\left(-8 a^2+8 a b+b^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \sin ^2(e+f x)+a}{a+b}\right)-\frac{1}{2} (a+b) \sec ^4(e+f x) ((8 a+3 b) \cos (2 (e+f x))+4 a-b)}{8 f (a+b)^3 \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)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[e + f*x]^2)/(a + b)] - ((a + b)*(4*a - b + (8*a + 3*b)*Cos[2*(e + f*x)])*Sec[e + f*x]^4)/2)/(8*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2])","C",1
522,1,75,118,0.1135799,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{(2 a-b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \sin ^2(e+f x)+a}{a+b}\right)+(a+b) \sec ^2(e+f x)}{2 f (a+b)^2 \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)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[e + f*x]^2)/(a + b)] + (a + b)*Sec[e + f*x]^2)/(2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2])","C",1
523,1,54,63,0.0710083,"\int \frac{\tan (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]/(a + b*Sin[e + f*x]^2)^(3/2),x]","-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1-\frac{b \cos ^2(e+f x)}{a+b}\right)}{f (a+b) \sqrt{a-b \cos ^2(e+f x)+b}}","\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,"-(Hypergeometric2F1[-1/2, 1, 1/2, 1 - (b*Cos[e + f*x]^2)/(a + b)]/((a + b)*f*Sqrt[a + b - b*Cos[e + f*x]^2]))","C",1
524,1,46,57,0.0515255,"\int \frac{\cot (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \sin ^2(e+f x)}{a}+1\right)}{a f \sqrt{a+b \sin ^2(e+f 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}",1,"Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Sin[e + f*x]^2)/a]/(a*f*Sqrt[a + b*Sin[e + f*x]^2])","C",1
525,1,70,110,0.098865,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{-(2 a+3 b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \sin ^2(e+f x)}{a}+1\right)-a \csc ^2(e+f x)}{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{2 a+3 b}{2 a^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\csc ^2(e+f x)}{2 a f \sqrt{a+b \sin ^2(e+f x)}}",1,"(-(a*Csc[e + f*x]^2) - (2*a + 3*b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Sin[e + f*x]^2)/a])/(2*a^2*f*Sqrt[a + b*Sin[e + f*x]^2])","C",1
526,1,94,167,0.3256566,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\left(8 a^2+24 a b+15 b^2\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \sin ^2(e+f x)}{a}+1\right)+a \csc ^2(e+f x) \left(-2 a \csc ^2(e+f x)+8 a+5 b\right)}{8 a^3 f \sqrt{a+b \sin ^2(e+f x)}}","\frac{(8 a+5 b) \csc ^2(e+f x)}{8 a^2 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^2+24 a b+15 b^2}{8 a^3 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,"(a*Csc[e + f*x]^2*(8*a + 5*b - 2*a*Csc[e + f*x]^2) + (8*a^2 + 24*a*b + 15*b^2)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Sin[e + f*x]^2)/a])/(8*a^3*f*Sqrt[a + b*Sin[e + f*x]^2])","C",1
527,1,197,292,2.2776581,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{-\frac{\tan (e+f x) \sec ^2(e+f x) \left(4 \left(4 a^2-3 a b+b^2\right) \cos (2 (e+f x))+8 a^2+b (b-7 a) \cos (4 (e+f x))-21 a b-5 b^2\right)}{2 \sqrt{2}}-8 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 a (7 a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 f (a+b)^3 \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(2*a*(7*a - b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 8*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] - ((8*a^2 - 21*a*b - 5*b^2 + 4*(4*a^2 - 3*a*b + b^2)*Cos[2*(e + f*x)] + b*(-7*a + b)*Cos[4*(e + f*x)])*Sec[e + f*x]^2*Tan[e + f*x])/(2*Sqrt[2]))/(6*(a + b)^3*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
528,1,145,224,0.8916207,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{2 \tan (e+f x) (a-b \cos (2 (e+f x)))+\sqrt{2} (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 \sqrt{2} a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{\sqrt{2} f (a+b)^2 \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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*Sqrt[2]*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + Sqrt[2]*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)] + 2*(a - b*Cos[2*(e + f*x)])*Tan[e + f*x])/(Sqrt[2]*(a + b)^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
529,1,90,101,0.1478368,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(-3/2),x]","\frac{2 a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)+\sqrt{2} b \sin (2 (e+f x))}{2 a f (a+b) \sqrt{2 a-b \cos (2 (e+f x))+b}}","\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,"(2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + Sqrt[2]*b*Sin[2*(e + f*x)])/(2*a*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
530,1,142,209,0.752607,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{-2 \cot (e+f x) (a-b \cos (2 (e+f x))+b)+\sqrt{2} a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 \sqrt{2} a \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{\sqrt{2} a^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","-\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,"(-2*(a + b - b*Cos[2*(e + f*x)])*Cot[e + f*x] - 2*Sqrt[2]*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + Sqrt[2]*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(Sqrt[2]*a^2*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
531,1,199,297,3.8241187,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{3/2}} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]","\frac{\frac{\cot (e+f x) \csc ^2(e+f x) \left(-4 \left(4 a^2+11 a b+8 b^2\right) \cos (2 (e+f x))+8 a^2+b (7 a+8 b) \cos (4 (e+f x))+37 a b+24 b^2\right)}{2 \sqrt{2}}-8 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 a (7 a+8 b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{6 a^3 f \sqrt{2 a-b \cos (2 (e+f x))+b}}","\frac{(7 a+8 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 f}+\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{(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{(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt{a+b \sin ^2(e+f x)}}",1,"(((8*a^2 + 37*a*b + 24*b^2 - 4*(4*a^2 + 11*a*b + 8*b^2)*Cos[2*(e + f*x)] + b*(7*a + 8*b)*Cos[4*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^2)/(2*Sqrt[2]) + 2*a*(7*a + 8*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 8*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(6*a^3*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])","A",1
532,1,107,218,0.4803106,"\int \frac{\tan ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\left(-8 a^2+24 a b-3 b^2\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \sin ^2(e+f x)+a}{a+b}\right)-\frac{3}{2} (a+b) \sec ^4(e+f x) ((8 a+b) \cos (2 (e+f x))+4 a-3 b)}{24 f (a+b)^3 \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)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sin[e + f*x]^2)/(a + b)] - (3*(a + b)*(4*a - 3*b + (8*a + b)*Cos[2*(e + f*x)])*Sec[e + f*x]^4)/2)/(24*(a + b)^3*f*(a + b*Sin[e + f*x]^2)^(3/2))","C",1
533,1,76,153,0.1188803,"\int \frac{\tan ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{(2 a-3 b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \sin ^2(e+f x)+a}{a+b}\right)+3 (a+b) \sec ^2(e+f x)}{6 f (a+b)^2 \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)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sin[e + f*x]^2)/(a + b)] + 3*(a + b)*Sec[e + f*x]^2)/(6*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2))","C",1
534,1,56,91,0.0820776,"\int \frac{\tan (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};1-\frac{b \cos ^2(e+f x)}{a+b}\right)}{3 f (a+b) \left(a-b \cos ^2(e+f x)+b\right)^{3/2}}","-\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,"-1/3*Hypergeometric2F1[-3/2, 1, -1/2, 1 - (b*Cos[e + f*x]^2)/(a + b)]/((a + b)*f*(a + b - b*Cos[e + f*x]^2)^(3/2))","C",1
535,1,49,83,0.0564503,"\int \frac{\cot (e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \sin ^2(e+f x)}{a}+1\right)}{3 a f \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}}\right)}{a^{5/2} f}+\frac{1}{a^2 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{1}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Sin[e + f*x]^2)/a]/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2))","C",1
536,1,69,143,0.2541169,"\int \frac{\cot ^3(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2),x]","-\frac{(2 a+5 b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \sin ^2(e+f x)}{a}+1\right)+3 a \csc ^2(e+f x)}{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{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{\csc ^2(e+f x)}{2 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"-1/6*(3*a*Csc[e + f*x]^2 + (2*a + 5*b)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Sin[e + f*x]^2)/a])/(a^2*f*(a + b*Sin[e + f*x]^2)^(3/2))","C",1
537,1,117,208,0.8399807,"\int \frac{\cot ^5(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{\left(8 a^2+40 a b+35 b^2\right) \csc ^2(e+f x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \sin ^2(e+f x)}{a}+1\right)+3 a \csc ^4(e+f x) \left(-2 a \csc ^2(e+f x)+8 a+7 b\right)}{24 a^3 f \sqrt{a+b \sin ^2(e+f x)} \left(a \csc ^2(e+f x)+b\right)}","\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{\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^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{\csc ^4(e+f x)}{4 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"(3*a*Csc[e + f*x]^4*(8*a + 7*b - 2*a*Csc[e + f*x]^2) + (8*a^2 + 40*a*b + 35*b^2)*Csc[e + f*x]^2*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Sin[e + f*x]^2)/a])/(24*a^3*f*(b + a*Csc[e + f*x]^2)*Sqrt[a + b*Sin[e + f*x]^2])","C",1
538,1,235,348,3.4196947,"\int \frac{\tan ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{2 a b \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} \left(\left(-5 a^2-2 a b+3 b^2\right) F\left(e+f x\left|-\frac{b}{a}\right.\right)+8 a (a-b) E\left(e+f x\left|-\frac{b}{a}\right.\right)\right)+\sqrt{2} b \left(2 a b (a+b) \sin (2 (e+f x))+4 b (a-b) \sin (2 (e+f x)) (2 a-b \cos (2 (e+f x))+b)-4 (a-b) \tan (e+f x) (2 a-b \cos (2 (e+f x))+b)^2+(a+b) \tan (e+f x) \sec ^2(e+f x) (2 a-b \cos (2 (e+f x))+b)^2\right)}{6 b f (a+b)^4 (2 a-b \cos (2 (e+f x))+b)^{3/2}}","-\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,"(2*a*b*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*(8*a*(a - b)*EllipticE[e + f*x, -(b/a)] + (-5*a^2 - 2*a*b + 3*b^2)*EllipticF[e + f*x, -(b/a)]) + Sqrt[2]*b*(2*a*b*(a + b)*Sin[2*(e + f*x)] + 4*(a - b)*b*(2*a + b - b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)] - 4*(a - b)*(2*a + b - b*Cos[2*(e + f*x)])^2*Tan[e + f*x] + (a + b)*(2*a + b - b*Cos[2*(e + f*x)])^2*Sec[e + f*x]^2*Tan[e + f*x]))/(6*b*(a + b)^4*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
539,1,199,292,2.6597015,"\int \frac{\tan ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Tan[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{8 a^2 (a+b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a^2 (7 a-b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)+\frac{\tan (e+f x) \left(24 a^3+4 a^2 b+b^2 (7 a-b) \cos (4 (e+f x))+5 a b^2-4 a b (11 a+3 b) \cos (2 (e+f x))+b^3\right)}{\sqrt{2}}}{6 a f (a+b)^3 (2 a-b \cos (2 (e+f x))+b)^{3/2}}","\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,"(-2*a^2*(7*a - b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] + 8*a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] + ((24*a^3 + 4*a^2*b + 5*a*b^2 + b^3 - 4*a*b*(11*a + 3*b)*Cos[2*(e + f*x)] + (7*a - b)*b^2*Cos[4*(e + f*x)])*Tan[e + f*x])/Sqrt[2])/(6*a*(a + b)^3*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
540,1,172,223,1.3511341,"\int \frac{1}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[(a + b*Sin[e + f*x]^2)^(-5/2),x]","\frac{-\sqrt{2} b \sin (2 (e+f x)) \left(-5 a^2+b (2 a+b) \cos (2 (e+f x))-5 a b-b^2\right)-a^2 (a+b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)+2 a^2 (2 a+b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)}{3 a^2 f (a+b)^2 (2 a-b \cos (2 (e+f x))+b)^{3/2}}","\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,"(2*a^2*(2*a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] - a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)] - Sqrt[2]*b*(-5*a^2 - 5*a*b - b^2 + b*(2*a + b)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(3*a^2*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
541,1,209,287,2.6608767,"\int \frac{\cot ^2(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{8 a^2 (a+b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} F\left(e+f x\left|-\frac{b}{a}\right.\right)-2 a^2 (7 a+8 b) \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} E\left(e+f x\left|-\frac{b}{a}\right.\right)-\frac{\cot (e+f x) \left(24 a^3-4 b \left(11 a^2+19 a b+8 b^2\right) \cos (2 (e+f x))+68 a^2 b+b^2 (7 a+8 b) \cos (4 (e+f x))+69 a b^2+24 b^3\right)}{\sqrt{2}}}{6 a^3 f (a+b) (2 a-b \cos (2 (e+f x))+b)^{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{(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{(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{\cot (e+f x)}{3 a f \left(a+b \sin ^2(e+f x)\right)^{3/2}}",1,"(-(((24*a^3 + 68*a^2*b + 69*a*b^2 + 24*b^3 - 4*b*(11*a^2 + 19*a*b + 8*b^2)*Cos[2*(e + f*x)] + b^2*(7*a + 8*b)*Cos[4*(e + f*x)])*Cot[e + f*x])/Sqrt[2]) - 2*a^2*(7*a + 8*b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x, -(b/a)] + 8*a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)])/(6*a^3*(a + b)*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
542,1,226,348,3.0671623,"\int \frac{\cot ^4(e+f x)}{\left(a+b \sin ^2(e+f x)\right)^{5/2}} \, dx","Integrate[Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2),x]","\frac{2 a^2 b \left(\frac{2 a-b \cos (2 (e+f x))+b}{a}\right)^{3/2} \left(8 (a+2 b) E\left(e+f x\left|-\frac{b}{a}\right.\right)-(5 a+8 b) F\left(e+f x\left|-\frac{b}{a}\right.\right)\right)+\sqrt{2} b \left(2 a b (a+b) \sin (2 (e+f x))+4 b (a+2 b) \sin (2 (e+f x)) (2 a-b \cos (2 (e+f x))+b)+4 (a+2 b) \cot (e+f x) (2 a-b \cos (2 (e+f x))+b)^2-a \cot (e+f x) \csc ^2(e+f x) (2 a-b \cos (2 (e+f x))+b)^2\right)}{6 a^4 b f (2 a-b \cos (2 (e+f x))+b)^{3/2}}","\frac{8 (a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^4 f}+\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{(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{(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{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{(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,"(2*a^2*b*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*(8*(a + 2*b)*EllipticE[e + f*x, -(b/a)] - (5*a + 8*b)*EllipticF[e + f*x, -(b/a)]) + Sqrt[2]*b*(4*(a + 2*b)*(2*a + b - b*Cos[2*(e + f*x)])^2*Cot[e + f*x] - a*(2*a + b - b*Cos[2*(e + f*x)])^2*Cot[e + f*x]*Csc[e + f*x]^2 + 2*a*b*(a + b)*Sin[2*(e + f*x)] + 4*b*(a + 2*b)*(2*a + b - b*Cos[2*(e + f*x)])*Sin[2*(e + f*x)]))/(6*a^4*b*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))","A",1
543,1,121,120,0.4952817,"\int \left(a+b \sin ^2(e+f x)\right)^p (d \tan (e+f x))^m \, dx","Integrate[(a + b*Sin[e + f*x]^2)^p*(d*Tan[e + f*x])^m,x]","\frac{\tan (e+f x) \cos ^2(e+f x)^{\frac{m+1}{2}} (d \tan (e+f x))^m \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)}{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*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(f*(1 + m)*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",1
544,1,83,102,0.2510978,"\int \left(a+b \sin ^2(c+d x)\right)^p \tan ^3(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x]^3,x]","\frac{\left(a+b \sin ^2(c+d x)\right)^{p+1} \left((p+1) (a+b) \sec ^2(c+d x)-(a+b p+b) \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^2(c+d x)+a}{a+b}\right)\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)*(1 + p)*Sec[c + d*x]^2)*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*(a + b)^2*d*(1 + p))","A",1
545,1,61,59,0.0624474,"\int \left(a+b \sin ^2(c+d x)\right)^p \tan (c+d x) \, dx","Integrate[(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x],x]","\frac{\left(a-b \cos ^2(c+d x)+b\right)^{p+1} \, _2F_1\left(1,p+1;p+2;1-\frac{b \cos ^2(c+d x)}{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,"((a + b - b*Cos[c + d*x]^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (b*Cos[c + d*x]^2)/(a + b)])/(2*(a + b)*d*(1 + p))","A",1
546,1,54,54,0.0556408,"\int \cot (c+d x) \left(a+b \sin ^2(c+d x)\right)^p \, dx","Integrate[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,"-1/2*(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^2)/a]*(a + b*Sin[c + d*x]^2)^(1 + p))/(a*d*(1 + p))","A",1
547,1,73,95,0.4290528,"\int \cot ^3(c+d x) \left(a+b \sin ^2(c+d x)\right)^p \, dx","Integrate[Cot[c + d*x]^3*(a + b*Sin[c + d*x]^2)^p,x]","-\frac{\left(a+b \sin ^2(c+d x)\right)^{p+1} \left(\frac{(b p-a) \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^2(c+d x)}{a}+1\right)}{p+1}+a \csc ^2(c+d x)\right)}{2 a^2 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,"-1/2*((a*Csc[c + d*x]^2 + ((-a + b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^2)/a])/(1 + p))*(a + b*Sin[c + d*x]^2)^(1 + p))/(a^2*d)","A",1
548,1,102,101,4.504196,"\int \left(a+b \sin ^2(c+d x)\right)^p \tan ^4(c+d x) \, dx","Integrate[(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{a+b \sin ^2(c+d x)}{a}\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*((a + b*Sin[c + d*x]^2)/a)^p)","A",1
549,1,102,101,0.3067461,"\int \left(a+b \sin ^2(c+d x)\right)^p \tan ^2(c+d x) \, dx","Integrate[(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{a+b \sin ^2(c+d x)}{a}\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*((a + b*Sin[c + d*x]^2)/a)^p)","A",1
550,1,98,97,0.2028325,"\int \cot ^2(c+d x) \left(a+b \sin ^2(c+d x)\right)^p \, dx","Integrate[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{a+b \sin ^2(c+d x)}{a}\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*((a + b*Sin[c + d*x]^2)/a)^p))","A",1
551,1,102,101,3.3563846,"\int \cot ^4(c+d x) \left(a+b \sin ^2(c+d x)\right)^p \, dx","Integrate[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{a+b \sin ^2(c+d x)}{a}\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,"-1/3*(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)/(d*((a + b*Sin[c + d*x]^2)/a)^p)","A",1
552,1,143,153,0.3117991,"\int \frac{\cot ^3(x)}{a+b \sin ^3(x)} \, dx","Integrate[Cot[x]^3/(a + b*Sin[x]^3),x]","\frac{2 \left(a^{2/3}-(-1)^{2/3} b^{2/3}\right) \log \left(-(-1)^{2/3} \sqrt[3]{a}-\sqrt[3]{b} \sin (x)\right)+2 \left(a^{2/3}-b^{2/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sin (x)\right)+2 \left(a^{2/3}+\sqrt[3]{-1} b^{2/3}\right) \log \left(\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b} \sin (x)\right)-3 a^{2/3} \csc ^2(x)-6 a^{2/3} \log (\sin (x))}{6 a^{5/3}}","\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,"(-3*a^(2/3)*Csc[x]^2 - 6*a^(2/3)*Log[Sin[x]] + 2*(a^(2/3) - (-1)^(2/3)*b^(2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sin[x]] + 2*(a^(2/3) - b^(2/3))*Log[a^(1/3) + b^(1/3)*Sin[x]] + 2*(a^(2/3) + (-1)^(1/3)*b^(2/3))*Log[a^(1/3) + (-1)^(2/3)*b^(1/3)*Sin[x]])/(6*a^(5/3))","A",1
553,1,45,45,0.0278398,"\int \cot (x) \sqrt{a+b \sin ^3(x)} \, dx","Integrate[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",1
554,1,28,28,0.0177946,"\int \frac{\cot (x)}{\sqrt{a+b \sin ^3(x)}} \, dx","Integrate[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",1
555,1,55,59,0.0527104,"\int \cot (c+d x) \sqrt{a+b \sin ^4(c+d x)} \, dx","Integrate[Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^4(c+d x)}}{\sqrt{a}}\right)-\sqrt{a+b \sin ^4(c+d x)}}{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,"-1/2*(Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]] - Sqrt[a + b*Sin[c + d*x]^4])/d","A",1
556,1,85,89,0.1970633,"\int \frac{\tan ^3(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Tan[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{\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)}{(a+b)^{3/2}}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin ^4(c+d x)}}{a+b}}{2 d}","\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,"-1/2*((a*ArcTanh[(a + b*Sin[c + d*x]^2)/(Sqrt[a + b]*Sqrt[a + b*Sin[c + d*x]^4])])/(a + b)^(3/2) - (Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]^4])/(a + b))/d","A",1
557,1,65,51,0.0902437,"\int \frac{\tan (c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Tan[c + d*x]/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\tanh ^{-1}\left(\frac{a-b \cos ^2(c+d x)+b}{\sqrt{a+b} \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\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 - 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])]/(2*Sqrt[a + b]*d)","A",1
558,1,35,35,0.0236825,"\int \frac{\cot (c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[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,"-1/2*ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]]/(Sqrt[a]*d)","A",1
559,1,66,70,0.0777443,"\int \frac{\cot ^3(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Cot[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^4(c+d x)}}{\sqrt{a}}\right)-\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,"(Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]] - Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]^4])/(2*a*d)","A",1
560,1,141,108,2.8954894,"\int \frac{\cot ^5(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Cot[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin ^4(c+d x)}}{\sqrt{a}}\right)-4 a \csc ^2(c+d x) \sqrt{a+b \sin ^4(c+d x)}+b \sqrt{a+b \sin ^4(c+d x)} \left(\frac{a \csc ^4(c+d x)}{b}-\frac{\tanh ^{-1}\left(\sqrt{\frac{b \sin ^4(c+d x)}{a}+1}\right)}{\sqrt{\frac{b \sin ^4(c+d x)}{a}+1}}\right)}{4 a^2 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,"-1/4*(2*a^(3/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]] - 4*a*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]^4] + b*Sqrt[a + b*Sin[c + d*x]^4]*((a*Csc[c + d*x]^4)/b - ArcTanh[Sqrt[1 + (b*Sin[c + d*x]^4)/a]]/Sqrt[1 + (b*Sin[c + d*x]^4)/a]))/(a^2*d)","A",1
561,1,291,411,6.1485077,"\int \frac{\tan ^2(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Tan[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{2 i \sqrt{2} \sqrt{a} \cos ^2(c+d x) \sqrt{1+\left(1-\frac{i \sqrt{b}}{\sqrt{a}}\right) \tan ^2(c+d x)} \sqrt{1+\left(1+\frac{i \sqrt{b}}{\sqrt{a}}\right) \tan ^2(c+d x)} \left(E\left(i \sinh ^{-1}\left(\sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)|\frac{\sqrt{a}+i \sqrt{b}}{\sqrt{a}-i \sqrt{b}}\right)-F\left(i \sinh ^{-1}\left(\sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)|\frac{\sqrt{a}+i \sqrt{b}}{\sqrt{a}-i \sqrt{b}}\right)\right)}{d \left(\sqrt{a}+i \sqrt{b}\right) \sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b}}","\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,"((-2*I)*Sqrt[2]*Sqrt[a]*Cos[c + d*x]^2*(EllipticE[I*ArcSinh[Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], (Sqrt[a] + I*Sqrt[b])/(Sqrt[a] - I*Sqrt[b])] - EllipticF[I*ArcSinh[Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], (Sqrt[a] + I*Sqrt[b])/(Sqrt[a] - I*Sqrt[b])])*Sqrt[1 + (1 - (I*Sqrt[b])/Sqrt[a])*Tan[c + d*x]^2]*Sqrt[1 + (1 + (I*Sqrt[b])/Sqrt[a])*Tan[c + d*x]^2])/((Sqrt[a] + I*Sqrt[b])*Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*d*Sqrt[8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)]])","C",1
562,1,304,162,2.742703,"\int \frac{1}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Sin[c + d*x]^4],x]","\frac{2 \sqrt{2} \left(\sqrt{b}+i \sqrt{a}\right) \sin ^2(c+d x) \tan (c+d x) \left(2 \sqrt{a}+i \sqrt{b} \cos (2 (c+d x))-i \sqrt{b}\right) \left(2 i \sqrt{a}+\sqrt{b} \cos (2 (c+d x))-\sqrt{b}\right) \sqrt{\csc ^2(c+d x) \left(-\frac{2 i \sqrt{a}}{\sqrt{b}}-\cos (2 (c+d x))+1\right)} \sqrt{\frac{\cot ^2(c+d x) \left(-a \csc ^2(c+d x)+i \sqrt{a} \sqrt{b}\right)}{\left(\sqrt{a}-i \sqrt{b}\right)^2}} F\left(\sin ^{-1}\left(\sqrt{\frac{\sqrt{a} \csc ^2(c+d x)-i \sqrt{b}}{\sqrt{a}-i \sqrt{b}}}\right)|\frac{i \sqrt{a}}{2 \sqrt{b}}+\frac{1}{2}\right)}{\sqrt{a} d (8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^{3/2}}","\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,"(2*Sqrt[2]*(I*Sqrt[a] + Sqrt[b])*(2*Sqrt[a] - I*Sqrt[b] + I*Sqrt[b]*Cos[2*(c + d*x)])*((2*I)*Sqrt[a] - Sqrt[b] + Sqrt[b]*Cos[2*(c + d*x)])*Sqrt[(1 - ((2*I)*Sqrt[a])/Sqrt[b] - Cos[2*(c + d*x)])*Csc[c + d*x]^2]*Sqrt[(Cot[c + d*x]^2*(I*Sqrt[a]*Sqrt[b] - a*Csc[c + d*x]^2))/(Sqrt[a] - I*Sqrt[b])^2]*EllipticF[ArcSin[Sqrt[((-I)*Sqrt[b] + Sqrt[a]*Csc[c + d*x]^2)/(Sqrt[a] - I*Sqrt[b])]], 1/2 + ((I/2)*Sqrt[a])/Sqrt[b]]*Sin[c + d*x]^2*Tan[c + d*x])/(Sqrt[a]*d*(8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^(3/2))","C",1
563,1,378,477,11.3004263,"\int \frac{\cot ^2(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx","Integrate[Cot[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4],x]","-\frac{\cot (c+d x) \sqrt{8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b}}{2 \sqrt{2} a d}-\frac{\cos ^4(c+d x) \left(\frac{\left(\sqrt{a} \sqrt{b}+i a\right) \sec ^2(c+d x) \sqrt{1+\left(1-\frac{i \sqrt{b}}{\sqrt{a}}\right) \tan ^2(c+d x)} \sqrt{1+\left(1+\frac{i \sqrt{b}}{\sqrt{a}}\right) \tan ^2(c+d x)} \left(E\left(i \sinh ^{-1}\left(\sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)|\frac{\sqrt{a}+i \sqrt{b}}{\sqrt{a}-i \sqrt{b}}\right)-F\left(i \sinh ^{-1}\left(\sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}} \tan (c+d x)\right)|\frac{\sqrt{a}+i \sqrt{b}}{\sqrt{a}-i \sqrt{b}}\right)\right)}{\sqrt{1-\frac{i \sqrt{b}}{\sqrt{a}}}}+a \tan (c+d x) \sec ^4(c+d x)+b \tan ^5(c+d x)\right)}{a d \sqrt{\cos ^4(c+d x) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}}","\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,"-1/2*(Sqrt[8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)]]*Cot[c + d*x])/(Sqrt[2]*a*d) - (Cos[c + d*x]^4*(a*Sec[c + d*x]^4*Tan[c + d*x] + b*Tan[c + d*x]^5 + ((I*a + Sqrt[a]*Sqrt[b])*(EllipticE[I*ArcSinh[Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], (Sqrt[a] + I*Sqrt[b])/(Sqrt[a] - I*Sqrt[b])] - EllipticF[I*ArcSinh[Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]*Tan[c + d*x]], (Sqrt[a] + I*Sqrt[b])/(Sqrt[a] - I*Sqrt[b])])*Sec[c + d*x]^2*Sqrt[1 + (1 - (I*Sqrt[b])/Sqrt[a])*Tan[c + d*x]^2]*Sqrt[1 + (1 + (I*Sqrt[b])/Sqrt[a])*Tan[c + d*x]^2])/Sqrt[1 - (I*Sqrt[b])/Sqrt[a]]))/(a*d*Sqrt[Cos[c + d*x]^4*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)])","C",1
564,0,0,26,5.7507459,"\int \left(a+b \sin ^4(c+d x)\right)^p \tan ^m(c+d x) \, dx","Integrate[(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,"Integrate[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^m, x]","A",-1
565,1,922,279,18.5622024,"\int \left(a+b \sin ^4(c+d x)\right)^p \tan ^3(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^3,x]","-\frac{2 \left(\sqrt{-a b}-b\right) \left(b+\sqrt{-a b}\right) (2 p-1) F_1\left(-2 p;-p,-p;1-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right) \cos ^2(c+d x) \left(\left(a-\sqrt{-a b}\right) \cot ^2(c+d x)+a+b\right) \left(\left(a+\sqrt{-a b}\right) \cot ^2(c+d x)+a+b\right) \sin ^4(c+d x) \left(b \sin ^4(c+d x)+a\right)^p}{(a+b)^2 d p \left(b (2 p-1) F_1\left(-2 p;-p,-p;1-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right) \cos ^2(c+d x)+\left(b+\sqrt{-a b}\right) p F_1\left(1-2 p;1-p,-p;2-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right)-\left(\sqrt{-a b}-b\right) p F_1\left(1-2 p;-p,1-p;2-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right)\right) (8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))}-\frac{\left(\sqrt{-a b}-b\right) \left(b+\sqrt{-a b}\right) (p-1) F_1\left(1-2 p;-p,-p;2-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right) \sec ^2(c+d x) \left(-\left((a+b) \tan ^2(c+d x)\right)-a+\sqrt{-a b}\right) \left((a+b) \tan ^2(c+d x)+a+\sqrt{-a b}\right) \left(b \sin ^4(c+d x)+a\right)^p}{(a+b)^2 d (2 p-1) \left(p \left(\left(b+\sqrt{-a b}\right) F_1\left(2-2 p;1-p,-p;3-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right)+\left(b-\sqrt{-a b}\right) F_1\left(2-2 p;-p,1-p;3-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right)\right) \sec ^2(c+d x)+2 b (p-1) F_1\left(1-2 p;-p,-p;2-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right)\right) \left((a+b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right)}","-\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{(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{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{\sec ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^{p+1}}{2 d (a+b)}",1,"(-2*(-b + Sqrt[-(a*b)])*(b + Sqrt[-(a*b)])*(-1 + 2*p)*AppellF1[-2*p, -p, -p, 1 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])]*Cos[c + d*x]^2*(a + b + (a - Sqrt[-(a*b)])*Cot[c + d*x]^2)*(a + b + (a + Sqrt[-(a*b)])*Cot[c + d*x]^2)*Sin[c + d*x]^4*(a + b*Sin[c + d*x]^4)^p)/((a + b)^2*d*p*((b + Sqrt[-(a*b)])*p*AppellF1[1 - 2*p, 1 - p, -p, 2 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])] - (-b + Sqrt[-(a*b)])*p*AppellF1[1 - 2*p, -p, 1 - p, 2 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])] + b*(-1 + 2*p)*AppellF1[-2*p, -p, -p, 1 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])]*Cos[c + d*x]^2)*(8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])) - ((-b + Sqrt[-(a*b)])*(b + Sqrt[-(a*b)])*(-1 + p)*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])]*Sec[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p*(-a + Sqrt[-(a*b)] - (a + b)*Tan[c + d*x]^2)*(a + Sqrt[-(a*b)] + (a + b)*Tan[c + d*x]^2))/((a + b)^2*d*(-1 + 2*p)*(2*b*(-1 + p)*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])] + p*((b + Sqrt[-(a*b)])*AppellF1[2 - 2*p, 1 - p, -p, 3 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])] + (b - Sqrt[-(a*b)])*AppellF1[2 - 2*p, -p, 1 - p, 3 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])])*Sec[c + d*x]^2)*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4))","B",0
566,1,463,141,9.900974,"\int \left(a+b \sin ^4(c+d x)\right)^p \tan (c+d x) \, dx","Integrate[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x],x]","\frac{2 (2 p-1) \left(\sqrt{-a b}-b\right) \left(\sqrt{-a b}+b\right) \sin ^4(c+d x) \cos ^2(c+d x) \left(\left(a-\sqrt{-a b}\right) \cot ^2(c+d x)+a+b\right) \left(\left(\sqrt{-a b}+a\right) \cot ^2(c+d x)+a+b\right) \left(a+b \sin ^4(c+d x)\right)^p F_1\left(-2 p;-p,-p;1-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right)}{d p (a+b)^2 (8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b) \left(p \left(\sqrt{-a b}+b\right) F_1\left(1-2 p;1-p,-p;2-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right)-p \left(\sqrt{-a b}-b\right) F_1\left(1-2 p;-p,1-p;2-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right)+b (2 p-1) \cos ^2(c+d x) F_1\left(-2 p;-p,-p;1-2 p;-\frac{(a+b) \sec ^2(c+d x)}{\sqrt{-a b}-b},\frac{(a+b) \sec ^2(c+d x)}{b+\sqrt{-a b}}\right)\right)}","\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,"(2*(-b + Sqrt[-(a*b)])*(b + Sqrt[-(a*b)])*(-1 + 2*p)*AppellF1[-2*p, -p, -p, 1 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])]*Cos[c + d*x]^2*(a + b + (a - Sqrt[-(a*b)])*Cot[c + d*x]^2)*(a + b + (a + Sqrt[-(a*b)])*Cot[c + d*x]^2)*Sin[c + d*x]^4*(a + b*Sin[c + d*x]^4)^p)/((a + b)^2*d*p*((b + Sqrt[-(a*b)])*p*AppellF1[1 - 2*p, 1 - p, -p, 2 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])] - (-b + Sqrt[-(a*b)])*p*AppellF1[1 - 2*p, -p, 1 - p, 2 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])] + b*(-1 + 2*p)*AppellF1[-2*p, -p, -p, 1 - 2*p, -(((a + b)*Sec[c + d*x]^2)/(-b + Sqrt[-(a*b)])), ((a + b)*Sec[c + d*x]^2)/(b + Sqrt[-(a*b)])]*Cos[c + d*x]^2)*(8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)]))","B",0
567,1,54,54,0.0900168,"\int \cot (c+d x) \left(a+b \sin ^4(c+d x)\right)^p \, dx","Integrate[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,"-1/4*(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^4)/a]*(a + b*Sin[c + d*x]^4)^(1 + p))/(a*d*(1 + p))","A",1
568,1,119,127,0.6217009,"\int \cot ^3(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \, dx","Integrate[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 \left(\frac{\left(a+b \sin ^4(c+d x)\right) \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^4(c+d x)}{a}+1\right)}{a (p+1)}-2 \csc ^2(c+d x) \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)\right)}{4 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,"((a + b*Sin[c + d*x]^4)^p*((Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^4)/a]*(a + b*Sin[c + d*x]^4))/(a*(1 + p)) - (2*Csc[c + d*x]^2*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sin[c + d*x]^4)/a)])/(1 + (b*Sin[c + d*x]^4)/a)^p))/(4*d)","A",1
569,0,0,26,41.6502466,"\int \left(a+b \sin ^4(c+d x)\right)^p \tan ^4(c+d x) \, dx","Integrate[(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,"Integrate[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^4, x]","A",-1
570,0,0,26,1.8258899,"\int \left(a+b \sin ^4(c+d x)\right)^p \tan ^2(c+d x) \, dx","Integrate[(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,"Integrate[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^2, x]","A",-1
571,0,0,17,0.2007315,"\int \left(a+b \sin ^4(c+d x)\right)^p \, dx","Integrate[(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,"Integrate[(a + b*Sin[c + d*x]^4)^p, x]","A",-1
572,0,0,26,1.5036082,"\int \cot ^2(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \, dx","Integrate[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,"Integrate[Cot[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p, x]","A",-1
573,0,0,26,35.1451298,"\int \cot ^4(c+d x) \left(a+b \sin ^4(c+d x)\right)^p \, dx","Integrate[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,"Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^4)^p, x]","A",-1
574,1,3544,306,19.6009053,"\int \left(a+b \sin ^n(c+d x)\right)^3 \tan ^m(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x]^n)^3*Tan[c + d*x]^m,x]","\text{Result too large to show}","\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^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{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,"(2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*((a^3*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a^2*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + 2*n) + (b*AppellF1[(1 + m + 3*n)/2, m, 1 + 3*n, (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 3*n))))*Tan[(c + d*x)/2]*Tan[c + d*x]^m*(a^3*Tan[c + d*x]^m + 3*a^2*b*Sin[c + d*x]^n*Tan[c + d*x]^m + 3*a*b^2*Sin[c + d*x]^(2*n)*Tan[c + d*x]^m + b^3*Sin[c + d*x]^(3*n)*Tan[c + d*x]^m))/(d*(2*m*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*Sec[c + d*x]^2*((a^3*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a^2*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + 2*n) + (b*AppellF1[(1 + m + 3*n)/2, m, 1 + 3*n, (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 3*n))))*Tan[(c + d*x)/2]*Tan[c + d*x]^(-1 + m) + Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*((a^3*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a^2*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + 2*n) + (b*AppellF1[(1 + m + 3*n)/2, m, 1 + 3*n, (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 3*n))))*Tan[c + d*x]^m + 2*m*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(-1 + m)*((a^3*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a^2*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + 2*n) + (b*AppellF1[(1 + m + 3*n)/2, m, 1 + 3*n, (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 3*n))))*Tan[(c + d*x)/2]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*Tan[c + d*x]^m + 2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*Tan[(c + d*x)/2]*(b*n*Cos[c + d*x]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^(-1 + n)*((3*a^2*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + 2*n) + (b*AppellF1[(1 + m + 3*n)/2, m, 1 + 3*n, (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 3*n))) + b*n*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a^2*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + 2*n) + (b*AppellF1[(1 + m + 3*n)/2, m, 1 + 3*n, (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 3*n)))*Tan[(c + d*x)/2] + (a^3*(-(((1 + m)*AppellF1[1 + (1 + m)/2, m, 2, 1 + (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m)) + (m*(1 + m)*AppellF1[1 + (1 + m)/2, 1 + m, 1, 1 + (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m)))/(1 + m) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*(b*n*Cos[c + d*x]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^(-1 + n)*((3*a*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + 2*n) + (b*AppellF1[(1 + m + 3*n)/2, m, 1 + 3*n, (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 3*n)) + b*n*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((3*a*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + 2*n) + (b*AppellF1[(1 + m + 3*n)/2, m, 1 + 3*n, (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 3*n))*Tan[(c + d*x)/2] + (3*a^2*(-(((1 + n)*(1 + m + n)*AppellF1[1 + (1 + m + n)/2, m, 2 + n, 1 + (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m + n)) + (m*(1 + m + n)*AppellF1[1 + (1 + m + n)/2, 1 + m, 1 + n, 1 + (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m + n)))/(1 + m + n) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((b*n*AppellF1[(1 + m + 3*n)/2, m, 1 + 3*n, (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^(-1 + n))/(1 + m + 3*n) + (b*n*AppellF1[(1 + m + 3*n)/2, m, 1 + 3*n, (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*Tan[(c + d*x)/2])/(1 + m + 3*n) + (3*a*(-(((1/2 + m/2 + n)*(1 + 2*n)*AppellF1[3/2 + m/2 + n, m, 2 + 2*n, 5/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3/2 + m/2 + n)) + (m*(1/2 + m/2 + n)*AppellF1[3/2 + m/2 + n, 1 + m, 1 + 2*n, 5/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3/2 + m/2 + n)))/(1 + m + 2*n) + (b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*(-(((1 + 3*n)*(1 + m + 3*n)*AppellF1[1 + (1 + m + 3*n)/2, m, 2 + 3*n, 1 + (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m + 3*n)) + (m*(1 + m + 3*n)*AppellF1[1 + (1 + m + 3*n)/2, 1 + m, 1 + 3*n, 1 + (3 + m + 3*n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m + 3*n)))/(1 + m + 3*n))))*Tan[c + d*x]^m))","C",0
575,1,2368,215,15.3748159,"\int \left(a+b \sin ^n(c+d x)\right)^2 \tan ^m(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x]^n)^2*Tan[c + d*x]^m,x]","\text{Result too large to show}","\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,"(2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*((a^2*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((2*a*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + (b*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 2*n)))*Tan[(c + d*x)/2]*Tan[c + d*x]^m*(a^2*Tan[c + d*x]^m + 2*a*b*Sin[c + d*x]^n*Tan[c + d*x]^m + b^2*Sin[c + d*x]^(2*n)*Tan[c + d*x]^m))/(d*(2*m*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*Sec[c + d*x]^2*((a^2*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((2*a*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + (b*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 2*n)))*Tan[(c + d*x)/2]*Tan[c + d*x]^(-1 + m) + Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*((a^2*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((2*a*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + (b*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 2*n)))*Tan[c + d*x]^m + 2*m*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(-1 + m)*((a^2*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((2*a*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + (b*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 2*n)))*Tan[(c + d*x)/2]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*Tan[c + d*x]^m + 2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*Tan[(c + d*x)/2]*(b*n*Cos[c + d*x]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^(-1 + n)*((2*a*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + (b*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 2*n)) + b*n*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((2*a*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(1 + m + n) + (b*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)/(1 + m + 2*n))*Tan[(c + d*x)/2] + (a^2*(-(((1 + m)*AppellF1[1 + (1 + m)/2, m, 2, 1 + (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m)) + (m*(1 + m)*AppellF1[1 + (1 + m)/2, 1 + m, 1, 1 + (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m)))/(1 + m) + b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*((b*n*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^(-1 + n))/(1 + m + 2*n) + (b*n*AppellF1[1/2 + m/2 + n, m, 1 + 2*n, 3/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*Tan[(c + d*x)/2])/(1 + m + 2*n) + (b*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*(-(((1/2 + m/2 + n)*(1 + 2*n)*AppellF1[3/2 + m/2 + n, m, 2 + 2*n, 5/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3/2 + m/2 + n)) + (m*(1/2 + m/2 + n)*AppellF1[3/2 + m/2 + n, 1 + m, 1 + 2*n, 5/2 + m/2 + n, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3/2 + m/2 + n)))/(1 + m + 2*n) + (2*a*(-(((1 + n)*(1 + m + n)*AppellF1[1 + (1 + m + n)/2, m, 2 + n, 1 + (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m + n)) + (m*(1 + m + n)*AppellF1[1 + (1 + m + n)/2, 1 + m, 1 + n, 1 + (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m + n)))/(1 + m + n)))*Tan[c + d*x]^m))","C",0
576,1,1395,124,13.5011551,"\int \left(a+b \sin ^n(c+d x)\right) \tan ^m(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x]^n)*Tan[c + d*x]^m,x]","\frac{2 \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^m \left(b (m+1) F_1\left(\frac{1}{2} (m+n+1);m,n+1;\frac{1}{2} (m+n+3);\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^n \sin ^n(c+d x)+a (m+n+1) F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan \left(\frac{1}{2} (c+d x)\right) \tan ^m(c+d x) \left(b \sin ^n(c+d x) \tan ^m(c+d x)+a \tan ^m(c+d x)\right)}{d (m+1) (m+n+1) \left(\frac{2 m \left(b (m+1) F_1\left(\frac{1}{2} (m+n+1);m,n+1;\frac{1}{2} (m+n+3);\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^n \sin ^n(c+d x)+a (m+n+1) F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \tan \left(\frac{1}{2} (c+d x)\right)-\sec ^2\left(\frac{1}{2} (c+d x)\right) \sin (c+d x)\right) \tan ^m(c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{m-1}}{(m+1) (m+n+1)}+\frac{2 m \sec ^2(c+d x) \left(b (m+1) F_1\left(\frac{1}{2} (m+n+1);m,n+1;\frac{1}{2} (m+n+3);\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^n \sin ^n(c+d x)+a (m+n+1) F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan \left(\frac{1}{2} (c+d x)\right) \tan ^{m-1}(c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^m}{(m+1) (m+n+1)}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) \left(b (m+1) F_1\left(\frac{1}{2} (m+n+1);m,n+1;\frac{1}{2} (m+n+3);\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^n \sin ^n(c+d x)+a (m+n+1) F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan ^m(c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^m}{(m+1) (m+n+1)}+\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \left(b (m+1) n F_1\left(\frac{1}{2} (m+n+1);m,n+1;\frac{1}{2} (m+n+3);\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)^n \sin ^{n-1}(c+d x)+b (m+1) n F_1\left(\frac{1}{2} (m+n+1);m,n+1;\frac{1}{2} (m+n+3);\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^n \tan \left(\frac{1}{2} (c+d x)\right) \sin ^n(c+d x)+b (m+1) \sec ^2\left(\frac{1}{2} (c+d x)\right)^n \left(\frac{m (m+n+1) F_1\left(\frac{1}{2} (m+n+1)+1;m+1,n+1;\frac{1}{2} (m+n+3)+1;\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \tan \left(\frac{1}{2} (c+d x)\right)}{m+n+3}-\frac{(n+1) (m+n+1) F_1\left(\frac{1}{2} (m+n+1)+1;m,n+2;\frac{1}{2} (m+n+3)+1;\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \tan \left(\frac{1}{2} (c+d x)\right)}{m+n+3}\right) \sin ^n(c+d x)+a (m+n+1) \left(\frac{m (m+1) F_1\left(\frac{m+1}{2}+1;m+1,1;\frac{m+3}{2}+1;\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \tan \left(\frac{1}{2} (c+d x)\right)}{m+3}-\frac{(m+1) F_1\left(\frac{m+1}{2}+1;m,2;\frac{m+3}{2}+1;\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \tan \left(\frac{1}{2} (c+d x)\right)}{m+3}\right)\right) \tan ^m(c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^m}{(m+1) (m+n+1)}\right)}","\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,"(2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*(a*(1 + m + n)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + b*(1 + m)*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)*Tan[(c + d*x)/2]*Tan[c + d*x]^m*(a*Tan[c + d*x]^m + b*Sin[c + d*x]^n*Tan[c + d*x]^m))/(d*(1 + m)*(1 + m + n)*((2*m*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*Sec[c + d*x]^2*(a*(1 + m + n)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + b*(1 + m)*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)*Tan[(c + d*x)/2]*Tan[c + d*x]^(-1 + m))/((1 + m)*(1 + m + n)) + (Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*(a*(1 + m + n)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + b*(1 + m)*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)*Tan[c + d*x]^m)/((1 + m)*(1 + m + n)) + (2*m*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(-1 + m)*(a*(1 + m + n)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + b*(1 + m)*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n)*Tan[(c + d*x)/2]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*Tan[c + d*x]^m)/((1 + m)*(1 + m + n)) + (2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*Tan[(c + d*x)/2]*(b*(1 + m)*n*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^(-1 + n) + b*(1 + m)*n*AppellF1[(1 + m + n)/2, m, 1 + n, (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*Tan[(c + d*x)/2] + a*(1 + m + n)*(-(((1 + m)*AppellF1[1 + (1 + m)/2, m, 2, 1 + (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m)) + (m*(1 + m)*AppellF1[1 + (1 + m)/2, 1 + m, 1, 1 + (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m)) + b*(1 + m)*(Sec[(c + d*x)/2]^2)^n*Sin[c + d*x]^n*(-(((1 + n)*(1 + m + n)*AppellF1[1 + (1 + m + n)/2, m, 2 + n, 1 + (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m + n)) + (m*(1 + m + n)*AppellF1[1 + (1 + m + n)/2, 1 + m, 1 + n, 1 + (3 + m + n)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3 + m + n)))*Tan[c + d*x]^m)/((1 + m)*(1 + m + n))))","C",0
577,0,0,26,2.4512442,"\int \frac{\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx","Integrate[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,"Integrate[Tan[c + d*x]^m/(a + b*Sin[c + d*x]^n), x]","A",-1
578,0,0,26,24.1086292,"\int \frac{\tan ^m(c+d x)}{\left(a+b \sin ^n(c+d x)\right)^2} \, dx","Integrate[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,"Integrate[Tan[c + d*x]^m/(a + b*Sin[c + d*x]^n)^2, x]","A",-1
579,1,45,47,0.0277447,"\int \cot (x) \sqrt{a+b \sin ^n(x)} \, dx","Integrate[Cot[x]*Sqrt[a + b*Sin[x]^n],x]","\frac{2 \sqrt{a+b \sin ^n(x)}-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]] + 2*Sqrt[a + b*Sin[x]^n])/n","A",1
580,1,29,29,0.0143662,"\int \frac{\cot (x)}{\sqrt{a+b \sin ^n(x)}} \, dx","Integrate[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",1
581,0,0,26,4.097216,"\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^m(c+d x) \, dx","Integrate[(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,"Integrate[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^m, x]","A",-1
582,0,0,26,22.8745603,"\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^3(c+d x) \, dx","Integrate[(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,"Integrate[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^3, x]","A",-1
583,0,0,24,0.964095,"\int \left(a+b \sin ^n(c+d x)\right)^p \tan (c+d x) \, dx","Integrate[(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,"Integrate[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x], x]","A",-1
584,1,55,55,0.0500069,"\int \cot (c+d x) \left(a+b \sin ^n(c+d x)\right)^p \, dx","Integrate[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",1
585,1,129,136,1.0316622,"\int \cot ^3(c+d x) \left(a+b \sin ^n(c+d x)\right)^p \, dx","Integrate[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 \left(\frac{2 \left(a+b \sin ^n(c+d x)\right) \, _2F_1\left(1,p+1;p+2;\frac{b \sin ^n(c+d x)}{a}+1\right)}{a n (p+1)}-\csc ^2(c+d x) \left(\frac{b \sin ^n(c+d x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{2}{n},-p;\frac{n-2}{n};-\frac{b \sin ^n(c+d x)}{a}\right)\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,"((a + b*Sin[c + d*x]^n)^p*((2*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^n)/a]*(a + b*Sin[c + d*x]^n))/(a*n*(1 + p)) - (Csc[c + d*x]^2*Hypergeometric2F1[-2/n, -p, (-2 + n)/n, -((b*Sin[c + d*x]^n)/a)])/(1 + (b*Sin[c + d*x]^n)/a)^p))/(2*d)","A",1
586,0,0,26,25.2544051,"\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^4(c+d x) \, dx","Integrate[(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,"Integrate[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^4, x]","A",-1
587,0,0,26,2.7519756,"\int \left(a+b \sin ^n(c+d x)\right)^p \tan ^2(c+d x) \, dx","Integrate[(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,"Integrate[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^2, x]","A",-1
588,0,0,17,0.6613001,"\int \left(a+b \sin ^n(c+d x)\right)^p \, dx","Integrate[(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,"Integrate[(a + b*Sin[c + d*x]^n)^p, x]","A",-1
589,0,0,26,2.0916376,"\int \cot ^2(c+d x) \left(a+b \sin ^n(c+d x)\right)^p \, dx","Integrate[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,"Integrate[Cot[c + d*x]^2*(a + b*Sin[c + d*x]^n)^p, x]","A",-1
590,0,0,26,36.6772043,"\int \cot ^4(c+d x) \left(a+b \sin ^n(c+d x)\right)^p \, dx","Integrate[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,"Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^n)^p, x]","A",-1
591,1,102,107,0.1826739,"\int \frac{a+b \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt{d \sin (e+f x)}} \, dx","Integrate[(a + b*Sin[e + f*x]^2)/((g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]]),x]","\frac{2 \cos ^2(e+f x)^{3/4} \left(5 a \sin (e+f x) \, _2F_1\left(\frac{1}{4},\frac{7}{4};\frac{5}{4};\sin ^2(e+f x)\right)+b \sin ^3(e+f x) \, _2F_1\left(\frac{5}{4},\frac{7}{4};\frac{9}{4};\sin ^2(e+f x)\right)\right)}{5 f g \sqrt{d \sin (e+f x)} (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*(Cos[e + f*x]^2)^(3/4)*(5*a*Hypergeometric2F1[1/4, 7/4, 5/4, Sin[e + f*x]^2]*Sin[e + f*x] + b*Hypergeometric2F1[5/4, 7/4, 9/4, Sin[e + f*x]^2]*Sin[e + f*x]^3))/(5*f*g*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])","C",1
592,1,135,137,0.8425706,"\int (c \cos (e+f x))^m (d \sin (e+f x))^n \left(a+b \sin ^2(e+f x)\right)^p \, dx","Integrate[(c*Cos[e + f*x])^m*(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \cos ^2(e+f x)^{\frac{1-m}{2}} (c \cos (e+f x))^m (d \sin (e+f x))^n \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)}{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,"(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])^m*(Cos[e + f*x]^2)^((1 - m)/2)*(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(f*(1 + n)*(1 + (b*Sin[e + f*x]^2)/a)^p)","A",1
593,1,325,79,1.6684678,"\int \sqrt{a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx","Integrate[Sqrt[a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2],x]","-\frac{\left(\left(b^2-c^2\right) \sin (2 (e+f x))+2 b c \cos (2 (e+f x))\right) \sqrt{2 a+\left(c^2-b^2\right) \cos (2 (e+f x))+b^2+2 b c \sin (2 (e+f x))+c^2} E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\left(b^2-c^2\right) \cos (2 (e+f x))-2 b c \sin (2 (e+f x))+\sqrt{\left(b^2+c^2\right)^2}}{\sqrt{\left(b^2+c^2\right)^2}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{\left(b^2+c^2\right)^2}}{b^2+c^2+2 a+\sqrt{\left(b^2+c^2\right)^2}}\right)}{\sqrt{2} f \sqrt{\left(b^2+c^2\right)^2} \sqrt{\frac{\left(\left(b^2-c^2\right) \sin (2 (e+f x))+2 b c \cos (2 (e+f x))\right)^2}{\left(b^2+c^2\right)^2}} \sqrt{\frac{2 a+\left(c^2-b^2\right) \cos (2 (e+f x))+b^2+2 b c \sin (2 (e+f x))+c^2}{2 a+\sqrt{\left(b^2+c^2\right)^2}+b^2+c^2}}}","\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,"-((EllipticE[ArcSin[Sqrt[(Sqrt[(b^2 + c^2)^2] + (b^2 - c^2)*Cos[2*(e + f*x)] - 2*b*c*Sin[2*(e + f*x)])/Sqrt[(b^2 + c^2)^2]]/Sqrt[2]], (2*Sqrt[(b^2 + c^2)^2])/(2*a + b^2 + c^2 + Sqrt[(b^2 + c^2)^2])]*Sqrt[2*a + b^2 + c^2 + (-b^2 + c^2)*Cos[2*(e + f*x)] + 2*b*c*Sin[2*(e + f*x)]]*(2*b*c*Cos[2*(e + f*x)] + (b^2 - c^2)*Sin[2*(e + f*x)]))/(Sqrt[2]*Sqrt[(b^2 + c^2)^2]*f*Sqrt[(2*a + b^2 + c^2 + (-b^2 + c^2)*Cos[2*(e + f*x)] + 2*b*c*Sin[2*(e + f*x)])/(2*a + b^2 + c^2 + Sqrt[(b^2 + c^2)^2])]*Sqrt[(2*b*c*Cos[2*(e + f*x)] + (b^2 - c^2)*Sin[2*(e + f*x)])^2/(b^2 + c^2)^2]))","B",1
594,1,529,79,1.5895725,"\int \frac{1}{\sqrt{a+(c \cos (e+f x)+b \sin (e+f x))^2}} \, dx","Integrate[1/Sqrt[a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2],x]","\frac{\sqrt{2} \sec \left(\tan ^{-1}\left(\frac{c^2-b^2}{2 b c}\right)+2 (e+f x)\right) \sqrt{-\frac{b c \sqrt{\frac{\left(b^2+c^2\right)^2}{b^2 c^2}} \left(\sin \left(\tan ^{-1}\left(\frac{c^2-b^2}{2 b c}\right)+2 (e+f x)\right)-1\right)}{2 a+b c \sqrt{\frac{\left(b^2+c^2\right)^2}{b^2 c^2}}+b^2+c^2}} \sqrt{-\frac{b c \sqrt{\frac{\left(b^2+c^2\right)^2}{b^2 c^2}} \left(\sin \left(\tan ^{-1}\left(\frac{c^2-b^2}{2 b c}\right)+2 (e+f x)\right)+1\right)}{2 a-b c \sqrt{\frac{\left(b^2+c^2\right)^2}{b^2 c^2}}+b^2+c^2}} \sqrt{2 a+b c \sqrt{\frac{\left(b^2+c^2\right)^2}{b^2 c^2}} \sin \left(\tan ^{-1}\left(\frac{c^2-b^2}{2 b c}\right)+2 (e+f x)\right)+b^2+c^2} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{b^2+c \sqrt{\frac{\left(b^2+c^2\right)^2}{b^2 c^2}} \sin \left(2 (e+f x)+\tan ^{-1}\left(\frac{c^2-b^2}{2 b c}\right)\right) b+c^2+2 a}{b^2-c \sqrt{\frac{\left(b^2+c^2\right)^2}{b^2 c^2}} b+c^2+2 a},\frac{b^2+c \sqrt{\frac{\left(b^2+c^2\right)^2}{b^2 c^2}} \sin \left(2 (e+f x)+\tan ^{-1}\left(\frac{c^2-b^2}{2 b c}\right)\right) b+c^2+2 a}{b^2+c \sqrt{\frac{\left(b^2+c^2\right)^2}{b^2 c^2}} b+c^2+2 a}\right)}{b c f \sqrt{\frac{\left(b^2+c^2\right)^2}{b^2 c^2}}}","\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,"(Sqrt[2]*AppellF1[1/2, 1/2, 1/2, 3/2, (2*a + b^2 + c^2 + b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]*Sin[2*(e + f*x) + ArcTan[(-b^2 + c^2)/(2*b*c)]])/(2*a + b^2 + c^2 - b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]), (2*a + b^2 + c^2 + b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]*Sin[2*(e + f*x) + ArcTan[(-b^2 + c^2)/(2*b*c)]])/(2*a + b^2 + c^2 + b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)])]*Sec[2*(e + f*x) + ArcTan[(-b^2 + c^2)/(2*b*c)]]*Sqrt[-((b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]*(-1 + Sin[2*(e + f*x) + ArcTan[(-b^2 + c^2)/(2*b*c)]]))/(2*a + b^2 + c^2 + b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]))]*Sqrt[-((b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]*(1 + Sin[2*(e + f*x) + ArcTan[(-b^2 + c^2)/(2*b*c)]]))/(2*a + b^2 + c^2 - b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]))]*Sqrt[2*a + b^2 + c^2 + b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]*Sin[2*(e + f*x) + ArcTan[(-b^2 + c^2)/(2*b*c)]]])/(b*c*Sqrt[(b^2 + c^2)^2/(b^2*c^2)]*f)","C",0