1,1,92,0,0.3977358,"\int \cos ^2(e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx","Int[Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{6 c f}-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{15 c f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{6 c f}-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{15 c f \sqrt{a \sin (e+f x)+a}}",1,"-(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(15*c*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(6*c*f)","A",3,3,38,0.07895,1,"{2841, 2740, 2738}"
2,1,92,0,0.3909009,"\int \cos ^2(e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx","Int[Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2),x]","-\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{5 c f}-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{10 c f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{5 c f}-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{10 c f \sqrt{a \sin (e+f x)+a}}",1,"-(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(10*c*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(5*c*f)","A",3,3,38,0.07895,1,"{2841, 2740, 2738}"
3,1,92,0,0.3900514,"\int \cos ^2(e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx","Int[Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{4 c f}-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{6 c f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{4 c f}-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{6 c f \sqrt{a \sin (e+f x)+a}}",1,"-(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(6*c*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(4*c*f)","A",3,3,38,0.07895,1,"{2841, 2740, 2738}"
4,1,92,0,0.3717078,"\int \cos ^2(e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)} \, dx","Int[Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]],x]","-\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 c f}-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 c f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 c f}-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 c f \sqrt{a \sin (e+f x)+a}}",1,"-(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(3*c*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))/(3*c*f)","A",3,3,38,0.07895,1,"{2841, 2740, 2738}"
5,1,45,0,0.2836451,"\int \frac{\cos ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/Sqrt[c - c*Sin[e + f*x]],x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 a f \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 a f \sqrt{c-c \sin (e+f x)}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*a*f*Sqrt[c - c*Sin[e + f*x]])","A",2,2,38,0.05263,1,"{2841, 2738}"
6,1,99,0,0.4249997,"\int \frac{\cos ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{2 a \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","-\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{2 a \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(-2*a*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]])","A",5,5,38,0.1316,1,"{2841, 2740, 2737, 2667, 31}"
7,1,97,0,0.4309974,"\int \frac{\cos ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f (c-c \sin (e+f x))^{3/2}}","\frac{a \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f (c-c \sin (e+f x))^{3/2}}",1,"(Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*(c - c*Sin[e + f*x])^(3/2)) + (a*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,5,38,0.1316,1,"{2841, 2739, 2737, 2667, 31}"
8,1,48,0,0.3187284,"\int \frac{\cos ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(7/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 a c f (c-c \sin (e+f x))^{5/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 a c f (c-c \sin (e+f x))^{5/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(4*a*c*f*(c - c*Sin[e + f*x])^(5/2))","A",2,2,38,0.05263,1,"{2841, 2742}"
9,1,140,0,0.5243381,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{4 a^2 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{105 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{7 c f}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{21 c f}","-\frac{4 a^2 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{105 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{7 c f}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{21 c f}",1,"(-4*a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(105*c*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(21*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(7*c*f)","A",4,3,38,0.07895,1,"{2841, 2740, 2738}"
10,1,140,0,0.5291824,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2),x]","-\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{15 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{6 c f}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{15 c f}","-\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{15 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{6 c f}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{15 c f}",1,"-(a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(15*c*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(15*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(6*c*f)","A",4,3,38,0.07895,1,"{2841, 2740, 2738}"
11,1,140,0,0.520556,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{2 a^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{15 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}{5 c f}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{5 c f}","-\frac{2 a^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{15 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}{5 c f}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{5 c f}",1,"(-2*a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(15*c*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(5*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(5*c*f)","A",4,3,38,0.07895,1,"{2841, 2740, 2738}"
12,1,92,0,0.390201,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]],x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}{4 a f}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 a f \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}{4 a f}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 a f \sqrt{c-c \sin (e+f x)}}",1,"(c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6*a*f*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(4*a*f)","A",3,3,38,0.07895,1,"{2841, 2740, 2738}"
13,1,45,0,0.3105877,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/Sqrt[c - c*Sin[e + f*x]],x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 a f \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 a f \sqrt{c-c \sin (e+f x)}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*a*f*Sqrt[c - c*Sin[e + f*x]])","A",2,2,38,0.05263,1,"{2841, 2738}"
14,1,147,0,0.5463711,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{4 a^2 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}","-\frac{4 a^2 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}",1,"(-4*a^2*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*Sqrt[c - c*Sin[e + f*x]])","A",6,5,38,0.1316,1,"{2841, 2740, 2737, 2667, 31}"
15,1,144,0,0.5472703,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{4 a^2 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c f (c-c \sin (e+f x))^{3/2}}","\frac{4 a^2 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c f (c-c \sin (e+f x))^{3/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c*f*(c - c*Sin[e + f*x])^(3/2)) + (4*a^2*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*Sqrt[c - c*Sin[e + f*x]])","A",6,6,38,0.1579,1,"{2841, 2739, 2740, 2737, 2667, 31}"
16,1,147,0,0.5655603,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(7/2),x]","-\frac{a^2 \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f (c-c \sin (e+f x))^{5/2}}","-\frac{a^2 \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f (c-c \sin (e+f x))^{5/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*(c - c*Sin[e + f*x])^(5/2)) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*(c - c*Sin[e + f*x])^(3/2)) - (a^2*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",6,5,38,0.1316,1,"{2841, 2739, 2737, 2667, 31}"
17,1,48,0,0.3374323,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(9/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 a c f (c-c \sin (e+f x))^{7/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 a c f (c-c \sin (e+f x))^{7/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6*a*c*f*(c - c*Sin[e + f*x])^(7/2))","A",2,2,38,0.05263,1,"{2841, 2742}"
18,1,97,0,0.435671,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(11/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{48 a c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 a c f (c-c \sin (e+f x))^{9/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{48 a c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 a c f (c-c \sin (e+f x))^{9/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(8*a*c*f*(c - c*Sin[e + f*x])^(9/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(48*a*c^2*f*(c - c*Sin[e + f*x])^(7/2))","A",3,3,38,0.07895,1,"{2841, 2743, 2742}"
19,1,188,0,0.6241608,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{14 c f}-\frac{a^3 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{35 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{8 c f}-\frac{3 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{28 c f}","-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{14 c f}-\frac{a^3 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{35 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{8 c f}-\frac{3 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{28 c f}",1,"-(a^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(35*c*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(14*c*f) - (3*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(28*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(9/2))/(8*c*f)","A",5,3,38,0.07895,1,"{2841, 2740, 2738}"
20,1,188,0,0.6201871,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2),x]","-\frac{4 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{35 c f}-\frac{2 a^3 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 c f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{7 c f}","-\frac{4 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{35 c f}-\frac{2 a^3 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 c f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{7 c f}",1,"(-2*a^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(35*c*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(35*c*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(7*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(7*c*f)","A",5,3,38,0.07895,1,"{2841, 2740, 2738}"
21,1,140,0,0.5166096,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2),x]","\frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 a f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}+\frac{2 c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{15 a f}","\frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 a f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}+\frac{2 c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{15 a f}",1,"(c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(15*a*f*Sqrt[c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(15*a*f) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2))/(6*a*f)","A",4,3,38,0.07895,1,"{2841, 2740, 2738}"
22,1,92,0,0.3950889,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]],x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{5 a f}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 a f \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{5 a f}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 a f \sqrt{c-c \sin (e+f x)}}",1,"(c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*a*f*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(5*a*f)","A",3,3,38,0.07895,1,"{2841, 2740, 2738}"
23,1,45,0,0.3068688,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/Sqrt[c - c*Sin[e + f*x]],x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 a f \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 a f \sqrt{c-c \sin (e+f x)}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*a*f*Sqrt[c - c*Sin[e + f*x]])","A",2,2,38,0.05263,1,"{2841, 2738}"
24,1,193,0,0.64229,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{4 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{8 a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c f \sqrt{c-c \sin (e+f x)}}","-\frac{4 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{8 a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c f \sqrt{c-c \sin (e+f x)}}",1,"(-8*a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (4*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c*f*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c*f*Sqrt[c - c*Sin[e + f*x]])","A",7,5,38,0.1316,1,"{2841, 2740, 2737, 2667, 31}"
25,1,192,0,0.6463813,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{6 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{12 a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{3 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{c f (c-c \sin (e+f x))^{3/2}}","\frac{6 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{12 a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{3 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{c f (c-c \sin (e+f x))^{3/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(c*f*(c - c*Sin[e + f*x])^(3/2)) + (12*a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*Sqrt[c - c*Sin[e + f*x]]) + (3*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c^2*f*Sqrt[c - c*Sin[e + f*x]])","A",7,6,38,0.1579,1,"{2841, 2739, 2740, 2737, 2667, 31}"
26,1,195,0,0.6569136,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(7/2),x]","-\frac{3 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^3 f \sqrt{c-c \sin (e+f x)}}-\frac{6 a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{3 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 c f (c-c \sin (e+f x))^{5/2}}","-\frac{3 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^3 f \sqrt{c-c \sin (e+f x)}}-\frac{6 a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{3 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 c f (c-c \sin (e+f x))^{5/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(2*c*f*(c - c*Sin[e + f*x])^(5/2)) - (3*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c^2*f*(c - c*Sin[e + f*x])^(3/2)) - (6*a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (3*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^3*f*Sqrt[c - c*Sin[e + f*x]])","A",7,6,38,0.1579,1,"{2841, 2739, 2740, 2737, 2667, 31}"
27,1,193,0,0.6626757,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(9/2),x]","\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}}","\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c*f*(c - c*Sin[e + f*x])^(7/2)) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c^2*f*(c - c*Sin[e + f*x])^(5/2)) + (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^3*f*(c - c*Sin[e + f*x])^(3/2)) + (a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",7,5,38,0.1316,1,"{2841, 2739, 2737, 2667, 31}"
28,1,48,0,0.3394026,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(11/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 a c f (c-c \sin (e+f x))^{9/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8 a c f (c-c \sin (e+f x))^{9/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(8*a*c*f*(c - c*Sin[e + f*x])^(9/2))","A",2,2,38,0.05263,1,"{2841, 2742}"
29,1,97,0,0.4377021,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(13/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{80 a c^2 f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 a c f (c-c \sin (e+f x))^{11/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{80 a c^2 f (c-c \sin (e+f x))^{9/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 a c f (c-c \sin (e+f x))^{11/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*a*c*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(80*a*c^2*f*(c - c*Sin[e + f*x])^(9/2))","A",3,3,38,0.07895,1,"{2841, 2743, 2742}"
30,1,236,0,0.7246637,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2),x]","-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{11/2}}{105 c f}-\frac{4 a^4 \cos (e+f x) (c-c \sin (e+f x))^{11/2}}{315 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}-\frac{4 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f}","-\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{11/2}}{105 c f}-\frac{4 a^4 \cos (e+f x) (c-c \sin (e+f x))^{11/2}}{315 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}-\frac{4 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f}",1,"(-4*a^4*Cos[e + f*x]*(c - c*Sin[e + f*x])^(11/2))/(315*c*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(11/2))/(105*c*f) - (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(11/2))/(15*c*f) - (4*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(11/2))/(45*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(11/2))/(10*c*f)","A",6,3,38,0.07895,1,"{2841, 2740, 2738}"
31,1,236,0,0.7348532,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{21 c f}-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{63 c f}-\frac{8 a^4 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{9/2}}{9 c f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{9 c f}","-\frac{2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{21 c f}-\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{63 c f}-\frac{8 a^4 \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 c f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{9/2}}{9 c f}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{9 c f}",1,"(-8*a^4*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(315*c*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(63*c*f) - (2*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(21*c*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(9/2))/(9*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2))/(9*c*f)","A",6,3,38,0.07895,1,"{2841, 2740, 2738}"
32,1,188,0,0.6195126,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2),x]","\frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt{c-c \sin (e+f x)}}{14 a f}+\frac{c^3 \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{35 a f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}+\frac{3 c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f}","\frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt{c-c \sin (e+f x)}}{14 a f}+\frac{c^3 \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{35 a f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{5/2}}{8 a f}+\frac{3 c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{3/2}}{28 a f}",1,"(c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(35*a*f*Sqrt[c - c*Sin[e + f*x]]) + (c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*Sqrt[c - c*Sin[e + f*x]])/(14*a*f) + (3*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*(c - c*Sin[e + f*x])^(3/2))/(28*a*f) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*(c - c*Sin[e + f*x])^(5/2))/(8*a*f)","A",5,3,38,0.07895,1,"{2841, 2740, 2738}"
33,1,140,0,0.515013,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2),x]","\frac{4 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{105 a f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{3/2}}{7 a f}+\frac{2 c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt{c-c \sin (e+f x)}}{21 a f}","\frac{4 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{105 a f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{3/2}}{7 a f}+\frac{2 c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt{c-c \sin (e+f x)}}{21 a f}",1,"(4*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(105*a*f*Sqrt[c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*Sqrt[c - c*Sin[e + f*x]])/(21*a*f) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*(c - c*Sin[e + f*x])^(3/2))/(7*a*f)","A",4,3,38,0.07895,1,"{2841, 2740, 2738}"
34,1,92,0,0.3864806,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]],x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt{c-c \sin (e+f x)}}{6 a f}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{15 a f \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt{c-c \sin (e+f x)}}{6 a f}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{15 a f \sqrt{c-c \sin (e+f x)}}",1,"(c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(15*a*f*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*Sqrt[c - c*Sin[e + f*x]])/(6*a*f)","A",3,3,38,0.07895,1,"{2841, 2740, 2738}"
35,1,45,0,0.3074747,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/Sqrt[c - c*Sin[e + f*x]],x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{5 a f \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{5 a f \sqrt{c-c \sin (e+f x)}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(5*a*f*Sqrt[c - c*Sin[e + f*x]])","A",2,2,38,0.05263,1,"{2841, 2738}"
36,1,241,0,0.7595959,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{8 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{16 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c f \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 c f \sqrt{c-c \sin (e+f x)}}","-\frac{8 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c f \sqrt{c-c \sin (e+f x)}}-\frac{16 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c f \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 c f \sqrt{c-c \sin (e+f x)}}",1,"(-16*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (8*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c*f*Sqrt[c - c*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c*f*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*c*f*Sqrt[c - c*Sin[e + f*x]])","A",8,5,38,0.1316,1,"{2841, 2740, 2737, 2667, 31}"
37,1,238,0,0.7536043,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{16 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{32 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{4 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{c f (c-c \sin (e+f x))^{3/2}}","\frac{16 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{32 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{4 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{c f (c-c \sin (e+f x))^{3/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(c*f*(c - c*Sin[e + f*x])^(3/2)) + (32*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (16*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*Sqrt[c - c*Sin[e + f*x]]) + (4*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c^2*f*Sqrt[c - c*Sin[e + f*x]]) + (4*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c^2*f*Sqrt[c - c*Sin[e + f*x]])","A",8,6,38,0.1579,1,"{2841, 2739, 2740, 2737, 2667, 31}"
38,1,239,0,0.7636423,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(7/2),x]","-\frac{12 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^3 f \sqrt{c-c \sin (e+f x)}}-\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^3 f \sqrt{c-c \sin (e+f x)}}-\frac{24 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 c f (c-c \sin (e+f x))^{5/2}}","-\frac{12 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^3 f \sqrt{c-c \sin (e+f x)}}-\frac{3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^3 f \sqrt{c-c \sin (e+f x)}}-\frac{24 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{c^2 f (c-c \sin (e+f x))^{3/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 c f (c-c \sin (e+f x))^{5/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(2*c*f*(c - c*Sin[e + f*x])^(5/2)) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(c^2*f*(c - c*Sin[e + f*x])^(3/2)) - (24*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (12*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^3*f*Sqrt[c - c*Sin[e + f*x]]) - (3*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c^3*f*Sqrt[c - c*Sin[e + f*x]])","A",8,6,38,0.1579,1,"{2841, 2739, 2740, 2737, 2667, 31}"
39,1,241,0,0.7810302,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(9/2),x]","\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^4 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}","\frac{4 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^4 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(3*c*f*(c - c*Sin[e + f*x])^(7/2)) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c^2*f*(c - c*Sin[e + f*x])^(5/2)) + (2*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c^3*f*(c - c*Sin[e + f*x])^(3/2)) + (8*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (4*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^4*f*Sqrt[c - c*Sin[e + f*x]])","A",8,6,38,0.1579,1,"{2841, 2739, 2740, 2737, 2667, 31}"
40,1,243,0,0.7849874,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(11/2),x]","-\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^4 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac{a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}","-\frac{a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^4 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac{a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*c*f*(c - c*Sin[e + f*x])^(9/2)) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c^2*f*(c - c*Sin[e + f*x])^(7/2)) + (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c^3*f*(c - c*Sin[e + f*x])^(5/2)) - (a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^4*f*(c - c*Sin[e + f*x])^(3/2)) - (a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",8,5,38,0.1316,1,"{2841, 2739, 2737, 2667, 31}"
41,1,48,0,0.338952,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(13/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{10 a c f (c-c \sin (e+f x))^{11/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{10 a c f (c-c \sin (e+f x))^{11/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(10*a*c*f*(c - c*Sin[e + f*x])^(11/2))","A",2,2,38,0.05263,1,"{2841, 2742}"
42,1,97,0,0.4444689,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(15/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{120 a c^2 f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{12 a c f (c-c \sin (e+f x))^{13/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{120 a c^2 f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{12 a c f (c-c \sin (e+f x))^{13/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(12*a*c*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(120*a*c^2*f*(c - c*Sin[e + f*x])^(11/2))","A",3,3,38,0.07895,1,"{2841, 2743, 2742}"
43,1,145,0,0.5389831,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(17/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{840 a c^3 f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{840 a c^3 f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(14*a*c*f*(c - c*Sin[e + f*x])^(15/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(84*a*c^2*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(840*a*c^3*f*(c - c*Sin[e + f*x])^(11/2))","A",4,3,38,0.07895,1,"{2841, 2743, 2742}"
44,1,45,0,0.3057206,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(5/2))/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 c f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 c f \sqrt{a \sin (e+f x)+a}}",1,"-(Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*c*f*Sqrt[a + a*Sin[e + f*x]])","A",2,2,38,0.05263,1,"{2841, 2738}"
45,1,45,0,0.3091106,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(3/2))/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt{a \sin (e+f x)+a}}",1,"-(Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*c*f*Sqrt[a + a*Sin[e + f*x]])","A",2,2,38,0.05263,1,"{2841, 2738}"
46,1,45,0,0.2838377,"\int \frac{\cos ^2(e+f x) \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[(Cos[e + f*x]^2*Sqrt[c - c*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt{a \sin (e+f x)+a}}",1,"-(Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*c*f*Sqrt[a + a*Sin[e + f*x]])","A",2,2,38,0.05263,1,"{2841, 2738}"
47,1,43,0,0.288419,"\int \frac{\cos ^2(e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx","Int[Cos[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{c f \sqrt{a \sin (e+f x)+a}}","-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{c f \sqrt{a \sin (e+f x)+a}}",1,"-((Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]))","A",2,2,38,0.05263,1,"{2841, 2738}"
48,1,54,0,0.3397935,"\int \frac{\cos ^2(e+f x)}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx","Int[Cos[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)),x]","-\frac{\cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","-\frac{\cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"-((Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]))","A",4,4,38,0.1053,1,"{2841, 2737, 2667, 31}"
49,1,42,0,0.3213845,"\int \frac{\cos ^2(e+f x)}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx","Int[Cos[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\cos (e+f x)}{c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}","\frac{\cos (e+f x)}{c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}",1,"Cos[e + f*x]/(c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))","A",2,2,38,0.05263,1,"{2841, 2738}"
50,1,239,0,0.744696,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(3/2),x]","\frac{8 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{16 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a \sin (e+f x)+a}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt{a \sin (e+f x)+a}}","\frac{8 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{16 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a \sin (e+f x)+a}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt{a \sin (e+f x)+a}}",1,"(16*c^4*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (8*c^3*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (2*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a*f*Sqrt[a + a*Sin[e + f*x]]) + (Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*a*f*Sqrt[a + a*Sin[e + f*x]])","A",8,5,38,0.1316,1,"{2841, 2740, 2737, 2667, 31}"
51,1,190,0,0.6394991,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(3/2),x]","\frac{4 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{8 c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a \sin (e+f x)+a}}","\frac{4 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{8 c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt{a \sin (e+f x)+a}}",1,"(8*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (4*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a*f*Sqrt[a + a*Sin[e + f*x]])","A",7,5,38,0.1316,1,"{2841, 2740, 2737, 2667, 31}"
52,1,145,0,0.5291179,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(3/2),x]","\frac{4 c^2 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f \sqrt{a \sin (e+f x)+a}}","\frac{4 c^2 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"(4*c^2*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*a*f*Sqrt[a + a*Sin[e + f*x]])","A",6,5,38,0.1316,1,"{2841, 2740, 2737, 2667, 31}"
53,1,96,0,0.4115277,"\int \frac{\cos ^2(e+f x) \sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[(Cos[e + f*x]^2*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{2 c \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}+\frac{2 c \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(2*c*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]])","A",5,5,38,0.1316,1,"{2841, 2740, 2737, 2667, 31}"
54,1,51,0,0.3332228,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}} \, dx","Int[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,4,38,0.1053,1,"{2841, 2737, 2667, 31}"
55,1,52,0,0.339742,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx","Int[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{a c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{a c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(a*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",3,3,38,0.07895,1,"{2841, 2741, 3770}"
56,1,104,0,0.4358425,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}} \, dx","Int[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x)}{2 a c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x)}{2 a c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}",1,"Cos[e + f*x]/(2*a*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,4,38,0.1053,1,"{2841, 2743, 2741, 3770}"
57,1,285,0,0.8649438,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{40 c^4 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{10 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{10 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{80 c^5 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{5 c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a \sin (e+f x)+a)^{3/2}}","-\frac{40 c^4 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{10 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{10 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{80 c^5 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{5 c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a \sin (e+f x)+a)^{3/2}}",1,"(-80*c^5*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (40*c^4*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (10*c^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (10*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (5*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(a*f*(a + a*Sin[e + f*x])^(3/2))","A",9,6,38,0.1579,1,"{2841, 2739, 2740, 2737, 2667, 31}"
58,1,237,0,0.7519054,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{16 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{4 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{32 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{a f (a \sin (e+f x)+a)^{3/2}}","-\frac{16 c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{4 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{32 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{a f (a \sin (e+f x)+a)^{3/2}}",1,"(-32*c^4*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (16*c^3*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (4*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (4*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(a*f*(a + a*Sin[e + f*x])^(3/2))","A",8,6,38,0.1579,1,"{2841, 2739, 2740, 2737, 2667, 31}"
59,1,191,0,0.6388001,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{6 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{12 c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{3 c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a \sin (e+f x)+a)^{3/2}}","-\frac{6 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{12 c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{3 c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a \sin (e+f x)+a)^{3/2}}",1,"(-12*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (6*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (3*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(a*f*(a + a*Sin[e + f*x])^(3/2))","A",7,6,38,0.1579,1,"{2841, 2739, 2740, 2737, 2667, 31}"
60,1,143,0,0.5384206,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{4 c^2 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 c \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f (a \sin (e+f x)+a)^{3/2}}","-\frac{4 c^2 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 c \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}-\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f (a \sin (e+f x)+a)^{3/2}}",1,"(-4*c^2*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a*f*(a + a*Sin[e + f*x])^(3/2))","A",6,6,38,0.1579,1,"{2841, 2739, 2740, 2737, 2667, 31}"
61,1,97,0,0.4233929,"\int \frac{\cos ^2(e+f x) \sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[(Cos[e + f*x]^2*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{c \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a \sin (e+f x)+a)^{3/2}}","-\frac{c \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a \sin (e+f x)+a)^{3/2}}",1,"-((c*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*(a + a*Sin[e + f*x])^(3/2))","A",5,5,38,0.1316,1,"{2841, 2739, 2737, 2667, 31}"
62,1,43,0,0.3208391,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}} \, dx","Int[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{\cos (e+f x)}{a f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}","-\frac{\cos (e+f x)}{a f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}",1,"-(Cos[e + f*x]/(a*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]))","A",2,2,38,0.05263,1,"{2841, 2738}"
63,1,104,0,0.4438443,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx","Int[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a^2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{2 a c f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a^2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{2 a c f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}",1,"-Cos[e + f*x]/(2*a*c*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,4,38,0.1053,1,"{2841, 2743, 2741, 3770}"
64,1,152,0,0.5404043,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}} \, dx","Int[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a^2 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x)}{2 a^2 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{2 a c f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}","\frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a^2 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x)}{2 a^2 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{2 a c f (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}",1,"-Cos[e + f*x]/(2*a*c*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)) + Cos[e + f*x]/(2*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a^2*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,4,38,0.1053,1,"{2841, 2743, 2741, 3770}"
65,1,114,0,0.3007453,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{c^2 2^{n+\frac{3}{2}} \cos ^3(e+f x) (1-\sin (e+f x))^{\frac{1}{2}-n} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2} (2 m+3),\frac{1}{2} (-2 n-1);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}","\frac{c^2 2^{n+\frac{3}{2}} \cos ^3(e+f x) (1-\sin (e+f x))^{\frac{1}{2}-n} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} \, _2F_1\left(\frac{1}{2} (2 m+3),\frac{1}{2} (-2 n-1);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}",1,"(2^(3/2 + n)*c^2*Cos[e + f*x]^3*Hypergeometric2F1[(3 + 2*m)/2, (-1 - 2*n)/2, (5 + 2*m)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(1/2 - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 + n))/(f*(3 + 2*m))","A",5,5,34,0.1471,1,"{2841, 2745, 2689, 70, 69}"
66,1,86,0,0.2119007,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3,x]","-\frac{a^4 c^3 2^{m+\frac{3}{2}} \cos ^9(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-4} \, _2F_1\left(\frac{9}{2},-m-\frac{1}{2};\frac{11}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{9 f}","-\frac{a^4 c^3 2^{m+\frac{3}{2}} \cos ^9(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-4} \, _2F_1\left(\frac{9}{2},-m-\frac{1}{2};\frac{11}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{9 f}",1,"-(2^(3/2 + m)*a^4*c^3*Cos[e + f*x]^9*Hypergeometric2F1[9/2, -1/2 - m, 11/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^(-4 + m))/(9*f)","A",4,4,34,0.1176,1,"{2840, 2689, 70, 69}"
67,1,86,0,0.2092844,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^2 \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^2,x]","-\frac{a^3 c^2 2^{m+\frac{3}{2}} \cos ^7(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{7}{2},-m-\frac{1}{2};\frac{9}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{7 f}","-\frac{a^3 c^2 2^{m+\frac{3}{2}} \cos ^7(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{7}{2},-m-\frac{1}{2};\frac{9}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{7 f}",1,"-(2^(3/2 + m)*a^3*c^2*Cos[e + f*x]^7*Hypergeometric2F1[7/2, -1/2 - m, 9/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^(-3 + m))/(7*f)","A",4,4,34,0.1176,1,"{2840, 2689, 70, 69}"
68,1,84,0,0.1558044,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x)) \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x]),x]","-\frac{a^2 c 2^{m+\frac{3}{2}} \cos ^5(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(\frac{5}{2},-m-\frac{1}{2};\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 f}","-\frac{a^2 c 2^{m+\frac{3}{2}} \cos ^5(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(\frac{5}{2},-m-\frac{1}{2};\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 f}",1,"-(2^(3/2 + m)*a^2*c*Cos[e + f*x]^5*Hypergeometric2F1[5/2, -1/2 - m, 7/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^(-2 + m))/(5*f)","A",4,4,32,0.1250,1,"{2840, 2689, 70, 69}"
69,1,81,0,0.0687709,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m,x]","-\frac{a 2^{m+\frac{3}{2}} \cos ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f}","-\frac{a 2^{m+\frac{3}{2}} \cos ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f}",1,"-(2^(3/2 + m)*a*Cos[e + f*x]^3*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^(-1 + m))/(3*f)","A",3,3,21,0.1429,1,"{2689, 70, 69}"
70,1,77,0,0.1648821,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x]),x]","-\frac{2^{m+\frac{3}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m-\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{c f}","-\frac{2^{m+\frac{3}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},-m-\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{c f}",1,"-((2^(3/2 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, -1/2 - m, 3/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^m)/(c*f))","A",3,3,34,0.08824,1,"{2840, 2652, 2651}"
71,1,81,0,0.2056792,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^2} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^2,x]","\frac{2^{m+\frac{3}{2}} \sec (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(-\frac{1}{2},-m-\frac{1}{2};\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{a c^2 f}","\frac{2^{m+\frac{3}{2}} \sec (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(-\frac{1}{2},-m-\frac{1}{2};\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{a c^2 f}",1,"(2^(3/2 + m)*Hypergeometric2F1[-1/2, -1/2 - m, 1/2, (1 - Sin[e + f*x])/2]*Sec[e + f*x]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^(1 + m))/(a*c^2*f)","A",4,4,34,0.1176,1,"{2840, 2689, 70, 69}"
72,1,86,0,0.212883,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^3,x]","\frac{2^{m+\frac{3}{2}} \sec ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left(-\frac{3}{2},-m-\frac{1}{2};-\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 a^2 c^3 f}","\frac{2^{m+\frac{3}{2}} \sec ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left(-\frac{3}{2},-m-\frac{1}{2};-\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 a^2 c^3 f}",1,"(2^(3/2 + m)*Hypergeometric2F1[-3/2, -1/2 - m, -1/2, (1 - Sin[e + f*x])/2]*Sec[e + f*x]^3*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^(2 + m))/(3*a^2*c^3*f)","A",4,4,34,0.1176,1,"{2840, 2689, 70, 69}"
73,1,244,0,0.6154316,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2),x]","\frac{192 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9) \left(4 m^2+24 m+35\right)}+\frac{768 c^3 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+7) (2 m+9) \left(4 m^2+16 m+15\right) \sqrt{c-c \sin (e+f x)}}+\frac{24 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{a f \left(4 m^2+32 m+63\right)}+\frac{2 \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9)}","\frac{192 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9) \left(4 m^2+24 m+35\right)}+\frac{768 c^3 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+7) (2 m+9) \left(4 m^2+16 m+15\right) \sqrt{c-c \sin (e+f x)}}+\frac{24 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{a f \left(4 m^2+32 m+63\right)}+\frac{2 \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9)}",1,"(768*c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(7 + 2*m)*(9 + 2*m)*(15 + 16*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (192*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c - c*Sin[e + f*x]])/(a*f*(9 + 2*m)*(35 + 24*m + 4*m^2)) + (24*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(3/2))/(a*f*(63 + 32*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(5/2))/(a*f*(9 + 2*m))","A",5,3,36,0.08333,1,"{2841, 2740, 2738}"
74,1,172,0,0.467299,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2),x]","\frac{64 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+7) \left(4 m^2+16 m+15\right) \sqrt{c-c \sin (e+f x)}}+\frac{16 c \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{a f \left(4 m^2+24 m+35\right)}+\frac{2 \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+7)}","\frac{64 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+7) \left(4 m^2+16 m+15\right) \sqrt{c-c \sin (e+f x)}}+\frac{16 c \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{a f \left(4 m^2+24 m+35\right)}+\frac{2 \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+7)}",1,"(64*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(7 + 2*m)*(15 + 16*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (16*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c - c*Sin[e + f*x]])/(a*f*(35 + 24*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(3/2))/(a*f*(7 + 2*m))","A",4,3,36,0.08333,1,"{2841, 2740, 2738}"
75,1,107,0,0.3541113,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]],x]","\frac{8 c \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f \left(4 m^2+16 m+15\right) \sqrt{c-c \sin (e+f x)}}+\frac{2 \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+5)}","\frac{8 c \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f \left(4 m^2+16 m+15\right) \sqrt{c-c \sin (e+f x)}}+\frac{2 \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+5)}",1,"(8*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(15 + 16*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c - c*Sin[e + f*x]])/(a*f*(5 + 2*m))","A",3,3,36,0.08333,1,"{2841, 2740, 2738}"
76,1,50,0,0.2616613,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}}","\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}}",1,"(2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",2,2,36,0.05556,1,"{2841, 2738}"
77,1,76,0,0.3617426,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(3/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(1,m+\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{a c f (2 m+3) \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(1,m+\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{a c f (2 m+3) \sqrt{c-c \sin (e+f x)}}",1,"(Cos[e + f*x]*Hypergeometric2F1[1, 3/2 + m, 5/2 + m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^(1 + m))/(a*c*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",4,4,36,0.1111,1,"{2841, 2745, 2667, 68}"
78,1,79,0,0.3723593,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(5/2),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(2,m+\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{2 a c^2 f (2 m+3) \sqrt{c-c \sin (e+f x)}}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(2,m+\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{2 a c^2 f (2 m+3) \sqrt{c-c \sin (e+f x)}}",1,"(Cos[e + f*x]*Hypergeometric2F1[2, 3/2 + m, 5/2 + m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^(1 + m))/(2*a*c^2*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",4,4,36,0.1111,1,"{2841, 2745, 2667, 68}"
79,1,50,0,0.2543027,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}}","\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}}",1,"(2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",2,2,36,0.05556,1,"{2841, 2738}"
80,1,50,0,0.2537445,"\int \frac{\cos ^2(e+f x) (c+c \sin (e+f x))^m}{\sqrt{a-a \sin (e+f x)}} \, dx","Int[(Cos[e + f*x]^2*(c + c*Sin[e + f*x])^m)/Sqrt[a - a*Sin[e + f*x]],x]","\frac{2 \cos (e+f x) (c \sin (e+f x)+c)^{m+1}}{c f (2 m+3) \sqrt{a-a \sin (e+f x)}}","\frac{2 \cos (e+f x) (c \sin (e+f x)+c)^{m+1}}{c f (2 m+3) \sqrt{a-a \sin (e+f x)}}",1,"(2*Cos[e + f*x]*(c + c*Sin[e + f*x])^(1 + m))/(c*f*(3 + 2*m)*Sqrt[a - a*Sin[e + f*x]])","A",2,2,36,0.05556,1,"{2841, 2738}"
81,1,182,0,0.4484374,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-5 - m),x]","\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-3}}{a c^2 f \left(4 m^2+24 m+35\right)}+\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-2}}{a c^3 f (2 m+7) \left(4 m^2+16 m+15\right)}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-4}}{a c f (2 m+7)}","\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-3}}{a c^2 f \left(4 m^2+24 m+35\right)}+\frac{2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-2}}{a c^3 f (2 m+7) \left(4 m^2+16 m+15\right)}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-4}}{a c f (2 m+7)}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-4 - m))/(a*c*f*(7 + 2*m)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-3 - m))/(a*c^2*f*(35 + 24*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-2 - m))/(a*c^3*f*(7 + 2*m)*(15 + 16*m + 4*m^2))","A",4,3,38,0.07895,1,"{2841, 2743, 2742}"
82,1,114,0,0.3345018,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-4-m} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-4 - m),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-2}}{a c^2 f \left(4 m^2+16 m+15\right)}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-3}}{a c f (2 m+5)}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-2}}{a c^2 f \left(4 m^2+16 m+15\right)}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-3}}{a c f (2 m+5)}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-3 - m))/(a*c*f*(5 + 2*m)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-2 - m))/(a*c^2*f*(15 + 16*m + 4*m^2))","A",3,3,38,0.07895,1,"{2841, 2743, 2742}"
83,1,54,0,0.2510783,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m),x]","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-2}}{a c f (2 m+3)}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-2}}{a c f (2 m+3)}",1,"(Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-2 - m))/(a*c*f*(3 + 2*m))","A",2,2,38,0.05263,1,"{2841, 2742}"
84,1,113,0,0.3797981,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m),x]","\frac{2^{-m-\frac{1}{2}} \cos ^3(e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2} \, _2F_1\left(\frac{1}{2} (2 m+3),\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}","\frac{2^{-m-\frac{1}{2}} \cos ^3(e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2} \, _2F_1\left(\frac{1}{2} (2 m+3),\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}",1,"(2^(-1/2 - m)*Cos[e + f*x]^3*Hypergeometric2F1[(3 + 2*m)/2, (3 + 2*m)/2, (5 + 2*m)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m))","A",5,5,38,0.1316,1,"{2841, 2745, 2689, 70, 69}"
85,1,114,0,0.371424,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m),x]","\frac{c 2^{\frac{1}{2}-m} \cos ^3(e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2} \, _2F_1\left(\frac{1}{2} (2 m+1),\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}","\frac{c 2^{\frac{1}{2}-m} \cos ^3(e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2} \, _2F_1\left(\frac{1}{2} (2 m+1),\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}",1,"(2^(1/2 - m)*c*Cos[e + f*x]^3*Hypergeometric2F1[(1 + 2*m)/2, (3 + 2*m)/2, (5 + 2*m)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m))","A",5,5,38,0.1316,1,"{2841, 2745, 2689, 70, 69}"
86,1,116,0,0.323419,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-m} \, dx","Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^m,x]","\frac{c^2 2^{\frac{3}{2}-m} \cos ^3(e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2} \, _2F_1\left(\frac{1}{2} (2 m-1),\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}","\frac{c^2 2^{\frac{3}{2}-m} \cos ^3(e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2} \, _2F_1\left(\frac{1}{2} (2 m-1),\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}",1,"(2^(3/2 - m)*c^2*Cos[e + f*x]^3*Hypergeometric2F1[(-1 + 2*m)/2, (3 + 2*m)/2, (5 + 2*m)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m))","A",5,5,36,0.1389,1,"{2841, 2745, 2689, 70, 69}"
87,1,116,0,0.3676602,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 - m),x]","\frac{c^3 2^{\frac{5}{2}-m} \cos ^3(e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2} \, _2F_1\left(\frac{1}{2} (2 m-3),\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}","\frac{c^3 2^{\frac{5}{2}-m} \cos ^3(e+f x) (1-\sin (e+f x))^{m+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2} \, _2F_1\left(\frac{1}{2} (2 m-3),\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m+3)}",1,"(2^(5/2 - m)*c^3*Cos[e + f*x]^3*Hypergeometric2F1[(-3 + 2*m)/2, (3 + 2*m)/2, (5 + 2*m)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m))","A",5,5,38,0.1316,1,"{2841, 2745, 2689, 70, 69}"
88,1,343,0,1.7431646,"\int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx","Int[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2),x]","\frac{2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^3 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{a \sin (e+f x)+a}}+\frac{10 a c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt{a \sin (e+f x)+a}}+\frac{2 a c^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt{a \sin (e+f x)+a}}","\frac{2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^3 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{a \sin (e+f x)+a}}+\frac{10 a c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt{a \sin (e+f x)+a}}+\frac{2 a c^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt{a \sin (e+f x)+a}}",1,"(2*a*c^4*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a*c^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(7*f*g*Sqrt[a + a*Sin[e + f*x]]) + (10*a*c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(77*f*g*Sqrt[a + a*Sin[e + f*x]]) + (2*a*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(33*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(11*f*g*Sqrt[a + a*Sin[e + f*x]])","A",8,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
89,1,290,0,1.4283767,"\int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx","Int[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2),x]","\frac{22 a c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{22 a c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{105 f g \sqrt{a \sin (e+f x)+a}}+\frac{22 a c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a}}","\frac{22 a c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{22 a c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{105 f g \sqrt{a \sin (e+f x)+a}}+\frac{22 a c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a}}",1,"(22*a*c^3*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(105*f*g*Sqrt[a + a*Sin[e + f*x]]) + (2*a*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(21*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]])","A",7,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
90,1,235,0,1.1290667,"\int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx","Int[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2),x]","\frac{2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 a c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{a \sin (e+f x)+a}}+\frac{6 a c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{a \sin (e+f x)+a}}","\frac{2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 a c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{a \sin (e+f x)+a}}+\frac{6 a c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{a \sin (e+f x)+a}}",1,"(2*a*c^2*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(35*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[a + a*Sin[e + f*x]])","A",6,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
91,1,178,0,0.8028534,"\int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)} \, dx","Int[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]],x]","-\frac{2 a \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a}}+\frac{2 a c (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 a c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","-\frac{2 a \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a}}+\frac{2 a c (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 a c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(2*a*c*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f*g*Sqrt[a + a*Sin[e + f*x]])","A",5,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
92,1,122,0,0.5695293,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 a g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{2 a g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(-2*a*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
93,1,123,0,0.5818523,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(3/2),x]","\frac{4 a (g \cos (e+f x))^{5/2}}{f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{6 a g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{4 a (g \cos (e+f x))^{5/2}}{f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{6 a g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(4*a*(g*Cos[e + f*x])^(5/2))/(f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (6*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,4,42,0.09524,1,"{2850, 2842, 2640, 2639}"
94,1,182,0,0.8786372,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(5/2),x]","\frac{6 a g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{6 a (g \cos (e+f x))^{5/2}}{5 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{4 a (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}","\frac{6 a g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{6 a (g \cos (e+f x))^{5/2}}{5 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{4 a (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (6*a*(g*Cos[e + f*x])^(5/2))/(5*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (6*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
95,1,237,0,1.1649499,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(7/2),x]","-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{2 a g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}","-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{2 a g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(15*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(15*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (2*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",6,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
96,1,292,0,1.4598178,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(9/2),x]","-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{2 a g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{65 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}","-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{2 a g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{65 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2))/(13*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(39*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(65*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(65*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (2*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(65*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",7,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
97,1,352,0,1.7269353,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2),x]","\frac{14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a}}+\frac{14 a^2 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{a \sin (e+f x)+a}}-\frac{14 a^2 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{99 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}","\frac{14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a}}+\frac{14 a^2 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{a \sin (e+f x)+a}}-\frac{14 a^2 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{99 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}",1,"(14*a^2*c^3*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (14*a^2*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^2*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(15*f*g*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(33*f*g*Sqrt[a + a*Sin[e + f*x]]) - (14*a^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(99*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(11*f*g)","A",8,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
98,1,295,0,1.503205,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2),x]","\frac{14 a^2 c^2 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{14 a^2 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{9 f g}","\frac{14 a^2 c^2 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{14 a^2 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{9 f g}",1,"(14*a^2*c^2*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (14*a^2*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^2*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(15*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))/(9*f*g)","A",7,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
99,1,235,0,1.1388568,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]],x]","-\frac{2 a^2 c (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 a^2 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{6 a c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{c-c \sin (e+f x)}}","-\frac{2 a^2 c (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 a^2 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{6 a c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{c-c \sin (e+f x)}}",1,"(-2*a^2*c*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a^2*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (6*a*c*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(35*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[c - c*Sin[e + f*x]])","A",6,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
100,1,180,0,0.8499264,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{14 a^2 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt{c-c \sin (e+f x)}}","-\frac{14 a^2 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt{c-c \sin (e+f x)}}",1,"(-14*a^2*(g*Cos[e + f*x])^(5/2))/(15*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*f*g*Sqrt[c - c*Sin[e + f*x]])","A",5,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
101,1,182,0,0.8757323,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(3/2),x]","\frac{14 a^2 (g \cos (e+f x))^{5/2}}{3 c f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}","\frac{14 a^2 (g \cos (e+f x))^{5/2}}{3 c f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(f*g*(c - c*Sin[e + f*x])^(3/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(3*c*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
102,1,186,0,0.8867998,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{42 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{5 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}","\frac{42 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{5 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*f*g*(c - c*Sin[e + f*x])^(5/2)) - (28*a^2*(g*Cos[e + f*x])^(5/2))/(5*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (42*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,4,42,0.09524,1,"{2850, 2842, 2640, 2639}"
103,1,243,0,1.1942824,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(7/2),x]","\frac{14 a^2 (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{45 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}","\frac{14 a^2 (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{45 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(9*f*g*(c - c*Sin[e + f*x])^(7/2)) - (28*a^2*(g*Cos[e + f*x])^(5/2))/(45*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(15*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",6,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
104,1,300,0,1.4968969,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(9/2),x]","\frac{14 a^2 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{195 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{195 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{117 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}","\frac{14 a^2 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{195 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{195 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{117 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - (28*a^2*(g*Cos[e + f*x])^(5/2))/(117*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(195*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(195*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(195*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",7,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
105,1,357,0,1.8083021,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(11/2),x]","\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{1105 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}","\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{1105 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (28*a^2*(g*Cos[e + f*x])^(5/2))/(221*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(663*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(1105*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(1105*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(1105*c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",8,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
106,1,406,0,2.0887765,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2),x]","\frac{22 a^3 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{195 f g \sqrt{a \sin (e+f x)+a}}+\frac{154 a^3 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{154 a^3 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{195 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a^3 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{117 f g \sqrt{a \sin (e+f x)+a}}+\frac{2 a^3 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{39 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}-\frac{2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}","\frac{22 a^3 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{195 f g \sqrt{a \sin (e+f x)+a}}+\frac{154 a^3 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{154 a^3 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{195 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a^3 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{117 f g \sqrt{a \sin (e+f x)+a}}+\frac{2 a^3 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{39 f g \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}-\frac{2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}",1,"(154*a^3*c^3*(g*Cos[e + f*x])^(5/2))/(585*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (154*a^3*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(195*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a^3*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(195*f*g*Sqrt[a + a*Sin[e + f*x]]) + (2*a^3*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(39*f*g*Sqrt[a + a*Sin[e + f*x]]) - (14*a^3*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(117*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(13*f*g) - (2*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(13*f*g)","A",9,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
107,1,352,0,1.7572753,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{14 a^3 c^2 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{15 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 a^3 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{99 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}","-\frac{14 a^3 c^2 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{15 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 a^3 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{99 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}",1,"(-14*a^3*c^2*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (14*a^3*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a^2*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(15*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(33*f*g*Sqrt[c - c*Sin[e + f*x]]) + (14*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(99*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(11*f*g)","A",8,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
108,1,290,0,1.4201082,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]],x]","-\frac{22 a^3 c (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^2 c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{105 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^3 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}","-\frac{22 a^3 c (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^2 c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{105 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^3 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}",1,"(-22*a^3*c*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a^3*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (22*a^2*c*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(105*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(21*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(9*f*g*Sqrt[c - c*Sin[e + f*x]])","A",7,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
109,1,234,0,1.1424841,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{22 a^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}}","-\frac{22 a^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}}",1,"(-22*a^3*(g*Cos[e + f*x])^(5/2))/(15*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (22*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(35*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[c - c*Sin[e + f*x]])","A",6,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
110,1,241,0,1.1402946,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(3/2),x]","\frac{154 a^3 (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{22 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 c f g \sqrt{c-c \sin (e+f x)}}-\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}","\frac{154 a^3 (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{22 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 c f g \sqrt{c-c \sin (e+f x)}}-\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(f*g*(c - c*Sin[e + f*x])^(3/2)) + (154*a^3*(g*Cos[e + f*x])^(5/2))/(15*c*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*c*f*g*Sqrt[c - c*Sin[e + f*x]])","A",6,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
111,1,243,0,1.1745421,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(5/2),x]","-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 c f g (c-c \sin (e+f x))^{3/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}","-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 c f g (c-c \sin (e+f x))^{3/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(5*f*g*(c - c*Sin[e + f*x])^(5/2)) - (44*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*c*f*g*(c - c*Sin[e + f*x])^(3/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(15*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",6,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
112,1,243,0,1.1968353,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(7/2),x]","\frac{308 a^3 (g \cos (e+f x))^{5/2}}{45 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{45 c f g (c-c \sin (e+f x))^{5/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}","\frac{308 a^3 (g \cos (e+f x))^{5/2}}{45 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{45 c f g (c-c \sin (e+f x))^{5/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(9*f*g*(c - c*Sin[e + f*x])^(7/2)) - (44*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(45*c*f*g*(c - c*Sin[e + f*x])^(5/2)) + (308*a^3*(g*Cos[e + f*x])^(5/2))/(45*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",6,4,42,0.09524,1,"{2850, 2842, 2640, 2639}"
113,1,300,0,1.4831446,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(9/2),x]","-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{585 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{195 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{117 c f g (c-c \sin (e+f x))^{7/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}","-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{585 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{195 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{117 c f g (c-c \sin (e+f x))^{7/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - (44*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(117*c*f*g*(c - c*Sin[e + f*x])^(7/2)) + (308*a^3*(g*Cos[e + f*x])^(5/2))/(585*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(195*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(195*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",7,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
114,1,357,0,1.7865944,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(11/2),x]","-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{1989 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{3315 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}","-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{1989 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{3315 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (44*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(221*c*f*g*(c - c*Sin[e + f*x])^(9/2)) + (308*a^3*(g*Cos[e + f*x])^(5/2))/(1989*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(3315*c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",8,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
115,1,414,0,2.0799801,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(13/2),x]","-\frac{22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{22 a^3 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac{22 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{3315 c^6 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}","-\frac{22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{22 a^3 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac{22 a^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{3315 c^6 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(21*f*g*(c - c*Sin[e + f*x])^(13/2)) - (44*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(357*c*f*g*(c - c*Sin[e + f*x])^(11/2)) + (44*a^3*(g*Cos[e + f*x])^(5/2))/(663*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) - (22*a^3*(g*Cos[e + f*x])^(5/2))/(1989*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (22*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (22*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (22*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(3315*c^6*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",9,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
116,1,463,0,2.3899116,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2),x]","-\frac{154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 c^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{195 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^3 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{39 f g \sqrt{c-c \sin (e+f x)}}+\frac{154 a^4 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{195 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^3 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{585 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{195 f g}+\frac{14 c^3 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}","-\frac{154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 c^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{195 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^3 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{39 f g \sqrt{c-c \sin (e+f x)}}+\frac{154 a^4 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{195 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^3 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{585 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{195 f g}+\frac{14 c^3 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{195 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}",1,"(-154*a^4*c^3*(g*Cos[e + f*x])^(5/2))/(585*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (154*a^4*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(195*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (22*a^3*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(195*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a^2*c^3*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(39*f*g*Sqrt[c - c*Sin[e + f*x]]) - (14*a*c^3*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(585*f*g*Sqrt[c - c*Sin[e + f*x]]) + (14*c^3*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2))/(195*f*g*Sqrt[c - c*Sin[e + f*x]]) + (22*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(195*f*g) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2))/(15*f*g)","A",10,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
117,1,409,0,2.0422868,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 a^4 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{13 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^2 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{429 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}","-\frac{14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 a^4 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{13 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 a c^2 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{429 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}",1,"(-14*a^4*c^2*(g*Cos[e + f*x])^(5/2))/(39*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (14*a^4*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(13*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a^3*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(13*f*g*Sqrt[c - c*Sin[e + f*x]]) - (10*a^2*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(143*f*g*Sqrt[c - c*Sin[e + f*x]]) - (14*a*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(429*f*g*Sqrt[c - c*Sin[e + f*x]]) + (14*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2))/(143*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(13*f*g)","A",9,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
118,1,343,0,1.7198946,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]],x]","-\frac{2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 a^4 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt{c-c \sin (e+f x)}}","-\frac{2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 a^4 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt{c-c \sin (e+f x)}}",1,"(-2*a^4*c*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^4*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a^3*c*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(7*f*g*Sqrt[c - c*Sin[e + f*x]]) - (10*a^2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(77*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(33*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2))/(11*f*g*Sqrt[c - c*Sin[e + f*x]])","A",8,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
119,1,288,0,1.4371285,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}","-\frac{22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}",1,"(-22*a^4*(g*Cos[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (22*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(21*f*g*Sqrt[c - c*Sin[e + f*x]]) - (10*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(21*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(9*f*g*Sqrt[c - c*Sin[e + f*x]])","A",7,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
120,1,294,0,1.455294,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(3/2),x]","\frac{22 a^4 (g \cos (e+f x))^{5/2}}{c f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{66 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{7 c f g \sqrt{c-c \sin (e+f x)}}+\frac{30 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 c f g \sqrt{c-c \sin (e+f x)}}-\frac{66 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}","\frac{22 a^4 (g \cos (e+f x))^{5/2}}{c f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{66 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{7 c f g \sqrt{c-c \sin (e+f x)}}+\frac{30 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 c f g \sqrt{c-c \sin (e+f x)}}-\frac{66 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(f*g*(c - c*Sin[e + f*x])^(3/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(c*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (66*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (66*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(7*c*f*g*Sqrt[c - c*Sin[e + f*x]]) + (30*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(7*c*f*g*Sqrt[c - c*Sin[e + f*x]])","A",7,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
121,1,298,0,1.4881622,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(5/2),x]","-\frac{154 a^4 (g \cos (e+f x))^{5/2}}{5 c^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{66 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 c^2 f g \sqrt{c-c \sin (e+f x)}}+\frac{462 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{12 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{c f g (c-c \sin (e+f x))^{3/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}","-\frac{154 a^4 (g \cos (e+f x))^{5/2}}{5 c^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{66 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 c^2 f g \sqrt{c-c \sin (e+f x)}}+\frac{462 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{12 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{c f g (c-c \sin (e+f x))^{3/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(5*f*g*(c - c*Sin[e + f*x])^(5/2)) - (12*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(c*f*g*(c - c*Sin[e + f*x])^(3/2)) - (154*a^4*(g*Cos[e + f*x])^(5/2))/(5*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (462*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (66*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*c^2*f*g*Sqrt[c - c*Sin[e + f*x]])","A",7,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
122,1,300,0,1.5410413,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(7/2),x]","\frac{154 a^4 (g \cos (e+f x))^{5/2}}{9 c^3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{44 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{3 c^2 f g (c-c \sin (e+f x))^{3/2}}-\frac{154 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{3 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{3 c f g (c-c \sin (e+f x))^{5/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}","\frac{154 a^4 (g \cos (e+f x))^{5/2}}{9 c^3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{44 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{3 c^2 f g (c-c \sin (e+f x))^{3/2}}-\frac{154 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{3 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{3 c f g (c-c \sin (e+f x))^{5/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(9*f*g*(c - c*Sin[e + f*x])^(7/2)) - (4*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(3*c*f*g*(c - c*Sin[e + f*x])^(5/2)) + (44*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(3*c^2*f*g*(c - c*Sin[e + f*x])^(3/2)) + (154*a^4*(g*Cos[e + f*x])^(5/2))/(9*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (154*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(3*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",7,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
123,1,300,0,1.5335679,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(9/2),x]","-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{39 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{44 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{39 c^2 f g (c-c \sin (e+f x))^{5/2}}+\frac{154 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{13 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{20 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{39 c f g (c-c \sin (e+f x))^{7/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}","-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{39 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{44 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{39 c^2 f g (c-c \sin (e+f x))^{5/2}}+\frac{154 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{13 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{20 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{39 c f g (c-c \sin (e+f x))^{7/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - (20*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(39*c*f*g*(c - c*Sin[e + f*x])^(7/2)) + (44*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(39*c^2*f*g*(c - c*Sin[e + f*x])^(5/2)) - (308*a^4*(g*Cos[e + f*x])^(5/2))/(39*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (154*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(13*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",7,4,42,0.09524,1,"{2850, 2842, 2640, 2639}"
124,1,357,0,1.8396324,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(11/2),x]","\frac{154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{220 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac{154 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{221 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{60 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}","\frac{154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{220 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac{154 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{221 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{60 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (60*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(221*c*f*g*(c - c*Sin[e + f*x])^(9/2)) + (220*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(663*c^2*f*g*(c - c*Sin[e + f*x])^(7/2)) - (308*a^4*(g*Cos[e + f*x])^(5/2))/(663*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (154*a^4*(g*Cos[e + f*x])^(5/2))/(221*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (154*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(221*c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",8,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
125,1,414,0,2.1713719,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(13/2),x]","\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{220 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{663 c^6 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{20 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}","\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{220 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{663 c^6 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{20 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(21*f*g*(c - c*Sin[e + f*x])^(13/2)) - (20*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(119*c*f*g*(c - c*Sin[e + f*x])^(11/2)) + (220*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(1547*c^2*f*g*(c - c*Sin[e + f*x])^(9/2)) - (220*a^4*(g*Cos[e + f*x])^(5/2))/(1989*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(663*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(663*c^5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (22*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(663*c^6*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",9,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
126,1,471,0,2.4867861,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(15/2),x]","\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac{44 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5525 c^7 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}","\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac{44 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5525 c^7 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}",1,"(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(25*f*g*(c - c*Sin[e + f*x])^(15/2)) - (4*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(35*c*f*g*(c - c*Sin[e + f*x])^(13/2)) + (44*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(595*c^2*f*g*(c - c*Sin[e + f*x])^(11/2)) - (44*a^4*(g*Cos[e + f*x])^(5/2))/(1105*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(3315*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(5525*c^5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(5525*c^6*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (22*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5525*c^7*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",10,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
127,1,234,0,1.1208569,"\int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/Sqrt[a + a*Sin[e + f*x]],x]","\frac{22 c^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{a \sin (e+f x)+a}}+\frac{22 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{a \sin (e+f x)+a}}","\frac{22 c^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{22 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{a \sin (e+f x)+a}}+\frac{22 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{a \sin (e+f x)+a}}",1,"(22*c^3*(g*Cos[e + f*x])^(5/2))/(15*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(35*f*g*Sqrt[a + a*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[a + a*Sin[e + f*x]])","A",6,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
128,1,180,0,0.8380017,"\int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2))/Sqrt[a + a*Sin[e + f*x]],x]","\frac{14 c^2 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a}}","\frac{14 c^2 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a}}",1,"(14*c^2*(g*Cos[e + f*x])^(5/2))/(15*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (14*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f*g*Sqrt[a + a*Sin[e + f*x]])","A",5,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
129,1,122,0,0.539677,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]],x]","\frac{2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(2*c*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,4,42,0.09524,1,"{2851, 2842, 2640, 2639}"
130,1,68,0,0.2794413,"\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",3,3,42,0.07143,1,"{2842, 2640, 2639}"
131,1,121,0,0.5700692,"\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{2 (g \cos (e+f x))^{5/2}}{f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{2 (g \cos (e+f x))^{5/2}}{f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(2*(g*Cos[e + f*x])^(5/2))/(f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,4,42,0.09524,1,"{2852, 2842, 2640, 2639}"
132,1,179,0,0.8597446,"\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)),x]","-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{5 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}","-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{5 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}",1,"(2*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (2*(g*Cos[e + f*x])^(5/2))/(5*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,4,42,0.09524,1,"{2852, 2842, 2640, 2639}"
133,1,233,0,1.1492568,"\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)),x]","\frac{2 (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}","\frac{2 (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{15 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}",1,"(2*(g*Cos[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (2*(g*Cos[e + f*x])^(5/2))/(15*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (2*(g*Cos[e + f*x])^(5/2))/(15*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",6,4,42,0.09524,1,"{2852, 2842, 2640, 2639}"
134,1,294,0,1.4198458,"\int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{22 c^4 (g \cos (e+f x))^{5/2}}{a f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{66 c^3 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{7 a f g \sqrt{a \sin (e+f x)+a}}-\frac{30 c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 a f g \sqrt{a \sin (e+f x)+a}}-\frac{66 c^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}","-\frac{22 c^4 (g \cos (e+f x))^{5/2}}{a f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{66 c^3 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{7 a f g \sqrt{a \sin (e+f x)+a}}-\frac{30 c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 a f g \sqrt{a \sin (e+f x)+a}}-\frac{66 c^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}",1,"(-22*c^4*(g*Cos[e + f*x])^(5/2))/(a*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (66*c^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (66*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(7*a*f*g*Sqrt[a + a*Sin[e + f*x]]) - (30*c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*a*f*g*Sqrt[a + a*Sin[e + f*x]]) - (4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2))","A",7,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
135,1,241,0,1.1363293,"\int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{154 c^3 (g \cos (e+f x))^{5/2}}{15 a f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 a f g \sqrt{a \sin (e+f x)+a}}-\frac{154 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}","-\frac{154 c^3 (g \cos (e+f x))^{5/2}}{15 a f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 a f g \sqrt{a \sin (e+f x)+a}}-\frac{154 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}",1,"(-154*c^3*(g*Cos[e + f*x])^(5/2))/(15*a*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (154*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (22*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*a*f*g*Sqrt[a + a*Sin[e + f*x]]) - (4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(f*g*(a + a*Sin[e + f*x])^(3/2))","A",6,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
136,1,182,0,0.8443608,"\int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{14 c^2 (g \cos (e+f x))^{5/2}}{3 a f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}","-\frac{14 c^2 (g \cos (e+f x))^{5/2}}{3 a f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{14 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}",1,"(-14*c^2*(g*Cos[e + f*x])^(5/2))/(3*a*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (14*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (4*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(f*g*(a + a*Sin[e + f*x])^(3/2))","A",5,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
137,1,123,0,0.5609329,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{4 c (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{6 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","-\frac{4 c (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{6 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(-4*c*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) - (6*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,4,42,0.09524,1,"{2850, 2842, 2640, 2639}"
138,1,121,0,0.5636754,"\int \frac{(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,4,42,0.09524,1,"{2852, 2842, 2640, 2639}"
139,1,176,0,0.8787351,"\int \frac{(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{2 (g \cos (e+f x))^{5/2}}{a f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}","\frac{2 (g \cos (e+f x))^{5/2}}{a f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}",1,"(-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)) + (2*(g*Cos[e + f*x])^(5/2))/(a*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,4,42,0.09524,1,"{2852, 2842, 2640, 2639}"
140,1,237,0,1.1588373,"\int \frac{(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)),x]","-\frac{6 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 (g \cos (e+f x))^{5/2}}{5 a c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{6 (g \cos (e+f x))^{5/2}}{5 a f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}","-\frac{6 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 (g \cos (e+f x))^{5/2}}{5 a c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{6 (g \cos (e+f x))^{5/2}}{5 a f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}",1,"(-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)) + (6*(g*Cos[e + f*x])^(5/2))/(5*a*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (6*(g*Cos[e + f*x])^(5/2))/(5*a*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (6*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",6,4,42,0.09524,1,"{2852, 2842, 2640, 2639}"
141,1,294,0,1.4740136,"\int \frac{(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2)),x]","\frac{2 (g \cos (e+f x))^{5/2}}{3 a c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{3 a c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{3 a c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{10 (g \cos (e+f x))^{5/2}}{9 a f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}","\frac{2 (g \cos (e+f x))^{5/2}}{3 a c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{3 a c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{3 a c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{10 (g \cos (e+f x))^{5/2}}{9 a f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}",1,"(-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2)) + (10*(g*Cos[e + f*x])^(5/2))/(9*a*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (2*(g*Cos[e + f*x])^(5/2))/(3*a*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (2*(g*Cos[e + f*x])^(5/2))/(3*a*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(3*a*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",7,4,42,0.09524,1,"{2852, 2842, 2640, 2639}"
142,1,357,0,1.7283289,"\int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^(5/2),x]","\frac{418 c^5 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{1254 c^4 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{114 c^3 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{1254 c^5 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{76 c^2 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2}}-\frac{4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}","\frac{418 c^5 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{1254 c^4 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{114 c^3 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{1254 c^5 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{76 c^2 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2}}-\frac{4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}",1,"(418*c^5*(g*Cos[e + f*x])^(5/2))/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (1254*c^5*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (1254*c^4*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(35*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]) + (114*c^3*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]) + (76*c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)) - (4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2))","A",8,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
143,1,298,0,1.4243596,"\int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(5/2),x]","\frac{154 c^4 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{66 c^3 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{462 c^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{12 c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{a f g (a \sin (e+f x)+a)^{3/2}}-\frac{4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}","\frac{154 c^4 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{66 c^3 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{462 c^4 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{12 c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{a f g (a \sin (e+f x)+a)^{3/2}}-\frac{4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}",1,"(154*c^4*(g*Cos[e + f*x])^(5/2))/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (462*c^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (66*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]) + (12*c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(a*f*g*(a + a*Sin[e + f*x])^(3/2)) - (4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2))","A",7,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
144,1,243,0,1.1357605,"\int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(5/2),x]","\frac{154 c^3 (g \cos (e+f x))^{5/2}}{15 a^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{154 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{44 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2}}-\frac{4 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}","\frac{154 c^3 (g \cos (e+f x))^{5/2}}{15 a^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{154 c^3 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{44 c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2}}-\frac{4 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}",1,"(154*c^3*(g*Cos[e + f*x])^(5/2))/(15*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (154*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (44*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)) - (4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2))","A",6,5,42,0.1190,1,"{2850, 2851, 2842, 2640, 2639}"
145,1,186,0,0.8518114,"\int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(5/2),x]","\frac{42 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{28 c^2 (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{4 c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}","\frac{42 c^2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{28 c^2 (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{4 c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}}",1,"(28*c^2*(g*Cos[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) + (42*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (4*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f*g*(a + a*Sin[e + f*x])^(5/2))","A",5,4,42,0.09524,1,"{2850, 2842, 2640, 2639}"
146,1,182,0,0.8317722,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{5/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(5/2),x]","\frac{6 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 c (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{4 c (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}","\frac{6 c g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 c (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{4 c (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}",1,"(-4*c*(g*Cos[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]) + (6*c*(g*Cos[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) + (6*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,5,42,0.1190,1,"{2850, 2852, 2842, 2640, 2639}"
147,1,179,0,0.8420566,"\int \frac{(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}","-\frac{2 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}",1,"(-2*(g*Cos[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]) - (2*(g*Cos[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",5,4,42,0.09524,1,"{2852, 2842, 2640, 2639}"
148,1,237,0,1.1609631,"\int \frac{(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)),x]","-\frac{6 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{6 (g \cos (e+f x))^{5/2}}{5 a c f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{6 (g \cos (e+f x))^{5/2}}{5 c f g (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2}}","-\frac{6 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{6 (g \cos (e+f x))^{5/2}}{5 a c f g (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{6 (g \cos (e+f x))^{5/2}}{5 c f g (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2}}",1,"(2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)) - (6*(g*Cos[e + f*x])^(5/2))/(5*c*f*g*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]) - (6*(g*Cos[e + f*x])^(5/2))/(5*a*c*f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) - (6*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",6,4,42,0.09524,1,"{2852, 2842, 2640, 2639}"
149,1,291,0,1.4723898,"\int \frac{(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)),x]","-\frac{6 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 (g \cos (e+f x))^{5/2}}{5 a^2 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{6 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{a f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}","-\frac{6 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 a^2 c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 (g \cos (e+f x))^{5/2}}{5 a^2 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{6 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{a f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}",1,"(-2*(g*Cos[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)) - (2*(g*Cos[e + f*x])^(5/2))/(a*f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)) + (6*(g*Cos[e + f*x])^(5/2))/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (6*(g*Cos[e + f*x])^(5/2))/(5*a^2*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (6*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",7,4,42,0.09524,1,"{2852, 2842, 2640, 2639}"
150,1,350,0,1.7943386,"\int \frac{(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2)),x]","\frac{14 (g \cos (e+f x))^{5/2}}{15 a^2 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{14 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 a^2 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{14 (g \cos (e+f x))^{5/2}}{15 a^2 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{14 (g \cos (e+f x))^{5/2}}{9 a^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{14 (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}","\frac{14 (g \cos (e+f x))^{5/2}}{15 a^2 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{14 g \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 a^2 c^3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{14 (g \cos (e+f x))^{5/2}}{15 a^2 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{14 (g \cos (e+f x))^{5/2}}{9 a^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{14 (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}-\frac{2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}",1,"(-2*(g*Cos[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2)) - (14*(g*Cos[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2)) + (14*(g*Cos[e + f*x])^(5/2))/(9*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (14*(g*Cos[e + f*x])^(5/2))/(15*a^2*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (14*(g*Cos[e + f*x])^(5/2))/(15*a^2*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (14*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*a^2*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",8,4,42,0.09524,1,"{2852, 2842, 2640, 2639}"
151,1,119,0,0.2926053,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{c 2^{n+\frac{9}{4}} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{-n-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} \, _2F_1\left(\frac{1}{4} (4 m+5),\frac{1}{4} (-4 n-1);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (4 m+5)}","\frac{c 2^{n+\frac{9}{4}} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{-n-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} \, _2F_1\left(\frac{1}{4} (4 m+5),\frac{1}{4} (-4 n-1);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (4 m+5)}",1,"(2^(9/4 + n)*c*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(5 + 4*m)/4, (-1 - 4*n)/4, (9 + 4*m)/4, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(-1/4 - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(f*g*(5 + 4*m))","A",4,4,38,0.1053,1,"{2853, 2689, 70, 69}"
152,1,93,0,0.2849448,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3,x]","-\frac{a^4 c^3 2^{m+\frac{9}{4}} (g \cos (e+f x))^{17/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-4} \, _2F_1\left(\frac{17}{4},-m-\frac{1}{4};\frac{21}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{17 f g^7}","-\frac{a^4 c^3 2^{m+\frac{9}{4}} (g \cos (e+f x))^{17/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-4} \, _2F_1\left(\frac{17}{4},-m-\frac{1}{4};\frac{21}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{17 f g^7}",1,"-(2^(9/4 + m)*a^4*c^3*(g*Cos[e + f*x])^(17/2)*Hypergeometric2F1[17/4, -1/4 - m, 21/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(-4 + m))/(17*f*g^7)","A",4,4,38,0.1053,1,"{2840, 2689, 70, 69}"
153,1,93,0,0.2755923,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^2 \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^2,x]","-\frac{a^3 c^2 2^{m+\frac{9}{4}} (g \cos (e+f x))^{13/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{13}{4},-m-\frac{1}{4};\frac{17}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{13 f g^5}","-\frac{a^3 c^2 2^{m+\frac{9}{4}} (g \cos (e+f x))^{13/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{13}{4},-m-\frac{1}{4};\frac{17}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{13 f g^5}",1,"-(2^(9/4 + m)*a^3*c^2*(g*Cos[e + f*x])^(13/2)*Hypergeometric2F1[13/4, -1/4 - m, 17/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(-3 + m))/(13*f*g^5)","A",4,4,38,0.1053,1,"{2840, 2689, 70, 69}"
154,1,91,0,0.2076806,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x)) \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x]),x]","-\frac{a^2 c 2^{m+\frac{9}{4}} (g \cos (e+f x))^{9/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(\frac{9}{4},-m-\frac{1}{4};\frac{13}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{9 f g^3}","-\frac{a^2 c 2^{m+\frac{9}{4}} (g \cos (e+f x))^{9/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(\frac{9}{4},-m-\frac{1}{4};\frac{13}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{9 f g^3}",1,"-(2^(9/4 + m)*a^2*c*(g*Cos[e + f*x])^(9/2)*Hypergeometric2F1[9/4, -1/4 - m, 13/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(-2 + m))/(9*f*g^3)","A",4,4,36,0.1111,1,"{2840, 2689, 70, 69}"
155,1,88,0,0.0855542,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m,x]","-\frac{a 2^{m+\frac{9}{4}} (g \cos (e+f x))^{5/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{5}{4},-m-\frac{1}{4};\frac{9}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{5 f g}","-\frac{a 2^{m+\frac{9}{4}} (g \cos (e+f x))^{5/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{5}{4},-m-\frac{1}{4};\frac{9}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{5 f g}",1,"-(2^(9/4 + m)*a*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[5/4, -1/4 - m, 9/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(-1 + m))/(5*f*g)","A",3,3,25,0.1200,1,"{2689, 70, 69}"
156,1,84,0,0.2747949,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x]),x]","-\frac{g 2^{m+\frac{9}{4}} \sqrt{g \cos (e+f x)} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{4},-m-\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{c f}","-\frac{g 2^{m+\frac{9}{4}} \sqrt{g \cos (e+f x)} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^m \, _2F_1\left(\frac{1}{4},-m-\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{c f}",1,"-((2^(9/4 + m)*g*Sqrt[g*Cos[e + f*x]]*Hypergeometric2F1[1/4, -1/4 - m, 5/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^m)/(c*f))","A",4,4,38,0.1053,1,"{2840, 2689, 70, 69}"
157,1,93,0,0.2805694,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^2} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^2,x]","\frac{g^3 2^{m+\frac{9}{4}} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(-\frac{3}{4},-m-\frac{1}{4};\frac{1}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{3 a c^2 f (g \cos (e+f x))^{3/2}}","\frac{g^3 2^{m+\frac{9}{4}} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(-\frac{3}{4},-m-\frac{1}{4};\frac{1}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{3 a c^2 f (g \cos (e+f x))^{3/2}}",1,"(2^(9/4 + m)*g^3*Hypergeometric2F1[-3/4, -1/4 - m, 1/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(1 + m))/(3*a*c^2*f*(g*Cos[e + f*x])^(3/2))","A",4,4,38,0.1053,1,"{2840, 2689, 70, 69}"
158,1,93,0,0.2825661,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^3,x]","\frac{g^5 2^{m+\frac{9}{4}} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left(-\frac{7}{4},-m-\frac{1}{4};-\frac{3}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{7 a^2 c^3 f (g \cos (e+f x))^{7/2}}","\frac{g^5 2^{m+\frac{9}{4}} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left(-\frac{7}{4},-m-\frac{1}{4};-\frac{3}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{7 a^2 c^3 f (g \cos (e+f x))^{7/2}}",1,"(2^(9/4 + m)*g^5*Hypergeometric2F1[-7/4, -1/4 - m, -3/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(2 + m))/(7*a^2*c^3*f*(g*Cos[e + f*x])^(7/2))","A",4,4,38,0.1053,1,"{2840, 2689, 70, 69}"
159,1,114,0,0.3569737,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2),x]","-\frac{a^3 c^2 2^{m+\frac{9}{4}} \sec (e+f x) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{15/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{15}{4},-m-\frac{1}{4};\frac{19}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{15 f g^6}","-\frac{a^3 c^2 2^{m+\frac{9}{4}} \sec (e+f x) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{15/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{15}{4},-m-\frac{1}{4};\frac{19}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{15 f g^6}",1,"-(2^(9/4 + m)*a^3*c^2*(g*Cos[e + f*x])^(15/2)*Hypergeometric2F1[15/4, -1/4 - m, 19/4, (1 - Sin[e + f*x])/2]*Sec[e + f*x]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(-3 + m)*Sqrt[c - c*Sin[e + f*x]])/(15*f*g^6)","A",4,4,40,0.1000,1,"{2853, 2689, 70, 69}"
160,1,112,0,0.3506611,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{a^2 c 2^{m+\frac{9}{4}} \sec (e+f x) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{11/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(\frac{11}{4},-m-\frac{1}{4};\frac{15}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{11 f g^4}","-\frac{a^2 c 2^{m+\frac{9}{4}} \sec (e+f x) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{11/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(\frac{11}{4},-m-\frac{1}{4};\frac{15}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{11 f g^4}",1,"-(2^(9/4 + m)*a^2*c*(g*Cos[e + f*x])^(11/2)*Hypergeometric2F1[11/4, -1/4 - m, 15/4, (1 - Sin[e + f*x])/2]*Sec[e + f*x]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(-2 + m)*Sqrt[c - c*Sin[e + f*x]])/(11*f*g^4)","A",4,4,40,0.1000,1,"{2853, 2689, 70, 69}"
161,1,109,0,0.3187755,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a 2^{m+\frac{9}{4}} \sec (e+f x) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{7/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{7}{4},-m-\frac{1}{4};\frac{11}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{7 f g^2}","-\frac{a 2^{m+\frac{9}{4}} \sec (e+f x) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{7/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{7}{4},-m-\frac{1}{4};\frac{11}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{7 f g^2}",1,"-(2^(9/4 + m)*a*(g*Cos[e + f*x])^(7/2)*Hypergeometric2F1[7/4, -1/4 - m, 11/4, (1 - Sin[e + f*x])/2]*Sec[e + f*x]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(-1 + m)*Sqrt[c - c*Sin[e + f*x]])/(7*f*g^2)","A",4,4,40,0.1000,1,"{2853, 2689, 70, 69}"
162,1,106,0,0.3257168,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a 2^{m+\frac{9}{4}} \cos (e+f x) (g \cos (e+f x))^{3/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{3}{4},-m-\frac{1}{4};\frac{7}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f \sqrt{c-c \sin (e+f x)}}","-\frac{a 2^{m+\frac{9}{4}} \cos (e+f x) (g \cos (e+f x))^{3/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{3}{4},-m-\frac{1}{4};\frac{7}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f \sqrt{c-c \sin (e+f x)}}",1,"-(2^(9/4 + m)*a*Cos[e + f*x]*(g*Cos[e + f*x])^(3/2)*Hypergeometric2F1[3/4, -1/4 - m, 7/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(-1 + m))/(3*f*Sqrt[c - c*Sin[e + f*x]])","A",4,4,40,0.1000,1,"{2853, 2689, 70, 69}"
163,1,106,0,0.3665579,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(3/2),x]","\frac{g^2 2^{m+\frac{9}{4}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^m \, _2F_1\left(-\frac{1}{4},-m-\frac{1}{4};\frac{3}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{c f \sqrt{c-c \sin (e+f x)} \sqrt{g \cos (e+f x)}}","\frac{g^2 2^{m+\frac{9}{4}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^m \, _2F_1\left(-\frac{1}{4},-m-\frac{1}{4};\frac{3}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{c f \sqrt{c-c \sin (e+f x)} \sqrt{g \cos (e+f x)}}",1,"(2^(9/4 + m)*g^2*Cos[e + f*x]*Hypergeometric2F1[-1/4, -1/4 - m, 3/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^m)/(c*f*Sqrt[g*Cos[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])","A",4,4,40,0.1000,1,"{2853, 2689, 70, 69}"
164,1,114,0,0.3572186,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(5/2),x]","\frac{g^4 2^{m+\frac{9}{4}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(-\frac{5}{4},-m-\frac{1}{4};-\frac{1}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{5 a c^2 f \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}","\frac{g^4 2^{m+\frac{9}{4}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(-\frac{5}{4},-m-\frac{1}{4};-\frac{1}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{5 a c^2 f \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}",1,"(2^(9/4 + m)*g^4*Cos[e + f*x]*Hypergeometric2F1[-5/4, -1/4 - m, -1/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(1 + m))/(5*a*c^2*f*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])","A",4,4,40,0.1000,1,"{2853, 2689, 70, 69}"
165,1,106,0,0.3109177,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a 2^{m+\frac{9}{4}} \cos (e+f x) (g \cos (e+f x))^{3/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{3}{4},-m-\frac{1}{4};\frac{7}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f \sqrt{c-c \sin (e+f x)}}","-\frac{a 2^{m+\frac{9}{4}} \cos (e+f x) (g \cos (e+f x))^{3/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{3}{4},-m-\frac{1}{4};\frac{7}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f \sqrt{c-c \sin (e+f x)}}",1,"-(2^(9/4 + m)*a*Cos[e + f*x]*(g*Cos[e + f*x])^(3/2)*Hypergeometric2F1[3/4, -1/4 - m, 7/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a + a*Sin[e + f*x])^(-1 + m))/(3*f*Sqrt[c - c*Sin[e + f*x]])","A",4,4,40,0.1000,1,"{2853, 2689, 70, 69}"
166,1,106,0,0.3132941,"\int \frac{(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt{a-a \sin (e+f x)}} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(c + c*Sin[e + f*x])^m)/Sqrt[a - a*Sin[e + f*x]],x]","-\frac{c 2^{m+\frac{9}{4}} \cos (e+f x) (g \cos (e+f x))^{3/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (c \sin (e+f x)+c)^{m-1} \, _2F_1\left(\frac{3}{4},-m-\frac{1}{4};\frac{7}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f \sqrt{a-a \sin (e+f x)}}","-\frac{c 2^{m+\frac{9}{4}} \cos (e+f x) (g \cos (e+f x))^{3/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (c \sin (e+f x)+c)^{m-1} \, _2F_1\left(\frac{3}{4},-m-\frac{1}{4};\frac{7}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f \sqrt{a-a \sin (e+f x)}}",1,"-(2^(9/4 + m)*c*Cos[e + f*x]*(g*Cos[e + f*x])^(3/2)*Hypergeometric2F1[3/4, -1/4 - m, 7/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(c + c*Sin[e + f*x])^(-1 + m))/(3*f*Sqrt[a - a*Sin[e + f*x]])","A",4,4,40,0.1000,1,"{2853, 2689, 70, 69}"
167,1,123,0,0.3696045,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m),x]","\frac{2^{-m-\frac{3}{4}} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m+5),\frac{1}{4} (4 m+11);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{c^2 f g (4 m+5)}","\frac{2^{-m-\frac{3}{4}} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m+5),\frac{1}{4} (4 m+11);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{c^2 f g (4 m+5)}",1,"(2^(-3/4 - m)*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(5 + 4*m)/4, (11 + 4*m)/4, (9 + 4*m)/4, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(-1/4 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^2*f*g*(5 + 4*m))","A",4,4,42,0.09524,1,"{2853, 2689, 70, 69}"
168,1,123,0,0.3666302,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m),x]","\frac{2^{\frac{1}{4}-m} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m+5),\frac{1}{4} (4 m+7);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{c f g (4 m+5)}","\frac{2^{\frac{1}{4}-m} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m+5),\frac{1}{4} (4 m+7);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{c f g (4 m+5)}",1,"(2^(1/4 - m)*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(5 + 4*m)/4, (7 + 4*m)/4, (9 + 4*m)/4, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(-1/4 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c*f*g*(5 + 4*m))","A",4,4,42,0.09524,1,"{2853, 2689, 70, 69}"
169,1,120,0,0.3592506,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m),x]","\frac{2^{\frac{5}{4}-m} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m+3),\frac{1}{4} (4 m+5);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (4 m+5)}","\frac{2^{\frac{5}{4}-m} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m+3),\frac{1}{4} (4 m+5);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (4 m+5)}",1,"(2^(5/4 - m)*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(3 + 4*m)/4, (5 + 4*m)/4, (9 + 4*m)/4, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(-1/4 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*g*(5 + 4*m))","A",4,4,42,0.09524,1,"{2853, 2689, 70, 69}"
170,1,121,0,0.3107354,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-m} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^m,x]","\frac{c 2^{\frac{9}{4}-m} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m-1),\frac{1}{4} (4 m+5);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (4 m+5)}","\frac{c 2^{\frac{9}{4}-m} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m-1),\frac{1}{4} (4 m+5);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (4 m+5)}",1,"(2^(9/4 - m)*c*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(-1 + 4*m)/4, (5 + 4*m)/4, (9 + 4*m)/4, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(-1/4 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*g*(5 + 4*m))","A",4,4,40,0.1000,1,"{2853, 2689, 70, 69}"
171,1,123,0,0.3410854,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 - m),x]","\frac{c^2 2^{\frac{13}{4}-m} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m-5),\frac{1}{4} (4 m+5);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (4 m+5)}","\frac{c^2 2^{\frac{13}{4}-m} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m-5),\frac{1}{4} (4 m+5);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (4 m+5)}",1,"(2^(13/4 - m)*c^2*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(-5 + 4*m)/4, (5 + 4*m)/4, (9 + 4*m)/4, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(-1/4 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*g*(5 + 4*m))","A",4,4,42,0.09524,1,"{2853, 2689, 70, 69}"
172,1,123,0,0.3534968,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2-m} \, dx","Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(2 - m),x]","\frac{c^3 2^{\frac{17}{4}-m} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m-9),\frac{1}{4} (4 m+5);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (4 m+5)}","\frac{c^3 2^{\frac{17}{4}-m} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left(\frac{1}{4} (4 m-9),\frac{1}{4} (4 m+5);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (4 m+5)}",1,"(2^(17/4 - m)*c^3*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(-9 + 4*m)/4, (5 + 4*m)/4, (9 + 4*m)/4, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^(-1/4 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*g*(5 + 4*m))","A",4,4,42,0.09524,1,"{2853, 2689, 70, 69}"
173,1,135,0,0.2865583,"\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{c 2^{n+\frac{p}{2}+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{p+1} (1-\sin (e+f x))^{\frac{1}{2} (-2 n-p+1)} \, _2F_1\left(\frac{1}{2} (-2 n-p+1),\frac{1}{2} (2 m+p+1);\frac{1}{2} (2 m+p+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (2 m+p+1)}","\frac{c 2^{n+\frac{p}{2}+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{p+1} (1-\sin (e+f x))^{\frac{1}{2} (-2 n-p+1)} \, _2F_1\left(\frac{1}{2} (-2 n-p+1),\frac{1}{2} (2 m+p+1);\frac{1}{2} (2 m+p+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (2 m+p+1)}",1,"(2^(1/2 + n + p/2)*c*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1 - 2*n - p)/2, (1 + 2*m + p)/2, (3 + 2*m + p)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^((1 - 2*n - p)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(f*g*(1 + 2*m + p))","A",4,4,36,0.1111,1,"{2853, 2689, 70, 69}"
174,1,57,0,0.227553,"\int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+m} \, dx","Int[(g*Cos[e + f*x])^(1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + m),x]","-\frac{g \log (1-\sin (e+f x)) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m}}{c f}","-\frac{g \log (1-\sin (e+f x)) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m}}{c f}",1,"-((g*Log[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^m)/(c*f*(g*Cos[e + f*x])^(2*m)))","A",4,4,42,0.09524,1,"{2843, 12, 2667, 31}"
175,1,203,0,0.6783717,"\int (g \cos (e+f x))^{5-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^(5 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","-\frac{8 a^3 (a \sin (e+f x)+a)^{m-3} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+3) (-m+n+4) (-m+n+5)}-\frac{4 a^2 (a \sin (e+f x)+a)^{m-2} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+4) (-m+n+5)}-\frac{a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+5)}","-\frac{8 a^3 (a \sin (e+f x)+a)^{m-3} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+3) (-m+n+4) (-m+n+5)}-\frac{4 a^2 (a \sin (e+f x)+a)^{m-2} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+4) (-m+n+5)}-\frac{a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{6-2 m}}{f g (-m+n+5)}",1,"(-8*a^3*(g*Cos[e + f*x])^(6 - 2*m)*(a + a*Sin[e + f*x])^(-3 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(3 - m + n)*(4 - m + n)*(5 - m + n)) - (4*a^2*(g*Cos[e + f*x])^(6 - 2*m)*(a + a*Sin[e + f*x])^(-2 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(4 - m + n)*(5 - m + n)) - (a*(g*Cos[e + f*x])^(6 - 2*m)*(a + a*Sin[e + f*x])^(-1 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(5 - m + n))","A",3,2,40,0.05000,1,"{2846, 2844}"
176,1,127,0,0.4076575,"\int (g \cos (e+f x))^{3-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^(3 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","-\frac{2 a^2 (a \sin (e+f x)+a)^{m-2} (c-c \sin (e+f x))^n (g \cos (e+f x))^{4-2 m}}{f g (-m+n+2) (-m+n+3)}-\frac{a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{4-2 m}}{f g (-m+n+3)}","-\frac{2 a^2 (a \sin (e+f x)+a)^{m-2} (c-c \sin (e+f x))^n (g \cos (e+f x))^{4-2 m}}{f g (-m+n+2) (-m+n+3)}-\frac{a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{4-2 m}}{f g (-m+n+3)}",1,"(-2*a^2*(g*Cos[e + f*x])^(4 - 2*m)*(a + a*Sin[e + f*x])^(-2 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(2 - m + n)*(3 - m + n)) - (a*(g*Cos[e + f*x])^(4 - 2*m)*(a + a*Sin[e + f*x])^(-1 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(3 - m + n))","A",2,2,40,0.05000,1,"{2846, 2844}"
177,1,58,0,0.163097,"\int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^(1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","-\frac{a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{2-2 m}}{f g (-m+n+1)}","-\frac{a (a \sin (e+f x)+a)^{m-1} (c-c \sin (e+f x))^n (g \cos (e+f x))^{2-2 m}}{f g (-m+n+1)}",1,"-((a*(g*Cos[e + f*x])^(2 - 2*m)*(a + a*Sin[e + f*x])^(-1 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(1 - m + n)))","A",1,1,40,0.02500,1,"{2844}"
178,1,81,0,0.2301534,"\int (g \cos (e+f x))^{-1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^(-1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-2 m} \, _2F_1\left(1,n-m;-m+n+1;\frac{1}{2} (1-\sin (e+f x))\right)}{2 f g (m-n)}","\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-2 m} \, _2F_1\left(1,n-m;-m+n+1;\frac{1}{2} (1-\sin (e+f x))\right)}{2 f g (m-n)}",1,"(Hypergeometric2F1[1, -m + n, 1 - m + n, (1 - Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(2*f*g*(m - n)*(g*Cos[e + f*x])^(2*m))","A",4,4,40,0.1000,1,"{2853, 12, 2667, 68}"
179,1,85,0,0.2440951,"\int (g \cos (e+f x))^{-3-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^(-3 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{c (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-2 m} \, _2F_1\left(2,-m+n-1;n-m;\frac{1}{2} (1-\sin (e+f x))\right)}{4 f g^3 (m-n+1)}","\frac{c (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-2 m} \, _2F_1\left(2,-m+n-1;n-m;\frac{1}{2} (1-\sin (e+f x))\right)}{4 f g^3 (m-n+1)}",1,"(c*Hypergeometric2F1[2, -1 - m + n, -m + n, (1 - Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(4*f*g^3*(1 + m - n)*(g*Cos[e + f*x])^(2*m))","A",4,4,40,0.1000,1,"{2853, 12, 2667, 68}"
180,1,88,0,0.2442163,"\int (g \cos (e+f x))^{-5-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^(-5 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{c^2 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-2 m} \, _2F_1\left(3,-m+n-2;-m+n-1;\frac{1}{2} (1-\sin (e+f x))\right)}{8 f g^5 (m-n+2)}","\frac{c^2 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-2 m} \, _2F_1\left(3,-m+n-2;-m+n-1;\frac{1}{2} (1-\sin (e+f x))\right)}{8 f g^5 (m-n+2)}",1,"(c^2*Hypergeometric2F1[3, -2 - m + n, -1 - m + n, (1 - Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 + n))/(8*f*g^5*(2 + m - n)*(g*Cos[e + f*x])^(2*m))","A",4,4,40,0.1000,1,"{2853, 12, 2667, 68}"
181,1,51,0,0.1720965,"\int (g \cos (e+f x))^{-1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m \, dx","Int[(g*Cos[e + f*x])^(-1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^m,x]","\frac{\tanh ^{-1}(\sin (e+f x)) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m}}{f g}","\frac{\tanh ^{-1}(\sin (e+f x)) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m}}{f g}",1,"(ArcTanh[Sin[e + f*x]]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^m)/(f*g*(g*Cos[e + f*x])^(2*m))","A",3,3,40,0.07500,1,"{2847, 12, 3770}"
182,1,134,0,0.3824414,"\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3+n} \, dx","Int[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3 + n),x]","\frac{c^3 2^{-\frac{m}{2}+\frac{n}{2}+3} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (1-\sin (e+f x))^{\frac{m-n}{2}} (g \cos (e+f x))^{-m-n} \, _2F_1\left(\frac{1}{2} (m-n-4),\frac{m-n}{2};\frac{1}{2} (m-n+2);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (m-n)}","\frac{c^3 2^{-\frac{m}{2}+\frac{n}{2}+3} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (1-\sin (e+f x))^{\frac{m-n}{2}} (g \cos (e+f x))^{-m-n} \, _2F_1\left(\frac{1}{2} (m-n-4),\frac{m-n}{2};\frac{1}{2} (m-n+2);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (m-n)}",1,"(2^(3 - m/2 + n/2)*c^3*(g*Cos[e + f*x])^(-m - n)*Hypergeometric2F1[(-4 + m - n)/2, (m - n)/2, (2 + m - n)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^((m - n)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*g*(m - n))","A",4,4,45,0.08889,1,"{2853, 2689, 70, 69}"
183,1,134,0,0.3737654,"\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2+n} \, dx","Int[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(2 + n),x]","\frac{c^2 2^{-\frac{m}{2}+\frac{n}{2}+2} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (1-\sin (e+f x))^{\frac{m-n}{2}} (g \cos (e+f x))^{-m-n} \, _2F_1\left(\frac{1}{2} (m-n-2),\frac{m-n}{2};\frac{1}{2} (m-n+2);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (m-n)}","\frac{c^2 2^{-\frac{m}{2}+\frac{n}{2}+2} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (1-\sin (e+f x))^{\frac{m-n}{2}} (g \cos (e+f x))^{-m-n} \, _2F_1\left(\frac{1}{2} (m-n-2),\frac{m-n}{2};\frac{1}{2} (m-n+2);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (m-n)}",1,"(2^(2 - m/2 + n/2)*c^2*(g*Cos[e + f*x])^(-m - n)*Hypergeometric2F1[(-2 + m - n)/2, (m - n)/2, (2 + m - n)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^((m - n)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*g*(m - n))","A",4,4,45,0.08889,1,"{2853, 2689, 70, 69}"
184,1,131,0,0.3693697,"\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1+n} \, dx","Int[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 + n),x]","\frac{c 2^{-\frac{m}{2}+\frac{n}{2}+1} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (1-\sin (e+f x))^{\frac{m-n}{2}} (g \cos (e+f x))^{-m-n} \, _2F_1\left(\frac{m-n}{2},\frac{m-n}{2};\frac{1}{2} (m-n+2);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (m-n)}","\frac{c 2^{-\frac{m}{2}+\frac{n}{2}+1} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (1-\sin (e+f x))^{\frac{m-n}{2}} (g \cos (e+f x))^{-m-n} \, _2F_1\left(\frac{m-n}{2},\frac{m-n}{2};\frac{1}{2} (m-n+2);\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (m-n)}",1,"(2^(1 - m/2 + n/2)*c*(g*Cos[e + f*x])^(-m - n)*Hypergeometric2F1[(m - n)/2, (m - n)/2, (2 + m - n)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^((m - n)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*g*(m - n))","A",4,4,45,0.08889,1,"{2853, 2689, 70, 69}"
185,1,55,0,0.1712022,"\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{f g (m-n)}","\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{f g (m-n)}",1,"((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*g*(m - n))","A",1,1,43,0.02326,1,"{2848}"
186,1,125,0,0.4106864,"\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n} \, dx","Int[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n),x]","\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{f g (m-n+2)}+\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c f g (m-n) (m-n+2)}","\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{f g (m-n+2)}+\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c f g (m-n) (m-n+2)}",1,"((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(f*g*(2 + m - n)) + ((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(c*f*g*(m - n)*(2 + m - n))","A",2,2,45,0.04444,1,"{2849, 2848}"
187,1,204,0,0.6653823,"\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2+n} \, dx","Int[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 + n),x]","\frac{2 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c^2 f g (m-n) (m-n+2) (m-n+4)}+\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}+\frac{2 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{c f g (m-n+2) (m-n+4)}","\frac{2 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c^2 f g (m-n) (m-n+2) (m-n+4)}+\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{f g (m-n+4)}+\frac{2 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{c f g (m-n+2) (m-n+4)}",1,"((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 + n))/(f*g*(4 + m - n)) + (2*(g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(c*f*g*(2 + m - n)*(4 + m - n)) + (2*(g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(c^2*f*g*(m - n)*(2 + m - n)*(4 + m - n))","A",3,2,45,0.04444,1,"{2849, 2848}"
188,1,290,0,0.9372378,"\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3+n} \, dx","Int[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 + n),x]","\frac{6 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{c^2 f g (m-n+2) (m-n+4) (m-n+6)}+\frac{6 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c^3 f g (m-n) (m-n+2) (m-n+4) (m-n+6)}+\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}+\frac{3 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{c f g (m-n+4) (m-n+6)}","\frac{6 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{-m-n}}{c^2 f g (m-n+2) (m-n+4) (m-n+6)}+\frac{6 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n}}{c^3 f g (m-n) (m-n+2) (m-n+4) (m-n+6)}+\frac{(a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-3} (g \cos (e+f x))^{-m-n}}{f g (m-n+6)}+\frac{3 (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} (g \cos (e+f x))^{-m-n}}{c f g (m-n+4) (m-n+6)}",1,"((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 + n))/(f*g*(6 + m - n)) + (3*(g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 + n))/(c*f*g*(4 + m - n)*(6 + m - n)) + (6*(g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(c^2*f*g*(2 + m - n)*(4 + m - n)*(6 + m - n)) + (6*(g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(c^3*f*g*(m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n))","A",4,2,45,0.04444,1,"{2849, 2848}"
189,1,138,0,0.4602266,"\int (g \sec (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Int[(g*Sec[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{c 2^{n-\frac{p}{2}+\frac{1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \sec (e+f x))^p (1-\sin (e+f x))^{\frac{1}{2} (-2 n+p+1)} \, _2F_1\left(\frac{1}{2} (2 m-p+1),\frac{1}{2} (-2 n+p+1);\frac{1}{2} (2 m-p+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m-p+1)}","\frac{c 2^{n-\frac{p}{2}+\frac{1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \sec (e+f x))^p (1-\sin (e+f x))^{\frac{1}{2} (-2 n+p+1)} \, _2F_1\left(\frac{1}{2} (2 m-p+1),\frac{1}{2} (-2 n+p+1);\frac{1}{2} (2 m-p+3);\frac{1}{2} (\sin (e+f x)+1)\right)}{f (2 m-p+1)}",1,"(2^(1/2 + n - p/2)*c*Cos[e + f*x]*Hypergeometric2F1[(1 + 2*m - p)/2, (1 - 2*n + p)/2, (3 + 2*m - p)/2, (1 + Sin[e + f*x])/2]*(g*Sec[e + f*x])^p*(1 - Sin[e + f*x])^((1 - 2*n + p)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(f*(1 + 2*m - p))","A",5,5,36,0.1389,1,"{2926, 2853, 2689, 70, 69}"
190,1,33,0,0.0454494,"\int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{3 d}","\frac{a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{3 d}",1,"(a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^4)/(4*d)","A",4,3,25,0.1200,1,"{2833, 12, 43}"
191,1,33,0,0.0358782,"\int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^2(c+d x)}{2 d}","\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^2(c+d x)}{2 d}",1,"(a*Sin[c + d*x]^2)/(2*d) + (a*Sin[c + d*x]^3)/(3*d)","A",4,3,23,0.1304,1,"{2833, 12, 43}"
192,1,24,0,0.0218486,"\int \cot (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}","\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"(a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d","A",3,2,17,0.1176,1,"{2707, 43}"
193,1,25,0,0.0360492,"\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \log (\sin (c+d x))}{d}-\frac{a \csc (c+d x)}{d}","\frac{a \log (\sin (c+d x))}{d}-\frac{a \csc (c+d x)}{d}",1,"-((a*Csc[c + d*x])/d) + (a*Log[Sin[c + d*x]])/d","A",4,3,23,0.1304,1,"{2833, 12, 43}"
194,1,30,0,0.0417927,"\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^2}{2 a d}","-\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^2}{2 a d}",1,"-(Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2)/(2*a*d)","A",3,3,25,0.1200,1,"{2833, 12, 37}"
195,1,33,0,0.0427132,"\int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}","-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}",1,"-(a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d)","A",4,3,25,0.1200,1,"{2833, 12, 43}"
196,1,33,0,0.0426549,"\int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{3 d}","-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{3 d}",1,"-(a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d)","A",4,3,25,0.1200,1,"{2833, 12, 43}"
197,1,55,0,0.0668741,"\int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^5(c+d x)}{5 d}+\frac{a^2 \sin ^4(c+d x)}{2 d}+\frac{a^2 \sin ^3(c+d x)}{3 d}","\frac{a^2 \sin ^5(c+d x)}{5 d}+\frac{a^2 \sin ^4(c+d x)}{2 d}+\frac{a^2 \sin ^3(c+d x)}{3 d}",1,"(a^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^4)/(2*d) + (a^2*Sin[c + d*x]^5)/(5*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
198,1,55,0,0.0482555,"\int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^4(c+d x)}{4 d}+\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin ^2(c+d x)}{2 d}","\frac{a^2 \sin ^4(c+d x)}{4 d}+\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin ^2(c+d x)}{2 d}",1,"(a^2*Sin[c + d*x]^2)/(2*d) + (2*a^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^4)/(4*d)","A",4,3,25,0.1200,1,"{2833, 12, 43}"
199,1,47,0,0.0385452,"\int \cot (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \log (\sin (c+d x))}{d}","\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \log (\sin (c+d x))}{d}",1,"(a^2*Log[Sin[c + d*x]])/d + (2*a^2*Sin[c + d*x])/d + (a^2*Sin[c + d*x]^2)/(2*d)","A",3,2,19,0.1053,1,"{2707, 43}"
200,1,43,0,0.0546732,"\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d}","\frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d}",1,"-((a^2*Csc[c + d*x])/d) + (2*a^2*Log[Sin[c + d*x]])/d + (a^2*Sin[c + d*x])/d","A",4,3,25,0.1200,1,"{2833, 12, 43}"
201,1,47,0,0.0648706,"\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{a^2 \log (\sin (c+d x))}{d}","-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{a^2 \log (\sin (c+d x))}{d}",1,"(-2*a^2*Csc[c + d*x])/d - (a^2*Csc[c + d*x]^2)/(2*d) + (a^2*Log[Sin[c + d*x]])/d","A",4,3,27,0.1111,1,"{2833, 12, 43}"
202,1,30,0,0.0581122,"\int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^3(c+d x) (a \sin (c+d x)+a)^3}{3 a d}","-\frac{\csc ^3(c+d x) (a \sin (c+d x)+a)^3}{3 a d}",1,"-(Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(3*a*d)","A",3,3,27,0.1111,1,"{2833, 12, 37}"
203,1,55,0,0.0663529,"\int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc ^4(c+d x)}{4 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{2 d}","-\frac{a^2 \csc ^4(c+d x)}{4 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{2 d}",1,"-(a^2*Csc[c + d*x]^2)/(2*d) - (2*a^2*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
204,1,55,0,0.0666974,"\int \cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc ^5(c+d x)}{5 d}-\frac{a^2 \csc ^4(c+d x)}{2 d}-\frac{a^2 \csc ^3(c+d x)}{3 d}","-\frac{a^2 \csc ^5(c+d x)}{5 d}-\frac{a^2 \csc ^4(c+d x)}{2 d}-\frac{a^2 \csc ^3(c+d x)}{3 d}",1,"-(a^2*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(2*d) - (a^2*Csc[c + d*x]^5)/(5*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
205,1,55,0,0.0660291,"\int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc ^6(c+d x)}{6 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{a^2 \csc ^4(c+d x)}{4 d}","-\frac{a^2 \csc ^6(c+d x)}{6 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{a^2 \csc ^4(c+d x)}{4 d}",1,"-(a^2*Csc[c + d*x]^4)/(4*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) - (a^2*Csc[c + d*x]^6)/(6*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
206,1,73,0,0.0738102,"\int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^7(c+d x)}{7 d}+\frac{a^3 \sin ^6(c+d x)}{2 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{a^3 \sin ^4(c+d x)}{4 d}","\frac{a^3 \sin ^7(c+d x)}{7 d}+\frac{a^3 \sin ^6(c+d x)}{2 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{a^3 \sin ^4(c+d x)}{4 d}",1,"(a^3*Sin[c + d*x]^4)/(4*d) + (3*a^3*Sin[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x]^6)/(2*d) + (a^3*Sin[c + d*x]^7)/(7*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
207,1,73,0,0.0706032,"\int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^6(c+d x)}{6 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{3 a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{3 d}","\frac{a^3 \sin ^6(c+d x)}{6 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{3 a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{3 d}",1,"(a^3*Sin[c + d*x]^3)/(3*d) + (3*a^3*Sin[c + d*x]^4)/(4*d) + (3*a^3*Sin[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x]^6)/(6*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
208,1,45,0,0.0453911,"\int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{(a \sin (c+d x)+a)^5}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^4}{4 a d}","\frac{(a \sin (c+d x)+a)^5}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^4}{4 a d}",1,"-(a + a*Sin[c + d*x])^4/(4*a*d) + (a + a*Sin[c + d*x])^5/(5*a^2*d)","A",4,3,25,0.1200,1,"{2833, 12, 43}"
209,1,65,0,0.0438446,"\int \cot (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}","\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}",1,"(a^3*Log[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (3*a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/(3*d)","A",3,2,19,0.1053,1,"{2707, 43}"
210,1,62,0,0.0618724,"\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d}","\frac{a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d}",1,"-((a^3*Csc[c + d*x])/d) + (3*a^3*Log[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (a^3*Sin[c + d*x]^2)/(2*d)","A",4,3,25,0.1200,1,"{2833, 12, 43}"
211,1,61,0,0.0691863,"\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d}","\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d}",1,"(-3*a^3*Csc[c + d*x])/d - (a^3*Csc[c + d*x]^2)/(2*d) + (3*a^3*Log[Sin[c + d*x]])/d + (a^3*Sin[c + d*x])/d","A",4,3,27,0.1111,1,"{2833, 12, 43}"
212,1,65,0,0.0707239,"\int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}","-\frac{a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}",1,"(-3*a^3*Csc[c + d*x])/d - (3*a^3*Csc[c + d*x]^2)/(2*d) - (a^3*Csc[c + d*x]^3)/(3*d) + (a^3*Log[Sin[c + d*x]])/d","A",4,3,27,0.1111,1,"{2833, 12, 43}"
213,1,30,0,0.0568223,"\int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{4 a d}","-\frac{\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{4 a d}",1,"-(Csc[c + d*x]^4*(a + a*Sin[c + d*x])^4)/(4*a*d)","A",3,3,27,0.1111,1,"{2833, 12, 37}"
214,1,61,0,0.0649166,"\int \cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{20 a d}-\frac{\csc ^5(c+d x) (a \sin (c+d x)+a)^4}{5 a d}","\frac{\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{20 a d}-\frac{\csc ^5(c+d x) (a \sin (c+d x)+a)^4}{5 a d}",1,"(Csc[c + d*x]^4*(a + a*Sin[c + d*x])^4)/(20*a*d) - (Csc[c + d*x]^5*(a + a*Sin[c + d*x])^4)/(5*a*d)","A",4,4,27,0.1481,1,"{2833, 12, 45, 37}"
215,1,73,0,0.0719408,"\int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \csc ^6(c+d x)}{6 d}-\frac{3 a^3 \csc ^5(c+d x)}{5 d}-\frac{3 a^3 \csc ^4(c+d x)}{4 d}-\frac{a^3 \csc ^3(c+d x)}{3 d}","-\frac{a^3 \csc ^6(c+d x)}{6 d}-\frac{3 a^3 \csc ^5(c+d x)}{5 d}-\frac{3 a^3 \csc ^4(c+d x)}{4 d}-\frac{a^3 \csc ^3(c+d x)}{3 d}",1,"-(a^3*Csc[c + d*x]^3)/(3*d) - (3*a^3*Csc[c + d*x]^4)/(4*d) - (3*a^3*Csc[c + d*x]^5)/(5*d) - (a^3*Csc[c + d*x]^6)/(6*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
216,1,73,0,0.0728913,"\int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \csc ^7(c+d x)}{7 d}-\frac{a^3 \csc ^6(c+d x)}{2 d}-\frac{3 a^3 \csc ^5(c+d x)}{5 d}-\frac{a^3 \csc ^4(c+d x)}{4 d}","-\frac{a^3 \csc ^7(c+d x)}{7 d}-\frac{a^3 \csc ^6(c+d x)}{2 d}-\frac{3 a^3 \csc ^5(c+d x)}{5 d}-\frac{a^3 \csc ^4(c+d x)}{4 d}",1,"-(a^3*Csc[c + d*x]^4)/(4*d) - (3*a^3*Csc[c + d*x]^5)/(5*d) - (a^3*Csc[c + d*x]^6)/(2*d) - (a^3*Csc[c + d*x]^7)/(7*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
217,1,91,0,0.0849619,"\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \sin ^9(c+d x)}{9 d}+\frac{a^4 \sin ^8(c+d x)}{2 d}+\frac{6 a^4 \sin ^7(c+d x)}{7 d}+\frac{2 a^4 \sin ^6(c+d x)}{3 d}+\frac{a^4 \sin ^5(c+d x)}{5 d}","\frac{a^4 \sin ^9(c+d x)}{9 d}+\frac{a^4 \sin ^8(c+d x)}{2 d}+\frac{6 a^4 \sin ^7(c+d x)}{7 d}+\frac{2 a^4 \sin ^6(c+d x)}{3 d}+\frac{a^4 \sin ^5(c+d x)}{5 d}",1,"(a^4*Sin[c + d*x]^5)/(5*d) + (2*a^4*Sin[c + d*x]^6)/(3*d) + (6*a^4*Sin[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x]^8)/(2*d) + (a^4*Sin[c + d*x]^9)/(9*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
218,1,88,0,0.0809407,"\int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \sin ^8(c+d x)}{8 d}+\frac{4 a^4 \sin ^7(c+d x)}{7 d}+\frac{a^4 \sin ^6(c+d x)}{d}+\frac{4 a^4 \sin ^5(c+d x)}{5 d}+\frac{a^4 \sin ^4(c+d x)}{4 d}","\frac{a^4 \sin ^8(c+d x)}{8 d}+\frac{4 a^4 \sin ^7(c+d x)}{7 d}+\frac{a^4 \sin ^6(c+d x)}{d}+\frac{4 a^4 \sin ^5(c+d x)}{5 d}+\frac{a^4 \sin ^4(c+d x)}{4 d}",1,"(a^4*Sin[c + d*x]^4)/(4*d) + (4*a^4*Sin[c + d*x]^5)/(5*d) + (a^4*Sin[c + d*x]^6)/d + (4*a^4*Sin[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x]^8)/(8*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
219,1,67,0,0.0753,"\int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","\frac{(a \sin (c+d x)+a)^7}{7 a^3 d}-\frac{(a \sin (c+d x)+a)^6}{3 a^2 d}+\frac{(a \sin (c+d x)+a)^5}{5 a d}","\frac{(a \sin (c+d x)+a)^7}{7 a^3 d}-\frac{(a \sin (c+d x)+a)^6}{3 a^2 d}+\frac{(a \sin (c+d x)+a)^5}{5 a d}",1,"(a + a*Sin[c + d*x])^5/(5*a*d) - (a + a*Sin[c + d*x])^6/(3*a^2*d) + (a + a*Sin[c + d*x])^7/(7*a^3*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
220,1,45,0,0.0448015,"\int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x])^4,x]","\frac{(a \sin (c+d x)+a)^6}{6 a^2 d}-\frac{(a \sin (c+d x)+a)^5}{5 a d}","\frac{(a \sin (c+d x)+a)^6}{6 a^2 d}-\frac{(a \sin (c+d x)+a)^5}{5 a d}",1,"-(a + a*Sin[c + d*x])^5/(5*a*d) + (a + a*Sin[c + d*x])^6/(6*a^2*d)","A",4,3,25,0.1200,1,"{2833, 12, 43}"
221,1,81,0,0.0472288,"\int \cot (c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cot[c + d*x]*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{3 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \log (\sin (c+d x))}{d}","\frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{3 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \log (\sin (c+d x))}{d}",1,"(a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (3*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^4)/(4*d)","A",3,2,19,0.1053,1,"{2707, 43}"
222,1,78,0,0.0662441,"\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{2 a^4 \sin ^2(c+d x)}{d}+\frac{6 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc (c+d x)}{d}+\frac{4 a^4 \log (\sin (c+d x))}{d}","\frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{2 a^4 \sin ^2(c+d x)}{d}+\frac{6 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc (c+d x)}{d}+\frac{4 a^4 \log (\sin (c+d x))}{d}",1,"-((a^4*Csc[c + d*x])/d) + (4*a^4*Log[Sin[c + d*x]])/d + (6*a^4*Sin[c + d*x])/d + (2*a^4*Sin[c + d*x]^2)/d + (a^4*Sin[c + d*x]^3)/(3*d)","A",4,3,25,0.1200,1,"{2833, 12, 43}"
223,1,80,0,0.0761459,"\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \sin ^2(c+d x)}{2 d}+\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^2(c+d x)}{2 d}-\frac{4 a^4 \csc (c+d x)}{d}+\frac{6 a^4 \log (\sin (c+d x))}{d}","\frac{a^4 \sin ^2(c+d x)}{2 d}+\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^2(c+d x)}{2 d}-\frac{4 a^4 \csc (c+d x)}{d}+\frac{6 a^4 \log (\sin (c+d x))}{d}",1,"(-4*a^4*Csc[c + d*x])/d - (a^4*Csc[c + d*x]^2)/(2*d) + (6*a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (a^4*Sin[c + d*x]^2)/(2*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
224,1,85,0,0.0866874,"\int \frac{\cos (c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^4(c+d x)}{4 a d}-\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x)+1)}{a d}","\frac{\sin ^4(c+d x)}{4 a d}-\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x)+1)}{a d}",1,"Log[1 + Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d) + Sin[c + d*x]^2/(2*a*d) - Sin[c + d*x]^3/(3*a*d) + Sin[c + d*x]^4/(4*a*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
225,1,67,0,0.0781027,"\int \frac{\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{2 a d}+\frac{\sin (c+d x)}{a d}-\frac{\log (\sin (c+d x)+1)}{a d}","\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{2 a d}+\frac{\sin (c+d x)}{a d}-\frac{\log (\sin (c+d x)+1)}{a d}",1,"-(Log[1 + Sin[c + d*x]]/(a*d)) + Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d) + Sin[c + d*x]^3/(3*a*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
226,1,49,0,0.0699884,"\int \frac{\cos (c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x)+1)}{a d}","\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x)+1)}{a d}",1,"Log[1 + Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d) + Sin[c + d*x]^2/(2*a*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
227,1,31,0,0.0460874,"\int \frac{\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\sin (c+d x)}{a d}-\frac{\log (\sin (c+d x)+1)}{a d}","\frac{\sin (c+d x)}{a d}-\frac{\log (\sin (c+d x)+1)}{a d}",1,"-(Log[1 + Sin[c + d*x]]/(a*d)) + Sin[c + d*x]/(a*d)","A",4,3,25,0.1200,1,"{2833, 12, 43}"
228,1,32,0,0.0359275,"\int \frac{\cot (c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]/(a + a*Sin[c + d*x]),x]","\frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d}","\frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d}",1,"Log[Sin[c + d*x]]/(a*d) - Log[1 + Sin[c + d*x]]/(a*d)","A",4,4,19,0.2105,1,"{2707, 36, 29, 31}"
229,1,46,0,0.0617029,"\int \frac{\cot (c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{\log (\sin (c+d x)+1)}{a d}","-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{\log (\sin (c+d x)+1)}{a d}",1,"-(Csc[c + d*x]/(a*d)) - Log[Sin[c + d*x]]/(a*d) + Log[1 + Sin[c + d*x]]/(a*d)","A",4,3,25,0.1200,1,"{2833, 12, 44}"
230,1,63,0,0.0786564,"\int \frac{\cot (c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d}","-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{\log (\sin (c+d x)+1)}{a d}",1,"Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) + Log[Sin[c + d*x]]/(a*d) - Log[1 + Sin[c + d*x]]/(a*d)","A",4,3,27,0.1111,1,"{2833, 12, 44}"
231,1,82,0,0.082485,"\int \frac{\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{\log (\sin (c+d x)+1)}{a d}","-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{\log (\sin (c+d x)+1)}{a d}",1,"-(Csc[c + d*x]/(a*d)) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) - Log[Sin[c + d*x]]/(a*d) + Log[1 + Sin[c + d*x]]/(a*d)","A",4,3,27,0.1111,1,"{2833, 12, 44}"
232,1,87,0,0.0885047,"\int \frac{\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^3(c+d x)}{3 a^2 d}-\frac{\sin ^2(c+d x)}{a^2 d}+\frac{3 \sin (c+d x)}{a^2 d}-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{4 \log (\sin (c+d x)+1)}{a^2 d}","\frac{\sin ^3(c+d x)}{3 a^2 d}-\frac{\sin ^2(c+d x)}{a^2 d}+\frac{3 \sin (c+d x)}{a^2 d}-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{4 \log (\sin (c+d x)+1)}{a^2 d}",1,"(-4*Log[1 + Sin[c + d*x]])/(a^2*d) + (3*Sin[c + d*x])/(a^2*d) - Sin[c + d*x]^2/(a^2*d) + Sin[c + d*x]^3/(3*a^2*d) - 1/(d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
233,1,70,0,0.0801053,"\int \frac{\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^2(c+d x)}{2 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{3 \log (\sin (c+d x)+1)}{a^2 d}","\frac{\sin ^2(c+d x)}{2 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{3 \log (\sin (c+d x)+1)}{a^2 d}",1,"(3*Log[1 + Sin[c + d*x]])/(a^2*d) - (2*Sin[c + d*x])/(a^2*d) + Sin[c + d*x]^2/(2*a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
234,1,52,0,0.0725106,"\int \frac{\cos (c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin (c+d x)}{a^2 d}-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{2 \log (\sin (c+d x)+1)}{a^2 d}","\frac{\sin (c+d x)}{a^2 d}-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{2 \log (\sin (c+d x)+1)}{a^2 d}",1,"(-2*Log[1 + Sin[c + d*x]])/(a^2*d) + Sin[c + d*x]/(a^2*d) - 1/(d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
235,1,37,0,0.048023,"\int \frac{\cos (c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\log (\sin (c+d x)+1)}{a^2 d}","\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\log (\sin (c+d x)+1)}{a^2 d}",1,"Log[1 + Sin[c + d*x]]/(a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,25,0.1200,1,"{2833, 12, 43}"
236,1,52,0,0.0485757,"\int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]/(a + a*Sin[c + d*x])^2,x]","\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\log (\sin (c+d x))}{a^2 d}-\frac{\log (\sin (c+d x)+1)}{a^2 d}","\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\log (\sin (c+d x))}{a^2 d}-\frac{\log (\sin (c+d x)+1)}{a^2 d}",1,"Log[Sin[c + d*x]]/(a^2*d) - Log[1 + Sin[c + d*x]]/(a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2707, 44}"
237,1,68,0,0.0741999,"\int \frac{\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\csc (c+d x)}{a^2 d}-\frac{2 \log (\sin (c+d x))}{a^2 d}+\frac{2 \log (\sin (c+d x)+1)}{a^2 d}","-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\csc (c+d x)}{a^2 d}-\frac{2 \log (\sin (c+d x))}{a^2 d}+\frac{2 \log (\sin (c+d x)+1)}{a^2 d}",1,"-(Csc[c + d*x]/(a^2*d)) - (2*Log[Sin[c + d*x]])/(a^2*d) + (2*Log[1 + Sin[c + d*x]])/(a^2*d) - 1/(d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,25,0.1200,1,"{2833, 12, 44}"
238,1,85,0,0.0874987,"\int \frac{\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}+\frac{3 \log (\sin (c+d x))}{a^2 d}-\frac{3 \log (\sin (c+d x)+1)}{a^2 d}","\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}+\frac{3 \log (\sin (c+d x))}{a^2 d}-\frac{3 \log (\sin (c+d x)+1)}{a^2 d}",1,"(2*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a^2*d) + (3*Log[Sin[c + d*x]])/(a^2*d) - (3*Log[1 + Sin[c + d*x]])/(a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 44}"
239,1,101,0,0.0986431,"\int \frac{\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\csc ^3(c+d x)}{3 a^2 d}+\frac{\csc ^2(c+d x)}{a^2 d}-\frac{3 \csc (c+d x)}{a^2 d}-\frac{4 \log (\sin (c+d x))}{a^2 d}+\frac{4 \log (\sin (c+d x)+1)}{a^2 d}","-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\csc ^3(c+d x)}{3 a^2 d}+\frac{\csc ^2(c+d x)}{a^2 d}-\frac{3 \csc (c+d x)}{a^2 d}-\frac{4 \log (\sin (c+d x))}{a^2 d}+\frac{4 \log (\sin (c+d x)+1)}{a^2 d}",1,"(-3*Csc[c + d*x])/(a^2*d) + Csc[c + d*x]^2/(a^2*d) - Csc[c + d*x]^3/(3*a^2*d) - (4*Log[Sin[c + d*x]])/(a^2*d) + (4*Log[1 + Sin[c + d*x]])/(a^2*d) - 1/(d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 44}"
240,1,111,0,0.1064298,"\int \frac{\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]","\frac{\sin ^3(c+d x)}{3 a^3 d}-\frac{3 \sin ^2(c+d x)}{2 a^3 d}+\frac{6 \sin (c+d x)}{a^3 d}-\frac{5}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{10 \log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2}","\frac{\sin ^3(c+d x)}{3 a^3 d}-\frac{3 \sin ^2(c+d x)}{2 a^3 d}+\frac{6 \sin (c+d x)}{a^3 d}-\frac{5}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{10 \log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2}",1,"(-10*Log[1 + Sin[c + d*x]])/(a^3*d) + (6*Sin[c + d*x])/(a^3*d) - (3*Sin[c + d*x]^2)/(2*a^3*d) + Sin[c + d*x]^3/(3*a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) - 5/(d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
241,1,93,0,0.0963241,"\int \frac{\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","\frac{\sin ^2(c+d x)}{2 a^3 d}-\frac{3 \sin (c+d x)}{a^3 d}+\frac{4}{d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{6 \log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2}","\frac{\sin ^2(c+d x)}{2 a^3 d}-\frac{3 \sin (c+d x)}{a^3 d}+\frac{4}{d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{6 \log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2}",1,"(6*Log[1 + Sin[c + d*x]])/(a^3*d) - (3*Sin[c + d*x])/(a^3*d) + Sin[c + d*x]^2/(2*a^3*d) - 1/(2*a*d*(a + a*Sin[c + d*x])^2) + 4/(d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
242,1,74,0,0.0896497,"\int \frac{\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{\sin (c+d x)}{a^3 d}-\frac{3}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{3 \log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2}","\frac{\sin (c+d x)}{a^3 d}-\frac{3}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{3 \log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2}",1,"(-3*Log[1 + Sin[c + d*x]])/(a^3*d) + Sin[c + d*x]/(a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) - 3/(d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
243,1,60,0,0.0805542,"\int \frac{\cos (c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{2}{d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2}","\frac{2}{d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2}",1,"Log[1 + Sin[c + d*x]]/(a^3*d) - 1/(2*a*d*(a + a*Sin[c + d*x])^2) + 2/(d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
244,1,30,0,0.0458973,"\int \frac{\cos (c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{\sin ^2(c+d x)}{2 a d (a \sin (c+d x)+a)^2}","\frac{\sin ^2(c+d x)}{2 a d (a \sin (c+d x)+a)^2}",1,"Sin[c + d*x]^2/(2*a*d*(a + a*Sin[c + d*x])^2)","A",3,3,25,0.1200,1,"{2833, 12, 37}"
245,1,74,0,0.0576459,"\int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]/(a + a*Sin[c + d*x])^3,x]","\frac{1}{d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{\log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2}","\frac{1}{d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{\log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2}",1,"Log[Sin[c + d*x]]/(a^3*d) - Log[1 + Sin[c + d*x]]/(a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) + 1/(d*(a^3 + a^3*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2707, 44}"
246,1,90,0,0.0836685,"\int \frac{\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{2}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\csc (c+d x)}{a^3 d}-\frac{3 \log (\sin (c+d x))}{a^3 d}+\frac{3 \log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2}","-\frac{2}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\csc (c+d x)}{a^3 d}-\frac{3 \log (\sin (c+d x))}{a^3 d}+\frac{3 \log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2}",1,"-(Csc[c + d*x]/(a^3*d)) - (3*Log[Sin[c + d*x]])/(a^3*d) + (3*Log[1 + Sin[c + d*x]])/(a^3*d) - 1/(2*a*d*(a + a*Sin[c + d*x])^2) - 2/(d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,25,0.1200,1,"{2833, 12, 44}"
247,1,108,0,0.1016406,"\int \frac{\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{3}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\csc ^2(c+d x)}{2 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}+\frac{6 \log (\sin (c+d x))}{a^3 d}-\frac{6 \log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2}","\frac{3}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\csc ^2(c+d x)}{2 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}+\frac{6 \log (\sin (c+d x))}{a^3 d}-\frac{6 \log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2}",1,"(3*Csc[c + d*x])/(a^3*d) - Csc[c + d*x]^2/(2*a^3*d) + (6*Log[Sin[c + d*x]])/(a^3*d) - (6*Log[1 + Sin[c + d*x]])/(a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) + 3/(d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 44}"
248,1,126,0,0.1127255,"\int \frac{\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","-\frac{4}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc ^2(c+d x)}{2 a^3 d}-\frac{6 \csc (c+d x)}{a^3 d}-\frac{10 \log (\sin (c+d x))}{a^3 d}+\frac{10 \log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2}","-\frac{4}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc ^2(c+d x)}{2 a^3 d}-\frac{6 \csc (c+d x)}{a^3 d}-\frac{10 \log (\sin (c+d x))}{a^3 d}+\frac{10 \log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2}",1,"(-6*Csc[c + d*x])/(a^3*d) + (3*Csc[c + d*x]^2)/(2*a^3*d) - Csc[c + d*x]^3/(3*a^3*d) - (10*Log[Sin[c + d*x]])/(a^3*d) + (10*Log[1 + Sin[c + d*x]])/(a^3*d) - 1/(2*a*d*(a + a*Sin[c + d*x])^2) - 4/(d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 44}"
249,1,116,0,0.1041873,"\int \frac{\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^4,x]","\frac{\sin ^2(c+d x)}{2 a^4 d}-\frac{4 \sin (c+d x)}{a^4 d}+\frac{10}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{5}{2 d \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{10 \log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3}","\frac{\sin ^2(c+d x)}{2 a^4 d}-\frac{4 \sin (c+d x)}{a^4 d}+\frac{10}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{5}{2 d \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{10 \log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3}",1,"(10*Log[1 + Sin[c + d*x]])/(a^4*d) - (4*Sin[c + d*x])/(a^4*d) + Sin[c + d*x]^2/(2*a^4*d) + 1/(3*a*d*(a + a*Sin[c + d*x])^3) - 5/(2*d*(a^2 + a^2*Sin[c + d*x])^2) + 10/(d*(a^4 + a^4*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
250,1,95,0,0.0965907,"\int \frac{\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^4,x]","\frac{\sin (c+d x)}{a^4 d}-\frac{6}{d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{2}{d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{4 \log (\sin (c+d x)+1)}{a^4 d}-\frac{1}{3 a d (a \sin (c+d x)+a)^3}","\frac{\sin (c+d x)}{a^4 d}-\frac{6}{d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{2}{d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{4 \log (\sin (c+d x)+1)}{a^4 d}-\frac{1}{3 a d (a \sin (c+d x)+a)^3}",1,"(-4*Log[1 + Sin[c + d*x]])/(a^4*d) + Sin[c + d*x]/(a^4*d) - 1/(3*a*d*(a + a*Sin[c + d*x])^3) + 2/(d*(a^2 + a^2*Sin[c + d*x])^2) - 6/(d*(a^4 + a^4*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
251,1,83,0,0.0928894,"\int \frac{\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^4,x]","\frac{3}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{3}{2 d \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{\log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3}","\frac{3}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{3}{2 d \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{\log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3}",1,"Log[1 + Sin[c + d*x]]/(a^4*d) + 1/(3*a*d*(a + a*Sin[c + d*x])^3) - 3/(2*d*(a^2 + a^2*Sin[c + d*x])^2) + 3/(d*(a^4 + a^4*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
252,1,30,0,0.065052,"\int \frac{\cos (c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^4,x]","\frac{\sin ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^3}","\frac{\sin ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^3}",1,"Sin[c + d*x]^3/(3*a*d*(a + a*Sin[c + d*x])^3)","A",3,3,27,0.1111,1,"{2833, 12, 37}"
253,1,46,0,0.0520895,"\int \frac{\cos (c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x])/(a + a*Sin[c + d*x])^4,x]","\frac{1}{3 a d (a \sin (c+d x)+a)^3}-\frac{1}{2 d \left(a^2 \sin (c+d x)+a^2\right)^2}","\frac{1}{3 a d (a \sin (c+d x)+a)^3}-\frac{1}{2 d \left(a^2 \sin (c+d x)+a^2\right)^2}",1,"1/(3*a*d*(a + a*Sin[c + d*x])^3) - 1/(2*d*(a^2 + a^2*Sin[c + d*x])^2)","A",4,3,25,0.1200,1,"{2833, 12, 43}"
254,1,97,0,0.064456,"\int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Cot[c + d*x]/(a + a*Sin[c + d*x])^4,x]","\frac{1}{d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{1}{2 d \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{\log (\sin (c+d x))}{a^4 d}-\frac{\log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3}","\frac{1}{d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{1}{2 d \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{\log (\sin (c+d x))}{a^4 d}-\frac{\log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3}",1,"Log[Sin[c + d*x]]/(a^4*d) - Log[1 + Sin[c + d*x]]/(a^4*d) + 1/(3*a*d*(a + a*Sin[c + d*x])^3) + 1/(2*d*(a^2 + a^2*Sin[c + d*x])^2) + 1/(d*(a^4 + a^4*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2707, 44}"
255,1,111,0,0.0936986,"\int \frac{\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x])^4,x]","-\frac{3}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{\csc (c+d x)}{a^4 d}-\frac{4 \log (\sin (c+d x))}{a^4 d}+\frac{4 \log (\sin (c+d x)+1)}{a^4 d}-\frac{1}{3 a d (a \sin (c+d x)+a)^3}","-\frac{3}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{\csc (c+d x)}{a^4 d}-\frac{4 \log (\sin (c+d x))}{a^4 d}+\frac{4 \log (\sin (c+d x)+1)}{a^4 d}-\frac{1}{3 a d (a \sin (c+d x)+a)^3}",1,"-(Csc[c + d*x]/(a^4*d)) - (4*Log[Sin[c + d*x]])/(a^4*d) + (4*Log[1 + Sin[c + d*x]])/(a^4*d) - 1/(3*a*d*(a + a*Sin[c + d*x])^3) - 1/(d*(a^2 + a^2*Sin[c + d*x])^2) - 3/(d*(a^4 + a^4*Sin[c + d*x]))","A",4,3,25,0.1200,1,"{2833, 12, 44}"
256,1,131,0,0.1176876,"\int \frac{\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^4,x]","\frac{6}{d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{3}{2 d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{\csc ^2(c+d x)}{2 a^4 d}+\frac{4 \csc (c+d x)}{a^4 d}+\frac{10 \log (\sin (c+d x))}{a^4 d}-\frac{10 \log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3}","\frac{6}{d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{3}{2 d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{\csc ^2(c+d x)}{2 a^4 d}+\frac{4 \csc (c+d x)}{a^4 d}+\frac{10 \log (\sin (c+d x))}{a^4 d}-\frac{10 \log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3}",1,"(4*Csc[c + d*x])/(a^4*d) - Csc[c + d*x]^2/(2*a^4*d) + (10*Log[Sin[c + d*x]])/(a^4*d) - (10*Log[1 + Sin[c + d*x]])/(a^4*d) + 1/(3*a*d*(a + a*Sin[c + d*x])^3) + 3/(2*d*(a^2 + a^2*Sin[c + d*x])^2) + 6/(d*(a^4 + a^4*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2833, 12, 44}"
257,1,51,0,0.0616903,"\int \cot (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 \sqrt{a \sin (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{a}}\right)}{d}","\frac{2 \sqrt{a \sin (c+d x)+a}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{a}}\right)}{d}",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[a + a*Sin[c + d*x]])/d","A",4,4,21,0.1905,1,"{2707, 50, 63, 207}"
258,1,114,0,0.1156885,"\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{4 a^4 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{6 a^4 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{4 a^4 \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{a^4 \sin ^{n+5}(c+d x)}{d (n+5)}","\frac{a^4 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{4 a^4 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{6 a^4 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{4 a^4 \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{a^4 \sin ^{n+5}(c+d x)}{d (n+5)}",1,"(a^4*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (4*a^4*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (6*a^4*Sin[c + d*x]^(3 + n))/(d*(3 + n)) + (4*a^4*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (a^4*Sin[c + d*x]^(5 + n))/(d*(5 + n))","A",3,2,27,0.07407,1,"{2833, 43}"
259,1,91,0,0.0953491,"\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{3 a^3 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{a^3 \sin ^{n+4}(c+d x)}{d (n+4)}","\frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{3 a^3 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{a^3 \sin ^{n+4}(c+d x)}{d (n+4)}",1,"(a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (3*a^3*Sin[c + d*x]^(3 + n))/(d*(3 + n)) + (a^3*Sin[c + d*x]^(4 + n))/(d*(4 + n))","A",3,2,27,0.07407,1,"{2833, 43}"
260,1,68,0,0.0862026,"\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{a^2 \sin ^{n+3}(c+d x)}{d (n+3)}","\frac{a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{a^2 \sin ^{n+3}(c+d x)}{d (n+3)}",1,"(a^2*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (2*a^2*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (a^2*Sin[c + d*x]^(3 + n))/(d*(3 + n))","A",3,2,27,0.07407,1,"{2833, 43}"
261,1,41,0,0.0524414,"\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{a \sin ^{n+2}(c+d x)}{d (n+2)}","\frac{a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{a \sin ^{n+2}(c+d x)}{d (n+2)}",1,"(a*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (a*Sin[c + d*x]^(2 + n))/(d*(2 + n))","A",3,2,25,0.08000,1,"{2833, 43}"
262,1,38,0,0.0748402,"\int \frac{\cos (c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^n)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a d (n+1)}","\frac{\sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a d (n+1)}",1,"(Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a*d*(1 + n))","A",2,2,27,0.07407,1,"{2833, 64}"
263,1,38,0,0.0738427,"\int \frac{\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;-\sin (c+d x))}{a^2 d (n+1)}","\frac{\sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;-\sin (c+d x))}{a^2 d (n+1)}",1,"(Hypergeometric2F1[2, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^2*d*(1 + n))","A",2,2,27,0.07407,1,"{2833, 64}"
264,1,38,0,0.0741083,"\int \frac{\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^3,x]","\frac{\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)}","\frac{\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)}",1,"(Hypergeometric2F1[3, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^3*d*(1 + n))","A",2,2,27,0.07407,1,"{2833, 64}"
265,1,38,0,0.0751362,"\int \frac{\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^4,x]","\frac{\sin ^{n+1}(c+d x) \, _2F_1(4,n+1;n+2;-\sin (c+d x))}{a^4 d (n+1)}","\frac{\sin ^{n+1}(c+d x) \, _2F_1(4,n+1;n+2;-\sin (c+d x))}{a^4 d (n+1)}",1,"(Hypergeometric2F1[4, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^4*d*(1 + n))","A",2,2,27,0.07407,1,"{2833, 64}"
266,1,105,0,0.165903,"\int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \sin ^3(c+d x) \cos ^3(c+d x)}{6 d}-\frac{a \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{a x}{16}","\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \sin ^3(c+d x) \cos ^3(c+d x)}{6 d}-\frac{a \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{a x}{16}",1,"(a*x)/16 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*d)","A",8,6,27,0.2222,1,"{2838, 2565, 14, 2568, 2635, 8}"
267,1,81,0,0.1281646,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a x}{8}","\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a x}{8}",1,"(a*x)/8 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",7,6,27,0.2222,1,"{2838, 2568, 2635, 8, 2565, 14}"
268,1,65,0,0.0929874,"\int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a x}{8}","-\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a x}{8}",1,"(a*x)/8 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",6,6,25,0.2400,1,"{2838, 2565, 30, 2568, 2635, 8}"
269,1,51,0,0.0637439,"\int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \cos (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a x}{2}","\frac{a \cos (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a x}{2}",1,"(a*x)/2 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",6,6,23,0.2609,1,"{2838, 2592, 321, 206, 2635, 8}"
270,1,41,0,0.0550534,"\int \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \cos (c+d x)}{d}-\frac{a \cot (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-a x","\frac{a \cos (c+d x)}{d}-\frac{a \cot (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-a x",1,"-(a*x) - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d - (a*Cot[c + d*x])/d","A",7,6,19,0.3158,1,"{2710, 2592, 321, 206, 3473, 8}"
271,1,52,0,0.0771847,"\int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \cot (c+d x)}{d}+\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-a x","-\frac{a \cot (c+d x)}{d}+\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-a x",1,"-(a*x) + (a*ArcTanh[Cos[c + d*x]])/(2*d) - (a*Cot[c + d*x])/d - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)","A",5,5,25,0.2000,1,"{2838, 2611, 3770, 3473, 8}"
272,1,52,0,0.1083508,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}",1,"(a*ArcTanh[Cos[c + d*x]])/(2*d) - (a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)","A",5,5,27,0.1852,1,"{2838, 2607, 30, 2611, 3770}"
273,1,74,0,0.1262059,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a \cot (c+d x) \csc (c+d x)}{8 d}","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a \cot (c+d x) \csc (c+d x)}{8 d}",1,"(a*ArcTanh[Cos[c + d*x]])/(8*d) - (a*Cot[c + d*x]^3)/(3*d) + (a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)","A",6,6,27,0.2222,1,"{2838, 2611, 3768, 3770, 2607, 30}"
274,1,90,0,0.1286289,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a \cot (c+d x) \csc (c+d x)}{8 d}","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a \cot (c+d x) \csc (c+d x)}{8 d}",1,"(a*ArcTanh[Cos[c + d*x]])/(8*d) - (a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^5)/(5*d) + (a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)","A",7,6,27,0.2222,1,"{2838, 2607, 14, 2611, 3768, 3770}"
275,1,135,0,0.2480581,"\int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos ^7(c+d x)}{7 d}+\frac{3 a^2 \cos ^5(c+d x)}{5 d}-\frac{2 a^2 \cos ^3(c+d x)}{3 d}-\frac{a^2 \sin ^3(c+d x) \cos ^3(c+d x)}{3 d}-\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a^2 x}{8}","-\frac{a^2 \cos ^7(c+d x)}{7 d}+\frac{3 a^2 \cos ^5(c+d x)}{5 d}-\frac{2 a^2 \cos ^3(c+d x)}{3 d}-\frac{a^2 \sin ^3(c+d x) \cos ^3(c+d x)}{3 d}-\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a^2 x}{8}",1,"(a^2*x)/8 - (2*a^2*Cos[c + d*x]^3)/(3*d) + (3*a^2*Cos[c + d*x]^5)/(5*d) - (a^2*Cos[c + d*x]^7)/(7*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x]^3)/(3*d)","A",12,7,29,0.2414,1,"{2873, 2565, 14, 2568, 2635, 8, 270}"
276,1,103,0,0.1678512,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos ^5(c+d x)}{10 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{3 a^2 x}{16}-\frac{\cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 a d}","-\frac{a^2 \cos ^5(c+d x)}{10 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{3 a^2 x}{16}-\frac{\cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 a d}",1,"(3*a^2*x)/16 - (a^2*Cos[c + d*x]^5)/(10*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(6*a*d)","A",5,4,29,0.1379,1,"{2870, 2669, 2635, 8}"
277,1,105,0,0.1300625,"\int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos ^3(c+d x)}{6 d}-\frac{\cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{10 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{a^2 x}{4}-\frac{\cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}","\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{2 a^2 \cos ^3(c+d x)}{3 d}-\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{a^2 x}{4}",1,"(a^2*x)/4 - (a^2*Cos[c + d*x]^3)/(6*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) - (Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(5*d) - (Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x]))/(10*d)","A",5,5,27,0.1852,1,"{2860, 2678, 2669, 2635, 8}"
278,1,71,0,0.1202897,"\int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+a^2 x","-\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+a^2 x",1,"a^2*x - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d - (a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/d","A",9,8,25,0.3200,1,"{2873, 2635, 8, 2592, 321, 206, 2565, 30}"
279,1,74,0,0.104235,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 x}{2}","\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 x}{2}",1,"-(a^2*x)/2 - (2*a^2*ArcTanh[Cos[c + d*x]])/d + (2*a^2*Cos[c + d*x])/d - (a^2*Cot[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",8,6,21,0.2857,1,"{2709, 3770, 3767, 8, 2638, 2635}"
280,1,73,0,0.1296192,"\int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d}-2 a^2 x","\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d}-2 a^2 x",1,"-2*a^2*x - (a^2*ArcTanh[Cos[c + d*x]])/(2*d) + (a^2*Cos[c + d*x])/d - (2*a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d)","A",7,6,27,0.2222,1,"{2872, 3767, 8, 3768, 3770, 2638}"
281,1,73,0,0.2214748,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-a^2 x","-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-a^2 x",1,"-(a^2*x) + (a^2*ArcTanh[Cos[c + d*x]])/d - (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]*Csc[c + d*x])/d","A",8,7,29,0.2414,1,"{2873, 3473, 8, 2611, 3770, 2607, 30}"
282,1,82,0,0.2008956,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}","-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}",1,"(5*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (2*a^2*Cot[c + d*x]^3)/(3*d) - (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)","A",9,6,29,0.2069,1,"{2873, 2611, 3770, 2607, 30, 3768}"
283,1,100,0,0.2096619,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d}+\frac{a^2 \cot (c+d x) \csc (c+d x)}{4 d}","-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d}+\frac{a^2 \cot (c+d x) \csc (c+d x)}{4 d}",1,"(a^2*ArcTanh[Cos[c + d*x]])/(4*d) - (2*a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) + (a^2*Cot[c + d*x]*Csc[c + d*x])/(4*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(2*d)","A",10,7,29,0.2414,1,"{2873, 2607, 30, 2611, 3768, 3770, 14}"
284,1,124,0,0.2556392,"\int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{16 d}","-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{16 d}",1,"(3*a^2*ArcTanh[Cos[c + d*x]])/(16*d) - (2*a^2*Cot[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)","A",12,6,29,0.2069,1,"{2873, 2611, 3768, 3770, 2607, 14}"
285,1,132,0,0.3045095,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^5(c+d x)}{d}-\frac{4 a^3 \cos ^3(c+d x)}{3 d}-\frac{a^3 \sin ^3(c+d x) \cos ^3(c+d x)}{2 d}-\frac{5 a^3 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{5 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^3 x}{16}","-\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^5(c+d x)}{d}-\frac{4 a^3 \cos ^3(c+d x)}{3 d}-\frac{a^3 \sin ^3(c+d x) \cos ^3(c+d x)}{2 d}-\frac{5 a^3 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{5 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^3 x}{16}",1,"(5*a^3*x)/16 - (4*a^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cos[c + d*x]^5)/d - (a^3*Cos[c + d*x]^7)/(7*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (5*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (a^3*Cos[c + d*x]^3*Sin[c + d*x]^3)/(2*d)","A",15,7,29,0.2414,1,"{2873, 2568, 2635, 8, 2565, 14, 270}"
286,1,133,0,0.1813153,"\int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]","-\frac{7 a^3 \cos ^3(c+d x)}{24 d}-\frac{7 \cos ^3(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{40 d}+\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{7 a^3 x}{16}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{10 d}-\frac{\cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}","\frac{3 a^3 \cos ^5(c+d x)}{5 d}-\frac{4 a^3 \cos ^3(c+d x)}{3 d}-\frac{a^3 \sin ^3(c+d x) \cos ^3(c+d x)}{6 d}-\frac{7 a^3 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{7 a^3 x}{16}",1,"(7*a^3*x)/16 - (7*a^3*Cos[c + d*x]^3)/(24*d) + (7*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(10*d) - (Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(6*d) - (7*Cos[c + d*x]^3*(a^3 + a^3*Sin[c + d*x]))/(40*d)","A",6,5,27,0.1852,1,"{2860, 2678, 2669, 2635, 8}"
287,1,99,0,0.1697191,"\int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos ^3(c+d x)}{d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{13 a^3 x}{8}","-\frac{a^3 \cos ^3(c+d x)}{d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{13 a^3 x}{8}",1,"(13*a^3*x)/8 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/d + (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^3*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",12,9,25,0.3600,1,"{2873, 2635, 8, 2592, 321, 206, 2565, 30, 2568}"
288,1,92,0,0.1379394,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 x}{2}","-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 x}{2}",1,"(a^3*x)/2 - (3*a^3*ArcTanh[Cos[c + d*x]])/d + (3*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",10,7,21,0.3333,1,"{2709, 3770, 3767, 8, 2638, 2635, 2633}"
289,1,98,0,0.1494916,"\int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{3 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{5 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{5 a^3 x}{2}","\frac{3 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{5 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{5 a^3 x}{2}",1,"(-5*a^3*x)/2 - (5*a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (3*a^3*Cos[c + d*x])/d - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",10,7,27,0.2593,1,"{2872, 3770, 3767, 8, 3768, 2638, 2635}"
290,1,91,0,0.1708308,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a^3 \cot (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-3 a^3 x","\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a^3 \cot (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-3 a^3 x",1,"-3*a^3*x + (a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (a^3*Cos[c + d*x])/d - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d)","A",10,6,29,0.2069,1,"{2872, 3770, 3767, 8, 3768, 2638}"
291,1,100,0,0.2212403,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot ^3(c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{13 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-a^3 x","-\frac{a^3 \cot ^3(c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{13 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-a^3 x",1,"-(a^3*x) + (13*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/d - (11*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)","A",11,8,29,0.2759,1,"{2873, 3473, 8, 2611, 3770, 2607, 30, 3768}"
292,1,100,0,0.2402124,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{7 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{8 d}","-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{7 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{8 d}",1,"(7*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (4*a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)","A",12,7,29,0.2414,1,"{2873, 2611, 3770, 2607, 30, 3768, 14}"
293,1,124,0,0.2819182,"\int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{7 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{7 a^3 \cot (c+d x) \csc (c+d x)}{16 d}","-\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{7 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{7 a^3 \cot (c+d x) \csc (c+d x)}{16 d}",1,"(7*a^3*ArcTanh[Cos[c + d*x]])/(16*d) - (4*a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]^5)/(5*d) + (7*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (17*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)","A",14,7,29,0.2414,1,"{2873, 2607, 30, 2611, 3768, 3770, 14}"
294,1,137,0,0.1574584,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","-\frac{7 a^4 \cos ^3(c+d x)}{8 d}-\frac{3 \cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^2}{10 d}-\frac{21 \cos ^3(c+d x) \left(a^4 \sin (c+d x)+a^4\right)}{40 d}+\frac{21 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{21 a^4 x}{16}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}","-\frac{7 a^4 \cos ^3(c+d x)}{8 d}-\frac{3 \cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^2}{10 d}-\frac{21 \cos ^3(c+d x) \left(a^4 \sin (c+d x)+a^4\right)}{40 d}+\frac{21 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{21 a^4 x}{16}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}",1,"(21*a^4*x)/16 - (7*a^4*Cos[c + d*x]^3)/(8*d) + (21*a^4*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(6*d) - (3*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^2)/(10*d) - (21*Cos[c + d*x]^3*(a^4 + a^4*Sin[c + d*x]))/(40*d)","A",6,4,21,0.1905,1,"{2678, 2669, 2635, 8}"
295,1,117,0,0.1994989,"\int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \cos ^5(c+d x)}{5 d}-\frac{7 a^4 \cos ^3(c+d x)}{3 d}+\frac{a^4 \cos (c+d x)}{d}-\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac{5 a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{5 a^4 x}{2}","\frac{a^4 \cos ^5(c+d x)}{5 d}-\frac{7 a^4 \cos ^3(c+d x)}{3 d}+\frac{a^4 \cos (c+d x)}{d}-\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac{5 a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{5 a^4 x}{2}",1,"(5*a^4*x)/2 - (a^4*ArcTanh[Cos[c + d*x]])/d + (a^4*Cos[c + d*x])/d - (7*a^4*Cos[c + d*x]^3)/(3*d) + (a^4*Cos[c + d*x]^5)/(5*d) + (5*a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a^4*Cos[c + d*x]^3*Sin[c + d*x])/d","A",15,10,25,0.4000,1,"{2873, 2635, 8, 2592, 321, 206, 2565, 30, 2568, 14}"
296,1,116,0,0.1682837,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","-\frac{4 a^4 \cos ^3(c+d x)}{3 d}+\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \cot (c+d x)}{d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{23 a^4 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{4 a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{17 a^4 x}{8}","-\frac{4 a^4 \cos ^3(c+d x)}{3 d}+\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \cot (c+d x)}{d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{23 a^4 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{4 a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{17 a^4 x}{8}",1,"(17*a^4*x)/8 - (4*a^4*ArcTanh[Cos[c + d*x]])/d + (4*a^4*Cos[c + d*x])/d - (4*a^4*Cos[c + d*x]^3)/(3*d) - (a^4*Cot[c + d*x])/d + (23*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",12,6,21,0.2857,1,"{2709, 3770, 3767, 8, 2635, 2633}"
297,1,104,0,0.1356556,"\int \frac{\cos ^2(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\cos ^5(c+d x)}{5 a d}-\frac{2 \cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x}{8 a}","\frac{\cos ^5(c+d x)}{5 a d}-\frac{2 \cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x}{8 a}",1,"(3*x)/(8*a) + Cos[c + d*x]/(a*d) - (2*Cos[c + d*x]^3)/(3*a*d) + Cos[c + d*x]^5/(5*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a*d)","A",6,4,29,0.1379,1,"{2839, 2635, 8, 2633}"
298,1,87,0,0.1295378,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\cos ^3(c+d x)}{3 a d}-\frac{\cos (c+d x)}{a d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{3 x}{8 a}","\frac{\cos ^3(c+d x)}{3 a d}-\frac{\cos (c+d x)}{a d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{3 x}{8 a}",1,"(-3*x)/(8*a) - Cos[c + d*x]/(a*d) + Cos[c + d*x]^3/(3*a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a*d)","A",6,4,29,0.1379,1,"{2839, 2633, 2635, 8}"
299,1,62,0,0.1173356,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x}{2 a}","-\frac{\cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x}{2 a}",1,"x/(2*a) + Cos[c + d*x]/(a*d) - Cos[c + d*x]^3/(3*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)","A",5,4,29,0.1379,1,"{2839, 2635, 8, 2633}"
300,1,45,0,0.072349,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\cos (c+d x)}{a d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x}{2 a}","-\frac{\cos (c+d x)}{a d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x}{2 a}",1,"-x/(2*a) - Cos[c + d*x]/(a*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)","A",4,4,27,0.1481,1,"{2839, 2638, 2635, 8}"
301,1,22,0,0.0715916,"\int \frac{\cos (c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{a}","-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{a}",1,"-(x/a) - ArcTanh[Cos[c + d*x]]/(a*d)","A",3,3,25,0.1200,1,"{2839, 3770, 8}"
302,1,29,0,0.0549483,"\int \frac{\cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^2/(a + a*Sin[c + d*x]),x]","\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\cot (c+d x)}{a d}","\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\cot (c+d x)}{a d}",1,"ArcTanh[Cos[c + d*x]]/(a*d) - Cot[c + d*x]/(a*d)","A",4,4,21,0.1905,1,"{2706, 3767, 8, 3770}"
303,1,53,0,0.1026794,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\cot (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}","\frac{\cot (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}",1,"-ArcTanh[Cos[c + d*x]]/(2*a*d) + Cot[c + d*x]/(a*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)","A",5,5,27,0.1852,1,"{2839, 3768, 3770, 3767, 8}"
304,1,72,0,0.1231045,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d}","-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d}",1,"ArcTanh[Cos[c + d*x]]/(2*a*d) - Cot[c + d*x]/(a*d) - Cot[c + d*x]^3/(3*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)","A",5,4,29,0.1379,1,"{2839, 3767, 3768, 3770}"
305,1,95,0,0.1344196,"\int \frac{\cot ^2(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cot (c+d x)}{a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}","\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cot (c+d x)}{a 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,"(-3*ArcTanh[Cos[c + d*x]])/(8*a*d) + Cot[c + d*x]/(a*d) + Cot[c + d*x]^3/(3*a*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)","A",6,4,29,0.1379,1,"{2839, 3768, 3770, 3767}"
306,1,114,0,0.1377073,"\int \frac{\cot ^2(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^5(c+d x)}{5 a d}-\frac{2 \cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}","-\frac{\cot ^5(c+d x)}{5 a d}-\frac{2 \cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a 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,"(3*ArcTanh[Cos[c + d*x]])/(8*a*d) - Cot[c + d*x]/(a*d) - (2*Cot[c + d*x]^3)/(3*a*d) - Cot[c + d*x]^5/(5*a*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)","A",6,4,29,0.1379,1,"{2839, 3767, 3768, 3770}"
307,1,111,0,0.2454976,"\int \frac{\cos ^2(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cos ^3(c+d x)}{3 a^2 d}-\frac{4 \cos (c+d x)}{a^2 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a^2 d}+\frac{11 \sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}-\frac{27 x}{8 a^2}","\frac{2 \cos ^3(c+d x)}{3 a^2 d}-\frac{4 \cos (c+d x)}{a^2 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a^2 d}+\frac{11 \sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}-\frac{27 x}{8 a^2}",1,"(-27*x)/(8*a^2) - (4*Cos[c + d*x])/(a^2*d) + (2*Cos[c + d*x]^3)/(3*a^2*d) + (11*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^2*d) - (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))","A",12,7,29,0.2414,1,"{2874, 2966, 2638, 2635, 8, 2633, 2648}"
308,1,83,0,0.2215775,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cos ^3(c+d x)}{3 a^2 d}+\frac{3 \cos (c+d x)}{a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{a^2 d}+\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}+\frac{3 x}{a^2}","-\frac{\cos ^3(c+d x)}{3 a^2 d}+\frac{3 \cos (c+d x)}{a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{a^2 d}+\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}+\frac{3 x}{a^2}",1,"(3*x)/a^2 + (3*Cos[c + d*x])/(a^2*d) - Cos[c + d*x]^3/(3*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(a^2*d) + (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))","A",9,7,29,0.2414,1,"{2874, 2966, 2638, 2635, 8, 2633, 2648}"
309,1,69,0,0.2716321,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \cos (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}-\frac{5 x}{2 a^2}","-\frac{2 \cos (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}-\frac{5 x}{2 a^2}",1,"(-5*x)/(2*a^2) - (2*Cos[c + d*x])/(a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))","A",8,7,29,0.2414,1,"{2874, 2950, 2709, 2638, 2635, 8, 2648}"
310,1,47,0,0.0724305,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{\cos (c+d x)}{a^2 d}+\frac{2 \cos (c+d x)}{d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{2 x}{a^2}","\frac{\cos (c+d x)}{a^2 d}+\frac{2 \cos (c+d x)}{d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{2 x}{a^2}",1,"(2*x)/a^2 + Cos[c + d*x]/(a^2*d) + (2*Cos[c + d*x])/(d*(a^2 + a^2*Sin[c + d*x]))","A",3,2,27,0.07407,1,"{2857, 2638}"
311,1,40,0,0.1571681,"\int \frac{\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}","\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}",1,"-(ArcTanh[Cos[c + d*x]]/(a^2*d)) + (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))","A",5,4,25,0.1600,1,"{2874, 2966, 3770, 2648}"
312,1,54,0,0.0962669,"\int \frac{\cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^2/(a + a*Sin[c + d*x])^2,x]","-\frac{\cot (c+d x)}{a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{2 \cot (c+d x)}{a^2 d (\csc (c+d x)+1)}","-\frac{\cot (c+d x)}{a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{2 \cot (c+d x)}{a^2 d (\csc (c+d x)+1)}",1,"(2*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a^2*d) - (2*Cot[c + d*x])/(a^2*d*(1 + Csc[c + d*x]))","A",7,5,21,0.2381,1,"{2709, 3770, 3767, 8, 3777}"
313,1,78,0,0.2201329,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cot (c+d x)}{a^2 d}+\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}","\frac{2 \cot (c+d x)}{a^2 d}+\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}",1,"(-5*ArcTanh[Cos[c + d*x]])/(2*a^2*d) + (2*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) + (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))","A",9,7,27,0.2593,1,"{2874, 2966, 3770, 3767, 8, 3768, 2648}"
314,1,91,0,0.2512147,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{3 \cot (c+d x)}{a^2 d}-\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d}","-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{3 \cot (c+d x)}{a^2 d}-\frac{2 \cos (c+d x)}{a^2 d (\sin (c+d x)+1)}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d}",1,"(3*ArcTanh[Cos[c + d*x]])/(a^2*d) - (3*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(a^2*d) - (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))","A",11,7,29,0.2414,1,"{2874, 2966, 3770, 3767, 8, 3768, 2648}"
315,1,97,0,0.2633034,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","-\frac{3 \cos (c+d x)}{a^3 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{19 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}+\frac{2 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)^2}-\frac{11 x}{2 a^3}","-\frac{3 \cos (c+d x)}{a^3 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{19 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}+\frac{2 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)^2}-\frac{11 x}{2 a^3}",1,"(-11*x)/(2*a^3) - (3*Cos[c + d*x])/(a^3*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + (2*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x])^2) - (19*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x]))","A",9,7,29,0.2414,1,"{2874, 2966, 2638, 2635, 8, 2650, 2648}"
316,1,76,0,0.1812967,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{3 \cos (c+d x)}{a^3 d}+\frac{3 x}{a^3}+\frac{2 \cos ^3(c+d x)}{a d (a \sin (c+d x)+a)^2}-\frac{\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3}","\frac{3 \cos (c+d x)}{a^3 d}+\frac{3 x}{a^3}+\frac{2 \cos ^3(c+d x)}{a d (a \sin (c+d x)+a)^2}-\frac{\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3}",1,"(3*x)/a^3 + (3*Cos[c + d*x])/(a^3*d) - Cos[c + d*x]^3/(3*d*(a + a*Sin[c + d*x])^3) + (2*Cos[c + d*x]^3)/(a*d*(a + a*Sin[c + d*x])^2)","A",4,4,29,0.1379,1,"{2871, 2680, 2682, 8}"
317,1,61,0,0.1086828,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{7 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}-\frac{x}{a^3}+\frac{2 \cos (c+d x)}{3 a d (a \sin (c+d x)+a)^2}","-\frac{7 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}-\frac{x}{a^3}+\frac{2 \cos (c+d x)}{3 a d (a \sin (c+d x)+a)^2}",1,"-(x/a^3) - (7*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x])) + (2*Cos[c + d*x])/(3*a*d*(a + a*Sin[c + d*x])^2)","A",3,3,27,0.1111,1,"{2857, 2735, 2648}"
318,1,68,0,0.1932748,"\int \frac{\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{5 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}+\frac{2 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)^2}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}","\frac{5 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}+\frac{2 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)^2}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}",1,"-(ArcTanh[Cos[c + d*x]]/(a^3*d)) + (2*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x])^2) + (5*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x]))","A",7,5,25,0.2000,1,"{2874, 2966, 3770, 2650, 2648}"
319,1,82,0,0.2077041,"\int \frac{\cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^2/(a + a*Sin[c + d*x])^3,x]","-\frac{\cot (c+d x)}{a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{13 \cot (c+d x)}{3 a^3 d (\csc (c+d x)+1)}+\frac{2 \cot (c+d x)}{3 a^3 d (\csc (c+d x)+1)^2}","-\frac{\cot (c+d x)}{a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{13 \cot (c+d x)}{3 a^3 d (\csc (c+d x)+1)}+\frac{2 \cot (c+d x)}{3 a^3 d (\csc (c+d x)+1)^2}",1,"(3*ArcTanh[Cos[c + d*x]])/(a^3*d) - Cot[c + d*x]/(a^3*d) + (2*Cot[c + d*x])/(3*a^3*d*(1 + Csc[c + d*x])^2) - (13*Cot[c + d*x])/(3*a^3*d*(1 + Csc[c + d*x]))","A",10,7,21,0.3333,1,"{2709, 3770, 3767, 8, 3777, 3919, 3794}"
320,1,106,0,0.2685781,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{3 \cot (c+d x)}{a^3 d}+\frac{17 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}+\frac{2 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)^2}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}","\frac{3 \cot (c+d x)}{a^3 d}+\frac{17 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}+\frac{2 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)^2}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}",1,"(-11*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (3*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + (2*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x])^2) + (17*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x]))","A",11,8,27,0.2963,1,"{2874, 2966, 3770, 3767, 8, 3768, 2650, 2648}"
321,1,144,0,0.1623124,"\int \frac{\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx","Int[(Cos[e + f*x]^2*Sin[e + f*x])/(a + a*Sin[e + f*x])^6,x]","\frac{4 \cos (e+f x)}{315 f \left(a^6 \sin (e+f x)+a^6\right)}+\frac{4 \cos (e+f x)}{315 f \left(a^3 \sin (e+f x)+a^3\right)^2}+\frac{2 \cos (e+f x)}{105 f \left(a^2 \sin (e+f x)+a^2\right)^3}-\frac{19 \cos (e+f x)}{63 a^2 f (a \sin (e+f x)+a)^4}+\frac{2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}","\frac{4 \cos (e+f x)}{315 f \left(a^6 \sin (e+f x)+a^6\right)}+\frac{4 \cos (e+f x)}{315 f \left(a^3 \sin (e+f x)+a^3\right)^2}+\frac{2 \cos (e+f x)}{105 f \left(a^2 \sin (e+f x)+a^2\right)^3}-\frac{19 \cos (e+f x)}{63 a^2 f (a \sin (e+f x)+a)^4}+\frac{2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5}",1,"(2*Cos[e + f*x])/(9*a*f*(a + a*Sin[e + f*x])^5) - (19*Cos[e + f*x])/(63*a^2*f*(a + a*Sin[e + f*x])^4) + (2*Cos[e + f*x])/(105*f*(a^2 + a^2*Sin[e + f*x])^3) + (4*Cos[e + f*x])/(315*f*(a^3 + a^3*Sin[e + f*x])^2) + (4*Cos[e + f*x])/(315*f*(a^6 + a^6*Sin[e + f*x]))","A",5,4,27,0.1481,1,"{2857, 2750, 2650, 2648}"
322,1,193,0,0.5683017,"\int \cos ^2(c+d x) \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}+\frac{2 a \sin ^4(c+d x) \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{38 a \sin ^3(c+d x) \cos (c+d x)}{693 d \sqrt{a \sin (c+d x)+a}}-\frac{76 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{1155 a d}+\frac{152 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3465 d}-\frac{76 a \cos (c+d x)}{495 d \sqrt{a \sin (c+d x)+a}}","\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}+\frac{2 a \sin ^4(c+d x) \cos (c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{38 a \sin ^3(c+d x) \cos (c+d x)}{693 d \sqrt{a \sin (c+d x)+a}}-\frac{76 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{1155 a d}+\frac{152 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3465 d}-\frac{76 a \cos (c+d x)}{495 d \sqrt{a \sin (c+d x)+a}}",1,"(-76*a*Cos[c + d*x])/(495*d*Sqrt[a + a*Sin[c + d*x]]) - (38*a*Cos[c + d*x]*Sin[c + d*x]^3)/(693*d*Sqrt[a + a*Sin[c + d*x]]) + (2*a*Cos[c + d*x]*Sin[c + d*x]^4)/(99*d*Sqrt[a + a*Sin[c + d*x]]) + (152*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3465*d) + (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(11*d) - (76*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(1155*a*d)","A",7,7,31,0.2258,1,"{2879, 2976, 2981, 2770, 2759, 2751, 2646}"
323,1,124,0,0.3580433,"\int \cos ^2(c+d x) \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{8 a^2 \cos ^3(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 a d}+\frac{4 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{21 d}-\frac{2 a \cos ^3(c+d x)}{21 d \sqrt{a \sin (c+d x)+a}}","-\frac{8 a^2 \cos ^3(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 a d}+\frac{4 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{21 d}-\frac{2 a \cos ^3(c+d x)}{21 d \sqrt{a \sin (c+d x)+a}}",1,"(-8*a^2*Cos[c + d*x]^3)/(63*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^3)/(21*d*Sqrt[a + a*Sin[c + d*x]]) + (4*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(21*d) - (2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(9*a*d)","A",4,4,31,0.1290,1,"{2878, 2856, 2674, 2673}"
324,1,92,0,0.1883396,"\int \cos ^2(c+d x) \sin (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{8 a^2 \cos ^3(c+d x)}{105 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{7 d}-\frac{2 a \cos ^3(c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}","-\frac{8 a^2 \cos ^3(c+d x)}{105 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{7 d}-\frac{2 a \cos ^3(c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}",1,"(-8*a^2*Cos[c + d*x]^3)/(105*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^3)/(35*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(7*d)","A",3,3,29,0.1034,1,"{2856, 2674, 2673}"
325,1,93,0,0.3393287,"\int \cos (c+d x) \cot (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}+\frac{2 a \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}","\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}+\frac{2 a \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"(-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d + (2*a*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)","A",5,5,27,0.1852,1,"{2874, 2976, 2981, 2773, 206}"
326,1,89,0,0.1987866,"\int \cot ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","\frac{3 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}","\frac{3 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (3*a*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d","A",4,4,23,0.1739,1,"{2716, 2981, 2773, 206}"
327,1,101,0,0.3971619,"\int \cot ^2(c+d x) \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{a \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{5 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}","-\frac{a \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{5 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}",1,"(5*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) - (a*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*d)","A",5,5,29,0.1724,1,"{2874, 2975, 2980, 2773, 206}"
328,1,137,0,0.4799411,"\int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","\frac{3 a \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}","\frac{3 a \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}",1,"(3*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) + (3*a*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(3*d)","A",6,6,31,0.1935,1,"{2874, 2975, 2980, 2772, 2773, 206}"
329,1,233,0,0.7203703,"\int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{38 a^2 \sin ^4(c+d x) \cos (c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}-\frac{862 a^2 \sin ^3(c+d x) \cos (c+d x)}{9009 d \sqrt{a \sin (c+d x)+a}}-\frac{1724 a^2 \cos (c+d x)}{6435 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}+\frac{6 a \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}-\frac{1724 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{15015 d}+\frac{3448 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{45045 d}","-\frac{38 a^2 \sin ^4(c+d x) \cos (c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}-\frac{862 a^2 \sin ^3(c+d x) \cos (c+d x)}{9009 d \sqrt{a \sin (c+d x)+a}}-\frac{1724 a^2 \cos (c+d x)}{6435 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}+\frac{6 a \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}-\frac{1724 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{15015 d}+\frac{3448 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{45045 d}",1,"(-1724*a^2*Cos[c + d*x])/(6435*d*Sqrt[a + a*Sin[c + d*x]]) - (862*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(9009*d*Sqrt[a + a*Sin[c + d*x]]) - (38*a^2*Cos[c + d*x]*Sin[c + d*x]^4)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) + (3448*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(45045*d) + (6*a*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(143*d) - (1724*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(15015*d) + (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(13*d)","A",8,7,31,0.2258,1,"{2879, 2976, 2981, 2770, 2759, 2751, 2646}"
330,1,156,0,0.4315416,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{48 a^2 \cos ^3(c+d x)}{385 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^3(c+d x)}{385 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{11 a d}+\frac{4 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{33 d}-\frac{6 a \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{77 d}","-\frac{48 a^2 \cos ^3(c+d x)}{385 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^3(c+d x)}{385 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{11 a d}+\frac{4 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{33 d}-\frac{6 a \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{77 d}",1,"(-64*a^3*Cos[c + d*x]^3)/(385*d*(a + a*Sin[c + d*x])^(3/2)) - (48*a^2*Cos[c + d*x]^3)/(385*d*Sqrt[a + a*Sin[c + d*x]]) - (6*a*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(77*d) + (4*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(33*d) - (2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2))/(11*a*d)","A",5,4,31,0.1290,1,"{2878, 2856, 2674, 2673}"
331,1,124,0,0.2550755,"\int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{16 a^2 \cos ^3(c+d x)}{105 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^3(c+d x)}{315 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 d}-\frac{2 a \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{21 d}","-\frac{16 a^2 \cos ^3(c+d x)}{105 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^3(c+d x)}{315 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 d}-\frac{2 a \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{21 d}",1,"(-64*a^3*Cos[c + d*x]^3)/(315*d*(a + a*Sin[c + d*x])^(3/2)) - (16*a^2*Cos[c + d*x]^3)/(105*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(21*d) - (2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(9*d)","A",4,3,29,0.1034,1,"{2856, 2674, 2673}"
332,1,123,0,0.4594994,"\int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 a^2 \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}+\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}+\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}","-\frac{2 a^2 \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}+\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}+\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}",1,"(-2*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d - (2*a^2*Cos[c + d*x])/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*d) + (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d)","A",6,5,27,0.1852,1,"{2874, 2976, 2981, 2773, 206}"
333,1,121,0,0.3080131,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","\frac{11 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}+\frac{5 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}","\frac{11 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}+\frac{5 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}",1,"(-3*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d + (11*a^2*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (5*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (Cot[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/d","A",5,5,23,0.2174,1,"{2716, 2976, 2981, 2773, 206}"
334,1,131,0,0.5184666,"\int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","\frac{13 a^2 \cos (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{3 a \cot (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}","\frac{13 a^2 \cos (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{3 a \cot (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}",1,"(a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) + (13*a^2*Cos[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(2*d)","A",6,5,29,0.1724,1,"{2874, 2975, 2981, 2773, 206}"
335,1,139,0,0.5733367,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","\frac{5 a^2 \cot (c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{13 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}","\frac{5 a^2 \cot (c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{13 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}",1,"(13*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) + (5*a^2*Cot[c + d*x])/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(3*d)","A",6,5,31,0.1613,1,"{2874, 2975, 2980, 2773, 206}"
336,1,158,0,0.4035855,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^3)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 a^2 d}+\frac{2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}+\frac{8 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 a d}-\frac{4 \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}","-\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 a^2 d}+\frac{2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}+\frac{8 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 a d}-\frac{4 \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}",1,"(-4*Cos[c + d*x])/(45*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sin[c + d*x]^4)/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (8*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*a*d) - (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*a^2*d)","A",6,6,31,0.1935,1,"{2879, 2981, 2770, 2759, 2751, 2646}"
337,1,92,0,0.3427928,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{7 a d}+\frac{12 \cos ^3(c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{22 a \cos ^3(c+d x)}{105 d (a \sin (c+d x)+a)^{3/2}}","-\frac{2 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{7 a d}+\frac{12 \cos ^3(c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{22 a \cos ^3(c+d x)}{105 d (a \sin (c+d x)+a)^{3/2}}",1,"(-22*a*Cos[c + d*x]^3)/(105*d*(a + a*Sin[c + d*x])^(3/2)) + (12*Cos[c + d*x]^3)/(35*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(7*a*d)","A",4,4,31,0.1290,1,"{2877, 2856, 2674, 2673}"
338,1,60,0,0.1268181,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 a \cos ^3(c+d x)}{15 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}","\frac{2 a \cos ^3(c+d x)}{15 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 \cos ^3(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}",1,"(2*a*Cos[c + d*x]^3)/(15*d*(a + a*Sin[c + d*x])^(3/2)) - (2*Cos[c + d*x]^3)/(5*d*Sqrt[a + a*Sin[c + d*x]])","A",2,2,29,0.06897,1,"{2856, 2673}"
339,1,63,0,0.2162035,"\int \frac{\cos (c+d x) \cot (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}","\frac{2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(-2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(Sqrt[a]*d) + (2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])","A",4,4,27,0.1481,1,"{2874, 2981, 2773, 206}"
340,1,62,0,0.1006308,"\int \frac{\cot ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Cot[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}",1,"ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(Sqrt[a]*d) - Cot[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]])","A",4,4,23,0.1739,1,"{2716, 21, 2773, 206}"
341,1,100,0,0.3165272,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}","\frac{\cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}",1,"ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(4*Sqrt[a]*d) + Cot[c + d*x]/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])","A",5,5,29,0.1724,1,"{2874, 2980, 2772, 2773, 206}"
342,1,135,0,0.4031466,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}","\frac{\cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}",1,"ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(8*Sqrt[a]*d) + Cot[c + d*x]/(8*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*d*Sqrt[a + a*Sin[c + d*x]])","A",6,5,31,0.1613,1,"{2874, 2980, 2772, 2773, 206}"
343,1,184,0,0.5892252,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^(3/2),x]","\frac{76 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{105 a^2 d}+\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}+\frac{2 \sin ^3(c+d x) \cos (c+d x)}{7 a d \sqrt{a \sin (c+d x)+a}}-\frac{16 \sin ^2(c+d x) \cos (c+d x)}{35 a d \sqrt{a \sin (c+d x)+a}}-\frac{344 \cos (c+d x)}{105 a d \sqrt{a \sin (c+d x)+a}}","\frac{76 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{105 a^2 d}+\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}+\frac{2 \sin ^3(c+d x) \cos (c+d x)}{7 a d \sqrt{a \sin (c+d x)+a}}-\frac{16 \sin ^2(c+d x) \cos (c+d x)}{35 a d \sqrt{a \sin (c+d x)+a}}-\frac{344 \cos (c+d x)}{105 a d \sqrt{a \sin (c+d x)+a}}",1,"(2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (344*Cos[c + d*x])/(105*a*d*Sqrt[a + a*Sin[c + d*x]]) - (16*Cos[c + d*x]*Sin[c + d*x]^2)/(35*a*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sin[c + d*x]^3)/(7*a*d*Sqrt[a + a*Sin[c + d*x]]) + (76*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*a^2*d)","A",8,7,31,0.2258,1,"{2879, 2983, 2968, 3023, 2751, 2649, 206}"
344,1,140,0,0.3482671,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{4 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 a^2 d}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{2 \cos ^3(c+d x)}{5 a d \sqrt{a \sin (c+d x)+a}}+\frac{18 \cos (c+d x)}{5 a d \sqrt{a \sin (c+d x)+a}}","-\frac{4 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 a^2 d}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{2 \cos ^3(c+d x)}{5 a d \sqrt{a \sin (c+d x)+a}}+\frac{18 \cos (c+d x)}{5 a d \sqrt{a \sin (c+d x)+a}}",1,"(-2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) + (18*Cos[c + d*x])/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]^3)/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) - (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*a^2*d)","A",5,5,31,0.1613,1,"{2878, 2858, 2751, 2649, 206}"
345,1,108,0,0.1575635,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a^2 d}+\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{10 \cos (c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}","\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a^2 d}+\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{10 \cos (c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}",1,"(2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (10*Cos[c + d*x])/(3*a*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*a^2*d)","A",4,4,29,0.1379,1,"{2858, 2751, 2649, 206}"
346,1,85,0,0.258633,"\int \frac{\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}","\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}",1,"(-2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d) + (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d)","A",6,5,27,0.1852,1,"{2874, 2985, 2649, 206, 2773}"
347,1,113,0,0.2238294,"\int \frac{\cot ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2),x]","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\cot (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\cot (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}",1,"(3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - Cot[c + d*x]/(a*d*Sqrt[a + a*Sin[c + d*x]])","A",6,5,23,0.2174,1,"{2715, 2985, 2649, 206, 2773}"
348,1,153,0,0.5440839,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{3/2} d}+\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}+\frac{5 \cot (c+d x)}{4 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d \sqrt{a \sin (c+d x)+a}}","-\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{3/2} d}+\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}+\frac{5 \cot (c+d x)}{4 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d \sqrt{a \sin (c+d x)+a}}",1,"(-11*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(3/2)*d) + (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) + (5*Cot[c + d*x])/(4*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*Sqrt[a + a*Sin[c + d*x]])","A",8,6,29,0.2069,1,"{2874, 2984, 2985, 2649, 206, 2773}"
349,1,191,0,0.7144883,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]","\frac{23 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 a^{3/2} d}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{9 \cot (c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}+\frac{7 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt{a \sin (c+d x)+a}}","\frac{23 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 a^{3/2} d}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{9 \cot (c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}+\frac{7 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt{a \sin (c+d x)+a}}",1,"(23*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*a^(3/2)*d) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (9*Cot[c + d*x])/(8*a*d*Sqrt[a + a*Sin[c + d*x]]) + (7*Cot[c + d*x]*Csc[c + d*x])/(12*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*Sqrt[a + a*Sin[c + d*x]])","A",9,6,31,0.1935,1,"{2874, 2984, 2985, 2649, 206, 2773}"
350,1,65,0,0.0688219,"\int \cos ^3(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^6(c+d x)}{6 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^4(c+d x)}{4 d}","-\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^6(c+d x)}{6 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^4(c+d x)}{4 d}",1,"(a*Sin[c + d*x]^4)/(4*d) + (a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^6)/(6*d) - (a*Sin[c + d*x]^7)/(7*d)","A",4,3,27,0.1111,1,"{2836, 12, 75}"
351,1,65,0,0.072234,"\int \cos ^3(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^6(c+d x)}{6 d}-\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{3 d}","-\frac{a \sin ^6(c+d x)}{6 d}-\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{3 d}",1,"(a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^4)/(4*d) - (a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^6)/(6*d)","A",4,3,27,0.1111,1,"{2836, 12, 75}"
352,1,49,0,0.0811166,"\int \cos ^3(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \cos ^4(c+d x)}{4 d}","-\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \cos ^4(c+d x)}{4 d}",1,"-(a*Cos[c + d*x]^4)/(4*d) + (a*Sin[c + d*x]^3)/(3*d) - (a*Sin[c + d*x]^5)/(5*d)","A",6,5,25,0.2000,1,"{2834, 2565, 30, 2564, 14}"
353,1,45,0,0.0332209,"\int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{2 (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac{(a \sin (c+d x)+a)^4}{4 a^3 d}","\frac{2 (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac{(a \sin (c+d x)+a)^4}{4 a^3 d}",1,"(2*(a + a*Sin[c + d*x])^3)/(3*a^2*d) - (a + a*Sin[c + d*x])^4/(4*a^3*d)","A",3,2,19,0.1053,1,"{2667, 43}"
354,1,56,0,0.0552297,"\int \cos ^2(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}","-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"(a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d) - (a*Sin[c + d*x]^3)/(3*d)","A",4,3,25,0.1200,1,"{2836, 12, 75}"
355,1,53,0,0.057603,"\int \cos (c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}","-\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"-((a*Csc[c + d*x])/d) + (a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d)","A",4,3,25,0.1200,1,"{2836, 12, 75}"
356,1,54,0,0.0341207,"\int \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d}","-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d}",1,"-((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d","A",3,2,19,0.1053,1,"{2707, 75}"
357,1,37,0,0.0970588,"\int \frac{\cos ^3(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^4(c+d x)}{4 a d}","\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^4(c+d x)}{4 a d}",1,"Sin[c + d*x]^3/(3*a*d) - Sin[c + d*x]^4/(4*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 43}"
358,1,37,0,0.0803932,"\int \frac{\cos ^3(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin ^3(c+d x)}{3 a d}","\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin ^3(c+d x)}{3 a d}",1,"Sin[c + d*x]^2/(2*a*d) - Sin[c + d*x]^3/(3*a*d)","A",5,3,27,0.1111,1,"{2835, 2564, 30}"
359,1,32,0,0.0416463,"\int \frac{\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^3/(a + a*Sin[c + d*x]),x]","\frac{\sin (c+d x)}{a d}-\frac{\sin ^2(c+d x)}{2 a d}","\frac{\sin (c+d x)}{a d}-\frac{\sin ^2(c+d x)}{2 a d}",1,"Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d)","A",2,1,21,0.04762,1,"{2667}"
360,1,29,0,0.0768426,"\int \frac{\cos ^2(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\log (\sin (c+d x))}{a d}-\frac{\sin (c+d x)}{a d}","\frac{\log (\sin (c+d x))}{a d}-\frac{\sin (c+d x)}{a d}",1,"Log[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d)","A",4,3,27,0.1111,1,"{2836, 12, 43}"
361,1,30,0,0.0792449,"\int \frac{\cos (c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}","-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}",1,"-(Csc[c + d*x]/(a*d)) - Log[Sin[c + d*x]]/(a*d)","A",4,3,27,0.1111,1,"{2836, 12, 43}"
362,1,32,0,0.06456,"\int \frac{\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^3/(a + a*Sin[c + d*x]),x]","\frac{\csc (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}","\frac{\csc (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}",1,"Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d)","A",5,4,21,0.1905,1,"{2706, 2606, 30, 8}"
363,1,37,0,0.0809194,"\int \frac{\cot ^3(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^3*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc ^3(c+d x)}{3 a d}","\frac{\csc ^2(c+d x)}{2 a d}-\frac{\csc ^3(c+d x)}{3 a d}",1,"Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d)","A",4,3,27,0.1111,1,"{2836, 12, 43}"
364,1,37,0,0.0976363,"\int \frac{\cot ^3(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^3*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\csc ^3(c+d x)}{3 a d}-\frac{\csc ^4(c+d x)}{4 a d}","\frac{\csc ^3(c+d x)}{3 a d}-\frac{\csc ^4(c+d x)}{4 a d}",1,"Csc[c + d*x]^3/(3*a*d) - Csc[c + d*x]^4/(4*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 43}"
365,1,143,0,0.1690908,"\int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \cos ^9(c+d x)}{9 d}+\frac{2 a \cos ^7(c+d x)}{7 d}-\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{a \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a x}{128}","-\frac{a \cos ^9(c+d x)}{9 d}+\frac{2 a \cos ^7(c+d x)}{7 d}-\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{a \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a x}{128}",1,"(3*a*x)/128 - (a*Cos[c + d*x]^5)/(5*d) + (2*a*Cos[c + d*x]^7)/(7*d) - (a*Cos[c + d*x]^9)/(9*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)","A",9,6,27,0.2222,1,"{2838, 2568, 2635, 8, 2565, 270}"
366,1,127,0,0.1648288,"\int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{a \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a x}{128}","\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{a \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a x}{128}",1,"(3*a*x)/128 - (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]^7)/(7*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)","A",9,6,27,0.2222,1,"{2838, 2565, 14, 2568, 2635, 8}"
367,1,103,0,0.1325115,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{a x}{16}","\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{a x}{16}",1,"(a*x)/16 - (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)","A",8,6,27,0.2222,1,"{2838, 2568, 2635, 8, 2565, 14}"
368,1,87,0,0.0993137,"\int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{a x}{16}","-\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{a x}{16}",1,"(a*x)/16 - (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)","A",7,6,25,0.2400,1,"{2838, 2565, 30, 2568, 2635, 8}"
369,1,89,0,0.086916,"\int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a x}{8}","\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a x}{8}",1,"(3*a*x)/8 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",8,6,25,0.2400,1,"{2838, 2592, 302, 206, 2635, 8}"
370,1,83,0,0.1136012,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}-\frac{3 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 a x}{2}","\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}-\frac{3 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 a x}{2}",1,"(-3*a*x)/2 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) - (3*a*Cot[c + d*x])/(2*d) + (a*Cos[c + d*x]^2*Cot[c + d*x])/(2*d)","A",9,8,27,0.2963,1,"{2838, 2591, 288, 321, 203, 2592, 302, 206}"
371,1,94,0,0.109099,"\int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{3 a \cos (c+d x)}{2 d}-\frac{3 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a x}{2}","-\frac{3 a \cos (c+d x)}{2 d}-\frac{3 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a x}{2}",1,"(-3*a*x)/2 + (3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a*Cos[c + d*x])/(2*d) - (3*a*Cot[c + d*x])/(2*d) + (a*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (a*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d)","A",9,7,25,0.2800,1,"{2838, 2592, 288, 321, 206, 2591, 203}"
372,1,82,0,0.0746513,"\int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{3 a \cos (c+d x)}{2 d}-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \cot (c+d x)}{d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}+a x","-\frac{3 a \cos (c+d x)}{2 d}-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \cot (c+d x)}{d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}+a x",1,"a*x + (3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a*Cos[c + d*x])/(2*d) + (a*Cot[c + d*x])/d - (a*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d)","A",9,7,19,0.3684,1,"{2710, 2592, 288, 321, 206, 3473, 8}"
373,1,88,0,0.0994979,"\int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \cot (c+d x)}{d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}+a x","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \cot (c+d x)}{d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}+a x",1,"a*x - (3*a*ArcTanh[Cos[c + d*x]])/(8*d) + (a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/(3*d) + (3*a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x])/(4*d)","A",7,5,25,0.2000,1,"{2838, 2611, 3770, 3473, 8}"
374,1,74,0,0.1176726,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}",1,"(-3*a*ArcTanh[Cos[c + d*x]])/(8*d) - (a*Cot[c + d*x]^5)/(5*d) + (3*a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x])/(4*d)","A",6,5,27,0.1852,1,"{2838, 2607, 30, 2611, 3770}"
375,1,98,0,0.1523389,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a \cot (c+d x) \csc (c+d x)}{16 d}","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a \cot (c+d x) \csc (c+d x)}{16 d}",1,"-(a*ArcTanh[Cos[c + d*x]])/(16*d) - (a*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)","A",7,6,27,0.2222,1,"{2838, 2611, 3768, 3770, 2607, 30}"
376,1,114,0,0.1543837,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a \cot (c+d x) \csc (c+d x)}{16 d}","-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a \cot (c+d x) \csc (c+d x)}{16 d}",1,"-(a*ArcTanh[Cos[c + d*x]])/(16*d) - (a*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)","A",8,6,27,0.2222,1,"{2838, 2607, 14, 2611, 3768, 3770}"
377,1,136,0,0.1713368,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{128 d}","-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{128 d}",1,"(-3*a*ArcTanh[Cos[c + d*x]])/(128*d) - (a*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]^7)/(7*d) - (3*a*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (a*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)","A",9,6,27,0.2222,1,"{2838, 2611, 3768, 3770, 2607, 14}"
378,1,185,0,0.3392053,"\int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cos ^9(c+d x)}{9 d}+\frac{4 a^2 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac{3 a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac{3 a^2 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{3 a^2 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{9 a^2 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{9 a^2 x}{256}","-\frac{2 a^2 \cos ^9(c+d x)}{9 d}+\frac{4 a^2 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac{3 a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac{3 a^2 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{3 a^2 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{9 a^2 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{9 a^2 x}{256}",1,"(9*a^2*x)/256 - (2*a^2*Cos[c + d*x]^5)/(5*d) + (4*a^2*Cos[c + d*x]^7)/(7*d) - (2*a^2*Cos[c + d*x]^9)/(9*d) + (9*a^2*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (3*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) - (3*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(32*d) - (3*a^2*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*d)","A",16,6,29,0.2069,1,"{2873, 2568, 2635, 8, 2565, 270}"
379,1,159,0,0.2549252,"\int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos ^9(c+d x)}{9 d}+\frac{3 a^2 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{4 d}-\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{3 a^2 x}{64}","-\frac{a^2 \cos ^9(c+d x)}{9 d}+\frac{3 a^2 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{4 d}-\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{3 a^2 x}{64}",1,"(3*a^2*x)/64 - (2*a^2*Cos[c + d*x]^5)/(5*d) + (3*a^2*Cos[c + d*x]^7)/(7*d) - (a^2*Cos[c + d*x]^9)/(9*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(32*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(8*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x]^3)/(4*d)","A",13,7,29,0.2414,1,"{2873, 2565, 14, 2568, 2635, 8, 270}"
380,1,141,0,0.2578263,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{11 a^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{11 a^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{11 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{11 a^2 x}{128}","\frac{2 a^2 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{11 a^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{11 a^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{11 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{11 a^2 x}{128}",1,"(11*a^2*x)/128 - (2*a^2*Cos[c + d*x]^5)/(5*d) + (2*a^2*Cos[c + d*x]^7)/(7*d) + (11*a^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (11*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) - (11*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)","A",14,6,29,0.2069,1,"{2873, 2568, 2635, 8, 2565, 14}"
381,1,129,0,0.1389629,"\int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos ^5(c+d x)}{15 d}-\frac{\cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{21 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a^2 x}{8}-\frac{\cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}","-\frac{a^2 \cos ^5(c+d x)}{15 d}-\frac{\cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{21 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a^2 x}{8}-\frac{\cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}",1,"(a^2*x)/8 - (a^2*Cos[c + d*x]^5)/(15*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(12*d) - (Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(7*d) - (Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x]))/(21*d)","A",6,5,27,0.1852,1,"{2860, 2678, 2669, 2635, 8}"
382,1,119,0,0.1428978,"\int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos ^5(c+d x)}{5 d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{4 d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^2 x}{4}","-\frac{a^2 \cos ^5(c+d x)}{5 d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{4 d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^2 x}{4}",1,"(3*a^2*x)/4 - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) - (a^2*Cos[c + d*x]^5)/(5*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(2*d)","A",11,8,27,0.2963,1,"{2873, 2635, 8, 2592, 302, 206, 2565, 30}"
383,1,116,0,0.1968788,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \cos ^3(c+d x)}{3 d}+\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}-\frac{a^2 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{9 a^2 x}{8}","\frac{2 a^2 \cos ^3(c+d x)}{3 d}+\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}-\frac{a^2 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{9 a^2 x}{8}",1,"(-9*a^2*x)/8 - (2*a^2*ArcTanh[Cos[c + d*x]])/d + (2*a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",13,7,29,0.2414,1,"{2872, 3770, 3767, 8, 2638, 2635, 2633}"
384,1,98,0,0.1570489,"\int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \cos ^3(c+d x)}{3 d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d}-3 a^2 x","\frac{a^2 \cos ^3(c+d x)}{3 d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d}-3 a^2 x",1,"-3*a^2*x + (a^2*ArcTanh[Cos[c + d*x]])/(2*d) + (a^2*Cos[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (a^2*Cos[c + d*x]*Sin[c + d*x])/d","A",12,8,27,0.2963,1,"{2872, 3770, 3767, 8, 3768, 2638, 2635, 2633}"
385,1,98,0,0.1551117,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^2 x}{2}","-\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^2 x}{2}",1,"-(a^2*x)/2 + (3*a^2*ArcTanh[Cos[c + d*x]])/d - (2*a^2*Cos[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]*Csc[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",12,7,21,0.3333,1,"{2709, 3770, 3767, 8, 3768, 2638, 2635}"
386,1,116,0,0.1826473,"\int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{2 a^2 \cot (c+d x)}{d}+\frac{9 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a^2 \cot (c+d x) \csc (c+d x)}{8 d}+2 a^2 x","-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{2 a^2 \cot (c+d x)}{d}+\frac{9 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a^2 \cot (c+d x) \csc (c+d x)}{8 d}+2 a^2 x",1,"2*a^2*x + (9*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (a^2*Cos[c + d*x])/d + (2*a^2*Cot[c + d*x])/d - (2*a^2*Cot[c + d*x]^3)/(3*d) + (a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)","A",13,6,27,0.2222,1,"{2872, 3770, 3767, 8, 3768, 2638}"
387,1,118,0,0.1911948,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{a^2 \cot ^3(c+d x) \csc (c+d x)}{2 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}+a^2 x","-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{a^2 \cot ^3(c+d x) \csc (c+d x)}{2 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}+a^2 x",1,"a^2*x - (3*a^2*ArcTanh[Cos[c + d*x]])/(4*d) + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(4*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x])/(2*d)","A",10,7,29,0.2414,1,"{2873, 3473, 8, 2611, 3770, 2607, 30}"
388,1,132,0,0.2617613,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{7 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac{a^2 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{16 d}","-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{7 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac{a^2 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{16 d}",1,"(-7*a^2*ArcTanh[Cos[c + d*x]])/(16*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) + (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)","A",11,6,29,0.2069,1,"{2873, 2611, 3770, 2607, 30, 3768}"
389,1,176,0,0.3203702,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{11 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{128 d}","-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{11 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{128 d}",1,"(-11*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (11*a^2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)","A",14,6,29,0.2069,1,"{2873, 2611, 3768, 3770, 2607, 14}"
390,1,168,0,0.274119,"\int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^9(c+d x)}{9 d}-\frac{3 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{64 d}","-\frac{a^2 \cot ^9(c+d x)}{9 d}-\frac{3 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{64 d}",1,"(-3*a^2*ArcTanh[Cos[c + d*x]])/(64*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (3*a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(9*d) - (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(64*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(32*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(8*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(4*d)","A",13,7,29,0.2414,1,"{2873, 2607, 14, 2611, 3768, 3770, 270}"
391,1,218,0,0.352659,"\int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{4 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{9 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac{9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{9 a^2 \cot (c+d x) \csc (c+d x)}{256 d}","-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{4 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{9 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac{9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{9 a^2 \cot (c+d x) \csc (c+d x)}{256 d}",1,"(-9*a^2*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (4*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) - (9*a^2*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (3*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (9*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x]^7)/(80*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*d)","A",16,6,29,0.2069,1,"{2873, 2611, 3768, 3770, 2607, 270}"
392,1,203,0,0.3931109,"\int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \cos ^{11}(c+d x)}{11 d}-\frac{2 a^3 \cos ^9(c+d x)}{3 d}+\frac{9 a^3 \cos ^7(c+d x)}{7 d}-\frac{4 a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac{5 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac{5 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{5 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{15 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{15 a^3 x}{256}","\frac{a^3 \cos ^{11}(c+d x)}{11 d}-\frac{2 a^3 \cos ^9(c+d x)}{3 d}+\frac{9 a^3 \cos ^7(c+d x)}{7 d}-\frac{4 a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac{5 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac{5 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{5 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{15 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{15 a^3 x}{256}",1,"(15*a^3*x)/256 - (4*a^3*Cos[c + d*x]^5)/(5*d) + (9*a^3*Cos[c + d*x]^7)/(7*d) - (2*a^3*Cos[c + d*x]^9)/(3*d) + (a^3*Cos[c + d*x]^11)/(11*d) + (15*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (5*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) - (5*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(32*d) - (5*a^3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*d) - (3*a^3*Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*d)","A",19,6,29,0.2069,1,"{2873, 2568, 2635, 8, 2565, 270}"
393,1,182,0,0.3823665,"\int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos ^9(c+d x)}{3 d}+\frac{a^3 \cos ^7(c+d x)}{d}-\frac{4 a^3 \cos ^5(c+d x)}{5 d}-\frac{a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac{7 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac{7 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{7 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{21 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{21 a^3 x}{256}","-\frac{a^3 \cos ^9(c+d x)}{3 d}+\frac{a^3 \cos ^7(c+d x)}{d}-\frac{4 a^3 \cos ^5(c+d x)}{5 d}-\frac{a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac{7 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac{7 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{7 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{21 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{21 a^3 x}{256}",1,"(21*a^3*x)/256 - (4*a^3*Cos[c + d*x]^5)/(5*d) + (a^3*Cos[c + d*x]^7)/d - (a^3*Cos[c + d*x]^9)/(3*d) + (21*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (7*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) - (7*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(32*d) - (7*a^3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*d)","A",19,7,29,0.2414,1,"{2873, 2565, 14, 2568, 2635, 8, 270}"
394,1,159,0,0.3230358,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos ^9(c+d x)}{9 d}+\frac{5 a^3 \cos ^7(c+d x)}{7 d}-\frac{4 a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{17 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{17 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{17 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{17 a^3 x}{128}","-\frac{a^3 \cos ^9(c+d x)}{9 d}+\frac{5 a^3 \cos ^7(c+d x)}{7 d}-\frac{4 a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{17 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{17 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{17 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{17 a^3 x}{128}",1,"(17*a^3*x)/128 - (4*a^3*Cos[c + d*x]^5)/(5*d) + (5*a^3*Cos[c + d*x]^7)/(7*d) - (a^3*Cos[c + d*x]^9)/(9*d) + (17*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (17*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) - (17*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (3*a^3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)","A",17,7,29,0.2414,1,"{2873, 2568, 2635, 8, 2565, 14, 270}"
395,1,157,0,0.1938719,"\int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]","-\frac{9 a^3 \cos ^5(c+d x)}{80 d}-\frac{9 \cos ^5(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{112 d}+\frac{9 a^3 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{27 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{27 a^3 x}{128}-\frac{\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}-\frac{3 a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{56 d}","-\frac{9 a^3 \cos ^5(c+d x)}{80 d}-\frac{9 \cos ^5(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{112 d}+\frac{9 a^3 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{27 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{27 a^3 x}{128}-\frac{\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}-\frac{3 a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{56 d}",1,"(27*a^3*x)/128 - (9*a^3*Cos[c + d*x]^5)/(80*d) + (27*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (9*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (3*a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(56*d) - (Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3)/(8*d) - (9*Cos[c + d*x]^5*(a^3 + a^3*Sin[c + d*x]))/(112*d)","A",7,5,27,0.1852,1,"{2860, 2678, 2669, 2635, 8}"
396,1,143,0,0.2021891,"\int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{19 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{19 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{19 a^3 x}{16}","-\frac{3 a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{19 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{19 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{19 a^3 x}{16}",1,"(19*a^3*x)/16 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) - (3*a^3*Cos[c + d*x]^5)/(5*d) + (19*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (19*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)","A",15,9,27,0.3333,1,"{2873, 2635, 8, 2592, 302, 206, 2565, 30, 2568}"
397,1,131,0,0.1964659,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}-\frac{3 a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{11 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 a^3 x}{8}","-\frac{a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}-\frac{3 a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{11 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 a^3 x}{8}",1,"(-3*a^3*x)/8 - (3*a^3*ArcTanh[Cos[c + d*x]])/d + (3*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/d - (a^3*Cos[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x])/d + (11*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (3*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",15,7,29,0.2414,1,"{2872, 3770, 3767, 8, 2638, 2635, 2633}"
398,1,137,0,0.1826179,"\int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \cos ^3(c+d x)}{d}+\frac{2 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{33 a^3 x}{8}","\frac{a^3 \cos ^3(c+d x)}{d}+\frac{2 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{33 a^3 x}{8}",1,"(-33*a^3*x)/8 - (3*a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (2*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/d - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (7*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",15,8,27,0.2963,1,"{2872, 3770, 3767, 8, 3768, 2638, 2635, 2633}"
399,1,134,0,0.1776027,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{2 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{2 a^3 \cot (c+d x)}{d}-\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{7 a^3 x}{2}","\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{2 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{2 a^3 \cot (c+d x)}{d}-\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{7 a^3 x}{2}",1,"(-7*a^3*x)/2 + (7*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (2*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) - (2*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",14,8,21,0.3810,1,"{2709, 3770, 3767, 8, 3768, 2638, 2635, 2633}"
400,1,138,0,0.1922187,"\int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{d}+\frac{2 a^3 \cot (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{33 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{7 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{3 a^3 x}{2}","-\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{d}+\frac{2 a^3 \cot (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{33 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{7 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{3 a^3 x}{2}",1,"(3*a^3*x)/2 + (33*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (3*a^3*Cos[c + d*x])/d + (2*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/d - (7*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) - (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",15,7,27,0.2593,1,"{2872, 3770, 3767, 8, 3768, 2638, 2635}"
401,1,132,0,0.216112,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{d}+\frac{3 a^3 \cot (c+d x)}{d}+\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+3 a^3 x","-\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{d}+\frac{3 a^3 \cot (c+d x)}{d}+\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+3 a^3 x",1,"3*a^3*x + (3*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (a^3*Cos[c + d*x])/d + (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/d - (a^3*Cot[c + d*x]^5)/(5*d) + (11*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)","A",15,6,29,0.2069,1,"{2872, 3770, 3767, 8, 3768, 2638}"
402,1,168,0,0.274657,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{19 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac{3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+a^3 x","-\frac{3 a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}-\frac{19 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac{3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+a^3 x",1,"a^3*x - (19*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]^5)/(5*d) + (17*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)","A",14,8,29,0.2759,1,"{2873, 3473, 8, 2611, 3770, 2607, 30, 3768}"
403,1,150,0,0.2842785,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}-\frac{a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{16 d}","-\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}-\frac{a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{16 d}",1,"(-9*a^3*ArcTanh[Cos[c + d*x]])/(16*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(2*d)","A",14,7,29,0.2414,1,"{2873, 2611, 3770, 2607, 30, 3768, 14}"
404,1,176,0,0.3381763,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{27 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{128 d}","-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{27 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{128 d}",1,"(-27*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x]^7)/(7*d) - (27*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (23*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(2*d) + (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)","A",16,7,29,0.2414,1,"{2873, 2607, 30, 2611, 3768, 3770, 14}"
405,1,194,0,0.3508668,"\int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{5 a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{17 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{128 d}","-\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{5 a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{17 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{128 d}",1,"(-17*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (5*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/(9*d) - (17*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)","A",17,7,29,0.2414,1,"{2873, 2611, 3768, 3770, 2607, 14, 270}"
406,1,216,0,0.3898454,"\int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{a^3 \cot ^7(c+d x)}{d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{21 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac{29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}","-\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{a^3 \cot ^7(c+d x)}{d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{21 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac{29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}",1,"(-21*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/d - (a^3*Cot[c + d*x]^9)/(3*d) - (21*a^3*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (7*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (29*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x]^7)/(80*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*d)","A",19,7,29,0.2414,1,"{2873, 2607, 14, 2611, 3768, 3770, 270}"
407,1,187,0,0.2342333,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","-\frac{11 a^4 \cos ^7(c+d x)}{112 d}-\frac{\cos ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^2}{18 d}-\frac{11 \cos ^7(c+d x) \left(a^4 \sin (c+d x)+a^4\right)}{144 d}+\frac{11 a^4 \sin (c+d x) \cos ^5(c+d x)}{96 d}+\frac{55 a^4 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{55 a^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{55 a^4 x}{256}-\frac{\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}","-\frac{11 a^4 \cos ^7(c+d x)}{112 d}-\frac{\cos ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^2}{18 d}-\frac{11 \cos ^7(c+d x) \left(a^4 \sin (c+d x)+a^4\right)}{144 d}+\frac{11 a^4 \sin (c+d x) \cos ^5(c+d x)}{96 d}+\frac{55 a^4 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{55 a^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{55 a^4 x}{256}-\frac{\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}",1,"(55*a^4*x)/256 - (11*a^4*Cos[c + d*x]^7)/(112*d) + (55*a^4*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (55*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (11*a^4*Cos[c + d*x]^5*Sin[c + d*x])/(96*d) - (Cos[c + d*x]^5*(a + a*Sin[c + d*x])^5)/(10*a*d) - (Cos[c + d*x]^7*(a^2 + a^2*Sin[c + d*x])^2)/(18*d) - (11*Cos[c + d*x]^7*(a^4 + a^4*Sin[c + d*x]))/(144*d)","A",8,5,29,0.1724,1,"{2870, 2678, 2669, 2635, 8}"
408,1,140,0,0.228222,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]","\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{19 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{61 a^4 x}{8}","\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{5 a^4 \cot (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{19 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{2 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{61 a^4 x}{8}",1,"(-61*a^4*x)/8 + (2*a^4*ArcTanh[Cos[c + d*x]])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) - (5*a^4*Cot[c + d*x])/d - (a^4*Cot[c + d*x]^3)/(3*d) - (2*a^4*Cot[c + d*x]*Csc[c + d*x])/d - (19*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",17,8,21,0.3810,1,"{2709, 3770, 3767, 8, 3768, 2638, 2635, 2633}"
409,1,135,0,0.1976975,"\int \frac{\cos ^4(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\cos ^7(c+d x)}{7 a d}-\frac{2 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}-\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos (c+d x)}{16 a d}+\frac{x}{16 a}","\frac{\cos ^7(c+d x)}{7 a d}-\frac{2 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}-\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos (c+d x)}{16 a d}+\frac{x}{16 a}",1,"x/(16*a) + Cos[c + d*x]^3/(3*a*d) - (2*Cos[c + d*x]^5)/(5*a*d) + Cos[c + d*x]^7/(7*a*d) + (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(8*a*d) - (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a*d)","A",8,6,29,0.2069,1,"{2839, 2568, 2635, 8, 2565, 270}"
410,1,117,0,0.1911694,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos ^3(c+d x)}{3 a d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos (c+d x)}{16 a d}-\frac{x}{16 a}","\frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos ^3(c+d x)}{3 a d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos (c+d x)}{16 a d}-\frac{x}{16 a}",1,"-x/(16*a) - Cos[c + d*x]^3/(3*a*d) + Cos[c + d*x]^5/(5*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a*d)","A",8,6,29,0.2069,1,"{2839, 2565, 14, 2568, 2635, 8}"
411,1,91,0,0.1618901,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{\sin (c+d x) \cos (c+d x)}{8 a d}+\frac{x}{8 a}","-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{\sin (c+d x) \cos (c+d x)}{8 a d}+\frac{x}{8 a}",1,"x/(8*a) + Cos[c + d*x]^3/(3*a*d) - Cos[c + d*x]^5/(5*a*d) + (Cos[c + d*x]*Sin[c + d*x])/(8*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)","A",7,6,29,0.2069,1,"{2839, 2568, 2635, 8, 2565, 14}"
412,1,73,0,0.112174,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\cos ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a d}-\frac{x}{8 a}","-\frac{\cos ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a d}-\frac{x}{8 a}",1,"-x/(8*a) - Cos[c + d*x]^3/(3*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)","A",6,6,27,0.2222,1,"{2839, 2565, 30, 2568, 2635, 8}"
413,1,59,0,0.0980408,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{2 a}","\frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{2 a}",1,"-x/(2*a) - ArcTanh[Cos[c + d*x]]/(a*d) + Cos[c + d*x]/(a*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)","A",6,6,27,0.2222,1,"{2839, 2592, 321, 206, 2635, 8}"
414,1,49,0,0.1163467,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\cos (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{a}","-\frac{\cos (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{a}",1,"-(x/a) + ArcTanh[Cos[c + d*x]]/(a*d) - Cos[c + d*x]/(a*d) - Cot[c + d*x]/(a*d)","A",6,6,29,0.2069,1,"{2839, 3473, 8, 2592, 321, 206}"
415,1,58,0,0.1058614,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\cot (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}+\frac{x}{a}","\frac{\cot (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}+\frac{x}{a}",1,"x/a + ArcTanh[Cos[c + d*x]]/(2*a*d) + Cot[c + d*x]/(a*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)","A",5,5,27,0.1852,1,"{2839, 2611, 3770, 3473, 8}"
416,1,58,0,0.0884282,"\int \frac{\cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d}","-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d}",1,"-ArcTanh[Cos[c + d*x]]/(2*a*d) - Cot[c + d*x]^3/(3*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)","A",5,5,21,0.2381,1,"{2706, 2607, 30, 2611, 3770}"
417,1,82,0,0.1478184,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\cot ^3(c+d x)}{3 a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a d}","\frac{\cot ^3(c+d x)}{3 a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a d}",1,"ArcTanh[Cos[c + d*x]]/(8*a*d) + Cot[c + d*x]^3/(3*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(8*a*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)","A",6,6,27,0.2222,1,"{2839, 2611, 3768, 3770, 2607, 30}"
418,1,100,0,0.1719769,"\int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{\cot (c+d x) \csc (c+d x)}{8 a d}","-\frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{\cot (c+d x) \csc (c+d x)}{8 a d}",1,"-ArcTanh[Cos[c + d*x]]/(8*a*d) - Cot[c + d*x]^3/(3*a*d) - Cot[c + d*x]^5/(5*a*d) - (Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)","A",7,6,29,0.2069,1,"{2839, 2607, 14, 2611, 3768, 3770}"
419,1,124,0,0.1803284,"\int \frac{\cot ^4(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{24 a d}+\frac{\cot (c+d x) \csc (c+d x)}{16 a d}","\frac{\cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{24 a d}+\frac{\cot (c+d x) \csc (c+d x)}{16 a d}",1,"ArcTanh[Cos[c + d*x]]/(16*a*d) + Cot[c + d*x]^3/(3*a*d) + Cot[c + d*x]^5/(5*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(16*a*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(24*a*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a*d)","A",8,6,29,0.2069,1,"{2839, 2611, 3768, 3770, 2607, 14}"
420,1,147,0,0.2235456,"\int \frac{\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]","\frac{\cos ^7(c+d x)}{7 a^2 d}-\frac{4 \cos ^5(c+d x)}{5 a^2 d}+\frac{5 \cos ^3(c+d x)}{3 a^2 d}-\frac{2 \cos (c+d x)}{a^2 d}+\frac{\sin ^5(c+d x) \cos (c+d x)}{3 a^2 d}+\frac{5 \sin ^3(c+d x) \cos (c+d x)}{12 a^2 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{5 x}{8 a^2}","\frac{\cos ^7(c+d x)}{7 a^2 d}-\frac{4 \cos ^5(c+d x)}{5 a^2 d}+\frac{5 \cos ^3(c+d x)}{3 a^2 d}-\frac{2 \cos (c+d x)}{a^2 d}+\frac{\sin ^5(c+d x) \cos (c+d x)}{3 a^2 d}+\frac{5 \sin ^3(c+d x) \cos (c+d x)}{12 a^2 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{5 x}{8 a^2}",1,"(-5*x)/(8*a^2) - (2*Cos[c + d*x])/(a^2*d) + (5*Cos[c + d*x]^3)/(3*a^2*d) - (4*Cos[c + d*x]^5)/(5*a^2*d) + Cos[c + d*x]^7/(7*a^2*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (5*Cos[c + d*x]*Sin[c + d*x]^3)/(12*a^2*d) + (Cos[c + d*x]*Sin[c + d*x]^5)/(3*a^2*d)","A",11,5,29,0.1724,1,"{2869, 2757, 2633, 2635, 8}"
421,1,129,0,0.2250786,"\int \frac{\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cos ^5(c+d x)}{5 a^2 d}-\frac{4 \cos ^3(c+d x)}{3 a^2 d}+\frac{2 \cos (c+d x)}{a^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x)}{6 a^2 d}-\frac{11 \sin ^3(c+d x) \cos (c+d x)}{24 a^2 d}-\frac{11 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{11 x}{16 a^2}","\frac{2 \cos ^5(c+d x)}{5 a^2 d}-\frac{4 \cos ^3(c+d x)}{3 a^2 d}+\frac{2 \cos (c+d x)}{a^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x)}{6 a^2 d}-\frac{11 \sin ^3(c+d x) \cos (c+d x)}{24 a^2 d}-\frac{11 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{11 x}{16 a^2}",1,"(11*x)/(16*a^2) + (2*Cos[c + d*x])/(a^2*d) - (4*Cos[c + d*x]^3)/(3*a^2*d) + (2*Cos[c + d*x]^5)/(5*a^2*d) - (11*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) - (11*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^2*d) - (Cos[c + d*x]*Sin[c + d*x]^5)/(6*a^2*d)","A",12,5,29,0.1724,1,"{2869, 2757, 2635, 8, 2633}"
422,1,102,0,0.199063,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cos ^5(c+d x)}{5 a^2 d}+\frac{\cos ^3(c+d x)}{a^2 d}-\frac{2 \cos (c+d x)}{a^2 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{2 a^2 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{4 a^2 d}-\frac{3 x}{4 a^2}","-\frac{\cos ^5(c+d x)}{5 a^2 d}+\frac{\cos ^3(c+d x)}{a^2 d}-\frac{2 \cos (c+d x)}{a^2 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{2 a^2 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{4 a^2 d}-\frac{3 x}{4 a^2}",1,"(-3*x)/(4*a^2) - (2*Cos[c + d*x])/(a^2*d) + Cos[c + d*x]^3/(a^2*d) - Cos[c + d*x]^5/(5*a^2*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(4*a^2*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(2*a^2*d)","A",10,5,29,0.1724,1,"{2869, 2757, 2633, 2635, 8}"
423,1,87,0,0.1975731,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \cos ^3(c+d x)}{3 a^2 d}+\frac{2 \cos (c+d x)}{a^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a^2 d}-\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{7 x}{8 a^2}","-\frac{2 \cos ^3(c+d x)}{3 a^2 d}+\frac{2 \cos (c+d x)}{a^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a^2 d}-\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{7 x}{8 a^2}",1,"(7*x)/(8*a^2) + (2*Cos[c + d*x])/(a^2*d) - (2*Cos[c + d*x]^3)/(3*a^2*d) - (7*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^2*d)","A",10,5,29,0.1724,1,"{2869, 2757, 2635, 8, 2633}"
424,1,70,0,0.1095223,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \cos ^3(c+d x)}{3 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{x}{a^2}-\frac{\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2}","-\frac{2 \cos ^3(c+d x)}{3 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{x}{a^2}-\frac{\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2}",1,"-(x/a^2) - (2*Cos[c + d*x]^3)/(3*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(a^2*d) - Cos[c + d*x]^5/(d*(a + a*Sin[c + d*x])^2)","A",4,4,27,0.1481,1,"{2859, 2682, 2635, 8}"
425,1,36,0,0.1293017,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\cos (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{2 x}{a^2}","-\frac{\cos (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{2 x}{a^2}",1,"(-2*x)/a^2 - ArcTanh[Cos[c + d*x]]/(a^2*d) - Cos[c + d*x]/(a^2*d)","A",4,4,27,0.1481,1,"{2869, 2746, 2735, 3770}"
426,1,35,0,0.1489474,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cot (c+d x)}{a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{x}{a^2}","-\frac{\cot (c+d x)}{a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{x}{a^2}",1,"x/a^2 + (2*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a^2*d)","A",6,5,29,0.1724,1,"{2869, 2757, 3770, 3767, 8}"
427,1,54,0,0.1514064,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cot (c+d x)}{a^2 d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}","\frac{2 \cot (c+d x)}{a^2 d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}",1,"(-3*ArcTanh[Cos[c + d*x]])/(2*a^2*d) + (2*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d)","A",8,6,27,0.2222,1,"{2869, 2757, 3770, 3767, 8, 3768}"
428,1,66,0,0.1261689,"\int \frac{\cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^2,x]","-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d}","-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d}",1,"ArcTanh[Cos[c + d*x]]/(a^2*d) - (2*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(a^2*d)","A",9,6,21,0.2857,1,"{2708, 2757, 3767, 8, 3768, 3770}"
429,1,96,0,0.1872412,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{7 \cot (c+d x) \csc (c+d x)}{8 a^2 d}","\frac{2 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{7 \cot (c+d x) \csc (c+d x)}{8 a^2 d}",1,"(-7*ArcTanh[Cos[c + d*x]])/(8*a^2*d) + (2*Cot[c + d*x])/(a^2*d) + (2*Cot[c + d*x]^3)/(3*a^2*d) - (7*Cot[c + d*x]*Csc[c + d*x])/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d)","A",10,5,27,0.1852,1,"{2869, 2757, 3768, 3770, 3767}"
430,1,112,0,0.240746,"\int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{a^2 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{4 a^2 d}","-\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{a^2 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{4 a^2 d}",1,"(3*ArcTanh[Cos[c + d*x]])/(4*a^2*d) - (2*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(a^2*d) - Cot[c + d*x]^5/(5*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(4*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(2*a^2*d)","A",10,5,29,0.1724,1,"{2869, 2757, 3767, 3768, 3770}"
431,1,138,0,0.2481817,"\int \frac{\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{16 a^2 d}","\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{4 \cot ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac{11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{16 a^2 d}",1,"(-11*ArcTanh[Cos[c + d*x]])/(16*a^2*d) + (2*Cot[c + d*x])/(a^2*d) + (4*Cot[c + d*x]^3)/(3*a^2*d) + (2*Cot[c + d*x]^5)/(5*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^2*d)","A",12,5,29,0.1724,1,"{2869, 2757, 3768, 3770, 3767}"
432,1,109,0,0.2592078,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","-\frac{\cos ^3(c+d x)}{a^3 d}+\frac{7 \cos (c+d x)}{a^3 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac{19 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}+\frac{51 x}{8 a^3}","-\frac{\cos ^3(c+d x)}{a^3 d}+\frac{7 \cos (c+d x)}{a^3 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac{19 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}+\frac{51 x}{8 a^3}",1,"(51*x)/(8*a^3) + (7*Cos[c + d*x])/(a^3*d) - Cos[c + d*x]^3/(a^3*d) - (19*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^3*d) + (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))","A",12,7,29,0.2414,1,"{2875, 2872, 2638, 2635, 8, 2633, 2648}"
433,1,87,0,0.226797,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\cos ^3(c+d x)}{3 a^3 d}-\frac{5 \cos (c+d x)}{a^3 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac{11 x}{2 a^3}","\frac{\cos ^3(c+d x)}{3 a^3 d}-\frac{5 \cos (c+d x)}{a^3 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac{11 x}{2 a^3}",1,"(-11*x)/(2*a^3) - (5*Cos[c + d*x])/(a^3*d) + Cos[c + d*x]^3/(3*a^3*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))","A",9,7,29,0.2414,1,"{2875, 2709, 2638, 2635, 8, 2633, 2648}"
434,1,80,0,0.1390811,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{9 \cos (c+d x)}{2 a^3 d}+\frac{3 \cos ^3(c+d x)}{2 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{9 x}{2 a^3}+\frac{\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^3}","\frac{9 \cos (c+d x)}{2 a^3 d}+\frac{3 \cos ^3(c+d x)}{2 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{9 x}{2 a^3}+\frac{\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^3}",1,"(9*x)/(2*a^3) + (9*Cos[c + d*x])/(2*a^3*d) + Cos[c + d*x]^5/(d*(a + a*Sin[c + d*x])^3) + (3*Cos[c + d*x]^3)/(2*d*(a^3 + a^3*Sin[c + d*x]))","A",4,4,27,0.1481,1,"{2859, 2679, 2682, 8}"
435,1,45,0,0.1828991,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{x}{a^3}","\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{x}{a^3}",1,"x/a^3 - ArcTanh[Cos[c + d*x]]/(a^3*d) + (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))","A",5,4,27,0.1481,1,"{2875, 2872, 3770, 2648}"
436,1,54,0,0.2355385,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}","-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}",1,"(3*ArcTanh[Cos[c + d*x]])/(a^3*d) - Cot[c + d*x]/(a^3*d) - (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))","A",7,6,29,0.2069,1,"{2875, 2872, 3770, 3767, 8, 2648}"
437,1,78,0,0.2485543,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{3 \cot (c+d x)}{a^3 d}+\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac{9 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}","\frac{3 \cot (c+d x)}{a^3 d}+\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac{9 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}",1,"(-9*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (3*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))","A",9,7,27,0.2593,1,"{2875, 2872, 3770, 3767, 8, 3768, 2648}"
438,1,96,0,0.1548573,"\int \frac{\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]","-\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{5 \cot (c+d x)}{a^3 d}+\frac{11 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{4 \cot (c+d x)}{a^3 d (\csc (c+d x)+1)}","-\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{5 \cot (c+d x)}{a^3 d}+\frac{11 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{4 \cot (c+d x)}{a^3 d (\csc (c+d x)+1)}",1,"(11*ArcTanh[Cos[c + d*x]])/(2*a^3*d) - (5*Cot[c + d*x])/(a^3*d) - Cot[c + d*x]^3/(3*a^3*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) - (4*Cot[c + d*x])/(a^3*d*(1 + Csc[c + d*x]))","A",11,6,21,0.2857,1,"{2709, 3770, 3767, 8, 3768, 3777}"
439,1,117,0,0.3003092,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{\cot ^3(c+d x)}{a^3 d}+\frac{7 \cot (c+d x)}{a^3 d}+\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac{51 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac{19 \cot (c+d x) \csc (c+d x)}{8 a^3 d}","\frac{\cot ^3(c+d x)}{a^3 d}+\frac{7 \cot (c+d x)}{a^3 d}+\frac{4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}-\frac{51 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac{19 \cot (c+d x) \csc (c+d x)}{8 a^3 d}",1,"(-51*ArcTanh[Cos[c + d*x]])/(8*a^3*d) + (7*Cot[c + d*x])/(a^3*d) + Cot[c + d*x]^3/(a^3*d) - (19*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d) + (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))","A",14,7,27,0.2593,1,"{2875, 2872, 3770, 3767, 8, 3768, 2648}"
440,1,58,0,0.1057192,"\int \frac{\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx","Int[(Cos[e + f*x]^4*Sin[e + f*x])/(a + a*Sin[e + f*x])^6,x]","\frac{\cos ^5(e+f x)}{7 f (a \sin (e+f x)+a)^6}-\frac{6 \cos ^5(e+f x)}{35 a f (a \sin (e+f x)+a)^5}","\frac{\cos ^5(e+f x)}{7 f (a \sin (e+f x)+a)^6}-\frac{6 \cos ^5(e+f x)}{35 a f (a \sin (e+f x)+a)^5}",1,"Cos[e + f*x]^5/(7*f*(a + a*Sin[e + f*x])^6) - (6*Cos[e + f*x]^5)/(35*a*f*(a + a*Sin[e + f*x])^5)","A",2,2,27,0.07407,1,"{2859, 2671}"
441,1,131,0,0.4589591,"\int \frac{\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx","Int[(Cos[e + f*x]^4*Sin[e + f*x]^2)/(a + a*Sin[e + f*x])^7,x]","-\frac{47 \cos (e+f x)}{315 a^7 f (\sin (e+f x)+1)}+\frac{268 \cos (e+f x)}{315 a^7 f (\sin (e+f x)+1)^2}-\frac{181 \cos (e+f x)}{105 a^7 f (\sin (e+f x)+1)^3}+\frac{92 \cos (e+f x)}{63 a^7 f (\sin (e+f x)+1)^4}-\frac{4 \cos (e+f x)}{9 a^7 f (\sin (e+f x)+1)^5}","-\frac{47 \cos ^5(e+f x)}{315 a^2 f (a \sin (e+f x)+a)^5}-\frac{a \cos ^7(e+f x)}{18 f (a \sin (e+f x)+a)^8}+\frac{25 \cos ^5(e+f x)}{126 a f (a \sin (e+f x)+a)^6}",1,"(-4*Cos[e + f*x])/(9*a^7*f*(1 + Sin[e + f*x])^5) + (92*Cos[e + f*x])/(63*a^7*f*(1 + Sin[e + f*x])^4) - (181*Cos[e + f*x])/(105*a^7*f*(1 + Sin[e + f*x])^3) + (268*Cos[e + f*x])/(315*a^7*f*(1 + Sin[e + f*x])^2) - (47*Cos[e + f*x])/(315*a^7*f*(1 + Sin[e + f*x]))","A",18,4,29,0.1379,1,"{2875, 2872, 2650, 2648}"
442,1,157,0,0.5695796,"\int \frac{\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx","Int[(Cos[e + f*x]^4*Sin[e + f*x]^3)/(a + a*Sin[e + f*x])^8,x]","-\frac{152 \cos (e+f x)}{1155 a^8 f (\sin (e+f x)+1)}+\frac{1003 \cos (e+f x)}{1155 a^8 f (\sin (e+f x)+1)^2}-\frac{846 \cos (e+f x)}{385 a^8 f (\sin (e+f x)+1)^3}+\frac{617 \cos (e+f x)}{231 a^8 f (\sin (e+f x)+1)^4}-\frac{52 \cos (e+f x)}{33 a^8 f (\sin (e+f x)+1)^5}+\frac{4 \cos (e+f x)}{11 a^8 f (\sin (e+f x)+1)^6}","-\frac{152 \cos (e+f x)}{1155 a^8 f (\sin (e+f x)+1)}+\frac{1003 \cos (e+f x)}{1155 a^8 f (\sin (e+f x)+1)^2}-\frac{846 \cos (e+f x)}{385 a^8 f (\sin (e+f x)+1)^3}+\frac{617 \cos (e+f x)}{231 a^8 f (\sin (e+f x)+1)^4}-\frac{52 \cos (e+f x)}{33 a^8 f (\sin (e+f x)+1)^5}+\frac{4 \cos (e+f x)}{11 a^8 f (\sin (e+f x)+1)^6}",1,"(4*Cos[e + f*x])/(11*a^8*f*(1 + Sin[e + f*x])^6) - (52*Cos[e + f*x])/(33*a^8*f*(1 + Sin[e + f*x])^5) + (617*Cos[e + f*x])/(231*a^8*f*(1 + Sin[e + f*x])^4) - (846*Cos[e + f*x])/(385*a^8*f*(1 + Sin[e + f*x])^3) + (1003*Cos[e + f*x])/(1155*a^8*f*(1 + Sin[e + f*x])^2) - (152*Cos[e + f*x])/(1155*a^8*f*(1 + Sin[e + f*x]))","A",24,4,29,0.1379,1,"{2875, 2872, 2650, 2648}"
443,1,156,0,0.424233,"\int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{368 a^2 \cos ^5(c+d x)}{9009 d (a \sin (c+d x)+a)^{3/2}}-\frac{1472 a^3 \cos ^5(c+d x)}{45045 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{13 a d}+\frac{20 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}-\frac{46 a \cos ^5(c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}","-\frac{368 a^2 \cos ^5(c+d x)}{9009 d (a \sin (c+d x)+a)^{3/2}}-\frac{1472 a^3 \cos ^5(c+d x)}{45045 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{13 a d}+\frac{20 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}-\frac{46 a \cos ^5(c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}",1,"(-1472*a^3*Cos[c + d*x]^5)/(45045*d*(a + a*Sin[c + d*x])^(5/2)) - (368*a^2*Cos[c + d*x]^5)/(9009*d*(a + a*Sin[c + d*x])^(3/2)) - (46*a*Cos[c + d*x]^5)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) + (20*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(143*d) - (2*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(13*a*d)","A",5,4,31,0.1290,1,"{2878, 2856, 2674, 2673}"
444,1,124,0,0.2565453,"\int \cos ^4(c+d x) \sin (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{16 a^2 \cos ^5(c+d x)}{693 d (a \sin (c+d x)+a)^{3/2}}-\frac{64 a^3 \cos ^5(c+d x)}{3465 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}-\frac{2 a \cos ^5(c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}","-\frac{16 a^2 \cos ^5(c+d x)}{693 d (a \sin (c+d x)+a)^{3/2}}-\frac{64 a^3 \cos ^5(c+d x)}{3465 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}-\frac{2 a \cos ^5(c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}",1,"(-64*a^3*Cos[c + d*x]^5)/(3465*d*(a + a*Sin[c + d*x])^(5/2)) - (16*a^2*Cos[c + d*x]^5)/(693*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^5)/(99*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(11*d)","A",4,3,29,0.1034,1,"{2856, 2674, 2673}"
445,1,159,0,0.491639,"\int \cos ^3(c+d x) \cot (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt{a \sin (c+d x)+a}}-\frac{12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac{164 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{105 d}+\frac{8 a \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}","-\frac{2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt{a \sin (c+d x)+a}}-\frac{12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac{164 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{105 d}+\frac{8 a \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"(-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d + (8*a*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sin[c + d*x]^3)/(7*d*Sqrt[a + a*Sin[c + d*x]]) + (164*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*d) - (12*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*a*d)","A",9,9,29,0.3103,1,"{2881, 2770, 2759, 2751, 2646, 3046, 2981, 2773, 206}"
446,1,148,0,0.4771775,"\int \cos ^2(c+d x) \cot ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}+\frac{4 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}+\frac{61 a \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}","-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 a d}+\frac{4 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}+\frac{61 a \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}",1,"-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (61*a*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) + (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*d) - (Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*a*d)","A",8,8,31,0.2581,1,"{2881, 2759, 2751, 2646, 3044, 2981, 2773, 206}"
447,1,156,0,0.4115931,"\int \cos (c+d x) \cot ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{2 a \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{a \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{13 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{2 a \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{a \cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{13 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}",1,"(13*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) - (2*a*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*d)","A",7,7,29,0.2414,1,"{2881, 2751, 2646, 3044, 2980, 2773, 206}"
448,1,163,0,0.3885025,"\int \cot ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{11 a \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}+\frac{11 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}","-\frac{2 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{11 a \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}+\frac{11 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}",1,"(11*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) - (2*a*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) + (11*a*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(3*d)","A",7,7,23,0.3043,1,"{2718, 2646, 3044, 2980, 2772, 2773, 206}"
449,1,173,0,0.5379998,"\int \cot ^4(c+d x) \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","\frac{61 a \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{67 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}}","\frac{61 a \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{67 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{61 a \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}}",1,"(-67*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*d) + (61*a*Cot[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (61*a*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(4*d)","A",9,6,29,0.2069,1,"{2881, 2773, 206, 3044, 2980, 2772}"
450,1,209,0,0.6905558,"\int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{31 a \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{31 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{128 d}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a \sin (c+d x)+a}}","-\frac{31 a \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{31 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{128 d}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt{a \sin (c+d x)+a}}+\frac{97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt{a \sin (c+d x)+a}}",1,"(-31*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*d) - (31*a*Cot[c + d*x])/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (97*a*Cot[c + d*x]*Csc[c + d*x])/(192*d*Sqrt[a + a*Sin[c + d*x]]) + (97*a*Cot[c + d*x]*Csc[c + d*x]^2)/(240*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(5*d)","A",11,6,31,0.1935,1,"{2881, 2772, 2773, 206, 3044, 2980}"
451,1,245,0,0.8115909,"\int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{55 a \cot (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{55 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{512 d}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{6 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt{a \sin (c+d x)+a}}+\frac{47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt{a \sin (c+d x)+a}}+\frac{329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt{a \sin (c+d x)+a}}-\frac{55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt{a \sin (c+d x)+a}}","-\frac{55 a \cot (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{55 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{512 d}-\frac{\cot (c+d x) \csc ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{6 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{60 d \sqrt{a \sin (c+d x)+a}}+\frac{47 a \cot (c+d x) \csc ^3(c+d x)}{160 d \sqrt{a \sin (c+d x)+a}}+\frac{329 a \cot (c+d x) \csc ^2(c+d x)}{960 d \sqrt{a \sin (c+d x)+a}}-\frac{55 a \cot (c+d x) \csc (c+d x)}{768 d \sqrt{a \sin (c+d x)+a}}",1,"(-55*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(512*d) - (55*a*Cot[c + d*x])/(512*d*Sqrt[a + a*Sin[c + d*x]]) - (55*a*Cot[c + d*x]*Csc[c + d*x])/(768*d*Sqrt[a + a*Sin[c + d*x]]) + (329*a*Cot[c + d*x]*Csc[c + d*x]^2)/(960*d*Sqrt[a + a*Sin[c + d*x]]) + (47*a*Cot[c + d*x]*Csc[c + d*x]^3)/(160*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(60*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(6*d)","A",13,6,31,0.1935,1,"{2881, 2772, 2773, 206, 3044, 2980}"
452,1,281,0,0.9476577,"\int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{61 a \cot (c+d x)}{1024 d \sqrt{a \sin (c+d x)+a}}-\frac{61 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{1024 d}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a \sin (c+d x)+a}}{7 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a \sin (c+d x)+a}}+\frac{193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt{a \sin (c+d x)+a}}+\frac{579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a \sin (c+d x)+a}}-\frac{61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt{a \sin (c+d x)+a}}-\frac{61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt{a \sin (c+d x)+a}}","-\frac{61 a \cot (c+d x)}{1024 d \sqrt{a \sin (c+d x)+a}}-\frac{61 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{1024 d}-\frac{\cot (c+d x) \csc ^6(c+d x) \sqrt{a \sin (c+d x)+a}}{7 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt{a \sin (c+d x)+a}}+\frac{193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt{a \sin (c+d x)+a}}+\frac{579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a \sin (c+d x)+a}}-\frac{61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt{a \sin (c+d x)+a}}-\frac{61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt{a \sin (c+d x)+a}}",1,"(-61*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(1024*d) - (61*a*Cot[c + d*x])/(1024*d*Sqrt[a + a*Sin[c + d*x]]) - (61*a*Cot[c + d*x]*Csc[c + d*x])/(1536*d*Sqrt[a + a*Sin[c + d*x]]) - (61*a*Cot[c + d*x]*Csc[c + d*x]^2)/(1920*d*Sqrt[a + a*Sin[c + d*x]]) + (579*a*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*d*Sqrt[a + a*Sin[c + d*x]]) + (193*a*Cot[c + d*x]*Csc[c + d*x]^4)/(840*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^5)/(84*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]])/(7*d)","A",15,6,31,0.1935,1,"{2881, 2772, 2773, 206, 3044, 2980}"
453,1,188,0,0.5147895,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{56 a^2 \cos ^5(c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^5(c+d x)}{1287 d (a \sin (c+d x)+a)^{3/2}}-\frac{256 a^4 \cos ^5(c+d x)}{6435 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{15 a d}+\frac{4 \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{39 d}-\frac{14 a \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{429 d}","-\frac{56 a^2 \cos ^5(c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^5(c+d x)}{1287 d (a \sin (c+d x)+a)^{3/2}}-\frac{256 a^4 \cos ^5(c+d x)}{6435 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{15 a d}+\frac{4 \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{39 d}-\frac{14 a \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{429 d}",1,"(-256*a^4*Cos[c + d*x]^5)/(6435*d*(a + a*Sin[c + d*x])^(5/2)) - (64*a^3*Cos[c + d*x]^5)/(1287*d*(a + a*Sin[c + d*x])^(3/2)) - (56*a^2*Cos[c + d*x]^5)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) - (14*a*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(429*d) + (4*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(39*d) - (2*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2))/(15*a*d)","A",6,4,31,0.1290,1,"{2878, 2856, 2674, 2673}"
454,1,156,0,0.3247183,"\int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{8 a^2 \cos ^5(c+d x)}{143 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^5(c+d x)}{1001 d (a \sin (c+d x)+a)^{3/2}}-\frac{256 a^4 \cos ^5(c+d x)}{5005 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}-\frac{6 a \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}","-\frac{8 a^2 \cos ^5(c+d x)}{143 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^5(c+d x)}{1001 d (a \sin (c+d x)+a)^{3/2}}-\frac{256 a^4 \cos ^5(c+d x)}{5005 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}-\frac{6 a \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}",1,"(-256*a^4*Cos[c + d*x]^5)/(5005*d*(a + a*Sin[c + d*x])^(5/2)) - (64*a^3*Cos[c + d*x]^5)/(1001*d*(a + a*Sin[c + d*x])^(3/2)) - (8*a^2*Cos[c + d*x]^5)/(143*d*Sqrt[a + a*Sin[c + d*x]]) - (6*a*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(143*d) - (2*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(13*d)","A",5,3,29,0.1034,1,"{2856, 2674, 2673}"
455,1,199,0,0.7107983,"\int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{34 a^2 \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{14 a^2 \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}+\frac{16 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 d}+\frac{388 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d}","-\frac{2 a^2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}-\frac{34 a^2 \sin ^3(c+d x) \cos (c+d x)}{63 d \sqrt{a \sin (c+d x)+a}}-\frac{14 a^2 \cos (c+d x)}{45 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}+\frac{16 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{105 d}+\frac{388 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 d}",1,"(-2*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d - (14*a^2*Cos[c + d*x])/(45*d*Sqrt[a + a*Sin[c + d*x]]) - (34*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sin[c + d*x]^4)/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (388*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*d) + (16*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*d)","A",12,12,29,0.4138,1,"{2881, 2763, 21, 2770, 2759, 2751, 2646, 3046, 2976, 2981, 2773, 206}"
456,1,178,0,0.6491816,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","\frac{171 a^2 \cos (c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}+\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}+\frac{69 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}","\frac{171 a^2 \cos (c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}+\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}+\frac{69 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}",1,"(-3*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d + (171*a^2*Cos[c + d*x])/(35*d*Sqrt[a + a*Sin[c + d*x]]) + (69*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(35*d) + (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*d) - (Cot[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/d - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(7*a*d)","A",10,10,31,0.3226,1,"{2881, 2759, 2751, 2647, 2646, 3044, 2976, 2981, 2773, 206}"
457,1,186,0,0.5714725,"\int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","\frac{73 a^2 \cos (c+d x)}{20 d \sqrt{a \sin (c+d x)+a}}+\frac{9 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac{3 a \cot (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}","\frac{73 a^2 \cos (c+d x)}{20 d \sqrt{a \sin (c+d x)+a}}+\frac{9 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 d}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}-\frac{2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac{3 a \cot (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}",1,"(9*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) + (73*a^2*Cos[c + d*x])/(20*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*d) - (3*a*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(2*d)","A",9,9,29,0.3103,1,"{2881, 2751, 2647, 2646, 3044, 2975, 2981, 2773, 206}"
458,1,197,0,0.4997659,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{8 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\frac{29 a^2 \cot (c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{37 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}","-\frac{8 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\frac{29 a^2 \cot (c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{37 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 d}-\frac{2 a \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{a \cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}",1,"(37*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) - (8*a^2*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (29*a^2*Cot[c + d*x])/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(3*d)","A",8,8,23,0.3478,1,"{2718, 2647, 2646, 3044, 2975, 2980, 2773, 206}"
459,1,205,0,0.7030112,"\int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 a^2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{149 a^2 \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}+\frac{21 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 d}+\frac{19 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}-\frac{a \cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{8 d}","-\frac{2 a^2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{149 a^2 \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}+\frac{21 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 d}+\frac{19 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}-\frac{a \cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{8 d}",1,"(21*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*d) - (2*a^2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) + (149*a^2*Cot[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (19*a^2*Cot[c + d*x]*Csc[c + d*x])/(32*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(8*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(4*d)","A",11,9,29,0.3103,1,"{2881, 2763, 21, 2773, 206, 3044, 2975, 2980, 2772}"
460,1,215,0,0.8079507,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","\frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{165 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{128 d}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a \sin (c+d x)+a}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{40 d}","\frac{91 a^2 \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{165 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{128 d}+\frac{31 a^2 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a \sin (c+d x)+a}}+\frac{73 a^2 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac{3 a \cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{40 d}",1,"(-165*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*d) + (91*a^2*Cot[c + d*x])/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (73*a^2*Cot[c + d*x]*Csc[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (31*a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(80*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a*Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(40*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(5*d)","A",12,9,31,0.2903,1,"{2881, 2762, 21, 2773, 206, 3044, 2975, 2980, 2772}"
461,1,253,0,0.9185463,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{179 a^2 \cot (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{179 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{512 d}+\frac{137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt{a \sin (c+d x)+a}}+\frac{239 a^2 \cot (c+d x) \csc ^2(c+d x)}{320 d \sqrt{a \sin (c+d x)+a}}+\frac{111 a^2 \cot (c+d x) \csc (c+d x)}{256 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{6 d}-\frac{a \cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{20 d}","-\frac{179 a^2 \cot (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{179 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{512 d}+\frac{137 a^2 \cot (c+d x) \csc ^3(c+d x)}{480 d \sqrt{a \sin (c+d x)+a}}+\frac{239 a^2 \cot (c+d x) \csc ^2(c+d x)}{320 d \sqrt{a \sin (c+d x)+a}}+\frac{111 a^2 \cot (c+d x) \csc (c+d x)}{256 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{6 d}-\frac{a \cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{20 d}",1,"(-179*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(512*d) - (179*a^2*Cot[c + d*x])/(512*d*Sqrt[a + a*Sin[c + d*x]]) + (111*a^2*Cot[c + d*x]*Csc[c + d*x])/(256*d*Sqrt[a + a*Sin[c + d*x]]) + (239*a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(320*d*Sqrt[a + a*Sin[c + d*x]]) + (137*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(480*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(20*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(6*d)","A",14,9,31,0.2903,1,"{2881, 2762, 21, 2772, 2773, 206, 3044, 2975, 2980}"
462,1,291,0,1.0608306,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{171 a^2 \cot (c+d x)}{1024 d \sqrt{a \sin (c+d x)+a}}-\frac{171 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{1024 d}+\frac{9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a \sin (c+d x)+a}}+\frac{199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt{a \sin (c+d x)+a}}-\frac{57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^6(c+d x) (a \sin (c+d x)+a)^{3/2}}{7 d}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{28 d}","-\frac{171 a^2 \cot (c+d x)}{1024 d \sqrt{a \sin (c+d x)+a}}-\frac{171 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{1024 d}+\frac{9 a^2 \cot (c+d x) \csc ^4(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{1237 a^2 \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt{a \sin (c+d x)+a}}+\frac{199 a^2 \cot (c+d x) \csc ^2(c+d x)}{640 d \sqrt{a \sin (c+d x)+a}}-\frac{57 a^2 \cot (c+d x) \csc (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^6(c+d x) (a \sin (c+d x)+a)^{3/2}}{7 d}-\frac{a \cot (c+d x) \csc ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{28 d}",1,"(-171*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(1024*d) - (171*a^2*Cot[c + d*x])/(1024*d*Sqrt[a + a*Sin[c + d*x]]) - (57*a^2*Cot[c + d*x]*Csc[c + d*x])/(512*d*Sqrt[a + a*Sin[c + d*x]]) + (199*a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(640*d*Sqrt[a + a*Sin[c + d*x]]) + (1237*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*d*Sqrt[a + a*Sin[c + d*x]]) + (9*a^2*Cot[c + d*x]*Csc[c + d*x]^4)/(40*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(28*d) - (Cot[c + d*x]*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2))/(7*d)","A",16,9,31,0.2903,1,"{2881, 2762, 21, 2772, 2773, 206, 3044, 2975, 2980}"
463,1,329,0,1.1873497,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{1587 a^2 \cot (c+d x)}{16384 d \sqrt{a \sin (c+d x)+a}}-\frac{1587 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{16384 d}+\frac{83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt{a \sin (c+d x)+a}}+\frac{1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt{a \sin (c+d x)+a}}+\frac{8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt{a \sin (c+d x)+a}}-\frac{529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt{a \sin (c+d x)+a}}-\frac{529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}-\frac{3 a \cot (c+d x) \csc ^6(c+d x) \sqrt{a \sin (c+d x)+a}}{112 d}","-\frac{1587 a^2 \cot (c+d x)}{16384 d \sqrt{a \sin (c+d x)+a}}-\frac{1587 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{16384 d}+\frac{83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt{a \sin (c+d x)+a}}+\frac{1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt{a \sin (c+d x)+a}}+\frac{8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt{a \sin (c+d x)+a}}-\frac{529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt{a \sin (c+d x)+a}}-\frac{529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}-\frac{3 a \cot (c+d x) \csc ^6(c+d x) \sqrt{a \sin (c+d x)+a}}{112 d}",1,"(-1587*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(16384*d) - (1587*a^2*Cot[c + d*x])/(16384*d*Sqrt[a + a*Sin[c + d*x]]) - (529*a^2*Cot[c + d*x]*Csc[c + d*x])/(8192*d*Sqrt[a + a*Sin[c + d*x]]) - (529*a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(10240*d*Sqrt[a + a*Sin[c + d*x]]) + (8653*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(35840*d*Sqrt[a + a*Sin[c + d*x]]) + (1957*a^2*Cot[c + d*x]*Csc[c + d*x]^4)/(4480*d*Sqrt[a + a*Sin[c + d*x]]) + (83*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(448*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a*Cot[c + d*x]*Csc[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]])/(112*d) - (Cot[c + d*x]*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^(3/2))/(8*d)","A",18,9,31,0.2903,1,"{2881, 2762, 21, 2772, 2773, 206, 3044, 2975, 2980}"
464,1,124,0,0.4057192,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{152 a^2 \cos ^5(c+d x)}{3465 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{11 a d}+\frac{20 \cos ^5(c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{38 a \cos ^5(c+d x)}{693 d (a \sin (c+d x)+a)^{3/2}}","-\frac{152 a^2 \cos ^5(c+d x)}{3465 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{11 a d}+\frac{20 \cos ^5(c+d x)}{99 d \sqrt{a \sin (c+d x)+a}}-\frac{38 a \cos ^5(c+d x)}{693 d (a \sin (c+d x)+a)^{3/2}}",1,"(-152*a^2*Cos[c + d*x]^5)/(3465*d*(a + a*Sin[c + d*x])^(5/2)) - (38*a*Cos[c + d*x]^5)/(693*d*(a + a*Sin[c + d*x])^(3/2)) + (20*Cos[c + d*x]^5)/(99*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(11*a*d)","A",5,4,31,0.1290,1,"{2877, 2856, 2674, 2673}"
465,1,92,0,0.19254,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","\frac{8 a^2 \cos ^5(c+d x)}{315 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}+\frac{2 a \cos ^5(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}","\frac{8 a^2 \cos ^5(c+d x)}{315 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}+\frac{2 a \cos ^5(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}",1,"(8*a^2*Cos[c + d*x]^5)/(315*d*(a + a*Sin[c + d*x])^(5/2)) + (2*a*Cos[c + d*x]^5)/(63*d*(a + a*Sin[c + d*x])^(3/2)) - (2*Cos[c + d*x]^5)/(9*d*Sqrt[a + a*Sin[c + d*x]])","A",3,3,29,0.1034,1,"{2856, 2674, 2673}"
466,1,130,0,0.5644648,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 a d}+\frac{32 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}","-\frac{2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 a d}+\frac{32 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(-2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(Sqrt[a]*d) + (32*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^2)/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*a*d)","A",13,10,29,0.3448,1,"{2881, 2778, 2968, 3023, 2751, 2649, 206, 3046, 2985, 2773}"
467,1,119,0,0.5033687,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a d}+\frac{4 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a d}+\frac{4 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{a} d}",1,"ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(Sqrt[a]*d) + (4*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - Cot[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*a*d)","A",11,8,31,0.2581,1,"{2881, 2759, 2751, 2649, 206, 3044, 2985, 2773}"
468,1,125,0,0.5430915,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}","-\frac{2 \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\cot (c+d x) \csc (c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}",1,"(9*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*Sqrt[a]*d) - (2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) + Cot[c + d*x]/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])","A",11,8,29,0.2759,1,"{2881, 2751, 2649, 206, 3044, 2984, 2985, 2773}"
469,1,135,0,0.596962,"\int \frac{\cot ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Cot[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]],x]","\frac{9 \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}","\frac{9 \cot (c+d x)}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}",1,"(-7*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*Sqrt[a]*d) + (9*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*d*Sqrt[a + a*Sin[c + d*x]])","A",11,7,23,0.3043,1,"{2718, 2649, 206, 3044, 2984, 2985, 2773}"
470,1,170,0,0.8867586,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{11 \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{53 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}}","-\frac{11 \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}+\frac{53 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}}",1,"(-11*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*Sqrt[a]*d) - (11*Cot[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (53*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*d*Sqrt[a + a*Sin[c + d*x]])","A",15,8,29,0.2759,1,"{2881, 2780, 2649, 206, 2773, 3044, 2984, 2985}"
471,1,205,0,1.1430471,"\int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{9 \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{128 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a \sin (c+d x)+a}}-\frac{3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}","-\frac{9 \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{128 \sqrt{a} d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}+\frac{29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt{a \sin (c+d x)+a}}-\frac{3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}",1,"(-9*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*Sqrt[a]*d) - (9*Cot[c + d*x])/(128*d*Sqrt[a + a*Sin[c + d*x]]) - (3*Cot[c + d*x]*Csc[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (29*Cot[c + d*x]*Csc[c + d*x]^2)/(80*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*d*Sqrt[a + a*Sin[c + d*x]])","A",17,8,31,0.2581,1,"{2881, 2779, 2985, 2649, 206, 2773, 3044, 2984}"
472,1,205,0,0.7760405,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{11 a^2 d}-\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{385 a^3 d}+\frac{8 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{1155 a^2 d}+\frac{14 \sin ^4(c+d x) \cos (c+d x)}{33 a d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sin ^3(c+d x) \cos (c+d x)}{231 a d \sqrt{a \sin (c+d x)+a}}-\frac{4 \cos (c+d x)}{165 a d \sqrt{a \sin (c+d x)+a}}","-\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{11 a^2 d}-\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{385 a^3 d}+\frac{8 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{1155 a^2 d}+\frac{14 \sin ^4(c+d x) \cos (c+d x)}{33 a d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sin ^3(c+d x) \cos (c+d x)}{231 a d \sqrt{a \sin (c+d x)+a}}-\frac{4 \cos (c+d x)}{165 a d \sqrt{a \sin (c+d x)+a}}",1,"(-4*Cos[c + d*x])/(165*a*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^3)/(231*a*d*Sqrt[a + a*Sin[c + d*x]]) + (14*Cos[c + d*x]*Sin[c + d*x]^4)/(33*a*d*Sqrt[a + a*Sin[c + d*x]]) + (8*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(1155*a^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(11*a^2*d) - (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(385*a^3*d)","A",12,7,31,0.2258,1,"{2880, 2770, 2759, 2751, 2646, 3046, 2981}"
473,1,92,0,0.3699683,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 \cos ^5(c+d x)}{9 a d \sqrt{a \sin (c+d x)+a}}+\frac{20 \cos ^5(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}-\frac{46 a \cos ^5(c+d x)}{315 d (a \sin (c+d x)+a)^{5/2}}","-\frac{2 \cos ^5(c+d x)}{9 a d \sqrt{a \sin (c+d x)+a}}+\frac{20 \cos ^5(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}-\frac{46 a \cos ^5(c+d x)}{315 d (a \sin (c+d x)+a)^{5/2}}",1,"(-46*a*Cos[c + d*x]^5)/(315*d*(a + a*Sin[c + d*x])^(5/2)) + (20*Cos[c + d*x]^5)/(63*d*(a + a*Sin[c + d*x])^(3/2)) - (2*Cos[c + d*x]^5)/(9*a*d*Sqrt[a + a*Sin[c + d*x]])","A",4,4,31,0.1290,1,"{2877, 2856, 2674, 2673}"
474,1,60,0,0.150723,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","\frac{6 a \cos ^5(c+d x)}{35 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}","\frac{6 a \cos ^5(c+d x)}{35 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^5(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}",1,"(6*a*Cos[c + d*x]^5)/(35*d*(a + a*Sin[c + d*x])^(5/2)) - (2*Cos[c + d*x]^5)/(7*d*(a + a*Sin[c + d*x])^(3/2))","A",2,2,29,0.06897,1,"{2856, 2673}"
475,1,98,0,0.3623247,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a^2 d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}+\frac{10 \cos (c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}","-\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a^2 d}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}+\frac{10 \cos (c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}",1,"(-2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d) + (10*Cos[c + d*x])/(3*a*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*a^2*d)","A",6,6,29,0.2069,1,"{2880, 2646, 3046, 2981, 2773, 206}"
476,1,94,0,0.4024216,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{\cot (c+d x) \sqrt{a \sin (c+d x)+a}}{a^2 d}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}","-\frac{\cot (c+d x) \sqrt{a \sin (c+d x)+a}}{a^2 d}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}",1,"(3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d) - Cos[c + d*x]/(a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(a^2*d)","A",9,6,31,0.1935,1,"{2880, 2773, 206, 3044, 21, 2763}"
477,1,106,0,0.4875943,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{2 a^2 d}+\frac{7 \cot (c+d x)}{4 a d \sqrt{a \sin (c+d x)+a}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a \sin (c+d x)+a}}{2 a^2 d}+\frac{7 \cot (c+d x)}{4 a d \sqrt{a \sin (c+d x)+a}}",1,"(-3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(3/2)*d) + (7*Cot[c + d*x])/(4*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*a^2*d)","A",8,6,29,0.2069,1,"{2880, 2772, 2773, 206, 3044, 2980}"
478,1,144,0,0.5393317,"\int \frac{\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 a^{3/2} d}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{3 a^2 d}-\frac{\cot (c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}+\frac{11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt{a \sin (c+d x)+a}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 a^{3/2} d}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{3 a^2 d}-\frac{\cot (c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}+\frac{11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt{a \sin (c+d x)+a}}",1,"-ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(8*a^(3/2)*d) - Cot[c + d*x]/(8*a*d*Sqrt[a + a*Sin[c + d*x]]) + (11*Cot[c + d*x]*Csc[c + d*x])/(12*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(3*a^2*d)","A",10,6,23,0.2609,1,"{2717, 2772, 2773, 206, 3044, 2980}"
479,1,182,0,0.7274328,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 a^{3/2} d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 a^2 d}-\frac{3 \cot (c+d x)}{64 a d \sqrt{a \sin (c+d x)+a}}+\frac{5 \cot (c+d x) \csc ^2(c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{32 a d \sqrt{a \sin (c+d x)+a}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 a^{3/2} d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 a^2 d}-\frac{3 \cot (c+d x)}{64 a d \sqrt{a \sin (c+d x)+a}}+\frac{5 \cot (c+d x) \csc ^2(c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{32 a d \sqrt{a \sin (c+d x)+a}}",1,"(-3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*a^(3/2)*d) - (3*Cot[c + d*x])/(64*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(32*a*d*Sqrt[a + a*Sin[c + d*x]]) + (5*Cot[c + d*x]*Csc[c + d*x]^2)/(8*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(4*a^2*d)","A",12,6,29,0.2069,1,"{2880, 2772, 2773, 206, 3044, 2980}"
480,1,220,0,0.8759788,"\int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{128 a^{3/2} d}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{5 a^2 d}-\frac{3 \cot (c+d x)}{128 a d \sqrt{a \sin (c+d x)+a}}+\frac{19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{64 a d \sqrt{a \sin (c+d x)+a}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{128 a^{3/2} d}-\frac{\cot (c+d x) \csc ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{5 a^2 d}-\frac{3 \cot (c+d x)}{128 a d \sqrt{a \sin (c+d x)+a}}+\frac{19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{64 a d \sqrt{a \sin (c+d x)+a}}",1,"(-3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*a^(3/2)*d) - (3*Cot[c + d*x])/(128*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(64*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(80*a*d*Sqrt[a + a*Sin[c + d*x]]) + (19*Cot[c + d*x]*Csc[c + d*x]^3)/(40*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(5*a^2*d)","A",14,6,31,0.1935,1,"{2880, 2772, 2773, 206, 3044, 2980}"
481,1,260,0,1.3590357,"\int \frac{\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \sin ^5(c+d x) \cos (c+d x)}{11 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{46 \sin ^4(c+d x) \cos (c+d x)}{99 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{424 \sin ^3(c+d x) \cos (c+d x)}{693 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{200 \sin ^2(c+d x) \cos (c+d x)}{231 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{1048 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{693 a^3 d}+\frac{4496 \cos (c+d x)}{693 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}","-\frac{2 \sin ^5(c+d x) \cos (c+d x)}{11 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{46 \sin ^4(c+d x) \cos (c+d x)}{99 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{424 \sin ^3(c+d x) \cos (c+d x)}{693 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{200 \sin ^2(c+d x) \cos (c+d x)}{231 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{1048 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{693 a^3 d}+\frac{4496 \cos (c+d x)}{693 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}",1,"(-4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (4496*Cos[c + d*x])/(693*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (200*Cos[c + d*x]*Sin[c + d*x]^2)/(231*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (424*Cos[c + d*x]*Sin[c + d*x]^3)/(693*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (46*Cos[c + d*x]*Sin[c + d*x]^4)/(99*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^5)/(11*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (1048*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(693*a^3*d)","A",18,9,31,0.2903,1,"{2880, 2778, 2983, 2968, 3023, 2751, 2649, 206, 3046}"
482,1,222,0,1.0798704,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \sin ^4(c+d x) \cos (c+d x)}{9 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{38 \sin ^3(c+d x) \cos (c+d x)}{63 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{92 \sin ^2(c+d x) \cos (c+d x)}{105 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{472 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 a^3 d}-\frac{2048 \cos (c+d x)}{315 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}","-\frac{2 \sin ^4(c+d x) \cos (c+d x)}{9 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{38 \sin ^3(c+d x) \cos (c+d x)}{63 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{92 \sin ^2(c+d x) \cos (c+d x)}{105 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{472 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 a^3 d}-\frac{2048 \cos (c+d x)}{315 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}",1,"(4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) - (2048*Cos[c + d*x])/(315*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (92*Cos[c + d*x]*Sin[c + d*x]^2)/(105*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (38*Cos[c + d*x]*Sin[c + d*x]^3)/(63*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^4)/(9*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (472*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*a^3*d)","A",16,9,31,0.2903,1,"{2880, 2778, 2983, 2968, 3023, 2751, 2649, 206, 3046}"
483,1,169,0,0.428343,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^(5/2),x]","\frac{4 \cos (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{2 \cos ^5(c+d x)}{7 a d (a \sin (c+d x)+a)^{3/2}}+\frac{4 \cos ^5(c+d x)}{7 d (a \sin (c+d x)+a)^{5/2}}+\frac{2 \cos ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^{3/2}}","\frac{4 \cos (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{2 \cos ^5(c+d x)}{7 a d (a \sin (c+d x)+a)^{3/2}}+\frac{4 \cos ^5(c+d x)}{7 d (a \sin (c+d x)+a)^{5/2}}+\frac{2 \cos ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^{3/2}}",1,"(-4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (4*Cos[c + d*x]^5)/(7*d*(a + a*Sin[c + d*x])^(5/2)) + (2*Cos[c + d*x]^3)/(3*a*d*(a + a*Sin[c + d*x])^(3/2)) - (2*Cos[c + d*x]^5)/(7*a*d*(a + a*Sin[c + d*x])^(3/2)) + (4*Cos[c + d*x])/(a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",6,5,31,0.1613,1,"{2878, 2860, 2679, 2649, 206}"
484,1,137,0,0.2345601,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{4 \cos (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{2 \cos ^5(c+d x)}{5 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^{3/2}}","-\frac{4 \cos (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{2 \cos ^5(c+d x)}{5 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 \cos ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^{3/2}}",1,"(4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) - (2*Cos[c + d*x]^5)/(5*d*(a + a*Sin[c + d*x])^(5/2)) - (2*Cos[c + d*x]^3)/(3*a*d*(a + a*Sin[c + d*x])^(3/2)) - (4*Cos[c + d*x])/(a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",5,4,29,0.1379,1,"{2860, 2679, 2649, 206}"
485,1,113,0,0.3913262,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \cos (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}+\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}","-\frac{2 \cos (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}+\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}",1,"(-2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(5/2)*d) + (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) - (2*Cos[c + d*x])/(a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",9,6,29,0.2069,1,"{2880, 2649, 206, 3046, 2985, 2773}"
486,1,113,0,0.5240053,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{\cot (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}","-\frac{\cot (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}",1,"(5*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(5/2)*d) - (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) - Cot[c + d*x]/(a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",12,7,31,0.2258,1,"{2880, 2780, 2649, 206, 2773, 3044, 2985}"
487,1,153,0,0.7329413,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^(5/2),x]","\frac{9 \cot (c+d x)}{4 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{23 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{5/2} d}+\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d \sqrt{a \sin (c+d x)+a}}","\frac{9 \cot (c+d x)}{4 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{23 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{4 a^{5/2} d}+\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d \sqrt{a \sin (c+d x)+a}}",1,"(-23*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(5/2)*d) + (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (9*Cot[c + d*x])/(4*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",14,8,29,0.2759,1,"{2880, 2779, 2985, 2649, 206, 2773, 3044, 2984}"
488,1,191,0,0.9472771,"\int \frac{\cot ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{19 \cot (c+d x)}{8 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{45 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 a^{5/2} d}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{13 \cot (c+d x) \csc (c+d x)}{12 a^2 d \sqrt{a \sin (c+d x)+a}}","-\frac{19 \cot (c+d x)}{8 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{45 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{8 a^{5/2} d}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{13 \cot (c+d x) \csc (c+d x)}{12 a^2 d \sqrt{a \sin (c+d x)+a}}",1,"(45*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*a^(5/2)*d) - (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) - (19*Cot[c + d*x])/(8*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (13*Cot[c + d*x]*Csc[c + d*x])/(12*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",16,8,23,0.3478,1,"{2717, 2779, 2984, 2985, 2649, 206, 2773, 3044}"
489,1,229,0,1.3135961,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^(5/2),x]","\frac{149 \cot (c+d x)}{64 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{363 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 a^{5/2} d}+\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt{a \sin (c+d x)+a}}","\frac{149 \cot (c+d x)}{64 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{363 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right)}{64 a^{5/2} d}+\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt{a \sin (c+d x)+a}}",1,"(-363*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*a^(5/2)*d) + (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (149*Cot[c + d*x])/(64*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (107*Cot[c + d*x]*Csc[c + d*x])/(96*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (17*Cot[c + d*x]*Csc[c + d*x]^2)/(24*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",18,8,29,0.2759,1,"{2880, 2779, 2984, 2985, 2649, 206, 2773, 3044}"
490,1,200,0,0.232272,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{2 a^2 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}+\frac{a^2 \cos (c+d x) \sin ^{n+3}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right)}{d (n+3) \sqrt{\cos ^2(c+d x)}}","\frac{a^2 \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{2 a^2 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}+\frac{a^2 \cos (c+d x) \sin ^{n+3}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right)}{d (n+3) \sqrt{\cos ^2(c+d x)}}",1,"(a^2*Cos[c + d*x]*Hypergeometric2F1[-3/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (2*a^2*Cos[c + d*x]*Hypergeometric2F1[-3/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2]) + (a^2*Cos[c + d*x]*Hypergeometric2F1[-3/2, (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(3 + n))/(d*(3 + n)*Sqrt[Cos[c + d*x]^2])","A",5,2,29,0.06897,1,"{2873, 2577}"
491,1,129,0,0.1347767,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]","\frac{a \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{a \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}","\frac{a \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{a \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}",1,"(a*Cos[c + d*x]*Hypergeometric2F1[-3/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (a*Cos[c + d*x]*Hypergeometric2F1[-3/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2])","A",3,2,27,0.07407,1,"{2838, 2577}"
492,1,134,0,0.1760566,"\int \frac{\cos ^4(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^n)/(a + a*Sin[c + d*x]),x]","\frac{\cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{a d (n+1) \sqrt{\cos ^2(c+d x)}}-\frac{\cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{a d (n+2) \sqrt{\cos ^2(c+d x)}}","\frac{\cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{a d (n+1) \sqrt{\cos ^2(c+d x)}}-\frac{\cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{a d (n+2) \sqrt{\cos ^2(c+d x)}}",1,"(Cos[c + d*x]*Hypergeometric2F1[-1/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(a*d*(1 + n)*Sqrt[Cos[c + d*x]^2]) - (Cos[c + d*x]*Hypergeometric2F1[-1/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(a*d*(2 + n)*Sqrt[Cos[c + d*x]^2])","A",3,2,29,0.06897,1,"{2839, 2577}"
493,1,173,0,0.2422261,"\int \frac{\cos ^4(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^2,x]","\frac{(2 n+3) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{a^2 d (n+1) (n+2) \sqrt{\cos ^2(c+d x)}}-\frac{2 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{a^2 d (n+2) \sqrt{\cos ^2(c+d x)}}-\frac{\cos (c+d x) \sin ^{n+1}(c+d x)}{a^2 d (n+2)}","\frac{(2 n+3) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{a^2 d (n+1) (n+2) \sqrt{\cos ^2(c+d x)}}-\frac{2 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{a^2 d (n+2) \sqrt{\cos ^2(c+d x)}}-\frac{\cos (c+d x) \sin ^{n+1}(c+d x)}{a^2 d (n+2)}",1,"-((Cos[c + d*x]*Sin[c + d*x]^(1 + n))/(a^2*d*(2 + n))) + ((3 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(a^2*d*(1 + n)*(2 + n)*Sqrt[Cos[c + d*x]^2]) - (2*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(a^2*d*(2 + n)*Sqrt[Cos[c + d*x]^2])","A",5,4,29,0.1379,1,"{2869, 2763, 2748, 2643}"
494,1,97,0,0.0894381,"\int \cos ^5(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^{11}(c+d x)}{11 d}+\frac{a \sin ^{10}(c+d x)}{10 d}-\frac{2 a \sin ^9(c+d x)}{9 d}-\frac{a \sin ^8(c+d x)}{4 d}+\frac{a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^6(c+d x)}{6 d}","\frac{a \sin ^{11}(c+d x)}{11 d}+\frac{a \sin ^{10}(c+d x)}{10 d}-\frac{2 a \sin ^9(c+d x)}{9 d}-\frac{a \sin ^8(c+d x)}{4 d}+\frac{a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^6(c+d x)}{6 d}",1,"(a*Sin[c + d*x]^6)/(6*d) + (a*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^8)/(4*d) - (2*a*Sin[c + d*x]^9)/(9*d) + (a*Sin[c + d*x]^10)/(10*d) + (a*Sin[c + d*x]^11)/(11*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
495,1,97,0,0.0844256,"\int \cos ^5(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^{10}(c+d x)}{10 d}+\frac{a \sin ^9(c+d x)}{9 d}-\frac{a \sin ^8(c+d x)}{4 d}-\frac{2 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^6(c+d x)}{6 d}+\frac{a \sin ^5(c+d x)}{5 d}","\frac{a \sin ^{10}(c+d x)}{10 d}+\frac{a \sin ^9(c+d x)}{9 d}-\frac{a \sin ^8(c+d x)}{4 d}-\frac{2 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^6(c+d x)}{6 d}+\frac{a \sin ^5(c+d x)}{5 d}",1,"(a*Sin[c + d*x]^5)/(5*d) + (a*Sin[c + d*x]^6)/(6*d) - (2*a*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^8)/(4*d) + (a*Sin[c + d*x]^9)/(9*d) + (a*Sin[c + d*x]^10)/(10*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
496,1,81,0,0.1269581,"\int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^9(c+d x)}{9 d}-\frac{2 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \cos ^8(c+d x)}{8 d}-\frac{a \cos ^6(c+d x)}{6 d}","\frac{a \sin ^9(c+d x)}{9 d}-\frac{2 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \cos ^8(c+d x)}{8 d}-\frac{a \cos ^6(c+d x)}{6 d}",1,"-(a*Cos[c + d*x]^6)/(6*d) + (a*Cos[c + d*x]^8)/(8*d) + (a*Sin[c + d*x]^5)/(5*d) - (2*a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^9)/(9*d)","A",7,5,27,0.1852,1,"{2834, 2565, 14, 2564, 270}"
497,1,81,0,0.1256655,"\int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^7(c+d x)}{7 d}-\frac{2 a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \cos ^8(c+d x)}{8 d}-\frac{a \cos ^6(c+d x)}{6 d}","\frac{a \sin ^7(c+d x)}{7 d}-\frac{2 a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \cos ^8(c+d x)}{8 d}-\frac{a \cos ^6(c+d x)}{6 d}",1,"-(a*Cos[c + d*x]^6)/(6*d) + (a*Cos[c + d*x]^8)/(8*d) + (a*Sin[c + d*x]^3)/(3*d) - (2*a*Sin[c + d*x]^5)/(5*d) + (a*Sin[c + d*x]^7)/(7*d)","A",7,5,27,0.1852,1,"{2834, 2564, 270, 2565, 14}"
498,1,65,0,0.0908986,"\int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^7(c+d x)}{7 d}-\frac{2 a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \cos ^6(c+d x)}{6 d}","\frac{a \sin ^7(c+d x)}{7 d}-\frac{2 a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \cos ^6(c+d x)}{6 d}",1,"-(a*Cos[c + d*x]^6)/(6*d) + (a*Sin[c + d*x]^3)/(3*d) - (2*a*Sin[c + d*x]^5)/(5*d) + (a*Sin[c + d*x]^7)/(7*d)","A",6,5,25,0.2000,1,"{2834, 2565, 30, 2564, 270}"
499,1,86,0,0.0666494,"\int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^4(c+d x)}{4 d}-\frac{2 a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}","\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^4(c+d x)}{4 d}-\frac{2 a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"(a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^4)/(4*d) + (a*Sin[c + d*x]^5)/(5*d)","A",4,3,25,0.1200,1,"{2836, 12, 88}"
500,1,83,0,0.078624,"\int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}","\frac{a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"-((a*Csc[c + d*x])/d) + (a*Log[Sin[c + d*x]])/d - (2*a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/d + (a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^4)/(4*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
501,1,86,0,0.0793945,"\int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^2(c+d x)}{2 d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{2 a \log (\sin (c+d x))}{d}","\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^2(c+d x)}{2 d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{2 a \log (\sin (c+d x))}{d}",1,"-((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (2*a*Log[Sin[c + d*x]])/d - (2*a*Sin[c + d*x])/d + (a*Sin[c + d*x]^2)/(2*d) + (a*Sin[c + d*x]^3)/(3*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
502,1,85,0,0.0720851,"\int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}+\frac{2 a \csc (c+d x)}{d}-\frac{2 a \log (\sin (c+d x))}{d}","\frac{a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}+\frac{2 a \csc (c+d x)}{d}-\frac{2 a \log (\sin (c+d x))}{d}",1,"(2*a*Csc[c + d*x])/d - (a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (2*a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d + (a*Sin[c + d*x]^2)/(2*d)","A",4,3,25,0.1200,1,"{2836, 12, 88}"
503,1,81,0,0.0440546,"\int \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a \sin (c+d x)}{d}-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{2 a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}","\frac{a \sin (c+d x)}{d}-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{2 a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"(2*a*Csc[c + d*x])/d + (a*Csc[c + d*x]^2)/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) + (a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d","A",3,2,19,0.1053,1,"{2707, 88}"
504,1,86,0,0.0705625,"\int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^4(c+d x)}{4 d}+\frac{2 a \csc ^3(c+d x)}{3 d}+\frac{a \csc ^2(c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}","-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^4(c+d x)}{4 d}+\frac{2 a \csc ^3(c+d x)}{3 d}+\frac{a \csc ^2(c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"-((a*Csc[c + d*x])/d) + (a*Csc[c + d*x]^2)/d + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^5)/(5*d) + (a*Log[Sin[c + d*x]])/d","A",4,3,25,0.1200,1,"{2836, 12, 88}"
505,1,61,0,0.103967,"\int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}","-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}",1,"-(a*Cot[c + d*x]^6)/(6*d) - (a*Csc[c + d*x])/d + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d)","A",6,5,27,0.1852,1,"{2834, 2607, 30, 2606, 194}"
506,1,65,0,0.1155766,"\int \cot ^5(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}","-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}",1,"-(a*Cot[c + d*x]^6)/(6*d) - (a*Csc[c + d*x]^3)/(3*d) + (2*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d)","A",6,5,27,0.1852,1,"{2834, 2606, 270, 2607, 30}"
507,1,81,0,0.1211715,"\int \cot ^5(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}","-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}",1,"-(a*Cot[c + d*x]^6)/(6*d) - (a*Cot[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^3)/(3*d) + (2*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d)","A",7,5,27,0.1852,1,"{2834, 2607, 14, 2606, 270}"
508,1,81,0,0.1216227,"\int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{2 a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}","-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^6(c+d x)}{6 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{2 a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}",1,"-(a*Cot[c + d*x]^6)/(6*d) - (a*Cot[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^5)/(5*d) + (2*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d)","A",7,5,27,0.1852,1,"{2834, 2606, 270, 2607, 14}"
509,1,97,0,0.0817221,"\int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^6*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^{10}(c+d x)}{10 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{a \csc ^8(c+d x)}{4 d}+\frac{2 a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}","-\frac{a \csc ^{10}(c+d x)}{10 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{a \csc ^8(c+d x)}{4 d}+\frac{2 a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}",1,"-(a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(6*d) + (2*a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) - (a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
510,1,97,0,0.0818183,"\int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^7*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^{11}(c+d x)}{11 d}-\frac{a \csc ^{10}(c+d x)}{10 d}+\frac{2 a \csc ^9(c+d x)}{9 d}+\frac{a \csc ^8(c+d x)}{4 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^6(c+d x)}{6 d}","-\frac{a \csc ^{11}(c+d x)}{11 d}-\frac{a \csc ^{10}(c+d x)}{10 d}+\frac{2 a \csc ^9(c+d x)}{9 d}+\frac{a \csc ^8(c+d x)}{4 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^6(c+d x)}{6 d}",1,"-(a*Csc[c + d*x]^6)/(6*d) - (a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) + (2*a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
511,1,127,0,0.1252191,"\int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^{10}(c+d x)}{10 d}+\frac{2 a^2 \sin ^9(c+d x)}{9 d}-\frac{a^2 \sin ^8(c+d x)}{8 d}-\frac{4 a^2 \sin ^7(c+d x)}{7 d}-\frac{a^2 \sin ^6(c+d x)}{6 d}+\frac{2 a^2 \sin ^5(c+d x)}{5 d}+\frac{a^2 \sin ^4(c+d x)}{4 d}","\frac{a^2 \sin ^{10}(c+d x)}{10 d}+\frac{2 a^2 \sin ^9(c+d x)}{9 d}-\frac{a^2 \sin ^8(c+d x)}{8 d}-\frac{4 a^2 \sin ^7(c+d x)}{7 d}-\frac{a^2 \sin ^6(c+d x)}{6 d}+\frac{2 a^2 \sin ^5(c+d x)}{5 d}+\frac{a^2 \sin ^4(c+d x)}{4 d}",1,"(a^2*Sin[c + d*x]^4)/(4*d) + (2*a^2*Sin[c + d*x]^5)/(5*d) - (a^2*Sin[c + d*x]^6)/(6*d) - (4*a^2*Sin[c + d*x]^7)/(7*d) - (a^2*Sin[c + d*x]^8)/(8*d) + (2*a^2*Sin[c + d*x]^9)/(9*d) + (a^2*Sin[c + d*x]^10)/(10*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
512,1,109,0,0.1258125,"\int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{(a \sin (c+d x)+a)^9}{9 a^7 d}-\frac{3 (a \sin (c+d x)+a)^8}{4 a^6 d}+\frac{13 (a \sin (c+d x)+a)^7}{7 a^5 d}-\frac{2 (a \sin (c+d x)+a)^6}{a^4 d}+\frac{4 (a \sin (c+d x)+a)^5}{5 a^3 d}","\frac{(a \sin (c+d x)+a)^9}{9 a^7 d}-\frac{3 (a \sin (c+d x)+a)^8}{4 a^6 d}+\frac{13 (a \sin (c+d x)+a)^7}{7 a^5 d}-\frac{2 (a \sin (c+d x)+a)^6}{a^4 d}+\frac{4 (a \sin (c+d x)+a)^5}{5 a^3 d}",1,"(4*(a + a*Sin[c + d*x])^5)/(5*a^3*d) - (2*(a + a*Sin[c + d*x])^6)/(a^4*d) + (13*(a + a*Sin[c + d*x])^7)/(7*a^5*d) - (3*(a + a*Sin[c + d*x])^8)/(4*a^6*d) + (a + a*Sin[c + d*x])^9/(9*a^7*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
513,1,89,0,0.0843447,"\int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{(a \sin (c+d x)+a)^8}{8 a^6 d}-\frac{5 (a \sin (c+d x)+a)^7}{7 a^5 d}+\frac{4 (a \sin (c+d x)+a)^6}{3 a^4 d}-\frac{4 (a \sin (c+d x)+a)^5}{5 a^3 d}","\frac{(a \sin (c+d x)+a)^8}{8 a^6 d}-\frac{5 (a \sin (c+d x)+a)^7}{7 a^5 d}+\frac{4 (a \sin (c+d x)+a)^6}{3 a^4 d}-\frac{4 (a \sin (c+d x)+a)^5}{5 a^3 d}",1,"(-4*(a + a*Sin[c + d*x])^5)/(5*a^3*d) + (4*(a + a*Sin[c + d*x])^6)/(3*a^4*d) - (5*(a + a*Sin[c + d*x])^7)/(7*a^5*d) + (a + a*Sin[c + d*x])^8/(8*a^6*d)","A",4,3,27,0.1111,1,"{2836, 12, 77}"
514,1,119,0,0.1006462,"\int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Cot[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^6(c+d x)}{6 d}+\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{a^2 \sin ^4(c+d x)}{4 d}-\frac{4 a^2 \sin ^3(c+d x)}{3 d}-\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \log (\sin (c+d x))}{d}","\frac{a^2 \sin ^6(c+d x)}{6 d}+\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{a^2 \sin ^4(c+d x)}{4 d}-\frac{4 a^2 \sin ^3(c+d x)}{3 d}-\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \log (\sin (c+d x))}{d}",1,"(a^2*Log[Sin[c + d*x]])/d + (2*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d) - (4*a^2*Sin[c + d*x]^3)/(3*d) - (a^2*Sin[c + d*x]^4)/(4*d) + (2*a^2*Sin[c + d*x]^5)/(5*d) + (a^2*Sin[c + d*x]^6)/(6*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
515,1,114,0,0.1184942,"\int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^5(c+d x)}{5 d}+\frac{a^2 \sin ^4(c+d x)}{2 d}-\frac{a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin ^2(c+d x)}{d}-\frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d}","\frac{a^2 \sin ^5(c+d x)}{5 d}+\frac{a^2 \sin ^4(c+d x)}{2 d}-\frac{a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin ^2(c+d x)}{d}-\frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d}",1,"-((a^2*Csc[c + d*x])/d) + (2*a^2*Log[Sin[c + d*x]])/d - (a^2*Sin[c + d*x])/d - (2*a^2*Sin[c + d*x]^2)/d - (a^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^4)/(2*d) + (a^2*Sin[c + d*x]^5)/(5*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
516,1,116,0,0.1177505,"\int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^4(c+d x)}{4 d}+\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{a^2 \sin ^2(c+d x)}{2 d}-\frac{4 a^2 \sin (c+d x)}{d}-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a^2 \csc (c+d x)}{d}-\frac{a^2 \log (\sin (c+d x))}{d}","\frac{a^2 \sin ^4(c+d x)}{4 d}+\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{a^2 \sin ^2(c+d x)}{2 d}-\frac{4 a^2 \sin (c+d x)}{d}-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a^2 \csc (c+d x)}{d}-\frac{a^2 \log (\sin (c+d x))}{d}",1,"(-2*a^2*Csc[c + d*x])/d - (a^2*Csc[c + d*x]^2)/(2*d) - (a^2*Log[Sin[c + d*x]])/d - (4*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d) + (2*a^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^4)/(4*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
517,1,110,0,0.1014015,"\int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin ^2(c+d x)}{d}-\frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{d}+\frac{a^2 \csc (c+d x)}{d}-\frac{4 a^2 \log (\sin (c+d x))}{d}","\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin ^2(c+d x)}{d}-\frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{d}+\frac{a^2 \csc (c+d x)}{d}-\frac{4 a^2 \log (\sin (c+d x))}{d}",1,"(a^2*Csc[c + d*x])/d - (a^2*Csc[c + d*x]^2)/d - (a^2*Csc[c + d*x]^3)/(3*d) - (4*a^2*Log[Sin[c + d*x]])/d - (a^2*Sin[c + d*x])/d + (a^2*Sin[c + d*x]^2)/d + (a^2*Sin[c + d*x]^3)/(3*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
518,1,116,0,0.0671104,"\int \cot ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{2 a^2 \sin (c+d x)}{d}-\frac{a^2 \csc ^4(c+d x)}{4 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}+\frac{a^2 \csc ^2(c+d x)}{2 d}+\frac{4 a^2 \csc (c+d x)}{d}-\frac{a^2 \log (\sin (c+d x))}{d}","\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{2 a^2 \sin (c+d x)}{d}-\frac{a^2 \csc ^4(c+d x)}{4 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}+\frac{a^2 \csc ^2(c+d x)}{2 d}+\frac{4 a^2 \csc (c+d x)}{d}-\frac{a^2 \log (\sin (c+d x))}{d}",1,"(4*a^2*Csc[c + d*x])/d + (a^2*Csc[c + d*x]^2)/(2*d) - (2*a^2*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*d) - (a^2*Log[Sin[c + d*x]])/d + (2*a^2*Sin[c + d*x])/d + (a^2*Sin[c + d*x]^2)/(2*d)","A",3,2,21,0.09524,1,"{2707, 88}"
519,1,112,0,0.1018286,"\int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc ^5(c+d x)}{5 d}-\frac{a^2 \csc ^4(c+d x)}{2 d}+\frac{a^2 \csc ^3(c+d x)}{3 d}+\frac{2 a^2 \csc ^2(c+d x)}{d}+\frac{a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d}","\frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc ^5(c+d x)}{5 d}-\frac{a^2 \csc ^4(c+d x)}{2 d}+\frac{a^2 \csc ^3(c+d x)}{3 d}+\frac{2 a^2 \csc ^2(c+d x)}{d}+\frac{a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d}",1,"(a^2*Csc[c + d*x])/d + (2*a^2*Csc[c + d*x]^2)/d + (a^2*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(2*d) - (a^2*Csc[c + d*x]^5)/(5*d) + (2*a^2*Log[Sin[c + d*x]])/d + (a^2*Sin[c + d*x])/d","A",4,3,27,0.1111,1,"{2836, 12, 88}"
520,1,119,0,0.1209959,"\int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc ^6(c+d x)}{6 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}+\frac{a^2 \csc ^4(c+d x)}{4 d}+\frac{4 a^2 \csc ^3(c+d x)}{3 d}+\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{a^2 \log (\sin (c+d x))}{d}","-\frac{a^2 \csc ^6(c+d x)}{6 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}+\frac{a^2 \csc ^4(c+d x)}{4 d}+\frac{4 a^2 \csc ^3(c+d x)}{3 d}+\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{a^2 \log (\sin (c+d x))}{d}",1,"(-2*a^2*Csc[c + d*x])/d + (a^2*Csc[c + d*x]^2)/(2*d) + (4*a^2*Csc[c + d*x]^3)/(3*d) + (a^2*Csc[c + d*x]^4)/(4*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) - (a^2*Csc[c + d*x]^6)/(6*d) + (a^2*Log[Sin[c + d*x]])/d","A",4,3,29,0.1034,1,"{2836, 12, 88}"
521,1,111,0,0.1280788,"\int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{(a \sin (c+d x)+a)^{10}}{10 a^7 d}-\frac{2 (a \sin (c+d x)+a)^9}{3 a^6 d}+\frac{13 (a \sin (c+d x)+a)^8}{8 a^5 d}-\frac{12 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d}","\frac{(a \sin (c+d x)+a)^{10}}{10 a^7 d}-\frac{2 (a \sin (c+d x)+a)^9}{3 a^6 d}+\frac{13 (a \sin (c+d x)+a)^8}{8 a^5 d}-\frac{12 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d}",1,"(2*(a + a*Sin[c + d*x])^6)/(3*a^3*d) - (12*(a + a*Sin[c + d*x])^7)/(7*a^4*d) + (13*(a + a*Sin[c + d*x])^8)/(8*a^5*d) - (2*(a + a*Sin[c + d*x])^9)/(3*a^6*d) + (a + a*Sin[c + d*x])^10/(10*a^7*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
522,1,89,0,0.0834011,"\int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{(a \sin (c+d x)+a)^9}{9 a^6 d}-\frac{5 (a \sin (c+d x)+a)^8}{8 a^5 d}+\frac{8 (a \sin (c+d x)+a)^7}{7 a^4 d}-\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d}","\frac{(a \sin (c+d x)+a)^9}{9 a^6 d}-\frac{5 (a \sin (c+d x)+a)^8}{8 a^5 d}+\frac{8 (a \sin (c+d x)+a)^7}{7 a^4 d}-\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d}",1,"(-2*(a + a*Sin[c + d*x])^6)/(3*a^3*d) + (8*(a + a*Sin[c + d*x])^7)/(7*a^4*d) - (5*(a + a*Sin[c + d*x])^8)/(8*a^5*d) + (a + a*Sin[c + d*x])^9/(9*a^6*d)","A",4,3,27,0.1111,1,"{2836, 12, 77}"
523,1,137,0,0.1060167,"\int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^7(c+d x)}{7 d}+\frac{a^3 \sin ^6(c+d x)}{2 d}+\frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{5 a^3 \sin ^4(c+d x)}{4 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}","\frac{a^3 \sin ^7(c+d x)}{7 d}+\frac{a^3 \sin ^6(c+d x)}{2 d}+\frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{5 a^3 \sin ^4(c+d x)}{4 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}",1,"(a^3*Log[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (a^3*Sin[c + d*x]^2)/(2*d) - (5*a^3*Sin[c + d*x]^3)/(3*d) - (5*a^3*Sin[c + d*x]^4)/(4*d) + (a^3*Sin[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x]^6)/(2*d) + (a^3*Sin[c + d*x]^7)/(7*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
524,1,133,0,0.1234466,"\int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^6(c+d x)}{6 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{a^3 \sin ^4(c+d x)}{4 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}-\frac{5 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d}","\frac{a^3 \sin ^6(c+d x)}{6 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{a^3 \sin ^4(c+d x)}{4 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}-\frac{5 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d}",1,"-((a^3*Csc[c + d*x])/d) + (3*a^3*Log[Sin[c + d*x]])/d + (a^3*Sin[c + d*x])/d - (5*a^3*Sin[c + d*x]^2)/(2*d) - (5*a^3*Sin[c + d*x]^3)/(3*d) + (a^3*Sin[c + d*x]^4)/(4*d) + (3*a^3*Sin[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x]^6)/(6*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
525,1,133,0,0.1275739,"\int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^5(c+d x)}{5 d}+\frac{3 a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{5 a^3 \sin ^2(c+d x)}{2 d}-\frac{5 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}","\frac{a^3 \sin ^5(c+d x)}{5 d}+\frac{3 a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{5 a^3 \sin ^2(c+d x)}{2 d}-\frac{5 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}",1,"(-3*a^3*Csc[c + d*x])/d - (a^3*Csc[c + d*x]^2)/(2*d) + (a^3*Log[Sin[c + d*x]])/d - (5*a^3*Sin[c + d*x])/d - (5*a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/(3*d) + (3*a^3*Sin[c + d*x]^4)/(4*d) + (a^3*Sin[c + d*x]^5)/(5*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
526,1,131,0,0.109789,"\int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{d}+\frac{a^3 \sin ^2(c+d x)}{2 d}-\frac{5 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc ^2(c+d x)}{2 d}-\frac{a^3 \csc (c+d x)}{d}-\frac{5 a^3 \log (\sin (c+d x))}{d}","\frac{a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{d}+\frac{a^3 \sin ^2(c+d x)}{2 d}-\frac{5 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^3(c+d x)}{3 d}-\frac{3 a^3 \csc ^2(c+d x)}{2 d}-\frac{a^3 \csc (c+d x)}{d}-\frac{5 a^3 \log (\sin (c+d x))}{d}",1,"-((a^3*Csc[c + d*x])/d) - (3*a^3*Csc[c + d*x]^2)/(2*d) - (a^3*Csc[c + d*x]^3)/(3*d) - (5*a^3*Log[Sin[c + d*x]])/d - (5*a^3*Sin[c + d*x])/d + (a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/d + (a^3*Sin[c + d*x]^4)/(4*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
527,1,131,0,0.070055,"\int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^4(c+d x)}{4 d}-\frac{a^3 \csc ^3(c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}+\frac{5 a^3 \csc (c+d x)}{d}-\frac{5 a^3 \log (\sin (c+d x))}{d}","\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^4(c+d x)}{4 d}-\frac{a^3 \csc ^3(c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}+\frac{5 a^3 \csc (c+d x)}{d}-\frac{5 a^3 \log (\sin (c+d x))}{d}",1,"(5*a^3*Csc[c + d*x])/d - (a^3*Csc[c + d*x]^2)/(2*d) - (a^3*Csc[c + d*x]^3)/d - (a^3*Csc[c + d*x]^4)/(4*d) - (5*a^3*Log[Sin[c + d*x]])/d + (a^3*Sin[c + d*x])/d + (3*a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/(3*d)","A",3,2,21,0.09524,1,"{2707, 88}"
528,1,133,0,0.110436,"\int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^5(c+d x)}{5 d}-\frac{3 a^3 \csc ^4(c+d x)}{4 d}-\frac{a^3 \csc ^3(c+d x)}{3 d}+\frac{5 a^3 \csc ^2(c+d x)}{2 d}+\frac{5 a^3 \csc (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}","\frac{a^3 \sin ^2(c+d x)}{2 d}+\frac{3 a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^5(c+d x)}{5 d}-\frac{3 a^3 \csc ^4(c+d x)}{4 d}-\frac{a^3 \csc ^3(c+d x)}{3 d}+\frac{5 a^3 \csc ^2(c+d x)}{2 d}+\frac{5 a^3 \csc (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}",1,"(5*a^3*Csc[c + d*x])/d + (5*a^3*Csc[c + d*x]^2)/(2*d) - (a^3*Csc[c + d*x]^3)/(3*d) - (3*a^3*Csc[c + d*x]^4)/(4*d) - (a^3*Csc[c + d*x]^5)/(5*d) + (a^3*Log[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (a^3*Sin[c + d*x]^2)/(2*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
529,1,133,0,0.1262466,"\int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^6(c+d x)}{6 d}-\frac{3 a^3 \csc ^5(c+d x)}{5 d}-\frac{a^3 \csc ^4(c+d x)}{4 d}+\frac{5 a^3 \csc ^3(c+d x)}{3 d}+\frac{5 a^3 \csc ^2(c+d x)}{2 d}-\frac{a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d}","\frac{a^3 \sin (c+d x)}{d}-\frac{a^3 \csc ^6(c+d x)}{6 d}-\frac{3 a^3 \csc ^5(c+d x)}{5 d}-\frac{a^3 \csc ^4(c+d x)}{4 d}+\frac{5 a^3 \csc ^3(c+d x)}{3 d}+\frac{5 a^3 \csc ^2(c+d x)}{2 d}-\frac{a^3 \csc (c+d x)}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d}",1,"-((a^3*Csc[c + d*x])/d) + (5*a^3*Csc[c + d*x]^2)/(2*d) + (5*a^3*Csc[c + d*x]^3)/(3*d) - (a^3*Csc[c + d*x]^4)/(4*d) - (3*a^3*Csc[c + d*x]^5)/(5*d) - (a^3*Csc[c + d*x]^6)/(6*d) + (3*a^3*Log[Sin[c + d*x]])/d + (a^3*Sin[c + d*x])/d","A",4,3,29,0.1034,1,"{2836, 12, 88}"
530,1,145,0,0.1206831,"\int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \sin ^5(c+d x)}{5 d}+\frac{a^4 \sin ^4(c+d x)}{d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{2 a^4 \sin ^2(c+d x)}{d}-\frac{10 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^3(c+d x)}{3 d}-\frac{2 a^4 \csc ^2(c+d x)}{d}-\frac{4 a^4 \csc (c+d x)}{d}-\frac{4 a^4 \log (\sin (c+d x))}{d}","\frac{a^4 \sin ^5(c+d x)}{5 d}+\frac{a^4 \sin ^4(c+d x)}{d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{2 a^4 \sin ^2(c+d x)}{d}-\frac{10 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^3(c+d x)}{3 d}-\frac{2 a^4 \csc ^2(c+d x)}{d}-\frac{4 a^4 \csc (c+d x)}{d}-\frac{4 a^4 \log (\sin (c+d x))}{d}",1,"(-4*a^4*Csc[c + d*x])/d - (2*a^4*Csc[c + d*x]^2)/d - (a^4*Csc[c + d*x]^3)/(3*d) - (4*a^4*Log[Sin[c + d*x]])/d - (10*a^4*Sin[c + d*x])/d - (2*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^4)/d + (a^4*Sin[c + d*x]^5)/(5*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
531,1,148,0,0.0788179,"\int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{2 a^4 \sin ^2(c+d x)}{d}-\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^4(c+d x)}{4 d}-\frac{4 a^4 \csc ^3(c+d x)}{3 d}-\frac{2 a^4 \csc ^2(c+d x)}{d}+\frac{4 a^4 \csc (c+d x)}{d}-\frac{10 a^4 \log (\sin (c+d x))}{d}","\frac{a^4 \sin ^4(c+d x)}{4 d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{2 a^4 \sin ^2(c+d x)}{d}-\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^4(c+d x)}{4 d}-\frac{4 a^4 \csc ^3(c+d x)}{3 d}-\frac{2 a^4 \csc ^2(c+d x)}{d}+\frac{4 a^4 \csc (c+d x)}{d}-\frac{10 a^4 \log (\sin (c+d x))}{d}",1,"(4*a^4*Csc[c + d*x])/d - (2*a^4*Csc[c + d*x]^2)/d - (4*a^4*Csc[c + d*x]^3)/(3*d) - (a^4*Csc[c + d*x]^4)/(4*d) - (10*a^4*Log[Sin[c + d*x]])/d - (4*a^4*Sin[c + d*x])/d + (2*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^4)/(4*d)","A",3,2,21,0.09524,1,"{2707, 88}"
532,1,146,0,0.1163066,"\int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{2 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^5(c+d x)}{5 d}-\frac{a^4 \csc ^4(c+d x)}{d}-\frac{4 a^4 \csc ^3(c+d x)}{3 d}+\frac{2 a^4 \csc ^2(c+d x)}{d}+\frac{10 a^4 \csc (c+d x)}{d}-\frac{4 a^4 \log (\sin (c+d x))}{d}","\frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{2 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^5(c+d x)}{5 d}-\frac{a^4 \csc ^4(c+d x)}{d}-\frac{4 a^4 \csc ^3(c+d x)}{3 d}+\frac{2 a^4 \csc ^2(c+d x)}{d}+\frac{10 a^4 \csc (c+d x)}{d}-\frac{4 a^4 \log (\sin (c+d x))}{d}",1,"(10*a^4*Csc[c + d*x])/d + (2*a^4*Csc[c + d*x]^2)/d - (4*a^4*Csc[c + d*x]^3)/(3*d) - (a^4*Csc[c + d*x]^4)/d - (a^4*Csc[c + d*x]^5)/(5*d) - (4*a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (2*a^4*Sin[c + d*x]^2)/d + (a^4*Sin[c + d*x]^3)/(3*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
533,1,73,0,0.1134629,"\int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^7(c+d x)}{7 a d}-\frac{\sin ^6(c+d x)}{6 a d}-\frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^4(c+d x)}{4 a d}","\frac{\sin ^7(c+d x)}{7 a d}-\frac{\sin ^6(c+d x)}{6 a d}-\frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^4(c+d x)}{4 a d}",1,"Sin[c + d*x]^4/(4*a*d) - Sin[c + d*x]^5/(5*a*d) - Sin[c + d*x]^6/(6*a*d) + Sin[c + d*x]^7/(7*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 75}"
534,1,73,0,0.155688,"\int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^6(c+d x)}{6 a d}-\frac{\sin ^5(c+d x)}{5 a d}-\frac{\sin ^4(c+d x)}{4 a d}+\frac{\sin ^3(c+d x)}{3 a d}","\frac{\sin ^6(c+d x)}{6 a d}-\frac{\sin ^5(c+d x)}{5 a d}-\frac{\sin ^4(c+d x)}{4 a d}+\frac{\sin ^3(c+d x)}{3 a d}",1,"Sin[c + d*x]^3/(3*a*d) - Sin[c + d*x]^4/(4*a*d) - Sin[c + d*x]^5/(5*a*d) + Sin[c + d*x]^6/(6*a*d)","A",7,3,29,0.1034,1,"{2835, 2564, 14}"
535,1,55,0,0.1072021,"\int \frac{\cos ^5(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\sin ^5(c+d x)}{5 a d}-\frac{\sin ^3(c+d x)}{3 a d}-\frac{\cos ^4(c+d x)}{4 a d}","\frac{\sin ^5(c+d x)}{5 a d}-\frac{\sin ^3(c+d x)}{3 a d}-\frac{\cos ^4(c+d x)}{4 a d}",1,"-Cos[c + d*x]^4/(4*a*d) - Sin[c + d*x]^3/(3*a*d) + Sin[c + d*x]^5/(5*a*d)","A",6,5,27,0.1852,1,"{2835, 2565, 30, 2564, 14}"
536,1,65,0,0.0891006,"\int \frac{\cos ^4(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}","\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}",1,"Log[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d) + Sin[c + d*x]^3/(3*a*d)","A",4,3,27,0.1111,1,"{2836, 12, 75}"
537,1,62,0,0.1084349,"\int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}","\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}",1,"-(Csc[c + d*x]/(a*d)) - Log[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d) + Sin[c + d*x]^2/(2*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 75}"
538,1,60,0,0.1055761,"\int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\sin (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}","\frac{\sin (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}",1,"Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - Log[Sin[c + d*x]]/(a*d) + Sin[c + d*x]/(a*d)","A",4,3,29,0.1034,1,"{2836, 12, 75}"
539,1,64,0,0.092183,"\int \frac{\cos (c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}","-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}",1,"Csc[c + d*x]/(a*d) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) + Log[Sin[c + d*x]]/(a*d)","A",4,3,27,0.1111,1,"{2836, 12, 75}"
540,1,51,0,0.0877663,"\int \frac{\cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^5/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^4(c+d x)}{4 a d}+\frac{\csc ^3(c+d x)}{3 a d}-\frac{\csc (c+d x)}{a d}","-\frac{\cot ^4(c+d x)}{4 a d}+\frac{\csc ^3(c+d x)}{3 a d}-\frac{\csc (c+d x)}{a d}",1,"-Cot[c + d*x]^4/(4*a*d) - Csc[c + d*x]/(a*d) + Csc[c + d*x]^3/(3*a*d)","A",5,4,21,0.1905,1,"{2706, 2607, 30, 2606}"
541,1,55,0,0.1355517,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^5*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\cot ^4(c+d x)}{4 a d}-\frac{\csc ^5(c+d x)}{5 a d}+\frac{\csc ^3(c+d x)}{3 a d}","\frac{\cot ^4(c+d x)}{4 a d}-\frac{\csc ^5(c+d x)}{5 a d}+\frac{\csc ^3(c+d x)}{3 a d}",1,"Cot[c + d*x]^4/(4*a*d) + Csc[c + d*x]^3/(3*a*d) - Csc[c + d*x]^5/(5*a*d)","A",6,5,27,0.1852,1,"{2835, 2606, 14, 2607, 30}"
542,1,73,0,0.1113923,"\int \frac{\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^5*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}+\frac{\csc ^4(c+d x)}{4 a d}-\frac{\csc ^3(c+d x)}{3 a d}","-\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}+\frac{\csc ^4(c+d x)}{4 a d}-\frac{\csc ^3(c+d x)}{3 a d}",1,"-Csc[c + d*x]^3/(3*a*d) + Csc[c + d*x]^4/(4*a*d) + Csc[c + d*x]^5/(5*a*d) - Csc[c + d*x]^6/(6*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 75}"
543,1,73,0,0.1091195,"\int \frac{\cot ^5(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^5*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^7(c+d x)}{7 a d}+\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{\csc ^4(c+d x)}{4 a d}","-\frac{\csc ^7(c+d x)}{7 a d}+\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{\csc ^4(c+d x)}{4 a d}",1,"-Csc[c + d*x]^4/(4*a*d) + Csc[c + d*x]^5/(5*a*d) + Csc[c + d*x]^6/(6*a*d) - Csc[c + d*x]^7/(7*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 75}"
544,1,55,0,0.1064262,"\int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^6(c+d x)}{6 a^2 d}-\frac{2 \sin ^5(c+d x)}{5 a^2 d}+\frac{\sin ^4(c+d x)}{4 a^2 d}","\frac{\sin ^6(c+d x)}{6 a^2 d}-\frac{2 \sin ^5(c+d x)}{5 a^2 d}+\frac{\sin ^4(c+d x)}{4 a^2 d}",1,"Sin[c + d*x]^4/(4*a^2*d) - (2*Sin[c + d*x]^5)/(5*a^2*d) + Sin[c + d*x]^6/(6*a^2*d)","A",4,3,29,0.1034,1,"{2836, 12, 43}"
545,1,55,0,0.104194,"\int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^5(c+d x)}{5 a^2 d}-\frac{\sin ^4(c+d x)}{2 a^2 d}+\frac{\sin ^3(c+d x)}{3 a^2 d}","\frac{\sin ^5(c+d x)}{5 a^2 d}-\frac{\sin ^4(c+d x)}{2 a^2 d}+\frac{\sin ^3(c+d x)}{3 a^2 d}",1,"Sin[c + d*x]^3/(3*a^2*d) - Sin[c + d*x]^4/(2*a^2*d) + Sin[c + d*x]^5/(5*a^2*d)","A",4,3,29,0.1034,1,"{2836, 12, 43}"
546,1,55,0,0.0675053,"\int \frac{\cos ^5(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^4(c+d x)}{4 a^2 d}-\frac{2 \sin ^3(c+d x)}{3 a^2 d}+\frac{\sin ^2(c+d x)}{2 a^2 d}","\frac{\sin ^4(c+d x)}{4 a^2 d}-\frac{2 \sin ^3(c+d x)}{3 a^2 d}+\frac{\sin ^2(c+d x)}{2 a^2 d}",1,"Sin[c + d*x]^2/(2*a^2*d) - (2*Sin[c + d*x]^3)/(3*a^2*d) + Sin[c + d*x]^4/(4*a^2*d)","A",4,3,27,0.1111,1,"{2836, 12, 43}"
547,1,47,0,0.0818374,"\int \frac{\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^2(c+d x)}{2 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{\log (\sin (c+d x))}{a^2 d}","\frac{\sin ^2(c+d x)}{2 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{\log (\sin (c+d x))}{a^2 d}",1,"Log[Sin[c + d*x]]/(a^2*d) - (2*Sin[c + d*x])/(a^2*d) + Sin[c + d*x]^2/(2*a^2*d)","A",4,3,27,0.1111,1,"{2836, 12, 43}"
548,1,43,0,0.0993012,"\int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin (c+d x)}{a^2 d}-\frac{\csc (c+d x)}{a^2 d}-\frac{2 \log (\sin (c+d x))}{a^2 d}","\frac{\sin (c+d x)}{a^2 d}-\frac{\csc (c+d x)}{a^2 d}-\frac{2 \log (\sin (c+d x))}{a^2 d}",1,"-(Csc[c + d*x]/(a^2*d)) - (2*Log[Sin[c + d*x]])/(a^2*d) + Sin[c + d*x]/(a^2*d)","A",4,3,29,0.1034,1,"{2836, 12, 43}"
549,1,47,0,0.1007665,"\int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}+\frac{\log (\sin (c+d x))}{a^2 d}","-\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}+\frac{\log (\sin (c+d x))}{a^2 d}",1,"(2*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a^2*d) + Log[Sin[c + d*x]]/(a^2*d)","A",4,3,29,0.1034,1,"{2836, 12, 43}"
550,1,31,0,0.079296,"\int \frac{\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^3(c+d x) (a-a \sin (c+d x))^3}{3 a^5 d}","-\frac{\csc ^3(c+d x) (a-a \sin (c+d x))^3}{3 a^5 d}",1,"-(Csc[c + d*x]^3*(a - a*Sin[c + d*x])^3)/(3*a^5*d)","A",3,3,27,0.1111,1,"{2836, 12, 37}"
551,1,55,0,0.0460195,"\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}","-\frac{\csc ^4(c+d x)}{4 a^2 d}+\frac{2 \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}",1,"-Csc[c + d*x]^2/(2*a^2*d) + (2*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(4*a^2*d)","A",3,2,21,0.09524,1,"{2707, 43}"
552,1,55,0,0.0849527,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^5*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^5(c+d x)}{5 a^2 d}+\frac{\csc ^4(c+d x)}{2 a^2 d}-\frac{\csc ^3(c+d x)}{3 a^2 d}","-\frac{\csc ^5(c+d x)}{5 a^2 d}+\frac{\csc ^4(c+d x)}{2 a^2 d}-\frac{\csc ^3(c+d x)}{3 a^2 d}",1,"-Csc[c + d*x]^3/(3*a^2*d) + Csc[c + d*x]^4/(2*a^2*d) - Csc[c + d*x]^5/(5*a^2*d)","A",4,3,27,0.1111,1,"{2836, 12, 43}"
553,1,55,0,0.1026009,"\int \frac{\cot ^5(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^5*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^6(c+d x)}{6 a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^4(c+d x)}{4 a^2 d}","-\frac{\csc ^6(c+d x)}{6 a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^4(c+d x)}{4 a^2 d}",1,"-Csc[c + d*x]^4/(4*a^2*d) + (2*Csc[c + d*x]^5)/(5*a^2*d) - Csc[c + d*x]^6/(6*a^2*d)","A",4,3,29,0.1034,1,"{2836, 12, 43}"
554,1,102,0,0.1255686,"\int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{\sin ^5(c+d x)}{5 a^3 d}-\frac{3 \sin ^4(c+d x)}{4 a^3 d}+\frac{4 \sin ^3(c+d x)}{3 a^3 d}-\frac{2 \sin ^2(c+d x)}{a^3 d}+\frac{4 \sin (c+d x)}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}","\frac{\sin ^5(c+d x)}{5 a^3 d}-\frac{3 \sin ^4(c+d x)}{4 a^3 d}+\frac{4 \sin ^3(c+d x)}{3 a^3 d}-\frac{2 \sin ^2(c+d x)}{a^3 d}+\frac{4 \sin (c+d x)}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"(-4*Log[1 + Sin[c + d*x]])/(a^3*d) + (4*Sin[c + d*x])/(a^3*d) - (2*Sin[c + d*x]^2)/(a^3*d) + (4*Sin[c + d*x]^3)/(3*a^3*d) - (3*Sin[c + d*x]^4)/(4*a^3*d) + Sin[c + d*x]^5/(5*a^3*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
555,1,82,0,0.1184766,"\int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\sin ^4(c+d x)}{4 a^3 d}-\frac{\sin ^3(c+d x)}{a^3 d}+\frac{2 \sin ^2(c+d x)}{a^3 d}-\frac{4 \sin (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d}","\frac{\sin ^4(c+d x)}{4 a^3 d}-\frac{\sin ^3(c+d x)}{a^3 d}+\frac{2 \sin ^2(c+d x)}{a^3 d}-\frac{4 \sin (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"(4*Log[1 + Sin[c + d*x]])/(a^3*d) - (4*Sin[c + d*x])/(a^3*d) + (2*Sin[c + d*x]^2)/(a^3*d) - Sin[c + d*x]^3/(a^3*d) + Sin[c + d*x]^4/(4*a^3*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
556,1,68,0,0.0763898,"\int \frac{\cos ^5(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{\sin ^3(c+d x)}{3 a^3 d}-\frac{3 \sin ^2(c+d x)}{2 a^3 d}+\frac{4 \sin (c+d x)}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}","\frac{\sin ^3(c+d x)}{3 a^3 d}-\frac{3 \sin ^2(c+d x)}{2 a^3 d}+\frac{4 \sin (c+d x)}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"(-4*Log[1 + Sin[c + d*x]])/(a^3*d) + (4*Sin[c + d*x])/(a^3*d) - (3*Sin[c + d*x]^2)/(2*a^3*d) + Sin[c + d*x]^3/(3*a^3*d)","A",4,3,27,0.1111,1,"{2836, 12, 77}"
557,1,45,0,0.08554,"\int \frac{\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{\sin (c+d x)}{a^3 d}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}","\frac{\sin (c+d x)}{a^3 d}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"Log[Sin[c + d*x]]/(a^3*d) - (4*Log[1 + Sin[c + d*x]])/(a^3*d) + Sin[c + d*x]/(a^3*d)","A",4,3,27,0.1111,1,"{2836, 12, 72}"
558,1,47,0,0.1057449,"\int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc (c+d x)}{a^3 d}-\frac{3 \log (\sin (c+d x))}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d}","-\frac{\csc (c+d x)}{a^3 d}-\frac{3 \log (\sin (c+d x))}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"-(Csc[c + d*x]/(a^3*d)) - (3*Log[Sin[c + d*x]])/(a^3*d) + (4*Log[1 + Sin[c + d*x]])/(a^3*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
559,1,65,0,0.1131561,"\int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^2(c+d x)}{2 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}","-\frac{\csc ^2(c+d x)}{2 a^3 d}+\frac{3 \csc (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"(3*Csc[c + d*x])/(a^3*d) - Csc[c + d*x]^2/(2*a^3*d) + (4*Log[Sin[c + d*x]])/(a^3*d) - (4*Log[1 + Sin[c + d*x]])/(a^3*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
560,1,83,0,0.1004395,"\int \frac{\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc ^2(c+d x)}{2 a^3 d}-\frac{4 \csc (c+d x)}{a^3 d}-\frac{4 \log (\sin (c+d x))}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d}","-\frac{\csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc ^2(c+d x)}{2 a^3 d}-\frac{4 \csc (c+d x)}{a^3 d}-\frac{4 \log (\sin (c+d x))}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"(-4*Csc[c + d*x])/(a^3*d) + (3*Csc[c + d*x]^2)/(2*a^3*d) - Csc[c + d*x]^3/(3*a^3*d) - (4*Log[Sin[c + d*x]])/(a^3*d) + (4*Log[1 + Sin[c + d*x]])/(a^3*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
561,1,96,0,0.0677812,"\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{a^3 d}-\frac{2 \csc ^2(c+d x)}{a^3 d}+\frac{4 \csc (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}","-\frac{\csc ^4(c+d x)}{4 a^3 d}+\frac{\csc ^3(c+d x)}{a^3 d}-\frac{2 \csc ^2(c+d x)}{a^3 d}+\frac{4 \csc (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x))}{a^3 d}-\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"(4*Csc[c + d*x])/(a^3*d) - (2*Csc[c + d*x]^2)/(a^3*d) + Csc[c + d*x]^3/(a^3*d) - Csc[c + d*x]^4/(4*a^3*d) + (4*Log[Sin[c + d*x]])/(a^3*d) - (4*Log[1 + Sin[c + d*x]])/(a^3*d)","A",3,2,21,0.09524,1,"{2707, 88}"
562,1,117,0,0.1194614,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cot[c + d*x]^5*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^5(c+d x)}{5 a^3 d}+\frac{3 \csc ^4(c+d x)}{4 a^3 d}-\frac{4 \csc ^3(c+d x)}{3 a^3 d}+\frac{2 \csc ^2(c+d x)}{a^3 d}-\frac{4 \csc (c+d x)}{a^3 d}-\frac{4 \log (\sin (c+d x))}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d}","-\frac{\csc ^5(c+d x)}{5 a^3 d}+\frac{3 \csc ^4(c+d x)}{4 a^3 d}-\frac{4 \csc ^3(c+d x)}{3 a^3 d}+\frac{2 \csc ^2(c+d x)}{a^3 d}-\frac{4 \csc (c+d x)}{a^3 d}-\frac{4 \log (\sin (c+d x))}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"(-4*Csc[c + d*x])/(a^3*d) + (2*Csc[c + d*x]^2)/(a^3*d) - (4*Csc[c + d*x]^3)/(3*a^3*d) + (3*Csc[c + d*x]^4)/(4*a^3*d) - Csc[c + d*x]^5/(5*a^3*d) - (4*Log[Sin[c + d*x]])/(a^3*d) + (4*Log[1 + Sin[c + d*x]])/(a^3*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
563,1,120,0,0.0826289,"\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^4,x]","\frac{4}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{\csc ^4(c+d x)}{4 a^4 d}+\frac{4 \csc ^3(c+d x)}{3 a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{12 \csc (c+d x)}{a^4 d}+\frac{16 \log (\sin (c+d x))}{a^4 d}-\frac{16 \log (\sin (c+d x)+1)}{a^4 d}","\frac{4}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{\csc ^4(c+d x)}{4 a^4 d}+\frac{4 \csc ^3(c+d x)}{3 a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{12 \csc (c+d x)}{a^4 d}+\frac{16 \log (\sin (c+d x))}{a^4 d}-\frac{16 \log (\sin (c+d x)+1)}{a^4 d}",1,"(12*Csc[c + d*x])/(a^4*d) - (4*Csc[c + d*x]^2)/(a^4*d) + (4*Csc[c + d*x]^3)/(3*a^4*d) - Csc[c + d*x]^4/(4*a^4*d) + (16*Log[Sin[c + d*x]])/(a^4*d) - (16*Log[1 + Sin[c + d*x]])/(a^4*d) + 4/(d*(a^4 + a^4*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 88}"
564,1,135,0,0.1323664,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Cot[c + d*x]^5*Csc[c + d*x])/(a + a*Sin[c + d*x])^4,x]","-\frac{4}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{\csc ^5(c+d x)}{5 a^4 d}+\frac{\csc ^4(c+d x)}{a^4 d}-\frac{8 \csc ^3(c+d x)}{3 a^4 d}+\frac{6 \csc ^2(c+d x)}{a^4 d}-\frac{16 \csc (c+d x)}{a^4 d}-\frac{20 \log (\sin (c+d x))}{a^4 d}+\frac{20 \log (\sin (c+d x)+1)}{a^4 d}","-\frac{4}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{\csc ^5(c+d x)}{5 a^4 d}+\frac{\csc ^4(c+d x)}{a^4 d}-\frac{8 \csc ^3(c+d x)}{3 a^4 d}+\frac{6 \csc ^2(c+d x)}{a^4 d}-\frac{16 \csc (c+d x)}{a^4 d}-\frac{20 \log (\sin (c+d x))}{a^4 d}+\frac{20 \log (\sin (c+d x)+1)}{a^4 d}",1,"(-16*Csc[c + d*x])/(a^4*d) + (6*Csc[c + d*x]^2)/(a^4*d) - (8*Csc[c + d*x]^3)/(3*a^4*d) + Csc[c + d*x]^4/(a^4*d) - Csc[c + d*x]^5/(5*a^4*d) - (20*Log[Sin[c + d*x]])/(a^4*d) + (20*Log[1 + Sin[c + d*x]])/(a^4*d) - 4/(d*(a^4 + a^4*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 88}"
565,1,181,0,0.180944,"\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{a^3 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{5 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{5 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{3 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}+\frac{a^3 \sin ^{n+8}(c+d x)}{d (n+8)}","\frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{a^3 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{5 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{5 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{3 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}+\frac{a^3 \sin ^{n+8}(c+d x)}{d (n+8)}",1,"(a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (a^3*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (5*a^3*Sin[c + d*x]^(4 + n))/(d*(4 + n)) - (5*a^3*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (a^3*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (3*a^3*Sin[c + d*x]^(7 + n))/(d*(7 + n)) + (a^3*Sin[c + d*x]^(8 + n))/(d*(8 + n))","A",3,2,29,0.06897,1,"{2836, 88}"
566,1,160,0,0.1667576,"\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{a^2 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{4 a^2 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{a^2 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{2 a^2 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{a^2 \sin ^{n+7}(c+d x)}{d (n+7)}","\frac{a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{a^2 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{4 a^2 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{a^2 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{2 a^2 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{a^2 \sin ^{n+7}(c+d x)}{d (n+7)}",1,"(a^2*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (2*a^2*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (a^2*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (4*a^2*Sin[c + d*x]^(4 + n))/(d*(4 + n)) - (a^2*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (2*a^2*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (a^2*Sin[c + d*x]^(7 + n))/(d*(7 + n))","A",3,2,29,0.06897,1,"{2836, 88}"
567,1,123,0,0.1191844,"\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{2 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{2 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{a \sin ^{n+6}(c+d x)}{d (n+6)}","\frac{a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{2 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{2 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{a \sin ^{n+6}(c+d x)}{d (n+6)}",1,"(a*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (a*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (2*a*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (2*a*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (a*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (a*Sin[c + d*x]^(6 + n))/(d*(6 + n))","A",3,2,27,0.07407,1,"{2836, 88}"
568,1,91,0,0.1394788,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^n)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^{n+1}(c+d x)}{a d (n+1)}-\frac{\sin ^{n+2}(c+d x)}{a d (n+2)}-\frac{\sin ^{n+3}(c+d x)}{a d (n+3)}+\frac{\sin ^{n+4}(c+d x)}{a d (n+4)}","\frac{\sin ^{n+1}(c+d x)}{a d (n+1)}-\frac{\sin ^{n+2}(c+d x)}{a d (n+2)}-\frac{\sin ^{n+3}(c+d x)}{a d (n+3)}+\frac{\sin ^{n+4}(c+d x)}{a d (n+4)}",1,"Sin[c + d*x]^(1 + n)/(a*d*(1 + n)) - Sin[c + d*x]^(2 + n)/(a*d*(2 + n)) - Sin[c + d*x]^(3 + n)/(a*d*(3 + n)) + Sin[c + d*x]^(4 + n)/(a*d*(4 + n))","A",3,2,29,0.06897,1,"{2836, 75}"
569,1,68,0,0.1272348,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^{n+1}(c+d x)}{a^2 d (n+1)}-\frac{2 \sin ^{n+2}(c+d x)}{a^2 d (n+2)}+\frac{\sin ^{n+3}(c+d x)}{a^2 d (n+3)}","\frac{\sin ^{n+1}(c+d x)}{a^2 d (n+1)}-\frac{2 \sin ^{n+2}(c+d x)}{a^2 d (n+2)}+\frac{\sin ^{n+3}(c+d x)}{a^2 d (n+3)}",1,"Sin[c + d*x]^(1 + n)/(a^2*d*(1 + n)) - (2*Sin[c + d*x]^(2 + n))/(a^2*d*(2 + n)) + Sin[c + d*x]^(3 + n)/(a^2*d*(3 + n))","A",3,2,29,0.06897,1,"{2836, 43}"
570,1,85,0,0.1446039,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^3,x]","\frac{4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)}-\frac{3 \sin ^{n+1}(c+d x)}{a^3 d (n+1)}+\frac{\sin ^{n+2}(c+d x)}{a^3 d (n+2)}","\frac{4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)}-\frac{3 \sin ^{n+1}(c+d x)}{a^3 d (n+1)}+\frac{\sin ^{n+2}(c+d x)}{a^3 d (n+2)}",1,"(-3*Sin[c + d*x]^(1 + n))/(a^3*d*(1 + n)) + (4*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^3*d*(1 + n)) + Sin[c + d*x]^(2 + n)/(a^3*d*(2 + n))","A",4,3,29,0.1034,1,"{2836, 88, 64}"
571,1,88,0,0.1374556,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^4,x]","-\frac{4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d}+\frac{\sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac{4 \sin ^{n+1}(c+d x)}{d \left(a^4 \sin (c+d x)+a^4\right)}","-\frac{4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d}+\frac{\sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac{4 \sin ^{n+1}(c+d x)}{d \left(a^4 \sin (c+d x)+a^4\right)}",1,"Sin[c + d*x]^(1 + n)/(a^4*d*(1 + n)) - (4*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^4*d) + (4*Sin[c + d*x]^(1 + n))/(d*(a^4 + a^4*Sin[c + d*x]))","A",4,4,29,0.1379,1,"{2836, 89, 80, 64}"
572,1,165,0,0.200495,"\int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \cos ^{11}(c+d x)}{11 d}+\frac{2 a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac{3 a x}{256}","-\frac{a \cos ^{11}(c+d x)}{11 d}+\frac{2 a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac{3 a x}{256}",1,"(3*a*x)/256 - (a*Cos[c + d*x]^7)/(7*d) + (2*a*Cos[c + d*x]^9)/(9*d) - (a*Cos[c + d*x]^11)/(11*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (3*a*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (a*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*d)","A",10,6,27,0.2222,1,"{2838, 2568, 2635, 8, 2565, 270}"
573,1,149,0,0.2061552,"\int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac{3 a x}{256}","\frac{a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac{3 a x}{256}",1,"(3*a*x)/256 - (a*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]^9)/(9*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (3*a*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (a*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*d)","A",10,6,27,0.2222,1,"{2838, 2565, 14, 2568, 2635, 8}"
574,1,125,0,0.1549139,"\int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5 a x}{128}","\frac{a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5 a x}{128}",1,"(5*a*x)/128 - (a*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]^9)/(9*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)","A",9,6,27,0.2222,1,"{2838, 2568, 2635, 8, 2565, 14}"
575,1,109,0,0.1195031,"\int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5 a x}{128}","-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5 a x}{128}",1,"(5*a*x)/128 - (a*Cos[c + d*x]^7)/(7*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)","A",8,6,25,0.2400,1,"{2838, 2565, 30, 2568, 2635, 8}"
576,1,127,0,0.1092116,"\int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{5 a x}{16}","\frac{a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{5 a x}{16}",1,"(5*a*x)/16 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]^5)/(5*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)","A",9,6,25,0.2400,1,"{2838, 2592, 302, 206, 2635, 8}"
577,1,121,0,0.1337986,"\int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}-\frac{15 a \cot (c+d x)}{8 d}+\frac{a \cos ^4(c+d x) \cot (c+d x)}{4 d}+\frac{5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{15 a x}{8}","\frac{a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}-\frac{15 a \cot (c+d x)}{8 d}+\frac{a \cos ^4(c+d x) \cot (c+d x)}{4 d}+\frac{5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{15 a x}{8}",1,"(-15*a*x)/8 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]^5)/(5*d) - (15*a*Cot[c + d*x])/(8*d) + (5*a*Cos[c + d*x]^2*Cot[c + d*x])/(8*d) + (a*Cos[c + d*x]^4*Cot[c + d*x])/(4*d)","A",10,8,27,0.2963,1,"{2838, 2591, 288, 321, 203, 2592, 302, 206}"
578,1,134,0,0.1437256,"\int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{5 a \cos ^3(c+d x)}{6 d}-\frac{5 a \cos (c+d x)}{2 d}-\frac{15 a \cot (c+d x)}{8 d}-\frac{a \cos ^3(c+d x) \cot ^2(c+d x)}{2 d}+\frac{a \cos ^4(c+d x) \cot (c+d x)}{4 d}+\frac{5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{15 a x}{8}","-\frac{5 a \cos ^3(c+d x)}{6 d}-\frac{5 a \cos (c+d x)}{2 d}-\frac{15 a \cot (c+d x)}{8 d}-\frac{a \cos ^3(c+d x) \cot ^2(c+d x)}{2 d}+\frac{a \cos ^4(c+d x) \cot (c+d x)}{4 d}+\frac{5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{15 a x}{8}",1,"(-15*a*x)/8 + (5*a*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a*Cos[c + d*x])/(2*d) - (5*a*Cos[c + d*x]^3)/(6*d) - (15*a*Cot[c + d*x])/(8*d) + (5*a*Cos[c + d*x]^2*Cot[c + d*x])/(8*d) + (a*Cos[c + d*x]^4*Cot[c + d*x])/(4*d) - (a*Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*d)","A",11,8,27,0.2963,1,"{2838, 2592, 288, 302, 206, 2591, 321, 203}"
579,1,130,0,0.1396665,"\int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{5 a \cos ^3(c+d x)}{6 d}-\frac{5 a \cos (c+d x)}{2 d}-\frac{5 a \cot ^3(c+d x)}{6 d}+\frac{5 a \cot (c+d x)}{2 d}-\frac{a \cos ^3(c+d x) \cot ^2(c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{5 a x}{2}","-\frac{5 a \cos ^3(c+d x)}{6 d}-\frac{5 a \cos (c+d x)}{2 d}-\frac{5 a \cot ^3(c+d x)}{6 d}+\frac{5 a \cot (c+d x)}{2 d}-\frac{a \cos ^3(c+d x) \cot ^2(c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{5 a x}{2}",1,"(5*a*x)/2 + (5*a*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a*Cos[c + d*x])/(2*d) - (5*a*Cos[c + d*x]^3)/(6*d) + (5*a*Cot[c + d*x])/(2*d) - (a*Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*d) - (5*a*Cot[c + d*x]^3)/(6*d) + (a*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d)","A",11,7,27,0.2593,1,"{2838, 2591, 288, 302, 203, 2592, 206}"
580,1,134,0,0.1260654,"\int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{15 a \cos (c+d x)}{8 d}-\frac{5 a \cot ^3(c+d x)}{6 d}+\frac{5 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{15 a \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{5 a x}{2}","\frac{15 a \cos (c+d x)}{8 d}-\frac{5 a \cot ^3(c+d x)}{6 d}+\frac{5 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{15 a \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{5 a x}{2}",1,"(5*a*x)/2 - (15*a*ArcTanh[Cos[c + d*x]])/(8*d) + (15*a*Cos[c + d*x])/(8*d) + (5*a*Cot[c + d*x])/(2*d) + (5*a*Cos[c + d*x]*Cot[c + d*x]^2)/(8*d) - (5*a*Cot[c + d*x]^3)/(6*d) + (a*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d) - (a*Cos[c + d*x]*Cot[c + d*x]^4)/(4*d)","A",11,8,25,0.3200,1,"{2838, 2592, 288, 321, 206, 2591, 302, 203}"
581,1,122,0,0.0991391,"\int \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^6*(a + a*Sin[c + d*x]),x]","\frac{15 a \cos (c+d x)}{8 d}-\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{15 a \tanh ^{-1}(\cos (c+d x))}{8 d}-a x","\frac{15 a \cos (c+d x)}{8 d}-\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac{5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac{15 a \tanh ^{-1}(\cos (c+d x))}{8 d}-a x",1,"-(a*x) - (15*a*ArcTanh[Cos[c + d*x]])/(8*d) + (15*a*Cos[c + d*x])/(8*d) - (a*Cot[c + d*x])/d + (5*a*Cos[c + d*x]*Cot[c + d*x]^2)/(8*d) + (a*Cot[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]*Cot[c + d*x]^4)/(4*d) - (a*Cot[c + d*x]^5)/(5*d)","A",11,7,19,0.3684,1,"{2710, 2592, 288, 321, 206, 3473, 8}"
582,1,128,0,0.1318582,"\int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}-a x","-\frac{a \cot ^5(c+d x)}{5 d}+\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}-a x",1,"-(a*x) + (5*a*ArcTanh[Cos[c + d*x]])/(16*d) - (a*Cot[c + d*x])/d + (a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^5)/(5*d) - (5*a*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (5*a*Cot[c + d*x]^3*Csc[c + d*x])/(24*d) - (a*Cot[c + d*x]^5*Csc[c + d*x])/(6*d)","A",9,5,25,0.2000,1,"{2838, 2611, 3770, 3473, 8}"
583,1,96,0,0.1404508,"\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}","-\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}",1,"(5*a*ArcTanh[Cos[c + d*x]])/(16*d) - (a*Cot[c + d*x]^7)/(7*d) - (5*a*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (5*a*Cot[c + d*x]^3*Csc[c + d*x])/(24*d) - (a*Cot[c + d*x]^5*Csc[c + d*x])/(6*d)","A",7,5,27,0.1852,1,"{2838, 2607, 30, 2611, 3770}"
584,1,122,0,0.1839887,"\int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a \cot (c+d x) \csc (c+d x)}{128 d}","-\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a \cot (c+d x) \csc (c+d x)}{128 d}",1,"(5*a*ArcTanh[Cos[c + d*x]])/(128*d) - (a*Cot[c + d*x]^7)/(7*d) + (5*a*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (5*a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)","A",8,6,27,0.2222,1,"{2838, 2611, 3768, 3770, 2607, 30}"
585,1,138,0,0.1989583,"\int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^9(c+d x)}{9 d}-\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a \cot (c+d x) \csc (c+d x)}{128 d}","-\frac{a \cot ^9(c+d x)}{9 d}-\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a \cot (c+d x) \csc (c+d x)}{128 d}",1,"(5*a*ArcTanh[Cos[c + d*x]])/(128*d) - (a*Cot[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]^9)/(9*d) + (5*a*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (5*a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)","A",9,6,27,0.2222,1,"{2838, 2607, 14, 2611, 3768, 3770}"
586,1,160,0,0.2092964,"\int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^9(c+d x)}{9 d}-\frac{a \cot ^7(c+d x)}{7 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{256 d}","-\frac{a \cot ^9(c+d x)}{9 d}-\frac{a \cot ^7(c+d x)}{7 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{256 d}",1,"(3*a*ArcTanh[Cos[c + d*x]])/(256*d) - (a*Cot[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]^9)/(9*d) + (3*a*Cot[c + d*x]*Csc[c + d*x])/(256*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)","A",10,6,27,0.2222,1,"{2838, 2611, 3768, 3770, 2607, 14}"
587,1,176,0,0.2179953,"\int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^{11}(c+d x)}{11 d}-\frac{2 a \cot ^9(c+d x)}{9 d}-\frac{a \cot ^7(c+d x)}{7 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{256 d}","-\frac{a \cot ^{11}(c+d x)}{11 d}-\frac{2 a \cot ^9(c+d x)}{9 d}-\frac{a \cot ^7(c+d x)}{7 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{256 d}",1,"(3*a*ArcTanh[Cos[c + d*x]])/(256*d) - (a*Cot[c + d*x]^7)/(7*d) - (2*a*Cot[c + d*x]^9)/(9*d) - (a*Cot[c + d*x]^11)/(11*d) + (3*a*Cot[c + d*x]*Csc[c + d*x])/(256*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)","A",10,6,27,0.2222,1,"{2838, 2607, 270, 2611, 3768, 3770}"
588,1,209,0,0.3957164,"\int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac{4 a^2 \cos ^9(c+d x)}{9 d}-\frac{2 a^2 \cos ^7(c+d x)}{7 d}-\frac{a^2 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac{17 a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac{17 a^2 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{17 a^2 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{17 a^2 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{17 a^2 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{17 a^2 x}{1024}","-\frac{2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac{4 a^2 \cos ^9(c+d x)}{9 d}-\frac{2 a^2 \cos ^7(c+d x)}{7 d}-\frac{a^2 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac{17 a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac{17 a^2 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{17 a^2 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{17 a^2 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{17 a^2 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{17 a^2 x}{1024}",1,"(17*a^2*x)/1024 - (2*a^2*Cos[c + d*x]^7)/(7*d) + (4*a^2*Cos[c + d*x]^9)/(9*d) - (2*a^2*Cos[c + d*x]^11)/(11*d) + (17*a^2*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (17*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + (17*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(1920*d) - (17*a^2*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (17*a^2*Cos[c + d*x]^7*Sin[c + d*x]^3)/(120*d) - (a^2*Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*d)","A",18,6,29,0.2069,1,"{2873, 2568, 2635, 8, 2565, 270}"
589,1,183,0,0.2663282,"\int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos ^{11}(c+d x)}{11 d}+\frac{a^2 \cos ^9(c+d x)}{3 d}-\frac{2 a^2 \cos ^7(c+d x)}{7 d}-\frac{a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{5 d}-\frac{3 a^2 \sin (c+d x) \cos ^7(c+d x)}{40 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{80 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a^2 x}{128}","-\frac{a^2 \cos ^{11}(c+d x)}{11 d}+\frac{a^2 \cos ^9(c+d x)}{3 d}-\frac{2 a^2 \cos ^7(c+d x)}{7 d}-\frac{a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{5 d}-\frac{3 a^2 \sin (c+d x) \cos ^7(c+d x)}{40 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{80 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a^2 x}{128}",1,"(3*a^2*x)/128 - (2*a^2*Cos[c + d*x]^7)/(7*d) + (a^2*Cos[c + d*x]^9)/(3*d) - (a^2*Cos[c + d*x]^11)/(11*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(80*d) - (3*a^2*Cos[c + d*x]^7*Sin[c + d*x])/(40*d) - (a^2*Cos[c + d*x]^7*Sin[c + d*x]^3)/(5*d)","A",14,7,29,0.2414,1,"{2873, 2565, 14, 2568, 2635, 8, 270}"
590,1,165,0,0.2955178,"\int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \cos ^9(c+d x)}{9 d}-\frac{2 a^2 \cos ^7(c+d x)}{7 d}-\frac{a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{13 a^2 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{13 a^2 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac{13 a^2 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{13 a^2 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{13 a^2 x}{256}","\frac{2 a^2 \cos ^9(c+d x)}{9 d}-\frac{2 a^2 \cos ^7(c+d x)}{7 d}-\frac{a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{13 a^2 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{13 a^2 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac{13 a^2 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{13 a^2 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{13 a^2 x}{256}",1,"(13*a^2*x)/256 - (2*a^2*Cos[c + d*x]^7)/(7*d) + (2*a^2*Cos[c + d*x]^9)/(9*d) + (13*a^2*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (13*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (13*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(480*d) - (13*a^2*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (a^2*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*d)","A",16,6,29,0.2069,1,"{2873, 2568, 2635, 8, 2565, 14}"
591,1,153,0,0.1536728,"\int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos ^7(c+d x)}{28 d}-\frac{\cos ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{36 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac{5 a^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{5 a^2 x}{64}-\frac{\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}","-\frac{a^2 \cos ^7(c+d x)}{28 d}-\frac{\cos ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{36 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac{5 a^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{5 a^2 x}{64}-\frac{\cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}",1,"(5*a^2*x)/64 - (a^2*Cos[c + d*x]^7)/(28*d) + (5*a^2*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (5*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(96*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(24*d) - (Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(9*d) - (Cos[c + d*x]^7*(a^2 + a^2*Sin[c + d*x]))/(36*d)","A",7,5,27,0.1852,1,"{2860, 2678, 2669, 2635, 8}"
592,1,161,0,0.1644333,"\int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*Cot[c + d*x]*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos ^7(c+d x)}{7 d}+\frac{a^2 \cos ^5(c+d x)}{5 d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac{5 a^2 \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{5 a^2 x}{8}","-\frac{a^2 \cos ^7(c+d x)}{7 d}+\frac{a^2 \cos ^5(c+d x)}{5 d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac{5 a^2 \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{5 a^2 x}{8}",1,"(5*a^2*x)/8 - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cos[c + d*x]^5)/(5*d) - (a^2*Cos[c + d*x]^7)/(7*d) + (5*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(12*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(3*d)","A",12,8,27,0.2963,1,"{2873, 2635, 8, 2592, 302, 206, 2565, 30}"
593,1,158,0,0.226387,"\int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \cos ^5(c+d x)}{5 d}+\frac{2 a^2 \cos ^3(c+d x)}{3 d}+\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{7 a^2 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{25 a^2 x}{16}","\frac{2 a^2 \cos ^5(c+d x)}{5 d}+\frac{2 a^2 \cos ^3(c+d x)}{3 d}+\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{7 a^2 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{25 a^2 x}{16}",1,"(-25*a^2*x)/16 - (2*a^2*ArcTanh[Cos[c + d*x]])/d + (2*a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x]^3)/(3*d) + (2*a^2*Cos[c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x])/d - (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (7*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^2*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)","A",17,7,29,0.2414,1,"{2872, 3770, 3767, 8, 2638, 2633, 2635}"
594,1,140,0,0.209124,"\int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{a^2 \sin ^3(c+d x) \cos (c+d x)}{2 d}-\frac{9 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{15 a^2 x}{4}","\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{a^2 \sin ^3(c+d x) \cos (c+d x)}{2 d}-\frac{9 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{15 a^2 x}{4}",1,"(-15*a^2*x)/4 + (3*a^2*ArcTanh[Cos[c + d*x]])/(2*d) - (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (9*a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(2*d)","A",16,7,29,0.2414,1,"{2872, 3770, 3767, 8, 3768, 2635, 2633}"
595,1,153,0,0.215374,"\int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cos ^3(c+d x)}{3 d}-\frac{4 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}+\frac{5 a^2 x}{8}","-\frac{2 a^2 \cos ^3(c+d x)}{3 d}-\frac{4 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}+\frac{5 a^2 x}{8}",1,"(5*a^2*x)/8 + (5*a^2*ArcTanh[Cos[c + d*x]])/d - (4*a^2*Cos[c + d*x])/d - (2*a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]*Csc[c + d*x])/d - (5*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",17,8,29,0.2759,1,"{2872, 3770, 3767, 8, 3768, 2638, 2635, 2633}"
596,1,153,0,0.1970782,"\int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cos ^3(c+d x)}{3 d}-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{4 a^2 \cot (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}+5 a^2 x","-\frac{a^2 \cos ^3(c+d x)}{3 d}-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{4 a^2 \cot (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}+5 a^2 x",1,"5*a^2*x + (5*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (a^2*Cos[c + d*x])/d - (a^2*Cos[c + d*x]^3)/(3*d) + (4*a^2*Cot[c + d*x])/d - (2*a^2*Cot[c + d*x]^3)/(3*d) + (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/d","A",16,8,27,0.2963,1,"{2872, 3767, 8, 3768, 3770, 2638, 2635, 2633}"
597,1,139,0,0.2494141,"\int \cot ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{15 a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d}+\frac{9 a^2 \cot (c+d x) \csc (c+d x)}{4 d}+\frac{3 a^2 x}{2}","\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{15 a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d}+\frac{9 a^2 \cot (c+d x) \csc (c+d x)}{4 d}+\frac{3 a^2 x}{2}",1,"(3*a^2*x)/2 - (15*a^2*ArcTanh[Cos[c + d*x]])/(4*d) + (2*a^2*Cos[c + d*x])/d + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^5)/(5*d) + (9*a^2*Cot[c + d*x]*Csc[c + d*x])/(4*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(2*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",15,7,21,0.3333,1,"{2709, 3770, 3768, 3767, 2638, 2635, 8}"
598,1,157,0,0.2345264,"\int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{25 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-2 a^2 x","\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{25 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-2 a^2 x",1,"-2*a^2*x - (25*a^2*ArcTanh[Cos[c + d*x]])/(16*d) + (a^2*Cos[c + d*x])/d - (2*a^2*Cot[c + d*x])/d + (2*a^2*Cot[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)","A",17,6,27,0.2222,1,"{2872, 3770, 3767, 8, 3768, 2638}"
599,1,162,0,0.2278968,"\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}+\frac{5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-a^2 x","-\frac{a^2 \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}+\frac{5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-a^2 x",1,"-(a^2*x) + (5*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x]^7)/(7*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x])/(12*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x])/(3*d)","A",12,7,29,0.2414,1,"{2873, 3473, 8, 2611, 3770, 2607, 30}"
600,1,182,0,0.3133346,"\int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{45 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{35 a^2 \cot (c+d x) \csc (c+d x)}{128 d}","-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{45 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{35 a^2 \cot (c+d x) \csc (c+d x)}{128 d}",1,"(45*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (35*a^2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x])/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x])/(6*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)","A",13,6,29,0.2069,1,"{2873, 2611, 3770, 2607, 30, 3768}"
601,1,152,0,0.2851754,"\int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{64 d}","-\frac{a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{64 d}",1,"(5*a^2*ArcTanh[Cos[c + d*x]])/(64*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(9*d) + (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(64*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(32*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3)/(4*d)","A",12,7,29,0.2414,1,"{2873, 2607, 30, 2611, 3768, 3770, 14}"
602,1,228,0,0.3909195,"\int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{13 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{13 a^2 \cot (c+d x) \csc (c+d x)}{256 d}","-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{13 a^2 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{9 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{13 a^2 \cot (c+d x) \csc (c+d x)}{256 d}",1,"(13*a^2*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) + (13*a^2*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (9*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)","A",16,6,29,0.2069,1,"{2873, 2611, 3768, 3770, 2607, 14}"
603,1,194,0,0.3016308,"\int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^{11}(c+d x)}{11 d}-\frac{a^2 \cot ^9(c+d x)}{3 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{128 d}","-\frac{a^2 \cot ^{11}(c+d x)}{11 d}-\frac{a^2 \cot ^9(c+d x)}{3 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{128 d}",1,"(3*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(3*d) - (a^2*Cot[c + d*x]^11)/(11*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^5)/(5*d)","A",14,7,29,0.2414,1,"{2873, 2607, 14, 2611, 3768, 3770, 270}"
604,1,270,0,0.4261465,"\int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]","-\frac{2 a^2 \cot ^{11}(c+d x)}{11 d}-\frac{4 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{17 a^2 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac{17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d}","-\frac{2 a^2 \cot ^{11}(c+d x)}{11 d}-\frac{4 a^2 \cot ^9(c+d x)}{9 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{17 a^2 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^2 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{a^2 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{11 a^2 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{17 a^2 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac{17 a^2 \cot (c+d x) \csc (c+d x)}{1024 d}",1,"(17*a^2*ArcTanh[Cos[c + d*x]])/(1024*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (4*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2*Cot[c + d*x]^11)/(11*d) + (17*a^2*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (17*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(1536*d) - (11*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(384*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^7)/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^7)/(12*d)","A",18,6,29,0.2069,1,"{2873, 2611, 3768, 3770, 2607, 270}"
605,1,224,0,0.4228323,"\int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \cos ^{13}(c+d x)}{13 d}-\frac{6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{a^3 \cos ^9(c+d x)}{d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{4 d}-\frac{9 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{40 d}-\frac{27 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{9 a^3 \sin (c+d x) \cos ^5(c+d x)}{640 d}+\frac{9 a^3 \sin (c+d x) \cos ^3(c+d x)}{512 d}+\frac{27 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{27 a^3 x}{1024}","\frac{a^3 \cos ^{13}(c+d x)}{13 d}-\frac{6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{a^3 \cos ^9(c+d x)}{d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{4 d}-\frac{9 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{40 d}-\frac{27 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{9 a^3 \sin (c+d x) \cos ^5(c+d x)}{640 d}+\frac{9 a^3 \sin (c+d x) \cos ^3(c+d x)}{512 d}+\frac{27 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{27 a^3 x}{1024}",1,"(27*a^3*x)/1024 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (a^3*Cos[c + d*x]^9)/d - (6*a^3*Cos[c + d*x]^11)/(11*d) + (a^3*Cos[c + d*x]^13)/(13*d) + (27*a^3*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (9*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(512*d) + (9*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(640*d) - (27*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (9*a^3*Cos[c + d*x]^7*Sin[c + d*x]^3)/(40*d) - (a^3*Cos[c + d*x]^7*Sin[c + d*x]^5)/(4*d)","A",21,6,29,0.2069,1,"{2873, 2568, 2635, 8, 2565, 270}"
606,1,209,0,0.4125022,"\int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{7 a^3 \cos ^9(c+d x)}{9 d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac{41 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac{41 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{41 a^3 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{41 a^3 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{41 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{41 a^3 x}{1024}","-\frac{3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{7 a^3 \cos ^9(c+d x)}{9 d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac{41 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac{41 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{41 a^3 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{41 a^3 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{41 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{41 a^3 x}{1024}",1,"(41*a^3*x)/1024 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (7*a^3*Cos[c + d*x]^9)/(9*d) - (3*a^3*Cos[c + d*x]^11)/(11*d) + (41*a^3*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (41*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + (41*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(1920*d) - (41*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (41*a^3*Cos[c + d*x]^7*Sin[c + d*x]^3)/(120*d) - (a^3*Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*d)","A",21,7,29,0.2414,1,"{2873, 2565, 14, 2568, 2635, 8, 270}"
607,1,183,0,0.3335027,"\int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos ^{11}(c+d x)}{11 d}+\frac{5 a^3 \cos ^9(c+d x)}{9 d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{3 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{19 a^3 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{19 a^3 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac{19 a^3 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{19 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{19 a^3 x}{256}","-\frac{a^3 \cos ^{11}(c+d x)}{11 d}+\frac{5 a^3 \cos ^9(c+d x)}{9 d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{3 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{19 a^3 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{19 a^3 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac{19 a^3 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{19 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{19 a^3 x}{256}",1,"(19*a^3*x)/256 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (5*a^3*Cos[c + d*x]^9)/(9*d) - (a^3*Cos[c + d*x]^11)/(11*d) + (19*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (19*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (19*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(480*d) - (19*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (3*a^3*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*d)","A",19,7,29,0.2414,1,"{2873, 2568, 2635, 8, 2565, 14, 270}"
608,1,181,0,0.203746,"\int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]","-\frac{33 a^3 \cos ^7(c+d x)}{560 d}-\frac{11 \cos ^7(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{240 d}+\frac{11 a^3 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{11 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{33 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{33 a^3 x}{256}-\frac{\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}-\frac{a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{30 d}","-\frac{33 a^3 \cos ^7(c+d x)}{560 d}-\frac{11 \cos ^7(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{240 d}+\frac{11 a^3 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{11 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{33 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{33 a^3 x}{256}-\frac{\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}-\frac{a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{30 d}",1,"(33*a^3*x)/256 - (33*a^3*Cos[c + d*x]^7)/(560*d) + (33*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (11*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (11*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (a*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(30*d) - (Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3)/(10*d) - (11*Cos[c + d*x]^7*(a^3 + a^3*Sin[c + d*x]))/(240*d)","A",8,5,27,0.1852,1,"{2860, 2678, 2669, 2635, 8}"
609,1,185,0,0.2358423,"\int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^5*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{25 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{125 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{125 a^3 \sin (c+d x) \cos (c+d x)}{128 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{125 a^3 x}{128}","-\frac{3 a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{25 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{125 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{125 a^3 \sin (c+d x) \cos (c+d x)}{128 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{125 a^3 x}{128}",1,"(125*a^3*x)/128 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cos[c + d*x]^5)/(5*d) - (3*a^3*Cos[c + d*x]^7)/(7*d) + (125*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (125*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (25*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a^3*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)","A",17,9,27,0.3333,1,"{2873, 2635, 8, 2592, 302, 206, 2565, 30, 2568}"
610,1,173,0,0.2439499,"\int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{3 a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \sin ^5(c+d x) \cos (c+d x)}{2 d}-\frac{11 a^3 \sin ^3(c+d x) \cos (c+d x)}{8 d}+\frac{15 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{15 a^3 x}{16}","-\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{3 a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \sin ^5(c+d x) \cos (c+d x)}{2 d}-\frac{11 a^3 \sin ^3(c+d x) \cos (c+d x)}{8 d}+\frac{15 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{15 a^3 x}{16}",1,"(-15*a^3*x)/16 - (3*a^3*ArcTanh[Cos[c + d*x]])/d + (3*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/d + (3*a^3*Cos[c + d*x]^5)/(5*d) - (a^3*Cos[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x])/d + (15*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (11*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(8*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^5)/(2*d)","A",19,7,29,0.2414,1,"{2872, 3770, 3767, 8, 2638, 2635, 2633}"
611,1,181,0,0.2529707,"\int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{3 a^3 \cos ^5(c+d x)}{5 d}+\frac{2 a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}+\frac{a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac{5 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{43 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{85 a^3 x}{16}","\frac{3 a^3 \cos ^5(c+d x)}{5 d}+\frac{2 a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}+\frac{a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac{5 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{43 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{85 a^3 x}{16}",1,"(-85*a^3*x)/16 - (a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (a^3*Cos[c + d*x])/d + (2*a^3*Cos[c + d*x]^3)/(3*d) + (3*a^3*Cos[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (43*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)","A",17,8,29,0.2759,1,"{2872, 3767, 8, 3768, 3770, 2638, 2635, 2633}"
612,1,176,0,0.2111571,"\int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{2 a^3 \cos ^3(c+d x)}{3 d}-\frac{5 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{23 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{13 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{25 a^3 x}{8}","\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{2 a^3 \cos ^3(c+d x)}{3 d}-\frac{5 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{23 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{13 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{25 a^3 x}{8}",1,"(-25*a^3*x)/8 + (13*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a^3*Cos[c + d*x])/d - (2*a^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cos[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (23*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (3*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",15,8,29,0.2759,1,"{2872, 3770, 3768, 3767, 2638, 2635, 8, 2633}"
613,1,178,0,0.2187433,"\int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos ^3(c+d x)}{d}-\frac{5 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{d}+\frac{5 a^3 \cot (c+d x)}{d}+\frac{a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{45 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{45 a^3 x}{8}","-\frac{a^3 \cos ^3(c+d x)}{d}-\frac{5 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{d}+\frac{5 a^3 \cot (c+d x)}{d}+\frac{a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{45 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{45 a^3 x}{8}",1,"(45*a^3*x)/8 + (45*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (5*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/d + (5*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/d - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",16,8,27,0.2963,1,"{2872, 3770, 3767, 8, 3768, 2638, 2633, 2635}"
614,1,175,0,0.2989436,"\int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}+\frac{5 a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{25 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{13 a^3 x}{2}","-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}+\frac{5 a^3 \cot (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{25 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{13 a^3 x}{2}",1,"(13*a^3*x)/2 - (25*a^3*ArcTanh[Cos[c + d*x]])/(8*d) + (a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) + (5*a^3*Cot[c + d*x])/d - (2*a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (23*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",16,7,21,0.3333,1,"{2709, 3770, 3767, 8, 3768, 2635, 2633}"
615,1,182,0,0.2481425,"\int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{3 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{85 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{a^3 x}{2}","\frac{3 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{85 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{a^3 x}{2}",1,"-(a^3*x)/2 - (85*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (3*a^3*Cos[c + d*x])/d - (a^3*Cot[c + d*x])/d + (2*a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]^5)/(5*d) + (43*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (5*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d) + (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",18,7,27,0.2593,1,"{2872, 3770, 3767, 8, 3768, 2638, 2635}"
616,1,172,0,0.2898301,"\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{15 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac{11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-3 a^3 x","\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \cot ^7(c+d x)}{7 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{15 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{2 d}+\frac{11 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-3 a^3 x",1,"-3*a^3*x - (15*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (a^3*Cos[c + d*x])/d - (3*a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/d - (3*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) - (15*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (11*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(2*d)","A",18,6,29,0.2069,1,"{2872, 3767, 8, 3768, 3770, 2638}"
617,1,238,0,0.3551594,"\int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{125 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-a^3 x","-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{125 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-a^3 x",1,"-(a^3*x) + (125*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x]^7)/(7*d) - (115*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x])/(2*d) - (5*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)","A",17,8,29,0.2759,1,"{2873, 3473, 8, 2611, 3770, 2607, 30, 3768}"
618,1,200,0,0.3585016,"\int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{55 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{25 a^3 \cot (c+d x) \csc (c+d x)}{128 d}","-\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{55 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac{15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{25 a^3 \cot (c+d x) \csc (c+d x)}{128 d}",1,"(55*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/(9*d) - (25*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x])/(24*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x])/(6*d) - (15*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(16*d) - (3*a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)","A",16,7,29,0.2414,1,"{2873, 2611, 3770, 2607, 30, 3768, 14}"
619,1,228,0,0.4271529,"\int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{33 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{29 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{33 a^3 \cot (c+d x) \csc (c+d x)}{256 d}","-\frac{a^3 \cot ^9(c+d x)}{3 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{33 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{29 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{33 a^3 \cot (c+d x) \csc (c+d x)}{256 d}",1,"(33*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/(3*d) + (33*a^3*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (29*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(16*d) - (3*a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)","A",18,7,29,0.2414,1,"{2873, 2607, 30, 2611, 3768, 3770, 14}"
620,1,246,0,0.4357412,"\int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot ^{11}(c+d x)}{11 d}-\frac{5 a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{19 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{19 a^3 \cot (c+d x) \csc (c+d x)}{256 d}","-\frac{a^3 \cot ^{11}(c+d x)}{11 d}-\frac{5 a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{19 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}-\frac{7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{19 a^3 \cot (c+d x) \csc (c+d x)}{256 d}",1,"(19*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (5*a^3*Cot[c + d*x]^9)/(9*d) - (a^3*Cot[c + d*x]^11)/(11*d) + (19*a^3*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (7*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (3*a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)","A",19,7,29,0.2414,1,"{2873, 2611, 3768, 3770, 2607, 14, 270}"
621,1,270,0,0.4641418,"\int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{7 a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{41 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac{41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}","-\frac{3 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{7 a^3 \cot ^9(c+d x)}{9 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{41 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{12 d}-\frac{3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{24 d}+\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{35 a^3 \cot (c+d x) \csc ^5(c+d x)}{384 d}+\frac{41 a^3 \cot (c+d x) \csc ^3(c+d x)}{1536 d}+\frac{41 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}",1,"(41*a^3*ArcTanh[Cos[c + d*x]])/(1024*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (7*a^3*Cot[c + d*x]^9)/(9*d) - (3*a^3*Cot[c + d*x]^11)/(11*d) + (41*a^3*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (41*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(1536*d) - (35*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(384*d) + (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (3*a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^3*Cot[c + d*x]^3*Csc[c + d*x]^7)/(24*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^7)/(12*d)","A",21,7,29,0.2414,1,"{2873, 2607, 14, 2611, 3768, 3770, 270}"
622,1,286,0,0.462119,"\int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^8*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot ^{13}(c+d x)}{13 d}-\frac{6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{a^3 \cot ^9(c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{27 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac{9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}+\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}","-\frac{a^3 \cot ^{13}(c+d x)}{13 d}-\frac{6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{a^3 \cot ^9(c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{27 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac{9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}+\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}",1,"(27*a^3*ArcTanh[Cos[c + d*x]])/(1024*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/d - (6*a^3*Cot[c + d*x]^11)/(11*d) - (a^3*Cot[c + d*x]^13)/(13*d) + (27*a^3*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (9*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(512*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(128*d) + (a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^3*Cot[c + d*x]^3*Csc[c + d*x]^7)/(8*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^7)/(4*d)","A",21,6,29,0.2069,1,"{2873, 2611, 3768, 3770, 2607, 270}"
623,1,178,0,0.2812261,"\int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]","\frac{4 a^4 \cos ^5(c+d x)}{5 d}-\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{4 a^4 \cot (c+d x)}{d}+\frac{a^4 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac{23 a^4 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{89 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{6 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{135 a^4 x}{16}","\frac{4 a^4 \cos ^5(c+d x)}{5 d}-\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{4 a^4 \cot (c+d x)}{d}+\frac{a^4 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac{23 a^4 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{89 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{6 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac{135 a^4 x}{16}",1,"(-135*a^4*x)/16 + (6*a^4*ArcTanh[Cos[c + d*x]])/d - (4*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x]^5)/(5*d) - (4*a^4*Cot[c + d*x])/d - (a^4*Cot[c + d*x]^3)/(3*d) - (2*a^4*Cot[c + d*x]*Csc[c + d*x])/d - (89*a^4*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (23*a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^4*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)","A",22,7,29,0.2414,1,"{2872, 3770, 3767, 8, 3768, 2635, 2633}"
624,1,159,0,0.2158667,"\int \frac{\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\cos ^9(c+d x)}{9 a d}-\frac{2 \cos ^7(c+d x)}{7 a d}+\frac{\cos ^5(c+d x)}{5 a d}-\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{64 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{128 a d}+\frac{3 x}{128 a}","\frac{\cos ^9(c+d x)}{9 a d}-\frac{2 \cos ^7(c+d x)}{7 a d}+\frac{\cos ^5(c+d x)}{5 a d}-\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{64 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{128 a d}+\frac{3 x}{128 a}",1,"(3*x)/(128*a) + Cos[c + d*x]^5/(5*a*d) - (2*Cos[c + d*x]^7)/(7*a*d) + Cos[c + d*x]^9/(9*a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(64*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(16*a*d) - (Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*a*d)","A",9,6,29,0.2069,1,"{2839, 2568, 2635, 8, 2565, 270}"
625,1,141,0,0.2146733,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\cos ^7(c+d x)}{7 a d}-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac{3 x}{128 a}","\frac{\cos ^7(c+d x)}{7 a d}-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac{3 x}{128 a}",1,"(-3*x)/(128*a) - Cos[c + d*x]^5/(5*a*d) + Cos[c + d*x]^7/(7*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(64*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(16*a*d) + (Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*a*d)","A",9,6,29,0.2069,1,"{2839, 2565, 14, 2568, 2635, 8}"
626,1,115,0,0.1771707,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\cos ^5(c+d x)}{5 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{24 a d}+\frac{\sin (c+d x) \cos (c+d x)}{16 a d}+\frac{x}{16 a}","-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\cos ^5(c+d x)}{5 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{24 a d}+\frac{\sin (c+d x) \cos (c+d x)}{16 a d}+\frac{x}{16 a}",1,"x/(16*a) + Cos[c + d*x]^5/(5*a*d) - Cos[c + d*x]^7/(7*a*d) + (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(24*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(6*a*d)","A",8,6,29,0.2069,1,"{2839, 2568, 2635, 8, 2565, 14}"
627,1,97,0,0.1261624,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{24 a d}-\frac{\sin (c+d x) \cos (c+d x)}{16 a d}-\frac{x}{16 a}","-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{24 a d}-\frac{\sin (c+d x) \cos (c+d x)}{16 a d}-\frac{x}{16 a}",1,"-x/(16*a) - Cos[c + d*x]^5/(5*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(24*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(6*a*d)","A",7,6,27,0.2222,1,"{2839, 2565, 30, 2568, 2635, 8}"
628,1,101,0,0.1231016,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{3 x}{8 a}","\frac{\cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{3 x}{8 a}",1,"(-3*x)/(8*a) - ArcTanh[Cos[c + d*x]]/(a*d) + Cos[c + d*x]/(a*d) + Cos[c + d*x]^3/(3*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)","A",8,6,27,0.2222,1,"{2839, 2592, 302, 206, 2635, 8}"
629,1,95,0,0.1561554,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\cos ^3(c+d x)}{3 a d}-\frac{\cos (c+d x)}{a d}-\frac{3 \cot (c+d x)}{2 a d}+\frac{\cos ^2(c+d x) \cot (c+d x)}{2 a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{3 x}{2 a}","-\frac{\cos ^3(c+d x)}{3 a d}-\frac{\cos (c+d x)}{a d}-\frac{3 \cot (c+d x)}{2 a d}+\frac{\cos ^2(c+d x) \cot (c+d x)}{2 a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{3 x}{2 a}",1,"(-3*x)/(2*a) + ArcTanh[Cos[c + d*x]]/(a*d) - Cos[c + d*x]/(a*d) - Cos[c + d*x]^3/(3*a*d) - (3*Cot[c + d*x])/(2*a*d) + (Cos[c + d*x]^2*Cot[c + d*x])/(2*a*d)","A",9,8,29,0.2759,1,"{2839, 2591, 288, 321, 203, 2592, 302, 206}"
630,1,106,0,0.1613828,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{3 \cos (c+d x)}{2 a d}+\frac{3 \cot (c+d x)}{2 a d}-\frac{\cos ^2(c+d x) \cot (c+d x)}{2 a d}-\frac{\cos (c+d x) \cot ^2(c+d x)}{2 a d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{3 x}{2 a}","-\frac{3 \cos (c+d x)}{2 a d}+\frac{3 \cot (c+d x)}{2 a d}-\frac{\cos ^2(c+d x) \cot (c+d x)}{2 a d}-\frac{\cos (c+d x) \cot ^2(c+d x)}{2 a d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{3 x}{2 a}",1,"(3*x)/(2*a) + (3*ArcTanh[Cos[c + d*x]])/(2*a*d) - (3*Cos[c + d*x])/(2*a*d) + (3*Cot[c + d*x])/(2*a*d) - (Cos[c + d*x]^2*Cot[c + d*x])/(2*a*d) - (Cos[c + d*x]*Cot[c + d*x]^2)/(2*a*d)","A",9,7,29,0.2414,1,"{2839, 2592, 288, 321, 206, 2591, 203}"
631,1,94,0,0.137807,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{3 \cos (c+d x)}{2 a d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cot (c+d x)}{a d}+\frac{\cos (c+d x) \cot ^2(c+d x)}{2 a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{x}{a}","\frac{3 \cos (c+d x)}{2 a d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cot (c+d x)}{a d}+\frac{\cos (c+d x) \cot ^2(c+d x)}{2 a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{x}{a}",1,"x/a - (3*ArcTanh[Cos[c + d*x]])/(2*a*d) + (3*Cos[c + d*x])/(2*a*d) + Cot[c + d*x]/(a*d) + (Cos[c + d*x]*Cot[c + d*x]^2)/(2*a*d) - Cot[c + d*x]^3/(3*a*d)","A",8,7,29,0.2414,1,"{2839, 3473, 8, 2592, 288, 321, 206}"
632,1,102,0,0.1333479,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\cot ^3(c+d x) \csc (c+d x)}{4 a d}+\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac{x}{a}","\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\cot ^3(c+d x) \csc (c+d x)}{4 a d}+\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac{x}{a}",1,"-(x/a) - (3*ArcTanh[Cos[c + d*x]])/(8*a*d) - Cot[c + d*x]/(a*d) + Cot[c + d*x]^3/(3*a*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) - (Cot[c + d*x]^3*Csc[c + d*x])/(4*a*d)","A",7,5,27,0.1852,1,"{2839, 2611, 3770, 3473, 8}"
633,1,82,0,0.1108027,"\int \frac{\cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^6/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^5(c+d x)}{5 a d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot ^3(c+d x) \csc (c+d x)}{4 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}","-\frac{\cot ^5(c+d x)}{5 a d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot ^3(c+d x) \csc (c+d x)}{4 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}",1,"(3*ArcTanh[Cos[c + d*x]])/(8*a*d) - Cot[c + d*x]^5/(5*a*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + (Cot[c + d*x]^3*Csc[c + d*x])/(4*a*d)","A",6,5,21,0.2381,1,"{2706, 2607, 30, 2611, 3770}"
634,1,135,0,0.347352,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cos ^7(c+d x)}{7 a^2 d}+\frac{3 \cos ^5(c+d x)}{5 a^2 d}-\frac{2 \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{x}{8 a^2}","-\frac{\cos ^7(c+d x)}{7 a^2 d}+\frac{3 \cos ^5(c+d x)}{5 a^2 d}-\frac{2 \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{3 a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{x}{8 a^2}",1,"-x/(8*a^2) - (2*Cos[c + d*x]^3)/(3*a^2*d) + (3*Cos[c + d*x]^5)/(5*a^2*d) - Cos[c + d*x]^7/(7*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x]^3)/(3*a^2*d)","A",13,8,29,0.2759,1,"{2875, 2873, 2565, 14, 2568, 2635, 8, 270}"
635,1,104,0,0.2682695,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\cos ^5(c+d x)}{10 a^2 d}+\frac{\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a^2 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{3 x}{16 a^2}","\frac{\cos ^5(c+d x)}{10 a^2 d}+\frac{\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a^2 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{3 x}{16 a^2}",1,"(3*x)/(16*a^2) + Cos[c + d*x]^5/(10*a^2*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*(a - a*Sin[c + d*x])^3)/(6*a^5*d)","A",6,5,29,0.1724,1,"{2875, 2870, 2669, 2635, 8}"
636,1,100,0,0.1262122,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \cos ^5(c+d x)}{15 a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{6 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{4 a^2 d}-\frac{x}{4 a^2}-\frac{\cos ^7(c+d x)}{3 d (a \sin (c+d x)+a)^2}","-\frac{2 \cos ^5(c+d x)}{15 a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{6 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{4 a^2 d}-\frac{x}{4 a^2}-\frac{\cos ^7(c+d x)}{3 d (a \sin (c+d x)+a)^2}",1,"-x/(4*a^2) - (2*Cos[c + d*x]^5)/(15*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(4*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(6*a^2*d) - Cos[c + d*x]^7/(3*d*(a + a*Sin[c + d*x])^2)","A",5,4,27,0.1481,1,"{2859, 2682, 2635, 8}"
637,1,73,0,0.2014293,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\cos ^3(c+d x)}{3 a^2 d}+\frac{\cos (c+d x)}{a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{x}{a^2}","-\frac{\cos ^3(c+d x)}{3 a^2 d}+\frac{\cos (c+d x)}{a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{x}{a^2}",1,"-(x/a^2) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(a^2*d) - Cos[c + d*x]^3/(3*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(a^2*d)","A",10,9,27,0.3333,1,"{2875, 2873, 2635, 8, 2592, 321, 206, 2565, 30}"
638,1,74,0,0.2030365,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \cos (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{x}{2 a^2}","-\frac{2 \cos (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{x}{2 a^2}",1,"-x/(2*a^2) + (2*ArcTanh[Cos[c + d*x]])/(a^2*d) - (2*Cos[c + d*x])/(a^2*d) - Cot[c + d*x]/(a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d)","A",9,7,29,0.2414,1,"{2875, 2709, 3770, 3767, 8, 2638, 2635}"
639,1,73,0,0.224917,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{\cos (c+d x)}{a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{2 x}{a^2}","\frac{\cos (c+d x)}{a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{2 x}{a^2}",1,"(2*x)/a^2 - ArcTanh[Cos[c + d*x]]/(2*a^2*d) + Cos[c + d*x]/(a^2*d) + (2*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d)","A",8,7,29,0.2414,1,"{2875, 2872, 3767, 8, 3768, 3770, 2638}"
640,1,73,0,0.3152565,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac{x}{a^2}","-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac{x}{a^2}",1,"-(x/a^2) - ArcTanh[Cos[c + d*x]]/(a^2*d) - Cot[c + d*x]/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(a^2*d)","A",9,8,29,0.2759,1,"{2875, 2873, 3473, 8, 2611, 3770, 2607, 30}"
641,1,82,0,0.300256,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cot ^3(c+d x)}{3 a^2 d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a^2 d}","\frac{2 \cot ^3(c+d x)}{3 a^2 d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a^2 d}",1,"(5*ArcTanh[Cos[c + d*x]])/(8*a^2*d) + (2*Cot[c + d*x]^3)/(3*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d)","A",10,7,27,0.2593,1,"{2875, 2873, 2611, 3770, 2607, 30, 3768}"
642,1,100,0,0.1471897,"\int \frac{\cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^2,x]","-\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{2 \cot ^3(c+d x)}{3 a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{4 a^2 d}","-\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{2 \cot ^3(c+d x)}{3 a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{4 a^2 d}",1,"-ArcTanh[Cos[c + d*x]]/(4*a^2*d) - (2*Cot[c + d*x]^3)/(3*a^2*d) - Cot[c + d*x]^5/(5*a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(4*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(2*a^2*d)","A",11,5,21,0.2381,1,"{2709, 3767, 8, 3768, 3770}"
643,1,124,0,0.3319696,"\int \frac{\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^6*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{2 \cot ^3(c+d x)}{3 a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac{5 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{16 a^2 d}","\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{2 \cot ^3(c+d x)}{3 a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac{5 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{16 a^2 d}",1,"(3*ArcTanh[Cos[c + d*x]])/(16*a^2*d) + (2*Cot[c + d*x]^3)/(3*a^2*d) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d) - (5*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^2*d)","A",13,7,27,0.2593,1,"{2875, 2873, 2611, 3768, 3770, 2607, 14}"
644,1,129,0,0.2420902,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","-\frac{3 \cos ^5(c+d x)}{5 a^3 d}+\frac{7 \cos ^3(c+d x)}{3 a^3 d}-\frac{4 \cos (c+d x)}{a^3 d}+\frac{\sin ^5(c+d x) \cos (c+d x)}{6 a^3 d}+\frac{23 \sin ^3(c+d x) \cos (c+d x)}{24 a^3 d}+\frac{23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac{23 x}{16 a^3}","-\frac{3 \cos ^5(c+d x)}{5 a^3 d}+\frac{7 \cos ^3(c+d x)}{3 a^3 d}-\frac{4 \cos (c+d x)}{a^3 d}+\frac{\sin ^5(c+d x) \cos (c+d x)}{6 a^3 d}+\frac{23 \sin ^3(c+d x) \cos (c+d x)}{24 a^3 d}+\frac{23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac{23 x}{16 a^3}",1,"(-23*x)/(16*a^3) - (4*Cos[c + d*x])/(a^3*d) + (7*Cos[c + d*x]^3)/(3*a^3*d) - (3*Cos[c + d*x]^5)/(5*a^3*d) + (23*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) + (23*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^3*d) + (Cos[c + d*x]*Sin[c + d*x]^5)/(6*a^3*d)","A",14,5,29,0.1724,1,"{2869, 2757, 2633, 2635, 8}"
645,1,105,0,0.2187232,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\cos ^5(c+d x)}{5 a^3 d}-\frac{5 \cos ^3(c+d x)}{3 a^3 d}+\frac{4 \cos (c+d x)}{a^3 d}-\frac{3 \sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac{13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{13 x}{8 a^3}","\frac{\cos ^5(c+d x)}{5 a^3 d}-\frac{5 \cos ^3(c+d x)}{3 a^3 d}+\frac{4 \cos (c+d x)}{a^3 d}-\frac{3 \sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}-\frac{13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{13 x}{8 a^3}",1,"(13*x)/(8*a^3) + (4*Cos[c + d*x])/(a^3*d) - (5*Cos[c + d*x]^3)/(3*a^3*d) + Cos[c + d*x]^5/(5*a^3*d) - (13*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - (3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^3*d)","A",12,5,29,0.1724,1,"{2869, 2757, 2635, 8, 2633}"
646,1,105,0,0.1655661,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{5 \cos ^3(c+d x)}{4 a^3 d}-\frac{3 \cos ^5(c+d x)}{4 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{15 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{15 x}{8 a^3}-\frac{\cos ^7(c+d x)}{d (a \sin (c+d x)+a)^3}","\frac{\cos ^3(c+d x)}{a^3 d}-\frac{4 \cos (c+d x)}{a^3 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}+\frac{15 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{15 x}{8 a^3}",1,"(-15*x)/(8*a^3) - (5*Cos[c + d*x]^3)/(4*a^3*d) - (15*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - Cos[c + d*x]^7/(d*(a + a*Sin[c + d*x])^3) - (3*Cos[c + d*x]^5)/(4*d*(a^3 + a^3*Sin[c + d*x]))","A",5,5,27,0.1852,1,"{2859, 2679, 2682, 2635, 8}"
647,1,60,0,0.1471545,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^5*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{3 \cos (c+d x)}{a^3 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{7 x}{2 a^3}","-\frac{3 \cos (c+d x)}{a^3 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{7 x}{2 a^3}",1,"(-7*x)/(2*a^3) - ArcTanh[Cos[c + d*x]]/(a^3*d) - (3*Cos[c + d*x])/(a^3*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d)","A",7,6,27,0.2222,1,"{2869, 2757, 3770, 2638, 2635, 8}"
648,1,49,0,0.1632867,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\cos (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{3 x}{a^3}","\frac{\cos (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{3 x}{a^3}",1,"(3*x)/a^3 + (3*ArcTanh[Cos[c + d*x]])/(a^3*d) + Cos[c + d*x]/(a^3*d) - Cot[c + d*x]/(a^3*d)","A",7,6,29,0.2069,1,"{2869, 2757, 3770, 3767, 8, 2638}"
649,1,60,0,0.1751505,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{3 \cot (c+d x)}{a^3 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{x}{a^3}","\frac{3 \cot (c+d x)}{a^3 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{x}{a^3}",1,"-(x/a^3) - (7*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (3*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d)","A",8,6,29,0.2069,1,"{2869, 2757, 3770, 3767, 8, 3768}"
650,1,72,0,0.1899609,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","-\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{4 \cot (c+d x)}{a^3 d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}","-\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{4 \cot (c+d x)}{a^3 d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}",1,"(5*ArcTanh[Cos[c + d*x]])/(2*a^3*d) - (4*Cot[c + d*x])/(a^3*d) - Cot[c + d*x]^3/(3*a^3*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d)","A",10,6,29,0.2069,1,"{2869, 2757, 3770, 3767, 8, 3768}"
651,1,93,0,0.2045194,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]","\frac{\cot ^3(c+d x)}{a^3 d}+\frac{4 \cot (c+d x)}{a^3 d}-\frac{15 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac{15 \cot (c+d x) \csc (c+d x)}{8 a^3 d}","\frac{\cot ^3(c+d x)}{a^3 d}+\frac{4 \cot (c+d x)}{a^3 d}-\frac{15 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac{15 \cot (c+d x) \csc (c+d x)}{8 a^3 d}",1,"(-15*ArcTanh[Cos[c + d*x]])/(8*a^3*d) + (4*Cot[c + d*x])/(a^3*d) + Cot[c + d*x]^3/(a^3*d) - (15*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d)","A",12,6,27,0.2222,1,"{2869, 2757, 3767, 8, 3768, 3770}"
652,1,114,0,0.1769097,"\int \frac{\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^3,x]","-\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{5 \cot ^3(c+d x)}{3 a^3 d}-\frac{4 \cot (c+d x)}{a^3 d}+\frac{13 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac{13 \cot (c+d x) \csc (c+d x)}{8 a^3 d}","-\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{5 \cot ^3(c+d x)}{3 a^3 d}-\frac{4 \cot (c+d x)}{a^3 d}+\frac{13 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac{13 \cot (c+d x) \csc (c+d x)}{8 a^3 d}",1,"(13*ArcTanh[Cos[c + d*x]])/(8*a^3*d) - (4*Cot[c + d*x])/(a^3*d) - (5*Cot[c + d*x]^3)/(3*a^3*d) - Cot[c + d*x]^5/(5*a^3*d) + (13*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) + (3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d)","A",12,5,21,0.2381,1,"{2708, 2757, 3768, 3770, 3767}"
653,1,267,0,0.2848839,"\int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^3 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^3 \cos (c+d x) \sin ^{n+3}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right)}{d (n+3) \sqrt{\cos ^2(c+d x)}}+\frac{a^3 \cos (c+d x) \sin ^{n+4}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+4}{2};\frac{n+6}{2};\sin ^2(c+d x)\right)}{d (n+4) \sqrt{\cos ^2(c+d x)}}","\frac{a^3 \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^3 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^3 \cos (c+d x) \sin ^{n+3}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right)}{d (n+3) \sqrt{\cos ^2(c+d x)}}+\frac{a^3 \cos (c+d x) \sin ^{n+4}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+4}{2};\frac{n+6}{2};\sin ^2(c+d x)\right)}{d (n+4) \sqrt{\cos ^2(c+d x)}}",1,"(a^3*Cos[c + d*x]*Hypergeometric2F1[-5/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (3*a^3*Cos[c + d*x]*Hypergeometric2F1[-5/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2]) + (3*a^3*Cos[c + d*x]*Hypergeometric2F1[-5/2, (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(3 + n))/(d*(3 + n)*Sqrt[Cos[c + d*x]^2]) + (a^3*Cos[c + d*x]*Hypergeometric2F1[-5/2, (4 + n)/2, (6 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(4 + n))/(d*(4 + n)*Sqrt[Cos[c + d*x]^2])","A",6,2,29,0.06897,1,"{2873, 2577}"
654,1,200,0,0.2422366,"\int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{2 a^2 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}+\frac{a^2 \cos (c+d x) \sin ^{n+3}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right)}{d (n+3) \sqrt{\cos ^2(c+d x)}}","\frac{a^2 \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{2 a^2 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}+\frac{a^2 \cos (c+d x) \sin ^{n+3}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right)}{d (n+3) \sqrt{\cos ^2(c+d x)}}",1,"(a^2*Cos[c + d*x]*Hypergeometric2F1[-5/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (2*a^2*Cos[c + d*x]*Hypergeometric2F1[-5/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2]) + (a^2*Cos[c + d*x]*Hypergeometric2F1[-5/2, (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(3 + n))/(d*(3 + n)*Sqrt[Cos[c + d*x]^2])","A",5,2,29,0.06897,1,"{2873, 2577}"
655,1,129,0,0.1368532,"\int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]","\frac{a \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{a \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}","\frac{a \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{a \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}",1,"(a*Cos[c + d*x]*Hypergeometric2F1[-5/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (a*Cos[c + d*x]*Hypergeometric2F1[-5/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2])","A",3,2,27,0.07407,1,"{2838, 2577}"
656,1,129,0,0.1006813,"\int \cos ^7(c+d x) \sin ^6(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*Sin[c + d*x]^6*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^{14}(c+d x)}{14 d}-\frac{a \sin ^{13}(c+d x)}{13 d}+\frac{a \sin ^{12}(c+d x)}{4 d}+\frac{3 a \sin ^{11}(c+d x)}{11 d}-\frac{3 a \sin ^{10}(c+d x)}{10 d}-\frac{a \sin ^9(c+d x)}{3 d}+\frac{a \sin ^8(c+d x)}{8 d}+\frac{a \sin ^7(c+d x)}{7 d}","-\frac{a \sin ^{14}(c+d x)}{14 d}-\frac{a \sin ^{13}(c+d x)}{13 d}+\frac{a \sin ^{12}(c+d x)}{4 d}+\frac{3 a \sin ^{11}(c+d x)}{11 d}-\frac{3 a \sin ^{10}(c+d x)}{10 d}-\frac{a \sin ^9(c+d x)}{3 d}+\frac{a \sin ^8(c+d x)}{8 d}+\frac{a \sin ^7(c+d x)}{7 d}",1,"(a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^8)/(8*d) - (a*Sin[c + d*x]^9)/(3*d) - (3*a*Sin[c + d*x]^10)/(10*d) + (3*a*Sin[c + d*x]^11)/(11*d) + (a*Sin[c + d*x]^12)/(4*d) - (a*Sin[c + d*x]^13)/(13*d) - (a*Sin[c + d*x]^14)/(14*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
657,1,113,0,0.1398588,"\int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*Sin[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^{13}(c+d x)}{13 d}+\frac{3 a \sin ^{11}(c+d x)}{11 d}-\frac{a \sin ^9(c+d x)}{3 d}+\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \cos ^{12}(c+d x)}{12 d}+\frac{a \cos ^{10}(c+d x)}{5 d}-\frac{a \cos ^8(c+d x)}{8 d}","-\frac{a \sin ^{13}(c+d x)}{13 d}+\frac{3 a \sin ^{11}(c+d x)}{11 d}-\frac{a \sin ^9(c+d x)}{3 d}+\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \cos ^{12}(c+d x)}{12 d}+\frac{a \cos ^{10}(c+d x)}{5 d}-\frac{a \cos ^8(c+d x)}{8 d}",1,"-(a*Cos[c + d*x]^8)/(8*d) + (a*Cos[c + d*x]^10)/(5*d) - (a*Cos[c + d*x]^12)/(12*d) + (a*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^9)/(3*d) + (3*a*Sin[c + d*x]^11)/(11*d) - (a*Sin[c + d*x]^13)/(13*d)","A",8,6,27,0.2222,1,"{2834, 2565, 266, 43, 2564, 270}"
658,1,113,0,0.1366121,"\int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^{11}(c+d x)}{11 d}+\frac{a \sin ^9(c+d x)}{3 d}-\frac{3 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \cos ^{12}(c+d x)}{12 d}+\frac{a \cos ^{10}(c+d x)}{5 d}-\frac{a \cos ^8(c+d x)}{8 d}","-\frac{a \sin ^{11}(c+d x)}{11 d}+\frac{a \sin ^9(c+d x)}{3 d}-\frac{3 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \cos ^{12}(c+d x)}{12 d}+\frac{a \cos ^{10}(c+d x)}{5 d}-\frac{a \cos ^8(c+d x)}{8 d}",1,"-(a*Cos[c + d*x]^8)/(8*d) + (a*Cos[c + d*x]^10)/(5*d) - (a*Cos[c + d*x]^12)/(12*d) + (a*Sin[c + d*x]^5)/(5*d) - (3*a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^9)/(3*d) - (a*Sin[c + d*x]^11)/(11*d)","A",8,6,27,0.2222,1,"{2834, 2564, 270, 2565, 266, 43}"
659,1,97,0,0.1316229,"\int \cos ^7(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^{11}(c+d x)}{11 d}+\frac{a \sin ^9(c+d x)}{3 d}-\frac{3 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \cos ^{10}(c+d x)}{10 d}-\frac{a \cos ^8(c+d x)}{8 d}","-\frac{a \sin ^{11}(c+d x)}{11 d}+\frac{a \sin ^9(c+d x)}{3 d}-\frac{3 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \cos ^{10}(c+d x)}{10 d}-\frac{a \cos ^8(c+d x)}{8 d}",1,"-(a*Cos[c + d*x]^8)/(8*d) + (a*Cos[c + d*x]^10)/(10*d) + (a*Sin[c + d*x]^5)/(5*d) - (3*a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^9)/(3*d) - (a*Sin[c + d*x]^11)/(11*d)","A",7,5,27,0.1852,1,"{2834, 2565, 14, 2564, 270}"
660,1,97,0,0.1282247,"\int \cos ^7(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^9(c+d x)}{9 d}+\frac{3 a \sin ^7(c+d x)}{7 d}-\frac{3 a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \cos ^{10}(c+d x)}{10 d}-\frac{a \cos ^8(c+d x)}{8 d}","-\frac{a \sin ^9(c+d x)}{9 d}+\frac{3 a \sin ^7(c+d x)}{7 d}-\frac{3 a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \cos ^{10}(c+d x)}{10 d}-\frac{a \cos ^8(c+d x)}{8 d}",1,"-(a*Cos[c + d*x]^8)/(8*d) + (a*Cos[c + d*x]^10)/(10*d) + (a*Sin[c + d*x]^3)/(3*d) - (3*a*Sin[c + d*x]^5)/(5*d) + (3*a*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^9)/(9*d)","A",7,5,27,0.1852,1,"{2834, 2564, 270, 2565, 14}"
661,1,81,0,0.08965,"\int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^9(c+d x)}{9 d}+\frac{3 a \sin ^7(c+d x)}{7 d}-\frac{3 a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \cos ^8(c+d x)}{8 d}","-\frac{a \sin ^9(c+d x)}{9 d}+\frac{3 a \sin ^7(c+d x)}{7 d}-\frac{3 a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \cos ^8(c+d x)}{8 d}",1,"-(a*Cos[c + d*x]^8)/(8*d) + (a*Sin[c + d*x]^3)/(3*d) - (3*a*Sin[c + d*x]^5)/(5*d) + (3*a*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^9)/(9*d)","A",6,5,25,0.2000,1,"{2834, 2565, 30, 2564, 270}"
662,1,118,0,0.0766889,"\int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^6*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^6(c+d x)}{6 d}+\frac{3 a \sin ^5(c+d x)}{5 d}+\frac{3 a \sin ^4(c+d x)}{4 d}-\frac{a \sin ^3(c+d x)}{d}-\frac{3 a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}","-\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^6(c+d x)}{6 d}+\frac{3 a \sin ^5(c+d x)}{5 d}+\frac{3 a \sin ^4(c+d x)}{4 d}-\frac{a \sin ^3(c+d x)}{d}-\frac{3 a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"(a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d - (3*a*Sin[c + d*x]^2)/(2*d) - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^4)/(4*d) + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^6)/(6*d) - (a*Sin[c + d*x]^7)/(7*d)","A",4,3,25,0.1200,1,"{2836, 12, 88}"
663,1,114,0,0.0874338,"\int \cos ^5(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^6(c+d x)}{6 d}-\frac{a \sin ^5(c+d x)}{5 d}+\frac{3 a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{d}-\frac{3 a \sin ^2(c+d x)}{2 d}-\frac{3 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}","-\frac{a \sin ^6(c+d x)}{6 d}-\frac{a \sin ^5(c+d x)}{5 d}+\frac{3 a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{d}-\frac{3 a \sin ^2(c+d x)}{2 d}-\frac{3 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"-((a*Csc[c + d*x])/d) + (a*Log[Sin[c + d*x]])/d - (3*a*Sin[c + d*x])/d - (3*a*Sin[c + d*x]^2)/(2*d) + (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^4)/(4*d) - (a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^6)/(6*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
664,1,115,0,0.0869306,"\int \cos ^4(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{d}+\frac{3 a \sin ^2(c+d x)}{2 d}-\frac{3 a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{3 a \log (\sin (c+d x))}{d}","-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^4(c+d x)}{4 d}+\frac{a \sin ^3(c+d x)}{d}+\frac{3 a \sin ^2(c+d x)}{2 d}-\frac{3 a \sin (c+d x)}{d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{3 a \log (\sin (c+d x))}{d}",1,"-((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (3*a*Log[Sin[c + d*x]])/d - (3*a*Sin[c + d*x])/d + (3*a*Sin[c + d*x]^2)/(2*d) + (a*Sin[c + d*x]^3)/d - (a*Sin[c + d*x]^4)/(4*d) - (a*Sin[c + d*x]^5)/(5*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
665,1,118,0,0.0864115,"\int \cos ^3(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^4(c+d x)}{4 d}-\frac{a \sin ^3(c+d x)}{3 d}+\frac{3 a \sin ^2(c+d x)}{2 d}+\frac{3 a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}+\frac{3 a \csc (c+d x)}{d}-\frac{3 a \log (\sin (c+d x))}{d}","-\frac{a \sin ^4(c+d x)}{4 d}-\frac{a \sin ^3(c+d x)}{3 d}+\frac{3 a \sin ^2(c+d x)}{2 d}+\frac{3 a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}+\frac{3 a \csc (c+d x)}{d}-\frac{3 a \log (\sin (c+d x))}{d}",1,"(3*a*Csc[c + d*x])/d - (a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (3*a*Log[Sin[c + d*x]])/d + (3*a*Sin[c + d*x])/d + (3*a*Sin[c + d*x]^2)/(2*d) - (a*Sin[c + d*x]^3)/(3*d) - (a*Sin[c + d*x]^4)/(4*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
666,1,118,0,0.0894501,"\int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{2 d}+\frac{3 a \sin (c+d x)}{d}-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{3 a \csc ^2(c+d x)}{2 d}+\frac{3 a \csc (c+d x)}{d}+\frac{3 a \log (\sin (c+d x))}{d}","-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x)}{2 d}+\frac{3 a \sin (c+d x)}{d}-\frac{a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{3 a \csc ^2(c+d x)}{2 d}+\frac{3 a \csc (c+d x)}{d}+\frac{3 a \log (\sin (c+d x))}{d}",1,"(3*a*Csc[c + d*x])/d + (3*a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) + (3*a*Log[Sin[c + d*x]])/d + (3*a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d) - (a*Sin[c + d*x]^3)/(3*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
667,1,115,0,0.0815798,"\int \cos (c+d x) \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^6*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}+\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}+\frac{3 a \log (\sin (c+d x))}{d}","-\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}+\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}+\frac{3 a \log (\sin (c+d x))}{d}",1,"(-3*a*Csc[c + d*x])/d + (3*a*Csc[c + d*x]^2)/(2*d) + (a*Csc[c + d*x]^3)/d - (a*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^5)/(5*d) + (3*a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d)","A",4,3,25,0.1200,1,"{2836, 12, 88}"
668,1,115,0,0.0530714,"\int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^7*(a + a*Sin[c + d*x]),x]","-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d}","-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d}",1,"(-3*a*Csc[c + d*x])/d - (3*a*Csc[c + d*x]^2)/(2*d) + (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(6*d) - (a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d","A",3,2,19,0.1053,1,"{2707, 88}"
669,1,119,0,0.0845058,"\int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^7*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^6(c+d x)}{6 d}+\frac{3 a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}+\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d}","-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^6(c+d x)}{6 d}+\frac{3 a \csc ^5(c+d x)}{5 d}+\frac{3 a \csc ^4(c+d x)}{4 d}-\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc ^2(c+d x)}{2 d}+\frac{a \csc (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d}",1,"(a*Csc[c + d*x])/d - (3*a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^4)/(4*d) + (3*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(6*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*Log[Sin[c + d*x]])/d","A",4,3,25,0.1200,1,"{2836, 12, 88}"
670,1,74,0,0.1072459,"\int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^7*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{a \csc (c+d x)}{d}","-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^7(c+d x)}{7 d}+\frac{3 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{d}+\frac{a \csc (c+d x)}{d}",1,"-(a*Cot[c + d*x]^8)/(8*d) + (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d)","A",6,5,27,0.1852,1,"{2834, 2607, 30, 2606, 194}"
671,1,81,0,0.1189104,"\int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^7*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{3 a \csc ^7(c+d x)}{7 d}-\frac{3 a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{3 d}","-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{3 a \csc ^7(c+d x)}{7 d}-\frac{3 a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{3 d}",1,"-(a*Cot[c + d*x]^8)/(8*d) + (a*Csc[c + d*x]^3)/(3*d) - (3*a*Csc[c + d*x]^5)/(5*d) + (3*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d)","A",6,5,27,0.1852,1,"{2834, 2606, 270, 2607, 30}"
672,1,97,0,0.1252649,"\int \cot ^7(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^7*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^{10}(c+d x)}{10 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{3 a \csc ^7(c+d x)}{7 d}-\frac{3 a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{3 d}","-\frac{a \cot ^{10}(c+d x)}{10 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^9(c+d x)}{9 d}+\frac{3 a \csc ^7(c+d x)}{7 d}-\frac{3 a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{3 d}",1,"-(a*Cot[c + d*x]^8)/(8*d) - (a*Cot[c + d*x]^10)/(10*d) + (a*Csc[c + d*x]^3)/(3*d) - (3*a*Csc[c + d*x]^5)/(5*d) + (3*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d)","A",7,5,27,0.1852,1,"{2834, 2607, 14, 2606, 270}"
673,1,97,0,0.124318,"\int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^7*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^{10}(c+d x)}{10 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{11}(c+d x)}{11 d}+\frac{a \csc ^9(c+d x)}{3 d}-\frac{3 a \csc ^7(c+d x)}{7 d}+\frac{a \csc ^5(c+d x)}{5 d}","-\frac{a \cot ^{10}(c+d x)}{10 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{11}(c+d x)}{11 d}+\frac{a \csc ^9(c+d x)}{3 d}-\frac{3 a \csc ^7(c+d x)}{7 d}+\frac{a \csc ^5(c+d x)}{5 d}",1,"-(a*Cot[c + d*x]^8)/(8*d) - (a*Cot[c + d*x]^10)/(10*d) + (a*Csc[c + d*x]^5)/(5*d) - (3*a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^9)/(3*d) - (a*Csc[c + d*x]^11)/(11*d)","A",7,5,27,0.1852,1,"{2834, 2606, 270, 2607, 14}"
674,1,113,0,0.1332334,"\int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^7*Csc[c + d*x]^6*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^{12}(c+d x)}{12 d}-\frac{a \cot ^{10}(c+d x)}{5 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{11}(c+d x)}{11 d}+\frac{a \csc ^9(c+d x)}{3 d}-\frac{3 a \csc ^7(c+d x)}{7 d}+\frac{a \csc ^5(c+d x)}{5 d}","-\frac{a \cot ^{12}(c+d x)}{12 d}-\frac{a \cot ^{10}(c+d x)}{5 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{11}(c+d x)}{11 d}+\frac{a \csc ^9(c+d x)}{3 d}-\frac{3 a \csc ^7(c+d x)}{7 d}+\frac{a \csc ^5(c+d x)}{5 d}",1,"-(a*Cot[c + d*x]^8)/(8*d) - (a*Cot[c + d*x]^10)/(5*d) - (a*Cot[c + d*x]^12)/(12*d) + (a*Csc[c + d*x]^5)/(5*d) - (3*a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^9)/(3*d) - (a*Csc[c + d*x]^11)/(11*d)","A",8,6,27,0.2222,1,"{2834, 2607, 266, 43, 2606, 270}"
675,1,113,0,0.1348779,"\int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^7*Csc[c + d*x]^7*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^{12}(c+d x)}{12 d}-\frac{a \cot ^{10}(c+d x)}{5 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{13}(c+d x)}{13 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{a \csc ^7(c+d x)}{7 d}","-\frac{a \cot ^{12}(c+d x)}{12 d}-\frac{a \cot ^{10}(c+d x)}{5 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{13}(c+d x)}{13 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{a \csc ^7(c+d x)}{7 d}",1,"-(a*Cot[c + d*x]^8)/(8*d) - (a*Cot[c + d*x]^10)/(5*d) - (a*Cot[c + d*x]^12)/(12*d) + (a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(3*d) + (3*a*Csc[c + d*x]^11)/(11*d) - (a*Csc[c + d*x]^13)/(13*d)","A",8,6,27,0.2222,1,"{2834, 2606, 270, 2607, 266, 43}"
676,1,129,0,0.098505,"\int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cot[c + d*x]^7*Csc[c + d*x]^8*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^{14}(c+d x)}{14 d}-\frac{a \csc ^{13}(c+d x)}{13 d}+\frac{a \csc ^{12}(c+d x)}{4 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{3 a \csc ^{10}(c+d x)}{10 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{a \csc ^8(c+d x)}{8 d}+\frac{a \csc ^7(c+d x)}{7 d}","-\frac{a \csc ^{14}(c+d x)}{14 d}-\frac{a \csc ^{13}(c+d x)}{13 d}+\frac{a \csc ^{12}(c+d x)}{4 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{3 a \csc ^{10}(c+d x)}{10 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{a \csc ^8(c+d x)}{8 d}+\frac{a \csc ^7(c+d x)}{7 d}",1,"(a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^9)/(3*d) - (3*a*Csc[c + d*x]^10)/(10*d) + (3*a*Csc[c + d*x]^11)/(11*d) + (a*Csc[c + d*x]^12)/(4*d) - (a*Csc[c + d*x]^13)/(13*d) - (a*Csc[c + d*x]^14)/(14*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
677,1,109,0,0.1276455,"\int \frac{\cos ^7(c+d x) \sin ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^7*Sin[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","-\frac{\sin ^{12}(c+d x)}{12 a d}+\frac{\sin ^{11}(c+d x)}{11 a d}+\frac{\sin ^{10}(c+d x)}{5 a d}-\frac{2 \sin ^9(c+d x)}{9 a d}-\frac{\sin ^8(c+d x)}{8 a d}+\frac{\sin ^7(c+d x)}{7 a d}","-\frac{\sin ^{12}(c+d x)}{12 a d}+\frac{\sin ^{11}(c+d x)}{11 a d}+\frac{\sin ^{10}(c+d x)}{5 a d}-\frac{2 \sin ^9(c+d x)}{9 a d}-\frac{\sin ^8(c+d x)}{8 a d}+\frac{\sin ^7(c+d x)}{7 a d}",1,"Sin[c + d*x]^7/(7*a*d) - Sin[c + d*x]^8/(8*a*d) - (2*Sin[c + d*x]^9)/(9*a*d) + Sin[c + d*x]^10/(5*a*d) + Sin[c + d*x]^11/(11*a*d) - Sin[c + d*x]^12/(12*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
678,1,109,0,0.1268672,"\int \frac{\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^7*Sin[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","-\frac{\sin ^{11}(c+d x)}{11 a d}+\frac{\sin ^{10}(c+d x)}{10 a d}+\frac{2 \sin ^9(c+d x)}{9 a d}-\frac{\sin ^8(c+d x)}{4 a d}-\frac{\sin ^7(c+d x)}{7 a d}+\frac{\sin ^6(c+d x)}{6 a d}","-\frac{\sin ^{11}(c+d x)}{11 a d}+\frac{\sin ^{10}(c+d x)}{10 a d}+\frac{2 \sin ^9(c+d x)}{9 a d}-\frac{\sin ^8(c+d x)}{4 a d}-\frac{\sin ^7(c+d x)}{7 a d}+\frac{\sin ^6(c+d x)}{6 a d}",1,"Sin[c + d*x]^6/(6*a*d) - Sin[c + d*x]^7/(7*a*d) - Sin[c + d*x]^8/(4*a*d) + (2*Sin[c + d*x]^9)/(9*a*d) + Sin[c + d*x]^10/(10*a*d) - Sin[c + d*x]^11/(11*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
679,1,109,0,0.1270471,"\int \frac{\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^7*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\sin ^{10}(c+d x)}{10 a d}+\frac{\sin ^9(c+d x)}{9 a d}+\frac{\sin ^8(c+d x)}{4 a d}-\frac{2 \sin ^7(c+d x)}{7 a d}-\frac{\sin ^6(c+d x)}{6 a d}+\frac{\sin ^5(c+d x)}{5 a d}","-\frac{\sin ^{10}(c+d x)}{10 a d}+\frac{\sin ^9(c+d x)}{9 a d}+\frac{\sin ^8(c+d x)}{4 a d}-\frac{2 \sin ^7(c+d x)}{7 a d}-\frac{\sin ^6(c+d x)}{6 a d}+\frac{\sin ^5(c+d x)}{5 a d}",1,"Sin[c + d*x]^5/(5*a*d) - Sin[c + d*x]^6/(6*a*d) - (2*Sin[c + d*x]^7)/(7*a*d) + Sin[c + d*x]^8/(4*a*d) + Sin[c + d*x]^9/(9*a*d) - Sin[c + d*x]^10/(10*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
680,1,91,0,0.1603859,"\int \frac{\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^7*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{\sin ^9(c+d x)}{9 a d}+\frac{2 \sin ^7(c+d x)}{7 a d}-\frac{\sin ^5(c+d x)}{5 a d}+\frac{\cos ^8(c+d x)}{8 a d}-\frac{\cos ^6(c+d x)}{6 a d}","-\frac{\sin ^9(c+d x)}{9 a d}+\frac{2 \sin ^7(c+d x)}{7 a d}-\frac{\sin ^5(c+d x)}{5 a d}+\frac{\cos ^8(c+d x)}{8 a d}-\frac{\cos ^6(c+d x)}{6 a d}",1,"-Cos[c + d*x]^6/(6*a*d) + Cos[c + d*x]^8/(8*a*d) - Sin[c + d*x]^5/(5*a*d) + (2*Sin[c + d*x]^7)/(7*a*d) - Sin[c + d*x]^9/(9*a*d)","A",7,5,29,0.1724,1,"{2835, 2565, 14, 2564, 270}"
681,1,91,0,0.161681,"\int \frac{\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^7*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^7(c+d x)}{7 a d}-\frac{2 \sin ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x)}{3 a d}-\frac{\cos ^8(c+d x)}{8 a d}+\frac{\cos ^6(c+d x)}{6 a d}","\frac{\sin ^7(c+d x)}{7 a d}-\frac{2 \sin ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x)}{3 a d}-\frac{\cos ^8(c+d x)}{8 a d}+\frac{\cos ^6(c+d x)}{6 a d}",1,"Cos[c + d*x]^6/(6*a*d) - Cos[c + d*x]^8/(8*a*d) + Sin[c + d*x]^3/(3*a*d) - (2*Sin[c + d*x]^5)/(5*a*d) + Sin[c + d*x]^7/(7*a*d)","A",7,5,29,0.1724,1,"{2835, 2564, 270, 2565, 14}"
682,1,73,0,0.1146993,"\int \frac{\cos ^7(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^7*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\sin ^7(c+d x)}{7 a d}+\frac{2 \sin ^5(c+d x)}{5 a d}-\frac{\sin ^3(c+d x)}{3 a d}-\frac{\cos ^6(c+d x)}{6 a d}","-\frac{\sin ^7(c+d x)}{7 a d}+\frac{2 \sin ^5(c+d x)}{5 a d}-\frac{\sin ^3(c+d x)}{3 a d}-\frac{\cos ^6(c+d x)}{6 a d}",1,"-Cos[c + d*x]^6/(6*a*d) - Sin[c + d*x]^3/(3*a*d) + (2*Sin[c + d*x]^5)/(5*a*d) - Sin[c + d*x]^7/(7*a*d)","A",6,5,27,0.1852,1,"{2835, 2565, 30, 2564, 270}"
683,1,68,0,0.0628442,"\int \frac{\cos ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^7/(a + a*Sin[c + d*x]),x]","-\frac{(a-a \sin (c+d x))^6}{6 a^7 d}+\frac{4 (a-a \sin (c+d x))^5}{5 a^6 d}-\frac{(a-a \sin (c+d x))^4}{a^5 d}","-\frac{(a-a \sin (c+d x))^6}{6 a^7 d}+\frac{4 (a-a \sin (c+d x))^5}{5 a^6 d}-\frac{(a-a \sin (c+d x))^4}{a^5 d}",1,"-((a - a*Sin[c + d*x])^4/(a^5*d)) + (4*(a - a*Sin[c + d*x])^5)/(5*a^6*d) - (a - a*Sin[c + d*x])^6/(6*a^7*d)","A",3,2,21,0.09524,1,"{2667, 43}"
684,1,99,0,0.1002602,"\int \frac{\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^6*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^4(c+d x)}{4 a d}+\frac{2 \sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}","-\frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^4(c+d x)}{4 a d}+\frac{2 \sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}",1,"Log[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(a*d) + (2*Sin[c + d*x]^3)/(3*a*d) + Sin[c + d*x]^4/(4*a*d) - Sin[c + d*x]^5/(5*a*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
685,1,95,0,0.1195033,"\int \frac{\cos ^5(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\sin ^4(c+d x)}{4 a d}+\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin ^2(c+d x)}{a d}-\frac{2 \sin (c+d x)}{a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}","-\frac{\sin ^4(c+d x)}{4 a d}+\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin ^2(c+d x)}{a d}-\frac{2 \sin (c+d x)}{a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}",1,"-(Csc[c + d*x]/(a*d)) - Log[Sin[c + d*x]]/(a*d) - (2*Sin[c + d*x])/(a*d) + Sin[c + d*x]^2/(a*d) + Sin[c + d*x]^3/(3*a*d) - Sin[c + d*x]^4/(4*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
686,1,97,0,0.1183778,"\int \frac{\cos ^4(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin ^2(c+d x)}{2 a d}+\frac{2 \sin (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}-\frac{2 \log (\sin (c+d x))}{a d}","-\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin ^2(c+d x)}{2 a d}+\frac{2 \sin (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}-\frac{2 \log (\sin (c+d x))}{a d}",1,"Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - (2*Log[Sin[c + d*x]])/(a*d) + (2*Sin[c + d*x])/(a*d) + Sin[c + d*x]^2/(2*a*d) - Sin[c + d*x]^3/(3*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
687,1,97,0,0.1200617,"\int \frac{\cos ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\sin ^2(c+d x)}{2 a d}+\frac{\sin (c+d x)}{a d}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}+\frac{2 \csc (c+d x)}{a d}+\frac{2 \log (\sin (c+d x))}{a d}","-\frac{\sin ^2(c+d x)}{2 a d}+\frac{\sin (c+d x)}{a d}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}+\frac{2 \csc (c+d x)}{a d}+\frac{2 \log (\sin (c+d x))}{a d}",1,"(2*Csc[c + d*x])/(a*d) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) + (2*Log[Sin[c + d*x]])/(a*d) + Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
688,1,94,0,0.1194475,"\int \frac{\cos ^2(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","-\frac{\sin (c+d x)}{a d}-\frac{\csc ^4(c+d x)}{4 a d}+\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{a d}-\frac{2 \csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}","-\frac{\sin (c+d x)}{a d}-\frac{\csc ^4(c+d x)}{4 a d}+\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{a d}-\frac{2 \csc (c+d x)}{a d}+\frac{\log (\sin (c+d x))}{a d}",1,"(-2*Csc[c + d*x])/(a*d) + Csc[c + d*x]^2/(a*d) + Csc[c + d*x]^3/(3*a*d) - Csc[c + d*x]^4/(4*a*d) + Log[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
689,1,100,0,0.1043562,"\int \frac{\cos (c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^5(c+d x)}{5 a d}+\frac{\csc ^4(c+d x)}{4 a d}+\frac{2 \csc ^3(c+d x)}{3 a d}-\frac{\csc ^2(c+d x)}{a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}","-\frac{\csc ^5(c+d x)}{5 a d}+\frac{\csc ^4(c+d x)}{4 a d}+\frac{2 \csc ^3(c+d x)}{3 a d}-\frac{\csc ^2(c+d x)}{a d}-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}",1,"-(Csc[c + d*x]/(a*d)) - Csc[c + d*x]^2/(a*d) + (2*Csc[c + d*x]^3)/(3*a*d) + Csc[c + d*x]^4/(4*a*d) - Csc[c + d*x]^5/(5*a*d) - Log[Sin[c + d*x]]/(a*d)","A",4,3,27,0.1111,1,"{2836, 12, 88}"
690,1,68,0,0.0895555,"\int \frac{\cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^7/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{2 \csc ^3(c+d x)}{3 a d}+\frac{\csc (c+d x)}{a d}","-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}-\frac{2 \csc ^3(c+d x)}{3 a d}+\frac{\csc (c+d x)}{a d}",1,"-Cot[c + d*x]^6/(6*a*d) + Csc[c + d*x]/(a*d) - (2*Csc[c + d*x]^3)/(3*a*d) + Csc[c + d*x]^5/(5*a*d)","A",6,5,21,0.2381,1,"{2706, 2607, 30, 2606, 194}"
691,1,73,0,0.1385356,"\int \frac{\cot ^7(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^7*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\cot ^6(c+d x)}{6 a d}-\frac{\csc ^7(c+d x)}{7 a d}+\frac{2 \csc ^5(c+d x)}{5 a d}-\frac{\csc ^3(c+d x)}{3 a d}","\frac{\cot ^6(c+d x)}{6 a d}-\frac{\csc ^7(c+d x)}{7 a d}+\frac{2 \csc ^5(c+d x)}{5 a d}-\frac{\csc ^3(c+d x)}{3 a d}",1,"Cot[c + d*x]^6/(6*a*d) - Csc[c + d*x]^3/(3*a*d) + (2*Csc[c + d*x]^5)/(5*a*d) - Csc[c + d*x]^7/(7*a*d)","A",6,5,27,0.1852,1,"{2835, 2606, 270, 2607, 30}"
692,1,91,0,0.1600918,"\int \frac{\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^7*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^8(c+d x)}{8 a d}-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^7(c+d x)}{7 a d}-\frac{2 \csc ^5(c+d x)}{5 a d}+\frac{\csc ^3(c+d x)}{3 a d}","-\frac{\cot ^8(c+d x)}{8 a d}-\frac{\cot ^6(c+d x)}{6 a d}+\frac{\csc ^7(c+d x)}{7 a d}-\frac{2 \csc ^5(c+d x)}{5 a d}+\frac{\csc ^3(c+d x)}{3 a d}",1,"-Cot[c + d*x]^6/(6*a*d) - Cot[c + d*x]^8/(8*a*d) + Csc[c + d*x]^3/(3*a*d) - (2*Csc[c + d*x]^5)/(5*a*d) + Csc[c + d*x]^7/(7*a*d)","A",7,5,29,0.1724,1,"{2835, 2607, 14, 2606, 270}"
693,1,91,0,0.1604353,"\int \frac{\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^7*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\cot ^8(c+d x)}{8 a d}+\frac{\cot ^6(c+d x)}{6 a d}-\frac{\csc ^9(c+d x)}{9 a d}+\frac{2 \csc ^7(c+d x)}{7 a d}-\frac{\csc ^5(c+d x)}{5 a d}","\frac{\cot ^8(c+d x)}{8 a d}+\frac{\cot ^6(c+d x)}{6 a d}-\frac{\csc ^9(c+d x)}{9 a d}+\frac{2 \csc ^7(c+d x)}{7 a d}-\frac{\csc ^5(c+d x)}{5 a d}",1,"Cot[c + d*x]^6/(6*a*d) + Cot[c + d*x]^8/(8*a*d) - Csc[c + d*x]^5/(5*a*d) + (2*Csc[c + d*x]^7)/(7*a*d) - Csc[c + d*x]^9/(9*a*d)","A",7,5,29,0.1724,1,"{2835, 2606, 270, 2607, 14}"
694,1,109,0,0.1211838,"\int \frac{\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^7*Csc[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^{10}(c+d x)}{10 a d}+\frac{\csc ^9(c+d x)}{9 a d}+\frac{\csc ^8(c+d x)}{4 a d}-\frac{2 \csc ^7(c+d x)}{7 a d}-\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}","-\frac{\csc ^{10}(c+d x)}{10 a d}+\frac{\csc ^9(c+d x)}{9 a d}+\frac{\csc ^8(c+d x)}{4 a d}-\frac{2 \csc ^7(c+d x)}{7 a d}-\frac{\csc ^6(c+d x)}{6 a d}+\frac{\csc ^5(c+d x)}{5 a d}",1,"Csc[c + d*x]^5/(5*a*d) - Csc[c + d*x]^6/(6*a*d) - (2*Csc[c + d*x]^7)/(7*a*d) + Csc[c + d*x]^8/(4*a*d) + Csc[c + d*x]^9/(9*a*d) - Csc[c + d*x]^10/(10*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
695,1,109,0,0.1229848,"\int \frac{\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^7*Csc[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^{11}(c+d x)}{11 a d}+\frac{\csc ^{10}(c+d x)}{10 a d}+\frac{2 \csc ^9(c+d x)}{9 a d}-\frac{\csc ^8(c+d x)}{4 a d}-\frac{\csc ^7(c+d x)}{7 a d}+\frac{\csc ^6(c+d x)}{6 a d}","-\frac{\csc ^{11}(c+d x)}{11 a d}+\frac{\csc ^{10}(c+d x)}{10 a d}+\frac{2 \csc ^9(c+d x)}{9 a d}-\frac{\csc ^8(c+d x)}{4 a d}-\frac{\csc ^7(c+d x)}{7 a d}+\frac{\csc ^6(c+d x)}{6 a d}",1,"Csc[c + d*x]^6/(6*a*d) - Csc[c + d*x]^7/(7*a*d) - Csc[c + d*x]^8/(4*a*d) + (2*Csc[c + d*x]^9)/(9*a*d) + Csc[c + d*x]^10/(10*a*d) - Csc[c + d*x]^11/(11*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
696,1,109,0,0.1219518,"\int \frac{\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^7*Csc[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^{12}(c+d x)}{12 a d}+\frac{\csc ^{11}(c+d x)}{11 a d}+\frac{\csc ^{10}(c+d x)}{5 a d}-\frac{2 \csc ^9(c+d x)}{9 a d}-\frac{\csc ^8(c+d x)}{8 a d}+\frac{\csc ^7(c+d x)}{7 a d}","-\frac{\csc ^{12}(c+d x)}{12 a d}+\frac{\csc ^{11}(c+d x)}{11 a d}+\frac{\csc ^{10}(c+d x)}{5 a d}-\frac{2 \csc ^9(c+d x)}{9 a d}-\frac{\csc ^8(c+d x)}{8 a d}+\frac{\csc ^7(c+d x)}{7 a d}",1,"Csc[c + d*x]^7/(7*a*d) - Csc[c + d*x]^8/(8*a*d) - (2*Csc[c + d*x]^9)/(9*a*d) + Csc[c + d*x]^10/(5*a*d) + Csc[c + d*x]^11/(11*a*d) - Csc[c + d*x]^12/(12*a*d)","A",4,3,29,0.1034,1,"{2836, 12, 88}"
697,1,184,0,0.1775368,"\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{8 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{6 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{6 a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{8 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}-\frac{3 a^3 \sin ^{n+9}(c+d x)}{d (n+9)}-\frac{a^3 \sin ^{n+10}(c+d x)}{d (n+10)}","\frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{8 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{6 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{6 a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{8 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}-\frac{3 a^3 \sin ^{n+9}(c+d x)}{d (n+9)}-\frac{a^3 \sin ^{n+10}(c+d x)}{d (n+10)}",1,"(a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (8*a^3*Sin[c + d*x]^(4 + n))/(d*(4 + n)) - (6*a^3*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (6*a^3*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (8*a^3*Sin[c + d*x]^(7 + n))/(d*(7 + n)) - (3*a^3*Sin[c + d*x]^(9 + n))/(d*(9 + n)) - (a^3*Sin[c + d*x]^(10 + n))/(d*(10 + n))","A",3,2,29,0.06897,1,"{2836, 88}"
698,1,184,0,0.1758987,"\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{2 a^2 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{6 a^2 \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{6 a^2 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{2 a^2 \sin ^{n+7}(c+d x)}{d (n+7)}-\frac{2 a^2 \sin ^{n+8}(c+d x)}{d (n+8)}-\frac{a^2 \sin ^{n+9}(c+d x)}{d (n+9)}","\frac{a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{2 a^2 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{6 a^2 \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{6 a^2 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{2 a^2 \sin ^{n+7}(c+d x)}{d (n+7)}-\frac{2 a^2 \sin ^{n+8}(c+d x)}{d (n+8)}-\frac{a^2 \sin ^{n+9}(c+d x)}{d (n+9)}",1,"(a^2*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (2*a^2*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (2*a^2*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (6*a^2*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (6*a^2*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (2*a^2*Sin[c + d*x]^(7 + n))/(d*(7 + n)) - (2*a^2*Sin[c + d*x]^(8 + n))/(d*(8 + n)) - (a^2*Sin[c + d*x]^(9 + n))/(d*(9 + n))","A",3,2,29,0.06897,1,"{2836, 88}"
699,1,167,0,0.1376656,"\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{3 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{3 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{3 a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{3 a \sin ^{n+6}(c+d x)}{d (n+6)}-\frac{a \sin ^{n+7}(c+d x)}{d (n+7)}-\frac{a \sin ^{n+8}(c+d x)}{d (n+8)}","\frac{a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{3 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{3 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{3 a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{3 a \sin ^{n+6}(c+d x)}{d (n+6)}-\frac{a \sin ^{n+7}(c+d x)}{d (n+7)}-\frac{a \sin ^{n+8}(c+d x)}{d (n+8)}",1,"(a*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (a*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (3*a*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (3*a*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (3*a*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (3*a*Sin[c + d*x]^(6 + n))/(d*(6 + n)) - (a*Sin[c + d*x]^(7 + n))/(d*(7 + n)) - (a*Sin[c + d*x]^(8 + n))/(d*(8 + n))","A",3,2,27,0.07407,1,"{2836, 88}"
700,1,137,0,0.1642547,"\int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^{n+1}(c+d x)}{a d (n+1)}-\frac{\sin ^{n+2}(c+d x)}{a d (n+2)}-\frac{2 \sin ^{n+3}(c+d x)}{a d (n+3)}+\frac{2 \sin ^{n+4}(c+d x)}{a d (n+4)}+\frac{\sin ^{n+5}(c+d x)}{a d (n+5)}-\frac{\sin ^{n+6}(c+d x)}{a d (n+6)}","\frac{\sin ^{n+1}(c+d x)}{a d (n+1)}-\frac{\sin ^{n+2}(c+d x)}{a d (n+2)}-\frac{2 \sin ^{n+3}(c+d x)}{a d (n+3)}+\frac{2 \sin ^{n+4}(c+d x)}{a d (n+4)}+\frac{\sin ^{n+5}(c+d x)}{a d (n+5)}-\frac{\sin ^{n+6}(c+d x)}{a d (n+6)}",1,"Sin[c + d*x]^(1 + n)/(a*d*(1 + n)) - Sin[c + d*x]^(2 + n)/(a*d*(2 + n)) - (2*Sin[c + d*x]^(3 + n))/(a*d*(3 + n)) + (2*Sin[c + d*x]^(4 + n))/(a*d*(4 + n)) + Sin[c + d*x]^(5 + n)/(a*d*(5 + n)) - Sin[c + d*x]^(6 + n)/(a*d*(6 + n))","A",3,2,29,0.06897,1,"{2836, 88}"
701,1,92,0,0.1428978,"\int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^{n+1}(c+d x)}{a^2 d (n+1)}-\frac{2 \sin ^{n+2}(c+d x)}{a^2 d (n+2)}+\frac{2 \sin ^{n+4}(c+d x)}{a^2 d (n+4)}-\frac{\sin ^{n+5}(c+d x)}{a^2 d (n+5)}","\frac{\sin ^{n+1}(c+d x)}{a^2 d (n+1)}-\frac{2 \sin ^{n+2}(c+d x)}{a^2 d (n+2)}+\frac{2 \sin ^{n+4}(c+d x)}{a^2 d (n+4)}-\frac{\sin ^{n+5}(c+d x)}{a^2 d (n+5)}",1,"Sin[c + d*x]^(1 + n)/(a^2*d*(1 + n)) - (2*Sin[c + d*x]^(2 + n))/(a^2*d*(2 + n)) + (2*Sin[c + d*x]^(4 + n))/(a^2*d*(4 + n)) - Sin[c + d*x]^(5 + n)/(a^2*d*(5 + n))","A",3,2,29,0.06897,1,"{2836, 75}"
702,1,92,0,0.1370753,"\int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^3,x]","\frac{\sin ^{n+1}(c+d x)}{a^3 d (n+1)}-\frac{3 \sin ^{n+2}(c+d x)}{a^3 d (n+2)}+\frac{3 \sin ^{n+3}(c+d x)}{a^3 d (n+3)}-\frac{\sin ^{n+4}(c+d x)}{a^3 d (n+4)}","\frac{\sin ^{n+1}(c+d x)}{a^3 d (n+1)}-\frac{3 \sin ^{n+2}(c+d x)}{a^3 d (n+2)}+\frac{3 \sin ^{n+3}(c+d x)}{a^3 d (n+3)}-\frac{\sin ^{n+4}(c+d x)}{a^3 d (n+4)}",1,"Sin[c + d*x]^(1 + n)/(a^3*d*(1 + n)) - (3*Sin[c + d*x]^(2 + n))/(a^3*d*(2 + n)) + (3*Sin[c + d*x]^(3 + n))/(a^3*d*(3 + n)) - Sin[c + d*x]^(4 + n)/(a^3*d*(4 + n))","A",3,2,29,0.06897,1,"{2836, 43}"
703,1,109,0,0.1746511,"\int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^4,x]","\frac{8 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d (n+1)}-\frac{7 \sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac{4 \sin ^{n+2}(c+d x)}{a^4 d (n+2)}-\frac{\sin ^{n+3}(c+d x)}{a^4 d (n+3)}","\frac{8 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d (n+1)}-\frac{7 \sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac{4 \sin ^{n+2}(c+d x)}{a^4 d (n+2)}-\frac{\sin ^{n+3}(c+d x)}{a^4 d (n+3)}",1,"(-7*Sin[c + d*x]^(1 + n))/(a^4*d*(1 + n)) + (8*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^4*d*(1 + n)) + (4*Sin[c + d*x]^(2 + n))/(a^4*d*(2 + n)) - Sin[c + d*x]^(3 + n)/(a^4*d*(3 + n))","A",8,4,29,0.1379,1,"{2836, 88, 43, 64}"
704,1,160,0,0.2038759,"\int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx","Int[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^5,x]","-\frac{4 (2 n+3) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^5 d (n+1)}+\frac{\sin ^{n+1}(c+d x) \left(a (2 n+7) \sin (c+d x)+a \left(8 n^2+30 n+27\right)\right)}{d \left(n^2+3 n+2\right) \left(a^6 \sin (c+d x)+a^6\right)}-\frac{(a-a \sin (c+d x))^2 \sin ^{n+1}(c+d x)}{d (n+2) \left(a^7 \sin (c+d x)+a^7\right)}","-\frac{4 (2 n+3) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^5 d (n+1)}+\frac{\sin ^{n+1}(c+d x) \left(a (2 n+7) \sin (c+d x)+a \left(8 n^2+30 n+27\right)\right)}{d \left(n^2+3 n+2\right) \left(a^6 \sin (c+d x)+a^6\right)}-\frac{(a-a \sin (c+d x))^2 \sin ^{n+1}(c+d x)}{d (n+2) \left(a^7 \sin (c+d x)+a^7\right)}",1,"(-4*(3 + 2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^5*d*(1 + n)) - (Sin[c + d*x]^(1 + n)*(a - a*Sin[c + d*x])^2)/(d*(2 + n)*(a^7 + a^7*Sin[c + d*x])) + (Sin[c + d*x]^(1 + n)*(a*(27 + 30*n + 8*n^2) + a*(7 + 2*n)*Sin[c + d*x]))/(d*(2 + 3*n + n^2)*(a^6 + a^6*Sin[c + d*x]))","A",4,4,29,0.1379,1,"{2836, 100, 146, 64}"
705,1,209,0,0.2818583,"\int \frac{\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","-\frac{\cos ^{11}(c+d x)}{11 a d}+\frac{2 \cos ^9(c+d x)}{9 a d}-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin ^5(c+d x) \cos ^7(c+d x)}{12 a d}+\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{24 a d}+\frac{\sin (c+d x) \cos ^7(c+d x)}{64 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{384 a d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{1536 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{1024 a d}-\frac{5 x}{1024 a}","-\frac{\cos ^{11}(c+d x)}{11 a d}+\frac{2 \cos ^9(c+d x)}{9 a d}-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin ^5(c+d x) \cos ^7(c+d x)}{12 a d}+\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{24 a d}+\frac{\sin (c+d x) \cos ^7(c+d x)}{64 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{384 a d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{1536 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{1024 a d}-\frac{5 x}{1024 a}",1,"(-5*x)/(1024*a) - Cos[c + d*x]^7/(7*a*d) + (2*Cos[c + d*x]^9)/(9*a*d) - Cos[c + d*x]^11/(11*a*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(1024*a*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(1536*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(384*a*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(64*a*d) + (Cos[c + d*x]^7*Sin[c + d*x]^3)/(24*a*d) + (Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*a*d)","A",11,6,29,0.2069,1,"{2839, 2565, 270, 2568, 2635, 8}"
706,1,183,0,0.2401737,"\int \frac{\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\cos ^{11}(c+d x)}{11 a d}-\frac{2 \cos ^9(c+d x)}{9 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}-\frac{3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{160 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{128 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{256 a d}+\frac{3 x}{256 a}","\frac{\cos ^{11}(c+d x)}{11 a d}-\frac{2 \cos ^9(c+d x)}{9 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}-\frac{3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{160 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{128 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{256 a d}+\frac{3 x}{256 a}",1,"(3*x)/(256*a) + Cos[c + d*x]^7/(7*a*d) - (2*Cos[c + d*x]^9)/(9*a*d) + Cos[c + d*x]^11/(11*a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(256*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(128*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(160*a*d) - (3*Cos[c + d*x]^7*Sin[c + d*x])/(80*a*d) - (Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*a*d)","A",10,6,29,0.2069,1,"{2839, 2568, 2635, 8, 2565, 270}"
707,1,165,0,0.2370259,"\int \frac{\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\cos ^9(c+d x)}{9 a d}-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}+\frac{3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{160 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{128 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{256 a d}-\frac{3 x}{256 a}","\frac{\cos ^9(c+d x)}{9 a d}-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin ^3(c+d x) \cos ^7(c+d x)}{10 a d}+\frac{3 \sin (c+d x) \cos ^7(c+d x)}{80 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{160 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{128 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{256 a d}-\frac{3 x}{256 a}",1,"(-3*x)/(256*a) - Cos[c + d*x]^7/(7*a*d) + Cos[c + d*x]^9/(9*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(256*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(128*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(160*a*d) + (3*Cos[c + d*x]^7*Sin[c + d*x])/(80*a*d) + (Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*a*d)","A",10,6,29,0.2069,1,"{2839, 2565, 14, 2568, 2635, 8}"
708,1,139,0,0.1966255,"\int \frac{\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\cos ^9(c+d x)}{9 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{\sin (c+d x) \cos ^7(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{48 a d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}+\frac{5 \sin (c+d x) \cos (c+d x)}{128 a d}+\frac{5 x}{128 a}","-\frac{\cos ^9(c+d x)}{9 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{\sin (c+d x) \cos ^7(c+d x)}{8 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{48 a d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}+\frac{5 \sin (c+d x) \cos (c+d x)}{128 a d}+\frac{5 x}{128 a}",1,"(5*x)/(128*a) + Cos[c + d*x]^7/(7*a*d) - Cos[c + d*x]^9/(9*a*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(192*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(48*a*d) - (Cos[c + d*x]^7*Sin[c + d*x])/(8*a*d)","A",9,6,29,0.2069,1,"{2839, 2568, 2635, 8, 2565, 14}"
709,1,121,0,0.1452365,"\int \frac{\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin (c+d x) \cos ^7(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{48 a d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac{5 x}{128 a}","-\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin (c+d x) \cos ^7(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{48 a d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac{5 x}{128 a}",1,"(-5*x)/(128*a) - Cos[c + d*x]^7/(7*a*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(192*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(48*a*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(8*a*d)","A",8,6,27,0.2222,1,"{2839, 2565, 30, 2568, 2635, 8}"
710,1,143,0,0.1459677,"\int \frac{\cos ^7(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^7*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{24 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{16 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{5 x}{16 a}","\frac{\cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{24 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{16 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{5 x}{16 a}",1,"(-5*x)/(16*a) - ArcTanh[Cos[c + d*x]]/(a*d) + Cos[c + d*x]/(a*d) + Cos[c + d*x]^3/(3*a*d) + Cos[c + d*x]^5/(5*a*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(16*a*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(24*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(6*a*d)","A",9,6,27,0.2222,1,"{2839, 2592, 302, 206, 2635, 8}"
711,1,137,0,0.173902,"\int \frac{\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^6*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos ^3(c+d x)}{3 a d}-\frac{\cos (c+d x)}{a d}-\frac{15 \cot (c+d x)}{8 a d}+\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}+\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{15 x}{8 a}","-\frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos ^3(c+d x)}{3 a d}-\frac{\cos (c+d x)}{a d}-\frac{15 \cot (c+d x)}{8 a d}+\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}+\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{15 x}{8 a}",1,"(-15*x)/(8*a) + ArcTanh[Cos[c + d*x]]/(a*d) - Cos[c + d*x]/(a*d) - Cos[c + d*x]^3/(3*a*d) - Cos[c + d*x]^5/(5*a*d) - (15*Cot[c + d*x])/(8*a*d) + (5*Cos[c + d*x]^2*Cot[c + d*x])/(8*a*d) + (Cos[c + d*x]^4*Cot[c + d*x])/(4*a*d)","A",10,8,29,0.2759,1,"{2839, 2591, 288, 321, 203, 2592, 302, 206}"
712,1,150,0,0.1886702,"\int \frac{\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{5 \cos ^3(c+d x)}{6 a d}-\frac{5 \cos (c+d x)}{2 a d}+\frac{15 \cot (c+d x)}{8 a d}-\frac{\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{15 x}{8 a}","-\frac{5 \cos ^3(c+d x)}{6 a d}-\frac{5 \cos (c+d x)}{2 a d}+\frac{15 \cot (c+d x)}{8 a d}-\frac{\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{15 x}{8 a}",1,"(15*x)/(8*a) + (5*ArcTanh[Cos[c + d*x]])/(2*a*d) - (5*Cos[c + d*x])/(2*a*d) - (5*Cos[c + d*x]^3)/(6*a*d) + (15*Cot[c + d*x])/(8*a*d) - (5*Cos[c + d*x]^2*Cot[c + d*x])/(8*a*d) - (Cos[c + d*x]^4*Cot[c + d*x])/(4*a*d) - (Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*a*d)","A",11,8,29,0.2759,1,"{2839, 2592, 288, 302, 206, 2591, 321, 203}"
713,1,146,0,0.1801875,"\int \frac{\cos ^4(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{5 \cos ^3(c+d x)}{6 a d}+\frac{5 \cos (c+d x)}{2 a d}-\frac{5 \cot ^3(c+d x)}{6 a d}+\frac{5 \cot (c+d x)}{2 a d}+\frac{\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac{\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{5 x}{2 a}","\frac{5 \cos ^3(c+d x)}{6 a d}+\frac{5 \cos (c+d x)}{2 a d}-\frac{5 \cot ^3(c+d x)}{6 a d}+\frac{5 \cot (c+d x)}{2 a d}+\frac{\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac{\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{5 x}{2 a}",1,"(5*x)/(2*a) - (5*ArcTanh[Cos[c + d*x]])/(2*a*d) + (5*Cos[c + d*x])/(2*a*d) + (5*Cos[c + d*x]^3)/(6*a*d) + (5*Cot[c + d*x])/(2*a*d) + (Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*a*d) - (5*Cot[c + d*x]^3)/(6*a*d) + (Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*a*d)","A",11,7,29,0.2414,1,"{2839, 2591, 288, 302, 203, 2592, 206}"
714,1,150,0,0.1769044,"\int \frac{\cos ^3(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","\frac{15 \cos (c+d x)}{8 a d}+\frac{5 \cot ^3(c+d x)}{6 a d}-\frac{5 \cot (c+d x)}{2 a d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac{\cos (c+d x) \cot ^4(c+d x)}{4 a d}+\frac{5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}-\frac{15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{5 x}{2 a}","\frac{15 \cos (c+d x)}{8 a d}+\frac{5 \cot ^3(c+d x)}{6 a d}-\frac{5 \cot (c+d x)}{2 a d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac{\cos (c+d x) \cot ^4(c+d x)}{4 a d}+\frac{5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}-\frac{15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{5 x}{2 a}",1,"(-5*x)/(2*a) - (15*ArcTanh[Cos[c + d*x]])/(8*a*d) + (15*Cos[c + d*x])/(8*a*d) - (5*Cot[c + d*x])/(2*a*d) + (5*Cos[c + d*x]*Cot[c + d*x]^2)/(8*a*d) + (5*Cot[c + d*x]^3)/(6*a*d) - (Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*a*d) - (Cos[c + d*x]*Cot[c + d*x]^4)/(4*a*d)","A",11,8,29,0.2759,1,"{2839, 2592, 288, 321, 206, 2591, 302, 203}"
715,1,138,0,0.1512575,"\int \frac{\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","-\frac{15 \cos (c+d x)}{8 a d}-\frac{\cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}+\frac{\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac{5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac{15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{x}{a}","-\frac{15 \cos (c+d x)}{8 a d}-\frac{\cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}+\frac{\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac{5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac{15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{x}{a}",1,"-(x/a) + (15*ArcTanh[Cos[c + d*x]])/(8*a*d) - (15*Cos[c + d*x])/(8*a*d) - Cot[c + d*x]/(a*d) - (5*Cos[c + d*x]*Cot[c + d*x]^2)/(8*a*d) + Cot[c + d*x]^3/(3*a*d) + (Cos[c + d*x]*Cot[c + d*x]^4)/(4*a*d) - Cot[c + d*x]^5/(5*a*d)","A",10,7,29,0.2414,1,"{2839, 3473, 8, 2592, 288, 321, 206}"
716,1,142,0,0.1786339,"\int \frac{\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cot (c+d x)}{a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d}+\frac{x}{a}","\frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cot (c+d x)}{a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d}+\frac{x}{a}",1,"x/a + (5*ArcTanh[Cos[c + d*x]])/(16*a*d) + Cot[c + d*x]/(a*d) - Cot[c + d*x]^3/(3*a*d) + Cot[c + d*x]^5/(5*a*d) - (5*Cot[c + d*x]*Csc[c + d*x])/(16*a*d) + (5*Cot[c + d*x]^3*Csc[c + d*x])/(24*a*d) - (Cot[c + d*x]^5*Csc[c + d*x])/(6*a*d)","A",9,5,27,0.1852,1,"{2839, 2611, 3770, 3473, 8}"
717,1,106,0,0.1475149,"\int \frac{\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cot[c + d*x]^8/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}-\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d}","-\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}-\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d}",1,"(-5*ArcTanh[Cos[c + d*x]])/(16*a*d) - Cot[c + d*x]^7/(7*a*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(16*a*d) - (5*Cot[c + d*x]^3*Csc[c + d*x])/(24*a*d) + (Cot[c + d*x]^5*Csc[c + d*x])/(6*a*d)","A",7,5,21,0.2381,1,"{2706, 2607, 30, 2611, 3770}"
718,1,134,0,0.218482,"\int \frac{\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^8*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\cot ^7(c+d x)}{7 a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{128 a d}-\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d}","\frac{\cot ^7(c+d x)}{7 a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{128 a d}-\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d}",1,"(5*ArcTanh[Cos[c + d*x]])/(128*a*d) + Cot[c + d*x]^7/(7*a*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(128*a*d) - (5*Cot[c + d*x]*Csc[c + d*x]^3)/(64*a*d) + (5*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*a*d) - (Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*a*d)","A",8,6,27,0.2222,1,"{2839, 2611, 3768, 3770, 2607, 30}"
719,1,152,0,0.2354241,"\int \frac{\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^8*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^9(c+d x)}{9 a d}-\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{128 a d}+\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d}","-\frac{\cot ^9(c+d x)}{9 a d}-\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{128 a d}+\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}+\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}-\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d}",1,"(-5*ArcTanh[Cos[c + d*x]])/(128*a*d) - Cot[c + d*x]^7/(7*a*d) - Cot[c + d*x]^9/(9*a*d) - (5*Cot[c + d*x]*Csc[c + d*x])/(128*a*d) + (5*Cot[c + d*x]*Csc[c + d*x]^3)/(64*a*d) - (5*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*a*d) + (Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*a*d)","A",9,6,29,0.2069,1,"{2839, 2607, 14, 2611, 3768, 3770}"
720,1,176,0,0.2476246,"\int \frac{\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^8*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\cot ^9(c+d x)}{9 a d}+\frac{\cot ^7(c+d x)}{7 a d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{256 a d}-\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac{3 \cot (c+d x) \csc (c+d x)}{256 a d}","\frac{\cot ^9(c+d x)}{9 a d}+\frac{\cot ^7(c+d x)}{7 a d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{256 a d}-\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac{3 \cot (c+d x) \csc (c+d x)}{256 a d}",1,"(3*ArcTanh[Cos[c + d*x]])/(256*a*d) + Cot[c + d*x]^7/(7*a*d) + Cot[c + d*x]^9/(9*a*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(256*a*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(128*a*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(32*a*d) + (Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*a*d) - (Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*a*d)","A",10,6,29,0.2069,1,"{2839, 2611, 3768, 3770, 2607, 14}"
721,1,194,0,0.2527387,"\int \frac{\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\cot ^{11}(c+d x)}{11 a d}-\frac{2 \cot ^9(c+d x)}{9 a d}-\frac{\cot ^7(c+d x)}{7 a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{256 a d}+\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{256 a d}","-\frac{\cot ^{11}(c+d x)}{11 a d}-\frac{2 \cot ^9(c+d x)}{9 a d}-\frac{\cot ^7(c+d x)}{7 a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{256 a d}+\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{32 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{256 a d}",1,"(-3*ArcTanh[Cos[c + d*x]])/(256*a*d) - Cot[c + d*x]^7/(7*a*d) - (2*Cot[c + d*x]^9)/(9*a*d) - Cot[c + d*x]^11/(11*a*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(256*a*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(128*a*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(32*a*d) - (Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*a*d) + (Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*a*d)","A",10,6,29,0.2069,1,"{2839, 2607, 270, 2611, 3768, 3770}"
722,1,203,0,0.4123953,"\int \frac{\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]","\frac{\cos ^{11}(c+d x)}{11 a^2 d}-\frac{4 \cos ^9(c+d x)}{9 a^2 d}+\frac{5 \cos ^7(c+d x)}{7 a^2 d}-\frac{2 \cos ^5(c+d x)}{5 a^2 d}+\frac{\sin ^5(c+d x) \cos ^5(c+d x)}{5 a^2 d}+\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a^2 d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{64 a^2 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{128 a^2 d}-\frac{3 x}{128 a^2}","\frac{\cos ^{11}(c+d x)}{11 a^2 d}-\frac{4 \cos ^9(c+d x)}{9 a^2 d}+\frac{5 \cos ^7(c+d x)}{7 a^2 d}-\frac{2 \cos ^5(c+d x)}{5 a^2 d}+\frac{\sin ^5(c+d x) \cos ^5(c+d x)}{5 a^2 d}+\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a^2 d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{64 a^2 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{128 a^2 d}-\frac{3 x}{128 a^2}",1,"(-3*x)/(128*a^2) - (2*Cos[c + d*x]^5)/(5*a^2*d) + (5*Cos[c + d*x]^7)/(7*a^2*d) - (4*Cos[c + d*x]^9)/(9*a^2*d) + Cos[c + d*x]^11/(11*a^2*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(128*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(64*a^2*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(16*a^2*d) + (Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*a^2*d) + (Cos[c + d*x]^5*Sin[c + d*x]^5)/(5*a^2*d)","A",15,7,29,0.2414,1,"{2875, 2873, 2565, 270, 2568, 2635, 8}"
723,1,185,0,0.4578524,"\int \frac{\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cos ^9(c+d x)}{9 a^2 d}-\frac{4 \cos ^7(c+d x)}{7 a^2 d}+\frac{2 \cos ^5(c+d x)}{5 a^2 d}-\frac{\sin ^5(c+d x) \cos ^5(c+d x)}{10 a^2 d}-\frac{3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac{3 \sin (c+d x) \cos ^5(c+d x)}{32 a^2 d}+\frac{3 \sin (c+d x) \cos ^3(c+d x)}{128 a^2 d}+\frac{9 \sin (c+d x) \cos (c+d x)}{256 a^2 d}+\frac{9 x}{256 a^2}","\frac{2 \cos ^9(c+d x)}{9 a^2 d}-\frac{4 \cos ^7(c+d x)}{7 a^2 d}+\frac{2 \cos ^5(c+d x)}{5 a^2 d}-\frac{\sin ^5(c+d x) \cos ^5(c+d x)}{10 a^2 d}-\frac{3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac{3 \sin (c+d x) \cos ^5(c+d x)}{32 a^2 d}+\frac{3 \sin (c+d x) \cos ^3(c+d x)}{128 a^2 d}+\frac{9 \sin (c+d x) \cos (c+d x)}{256 a^2 d}+\frac{9 x}{256 a^2}",1,"(9*x)/(256*a^2) + (2*Cos[c + d*x]^5)/(5*a^2*d) - (4*Cos[c + d*x]^7)/(7*a^2*d) + (2*Cos[c + d*x]^9)/(9*a^2*d) + (9*Cos[c + d*x]*Sin[c + d*x])/(256*a^2*d) + (3*Cos[c + d*x]^3*Sin[c + d*x])/(128*a^2*d) - (3*Cos[c + d*x]^5*Sin[c + d*x])/(32*a^2*d) - (3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*a^2*d)","A",17,7,29,0.2414,1,"{2875, 2873, 2568, 2635, 8, 2565, 270}"
724,1,159,0,0.3632481,"\int \frac{\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cos ^9(c+d x)}{9 a^2 d}+\frac{3 \cos ^7(c+d x)}{7 a^2 d}-\frac{2 \cos ^5(c+d x)}{5 a^2 d}+\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{4 a^2 d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{8 a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{32 a^2 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{64 a^2 d}-\frac{3 x}{64 a^2}","-\frac{\cos ^9(c+d x)}{9 a^2 d}+\frac{3 \cos ^7(c+d x)}{7 a^2 d}-\frac{2 \cos ^5(c+d x)}{5 a^2 d}+\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{4 a^2 d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{8 a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{32 a^2 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{64 a^2 d}-\frac{3 x}{64 a^2}",1,"(-3*x)/(64*a^2) - (2*Cos[c + d*x]^5)/(5*a^2*d) + (3*Cos[c + d*x]^7)/(7*a^2*d) - Cos[c + d*x]^9/(9*a^2*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(64*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(32*a^2*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^5*Sin[c + d*x]^3)/(4*a^2*d)","A",14,8,29,0.2759,1,"{2875, 2873, 2565, 14, 2568, 2635, 8, 270}"
725,1,141,0,0.3743033,"\int \frac{\cos ^8(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \cos ^7(c+d x)}{7 a^2 d}+\frac{2 \cos ^5(c+d x)}{5 a^2 d}-\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a^2 d}-\frac{11 \sin (c+d x) \cos ^5(c+d x)}{48 a^2 d}+\frac{11 \sin (c+d x) \cos ^3(c+d x)}{192 a^2 d}+\frac{11 \sin (c+d x) \cos (c+d x)}{128 a^2 d}+\frac{11 x}{128 a^2}","-\frac{2 \cos ^7(c+d x)}{7 a^2 d}+\frac{2 \cos ^5(c+d x)}{5 a^2 d}-\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a^2 d}-\frac{11 \sin (c+d x) \cos ^5(c+d x)}{48 a^2 d}+\frac{11 \sin (c+d x) \cos ^3(c+d x)}{192 a^2 d}+\frac{11 \sin (c+d x) \cos (c+d x)}{128 a^2 d}+\frac{11 x}{128 a^2}",1,"(11*x)/(128*a^2) + (2*Cos[c + d*x]^5)/(5*a^2*d) - (2*Cos[c + d*x]^7)/(7*a^2*d) + (11*Cos[c + d*x]*Sin[c + d*x])/(128*a^2*d) + (11*Cos[c + d*x]^3*Sin[c + d*x])/(192*a^2*d) - (11*Cos[c + d*x]^5*Sin[c + d*x])/(48*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*a^2*d)","A",15,7,29,0.2414,1,"{2875, 2873, 2568, 2635, 8, 2565, 14}"
726,1,124,0,0.1375287,"\int \frac{\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \cos ^7(c+d x)}{35 a^2 d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{15 a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{x}{8 a^2}-\frac{\cos ^9(c+d x)}{5 d (a \sin (c+d x)+a)^2}","-\frac{2 \cos ^7(c+d x)}{35 a^2 d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{15 a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}-\frac{x}{8 a^2}-\frac{\cos ^9(c+d x)}{5 d (a \sin (c+d x)+a)^2}",1,"-x/(8*a^2) - (2*Cos[c + d*x]^7)/(35*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(12*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(15*a^2*d) - Cos[c + d*x]^9/(5*d*(a + a*Sin[c + d*x])^2)","A",6,4,27,0.1481,1,"{2859, 2682, 2635, 8}"
727,1,119,0,0.236758,"\int \frac{\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^7*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\cos ^5(c+d x)}{5 a^2 d}+\frac{\cos ^3(c+d x)}{3 a^2 d}+\frac{\cos (c+d x)}{a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{2 a^2 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{4 a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{3 x}{4 a^2}","-\frac{\cos ^5(c+d x)}{5 a^2 d}+\frac{\cos ^3(c+d x)}{3 a^2 d}+\frac{\cos (c+d x)}{a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{2 a^2 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{4 a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{3 x}{4 a^2}",1,"(-3*x)/(4*a^2) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(a^2*d) + Cos[c + d*x]^3/(3*a^2*d) - Cos[c + d*x]^5/(5*a^2*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(4*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(2*a^2*d)","A",12,9,27,0.3333,1,"{2875, 2873, 2635, 8, 2592, 302, 206, 2565, 30}"
728,1,116,0,0.301899,"\int \frac{\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^6*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \cos ^3(c+d x)}{3 a^2 d}-\frac{2 \cos (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{9 x}{8 a^2}","-\frac{2 \cos ^3(c+d x)}{3 a^2 d}-\frac{2 \cos (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{9 x}{8 a^2}",1,"(-9*x)/(8*a^2) + (2*ArcTanh[Cos[c + d*x]])/(a^2*d) - (2*Cos[c + d*x])/(a^2*d) - (2*Cos[c + d*x]^3)/(3*a^2*d) - Cot[c + d*x]/(a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^2*d)","A",14,8,29,0.2759,1,"{2875, 2872, 3770, 3767, 8, 2638, 2635, 2633}"
729,1,97,0,0.2525079,"\int \frac{\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{\cos ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{a^2 d}+\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{3 x}{a^2}","\frac{\cos ^3(c+d x)}{3 a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{a^2 d}+\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{3 x}{a^2}",1,"(3*x)/a^2 + ArcTanh[Cos[c + d*x]]/(2*a^2*d) + Cos[c + d*x]^3/(3*a^2*d) + (2*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(a^2*d)","A",13,9,29,0.3103,1,"{2875, 2872, 3770, 3767, 8, 3768, 2638, 2635, 2633}"
730,1,97,0,0.2430233,"\int \frac{\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cos (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac{x}{2 a^2}","\frac{2 \cos (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac{x}{2 a^2}",1,"-x/(2*a^2) - (3*ArcTanh[Cos[c + d*x]])/(a^2*d) + (2*Cos[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d)","A",13,8,29,0.2759,1,"{2875, 2709, 3770, 3767, 8, 3768, 2638, 2635}"
731,1,116,0,0.281237,"\int \frac{\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cos (c+d x)}{a^2 d}+\frac{2 \cot ^3(c+d x)}{3 a^2 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{9 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac{2 x}{a^2}","-\frac{\cos (c+d x)}{a^2 d}+\frac{2 \cot ^3(c+d x)}{3 a^2 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{9 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac{2 x}{a^2}",1,"(-2*x)/a^2 + (9*ArcTanh[Cos[c + d*x]])/(8*a^2*d) - Cos[c + d*x]/(a^2*d) - (2*Cot[c + d*x])/(a^2*d) + (2*Cot[c + d*x]^3)/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d)","A",14,7,29,0.2414,1,"{2875, 2872, 3770, 3767, 8, 3768, 2638}"
732,1,118,0,0.3212978,"\int \frac{\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^6)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\cot (c+d x)}{a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac{\cot ^3(c+d x) \csc (c+d x)}{2 a^2 d}-\frac{3 \cot (c+d x) \csc (c+d x)}{4 a^2 d}+\frac{x}{a^2}","-\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\cot (c+d x)}{a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac{\cot ^3(c+d x) \csc (c+d x)}{2 a^2 d}-\frac{3 \cot (c+d x) \csc (c+d x)}{4 a^2 d}+\frac{x}{a^2}",1,"x/a^2 + (3*ArcTanh[Cos[c + d*x]])/(4*a^2*d) + Cot[c + d*x]/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) - Cot[c + d*x]^5/(5*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(4*a^2*d) + (Cot[c + d*x]^3*Csc[c + d*x])/(2*a^2*d)","A",11,8,29,0.2759,1,"{2875, 2873, 3473, 8, 2611, 3770, 2607, 30}"
733,1,132,0,0.3424498,"\int \frac{\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}-\frac{\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}-\frac{\cot ^3(c+d x) \csc (c+d x)}{4 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^2 d}+\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^2 d}","\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{16 a^2 d}-\frac{\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}-\frac{\cot ^3(c+d x) \csc (c+d x)}{4 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^2 d}+\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^2 d}",1,"(-7*ArcTanh[Cos[c + d*x]])/(16*a^2*d) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x])/(4*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*a^2*d)","A",12,7,27,0.2593,1,"{2875, 2873, 2611, 3770, 2607, 30, 3768}"
734,1,124,0,0.2599802,"\int \frac{\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^2,x]","-\frac{\cot ^7(c+d x)}{7 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{\tanh ^{-1}(\cos (c+d x))}{8 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}-\frac{7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a^2 d}","-\frac{\cot ^7(c+d x)}{7 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{\tanh ^{-1}(\cos (c+d x))}{8 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{3 a^2 d}-\frac{7 \cot (c+d x) \csc ^3(c+d x)}{12 a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a^2 d}",1,"ArcTanh[Cos[c + d*x]]/(8*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) - Cot[c + d*x]^7/(7*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(8*a^2*d) - (7*Cot[c + d*x]*Csc[c + d*x]^3)/(12*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(3*a^2*d)","A",19,5,21,0.2381,1,"{2709, 3767, 8, 3768, 3770}"
735,1,176,0,0.4073614,"\int \frac{\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^8*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cot ^7(c+d x)}{7 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}-\frac{\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{16 a^2 d}+\frac{7 \cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{128 a^2 d}","\frac{2 \cot ^7(c+d x)}{7 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}-\frac{\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{16 a^2 d}+\frac{7 \cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{128 a^2 d}",1,"(-11*ArcTanh[Cos[c + d*x]])/(128*a^2*d) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (2*Cot[c + d*x]^7)/(7*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x])/(128*a^2*d) + (7*Cot[c + d*x]*Csc[c + d*x]^3)/(64*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(16*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*a^2*d)","A",15,7,27,0.2593,1,"{2875, 2873, 2611, 3768, 3770, 2607, 14}"
736,1,168,0,0.3864138,"\int \frac{\cot ^8(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^8*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cot ^9(c+d x)}{9 a^2 d}-\frac{3 \cot ^7(c+d x)}{7 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{64 a^2 d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{32 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{64 a^2 d}","-\frac{\cot ^9(c+d x)}{9 a^2 d}-\frac{3 \cot ^7(c+d x)}{7 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{64 a^2 d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{32 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{64 a^2 d}",1,"(3*ArcTanh[Cos[c + d*x]])/(64*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) - (3*Cot[c + d*x]^7)/(7*a^2*d) - Cot[c + d*x]^9/(9*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(64*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(32*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(8*a^2*d) + (Cot[c + d*x]^3*Csc[c + d*x]^5)/(4*a^2*d)","A",14,8,29,0.2759,1,"{2875, 2873, 2607, 14, 2611, 3768, 3770, 270}"
737,1,218,0,0.4872132,"\int \frac{\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^8*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \cot ^9(c+d x)}{9 a^2 d}+\frac{4 \cot ^7(c+d x)}{7 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{9 \tanh ^{-1}(\cos (c+d x))}{256 a^2 d}-\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{3 \cot (c+d x) \csc ^7(c+d x)}{80 a^2 d}+\frac{9 \cot (c+d x) \csc ^5(c+d x)}{160 a^2 d}-\frac{3 \cot (c+d x) \csc ^3(c+d x)}{128 a^2 d}-\frac{9 \cot (c+d x) \csc (c+d x)}{256 a^2 d}","\frac{2 \cot ^9(c+d x)}{9 a^2 d}+\frac{4 \cot ^7(c+d x)}{7 a^2 d}+\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{9 \tanh ^{-1}(\cos (c+d x))}{256 a^2 d}-\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{10 a^2 d}-\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{3 \cot (c+d x) \csc ^7(c+d x)}{80 a^2 d}+\frac{9 \cot (c+d x) \csc ^5(c+d x)}{160 a^2 d}-\frac{3 \cot (c+d x) \csc ^3(c+d x)}{128 a^2 d}-\frac{9 \cot (c+d x) \csc (c+d x)}{256 a^2 d}",1,"(-9*ArcTanh[Cos[c + d*x]])/(256*a^2*d) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (4*Cot[c + d*x]^7)/(7*a^2*d) + (2*Cot[c + d*x]^9)/(9*a^2*d) - (9*Cot[c + d*x]*Csc[c + d*x])/(256*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*a^2*d) + (9*Cot[c + d*x]*Csc[c + d*x]^5)/(160*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x]^7)/(80*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*a^2*d)","A",17,7,29,0.2414,1,"{2875, 2873, 2611, 3768, 3770, 2607, 270}"
738,1,210,0,0.4321133,"\int \frac{\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cot ^{11}(c+d x)}{11 a^2 d}-\frac{4 \cot ^9(c+d x)}{9 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}+\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac{3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{128 a^2 d}","-\frac{\cot ^{11}(c+d x)}{11 a^2 d}-\frac{4 \cot ^9(c+d x)}{9 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}+\frac{\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac{3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{128 a^2 d}",1,"(3*ArcTanh[Cos[c + d*x]])/(128*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) - (5*Cot[c + d*x]^7)/(7*a^2*d) - (4*Cot[c + d*x]^9)/(9*a^2*d) - Cot[c + d*x]^11/(11*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(128*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(64*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(80*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x]^7)/(40*a^2*d) + (Cot[c + d*x]^3*Csc[c + d*x]^7)/(5*a^2*d)","A",15,7,29,0.2414,1,"{2875, 2873, 2607, 270, 2611, 3768, 3770}"
739,1,161,0,0.4770718,"\int \frac{\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","-\frac{3 \cos ^7(c+d x)}{7 a^3 d}+\frac{7 \cos ^5(c+d x)}{5 a^3 d}-\frac{4 \cos ^3(c+d x)}{3 a^3 d}+\frac{\sin ^5(c+d x) \cos ^3(c+d x)}{8 a^3 d}+\frac{29 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a^3 d}+\frac{29 \sin (c+d x) \cos ^3(c+d x)}{64 a^3 d}-\frac{29 \sin (c+d x) \cos (c+d x)}{128 a^3 d}-\frac{29 x}{128 a^3}","-\frac{3 \cos ^7(c+d x)}{7 a^3 d}+\frac{7 \cos ^5(c+d x)}{5 a^3 d}-\frac{4 \cos ^3(c+d x)}{3 a^3 d}+\frac{\sin ^5(c+d x) \cos ^3(c+d x)}{8 a^3 d}+\frac{29 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a^3 d}+\frac{29 \sin (c+d x) \cos ^3(c+d x)}{64 a^3 d}-\frac{29 \sin (c+d x) \cos (c+d x)}{128 a^3 d}-\frac{29 x}{128 a^3}",1,"(-29*x)/(128*a^3) - (4*Cos[c + d*x]^3)/(3*a^3*d) + (7*Cos[c + d*x]^5)/(5*a^3*d) - (3*Cos[c + d*x]^7)/(7*a^3*d) - (29*Cos[c + d*x]*Sin[c + d*x])/(128*a^3*d) + (29*Cos[c + d*x]^3*Sin[c + d*x])/(64*a^3*d) + (29*Cos[c + d*x]^3*Sin[c + d*x]^3)/(48*a^3*d) + (Cos[c + d*x]^3*Sin[c + d*x]^5)/(8*a^3*d)","A",18,8,29,0.2759,1,"{2875, 2873, 2565, 14, 2568, 2635, 8, 270}"
740,1,133,0,0.4013991,"\int \frac{\cos ^8(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\cos ^7(c+d x)}{7 a^3 d}-\frac{\cos ^5(c+d x)}{a^3 d}+\frac{4 \cos ^3(c+d x)}{3 a^3 d}-\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{2 a^3 d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{8 a^3 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{16 a^3 d}+\frac{5 x}{16 a^3}","\frac{\cos ^7(c+d x)}{7 a^3 d}-\frac{\cos ^5(c+d x)}{a^3 d}+\frac{4 \cos ^3(c+d x)}{3 a^3 d}-\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{2 a^3 d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{8 a^3 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{16 a^3 d}+\frac{5 x}{16 a^3}",1,"(5*x)/(16*a^3) + (4*Cos[c + d*x]^3)/(3*a^3*d) - Cos[c + d*x]^5/(a^3*d) + Cos[c + d*x]^7/(7*a^3*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(8*a^3*d) - (Cos[c + d*x]^3*Sin[c + d*x]^3)/(2*a^3*d)","A",16,8,29,0.2759,1,"{2875, 2873, 2568, 2635, 8, 2565, 14, 270}"
741,1,131,0,0.1698198,"\int \frac{\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{7 \cos ^5(c+d x)}{30 a^3 d}-\frac{\cos ^7(c+d x)}{6 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{7 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac{7 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac{7 x}{16 a^3}-\frac{\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}","-\frac{7 \cos ^5(c+d x)}{30 a^3 d}-\frac{\cos ^7(c+d x)}{6 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{7 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac{7 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac{7 x}{16 a^3}-\frac{\cos ^9(c+d x)}{3 d (a \sin (c+d x)+a)^3}",1,"(-7*x)/(16*a^3) - (7*Cos[c + d*x]^5)/(30*a^3*d) - (7*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) - (7*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^3*d) - Cos[c + d*x]^9/(3*d*(a + a*Sin[c + d*x])^3) - Cos[c + d*x]^7/(6*d*(a^3 + a^3*Sin[c + d*x]))","A",6,5,27,0.1852,1,"{2859, 2679, 2682, 2635, 8}"
742,1,99,0,0.2429142,"\int \frac{\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^7*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{\cos ^3(c+d x)}{a^3 d}+\frac{\cos (c+d x)}{a^3 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^3 d}-\frac{13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{13 x}{8 a^3}","-\frac{\cos ^3(c+d x)}{a^3 d}+\frac{\cos (c+d x)}{a^3 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^3 d}-\frac{13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{13 x}{8 a^3}",1,"(-13*x)/(8*a^3) - ArcTanh[Cos[c + d*x]]/(a^3*d) + Cos[c + d*x]/(a^3*d) - Cos[c + d*x]^3/(a^3*d) - (13*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^3*d)","A",13,10,27,0.3704,1,"{2875, 2873, 2635, 8, 2592, 321, 206, 2565, 30, 2568}"
743,1,92,0,0.222455,"\int \frac{\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^6*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\cos ^3(c+d x)}{3 a^3 d}-\frac{3 \cos (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{x}{2 a^3}","\frac{\cos ^3(c+d x)}{3 a^3 d}-\frac{3 \cos (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{x}{2 a^3}",1,"x/(2*a^3) + (3*ArcTanh[Cos[c + d*x]])/(a^3*d) - (3*Cos[c + d*x])/(a^3*d) + Cos[c + d*x]^3/(3*a^3*d) - Cot[c + d*x]/(a^3*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d)","A",11,8,29,0.2759,1,"{2875, 2709, 3770, 3767, 8, 2638, 2635, 2633}"
744,1,98,0,0.2469388,"\int \frac{\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^5*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{3 \cos (c+d x)}{a^3 d}+\frac{3 \cot (c+d x)}{a^3 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{5 x}{2 a^3}","\frac{3 \cos (c+d x)}{a^3 d}+\frac{3 \cot (c+d x)}{a^3 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{5 x}{2 a^3}",1,"(5*x)/(2*a^3) - (5*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (3*Cos[c + d*x])/(a^3*d) + (3*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d)","A",11,8,29,0.2759,1,"{2875, 2872, 3770, 3767, 8, 3768, 2638, 2635}"
745,1,92,0,0.2743051,"\int \frac{\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","-\frac{\cos (c+d x)}{a^3 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{3 \cot (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{3 x}{a^3}","-\frac{\cos (c+d x)}{a^3 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{3 \cot (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{3 x}{a^3}",1,"(-3*x)/a^3 - ArcTanh[Cos[c + d*x]]/(2*a^3*d) - Cos[c + d*x]/(a^3*d) - (3*Cot[c + d*x])/(a^3*d) - Cot[c + d*x]^3/(3*a^3*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d)","A",11,7,29,0.2414,1,"{2875, 2872, 3770, 3767, 8, 3768, 2638}"
746,1,97,0,0.3181651,"\int \frac{\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]","\frac{\cot ^3(c+d x)}{a^3 d}+\frac{\cot (c+d x)}{a^3 d}+\frac{13 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac{x}{a^3}","\frac{\cot ^3(c+d x)}{a^3 d}+\frac{\cot (c+d x)}{a^3 d}+\frac{13 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac{11 \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac{x}{a^3}",1,"x/a^3 + (13*ArcTanh[Cos[c + d*x]])/(8*a^3*d) + Cot[c + d*x]/(a^3*d) + Cot[c + d*x]^3/(a^3*d) - (11*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d)","A",12,9,29,0.3103,1,"{2875, 2873, 3473, 8, 2611, 3770, 2607, 30, 3768}"
747,1,100,0,0.3374837,"\int \frac{\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^6)/(a + a*Sin[c + d*x])^3,x]","-\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a^3 d}","-\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a^3 d}",1,"(-7*ArcTanh[Cos[c + d*x]])/(8*a^3*d) - (4*Cot[c + d*x]^3)/(3*a^3*d) - Cot[c + d*x]^5/(5*a^3*d) + (Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) + (3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d)","A",13,8,29,0.2759,1,"{2875, 2873, 2611, 3770, 2607, 30, 3768, 14}"
748,1,124,0,0.3638558,"\int \frac{\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^3,x]","\frac{3 \cot ^5(c+d x)}{5 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{7 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac{17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}+\frac{7 \cot (c+d x) \csc (c+d x)}{16 a^3 d}","\frac{3 \cot ^5(c+d x)}{5 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{7 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac{17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}+\frac{7 \cot (c+d x) \csc (c+d x)}{16 a^3 d}",1,"(7*ArcTanh[Cos[c + d*x]])/(16*a^3*d) + (4*Cot[c + d*x]^3)/(3*a^3*d) + (3*Cot[c + d*x]^5)/(5*a^3*d) + (7*Cot[c + d*x]*Csc[c + d*x])/(16*a^3*d) - (17*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^3*d)","A",15,8,27,0.2963,1,"{2875, 2873, 2607, 30, 2611, 3768, 3770, 14}"
749,1,140,0,0.231049,"\int \frac{\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]","-\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}","-\frac{\cot ^7(c+d x)}{7 a^3 d}-\frac{\cot ^5(c+d x)}{a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a^3 d}",1,"(-5*ArcTanh[Cos[c + d*x]])/(16*a^3*d) - (4*Cot[c + d*x]^3)/(3*a^3*d) - Cot[c + d*x]^5/(a^3*d) - Cot[c + d*x]^7/(7*a^3*d) - (5*Cot[c + d*x]*Csc[c + d*x])/(16*a^3*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^3*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(2*a^3*d)","A",17,4,21,0.1905,1,"{2709, 3768, 3770, 3767}"
750,1,166,0,0.4056744,"\int \frac{\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Cot[c + d*x]^8*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{3 \cot ^7(c+d x)}{7 a^3 d}+\frac{7 \cot ^5(c+d x)}{5 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{29 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac{23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}+\frac{29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}+\frac{29 \cot (c+d x) \csc (c+d x)}{128 a^3 d}","\frac{3 \cot ^7(c+d x)}{7 a^3 d}+\frac{7 \cot ^5(c+d x)}{5 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{29 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac{23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}+\frac{29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}+\frac{29 \cot (c+d x) \csc (c+d x)}{128 a^3 d}",1,"(29*ArcTanh[Cos[c + d*x]])/(128*a^3*d) + (4*Cot[c + d*x]^3)/(3*a^3*d) + (7*Cot[c + d*x]^5)/(5*a^3*d) + (3*Cot[c + d*x]^7)/(7*a^3*d) + (29*Cot[c + d*x]*Csc[c + d*x])/(128*a^3*d) + (29*Cot[c + d*x]*Csc[c + d*x]^3)/(192*a^3*d) - (23*Cot[c + d*x]*Csc[c + d*x]^5)/(48*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^7)/(8*a^3*d)","A",18,8,27,0.2963,1,"{2875, 2873, 2607, 14, 2611, 3768, 3770, 270}"
751,1,82,0,0.1307931,"\int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Int[Sin[c + d*x]^2*(a + a*Sin[c + d*x])*Tan[c + d*x]^2,x]","-\frac{a \cos ^3(c+d x)}{3 d}+\frac{2 a \cos (c+d x)}{d}+\frac{3 a \tan (c+d x)}{2 d}+\frac{a \sec (c+d x)}{d}-\frac{a \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 a x}{2}","-\frac{a \cos ^3(c+d x)}{3 d}+\frac{2 a \cos (c+d x)}{d}+\frac{3 a \tan (c+d x)}{2 d}+\frac{a \sec (c+d x)}{d}-\frac{a \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 a x}{2}",1,"(-3*a*x)/2 + (2*a*Cos[c + d*x])/d - (a*Cos[c + d*x]^3)/(3*d) + (a*Sec[c + d*x])/d + (3*a*Tan[c + d*x])/(2*d) - (a*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)","A",8,7,27,0.2593,1,"{2838, 2591, 288, 321, 203, 2590, 270}"
752,1,65,0,0.1023261,"\int \sin (c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Int[Sin[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{a \cos (c+d x)}{d}+\frac{3 a \tan (c+d x)}{2 d}+\frac{a \sec (c+d x)}{d}-\frac{a \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 a x}{2}","\frac{a \cos (c+d x)}{d}+\frac{3 a \tan (c+d x)}{2 d}+\frac{a \sec (c+d x)}{d}-\frac{a \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 a x}{2}",1,"(-3*a*x)/2 + (a*Cos[c + d*x])/d + (a*Sec[c + d*x])/d + (3*a*Tan[c + d*x])/(2*d) - (a*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)","A",8,7,25,0.2800,1,"{2838, 2590, 14, 2591, 288, 321, 203}"
753,1,39,0,0.1043196,"\int (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Int[(a + a*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{a \cos (c+d x)}{d}+\frac{a \cos (c+d x)}{d (1-\sin (c+d x))}-a x","\frac{a \cos (c+d x)}{d}+\frac{a \cos (c+d x)}{d (1-\sin (c+d x))}-a x",1,"-(a*x) + (a*Cos[c + d*x])/d + (a*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))","A",5,5,19,0.2632,1,"{2708, 2746, 12, 2735, 2648}"
754,1,27,0,0.0465878,"\int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x],x]","\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-a x","\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-a x",1,"-(a*x) + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",5,4,23,0.1739,1,"{2838, 2606, 8, 3473}"
755,1,36,0,0.0742867,"\int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}","\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}",1,"-((a*ArcTanh[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",6,6,25,0.2400,1,"{2838, 2622, 321, 207, 3767, 8}"
756,1,48,0,0.1088781,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}","\frac{a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}",1,"-((a*ArcTanh[Cos[c + d*x]])/d) - (a*Cot[c + d*x])/d + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",7,6,27,0.2222,1,"{2838, 2620, 14, 2622, 321, 207}"
757,1,75,0,0.1301296,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{3 a \sec (c+d x)}{2 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \csc ^2(c+d x) \sec (c+d x)}{2 d}","\frac{a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{3 a \sec (c+d x)}{2 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \csc ^2(c+d x) \sec (c+d x)}{2 d}",1,"(-3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (a*Cot[c + d*x])/d + (3*a*Sec[c + d*x])/(2*d) - (a*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (a*Tan[c + d*x])/d","A",8,7,27,0.2593,1,"{2838, 2622, 288, 321, 207, 2620, 14}"
758,1,91,0,0.1314791,"\int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \tan (c+d x)}{d}-\frac{a \cot ^3(c+d x)}{3 d}-\frac{2 a \cot (c+d x)}{d}+\frac{3 a \sec (c+d x)}{2 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \csc ^2(c+d x) \sec (c+d x)}{2 d}","\frac{a \tan (c+d x)}{d}-\frac{a \cot ^3(c+d x)}{3 d}-\frac{2 a \cot (c+d x)}{d}+\frac{3 a \sec (c+d x)}{2 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \csc ^2(c+d x) \sec (c+d x)}{2 d}",1,"(-3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (2*a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/(3*d) + (3*a*Sec[c + d*x])/(2*d) - (a*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (a*Tan[c + d*x])/d","A",8,7,27,0.2593,1,"{2838, 2620, 270, 2622, 288, 321, 207}"
759,1,89,0,0.2083725,"\int \sin (c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Int[Sin[c + d*x]*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","-\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{3 a^2 \cos (c+d x)}{d}+\frac{3 a^2 \tan (c+d x)}{d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \sin ^2(c+d x) \tan (c+d x)}{d}-3 a^2 x","-\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{3 a^2 \cos (c+d x)}{d}+\frac{3 a^2 \tan (c+d x)}{d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \sin ^2(c+d x) \tan (c+d x)}{d}-3 a^2 x",1,"-3*a^2*x + (3*a^2*Cos[c + d*x])/d - (a^2*Cos[c + d*x]^3)/(3*d) + (2*a^2*Sec[c + d*x])/d + (3*a^2*Tan[c + d*x])/d - (a^2*Sin[c + d*x]^2*Tan[c + d*x])/d","A",12,8,27,0.2963,1,"{2873, 2590, 14, 2591, 288, 321, 203, 270}"
760,1,71,0,0.0857938,"\int (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{2 a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{5 a^2 x}{2}","\frac{2 a^2 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{5 a^2 x}{2}",1,"(-5*a^2*x)/2 + (2*a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",6,5,21,0.2381,1,"{2709, 2648, 2638, 2635, 8}"
761,1,43,0,0.0568371,"\int \sec (c+d x) (a+a \sin (c+d x))^2 \tan (c+d x) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^2*Tan[c + d*x],x]","\frac{2 a^2 \cos (c+d x)}{d}-2 a^2 x+\frac{\sec (c+d x) (a \sin (c+d x)+a)^2}{d}","\frac{2 a^2 \cos (c+d x)}{d}-2 a^2 x+\frac{\sec (c+d x) (a \sin (c+d x)+a)^2}{d}",1,"-2*a^2*x + (2*a^2*Cos[c + d*x])/d + (Sec[c + d*x]*(a + a*Sin[c + d*x])^2)/d","A",3,2,25,0.08000,1,"{2855, 2638}"
762,1,44,0,0.1264337,"\int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \tan (c+d x)}{d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}","\frac{2 a^2 \tan (c+d x)}{d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}",1,"-((a^2*ArcTanh[Cos[c + d*x]])/d) + (2*a^2*Sec[c + d*x])/d + (2*a^2*Tan[c + d*x])/d","A",9,7,27,0.2593,1,"{2873, 3767, 8, 2622, 321, 207, 2606}"
763,1,58,0,0.2225252,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \tan (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}","\frac{2 a^2 \tan (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a^2 \sec (c+d x)}{d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}",1,"(-2*a^2*ArcTanh[Cos[c + d*x]])/d - (a^2*Cot[c + d*x])/d + (2*a^2*Sec[c + d*x])/d + (2*a^2*Tan[c + d*x])/d","A",10,8,29,0.2759,1,"{2873, 3767, 8, 2622, 321, 207, 2620, 14}"
764,1,86,0,0.210062,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{5 a^2 \sec (c+d x)}{2 d}-\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}","\frac{2 a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{5 a^2 \sec (c+d x)}{2 d}-\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}",1,"(-5*a^2*ArcTanh[Cos[c + d*x]])/(2*d) - (2*a^2*Cot[c + d*x])/d + (5*a^2*Sec[c + d*x])/(2*d) - (a^2*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (2*a^2*Tan[c + d*x])/d","A",12,7,29,0.2414,1,"{2873, 2622, 321, 207, 2620, 14, 288}"
765,1,111,0,0.1701079,"\int \sin (c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Int[Sin[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","-\frac{a^3 \cos ^3(c+d x)}{d}+\frac{7 a^3 \cos (c+d x)}{d}+\frac{a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{19 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{51 a^3 x}{8}","-\frac{a^3 \cos ^3(c+d x)}{d}+\frac{7 a^3 \cos (c+d x)}{d}+\frac{a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{19 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{51 a^3 x}{8}",1,"(-51*a^3*x)/8 + (7*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/d + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (19*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",11,6,27,0.2222,1,"{2872, 2648, 2638, 2635, 8, 2633}"
766,1,89,0,0.1240795,"\int (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{5 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{11 a^3 x}{2}","-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{5 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{11 a^3 x}{2}",1,"(-11*a^3*x)/2 + (5*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",8,6,21,0.2857,1,"{2709, 2648, 2638, 2635, 8, 2633}"
767,1,67,0,0.063891,"\int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]","\frac{6 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{9 a^3 x}{2}+\frac{\sec (c+d x) (a \sin (c+d x)+a)^3}{d}","\frac{6 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{9 a^3 x}{2}+\frac{\sec (c+d x) (a \sin (c+d x)+a)^3}{d}",1,"(-9*a^3*x)/2 + (6*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (Sec[c + d*x]*(a + a*Sin[c + d*x])^3)/d","A",2,2,25,0.08000,1,"{2855, 2644}"
768,1,48,0,0.1042007,"\int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+a^3 (-x)","\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+a^3 (-x)",1,"-(a^3*x) - (a^3*ArcTanh[Cos[c + d*x]])/d + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2872, 3770, 2648}"
769,1,56,0,0.1352015,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot (c+d x)}{d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}","-\frac{a^3 \cot (c+d x)}{d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}",1,"(-3*a^3*ArcTanh[Cos[c + d*x]])/d - (a^3*Cot[c + d*x])/d + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))","A",6,5,29,0.1724,1,"{2872, 3770, 3767, 8, 2648}"
770,1,80,0,0.1563334,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cot (c+d x)}{d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}","-\frac{3 a^3 \cot (c+d x)}{d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}",1,"(-9*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))","A",8,6,29,0.2069,1,"{2872, 3770, 3767, 8, 3768, 2648}"
771,1,98,0,0.1754064,"\int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{5 a^3 \cot (c+d x)}{d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{11 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}","-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{5 a^3 \cot (c+d x)}{d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{11 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}",1,"(-11*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))","A",10,6,29,0.2069,1,"{2872, 3770, 3767, 8, 3768, 2648}"
772,1,83,0,0.1423753,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\cos (c+d x)}{a d}+\frac{\tan ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}-\frac{\sec ^3(c+d x)}{3 a d}+\frac{2 \sec (c+d x)}{a d}+\frac{x}{a}","\frac{\cos (c+d x)}{a d}+\frac{\tan ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}-\frac{\sec ^3(c+d x)}{3 a d}+\frac{2 \sec (c+d x)}{a d}+\frac{x}{a}",1,"x/a + Cos[c + d*x]/(a*d) + (2*Sec[c + d*x])/(a*d) - Sec[c + d*x]^3/(3*a*d) - Tan[c + d*x]/(a*d) + Tan[c + d*x]^3/(3*a*d)","A",7,5,29,0.1724,1,"{2839, 3473, 8, 2590, 270}"
773,1,70,0,0.1107676,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\tan ^3(c+d x)}{3 a d}+\frac{\tan (c+d x)}{a d}+\frac{\sec ^3(c+d x)}{3 a d}-\frac{\sec (c+d x)}{a d}-\frac{x}{a}","-\frac{\tan ^3(c+d x)}{3 a d}+\frac{\tan (c+d x)}{a d}+\frac{\sec ^3(c+d x)}{3 a d}-\frac{\sec (c+d x)}{a d}-\frac{x}{a}",1,"-(x/a) - Sec[c + d*x]/(a*d) + Sec[c + d*x]^3/(3*a*d) + Tan[c + d*x]/(a*d) - Tan[c + d*x]^3/(3*a*d)","A",6,4,27,0.1481,1,"{2839, 2606, 3473, 8}"
774,1,50,0,0.0897598,"\int \frac{\tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Tan[c + d*x]^2/(a + a*Sin[c + d*x]),x]","\frac{\tan ^3(c+d x)}{3 a d}-\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}","\frac{\tan ^3(c+d x)}{3 a d}-\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}",1,"Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(3*a*d) + Tan[c + d*x]^3/(3*a*d)","A",5,4,21,0.1905,1,"{2706, 2607, 30, 2606}"
775,1,37,0,0.093388,"\int \frac{\sec (c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\sec ^3(c+d x)}{3 a d}-\frac{\tan ^3(c+d x)}{3 a d}","\frac{\sec ^3(c+d x)}{3 a d}-\frac{\tan ^3(c+d x)}{3 a d}",1,"Sec[c + d*x]^3/(3*a*d) - Tan[c + d*x]^3/(3*a*d)","A",5,4,25,0.1600,1,"{2839, 2606, 30, 2607}"
776,1,79,0,0.1198433,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\tan ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}+\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}","-\frac{\tan ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}+\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"-(ArcTanh[Cos[c + d*x]]/(a*d)) + Sec[c + d*x]/(a*d) + Sec[c + d*x]^3/(3*a*d) - Tan[c + d*x]/(a*d) - Tan[c + d*x]^3/(3*a*d)","A",7,5,27,0.1852,1,"{2839, 2622, 302, 207, 3767}"
777,1,93,0,0.1941315,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\tan ^3(c+d x)}{3 a d}+\frac{2 \tan (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}-\frac{\sec ^3(c+d x)}{3 a d}-\frac{\sec (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}","\frac{\tan ^3(c+d x)}{3 a d}+\frac{2 \tan (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}-\frac{\sec ^3(c+d x)}{3 a d}-\frac{\sec (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"ArcTanh[Cos[c + d*x]]/(a*d) - Cot[c + d*x]/(a*d) - Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(3*a*d) + (2*Tan[c + d*x])/(a*d) + Tan[c + d*x]^3/(3*a*d)","A",8,6,29,0.2069,1,"{2839, 2620, 270, 2622, 302, 207}"
778,1,149,0,0.3064879,"\int \frac{\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Sin[c + d*x]^4*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \cos (c+d x)}{a^2 d}+\frac{9 \tan ^5(c+d x)}{10 a^2 d}-\frac{3 \tan ^3(c+d x)}{2 a^2 d}+\frac{9 \tan (c+d x)}{2 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^3(c+d x)}{a^2 d}-\frac{6 \sec (c+d x)}{a^2 d}-\frac{\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}-\frac{9 x}{2 a^2}","-\frac{2 \cos (c+d x)}{a^2 d}+\frac{9 \tan ^5(c+d x)}{10 a^2 d}-\frac{3 \tan ^3(c+d x)}{2 a^2 d}+\frac{9 \tan (c+d x)}{2 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^3(c+d x)}{a^2 d}-\frac{6 \sec (c+d x)}{a^2 d}-\frac{\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}-\frac{9 x}{2 a^2}",1,"(-9*x)/(2*a^2) - (2*Cos[c + d*x])/(a^2*d) - (6*Sec[c + d*x])/(a^2*d) + (2*Sec[c + d*x]^3)/(a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) + (9*Tan[c + d*x])/(2*a^2*d) - (3*Tan[c + d*x]^3)/(2*a^2*d) + (9*Tan[c + d*x]^5)/(10*a^2*d) - (Sin[c + d*x]^2*Tan[c + d*x]^5)/(2*a^2*d)","A",15,10,29,0.3448,1,"{2875, 2710, 3473, 8, 2590, 270, 2591, 288, 302, 203}"
779,1,120,0,0.2778039,"\int \frac{\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Sin[c + d*x]^3*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\cos (c+d x)}{a^2 d}-\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \tan ^3(c+d x)}{3 a^2 d}-\frac{2 \tan (c+d x)}{a^2 d}+\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{5 \sec ^3(c+d x)}{3 a^2 d}+\frac{4 \sec (c+d x)}{a^2 d}+\frac{2 x}{a^2}","\frac{\cos (c+d x)}{a^2 d}-\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \tan ^3(c+d x)}{3 a^2 d}-\frac{2 \tan (c+d x)}{a^2 d}+\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{5 \sec ^3(c+d x)}{3 a^2 d}+\frac{4 \sec (c+d x)}{a^2 d}+\frac{2 x}{a^2}",1,"(2*x)/a^2 + Cos[c + d*x]/(a^2*d) + (4*Sec[c + d*x])/(a^2*d) - (5*Sec[c + d*x]^3)/(3*a^2*d) + (2*Sec[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x])/(a^2*d) + (2*Tan[c + d*x]^3)/(3*a^2*d) - (2*Tan[c + d*x]^5)/(5*a^2*d)","A",13,8,29,0.2759,1,"{2875, 2873, 2606, 194, 3473, 8, 2590, 270}"
780,1,106,0,0.2822277,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \tan ^5(c+d x)}{5 a^2 d}-\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{\tan (c+d x)}{a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{4 \sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{x}{a^2}","\frac{2 \tan ^5(c+d x)}{5 a^2 d}-\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{\tan (c+d x)}{a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{4 \sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{x}{a^2}",1,"-(x/a^2) - (2*Sec[c + d*x])/(a^2*d) + (4*Sec[c + d*x]^3)/(3*a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) + Tan[c + d*x]/(a^2*d) - Tan[c + d*x]^3/(3*a^2*d) + (2*Tan[c + d*x]^5)/(5*a^2*d)","A",12,8,29,0.2759,1,"{2875, 2873, 2607, 30, 2606, 194, 3473, 8}"
781,1,66,0,0.2586766,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{\sec ^3(c+d x)}{a^2 d}+\frac{\sec (c+d x)}{a^2 d}","-\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{\sec ^3(c+d x)}{a^2 d}+\frac{\sec (c+d x)}{a^2 d}",1,"Sec[c + d*x]/(a^2*d) - Sec[c + d*x]^3/(a^2*d) + (2*Sec[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x]^5)/(5*a^2*d)","A",11,7,27,0.2593,1,"{2875, 2873, 2606, 14, 2607, 30, 194}"
782,1,73,0,0.1897612,"\int \frac{\tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]^2/(a + a*Sin[c + d*x])^2,x]","\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{\tan ^3(c+d x)}{3 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^3(c+d x)}{3 a^2 d}","\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{\tan ^3(c+d x)}{3 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^3(c+d x)}{3 a^2 d}",1,"(2*Sec[c + d*x]^3)/(3*a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) + Tan[c + d*x]^3/(3*a^2*d) + (2*Tan[c + d*x]^5)/(5*a^2*d)","A",10,5,21,0.2381,1,"{2711, 2607, 14, 2606, 30}"
783,1,71,0,0.1280786,"\int \frac{\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{4 \tan (c+d x)}{15 a^2 d}-\frac{2 \sec (c+d x)}{15 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2}","\frac{4 \tan (c+d x)}{15 a^2 d}-\frac{2 \sec (c+d x)}{15 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2}",1,"Sec[c + d*x]/(5*d*(a + a*Sin[c + d*x])^2) - (2*Sec[c + d*x])/(15*d*(a^2 + a^2*Sin[c + d*x])) + (4*Tan[c + d*x])/(15*a^2*d)","A",4,4,25,0.1600,1,"{2859, 2672, 3767, 8}"
784,1,115,0,0.2662538,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \tan ^5(c+d x)}{5 a^2 d}-\frac{4 \tan ^3(c+d x)}{3 a^2 d}-\frac{2 \tan (c+d x)}{a^2 d}+\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}+\frac{\sec (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}","-\frac{2 \tan ^5(c+d x)}{5 a^2 d}-\frac{4 \tan ^3(c+d x)}{3 a^2 d}-\frac{2 \tan (c+d x)}{a^2 d}+\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}+\frac{\sec (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}",1,"-(ArcTanh[Cos[c + d*x]]/(a^2*d)) + Sec[c + d*x]/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d) + (2*Sec[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x])/(a^2*d) - (4*Tan[c + d*x]^3)/(3*a^2*d) - (2*Tan[c + d*x]^5)/(5*a^2*d)","A",11,8,27,0.2963,1,"{2875, 2873, 3767, 2622, 302, 207, 2606, 30}"
785,1,130,0,0.3103368,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{5 \tan ^3(c+d x)}{3 a^2 d}+\frac{4 \tan (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}","\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{5 \tan ^3(c+d x)}{3 a^2 d}+\frac{4 \tan (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}",1,"(2*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a^2*d) - (2*Sec[c + d*x])/(a^2*d) - (2*Sec[c + d*x]^3)/(3*a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) + (4*Tan[c + d*x])/(a^2*d) + (5*Tan[c + d*x]^3)/(3*a^2*d) + (2*Tan[c + d*x]^5)/(5*a^2*d)","A",12,8,29,0.2759,1,"{2875, 2873, 3767, 2622, 302, 207, 2620, 270}"
786,1,158,0,0.346225,"\int \frac{\csc ^3(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Csc[c + d*x]^3*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \tan ^5(c+d x)}{5 a^2 d}-\frac{2 \tan ^3(c+d x)}{a^2 d}-\frac{6 \tan (c+d x)}{a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{9 \sec ^5(c+d x)}{10 a^2 d}+\frac{3 \sec ^3(c+d x)}{2 a^2 d}+\frac{9 \sec (c+d x)}{2 a^2 d}-\frac{9 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\csc ^2(c+d x) \sec ^5(c+d x)}{2 a^2 d}","-\frac{2 \tan ^5(c+d x)}{5 a^2 d}-\frac{2 \tan ^3(c+d x)}{a^2 d}-\frac{6 \tan (c+d x)}{a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{9 \sec ^5(c+d x)}{10 a^2 d}+\frac{3 \sec ^3(c+d x)}{2 a^2 d}+\frac{9 \sec (c+d x)}{2 a^2 d}-\frac{9 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\csc ^2(c+d x) \sec ^5(c+d x)}{2 a^2 d}",1,"(-9*ArcTanh[Cos[c + d*x]])/(2*a^2*d) + (2*Cot[c + d*x])/(a^2*d) + (9*Sec[c + d*x])/(2*a^2*d) + (3*Sec[c + d*x]^3)/(2*a^2*d) + (9*Sec[c + d*x]^5)/(10*a^2*d) - (Csc[c + d*x]^2*Sec[c + d*x]^5)/(2*a^2*d) - (6*Tan[c + d*x])/(a^2*d) - (2*Tan[c + d*x]^3)/(a^2*d) - (2*Tan[c + d*x]^5)/(5*a^2*d)","A",15,8,29,0.2759,1,"{2875, 2873, 2622, 302, 207, 2620, 270, 288}"
787,1,151,0,0.3414598,"\int \frac{\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Sin[c + d*x]^4*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\cos (c+d x)}{a^3 d}+\frac{4 \tan ^7(c+d x)}{7 a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{13 \sec ^5(c+d x)}{5 a^3 d}-\frac{5 \sec ^3(c+d x)}{a^3 d}+\frac{7 \sec (c+d x)}{a^3 d}+\frac{3 x}{a^3}","\frac{\cos (c+d x)}{a^3 d}+\frac{4 \tan ^7(c+d x)}{7 a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{13 \sec ^5(c+d x)}{5 a^3 d}-\frac{5 \sec ^3(c+d x)}{a^3 d}+\frac{7 \sec (c+d x)}{a^3 d}+\frac{3 x}{a^3}",1,"(3*x)/a^3 + Cos[c + d*x]/(a^3*d) + (7*Sec[c + d*x])/(a^3*d) - (5*Sec[c + d*x]^3)/(a^3*d) + (13*Sec[c + d*x]^5)/(5*a^3*d) - (4*Sec[c + d*x]^7)/(7*a^3*d) - (3*Tan[c + d*x])/(a^3*d) + Tan[c + d*x]^3/(a^3*d) - (3*Tan[c + d*x]^5)/(5*a^3*d) + (4*Tan[c + d*x]^7)/(7*a^3*d)","A",16,10,29,0.3448,1,"{2875, 2873, 2607, 30, 2606, 194, 3473, 8, 2590, 270}"
788,1,142,0,0.3395149,"\int \frac{\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Sin[c + d*x]^3*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","-\frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{\tan ^5(c+d x)}{5 a^3 d}-\frac{\tan ^3(c+d x)}{3 a^3 d}+\frac{\tan (c+d x)}{a^3 d}+\frac{4 \sec ^7(c+d x)}{7 a^3 d}-\frac{11 \sec ^5(c+d x)}{5 a^3 d}+\frac{10 \sec ^3(c+d x)}{3 a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}-\frac{x}{a^3}","-\frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{\tan ^5(c+d x)}{5 a^3 d}-\frac{\tan ^3(c+d x)}{3 a^3 d}+\frac{\tan (c+d x)}{a^3 d}+\frac{4 \sec ^7(c+d x)}{7 a^3 d}-\frac{11 \sec ^5(c+d x)}{5 a^3 d}+\frac{10 \sec ^3(c+d x)}{3 a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}-\frac{x}{a^3}",1,"-(x/a^3) - (3*Sec[c + d*x])/(a^3*d) + (10*Sec[c + d*x]^3)/(3*a^3*d) - (11*Sec[c + d*x]^5)/(5*a^3*d) + (4*Sec[c + d*x]^7)/(7*a^3*d) + Tan[c + d*x]/(a^3*d) - Tan[c + d*x]^3/(3*a^3*d) + Tan[c + d*x]^5/(5*a^3*d) - (4*Tan[c + d*x]^7)/(7*a^3*d)","A",16,9,29,0.3103,1,"{2875, 2873, 2606, 270, 2607, 30, 194, 3473, 8}"
789,1,102,0,0.3286765,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{\tan ^5(c+d x)}{5 a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{9 \sec ^5(c+d x)}{5 a^3 d}-\frac{2 \sec ^3(c+d x)}{a^3 d}+\frac{\sec (c+d x)}{a^3 d}","\frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{\tan ^5(c+d x)}{5 a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{9 \sec ^5(c+d x)}{5 a^3 d}-\frac{2 \sec ^3(c+d x)}{a^3 d}+\frac{\sec (c+d x)}{a^3 d}",1,"Sec[c + d*x]/(a^3*d) - (2*Sec[c + d*x]^3)/(a^3*d) + (9*Sec[c + d*x]^5)/(5*a^3*d) - (4*Sec[c + d*x]^7)/(7*a^3*d) + Tan[c + d*x]^5/(5*a^3*d) + (4*Tan[c + d*x]^7)/(7*a^3*d)","A",14,8,29,0.2759,1,"{2875, 2873, 2607, 14, 2606, 270, 30, 194}"
790,1,88,0,0.3149144,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","-\frac{4 \tan ^7(c+d x)}{7 a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{4 \sec ^7(c+d x)}{7 a^3 d}-\frac{7 \sec ^5(c+d x)}{5 a^3 d}+\frac{\sec ^3(c+d x)}{a^3 d}","-\frac{4 \tan ^7(c+d x)}{7 a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{4 \sec ^7(c+d x)}{7 a^3 d}-\frac{7 \sec ^5(c+d x)}{5 a^3 d}+\frac{\sec ^3(c+d x)}{a^3 d}",1,"Sec[c + d*x]^3/(a^3*d) - (7*Sec[c + d*x]^5)/(5*a^3*d) + (4*Sec[c + d*x]^7)/(7*a^3*d) - (3*Tan[c + d*x]^5)/(5*a^3*d) - (4*Tan[c + d*x]^7)/(7*a^3*d)","A",14,7,27,0.2593,1,"{2875, 2873, 2606, 14, 2607, 270, 30}"
791,1,103,0,0.2433287,"\int \frac{\tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Tan[c + d*x]^2/(a + a*Sin[c + d*x])^3,x]","\frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{\tan ^5(c+d x)}{a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{\sec ^5(c+d x)}{a^3 d}-\frac{\sec ^3(c+d x)}{3 a^3 d}","\frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{\tan ^5(c+d x)}{a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{\sec ^5(c+d x)}{a^3 d}-\frac{\sec ^3(c+d x)}{3 a^3 d}",1,"-Sec[c + d*x]^3/(3*a^3*d) + Sec[c + d*x]^5/(a^3*d) - (4*Sec[c + d*x]^7)/(7*a^3*d) + Tan[c + d*x]^3/(3*a^3*d) + Tan[c + d*x]^5/(a^3*d) + (4*Tan[c + d*x]^7)/(7*a^3*d)","A",14,5,21,0.2381,1,"{2711, 2607, 270, 2606, 14}"
792,1,99,0,0.1419344,"\int \frac{\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{6 \tan (c+d x)}{35 a^3 d}-\frac{3 \sec (c+d x)}{35 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{3 \sec (c+d x)}{35 a d (a \sin (c+d x)+a)^2}+\frac{\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}","\frac{6 \tan (c+d x)}{35 a^3 d}-\frac{3 \sec (c+d x)}{35 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{3 \sec (c+d x)}{35 a d (a \sin (c+d x)+a)^2}+\frac{\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}",1,"Sec[c + d*x]/(7*d*(a + a*Sin[c + d*x])^3) - (3*Sec[c + d*x])/(35*a*d*(a + a*Sin[c + d*x])^2) - (3*Sec[c + d*x])/(35*d*(a^3 + a^3*Sin[c + d*x])) + (6*Tan[c + d*x])/(35*a^3*d)","A",5,4,25,0.1600,1,"{2859, 2672, 3767, 8}"
793,1,151,0,0.2933907,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","-\frac{4 \tan ^7(c+d x)}{7 a^3 d}-\frac{11 \tan ^5(c+d x)}{5 a^3 d}-\frac{10 \tan ^3(c+d x)}{3 a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}+\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{\sec ^5(c+d x)}{5 a^3 d}+\frac{\sec ^3(c+d x)}{3 a^3 d}+\frac{\sec (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}","-\frac{4 \tan ^7(c+d x)}{7 a^3 d}-\frac{11 \tan ^5(c+d x)}{5 a^3 d}-\frac{10 \tan ^3(c+d x)}{3 a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}+\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{\sec ^5(c+d x)}{5 a^3 d}+\frac{\sec ^3(c+d x)}{3 a^3 d}+\frac{\sec (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}",1,"-(ArcTanh[Cos[c + d*x]]/(a^3*d)) + Sec[c + d*x]/(a^3*d) + Sec[c + d*x]^3/(3*a^3*d) + Sec[c + d*x]^5/(5*a^3*d) + (4*Sec[c + d*x]^7)/(7*a^3*d) - (3*Tan[c + d*x])/(a^3*d) - (10*Tan[c + d*x]^3)/(3*a^3*d) - (11*Tan[c + d*x]^5)/(5*a^3*d) - (4*Tan[c + d*x]^7)/(7*a^3*d)","A",14,10,27,0.3704,1,"{2875, 2873, 3767, 2622, 302, 207, 2606, 30, 2607, 270}"
794,1,162,0,0.3470027,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{13 \tan ^5(c+d x)}{5 a^3 d}+\frac{5 \tan ^3(c+d x)}{a^3 d}+\frac{7 \tan (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \sec ^5(c+d x)}{5 a^3 d}-\frac{\sec ^3(c+d x)}{a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}","\frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{13 \tan ^5(c+d x)}{5 a^3 d}+\frac{5 \tan ^3(c+d x)}{a^3 d}+\frac{7 \tan (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \sec ^5(c+d x)}{5 a^3 d}-\frac{\sec ^3(c+d x)}{a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}",1,"(3*ArcTanh[Cos[c + d*x]])/(a^3*d) - Cot[c + d*x]/(a^3*d) - (3*Sec[c + d*x])/(a^3*d) - Sec[c + d*x]^3/(a^3*d) - (3*Sec[c + d*x]^5)/(5*a^3*d) - (4*Sec[c + d*x]^7)/(7*a^3*d) + (7*Tan[c + d*x])/(a^3*d) + (5*Tan[c + d*x]^3)/(a^3*d) + (13*Tan[c + d*x]^5)/(5*a^3*d) + (4*Tan[c + d*x]^7)/(7*a^3*d)","A",14,10,29,0.3448,1,"{2875, 2873, 3767, 2622, 302, 207, 2620, 270, 2606, 30}"
795,1,117,0,0.1345157,"\int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx","Int[Sin[c + d*x]^2*(a + a*Sin[c + d*x])*Tan[c + d*x]^4,x]","\frac{a \cos ^3(c+d x)}{3 d}-\frac{3 a \cos (c+d x)}{d}+\frac{5 a \tan ^3(c+d x)}{6 d}-\frac{5 a \tan (c+d x)}{2 d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{3 a \sec (c+d x)}{d}-\frac{a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{5 a x}{2}","\frac{a \cos ^3(c+d x)}{3 d}-\frac{3 a \cos (c+d x)}{d}+\frac{5 a \tan ^3(c+d x)}{6 d}-\frac{5 a \tan (c+d x)}{2 d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{3 a \sec (c+d x)}{d}-\frac{a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{5 a x}{2}",1,"(5*a*x)/2 - (3*a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) - (3*a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) - (5*a*Tan[c + d*x])/(2*d) + (5*a*Tan[c + d*x]^3)/(6*d) - (a*Sin[c + d*x]^2*Tan[c + d*x]^3)/(2*d)","A",9,7,27,0.2593,1,"{2838, 2591, 288, 302, 203, 2590, 270}"
796,1,101,0,0.1300548,"\int \sin (c+d x) (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx","Int[Sin[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^4,x]","-\frac{a \cos (c+d x)}{d}+\frac{5 a \tan ^3(c+d x)}{6 d}-\frac{5 a \tan (c+d x)}{2 d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{2 a \sec (c+d x)}{d}-\frac{a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{5 a x}{2}","-\frac{a \cos (c+d x)}{d}+\frac{5 a \tan ^3(c+d x)}{6 d}-\frac{5 a \tan (c+d x)}{2 d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{2 a \sec (c+d x)}{d}-\frac{a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{5 a x}{2}",1,"(5*a*x)/2 - (a*Cos[c + d*x])/d - (2*a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) - (5*a*Tan[c + d*x])/(2*d) + (5*a*Tan[c + d*x]^3)/(6*d) - (a*Sin[c + d*x]^2*Tan[c + d*x]^3)/(2*d)","A",9,7,25,0.2800,1,"{2838, 2590, 270, 2591, 288, 302, 203}"
797,1,72,0,0.0960119,"\int (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx","Int[(a + a*Sin[c + d*x])*Tan[c + d*x]^4,x]","-\frac{a \cos (c+d x)}{d}+\frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{2 a \sec (c+d x)}{d}+a x","-\frac{a \cos (c+d x)}{d}+\frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{2 a \sec (c+d x)}{d}+a x",1,"a*x - (a*Cos[c + d*x])/d - (2*a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)","A",8,5,19,0.2632,1,"{2710, 3473, 8, 2590, 270}"
798,1,60,0,0.0978578,"\int \sec (c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^3,x]","\frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{a \sec (c+d x)}{d}+a x","\frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{a \sec (c+d x)}{d}+a x",1,"a*x - (a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)","A",6,4,25,0.1600,1,"{2838, 2606, 3473, 8}"
799,1,45,0,0.1039105,"\int \sec ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{a \sec (c+d x)}{d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{a \sec (c+d x)}{d}",1,"-((a*Sec[c + d*x])/d) + (a*Sec[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^3)/(3*d)","A",5,4,27,0.1481,1,"{2838, 2607, 30, 2606}"
800,1,33,0,0.0838112,"\int \sec ^3(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])*Tan[c + d*x],x]","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \sec ^3(c+d x)}{3 d}",1,"(a*Sec[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^3)/(3*d)","A",5,4,25,0.1600,1,"{2838, 2606, 30, 2607}"
801,1,68,0,0.0874933,"\int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}+\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}+\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}",1,"-((a*ArcTanh[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) + (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)","A",7,5,25,0.2000,1,"{2838, 2622, 302, 207, 3767}"
802,1,81,0,0.1237523,"\int \csc ^2(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{a \tan ^3(c+d x)}{3 d}+\frac{2 a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}+\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{2 a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}+\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}",1,"-((a*ArcTanh[Cos[c + d*x]])/d) - (a*Cot[c + d*x])/d + (a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) + (2*a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)","A",8,6,27,0.2222,1,"{2838, 2620, 270, 2622, 302, 207}"
803,1,110,0,0.1364682,"\int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{a \tan ^3(c+d x)}{3 d}+\frac{2 a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{5 a \sec ^3(c+d x)}{6 d}+\frac{5 a \sec (c+d x)}{2 d}-\frac{5 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{2 a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{5 a \sec ^3(c+d x)}{6 d}+\frac{5 a \sec (c+d x)}{2 d}-\frac{5 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}",1,"(-5*a*ArcTanh[Cos[c + d*x]])/(2*d) - (a*Cot[c + d*x])/d + (5*a*Sec[c + d*x])/(2*d) + (5*a*Sec[c + d*x]^3)/(6*d) - (a*Csc[c + d*x]^2*Sec[c + d*x]^3)/(2*d) + (2*a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)","A",9,7,27,0.2593,1,"{2838, 2622, 288, 302, 207, 2620, 270}"
804,1,120,0,0.2053751,"\int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^4,x]","-\frac{16 a^2 \cos (c+d x)}{3 d}+\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac{8 a^2 \sin ^2(c+d x) \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^2 x}{2}","-\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{11 a^2 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{a^2 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{7 a^2 x}{2}",1,"(7*a^2*x)/2 - (16*a^2*Cos[c + d*x])/(3*d) - (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (8*a^2*Cos[c + d*x]*Sin[c + d*x]^2)/(3*d*(1 - Sin[c + d*x])) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(3*d*(a - a*Sin[c + d*x])^2)","A",4,4,21,0.1905,1,"{2708, 2765, 2977, 2734}"
805,1,86,0,0.245919,"\int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","-\frac{4 a^2 \cos (c+d x)}{3 d}+\frac{a^4 \sin ^2(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac{2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}+2 a^2 x","-\frac{4 a^2 \cos (c+d x)}{3 d}+\frac{a^4 \sin ^2(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac{2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}+2 a^2 x",1,"2*a^2*x - (4*a^2*Cos[c + d*x])/(3*d) - (2*a^2*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (a^4*Cos[c + d*x]*Sin[c + d*x]^2)/(3*d*(a - a*Sin[c + d*x])^2)","A",7,7,27,0.2593,1,"{2869, 2765, 2968, 3023, 12, 2735, 2648}"
806,1,63,0,0.2067847,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac{5 a^2 \cos (c+d x)}{3 d (1-\sin (c+d x))}+a^2 x","\frac{a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac{5 a^2 \cos (c+d x)}{3 d (1-\sin (c+d x))}+a^2 x",1,"a^2*x - (5*a^2*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) + (a^4*Cos[c + d*x])/(3*d*(a - a*Sin[c + d*x])^2)","A",4,4,29,0.1379,1,"{2869, 2758, 2735, 2648}"
807,1,60,0,0.0864827,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^2 \tan (c+d x) \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2*Tan[c + d*x],x]","-\frac{2 a^2 \tan (c+d x)}{3 d}-\frac{2 a^2 \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^2}{3 d}","-\frac{2 a^2 \tan (c+d x)}{3 d}-\frac{2 a^2 \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^2}{3 d}",1,"(-2*a^2*Sec[c + d*x])/(3*d) + (Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(3*d) - (2*a^2*Tan[c + d*x])/(3*d)","A",4,4,27,0.1481,1,"{2855, 2669, 3767, 8}"
808,1,73,0,0.1874614,"\int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{4 a^2 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}","\frac{a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{4 a^2 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}",1,"-((a^2*ArcTanh[Cos[c + d*x]])/d) + (4*a^2*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) + (a^4*Cos[c + d*x])/(3*d*(a - a*Sin[c + d*x])^2)","A",5,5,27,0.1852,1,"{2869, 2766, 2978, 12, 3770}"
809,1,87,0,0.2739095,"\int \csc ^2(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{10 a^2 \cot (c+d x)}{3 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^4 \cot (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{2 a^2 \cot (c+d x)}{d (1-\sin (c+d x))}","-\frac{10 a^2 \cot (c+d x)}{3 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^4 \cot (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{2 a^2 \cot (c+d x)}{d (1-\sin (c+d x))}",1,"(-2*a^2*ArcTanh[Cos[c + d*x]])/d - (10*a^2*Cot[c + d*x])/(3*d) + (2*a^2*Cot[c + d*x])/(d*(1 - Sin[c + d*x])) + (a^4*Cot[c + d*x])/(3*d*(a - a*Sin[c + d*x])^2)","A",7,7,29,0.2414,1,"{2869, 2766, 2978, 2748, 3767, 8, 3770}"
810,1,125,0,0.3079502,"\int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{16 a^2 \cot (c+d x)}{3 d}-\frac{7 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{a^4 \cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{8 a^2 \cot (c+d x) \csc (c+d x)}{3 d (1-\sin (c+d x))}","-\frac{16 a^2 \cot (c+d x)}{3 d}-\frac{7 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{7 a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{a^4 \cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{8 a^2 \cot (c+d x) \csc (c+d x)}{3 d (1-\sin (c+d x))}",1,"(-7*a^2*ArcTanh[Cos[c + d*x]])/(2*d) - (16*a^2*Cot[c + d*x])/(3*d) - (7*a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (8*a^2*Cot[c + d*x]*Csc[c + d*x])/(3*d*(1 - Sin[c + d*x])) + (a^4*Cot[c + d*x]*Csc[c + d*x])/(3*d*(a - a*Sin[c + d*x])^2)","A",8,8,29,0.2759,1,"{2869, 2766, 2978, 2748, 3768, 3770, 3767, 8}"
811,1,119,0,0.1937877,"\int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^4,x]","\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{6 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{25 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{17 a^3 x}{2}","\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{6 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{25 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{17 a^3 x}{2}",1,"(17*a^3*x)/2 - (6*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (25*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",10,7,21,0.3333,1,"{2709, 2650, 2648, 2638, 2635, 8, 2633}"
812,1,101,0,0.1622591,"\int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^3,x]","-\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{19 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{11 a^3 x}{2}","-\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{19 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{11 a^3 x}{2}",1,"(11*a^3*x)/2 - (3*a^3*Cos[c + d*x])/d + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (19*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",8,6,27,0.2222,1,"{2872, 2650, 2648, 2638, 2635, 8}"
813,1,77,0,0.218969,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","-\frac{3 a^3 \cos (c+d x)}{d}-\frac{2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}+3 a^3 x+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}","-\frac{3 a^3 \cos (c+d x)}{d}-\frac{2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}+3 a^3 x+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}",1,"3*a^3*x - (3*a^3*Cos[c + d*x])/d - (2*a^5*Cos[c + d*x]^3)/(d*(a - a*Sin[c + d*x])^2) + (Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(3*d)","A",5,5,29,0.1724,1,"{2871, 2670, 2680, 2682, 8}"
814,1,64,0,0.1388112,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]","-\frac{2 a^5 \cos (c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)}+a^3 x+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}","-\frac{2 a^5 \cos (c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)}+a^3 x+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}",1,"a^3*x + (Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(3*d) - (2*a^5*Cos[c + d*x])/(d*(a^2 - a^2*Sin[c + d*x]))","A",4,4,27,0.1481,1,"{2855, 2670, 2680, 8}"
815,1,72,0,0.1407257,"\int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{5 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}","\frac{5 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}",1,"-((a^3*ArcTanh[Cos[c + d*x]])/d) + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) + (5*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x]))","A",6,4,27,0.1481,1,"{2872, 3770, 2650, 2648}"
816,1,86,0,0.1699139,"\int \csc ^2(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot (c+d x)}{d}+\frac{11 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}","-\frac{a^3 \cot (c+d x)}{d}+\frac{11 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}",1,"(-3*a^3*ArcTanh[Cos[c + d*x]])/d - (a^3*Cot[c + d*x])/d + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) + (11*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x]))","A",8,6,29,0.2069,1,"{2872, 3770, 3767, 8, 2650, 2648}"
817,1,110,0,0.1903619,"\int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cot (c+d x)}{d}+\frac{17 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{11 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}","-\frac{3 a^3 \cot (c+d x)}{d}+\frac{17 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{11 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}",1,"(-11*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) + (17*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x]))","A",10,7,29,0.2414,1,"{2872, 3770, 3767, 8, 3768, 2650, 2648}"
818,1,128,0,0.2060431,"\int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^4*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{6 a^3 \cot (c+d x)}{d}+\frac{23 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{17 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}","-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{6 a^3 \cot (c+d x)}{d}+\frac{23 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{17 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}",1,"(-17*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (6*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) + (23*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x]))","A",12,7,29,0.2414,1,"{2872, 3770, 3767, 8, 3768, 2650, 2648}"
819,1,143,0,0.2037447,"\int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^4,x]","\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{16 a^4 \cos (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{35 a^4 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{56 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{163 a^4 x}{8}","\frac{4 a^4 \cos ^3(c+d x)}{3 d}-\frac{16 a^4 \cos (c+d x)}{d}-\frac{a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{35 a^4 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{56 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{163 a^4 x}{8}",1,"(163*a^4*x)/8 - (16*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) + (4*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (56*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (35*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",13,7,21,0.3333,1,"{2709, 2650, 2648, 2638, 2635, 8, 2633}"
820,1,101,0,0.1633202,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^4*Tan[c + d*x]^2,x]","-\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{17 a^4 x}{2}","-\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{17 a^4 x}{2}",1,"(17*a^4*x)/2 - (4*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (32*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",8,6,29,0.2069,1,"{2872, 2650, 2648, 2638, 2635, 8}"
821,1,117,0,0.1583518,"\int \frac{\sin ^2(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\cos (c+d x)}{a d}+\frac{\tan ^5(c+d x)}{5 a d}-\frac{\tan ^3(c+d x)}{3 a d}+\frac{\tan (c+d x)}{a d}-\frac{\sec ^5(c+d x)}{5 a d}+\frac{\sec ^3(c+d x)}{a d}-\frac{3 \sec (c+d x)}{a d}-\frac{x}{a}","-\frac{\cos (c+d x)}{a d}+\frac{\tan ^5(c+d x)}{5 a d}-\frac{\tan ^3(c+d x)}{3 a d}+\frac{\tan (c+d x)}{a d}-\frac{\sec ^5(c+d x)}{5 a d}+\frac{\sec ^3(c+d x)}{a d}-\frac{3 \sec (c+d x)}{a d}-\frac{x}{a}",1,"-(x/a) - Cos[c + d*x]/(a*d) - (3*Sec[c + d*x])/(a*d) + Sec[c + d*x]^3/(a*d) - Sec[c + d*x]^5/(5*a*d) + Tan[c + d*x]/(a*d) - Tan[c + d*x]^3/(3*a*d) + Tan[c + d*x]^5/(5*a*d)","A",8,5,29,0.1724,1,"{2839, 3473, 8, 2590, 270}"
822,1,105,0,0.1291491,"\int \frac{\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\tan ^5(c+d x)}{5 a d}+\frac{\tan ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}+\frac{\sec ^5(c+d x)}{5 a d}-\frac{2 \sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}+\frac{x}{a}","-\frac{\tan ^5(c+d x)}{5 a d}+\frac{\tan ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}+\frac{\sec ^5(c+d x)}{5 a d}-\frac{2 \sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}+\frac{x}{a}",1,"x/a + Sec[c + d*x]/(a*d) - (2*Sec[c + d*x]^3)/(3*a*d) + Sec[c + d*x]^5/(5*a*d) - Tan[c + d*x]/(a*d) + Tan[c + d*x]^3/(3*a*d) - Tan[c + d*x]^5/(5*a*d)","A",8,5,27,0.1852,1,"{2839, 2606, 194, 3473, 8}"
823,1,69,0,0.0956588,"\int \frac{\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Tan[c + d*x]^4/(a + a*Sin[c + d*x]),x]","\frac{\tan ^5(c+d x)}{5 a d}-\frac{\sec ^5(c+d x)}{5 a d}+\frac{2 \sec ^3(c+d x)}{3 a d}-\frac{\sec (c+d x)}{a d}","\frac{\tan ^5(c+d x)}{5 a d}-\frac{\sec ^5(c+d x)}{5 a d}+\frac{2 \sec ^3(c+d x)}{3 a d}-\frac{\sec (c+d x)}{a d}",1,"-(Sec[c + d*x]/(a*d)) + (2*Sec[c + d*x]^3)/(3*a*d) - Sec[c + d*x]^5/(5*a*d) + Tan[c + d*x]^5/(5*a*d)","A",6,5,21,0.2381,1,"{2706, 2607, 30, 2606, 194}"
824,1,55,0,0.1384808,"\int \frac{\sec (c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{\tan ^5(c+d x)}{5 a d}+\frac{\sec ^5(c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{3 a d}","-\frac{\tan ^5(c+d x)}{5 a d}+\frac{\sec ^5(c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{3 a d}",1,"-Sec[c + d*x]^3/(3*a*d) + Sec[c + d*x]^5/(5*a*d) - Tan[c + d*x]^5/(5*a*d)","A",6,5,27,0.1852,1,"{2839, 2606, 14, 2607, 30}"
825,1,73,0,0.1599387,"\int \frac{\sec ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\tan ^5(c+d x)}{5 a d}+\frac{\tan ^3(c+d x)}{3 a d}-\frac{\sec ^5(c+d x)}{5 a d}+\frac{\sec ^3(c+d x)}{3 a d}","\frac{\tan ^5(c+d x)}{5 a d}+\frac{\tan ^3(c+d x)}{3 a d}-\frac{\sec ^5(c+d x)}{5 a d}+\frac{\sec ^3(c+d x)}{3 a d}",1,"Sec[c + d*x]^3/(3*a*d) - Sec[c + d*x]^5/(5*a*d) + Tan[c + d*x]^3/(3*a*d) + Tan[c + d*x]^5/(5*a*d)","A",7,4,29,0.1379,1,"{2839, 2607, 14, 2606}"
826,1,55,0,0.1079345,"\int \frac{\sec ^3(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^3*Tan[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\tan ^5(c+d x)}{5 a d}-\frac{\tan ^3(c+d x)}{3 a d}+\frac{\sec ^5(c+d x)}{5 a d}","-\frac{\tan ^5(c+d x)}{5 a d}-\frac{\tan ^3(c+d x)}{3 a d}+\frac{\sec ^5(c+d x)}{5 a d}",1,"Sec[c + d*x]^5/(5*a*d) - Tan[c + d*x]^3/(3*a*d) - Tan[c + d*x]^5/(5*a*d)","A",6,5,27,0.1852,1,"{2839, 2606, 30, 2607, 14}"
827,1,115,0,0.124173,"\int \frac{\csc (c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\tan ^5(c+d x)}{5 a d}-\frac{2 \tan ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}+\frac{\sec ^5(c+d x)}{5 a d}+\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}","-\frac{\tan ^5(c+d x)}{5 a d}-\frac{2 \tan ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}+\frac{\sec ^5(c+d x)}{5 a d}+\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"-(ArcTanh[Cos[c + d*x]]/(a*d)) + Sec[c + d*x]/(a*d) + Sec[c + d*x]^3/(3*a*d) + Sec[c + d*x]^5/(5*a*d) - Tan[c + d*x]/(a*d) - (2*Tan[c + d*x]^3)/(3*a*d) - Tan[c + d*x]^5/(5*a*d)","A",7,5,27,0.1852,1,"{2839, 2622, 302, 207, 3767}"
828,1,126,0,0.1660083,"\int \frac{\csc ^2(c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\tan ^5(c+d x)}{5 a d}+\frac{\tan ^3(c+d x)}{a d}+\frac{3 \tan (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}-\frac{\sec ^5(c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{3 a d}-\frac{\sec (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}","\frac{\tan ^5(c+d x)}{5 a d}+\frac{\tan ^3(c+d x)}{a d}+\frac{3 \tan (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}-\frac{\sec ^5(c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{3 a d}-\frac{\sec (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"ArcTanh[Cos[c + d*x]]/(a*d) - Cot[c + d*x]/(a*d) - Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(3*a*d) - Sec[c + d*x]^5/(5*a*d) + (3*Tan[c + d*x])/(a*d) + Tan[c + d*x]^3/(a*d) + Tan[c + d*x]^5/(5*a*d)","A",8,6,29,0.2069,1,"{2839, 2620, 270, 2622, 302, 207}"
829,1,155,0,0.2897519,"\int \frac{\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Sin[c + d*x]^3*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{\cos (c+d x)}{a^2 d}-\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{2 \tan ^5(c+d x)}{5 a^2 d}-\frac{2 \tan ^3(c+d x)}{3 a^2 d}+\frac{2 \tan (c+d x)}{a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{7 \sec ^5(c+d x)}{5 a^2 d}+\frac{3 \sec ^3(c+d x)}{a^2 d}-\frac{5 \sec (c+d x)}{a^2 d}-\frac{2 x}{a^2}","-\frac{\cos (c+d x)}{a^2 d}-\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{2 \tan ^5(c+d x)}{5 a^2 d}-\frac{2 \tan ^3(c+d x)}{3 a^2 d}+\frac{2 \tan (c+d x)}{a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{7 \sec ^5(c+d x)}{5 a^2 d}+\frac{3 \sec ^3(c+d x)}{a^2 d}-\frac{5 \sec (c+d x)}{a^2 d}-\frac{2 x}{a^2}",1,"(-2*x)/a^2 - Cos[c + d*x]/(a^2*d) - (5*Sec[c + d*x])/(a^2*d) + (3*Sec[c + d*x]^3)/(a^2*d) - (7*Sec[c + d*x]^5)/(5*a^2*d) + (2*Sec[c + d*x]^7)/(7*a^2*d) + (2*Tan[c + d*x])/(a^2*d) - (2*Tan[c + d*x]^3)/(3*a^2*d) + (2*Tan[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x]^7)/(7*a^2*d)","A",14,8,29,0.2759,1,"{2875, 2873, 2606, 194, 3473, 8, 2590, 270}"
830,1,140,0,0.3001877,"\int \frac{\sin ^2(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \tan ^7(c+d x)}{7 a^2 d}-\frac{\tan ^5(c+d x)}{5 a^2 d}+\frac{\tan ^3(c+d x)}{3 a^2 d}-\frac{\tan (c+d x)}{a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{6 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{a^2 d}+\frac{2 \sec (c+d x)}{a^2 d}+\frac{x}{a^2}","\frac{2 \tan ^7(c+d x)}{7 a^2 d}-\frac{\tan ^5(c+d x)}{5 a^2 d}+\frac{\tan ^3(c+d x)}{3 a^2 d}-\frac{\tan (c+d x)}{a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{6 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{a^2 d}+\frac{2 \sec (c+d x)}{a^2 d}+\frac{x}{a^2}",1,"x/a^2 + (2*Sec[c + d*x])/(a^2*d) - (2*Sec[c + d*x]^3)/(a^2*d) + (6*Sec[c + d*x]^5)/(5*a^2*d) - (2*Sec[c + d*x]^7)/(7*a^2*d) - Tan[c + d*x]/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d) - Tan[c + d*x]^5/(5*a^2*d) + (2*Tan[c + d*x]^7)/(7*a^2*d)","A",13,8,29,0.2759,1,"{2875, 2873, 2607, 30, 2606, 194, 3473, 8}"
831,1,85,0,0.2715212,"\int \frac{\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{\sec ^5(c+d x)}{a^2 d}+\frac{4 \sec ^3(c+d x)}{3 a^2 d}-\frac{\sec (c+d x)}{a^2 d}","-\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{\sec ^5(c+d x)}{a^2 d}+\frac{4 \sec ^3(c+d x)}{3 a^2 d}-\frac{\sec (c+d x)}{a^2 d}",1,"-(Sec[c + d*x]/(a^2*d)) + (4*Sec[c + d*x]^3)/(3*a^2*d) - Sec[c + d*x]^5/(a^2*d) + (2*Sec[c + d*x]^7)/(7*a^2*d) - (2*Tan[c + d*x]^7)/(7*a^2*d)","A",11,7,27,0.2593,1,"{2875, 2873, 2606, 270, 2607, 30, 194}"
832,1,91,0,0.1566135,"\int \frac{\tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]^4/(a + a*Sin[c + d*x])^2,x]","\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{\tan ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{4 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}","\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{\tan ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{4 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}",1,"(-2*Sec[c + d*x]^3)/(3*a^2*d) + (4*Sec[c + d*x]^5)/(5*a^2*d) - (2*Sec[c + d*x]^7)/(7*a^2*d) + Tan[c + d*x]^5/(5*a^2*d) + (2*Tan[c + d*x]^7)/(7*a^2*d)","A",10,6,21,0.2857,1,"{2711, 2607, 14, 2606, 270, 30}"
833,1,91,0,0.288258,"\int \frac{\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \tan ^7(c+d x)}{7 a^2 d}-\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{3 \sec ^5(c+d x)}{5 a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}","-\frac{2 \tan ^7(c+d x)}{7 a^2 d}-\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{3 \sec ^5(c+d x)}{5 a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}",1,"Sec[c + d*x]^3/(3*a^2*d) - (3*Sec[c + d*x]^5)/(5*a^2*d) + (2*Sec[c + d*x]^7)/(7*a^2*d) - (2*Tan[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x]^7)/(7*a^2*d)","A",12,6,27,0.2222,1,"{2875, 2873, 2606, 14, 2607, 270}"
834,1,91,0,0.3057741,"\int \frac{\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{3 \tan ^5(c+d x)}{5 a^2 d}+\frac{\tan ^3(c+d x)}{3 a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{2 \sec ^5(c+d x)}{5 a^2 d}","\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{3 \tan ^5(c+d x)}{5 a^2 d}+\frac{\tan ^3(c+d x)}{3 a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{2 \sec ^5(c+d x)}{5 a^2 d}",1,"(2*Sec[c + d*x]^5)/(5*a^2*d) - (2*Sec[c + d*x]^7)/(7*a^2*d) + Tan[c + d*x]^3/(3*a^2*d) + (3*Tan[c + d*x]^5)/(5*a^2*d) + (2*Tan[c + d*x]^7)/(7*a^2*d)","A",12,6,29,0.2069,1,"{2875, 2873, 2607, 270, 2606, 14}"
835,1,93,0,0.1154501,"\int \frac{\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]^3*Tan[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{8 \tan ^3(c+d x)}{105 a^2 d}+\frac{8 \tan (c+d x)}{35 a^2 d}-\frac{2 \sec ^3(c+d x)}{35 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2}","\frac{8 \tan ^3(c+d x)}{105 a^2 d}+\frac{8 \tan (c+d x)}{35 a^2 d}-\frac{2 \sec ^3(c+d x)}{35 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2}",1,"Sec[c + d*x]^3/(7*d*(a + a*Sin[c + d*x])^2) - (2*Sec[c + d*x]^3)/(35*d*(a^2 + a^2*Sin[c + d*x])) + (8*Tan[c + d*x])/(35*a^2*d) + (8*Tan[c + d*x]^3)/(105*a^2*d)","A",4,3,27,0.1111,1,"{2859, 2672, 3767}"
836,1,149,0,0.252569,"\int \frac{\csc (c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \tan ^7(c+d x)}{7 a^2 d}-\frac{6 \tan ^5(c+d x)}{5 a^2 d}-\frac{2 \tan ^3(c+d x)}{a^2 d}-\frac{2 \tan (c+d x)}{a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{\sec ^5(c+d x)}{5 a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}+\frac{\sec (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}","-\frac{2 \tan ^7(c+d x)}{7 a^2 d}-\frac{6 \tan ^5(c+d x)}{5 a^2 d}-\frac{2 \tan ^3(c+d x)}{a^2 d}-\frac{2 \tan (c+d x)}{a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{\sec ^5(c+d x)}{5 a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}+\frac{\sec (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}",1,"-(ArcTanh[Cos[c + d*x]]/(a^2*d)) + Sec[c + d*x]/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d) + Sec[c + d*x]^5/(5*a^2*d) + (2*Sec[c + d*x]^7)/(7*a^2*d) - (2*Tan[c + d*x])/(a^2*d) - (2*Tan[c + d*x]^3)/(a^2*d) - (6*Tan[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x]^7)/(7*a^2*d)","A",11,8,27,0.2963,1,"{2875, 2873, 3767, 2622, 302, 207, 2606, 30}"
837,1,164,0,0.3299885,"\int \frac{\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{7 \tan ^5(c+d x)}{5 a^2 d}+\frac{3 \tan ^3(c+d x)}{a^2 d}+\frac{5 \tan (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}","\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{7 \tan ^5(c+d x)}{5 a^2 d}+\frac{3 \tan ^3(c+d x)}{a^2 d}+\frac{5 \tan (c+d x)}{a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}+\frac{2 \tanh ^{-1}(\cos (c+d x))}{a^2 d}",1,"(2*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a^2*d) - (2*Sec[c + d*x])/(a^2*d) - (2*Sec[c + d*x]^3)/(3*a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) - (2*Sec[c + d*x]^7)/(7*a^2*d) + (5*Tan[c + d*x])/(a^2*d) + (3*Tan[c + d*x]^3)/(a^2*d) + (7*Tan[c + d*x]^5)/(5*a^2*d) + (2*Tan[c + d*x]^7)/(7*a^2*d)","A",12,8,29,0.2759,1,"{2875, 2873, 3767, 2622, 302, 207, 2620, 270}"
838,1,194,0,0.3656038,"\int \frac{\csc ^3(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[(Csc[c + d*x]^3*Sec[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \tan ^7(c+d x)}{7 a^2 d}-\frac{8 \tan ^5(c+d x)}{5 a^2 d}-\frac{4 \tan ^3(c+d x)}{a^2 d}-\frac{8 \tan (c+d x)}{a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{11 \sec ^7(c+d x)}{14 a^2 d}+\frac{11 \sec ^5(c+d x)}{10 a^2 d}+\frac{11 \sec ^3(c+d x)}{6 a^2 d}+\frac{11 \sec (c+d x)}{2 a^2 d}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\csc ^2(c+d x) \sec ^7(c+d x)}{2 a^2 d}","-\frac{2 \tan ^7(c+d x)}{7 a^2 d}-\frac{8 \tan ^5(c+d x)}{5 a^2 d}-\frac{4 \tan ^3(c+d x)}{a^2 d}-\frac{8 \tan (c+d x)}{a^2 d}+\frac{2 \cot (c+d x)}{a^2 d}+\frac{11 \sec ^7(c+d x)}{14 a^2 d}+\frac{11 \sec ^5(c+d x)}{10 a^2 d}+\frac{11 \sec ^3(c+d x)}{6 a^2 d}+\frac{11 \sec (c+d x)}{2 a^2 d}-\frac{11 \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\csc ^2(c+d x) \sec ^7(c+d x)}{2 a^2 d}",1,"(-11*ArcTanh[Cos[c + d*x]])/(2*a^2*d) + (2*Cot[c + d*x])/(a^2*d) + (11*Sec[c + d*x])/(2*a^2*d) + (11*Sec[c + d*x]^3)/(6*a^2*d) + (11*Sec[c + d*x]^5)/(10*a^2*d) + (11*Sec[c + d*x]^7)/(14*a^2*d) - (Csc[c + d*x]^2*Sec[c + d*x]^7)/(2*a^2*d) - (8*Tan[c + d*x])/(a^2*d) - (4*Tan[c + d*x]^3)/(a^2*d) - (8*Tan[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x]^7)/(7*a^2*d)","A",15,8,29,0.2759,1,"{2875, 2873, 2622, 302, 207, 2620, 270, 288}"
839,1,178,0,0.3675967,"\int \frac{\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Sin[c + d*x]^3*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","-\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan (c+d x)}{a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{15 \sec ^7(c+d x)}{7 a^3 d}+\frac{21 \sec ^5(c+d x)}{5 a^3 d}-\frac{13 \sec ^3(c+d x)}{3 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{x}{a^3}","-\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan (c+d x)}{a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{15 \sec ^7(c+d x)}{7 a^3 d}+\frac{21 \sec ^5(c+d x)}{5 a^3 d}-\frac{13 \sec ^3(c+d x)}{3 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{x}{a^3}",1,"x/a^3 + (3*Sec[c + d*x])/(a^3*d) - (13*Sec[c + d*x]^3)/(3*a^3*d) + (21*Sec[c + d*x]^5)/(5*a^3*d) - (15*Sec[c + d*x]^7)/(7*a^3*d) + (4*Sec[c + d*x]^9)/(9*a^3*d) - Tan[c + d*x]/(a^3*d) + Tan[c + d*x]^3/(3*a^3*d) - Tan[c + d*x]^5/(5*a^3*d) + Tan[c + d*x]^7/(7*a^3*d) - (4*Tan[c + d*x]^9)/(9*a^3*d)","A",17,9,29,0.3103,1,"{2875, 2873, 2606, 270, 2607, 30, 194, 3473, 8}"
840,1,121,0,0.3467451,"\int \frac{\sin ^2(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{13 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \sec ^5(c+d x)}{a^3 d}+\frac{7 \sec ^3(c+d x)}{3 a^3 d}-\frac{\sec (c+d x)}{a^3 d}","\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{13 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \sec ^5(c+d x)}{a^3 d}+\frac{7 \sec ^3(c+d x)}{3 a^3 d}-\frac{\sec (c+d x)}{a^3 d}",1,"-(Sec[c + d*x]/(a^3*d)) + (7*Sec[c + d*x]^3)/(3*a^3*d) - (3*Sec[c + d*x]^5)/(a^3*d) + (13*Sec[c + d*x]^7)/(7*a^3*d) - (4*Sec[c + d*x]^9)/(9*a^3*d) + Tan[c + d*x]^7/(7*a^3*d) + (4*Tan[c + d*x]^9)/(9*a^3*d)","A",14,8,29,0.2759,1,"{2875, 2873, 2607, 14, 2606, 270, 30, 194}"
841,1,105,0,0.338446,"\int \frac{\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","-\frac{4 \tan ^9(c+d x)}{9 a^3 d}-\frac{3 \tan ^7(c+d x)}{7 a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{11 \sec ^7(c+d x)}{7 a^3 d}+\frac{2 \sec ^5(c+d x)}{a^3 d}-\frac{\sec ^3(c+d x)}{a^3 d}","-\frac{4 \tan ^9(c+d x)}{9 a^3 d}-\frac{3 \tan ^7(c+d x)}{7 a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{11 \sec ^7(c+d x)}{7 a^3 d}+\frac{2 \sec ^5(c+d x)}{a^3 d}-\frac{\sec ^3(c+d x)}{a^3 d}",1,"-(Sec[c + d*x]^3/(a^3*d)) + (2*Sec[c + d*x]^5)/(a^3*d) - (11*Sec[c + d*x]^7)/(7*a^3*d) + (4*Sec[c + d*x]^9)/(9*a^3*d) - (3*Tan[c + d*x]^7)/(7*a^3*d) - (4*Tan[c + d*x]^9)/(9*a^3*d)","A",14,7,27,0.2593,1,"{2875, 2873, 2606, 270, 2607, 14, 30}"
842,1,127,0,0.2232191,"\int \frac{\tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Tan[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]","\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{5 \tan ^7(c+d x)}{7 a^3 d}+\frac{\tan ^5(c+d x)}{5 a^3 d}-\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{9 \sec ^7(c+d x)}{7 a^3 d}-\frac{6 \sec ^5(c+d x)}{5 a^3 d}+\frac{\sec ^3(c+d x)}{3 a^3 d}","\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{5 \tan ^7(c+d x)}{7 a^3 d}+\frac{\tan ^5(c+d x)}{5 a^3 d}-\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{9 \sec ^7(c+d x)}{7 a^3 d}-\frac{6 \sec ^5(c+d x)}{5 a^3 d}+\frac{\sec ^3(c+d x)}{3 a^3 d}",1,"Sec[c + d*x]^3/(3*a^3*d) - (6*Sec[c + d*x]^5)/(5*a^3*d) + (9*Sec[c + d*x]^7)/(7*a^3*d) - (4*Sec[c + d*x]^9)/(9*a^3*d) + Tan[c + d*x]^5/(5*a^3*d) + (5*Tan[c + d*x]^7)/(7*a^3*d) + (4*Tan[c + d*x]^9)/(9*a^3*d)","A",14,5,21,0.2381,1,"{2711, 2607, 270, 2606, 14}"
843,1,105,0,0.3353795,"\int \frac{\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","-\frac{4 \tan ^9(c+d x)}{9 a^3 d}-\frac{\tan ^7(c+d x)}{a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{\sec ^7(c+d x)}{a^3 d}+\frac{3 \sec ^5(c+d x)}{5 a^3 d}","-\frac{4 \tan ^9(c+d x)}{9 a^3 d}-\frac{\tan ^7(c+d x)}{a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{\sec ^7(c+d x)}{a^3 d}+\frac{3 \sec ^5(c+d x)}{5 a^3 d}",1,"(3*Sec[c + d*x]^5)/(5*a^3*d) - Sec[c + d*x]^7/(a^3*d) + (4*Sec[c + d*x]^9)/(9*a^3*d) - (3*Tan[c + d*x]^5)/(5*a^3*d) - Tan[c + d*x]^7/(a^3*d) - (4*Tan[c + d*x]^9)/(9*a^3*d)","A",15,6,27,0.2222,1,"{2875, 2873, 2606, 14, 2607, 270}"
844,1,127,0,0.362324,"\int \frac{\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Sec[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{9 \tan ^7(c+d x)}{7 a^3 d}+\frac{6 \tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{5 \sec ^7(c+d x)}{7 a^3 d}-\frac{\sec ^5(c+d x)}{5 a^3 d}","\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{9 \tan ^7(c+d x)}{7 a^3 d}+\frac{6 \tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{5 \sec ^7(c+d x)}{7 a^3 d}-\frac{\sec ^5(c+d x)}{5 a^3 d}",1,"-Sec[c + d*x]^5/(5*a^3*d) + (5*Sec[c + d*x]^7)/(7*a^3*d) - (4*Sec[c + d*x]^9)/(9*a^3*d) + Tan[c + d*x]^3/(3*a^3*d) + (6*Tan[c + d*x]^5)/(5*a^3*d) + (9*Tan[c + d*x]^7)/(7*a^3*d) + (4*Tan[c + d*x]^9)/(9*a^3*d)","A",15,6,29,0.2069,1,"{2875, 2873, 2607, 270, 2606, 14}"
845,1,123,0,0.1609073,"\int \frac{\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Sec[c + d*x]^3*Tan[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{4 \tan ^3(c+d x)}{63 a^3 d}+\frac{4 \tan (c+d x)}{21 a^3 d}-\frac{\sec ^3(c+d x)}{21 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}+\frac{\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}","\frac{4 \tan ^3(c+d x)}{63 a^3 d}+\frac{4 \tan (c+d x)}{21 a^3 d}-\frac{\sec ^3(c+d x)}{21 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}+\frac{\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}",1,"Sec[c + d*x]^3/(9*d*(a + a*Sin[c + d*x])^3) - Sec[c + d*x]^3/(21*a*d*(a + a*Sin[c + d*x])^2) - Sec[c + d*x]^3/(21*d*(a^3 + a^3*Sin[c + d*x])) + (4*Tan[c + d*x])/(21*a^3*d) + (4*Tan[c + d*x]^3)/(63*a^3*d)","A",5,3,27,0.1111,1,"{2859, 2672, 3767}"
846,1,187,0,0.3601076,"\int \frac{\csc (c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","-\frac{4 \tan ^9(c+d x)}{9 a^3 d}-\frac{15 \tan ^7(c+d x)}{7 a^3 d}-\frac{21 \tan ^5(c+d x)}{5 a^3 d}-\frac{13 \tan ^3(c+d x)}{3 a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{\sec ^7(c+d x)}{7 a^3 d}+\frac{\sec ^5(c+d x)}{5 a^3 d}+\frac{\sec ^3(c+d x)}{3 a^3 d}+\frac{\sec (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}","-\frac{4 \tan ^9(c+d x)}{9 a^3 d}-\frac{15 \tan ^7(c+d x)}{7 a^3 d}-\frac{21 \tan ^5(c+d x)}{5 a^3 d}-\frac{13 \tan ^3(c+d x)}{3 a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{\sec ^7(c+d x)}{7 a^3 d}+\frac{\sec ^5(c+d x)}{5 a^3 d}+\frac{\sec ^3(c+d x)}{3 a^3 d}+\frac{\sec (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}",1,"-(ArcTanh[Cos[c + d*x]]/(a^3*d)) + Sec[c + d*x]/(a^3*d) + Sec[c + d*x]^3/(3*a^3*d) + Sec[c + d*x]^5/(5*a^3*d) + Sec[c + d*x]^7/(7*a^3*d) + (4*Sec[c + d*x]^9)/(9*a^3*d) - (3*Tan[c + d*x])/(a^3*d) - (13*Tan[c + d*x]^3)/(3*a^3*d) - (21*Tan[c + d*x]^5)/(5*a^3*d) - (15*Tan[c + d*x]^7)/(7*a^3*d) - (4*Tan[c + d*x]^9)/(9*a^3*d)","A",14,10,27,0.3704,1,"{2875, 2873, 3767, 2622, 302, 207, 2606, 30, 2607, 270}"
847,1,200,0,0.3934189,"\int \frac{\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{17 \tan ^7(c+d x)}{7 a^3 d}+\frac{28 \tan ^5(c+d x)}{5 a^3 d}+\frac{22 \tan ^3(c+d x)}{3 a^3 d}+\frac{8 \tan (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{3 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \sec ^5(c+d x)}{5 a^3 d}-\frac{\sec ^3(c+d x)}{a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}","\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{17 \tan ^7(c+d x)}{7 a^3 d}+\frac{28 \tan ^5(c+d x)}{5 a^3 d}+\frac{22 \tan ^3(c+d x)}{3 a^3 d}+\frac{8 \tan (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}-\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{3 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \sec ^5(c+d x)}{5 a^3 d}-\frac{\sec ^3(c+d x)}{a^3 d}-\frac{3 \sec (c+d x)}{a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}",1,"(3*ArcTanh[Cos[c + d*x]])/(a^3*d) - Cot[c + d*x]/(a^3*d) - (3*Sec[c + d*x])/(a^3*d) - Sec[c + d*x]^3/(a^3*d) - (3*Sec[c + d*x]^5)/(5*a^3*d) - (3*Sec[c + d*x]^7)/(7*a^3*d) - (4*Sec[c + d*x]^9)/(9*a^3*d) + (8*Tan[c + d*x])/(a^3*d) + (22*Tan[c + d*x]^3)/(3*a^3*d) + (28*Tan[c + d*x]^5)/(5*a^3*d) + (17*Tan[c + d*x]^7)/(7*a^3*d) + (4*Tan[c + d*x]^9)/(9*a^3*d)","A",14,10,29,0.3448,1,"{2875, 2873, 3767, 2622, 302, 207, 2620, 270, 2606, 30}"
848,1,145,0,0.3132902,"\int \frac{\tan ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[Tan[c + d*x]^4/(a + a*Sin[c + d*x])^4,x]","\frac{8 \tan ^{11}(c+d x)}{11 a^4 d}+\frac{16 \tan ^9(c+d x)}{9 a^4 d}+\frac{9 \tan ^7(c+d x)}{7 a^4 d}+\frac{\tan ^5(c+d x)}{5 a^4 d}-\frac{8 \sec ^{11}(c+d x)}{11 a^4 d}+\frac{20 \sec ^9(c+d x)}{9 a^4 d}-\frac{16 \sec ^7(c+d x)}{7 a^4 d}+\frac{4 \sec ^5(c+d x)}{5 a^4 d}","\frac{8 \tan ^{11}(c+d x)}{11 a^4 d}+\frac{16 \tan ^9(c+d x)}{9 a^4 d}+\frac{9 \tan ^7(c+d x)}{7 a^4 d}+\frac{\tan ^5(c+d x)}{5 a^4 d}-\frac{8 \sec ^{11}(c+d x)}{11 a^4 d}+\frac{20 \sec ^9(c+d x)}{9 a^4 d}-\frac{16 \sec ^7(c+d x)}{7 a^4 d}+\frac{4 \sec ^5(c+d x)}{5 a^4 d}",1,"(4*Sec[c + d*x]^5)/(5*a^4*d) - (16*Sec[c + d*x]^7)/(7*a^4*d) + (20*Sec[c + d*x]^9)/(9*a^4*d) - (8*Sec[c + d*x]^11)/(11*a^4*d) + Tan[c + d*x]^5/(5*a^4*d) + (9*Tan[c + d*x]^7)/(7*a^4*d) + (16*Tan[c + d*x]^9)/(9*a^4*d) + (8*Tan[c + d*x]^11)/(11*a^4*d)","A",17,5,21,0.2381,1,"{2711, 2607, 270, 2606, 14}"
849,1,145,0,0.4092202,"\int \frac{\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + a*Sin[c + d*x])^4,x]","-\frac{8 \tan ^{11}(c+d x)}{11 a^4 d}-\frac{20 \tan ^9(c+d x)}{9 a^4 d}-\frac{16 \tan ^7(c+d x)}{7 a^4 d}-\frac{4 \tan ^5(c+d x)}{5 a^4 d}+\frac{8 \sec ^{11}(c+d x)}{11 a^4 d}-\frac{16 \sec ^9(c+d x)}{9 a^4 d}+\frac{9 \sec ^7(c+d x)}{7 a^4 d}-\frac{\sec ^5(c+d x)}{5 a^4 d}","-\frac{8 \tan ^{11}(c+d x)}{11 a^4 d}-\frac{20 \tan ^9(c+d x)}{9 a^4 d}-\frac{16 \tan ^7(c+d x)}{7 a^4 d}-\frac{4 \tan ^5(c+d x)}{5 a^4 d}+\frac{8 \sec ^{11}(c+d x)}{11 a^4 d}-\frac{16 \sec ^9(c+d x)}{9 a^4 d}+\frac{9 \sec ^7(c+d x)}{7 a^4 d}-\frac{\sec ^5(c+d x)}{5 a^4 d}",1,"-Sec[c + d*x]^5/(5*a^4*d) + (9*Sec[c + d*x]^7)/(7*a^4*d) - (16*Sec[c + d*x]^9)/(9*a^4*d) + (8*Sec[c + d*x]^11)/(11*a^4*d) - (4*Tan[c + d*x]^5)/(5*a^4*d) - (16*Tan[c + d*x]^7)/(7*a^4*d) - (20*Tan[c + d*x]^9)/(9*a^4*d) - (8*Tan[c + d*x]^11)/(11*a^4*d)","A",18,6,27,0.2222,1,"{2875, 2873, 2606, 14, 2607, 270}"
850,1,184,0,0.3564503,"\int \frac{\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Int[(Sec[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^4,x]","\frac{8 \tan (c+d x)}{231 a^4 d}-\frac{4 \sec (c+d x)}{231 d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{4 \sec (c+d x)}{231 d \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{\sec ^3(c+d x)}{6 a d (a \sin (c+d x)+a)^3}-\frac{5 \sec (c+d x)}{231 a d (a \sin (c+d x)+a)^3}-\frac{\sec (c+d x)}{33 d (a \sin (c+d x)+a)^4}-\frac{a \sec (c+d x)}{22 d (a \sin (c+d x)+a)^5}","\frac{8 \tan ^{11}(c+d x)}{11 a^4 d}+\frac{8 \tan ^9(c+d x)}{3 a^4 d}+\frac{25 \tan ^7(c+d x)}{7 a^4 d}+\frac{2 \tan ^5(c+d x)}{a^4 d}+\frac{\tan ^3(c+d x)}{3 a^4 d}-\frac{8 \sec ^{11}(c+d x)}{11 a^4 d}+\frac{4 \sec ^9(c+d x)}{3 a^4 d}-\frac{4 \sec ^7(c+d x)}{7 a^4 d}",1,"-(a*Sec[c + d*x])/(22*d*(a + a*Sin[c + d*x])^5) - Sec[c + d*x]/(33*d*(a + a*Sin[c + d*x])^4) - (5*Sec[c + d*x])/(231*a*d*(a + a*Sin[c + d*x])^3) + Sec[c + d*x]^3/(6*a*d*(a + a*Sin[c + d*x])^3) - (4*Sec[c + d*x])/(231*d*(a^2 + a^2*Sin[c + d*x])^2) - (4*Sec[c + d*x])/(231*d*(a^4 + a^4*Sin[c + d*x])) + (8*Tan[c + d*x])/(231*a^4*d)","A",8,4,29,0.1379,1,"{2870, 2672, 3767, 8}"
851,1,133,0,0.1111468,"\int \sin (c+d x) (a+a \sin (c+d x)) \tan ^5(c+d x) \, dx","Int[Sin[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^5,x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{5 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{39 a \log (1-\sin (c+d x))}{16 d}-\frac{9 a \log (\sin (c+d x)+1)}{16 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{5 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{39 a \log (1-\sin (c+d x))}{16 d}-\frac{9 a \log (\sin (c+d x)+1)}{16 d}",1,"(-39*a*Log[1 - Sin[c + d*x]])/(16*d) - (9*a*Log[1 + Sin[c + d*x]])/(16*d) - (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) - (5*a^2)/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))","A",4,3,25,0.1200,1,"{2836, 12, 88}"
852,1,115,0,0.0680902,"\int (a+a \sin (c+d x)) \tan ^5(c+d x) \, dx","Int[(a + a*Sin[c + d*x])*Tan[c + d*x]^5,x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{a^2}{d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \sin (c+d x)}{d}-\frac{23 a \log (1-\sin (c+d x))}{16 d}+\frac{7 a \log (\sin (c+d x)+1)}{16 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{a^2}{d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \sin (c+d x)}{d}-\frac{23 a \log (1-\sin (c+d x))}{16 d}+\frac{7 a \log (\sin (c+d x)+1)}{16 d}",1,"(-23*a*Log[1 - Sin[c + d*x]])/(16*d) + (7*a*Log[1 + Sin[c + d*x]])/(16*d) - (a*Sin[c + d*x])/d + a^3/(8*d*(a - a*Sin[c + d*x])^2) - a^2/(d*(a - a*Sin[c + d*x])) + a^2/(8*d*(a + a*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{2707, 88}"
853,1,105,0,0.094418,"\int \sec (c+d x) (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^4,x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{3 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{11 a \log (1-\sin (c+d x))}{16 d}-\frac{5 a \log (\sin (c+d x)+1)}{16 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{3 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{11 a \log (1-\sin (c+d x))}{16 d}-\frac{5 a \log (\sin (c+d x)+1)}{16 d}",1,"(-11*a*Log[1 - Sin[c + d*x]])/(16*d) - (5*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) - (3*a^2)/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))","A",4,3,25,0.1200,1,"{2836, 12, 88}"
854,1,84,0,0.1015885,"\int \sec ^2(c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])*Tan[c + d*x]^3,x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{a^2}{2 d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{a^2}{2 d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"(3*a*ArcTanh[Sin[c + d*x]])/(8*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) - a^2/(2*d*(a - a*Sin[c + d*x])) + a^2/(8*d*(a + a*Sin[c + d*x]))","A",5,4,27,0.1481,1,"{2836, 12, 88, 206}"
855,1,84,0,0.0978452,"\int \sec ^3(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \tanh ^{-1}(\sin (c+d x))}{8 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}-\frac{a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"-(a*ArcTanh[Sin[c + d*x]])/(8*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) - a^2/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))","A",5,4,27,0.1481,1,"{2836, 12, 88, 206}"
856,1,61,0,0.0667725,"\int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])*Tan[c + d*x],x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \tanh ^{-1}(\sin (c+d x))}{8 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"-(a*ArcTanh[Sin[c + d*x]])/(8*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + a^2/(8*d*(a + a*Sin[c + d*x]))","A",5,4,25,0.1600,1,"{2836, 12, 77, 206}"
857,1,117,0,0.1072735,"\int \csc (c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{2 d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{11 a \log (1-\sin (c+d x))}{16 d}+\frac{a \log (\sin (c+d x))}{d}-\frac{5 a \log (\sin (c+d x)+1)}{16 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{2 d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{11 a \log (1-\sin (c+d x))}{16 d}+\frac{a \log (\sin (c+d x))}{d}-\frac{5 a \log (\sin (c+d x)+1)}{16 d}",1,"(-11*a*Log[1 - Sin[c + d*x]])/(16*d) + (a*Log[Sin[c + d*x]])/d - (5*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + a^2/(2*d*(a - a*Sin[c + d*x])) + a^2/(8*d*(a + a*Sin[c + d*x]))","A",4,3,25,0.1200,1,"{2836, 12, 88}"
858,1,129,0,0.1210798,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{3 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \csc (c+d x)}{d}-\frac{23 a \log (1-\sin (c+d x))}{16 d}+\frac{a \log (\sin (c+d x))}{d}+\frac{7 a \log (\sin (c+d x)+1)}{16 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{3 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \csc (c+d x)}{d}-\frac{23 a \log (1-\sin (c+d x))}{16 d}+\frac{a \log (\sin (c+d x))}{d}+\frac{7 a \log (\sin (c+d x)+1)}{16 d}",1,"-((a*Csc[c + d*x])/d) - (23*a*Log[1 - Sin[c + d*x]])/(16*d) + (a*Log[Sin[c + d*x]])/d + (7*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + (3*a^2)/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 88}"
859,1,143,0,0.132132,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{39 a \log (1-\sin (c+d x))}{16 d}+\frac{3 a \log (\sin (c+d x))}{d}-\frac{9 a \log (\sin (c+d x)+1)}{16 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{d (a-a \sin (c+d x))}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}-\frac{39 a \log (1-\sin (c+d x))}{16 d}+\frac{3 a \log (\sin (c+d x))}{d}-\frac{9 a \log (\sin (c+d x)+1)}{16 d}",1,"-((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (39*a*Log[1 - Sin[c + d*x]])/(16*d) + (3*a*Log[Sin[c + d*x]])/d - (9*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + a^2/(d*(a - a*Sin[c + d*x])) + a^2/(8*d*(a + a*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 88}"
860,1,162,0,0.1408661,"\int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Csc[c + d*x]^4*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{5 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{59 a \log (1-\sin (c+d x))}{16 d}+\frac{3 a \log (\sin (c+d x))}{d}+\frac{11 a \log (\sin (c+d x)+1)}{16 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{5 a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{3 a \csc (c+d x)}{d}-\frac{59 a \log (1-\sin (c+d x))}{16 d}+\frac{3 a \log (\sin (c+d x))}{d}+\frac{11 a \log (\sin (c+d x)+1)}{16 d}",1,"(-3*a*Csc[c + d*x])/d - (a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (59*a*Log[1 - Sin[c + d*x]])/(16*d) + (3*a*Log[Sin[c + d*x]])/d + (11*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + (5*a^2)/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 88}"
861,1,119,0,0.0878471,"\int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^5,x]","-\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{9 a^3}{4 d (a-a \sin (c+d x))}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{31 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d}","-\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{9 a^3}{4 d (a-a \sin (c+d x))}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{31 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d}",1,"(-31*a^2*Log[1 - Sin[c + d*x]])/(8*d) - (a^2*Log[1 + Sin[c + d*x]])/(8*d) - (2*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) - (9*a^3)/(4*d*(a - a*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 88}"
862,1,101,0,0.130506,"\int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^4,x]","\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{7 a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \sin (c+d x)}{d}-\frac{17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{a^2 \log (\sin (c+d x)+1)}{8 d}","\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{7 a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \sin (c+d x)}{d}-\frac{17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{a^2 \log (\sin (c+d x)+1)}{8 d}",1,"(-17*a^2*Log[1 - Sin[c + d*x]])/(8*d) + (a^2*Log[1 + Sin[c + d*x]])/(8*d) - (a^2*Sin[c + d*x])/d + a^4/(4*d*(a - a*Sin[c + d*x])^2) - (7*a^3)/(4*d*(a - a*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 88}"
863,1,87,0,0.1387008,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{5 a^3}{4 d (a-a \sin (c+d x))}-\frac{7 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d}","\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{5 a^3}{4 d (a-a \sin (c+d x))}-\frac{7 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d}",1,"(-7*a^2*Log[1 - Sin[c + d*x]])/(8*d) - (a^2*Log[1 + Sin[c + d*x]])/(8*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) - (5*a^3)/(4*d*(a - a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
864,1,64,0,0.1184801,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{3 a^3}{4 d (a-a \sin (c+d x))}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}","\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{3 a^3}{4 d (a-a \sin (c+d x))}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}",1,"(a^2*ArcTanh[Sin[c + d*x]])/(4*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) - (3*a^3)/(4*d*(a - a*Sin[c + d*x]))","A",5,4,29,0.1379,1,"{2836, 12, 88, 206}"
865,1,64,0,0.0869577,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^2 \tan (c+d x) \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2*Tan[c + d*x],x]","\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}","\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}",1,"-(a^2*ArcTanh[Sin[c + d*x]])/(4*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) - a^3/(4*d*(a - a*Sin[c + d*x]))","A",5,4,27,0.1481,1,"{2836, 12, 77, 206}"
866,1,101,0,0.1149255,"\int \csc (c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^4}{4 d (a-a \sin (c+d x))^2}+\frac{3 a^3}{4 d (a-a \sin (c+d x))}-\frac{7 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{a^2 \log (\sin (c+d x))}{d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d}","\frac{a^4}{4 d (a-a \sin (c+d x))^2}+\frac{3 a^3}{4 d (a-a \sin (c+d x))}-\frac{7 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{a^2 \log (\sin (c+d x))}{d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d}",1,"(-7*a^2*Log[1 - Sin[c + d*x]])/(8*d) + (a^2*Log[Sin[c + d*x]])/d - (a^2*Log[1 + Sin[c + d*x]])/(8*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) + (3*a^3)/(4*d*(a - a*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 72}"
867,1,116,0,0.1416459,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^4}{4 d (a-a \sin (c+d x))^2}+\frac{5 a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \csc (c+d x)}{d}-\frac{17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{2 a^2 \log (\sin (c+d x))}{d}+\frac{a^2 \log (\sin (c+d x)+1)}{8 d}","\frac{a^4}{4 d (a-a \sin (c+d x))^2}+\frac{5 a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \csc (c+d x)}{d}-\frac{17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{2 a^2 \log (\sin (c+d x))}{d}+\frac{a^2 \log (\sin (c+d x)+1)}{8 d}",1,"-((a^2*Csc[c + d*x])/d) - (17*a^2*Log[1 - Sin[c + d*x]])/(8*d) + (2*a^2*Log[Sin[c + d*x]])/d + (a^2*Log[1 + Sin[c + d*x]])/(8*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) + (5*a^3)/(4*d*(a - a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
868,1,134,0,0.1481363,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^4}{4 d (a-a \sin (c+d x))^2}+\frac{7 a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a^2 \csc (c+d x)}{d}-\frac{31 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{4 a^2 \log (\sin (c+d x))}{d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d}","\frac{a^4}{4 d (a-a \sin (c+d x))^2}+\frac{7 a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a^2 \csc (c+d x)}{d}-\frac{31 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{4 a^2 \log (\sin (c+d x))}{d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d}",1,"(-2*a^2*Csc[c + d*x])/d - (a^2*Csc[c + d*x]^2)/(2*d) - (31*a^2*Log[1 - Sin[c + d*x]])/(8*d) + (4*a^2*Log[Sin[c + d*x]])/d - (a^2*Log[1 + Sin[c + d*x]])/(8*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) + (7*a^3)/(4*d*(a - a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
869,1,150,0,0.164349,"\int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^4*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^4}{4 d (a-a \sin (c+d x))^2}+\frac{9 a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{d}-\frac{4 a^2 \csc (c+d x)}{d}-\frac{49 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{6 a^2 \log (\sin (c+d x))}{d}+\frac{a^2 \log (\sin (c+d x)+1)}{8 d}","\frac{a^4}{4 d (a-a \sin (c+d x))^2}+\frac{9 a^3}{4 d (a-a \sin (c+d x))}-\frac{a^2 \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^2(c+d x)}{d}-\frac{4 a^2 \csc (c+d x)}{d}-\frac{49 a^2 \log (1-\sin (c+d x))}{8 d}+\frac{6 a^2 \log (\sin (c+d x))}{d}+\frac{a^2 \log (\sin (c+d x)+1)}{8 d}",1,"(-4*a^2*Csc[c + d*x])/d - (a^2*Csc[c + d*x]^2)/d - (a^2*Csc[c + d*x]^3)/(3*d) - (49*a^2*Log[1 - Sin[c + d*x]])/(8*d) + (6*a^2*Log[Sin[c + d*x]])/d + (a^2*Log[1 + Sin[c + d*x]])/(8*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) + (9*a^3)/(4*d*(a - a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
870,1,114,0,0.0840697,"\int (a+a \sin (c+d x))^3 \tan ^5(c+d x) \, dx","Int[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^5,x]","-\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^5}{2 d (a-a \sin (c+d x))^2}-\frac{5 a^4}{d (a-a \sin (c+d x))}-\frac{6 a^3 \sin (c+d x)}{d}-\frac{10 a^3 \log (1-\sin (c+d x))}{d}","-\frac{a^3 \sin ^3(c+d x)}{3 d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^5}{2 d (a-a \sin (c+d x))^2}-\frac{5 a^4}{d (a-a \sin (c+d x))}-\frac{6 a^3 \sin (c+d x)}{d}-\frac{10 a^3 \log (1-\sin (c+d x))}{d}",1,"(-10*a^3*Log[1 - Sin[c + d*x]])/d - (6*a^3*Sin[c + d*x])/d - (3*a^3*Sin[c + d*x]^2)/(2*d) - (a^3*Sin[c + d*x]^3)/(3*d) + a^5/(2*d*(a - a*Sin[c + d*x])^2) - (5*a^4)/(d*(a - a*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2707, 43}"
871,1,96,0,0.1109842,"\int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^4,x]","-\frac{a^3 \sin ^2(c+d x)}{2 d}+\frac{a^5}{2 d (a-a \sin (c+d x))^2}-\frac{4 a^4}{d (a-a \sin (c+d x))}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{6 a^3 \log (1-\sin (c+d x))}{d}","-\frac{a^3 \sin ^2(c+d x)}{2 d}+\frac{a^5}{2 d (a-a \sin (c+d x))^2}-\frac{4 a^4}{d (a-a \sin (c+d x))}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{6 a^3 \log (1-\sin (c+d x))}{d}",1,"(-6*a^3*Log[1 - Sin[c + d*x]])/d - (3*a^3*Sin[c + d*x])/d - (a^3*Sin[c + d*x]^2)/(2*d) + a^5/(2*d*(a - a*Sin[c + d*x])^2) - (4*a^4)/(d*(a - a*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 43}"
872,1,78,0,0.1189989,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^3,x]","\frac{a^5}{2 d (a-a \sin (c+d x))^2}-\frac{3 a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \sin (c+d x)}{d}-\frac{3 a^3 \log (1-\sin (c+d x))}{d}","\frac{a^5}{2 d (a-a \sin (c+d x))^2}-\frac{3 a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \sin (c+d x)}{d}-\frac{3 a^3 \log (1-\sin (c+d x))}{d}",1,"(-3*a^3*Log[1 - Sin[c + d*x]])/d - (a^3*Sin[c + d*x])/d + a^5/(2*d*(a - a*Sin[c + d*x])^2) - (3*a^4)/(d*(a - a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 43}"
873,1,64,0,0.1100293,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{a^5}{2 d (a-a \sin (c+d x))^2}-\frac{2 a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \log (1-\sin (c+d x))}{d}","\frac{a^5}{2 d (a-a \sin (c+d x))^2}-\frac{2 a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \log (1-\sin (c+d x))}{d}",1,"-((a^3*Log[1 - Sin[c + d*x]])/d) + a^5/(2*d*(a - a*Sin[c + d*x])^2) - (2*a^4)/(d*(a - a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 43}"
874,1,31,0,0.059388,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]","\frac{a^5 \sin ^2(c+d x)}{2 d (a-a \sin (c+d x))^2}","\frac{a^5 \sin ^2(c+d x)}{2 d (a-a \sin (c+d x))^2}",1,"(a^5*Sin[c + d*x]^2)/(2*d*(a - a*Sin[c + d*x])^2)","A",3,3,27,0.1111,1,"{2836, 12, 37}"
875,1,77,0,0.1050165,"\int \csc (c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{a^5}{2 d (a-a \sin (c+d x))^2}+\frac{a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \log (1-\sin (c+d x))}{d}+\frac{a^3 \log (\sin (c+d x))}{d}","\frac{a^5}{2 d (a-a \sin (c+d x))^2}+\frac{a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \log (1-\sin (c+d x))}{d}+\frac{a^3 \log (\sin (c+d x))}{d}",1,"-((a^3*Log[1 - Sin[c + d*x]])/d) + (a^3*Log[Sin[c + d*x]])/d + a^5/(2*d*(a - a*Sin[c + d*x])^2) + a^4/(d*(a - a*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 44}"
876,1,93,0,0.1575219,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{a^5}{2 d (a-a \sin (c+d x))^2}+\frac{2 a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \csc (c+d x)}{d}-\frac{3 a^3 \log (1-\sin (c+d x))}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d}","\frac{a^5}{2 d (a-a \sin (c+d x))^2}+\frac{2 a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \csc (c+d x)}{d}-\frac{3 a^3 \log (1-\sin (c+d x))}{d}+\frac{3 a^3 \log (\sin (c+d x))}{d}",1,"-((a^3*Csc[c + d*x])/d) - (3*a^3*Log[1 - Sin[c + d*x]])/d + (3*a^3*Log[Sin[c + d*x]])/d + a^5/(2*d*(a - a*Sin[c + d*x])^2) + (2*a^4)/(d*(a - a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 44}"
877,1,111,0,0.1475728,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{a^5}{2 d (a-a \sin (c+d x))^2}+\frac{3 a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}-\frac{6 a^3 \log (1-\sin (c+d x))}{d}+\frac{6 a^3 \log (\sin (c+d x))}{d}","\frac{a^5}{2 d (a-a \sin (c+d x))^2}+\frac{3 a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}-\frac{6 a^3 \log (1-\sin (c+d x))}{d}+\frac{6 a^3 \log (\sin (c+d x))}{d}",1,"(-3*a^3*Csc[c + d*x])/d - (a^3*Csc[c + d*x]^2)/(2*d) - (6*a^3*Log[1 - Sin[c + d*x]])/d + (6*a^3*Log[Sin[c + d*x]])/d + a^5/(2*d*(a - a*Sin[c + d*x])^2) + (3*a^4)/(d*(a - a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 44}"
878,1,236,0,0.2520074,"\int \frac{\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^4*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{3 a^2}{16 d (a \sin (c+d x)+a)^3}+\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{2 a d}-\frac{17 a}{128 d (a-a \sin (c+d x))^2}+\frac{71 a}{64 d (a \sin (c+d x)+a)^2}+\frac{125}{128 d (a-a \sin (c+d x))}-\frac{5}{d (a \sin (c+d x)+a)}+\frac{5 \sin (c+d x)}{a d}+\frac{515 \log (1-\sin (c+d x))}{256 a d}-\frac{1795 \log (\sin (c+d x)+1)}{256 a d}","\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{3 a^2}{16 d (a \sin (c+d x)+a)^3}+\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^2(c+d x)}{2 a d}-\frac{17 a}{128 d (a-a \sin (c+d x))^2}+\frac{71 a}{64 d (a \sin (c+d x)+a)^2}+\frac{125}{128 d (a-a \sin (c+d x))}-\frac{5}{d (a \sin (c+d x)+a)}+\frac{5 \sin (c+d x)}{a d}+\frac{515 \log (1-\sin (c+d x))}{256 a d}-\frac{1795 \log (\sin (c+d x)+1)}{256 a d}",1,"(515*Log[1 - Sin[c + d*x]])/(256*a*d) - (1795*Log[1 + Sin[c + d*x]])/(256*a*d) + (5*Sin[c + d*x])/(a*d) - Sin[c + d*x]^2/(2*a*d) + Sin[c + d*x]^3/(3*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (17*a)/(128*d*(a - a*Sin[c + d*x])^2) + 125/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d*(a + a*Sin[c + d*x])^4) - (3*a^2)/(16*d*(a + a*Sin[c + d*x])^3) + (71*a)/(64*d*(a + a*Sin[c + d*x])^2) - 5/(d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
879,1,220,0,0.2285878,"\int \frac{\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^3*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","-\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{a^2}{6 d (a \sin (c+d x)+a)^3}+\frac{\sin ^2(c+d x)}{2 a d}-\frac{15 a}{128 d (a-a \sin (c+d x))^2}-\frac{55 a}{64 d (a \sin (c+d x)+a)^2}+\frac{95}{128 d (a-a \sin (c+d x))}+\frac{105}{32 d (a \sin (c+d x)+a)}-\frac{\sin (c+d x)}{a d}+\frac{325 \log (1-\sin (c+d x))}{256 a d}+\frac{955 \log (\sin (c+d x)+1)}{256 a d}","-\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{a^2}{6 d (a \sin (c+d x)+a)^3}+\frac{\sin ^2(c+d x)}{2 a d}-\frac{15 a}{128 d (a-a \sin (c+d x))^2}-\frac{55 a}{64 d (a \sin (c+d x)+a)^2}+\frac{95}{128 d (a-a \sin (c+d x))}+\frac{105}{32 d (a \sin (c+d x)+a)}-\frac{\sin (c+d x)}{a d}+\frac{325 \log (1-\sin (c+d x))}{256 a d}+\frac{955 \log (\sin (c+d x)+1)}{256 a d}",1,"(325*Log[1 - Sin[c + d*x]])/(256*a*d) + (955*Log[1 + Sin[c + d*x]])/(256*a*d) - Sin[c + d*x]/(a*d) + Sin[c + d*x]^2/(2*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (15*a)/(128*d*(a - a*Sin[c + d*x])^2) + 95/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) + a^2/(6*d*(a + a*Sin[c + d*x])^3) - (55*a)/(64*d*(a + a*Sin[c + d*x])^2) + 105/(32*d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
880,1,199,0,0.2123573,"\int \frac{\sin ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{7 a^2}{48 d (a \sin (c+d x)+a)^3}-\frac{13 a}{128 d (a-a \sin (c+d x))^2}+\frac{41 a}{64 d (a \sin (c+d x)+a)^2}+\frac{69}{128 d (a-a \sin (c+d x))}-\frac{2}{d (a \sin (c+d x)+a)}+\frac{\sin (c+d x)}{a d}+\frac{187 \log (1-\sin (c+d x))}{256 a d}-\frac{443 \log (\sin (c+d x)+1)}{256 a d}","\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{7 a^2}{48 d (a \sin (c+d x)+a)^3}-\frac{13 a}{128 d (a-a \sin (c+d x))^2}+\frac{41 a}{64 d (a \sin (c+d x)+a)^2}+\frac{69}{128 d (a-a \sin (c+d x))}-\frac{2}{d (a \sin (c+d x)+a)}+\frac{\sin (c+d x)}{a d}+\frac{187 \log (1-\sin (c+d x))}{256 a d}-\frac{443 \log (\sin (c+d x)+1)}{256 a d}",1,"(187*Log[1 - Sin[c + d*x]])/(256*a*d) - (443*Log[1 + Sin[c + d*x]])/(256*a*d) + Sin[c + d*x]/(a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (13*a)/(128*d*(a - a*Sin[c + d*x])^2) + 69/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d*(a + a*Sin[c + d*x])^4) - (7*a^2)/(48*d*(a + a*Sin[c + d*x])^3) + (41*a)/(64*d*(a + a*Sin[c + d*x])^2) - 2/(d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
881,1,188,0,0.1835773,"\int \frac{\sin (c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","-\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{a^2}{8 d (a \sin (c+d x)+a)^3}-\frac{11 a}{128 d (a-a \sin (c+d x))^2}-\frac{29 a}{64 d (a \sin (c+d x)+a)^2}+\frac{47}{128 d (a-a \sin (c+d x))}+\frac{35}{32 d (a \sin (c+d x)+a)}+\frac{93 \log (1-\sin (c+d x))}{256 a d}+\frac{163 \log (\sin (c+d x)+1)}{256 a d}","-\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{a^2}{8 d (a \sin (c+d x)+a)^3}-\frac{11 a}{128 d (a-a \sin (c+d x))^2}-\frac{29 a}{64 d (a \sin (c+d x)+a)^2}+\frac{47}{128 d (a-a \sin (c+d x))}+\frac{35}{32 d (a \sin (c+d x)+a)}+\frac{93 \log (1-\sin (c+d x))}{256 a d}+\frac{163 \log (\sin (c+d x)+1)}{256 a d}",1,"(93*Log[1 - Sin[c + d*x]])/(256*a*d) + (163*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (11*a)/(128*d*(a - a*Sin[c + d*x])^2) + 47/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) + a^2/(8*d*(a + a*Sin[c + d*x])^3) - (29*a)/(64*d*(a + a*Sin[c + d*x])^2) + 35/(32*d*(a + a*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 88}"
882,1,130,0,0.171751,"\int \frac{\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Tan[c + d*x]^7/(a + a*Sin[c + d*x]),x]","\frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^7(c+d x) \sec (c+d x)}{8 a d}+\frac{7 \tan ^5(c+d x) \sec (c+d x)}{48 a d}-\frac{35 \tan ^3(c+d x) \sec (c+d x)}{192 a d}+\frac{35 \tan (c+d x) \sec (c+d x)}{128 a d}","\frac{\tan ^8(c+d x)}{8 a d}-\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^7(c+d x) \sec (c+d x)}{8 a d}+\frac{7 \tan ^5(c+d x) \sec (c+d x)}{48 a d}-\frac{35 \tan ^3(c+d x) \sec (c+d x)}{192 a d}+\frac{35 \tan (c+d x) \sec (c+d x)}{128 a d}",1,"(-35*ArcTanh[Sin[c + d*x]])/(128*a*d) + (35*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (35*Sec[c + d*x]*Tan[c + d*x]^3)/(192*a*d) + (7*Sec[c + d*x]*Tan[c + d*x]^5)/(48*a*d) - (Sec[c + d*x]*Tan[c + d*x]^7)/(8*a*d) + Tan[c + d*x]^8/(8*a*d)","A",8,5,21,0.2381,1,"{2706, 2607, 30, 2611, 3770}"
883,1,134,0,0.2364829,"\int \frac{\sec (c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","-\frac{\tan ^8(c+d x)}{8 a d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac{\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}-\frac{5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}-\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d}","-\frac{\tan ^8(c+d x)}{8 a d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac{\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}-\frac{5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}-\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d}",1,"(-5*ArcTanh[Sin[c + d*x]])/(128*a*d) - (5*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) + (5*Sec[c + d*x]^3*Tan[c + d*x])/(64*a*d) - (5*Sec[c + d*x]^3*Tan[c + d*x]^3)/(48*a*d) + (Sec[c + d*x]^3*Tan[c + d*x]^5)/(8*a*d) - Tan[c + d*x]^8/(8*a*d)","A",8,6,27,0.2222,1,"{2835, 2611, 3768, 3770, 2607, 30}"
884,1,152,0,0.2446309,"\int \frac{\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^2*Tan[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","\frac{\tan ^8(c+d x)}{8 a d}+\frac{\tan ^6(c+d x)}{6 a d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}+\frac{5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}-\frac{5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d}","\frac{\tan ^8(c+d x)}{8 a d}+\frac{\tan ^6(c+d x)}{6 a d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}+\frac{5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}-\frac{5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d}",1,"(5*ArcTanh[Sin[c + d*x]])/(128*a*d) + (5*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (5*Sec[c + d*x]^3*Tan[c + d*x])/(64*a*d) + (5*Sec[c + d*x]^3*Tan[c + d*x]^3)/(48*a*d) - (Sec[c + d*x]^3*Tan[c + d*x]^5)/(8*a*d) + Tan[c + d*x]^6/(6*a*d) + Tan[c + d*x]^8/(8*a*d)","A",9,6,29,0.2069,1,"{2835, 2607, 14, 2611, 3768, 3770}"
885,1,150,0,0.2258594,"\int \frac{\sec ^3(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^3*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\tan ^8(c+d x)}{8 a d}-\frac{\tan ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{8 a d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{16 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{128 a d}","-\frac{\tan ^8(c+d x)}{8 a d}-\frac{\tan ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{8 a d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{16 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{128 a d}",1,"(3*ArcTanh[Sin[c + d*x]])/(128*a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(64*a*d) - (Sec[c + d*x]^5*Tan[c + d*x])/(16*a*d) + (Sec[c + d*x]^5*Tan[c + d*x]^3)/(8*a*d) - Tan[c + d*x]^6/(6*a*d) - Tan[c + d*x]^8/(8*a*d)","A",9,6,29,0.2069,1,"{2835, 2611, 3768, 3770, 2607, 14}"
886,1,150,0,0.224721,"\int \frac{\sec ^4(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^4*Tan[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\sec ^8(c+d x)}{8 a d}-\frac{\sec ^6(c+d x)}{6 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{8 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{16 a d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{64 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{128 a d}","\frac{\sec ^8(c+d x)}{8 a d}-\frac{\sec ^6(c+d x)}{6 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{8 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{16 a d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{64 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{128 a d}",1,"(-3*ArcTanh[Sin[c + d*x]])/(128*a*d) - Sec[c + d*x]^6/(6*a*d) + Sec[c + d*x]^8/(8*a*d) - (3*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(64*a*d) + (Sec[c + d*x]^5*Tan[c + d*x])/(16*a*d) - (Sec[c + d*x]^5*Tan[c + d*x]^3)/(8*a*d)","A",9,6,29,0.2069,1,"{2835, 2606, 14, 2611, 3768, 3770}"
887,1,148,0,0.1994726,"\int \frac{\sec ^5(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^5*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^8(c+d x)}{8 a d}+\frac{\sec ^6(c+d x)}{6 a d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac{\tan (c+d x) \sec ^7(c+d x)}{8 a d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{48 a d}-\frac{5 \tan (c+d x) \sec ^3(c+d x)}{192 a d}-\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d}","-\frac{\sec ^8(c+d x)}{8 a d}+\frac{\sec ^6(c+d x)}{6 a d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac{\tan (c+d x) \sec ^7(c+d x)}{8 a d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{48 a d}-\frac{5 \tan (c+d x) \sec ^3(c+d x)}{192 a d}-\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d}",1,"(-5*ArcTanh[Sin[c + d*x]])/(128*a*d) + Sec[c + d*x]^6/(6*a*d) - Sec[c + d*x]^8/(8*a*d) - (5*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (5*Sec[c + d*x]^3*Tan[c + d*x])/(192*a*d) - (Sec[c + d*x]^5*Tan[c + d*x])/(48*a*d) + (Sec[c + d*x]^7*Tan[c + d*x])/(8*a*d)","A",9,6,29,0.2069,1,"{2835, 2611, 3768, 3770, 2606, 14}"
888,1,130,0,0.1475179,"\int \frac{\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^6*Tan[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\sec ^8(c+d x)}{8 a d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan (c+d x) \sec ^7(c+d x)}{8 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{48 a d}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{192 a d}+\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d}","\frac{\sec ^8(c+d x)}{8 a d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{\tan (c+d x) \sec ^7(c+d x)}{8 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{48 a d}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{192 a d}+\frac{5 \tan (c+d x) \sec (c+d x)}{128 a d}",1,"(5*ArcTanh[Sin[c + d*x]])/(128*a*d) + Sec[c + d*x]^8/(8*a*d) + (5*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) + (5*Sec[c + d*x]^3*Tan[c + d*x])/(192*a*d) + (Sec[c + d*x]^5*Tan[c + d*x])/(48*a*d) - (Sec[c + d*x]^7*Tan[c + d*x])/(8*a*d)","A",8,6,27,0.2222,1,"{2835, 2606, 30, 2611, 3768, 3770}"
889,1,165,0,0.1258982,"\int \frac{\sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^7/(a + a*Sin[c + d*x]),x]","-\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{a^2}{24 d (a \sin (c+d x)+a)^3}+\frac{5 a}{128 d (a-a \sin (c+d x))^2}-\frac{5 a}{64 d (a \sin (c+d x)+a)^2}+\frac{15}{128 d (a-a \sin (c+d x))}-\frac{5}{32 d (a \sin (c+d x)+a)}+\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}","-\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{a^2}{24 d (a \sin (c+d x)+a)^3}+\frac{5 a}{128 d (a-a \sin (c+d x))^2}-\frac{5 a}{64 d (a \sin (c+d x)+a)^2}+\frac{15}{128 d (a-a \sin (c+d x))}-\frac{5}{32 d (a \sin (c+d x)+a)}+\frac{35 \tanh ^{-1}(\sin (c+d x))}{128 a d}",1,"(35*ArcTanh[Sin[c + d*x]])/(128*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (5*a)/(128*d*(a - a*Sin[c + d*x])^2) + 15/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) - a^2/(24*d*(a + a*Sin[c + d*x])^3) - (5*a)/(64*d*(a + a*Sin[c + d*x])^2) - 5/(32*d*(a + a*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
890,1,202,0,0.2006727,"\int \frac{\csc (c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{a^2}{16 d (a \sin (c+d x)+a)^3}+\frac{7 a}{128 d (a-a \sin (c+d x))^2}+\frac{11 a}{64 d (a \sin (c+d x)+a)^2}+\frac{29}{128 d (a-a \sin (c+d x))}+\frac{1}{2 d (a \sin (c+d x)+a)}-\frac{93 \log (1-\sin (c+d x))}{256 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{163 \log (\sin (c+d x)+1)}{256 a d}","\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{a^2}{16 d (a \sin (c+d x)+a)^3}+\frac{7 a}{128 d (a-a \sin (c+d x))^2}+\frac{11 a}{64 d (a \sin (c+d x)+a)^2}+\frac{29}{128 d (a-a \sin (c+d x))}+\frac{1}{2 d (a \sin (c+d x)+a)}-\frac{93 \log (1-\sin (c+d x))}{256 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{163 \log (\sin (c+d x)+1)}{256 a d}",1,"(-93*Log[1 - Sin[c + d*x]])/(256*a*d) + Log[Sin[c + d*x]]/(a*d) - (163*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (7*a)/(128*d*(a - a*Sin[c + d*x])^2) + 29/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d*(a + a*Sin[c + d*x])^4) + a^2/(16*d*(a + a*Sin[c + d*x])^3) + (11*a)/(64*d*(a + a*Sin[c + d*x])^2) + 1/(2*d*(a + a*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 88}"
891,1,217,0,0.2393938,"\int \frac{\csc ^2(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","-\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{a^2}{12 d (a \sin (c+d x)+a)^3}+\frac{9 a}{128 d (a-a \sin (c+d x))^2}-\frac{19 a}{64 d (a \sin (c+d x)+a)^2}+\frac{47}{128 d (a-a \sin (c+d x))}-\frac{35}{32 d (a \sin (c+d x)+a)}-\frac{\csc (c+d x)}{a d}-\frac{187 \log (1-\sin (c+d x))}{256 a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{443 \log (\sin (c+d x)+1)}{256 a d}","-\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{a^2}{12 d (a \sin (c+d x)+a)^3}+\frac{9 a}{128 d (a-a \sin (c+d x))^2}-\frac{19 a}{64 d (a \sin (c+d x)+a)^2}+\frac{47}{128 d (a-a \sin (c+d x))}-\frac{35}{32 d (a \sin (c+d x)+a)}-\frac{\csc (c+d x)}{a d}-\frac{187 \log (1-\sin (c+d x))}{256 a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{443 \log (\sin (c+d x)+1)}{256 a d}",1,"-(Csc[c + d*x]/(a*d)) - (187*Log[1 - Sin[c + d*x]])/(256*a*d) - Log[Sin[c + d*x]]/(a*d) + (443*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (9*a)/(128*d*(a - a*Sin[c + d*x])^2) + 47/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) - a^2/(12*d*(a + a*Sin[c + d*x])^3) - (19*a)/(64*d*(a + a*Sin[c + d*x])^2) - 35/(32*d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
892,1,232,0,0.2462262,"\int \frac{\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^3*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{5 a^2}{48 d (a \sin (c+d x)+a)^3}+\frac{11 a}{128 d (a-a \sin (c+d x))^2}+\frac{29 a}{64 d (a \sin (c+d x)+a)^2}+\frac{69}{128 d (a-a \sin (c+d x))}+\frac{2}{d (a \sin (c+d x)+a)}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}-\frac{325 \log (1-\sin (c+d x))}{256 a d}+\frac{5 \log (\sin (c+d x))}{a d}-\frac{955 \log (\sin (c+d x)+1)}{256 a d}","\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}+\frac{5 a^2}{48 d (a \sin (c+d x)+a)^3}+\frac{11 a}{128 d (a-a \sin (c+d x))^2}+\frac{29 a}{64 d (a \sin (c+d x)+a)^2}+\frac{69}{128 d (a-a \sin (c+d x))}+\frac{2}{d (a \sin (c+d x)+a)}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}-\frac{325 \log (1-\sin (c+d x))}{256 a d}+\frac{5 \log (\sin (c+d x))}{a d}-\frac{955 \log (\sin (c+d x)+1)}{256 a d}",1,"Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - (325*Log[1 - Sin[c + d*x]])/(256*a*d) + (5*Log[Sin[c + d*x]])/(a*d) - (955*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (11*a)/(128*d*(a - a*Sin[c + d*x])^2) + 69/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d*(a + a*Sin[c + d*x])^4) + (5*a^2)/(48*d*(a + a*Sin[c + d*x])^3) + (29*a)/(64*d*(a + a*Sin[c + d*x])^2) + 2/(d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
893,1,253,0,0.2641459,"\int \frac{\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^4*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","-\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{a^2}{8 d (a \sin (c+d x)+a)^3}+\frac{13 a}{128 d (a-a \sin (c+d x))^2}-\frac{41 a}{64 d (a \sin (c+d x)+a)^2}+\frac{95}{128 d (a-a \sin (c+d x))}-\frac{105}{32 d (a \sin (c+d x)+a)}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{5 \csc (c+d x)}{a d}-\frac{515 \log (1-\sin (c+d x))}{256 a d}-\frac{5 \log (\sin (c+d x))}{a d}+\frac{1795 \log (\sin (c+d x)+1)}{256 a d}","-\frac{a^3}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2}{96 d (a-a \sin (c+d x))^3}-\frac{a^2}{8 d (a \sin (c+d x)+a)^3}+\frac{13 a}{128 d (a-a \sin (c+d x))^2}-\frac{41 a}{64 d (a \sin (c+d x)+a)^2}+\frac{95}{128 d (a-a \sin (c+d x))}-\frac{105}{32 d (a \sin (c+d x)+a)}-\frac{\csc ^3(c+d x)}{3 a d}+\frac{\csc ^2(c+d x)}{2 a d}-\frac{5 \csc (c+d x)}{a d}-\frac{515 \log (1-\sin (c+d x))}{256 a d}-\frac{5 \log (\sin (c+d x))}{a d}+\frac{1795 \log (\sin (c+d x)+1)}{256 a d}",1,"(-5*Csc[c + d*x])/(a*d) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) - (515*Log[1 - Sin[c + d*x]])/(256*a*d) - (5*Log[Sin[c + d*x]])/(a*d) + (1795*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (13*a)/(128*d*(a - a*Sin[c + d*x])^2) + 95/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) - a^2/(8*d*(a + a*Sin[c + d*x])^3) - (41*a)/(64*d*(a + a*Sin[c + d*x])^2) - 105/(32*d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
894,1,91,0,0.2137397,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{2 a^2 \tan ^7(c+d x)}{7 d}+\frac{2 a^2 \tan ^5(c+d x)}{5 d}+\frac{2 a^2 \sec ^7(c+d x)}{7 d}-\frac{3 a^2 \sec ^5(c+d x)}{5 d}+\frac{a^2 \sec ^3(c+d x)}{3 d}","\frac{2 a^2 \tan ^7(c+d x)}{7 d}+\frac{2 a^2 \tan ^5(c+d x)}{5 d}+\frac{2 a^2 \sec ^7(c+d x)}{7 d}-\frac{3 a^2 \sec ^5(c+d x)}{5 d}+\frac{a^2 \sec ^3(c+d x)}{3 d}",1,"(a^2*Sec[c + d*x]^3)/(3*d) - (3*a^2*Sec[c + d*x]^5)/(5*d) + (2*a^2*Sec[c + d*x]^7)/(7*d) + (2*a^2*Tan[c + d*x]^5)/(5*d) + (2*a^2*Tan[c + d*x]^7)/(7*d)","A",11,5,29,0.1724,1,"{2873, 2606, 14, 2607, 270}"
895,1,264,0,0.2827151,"\int \frac{\sin ^3(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^3*Tan[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","-\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{19 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac{3 a^2}{64 d (a-a \sin (c+d x))^3}-\frac{53 a^2}{128 d (a \sin (c+d x)+a)^3}-\frac{\sin ^2(c+d x)}{2 a d}+\frac{141 a}{512 d (a-a \sin (c+d x))^2}+\frac{765 a}{512 d (a \sin (c+d x)+a)^2}-\frac{39}{32 d (a-a \sin (c+d x))}-\frac{1155}{256 d (a \sin (c+d x)+a)}+\frac{\sin (c+d x)}{a d}-\frac{843 \log (1-\sin (c+d x))}{512 a d}-\frac{2229 \log (\sin (c+d x)+1)}{512 a d}","-\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{19 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac{3 a^2}{64 d (a-a \sin (c+d x))^3}-\frac{53 a^2}{128 d (a \sin (c+d x)+a)^3}-\frac{\sin ^2(c+d x)}{2 a d}+\frac{141 a}{512 d (a-a \sin (c+d x))^2}+\frac{765 a}{512 d (a \sin (c+d x)+a)^2}-\frac{39}{32 d (a-a \sin (c+d x))}-\frac{1155}{256 d (a \sin (c+d x)+a)}+\frac{\sin (c+d x)}{a d}-\frac{843 \log (1-\sin (c+d x))}{512 a d}-\frac{2229 \log (\sin (c+d x)+1)}{512 a d}",1,"(-843*Log[1 - Sin[c + d*x]])/(512*a*d) - (2229*Log[1 + Sin[c + d*x]])/(512*a*d) + Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) - (3*a^2)/(64*d*(a - a*Sin[c + d*x])^3) + (141*a)/(512*d*(a - a*Sin[c + d*x])^2) - 39/(32*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) + (19*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (53*a^2)/(128*d*(a + a*Sin[c + d*x])^3) + (765*a)/(512*d*(a + a*Sin[c + d*x])^2) - 1155/(256*d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
896,1,247,0,0.2613197,"\int \frac{\sin ^2(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}-\frac{17 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac{a^2}{24 d (a-a \sin (c+d x))^3}+\frac{125 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{109 a}{512 d (a-a \sin (c+d x))^2}-\frac{515 a}{512 d (a \sin (c+d x)+a)^2}-\frac{203}{256 d (a-a \sin (c+d x))}+\frac{5}{2 d (a \sin (c+d x)+a)}-\frac{\sin (c+d x)}{a d}-\frac{437 \log (1-\sin (c+d x))}{512 a d}+\frac{949 \log (\sin (c+d x)+1)}{512 a d}","\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}-\frac{17 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac{a^2}{24 d (a-a \sin (c+d x))^3}+\frac{125 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{109 a}{512 d (a-a \sin (c+d x))^2}-\frac{515 a}{512 d (a \sin (c+d x)+a)^2}-\frac{203}{256 d (a-a \sin (c+d x))}+\frac{5}{2 d (a \sin (c+d x)+a)}-\frac{\sin (c+d x)}{a d}-\frac{437 \log (1-\sin (c+d x))}{512 a d}+\frac{949 \log (\sin (c+d x)+1)}{512 a d}",1,"(-437*Log[1 - Sin[c + d*x]])/(512*a*d) + (949*Log[1 + Sin[c + d*x]])/(512*a*d) - Sin[c + d*x]/(a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) - a^2/(24*d*(a - a*Sin[c + d*x])^3) + (109*a)/(512*d*(a - a*Sin[c + d*x])^2) - 203/(256*d*(a - a*Sin[c + d*x])) + a^4/(160*d*(a + a*Sin[c + d*x])^5) - (17*a^3)/(256*d*(a + a*Sin[c + d*x])^4) + (125*a^2)/(384*d*(a + a*Sin[c + d*x])^3) - (515*a)/(512*d*(a + a*Sin[c + d*x])^2) + 5/(2*d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
897,1,233,0,0.2345891,"\int \frac{\sin (c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","-\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{15 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac{7 a^2}{192 d (a-a \sin (c+d x))^3}-\frac{95 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{81 a}{512 d (a-a \sin (c+d x))^2}+\frac{325 a}{512 d (a \sin (c+d x)+a)^2}-\frac{61}{128 d (a-a \sin (c+d x))}-\frac{315}{256 d (a \sin (c+d x)+a)}-\frac{193 \log (1-\sin (c+d x))}{512 a d}-\frac{319 \log (\sin (c+d x)+1)}{512 a d}","-\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{15 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac{7 a^2}{192 d (a-a \sin (c+d x))^3}-\frac{95 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{81 a}{512 d (a-a \sin (c+d x))^2}+\frac{325 a}{512 d (a \sin (c+d x)+a)^2}-\frac{61}{128 d (a-a \sin (c+d x))}-\frac{315}{256 d (a \sin (c+d x)+a)}-\frac{193 \log (1-\sin (c+d x))}{512 a d}-\frac{319 \log (\sin (c+d x)+1)}{512 a d}",1,"(-193*Log[1 - Sin[c + d*x]])/(512*a*d) - (319*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) - (7*a^2)/(192*d*(a - a*Sin[c + d*x])^3) + (81*a)/(512*d*(a - a*Sin[c + d*x])^2) - 61/(128*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) + (15*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (95*a^2)/(384*d*(a + a*Sin[c + d*x])^3) + (325*a)/(512*d*(a + a*Sin[c + d*x])^2) - 315/(256*d*(a + a*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 88}"
898,1,154,0,0.1925619,"\int \frac{\tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Tan[c + d*x]^9/(a + a*Sin[c + d*x]),x]","\frac{\tan ^{10}(c+d x)}{10 a d}+\frac{63 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\tan ^9(c+d x) \sec (c+d x)}{10 a d}+\frac{9 \tan ^7(c+d x) \sec (c+d x)}{80 a d}-\frac{21 \tan ^5(c+d x) \sec (c+d x)}{160 a d}+\frac{21 \tan ^3(c+d x) \sec (c+d x)}{128 a d}-\frac{63 \tan (c+d x) \sec (c+d x)}{256 a d}","\frac{\tan ^{10}(c+d x)}{10 a d}+\frac{63 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\tan ^9(c+d x) \sec (c+d x)}{10 a d}+\frac{9 \tan ^7(c+d x) \sec (c+d x)}{80 a d}-\frac{21 \tan ^5(c+d x) \sec (c+d x)}{160 a d}+\frac{21 \tan ^3(c+d x) \sec (c+d x)}{128 a d}-\frac{63 \tan (c+d x) \sec (c+d x)}{256 a d}",1,"(63*ArcTanh[Sin[c + d*x]])/(256*a*d) - (63*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (21*Sec[c + d*x]*Tan[c + d*x]^3)/(128*a*d) - (21*Sec[c + d*x]*Tan[c + d*x]^5)/(160*a*d) + (9*Sec[c + d*x]*Tan[c + d*x]^7)/(80*a*d) - (Sec[c + d*x]*Tan[c + d*x]^9)/(10*a*d) + Tan[c + d*x]^10/(10*a*d)","A",9,5,21,0.2381,1,"{2706, 2607, 30, 2611, 3770}"
899,1,160,0,0.270722,"\int \frac{\sec (c+d x) \tan ^8(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x]^8)/(a + a*Sin[c + d*x]),x]","-\frac{\tan ^{10}(c+d x)}{10 a d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^7(c+d x) \sec ^3(c+d x)}{10 a d}-\frac{7 \tan ^5(c+d x) \sec ^3(c+d x)}{80 a d}+\frac{7 \tan ^3(c+d x) \sec ^3(c+d x)}{96 a d}-\frac{7 \tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac{7 \tan (c+d x) \sec (c+d x)}{256 a d}","-\frac{\tan ^{10}(c+d x)}{10 a d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^7(c+d x) \sec ^3(c+d x)}{10 a d}-\frac{7 \tan ^5(c+d x) \sec ^3(c+d x)}{80 a d}+\frac{7 \tan ^3(c+d x) \sec ^3(c+d x)}{96 a d}-\frac{7 \tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac{7 \tan (c+d x) \sec (c+d x)}{256 a d}",1,"(7*ArcTanh[Sin[c + d*x]])/(256*a*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) - (7*Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) + (7*Sec[c + d*x]^3*Tan[c + d*x]^3)/(96*a*d) - (7*Sec[c + d*x]^3*Tan[c + d*x]^5)/(80*a*d) + (Sec[c + d*x]^3*Tan[c + d*x]^7)/(10*a*d) - Tan[c + d*x]^10/(10*a*d)","A",9,6,27,0.2222,1,"{2835, 2611, 3768, 3770, 2607, 30}"
900,1,178,0,0.2804507,"\int \frac{\sec ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^2*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{\tan ^{10}(c+d x)}{10 a d}+\frac{\tan ^8(c+d x)}{8 a d}-\frac{7 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\tan ^7(c+d x) \sec ^3(c+d x)}{10 a d}+\frac{7 \tan ^5(c+d x) \sec ^3(c+d x)}{80 a d}-\frac{7 \tan ^3(c+d x) \sec ^3(c+d x)}{96 a d}+\frac{7 \tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac{7 \tan (c+d x) \sec (c+d x)}{256 a d}","\frac{\tan ^{10}(c+d x)}{10 a d}+\frac{\tan ^8(c+d x)}{8 a d}-\frac{7 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\tan ^7(c+d x) \sec ^3(c+d x)}{10 a d}+\frac{7 \tan ^5(c+d x) \sec ^3(c+d x)}{80 a d}-\frac{7 \tan ^3(c+d x) \sec ^3(c+d x)}{96 a d}+\frac{7 \tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac{7 \tan (c+d x) \sec (c+d x)}{256 a d}",1,"(-7*ArcTanh[Sin[c + d*x]])/(256*a*d) - (7*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (7*Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) - (7*Sec[c + d*x]^3*Tan[c + d*x]^3)/(96*a*d) + (7*Sec[c + d*x]^3*Tan[c + d*x]^5)/(80*a*d) - (Sec[c + d*x]^3*Tan[c + d*x]^7)/(10*a*d) + Tan[c + d*x]^8/(8*a*d) + Tan[c + d*x]^10/(10*a*d)","A",10,6,29,0.2069,1,"{2835, 2607, 14, 2611, 3768, 3770}"
901,1,176,0,0.2662182,"\int \frac{\sec ^3(c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^3*Tan[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","-\frac{\tan ^{10}(c+d x)}{10 a d}-\frac{\tan ^8(c+d x)}{8 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^5(c+d x) \sec ^5(c+d x)}{10 a d}-\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{16 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{32 a d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d}","-\frac{\tan ^{10}(c+d x)}{10 a d}-\frac{\tan ^8(c+d x)}{8 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^5(c+d x) \sec ^5(c+d x)}{10 a d}-\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{16 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{32 a d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d}",1,"(-3*ArcTanh[Sin[c + d*x]])/(256*a*d) - (3*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) + (Sec[c + d*x]^5*Tan[c + d*x])/(32*a*d) - (Sec[c + d*x]^5*Tan[c + d*x]^3)/(16*a*d) + (Sec[c + d*x]^5*Tan[c + d*x]^5)/(10*a*d) - Tan[c + d*x]^8/(8*a*d) - Tan[c + d*x]^10/(10*a*d)","A",10,6,29,0.2069,1,"{2835, 2611, 3768, 3770, 2607, 14}"
902,1,194,0,0.2731138,"\int \frac{\sec ^4(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^4*Tan[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","\frac{\sec ^{10}(c+d x)}{10 a d}-\frac{\sec ^8(c+d x)}{4 a d}+\frac{\sec ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\tan ^5(c+d x) \sec ^5(c+d x)}{10 a d}+\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{16 a d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{32 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d}","\frac{\sec ^{10}(c+d x)}{10 a d}-\frac{\sec ^8(c+d x)}{4 a d}+\frac{\sec ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\tan ^5(c+d x) \sec ^5(c+d x)}{10 a d}+\frac{\tan ^3(c+d x) \sec ^5(c+d x)}{16 a d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{32 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d}",1,"(3*ArcTanh[Sin[c + d*x]])/(256*a*d) + Sec[c + d*x]^6/(6*a*d) - Sec[c + d*x]^8/(4*a*d) + Sec[c + d*x]^10/(10*a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) - (Sec[c + d*x]^5*Tan[c + d*x])/(32*a*d) + (Sec[c + d*x]^5*Tan[c + d*x]^3)/(16*a*d) - (Sec[c + d*x]^5*Tan[c + d*x]^5)/(10*a*d)","A",11,7,29,0.2414,1,"{2835, 2606, 266, 43, 2611, 3768, 3770}"
903,1,192,0,0.2528895,"\int \frac{\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^5*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}-\frac{3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{160 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d}","-\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{\sec ^8(c+d x)}{4 a d}-\frac{\sec ^6(c+d x)}{6 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}-\frac{3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac{\tan (c+d x) \sec ^5(c+d x)}{160 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d}",1,"(3*ArcTanh[Sin[c + d*x]])/(256*a*d) - Sec[c + d*x]^6/(6*a*d) + Sec[c + d*x]^8/(4*a*d) - Sec[c + d*x]^10/(10*a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) + (Sec[c + d*x]^5*Tan[c + d*x])/(160*a*d) - (3*Sec[c + d*x]^7*Tan[c + d*x])/(80*a*d) + (Sec[c + d*x]^7*Tan[c + d*x]^3)/(10*a*d)","A",11,7,29,0.2414,1,"{2835, 2611, 3768, 3770, 2606, 266, 43}"
904,1,174,0,0.2415455,"\int \frac{\sec ^6(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^6*Tan[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\sec ^{10}(c+d x)}{10 a d}-\frac{\sec ^8(c+d x)}{8 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}+\frac{3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{160 a d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d}","\frac{\sec ^{10}(c+d x)}{10 a d}-\frac{\sec ^8(c+d x)}{8 a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}+\frac{3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{160 a d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{128 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{256 a d}",1,"(-3*ArcTanh[Sin[c + d*x]])/(256*a*d) - Sec[c + d*x]^8/(8*a*d) + Sec[c + d*x]^10/(10*a*d) - (3*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) - (Sec[c + d*x]^5*Tan[c + d*x])/(160*a*d) + (3*Sec[c + d*x]^7*Tan[c + d*x])/(80*a*d) - (Sec[c + d*x]^7*Tan[c + d*x]^3)/(10*a*d)","A",10,6,29,0.2069,1,"{2835, 2606, 14, 2611, 3768, 3770}"
905,1,172,0,0.2191428,"\int \frac{\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^7*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{\sec ^8(c+d x)}{8 a d}-\frac{7 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan (c+d x) \sec ^9(c+d x)}{10 a d}-\frac{\tan (c+d x) \sec ^7(c+d x)}{80 a d}-\frac{7 \tan (c+d x) \sec ^5(c+d x)}{480 a d}-\frac{7 \tan (c+d x) \sec ^3(c+d x)}{384 a d}-\frac{7 \tan (c+d x) \sec (c+d x)}{256 a d}","-\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{\sec ^8(c+d x)}{8 a d}-\frac{7 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{\tan (c+d x) \sec ^9(c+d x)}{10 a d}-\frac{\tan (c+d x) \sec ^7(c+d x)}{80 a d}-\frac{7 \tan (c+d x) \sec ^5(c+d x)}{480 a d}-\frac{7 \tan (c+d x) \sec ^3(c+d x)}{384 a d}-\frac{7 \tan (c+d x) \sec (c+d x)}{256 a d}",1,"(-7*ArcTanh[Sin[c + d*x]])/(256*a*d) + Sec[c + d*x]^8/(8*a*d) - Sec[c + d*x]^10/(10*a*d) - (7*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) - (7*Sec[c + d*x]^3*Tan[c + d*x])/(384*a*d) - (7*Sec[c + d*x]^5*Tan[c + d*x])/(480*a*d) - (Sec[c + d*x]^7*Tan[c + d*x])/(80*a*d) + (Sec[c + d*x]^9*Tan[c + d*x])/(10*a*d)","A",10,6,29,0.2069,1,"{2835, 2611, 3768, 3770, 2606, 14}"
906,1,154,0,0.1652159,"\int \frac{\sec ^8(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^8*Tan[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\tan (c+d x) \sec ^9(c+d x)}{10 a d}+\frac{\tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac{7 \tan (c+d x) \sec ^5(c+d x)}{480 a d}+\frac{7 \tan (c+d x) \sec ^3(c+d x)}{384 a d}+\frac{7 \tan (c+d x) \sec (c+d x)}{256 a d}","\frac{\sec ^{10}(c+d x)}{10 a d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{256 a d}-\frac{\tan (c+d x) \sec ^9(c+d x)}{10 a d}+\frac{\tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac{7 \tan (c+d x) \sec ^5(c+d x)}{480 a d}+\frac{7 \tan (c+d x) \sec ^3(c+d x)}{384 a d}+\frac{7 \tan (c+d x) \sec (c+d x)}{256 a d}",1,"(7*ArcTanh[Sin[c + d*x]])/(256*a*d) + Sec[c + d*x]^10/(10*a*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (7*Sec[c + d*x]^3*Tan[c + d*x])/(384*a*d) + (7*Sec[c + d*x]^5*Tan[c + d*x])/(480*a*d) + (Sec[c + d*x]^7*Tan[c + d*x])/(80*a*d) - (Sec[c + d*x]^9*Tan[c + d*x])/(10*a*d)","A",9,6,27,0.2222,1,"{2835, 2606, 30, 2611, 3768, 3770}"
907,1,210,0,0.1670366,"\int \frac{\sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^9/(a + a*Sin[c + d*x]),x]","-\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}-\frac{5 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{64 d (a-a \sin (c+d x))^3}-\frac{5 a^2}{128 d (a \sin (c+d x)+a)^3}+\frac{21 a}{512 d (a-a \sin (c+d x))^2}-\frac{35 a}{512 d (a \sin (c+d x)+a)^2}+\frac{7}{64 d (a-a \sin (c+d x))}-\frac{35}{256 d (a \sin (c+d x)+a)}+\frac{63 \tanh ^{-1}(\sin (c+d x))}{256 a d}","-\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}-\frac{5 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{64 d (a-a \sin (c+d x))^3}-\frac{5 a^2}{128 d (a \sin (c+d x)+a)^3}+\frac{21 a}{512 d (a-a \sin (c+d x))^2}-\frac{35 a}{512 d (a \sin (c+d x)+a)^2}+\frac{7}{64 d (a-a \sin (c+d x))}-\frac{35}{256 d (a \sin (c+d x)+a)}+\frac{63 \tanh ^{-1}(\sin (c+d x))}{256 a d}",1,"(63*ArcTanh[Sin[c + d*x]])/(256*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + a^2/(64*d*(a - a*Sin[c + d*x])^3) + (21*a)/(512*d*(a - a*Sin[c + d*x])^2) + 7/(64*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) - (5*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (5*a^2)/(128*d*(a + a*Sin[c + d*x])^3) - (35*a)/(512*d*(a + a*Sin[c + d*x])^2) - 35/(256*d*(a + a*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
908,1,247,0,0.2444145,"\int \frac{\csc (c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{7 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{48 d (a-a \sin (c+d x))^3}+\frac{29 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{37 a}{512 d (a-a \sin (c+d x))^2}+\frac{93 a}{512 d (a \sin (c+d x)+a)^2}+\frac{65}{256 d (a-a \sin (c+d x))}+\frac{1}{2 d (a \sin (c+d x)+a)}-\frac{193 \log (1-\sin (c+d x))}{512 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{319 \log (\sin (c+d x)+1)}{512 a d}","\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{7 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{48 d (a-a \sin (c+d x))^3}+\frac{29 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{37 a}{512 d (a-a \sin (c+d x))^2}+\frac{93 a}{512 d (a \sin (c+d x)+a)^2}+\frac{65}{256 d (a-a \sin (c+d x))}+\frac{1}{2 d (a \sin (c+d x)+a)}-\frac{193 \log (1-\sin (c+d x))}{512 a d}+\frac{\log (\sin (c+d x))}{a d}-\frac{319 \log (\sin (c+d x)+1)}{512 a d}",1,"(-193*Log[1 - Sin[c + d*x]])/(512*a*d) + Log[Sin[c + d*x]]/(a*d) - (319*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + a^2/(48*d*(a - a*Sin[c + d*x])^3) + (37*a)/(512*d*(a - a*Sin[c + d*x])^2) + 65/(256*d*(a - a*Sin[c + d*x])) + a^4/(160*d*(a + a*Sin[c + d*x])^5) + (7*a^3)/(256*d*(a + a*Sin[c + d*x])^4) + (29*a^2)/(384*d*(a + a*Sin[c + d*x])^3) + (93*a)/(512*d*(a + a*Sin[c + d*x])^2) + 1/(2*d*(a + a*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2836, 12, 88}"
909,1,262,0,0.2908251,"\int \frac{\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","-\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}-\frac{9 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{5 a^2}{192 d (a-a \sin (c+d x))^3}-\frac{47 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{57 a}{512 d (a-a \sin (c+d x))^2}-\frac{187 a}{512 d (a \sin (c+d x)+a)^2}+\frac{61}{128 d (a-a \sin (c+d x))}-\frac{315}{256 d (a \sin (c+d x)+a)}-\frac{\csc (c+d x)}{a d}-\frac{437 \log (1-\sin (c+d x))}{512 a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{949 \log (\sin (c+d x)+1)}{512 a d}","-\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}-\frac{9 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{5 a^2}{192 d (a-a \sin (c+d x))^3}-\frac{47 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac{57 a}{512 d (a-a \sin (c+d x))^2}-\frac{187 a}{512 d (a \sin (c+d x)+a)^2}+\frac{61}{128 d (a-a \sin (c+d x))}-\frac{315}{256 d (a \sin (c+d x)+a)}-\frac{\csc (c+d x)}{a d}-\frac{437 \log (1-\sin (c+d x))}{512 a d}-\frac{\log (\sin (c+d x))}{a d}+\frac{949 \log (\sin (c+d x)+1)}{512 a d}",1,"-(Csc[c + d*x]/(a*d)) - (437*Log[1 - Sin[c + d*x]])/(512*a*d) - Log[Sin[c + d*x]]/(a*d) + (949*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + (5*a^2)/(192*d*(a - a*Sin[c + d*x])^3) + (57*a)/(512*d*(a - a*Sin[c + d*x])^2) + 61/(128*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) - (9*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (47*a^2)/(384*d*(a + a*Sin[c + d*x])^3) - (187*a)/(512*d*(a + a*Sin[c + d*x])^2) - 315/(256*d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
910,1,279,0,0.3024059,"\int \frac{\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^3*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{11 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{32 d (a-a \sin (c+d x))^3}+\frac{23 a^2}{128 d (a \sin (c+d x)+a)^3}+\frac{81 a}{512 d (a-a \sin (c+d x))^2}+\frac{325 a}{512 d (a \sin (c+d x)+a)^2}+\frac{203}{256 d (a-a \sin (c+d x))}+\frac{5}{2 d (a \sin (c+d x)+a)}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}-\frac{843 \log (1-\sin (c+d x))}{512 a d}+\frac{6 \log (\sin (c+d x))}{a d}-\frac{2229 \log (\sin (c+d x)+1)}{512 a d}","\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{11 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{32 d (a-a \sin (c+d x))^3}+\frac{23 a^2}{128 d (a \sin (c+d x)+a)^3}+\frac{81 a}{512 d (a-a \sin (c+d x))^2}+\frac{325 a}{512 d (a \sin (c+d x)+a)^2}+\frac{203}{256 d (a-a \sin (c+d x))}+\frac{5}{2 d (a \sin (c+d x)+a)}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\csc (c+d x)}{a d}-\frac{843 \log (1-\sin (c+d x))}{512 a d}+\frac{6 \log (\sin (c+d x))}{a d}-\frac{2229 \log (\sin (c+d x)+1)}{512 a d}",1,"Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - (843*Log[1 - Sin[c + d*x]])/(512*a*d) + (6*Log[Sin[c + d*x]])/(a*d) - (2229*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + a^2/(32*d*(a - a*Sin[c + d*x])^3) + (81*a)/(512*d*(a - a*Sin[c + d*x])^2) + 203/(256*d*(a - a*Sin[c + d*x])) + a^4/(160*d*(a + a*Sin[c + d*x])^5) + (11*a^3)/(256*d*(a + a*Sin[c + d*x])^4) + (23*a^2)/(128*d*(a + a*Sin[c + d*x])^3) + (325*a)/(512*d*(a + a*Sin[c + d*x])^2) + 5/(2*d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 12, 88}"
911,1,127,0,0.3494342,"\int (g \sec (e+f x))^p (d \sin (e+f x))^n (a+a \sin (e+f x))^m \, dx","Int[(g*Sec[e + f*x])^p*(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^m,x]","\frac{\sec (e+f x) (1-\sin (e+f x))^{\frac{p+1}{2}} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{n+1} (g \sec (e+f x))^p (\sin (e+f x)+1)^{\frac{1}{2} (-2 m+p+1)} F_1\left(n+1;\frac{p+1}{2},\frac{1}{2} (-2 m+p+1);n+2;\sin (e+f x),-\sin (e+f x)\right)}{d f (n+1)}","\frac{\sec (e+f x) (1-\sin (e+f x))^{\frac{p+1}{2}} (a \sin (e+f x)+a)^m (d \sin (e+f x))^{n+1} (g \sec (e+f x))^p (\sin (e+f x)+1)^{\frac{1}{2} (-2 m+p+1)} F_1\left(n+1;\frac{p+1}{2},\frac{1}{2} (-2 m+p+1);n+2;\sin (e+f x),-\sin (e+f x)\right)}{d f (n+1)}",1,"(AppellF1[1 + n, (1 + p)/2, (1 - 2*m + p)/2, 2 + n, Sin[e + f*x], -Sin[e + f*x]]*Sec[e + f*x]*(g*Sec[e + f*x])^p*(1 - Sin[e + f*x])^((1 + p)/2)*(d*Sin[e + f*x])^(1 + n)*(1 + Sin[e + f*x])^((1 - 2*m + p)/2)*(a + a*Sin[e + f*x])^m)/(d*f*(1 + n))","A",5,4,33,0.1212,1,"{2926, 2886, 135, 133}"
912,1,88,0,0.1397696,"\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","\frac{(a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} \, _2F_1\left(m+1,-n;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (m+1)}","\frac{(a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} \, _2F_1\left(m+1,-n;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (m+1)}",1,"(Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(a*f*(1 + m)*((c + d*Sin[e + f*x])/(c - d))^n)","A",3,3,31,0.09677,1,"{2833, 70, 69}"
913,1,175,0,0.2058806,"\int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^4*(c + d*Sin[e + f*x])^n,x]","\frac{a^4 (c-d)^4 (c+d \sin (e+f x))^{n+1}}{d^5 f (n+1)}-\frac{4 a^4 (c-d)^3 (c+d \sin (e+f x))^{n+2}}{d^5 f (n+2)}+\frac{6 a^4 (c-d)^2 (c+d \sin (e+f x))^{n+3}}{d^5 f (n+3)}-\frac{4 a^4 (c-d) (c+d \sin (e+f x))^{n+4}}{d^5 f (n+4)}+\frac{a^4 (c+d \sin (e+f x))^{n+5}}{d^5 f (n+5)}","\frac{a^4 (c-d)^4 (c+d \sin (e+f x))^{n+1}}{d^5 f (n+1)}-\frac{4 a^4 (c-d)^3 (c+d \sin (e+f x))^{n+2}}{d^5 f (n+2)}+\frac{6 a^4 (c-d)^2 (c+d \sin (e+f x))^{n+3}}{d^5 f (n+3)}-\frac{4 a^4 (c-d) (c+d \sin (e+f x))^{n+4}}{d^5 f (n+4)}+\frac{a^4 (c+d \sin (e+f x))^{n+5}}{d^5 f (n+5)}",1,"(a^4*(c - d)^4*(c + d*Sin[e + f*x])^(1 + n))/(d^5*f*(1 + n)) - (4*a^4*(c - d)^3*(c + d*Sin[e + f*x])^(2 + n))/(d^5*f*(2 + n)) + (6*a^4*(c - d)^2*(c + d*Sin[e + f*x])^(3 + n))/(d^5*f*(3 + n)) - (4*a^4*(c - d)*(c + d*Sin[e + f*x])^(4 + n))/(d^5*f*(4 + n)) + (a^4*(c + d*Sin[e + f*x])^(5 + n))/(d^5*f*(5 + n))","A",3,2,31,0.06452,1,"{2833, 43}"
914,1,139,0,0.1654516,"\int \cos (e+f x) (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n,x]","-\frac{a^3 (c-d)^3 (c+d \sin (e+f x))^{n+1}}{d^4 f (n+1)}+\frac{3 a^3 (c-d)^2 (c+d \sin (e+f x))^{n+2}}{d^4 f (n+2)}-\frac{3 a^3 (c-d) (c+d \sin (e+f x))^{n+3}}{d^4 f (n+3)}+\frac{a^3 (c+d \sin (e+f x))^{n+4}}{d^4 f (n+4)}","-\frac{a^3 (c-d)^3 (c+d \sin (e+f x))^{n+1}}{d^4 f (n+1)}+\frac{3 a^3 (c-d)^2 (c+d \sin (e+f x))^{n+2}}{d^4 f (n+2)}-\frac{3 a^3 (c-d) (c+d \sin (e+f x))^{n+3}}{d^4 f (n+3)}+\frac{a^3 (c+d \sin (e+f x))^{n+4}}{d^4 f (n+4)}",1,"-((a^3*(c - d)^3*(c + d*Sin[e + f*x])^(1 + n))/(d^4*f*(1 + n))) + (3*a^3*(c - d)^2*(c + d*Sin[e + f*x])^(2 + n))/(d^4*f*(2 + n)) - (3*a^3*(c - d)*(c + d*Sin[e + f*x])^(3 + n))/(d^4*f*(3 + n)) + (a^3*(c + d*Sin[e + f*x])^(4 + n))/(d^4*f*(4 + n))","A",3,2,31,0.06452,1,"{2833, 43}"
915,1,101,0,0.1499433,"\int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","\frac{a^2 (c-d)^2 (c+d \sin (e+f x))^{n+1}}{d^3 f (n+1)}-\frac{2 a^2 (c-d) (c+d \sin (e+f x))^{n+2}}{d^3 f (n+2)}+\frac{a^2 (c+d \sin (e+f x))^{n+3}}{d^3 f (n+3)}","\frac{a^2 (c-d)^2 (c+d \sin (e+f x))^{n+1}}{d^3 f (n+1)}-\frac{2 a^2 (c-d) (c+d \sin (e+f x))^{n+2}}{d^3 f (n+2)}+\frac{a^2 (c+d \sin (e+f x))^{n+3}}{d^3 f (n+3)}",1,"(a^2*(c - d)^2*(c + d*Sin[e + f*x])^(1 + n))/(d^3*f*(1 + n)) - (2*a^2*(c - d)*(c + d*Sin[e + f*x])^(2 + n))/(d^3*f*(2 + n)) + (a^2*(c + d*Sin[e + f*x])^(3 + n))/(d^3*f*(3 + n))","A",3,2,31,0.06452,1,"{2833, 43}"
916,1,61,0,0.09559,"\int \cos (e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\frac{a (c+d \sin (e+f x))^{n+2}}{d^2 f (n+2)}-\frac{a (c-d) (c+d \sin (e+f x))^{n+1}}{d^2 f (n+1)}","\frac{a (c+d \sin (e+f x))^{n+2}}{d^2 f (n+2)}-\frac{a (c-d) (c+d \sin (e+f x))^{n+1}}{d^2 f (n+1)}",1,"-((a*(c - d)*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(1 + n))) + (a*(c + d*Sin[e + f*x])^(2 + n))/(d^2*f*(2 + n))","A",3,2,29,0.06897,1,"{2833, 43}"
917,1,60,0,0.1244843,"\int \frac{\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","Int[(Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]),x]","-\frac{(c+d \sin (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right)}{a f (n+1) (c-d)}","-\frac{(c+d \sin (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right)}{a f (n+1) (c-d)}",1,"-((Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Sin[e + f*x])/(c - d)]*(c + d*Sin[e + f*x])^(1 + n))/(a*(c - d)*f*(1 + n)))","A",2,2,31,0.06452,1,"{2833, 68}"
918,1,60,0,0.1093142,"\int \frac{\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Int[(Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2,x]","\frac{d (c+d \sin (e+f x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right)}{a^2 f (n+1) (c-d)^2}","\frac{d (c+d \sin (e+f x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right)}{a^2 f (n+1) (c-d)^2}",1,"(d*Hypergeometric2F1[2, 1 + n, 2 + n, (c + d*Sin[e + f*x])/(c - d)]*(c + d*Sin[e + f*x])^(1 + n))/(a^2*(c - d)^2*f*(1 + n))","A",2,2,31,0.06452,1,"{2833, 68}"
919,1,63,0,0.1172611,"\int \frac{\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","Int[(Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^3,x]","-\frac{d^2 (c+d \sin (e+f x))^{n+1} \, _2F_1\left(3,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right)}{a^3 f (n+1) (c-d)^3}","-\frac{d^2 (c+d \sin (e+f x))^{n+1} \, _2F_1\left(3,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right)}{a^3 f (n+1) (c-d)^3}",1,"-((d^2*Hypergeometric2F1[3, 1 + n, 2 + n, (c + d*Sin[e + f*x])/(c - d)]*(c + d*Sin[e + f*x])^(1 + n))/(a^3*(c - d)^3*f*(1 + n)))","A",2,2,31,0.06452,1,"{2833, 68}"
920,1,170,0,0.1818918,"\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx","Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^4,x]","\frac{6 d^2 (c-d)^2 (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{4 d^3 (c-d) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac{4 d (c-d)^3 (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{d^4 (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac{(c-d)^4 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}","\frac{6 d^2 (c-d)^2 (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{4 d^3 (c-d) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac{4 d (c-d)^3 (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{d^4 (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac{(c-d)^4 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}",1,"((c - d)^4*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (4*(c - d)^3*d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m)) + (6*(c - d)^2*d^2*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m)) + (4*(c - d)*d^3*(a + a*Sin[e + f*x])^(4 + m))/(a^4*f*(4 + m)) + (d^4*(a + a*Sin[e + f*x])^(5 + m))/(a^5*f*(5 + m))","A",3,2,31,0.06452,1,"{2833, 43}"
921,1,133,0,0.1400711,"\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx","Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^3,x]","\frac{3 d^2 (c-d) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{3 d (c-d)^2 (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{d^3 (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac{(c-d)^3 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}","\frac{3 d^2 (c-d) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{3 d (c-d)^2 (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{d^3 (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac{(c-d)^3 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}",1,"((c - d)^3*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (3*(c - d)^2*d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m)) + (3*(c - d)*d^2*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m)) + (d^3*(a + a*Sin[e + f*x])^(4 + m))/(a^4*f*(4 + m))","A",3,2,31,0.06452,1,"{2833, 43}"
922,1,96,0,0.1117225,"\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx","Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2,x]","\frac{2 d (c-d) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{d^2 (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{(c-d)^2 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}","\frac{2 d (c-d) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{d^2 (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{(c-d)^2 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}",1,"((c - d)^2*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (2*(c - d)*d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m)) + (d^2*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m))","A",3,2,31,0.06452,1,"{2833, 43}"
923,1,59,0,0.0662621,"\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx","Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x]),x]","\frac{d (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{(c-d) (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}","\frac{d (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{(c-d) (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}",1,"((c - d)*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m))","A",3,2,29,0.06897,1,"{2833, 43}"
924,1,59,0,0.1028362,"\int \frac{\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx","Int[(Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(c + d*Sin[e + f*x]),x]","\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (m+1) (c-d)}","\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (m+1) (c-d)}",1,"(Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m))/(a*(c - d)*f*(1 + m))","A",2,2,31,0.06452,1,"{2833, 68}"
925,1,59,0,0.0971774,"\int \frac{\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx","Int[(Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(c + d*Sin[e + f*x])^2,x]","\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left(2,m+1;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (m+1) (c-d)^2}","\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left(2,m+1;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (m+1) (c-d)^2}",1,"(Hypergeometric2F1[2, 1 + m, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m))/(a*(c - d)^2*f*(1 + m))","A",2,2,31,0.06452,1,"{2833, 68}"
926,1,59,0,0.1031916,"\int \frac{\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx","Int[(Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(c + d*Sin[e + f*x])^3,x]","\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left(3,m+1;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (m+1) (c-d)^3}","\frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left(3,m+1;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (m+1) (c-d)^3}",1,"(Hypergeometric2F1[3, 1 + m, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m))/(a*(c - d)^3*f*(1 + m))","A",2,2,31,0.06452,1,"{2833, 68}"
927,1,61,0,0.0778548,"\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^m,x]","\frac{(\sin (c+d x)+1)^{-m} \sin ^{n+1}(c+d x) (a \sin (c+d x)+a)^m \, _2F_1(-m,n+1;n+2;-\sin (c+d x))}{d (n+1)}","-\frac{\sin ^{n+1}(c+d x) (a \sin (c+d x)+a)^{m+1} \, _2F_1(1,m+n+2;m+2;\sin (c+d x)+1)}{a d (m+1)}",1,"(Hypergeometric2F1[-m, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + a*Sin[c + d*x])^m)/(d*(1 + n)*(1 + Sin[c + d*x])^m)","A",3,3,27,0.1111,1,"{2833, 66, 64}"
928,1,134,0,0.1192516,"\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^m,x]","-\frac{4 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}+\frac{6 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac{4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac{(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}","-\frac{4 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}+\frac{6 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac{4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac{(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}",1,"(a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m)) - (4*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*(2 + m)) + (6*(a + a*Sin[c + d*x])^(3 + m))/(a^3*d*(3 + m)) - (4*(a + a*Sin[c + d*x])^(4 + m))/(a^4*d*(4 + m)) + (a + a*Sin[c + d*x])^(5 + m)/(a^5*d*(5 + m))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
929,1,108,0,0.0969453,"\int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^m,x]","\frac{3 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{3 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}+\frac{(a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}-\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}","\frac{3 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{3 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}+\frac{(a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}-\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}",1,"-((a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m))) + (3*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*(2 + m)) - (3*(a + a*Sin[c + d*x])^(3 + m))/(a^3*d*(3 + m)) + (a + a*Sin[c + d*x])^(4 + m)/(a^4*d*(4 + m))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
930,1,80,0,0.0842618,"\int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^m,x]","-\frac{2 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}+\frac{(a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}+\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}","-\frac{2 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}+\frac{(a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}+\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}",1,"(a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m)) - (2*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*(2 + m)) + (a + a*Sin[c + d*x])^(3 + m)/(a^3*d*(3 + m))","A",4,3,27,0.1111,1,"{2833, 12, 43}"
931,1,54,0,0.0526407,"\int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x])^m,x]","\frac{(a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}","\frac{(a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}",1,"-((a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m))) + (a + a*Sin[c + d*x])^(2 + m)/(a^2*d*(2 + m))","A",4,3,25,0.1200,1,"{2833, 12, 43}"
932,1,43,0,0.0414507,"\int \cot (c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cot[c + d*x]*(a + a*Sin[c + d*x])^m,x]","-\frac{(a \sin (c+d x)+a)^{m+1} \, _2F_1(1,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)}","-\frac{(a \sin (c+d x)+a)^{m+1} \, _2F_1(1,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)}",1,"-((Hypergeometric2F1[1, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x])^(1 + m))/(a*d*(1 + m)))","A",2,2,19,0.1053,1,"{2707, 65}"
933,1,42,0,0.0579094,"\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^m,x]","\frac{(a \sin (c+d x)+a)^{m+1} \, _2F_1(2,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)}","\frac{(a \sin (c+d x)+a)^{m+1} \, _2F_1(2,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)}",1,"(Hypergeometric2F1[2, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x])^(1 + m))/(a*d*(1 + m))","A",3,3,25,0.1200,1,"{2833, 12, 65}"
934,1,43,0,0.0811075,"\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^m,x]","-\frac{(a \sin (c+d x)+a)^{m+1} \, _2F_1(3,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)}","-\frac{(a \sin (c+d x)+a)^{m+1} \, _2F_1(3,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)}",1,"-((Hypergeometric2F1[3, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x])^(1 + m))/(a*d*(1 + m)))","A",3,3,27,0.1111,1,"{2833, 12, 65}"
935,1,84,0,0.0918696,"\int \cos ^2(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","-\frac{a (4 c+d) \cos ^3(e+f x)}{12 f}+\frac{a (4 c+d) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a x (4 c+d)-\frac{d \cos ^3(e+f x) (a \sin (e+f x)+a)}{4 f}","-\frac{a (c+d) \cos ^3(e+f x)}{3 f}+\frac{a (4 c+d) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a x (4 c+d)-\frac{a d \sin (e+f x) \cos ^3(e+f x)}{4 f}",1,"(a*(4*c + d)*x)/8 - (a*(4*c + d)*Cos[e + f*x]^3)/(12*f) + (a*(4*c + d)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (d*Cos[e + f*x]^3*(a + a*Sin[e + f*x]))/(4*f)","A",4,4,29,0.1379,1,"{2860, 2669, 2635, 8}"
936,1,123,0,0.5513691,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))} \, dx","Int[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])),x]","\frac{2 \sqrt{c+d} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} \sqrt{d} f (c-d)}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f (c-d)}","\frac{2 \sqrt{c+d} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} \sqrt{d} f (c-d)}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right)}{a^{3/2} f (c-d)}",1,"(-2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*(c - d)*f) + (2*Sqrt[c + d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*(c - d)*Sqrt[d]*f)","A",6,6,35,0.1714,1,"{2916, 2985, 2649, 206, 2773, 208}"
937,1,141,0,0.6715452,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx","Int[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]),x]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{3/2} \sqrt{d} f}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{3/2} f \sqrt{c-d}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{3/2} \sqrt{d} f}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right)}{a^{3/2} f \sqrt{c-d}}",1,"(2*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(3/2)*Sqrt[d]*f) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(3/2)*Sqrt[c - d]*f)","A",6,6,37,0.1622,1,"{2916, 2982, 2782, 208, 2775, 205}"
938,1,135,0,0.2449266,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","\frac{2 \sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(m+\frac{3}{2};-\frac{1}{2},-n;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m+3) \sqrt{1-\sin (e+f x)}}","\frac{2 \sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(m+\frac{3}{2};-\frac{1}{2},-n;m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m+3) \sqrt{1-\sin (e+f x)}}",1,"(2*Sqrt[2]*AppellF1[3/2 + m, -1/2, -n, 5/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(a*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c - d))^n)","A",4,4,33,0.1212,1,"{2918, 140, 139, 138}"
939,1,119,0,0.1803053,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n,x]","-\frac{16 \sqrt{2} a^3 (1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};-\frac{7}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{3 f \sqrt{\sin (e+f x)+1}}","-\frac{16 \sqrt{2} a^3 (1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};-\frac{7}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{3 f \sqrt{\sin (e+f x)+1}}",1,"(-16*Sqrt[2]*a^3*AppellF1[3/2, -7/2, -n, 5/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(3*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,33,0.09091,1,"{2917, 139, 138}"
940,1,119,0,0.1782333,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","-\frac{8 \sqrt{2} a^2 (1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};-\frac{5}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{3 f \sqrt{\sin (e+f x)+1}}","-\frac{8 \sqrt{2} a^2 (1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};-\frac{5}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{3 f \sqrt{\sin (e+f x)+1}}",1,"(-8*Sqrt[2]*a^2*AppellF1[3/2, -5/2, -n, 5/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(3*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,33,0.09091,1,"{2917, 139, 138}"
941,1,117,0,0.1308255,"\int \cos ^2(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","-\frac{4 \sqrt{2} a (1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};-\frac{3}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{3 f \sqrt{\sin (e+f x)+1}}","-\frac{4 \sqrt{2} a (1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};-\frac{3}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{3 f \sqrt{\sin (e+f x)+1}}",1,"(-4*Sqrt[2]*a*AppellF1[3/2, -3/2, -n, 5/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(3*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,31,0.09677,1,"{2868, 139, 138}"
942,1,119,0,0.2288024,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","Int[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]),x]","-\frac{\sqrt{2} (1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{3 a f \sqrt{\sin (e+f x)+1}}","-\frac{\sqrt{2} (1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{3 a f \sqrt{\sin (e+f x)+1}}",1,"-(Sqrt[2]*AppellF1[3/2, 1/2, -n, 5/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(3*a*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",4,4,33,0.1212,1,"{2914, 2755, 139, 138}"
943,1,119,0,0.2160909,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Int[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2,x]","-\frac{(1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};\frac{3}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{3 \sqrt{2} a^2 f \sqrt{\sin (e+f x)+1}}","-\frac{(1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};\frac{3}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{3 \sqrt{2} a^2 f \sqrt{\sin (e+f x)+1}}",1,"-(AppellF1[3/2, 3/2, -n, 5/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(3*Sqrt[2]*a^2*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,33,0.09091,1,"{2917, 139, 138}"
944,1,119,0,0.2136852,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","Int[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^3,x]","-\frac{(1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};\frac{5}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{6 \sqrt{2} a^3 f \sqrt{\sin (e+f x)+1}}","-\frac{(1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{3}{2};\frac{5}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{6 \sqrt{2} a^3 f \sqrt{\sin (e+f x)+1}}",1,"-(AppellF1[3/2, 5/2, -n, 5/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(6*Sqrt[2]*a^3*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,33,0.09091,1,"{2917, 139, 138}"
945,1,135,0,0.2261703,"\int \cos ^4(e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^4*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","\frac{4 \sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^{m+2} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(m+\frac{5}{2};-\frac{3}{2},-n;m+\frac{7}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a^2 f (2 m+5) \sqrt{1-\sin (e+f x)}}","\frac{4 \sqrt{2} \cos (e+f x) (a \sin (e+f x)+a)^{m+2} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(m+\frac{5}{2};-\frac{3}{2},-n;m+\frac{7}{2};\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a^2 f (2 m+5) \sqrt{1-\sin (e+f x)}}",1,"(4*Sqrt[2]*AppellF1[5/2 + m, -3/2, -n, 7/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(2 + m)*(c + d*Sin[e + f*x])^n)/(a^2*f*(5 + 2*m)*Sqrt[1 - Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c - d))^n)","A",4,4,33,0.1212,1,"{2918, 140, 139, 138}"
946,1,121,0,0.177085,"\int \cos ^4(e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^4*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","-\frac{16 \sqrt{2} a^2 (1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};-\frac{7}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{5 f \sqrt{\sin (e+f x)+1}}","-\frac{16 \sqrt{2} a^2 (1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};-\frac{7}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{5 f \sqrt{\sin (e+f x)+1}}",1,"(-16*Sqrt[2]*a^2*AppellF1[5/2, -7/2, -n, 7/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(5*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,33,0.09091,1,"{2917, 139, 138}"
947,1,119,0,0.1410125,"\int \cos ^4(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^4*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","-\frac{8 \sqrt{2} a (1-\sin (e+f x)) \cos ^3(e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};-\frac{5}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{5 f (\sin (e+f x)+1)^{3/2}}","-\frac{8 \sqrt{2} a (1-\sin (e+f x)) \cos ^3(e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};-\frac{5}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{5 f (\sin (e+f x)+1)^{3/2}}",1,"(-8*Sqrt[2]*a*AppellF1[5/2, -5/2, -n, 7/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]^3*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(5*f*(1 + Sin[e + f*x])^(3/2)*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,31,0.09677,1,"{2868, 139, 138}"
948,1,121,0,0.1812312,"\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","Int[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]),x]","-\frac{2 \sqrt{2} (1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};-\frac{1}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{5 a f \sqrt{\sin (e+f x)+1}}","-\frac{2 \sqrt{2} (1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};-\frac{1}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{5 a f \sqrt{\sin (e+f x)+1}}",1,"(-2*Sqrt[2]*AppellF1[5/2, -1/2, -n, 7/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(5*a*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,33,0.09091,1,"{2917, 139, 138}"
949,1,121,0,0.244807,"\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Int[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2,x]","-\frac{\sqrt{2} (1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};\frac{1}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{5 a^2 f \sqrt{\sin (e+f x)+1}}","-\frac{\sqrt{2} (1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};\frac{1}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{5 a^2 f \sqrt{\sin (e+f x)+1}}",1,"-(Sqrt[2]*AppellF1[5/2, 1/2, -n, 7/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(5*a^2*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",4,4,33,0.1212,1,"{2914, 2784, 139, 138}"
950,1,121,0,0.18151,"\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","Int[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^3,x]","-\frac{(1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};\frac{3}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{5 \sqrt{2} a^3 f \sqrt{\sin (e+f x)+1}}","-\frac{(1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};\frac{3}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{5 \sqrt{2} a^3 f \sqrt{\sin (e+f x)+1}}",1,"-(AppellF1[5/2, 3/2, -n, 7/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(5*Sqrt[2]*a^3*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,33,0.09091,1,"{2917, 139, 138}"
951,1,121,0,0.1856336,"\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^4} \, dx","Int[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^4,x]","-\frac{(1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};\frac{5}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{10 \sqrt{2} a^4 f \sqrt{\sin (e+f x)+1}}","-\frac{(1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};\frac{5}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{10 \sqrt{2} a^4 f \sqrt{\sin (e+f x)+1}}",1,"-(AppellF1[5/2, 5/2, -n, 7/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(10*Sqrt[2]*a^4*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,33,0.09091,1,"{2917, 139, 138}"
952,1,121,0,0.1784092,"\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^5} \, dx","Int[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^5,x]","-\frac{(1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};\frac{7}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{20 \sqrt{2} a^5 f \sqrt{\sin (e+f x)+1}}","-\frac{(1-\sin (e+f x))^2 \cos (e+f x) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{5}{2};\frac{7}{2},-n;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{20 \sqrt{2} a^5 f \sqrt{\sin (e+f x)+1}}",1,"-(AppellF1[5/2, 7/2, -n, 7/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(20*Sqrt[2]*a^5*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",3,3,33,0.09091,1,"{2917, 139, 138}"
953,1,134,0,0.1420708,"\int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{(A-7 B) (a \sin (c+d x)+a)^8}{8 a^7 d}+\frac{6 (A-3 B) (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac{2 (3 A-5 B) (a \sin (c+d x)+a)^6}{3 a^5 d}+\frac{8 (A-B) (a \sin (c+d x)+a)^5}{5 a^4 d}-\frac{B (a \sin (c+d x)+a)^9}{9 a^8 d}","-\frac{(A-7 B) (a \sin (c+d x)+a)^8}{8 a^7 d}+\frac{6 (A-3 B) (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac{2 (3 A-5 B) (a \sin (c+d x)+a)^6}{3 a^5 d}+\frac{8 (A-B) (a \sin (c+d x)+a)^5}{5 a^4 d}-\frac{B (a \sin (c+d x)+a)^9}{9 a^8 d}",1,"(8*(A - B)*(a + a*Sin[c + d*x])^5)/(5*a^4*d) - (2*(3*A - 5*B)*(a + a*Sin[c + d*x])^6)/(3*a^5*d) + (6*(A - 3*B)*(a + a*Sin[c + d*x])^7)/(7*a^6*d) - ((A - 7*B)*(a + a*Sin[c + d*x])^8)/(8*a^7*d) - (B*(a + a*Sin[c + d*x])^9)/(9*a^8*d)","A",3,2,29,0.06897,1,"{2836, 77}"
954,1,102,0,0.1077098,"\int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{(A-5 B) (a \sin (c+d x)+a)^6}{6 a^5 d}-\frac{4 (A-2 B) (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{a^3 d}+\frac{B (a \sin (c+d x)+a)^7}{7 a^6 d}","\frac{(A-5 B) (a \sin (c+d x)+a)^6}{6 a^5 d}-\frac{4 (A-2 B) (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{a^3 d}+\frac{B (a \sin (c+d x)+a)^7}{7 a^6 d}",1,"((A - B)*(a + a*Sin[c + d*x])^4)/(a^3*d) - (4*(A - 2*B)*(a + a*Sin[c + d*x])^5)/(5*a^4*d) + ((A - 5*B)*(a + a*Sin[c + d*x])^6)/(6*a^5*d) + (B*(a + a*Sin[c + d*x])^7)/(7*a^6*d)","A",3,2,29,0.06897,1,"{2836, 77}"
955,1,78,0,0.0940424,"\int \cos ^3(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{(A-3 B) (a \sin (c+d x)+a)^4}{4 a^3 d}+\frac{2 (A-B) (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac{B (a \sin (c+d x)+a)^5}{5 a^4 d}","-\frac{(A-3 B) (a \sin (c+d x)+a)^4}{4 a^3 d}+\frac{2 (A-B) (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac{B (a \sin (c+d x)+a)^5}{5 a^4 d}",1,"(2*(A - B)*(a + a*Sin[c + d*x])^3)/(3*a^2*d) - ((A - 3*B)*(a + a*Sin[c + d*x])^4)/(4*a^3*d) - (B*(a + a*Sin[c + d*x])^5)/(5*a^4*d)","A",3,2,29,0.06897,1,"{2836, 77}"
956,1,49,0,0.0631775,"\int \cos (c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a (A+B) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}+\frac{a B \sin ^3(c+d x)}{3 d}","\frac{a (A+B) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}+\frac{a B \sin ^3(c+d x)}{3 d}",1,"(a*A*Sin[c + d*x])/d + (a*(A + B)*Sin[c + d*x]^2)/(2*d) + (a*B*Sin[c + d*x]^3)/(3*d)","A",3,2,27,0.07407,1,"{2833, 43}"
957,1,34,0,0.0695984,"\int \sec (c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{a (A+B) \log (1-\sin (c+d x))}{d}-\frac{a B \sin (c+d x)}{d}","-\frac{a (A+B) \log (1-\sin (c+d x))}{d}-\frac{a B \sin (c+d x)}{d}",1,"-((a*(A + B)*Log[1 - Sin[c + d*x]])/d) - (a*B*Sin[c + d*x])/d","A",3,2,27,0.07407,1,"{2836, 43}"
958,1,47,0,0.0864544,"\int \sec ^3(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a^2 (A+B)}{2 d (a-a \sin (c+d x))}+\frac{a (A-B) \tanh ^{-1}(\sin (c+d x))}{2 d}","\frac{a^2 (A+B)}{2 d (a-a \sin (c+d x))}+\frac{a (A-B) \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(a*(A - B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(A + B))/(2*d*(a - a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 77, 206}"
959,1,100,0,0.1220771,"\int \sec ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a^3 (A+B)}{8 d (a-a \sin (c+d x))^2}-\frac{a^2 (A-B)}{8 d (a \sin (c+d x)+a)}+\frac{a^2 A}{4 d (a-a \sin (c+d x))}+\frac{a (3 A-B) \tanh ^{-1}(\sin (c+d x))}{8 d}","\frac{a^3 (A+B)}{8 d (a-a \sin (c+d x))^2}-\frac{a^2 (A-B)}{8 d (a \sin (c+d x)+a)}+\frac{a^2 A}{4 d (a-a \sin (c+d x))}+\frac{a (3 A-B) \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"(a*(3*A - B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(A + B))/(8*d*(a - a*Sin[c + d*x])^2) + (a^2*A)/(4*d*(a - a*Sin[c + d*x])) - (a^2*(A - B))/(8*d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 77, 206}"
960,1,157,0,0.1678529,"\int \sec ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^7*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a^4 (A+B)}{24 d (a-a \sin (c+d x))^3}+\frac{a^3 (3 A+B)}{32 d (a-a \sin (c+d x))^2}-\frac{a^3 (A-B)}{32 d (a \sin (c+d x)+a)^2}-\frac{a^2 (2 A-B)}{16 d (a \sin (c+d x)+a)}+\frac{3 a^2 A}{16 d (a-a \sin (c+d x))}+\frac{a (5 A-B) \tanh ^{-1}(\sin (c+d x))}{16 d}","\frac{a^4 (A+B)}{24 d (a-a \sin (c+d x))^3}+\frac{a^3 (3 A+B)}{32 d (a-a \sin (c+d x))^2}-\frac{a^3 (A-B)}{32 d (a \sin (c+d x)+a)^2}-\frac{a^2 (2 A-B)}{16 d (a \sin (c+d x)+a)}+\frac{3 a^2 A}{16 d (a-a \sin (c+d x))}+\frac{a (5 A-B) \tanh ^{-1}(\sin (c+d x))}{16 d}",1,"(a*(5*A - B)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^4*(A + B))/(24*d*(a - a*Sin[c + d*x])^3) + (a^3*(3*A + B))/(32*d*(a - a*Sin[c + d*x])^2) + (3*a^2*A)/(16*d*(a - a*Sin[c + d*x])) - (a^3*(A - B))/(32*d*(a + a*Sin[c + d*x])^2) - (a^2*(2*A - B))/(16*d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 77, 206}"
961,1,138,0,0.1408024,"\int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac{a (8 A+B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a (8 A+B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a (8 A+B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} a x (8 A+B)-\frac{B \cos ^7(c+d x) (a \sin (c+d x)+a)}{8 d}","-\frac{a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac{a (8 A+B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a (8 A+B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a (8 A+B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} a x (8 A+B)-\frac{B \cos ^7(c+d x) (a \sin (c+d x)+a)}{8 d}",1,"(5*a*(8*A + B)*x)/128 - (a*(8*A + B)*Cos[c + d*x]^7)/(56*d) + (5*a*(8*A + B)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*a*(8*A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a*(8*A + B)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (B*Cos[c + d*x]^7*(a + a*Sin[c + d*x]))/(8*d)","A",6,4,29,0.1379,1,"{2860, 2669, 2635, 8}"
962,1,111,0,0.1126406,"\int \cos ^4(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{a (6 A+B) \cos ^5(c+d x)}{30 d}+\frac{a (6 A+B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a (6 A+B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a x (6 A+B)-\frac{B \cos ^5(c+d x) (a \sin (c+d x)+a)}{6 d}","-\frac{a (6 A+B) \cos ^5(c+d x)}{30 d}+\frac{a (6 A+B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a (6 A+B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a x (6 A+B)-\frac{B \cos ^5(c+d x) (a \sin (c+d x)+a)}{6 d}",1,"(a*(6*A + B)*x)/16 - (a*(6*A + B)*Cos[c + d*x]^5)/(30*d) + (a*(6*A + B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(6*A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (B*Cos[c + d*x]^5*(a + a*Sin[c + d*x]))/(6*d)","A",5,4,29,0.1379,1,"{2860, 2669, 2635, 8}"
963,1,84,0,0.0953011,"\int \cos ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{a (4 A+B) \cos ^3(c+d x)}{12 d}+\frac{a (4 A+B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+B)-\frac{B \cos ^3(c+d x) (a \sin (c+d x)+a)}{4 d}","-\frac{a (4 A+B) \cos ^3(c+d x)}{12 d}+\frac{a (4 A+B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+B)-\frac{B \cos ^3(c+d x) (a \sin (c+d x)+a)}{4 d}",1,"(a*(4*A + B)*x)/8 - (a*(4*A + B)*Cos[c + d*x]^3)/(12*d) + (a*(4*A + B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (B*Cos[c + d*x]^3*(a + a*Sin[c + d*x]))/(4*d)","A",4,4,29,0.1379,1,"{2860, 2669, 2635, 8}"
964,1,29,0,0.0488888,"\int \sec ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{(A+B) \sec (c+d x) (a \sin (c+d x)+a)}{d}-a B x","\frac{(A+B) \sec (c+d x) (a \sin (c+d x)+a)}{d}-a B x",1,"-(a*B*x) + ((A + B)*Sec[c + d*x]*(a + a*Sin[c + d*x]))/d","A",2,2,29,0.06897,1,"{2855, 8}"
965,1,50,0,0.0678648,"\int \sec ^4(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a (2 A-B) \tan (c+d x)}{3 d}+\frac{(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)}{3 d}","\frac{a (2 A-B) \tan (c+d x)}{3 d}+\frac{(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)}{3 d}",1,"((A + B)*Sec[c + d*x]^3*(a + a*Sin[c + d*x]))/(3*d) + (a*(2*A - B)*Tan[c + d*x])/(3*d)","A",3,3,29,0.1034,1,"{2855, 3767, 8}"
966,1,73,0,0.0729733,"\int \sec ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^6*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a (4 A-B) \tan ^3(c+d x)}{15 d}+\frac{a (4 A-B) \tan (c+d x)}{5 d}+\frac{(A+B) \sec ^5(c+d x) (a \sin (c+d x)+a)}{5 d}","\frac{a (4 A-B) \tan ^3(c+d x)}{15 d}+\frac{a (4 A-B) \tan (c+d x)}{5 d}+\frac{(A+B) \sec ^5(c+d x) (a \sin (c+d x)+a)}{5 d}",1,"((A + B)*Sec[c + d*x]^5*(a + a*Sin[c + d*x]))/(5*d) + (a*(4*A - B)*Tan[c + d*x])/(5*d) + (a*(4*A - B)*Tan[c + d*x]^3)/(15*d)","A",3,2,29,0.06897,1,"{2855, 3767}"
967,1,96,0,0.0797963,"\int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^8*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a (6 A-B) \tan ^5(c+d x)}{35 d}+\frac{2 a (6 A-B) \tan ^3(c+d x)}{21 d}+\frac{a (6 A-B) \tan (c+d x)}{7 d}+\frac{(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)}{7 d}","\frac{a (6 A-B) \tan ^5(c+d x)}{35 d}+\frac{2 a (6 A-B) \tan ^3(c+d x)}{21 d}+\frac{a (6 A-B) \tan (c+d x)}{7 d}+\frac{(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)}{7 d}",1,"((A + B)*Sec[c + d*x]^7*(a + a*Sin[c + d*x]))/(7*d) + (a*(6*A - B)*Tan[c + d*x])/(7*d) + (2*a*(6*A - B)*Tan[c + d*x]^3)/(21*d) + (a*(6*A - B)*Tan[c + d*x]^5)/(35*d)","A",3,2,29,0.06897,1,"{2855, 3767}"
968,1,119,0,0.0865134,"\int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^10*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a (8 A-B) \tan ^7(c+d x)}{63 d}+\frac{a (8 A-B) \tan ^5(c+d x)}{15 d}+\frac{a (8 A-B) \tan ^3(c+d x)}{9 d}+\frac{a (8 A-B) \tan (c+d x)}{9 d}+\frac{(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)}{9 d}","\frac{a (8 A-B) \tan ^7(c+d x)}{63 d}+\frac{a (8 A-B) \tan ^5(c+d x)}{15 d}+\frac{a (8 A-B) \tan ^3(c+d x)}{9 d}+\frac{a (8 A-B) \tan (c+d x)}{9 d}+\frac{(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)}{9 d}",1,"((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x]))/(9*d) + (a*(8*A - B)*Tan[c + d*x])/(9*d) + (a*(8*A - B)*Tan[c + d*x]^3)/(9*d) + (a*(8*A - B)*Tan[c + d*x]^5)/(15*d) + (a*(8*A - B)*Tan[c + d*x]^7)/(63*d)","A",3,2,29,0.06897,1,"{2855, 3767}"
969,1,134,0,0.1777662,"\int \cos ^7(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{(A-7 B) (a \sin (c+d x)+a)^9}{9 a^7 d}+\frac{3 (A-3 B) (a \sin (c+d x)+a)^8}{4 a^6 d}-\frac{4 (3 A-5 B) (a \sin (c+d x)+a)^7}{7 a^5 d}+\frac{4 (A-B) (a \sin (c+d x)+a)^6}{3 a^4 d}-\frac{B (a \sin (c+d x)+a)^{10}}{10 a^8 d}","-\frac{(A-7 B) (a \sin (c+d x)+a)^9}{9 a^7 d}+\frac{3 (A-3 B) (a \sin (c+d x)+a)^8}{4 a^6 d}-\frac{4 (3 A-5 B) (a \sin (c+d x)+a)^7}{7 a^5 d}+\frac{4 (A-B) (a \sin (c+d x)+a)^6}{3 a^4 d}-\frac{B (a \sin (c+d x)+a)^{10}}{10 a^8 d}",1,"(4*(A - B)*(a + a*Sin[c + d*x])^6)/(3*a^4*d) - (4*(3*A - 5*B)*(a + a*Sin[c + d*x])^7)/(7*a^5*d) + (3*(A - 3*B)*(a + a*Sin[c + d*x])^8)/(4*a^6*d) - ((A - 7*B)*(a + a*Sin[c + d*x])^9)/(9*a^7*d) - (B*(a + a*Sin[c + d*x])^10)/(10*a^8*d)","A",3,2,31,0.06452,1,"{2836, 77}"
970,1,105,0,0.1486387,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{(A-5 B) (a \sin (c+d x)+a)^7}{7 a^5 d}-\frac{2 (A-2 B) (a \sin (c+d x)+a)^6}{3 a^4 d}+\frac{4 (A-B) (a \sin (c+d x)+a)^5}{5 a^3 d}+\frac{B (a \sin (c+d x)+a)^8}{8 a^6 d}","\frac{(A-5 B) (a \sin (c+d x)+a)^7}{7 a^5 d}-\frac{2 (A-2 B) (a \sin (c+d x)+a)^6}{3 a^4 d}+\frac{4 (A-B) (a \sin (c+d x)+a)^5}{5 a^3 d}+\frac{B (a \sin (c+d x)+a)^8}{8 a^6 d}",1,"(4*(A - B)*(a + a*Sin[c + d*x])^5)/(5*a^3*d) - (2*(A - 2*B)*(a + a*Sin[c + d*x])^6)/(3*a^4*d) + ((A - 5*B)*(a + a*Sin[c + d*x])^7)/(7*a^5*d) + (B*(a + a*Sin[c + d*x])^8)/(8*a^6*d)","A",3,2,31,0.06452,1,"{2836, 77}"
971,1,78,0,0.1091705,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{(A-3 B) (a \sin (c+d x)+a)^5}{5 a^3 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{2 a^2 d}-\frac{B (a \sin (c+d x)+a)^6}{6 a^4 d}","-\frac{(A-3 B) (a \sin (c+d x)+a)^5}{5 a^3 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{2 a^2 d}-\frac{B (a \sin (c+d x)+a)^6}{6 a^4 d}",1,"((A - B)*(a + a*Sin[c + d*x])^4)/(2*a^2*d) - ((A - 3*B)*(a + a*Sin[c + d*x])^5)/(5*a^3*d) - (B*(a + a*Sin[c + d*x])^6)/(6*a^4*d)","A",3,2,31,0.06452,1,"{2836, 77}"
972,1,51,0,0.069862,"\int \cos (c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{B (a \sin (c+d x)+a)^4}{4 a^2 d}+\frac{(A-B) (a \sin (c+d x)+a)^3}{3 a d}","\frac{B (a \sin (c+d x)+a)^4}{4 a^2 d}+\frac{(A-B) (a \sin (c+d x)+a)^3}{3 a d}",1,"((A - B)*(a + a*Sin[c + d*x])^3)/(3*a*d) + (B*(a + a*Sin[c + d*x])^4)/(4*a^2*d)","A",3,2,29,0.06897,1,"{2833, 43}"
973,1,60,0,0.0962114,"\int \sec (c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{a^2 (A+B) \sin (c+d x)}{d}-\frac{2 a^2 (A+B) \log (1-\sin (c+d x))}{d}-\frac{B (a \sin (c+d x)+a)^2}{2 d}","-\frac{a^2 (A+B) \sin (c+d x)}{d}-\frac{2 a^2 (A+B) \log (1-\sin (c+d x))}{d}-\frac{B (a \sin (c+d x)+a)^2}{2 d}",1,"(-2*a^2*(A + B)*Log[1 - Sin[c + d*x]])/d - (a^2*(A + B)*Sin[c + d*x])/d - (B*(a + a*Sin[c + d*x])^2)/(2*d)","A",3,2,29,0.06897,1,"{2836, 77}"
974,1,43,0,0.0903995,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^3 (A+B)}{d (a-a \sin (c+d x))}+\frac{a^2 B \log (1-\sin (c+d x))}{d}","\frac{a^3 (A+B)}{d (a-a \sin (c+d x))}+\frac{a^2 B \log (1-\sin (c+d x))}{d}",1,"(a^2*B*Log[1 - Sin[c + d*x]])/d + (a^3*(A + B))/(d*(a - a*Sin[c + d*x]))","A",3,2,31,0.06452,1,"{2836, 43}"
975,1,77,0,0.1189007,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^4 (A+B)}{4 d (a-a \sin (c+d x))^2}+\frac{a^3 (A-B)}{4 d (a-a \sin (c+d x))}+\frac{a^2 (A-B) \tanh ^{-1}(\sin (c+d x))}{4 d}","\frac{a^4 (A+B)}{4 d (a-a \sin (c+d x))^2}+\frac{a^3 (A-B)}{4 d (a-a \sin (c+d x))}+\frac{a^2 (A-B) \tanh ^{-1}(\sin (c+d x))}{4 d}",1,"(a^2*(A - B)*ArcTanh[Sin[c + d*x]])/(4*d) + (a^4*(A + B))/(4*d*(a - a*Sin[c + d*x])^2) + (a^3*(A - B))/(4*d*(a - a*Sin[c + d*x]))","A",4,3,31,0.09677,1,"{2836, 77, 206}"
976,1,132,0,0.1567477,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^5 (A+B)}{12 d (a-a \sin (c+d x))^3}+\frac{a^3 (3 A-B)}{16 d (a-a \sin (c+d x))}-\frac{a^3 (A-B)}{16 d (a \sin (c+d x)+a)}+\frac{a^2 (2 A-B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 A}{8 d (a-a \sin (c+d x))^2}","\frac{a^5 (A+B)}{12 d (a-a \sin (c+d x))^3}+\frac{a^3 (3 A-B)}{16 d (a-a \sin (c+d x))}-\frac{a^3 (A-B)}{16 d (a \sin (c+d x)+a)}+\frac{a^2 (2 A-B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 A}{8 d (a-a \sin (c+d x))^2}",1,"(a^2*(2*A - B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^5*(A + B))/(12*d*(a - a*Sin[c + d*x])^3) + (a^4*A)/(8*d*(a - a*Sin[c + d*x])^2) + (a^3*(3*A - B))/(16*d*(a - a*Sin[c + d*x])) - (a^3*(A - B))/(16*d*(a + a*Sin[c + d*x]))","A",4,3,31,0.09677,1,"{2836, 77, 206}"
977,1,196,0,0.2110178,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}-\frac{(9 A+2 B) \cos ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{72 d}+\frac{a^2 (9 A+2 B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a^2 (9 A+2 B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a^2 (9 A+2 B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} a^2 x (9 A+2 B)-\frac{B \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}","-\frac{a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}-\frac{(9 A+2 B) \cos ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{72 d}+\frac{a^2 (9 A+2 B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a^2 (9 A+2 B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a^2 (9 A+2 B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} a^2 x (9 A+2 B)-\frac{B \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}",1,"(5*a^2*(9*A + 2*B)*x)/128 - (a^2*(9*A + 2*B)*Cos[c + d*x]^7)/(56*d) + (5*a^2*(9*A + 2*B)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*a^2*(9*A + 2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a^2*(9*A + 2*B)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (B*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(9*d) - ((9*A + 2*B)*Cos[c + d*x]^7*(a^2 + a^2*Sin[c + d*x]))/(72*d)","A",7,5,31,0.1613,1,"{2860, 2678, 2669, 2635, 8}"
978,1,165,0,0.1892126,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}-\frac{(7 A+2 B) \cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{42 d}+\frac{a^2 (7 A+2 B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a^2 (7 A+2 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (7 A+2 B)-\frac{B \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}","-\frac{a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}-\frac{(7 A+2 B) \cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{42 d}+\frac{a^2 (7 A+2 B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a^2 (7 A+2 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (7 A+2 B)-\frac{B \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}",1,"(a^2*(7*A + 2*B)*x)/16 - (a^2*(7*A + 2*B)*Cos[c + d*x]^5)/(30*d) + (a^2*(7*A + 2*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(7*A + 2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (B*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(7*d) - ((7*A + 2*B)*Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x]))/(42*d)","A",6,5,31,0.1613,1,"{2860, 2678, 2669, 2635, 8}"
979,1,134,0,0.1684602,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}-\frac{(5 A+2 B) \cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{20 d}+\frac{a^2 (5 A+2 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (5 A+2 B)-\frac{B \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}","-\frac{a^2 (5 A+2 B) \cos ^3(c+d x)}{12 d}-\frac{(5 A+2 B) \cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{20 d}+\frac{a^2 (5 A+2 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (5 A+2 B)-\frac{B \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}",1,"(a^2*(5*A + 2*B)*x)/8 - (a^2*(5*A + 2*B)*Cos[c + d*x]^3)/(12*d) + (a^2*(5*A + 2*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (B*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(5*d) - ((5*A + 2*B)*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x]))/(20*d)","A",5,5,31,0.1613,1,"{2860, 2678, 2669, 2635, 8}"
980,1,55,0,0.0933537,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 (A+2 B) \cos (c+d x)}{d}+a^2 x (-(A+2 B))+\frac{(A+B) \sec (c+d x) (a \sin (c+d x)+a)^2}{d}","\frac{a^2 (A+2 B) \cos (c+d x)}{d}+a^2 x (-(A+2 B))+\frac{(A+B) \sec (c+d x) (a \sin (c+d x)+a)^2}{d}",1,"-(a^2*(A + 2*B)*x) + (a^2*(A + 2*B)*Cos[c + d*x])/d + ((A + B)*Sec[c + d*x]*(a + a*Sin[c + d*x])^2)/d","A",3,2,31,0.06452,1,"{2855, 2638}"
981,1,73,0,0.1158499,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 (A-2 B) \tan (c+d x)}{3 d}+\frac{a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac{(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)^2}{3 d}","\frac{a^2 (A-2 B) \tan (c+d x)}{3 d}+\frac{a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac{(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)^2}{3 d}",1,"(a^2*(A - 2*B)*Sec[c + d*x])/(3*d) + ((A + B)*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(3*d) + (a^2*(A - 2*B)*Tan[c + d*x])/(3*d)","A",4,4,31,0.1290,1,"{2855, 2669, 3767, 8}"
982,1,104,0,0.1248307,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 (3 A-2 B) \tan ^3(c+d x)}{15 d}+\frac{a^2 (3 A-2 B) \tan (c+d x)}{5 d}+\frac{a^2 (3 A-2 B) \sec ^3(c+d x)}{15 d}+\frac{(A+B) \sec ^5(c+d x) (a \sin (c+d x)+a)^2}{5 d}","\frac{a^2 (3 A-2 B) \tan ^3(c+d x)}{15 d}+\frac{a^2 (3 A-2 B) \tan (c+d x)}{5 d}+\frac{a^2 (3 A-2 B) \sec ^3(c+d x)}{15 d}+\frac{(A+B) \sec ^5(c+d x) (a \sin (c+d x)+a)^2}{5 d}",1,"(a^2*(3*A - 2*B)*Sec[c + d*x]^3)/(15*d) + ((A + B)*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(5*d) + (a^2*(3*A - 2*B)*Tan[c + d*x])/(5*d) + (a^2*(3*A - 2*B)*Tan[c + d*x]^3)/(15*d)","A",4,3,31,0.09677,1,"{2855, 2669, 3767}"
983,1,129,0,0.133749,"\int \sec ^8(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^8*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 (5 A-2 B) \tan ^5(c+d x)}{35 d}+\frac{2 a^2 (5 A-2 B) \tan ^3(c+d x)}{21 d}+\frac{a^2 (5 A-2 B) \tan (c+d x)}{7 d}+\frac{a^2 (5 A-2 B) \sec ^5(c+d x)}{35 d}+\frac{(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)^2}{7 d}","\frac{a^2 (5 A-2 B) \tan ^5(c+d x)}{35 d}+\frac{2 a^2 (5 A-2 B) \tan ^3(c+d x)}{21 d}+\frac{a^2 (5 A-2 B) \tan (c+d x)}{7 d}+\frac{a^2 (5 A-2 B) \sec ^5(c+d x)}{35 d}+\frac{(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)^2}{7 d}",1,"(a^2*(5*A - 2*B)*Sec[c + d*x]^5)/(35*d) + ((A + B)*Sec[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(7*d) + (a^2*(5*A - 2*B)*Tan[c + d*x])/(7*d) + (2*a^2*(5*A - 2*B)*Tan[c + d*x]^3)/(21*d) + (a^2*(5*A - 2*B)*Tan[c + d*x]^5)/(35*d)","A",4,3,31,0.09677,1,"{2855, 2669, 3767}"
984,1,154,0,0.141179,"\int \sec ^{10}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^10*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 (7 A-2 B) \tan ^7(c+d x)}{63 d}+\frac{a^2 (7 A-2 B) \tan ^5(c+d x)}{15 d}+\frac{a^2 (7 A-2 B) \tan ^3(c+d x)}{9 d}+\frac{a^2 (7 A-2 B) \tan (c+d x)}{9 d}+\frac{a^2 (7 A-2 B) \sec ^7(c+d x)}{63 d}+\frac{(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^2}{9 d}","\frac{a^2 (7 A-2 B) \tan ^7(c+d x)}{63 d}+\frac{a^2 (7 A-2 B) \tan ^5(c+d x)}{15 d}+\frac{a^2 (7 A-2 B) \tan ^3(c+d x)}{9 d}+\frac{a^2 (7 A-2 B) \tan (c+d x)}{9 d}+\frac{a^2 (7 A-2 B) \sec ^7(c+d x)}{63 d}+\frac{(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^2}{9 d}",1,"(a^2*(7*A - 2*B)*Sec[c + d*x]^7)/(63*d) + ((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x])^2)/(9*d) + (a^2*(7*A - 2*B)*Tan[c + d*x])/(9*d) + (a^2*(7*A - 2*B)*Tan[c + d*x]^3)/(9*d) + (a^2*(7*A - 2*B)*Tan[c + d*x]^5)/(15*d) + (a^2*(7*A - 2*B)*Tan[c + d*x]^7)/(63*d)","A",4,3,31,0.09677,1,"{2855, 2669, 3767}"
985,1,179,0,0.1507618,"\int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^12*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 (9 A-2 B) \tan ^9(c+d x)}{99 d}+\frac{4 a^2 (9 A-2 B) \tan ^7(c+d x)}{77 d}+\frac{6 a^2 (9 A-2 B) \tan ^5(c+d x)}{55 d}+\frac{4 a^2 (9 A-2 B) \tan ^3(c+d x)}{33 d}+\frac{a^2 (9 A-2 B) \tan (c+d x)}{11 d}+\frac{a^2 (9 A-2 B) \sec ^9(c+d x)}{99 d}+\frac{(A+B) \sec ^{11}(c+d x) (a \sin (c+d x)+a)^2}{11 d}","\frac{a^2 (9 A-2 B) \tan ^9(c+d x)}{99 d}+\frac{4 a^2 (9 A-2 B) \tan ^7(c+d x)}{77 d}+\frac{6 a^2 (9 A-2 B) \tan ^5(c+d x)}{55 d}+\frac{4 a^2 (9 A-2 B) \tan ^3(c+d x)}{33 d}+\frac{a^2 (9 A-2 B) \tan (c+d x)}{11 d}+\frac{a^2 (9 A-2 B) \sec ^9(c+d x)}{99 d}+\frac{(A+B) \sec ^{11}(c+d x) (a \sin (c+d x)+a)^2}{11 d}",1,"(a^2*(9*A - 2*B)*Sec[c + d*x]^9)/(99*d) + ((A + B)*Sec[c + d*x]^11*(a + a*Sin[c + d*x])^2)/(11*d) + (a^2*(9*A - 2*B)*Tan[c + d*x])/(11*d) + (4*a^2*(9*A - 2*B)*Tan[c + d*x]^3)/(33*d) + (6*a^2*(9*A - 2*B)*Tan[c + d*x]^5)/(55*d) + (4*a^2*(9*A - 2*B)*Tan[c + d*x]^7)/(77*d) + (a^2*(9*A - 2*B)*Tan[c + d*x]^9)/(99*d)","A",4,3,31,0.09677,1,"{2855, 2669, 3767}"
986,1,134,0,0.1823231,"\int \cos ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{(A-7 B) (a \sin (c+d x)+a)^{10}}{10 a^7 d}+\frac{2 (A-3 B) (a \sin (c+d x)+a)^9}{3 a^6 d}-\frac{(3 A-5 B) (a \sin (c+d x)+a)^8}{2 a^5 d}+\frac{8 (A-B) (a \sin (c+d x)+a)^7}{7 a^4 d}-\frac{B (a \sin (c+d x)+a)^{11}}{11 a^8 d}","-\frac{(A-7 B) (a \sin (c+d x)+a)^{10}}{10 a^7 d}+\frac{2 (A-3 B) (a \sin (c+d x)+a)^9}{3 a^6 d}-\frac{(3 A-5 B) (a \sin (c+d x)+a)^8}{2 a^5 d}+\frac{8 (A-B) (a \sin (c+d x)+a)^7}{7 a^4 d}-\frac{B (a \sin (c+d x)+a)^{11}}{11 a^8 d}",1,"(8*(A - B)*(a + a*Sin[c + d*x])^7)/(7*a^4*d) - ((3*A - 5*B)*(a + a*Sin[c + d*x])^8)/(2*a^5*d) + (2*(A - 3*B)*(a + a*Sin[c + d*x])^9)/(3*a^6*d) - ((A - 7*B)*(a + a*Sin[c + d*x])^10)/(10*a^7*d) - (B*(a + a*Sin[c + d*x])^11)/(11*a^8*d)","A",3,2,31,0.06452,1,"{2836, 77}"
987,1,105,0,0.1522589,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{(A-5 B) (a \sin (c+d x)+a)^8}{8 a^5 d}-\frac{4 (A-2 B) (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac{2 (A-B) (a \sin (c+d x)+a)^6}{3 a^3 d}+\frac{B (a \sin (c+d x)+a)^9}{9 a^6 d}","\frac{(A-5 B) (a \sin (c+d x)+a)^8}{8 a^5 d}-\frac{4 (A-2 B) (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac{2 (A-B) (a \sin (c+d x)+a)^6}{3 a^3 d}+\frac{B (a \sin (c+d x)+a)^9}{9 a^6 d}",1,"(2*(A - B)*(a + a*Sin[c + d*x])^6)/(3*a^3*d) - (4*(A - 2*B)*(a + a*Sin[c + d*x])^7)/(7*a^4*d) + ((A - 5*B)*(a + a*Sin[c + d*x])^8)/(8*a^5*d) + (B*(a + a*Sin[c + d*x])^9)/(9*a^6*d)","A",3,2,31,0.06452,1,"{2836, 77}"
988,1,78,0,0.1335224,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{(A-3 B) (a \sin (c+d x)+a)^6}{6 a^3 d}+\frac{2 (A-B) (a \sin (c+d x)+a)^5}{5 a^2 d}-\frac{B (a \sin (c+d x)+a)^7}{7 a^4 d}","-\frac{(A-3 B) (a \sin (c+d x)+a)^6}{6 a^3 d}+\frac{2 (A-B) (a \sin (c+d x)+a)^5}{5 a^2 d}-\frac{B (a \sin (c+d x)+a)^7}{7 a^4 d}",1,"(2*(A - B)*(a + a*Sin[c + d*x])^5)/(5*a^2*d) - ((A - 3*B)*(a + a*Sin[c + d*x])^6)/(6*a^3*d) - (B*(a + a*Sin[c + d*x])^7)/(7*a^4*d)","A",3,2,31,0.06452,1,"{2836, 77}"
989,1,51,0,0.062236,"\int \cos (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{B (a \sin (c+d x)+a)^5}{5 a^2 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{4 a d}","\frac{B (a \sin (c+d x)+a)^5}{5 a^2 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{4 a d}",1,"((A - B)*(a + a*Sin[c + d*x])^4)/(4*a*d) + (B*(a + a*Sin[c + d*x])^5)/(5*a^2*d)","A",3,2,29,0.06897,1,"{2833, 43}"
990,1,81,0,0.0947624,"\int \sec (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{a^3 (A+B) \sin ^2(c+d x)}{2 d}-\frac{3 a^3 (A+B) \sin (c+d x)}{d}-\frac{4 a^3 (A+B) \log (1-\sin (c+d x))}{d}-\frac{B (a \sin (c+d x)+a)^3}{3 d}","-\frac{a^3 (A+B) \sin ^2(c+d x)}{2 d}-\frac{3 a^3 (A+B) \sin (c+d x)}{d}-\frac{4 a^3 (A+B) \log (1-\sin (c+d x))}{d}-\frac{B (a \sin (c+d x)+a)^3}{3 d}",1,"(-4*a^3*(A + B)*Log[1 - Sin[c + d*x]])/d - (3*a^3*(A + B)*Sin[c + d*x])/d - (a^3*(A + B)*Sin[c + d*x]^2)/(2*d) - (B*(a + a*Sin[c + d*x])^3)/(3*d)","A",3,2,29,0.06897,1,"{2836, 77}"
991,1,62,0,0.1085733,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{2 a^4 (A+B)}{d (a-a \sin (c+d x))}+\frac{a^3 (A+3 B) \log (1-\sin (c+d x))}{d}+\frac{a^3 B \sin (c+d x)}{d}","\frac{2 a^4 (A+B)}{d (a-a \sin (c+d x))}+\frac{a^3 (A+3 B) \log (1-\sin (c+d x))}{d}+\frac{a^3 B \sin (c+d x)}{d}",1,"(a^3*(A + 3*B)*Log[1 - Sin[c + d*x]])/d + (a^3*B*Sin[c + d*x])/d + (2*a^4*(A + B))/(d*(a - a*Sin[c + d*x]))","A",3,2,31,0.06452,1,"{2836, 77}"
992,1,43,0,0.0817873,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^3 (a A+a B \sin (c+d x))^2}{2 d (A+B) (a-a \sin (c+d x))^2}","\frac{a^3 (a A+a B \sin (c+d x))^2}{2 d (A+B) (a-a \sin (c+d x))^2}",1,"(a^3*(a*A + a*B*Sin[c + d*x])^2)/(2*(A + B)*d*(a - a*Sin[c + d*x])^2)","A",2,2,31,0.06452,1,"{2836, 37}"
993,1,105,0,0.1351484,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^6 (A+B)}{6 d (a-a \sin (c+d x))^3}+\frac{a^5 (A-B)}{8 d (a-a \sin (c+d x))^2}+\frac{a^4 (A-B)}{8 d (a-a \sin (c+d x))}+\frac{a^3 (A-B) \tanh ^{-1}(\sin (c+d x))}{8 d}","\frac{a^6 (A+B)}{6 d (a-a \sin (c+d x))^3}+\frac{a^5 (A-B)}{8 d (a-a \sin (c+d x))^2}+\frac{a^4 (A-B)}{8 d (a-a \sin (c+d x))}+\frac{a^3 (A-B) \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"(a^3*(A - B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^6*(A + B))/(6*d*(a - a*Sin[c + d*x])^3) + (a^5*(A - B))/(8*d*(a - a*Sin[c + d*x])^2) + (a^4*(A - B))/(8*d*(a - a*Sin[c + d*x]))","A",4,3,31,0.09677,1,"{2836, 77, 206}"
994,1,162,0,0.1862582,"\int \sec ^9(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^9*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^7 (A+B)}{16 d (a-a \sin (c+d x))^4}+\frac{a^5 (3 A-B)}{32 d (a-a \sin (c+d x))^2}+\frac{a^4 (2 A-B)}{16 d (a-a \sin (c+d x))}-\frac{a^4 (A-B)}{32 d (a \sin (c+d x)+a)}+\frac{a^3 (5 A-3 B) \tanh ^{-1}(\sin (c+d x))}{32 d}+\frac{a^6 A}{12 d (a-a \sin (c+d x))^3}","\frac{a^7 (A+B)}{16 d (a-a \sin (c+d x))^4}+\frac{a^5 (3 A-B)}{32 d (a-a \sin (c+d x))^2}+\frac{a^4 (2 A-B)}{16 d (a-a \sin (c+d x))}-\frac{a^4 (A-B)}{32 d (a \sin (c+d x)+a)}+\frac{a^3 (5 A-3 B) \tanh ^{-1}(\sin (c+d x))}{32 d}+\frac{a^6 A}{12 d (a-a \sin (c+d x))^3}",1,"(a^3*(5*A - 3*B)*ArcTanh[Sin[c + d*x]])/(32*d) + (a^7*(A + B))/(16*d*(a - a*Sin[c + d*x])^4) + (a^6*A)/(12*d*(a - a*Sin[c + d*x])^3) + (a^5*(3*A - B))/(32*d*(a - a*Sin[c + d*x])^2) + (a^4*(2*A - B))/(16*d*(a - a*Sin[c + d*x])) - (a^4*(A - B))/(32*d*(a + a*Sin[c + d*x]))","A",4,3,31,0.09677,1,"{2836, 77, 206}"
995,1,231,0,0.2681841,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}-\frac{11 (10 A+3 B) \cos ^7(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{720 d}+\frac{11 a^3 (10 A+3 B) \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac{11 a^3 (10 A+3 B) \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{11 a^3 (10 A+3 B) \sin (c+d x) \cos (c+d x)}{256 d}+\frac{11}{256} a^3 x (10 A+3 B)-\frac{a (10 A+3 B) \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{90 d}-\frac{B \cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}","-\frac{11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}-\frac{11 (10 A+3 B) \cos ^7(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{720 d}+\frac{11 a^3 (10 A+3 B) \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac{11 a^3 (10 A+3 B) \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{11 a^3 (10 A+3 B) \sin (c+d x) \cos (c+d x)}{256 d}+\frac{11}{256} a^3 x (10 A+3 B)-\frac{a (10 A+3 B) \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{90 d}-\frac{B \cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}",1,"(11*a^3*(10*A + 3*B)*x)/256 - (11*a^3*(10*A + 3*B)*Cos[c + d*x]^7)/(560*d) + (11*a^3*(10*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (11*a^3*(10*A + 3*B)*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (11*a^3*(10*A + 3*B)*Cos[c + d*x]^5*Sin[c + d*x])/(480*d) - (a*(10*A + 3*B)*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(90*d) - (B*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3)/(10*d) - (11*(10*A + 3*B)*Cos[c + d*x]^7*(a^3 + a^3*Sin[c + d*x]))/(720*d)","A",8,5,31,0.1613,1,"{2860, 2678, 2669, 2635, 8}"
996,1,200,0,0.2389756,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}-\frac{3 (8 A+3 B) \cos ^5(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{112 d}+\frac{3 a^3 (8 A+3 B) \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{9 a^3 (8 A+3 B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{9}{128} a^3 x (8 A+3 B)-\frac{a (8 A+3 B) \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{56 d}-\frac{B \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}","-\frac{3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}-\frac{3 (8 A+3 B) \cos ^5(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{112 d}+\frac{3 a^3 (8 A+3 B) \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{9 a^3 (8 A+3 B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{9}{128} a^3 x (8 A+3 B)-\frac{a (8 A+3 B) \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{56 d}-\frac{B \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}",1,"(9*a^3*(8*A + 3*B)*x)/128 - (3*a^3*(8*A + 3*B)*Cos[c + d*x]^5)/(80*d) + (9*a^3*(8*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (3*a^3*(8*A + 3*B)*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (a*(8*A + 3*B)*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(56*d) - (B*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3)/(8*d) - (3*(8*A + 3*B)*Cos[c + d*x]^5*(a^3 + a^3*Sin[c + d*x]))/(112*d)","A",7,5,31,0.1613,1,"{2860, 2678, 2669, 2635, 8}"
997,1,159,0,0.2174302,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{7 a^3 (2 A+B) \cos ^3(c+d x)}{24 d}-\frac{7 (2 A+B) \cos ^3(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{40 d}+\frac{7 a^3 (2 A+B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{7}{16} a^3 x (2 A+B)-\frac{a (2 A+B) \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{10 d}-\frac{B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}","-\frac{7 a^3 (2 A+B) \cos ^3(c+d x)}{24 d}-\frac{7 (2 A+B) \cos ^3(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{40 d}+\frac{7 a^3 (2 A+B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{7}{16} a^3 x (2 A+B)-\frac{a (2 A+B) \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{10 d}-\frac{B \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}",1,"(7*a^3*(2*A + B)*x)/16 - (7*a^3*(2*A + B)*Cos[c + d*x]^3)/(24*d) + (7*a^3*(2*A + B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a*(2*A + B)*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(10*d) - (B*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(6*d) - (7*(2*A + B)*Cos[c + d*x]^3*(a^3 + a^3*Sin[c + d*x]))/(40*d)","A",6,5,31,0.1613,1,"{2860, 2678, 2669, 2635, 8}"
998,1,91,0,0.1048555,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{2 a^3 (2 A+3 B) \cos (c+d x)}{d}+\frac{a^3 (2 A+3 B) \sin (c+d x) \cos (c+d x)}{2 d}-\frac{3}{2} a^3 x (2 A+3 B)+\frac{(A+B) \sec (c+d x) (a \sin (c+d x)+a)^3}{d}","\frac{2 a^3 (2 A+3 B) \cos (c+d x)}{d}+\frac{a^3 (2 A+3 B) \sin (c+d x) \cos (c+d x)}{2 d}-\frac{3}{2} a^3 x (2 A+3 B)+\frac{(A+B) \sec (c+d x) (a \sin (c+d x)+a)^3}{d}",1,"(-3*a^3*(2*A + 3*B)*x)/2 + (2*a^3*(2*A + 3*B)*Cos[c + d*x])/d + (a^3*(2*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + ((A + B)*Sec[c + d*x]*(a + a*Sin[c + d*x])^3)/d","A",2,2,31,0.06452,1,"{2855, 2644}"
999,1,69,0,0.1540122,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{2 a^5 B \cos (c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)}+a^3 B x+\frac{(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}","-\frac{2 a^5 B \cos (c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)}+a^3 B x+\frac{(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}",1,"a^3*B*x + ((A + B)*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(3*d) - (2*a^5*B*Cos[c + d*x])/(d*(a^2 - a^2*Sin[c + d*x]))","A",4,4,31,0.1290,1,"{2855, 2670, 2680, 8}"
1000,1,107,0,0.1537757,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^5 (2 A-3 B) \cos (c+d x)}{15 d \left(a^2-a^2 \sin (c+d x)\right)}+\frac{a^5 (2 A-3 B) \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac{(A+B) \sec ^5(c+d x) (a \sin (c+d x)+a)^3}{5 d}","\frac{a^5 (2 A-3 B) \cos (c+d x)}{15 d \left(a^2-a^2 \sin (c+d x)\right)}+\frac{a^5 (2 A-3 B) \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac{(A+B) \sec ^5(c+d x) (a \sin (c+d x)+a)^3}{5 d}",1,"(a^5*(2*A - 3*B)*Cos[c + d*x])/(15*d*(a - a*Sin[c + d*x])^2) + ((A + B)*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3)/(5*d) + (a^5*(2*A - 3*B)*Cos[c + d*x])/(15*d*(a^2 - a^2*Sin[c + d*x]))","A",4,4,31,0.1290,1,"{2855, 2670, 2650, 2648}"
1001,1,115,0,0.1456094,"\int \sec ^8(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^8*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^3 (4 A-3 B) \tan ^3(c+d x)}{35 d}+\frac{3 a^3 (4 A-3 B) \tan (c+d x)}{35 d}+\frac{2 (4 A-3 B) \sec ^5(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{35 d}+\frac{(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)^3}{7 d}","\frac{a^3 (4 A-3 B) \tan ^3(c+d x)}{35 d}+\frac{3 a^3 (4 A-3 B) \tan (c+d x)}{35 d}+\frac{2 (4 A-3 B) \sec ^5(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{35 d}+\frac{(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)^3}{7 d}",1,"((A + B)*Sec[c + d*x]^7*(a + a*Sin[c + d*x])^3)/(7*d) + (2*(4*A - 3*B)*Sec[c + d*x]^5*(a^3 + a^3*Sin[c + d*x]))/(35*d) + (3*a^3*(4*A - 3*B)*Tan[c + d*x])/(35*d) + (a^3*(4*A - 3*B)*Tan[c + d*x]^3)/(35*d)","A",4,3,31,0.09677,1,"{2855, 2676, 3767}"
1002,1,140,0,0.1514008,"\int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^10*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^3 (2 A-B) \tan ^5(c+d x)}{21 d}+\frac{10 a^3 (2 A-B) \tan ^3(c+d x)}{63 d}+\frac{5 a^3 (2 A-B) \tan (c+d x)}{21 d}+\frac{2 (2 A-B) \sec ^7(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{21 d}+\frac{(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^3}{9 d}","\frac{a^3 (2 A-B) \tan ^5(c+d x)}{21 d}+\frac{10 a^3 (2 A-B) \tan ^3(c+d x)}{63 d}+\frac{5 a^3 (2 A-B) \tan (c+d x)}{21 d}+\frac{2 (2 A-B) \sec ^7(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{21 d}+\frac{(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^3}{9 d}",1,"((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x])^3)/(9*d) + (2*(2*A - B)*Sec[c + d*x]^7*(a^3 + a^3*Sin[c + d*x]))/(21*d) + (5*a^3*(2*A - B)*Tan[c + d*x])/(21*d) + (10*a^3*(2*A - B)*Tan[c + d*x]^3)/(63*d) + (a^3*(2*A - B)*Tan[c + d*x]^5)/(21*d)","A",4,3,31,0.09677,1,"{2855, 2676, 3767}"
1003,1,105,0,0.1468904,"\int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","-\frac{(A+5 B) (a-a \sin (c+d x))^6}{6 a^7 d}+\frac{4 (A+2 B) (a-a \sin (c+d x))^5}{5 a^6 d}-\frac{(A+B) (a-a \sin (c+d x))^4}{a^5 d}+\frac{B (a-a \sin (c+d x))^7}{7 a^8 d}","-\frac{(A+5 B) (a-a \sin (c+d x))^6}{6 a^7 d}+\frac{4 (A+2 B) (a-a \sin (c+d x))^5}{5 a^6 d}-\frac{(A+B) (a-a \sin (c+d x))^4}{a^5 d}+\frac{B (a-a \sin (c+d x))^7}{7 a^8 d}",1,"-(((A + B)*(a - a*Sin[c + d*x])^4)/(a^5*d)) + (4*(A + 2*B)*(a - a*Sin[c + d*x])^5)/(5*a^6*d) - ((A + 5*B)*(a - a*Sin[c + d*x])^6)/(6*a^7*d) + (B*(a - a*Sin[c + d*x])^7)/(7*a^8*d)","A",3,2,31,0.06452,1,"{2836, 77}"
1004,1,79,0,0.1161783,"\int \frac{\cos ^5(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","\frac{(A+3 B) (a-a \sin (c+d x))^4}{4 a^5 d}-\frac{2 (A+B) (a-a \sin (c+d x))^3}{3 a^4 d}-\frac{B (a-a \sin (c+d x))^5}{5 a^6 d}","\frac{(A+3 B) (a-a \sin (c+d x))^4}{4 a^5 d}-\frac{2 (A+B) (a-a \sin (c+d x))^3}{3 a^4 d}-\frac{B (a-a \sin (c+d x))^5}{5 a^6 d}",1,"(-2*(A + B)*(a - a*Sin[c + d*x])^3)/(3*a^4*d) + ((A + 3*B)*(a - a*Sin[c + d*x])^4)/(4*a^5*d) - (B*(a - a*Sin[c + d*x])^5)/(5*a^6*d)","A",3,2,31,0.06452,1,"{2836, 77}"
1005,1,57,0,0.0954406,"\int \frac{\cos ^3(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","-\frac{(A-B) \sin ^2(c+d x)}{2 a d}+\frac{A \sin (c+d x)}{a d}-\frac{B \sin ^3(c+d x)}{3 a d}","-\frac{(A-B) \sin ^2(c+d x)}{2 a d}+\frac{A \sin (c+d x)}{a d}-\frac{B \sin ^3(c+d x)}{3 a d}",1,"(A*Sin[c + d*x])/(a*d) - ((A - B)*Sin[c + d*x]^2)/(2*a*d) - (B*Sin[c + d*x]^3)/(3*a*d)","A",3,2,31,0.06452,1,"{2836, 43}"
1006,1,36,0,0.0817913,"\int \frac{\cos (c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","\frac{(A-B) \log (\sin (c+d x)+1)}{a d}+\frac{B \sin (c+d x)}{a d}","\frac{(A-B) \log (\sin (c+d x)+1)}{a d}+\frac{B \sin (c+d x)}{a d}",1,"((A - B)*Log[1 + Sin[c + d*x]])/(a*d) + (B*Sin[c + d*x])/(a*d)","A",3,2,29,0.06897,1,"{2833, 43}"
1007,1,45,0,0.0941355,"\int \frac{\sec (c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","\frac{(A+B) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{A-B}{2 d (a \sin (c+d x)+a)}","\frac{(A+B) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{A-B}{2 d (a \sin (c+d x)+a)}",1,"((A + B)*ArcTanh[Sin[c + d*x]])/(2*a*d) - (A - B)/(2*d*(a + a*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 77, 206}"
1008,1,91,0,0.1382038,"\int \frac{\sec ^3(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","\frac{A+B}{8 d (a-a \sin (c+d x))}-\frac{a (A-B)}{8 d (a \sin (c+d x)+a)^2}+\frac{(3 A+B) \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{A}{4 d (a \sin (c+d x)+a)}","\frac{A+B}{8 d (a-a \sin (c+d x))}-\frac{a (A-B)}{8 d (a \sin (c+d x)+a)^2}+\frac{(3 A+B) \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{A}{4 d (a \sin (c+d x)+a)}",1,"((3*A + B)*ArcTanh[Sin[c + d*x]])/(8*a*d) + (A + B)/(8*d*(a - a*Sin[c + d*x])) - (a*(A - B))/(8*d*(a + a*Sin[c + d*x])^2) - A/(4*d*(a + a*Sin[c + d*x]))","A",4,3,31,0.09677,1,"{2836, 77, 206}"
1009,1,146,0,0.1887799,"\int \frac{\sec ^5(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","-\frac{a^2 (A-B)}{24 d (a \sin (c+d x)+a)^3}+\frac{a (A+B)}{32 d (a-a \sin (c+d x))^2}-\frac{a (3 A-B)}{32 d (a \sin (c+d x)+a)^2}+\frac{2 A+B}{16 d (a-a \sin (c+d x))}+\frac{(5 A+B) \tanh ^{-1}(\sin (c+d x))}{16 a d}-\frac{3 A}{16 d (a \sin (c+d x)+a)}","-\frac{a^2 (A-B)}{24 d (a \sin (c+d x)+a)^3}+\frac{a (A+B)}{32 d (a-a \sin (c+d x))^2}-\frac{a (3 A-B)}{32 d (a \sin (c+d x)+a)^2}+\frac{2 A+B}{16 d (a-a \sin (c+d x))}+\frac{(5 A+B) \tanh ^{-1}(\sin (c+d x))}{16 a d}-\frac{3 A}{16 d (a \sin (c+d x)+a)}",1,"((5*A + B)*ArcTanh[Sin[c + d*x]])/(16*a*d) + (a*(A + B))/(32*d*(a - a*Sin[c + d*x])^2) + (2*A + B)/(16*d*(a - a*Sin[c + d*x])) - (a^2*(A - B))/(24*d*(a + a*Sin[c + d*x])^3) - (a*(3*A - B))/(32*d*(a + a*Sin[c + d*x])^2) - (3*A)/(16*d*(a + a*Sin[c + d*x]))","A",4,3,31,0.09677,1,"{2836, 77, 206}"
1010,1,205,0,0.2485322,"\int \frac{\sec ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","-\frac{a^3 (A-B)}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2 (A+B)}{96 d (a-a \sin (c+d x))^3}-\frac{a^2 (2 A-B)}{48 d (a \sin (c+d x)+a)^3}+\frac{a (5 A+3 B)}{128 d (a-a \sin (c+d x))^2}-\frac{a (5 A-B)}{64 d (a \sin (c+d x)+a)^2}+\frac{5 (3 A+B)}{128 d (a-a \sin (c+d x))}+\frac{5 (7 A+B) \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{5 A}{32 d (a \sin (c+d x)+a)}","-\frac{a^3 (A-B)}{64 d (a \sin (c+d x)+a)^4}+\frac{a^2 (A+B)}{96 d (a-a \sin (c+d x))^3}-\frac{a^2 (2 A-B)}{48 d (a \sin (c+d x)+a)^3}+\frac{a (5 A+3 B)}{128 d (a-a \sin (c+d x))^2}-\frac{a (5 A-B)}{64 d (a \sin (c+d x)+a)^2}+\frac{5 (3 A+B)}{128 d (a-a \sin (c+d x))}+\frac{5 (7 A+B) \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac{5 A}{32 d (a \sin (c+d x)+a)}",1,"(5*(7*A + B)*ArcTanh[Sin[c + d*x]])/(128*a*d) + (a^2*(A + B))/(96*d*(a - a*Sin[c + d*x])^3) + (a*(5*A + 3*B))/(128*d*(a - a*Sin[c + d*x])^2) + (5*(3*A + B))/(128*d*(a - a*Sin[c + d*x])) - (a^3*(A - B))/(64*d*(a + a*Sin[c + d*x])^4) - (a^2*(2*A - B))/(48*d*(a + a*Sin[c + d*x])^3) - (a*(5*A - B))/(64*d*(a + a*Sin[c + d*x])^2) - (5*A)/(32*d*(a + a*Sin[c + d*x]))","A",4,3,31,0.09677,1,"{2836, 77, 206}"
1011,1,79,0,0.1249735,"\int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","\frac{(A+3 B) (a-a \sin (c+d x))^5}{5 a^7 d}-\frac{(A+B) (a-a \sin (c+d x))^4}{2 a^6 d}-\frac{B (a-a \sin (c+d x))^6}{6 a^8 d}","\frac{(A+3 B) (a-a \sin (c+d x))^5}{5 a^7 d}-\frac{(A+B) (a-a \sin (c+d x))^4}{2 a^6 d}-\frac{B (a-a \sin (c+d x))^6}{6 a^8 d}",1,"-((A + B)*(a - a*Sin[c + d*x])^4)/(2*a^6*d) + ((A + 3*B)*(a - a*Sin[c + d*x])^5)/(5*a^7*d) - (B*(a - a*Sin[c + d*x])^6)/(6*a^8*d)","A",3,2,31,0.06452,1,"{2836, 77}"
1012,1,51,0,0.1028888,"\int \frac{\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","\frac{B (a-a \sin (c+d x))^4}{4 a^6 d}-\frac{(A+B) (a-a \sin (c+d x))^3}{3 a^5 d}","\frac{B (a-a \sin (c+d x))^4}{4 a^6 d}-\frac{(A+B) (a-a \sin (c+d x))^3}{3 a^5 d}",1,"-((A + B)*(a - a*Sin[c + d*x])^3)/(3*a^5*d) + (B*(a - a*Sin[c + d*x])^4)/(4*a^6*d)","A",3,2,31,0.06452,1,"{2836, 43}"
1013,1,66,0,0.1073435,"\int \frac{\cos ^3(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","-\frac{(A-B) \sin (c+d x)}{a^2 d}+\frac{2 (A-B) \log (\sin (c+d x)+1)}{a^2 d}-\frac{B (a-a \sin (c+d x))^2}{2 a^4 d}","-\frac{(A-B) \sin (c+d x)}{a^2 d}+\frac{2 (A-B) \log (\sin (c+d x)+1)}{a^2 d}-\frac{B (a-a \sin (c+d x))^2}{2 a^4 d}",1,"(2*(A - B)*Log[1 + Sin[c + d*x]])/(a^2*d) - ((A - B)*Sin[c + d*x])/(a^2*d) - (B*(a - a*Sin[c + d*x])^2)/(2*a^4*d)","A",3,2,31,0.06452,1,"{2836, 77}"
1014,1,44,0,0.0678509,"\int \frac{\cos (c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","\frac{B \log (\sin (c+d x)+1)}{a^2 d}-\frac{A-B}{d \left(a^2 \sin (c+d x)+a^2\right)}","\frac{B \log (\sin (c+d x)+1)}{a^2 d}-\frac{A-B}{d \left(a^2 \sin (c+d x)+a^2\right)}",1,"(B*Log[1 + Sin[c + d*x]])/(a^2*d) - (A - B)/(d*(a^2 + a^2*Sin[c + d*x]))","A",3,2,29,0.06897,1,"{2833, 43}"
1015,1,71,0,0.1074998,"\int \frac{\sec (c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","-\frac{A+B}{4 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{(A+B) \tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac{A-B}{4 d (a \sin (c+d x)+a)^2}","-\frac{A+B}{4 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{(A+B) \tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac{A-B}{4 d (a \sin (c+d x)+a)^2}",1,"((A + B)*ArcTanh[Sin[c + d*x]])/(4*a^2*d) - (A - B)/(4*d*(a + a*Sin[c + d*x])^2) - (A + B)/(4*d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2836, 77, 206}"
1016,1,123,0,0.1546159,"\int \frac{\sec ^3(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","\frac{A+B}{16 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{3 A+B}{16 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{(2 A+B) \tanh ^{-1}(\sin (c+d x))}{8 a^2 d}-\frac{a (A-B)}{12 d (a \sin (c+d x)+a)^3}-\frac{A}{8 d (a \sin (c+d x)+a)^2}","\frac{A+B}{16 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{3 A+B}{16 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{(2 A+B) \tanh ^{-1}(\sin (c+d x))}{8 a^2 d}-\frac{a (A-B)}{12 d (a \sin (c+d x)+a)^3}-\frac{A}{8 d (a \sin (c+d x)+a)^2}",1,"((2*A + B)*ArcTanh[Sin[c + d*x]])/(8*a^2*d) - (a*(A - B))/(12*d*(a + a*Sin[c + d*x])^3) - A/(8*d*(a + a*Sin[c + d*x])^2) + (A + B)/(16*d*(a^2 - a^2*Sin[c + d*x])) - (3*A + B)/(16*d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,31,0.09677,1,"{2836, 77, 206}"
1017,1,179,0,0.2057362,"\int \frac{\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 (A-B)}{32 d (a \sin (c+d x)+a)^4}+\frac{5 A+3 B}{64 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{5 A+B}{32 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{5 (3 A+B) \tanh ^{-1}(\sin (c+d x))}{64 a^2 d}-\frac{a (3 A-B)}{48 d (a \sin (c+d x)+a)^3}+\frac{A+B}{64 d (a-a \sin (c+d x))^2}-\frac{3 A}{32 d (a \sin (c+d x)+a)^2}","-\frac{a^2 (A-B)}{32 d (a \sin (c+d x)+a)^4}+\frac{5 A+3 B}{64 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{5 A+B}{32 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{5 (3 A+B) \tanh ^{-1}(\sin (c+d x))}{64 a^2 d}-\frac{a (3 A-B)}{48 d (a \sin (c+d x)+a)^3}+\frac{A+B}{64 d (a-a \sin (c+d x))^2}-\frac{3 A}{32 d (a \sin (c+d x)+a)^2}",1,"(5*(3*A + B)*ArcTanh[Sin[c + d*x]])/(64*a^2*d) + (A + B)/(64*d*(a - a*Sin[c + d*x])^2) - (a^2*(A - B))/(32*d*(a + a*Sin[c + d*x])^4) - (a*(3*A - B))/(48*d*(a + a*Sin[c + d*x])^3) - (3*A)/(32*d*(a + a*Sin[c + d*x])^2) + (5*A + 3*B)/(64*d*(a^2 - a^2*Sin[c + d*x])) - (5*A + B)/(32*d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,31,0.09677,1,"{2836, 77, 206}"
1018,1,236,0,0.2785529,"\int \frac{\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","-\frac{a^3 (A-B)}{80 d (a \sin (c+d x)+a)^5}-\frac{a^2 (2 A-B)}{64 d (a \sin (c+d x)+a)^4}+\frac{3 (7 A+3 B)}{256 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{5 (7 A+B)}{256 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{7 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{128 a^2 d}+\frac{a (A+B)}{192 d (a-a \sin (c+d x))^3}-\frac{a (5 A-B)}{96 d (a \sin (c+d x)+a)^3}+\frac{3 A+2 B}{128 d (a-a \sin (c+d x))^2}-\frac{5 A}{64 d (a \sin (c+d x)+a)^2}","-\frac{a^3 (A-B)}{80 d (a \sin (c+d x)+a)^5}-\frac{a^2 (2 A-B)}{64 d (a \sin (c+d x)+a)^4}+\frac{3 (7 A+3 B)}{256 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{5 (7 A+B)}{256 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{7 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{128 a^2 d}+\frac{a (A+B)}{192 d (a-a \sin (c+d x))^3}-\frac{a (5 A-B)}{96 d (a \sin (c+d x)+a)^3}+\frac{3 A+2 B}{128 d (a-a \sin (c+d x))^2}-\frac{5 A}{64 d (a \sin (c+d x)+a)^2}",1,"(7*(4*A + B)*ArcTanh[Sin[c + d*x]])/(128*a^2*d) + (a*(A + B))/(192*d*(a - a*Sin[c + d*x])^3) + (3*A + 2*B)/(128*d*(a - a*Sin[c + d*x])^2) - (a^3*(A - B))/(80*d*(a + a*Sin[c + d*x])^5) - (a^2*(2*A - B))/(64*d*(a + a*Sin[c + d*x])^4) - (a*(5*A - B))/(96*d*(a + a*Sin[c + d*x])^3) - (5*A)/(64*d*(a + a*Sin[c + d*x])^2) + (3*(7*A + 3*B))/(256*d*(a^2 - a^2*Sin[c + d*x])) - (5*(7*A + B))/(256*d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,31,0.09677,1,"{2836, 77, 206}"
1019,1,170,0,0.269419,"\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{a 2^{\frac{1}{2} (2 m+p+1)} (A (m+p+1)+B m) (a \sin (e+f x)+a)^{m-1} (g \cos (e+f x))^{p+1} (\sin (e+f x)+1)^{\frac{1}{2} (-2 m-p+1)} \, _2F_1\left(\frac{1}{2} (-2 m-p+1),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f g (p+1) (m+p+1)}-\frac{B (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g (m+p+1)}","-\frac{a 2^{\frac{1}{2} (2 m+p+1)} (A (m+p+1)+B m) (a \sin (e+f x)+a)^{m-1} (g \cos (e+f x))^{p+1} (\sin (e+f x)+1)^{\frac{1}{2} (-2 m-p+1)} \, _2F_1\left(\frac{1}{2} (-2 m-p+1),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f g (p+1) (m+p+1)}-\frac{B (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g (m+p+1)}",1,"-((2^((1 + 2*m + p)/2)*a*(B*m + A*(1 + m + p))*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1 - 2*m - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^((1 - 2*m - p)/2)*(a + a*Sin[e + f*x])^(-1 + m))/(f*g*(1 + p)*(1 + m + p))) - (B*(g*Cos[e + f*x])^(1 + p)*(a + a*Sin[e + f*x])^m)/(f*g*(1 + m + p))","A",4,4,33,0.1212,1,"{2860, 2689, 70, 69}"
1020,1,159,0,0.1661942,"\int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Cos[e + f*x]^7*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{8 (A-B) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}-\frac{4 (3 A-5 B) (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac{6 (A-3 B) (a \sin (e+f x)+a)^{m+6}}{a^6 f (m+6)}-\frac{(A-7 B) (a \sin (e+f x)+a)^{m+7}}{a^7 f (m+7)}-\frac{B (a \sin (e+f x)+a)^{m+8}}{a^8 f (m+8)}","\frac{8 (A-B) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}-\frac{4 (3 A-5 B) (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac{6 (A-3 B) (a \sin (e+f x)+a)^{m+6}}{a^6 f (m+6)}-\frac{(A-7 B) (a \sin (e+f x)+a)^{m+7}}{a^7 f (m+7)}-\frac{B (a \sin (e+f x)+a)^{m+8}}{a^8 f (m+8)}",1,"(8*(A - B)*(a + a*Sin[e + f*x])^(4 + m))/(a^4*f*(4 + m)) - (4*(3*A - 5*B)*(a + a*Sin[e + f*x])^(5 + m))/(a^5*f*(5 + m)) + (6*(A - 3*B)*(a + a*Sin[e + f*x])^(6 + m))/(a^6*f*(6 + m)) - ((A - 7*B)*(a + a*Sin[e + f*x])^(7 + m))/(a^7*f*(7 + m)) - (B*(a + a*Sin[e + f*x])^(8 + m))/(a^8*f*(8 + m))","A",3,2,31,0.06452,1,"{2836, 77}"
1021,1,123,0,0.1353048,"\int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Cos[e + f*x]^5*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{4 (A-B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}-\frac{4 (A-2 B) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac{(A-5 B) (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac{B (a \sin (e+f x)+a)^{m+6}}{a^6 f (m+6)}","\frac{4 (A-B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}-\frac{4 (A-2 B) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac{(A-5 B) (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac{B (a \sin (e+f x)+a)^{m+6}}{a^6 f (m+6)}",1,"(4*(A - B)*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m)) - (4*(A - 2*B)*(a + a*Sin[e + f*x])^(4 + m))/(a^4*f*(4 + m)) + ((A - 5*B)*(a + a*Sin[e + f*x])^(5 + m))/(a^5*f*(5 + m)) + (B*(a + a*Sin[e + f*x])^(6 + m))/(a^6*f*(6 + m))","A",3,2,31,0.06452,1,"{2836, 77}"
1022,1,93,0,0.1176405,"\int \cos ^3(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Cos[e + f*x]^3*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{2 (A-B) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}-\frac{(A-3 B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}-\frac{B (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}","\frac{2 (A-B) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}-\frac{(A-3 B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}-\frac{B (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}",1,"(2*(A - B)*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m)) - ((A - 3*B)*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m)) - (B*(a + a*Sin[e + f*x])^(4 + m))/(a^4*f*(4 + m))","A",3,2,31,0.06452,1,"{2836, 77}"
1023,1,59,0,0.0705771,"\int \cos (e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{B (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{(A-B) (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}","\frac{B (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{(A-B) (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}",1,"((A - B)*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (B*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m))","A",3,2,29,0.06897,1,"{2833, 43}"
1024,1,80,0,0.1058481,"\int \sec (e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Sec[e + f*x]*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{(A+B) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{1}{2} (\sin (e+f x)+1)\right)}{4 a f (m+1)}+\frac{(A-B) (a \sin (e+f x)+a)^m}{2 f m}","\frac{(A+B) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{1}{2} (\sin (e+f x)+1)\right)}{4 a f (m+1)}+\frac{(A-B) (a \sin (e+f x)+a)^m}{2 f m}",1,"((A - B)*(a + a*Sin[e + f*x])^m)/(2*f*m) + ((A + B)*Hypergeometric2F1[1, 1 + m, 2 + m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^(1 + m))/(4*a*f*(1 + m))","A",3,3,29,0.1034,1,"{2836, 79, 68}"
1025,1,100,0,0.1328011,"\int \sec ^3(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Sec[e + f*x]^3*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{a^2 (A+B) (a \sin (e+f x)+a)^{m-1}}{2 f (a-a \sin (e+f x))}-\frac{a (A (2-m)-B m) (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(1,m-1;m;\frac{1}{2} (\sin (e+f x)+1)\right)}{4 f (1-m)}","\frac{a^2 (A+B) (a \sin (e+f x)+a)^{m-1}}{2 f (a-a \sin (e+f x))}-\frac{a (A (2-m)-B m) (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(1,m-1;m;\frac{1}{2} (\sin (e+f x)+1)\right)}{4 f (1-m)}",1,"-(a*(A*(2 - m) - B*m)*Hypergeometric2F1[1, -1 + m, m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^(-1 + m))/(4*f*(1 - m)) + (a^2*(A + B)*(a + a*Sin[e + f*x])^(-1 + m))/(2*f*(a - a*Sin[e + f*x]))","A",3,3,31,0.09677,1,"{2836, 78, 68}"
1026,1,104,0,0.1393985,"\int \sec ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Sec[e + f*x]^5*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{a^4 (A+B) (a \sin (e+f x)+a)^{m-2}}{4 f (a-a \sin (e+f x))^2}-\frac{a^2 (A (4-m)-B m) (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(2,m-2;m-1;\frac{1}{2} (\sin (e+f x)+1)\right)}{16 f (2-m)}","\frac{a^4 (A+B) (a \sin (e+f x)+a)^{m-2}}{4 f (a-a \sin (e+f x))^2}-\frac{a^2 (A (4-m)-B m) (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(2,m-2;m-1;\frac{1}{2} (\sin (e+f x)+1)\right)}{16 f (2-m)}",1,"-(a^2*(A*(4 - m) - B*m)*Hypergeometric2F1[2, -2 + m, -1 + m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^(-2 + m))/(16*f*(2 - m)) + (a^4*(A + B)*(a + a*Sin[e + f*x])^(-2 + m))/(4*f*(a - a*Sin[e + f*x])^2)","A",3,3,31,0.09677,1,"{2836, 78, 68}"
1027,1,129,0,0.1987945,"\int \cos ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Cos[e + f*x]^6*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{a^3 2^{m+\frac{7}{2}} (A (m+7)+B m) \cos ^7(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{7 f (m+7)}-\frac{B \cos ^7(e+f x) (a \sin (e+f x)+a)^m}{f (m+7)}","-\frac{a^3 2^{m+\frac{7}{2}} (A (m+7)+B m) \cos ^7(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{7 f (m+7)}-\frac{B \cos ^7(e+f x) (a \sin (e+f x)+a)^m}{f (m+7)}",1,"-(2^(7/2 + m)*a^3*(B*m + A*(7 + m))*Cos[e + f*x]^7*Hypergeometric2F1[7/2, -5/2 - m, 9/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^(-3 + m))/(7*f*(7 + m)) - (B*Cos[e + f*x]^7*(a + a*Sin[e + f*x])^m)/(f*(7 + m))","A",4,4,31,0.1290,1,"{2860, 2689, 70, 69}"
1028,1,129,0,0.1979285,"\int \cos ^4(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Cos[e + f*x]^4*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{a^2 2^{m+\frac{5}{2}} (A (m+5)+B m) \cos ^5(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 f (m+5)}-\frac{B \cos ^5(e+f x) (a \sin (e+f x)+a)^m}{f (m+5)}","-\frac{a^2 2^{m+\frac{5}{2}} (A (m+5)+B m) \cos ^5(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left(\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 f (m+5)}-\frac{B \cos ^5(e+f x) (a \sin (e+f x)+a)^m}{f (m+5)}",1,"-(2^(5/2 + m)*a^2*(B*m + A*(5 + m))*Cos[e + f*x]^5*Hypergeometric2F1[5/2, -3/2 - m, 7/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^(-2 + m))/(5*f*(5 + m)) - (B*Cos[e + f*x]^5*(a + a*Sin[e + f*x])^m)/(f*(5 + m))","A",4,4,31,0.1290,1,"{2860, 2689, 70, 69}"
1029,1,127,0,0.1854906,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{a 2^{m+\frac{3}{2}} (A (m+3)+B m) \cos ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f (m+3)}-\frac{B \cos ^3(e+f x) (a \sin (e+f x)+a)^m}{f (m+3)}","-\frac{a 2^{m+\frac{3}{2}} (A (m+3)+B m) \cos ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 f (m+3)}-\frac{B \cos ^3(e+f x) (a \sin (e+f x)+a)^m}{f (m+3)}",1,"-(2^(3/2 + m)*a*(B*m + A*(3 + m))*Cos[e + f*x]^3*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^(-1 + m))/(3*f*(3 + m)) - (B*Cos[e + f*x]^3*(a + a*Sin[e + f*x])^m)/(f*(3 + m))","A",4,4,31,0.1290,1,"{2860, 2689, 70, 69}"
1030,1,123,0,0.1874092,"\int \sec ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Sec[e + f*x]^2*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{2^{m-\frac{1}{2}} (A (1-m)-B m) \sec (e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (1-m)}+\frac{B \sec (e+f x) (a \sin (e+f x)+a)^m}{f (1-m)}","\frac{2^{m-\frac{1}{2}} (A (1-m)-B m) \sec (e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^m \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{f (1-m)}+\frac{B \sec (e+f x) (a \sin (e+f x)+a)^m}{f (1-m)}",1,"(B*Sec[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 - m)) + (2^(-1/2 + m)*(A*(1 - m) - B*m)*Hypergeometric2F1[-1/2, 3/2 - m, 1/2, (1 - Sin[e + f*x])/2]*Sec[e + f*x]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 - m))","A",4,4,31,0.1290,1,"{2860, 2689, 70, 69}"
1031,1,135,0,0.1962212,"\int \sec ^4(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Sec[e + f*x]^4*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{2^{m-\frac{3}{2}} (A (3-m)-B m) \sec ^3(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 a f (3-m)}+\frac{B \sec ^3(e+f x) (a \sin (e+f x)+a)^m}{f (3-m)}","\frac{2^{m-\frac{3}{2}} (A (3-m)-B m) \sec ^3(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left(-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 a f (3-m)}+\frac{B \sec ^3(e+f x) (a \sin (e+f x)+a)^m}{f (3-m)}",1,"(B*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^m)/(f*(3 - m)) + (2^(-3/2 + m)*(A*(3 - m) - B*m)*Hypergeometric2F1[-3/2, 5/2 - m, -1/2, (1 - Sin[e + f*x])/2]*Sec[e + f*x]^3*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(1 + m))/(3*a*f*(3 - m))","A",4,4,31,0.1290,1,"{2860, 2689, 70, 69}"
1032,1,135,0,0.1930753,"\int \sec ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[Sec[e + f*x]^6*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{2^{m-\frac{5}{2}} (A (5-m)-B m) \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left(-\frac{5}{2},\frac{7}{2}-m;-\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 a^2 f (5-m)}+\frac{B \sec ^5(e+f x) (a \sin (e+f x)+a)^m}{f (5-m)}","\frac{2^{m-\frac{5}{2}} (A (5-m)-B m) \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac{1}{2}-m} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left(-\frac{5}{2},\frac{7}{2}-m;-\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{5 a^2 f (5-m)}+\frac{B \sec ^5(e+f x) (a \sin (e+f x)+a)^m}{f (5-m)}",1,"(B*Sec[e + f*x]^5*(a + a*Sin[e + f*x])^m)/(f*(5 - m)) + (2^(-5/2 + m)*(A*(5 - m) - B*m)*Hypergeometric2F1[-5/2, 7/2 - m, -3/2, (1 - Sin[e + f*x])/2]*Sec[e + f*x]^5*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(2 + m))/(5*a^2*f*(5 - m))","A",4,4,31,0.1290,1,"{2860, 2689, 70, 69}"
1033,1,239,0,0.4409744,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-p} \, dx","Int[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-4 - p),x]","\frac{2 (3 A-B (p+4)) (c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{c^2 f g (p+3) (p+5) (p+7)}+\frac{2 (3 A-B (p+4)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{c^3 f g (p+1) (p+3) (p+5) (p+7)}+\frac{(A+B) (c-c \sin (e+f x))^{-p-4} (g \cos (e+f x))^{p+1}}{f g (p+7)}+\frac{(3 A-B (p+4)) (c-c \sin (e+f x))^{-p-3} (g \cos (e+f x))^{p+1}}{c f g (p+5) (p+7)}","\frac{2 (3 A-B (p+4)) (c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{c^2 f g (p+3) (p+5) (p+7)}+\frac{2 (3 A-B (p+4)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{c^3 f g (p+1) (p+3) (p+5) (p+7)}+\frac{(A+B) (c-c \sin (e+f x))^{-p-4} (g \cos (e+f x))^{p+1}}{f g (p+7)}+\frac{(3 A-B (p+4)) (c-c \sin (e+f x))^{-p-3} (g \cos (e+f x))^{p+1}}{c f g (p+5) (p+7)}",1,"((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-4 - p))/(f*g*(7 + p)) + ((3*A - B*(4 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-3 - p))/(c*f*g*(5 + p)*(7 + p)) + (2*(3*A - B*(4 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-2 - p))/(c^2*f*g*(3 + p)*(5 + p)*(7 + p)) + (2*(3*A - B*(4 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(c^3*f*g*(1 + p)*(3 + p)*(5 + p)*(7 + p))","A",4,3,38,0.07895,1,"{2859, 2672, 2671}"
1034,1,168,0,0.3060623,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-3-p} \, dx","Int[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-3 - p),x]","\frac{(2 A-B (p+3)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{c^2 f g (p+1) (p+3) (p+5)}+\frac{(A+B) (c-c \sin (e+f x))^{-p-3} (g \cos (e+f x))^{p+1}}{f g (p+5)}+\frac{(2 A-B (p+3)) (c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{c f g (p+3) (p+5)}","\frac{(2 A-B (p+3)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{c^2 f g (p+1) (p+3) (p+5)}+\frac{(A+B) (c-c \sin (e+f x))^{-p-3} (g \cos (e+f x))^{p+1}}{f g (p+5)}+\frac{(2 A-B (p+3)) (c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{c f g (p+3) (p+5)}",1,"((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-3 - p))/(f*g*(5 + p)) + ((2*A - B*(3 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-2 - p))/(c*f*g*(3 + p)*(5 + p)) + ((2*A - B*(3 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(c^2*f*g*(1 + p)*(3 + p)*(5 + p))","A",3,3,38,0.07895,1,"{2859, 2672, 2671}"
1035,1,102,0,0.2121998,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-2-p} \, dx","Int[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-2 - p),x]","\frac{(A+B) (c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{f g (p+3)}+\frac{(A-B (p+2)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{c f g (p+1) (p+3)}","\frac{(A+B) (c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{f g (p+3)}+\frac{(A-B (p+2)) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{c f g (p+1) (p+3)}",1,"((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-2 - p))/(f*g*(3 + p)) + ((A - B*(2 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(c*f*g*(1 + p)*(3 + p))","A",2,2,38,0.05263,1,"{2859, 2671}"
1036,1,151,0,0.2642587,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx","Int[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-1 - p),x]","\frac{(A+B) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{f g (p+1)}-\frac{B 2^{\frac{1}{2}-\frac{p}{2}} (1-\sin (e+f x))^{\frac{p+1}{2}} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (p+1)}","\frac{(A+B) (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{f g (p+1)}-\frac{B 2^{\frac{1}{2}-\frac{p}{2}} (1-\sin (e+f x))^{\frac{p+1}{2}} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (p+1)}",1,"((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(f*g*(1 + p)) - (2^(1/2 - p/2)*B*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1 + p)/2, (1 + p)/2, (3 + p)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^((1 + p)/2)*(c - c*Sin[e + f*x])^(-1 - p))/(f*g*(1 + p))","A",4,4,38,0.1053,1,"{2859, 2689, 70, 69}"
1037,1,147,0,0.2090819,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-p} \, dx","Int[((g*Cos[e + f*x])^p*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^p,x]","\frac{c 2^{\frac{1}{2}-\frac{p}{2}} (A+B p) (1-\sin (e+f x))^{\frac{p+1}{2}} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (p+1)}-\frac{B (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^{p+1}}{f g}","\frac{c 2^{\frac{1}{2}-\frac{p}{2}} (A+B p) (1-\sin (e+f x))^{\frac{p+1}{2}} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1} \, _2F_1\left(\frac{p+1}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (p+1)}-\frac{B (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^{p+1}}{f g}",1,"(2^(1/2 - p/2)*c*(A + B*p)*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1 + p)/2, (1 + p)/2, (3 + p)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^((1 + p)/2)*(c - c*Sin[e + f*x])^(-1 - p))/(f*g*(1 + p)) - (B*(g*Cos[e + f*x])^(1 + p))/(f*g*(c - c*Sin[e + f*x])^p)","A",4,4,36,0.1111,1,"{2860, 2689, 70, 69}"
1038,1,160,0,0.2724166,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{1-p} \, dx","Int[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1 - p),x]","\frac{c^2 2^{\frac{1}{2}-\frac{p}{2}} (2 A-B (1-p)) (1-\sin (e+f x))^{\frac{p+1}{2}} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1} \, _2F_1\left(\frac{p-1}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (p+1)}-\frac{B (c-c \sin (e+f x))^{1-p} (g \cos (e+f x))^{p+1}}{2 f g}","\frac{c^2 2^{\frac{1}{2}-\frac{p}{2}} (2 A-B (1-p)) (1-\sin (e+f x))^{\frac{p+1}{2}} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1} \, _2F_1\left(\frac{p-1}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{f g (p+1)}-\frac{B (c-c \sin (e+f x))^{1-p} (g \cos (e+f x))^{p+1}}{2 f g}",1,"(2^(1/2 - p/2)*c^2*(2*A - B*(1 - p))*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(-1 + p)/2, (1 + p)/2, (3 + p)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^((1 + p)/2)*(c - c*Sin[e + f*x])^(-1 - p))/(f*g*(1 + p)) - (B*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(1 - p))/(2*f*g)","A",4,4,38,0.1053,1,"{2860, 2689, 70, 69}"
1039,1,163,0,0.2756052,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{2-p} \, dx","Int[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(2 - p),x]","\frac{c^3 2^{\frac{5}{2}-\frac{p}{2}} (3 A-B (2-p)) (1-\sin (e+f x))^{\frac{p+1}{2}} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1} \, _2F_1\left(\frac{p-3}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{3 f g (p+1)}-\frac{B (c-c \sin (e+f x))^{2-p} (g \cos (e+f x))^{p+1}}{3 f g}","\frac{c^3 2^{\frac{5}{2}-\frac{p}{2}} (3 A-B (2-p)) (1-\sin (e+f x))^{\frac{p+1}{2}} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1} \, _2F_1\left(\frac{p-3}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)}{3 f g (p+1)}-\frac{B (c-c \sin (e+f x))^{2-p} (g \cos (e+f x))^{p+1}}{3 f g}",1,"(2^(5/2 - p/2)*c^3*(3*A - B*(2 - p))*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(-3 + p)/2, (1 + p)/2, (3 + p)/2, (1 + Sin[e + f*x])/2]*(1 - Sin[e + f*x])^((1 + p)/2)*(c - c*Sin[e + f*x])^(-1 - p))/(3*f*g*(1 + p)) - (B*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(2 - p))/(3*f*g)","A",4,4,38,0.1053,1,"{2860, 2689, 70, 69}"
1040,1,32,0,0.117022,"\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A m-A (1+m+p) \sin (e+f x)) \, dx","Int[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(A*m - A*(1 + m + p)*Sin[e + f*x]),x]","\frac{A (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g}","\frac{A (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g}",1,"(A*(g*Cos[e + f*x])^(1 + p)*(a + a*Sin[e + f*x])^m)/(f*g)","A",1,1,40,0.02500,1,"{2854}"
1041,1,34,0,0.1154905,"\int (g \cos (e+f x))^p (a-a \sin (e+f x))^m (A m+A (1+m+p) \sin (e+f x)) \, dx","Int[(g*Cos[e + f*x])^p*(a - a*Sin[e + f*x])^m*(A*m + A*(1 + m + p)*Sin[e + f*x]),x]","-\frac{A (a-a \sin (e+f x))^m (g \cos (e+f x))^{p+1}}{f g}","-\frac{A (a-a \sin (e+f x))^m (g \cos (e+f x))^{p+1}}{f g}",1,"-((A*(g*Cos[e + f*x])^(1 + p)*(a - a*Sin[e + f*x])^m)/(f*g))","A",1,1,40,0.02500,1,"{2854}"
1042,1,168,0,0.2802904,"\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","\frac{g 2^{\frac{p+1}{2}} (1-\sin (e+f x))^{\frac{1-p}{2}} (a \sin (e+f x)+a)^{m+1} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(\frac{1}{2} (2 m+p+1);\frac{1-p}{2},-n;\frac{1}{2} (2 m+p+3);\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m+p+1)}","\frac{g 2^{\frac{p+1}{2}} (1-\sin (e+f x))^{\frac{1-p}{2}} (a \sin (e+f x)+a)^{m+1} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(\frac{1}{2} (2 m+p+1);\frac{1-p}{2},-n;\frac{1}{2} (2 m+p+3);\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m+p+1)}",1,"(2^((1 + p)/2)*g*AppellF1[(1 + 2*m + p)/2, (1 - p)/2, -n, (3 + 2*m + p)/2, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])^((1 - p)/2)*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(a*f*(1 + 2*m + p)*((c + d*Sin[e + f*x])/(c - d))^n)","A",4,4,35,0.1143,1,"{2921, 140, 139, 138}"
1043,1,153,0,0.222132,"\int (g \cos (e+f x))^p (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","-\frac{a^2 g 2^{\frac{p+5}{2}} (1-\sin (e+f x)) (\sin (e+f x)+1)^{\frac{1-p}{2}} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{1}{2} (-p-3),-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f (p+1)}","-\frac{a^2 2^{\frac{p}{2}+\frac{5}{2}} (\sin (e+f x)+1)^{\frac{1}{2} (-p-5)+2} (g \cos (e+f x))^{p+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{1}{2} (-p-3),-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f g (p+1)}",1,"-((2^((5 + p)/2)*a^2*g*AppellF1[(1 + p)/2, (-3 - p)/2, -n, (3 + p)/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(f*(1 + p)*((c + d*Sin[e + f*x])/(c + d))^n))","A",3,3,35,0.08571,1,"{2920, 139, 138}"
1044,1,151,0,0.1651321,"\int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Int[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","-\frac{a g 2^{\frac{p+3}{2}} (1-\sin (e+f x)) (\sin (e+f x)+1)^{\frac{1-p}{2}} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{1}{2} (-p-1),-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f (p+1)}","-\frac{a 2^{\frac{p}{2}+\frac{3}{2}} (\sin (e+f x)+1)^{\frac{1}{2} (-p-1)} (g \cos (e+f x))^{p+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{1}{2} (-p-1),-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{f g (p+1)}",1,"-((2^((3 + p)/2)*a*g*AppellF1[(1 + p)/2, (-1 - p)/2, -n, (3 + p)/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(f*(1 + p)*((c + d*Sin[e + f*x])/(c + d))^n))","A",3,3,33,0.09091,1,"{2868, 139, 138}"
1045,1,155,0,0.2473442,"\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]),x]","-\frac{g 2^{\frac{p}{2}-\frac{1}{2}} (1-\sin (e+f x)) (\sin (e+f x)+1)^{\frac{1-p}{2}} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{3-p}{2},-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{a f (p+1)}","-\frac{2^{\frac{p}{2}-\frac{1}{2}} (\sin (e+f x)+1)^{\frac{1-p}{2}-1} (g \cos (e+f x))^{p+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{3-p}{2},-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{a f g (p+1)}",1,"-((2^(-1/2 + p/2)*g*AppellF1[(1 + p)/2, (3 - p)/2, -n, (3 + p)/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(a*f*(1 + p)*((c + d*Sin[e + f*x])/(c + d))^n))","A",3,3,35,0.08571,1,"{2920, 139, 138}"
1046,1,153,0,0.2299982,"\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Int[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2,x]","-\frac{g 2^{\frac{p-3}{2}} (1-\sin (e+f x)) (\sin (e+f x)+1)^{\frac{1-p}{2}} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{5-p}{2},-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{a^2 f (p+1)}","-\frac{2^{\frac{p}{2}-\frac{3}{2}} (\sin (e+f x)+1)^{\frac{3-p}{2}-2} (g \cos (e+f x))^{p+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{5-p}{2},-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{a^2 f g (p+1)}",1,"-((2^((-3 + p)/2)*g*AppellF1[(1 + p)/2, (5 - p)/2, -n, (3 + p)/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(a^2*f*(1 + p)*((c + d*Sin[e + f*x])/(c + d))^n))","A",3,3,35,0.08571,1,"{2920, 139, 138}"
1047,1,153,0,0.2326279,"\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","Int[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^3,x]","-\frac{g 2^{\frac{p-5}{2}} (1-\sin (e+f x)) (\sin (e+f x)+1)^{\frac{1-p}{2}} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{7-p}{2},-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{a^3 f (p+1)}","-\frac{2^{\frac{p}{2}-\frac{5}{2}} (\sin (e+f x)+1)^{\frac{5-p}{2}-3} (g \cos (e+f x))^{p+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{7-p}{2},-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{a^3 f g (p+1)}",1,"-((2^((-5 + p)/2)*g*AppellF1[(1 + p)/2, (7 - p)/2, -n, (3 + p)/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(a^3*f*(1 + p)*((c + d*Sin[e + f*x])/(c + d))^n))","A",3,3,35,0.08571,1,"{2920, 139, 138}"
1048,1,153,0,0.2338771,"\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^4} \, dx","Int[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^4,x]","-\frac{g 2^{\frac{p-7}{2}} (1-\sin (e+f x)) (\sin (e+f x)+1)^{\frac{1-p}{2}} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{9-p}{2},-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{a^4 f (p+1)}","-\frac{2^{\frac{p}{2}-\frac{7}{2}} (\sin (e+f x)+1)^{\frac{7-p}{2}-4} (g \cos (e+f x))^{p+1} (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{p+1}{2};\frac{9-p}{2},-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{a^4 f g (p+1)}",1,"-((2^((-7 + p)/2)*g*AppellF1[(1 + p)/2, (9 - p)/2, -n, (3 + p)/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(a^4*f*(1 + p)*((c + d*Sin[e + f*x])/(c + d))^n))","A",3,3,35,0.08571,1,"{2920, 139, 138}"
1049,1,175,0,0.4293059,"\int (g \sec (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Int[(g*Sec[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","\frac{2^{\frac{1}{2}-\frac{p}{2}} \sec (e+f x) (1-\sin (e+f x))^{\frac{p+1}{2}} (a \sin (e+f x)+a)^{m+1} (g \sec (e+f x))^p (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(\frac{1}{2} (2 m-p+1);\frac{p+1}{2},-n;\frac{1}{2} (2 m-p+3);\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m-p+1)}","\frac{2^{\frac{1}{2}-\frac{p}{2}} \sec (e+f x) (1-\sin (e+f x))^{\frac{p+1}{2}} (a \sin (e+f x)+a)^{m+1} (g \sec (e+f x))^p (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c-d}\right)^{-n} F_1\left(\frac{1}{2} (2 m-p+1);\frac{p+1}{2},-n;\frac{1}{2} (2 m-p+3);\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right)}{a f (2 m-p+1)}",1,"(2^(1/2 - p/2)*AppellF1[(1 + 2*m - p)/2, (1 + p)/2, -n, (3 + 2*m - p)/2, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Sec[e + f*x]*(g*Sec[e + f*x])^p*(1 - Sin[e + f*x])^((1 + p)/2)*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(a*f*(1 + 2*m - p)*((c + d*Sin[e + f*x])/(c - d))^n)","A",5,5,35,0.1429,1,"{2926, 2921, 140, 139, 138}"
1050,1,105,0,0.1639554,"\int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + b*Sin[c + d*x]),x]","\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \cos ^3(c+d x)}{3 d}-\frac{b \sin ^3(c+d x) \cos ^3(c+d x)}{6 d}-\frac{b \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{b \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b x}{16}","\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \cos ^3(c+d x)}{3 d}-\frac{b \sin ^3(c+d x) \cos ^3(c+d x)}{6 d}-\frac{b \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{b \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b x}{16}",1,"(b*x)/16 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]^5)/(5*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (b*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (b*Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*d)","A",8,6,27,0.2222,1,"{2838, 2565, 14, 2568, 2635, 8}"
1051,1,81,0,0.1306057,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a x}{8}+\frac{b \cos ^5(c+d x)}{5 d}-\frac{b \cos ^3(c+d x)}{3 d}","-\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a x}{8}+\frac{b \cos ^5(c+d x)}{5 d}-\frac{b \cos ^3(c+d x)}{3 d}",1,"(a*x)/8 - (b*Cos[c + d*x]^3)/(3*d) + (b*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",7,6,27,0.2222,1,"{2838, 2568, 2635, 8, 2565, 14}"
1052,1,65,0,0.0942328,"\int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]*(a + b*Sin[c + d*x]),x]","-\frac{a \cos ^3(c+d x)}{3 d}-\frac{b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{b x}{8}","-\frac{a \cos ^3(c+d x)}{3 d}-\frac{b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{b x}{8}",1,"(b*x)/8 - (a*Cos[c + d*x]^3)/(3*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",6,6,25,0.2400,1,"{2838, 2565, 30, 2568, 2635, 8}"
1053,1,51,0,0.0694966,"\int \cos (c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Cot[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{a \cos (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b x}{2}","\frac{a \cos (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b x}{2}",1,"(b*x)/2 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (b*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",6,6,23,0.2609,1,"{2838, 2592, 321, 206, 2635, 8}"
1054,1,41,0,0.0561516,"\int \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{a \cot (c+d x)}{d}-a x+\frac{b \cos (c+d x)}{d}-\frac{b \tanh ^{-1}(\cos (c+d x))}{d}","-\frac{a \cot (c+d x)}{d}-a x+\frac{b \cos (c+d x)}{d}-\frac{b \tanh ^{-1}(\cos (c+d x))}{d}",1,"-(a*x) - (b*ArcTanh[Cos[c + d*x]])/d + (b*Cos[c + d*x])/d - (a*Cot[c + d*x])/d","A",7,6,19,0.3158,1,"{2722, 2592, 321, 206, 3473, 8}"
1055,1,52,0,0.079695,"\int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-\frac{b \cot (c+d x)}{d}-b x","\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-\frac{b \cot (c+d x)}{d}-b x",1,"-(b*x) + (a*ArcTanh[Cos[c + d*x]])/(2*d) - (b*Cot[c + d*x])/d - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)","A",5,5,25,0.2000,1,"{2838, 2611, 3770, 3473, 8}"
1056,1,52,0,0.1090894,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b \cot (c+d x) \csc (c+d x)}{2 d}","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b \cot (c+d x) \csc (c+d x)}{2 d}",1,"(b*ArcTanh[Cos[c + d*x]])/(2*d) - (a*Cot[c + d*x]^3)/(3*d) - (b*Cot[c + d*x]*Csc[c + d*x])/(2*d)","A",5,5,27,0.1852,1,"{2838, 2607, 30, 2611, 3770}"
1057,1,74,0,0.127812,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^3*(a + b*Sin[c + d*x]),x]","\frac{a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot ^3(c+d x)}{3 d}","\frac{a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot ^3(c+d x)}{3 d}",1,"(a*ArcTanh[Cos[c + d*x]])/(8*d) - (b*Cot[c + d*x]^3)/(3*d) + (a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)","A",6,6,27,0.2222,1,"{2838, 2611, 3768, 3770, 2607, 30}"
1058,1,90,0,0.1334385,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{a \cot ^3(c+d x)}{3 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{b \cot (c+d x) \csc (c+d x)}{8 d}","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{a \cot ^3(c+d x)}{3 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{b \cot (c+d x) \csc (c+d x)}{8 d}",1,"(b*ArcTanh[Cos[c + d*x]])/(8*d) - (a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^5)/(5*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)","A",7,6,27,0.2222,1,"{2838, 2607, 14, 2611, 3768, 3770}"
1059,1,190,0,0.3821774,"\int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{\left(7 a^2+4 b^2\right) \cos ^3(c+d x)}{105 d}-\frac{\left(7 a^2+4 b^2\right) \cos (c+d x)}{35 d}+\frac{\left(2 a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{35 d}+\frac{a b \sin ^5(c+d x) \cos (c+d x)}{21 d}+\frac{\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{7 d}-\frac{a b \sin ^3(c+d x) \cos (c+d x)}{12 d}-\frac{a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a b x}{8}","\frac{\left(7 a^2+4 b^2\right) \cos ^3(c+d x)}{105 d}-\frac{\left(7 a^2+4 b^2\right) \cos (c+d x)}{35 d}+\frac{\left(2 a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{35 d}+\frac{a b \sin ^5(c+d x) \cos (c+d x)}{21 d}+\frac{\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{7 d}-\frac{a b \sin ^3(c+d x) \cos (c+d x)}{12 d}-\frac{a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a b x}{8}",1,"(a*b*x)/8 - ((7*a^2 + 4*b^2)*Cos[c + d*x])/(35*d) + ((7*a^2 + 4*b^2)*Cos[c + d*x]^3)/(105*d) - (a*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*b*Cos[c + d*x]*Sin[c + d*x]^3)/(12*d) + ((2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(35*d) + (a*b*Cos[c + d*x]*Sin[c + d*x]^5)/(21*d) + (Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(7*d)","A",10,8,29,0.2759,1,"{2889, 3050, 3033, 3023, 2748, 2633, 2635, 8}"
1060,1,163,0,0.3815831,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{\left(2 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(2 a^2+b^2\right)+\frac{2 a b \cos ^3(c+d x)}{15 d}-\frac{2 a b \cos (c+d x)}{5 d}+\frac{a b \sin ^4(c+d x) \cos (c+d x)}{15 d}+\frac{\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}","\frac{\left(2 a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{\left(2 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(2 a^2+b^2\right)+\frac{2 a b \cos ^3(c+d x)}{15 d}-\frac{2 a b \cos (c+d x)}{5 d}+\frac{a b \sin ^4(c+d x) \cos (c+d x)}{15 d}+\frac{\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}",1,"((2*a^2 + b^2)*x)/16 - (2*a*b*Cos[c + d*x])/(5*d) + (2*a*b*Cos[c + d*x]^3)/(15*d) - ((2*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a*b*Cos[c + d*x]*Sin[c + d*x]^4)/(15*d) + (Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(6*d)","A",9,8,29,0.2759,1,"{2889, 3050, 3033, 3023, 2748, 2635, 8, 2633}"
1061,1,106,0,0.1589245,"\int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2+4 b^2\right) \cos ^3(c+d x)}{30 d}-\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}-\frac{a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{a b x}{4}","-\frac{\left(a^2+4 b^2\right) \cos ^3(c+d x)}{30 d}-\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}-\frac{a \cos ^3(c+d x) (a+b \sin (c+d x))}{10 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{a b x}{4}",1,"(a*b*x)/4 - ((a^2 + 4*b^2)*Cos[c + d*x]^3)/(30*d) + (a*b*Cos[c + d*x]*Sin[c + d*x])/(4*d) - (a*Cos[c + d*x]^3*(a + b*Sin[c + d*x]))/(10*d) - (Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(5*d)","A",5,4,27,0.1481,1,"{2862, 2669, 2635, 8}"
1062,1,90,0,0.2530127,"\int \cos (c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \cos (c+d x)}{3 d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a b \sin (c+d x) \cos (c+d x)}{3 d}+\frac{\cos (c+d x) (a+b \sin (c+d x))^2}{3 d}+a b x","\frac{\left(2 a^2-b^2\right) \cos (c+d x)}{3 d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a b \sin (c+d x) \cos (c+d x)}{3 d}+\frac{\cos (c+d x) (a+b \sin (c+d x))^2}{3 d}+a b x",1,"a*b*x - (a^2*ArcTanh[Cos[c + d*x]])/d + ((2*a^2 - b^2)*Cos[c + d*x])/(3*d) + (a*b*Cos[c + d*x]*Sin[c + d*x])/(3*d) + (Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(3*d)","A",6,6,25,0.2400,1,"{2889, 3050, 3033, 3023, 2735, 3770}"
1063,1,78,0,0.0976003,"\int \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","-\frac{a^2 \cot (c+d x)}{d}+a^2 (-x)+\frac{2 a b \cos (c+d x)}{d}-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b^2 x}{2}","-\frac{a^2 \cot (c+d x)}{d}+a^2 (-x)+\frac{2 a b \cos (c+d x)}{d}-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b^2 x}{2}",1,"-(a^2*x) + (b^2*x)/2 - (2*a*b*ArcTanh[Cos[c + d*x]])/d + (2*a*b*Cos[c + d*x])/d - (a^2*Cot[c + d*x])/d + (b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",9,7,21,0.3333,1,"{2722, 2635, 8, 2592, 321, 206, 3473}"
1064,1,89,0,0.3057844,"\int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a b \cot (c+d x)}{d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-2 a b x+\frac{3 b^2 \cos (c+d x)}{2 d}","\frac{\left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a b \cot (c+d x)}{d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-2 a b x+\frac{3 b^2 \cos (c+d x)}{2 d}",1,"-2*a*b*x + ((a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) + (3*b^2*Cos[c + d*x])/(2*d) - (a*b*Cot[c + d*x])/d - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^2)/(2*d)","A",6,6,27,0.2222,1,"{2889, 3048, 3031, 3023, 2735, 3770}"
1065,1,96,0,0.389709,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \cot (c+d x)}{3 d}+\frac{a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}+b^2 (-x)","\frac{\left(a^2-2 b^2\right) \cot (c+d x)}{3 d}+\frac{a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}+b^2 (-x)",1,"-(b^2*x) + (a*b*ArcTanh[Cos[c + d*x]])/d + ((a^2 - 2*b^2)*Cot[c + d*x])/(3*d) - (a*b*Cot[c + d*x]*Csc[c + d*x])/(3*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(3*d)","A",6,6,29,0.2069,1,"{2889, 3048, 3031, 3021, 2735, 3770}"
1066,1,123,0,0.3645435,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2+4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{8 d}+\frac{2 a b \cot (c+d x)}{3 d}-\frac{a b \cot (c+d x) \csc ^2(c+d x)}{6 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}","\frac{\left(a^2+4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{8 d}+\frac{2 a b \cot (c+d x)}{3 d}-\frac{a b \cot (c+d x) \csc ^2(c+d x)}{6 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}",1,"((a^2 + 4*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) + (2*a*b*Cot[c + d*x])/(3*d) + ((a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^2)/(6*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(4*d)","A",8,8,29,0.2759,1,"{2889, 3048, 3031, 3021, 2748, 3767, 8, 3770}"
1067,1,148,0,0.3861624,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2+5 b^2\right) \cot (c+d x)}{15 d}+\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{15 d}+\frac{a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{a b \cot (c+d x) \csc ^3(c+d x)}{10 d}+\frac{a b \cot (c+d x) \csc (c+d x)}{4 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}","\frac{\left(2 a^2+5 b^2\right) \cot (c+d x)}{15 d}+\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{15 d}+\frac{a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{a b \cot (c+d x) \csc ^3(c+d x)}{10 d}+\frac{a b \cot (c+d x) \csc (c+d x)}{4 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{5 d}",1,"(a*b*ArcTanh[Cos[c + d*x]])/(4*d) + ((2*a^2 + 5*b^2)*Cot[c + d*x])/(15*d) + (a*b*Cot[c + d*x]*Csc[c + d*x])/(4*d) + ((a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^3)/(10*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(5*d)","A",9,9,29,0.3103,1,"{2889, 3048, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
1068,1,170,0,0.39605,"\int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{\left(a^2+2 b^2\right) \cot (c+d x) \csc (c+d x)}{16 d}+\frac{2 a b \cot ^3(c+d x)}{15 d}+\frac{2 a b \cot (c+d x)}{5 d}-\frac{a b \cot (c+d x) \csc ^4(c+d x)}{15 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2}{6 d}","\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{\left(a^2+2 b^2\right) \cot (c+d x) \csc (c+d x)}{16 d}+\frac{2 a b \cot ^3(c+d x)}{15 d}+\frac{2 a b \cot (c+d x)}{5 d}-\frac{a b \cot (c+d x) \csc ^4(c+d x)}{15 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2}{6 d}",1,"((a^2 + 2*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) + (2*a*b*Cot[c + d*x])/(5*d) + (2*a*b*Cot[c + d*x]^3)/(15*d) + ((a^2 + 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(16*d) + ((a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^4)/(15*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2)/(6*d)","A",9,8,29,0.2759,1,"{2889, 3048, 3031, 3021, 2748, 3767, 3768, 3770}"
1069,1,232,0,0.5718159,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{b \left(21 a^2+4 b^2\right) \cos ^3(c+d x)}{105 d}-\frac{b \left(21 a^2+4 b^2\right) \cos (c+d x)}{35 d}+\frac{b \left(a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{35 d}+\frac{a \left(2 a^2-7 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{56 d}-\frac{a \left(2 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a x \left(2 a^2+3 b^2\right)+\frac{\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}+\frac{a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{14 d}","\frac{b \left(21 a^2+4 b^2\right) \cos ^3(c+d x)}{105 d}-\frac{b \left(21 a^2+4 b^2\right) \cos (c+d x)}{35 d}+\frac{b \left(a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{35 d}+\frac{a \left(2 a^2-7 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{56 d}-\frac{a \left(2 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a x \left(2 a^2+3 b^2\right)+\frac{\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}+\frac{a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{14 d}",1,"(a*(2*a^2 + 3*b^2)*x)/16 - (b*(21*a^2 + 4*b^2)*Cos[c + d*x])/(35*d) + (b*(21*a^2 + 4*b^2)*Cos[c + d*x]^3)/(105*d) - (a*(2*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(2*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(56*d) + (b*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(35*d) + (a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(14*d) + (Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(7*d)","A",10,9,29,0.3103,1,"{2889, 3050, 3049, 3033, 3023, 2748, 2635, 8, 2633}"
1070,1,163,0,0.2945424,"\int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*Sin[c + d*x]*(a + b*Sin[c + d*x])^3,x]","-\frac{a \left(2 a^2+33 b^2\right) \cos ^3(c+d x)}{120 d}-\frac{\left(2 a^2+5 b^2\right) \cos ^3(c+d x) (a+b \sin (c+d x))}{40 d}+\frac{b \left(6 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} b x \left(6 a^2+b^2\right)-\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}-\frac{a \cos ^3(c+d x) (a+b \sin (c+d x))^2}{10 d}","-\frac{a \left(2 a^2+33 b^2\right) \cos ^3(c+d x)}{120 d}-\frac{\left(2 a^2+5 b^2\right) \cos ^3(c+d x) (a+b \sin (c+d x))}{40 d}+\frac{b \left(6 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} b x \left(6 a^2+b^2\right)-\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}-\frac{a \cos ^3(c+d x) (a+b \sin (c+d x))^2}{10 d}",1,"(b*(6*a^2 + b^2)*x)/16 - (a*(2*a^2 + 33*b^2)*Cos[c + d*x]^3)/(120*d) + (b*(6*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) - ((2*a^2 + 5*b^2)*Cos[c + d*x]^3*(a + b*Sin[c + d*x]))/(40*d) - (a*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(10*d) - (Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(6*d)","A",6,4,27,0.1481,1,"{2862, 2669, 2635, 8}"
1071,1,136,0,0.4130538,"\int \cos (c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{a \left(a^2-2 b^2\right) \cos (c+d x)}{2 d}+\frac{b \left(2 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} b x \left(12 a^2+b^2\right)-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a \cos (c+d x) (a+b \sin (c+d x))^2}{4 d}+\frac{\cos (c+d x) (a+b \sin (c+d x))^3}{4 d}","\frac{a \left(a^2-2 b^2\right) \cos (c+d x)}{2 d}+\frac{b \left(2 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} b x \left(12 a^2+b^2\right)-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a \cos (c+d x) (a+b \sin (c+d x))^2}{4 d}+\frac{\cos (c+d x) (a+b \sin (c+d x))^3}{4 d}",1,"(b*(12*a^2 + b^2)*x)/8 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a*(a^2 - 2*b^2)*Cos[c + d*x])/(2*d) + (b*(2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(4*d) + (Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(4*d)","A",7,7,25,0.2800,1,"{2889, 3050, 3049, 3033, 3023, 2735, 3770}"
1072,1,102,0,0.1420065,"\int \cot ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{3 a^2 b \cos (c+d x)}{d}-\frac{3 a^2 b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}+a^3 (-x)+\frac{3 a b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3}{2} a b^2 x-\frac{b^3 \cos ^3(c+d x)}{3 d}","\frac{3 a^2 b \cos (c+d x)}{d}-\frac{3 a^2 b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}+a^3 (-x)+\frac{3 a b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3}{2} a b^2 x-\frac{b^3 \cos ^3(c+d x)}{3 d}",1,"-(a^3*x) + (3*a*b^2*x)/2 - (3*a^2*b*ArcTanh[Cos[c + d*x]])/d + (3*a^2*b*Cos[c + d*x])/d - (b^3*Cos[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x])/d + (3*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",11,9,21,0.4286,1,"{2722, 2635, 8, 2592, 321, 206, 3473, 2565, 30}"
1073,1,138,0,0.4651987,"\int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{a \left(a^2-6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{1}{2} b x \left(6 a^2-b^2\right)+\frac{15 a b^2 \cos (c+d x)}{2 d}-\frac{3 b \cot (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}+\frac{5 b^3 \sin (c+d x) \cos (c+d x)}{2 d}","\frac{a \left(a^2-6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{1}{2} b x \left(6 a^2-b^2\right)+\frac{15 a b^2 \cos (c+d x)}{2 d}-\frac{3 b \cot (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}+\frac{5 b^3 \sin (c+d x) \cos (c+d x)}{2 d}",1,"-(b*(6*a^2 - b^2)*x)/2 + (a*(a^2 - 6*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) + (15*a*b^2*Cos[c + d*x])/(2*d) + (5*b^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (3*b*Cot[c + d*x]*(a + b*Sin[c + d*x])^2)/(2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^3)/(2*d)","A",7,7,27,0.2593,1,"{2889, 3048, 3047, 3033, 3023, 2735, 3770}"
1074,1,138,0,0.4826809,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{a \left(a^2-3 b^2\right) \cot (c+d x)}{3 d}+\frac{b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-3 a b^2 x-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{3 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}+\frac{11 b^3 \cos (c+d x)}{6 d}","\frac{a \left(a^2-3 b^2\right) \cot (c+d x)}{3 d}+\frac{b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-3 a b^2 x-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{3 d}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}+\frac{11 b^3 \cos (c+d x)}{6 d}",1,"-3*a*b^2*x + (b*(3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) + (11*b^3*Cos[c + d*x])/(6*d) + (a*(a^2 - 3*b^2)*Cot[c + d*x])/(3*d) - (b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^2)/(2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3)/(3*d)","A",7,7,29,0.2414,1,"{2889, 3048, 3047, 3031, 3023, 2735, 3770}"
1075,1,152,0,0.5106265,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","\frac{b \left(2 a^2-b^2\right) \cot (c+d x)}{2 d}+\frac{a \left(a^2+12 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{a \left(a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}+b^3 (-x)","\frac{b \left(2 a^2-b^2\right) \cot (c+d x)}{2 d}+\frac{a \left(a^2+12 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{a \left(a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}+b^3 (-x)",1,"-(b^3*x) + (a*(a^2 + 12*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) + (b*(2*a^2 - b^2)*Cot[c + d*x])/(2*d) + (a*(a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(4*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(4*d)","A",7,7,29,0.2414,1,"{2889, 3048, 3047, 3031, 3021, 2735, 3770}"
1076,1,183,0,0.568896,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","\frac{a \left(2 a^2+15 b^2\right) \cot (c+d x)}{15 d}+\frac{b \left(3 a^2+4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{a \left(2 a^2-3 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{30 d}+\frac{3 b \left(5 a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}-\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}","\frac{a \left(2 a^2+15 b^2\right) \cot (c+d x)}{15 d}+\frac{b \left(3 a^2+4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{a \left(2 a^2-3 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{30 d}+\frac{3 b \left(5 a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{40 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}-\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}",1,"(b*(3*a^2 + 4*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) + (a*(2*a^2 + 15*b^2)*Cot[c + d*x])/(15*d) + (3*b*(5*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(40*d) + (a*(2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*d) - (3*b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(20*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(5*d)","A",9,9,29,0.3103,1,"{2889, 3048, 3047, 3031, 3021, 2748, 3767, 8, 3770}"
1077,1,212,0,0.5982712,"\int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","\frac{b \left(6 a^2+5 b^2\right) \cot (c+d x)}{15 d}+\frac{a \left(a^2+6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{a \left(5 a^2-6 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{120 d}+\frac{b \left(3 a^2-b^2\right) \cot (c+d x) \csc ^2(c+d x)}{15 d}+\frac{a \left(a^2+6 b^2\right) \cot (c+d x) \csc (c+d x)}{16 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}-\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{10 d}","\frac{b \left(6 a^2+5 b^2\right) \cot (c+d x)}{15 d}+\frac{a \left(a^2+6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{a \left(5 a^2-6 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{120 d}+\frac{b \left(3 a^2-b^2\right) \cot (c+d x) \csc ^2(c+d x)}{15 d}+\frac{a \left(a^2+6 b^2\right) \cot (c+d x) \csc (c+d x)}{16 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 d}-\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{10 d}",1,"(a*(a^2 + 6*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) + (b*(6*a^2 + 5*b^2)*Cot[c + d*x])/(15*d) + (a*(a^2 + 6*b^2)*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (b*(3*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) + (a*(5*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(120*d) - (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(10*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(6*d)","A",10,10,29,0.3448,1,"{2889, 3048, 3047, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
1078,1,188,0,0.7415845,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","\frac{\left(12 a^2-b^2\right) \cos (c+d x)}{3 b^4 d}-\frac{2 a^2 \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d \sqrt{a^2-b^2}}+\frac{a x \left(4 a^2-b^2\right)}{b^5}-\frac{2 a \sin (c+d x) \cos (c+d x)}{b^3 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}+\frac{4 \sin ^2(c+d x) \cos (c+d x)}{3 b^2 d}","\frac{\left(12 a^2-b^2\right) \cos (c+d x)}{3 b^4 d}-\frac{2 a^2 \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d \sqrt{a^2-b^2}}+\frac{a x \left(4 a^2-b^2\right)}{b^5}-\frac{2 a \sin (c+d x) \cos (c+d x)}{b^3 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}+\frac{4 \sin ^2(c+d x) \cos (c+d x)}{3 b^2 d}",1,"(a*(4*a^2 - b^2)*x)/b^5 - (2*a^2*(4*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*d) + ((12*a^2 - b^2)*Cos[c + d*x])/(3*b^4*d) - (2*a*Cos[c + d*x]*Sin[c + d*x])/(b^3*d) + (4*Cos[c + d*x]*Sin[c + d*x]^2)/(3*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(b*d*(a + b*Sin[c + d*x]))","A",9,9,29,0.3103,1,"{2889, 3048, 3050, 3049, 3023, 2735, 2660, 618, 204}"
1079,1,153,0,0.4918382,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{2 a \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d \sqrt{a^2-b^2}}-\frac{x \left(6 a^2-b^2\right)}{2 b^4}-\frac{3 a \cos (c+d x)}{b^3 d}-\frac{\sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 b^2 d}","\frac{2 a \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d \sqrt{a^2-b^2}}-\frac{x \left(6 a^2-b^2\right)}{2 b^4}-\frac{3 a \cos (c+d x)}{b^3 d}-\frac{\sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 b^2 d}",1,"-((6*a^2 - b^2)*x)/(2*b^4) + (2*a*(3*a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*d) - (3*a*Cos[c + d*x])/(b^3*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^2)/(b*d*(a + b*Sin[c + d*x]))","A",8,8,29,0.2759,1,"{2889, 3048, 3050, 3023, 2735, 2660, 618, 204}"
1080,1,106,0,0.1525869,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 d \sqrt{a^2-b^2}}+\frac{\cos (c+d x) (2 a+b \sin (c+d x))}{b^2 d (a+b \sin (c+d x))}+\frac{2 a x}{b^3}","-\frac{2 \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 d \sqrt{a^2-b^2}}+\frac{\cos (c+d x) (2 a+b \sin (c+d x))}{b^2 d (a+b \sin (c+d x))}+\frac{2 a x}{b^3}",1,"(2*a*x)/b^3 - (2*(2*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*d) + (Cos[c + d*x]*(2*a + b*Sin[c + d*x]))/(b^2*d*(a + b*Sin[c + d*x]))","A",5,5,27,0.1852,1,"{2863, 2735, 2660, 618, 204}"
1081,1,92,0,0.2422281,"\int \frac{\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]","-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cos (c+d x)}{a d (a+b \sin (c+d x))}","-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cos (c+d x)}{a d (a+b \sin (c+d x))}",1,"(-2*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*Sqrt[a^2 - b^2]*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(a*d*(a + b*Sin[c + d*x]))","A",8,8,25,0.3200,1,"{2889, 3056, 12, 2747, 3770, 2660, 618, 204}"
1082,1,115,0,0.4351717,"\int \frac{\cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \sqrt{a^2-b^2}}+\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{\cot (c+d x)}{a d (a+b \sin (c+d x))}","-\frac{2 \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \sqrt{a^2-b^2}}+\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{\cot (c+d x)}{a d (a+b \sin (c+d x))}",1,"(-2*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2]*d) + (2*b*ArcTanh[Cos[c + d*x]])/(a^3*d) - (2*Cot[c + d*x])/(a^2*d) + Cot[c + d*x]/(a*d*(a + b*Sin[c + d*x]))","A",8,7,21,0.3333,1,"{2723, 3056, 3001, 3770, 2660, 618, 204}"
1083,1,157,0,0.7676188,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{2 b \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \sqrt{a^2-b^2}}+\frac{\left(a^2-6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{3 b \cot (c+d x)}{a^3 d}-\frac{3 \cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}","\frac{2 b \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \sqrt{a^2-b^2}}+\frac{\left(a^2-6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{3 b \cot (c+d x)}{a^3 d}-\frac{3 \cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}",1,"(2*b*(2*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]*d) + ((a^2 - 6*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^4*d) + (3*b*Cot[c + d*x])/(a^3*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(a*d*(a + b*Sin[c + d*x]))","A",9,8,27,0.2963,1,"{2889, 3056, 3055, 3001, 3770, 2660, 618, 204}"
1084,1,193,0,1.0384141,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","-\frac{2 b^2 \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}+\frac{\left(a^2-12 b^2\right) \cot (c+d x)}{3 a^4 d}-\frac{b \left(a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{2 b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}","-\frac{2 b^2 \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}+\frac{\left(a^2-12 b^2\right) \cot (c+d x)}{3 a^4 d}-\frac{b \left(a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{2 b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}",1,"(-2*b^2*(3*a^2 - 4*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) - (b*(a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + ((a^2 - 12*b^2)*Cot[c + d*x])/(3*a^4*d) + (2*b*Cot[c + d*x]*Csc[c + d*x])/(a^3*d) - (4*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^2)/(a*d*(a + b*Sin[c + d*x]))","A",10,8,29,0.2759,1,"{2889, 3056, 3055, 3001, 3770, 2660, 618, 204}"
1085,1,266,0,0.8778461,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]","-\frac{a \left(12 a^2-11 b^2\right) \cos (c+d x)}{2 b^4 d \left(a^2-b^2\right)}+\frac{a \left(-19 a^2 b^2+12 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d \left(a^2-b^2\right)^{3/2}}-\frac{\left(4 a^2-3 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{\left(6 a^2-5 b^2\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{x \left(12 a^2-b^2\right)}{2 b^5}-\frac{\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}","-\frac{a \left(12 a^2-11 b^2\right) \cos (c+d x)}{2 b^4 d \left(a^2-b^2\right)}+\frac{a \left(-19 a^2 b^2+12 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d \left(a^2-b^2\right)^{3/2}}-\frac{\left(4 a^2-3 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{\left(6 a^2-5 b^2\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{x \left(12 a^2-b^2\right)}{2 b^5}-\frac{\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}",1,"-((12*a^2 - b^2)*x)/(2*b^5) + (a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(3/2)*d) - (a*(12*a^2 - 11*b^2)*Cos[c + d*x])/(2*b^4*(a^2 - b^2)*d) + ((6*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(2*b*d*(a + b*Sin[c + d*x])^2) - ((4*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(2*b^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",9,8,29,0.2759,1,"{2889, 3048, 3049, 3023, 2735, 2660, 618, 204}"
1086,1,180,0,0.5623315,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","-\frac{\left(-9 a^2 b^2+6 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d \left(a^2-b^2\right)^{3/2}}+\frac{a \left(3 a^2-2 b^2\right) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 a x}{b^4}-\frac{\sin ^2(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}+\frac{3 \cos (c+d x)}{2 b^3 d}","-\frac{\left(-9 a^2 b^2+6 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d \left(a^2-b^2\right)^{3/2}}+\frac{a \left(3 a^2-2 b^2\right) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 a x}{b^4}-\frac{\sin ^2(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}+\frac{3 \cos (c+d x)}{2 b^3 d}",1,"(3*a*x)/b^4 - ((6*a^4 - 9*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)*d) + (3*Cos[c + d*x])/(2*b^3*d) - (Cos[c + d*x]*Sin[c + d*x]^2)/(2*b*d*(a + b*Sin[c + d*x])^2) + (a*(3*a^2 - 2*b^2)*Cos[c + d*x])/(2*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",8,8,29,0.2759,1,"{2889, 3048, 3032, 3023, 2735, 2660, 618, 204}"
1087,1,167,0,0.2790701,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{a \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 d \left(a^2-b^2\right)^{3/2}}-\frac{a \cos ^3(c+d x)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{\cos (c+d x) \left(2 \left(a^2-b^2\right)+a b \sin (c+d x)\right)}{2 b^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{x}{b^3}","\frac{a \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 d \left(a^2-b^2\right)^{3/2}}-\frac{a \cos ^3(c+d x)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{\cos (c+d x) \left(2 \left(a^2-b^2\right)+a b \sin (c+d x)\right)}{2 b^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{x}{b^3}",1,"-(x/b^3) + (a*(2*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*d) - (a*Cos[c + d*x]^3)/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (Cos[c + d*x]*(2*(a^2 - b^2) + a*b*Sin[c + d*x]))/(2*b^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",6,6,27,0.2222,1,"{2864, 2863, 2735, 2660, 618, 204}"
1088,1,154,0,0.4774508,"\int \frac{\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x])^3,x]","-\frac{b \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{3/2}}+\frac{\left(a^2-2 b^2\right) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{\cos (c+d x)}{2 a d (a+b \sin (c+d x))^2}","-\frac{b \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{3/2}}+\frac{\left(a^2-2 b^2\right) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{\cos (c+d x)}{2 a d (a+b \sin (c+d x))^2}",1,"-((b*(3*a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(3/2)*d)) - ArcTanh[Cos[c + d*x]]/(a^3*d) + Cos[c + d*x]/(2*a*d*(a + b*Sin[c + d*x])^2) + ((a^2 - 2*b^2)*Cos[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",8,7,25,0.2800,1,"{2889, 3056, 3001, 3770, 2660, 618, 204}"
1089,1,202,0,0.7948949,"\int \frac{\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","-\frac{\left(-9 a^2 b^2+2 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \left(a^2-b^2\right)^{3/2}}-\frac{\left(5 a^2-6 b^2\right) \cot (c+d x)}{2 a^3 d \left(a^2-b^2\right)}+\frac{\left(2 a^2-3 b^2\right) \cot (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}","-\frac{\left(-9 a^2 b^2+2 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \left(a^2-b^2\right)^{3/2}}-\frac{\left(5 a^2-6 b^2\right) \cot (c+d x)}{2 a^3 d \left(a^2-b^2\right)}+\frac{\left(2 a^2-3 b^2\right) \cot (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}",1,"-(((2*a^4 - 9*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(3/2)*d)) + (3*b*ArcTanh[Cos[c + d*x]])/(a^4*d) - ((5*a^2 - 6*b^2)*Cot[c + d*x])/(2*a^3*(a^2 - b^2)*d) + Cot[c + d*x]/(2*a*d*(a + b*Sin[c + d*x])^2) + ((2*a^2 - 3*b^2)*Cot[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",9,8,21,0.3810,1,"{2723, 3056, 3055, 3001, 3770, 2660, 618, 204}"
1090,1,269,0,1.1508145,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{b \left(-19 a^2 b^2+6 a^4+12 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \left(a^2-b^2\right)^{3/2}}+\frac{b \left(11 a^2-12 b^2\right) \cot (c+d x)}{2 a^4 d \left(a^2-b^2\right)}+\frac{\left(a^2-12 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}-\frac{\left(5 a^2-6 b^2\right) \cot (c+d x) \csc (c+d x)}{2 a^3 d \left(a^2-b^2\right)}+\frac{\left(3 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}","\frac{b \left(-19 a^2 b^2+6 a^4+12 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \left(a^2-b^2\right)^{3/2}}+\frac{b \left(11 a^2-12 b^2\right) \cot (c+d x)}{2 a^4 d \left(a^2-b^2\right)}+\frac{\left(a^2-12 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}-\frac{\left(5 a^2-6 b^2\right) \cot (c+d x) \csc (c+d x)}{2 a^3 d \left(a^2-b^2\right)}+\frac{\left(3 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}",1,"(b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(3/2)*d) + ((a^2 - 12*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^5*d) + (b*(11*a^2 - 12*b^2)*Cot[c + d*x])/(2*a^4*(a^2 - b^2)*d) - ((5*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*(a^2 - b^2)*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",10,8,27,0.2963,1,"{2889, 3056, 3055, 3001, 3770, 2660, 618, 204}"
1091,1,347,0,0.774143,"\int \frac{\cos ^2(e+f x)}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}} \, dx","Int[Cos[e + f*x]^2/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(5/2)),x]","\frac{4 b \cos (e+f x)}{3 a f \left(a^2-b^2\right) \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{4 \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 \sqrt{d} f \sqrt{a+b}}-\frac{4 b \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 \sqrt{d} f \sqrt{a+b}}+\frac{2 \cos (e+f x) \sqrt{d \sin (e+f x)}}{3 a d f (a+b \sin (e+f x))^{3/2}}","\frac{4 b \cos (e+f x)}{3 a f \left(a^2-b^2\right) \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{4 \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 \sqrt{d} f \sqrt{a+b}}-\frac{4 b \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 \sqrt{d} f \sqrt{a+b}}+\frac{2 \cos (e+f x) \sqrt{d \sin (e+f x)}}{3 a d f (a+b \sin (e+f x))^{3/2}}",1,"(2*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(3*a*d*f*(a + b*Sin[e + f*x])^(3/2)) + (4*b*Cos[e + f*x])/(3*a*(a^2 - b^2)*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (4*b*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(3*a^3*Sqrt[a + b]*Sqrt[d]*f) - (4*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(3*a^2*Sqrt[a + b]*Sqrt[d]*f)","A",5,5,35,0.1429,1,"{2887, 2800, 2998, 2816, 2994}"
1092,1,143,0,0.1909956,"\int \cos ^4(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + b*Sin[c + d*x]),x]","-\frac{a \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{a \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a x}{128}-\frac{b \cos ^9(c+d x)}{9 d}+\frac{2 b \cos ^7(c+d x)}{7 d}-\frac{b \cos ^5(c+d x)}{5 d}","-\frac{a \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{a \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a x}{128}-\frac{b \cos ^9(c+d x)}{9 d}+\frac{2 b \cos ^7(c+d x)}{7 d}-\frac{b \cos ^5(c+d x)}{5 d}",1,"(3*a*x)/128 - (b*Cos[c + d*x]^5)/(5*d) + (2*b*Cos[c + d*x]^7)/(7*d) - (b*Cos[c + d*x]^9)/(9*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)","A",9,6,27,0.2222,1,"{2838, 2568, 2635, 8, 2565, 270}"
1093,1,127,0,0.186913,"\int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + b*Sin[c + d*x]),x]","\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \cos ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{b \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 b x}{128}","\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \cos ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{b \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 b x}{128}",1,"(3*b*x)/128 - (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]^7)/(7*d) + (3*b*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (b*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (b*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (b*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)","A",9,6,27,0.2222,1,"{2838, 2565, 14, 2568, 2635, 8}"
1094,1,103,0,0.1484563,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{a x}{16}+\frac{b \cos ^7(c+d x)}{7 d}-\frac{b \cos ^5(c+d x)}{5 d}","-\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{a x}{16}+\frac{b \cos ^7(c+d x)}{7 d}-\frac{b \cos ^5(c+d x)}{5 d}",1,"(a*x)/16 - (b*Cos[c + d*x]^5)/(5*d) + (b*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)","A",8,6,27,0.2222,1,"{2838, 2568, 2635, 8, 2565, 14}"
1095,1,87,0,0.1109555,"\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x]),x]","-\frac{a \cos ^5(c+d x)}{5 d}-\frac{b \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{b \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b x}{16}","-\frac{a \cos ^5(c+d x)}{5 d}-\frac{b \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{b \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b x}{16}",1,"(b*x)/16 - (a*Cos[c + d*x]^5)/(5*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (b*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)","A",7,6,25,0.2400,1,"{2838, 2565, 30, 2568, 2635, 8}"
1096,1,89,0,0.0998257,"\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 b x}{8}","\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 b x}{8}",1,"(3*b*x)/8 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) + (3*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",8,6,25,0.2400,1,"{2838, 2592, 302, 206, 2635, 8}"
1097,1,83,0,0.1336411,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{3 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{3 a x}{2}+\frac{b \cos ^3(c+d x)}{3 d}+\frac{b \cos (c+d x)}{d}-\frac{b \tanh ^{-1}(\cos (c+d x))}{d}","-\frac{3 a \cot (c+d x)}{2 d}+\frac{a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{3 a x}{2}+\frac{b \cos ^3(c+d x)}{3 d}+\frac{b \cos (c+d x)}{d}-\frac{b \tanh ^{-1}(\cos (c+d x))}{d}",1,"(-3*a*x)/2 - (b*ArcTanh[Cos[c + d*x]])/d + (b*Cos[c + d*x])/d + (b*Cos[c + d*x]^3)/(3*d) - (3*a*Cot[c + d*x])/(2*d) + (a*Cos[c + d*x]^2*Cot[c + d*x])/(2*d)","A",9,8,27,0.2963,1,"{2838, 2591, 288, 321, 203, 2592, 302, 206}"
1098,1,94,0,0.1219868,"\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x]),x]","-\frac{3 a \cos (c+d x)}{2 d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 b \cot (c+d x)}{2 d}+\frac{b \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{3 b x}{2}","-\frac{3 a \cos (c+d x)}{2 d}-\frac{a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 b \cot (c+d x)}{2 d}+\frac{b \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{3 b x}{2}",1,"(-3*b*x)/2 + (3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a*Cos[c + d*x])/(2*d) - (3*b*Cot[c + d*x])/(2*d) + (b*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (a*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d)","A",9,7,25,0.2800,1,"{2838, 2592, 288, 321, 206, 2591, 203}"
1099,1,82,0,0.0847967,"\int \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \cot (c+d x)}{d}+a x-\frac{3 b \cos (c+d x)}{2 d}-\frac{b \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{2 d}","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \cot (c+d x)}{d}+a x-\frac{3 b \cos (c+d x)}{2 d}-\frac{b \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{2 d}",1,"a*x + (3*b*ArcTanh[Cos[c + d*x]])/(2*d) - (3*b*Cos[c + d*x])/(2*d) + (a*Cot[c + d*x])/d - (b*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d)","A",9,7,19,0.3684,1,"{2722, 2592, 288, 321, 206, 3473, 8}"
1100,1,88,0,0.1111626,"\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x]),x]","-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot ^3(c+d x)}{3 d}+\frac{b \cot (c+d x)}{d}+b x","-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot ^3(c+d x)}{3 d}+\frac{b \cot (c+d x)}{d}+b x",1,"b*x - (3*a*ArcTanh[Cos[c + d*x]])/(8*d) + (b*Cot[c + d*x])/d - (b*Cot[c + d*x]^3)/(3*d) + (3*a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x])/(4*d)","A",7,5,25,0.2000,1,"{2838, 2611, 3770, 3473, 8}"
1101,1,74,0,0.1329823,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{3 b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{b \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 b \cot (c+d x) \csc (c+d x)}{8 d}","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{3 b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{b \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac{3 b \cot (c+d x) \csc (c+d x)}{8 d}",1,"(-3*b*ArcTanh[Cos[c + d*x]])/(8*d) - (a*Cot[c + d*x]^5)/(5*d) + (3*b*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (b*Cot[c + d*x]^3*Csc[c + d*x])/(4*d)","A",6,5,27,0.1852,1,"{2838, 2607, 30, 2611, 3770}"
1102,1,98,0,0.1652853,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x]),x]","-\frac{a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a \cot (c+d x) \csc (c+d x)}{16 d}-\frac{b \cot ^5(c+d x)}{5 d}","-\frac{a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a \cot (c+d x) \csc (c+d x)}{16 d}-\frac{b \cot ^5(c+d x)}{5 d}",1,"-(a*ArcTanh[Cos[c + d*x]])/(16*d) - (b*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)","A",7,6,27,0.2222,1,"{2838, 2611, 3768, 3770, 2607, 30}"
1103,1,114,0,0.1731388,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{b \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{b \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{b \cot (c+d x) \csc (c+d x)}{16 d}","-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{b \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{b \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{b \cot (c+d x) \csc (c+d x)}{16 d}",1,"-(b*ArcTanh[Cos[c + d*x]])/(16*d) - (a*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]^7)/(7*d) - (b*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (b*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)","A",8,6,27,0.2222,1,"{2838, 2607, 14, 2611, 3768, 3770}"
1104,1,136,0,0.1841398,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x]),x]","-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac{b \cot ^7(c+d x)}{7 d}-\frac{b \cot ^5(c+d x)}{5 d}","-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac{b \cot ^7(c+d x)}{7 d}-\frac{b \cot ^5(c+d x)}{5 d}",1,"(-3*a*ArcTanh[Cos[c + d*x]])/(128*d) - (b*Cot[c + d*x]^5)/(5*d) - (b*Cot[c + d*x]^7)/(7*d) - (3*a*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (a*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)","A",9,6,27,0.2222,1,"{2838, 2611, 3768, 3770, 2607, 14}"
1105,1,301,0,0.6538379,"\int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{\left(9 a^2+4 b^2\right) \cos ^3(c+d x)}{315 d}-\frac{\left(9 a^2+4 b^2\right) \cos (c+d x)}{105 d}-\frac{a \left(10 a^2-29 b^2\right) \sin ^5(c+d x) \cos (c+d x)}{504 b d}-\frac{5 \left(3 a^2-8 b^2\right) \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}-\frac{\left(-44 a^2 b^2+15 a^4+6 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{630 b^2 d}+\frac{a \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac{a b \sin ^3(c+d x) \cos (c+d x)}{32 d}-\frac{3 a b \sin (c+d x) \cos (c+d x)}{64 d}+\frac{3 a b x}{64}","\frac{\left(9 a^2+4 b^2\right) \cos ^3(c+d x)}{315 d}-\frac{\left(9 a^2+4 b^2\right) \cos (c+d x)}{105 d}-\frac{a \left(10 a^2-29 b^2\right) \sin ^5(c+d x) \cos (c+d x)}{504 b d}-\frac{5 \left(3 a^2-8 b^2\right) \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}-\frac{\left(-44 a^2 b^2+15 a^4+6 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{630 b^2 d}+\frac{a \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac{a b \sin ^3(c+d x) \cos (c+d x)}{32 d}-\frac{3 a b \sin (c+d x) \cos (c+d x)}{64 d}+\frac{3 a b x}{64}",1,"(3*a*b*x)/64 - ((9*a^2 + 4*b^2)*Cos[c + d*x])/(105*d) + ((9*a^2 + 4*b^2)*Cos[c + d*x]^3)/(315*d) - (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(64*d) - (a*b*Cos[c + d*x]*Sin[c + d*x]^3)/(32*d) - ((15*a^4 - 44*a^2*b^2 + 6*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(630*b^2*d) - (a*(10*a^2 - 29*b^2)*Cos[c + d*x]*Sin[c + d*x]^5)/(504*b*d) - (5*(3*a^2 - 8*b^2)*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(252*b^2*d) + (a*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(12*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(9*b*d)","A",10,8,29,0.2759,1,"{2895, 3049, 3033, 3023, 2748, 2633, 2635, 8}"
1106,1,278,0,0.626873,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","-\frac{a \left(20 a^2-69 b^2\right) \sin ^4(c+d x) \cos (c+d x)}{840 b d}-\frac{\left(20 a^2-63 b^2\right) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{336 b^2 d}-\frac{\left(-140 a^2 b^2+40 a^4+21 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{1344 b^2 d}-\frac{\left(8 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} x \left(8 a^2+3 b^2\right)+\frac{5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{56 b^2 d}+\frac{2 a b \cos ^3(c+d x)}{35 d}-\frac{6 a b \cos (c+d x)}{35 d}-\frac{\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{8 b d}","-\frac{a \left(20 a^2-69 b^2\right) \sin ^4(c+d x) \cos (c+d x)}{840 b d}-\frac{\left(20 a^2-63 b^2\right) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{336 b^2 d}-\frac{\left(-140 a^2 b^2+40 a^4+21 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{1344 b^2 d}-\frac{\left(8 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} x \left(8 a^2+3 b^2\right)+\frac{5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{56 b^2 d}+\frac{2 a b \cos ^3(c+d x)}{35 d}-\frac{6 a b \cos (c+d x)}{35 d}-\frac{\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{8 b d}",1,"((8*a^2 + 3*b^2)*x)/128 - (6*a*b*Cos[c + d*x])/(35*d) + (2*a*b*Cos[c + d*x]^3)/(35*d) - ((8*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) - ((40*a^4 - 140*a^2*b^2 + 21*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(1344*b^2*d) - (a*(20*a^2 - 69*b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(840*b*d) - ((20*a^2 - 63*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(336*b^2*d) + (5*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(56*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(8*b*d)","A",9,8,29,0.2759,1,"{2895, 3049, 3033, 3023, 2748, 2635, 8, 2633}"
1107,1,129,0,0.1807536,"\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2+6 b^2\right) \cos ^5(c+d x)}{105 d}-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}-\frac{a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}+\frac{a b \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a b x}{8}","-\frac{\left(a^2+6 b^2\right) \cos ^5(c+d x)}{105 d}-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}-\frac{a \cos ^5(c+d x) (a+b \sin (c+d x))}{21 d}+\frac{a b \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a b x}{8}",1,"(a*b*x)/8 - ((a^2 + 6*b^2)*Cos[c + d*x]^5)/(105*d) + (a*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*b*Cos[c + d*x]^3*Sin[c + d*x])/(12*d) - (a*Cos[c + d*x]^5*(a + b*Sin[c + d*x]))/(21*d) - (Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2)/(7*d)","A",6,4,27,0.1481,1,"{2862, 2669, 2635, 8}"
1108,1,190,0,0.4473647,"\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(-14 a^2 b^2+a^4+3 b^4\right) \cos (c+d x)}{15 b^2 d}-\frac{\left(a^2-12 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^2}{30 b^2 d}-\frac{a \left(2 a^2-27 b^2\right) \sin (c+d x) \cos (c+d x)}{60 b d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a \cos (c+d x) (a+b \sin (c+d x))^3}{10 b^2 d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{5 b d}+\frac{3 a b x}{4}","\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a b \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a b x}{4}-\frac{b^2 \cos ^5(c+d x)}{5 d}",1,"(3*a*b*x)/4 - (a^2*ArcTanh[Cos[c + d*x]])/d - ((a^4 - 14*a^2*b^2 + 3*b^4)*Cos[c + d*x])/(15*b^2*d) - (a*(2*a^2 - 27*b^2)*Cos[c + d*x]*Sin[c + d*x])/(60*b*d) - ((a^2 - 12*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(30*b^2*d) + (a*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(10*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^3)/(5*b*d)","A",6,6,27,0.2222,1,"{2895, 3049, 3033, 3023, 2735, 3770}"
1109,1,181,0,0.5211785,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{a \left(a^2+28 b^2\right) \cos (c+d x)}{6 b d}+\frac{\left(a^2+12 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^2}{12 a b d}+\frac{\left(2 a^2+39 b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}-\frac{3}{8} x \left(4 a^2-b^2\right)-\frac{\cos (c+d x) (a+b \sin (c+d x))^3}{4 b d}-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^3}{a d}","\frac{a \left(a^2+28 b^2\right) \cos (c+d x)}{6 b d}+\frac{\left(a^2+12 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^2}{12 a b d}+\frac{\left(2 a^2+39 b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}-\frac{3}{8} x \left(4 a^2-b^2\right)-\frac{\cos (c+d x) (a+b \sin (c+d x))^3}{4 b d}-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^3}{a d}",1,"(-3*(4*a^2 - b^2)*x)/8 - (2*a*b*ArcTanh[Cos[c + d*x]])/d + (a*(a^2 + 28*b^2)*Cos[c + d*x])/(6*b*d) + ((2*a^2 + 39*b^2)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((a^2 + 12*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(12*a*b*d) - (Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(4*b*d) - (Cot[c + d*x]*(a + b*Sin[c + d*x])^3)/(a*d)","A",6,6,29,0.2069,1,"{2894, 3049, 3033, 3023, 2735, 3770}"
1110,1,189,0,0.4838689,"\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(4 a^2-23 b^2\right) \cos (c+d x)}{6 d}-\frac{\left(2 a^2-3 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac{b \left(a^2-3 b^2\right) \sin (c+d x) \cos (c+d x)}{3 a d}+\frac{\left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-3 a b x","-\frac{\left(4 a^2-23 b^2\right) \cos (c+d x)}{6 d}-\frac{\left(2 a^2-3 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac{b \left(a^2-3 b^2\right) \sin (c+d x) \cos (c+d x)}{3 a d}+\frac{\left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-3 a b x",1,"-3*a*b*x + ((3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) - ((4*a^2 - 23*b^2)*Cos[c + d*x])/(6*d) - (b*(a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(3*a*d) - ((2*a^2 - 3*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(6*a^2*d) - (b*Cot[c + d*x]*(a + b*Sin[c + d*x])^3)/(2*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^3)/(2*a*d)","A",6,6,27,0.2222,1,"{2893, 3049, 3033, 3023, 2735, 3770}"
1111,1,133,0,0.1624093,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{3 a b \cos (c+d x)}{d}-\frac{a b \cos (c+d x) \cot ^2(c+d x)}{d}+\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 b^2 \cot (c+d x)}{2 d}+\frac{b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{3 b^2 x}{2}","-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}+a^2 x-\frac{3 a b \cos (c+d x)}{d}-\frac{a b \cos (c+d x) \cot ^2(c+d x)}{d}+\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 b^2 \cot (c+d x)}{2 d}+\frac{b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{3 b^2 x}{2}",1,"a^2*x - (3*b^2*x)/2 + (3*a*b*ArcTanh[Cos[c + d*x]])/d - (3*a*b*Cos[c + d*x])/d + (a^2*Cot[c + d*x])/d - (3*b^2*Cot[c + d*x])/(2*d) + (b^2*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (a*b*Cos[c + d*x]*Cot[c + d*x]^2)/d - (a^2*Cot[c + d*x]^3)/(3*d)","A",13,9,21,0.4286,1,"{2722, 2591, 288, 321, 203, 2592, 206, 3473, 8}"
1112,1,178,0,0.4627775,"\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]","-\frac{b^2 \left(39 a^2+2 b^2\right) \cos (c+d x)}{24 a^2 d}-\frac{3 \left(a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}+\frac{17 a b \cot (c+d x)}{12 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac{5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+2 a b x","-\frac{b^2 \left(39 a^2+2 b^2\right) \cos (c+d x)}{24 a^2 d}-\frac{3 \left(a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}+\frac{17 a b \cot (c+d x)}{12 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac{5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+2 a b x",1,"2*a*b*x - (3*(a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) - (b^2*(39*a^2 + 2*b^2)*Cos[c + d*x])/(24*a^2*d) + (17*a*b*Cot[c + d*x])/(12*d) + (5*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^2)/(8*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3)/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(4*a*d)","A",6,6,27,0.2222,1,"{2893, 3047, 3031, 3023, 2735, 3770}"
1113,1,209,0,0.5159924,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(-14 a^2 b^2+3 a^4+b^4\right) \cot (c+d x)}{15 a^2 d}+\frac{b \left(27 a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{60 a d}+\frac{\left(12 a^2-b^2\right) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{30 a^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 a d}+b^2 x","-\frac{\left(-14 a^2 b^2+3 a^4+b^4\right) \cot (c+d x)}{15 a^2 d}+\frac{b \left(27 a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{60 a d}+\frac{\left(12 a^2-b^2\right) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{30 a^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 a d}+b^2 x",1,"b^2*x - (3*a*b*ArcTanh[Cos[c + d*x]])/(4*d) - ((3*a^4 - 14*a^2*b^2 + b^4)*Cot[c + d*x])/(15*a^2*d) + (b*(27*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(60*a*d) + ((12*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(30*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(10*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(5*a*d)","A",6,6,29,0.2069,1,"{2893, 3047, 3031, 3021, 2735, 3770}"
1114,1,236,0,0.603552,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2+6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{b \left(13 a^2-2 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{60 a d}-\frac{\left(-80 a^2 b^2+15 a^4+12 b^4\right) \cot (c+d x) \csc (c+d x)}{240 a^2 d}+\frac{\left(35 a^2-6 b^2\right) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac{2 a b \cot (c+d x)}{5 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}","-\frac{\left(a^2+6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{b \left(13 a^2-2 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{60 a d}-\frac{\left(-80 a^2 b^2+15 a^4+12 b^4\right) \cot (c+d x) \csc (c+d x)}{240 a^2 d}+\frac{\left(35 a^2-6 b^2\right) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac{2 a b \cot (c+d x)}{5 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}",1,"-((a^2 + 6*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) - (2*a*b*Cot[c + d*x])/(5*d) - ((15*a^4 - 80*a^2*b^2 + 12*b^4)*Cot[c + d*x]*Csc[c + d*x])/(240*a^2*d) + (b*(13*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(60*a*d) + ((35*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(120*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(10*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(6*a*d)","A",8,8,29,0.2759,1,"{2893, 3047, 3031, 3021, 2748, 3767, 8, 3770}"
1115,1,261,0,0.6286241,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(2 a^2+7 b^2\right) \cot (c+d x)}{35 d}+\frac{b \left(53 a^2-12 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{420 a d}-\frac{\left(-18 a^2 b^2+3 a^4+4 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac{2 \left(4 a^2-b^2\right) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac{2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac{a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}","-\frac{\left(2 a^2+7 b^2\right) \cot (c+d x)}{35 d}+\frac{b \left(53 a^2-12 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{420 a d}-\frac{\left(-18 a^2 b^2+3 a^4+4 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac{2 \left(4 a^2-b^2\right) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac{2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac{a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}",1,"-(a*b*ArcTanh[Cos[c + d*x]])/(8*d) - ((2*a^2 + 7*b^2)*Cot[c + d*x])/(35*d) - (a*b*Cot[c + d*x]*Csc[c + d*x])/(8*d) - ((3*a^4 - 18*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(105*a^2*d) + (b*(53*a^2 - 12*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(420*a*d) + (2*(4*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(35*a^2*d) + (2*b*Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(21*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^3)/(7*a*d)","A",9,9,29,0.3103,1,"{2893, 3047, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
1116,1,354,0,0.9277249,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{b \left(27 a^2+4 b^2\right) \cos ^3(c+d x)}{315 d}-\frac{b \left(27 a^2+4 b^2\right) \cos (c+d x)}{105 d}-\frac{\left(-93 a^2 b^2+20 a^4+24 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{2520 b d}-\frac{5 \left(a^2-4 b^2\right) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}-\frac{a \left(20 a^2-87 b^2\right) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac{a \left(-188 a^2 b^2+40 a^4+189 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{4032 b^2 d}-\frac{a \left(8 a^2+9 b^2\right) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a x \left(8 a^2+9 b^2\right)+\frac{5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}","\frac{b \left(27 a^2+4 b^2\right) \cos ^3(c+d x)}{315 d}-\frac{b \left(27 a^2+4 b^2\right) \cos (c+d x)}{105 d}-\frac{\left(-93 a^2 b^2+20 a^4+24 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{2520 b d}-\frac{5 \left(a^2-4 b^2\right) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}-\frac{a \left(20 a^2-87 b^2\right) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac{a \left(-188 a^2 b^2+40 a^4+189 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{4032 b^2 d}-\frac{a \left(8 a^2+9 b^2\right) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a x \left(8 a^2+9 b^2\right)+\frac{5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac{\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}",1,"(a*(8*a^2 + 9*b^2)*x)/128 - (b*(27*a^2 + 4*b^2)*Cos[c + d*x])/(105*d) + (b*(27*a^2 + 4*b^2)*Cos[c + d*x]^3)/(315*d) - (a*(8*a^2 + 9*b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (a*(40*a^4 - 188*a^2*b^2 + 189*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(4032*b^2*d) - ((20*a^4 - 93*a^2*b^2 + 24*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(2520*b*d) - (a*(20*a^2 - 87*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(1008*b^2*d) - (5*(a^2 - 4*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(126*b^2*d) + (5*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(72*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^4)/(9*b*d)","A",10,8,29,0.2759,1,"{2895, 3049, 3033, 3023, 2748, 2635, 8, 2633}"
1117,1,194,0,0.3270277,"\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^3,x]","-\frac{a \left(2 a^2+61 b^2\right) \cos ^5(c+d x)}{560 d}-\frac{\left(2 a^2+7 b^2\right) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}+\frac{b \left(8 a^2+b^2\right) \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 b \left(8 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3}{128} b x \left(8 a^2+b^2\right)-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}-\frac{3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}","-\frac{a \left(2 a^2+61 b^2\right) \cos ^5(c+d x)}{560 d}-\frac{\left(2 a^2+7 b^2\right) \cos ^5(c+d x) (a+b \sin (c+d x))}{112 d}+\frac{b \left(8 a^2+b^2\right) \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 b \left(8 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3}{128} b x \left(8 a^2+b^2\right)-\frac{\cos ^5(c+d x) (a+b \sin (c+d x))^3}{8 d}-\frac{3 a \cos ^5(c+d x) (a+b \sin (c+d x))^2}{56 d}",1,"(3*b*(8*a^2 + b^2)*x)/128 - (a*(2*a^2 + 61*b^2)*Cos[c + d*x]^5)/(560*d) + (3*b*(8*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (b*(8*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - ((2*a^2 + 7*b^2)*Cos[c + d*x]^5*(a + b*Sin[c + d*x]))/(112*d) - (3*a*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2)/(56*d) - (Cos[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(8*d)","A",7,4,27,0.1481,1,"{2862, 2669, 2635, 8}"
1118,1,250,0,0.6593571,"\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x])^3,x]","-\frac{a \left(-43 a^2 b^2+2 a^4+36 b^4\right) \cos (c+d x)}{60 b^2 d}-\frac{\left(2 a^2-35 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}-\frac{a \left(2 a^2-39 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^2}{120 b^2 d}-\frac{\left(-84 a^2 b^2+4 a^4+15 b^4\right) \sin (c+d x) \cos (c+d x)}{240 b d}+\frac{1}{16} b x \left(18 a^2+b^2\right)-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{6 b d}","-\frac{a \left(-43 a^2 b^2+2 a^4+36 b^4\right) \cos (c+d x)}{60 b^2 d}-\frac{\left(2 a^2-35 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^3}{120 b^2 d}-\frac{a \left(2 a^2-39 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^2}{120 b^2 d}-\frac{\left(-84 a^2 b^2+4 a^4+15 b^4\right) \sin (c+d x) \cos (c+d x)}{240 b d}+\frac{1}{16} b x \left(18 a^2+b^2\right)-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a \cos (c+d x) (a+b \sin (c+d x))^4}{15 b^2 d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{6 b d}",1,"(b*(18*a^2 + b^2)*x)/16 - (a^3*ArcTanh[Cos[c + d*x]])/d - (a*(2*a^4 - 43*a^2*b^2 + 36*b^4)*Cos[c + d*x])/(60*b^2*d) - ((4*a^4 - 84*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sin[c + d*x])/(240*b*d) - (a*(2*a^2 - 39*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(120*b^2*d) - ((2*a^2 - 35*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(120*b^2*d) + (a*Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(15*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^4)/(6*b*d)","A",7,6,27,0.2222,1,"{2895, 3049, 3033, 3023, 2735, 3770}"
1119,1,229,0,0.6772611,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{\left(56 a^2 b^2+a^4-2 b^4\right) \cos (c+d x)}{10 b d}+\frac{\left(a^2+20 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^3}{20 a b d}+\frac{\left(a^2+28 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^2}{20 b d}+\frac{a \left(2 a^2+83 b^2\right) \sin (c+d x) \cos (c+d x)}{40 d}-\frac{3}{8} a x \left(4 a^2-3 b^2\right)-\frac{3 a^2 b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{\cos (c+d x) (a+b \sin (c+d x))^4}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^4}{a d}","\frac{\left(56 a^2 b^2+a^4-2 b^4\right) \cos (c+d x)}{10 b d}+\frac{\left(a^2+20 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^3}{20 a b d}+\frac{\left(a^2+28 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^2}{20 b d}+\frac{a \left(2 a^2+83 b^2\right) \sin (c+d x) \cos (c+d x)}{40 d}-\frac{3}{8} a x \left(4 a^2-3 b^2\right)-\frac{3 a^2 b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{\cos (c+d x) (a+b \sin (c+d x))^4}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^4}{a d}",1,"(-3*a*(4*a^2 - 3*b^2)*x)/8 - (3*a^2*b*ArcTanh[Cos[c + d*x]])/d + ((a^4 + 56*a^2*b^2 - 2*b^4)*Cos[c + d*x])/(10*b*d) + (a*(2*a^2 + 83*b^2)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + ((a^2 + 28*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(20*b*d) + ((a^2 + 20*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(20*a*b*d) - (Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(5*b*d) - (Cot[c + d*x]*(a + b*Sin[c + d*x])^4)/(a*d)","A",7,6,29,0.2069,1,"{2894, 3049, 3033, 3023, 2735, 3770}"
1120,1,231,0,0.6917217,"\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","-\frac{a \left(a^2-17 b^2\right) \cos (c+d x)}{2 d}-\frac{\left(a^2-4 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac{\left(a^2-6 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac{b \left(2 a^2-21 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a \left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3}{8} b x \left(12 a^2-b^2\right)-\frac{b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}","-\frac{a \left(a^2-17 b^2\right) \cos (c+d x)}{2 d}-\frac{\left(a^2-4 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac{\left(a^2-6 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac{b \left(2 a^2-21 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a \left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3}{8} b x \left(12 a^2-b^2\right)-\frac{b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}",1,"(-3*b*(12*a^2 - b^2)*x)/8 + (3*a*(a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) - (a*(a^2 - 17*b^2)*Cos[c + d*x])/(2*d) - (b*(2*a^2 - 21*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - ((a^2 - 6*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(4*a*d) - ((a^2 - 4*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(4*a^2*d) - (b*Cot[c + d*x]*(a + b*Sin[c + d*x])^4)/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^4)/(2*a*d)","A",7,6,27,0.2222,1,"{2893, 3049, 3033, 3023, 2735, 3770}"
1121,1,194,0,0.2225303,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","-\frac{9 a^2 b \cos (c+d x)}{2 d}-\frac{3 a^2 b \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}+a^3 x-\frac{9 a b^2 \cot (c+d x)}{2 d}+\frac{3 a b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{9}{2} a b^2 x+\frac{b^3 \cos ^3(c+d x)}{3 d}+\frac{b^3 \cos (c+d x)}{d}-\frac{b^3 \tanh ^{-1}(\cos (c+d x))}{d}","-\frac{9 a^2 b \cos (c+d x)}{2 d}-\frac{3 a^2 b \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac{9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot ^3(c+d x)}{3 d}+\frac{a^3 \cot (c+d x)}{d}+a^3 x-\frac{9 a b^2 \cot (c+d x)}{2 d}+\frac{3 a b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac{9}{2} a b^2 x+\frac{b^3 \cos ^3(c+d x)}{3 d}+\frac{b^3 \cos (c+d x)}{d}-\frac{b^3 \tanh ^{-1}(\cos (c+d x))}{d}",1,"a^3*x - (9*a*b^2*x)/2 + (9*a^2*b*ArcTanh[Cos[c + d*x]])/(2*d) - (b^3*ArcTanh[Cos[c + d*x]])/d - (9*a^2*b*Cos[c + d*x])/(2*d) + (b^3*Cos[c + d*x])/d + (b^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x])/d - (9*a*b^2*Cot[c + d*x])/(2*d) + (3*a*b^2*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (3*a^2*b*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a^3*Cot[c + d*x]^3)/(3*d)","A",17,10,21,0.4762,1,"{2722, 2592, 302, 206, 2591, 288, 321, 203, 3473, 8}"
1122,1,187,0,0.6562898,"\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^3,x]","-\frac{b^2 \left(73 a^2-2 b^2\right) \cos (c+d x)}{8 a d}-\frac{3 a \left(a^2-12 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{3}{2} b x \left(2 a^2-b^2\right)+\frac{17 b \cot (c+d x) (a+b \sin (c+d x))^2}{8 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}+\frac{5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{8 d}-\frac{13 b^3 \sin (c+d x) \cos (c+d x)}{4 d}","-\frac{b^2 \left(73 a^2-2 b^2\right) \cos (c+d x)}{8 a d}-\frac{3 a \left(a^2-12 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{3}{2} b x \left(2 a^2-b^2\right)+\frac{17 b \cot (c+d x) (a+b \sin (c+d x))^2}{8 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{4 a d}+\frac{5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{8 d}-\frac{13 b^3 \sin (c+d x) \cos (c+d x)}{4 d}",1,"(3*b*(2*a^2 - b^2)*x)/2 - (3*a*(a^2 - 12*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) - (b^2*(73*a^2 - 2*b^2)*Cos[c + d*x])/(8*a*d) - (13*b^3*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (17*b*Cot[c + d*x]*(a + b*Sin[c + d*x])^2)/(8*d) + (5*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^3)/(8*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(4*a*d)","A",7,6,27,0.2222,1,"{2893, 3047, 3033, 3023, 2735, 3770}"
1123,1,227,0,0.7110533,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","-\frac{b^3 \left(83 a^2+2 b^2\right) \cos (c+d x)}{40 a^2 d}-\frac{a \left(4 a^2-29 b^2\right) \cot (c+d x)}{20 d}-\frac{3 b \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}+3 a b^2 x-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac{2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac{27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}","-\frac{b^3 \left(83 a^2+2 b^2\right) \cos (c+d x)}{40 a^2 d}-\frac{a \left(4 a^2-29 b^2\right) \cot (c+d x)}{20 d}-\frac{3 b \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}+3 a b^2 x-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac{2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac{27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}",1,"3*a*b^2*x - (3*b*(3*a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) - (b^3*(83*a^2 + 2*b^2)*Cos[c + d*x])/(40*a^2*d) - (a*(4*a^2 - 29*b^2)*Cot[c + d*x])/(20*d) + (27*b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^2)/(40*d) + (2*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3)/(5*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(20*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^4)/(5*a*d)","A",7,6,29,0.2069,1,"{2893, 3047, 3031, 3023, 2735, 3770}"
1124,1,275,0,0.755893,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","-\frac{b \left(-43 a^2 b^2+36 a^4+2 b^4\right) \cot (c+d x)}{60 a^2 d}-\frac{a \left(a^2+18 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{\left(-84 a^2 b^2+15 a^4+4 b^4\right) \cot (c+d x) \csc (c+d x)}{240 a d}+\frac{\left(35 a^2-2 b^2\right) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{120 a^2 d}+\frac{b \left(39 a^2-2 b^2\right) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}+b^3 x","-\frac{b \left(-43 a^2 b^2+36 a^4+2 b^4\right) \cot (c+d x)}{60 a^2 d}-\frac{a \left(a^2+18 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{\left(-84 a^2 b^2+15 a^4+4 b^4\right) \cot (c+d x) \csc (c+d x)}{240 a d}+\frac{\left(35 a^2-2 b^2\right) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{120 a^2 d}+\frac{b \left(39 a^2-2 b^2\right) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{15 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{6 a d}+b^3 x",1,"b^3*x - (a*(a^2 + 18*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) - (b*(36*a^4 - 43*a^2*b^2 + 2*b^4)*Cot[c + d*x])/(60*a^2*d) - ((15*a^4 - 84*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x])/(240*a*d) + (b*(39*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(120*a^2*d) + ((35*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(120*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^4)/(15*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^4)/(6*a*d)","A",7,6,29,0.2069,1,"{2893, 3047, 3031, 3021, 2735, 3770}"
1125,1,303,0,0.8602017,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","-\frac{a \left(2 a^2+21 b^2\right) \cot (c+d x)}{35 d}-\frac{3 b \left(a^2+2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{\left(-19 a^2 b^2+4 a^4+2 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{140 a d}-\frac{b \left(-116 a^2 b^2+105 a^4+12 b^4\right) \cot (c+d x) \csc (c+d x)}{560 a^2 d}+\frac{\left(8 a^2-b^2\right) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac{b \left(53 a^2-6 b^2\right) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}","-\frac{a \left(2 a^2+21 b^2\right) \cot (c+d x)}{35 d}-\frac{3 b \left(a^2+2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{\left(-19 a^2 b^2+4 a^4+2 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{140 a d}-\frac{b \left(-116 a^2 b^2+105 a^4+12 b^4\right) \cot (c+d x) \csc (c+d x)}{560 a^2 d}+\frac{\left(8 a^2-b^2\right) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac{b \left(53 a^2-6 b^2\right) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}",1,"(-3*b*(a^2 + 2*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) - (a*(2*a^2 + 21*b^2)*Cot[c + d*x])/(35*d) - (b*(105*a^4 - 116*a^2*b^2 + 12*b^4)*Cot[c + d*x]*Csc[c + d*x])/(560*a^2*d) - ((4*a^4 - 19*a^2*b^2 + 2*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(140*a*d) + (b*(53*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(280*a^2*d) + ((8*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(35*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^4)/(14*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^4)/(7*a*d)","A",9,8,29,0.2759,1,"{2893, 3047, 3031, 3021, 2748, 3767, 8, 3770}"
1126,1,334,0,0.9072716,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","-\frac{b \left(6 a^2+7 b^2\right) \cot (c+d x)}{35 d}-\frac{3 a \left(a^2+8 b^2\right) \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{\left(-148 a^2 b^2+35 a^4+24 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}-\frac{b \left(-25 a^2 b^2+24 a^4+4 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac{3 a \left(a^2+8 b^2\right) \cot (c+d x) \csc (c+d x)}{128 d}+\frac{\left(21 a^2-4 b^2\right) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac{3 b \left(23 a^2-4 b^2\right) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}","-\frac{b \left(6 a^2+7 b^2\right) \cot (c+d x)}{35 d}-\frac{3 a \left(a^2+8 b^2\right) \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{\left(-148 a^2 b^2+35 a^4+24 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}-\frac{b \left(-25 a^2 b^2+24 a^4+4 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac{3 a \left(a^2+8 b^2\right) \cot (c+d x) \csc (c+d x)}{128 d}+\frac{\left(21 a^2-4 b^2\right) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac{3 b \left(23 a^2-4 b^2\right) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}",1,"(-3*a*(a^2 + 8*b^2)*ArcTanh[Cos[c + d*x]])/(128*d) - (b*(6*a^2 + 7*b^2)*Cot[c + d*x])/(35*d) - (3*a*(a^2 + 8*b^2)*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (b*(24*a^4 - 25*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(280*a^2*d) - ((35*a^4 - 148*a^2*b^2 + 24*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*a*d) + (3*b*(23*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(560*a^2*d) + ((21*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(112*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^4)/(14*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^7*(a + b*Sin[c + d*x])^4)/(8*a*d)","A",10,9,29,0.3103,1,"{2893, 3047, 3031, 3021, 2748, 3768, 3770, 3767, 8}"
1127,1,307,0,1.0158446,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","-\frac{\left(-25 a^2 b^2+30 a^4+b^4\right) \cos (c+d x)}{5 b^6 d}+\frac{6 a^2 \left(-3 a^2 b^2+2 a^4+b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d \sqrt{a^2-b^2}}-\frac{\left(a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac{\left(3 a^2-2 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{2 a b^3 d}-\frac{\left(10 a^2-7 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{5 b^4 d}+\frac{3 a \left(4 a^2-3 b^2\right) \sin (c+d x) \cos (c+d x)}{4 b^5 d}-\frac{3 a x \left(-8 a^2 b^2+8 a^4+b^4\right)}{4 b^7}-\frac{\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}","-\frac{\left(-25 a^2 b^2+30 a^4+b^4\right) \cos (c+d x)}{5 b^6 d}+\frac{6 a^2 \left(-3 a^2 b^2+2 a^4+b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d \sqrt{a^2-b^2}}-\frac{\left(a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac{\left(3 a^2-2 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{2 a b^3 d}-\frac{\left(10 a^2-7 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{5 b^4 d}+\frac{3 a \left(4 a^2-3 b^2\right) \sin (c+d x) \cos (c+d x)}{4 b^5 d}-\frac{3 a x \left(-8 a^2 b^2+8 a^4+b^4\right)}{4 b^7}-\frac{\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}",1,"(-3*a*(8*a^4 - 8*a^2*b^2 + b^4)*x)/(4*b^7) + (6*a^2*(2*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) - ((30*a^4 - 25*a^2*b^2 + b^4)*Cos[c + d*x])/(5*b^6*d) + (3*a*(4*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(4*b^5*d) - ((10*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(5*b^4*d) + ((3*a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(2*a*b^3*d) - (Cos[c + d*x]*Sin[c + d*x]^4)/(5*b^2*d) - ((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(a*b^2*d*(a + b*Sin[c + d*x]))","A",9,7,29,0.2414,1,"{2892, 3049, 3023, 2735, 2660, 618, 204}"
1128,1,267,0,0.7581606,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{a \left(15 a^2-11 b^2\right) \cos (c+d x)}{3 b^5 d}-\frac{2 a \left(-7 a^2 b^2+5 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d \sqrt{a^2-b^2}}-\frac{\left(a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac{\left(5 a^2-3 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{3 a b^3 d}-\frac{\left(20 a^2-13 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 d}+\frac{x \left(-36 a^2 b^2+40 a^4+3 b^4\right)}{8 b^6}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b^2 d}","\frac{a \left(15 a^2-11 b^2\right) \cos (c+d x)}{3 b^5 d}-\frac{2 a \left(-7 a^2 b^2+5 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d \sqrt{a^2-b^2}}-\frac{\left(a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac{\left(5 a^2-3 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{3 a b^3 d}-\frac{\left(20 a^2-13 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 d}+\frac{x \left(-36 a^2 b^2+40 a^4+3 b^4\right)}{8 b^6}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b^2 d}",1,"((40*a^4 - 36*a^2*b^2 + 3*b^4)*x)/(8*b^6) - (2*a*(5*a^4 - 7*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^6*Sqrt[a^2 - b^2]*d) + (a*(15*a^2 - 11*b^2)*Cos[c + d*x])/(3*b^5*d) - ((20*a^2 - 13*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*d) + ((5*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(3*a*b^3*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*b^2*d) - ((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(a*b^2*d*(a + b*Sin[c + d*x]))","A",8,7,29,0.2414,1,"{2892, 3049, 3023, 2735, 2660, 618, 204}"
1129,1,163,0,0.3061502,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{2 \left(-5 a^2 b^2+4 a^4+b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d \sqrt{a^2-b^2}}-\frac{\cos (c+d x) \left(4 a^2-2 a b \sin (c+d x)-b^2\right)}{b^4 d}-\frac{a x \left(4 a^2-3 b^2\right)}{b^5}+\frac{\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}","\frac{2 \left(-5 a^2 b^2+4 a^4+b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d \sqrt{a^2-b^2}}-\frac{\cos (c+d x) \left(4 a^2-2 a b \sin (c+d x)-b^2\right)}{b^4 d}-\frac{a x \left(4 a^2-3 b^2\right)}{b^5}+\frac{\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}",1,"-((a*(4*a^2 - 3*b^2)*x)/b^5) + (2*(4*a^4 - 5*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*d) + (Cos[c + d*x]^3*(4*a + b*Sin[c + d*x]))/(3*b^2*d*(a + b*Sin[c + d*x])) - (Cos[c + d*x]*(4*a^2 - b^2 - 2*a*b*Sin[c + d*x]))/(b^4*d)","A",6,6,27,0.2222,1,"{2863, 2865, 2735, 2660, 618, 204}"
1130,1,137,0,0.2738739,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{2 \sqrt{a^2-b^2} \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^3 d}-\frac{\left(a^2-b^2\right) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{2 a x}{b^3}-\frac{\cos (c+d x)}{b^2 d}","\frac{2 \sqrt{a^2-b^2} \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^3 d}-\frac{\left(a^2-b^2\right) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{2 a x}{b^3}-\frac{\cos (c+d x)}{b^2 d}",1,"(-2*a*x)/b^3 + (2*Sqrt[a^2 - b^2]*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^3*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) - Cos[c + d*x]/(b^2*d) - ((a^2 - b^2)*Cos[c + d*x])/(a*b^2*d*(a + b*Sin[c + d*x]))","A",6,6,27,0.2222,1,"{2892, 3057, 2660, 618, 204, 3770}"
1131,1,154,0,0.3037157,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(a^2 b^2+a^4-2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^2 d \sqrt{a^2-b^2}}+\frac{\left(a^2-2 b^2\right) \cos (c+d x)}{a^2 b d (a+b \sin (c+d x))}+\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{\cot (c+d x)}{a d (a+b \sin (c+d x))}+\frac{x}{b^2}","-\frac{2 \left(a^2 b^2+a^4-2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^2 d \sqrt{a^2-b^2}}+\frac{\left(a^2-2 b^2\right) \cos (c+d x)}{a^2 b d (a+b \sin (c+d x))}+\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{\cot (c+d x)}{a d (a+b \sin (c+d x))}+\frac{x}{b^2}",1,"x/b^2 - (2*(a^4 + a^2*b^2 - 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b^2*Sqrt[a^2 - b^2]*d) + (2*b*ArcTanh[Cos[c + d*x]])/(a^3*d) + ((a^2 - 2*b^2)*Cos[c + d*x])/(a^2*b*d*(a + b*Sin[c + d*x])) - Cot[c + d*x]/(a*d*(a + b*Sin[c + d*x]))","A",6,6,29,0.2069,1,"{2890, 3057, 2660, 618, 204, 3770}"
1132,1,180,0,0.4496384,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","\frac{6 b \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d}-\frac{\left(a^2-3 b^2\right) \cot (c+d x)}{a^3 b d}+\frac{3 \left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{\left(2 a^2-3 b^2\right) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}","\frac{6 b \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d}-\frac{\left(a^2-b^2\right) \cos (c+d x)}{a^3 d (a+b \sin (c+d x))}+\frac{3 \left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\cos (c+d x)}{2 a^2 d \left(1-\cos ^2(c+d x)\right)}",1,"(6*b*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*d) + (3*(a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^4*d) - ((a^2 - 3*b^2)*Cot[c + d*x])/(a^3*b*d) + ((2*a^2 - 3*b^2)*Cot[c + d*x])/(2*a^2*b*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b*Sin[c + d*x]))","A",7,7,27,0.2593,1,"{2890, 3055, 3001, 3770, 2660, 618, 204}"
1133,1,238,0,0.7032522,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^2,x]","\frac{2 \left(-5 a^2 b^2+a^4+4 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}+\frac{\left(7 a^2-12 b^2\right) \cot (c+d x)}{3 a^4 d}-\frac{b \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}-\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac{\left(3 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}","\frac{2 \left(-5 a^2 b^2+a^4+4 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}+\frac{\left(7 a^2-12 b^2\right) \cot (c+d x)}{3 a^4 d}-\frac{b \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}-\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac{\left(3 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))}",1,"(2*(a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) - (b*(3*a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + ((7*a^2 - 12*b^2)*Cot[c + d*x])/(3*a^4*d) - ((a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a^3*b*d) + ((3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x]))","A",8,7,21,0.3333,1,"{2724, 3055, 3001, 3770, 2660, 618, 204}"
1134,1,292,0,1.047256,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]","-\frac{2 b \left(-7 a^2 b^2+2 a^4+5 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2}}-\frac{b \left(11 a^2-15 b^2\right) \cot (c+d x)}{3 a^5 d}-\frac{\left(-36 a^2 b^2+3 a^4+40 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left(3 a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac{\left(13 a^2-20 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac{\left(4 a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}","-\frac{2 b \left(-7 a^2 b^2+2 a^4+5 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2}}-\frac{b \left(11 a^2-15 b^2\right) \cot (c+d x)}{3 a^5 d}-\frac{\left(-36 a^2 b^2+3 a^4+40 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left(3 a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{3 a^3 b d}+\frac{\left(13 a^2-20 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac{\left(4 a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}",1,"(-2*b*(2*a^4 - 7*a^2*b^2 + 5*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) - ((3*a^4 - 36*a^2*b^2 + 40*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - (b*(11*a^2 - 15*b^2)*Cot[c + d*x])/(3*a^5*d) + ((13*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) - ((3*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^3*b*d) + ((4*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^2*b*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x]))","A",9,7,27,0.2593,1,"{2890, 3055, 3001, 3770, 2660, 618, 204}"
1135,1,331,0,1.0084613,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]","\frac{a \left(30 a^2-13 b^2\right) \cos (c+d x)}{2 b^6 d}-\frac{3 a \left(-11 a^2 b^2+10 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d \sqrt{a^2-b^2}}+\frac{\left(7 a^2-2 b^2\right) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\left(a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}-\frac{\left(15 a^2-4 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{4 a^2 b^3 d}+\frac{\left(10 a^2-3 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{2 a b^4 d}-\frac{3 \left(20 a^2-7 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^5 d}+\frac{3 x \left(-24 a^2 b^2+40 a^4+b^4\right)}{8 b^7}","\frac{a \left(30 a^2-13 b^2\right) \cos (c+d x)}{2 b^6 d}-\frac{3 a \left(-11 a^2 b^2+10 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d \sqrt{a^2-b^2}}+\frac{\left(7 a^2-2 b^2\right) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\left(a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}-\frac{\left(15 a^2-4 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{4 a^2 b^3 d}+\frac{\left(10 a^2-3 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{2 a b^4 d}-\frac{3 \left(20 a^2-7 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^5 d}+\frac{3 x \left(-24 a^2 b^2+40 a^4+b^4\right)}{8 b^7}",1,"(3*(40*a^4 - 24*a^2*b^2 + b^4)*x)/(8*b^7) - (3*a*(10*a^4 - 11*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) + (a*(30*a^2 - 13*b^2)*Cos[c + d*x])/(2*b^6*d) - (3*(20*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^5*d) + ((10*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(2*a*b^4*d) - ((15*a^2 - 4*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^2*b^3*d) - ((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(2*a*b^2*d*(a + b*Sin[c + d*x])^2) + ((7*a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(2*a^2*b^2*d*(a + b*Sin[c + d*x]))","A",9,7,29,0.2414,1,"{2891, 3049, 3023, 2735, 2660, 618, 204}"
1136,1,284,0,0.7382132,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","-\frac{\left(60 a^2-17 b^2\right) \cos (c+d x)}{6 b^5 d}+\frac{\left(-19 a^2 b^2+20 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d \sqrt{a^2-b^2}}+\frac{\left(6 a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\left(a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}-\frac{\left(20 a^2-3 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{6 a^2 b^3 d}+\frac{\left(5 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{a b^4 d}+\frac{a x \left(9-\frac{20 a^2}{b^2}\right)}{2 b^4}","-\frac{\left(60 a^2-17 b^2\right) \cos (c+d x)}{6 b^5 d}+\frac{\left(-19 a^2 b^2+20 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d \sqrt{a^2-b^2}}+\frac{\left(6 a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\left(a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}-\frac{\left(20 a^2-3 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{6 a^2 b^3 d}+\frac{\left(5 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{a b^4 d}+\frac{a x \left(9-\frac{20 a^2}{b^2}\right)}{2 b^4}",1,"(a*(9 - (20*a^2)/b^2)*x)/(2*b^4) + ((20*a^4 - 19*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^6*Sqrt[a^2 - b^2]*d) - ((60*a^2 - 17*b^2)*Cos[c + d*x])/(6*b^5*d) + ((5*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(a*b^4*d) - ((20*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(6*a^2*b^3*d) - ((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(2*a*b^2*d*(a + b*Sin[c + d*x])^2) + ((6*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(2*a^2*b^2*d*(a + b*Sin[c + d*x]))","A",8,7,29,0.2414,1,"{2891, 3049, 3023, 2735, 2660, 618, 204}"
1137,1,173,0,0.2749427,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]","-\frac{3 a \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d \sqrt{a^2-b^2}}+\frac{3 \cos (c+d x) \left(4 a^2+2 a b \sin (c+d x)-b^2\right)}{2 b^4 d (a+b \sin (c+d x))}+\frac{3 x \left(4 a^2-b^2\right)}{2 b^5}+\frac{\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}","-\frac{3 a \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d \sqrt{a^2-b^2}}+\frac{3 \cos (c+d x) \left(4 a^2+2 a b \sin (c+d x)-b^2\right)}{2 b^4 d (a+b \sin (c+d x))}+\frac{3 x \left(4 a^2-b^2\right)}{2 b^5}+\frac{\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}",1,"(3*(4*a^2 - b^2)*x)/(2*b^5) - (3*a*(4*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*d) + (Cos[c + d*x]^3*(2*a + b*Sin[c + d*x]))/(2*b^2*d*(a + b*Sin[c + d*x])^2) + (3*Cos[c + d*x]*(4*a^2 - b^2 + 2*a*b*Sin[c + d*x]))/(2*b^4*d*(a + b*Sin[c + d*x]))","A",6,5,27,0.1852,1,"{2863, 2735, 2660, 618, 204}"
1138,1,175,0,0.2878935,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x])^3,x]","-\frac{\left(-a^2 b^2+2 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^3 d \sqrt{a^2-b^2}}+\frac{\left(3 a^2+2 b^2\right) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\left(a^2-b^2\right) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{x}{b^3}","-\frac{\left(-a^2 b^2+2 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^3 d \sqrt{a^2-b^2}}+\frac{\left(3 a^2+2 b^2\right) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\left(a^2-b^2\right) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{x}{b^3}",1,"x/b^3 - ((2*a^4 - a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b^3*Sqrt[a^2 - b^2]*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) - ((a^2 - b^2)*Cos[c + d*x])/(2*a*b^2*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 + 2*b^2)*Cos[c + d*x])/(2*a^2*b^2*d*(a + b*Sin[c + d*x]))","A",6,6,27,0.2222,1,"{2891, 3057, 2660, 618, 204, 3770}"
1139,1,182,0,0.4716536,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","-\frac{3 \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \sqrt{a^2-b^2}}-\frac{\left(a^2+6 b^2\right) \cos (c+d x)}{2 a^3 b d (a+b \sin (c+d x))}+\frac{\left(a^2-3 b^2\right) \cos (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a d (a+b \sin (c+d x))^2}","-\frac{3 \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \sqrt{a^2-b^2}}-\frac{\left(a^2+6 b^2\right) \cos (c+d x)}{2 a^3 b d (a+b \sin (c+d x))}+\frac{\left(a^2-3 b^2\right) \cos (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a d (a+b \sin (c+d x))^2}",1,"(-3*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]*d) + (3*b*ArcTanh[Cos[c + d*x]])/(a^4*d) + ((a^2 - 3*b^2)*Cos[c + d*x])/(2*a^2*b*d*(a + b*Sin[c + d*x])^2) - Cot[c + d*x]/(a*d*(a + b*Sin[c + d*x])^2) - ((a^2 + 6*b^2)*Cos[c + d*x])/(2*a^3*b*d*(a + b*Sin[c + d*x]))","A",7,7,29,0.2414,1,"{2890, 3055, 3001, 3770, 2660, 618, 204}"
1140,1,218,0,0.7719708,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]","\frac{3 b \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}-\frac{\left(a^2-12 b^2\right) \cot (c+d x)}{2 a^4 b d}+\frac{3 \left(a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}+\frac{\left(a^2-2 b^2\right) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac{3 b \cot (c+d x)}{a^3 d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}","\frac{3 b \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}-\frac{\left(a^2-12 b^2\right) \cot (c+d x)}{2 a^4 b d}+\frac{3 \left(a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}+\frac{\left(a^2-2 b^2\right) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac{3 b \cot (c+d x)}{a^3 d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}",1,"(3*b*(3*a^2 - 4*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) + (3*(a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^5*d) - ((a^2 - 12*b^2)*Cot[c + d*x])/(2*a^4*b*d) + ((a^2 - 2*b^2)*Cot[c + d*x])/(2*a^2*b*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b*Sin[c + d*x])^2) - (3*b*Cot[c + d*x])/(a^3*d*(a + b*Sin[c + d*x]))","A",8,7,27,0.2593,1,"{2890, 3055, 3001, 3770, 2660, 618, 204}"
1141,1,289,0,1.0986495,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]","\frac{\left(-19 a^2 b^2+2 a^4+20 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2}}+\frac{\left(17 a^2-60 b^2\right) \cot (c+d x)}{6 a^5 d}-\frac{b \left(9 a^2-20 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}-\frac{\left(a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left(3 a^2-20 b^2\right) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac{\left(3 a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}","\frac{\left(-19 a^2 b^2+2 a^4+20 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2}}+\frac{\left(17 a^2-60 b^2\right) \cot (c+d x)}{6 a^5 d}-\frac{b \left(9 a^2-20 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}-\frac{\left(a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac{\left(3 a^2-20 b^2\right) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac{\left(3 a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}",1,"((2*a^4 - 19*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) - (b*(9*a^2 - 20*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^6*d) + ((17*a^2 - 60*b^2)*Cot[c + d*x])/(6*a^5*d) - ((a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a^4*b*d) + ((3*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(6*a^2*b*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(6*a^3*b*d*(a + b*Sin[c + d*x]))","A",9,7,21,0.3333,1,"{2724, 3055, 3001, 3770, 2660, 618, 204}"
1142,1,340,0,1.458667,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x])^3,x]","-\frac{3 b \left(-11 a^2 b^2+2 a^4+10 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^7 d \sqrt{a^2-b^2}}-\frac{b \left(13 a^2-30 b^2\right) \cot (c+d x)}{2 a^6 d}-\frac{3 \left(-24 a^2 b^2+a^4+40 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}-\frac{\left(3 a^2-10 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac{3 \left(7 a^2-20 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^5 d}+\frac{\left(4 a^2-15 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}","-\frac{3 b \left(-11 a^2 b^2+2 a^4+10 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^7 d \sqrt{a^2-b^2}}-\frac{b \left(13 a^2-30 b^2\right) \cot (c+d x)}{2 a^6 d}-\frac{3 \left(-24 a^2 b^2+a^4+40 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}-\frac{\left(3 a^2-10 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b d}+\frac{3 \left(7 a^2-20 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^5 d}+\frac{\left(4 a^2-15 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{4 a^3 b d (a+b \sin (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{4 a^2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}",1,"(-3*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^7*Sqrt[a^2 - b^2]*d) - (3*(a^4 - 24*a^2*b^2 + 40*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^7*d) - (b*(13*a^2 - 30*b^2)*Cot[c + d*x])/(2*a^6*d) + (3*(7*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^5*d) - ((3*a^2 - 10*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(2*a^4*b*d) + ((2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^2*b*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x])^2) + ((4*a^2 - 15*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^3*b*d*(a + b*Sin[c + d*x]))","A",10,7,27,0.2593,1,"{2890, 3055, 3001, 3770, 2660, 618, 204}"
1143,1,463,0,1.0476905,"\int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{10 \left(16 a^2-33 b^2\right) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{8 a \left(40 a^2-81 b^2\right) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{8 \left(-937 a^2 b^2+480 a^4+231 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{16 a \left(-279 a^2 b^2+160 a^4+27 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{16 a \left(-439 a^4 b^2+306 a^2 b^4+160 a^6-27 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left(-798 a^4 b^2+435 a^2 b^4+320 a^6-693 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}","-\frac{10 \left(16 a^2-33 b^2\right) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{8 a \left(40 a^2-81 b^2\right) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{8 \left(-937 a^2 b^2+480 a^4+231 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{16 a \left(-279 a^2 b^2+160 a^4+27 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{16 a \left(-439 a^4 b^2+306 a^2 b^4+160 a^6-27 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left(-798 a^4 b^2+435 a^2 b^4+320 a^6-693 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}",1,"(16*a*(160*a^4 - 279*a^2*b^2 + 27*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d) - (8*(480*a^4 - 937*a^2*b^2 + 231*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(45045*b^5*d) + (8*a*(40*a^2 - 81*b^2)*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(3003*b^4*d) - (10*(16*a^2 - 33*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(1287*b^3*d) + (20*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(143*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2))/(13*b*d) - (8*(320*a^6 - 798*a^4*b^2 + 435*a^2*b^4 - 693*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (16*a*(160*a^6 - 439*a^4*b^2 + 306*a^2*b^4 - 27*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(45045*b^6*d*Sqrt[a + b*Sin[c + d*x]])","A",10,9,31,0.2903,1,"{2895, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
1144,1,332,0,0.6337818,"\int \cos ^4(c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(8 a^2-7 a b \sin (c+d x)-9 b^2\right)}{693 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(-24 a b \left(a^2-2 b^2\right) \sin (c+d x)-69 a^2 b^2+32 a^4+45 b^4\right)}{3465 b^4 d}-\frac{8 \left(-101 a^4 b^2+114 a^2 b^4+32 a^6-45 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3465 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \left(-93 a^2 b^2+32 a^4+93 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3465 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}","-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(8 a^2-7 a b \sin (c+d x)-9 b^2\right)}{693 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(-24 a b \left(a^2-2 b^2\right) \sin (c+d x)-69 a^2 b^2+32 a^4+45 b^4\right)}{3465 b^4 d}-\frac{8 \left(-101 a^4 b^2+114 a^2 b^4+32 a^6-45 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3465 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \left(-93 a^2 b^2+32 a^4+93 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3465 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}",1,"(-2*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(11*d) + (8*a*(32*a^4 - 93*a^2*b^2 + 93*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*(32*a^6 - 101*a^4*b^2 + 114*a^2*b^4 - 45*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3465*b^5*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(8*a^2 - 9*b^2 - 7*a*b*Sin[c + d*x]))/(693*b^2*d) + (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^4 - 69*a^2*b^2 + 45*b^4 - 24*a*b*(a^2 - 2*b^2)*Sin[c + d*x]))/(3465*b^4*d)","A",8,7,29,0.2414,1,"{2862, 2865, 2752, 2663, 2661, 2655, 2653}"
1145,1,338,0,0.8922568,"\int \cos ^3(c+d x) \cot (c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{2 \left(8 a^2-45 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{105 b^2 d}-\frac{2 \left(-53 a^2 b^2+8 a^4-60 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 a \left(8 a^2-51 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 b^2 d}-\frac{2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}+\frac{2 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}","-\frac{2 \left(8 a^2-45 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{105 b^2 d}-\frac{2 \left(-53 a^2 b^2+8 a^4-60 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 a \left(8 a^2-51 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 b^2 d}-\frac{2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{7 b d}+\frac{2 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}",1,"(-2*(8*a^2 - 45*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(105*b^2*d) + (8*a*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(35*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(7*b*d) + (2*a*(8*a^2 - 51*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(105*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(8*a^4 - 53*a^2*b^2 - 60*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(105*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*a*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])","A",10,10,29,0.3448,1,"{2895, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1146,1,323,0,0.8755009,"\int \cos ^2(c+d x) \cot ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]","\frac{\left(4 a^2+15 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 a b d}+\frac{a \left(4 a^2+11 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(4 a^2+57 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{3/2}}{a d}+\frac{b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}","\frac{\left(4 a^2+15 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 a b d}+\frac{a \left(4 a^2+11 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(4 a^2+57 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{3/2}}{a d}+\frac{b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}",1,"((4*a^2 + 15*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15*a*b*d) - (2*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(5*b*d) - (Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(a*d) - ((4*a^2 + 57*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (a*(4*a^2 + 11*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])","A",10,10,31,0.3226,1,"{2894, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1147,1,345,0,0.8982985,"\int \cos (c+d x) \cot ^3(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{\left(8 a^2+3 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^2 d}-\frac{\left(8 a^2+31 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{12 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(8 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{12 a b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left(12 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \sin (c+d x)}}+\frac{b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}","-\frac{\left(8 a^2+3 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^2 d}-\frac{\left(8 a^2+31 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{12 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(8 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{12 a b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left(12 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \sin (c+d x)}}+\frac{b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}",1,"-((8*a^2 + 3*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(12*a^2*d) + (b*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(2*a*d) + ((8*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(12*a*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((8*a^2 + 31*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(12*b*d*Sqrt[a + b*Sin[c + d*x]]) - ((12*a^2 + b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a*d*Sqrt[a + b*Sin[c + d*x]])","A",10,10,29,0.3448,1,"{2893, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1148,1,351,0,0.8618729,"\int \cot ^4(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]],x]","\frac{\left(32 a^2-3 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\left(32 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{a+b \sin (c+d x)}}+\frac{\left(80 a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{b \left(12 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{3 a d}","\frac{\left(32 a^2-3 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\left(32 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{a+b \sin (c+d x)}}+\frac{\left(80 a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{b \left(12 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{3 a d}",1,"((32*a^2 - 3*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(24*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(3*a*d) + ((80*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(24*a^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((32*a^2 + b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(24*a*d*Sqrt[a + b*Sin[c + d*x]]) - (b*(12*a^2 - b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^2*d*Sqrt[a + b*Sin[c + d*x]])","A",10,10,23,0.4348,1,"{2725, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1149,1,412,0,1.2354191,"\int \cot ^4(c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]","\frac{b \left(68 a^2-15 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^3 d}+\frac{b \left(196 a^2+5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{192 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{b \left(68 a^2-15 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{192 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left(24 a^2 b^2+48 a^4-5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left(4 a^2-b^2\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{32 a^2 d}+\frac{5 b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d}","\frac{b \left(68 a^2-15 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^3 d}+\frac{b \left(196 a^2+5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{192 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{b \left(68 a^2-15 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{192 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left(24 a^2 b^2+48 a^4-5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left(4 a^2-b^2\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{32 a^2 d}+\frac{5 b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d}",1,"(b*(68*a^2 - 15*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(192*a^3*d) + (5*(4*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(32*a^2*d) + (5*b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(4*a*d) + (b*(68*a^2 - 15*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(192*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (b*(196*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(192*a^2*d*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 + 24*a^2*b^2 - 5*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a^3*d*Sqrt[a + b*Sin[c + d*x]])","A",11,11,29,0.3793,1,"{2893, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1150,1,484,0,1.6221131,"\int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{\left(332 a^2 b^2+384 a^4-105 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1920 a^4 d}+\frac{\left(116 a^2 b^2+384 a^4-35 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1920 a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(332 a^2 b^2+384 a^4-105 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1920 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{b \left(-24 a^2 b^2+48 a^4+7 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{128 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(96 a^2-35 b^2\right) \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{240 a^2 d}+\frac{b \left(108 a^2-35 b^2\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{960 a^3 d}+\frac{7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}","-\frac{\left(332 a^2 b^2+384 a^4-105 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1920 a^4 d}+\frac{\left(116 a^2 b^2+384 a^4-35 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1920 a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(332 a^2 b^2+384 a^4-105 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1920 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{b \left(-24 a^2 b^2+48 a^4+7 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{128 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(96 a^2-35 b^2\right) \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{240 a^2 d}+\frac{b \left(108 a^2-35 b^2\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{960 a^3 d}+\frac{7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}",1,"-((384*a^4 + 332*a^2*b^2 - 105*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(1920*a^4*d) + (b*(108*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(960*a^3*d) + ((96*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(240*a^2*d) + (7*b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(40*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2))/(5*a*d) - ((384*a^4 + 332*a^2*b^2 - 105*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(1920*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((384*a^4 + 116*a^2*b^2 - 35*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(1920*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + (b*(48*a^4 - 24*a^2*b^2 + 7*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(128*a^4*d*Sqrt[a + b*Sin[c + d*x]])","A",12,11,31,0.3548,1,"{2893, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1151,1,528,0,1.2081633,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(80 a^2-221 b^2\right) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{8 a \left(8 a^2-21 b^2\right) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac{8 \left(-375 a^2 b^2+160 a^4+117 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac{16 a \left(-47 a^2 b^2+32 a^4-27 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 \left(-174 a^4 b^2+81 a^2 b^4+64 a^6-195 b^6\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{8 \left(-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+64 a^8+195 b^8\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 a \left(-111 a^4 b^2+102 a^2 b^4+32 a^6-471 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}","-\frac{2 \left(80 a^2-221 b^2\right) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{8 a \left(8 a^2-21 b^2\right) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac{8 \left(-375 a^2 b^2+160 a^4+117 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac{16 a \left(-47 a^2 b^2+32 a^4-27 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 \left(-174 a^4 b^2+81 a^2 b^4+64 a^6-195 b^6\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{8 \left(-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+64 a^8+195 b^8\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 a \left(-111 a^4 b^2+102 a^2 b^4+32 a^6-471 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}",1,"(8*(64*a^6 - 174*a^4*b^2 + 81*a^2*b^4 - 195*b^6)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d) + (16*a*(32*a^4 - 47*a^2*b^2 - 27*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(45045*b^5*d) - (8*(160*a^4 - 375*a^2*b^2 + 117*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(45045*b^5*d) + (8*a*(8*a^2 - 21*b^2)*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(1287*b^4*d) - (2*(80*a^2 - 221*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/(2145*b^3*d) + (4*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(39*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(15*b*d) - (16*a*(32*a^6 - 111*a^4*b^2 + 102*a^2*b^4 - 471*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(64*a^8 - 238*a^6*b^2 + 255*a^4*b^4 - 276*a^2*b^6 + 195*b^8)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(45045*b^6*d*Sqrt[a + b*Sin[c + d*x]])","A",11,9,31,0.2903,1,"{2895, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653}"
1152,1,394,0,0.8586366,"\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a \left(2 a^2-5 b^2\right)-7 b \left(a^2+11 b^2\right) \sin (c+d x)\right)}{3003 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(a \left(-113 a^2 b^2+32 a^4+177 b^4\right)-3 b \left(-27 a^2 b^2+8 a^4-77 b^4\right) \sin (c+d x)\right)}{15015 b^4 d}-\frac{8 a \left(-145 a^4 b^2+290 a^2 b^4+32 a^6-177 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(-137 a^4 b^2+258 a^2 b^4+32 a^6+231 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac{6 a \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{143 d}","-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a \left(2 a^2-5 b^2\right)-7 b \left(a^2+11 b^2\right) \sin (c+d x)\right)}{3003 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(a \left(-113 a^2 b^2+32 a^4+177 b^4\right)-3 b \left(-27 a^2 b^2+8 a^4-77 b^4\right) \sin (c+d x)\right)}{15015 b^4 d}-\frac{8 a \left(-145 a^4 b^2+290 a^2 b^4+32 a^6-177 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(-137 a^4 b^2+258 a^2 b^4+32 a^6+231 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac{6 a \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{143 d}",1,"(-6*a*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(143*d) - (2*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2))/(13*d) + (8*(32*a^6 - 137*a^4*b^2 + 258*a^2*b^4 + 231*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(32*a^6 - 145*a^4*b^2 + 290*a^2*b^4 - 177*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15015*b^5*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(4*a*(2*a^2 - 5*b^2) - 7*b*(a^2 + 11*b^2)*Sin[c + d*x]))/(3003*b^2*d) + (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(32*a^4 - 113*a^2*b^2 + 177*b^4) - 3*b*(8*a^4 - 27*a^2*b^2 - 77*b^4)*Sin[c + d*x]))/(15015*b^4*d)","A",9,7,29,0.2414,1,"{2862, 2865, 2752, 2663, 2661, 2655, 2653}"
1153,1,390,0,1.1593209,"\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(8 a^2-77 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b^2 d}-\frac{2 a \left(8 a^2-87 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^2 d}-\frac{2 a \left(-95 a^2 b^2+8 a^4-228 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(-93 a^2 b^2+8 a^4+84 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 a^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 b^2 d}-\frac{2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{9 b d}","-\frac{2 \left(8 a^2-77 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b^2 d}-\frac{2 a \left(8 a^2-87 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^2 d}-\frac{2 a \left(-95 a^2 b^2+8 a^4-228 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(-93 a^2 b^2+8 a^4+84 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 a^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 b^2 d}-\frac{2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{9 b d}",1,"(-2*a*(8*a^2 - 87*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b^2*d) - (2*(8*a^2 - 77*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(315*b^2*d) + (8*a*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(63*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(9*b*d) + (2*(8*a^4 - 93*a^2*b^2 + 84*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*a*(8*a^4 - 95*a^2*b^2 - 228*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*a^2*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])","A",11,10,29,0.3448,1,"{2895, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1154,1,374,0,1.1653706,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]","\frac{\left(4 a^2+35 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 a b d}+\frac{\left(4 a^2+65 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{35 b d}+\frac{\left(61 a^2 b^2+4 a^4+40 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{a \left(4 a^2+167 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}+\frac{3 a b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}","\frac{\left(4 a^2+35 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 a b d}+\frac{\left(4 a^2+65 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{35 b d}+\frac{\left(61 a^2 b^2+4 a^4+40 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{a \left(4 a^2+167 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}+\frac{3 a b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}",1,"((4*a^2 + 65*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(35*b*d) + ((4*a^2 + 35*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(35*a*b*d) - (2*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(7*b*d) - (Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(a*d) - (a*(4*a^2 + 167*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(35*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^4 + 61*a^2*b^2 + 40*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(35*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (3*a*b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])","A",11,10,31,0.3226,1,"{2894, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1155,1,383,0,1.1647054,"\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{\left(8 a^2-5 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{20 a^2 d}-\frac{\left(8 a^2-15 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{20 a d}-\frac{a \left(8 a^2+37 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{20 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(8 a^2-81 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{20 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 \left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \sin (c+d x)}}-\frac{b \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 a d}","-\frac{\left(8 a^2-5 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{20 a^2 d}-\frac{\left(8 a^2-15 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{20 a d}-\frac{a \left(8 a^2+37 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{20 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(8 a^2-81 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{20 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 \left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \sin (c+d x)}}-\frac{b \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 a d}",1,"-((8*a^2 - 15*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(20*a*d) - ((8*a^2 - 5*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(20*a^2*d) - (b*Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(2*a*d) + ((8*a^2 - 81*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(20*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (a*(8*a^2 + 37*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(20*b*d*Sqrt[a + b*Sin[c + d*x]]) - (3*(4*a^2 - b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*d*Sqrt[a + b*Sin[c + d*x]])","A",11,10,29,0.3448,1,"{2893, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1156,1,386,0,1.0997392,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{b \left(16 a^2+b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{8 a^2 d}+\frac{\left(32 a^2+b^2\right) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\left(16 a^2+21 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(32 a^2-b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{b \left(36 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{a+b \sin (c+d x)}}+\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}","-\frac{b \left(16 a^2+b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{8 a^2 d}+\frac{\left(32 a^2+b^2\right) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\left(16 a^2+21 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(32 a^2-b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{b \left(36 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{a+b \sin (c+d x)}}+\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}",1,"-(b*(16*a^2 + b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(8*a^2*d) + ((32*a^2 + b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(24*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/(3*a*d) + ((32*a^2 - b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(8*a*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((16*a^2 + 21*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*d*Sqrt[a + b*Sin[c + d*x]]) - (b*(36*a^2 + b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a*d*Sqrt[a + b*Sin[c + d*x]])","A",11,11,23,0.4783,1,"{2725, 3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1157,1,408,0,1.2371759,"\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2),x]","\frac{b \left(68 a^2-3 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{64 a^2 d}-\frac{b \left(20 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \sin (c+d x)}}+\frac{b \left(236 a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{3 \left(-24 a^2 b^2+16 a^4+b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(20 a^2-b^2\right) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{32 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}","\frac{b \left(68 a^2-3 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{64 a^2 d}-\frac{b \left(20 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \sin (c+d x)}}+\frac{b \left(236 a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{3 \left(-24 a^2 b^2+16 a^4+b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(20 a^2-b^2\right) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{32 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}",1,"(b*(68*a^2 - 3*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(64*a^2*d) + ((20*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(32*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(4*a*d) + (b*(236*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(64*a^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (b*(20*a^2 + b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a*d*Sqrt[a + b*Sin[c + d*x]]) + (3*(16*a^4 - 24*a^2*b^2 + b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a^2*d*Sqrt[a + b*Sin[c + d*x]])","A",11,10,29,0.3448,1,"{2893, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1158,1,484,0,1.6515243,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{\left(-116 a^2 b^2+128 a^4+15 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^3 d}+\frac{\left(692 a^2 b^2+128 a^4+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{640 a^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(-116 a^2 b^2+128 a^4+15 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{640 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{3 b \left(8 a^2 b^2+48 a^4-b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{128 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(32 a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac{3 b \left(36 a^2-5 b^2\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{320 a^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}","-\frac{\left(-116 a^2 b^2+128 a^4+15 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^3 d}+\frac{\left(692 a^2 b^2+128 a^4+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{640 a^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(-116 a^2 b^2+128 a^4+15 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{640 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{3 b \left(8 a^2 b^2+48 a^4-b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{128 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(32 a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{80 a^2 d}+\frac{3 b \left(36 a^2-5 b^2\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{320 a^2 d}+\frac{b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}",1,"-((128*a^4 - 116*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(640*a^3*d) + (3*b*(36*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(320*a^2*d) + ((32*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(80*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(5*a*d) - ((128*a^4 - 116*a^2*b^2 + 15*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(640*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((128*a^4 + 692*a^2*b^2 + 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(640*a^2*d*Sqrt[a + b*Sin[c + d*x]]) + (3*b*(48*a^4 + 8*a^2*b^2 - b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(128*a^3*d*Sqrt[a + b*Sin[c + d*x]])","A",12,11,31,0.3548,1,"{2893, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1159,1,551,0,1.9839711,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{b \left(512 a^2 b^2+2064 a^4-105 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{7680 a^4 d}+\frac{b \left(176 a^2 b^2+2544 a^4-35 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{7680 a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{b \left(512 a^2 b^2+2064 a^4-105 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{7680 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left(144 a^4 b^2-36 a^2 b^4+64 a^6+7 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{512 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{7 \left(4 a^2-b^2\right) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac{b \left(156 a^2-35 b^2\right) \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{960 a^2 d}-\frac{\left(-168 a^2 b^2+240 a^4+35 b^4\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{3840 a^3 d}+\frac{7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}","-\frac{b \left(512 a^2 b^2+2064 a^4-105 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{7680 a^4 d}+\frac{b \left(176 a^2 b^2+2544 a^4-35 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{7680 a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{b \left(512 a^2 b^2+2064 a^4-105 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{7680 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left(144 a^4 b^2-36 a^2 b^4+64 a^6+7 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{512 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{7 \left(4 a^2-b^2\right) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{96 a^2 d}+\frac{b \left(156 a^2-35 b^2\right) \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{960 a^2 d}-\frac{\left(-168 a^2 b^2+240 a^4+35 b^4\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{3840 a^3 d}+\frac{7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}",1,"-(b*(2064*a^4 + 512*a^2*b^2 - 105*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(7680*a^4*d) - ((240*a^4 - 168*a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3840*a^3*d) + (b*(156*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(960*a^2*d) + (7*(4*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(96*a^2*d) + (7*b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(60*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2))/(6*a*d) - (b*(2064*a^4 + 512*a^2*b^2 - 105*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(7680*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (b*(2544*a^4 + 176*a^2*b^2 - 35*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(7680*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + ((64*a^6 + 144*a^4*b^2 - 36*a^2*b^4 + 7*b^6)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(512*a^4*d*Sqrt[a + b*Sin[c + d*x]])","A",13,11,31,0.3548,1,"{2893, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1160,1,451,0,1.0748682,"\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{2 \left(3 a^2+13 b^2\right) \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{429 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(-7 a b \left(a^2+63 b^2\right) \sin (c+d x)-33 a^2 b^2+8 a^4-39 b^4\right)}{9009 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(-24 a b \left(-5 a^2 b^2+a^4-60 b^4\right) \sin (c+d x)-165 a^4 b^2+450 a^2 b^4+32 a^6+195 b^6\right)}{45045 b^4 d}-\frac{8 \left(-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6+32 a^8-195 b^8\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \left(-189 a^4 b^2+570 a^2 b^4+32 a^6+1635 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac{2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}","-\frac{2 \left(3 a^2+13 b^2\right) \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{429 d}-\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(-7 a b \left(a^2+63 b^2\right) \sin (c+d x)-33 a^2 b^2+8 a^4-39 b^4\right)}{9009 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(-24 a b \left(-5 a^2 b^2+a^4-60 b^4\right) \sin (c+d x)-165 a^4 b^2+450 a^2 b^4+32 a^6+195 b^6\right)}{45045 b^4 d}-\frac{8 \left(-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6+32 a^8-195 b^8\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \left(-189 a^4 b^2+570 a^2 b^4+32 a^6+1635 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{45045 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{15 d}-\frac{2 a \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{39 d}",1,"(-2*(3*a^2 + 13*b^2)*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(429*d) - (2*a*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2))/(39*d) - (2*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2))/(15*d) + (8*a*(32*a^6 - 189*a^4*b^2 + 570*a^2*b^4 + 1635*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*(32*a^8 - 197*a^6*b^2 + 615*a^4*b^4 - 255*a^2*b^6 - 195*b^8)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(45045*b^5*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(8*a^4 - 33*a^2*b^2 - 39*b^4 - 7*a*b*(a^2 + 63*b^2)*Sin[c + d*x]))/(9009*b^2*d) + (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^6 - 165*a^4*b^2 + 450*a^2*b^4 + 195*b^6 - 24*a*b*(a^4 - 5*a^2*b^2 - 60*b^4)*Sin[c + d*x]))/(45045*b^4*d)","A",10,7,29,0.2414,1,"{2862, 2865, 2752, 2663, 2661, 2655, 2653}"
1161,1,447,0,1.4218262,"\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{2 \left(8 a^2-117 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}-\frac{2 a \left(8 a^2-131 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac{2 \left(-141 a^2 b^2+8 a^4+36 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^2 d}-\frac{2 \left(-149 a^4 b^2-516 a^2 b^4+8 a^6-36 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 a \left(-147 a^2 b^2+8 a^4+444 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 a^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}","-\frac{2 \left(8 a^2-117 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}-\frac{2 a \left(8 a^2-131 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac{2 \left(-141 a^2 b^2+8 a^4+36 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^2 d}-\frac{2 \left(-149 a^4 b^2-516 a^2 b^4+8 a^6-36 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 a \left(-147 a^2 b^2+8 a^4+444 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 a^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}",1,"(-2*(8*a^4 - 141*a^2*b^2 + 36*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(693*b^2*d) - (2*a*(8*a^2 - 131*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(693*b^2*d) - (2*(8*a^2 - 117*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(693*b^2*d) + (8*a*Cos[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(99*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(11*b*d) + (2*a*(8*a^4 - 147*a^2*b^2 + 444*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(693*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(8*a^6 - 149*a^4*b^2 - 516*a^2*b^4 - 36*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(693*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*a^3*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])","A",12,10,29,0.3448,1,"{2895, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1162,1,426,0,1.4217042,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2),x]","\frac{\left(4 a^2+63 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 a b d}+\frac{\left(20 a^2+469 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b d}+\frac{a \left(20 a^2+759 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b d}+\frac{a \left(739 a^2 b^2+20 a^4+816 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(1689 a^2 b^2+20 a^4-168 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{5 a^2 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d}","\frac{\left(4 a^2+63 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 a b d}+\frac{\left(20 a^2+469 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b d}+\frac{a \left(20 a^2+759 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b d}+\frac{a \left(739 a^2 b^2+20 a^4+816 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(1689 a^2 b^2+20 a^4-168 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{5 a^2 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d}",1,"(a*(20*a^2 + 759*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b*d) + ((20*a^2 + 469*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(315*b*d) + ((4*a^2 + 63*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(63*a*b*d) - (2*Cos[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(9*b*d) - (Cot[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(a*d) - ((20*a^4 + 1689*a^2*b^2 - 168*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (a*(20*a^4 + 739*a^2*b^2 + 816*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (5*a^2*b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])","A",12,10,31,0.3226,1,"{2894, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1163,1,430,0,1.423144,"\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{\left(8 a^2-21 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac{\left(8 a^2-35 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac{\left(8 a^2-73 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{28 d}-\frac{\left(3 a^2 b^2+8 a^4-32 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{28 b d \sqrt{a+b \sin (c+d x)}}+\frac{a \left(8 a^2-247 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{28 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 a \left(4 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \sin (c+d x)}}-\frac{3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}","-\frac{\left(8 a^2-21 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac{\left(8 a^2-35 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac{\left(8 a^2-73 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{28 d}-\frac{\left(3 a^2 b^2+8 a^4-32 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{28 b d \sqrt{a+b \sin (c+d x)}}+\frac{a \left(8 a^2-247 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{28 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 a \left(4 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \sin (c+d x)}}-\frac{3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}",1,"-((8*a^2 - 73*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(28*d) - ((8*a^2 - 35*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(28*a*d) - ((8*a^2 - 21*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(28*a^2*d) - (3*b*Cot[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(2*a*d) + (a*(8*a^2 - 247*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(28*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((8*a^4 + 3*a^2*b^2 - 32*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(28*b*d*Sqrt[a + b*Sin[c + d*x]]) - (3*a*(4*a^2 - 5*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*d*Sqrt[a + b*Sin[c + d*x]])","A",12,10,29,0.3448,1,"{2893, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1164,1,429,0,1.3553161,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{b \left(208 a^2-25 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}-\frac{b \left(96 a^2-25 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{40 a d}+\frac{\left(32 a^2-3 b^2\right) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac{a \left(96 a^2+179 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{40 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(176 a^2-167 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{40 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{5 b \left(12 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \sin (c+d x)}}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}","-\frac{b \left(208 a^2-25 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}-\frac{b \left(96 a^2-25 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{40 a d}+\frac{\left(32 a^2-3 b^2\right) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac{a \left(96 a^2+179 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{40 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(176 a^2-167 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{40 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{5 b \left(12 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \sin (c+d x)}}-\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}",1,"-(b*(96*a^2 - 25*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(40*a*d) - (b*(208*a^2 - 25*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(120*a^2*d) + ((32*a^2 - 3*b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(24*a^2*d) - (b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(7/2))/(3*a*d) + ((176*a^2 - 167*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(40*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (a*(96*a^2 + 179*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(40*d*Sqrt[a + b*Sin[c + d*x]]) - (5*b*(12*a^2 - b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*d*Sqrt[a + b*Sin[c + d*x]])","A",12,11,23,0.4783,1,"{2725, 3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1165,1,449,0,1.511367,"\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{b^2 \left(196 a^2+5 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{64 a^2 d}+\frac{5 b \left(68 a^2+b^2\right) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}-\frac{b \left(148 a^2+169 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 d \sqrt{a+b \sin (c+d x)}}+\frac{b \left(492 a^2-5 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left(-360 a^2 b^2+48 a^4-5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \sin (c+d x)}}+\frac{\left(60 a^2+b^2\right) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}","-\frac{b^2 \left(196 a^2+5 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{64 a^2 d}+\frac{5 b \left(68 a^2+b^2\right) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}-\frac{b \left(148 a^2+169 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 d \sqrt{a+b \sin (c+d x)}}+\frac{b \left(492 a^2-5 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left(-360 a^2 b^2+48 a^4-5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \sin (c+d x)}}+\frac{\left(60 a^2+b^2\right) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}",1,"-(b^2*(196*a^2 + 5*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(64*a^2*d) + (5*b*(68*a^2 + b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(192*a^2*d) + ((60*a^2 + b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(96*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(7/2))/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(7/2))/(4*a*d) + (b*(492*a^2 - 5*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(64*a*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (b*(148*a^2 + 169*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*d*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 - 360*a^2*b^2 - 5*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a*d*Sqrt[a + b*Sin[c + d*x]])","A",12,11,29,0.3793,1,"{2893, 3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1166,1,482,0,1.6234136,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{\left(-580 a^2 b^2+128 a^4+15 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^2 d}+\frac{\left(492 a^2 b^2+128 a^4-5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{640 a d \sqrt{a+b \sin (c+d x)}}-\frac{\left(-2476 a^2 b^2+128 a^4-15 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{640 a^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{3 b \left(-40 a^2 b^2+80 a^4+b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{128 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(32 a^2-b^2\right) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac{b \left(36 a^2-b^2\right) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}","-\frac{\left(-580 a^2 b^2+128 a^4+15 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^2 d}+\frac{\left(492 a^2 b^2+128 a^4-5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{640 a d \sqrt{a+b \sin (c+d x)}}-\frac{\left(-2476 a^2 b^2+128 a^4-15 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{640 a^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{3 b \left(-40 a^2 b^2+80 a^4+b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{128 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(32 a^2-b^2\right) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{80 a^2 d}+\frac{b \left(36 a^2-b^2\right) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{64 a^2 d}+\frac{3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{40 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}",1,"-((128*a^4 - 580*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(640*a^2*d) + (b*(36*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(64*a^2*d) + ((32*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/(80*a^2*d) + (3*b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(7/2))/(40*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(7/2))/(5*a*d) - ((128*a^4 - 2476*a^2*b^2 - 15*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(640*a^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((128*a^4 + 492*a^2*b^2 - 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(640*a*d*Sqrt[a + b*Sin[c + d*x]]) + (3*b*(80*a^4 - 40*a^2*b^2 + b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(128*a^2*d*Sqrt[a + b*Sin[c + d*x]])","A",12,10,31,0.3226,1,"{2893, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1167,1,551,0,1.9714238,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{b \left(-176 a^2 b^2+720 a^4+15 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1536 a^3 d}+\frac{b \left(1696 a^2 b^2+816 a^4+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1536 a^2 d \sqrt{a+b \sin (c+d x)}}-\frac{b \left(-176 a^2 b^2+720 a^4+15 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1536 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left(720 a^4 b^2+60 a^2 b^4+64 a^6-5 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{512 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(28 a^2-3 b^2\right) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \left(52 a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}-\frac{\left(-56 a^2 b^2+16 a^4+5 b^4\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{256 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}","-\frac{b \left(-176 a^2 b^2+720 a^4+15 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1536 a^3 d}+\frac{b \left(1696 a^2 b^2+816 a^4+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1536 a^2 d \sqrt{a+b \sin (c+d x)}}-\frac{b \left(-176 a^2 b^2+720 a^4+15 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1536 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left(720 a^4 b^2+60 a^2 b^4+64 a^6-5 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{512 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(28 a^2-3 b^2\right) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \left(52 a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}-\frac{\left(-56 a^2 b^2+16 a^4+5 b^4\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{256 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}",1,"-(b*(720*a^4 - 176*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(1536*a^3*d) - ((16*a^4 - 56*a^2*b^2 + 5*b^4)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(256*a^2*d) + (b*(52*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(192*a^2*d) + ((28*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(96*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(7/2))/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^(7/2))/(6*a*d) - (b*(720*a^4 - 176*a^2*b^2 + 15*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(1536*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (b*(816*a^4 + 1696*a^2*b^2 + 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(1536*a^2*d*Sqrt[a + b*Sin[c + d*x]]) + ((64*a^6 + 720*a^4*b^2 + 60*a^2*b^4 - 5*b^6)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(512*a^3*d*Sqrt[a + b*Sin[c + d*x]])","A",13,11,31,0.3548,1,"{2893, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1168,1,471,0,1.1847782,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{10 \left(8 a^2-11 b^2\right) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{429 b^3 d}+\frac{4 a \left(160 a^2-223 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3003 b^4 d}-\frac{8 \left(-683 a^2 b^2+480 a^4+77 b^4\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^5 d}+\frac{64 a \left(-118 a^2 b^2+80 a^4+17 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^6 d}-\frac{8 a \left(-2368 a^4 b^2+875 a^2 b^4+1280 a^6+213 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^7 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(-2048 a^4 b^2+453 a^2 b^4+1280 a^6+231 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{24 a \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \sin ^5(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d}","-\frac{10 \left(8 a^2-11 b^2\right) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{429 b^3 d}+\frac{4 a \left(160 a^2-223 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3003 b^4 d}-\frac{8 \left(-683 a^2 b^2+480 a^4+77 b^4\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^5 d}+\frac{64 a \left(-118 a^2 b^2+80 a^4+17 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^6 d}-\frac{8 a \left(-2368 a^4 b^2+875 a^2 b^4+1280 a^6+213 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^7 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(-2048 a^4 b^2+453 a^2 b^4+1280 a^6+231 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{24 a \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{143 b^2 d}-\frac{2 \sin ^5(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{13 b d}",1,"(64*a*(80*a^4 - 118*a^2*b^2 + 17*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^6*d) - (8*(480*a^4 - 683*a^2*b^2 + 77*b^4)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^5*d) + (4*a*(160*a^2 - 223*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(3003*b^4*d) - (10*(8*a^2 - 11*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(429*b^3*d) + (24*a*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]])/(143*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(13*b*d) + (8*(1280*a^6 - 2048*a^4*b^2 + 453*a^2*b^4 + 231*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^7*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(1280*a^6 - 2368*a^4*b^2 + 875*a^2*b^4 + 213*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15015*b^7*d*Sqrt[a + b*Sin[c + d*x]])","A",10,8,31,0.2581,1,"{2895, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
1169,1,405,0,0.8859412,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{2 \left(80 a^2-117 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^3 d}+\frac{8 a \left(120 a^2-179 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3465 b^4 d}-\frac{8 \left(-247 a^2 b^2+160 a^4+45 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3465 b^5 d}+\frac{8 \left(-614 a^4 b^2+249 a^2 b^4+320 a^6+45 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3465 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 a \left(-267 a^2 b^2+160 a^4+69 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3465 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 a \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{99 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{11 b d}","-\frac{2 \left(80 a^2-117 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^3 d}+\frac{8 a \left(120 a^2-179 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3465 b^4 d}-\frac{8 \left(-247 a^2 b^2+160 a^4+45 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3465 b^5 d}+\frac{8 \left(-614 a^4 b^2+249 a^2 b^4+320 a^6+45 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3465 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 a \left(-267 a^2 b^2+160 a^4+69 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3465 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 a \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{99 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{11 b d}",1,"(-8*(160*a^4 - 247*a^2*b^2 + 45*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^5*d) + (8*a*(120*a^2 - 179*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^4*d) - (2*(80*a^2 - 117*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(693*b^3*d) + (20*a*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(99*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]])/(11*b*d) - (16*a*(160*a^4 - 267*a^2*b^2 + 69*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(320*a^6 - 614*a^4*b^2 + 249*a^2*b^4 + 45*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3465*b^6*d*Sqrt[a + b*Sin[c + d*x]])","A",9,8,31,0.2581,1,"{2895, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
1170,1,283,0,0.4528639,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/Sqrt[a + b*Sin[c + d*x]],x]","\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(a \left(32 a^2-33 b^2\right)-3 b \left(8 a^2-7 b^2\right) \sin (c+d x)\right)}{315 b^4 d}-\frac{8 a \left(-65 a^2 b^2+32 a^4+33 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(-57 a^2 b^2+32 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^2 d}","\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(a \left(32 a^2-33 b^2\right)-3 b \left(8 a^2-7 b^2\right) \sin (c+d x)\right)}{315 b^4 d}-\frac{8 a \left(-65 a^2 b^2+32 a^4+33 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(-57 a^2 b^2+32 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^2 d}",1,"(-2*Cos[c + d*x]^3*(8*a - 7*b*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(63*b^2*d) + (8*(32*a^4 - 57*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^5*d*Sqrt[a + b*Sin[c + d*x]]) + (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(32*a^2 - 33*b^2) - 3*b*(8*a^2 - 7*b^2)*Sin[c + d*x]))/(315*b^4*d)","A",7,6,29,0.2069,1,"{2865, 2752, 2663, 2661, 2655, 2653}"
1171,1,288,0,0.6548777,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{2 a \left(8 a^2-23 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(8 a^2-21 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{8 a \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 b^2 d}-\frac{2 \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{5 b d}+\frac{2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}","-\frac{2 a \left(8 a^2-23 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(8 a^2-21 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{8 a \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 b^2 d}-\frac{2 \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{5 b d}+\frac{2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}",1,"(8*a*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(5*b*d) + (2*(8*a^2 - 21*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*a*(8*a^2 - 23*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])","A",9,9,29,0.3103,1,"{2895, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1172,1,285,0,0.6727466,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/Sqrt[a + b*Sin[c + d*x]],x]","\frac{\left(4 a^2-7 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(4 a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}-\frac{\cot (c+d x) \sqrt{a+b \sin (c+d x)}}{a d}-\frac{b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \sin (c+d x)}}","\frac{\left(4 a^2-7 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(4 a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}-\frac{\cot (c+d x) \sqrt{a+b \sin (c+d x)}}{a d}-\frac{b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \sin (c+d x)}}",1,"(-2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*b*d) - (Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(a*d) - ((4*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*a*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^2 - 7*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a*d*Sqrt[a + b*Sin[c + d*x]])","A",9,9,31,0.2903,1,"{2894, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1173,1,307,0,0.6651933,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{\left(8 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(8 a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^2 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 \left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{3 b \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{2 a d}","-\frac{\left(8 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(8 a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^2 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 \left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{3 b \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{2 a d}",1,"(3*b*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(2*a*d) + ((8*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(4*a^2*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((8*a^2 + b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a*b*d*Sqrt[a + b*Sin[c + d*x]]) - (3*(4*a^2 - b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a^2*d*Sqrt[a + b*Sin[c + d*x]])","A",9,9,29,0.3103,1,"{2893, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1174,1,353,0,0.8811759,"\int \frac{\cot ^4(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[Cot[c + d*x]^4/Sqrt[a + b*Sin[c + d*x]],x]","\frac{\left(32 a^2-15 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^3 d}+\frac{\left(16 a^2+5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(32 a^2-15 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{b \left(12 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{5 b \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3 a d}","\frac{\left(32 a^2-15 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^3 d}+\frac{\left(16 a^2+5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(32 a^2-15 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{b \left(12 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{5 b \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3 a d}",1,"((32*a^2 - 15*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(24*a^3*d) + (5*b*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(3*a*d) + ((32*a^2 - 15*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(24*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((16*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(24*a^2*d*Sqrt[a + b*Sin[c + d*x]]) + (b*(12*a^2 - 5*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^3*d*Sqrt[a + b*Sin[c + d*x]])","A",10,10,23,0.4348,1,"{2725, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1175,1,412,0,1.2443416,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x])/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{b \left(188 a^2-105 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^4 d}+\frac{b \left(68 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{192 a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{b \left(188 a^2-105 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{192 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left(-72 a^2 b^2+48 a^4+35 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left(12 a^2-7 b^2\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{96 a^3 d}+\frac{7 b \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d}","-\frac{b \left(188 a^2-105 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^4 d}+\frac{b \left(68 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{192 a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{b \left(188 a^2-105 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{192 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left(-72 a^2 b^2+48 a^4+35 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{64 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left(12 a^2-7 b^2\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{96 a^3 d}+\frac{7 b \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d}",1,"-(b*(188*a^2 - 105*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(192*a^4*d) + (5*(12*a^2 - 7*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(96*a^3*d) + (7*b*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(4*a*d) - (b*(188*a^2 - 105*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(192*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (b*(68*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(192*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 - 72*a^2*b^2 + 35*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a^4*d*Sqrt[a + b*Sin[c + d*x]])","A",11,10,29,0.3448,1,"{2893, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1176,1,466,0,1.2140167,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(40 a^2-33 b^2\right) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{33 a b^3 d}-\frac{20 \left(32 a^2-27 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{231 b^4 d}+\frac{8 a \left(480 a^2-419 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^5 d}-\frac{8 \left(-592 a^2 b^2+640 a^4+15 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^6 d}+\frac{8 \left(-1664 a^4 b^2+369 a^2 b^4+1280 a^6+15 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1155 b^7 d \sqrt{a+b \sin (c+d x)}}-\frac{8 a \left(-1344 a^2 b^2+1280 a^4+123 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1155 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d}","-\frac{2 \left(a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(40 a^2-33 b^2\right) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{33 a b^3 d}-\frac{20 \left(32 a^2-27 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{231 b^4 d}+\frac{8 a \left(480 a^2-419 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^5 d}-\frac{8 \left(-592 a^2 b^2+640 a^4+15 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^6 d}+\frac{8 \left(-1664 a^4 b^2+369 a^2 b^4+1280 a^6+15 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1155 b^7 d \sqrt{a+b \sin (c+d x)}}-\frac{8 a \left(-1344 a^2 b^2+1280 a^4+123 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1155 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d}",1,"(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (8*(640*a^4 - 592*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^6*d) + (8*a*(480*a^2 - 419*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^5*d) - (20*(32*a^2 - 27*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(231*b^4*d) + (2*(40*a^2 - 33*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(33*a*b^3*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]])/(11*b^2*d) - (8*a*(1280*a^4 - 1344*a^2*b^2 + 123*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^7*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(1280*a^6 - 1664*a^4*b^2 + 369*a^2*b^4 + 15*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(1155*b^7*d*Sqrt[a + b*Sin[c + d*x]])","A",10,8,31,0.2581,1,"{2892, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
1177,1,401,0,0.9037989,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(80 a^2-63 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^3 d}-\frac{16 \left(60 a^2-49 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^4 d}+\frac{8 a \left(160 a^2-139 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^5 d}-\frac{16 a \left(-199 a^2 b^2+160 a^4+39 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^6 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(-318 a^2 b^2+320 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{9 b^2 d}","-\frac{2 \left(a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(80 a^2-63 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^3 d}-\frac{16 \left(60 a^2-49 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^4 d}+\frac{8 a \left(160 a^2-139 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^5 d}-\frac{16 a \left(-199 a^2 b^2+160 a^4+39 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^6 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(-318 a^2 b^2+320 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{9 b^2 d}",1,"(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (8*a*(160*a^2 - 139*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b^5*d) - (16*(60*a^2 - 49*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b^4*d) + (2*(80*a^2 - 63*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(63*a*b^3*d) - (2*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(9*b^2*d) + (8*(320*a^4 - 318*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (16*a*(160*a^4 - 199*a^2*b^2 + 39*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^6*d*Sqrt[a + b*Sin[c + d*x]])","A",9,8,31,0.2581,1,"{2892, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
1178,1,261,0,0.4242174,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(32 a^2-24 a b \sin (c+d x)-5 b^2\right)}{35 b^4 d}+\frac{8 \left(-37 a^2 b^2+32 a^4+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^5 d \sqrt{a+b \sin (c+d x)}}-\frac{8 a \left(32 a^2-29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt{a+b \sin (c+d x)}}","-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(32 a^2-24 a b \sin (c+d x)-5 b^2\right)}{35 b^4 d}+\frac{8 \left(-37 a^2 b^2+32 a^4+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^5 d \sqrt{a+b \sin (c+d x)}}-\frac{8 a \left(32 a^2-29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt{a+b \sin (c+d x)}}",1,"(-8*a*(32*a^2 - 29*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(35*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(32*a^4 - 37*a^2*b^2 + 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(35*b^5*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]^3*(8*a + b*Sin[c + d*x]))/(7*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^2 - 5*b^2 - 24*a*b*Sin[c + d*x]))/(35*b^4*d)","A",7,7,29,0.2414,1,"{2863, 2865, 2752, 2663, 2661, 2655, 2653}"
1179,1,296,0,0.6782508,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(a^2-b^2\right) \cos (c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(8 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left(8 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b^2 d}+\frac{2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \sin (c+d x)}}","-\frac{2 \left(a^2-b^2\right) \cos (c+d x)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(8 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left(8 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b^2 d}+\frac{2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \sin (c+d x)}}",1,"(-2*(a^2 - b^2)*Cos[c + d*x])/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*b^2*d) - (2*(8*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*a*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (2*(8*a^2 - 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a*d*Sqrt[a + b*Sin[c + d*x]])","A",9,9,29,0.3103,1,"{2892, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1180,1,294,0,0.705431,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^(3/2),x]","\frac{\left(2 a^2-3 b^2\right) \cos (c+d x)}{a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a^2 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x)}{a d \sqrt{a+b \sin (c+d x)}}","\frac{\left(2 a^2-3 b^2\right) \cos (c+d x)}{a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a^2 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x)}{a d \sqrt{a+b \sin (c+d x)}}",1,"((2*a^2 - 3*b^2)*Cos[c + d*x])/(a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - Cot[c + d*x]/(a*d*Sqrt[a + b*Sin[c + d*x]]) + ((4*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(a^2*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((4*a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (3*b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a^2*d*Sqrt[a + b*Sin[c + d*x]])","A",9,9,31,0.2903,1,"{2890, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1181,1,366,0,0.9283401,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{\left(8 a^2-15 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^3 b d}+\frac{\left(4 a^2-5 b^2\right) \cot (c+d x)}{2 a^2 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(8 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\left(8 a^2-15 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^3 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 \left(4 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d \sqrt{a+b \sin (c+d x)}}","-\frac{\left(8 a^2-15 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^3 b d}+\frac{\left(4 a^2-5 b^2\right) \cot (c+d x)}{2 a^2 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(8 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\left(8 a^2-15 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^3 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 \left(4 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d \sqrt{a+b \sin (c+d x)}}",1,"((4*a^2 - 5*b^2)*Cot[c + d*x])/(2*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*Sqrt[a + b*Sin[c + d*x]]) - ((8*a^2 - 15*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(4*a^3*b*d) - ((8*a^2 - 15*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(4*a^3*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((8*a^2 - 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (3*(4*a^2 - 5*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a^3*d*Sqrt[a + b*Sin[c + d*x]])","A",10,10,29,0.3448,1,"{2890, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1182,1,416,0,1.1548105,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^(3/2),x]","\frac{5 \left(16 a^2-21 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^4 d}-\frac{\left(32 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left(16 a^2-21 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{b \left(36 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(24 a^2-35 b^2\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^3 b d}+\frac{\left(6 a^2-7 b^2\right) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}}","\frac{5 \left(16 a^2-21 b^2\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^4 d}-\frac{\left(32 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left(16 a^2-21 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{24 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{b \left(36 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(24 a^2-35 b^2\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^3 b d}+\frac{\left(6 a^2-7 b^2\right) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}}",1,"((6*a^2 - 7*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*Sqrt[a + b*Sin[c + d*x]]) + (5*(16*a^2 - 21*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(24*a^4*d) - ((24*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(12*a^3*b*d) + (5*(16*a^2 - 21*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(24*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((32*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(24*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + (b*(36*a^2 - 35*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^4*d*Sqrt[a + b*Sin[c + d*x]])","A",11,10,23,0.4348,1,"{2724, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1183,1,469,0,1.1808818,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 \left(13 a^2-5 b^2\right) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac{10 \left(8 a^2-3 b^2\right) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac{4 \left(160 a^2-63 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^4 d}-\frac{8 \left(480 a^2-203 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^5 d}+\frac{128 a \left(40 a^2-19 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^6 d}-\frac{8 a \left(-1088 a^2 b^2+1280 a^4+123 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^7 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(-768 a^2 b^2+1280 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","\frac{2 \left(13 a^2-5 b^2\right) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac{10 \left(8 a^2-3 b^2\right) \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac{4 \left(160 a^2-63 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{63 a b^4 d}-\frac{8 \left(480 a^2-203 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^5 d}+\frac{128 a \left(40 a^2-19 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{315 b^6 d}-\frac{8 a \left(-1088 a^2 b^2+1280 a^4+123 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^7 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(-768 a^2 b^2+1280 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^7 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(3*a*b^2*d*(a + b*Sin[c + d*x])^(3/2)) + (2*(13*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(3*a^2*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (128*a*(40*a^2 - 19*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b^6*d) - (8*(480*a^2 - 203*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b^5*d) + (4*(160*a^2 - 63*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(63*a*b^4*d) - (10*(8*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(9*a^2*b^3*d) + (8*(1280*a^4 - 768*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^7*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(1280*a^4 - 1088*a^2*b^2 + 123*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^7*d*Sqrt[a + b*Sin[c + d*x]])","A",10,8,31,0.2581,1,"{2891, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
1184,1,411,0,0.9181702,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 \left(11 a^2-3 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac{2 \left(80 a^2-21 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a^2 b^3 d}+\frac{8 \left(24 a^2-7 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a b^4 d}-\frac{8 \left(32 a^2-11 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 b^5 d}+\frac{8 \left(-46 a^2 b^2+64 a^4+3 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{21 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 a \left(32 a^2-15 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{21 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","\frac{2 \left(11 a^2-3 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) \sin ^3(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac{2 \left(80 a^2-21 b^2\right) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a^2 b^3 d}+\frac{8 \left(24 a^2-7 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 a b^4 d}-\frac{8 \left(32 a^2-11 b^2\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{21 b^5 d}+\frac{8 \left(-46 a^2 b^2+64 a^4+3 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{21 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 a \left(32 a^2-15 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{21 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*b^2*d*(a + b*Sin[c + d*x])^(3/2)) + (2*(11*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(3*a^2*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (8*(32*a^2 - 11*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(21*b^5*d) + (8*(24*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(21*a*b^4*d) - (2*(80*a^2 - 21*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(21*a^2*b^3*d) - (16*a*(32*a^2 - 15*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(21*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(64*a^4 - 46*a^2*b^2 + 3*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(21*b^6*d*Sqrt[a + b*Sin[c + d*x]])","A",9,8,31,0.2581,1,"{2891, 3049, 3023, 2752, 2663, 2661, 2655, 2653}"
1185,1,254,0,0.4404576,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^(5/2),x]","\frac{4 \cos (c+d x) \left(32 a^2+8 a b \sin (c+d x)-9 b^2\right)}{15 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{8 a \left(32 a^2-17 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(32 a^2-9 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos ^3(c+d x) (8 a+3 b \sin (c+d x))}{15 b^2 d (a+b \sin (c+d x))^{3/2}}","\frac{4 \cos (c+d x) \left(32 a^2+8 a b \sin (c+d x)-9 b^2\right)}{15 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{8 a \left(32 a^2-17 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(32 a^2-9 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos ^3(c+d x) (8 a+3 b \sin (c+d x))}{15 b^2 d (a+b \sin (c+d x))^{3/2}}",1,"(8*(32*a^2 - 9*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(32*a^2 - 17*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15*b^5*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]^3*(8*a + 3*b*Sin[c + d*x]))/(15*b^2*d*(a + b*Sin[c + d*x])^(3/2)) + (4*Cos[c + d*x]*(32*a^2 - 9*b^2 + 8*a*b*Sin[c + d*x]))/(15*b^4*d*Sqrt[a + b*Sin[c + d*x]])","A",7,6,29,0.2069,1,"{2863, 2752, 2663, 2661, 2655, 2653}"
1186,1,313,0,0.6806063,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 \left(5 a^2+3 b^2\right) \cos (c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac{2 \left(8 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(8 a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a^2 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \sin (c+d x)}}","\frac{2 \left(5 a^2+3 b^2\right) \cos (c+d x)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac{2 \left(8 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \left(8 a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a^2 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \sin (c+d x)}}",1,"(-2*(a^2 - b^2)*Cos[c + d*x])/(3*a*b^2*d*(a + b*Sin[c + d*x])^(3/2)) + (2*(5*a^2 + 3*b^2)*Cos[c + d*x])/(3*a^2*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (2*(8*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*a^2*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(8*a^2 + b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*a*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a^2*d*Sqrt[a + b*Sin[c + d*x]])","A",9,9,29,0.3103,1,"{2891, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1187,1,346,0,0.987787,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{\left(4 a^2+15 b^2\right) \cos (c+d x)}{3 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(2 a^2-5 b^2\right) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}+\frac{\left(4 a^2+5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(4 a^2+15 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a^3 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{5 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}","-\frac{\left(4 a^2+15 b^2\right) \cos (c+d x)}{3 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(2 a^2-5 b^2\right) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}+\frac{\left(4 a^2+5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a^2 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(4 a^2+15 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 a^3 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{5 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}",1,"((2*a^2 - 5*b^2)*Cos[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])^(3/2)) - Cot[c + d*x]/(a*d*(a + b*Sin[c + d*x])^(3/2)) - ((4*a^2 + 15*b^2)*Cos[c + d*x])/(3*a^3*b*d*Sqrt[a + b*Sin[c + d*x]]) - ((4*a^2 + 15*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*a^3*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*a^2*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (5*b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a^3*d*Sqrt[a + b*Sin[c + d*x]])","A",10,10,31,0.3226,1,"{2890, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1188,1,407,0,1.2652101,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{\left(8 a^2-105 b^2\right) \cos (c+d x)}{12 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(8 a^2-35 b^2\right) \cot (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(4 a^2-7 b^2\right) \cot (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^{3/2}}+\frac{\left(8 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}-\frac{\left(8 a^2-105 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{12 a^4 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left(12 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^{3/2}}","-\frac{\left(8 a^2-105 b^2\right) \cos (c+d x)}{12 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(8 a^2-35 b^2\right) \cot (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(4 a^2-7 b^2\right) \cot (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^{3/2}}+\frac{\left(8 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}-\frac{\left(8 a^2-105 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{12 a^4 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left(12 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{4 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^{3/2}}",1,"((4*a^2 - 7*b^2)*Cot[c + d*x])/(6*a^2*b*d*(a + b*Sin[c + d*x])^(3/2)) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b*Sin[c + d*x])^(3/2)) - ((8*a^2 - 105*b^2)*Cos[c + d*x])/(12*a^4*d*Sqrt[a + b*Sin[c + d*x]]) - ((8*a^2 - 35*b^2)*Cot[c + d*x])/(12*a^3*b*d*Sqrt[a + b*Sin[c + d*x]]) - ((8*a^2 - 105*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(12*a^4*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((8*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(12*a^3*b*d*Sqrt[a + b*Sin[c + d*x]]) - ((12*a^2 - 35*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a^4*d*Sqrt[a + b*Sin[c + d*x]])","A",11,10,29,0.3448,1,"{2890, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1189,1,458,0,1.563916,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2),x]","\frac{b \left(32 a^2-105 b^2\right) \cos (c+d x)}{8 a^5 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(16 a^2-35 b^2\right) \cot (c+d x)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(16 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(32 a^2-105 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{15 b \left(4 a^2-7 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^5 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(8 a^2-21 b^2\right) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(2 a^2-3 b^2\right) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}","\frac{b \left(32 a^2-105 b^2\right) \cos (c+d x)}{8 a^5 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(16 a^2-35 b^2\right) \cot (c+d x)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(16 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{\left(32 a^2-105 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^5 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{15 b \left(4 a^2-7 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{8 a^5 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(8 a^2-21 b^2\right) \cot (c+d x) \csc (c+d x)}{12 a^3 b d \sqrt{a+b \sin (c+d x)}}+\frac{\left(2 a^2-3 b^2\right) \cot (c+d x) \csc (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}",1,"((2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])^(3/2)) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^(3/2)) + (b*(32*a^2 - 105*b^2)*Cos[c + d*x])/(8*a^5*d*Sqrt[a + b*Sin[c + d*x]]) + ((16*a^2 - 35*b^2)*Cot[c + d*x])/(8*a^4*d*Sqrt[a + b*Sin[c + d*x]]) - ((8*a^2 - 21*b^2)*Cot[c + d*x]*Csc[c + d*x])/(12*a^3*b*d*Sqrt[a + b*Sin[c + d*x]]) + ((32*a^2 - 105*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(8*a^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((16*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^4*d*Sqrt[a + b*Sin[c + d*x]]) + (15*b*(4*a^2 - 7*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^5*d*Sqrt[a + b*Sin[c + d*x]])","A",12,10,23,0.4348,1,"{2724, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805}"
1190,1,510,0,1.9104415,"\int \frac{\cos ^4(e+f x)}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx","Int[Cos[e + f*x]^4/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(9/2)),x]","\frac{32 b \left(2 a^2-b^2\right) \cos (e+f x)}{35 a^3 f \left(a^2-b^2\right)^2 \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}+\frac{8 \left(a^2-2 b^2\right) \cos (e+f x) \sqrt{d \sin (e+f x)}}{35 a^3 d f \left(a^2-b^2\right) (a+b \sin (e+f x))^{3/2}}-\frac{8 \left(5 a^2-3 a b-4 b^2\right) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{35 a^4 \sqrt{d} f (a-b) (a+b)^{3/2}}-\frac{32 b \left(2 a^2-b^2\right) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{35 a^5 \sqrt{d} f (a-b) (a+b)^{3/2}}+\frac{12 \cos (e+f x) \sqrt{d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac{2 \cos ^3(e+f x) \sqrt{d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}","\frac{32 b \left(2 a^2-b^2\right) \cos (e+f x)}{35 a^3 f \left(a^2-b^2\right)^2 \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}+\frac{8 \left(a^2-2 b^2\right) \cos (e+f x) \sqrt{d \sin (e+f x)}}{35 a^3 d f \left(a^2-b^2\right) (a+b \sin (e+f x))^{3/2}}-\frac{8 \left(5 a^2-3 a b-4 b^2\right) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{35 a^4 \sqrt{d} f (a-b) (a+b)^{3/2}}-\frac{32 b \left(2 a^2-b^2\right) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{35 a^5 \sqrt{d} f (a-b) (a+b)^{3/2}}+\frac{12 \cos (e+f x) \sqrt{d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac{2 \cos ^3(e+f x) \sqrt{d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}",1,"(2*Cos[e + f*x]^3*Sqrt[d*Sin[e + f*x]])/(7*a*d*f*(a + b*Sin[e + f*x])^(7/2)) + (12*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(35*a^2*d*f*(a + b*Sin[e + f*x])^(5/2)) + (8*(a^2 - 2*b^2)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(35*a^3*(a^2 - b^2)*d*f*(a + b*Sin[e + f*x])^(3/2)) + (32*b*(2*a^2 - b^2)*Cos[e + f*x])/(35*a^3*(a^2 - b^2)^2*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (32*b*(2*a^2 - b^2)*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(35*a^5*(a - b)*(a + b)^(3/2)*Sqrt[d]*f) - (8*(5*a^2 - 3*a*b - 4*b^2)*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(35*a^4*(a - b)*(a + b)^(3/2)*Sqrt[d]*f)","A",8,7,35,0.2000,1,"{2887, 2889, 3056, 2993, 2998, 2816, 2994}"
1191,0,0,0,0.1488465,"\int \frac{\cos ^4(c+d x) \sqrt[3]{\sin (c+d x)}}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]],x]","\int \frac{\cos ^4(c+d x) \sqrt[3]{\sin (c+d x)}}{\sqrt{a+b \sin (c+d x)}} \, dx","\text{Int}\left(\frac{\sqrt[3]{\sin (c+d x)} \cos ^4(c+d x)}{\sqrt{a+b \sin (c+d x)}},x\right)",0,"Defer[Int][(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]], x]","A",0,0,0,0,-1,"{}"
1192,0,0,0,0.1008695,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p,x]","\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx","\text{Int}\left(\cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p,x\right)",0,"Defer[Int][Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p, x]","A",0,0,0,0,-1,"{}"
1193,0,0,0,0.1137246,"\int \cos ^4(c+d x) \sin ^{-3-p}(c+d x) (a+b \sin (c+d x))^p \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^(-3 - p)*(a + b*Sin[c + d*x])^p,x]","\int \cos ^4(c+d x) \sin ^{-3-p}(c+d x) (a+b \sin (c+d x))^p \, dx","\text{Int}\left(\cos ^4(c+d x) \sin ^{-p-3}(c+d x) (a+b \sin (c+d x))^p,x\right)",0,"Defer[Int][Cos[c + d*x]^4*Sin[c + d*x]^(-3 - p)*(a + b*Sin[c + d*x])^p, x]","A",0,0,0,0,-1,"{}"
1194,0,0,0,0.1121203,"\int \cos ^4(c+d x) \sin ^{-4-p}(c+d x) (a+b \sin (c+d x))^p \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^(-4 - p)*(a + b*Sin[c + d*x])^p,x]","\int \cos ^4(c+d x) \sin ^{-4-p}(c+d x) (a+b \sin (c+d x))^p \, dx","\text{Int}\left(\cos ^4(c+d x) \sin ^{-p-4}(c+d x) (a+b \sin (c+d x))^p,x\right)",0,"Defer[Int][Cos[c + d*x]^4*Sin[c + d*x]^(-4 - p)*(a + b*Sin[c + d*x])^p, x]","A",0,0,0,0,-1,"{}"
1195,1,623,0,1.7591641,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^3,x]","\frac{3 a \left(a^2 (n+6)+3 b^2 (n+1)\right) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) (n+2) (n+4) (n+6) \sqrt{\cos ^2(c+d x)}}+\frac{3 b \left(3 a^2 (n+7)+b^2 (n+2)\right) \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) (n+3) (n+5) (n+7) \sqrt{\cos ^2(c+d x)}}-\frac{3 a \left(a^2 \left(n^2+5 n+6\right)-b^2 \left(n^2+15 n+53\right)\right) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (n+4) (n+5) (n+6) (n+7)}-\frac{3 a \left(-2 a^2 b^2 \left(n^2+16 n+58\right)+2 a^4 \left(n^2+5 n+6\right)+3 b^4 \left(n^2+12 n+35\right)\right) \cos (c+d x) \sin ^{n+1}(c+d x)}{b^2 d (n+2) (n+4) (n+5) (n+6) (n+7)}-\frac{3 \left(-2 a^2 b^2 \left(n^2+16 n+57\right)+2 a^4 \left(n^2+5 n+6\right)+b^4 \left(n^2+10 n+24\right)\right) \cos (c+d x) \sin ^{n+2}(c+d x)}{b d (n+3) (n+4) (n+5) (n+6) (n+7)}-\frac{\left(a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (n+5) (n+6) (n+7)}+\frac{a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac{\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}","\frac{3 a \left(a^2 (n+6)+3 b^2 (n+1)\right) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) (n+2) (n+4) (n+6) \sqrt{\cos ^2(c+d x)}}+\frac{3 b \left(3 a^2 (n+7)+b^2 (n+2)\right) \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) (n+3) (n+5) (n+7) \sqrt{\cos ^2(c+d x)}}-\frac{3 a \left(a^2 \left(n^2+5 n+6\right)-b^2 \left(n^2+15 n+53\right)\right) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (n+4) (n+5) (n+6) (n+7)}-\frac{3 a \left(-2 a^2 b^2 \left(n^2+16 n+58\right)+2 a^4 \left(n^2+5 n+6\right)+3 b^4 \left(n^2+12 n+35\right)\right) \cos (c+d x) \sin ^{n+1}(c+d x)}{b^2 d (n+2) (n+4) (n+5) (n+6) (n+7)}-\frac{3 \left(-2 a^2 b^2 \left(n^2+16 n+57\right)+2 a^4 \left(n^2+5 n+6\right)+b^4 \left(n^2+10 n+24\right)\right) \cos (c+d x) \sin ^{n+2}(c+d x)}{b d (n+3) (n+4) (n+5) (n+6) (n+7)}-\frac{\left(a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (n+5) (n+6) (n+7)}+\frac{a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac{\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)}",1,"(-3*a*(2*a^4*(6 + 5*n + n^2) + 3*b^4*(35 + 12*n + n^2) - 2*a^2*b^2*(58 + 16*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(1 + n))/(b^2*d*(2 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)) + (3*a*(3*b^2*(1 + n) + a^2*(6 + n))*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*(2 + n)*(4 + n)*(6 + n)*Sqrt[Cos[c + d*x]^2]) - (3*(2*a^4*(6 + 5*n + n^2) + b^4*(24 + 10*n + n^2) - 2*a^2*b^2*(57 + 16*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(2 + n))/(b*d*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)) + (3*b*(b^2*(2 + n) + 3*a^2*(7 + n))*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*(3 + n)*(5 + n)*(7 + n)*Sqrt[Cos[c + d*x]^2]) - (3*a*(a^2*(6 + 5*n + n^2) - b^2*(53 + 15*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^2)/(b^2*d*(4 + n)*(5 + n)*(6 + n)*(7 + n)) - ((a^2*(2 + n)*(3 + n) - b^2*(6 + n)*(8 + n))*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^3)/(b^2*d*(5 + n)*(6 + n)*(7 + n)) + (a*(3 + n)*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^4)/(b^2*d*(6 + n)*(7 + n)) - (Cos[c + d*x]*Sin[c + d*x]^(2 + n)*(a + b*Sin[c + d*x])^4)/(b*d*(7 + n))","A",8,6,29,0.2069,1,"{2895, 3049, 3033, 3023, 2748, 2643}"
1196,1,487,0,1.1202946,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^2,x]","\frac{3 \left(a^2 (n+6)+b^2 (n+1)\right) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) (n+2) (n+4) (n+6) \sqrt{\cos ^2(c+d x)}}-\frac{\left(-2 a^2 b^2 \left(n^2+13 n+40\right)+2 a^4 \left(n^2+5 n+6\right)+3 b^4 (n+5)\right) \cos (c+d x) \sin ^{n+1}(c+d x)}{b^2 d (n+2) (n+4) (n+5) (n+6)}-\frac{2 a \left(a^2 \left(n^2+5 n+6\right)-b^2 \left(n^2+13 n+39\right)\right) \cos (c+d x) \sin ^{n+2}(c+d x)}{b d (n+3) (n+4) (n+5) (n+6)}-\frac{\left(a^2 (n+2) (n+3)-b^2 (n+5) (n+7)\right) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (n+4) (n+5) (n+6)}+\frac{a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (n+5) (n+6)}+\frac{6 a b \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) (n+3) (n+5) \sqrt{\cos ^2(c+d x)}}-\frac{\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^3}{b d (n+6)}","\frac{3 \left(a^2 (n+6)+b^2 (n+1)\right) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) (n+2) (n+4) (n+6) \sqrt{\cos ^2(c+d x)}}-\frac{\left(-2 a^2 b^2 \left(n^2+13 n+40\right)+2 a^4 \left(n^2+5 n+6\right)+3 b^4 (n+5)\right) \cos (c+d x) \sin ^{n+1}(c+d x)}{b^2 d (n+2) (n+4) (n+5) (n+6)}-\frac{2 a \left(a^2 \left(n^2+5 n+6\right)-b^2 \left(n^2+13 n+39\right)\right) \cos (c+d x) \sin ^{n+2}(c+d x)}{b d (n+3) (n+4) (n+5) (n+6)}-\frac{\left(a^2 (n+2) (n+3)-b^2 (n+5) (n+7)\right) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (n+4) (n+5) (n+6)}+\frac{a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (n+5) (n+6)}+\frac{6 a b \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) (n+3) (n+5) \sqrt{\cos ^2(c+d x)}}-\frac{\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^3}{b d (n+6)}",1,"-(((3*b^4*(5 + n) + 2*a^4*(6 + 5*n + n^2) - 2*a^2*b^2*(40 + 13*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(1 + n))/(b^2*d*(2 + n)*(4 + n)*(5 + n)*(6 + n))) + (3*(b^2*(1 + n) + a^2*(6 + n))*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*(2 + n)*(4 + n)*(6 + n)*Sqrt[Cos[c + d*x]^2]) - (2*a*(a^2*(6 + 5*n + n^2) - b^2*(39 + 13*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(2 + n))/(b*d*(3 + n)*(4 + n)*(5 + n)*(6 + n)) + (6*a*b*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*(3 + n)*(5 + n)*Sqrt[Cos[c + d*x]^2]) - ((a^2*(2 + n)*(3 + n) - b^2*(5 + n)*(7 + n))*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^2)/(b^2*d*(4 + n)*(5 + n)*(6 + n)) + (a*(3 + n)*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^3)/(b^2*d*(5 + n)*(6 + n)) - (Cos[c + d*x]*Sin[c + d*x]^(2 + n)*(a + b*Sin[c + d*x])^3)/(b*d*(6 + n))","A",7,6,29,0.2069,1,"{2895, 3049, 3033, 3023, 2748, 2643}"
1197,1,129,0,0.1475957,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x]),x]","\frac{a \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{b \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}","\frac{a \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{b \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2) \sqrt{\cos ^2(c+d x)}}",1,"(a*Cos[c + d*x]*Hypergeometric2F1[-3/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (b*Cos[c + d*x]*Hypergeometric2F1[-3/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2])","A",3,2,27,0.07407,1,"{2838, 2577}"
1198,1,97,0,0.1297534,"\int \cos ^5(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^5*(a + b*Sin[c + d*x]),x]","\frac{a \sin ^{10}(c+d x)}{10 d}-\frac{a \sin ^8(c+d x)}{4 d}+\frac{a \sin ^6(c+d x)}{6 d}+\frac{b \sin ^{11}(c+d x)}{11 d}-\frac{2 b \sin ^9(c+d x)}{9 d}+\frac{b \sin ^7(c+d x)}{7 d}","\frac{a \sin ^{10}(c+d x)}{10 d}-\frac{a \sin ^8(c+d x)}{4 d}+\frac{a \sin ^6(c+d x)}{6 d}+\frac{b \sin ^{11}(c+d x)}{11 d}-\frac{2 b \sin ^9(c+d x)}{9 d}+\frac{b \sin ^7(c+d x)}{7 d}",1,"(a*Sin[c + d*x]^6)/(6*d) + (b*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^8)/(4*d) - (2*b*Sin[c + d*x]^9)/(9*d) + (a*Sin[c + d*x]^10)/(10*d) + (b*Sin[c + d*x]^11)/(11*d)","A",4,3,27,0.1111,1,"{2837, 12, 766}"
1199,1,97,0,0.1204722,"\int \cos ^5(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^4*(a + b*Sin[c + d*x]),x]","\frac{a \sin ^9(c+d x)}{9 d}-\frac{2 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{b \sin ^{10}(c+d x)}{10 d}-\frac{b \sin ^8(c+d x)}{4 d}+\frac{b \sin ^6(c+d x)}{6 d}","\frac{a \sin ^9(c+d x)}{9 d}-\frac{2 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{b \sin ^{10}(c+d x)}{10 d}-\frac{b \sin ^8(c+d x)}{4 d}+\frac{b \sin ^6(c+d x)}{6 d}",1,"(a*Sin[c + d*x]^5)/(5*d) + (b*Sin[c + d*x]^6)/(6*d) - (2*a*Sin[c + d*x]^7)/(7*d) - (b*Sin[c + d*x]^8)/(4*d) + (a*Sin[c + d*x]^9)/(9*d) + (b*Sin[c + d*x]^10)/(10*d)","A",4,3,27,0.1111,1,"{2837, 12, 766}"
1200,1,81,0,0.1355532,"\int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + b*Sin[c + d*x]),x]","\frac{a \cos ^8(c+d x)}{8 d}-\frac{a \cos ^6(c+d x)}{6 d}+\frac{b \sin ^9(c+d x)}{9 d}-\frac{2 b \sin ^7(c+d x)}{7 d}+\frac{b \sin ^5(c+d x)}{5 d}","\frac{a \cos ^8(c+d x)}{8 d}-\frac{a \cos ^6(c+d x)}{6 d}+\frac{b \sin ^9(c+d x)}{9 d}-\frac{2 b \sin ^7(c+d x)}{7 d}+\frac{b \sin ^5(c+d x)}{5 d}",1,"-(a*Cos[c + d*x]^6)/(6*d) + (a*Cos[c + d*x]^8)/(8*d) + (b*Sin[c + d*x]^5)/(5*d) - (2*b*Sin[c + d*x]^7)/(7*d) + (b*Sin[c + d*x]^9)/(9*d)","A",7,5,27,0.1852,1,"{2834, 2565, 14, 2564, 270}"
1201,1,81,0,0.1393108,"\int \cos ^5(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + b*Sin[c + d*x]),x]","\frac{a \sin ^7(c+d x)}{7 d}-\frac{2 a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}+\frac{b \cos ^8(c+d x)}{8 d}-\frac{b \cos ^6(c+d x)}{6 d}","\frac{a \sin ^7(c+d x)}{7 d}-\frac{2 a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{3 d}+\frac{b \cos ^8(c+d x)}{8 d}-\frac{b \cos ^6(c+d x)}{6 d}",1,"-(b*Cos[c + d*x]^6)/(6*d) + (b*Cos[c + d*x]^8)/(8*d) + (a*Sin[c + d*x]^3)/(3*d) - (2*a*Sin[c + d*x]^5)/(5*d) + (a*Sin[c + d*x]^7)/(7*d)","A",7,5,27,0.1852,1,"{2834, 2564, 270, 2565, 14}"
1202,1,65,0,0.0977145,"\int \cos ^5(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]*(a + b*Sin[c + d*x]),x]","-\frac{a \cos ^6(c+d x)}{6 d}+\frac{b \sin ^7(c+d x)}{7 d}-\frac{2 b \sin ^5(c+d x)}{5 d}+\frac{b \sin ^3(c+d x)}{3 d}","-\frac{a \cos ^6(c+d x)}{6 d}+\frac{b \sin ^7(c+d x)}{7 d}-\frac{2 b \sin ^5(c+d x)}{5 d}+\frac{b \sin ^3(c+d x)}{3 d}",1,"-(a*Cos[c + d*x]^6)/(6*d) + (b*Sin[c + d*x]^3)/(3*d) - (2*b*Sin[c + d*x]^5)/(5*d) + (b*Sin[c + d*x]^7)/(7*d)","A",6,5,25,0.2000,1,"{2834, 2565, 30, 2564, 270}"
1203,1,86,0,0.0807859,"\int \cos ^4(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*Cot[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{a \sin ^4(c+d x)}{4 d}-\frac{a \sin ^2(c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}+\frac{b \sin ^5(c+d x)}{5 d}-\frac{2 b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d}","\frac{a \sin ^4(c+d x)}{4 d}-\frac{a \sin ^2(c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}+\frac{b \sin ^5(c+d x)}{5 d}-\frac{2 b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d}",1,"(a*Log[Sin[c + d*x]])/d + (b*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/d - (2*b*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^4)/(4*d) + (b*Sin[c + d*x]^5)/(5*d)","A",4,3,25,0.1200,1,"{2837, 12, 766}"
1204,1,83,0,0.0926954,"\int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]^2*(a + b*Sin[c + d*x]),x]","\frac{a \sin ^3(c+d x)}{3 d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{b \sin ^4(c+d x)}{4 d}-\frac{b \sin ^2(c+d x)}{d}+\frac{b \log (\sin (c+d x))}{d}","\frac{a \sin ^3(c+d x)}{3 d}-\frac{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{b \sin ^4(c+d x)}{4 d}-\frac{b \sin ^2(c+d x)}{d}+\frac{b \log (\sin (c+d x))}{d}",1,"-((a*Csc[c + d*x])/d) + (b*Log[Sin[c + d*x]])/d - (2*a*Sin[c + d*x])/d - (b*Sin[c + d*x]^2)/d + (a*Sin[c + d*x]^3)/(3*d) + (b*Sin[c + d*x]^4)/(4*d)","A",4,3,27,0.1111,1,"{2837, 12, 766}"
1205,1,86,0,0.0917852,"\int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^3*(a + b*Sin[c + d*x]),x]","\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{2 a \log (\sin (c+d x))}{d}+\frac{b \sin ^3(c+d x)}{3 d}-\frac{2 b \sin (c+d x)}{d}-\frac{b \csc (c+d x)}{d}","\frac{a \sin ^2(c+d x)}{2 d}-\frac{a \csc ^2(c+d x)}{2 d}-\frac{2 a \log (\sin (c+d x))}{d}+\frac{b \sin ^3(c+d x)}{3 d}-\frac{2 b \sin (c+d x)}{d}-\frac{b \csc (c+d x)}{d}",1,"-((b*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (2*a*Log[Sin[c + d*x]])/d - (2*b*Sin[c + d*x])/d + (a*Sin[c + d*x]^2)/(2*d) + (b*Sin[c + d*x]^3)/(3*d)","A",4,3,27,0.1111,1,"{2837, 12, 766}"
1206,1,85,0,0.0850271,"\int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^4*(a + b*Sin[c + d*x]),x]","\frac{a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{2 a \csc (c+d x)}{d}+\frac{b \sin ^2(c+d x)}{2 d}-\frac{b \csc ^2(c+d x)}{2 d}-\frac{2 b \log (\sin (c+d x))}{d}","\frac{a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{2 a \csc (c+d x)}{d}+\frac{b \sin ^2(c+d x)}{2 d}-\frac{b \csc ^2(c+d x)}{2 d}-\frac{2 b \log (\sin (c+d x))}{d}",1,"(2*a*Csc[c + d*x])/d - (b*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (2*b*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d + (b*Sin[c + d*x]^2)/(2*d)","A",4,3,25,0.1200,1,"{2837, 12, 766}"
1207,1,81,0,0.0529924,"\int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}+\frac{b \sin (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}+\frac{2 b \csc (c+d x)}{d}","-\frac{a \csc ^4(c+d x)}{4 d}+\frac{a \csc ^2(c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}+\frac{b \sin (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}+\frac{2 b \csc (c+d x)}{d}",1,"(2*b*Csc[c + d*x])/d + (a*Csc[c + d*x]^2)/d - (b*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) + (a*Log[Sin[c + d*x]])/d + (b*Sin[c + d*x])/d","A",3,2,19,0.1053,1,"{2721, 766}"
1208,1,86,0,0.0835282,"\int \cot ^5(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^5(c+d x)}{5 d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}-\frac{b \csc ^4(c+d x)}{4 d}+\frac{b \csc ^2(c+d x)}{d}+\frac{b \log (\sin (c+d x))}{d}","-\frac{a \csc ^5(c+d x)}{5 d}+\frac{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}-\frac{b \csc ^4(c+d x)}{4 d}+\frac{b \csc ^2(c+d x)}{d}+\frac{b \log (\sin (c+d x))}{d}",1,"-((a*Csc[c + d*x])/d) + (b*Csc[c + d*x]^2)/d + (2*a*Csc[c + d*x]^3)/(3*d) - (b*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^5)/(5*d) + (b*Log[Sin[c + d*x]])/d","A",4,3,25,0.1200,1,"{2837, 12, 766}"
1209,1,61,0,0.1123619,"\int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^6(c+d x)}{6 d}-\frac{b \csc ^5(c+d x)}{5 d}+\frac{2 b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}","-\frac{a \cot ^6(c+d x)}{6 d}-\frac{b \csc ^5(c+d x)}{5 d}+\frac{2 b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}",1,"-(a*Cot[c + d*x]^6)/(6*d) - (b*Csc[c + d*x])/d + (2*b*Csc[c + d*x]^3)/(3*d) - (b*Csc[c + d*x]^5)/(5*d)","A",6,5,27,0.1852,1,"{2834, 2607, 30, 2606, 194}"
1210,1,65,0,0.1224798,"\int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^3*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{b \cot ^6(c+d x)}{6 d}","-\frac{a \csc ^7(c+d x)}{7 d}+\frac{2 a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{b \cot ^6(c+d x)}{6 d}",1,"-(b*Cot[c + d*x]^6)/(6*d) - (a*Csc[c + d*x]^3)/(3*d) + (2*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d)","A",6,5,27,0.1852,1,"{2834, 2606, 270, 2607, 30}"
1211,1,81,0,0.1293114,"\int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^6(c+d x)}{6 d}-\frac{b \csc ^7(c+d x)}{7 d}+\frac{2 b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{3 d}","-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^6(c+d x)}{6 d}-\frac{b \csc ^7(c+d x)}{7 d}+\frac{2 b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{3 d}",1,"-(a*Cot[c + d*x]^6)/(6*d) - (a*Cot[c + d*x]^8)/(8*d) - (b*Csc[c + d*x]^3)/(3*d) + (2*b*Csc[c + d*x]^5)/(5*d) - (b*Csc[c + d*x]^7)/(7*d)","A",7,5,27,0.1852,1,"{2834, 2607, 14, 2606, 270}"
1212,1,81,0,0.1310183,"\int \cot ^5(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^5*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^9(c+d x)}{9 d}+\frac{2 a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{b \cot ^8(c+d x)}{8 d}-\frac{b \cot ^6(c+d x)}{6 d}","-\frac{a \csc ^9(c+d x)}{9 d}+\frac{2 a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{b \cot ^8(c+d x)}{8 d}-\frac{b \cot ^6(c+d x)}{6 d}",1,"-(b*Cot[c + d*x]^6)/(6*d) - (b*Cot[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^5)/(5*d) + (2*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d)","A",7,5,27,0.1852,1,"{2834, 2606, 270, 2607, 14}"
1213,1,97,0,0.0949734,"\int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^6*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^{10}(c+d x)}{10 d}+\frac{a \csc ^8(c+d x)}{4 d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{b \csc ^9(c+d x)}{9 d}+\frac{2 b \csc ^7(c+d x)}{7 d}-\frac{b \csc ^5(c+d x)}{5 d}","-\frac{a \csc ^{10}(c+d x)}{10 d}+\frac{a \csc ^8(c+d x)}{4 d}-\frac{a \csc ^6(c+d x)}{6 d}-\frac{b \csc ^9(c+d x)}{9 d}+\frac{2 b \csc ^7(c+d x)}{7 d}-\frac{b \csc ^5(c+d x)}{5 d}",1,"-(b*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(6*d) + (2*b*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) - (b*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d)","A",4,3,27,0.1111,1,"{2837, 12, 766}"
1214,1,97,0,0.0950315,"\int \cot ^5(c+d x) \csc ^7(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^7*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^{11}(c+d x)}{11 d}+\frac{2 a \csc ^9(c+d x)}{9 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{b \csc ^{10}(c+d x)}{10 d}+\frac{b \csc ^8(c+d x)}{4 d}-\frac{b \csc ^6(c+d x)}{6 d}","-\frac{a \csc ^{11}(c+d x)}{11 d}+\frac{2 a \csc ^9(c+d x)}{9 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{b \csc ^{10}(c+d x)}{10 d}+\frac{b \csc ^8(c+d x)}{4 d}-\frac{b \csc ^6(c+d x)}{6 d}",1,"-(b*Csc[c + d*x]^6)/(6*d) - (a*Csc[c + d*x]^7)/(7*d) + (b*Csc[c + d*x]^8)/(4*d) + (2*a*Csc[c + d*x]^9)/(9*d) - (b*Csc[c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*d)","A",4,3,27,0.1111,1,"{2837, 12, 766}"
1215,1,138,0,0.1878171,"\int \cos ^5(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \sin ^7(c+d x)}{7 d}-\frac{\left(2 a^2-b^2\right) \sin ^5(c+d x)}{5 d}+\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a b \sin ^8(c+d x)}{4 d}-\frac{2 a b \sin ^6(c+d x)}{3 d}+\frac{a b \sin ^4(c+d x)}{2 d}+\frac{b^2 \sin ^9(c+d x)}{9 d}","\frac{\left(a^2-2 b^2\right) \sin ^7(c+d x)}{7 d}-\frac{\left(2 a^2-b^2\right) \sin ^5(c+d x)}{5 d}+\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a b \sin ^8(c+d x)}{4 d}-\frac{2 a b \sin ^6(c+d x)}{3 d}+\frac{a b \sin ^4(c+d x)}{2 d}+\frac{b^2 \sin ^9(c+d x)}{9 d}",1,"(a^2*Sin[c + d*x]^3)/(3*d) + (a*b*Sin[c + d*x]^4)/(2*d) - ((2*a^2 - b^2)*Sin[c + d*x]^5)/(5*d) - (2*a*b*Sin[c + d*x]^6)/(3*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^7)/(7*d) + (a*b*Sin[c + d*x]^8)/(4*d) + (b^2*Sin[c + d*x]^9)/(9*d)","A",4,3,29,0.1034,1,"{2837, 12, 948}"
1216,1,138,0,0.1278398,"\int \cos ^5(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \sin ^6(c+d x)}{6 d}-\frac{\left(2 a^2-b^2\right) \sin ^4(c+d x)}{4 d}+\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{2 a b \sin ^7(c+d x)}{7 d}-\frac{4 a b \sin ^5(c+d x)}{5 d}+\frac{2 a b \sin ^3(c+d x)}{3 d}+\frac{b^2 \sin ^8(c+d x)}{8 d}","\frac{\left(a^2-2 b^2\right) \sin ^6(c+d x)}{6 d}-\frac{\left(2 a^2-b^2\right) \sin ^4(c+d x)}{4 d}+\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{2 a b \sin ^7(c+d x)}{7 d}-\frac{4 a b \sin ^5(c+d x)}{5 d}+\frac{2 a b \sin ^3(c+d x)}{3 d}+\frac{b^2 \sin ^8(c+d x)}{8 d}",1,"(a^2*Sin[c + d*x]^2)/(2*d) + (2*a*b*Sin[c + d*x]^3)/(3*d) - ((2*a^2 - b^2)*Sin[c + d*x]^4)/(4*d) - (4*a*b*Sin[c + d*x]^5)/(5*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^6)/(6*d) + (2*a*b*Sin[c + d*x]^7)/(7*d) + (b^2*Sin[c + d*x]^8)/(8*d)","A",4,3,27,0.1111,1,"{2837, 12, 772}"
1217,1,130,0,0.1335192,"\int \cos ^4(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Cot[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \sin ^4(c+d x)}{4 d}-\frac{\left(2 a^2-b^2\right) \sin ^2(c+d x)}{2 d}+\frac{a^2 \log (\sin (c+d x))}{d}+\frac{2 a b \sin ^5(c+d x)}{5 d}-\frac{4 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin ^6(c+d x)}{6 d}","\frac{\left(a^2-2 b^2\right) \sin ^4(c+d x)}{4 d}-\frac{\left(2 a^2-b^2\right) \sin ^2(c+d x)}{2 d}+\frac{a^2 \log (\sin (c+d x))}{d}+\frac{2 a b \sin ^5(c+d x)}{5 d}-\frac{4 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin ^6(c+d x)}{6 d}",1,"(a^2*Log[Sin[c + d*x]])/d + (2*a*b*Sin[c + d*x])/d - ((2*a^2 - b^2)*Sin[c + d*x]^2)/(2*d) - (4*a*b*Sin[c + d*x]^3)/(3*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^4)/(4*d) + (2*a*b*Sin[c + d*x]^5)/(5*d) + (b^2*Sin[c + d*x]^6)/(6*d)","A",4,3,27,0.1111,1,"{2837, 12, 948}"
1218,1,125,0,0.1592454,"\int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \sin ^3(c+d x)}{3 d}-\frac{\left(2 a^2-b^2\right) \sin (c+d x)}{d}-\frac{a^2 \csc (c+d x)}{d}+\frac{a b \sin ^4(c+d x)}{2 d}-\frac{2 a b \sin ^2(c+d x)}{d}+\frac{2 a b \log (\sin (c+d x))}{d}+\frac{b^2 \sin ^5(c+d x)}{5 d}","\frac{\left(a^2-2 b^2\right) \sin ^3(c+d x)}{3 d}-\frac{\left(2 a^2-b^2\right) \sin (c+d x)}{d}-\frac{a^2 \csc (c+d x)}{d}+\frac{a b \sin ^4(c+d x)}{2 d}-\frac{2 a b \sin ^2(c+d x)}{d}+\frac{2 a b \log (\sin (c+d x))}{d}+\frac{b^2 \sin ^5(c+d x)}{5 d}",1,"-((a^2*Csc[c + d*x])/d) + (2*a*b*Log[Sin[c + d*x]])/d - ((2*a^2 - b^2)*Sin[c + d*x])/d - (2*a*b*Sin[c + d*x]^2)/d + ((a^2 - 2*b^2)*Sin[c + d*x]^3)/(3*d) + (a*b*Sin[c + d*x]^4)/(2*d) + (b^2*Sin[c + d*x]^5)/(5*d)","A",4,3,29,0.1034,1,"{2837, 12, 948}"
1219,1,127,0,0.1572138,"\int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \sin ^2(c+d x)}{2 d}-\frac{\left(2 a^2-b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \csc ^2(c+d x)}{2 d}+\frac{2 a b \sin ^3(c+d x)}{3 d}-\frac{4 a b \sin (c+d x)}{d}-\frac{2 a b \csc (c+d x)}{d}+\frac{b^2 \sin ^4(c+d x)}{4 d}","\frac{\left(a^2-2 b^2\right) \sin ^2(c+d x)}{2 d}-\frac{\left(2 a^2-b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \csc ^2(c+d x)}{2 d}+\frac{2 a b \sin ^3(c+d x)}{3 d}-\frac{4 a b \sin (c+d x)}{d}-\frac{2 a b \csc (c+d x)}{d}+\frac{b^2 \sin ^4(c+d x)}{4 d}",1,"(-2*a*b*Csc[c + d*x])/d - (a^2*Csc[c + d*x]^2)/(2*d) - ((2*a^2 - b^2)*Log[Sin[c + d*x]])/d - (4*a*b*Sin[c + d*x])/d + ((a^2 - 2*b^2)*Sin[c + d*x]^2)/(2*d) + (2*a*b*Sin[c + d*x]^3)/(3*d) + (b^2*Sin[c + d*x]^4)/(4*d)","A",4,3,29,0.1034,1,"{2837, 12, 948}"
1220,1,120,0,0.1408225,"\int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{d}+\frac{\left(2 a^2-b^2\right) \csc (c+d x)}{d}-\frac{a^2 \csc ^3(c+d x)}{3 d}+\frac{a b \sin ^2(c+d x)}{d}-\frac{a b \csc ^2(c+d x)}{d}-\frac{4 a b \log (\sin (c+d x))}{d}+\frac{b^2 \sin ^3(c+d x)}{3 d}","\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{d}+\frac{\left(2 a^2-b^2\right) \csc (c+d x)}{d}-\frac{a^2 \csc ^3(c+d x)}{3 d}+\frac{a b \sin ^2(c+d x)}{d}-\frac{a b \csc ^2(c+d x)}{d}-\frac{4 a b \log (\sin (c+d x))}{d}+\frac{b^2 \sin ^3(c+d x)}{3 d}",1,"((2*a^2 - b^2)*Csc[c + d*x])/d - (a*b*Csc[c + d*x]^2)/d - (a^2*Csc[c + d*x]^3)/(3*d) - (4*a*b*Log[Sin[c + d*x]])/d + ((a^2 - 2*b^2)*Sin[c + d*x])/d + (a*b*Sin[c + d*x]^2)/d + (b^2*Sin[c + d*x]^3)/(3*d)","A",4,3,27,0.1111,1,"{2837, 12, 948}"
1221,1,126,0,0.0919727,"\int \cot ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 d}+\frac{\left(a^2-2 b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \csc ^4(c+d x)}{4 d}+\frac{2 a b \sin (c+d x)}{d}-\frac{2 a b \csc ^3(c+d x)}{3 d}+\frac{4 a b \csc (c+d x)}{d}+\frac{b^2 \sin ^2(c+d x)}{2 d}","\frac{\left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 d}+\frac{\left(a^2-2 b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \csc ^4(c+d x)}{4 d}+\frac{2 a b \sin (c+d x)}{d}-\frac{2 a b \csc ^3(c+d x)}{3 d}+\frac{4 a b \csc (c+d x)}{d}+\frac{b^2 \sin ^2(c+d x)}{2 d}",1,"(4*a*b*Csc[c + d*x])/d + ((2*a^2 - b^2)*Csc[c + d*x]^2)/(2*d) - (2*a*b*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*d) + ((a^2 - 2*b^2)*Log[Sin[c + d*x]])/d + (2*a*b*Sin[c + d*x])/d + (b^2*Sin[c + d*x]^2)/(2*d)","A",3,2,21,0.09524,1,"{2721, 948}"
1222,1,124,0,0.1403219,"\int \cot ^5(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \csc ^3(c+d x)}{3 d}-\frac{\left(a^2-2 b^2\right) \csc (c+d x)}{d}-\frac{a^2 \csc ^5(c+d x)}{5 d}-\frac{a b \csc ^4(c+d x)}{2 d}+\frac{2 a b \csc ^2(c+d x)}{d}+\frac{2 a b \log (\sin (c+d x))}{d}+\frac{b^2 \sin (c+d x)}{d}","\frac{\left(2 a^2-b^2\right) \csc ^3(c+d x)}{3 d}-\frac{\left(a^2-2 b^2\right) \csc (c+d x)}{d}-\frac{a^2 \csc ^5(c+d x)}{5 d}-\frac{a b \csc ^4(c+d x)}{2 d}+\frac{2 a b \csc ^2(c+d x)}{d}+\frac{2 a b \log (\sin (c+d x))}{d}+\frac{b^2 \sin (c+d x)}{d}",1,"-(((a^2 - 2*b^2)*Csc[c + d*x])/d) + (2*a*b*Csc[c + d*x]^2)/d + ((2*a^2 - b^2)*Csc[c + d*x]^3)/(3*d) - (a*b*Csc[c + d*x]^4)/(2*d) - (a^2*Csc[c + d*x]^5)/(5*d) + (2*a*b*Log[Sin[c + d*x]])/d + (b^2*Sin[c + d*x])/d","A",4,3,27,0.1111,1,"{2837, 12, 948}"
1223,1,130,0,0.1581913,"\int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \csc ^4(c+d x)}{4 d}-\frac{\left(a^2-2 b^2\right) \csc ^2(c+d x)}{2 d}-\frac{a^2 \csc ^6(c+d x)}{6 d}-\frac{2 a b \csc ^5(c+d x)}{5 d}+\frac{4 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{b^2 \log (\sin (c+d x))}{d}","\frac{\left(2 a^2-b^2\right) \csc ^4(c+d x)}{4 d}-\frac{\left(a^2-2 b^2\right) \csc ^2(c+d x)}{2 d}-\frac{a^2 \csc ^6(c+d x)}{6 d}-\frac{2 a b \csc ^5(c+d x)}{5 d}+\frac{4 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{b^2 \log (\sin (c+d x))}{d}",1,"(-2*a*b*Csc[c + d*x])/d - ((a^2 - 2*b^2)*Csc[c + d*x]^2)/(2*d) + (4*a*b*Csc[c + d*x]^3)/(3*d) + ((2*a^2 - b^2)*Csc[c + d*x]^4)/(4*d) - (2*a*b*Csc[c + d*x]^5)/(5*d) - (a^2*Csc[c + d*x]^6)/(6*d) + (b^2*Log[Sin[c + d*x]])/d","A",4,3,29,0.1034,1,"{2837, 12, 948}"
1224,1,129,0,0.1593145,"\int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \csc ^5(c+d x)}{5 d}-\frac{\left(a^2-2 b^2\right) \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^7(c+d x)}{7 d}-\frac{a b \csc ^6(c+d x)}{3 d}+\frac{a b \csc ^4(c+d x)}{d}-\frac{a b \csc ^2(c+d x)}{d}-\frac{b^2 \csc (c+d x)}{d}","\frac{\left(2 a^2-b^2\right) \csc ^5(c+d x)}{5 d}-\frac{\left(a^2-2 b^2\right) \csc ^3(c+d x)}{3 d}-\frac{a^2 \csc ^7(c+d x)}{7 d}-\frac{a b \csc ^6(c+d x)}{3 d}+\frac{a b \csc ^4(c+d x)}{d}-\frac{a b \csc ^2(c+d x)}{d}-\frac{b^2 \csc (c+d x)}{d}",1,"-((b^2*Csc[c + d*x])/d) - (a*b*Csc[c + d*x]^2)/d - ((a^2 - 2*b^2)*Csc[c + d*x]^3)/(3*d) + (a*b*Csc[c + d*x]^4)/d + ((2*a^2 - b^2)*Csc[c + d*x]^5)/(5*d) - (a*b*Csc[c + d*x]^6)/(3*d) - (a^2*Csc[c + d*x]^7)/(7*d)","A",4,3,29,0.1034,1,"{2837, 12, 948}"
1225,1,138,0,0.1610116,"\int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \csc ^6(c+d x)}{6 d}-\frac{\left(a^2-2 b^2\right) \csc ^4(c+d x)}{4 d}-\frac{a^2 \csc ^8(c+d x)}{8 d}-\frac{2 a b \csc ^7(c+d x)}{7 d}+\frac{4 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{b^2 \csc ^2(c+d x)}{2 d}","\frac{\left(2 a^2-b^2\right) \csc ^6(c+d x)}{6 d}-\frac{\left(a^2-2 b^2\right) \csc ^4(c+d x)}{4 d}-\frac{a^2 \csc ^8(c+d x)}{8 d}-\frac{2 a b \csc ^7(c+d x)}{7 d}+\frac{4 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{b^2 \csc ^2(c+d x)}{2 d}",1,"-(b^2*Csc[c + d*x]^2)/(2*d) - (2*a*b*Csc[c + d*x]^3)/(3*d) - ((a^2 - 2*b^2)*Csc[c + d*x]^4)/(4*d) + (4*a*b*Csc[c + d*x]^5)/(5*d) + ((2*a^2 - b^2)*Csc[c + d*x]^6)/(6*d) - (2*a*b*Csc[c + d*x]^7)/(7*d) - (a^2*Csc[c + d*x]^8)/(8*d)","A",4,3,29,0.1034,1,"{2837, 12, 948}"
1226,1,235,0,0.2805098,"\int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","\frac{\left(3 a^2-2 b^2\right) \sin ^4(c+d x)}{4 b^4 d}-\frac{4 a \left(a^2-b^2\right) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left(-6 a^2 b^2+5 a^4+b^4\right) \sin ^2(c+d x)}{2 b^6 d}-\frac{2 a \left(-4 a^2 b^2+3 a^4+b^4\right) \sin (c+d x)}{b^7 d}+\frac{a^3 \left(a^2-b^2\right)^2}{b^8 d (a+b \sin (c+d x))}+\frac{a^2 \left(-10 a^2 b^2+7 a^4+3 b^4\right) \log (a+b \sin (c+d x))}{b^8 d}-\frac{2 a \sin ^5(c+d x)}{5 b^3 d}+\frac{\sin ^6(c+d x)}{6 b^2 d}","\frac{\left(3 a^2-2 b^2\right) \sin ^4(c+d x)}{4 b^4 d}-\frac{4 a \left(a^2-b^2\right) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left(-6 a^2 b^2+5 a^4+b^4\right) \sin ^2(c+d x)}{2 b^6 d}-\frac{2 a \left(-4 a^2 b^2+3 a^4+b^4\right) \sin (c+d x)}{b^7 d}+\frac{a^3 \left(a^2-b^2\right)^2}{b^8 d (a+b \sin (c+d x))}+\frac{a^2 \left(-10 a^2 b^2+7 a^4+3 b^4\right) \log (a+b \sin (c+d x))}{b^8 d}-\frac{2 a \sin ^5(c+d x)}{5 b^3 d}+\frac{\sin ^6(c+d x)}{6 b^2 d}",1,"(a^2*(7*a^4 - 10*a^2*b^2 + 3*b^4)*Log[a + b*Sin[c + d*x]])/(b^8*d) - (2*a*(3*a^4 - 4*a^2*b^2 + b^4)*Sin[c + d*x])/(b^7*d) + ((5*a^4 - 6*a^2*b^2 + b^4)*Sin[c + d*x]^2)/(2*b^6*d) - (4*a*(a^2 - b^2)*Sin[c + d*x]^3)/(3*b^5*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x]^4)/(4*b^4*d) - (2*a*Sin[c + d*x]^5)/(5*b^3*d) + Sin[c + d*x]^6/(6*b^2*d) + (a^3*(a^2 - b^2)^2)/(b^8*d*(a + b*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2837, 12, 948}"
1227,1,193,0,0.239489,"\int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","-\frac{\left(2-\frac{3 a^2}{b^2}\right) \sin ^3(c+d x)}{3 b^2 d}-\frac{2 a \left(a^2-b^2\right) \sin ^2(c+d x)}{b^5 d}+\frac{\left(-6 a^2 b^2+5 a^4+b^4\right) \sin (c+d x)}{b^6 d}-\frac{a^2 \left(a^2-b^2\right)^2}{b^7 d (a+b \sin (c+d x))}-\frac{2 a \left(-4 a^2 b^2+3 a^4+b^4\right) \log (a+b \sin (c+d x))}{b^7 d}-\frac{a \sin ^4(c+d x)}{2 b^3 d}+\frac{\sin ^5(c+d x)}{5 b^2 d}","-\frac{\left(2-\frac{3 a^2}{b^2}\right) \sin ^3(c+d x)}{3 b^2 d}-\frac{2 a \left(a^2-b^2\right) \sin ^2(c+d x)}{b^5 d}+\frac{\left(-6 a^2 b^2+5 a^4+b^4\right) \sin (c+d x)}{b^6 d}-\frac{a^2 \left(a^2-b^2\right)^2}{b^7 d (a+b \sin (c+d x))}-\frac{2 a \left(-4 a^2 b^2+3 a^4+b^4\right) \log (a+b \sin (c+d x))}{b^7 d}-\frac{a \sin ^4(c+d x)}{2 b^3 d}+\frac{\sin ^5(c+d x)}{5 b^2 d}",1,"(-2*a*(3*a^4 - 4*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/(b^7*d) + ((5*a^4 - 6*a^2*b^2 + b^4)*Sin[c + d*x])/(b^6*d) - (2*a*(a^2 - b^2)*Sin[c + d*x]^2)/(b^5*d) - ((2 - (3*a^2)/b^2)*Sin[c + d*x]^3)/(3*b^2*d) - (a*Sin[c + d*x]^4)/(2*b^3*d) + Sin[c + d*x]^5/(5*b^2*d) - (a^2*(a^2 - b^2)^2)/(b^7*d*(a + b*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2837, 12, 948}"
1228,1,157,0,0.1546812,"\int \frac{\cos ^5(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{\left(3 a^2-2 b^2\right) \sin ^2(c+d x)}{2 b^4 d}-\frac{4 a \left(a^2-b^2\right) \sin (c+d x)}{b^5 d}+\frac{a \left(a^2-b^2\right)^2}{b^6 d (a+b \sin (c+d x))}+\frac{\left(-6 a^2 b^2+5 a^4+b^4\right) \log (a+b \sin (c+d x))}{b^6 d}-\frac{2 a \sin ^3(c+d x)}{3 b^3 d}+\frac{\sin ^4(c+d x)}{4 b^2 d}","\frac{\left(3 a^2-2 b^2\right) \sin ^2(c+d x)}{2 b^4 d}-\frac{4 a \left(a^2-b^2\right) \sin (c+d x)}{b^5 d}+\frac{a \left(a^2-b^2\right)^2}{b^6 d (a+b \sin (c+d x))}+\frac{\left(-6 a^2 b^2+5 a^4+b^4\right) \log (a+b \sin (c+d x))}{b^6 d}-\frac{2 a \sin ^3(c+d x)}{3 b^3 d}+\frac{\sin ^4(c+d x)}{4 b^2 d}",1,"((5*a^4 - 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/(b^6*d) - (4*a*(a^2 - b^2)*Sin[c + d*x])/(b^5*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x]^2)/(2*b^4*d) - (2*a*Sin[c + d*x]^3)/(3*b^3*d) + Sin[c + d*x]^4/(4*b^2*d) + (a*(a^2 - b^2)^2)/(b^6*d*(a + b*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2837, 12, 772}"
1229,1,120,0,0.1589056,"\int \frac{\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-b^2\right)^2}{a b^4 d (a+b \sin (c+d x))}+\frac{\left(3 a^2+b^2\right) \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{a^2 b^4 d}+\frac{\log (\sin (c+d x))}{a^2 d}-\frac{2 a \sin (c+d x)}{b^3 d}+\frac{\sin ^2(c+d x)}{2 b^2 d}","\frac{\left(a^2-b^2\right)^2}{a b^4 d (a+b \sin (c+d x))}+\frac{\left(3 a^2+b^2\right) \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{a^2 b^4 d}+\frac{\log (\sin (c+d x))}{a^2 d}-\frac{2 a \sin (c+d x)}{b^3 d}+\frac{\sin ^2(c+d x)}{2 b^2 d}",1,"Log[Sin[c + d*x]]/(a^2*d) + ((a^2 - b^2)*(3*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/(a^2*b^4*d) - (2*a*Sin[c + d*x])/(b^3*d) + Sin[c + d*x]^2/(2*b^2*d) + (a^2 - b^2)^2/(a*b^4*d*(a + b*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1230,1,109,0,0.1703325,"\int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2-b^2\right)^2}{a^2 b^3 d (a+b \sin (c+d x))}-\frac{2 \left(a^4-b^4\right) \log (a+b \sin (c+d x))}{a^3 b^3 d}-\frac{2 b \log (\sin (c+d x))}{a^3 d}-\frac{\csc (c+d x)}{a^2 d}+\frac{\sin (c+d x)}{b^2 d}","-\frac{\left(a^2-b^2\right)^2}{a^2 b^3 d (a+b \sin (c+d x))}-\frac{2 \left(a^4-b^4\right) \log (a+b \sin (c+d x))}{a^3 b^3 d}-\frac{2 b \log (\sin (c+d x))}{a^3 d}-\frac{\csc (c+d x)}{a^2 d}+\frac{\sin (c+d x)}{b^2 d}",1,"-(Csc[c + d*x]/(a^2*d)) - (2*b*Log[Sin[c + d*x]])/(a^3*d) - (2*(a^4 - b^4)*Log[a + b*Sin[c + d*x]])/(a^3*b^3*d) + Sin[c + d*x]/(b^2*d) - (a^2 - b^2)^2/(a^2*b^3*d*(a + b*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2837, 12, 894}"
1231,1,131,0,0.1998028,"\int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-b^2\right)^2}{a^3 b^2 d (a+b \sin (c+d x))}-\frac{\left(2 a^2-3 b^2\right) \log (\sin (c+d x))}{a^4 d}+\frac{\left(2 a^2 b^2+a^4-3 b^4\right) \log (a+b \sin (c+d x))}{a^4 b^2 d}+\frac{2 b \csc (c+d x)}{a^3 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}","\frac{\left(a^2-b^2\right)^2}{a^3 b^2 d (a+b \sin (c+d x))}-\frac{\left(2 a^2-3 b^2\right) \log (\sin (c+d x))}{a^4 d}+\frac{\left(2 a^2 b^2+a^4-3 b^4\right) \log (a+b \sin (c+d x))}{a^4 b^2 d}+\frac{2 b \csc (c+d x)}{a^3 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}",1,"(2*b*Csc[c + d*x])/(a^3*d) - Csc[c + d*x]^2/(2*a^2*d) - ((2*a^2 - 3*b^2)*Log[Sin[c + d*x]])/(a^4*d) + ((a^4 + 2*a^2*b^2 - 3*b^4)*Log[a + b*Sin[c + d*x]])/(a^4*b^2*d) + (a^2 - b^2)^2/(a^3*b^2*d*(a + b*Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2837, 12, 894}"
1232,1,147,0,0.2057354,"\int \frac{\cos (c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2-b^2\right)^2}{a^4 b d (a+b \sin (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \csc (c+d x)}{a^4 d}+\frac{4 b \left(a^2-b^2\right) \log (\sin (c+d x))}{a^5 d}-\frac{4 b \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{a^5 d}+\frac{b \csc ^2(c+d x)}{a^3 d}-\frac{\csc ^3(c+d x)}{3 a^2 d}","-\frac{\left(a^2-b^2\right)^2}{a^4 b d (a+b \sin (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \csc (c+d x)}{a^4 d}+\frac{4 b \left(a^2-b^2\right) \log (\sin (c+d x))}{a^5 d}-\frac{4 b \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{a^5 d}+\frac{b \csc ^2(c+d x)}{a^3 d}-\frac{\csc ^3(c+d x)}{3 a^2 d}",1,"((2*a^2 - 3*b^2)*Csc[c + d*x])/(a^4*d) + (b*Csc[c + d*x]^2)/(a^3*d) - Csc[c + d*x]^3/(3*a^2*d) + (4*b*(a^2 - b^2)*Log[Sin[c + d*x]])/(a^5*d) - (4*b*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(a^5*d) - (a^2 - b^2)^2/(a^4*b*d*(a + b*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1233,1,188,0,0.1705191,"\int \frac{\cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^5/(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-b^2\right)^2}{a^5 d (a+b \sin (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \csc ^2(c+d x)}{2 a^4 d}-\frac{4 b \left(a^2-b^2\right) \csc (c+d x)}{a^5 d}+\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \log (\sin (c+d x))}{a^6 d}-\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \log (a+b \sin (c+d x))}{a^6 d}+\frac{2 b \csc ^3(c+d x)}{3 a^3 d}-\frac{\csc ^4(c+d x)}{4 a^2 d}","\frac{\left(a^2-b^2\right)^2}{a^5 d (a+b \sin (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \csc ^2(c+d x)}{2 a^4 d}-\frac{4 b \left(a^2-b^2\right) \csc (c+d x)}{a^5 d}+\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \log (\sin (c+d x))}{a^6 d}-\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \log (a+b \sin (c+d x))}{a^6 d}+\frac{2 b \csc ^3(c+d x)}{3 a^3 d}-\frac{\csc ^4(c+d x)}{4 a^2 d}",1,"(-4*b*(a^2 - b^2)*Csc[c + d*x])/(a^5*d) + ((2*a^2 - 3*b^2)*Csc[c + d*x]^2)/(2*a^4*d) + (2*b*Csc[c + d*x]^3)/(3*a^3*d) - Csc[c + d*x]^4/(4*a^2*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*Log[Sin[c + d*x]])/(a^6*d) - ((a^4 - 6*a^2*b^2 + 5*b^4)*Log[a + b*Sin[c + d*x]])/(a^6*d) + (a^2 - b^2)^2/(a^5*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2721, 894}"
1234,1,226,0,0.2614492,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^5*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]","-\frac{b \left(a^2-b^2\right)^2}{a^6 d (a+b \sin (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \csc ^3(c+d x)}{3 a^4 d}-\frac{2 b \left(a^2-b^2\right) \csc ^2(c+d x)}{a^5 d}-\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \csc (c+d x)}{a^6 d}-\frac{2 b \left(-4 a^2 b^2+a^4+3 b^4\right) \log (\sin (c+d x))}{a^7 d}+\frac{2 b \left(-4 a^2 b^2+a^4+3 b^4\right) \log (a+b \sin (c+d x))}{a^7 d}+\frac{b \csc ^4(c+d x)}{2 a^3 d}-\frac{\csc ^5(c+d x)}{5 a^2 d}","-\frac{b \left(a^2-b^2\right)^2}{a^6 d (a+b \sin (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \csc ^3(c+d x)}{3 a^4 d}-\frac{2 b \left(a^2-b^2\right) \csc ^2(c+d x)}{a^5 d}-\frac{\left(-6 a^2 b^2+a^4+5 b^4\right) \csc (c+d x)}{a^6 d}-\frac{2 b \left(-4 a^2 b^2+a^4+3 b^4\right) \log (\sin (c+d x))}{a^7 d}+\frac{2 b \left(-4 a^2 b^2+a^4+3 b^4\right) \log (a+b \sin (c+d x))}{a^7 d}+\frac{b \csc ^4(c+d x)}{2 a^3 d}-\frac{\csc ^5(c+d x)}{5 a^2 d}",1,"-(((a^4 - 6*a^2*b^2 + 5*b^4)*Csc[c + d*x])/(a^6*d)) - (2*b*(a^2 - b^2)*Csc[c + d*x]^2)/(a^5*d) + ((2*a^2 - 3*b^2)*Csc[c + d*x]^3)/(3*a^4*d) + (b*Csc[c + d*x]^4)/(2*a^3*d) - Csc[c + d*x]^5/(5*a^2*d) - (2*b*(a^4 - 4*a^2*b^2 + 3*b^4)*Log[Sin[c + d*x]])/(a^7*d) + (2*b*(a^4 - 4*a^2*b^2 + 3*b^4)*Log[a + b*Sin[c + d*x]])/(a^7*d) - (b*(a^2 - b^2)^2)/(a^6*d*(a + b*Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1235,1,170,0,0.2131979,"\int \cos ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(2 a^2-b^2\right) \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{\left(a^2-2 b^2\right) \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{2 a b \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{4 a b \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{2 a b \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{b^2 \sin ^{n+7}(c+d x)}{d (n+7)}","-\frac{\left(2 a^2-b^2\right) \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{\left(a^2-2 b^2\right) \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{2 a b \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{4 a b \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{2 a b \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{b^2 \sin ^{n+7}(c+d x)}{d (n+7)}",1,"(a^2*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (2*a*b*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - ((2*a^2 - b^2)*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (4*a*b*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + ((a^2 - 2*b^2)*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (2*a*b*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (b^2*Sin[c + d*x]^(7 + n))/(d*(7 + n))","A",3,2,29,0.06897,1,"{2837, 948}"
1236,1,123,0,0.1371815,"\int \cos ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x]),x]","\frac{a \sin ^{n+1}(c+d x)}{d (n+1)}-\frac{2 a \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{b \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{2 b \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{b \sin ^{n+6}(c+d x)}{d (n+6)}","\frac{a \sin ^{n+1}(c+d x)}{d (n+1)}-\frac{2 a \sin ^{n+3}(c+d x)}{d (n+3)}+\frac{a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{b \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{2 b \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{b \sin ^{n+6}(c+d x)}{d (n+6)}",1,"(a*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (b*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (2*a*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (2*b*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (a*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (b*Sin[c + d*x]^(6 + n))/(d*(6 + n))","A",3,2,27,0.07407,1,"{2837, 766}"
1237,1,167,0,0.3383708,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^n)/(a + b*Sin[c + d*x]),x]","\frac{\left(a^2-b^2\right)^2 \sin ^{n+1}(c+d x) \, _2F_1\left(1,n+1;n+2;-\frac{b \sin (c+d x)}{a}\right)}{a b^4 d (n+1)}-\frac{a \left(a^2-2 b^2\right) \sin ^{n+1}(c+d x)}{b^4 d (n+1)}+\frac{\left(a^2-2 b^2\right) \sin ^{n+2}(c+d x)}{b^3 d (n+2)}-\frac{a \sin ^{n+3}(c+d x)}{b^2 d (n+3)}+\frac{\sin ^{n+4}(c+d x)}{b d (n+4)}","\frac{\left(a^2-b^2\right)^2 \sin ^{n+1}(c+d x) \, _2F_1\left(1,n+1;n+2;-\frac{b \sin (c+d x)}{a}\right)}{a b^4 d (n+1)}-\frac{a \left(a^2-2 b^2\right) \sin ^{n+1}(c+d x)}{b^4 d (n+1)}+\frac{\left(a^2-2 b^2\right) \sin ^{n+2}(c+d x)}{b^3 d (n+2)}-\frac{a \sin ^{n+3}(c+d x)}{b^2 d (n+3)}+\frac{\sin ^{n+4}(c+d x)}{b d (n+4)}",1,"-((a*(a^2 - 2*b^2)*Sin[c + d*x]^(1 + n))/(b^4*d*(1 + n))) + ((a^2 - b^2)^2*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Sin[c + d*x])/a)]*Sin[c + d*x]^(1 + n))/(a*b^4*d*(1 + n)) + ((a^2 - 2*b^2)*Sin[c + d*x]^(2 + n))/(b^3*d*(2 + n)) - (a*Sin[c + d*x]^(3 + n))/(b^2*d*(3 + n)) + Sin[c + d*x]^(4 + n)/(b*d*(4 + n))","A",5,4,29,0.1379,1,"{2837, 952, 1620, 64}"
1238,1,191,0,0.3649074,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^n)/(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-b^2\right) \left(b^2 n-a^2 (n+4)\right) \sin ^{n+1}(c+d x) \, _2F_1\left(1,n+1;n+2;-\frac{b \sin (c+d x)}{a}\right)}{a^2 b^4 d (n+1)}+\frac{\left(3 a^2-2 b^2\right) \sin ^{n+1}(c+d x)}{b^4 d (n+1)}+\frac{\left(a^2-b^2\right)^2 \sin ^{n+1}(c+d x)}{a b^4 d (a+b \sin (c+d x))}-\frac{2 a \sin ^{n+2}(c+d x)}{b^3 d (n+2)}+\frac{\sin ^{n+3}(c+d x)}{b^2 d (n+3)}","\frac{\left(a^2-b^2\right) \left(b^2 n-a^2 (n+4)\right) \sin ^{n+1}(c+d x) \, _2F_1\left(1,n+1;n+2;-\frac{b \sin (c+d x)}{a}\right)}{a^2 b^4 d (n+1)}+\frac{\left(3 a^2-2 b^2\right) \sin ^{n+1}(c+d x)}{b^4 d (n+1)}+\frac{\left(a^2-b^2\right)^2 \sin ^{n+1}(c+d x)}{a b^4 d (a+b \sin (c+d x))}-\frac{2 a \sin ^{n+2}(c+d x)}{b^3 d (n+2)}+\frac{\sin ^{n+3}(c+d x)}{b^2 d (n+3)}",1,"((3*a^2 - 2*b^2)*Sin[c + d*x]^(1 + n))/(b^4*d*(1 + n)) + ((a^2 - b^2)*(b^2*n - a^2*(4 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Sin[c + d*x])/a)]*Sin[c + d*x]^(1 + n))/(a^2*b^4*d*(1 + n)) - (2*a*Sin[c + d*x]^(2 + n))/(b^3*d*(2 + n)) + Sin[c + d*x]^(3 + n)/(b^2*d*(3 + n)) + ((a^2 - b^2)^2*Sin[c + d*x]^(1 + n))/(a*b^4*d*(a + b*Sin[c + d*x]))","A",5,4,29,0.1379,1,"{2837, 950, 1620, 64}"
1239,1,238,0,0.3719074,"\int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2+3 b^2\right) \cos ^{11}(c+d x)}{11 d}+\frac{\left(2 a^2+3 b^2\right) \cos ^9(c+d x)}{9 d}-\frac{\left(a^2+b^2\right) \cos ^7(c+d x)}{7 d}-\frac{a b \sin ^5(c+d x) \cos ^7(c+d x)}{6 d}-\frac{a b \sin ^3(c+d x) \cos ^7(c+d x)}{12 d}-\frac{a b \sin (c+d x) \cos ^7(c+d x)}{32 d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{192 d}+\frac{5 a b \sin (c+d x) \cos ^3(c+d x)}{768 d}+\frac{5 a b \sin (c+d x) \cos (c+d x)}{512 d}+\frac{5 a b x}{512}+\frac{b^2 \cos ^{13}(c+d x)}{13 d}","-\frac{\left(a^2+3 b^2\right) \cos ^{11}(c+d x)}{11 d}+\frac{\left(2 a^2+3 b^2\right) \cos ^9(c+d x)}{9 d}-\frac{\left(a^2+b^2\right) \cos ^7(c+d x)}{7 d}-\frac{a b \sin ^5(c+d x) \cos ^7(c+d x)}{6 d}-\frac{a b \sin ^3(c+d x) \cos ^7(c+d x)}{12 d}-\frac{a b \sin (c+d x) \cos ^7(c+d x)}{32 d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{192 d}+\frac{5 a b \sin (c+d x) \cos ^3(c+d x)}{768 d}+\frac{5 a b \sin (c+d x) \cos (c+d x)}{512 d}+\frac{5 a b x}{512}+\frac{b^2 \cos ^{13}(c+d x)}{13 d}",1,"(5*a*b*x)/512 - ((a^2 + b^2)*Cos[c + d*x]^7)/(7*d) + ((2*a^2 + 3*b^2)*Cos[c + d*x]^9)/(9*d) - ((a^2 + 3*b^2)*Cos[c + d*x]^11)/(11*d) + (b^2*Cos[c + d*x]^13)/(13*d) + (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(512*d) + (5*a*b*Cos[c + d*x]^3*Sin[c + d*x])/(768*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(192*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*x])/(32*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*x]^3)/(12*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*x]^5)/(6*d)","A",12,7,29,0.2414,1,"{2911, 2568, 2635, 8, 3201, 446, 77}"
1240,1,250,0,0.3547249,"\int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{\left(12 a^2+25 b^2\right) \sin (c+d x) \cos ^9(c+d x)}{120 d}-\frac{\left(44 a^2+45 b^2\right) \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{\left(12 a^2+5 b^2\right) \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{\left(12 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{\left(12 a^2+5 b^2\right) \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{x \left(12 a^2+5 b^2\right)}{1024}-\frac{2 a b \cos ^{11}(c+d x)}{11 d}+\frac{4 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^7(c+d x)}{7 d}-\frac{b^2 \sin (c+d x) \cos ^{11}(c+d x)}{12 d}","\frac{\left(12 a^2+25 b^2\right) \sin (c+d x) \cos ^9(c+d x)}{120 d}-\frac{\left(44 a^2+45 b^2\right) \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{\left(12 a^2+5 b^2\right) \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{\left(12 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{\left(12 a^2+5 b^2\right) \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{x \left(12 a^2+5 b^2\right)}{1024}-\frac{2 a b \cos ^{11}(c+d x)}{11 d}+\frac{4 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^7(c+d x)}{7 d}-\frac{b^2 \sin (c+d x) \cos ^{11}(c+d x)}{12 d}",1,"((12*a^2 + 5*b^2)*x)/1024 - (2*a*b*Cos[c + d*x]^7)/(7*d) + (4*a*b*Cos[c + d*x]^9)/(9*d) - (2*a*b*Cos[c + d*x]^11)/(11*d) + ((12*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + ((12*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + ((12*a^2 + 5*b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(1920*d) - ((44*a^2 + 45*b^2)*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) + ((12*a^2 + 25*b^2)*Cos[c + d*x]^9*Sin[c + d*x])/(120*d) - (b^2*Cos[c + d*x]^11*Sin[c + d*x])/(12*d)","A",12,9,29,0.3103,1,"{2911, 2565, 270, 3200, 455, 1157, 385, 199, 203}"
1241,1,187,0,0.3145483,"\int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2+2 b^2\right) \cos ^9(c+d x)}{9 d}-\frac{\left(a^2+b^2\right) \cos ^7(c+d x)}{7 d}-\frac{a b \sin ^3(c+d x) \cos ^7(c+d x)}{5 d}-\frac{3 a b \sin (c+d x) \cos ^7(c+d x)}{40 d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{80 d}+\frac{a b \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a b \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a b x}{128}-\frac{b^2 \cos ^{11}(c+d x)}{11 d}","\frac{\left(a^2+2 b^2\right) \cos ^9(c+d x)}{9 d}-\frac{\left(a^2+b^2\right) \cos ^7(c+d x)}{7 d}-\frac{a b \sin ^3(c+d x) \cos ^7(c+d x)}{5 d}-\frac{3 a b \sin (c+d x) \cos ^7(c+d x)}{40 d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{80 d}+\frac{a b \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 a b \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 a b x}{128}-\frac{b^2 \cos ^{11}(c+d x)}{11 d}",1,"(3*a*b*x)/128 - ((a^2 + b^2)*Cos[c + d*x]^7)/(7*d) + ((a^2 + 2*b^2)*Cos[c + d*x]^9)/(9*d) - (b^2*Cos[c + d*x]^11)/(11*d) + (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a*b*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(80*d) - (3*a*b*Cos[c + d*x]^7*Sin[c + d*x])/(40*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*x]^3)/(5*d)","A",11,7,29,0.2414,1,"{2911, 2568, 2635, 8, 3201, 446, 77}"
1242,1,201,0,0.26689,"\int \cos ^6(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(10 a^2+11 b^2\right) \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{\left(10 a^2+3 b^2\right) \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac{\left(10 a^2+3 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{\left(10 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{256 d}+\frac{1}{256} x \left(10 a^2+3 b^2\right)+\frac{2 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^7(c+d x)}{7 d}+\frac{b^2 \sin (c+d x) \cos ^9(c+d x)}{10 d}","-\frac{\left(10 a^2+11 b^2\right) \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{\left(10 a^2+3 b^2\right) \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac{\left(10 a^2+3 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{\left(10 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{256 d}+\frac{1}{256} x \left(10 a^2+3 b^2\right)+\frac{2 a b \cos ^9(c+d x)}{9 d}-\frac{2 a b \cos ^7(c+d x)}{7 d}+\frac{b^2 \sin (c+d x) \cos ^9(c+d x)}{10 d}",1,"((10*a^2 + 3*b^2)*x)/256 - (2*a*b*Cos[c + d*x]^7)/(7*d) + (2*a*b*Cos[c + d*x]^9)/(9*d) + ((10*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(256*d) + ((10*a^2 + 3*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + ((10*a^2 + 3*b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(480*d) - ((10*a^2 + 11*b^2)*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) + (b^2*Cos[c + d*x]^9*Sin[c + d*x])/(10*d)","A",11,8,29,0.2759,1,"{2911, 2565, 14, 3200, 455, 385, 199, 203}"
1243,1,152,0,0.2123794,"\int \cos ^6(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*Sin[c + d*x]*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2+8 b^2\right) \cos ^7(c+d x)}{252 d}-\frac{\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}-\frac{a \cos ^7(c+d x) (a+b \sin (c+d x))}{36 d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac{5 a b \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac{5 a b \sin (c+d x) \cos (c+d x)}{64 d}+\frac{5 a b x}{64}","-\frac{\left(a^2+8 b^2\right) \cos ^7(c+d x)}{252 d}-\frac{\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}-\frac{a \cos ^7(c+d x) (a+b \sin (c+d x))}{36 d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac{5 a b \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac{5 a b \sin (c+d x) \cos (c+d x)}{64 d}+\frac{5 a b x}{64}",1,"(5*a*b*x)/64 - ((a^2 + 8*b^2)*Cos[c + d*x]^7)/(252*d) + (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (5*a*b*Cos[c + d*x]^3*Sin[c + d*x])/(96*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(24*d) - (a*Cos[c + d*x]^7*(a + b*Sin[c + d*x]))/(36*d) - (Cos[c + d*x]^7*(a + b*Sin[c + d*x])^2)/(9*d)","A",7,4,27,0.1481,1,"{2862, 2669, 2635, 8}"
1244,1,157,0,0.2070317,"\int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*Cot[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{a^2 \cos ^5(c+d x)}{5 d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac{5 a b \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac{5 a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a b x}{8}-\frac{b^2 \cos ^7(c+d x)}{7 d}","\frac{a^2 \cos ^5(c+d x)}{5 d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac{5 a b \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac{5 a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a b x}{8}-\frac{b^2 \cos ^7(c+d x)}{7 d}",1,"(5*a*b*x)/8 - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cos[c + d*x]^5)/(5*d) - (b^2*Cos[c + d*x]^7)/(7*d) + (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a*b*Cos[c + d*x]^3*Sin[c + d*x])/(12*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(3*d)","A",9,5,27,0.1852,1,"{2911, 2635, 8, 14, 207}"
1245,1,178,0,0.4634828,"\int \cos ^4(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(6 a^2-5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}-\frac{\left(14 a^2-5 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}-\frac{5}{16} x \left(6 a^2-b^2\right)-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a b \cos ^5(c+d x)}{5 d}+\frac{2 a b \cos ^3(c+d x)}{3 d}+\frac{2 a b \cos (c+d x)}{d}-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}","-\frac{\left(6 a^2-5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}-\frac{\left(14 a^2-5 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}-\frac{5}{16} x \left(6 a^2-b^2\right)-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a b \cos ^5(c+d x)}{5 d}+\frac{2 a b \cos ^3(c+d x)}{3 d}+\frac{2 a b \cos (c+d x)}{d}-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}",1,"(-5*(6*a^2 - b^2)*x)/16 - (2*a*b*ArcTanh[Cos[c + d*x]])/d + (2*a*b*Cos[c + d*x])/d + (2*a*b*Cos[c + d*x]^3)/(3*d) + (2*a*b*Cos[c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x])/d - ((14*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) - ((6*a^2 - 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)","A",12,8,29,0.2759,1,"{2911, 2592, 302, 206, 434, 456, 453, 203}"
1246,1,180,0,0.3049694,"\int \cos ^3(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2-b^2\right) \cos ^3(c+d x)}{3 d}-\frac{\left(2 a^2-b^2\right) \cos (c+d x)}{d}+\frac{\left(5 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{15 a b \cot (c+d x)}{4 d}+\frac{a b \cos ^4(c+d x) \cot (c+d x)}{2 d}+\frac{5 a b \cos ^2(c+d x) \cot (c+d x)}{4 d}-\frac{15 a b x}{4}+\frac{b^2 \cos ^5(c+d x)}{5 d}","-\frac{\left(a^2-b^2\right) \cos ^3(c+d x)}{3 d}-\frac{\left(2 a^2-b^2\right) \cos (c+d x)}{d}+\frac{\left(5 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{15 a b \cot (c+d x)}{4 d}+\frac{a b \cos ^4(c+d x) \cot (c+d x)}{2 d}+\frac{5 a b \cos ^2(c+d x) \cot (c+d x)}{4 d}-\frac{15 a b x}{4}+\frac{b^2 \cos ^5(c+d x)}{5 d}",1,"(-15*a*b*x)/4 + ((5*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) - ((2*a^2 - b^2)*Cos[c + d*x])/d - ((a^2 - b^2)*Cos[c + d*x]^3)/(3*d) + (b^2*Cos[c + d*x]^5)/(5*d) - (15*a*b*Cot[c + d*x])/(4*d) + (5*a*b*Cos[c + d*x]^2*Cot[c + d*x])/(4*d) + (a*b*Cos[c + d*x]^4*Cot[c + d*x])/(2*d) - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d)","A",11,8,29,0.2759,1,"{2911, 2591, 288, 321, 203, 455, 1810, 206}"
1247,1,177,0,0.4396282,"\int \cos ^2(c+d x) \cot ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \cot (c+d x)}{d}+\frac{\left(4 a^2-7 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} x \left(4 a^2-3 b^2\right)-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{5 a b \cos ^3(c+d x)}{3 d}-\frac{5 a b \cos (c+d x)}{d}-\frac{a b \cos ^3(c+d x) \cot ^2(c+d x)}{d}+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}","\frac{\left(2 a^2-b^2\right) \cot (c+d x)}{d}+\frac{\left(4 a^2-7 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} x \left(4 a^2-3 b^2\right)-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{5 a b \cos ^3(c+d x)}{3 d}-\frac{5 a b \cos (c+d x)}{d}-\frac{a b \cos ^3(c+d x) \cot ^2(c+d x)}{d}+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(5*(4*a^2 - 3*b^2)*x)/8 + (5*a*b*ArcTanh[Cos[c + d*x]])/d - (5*a*b*Cos[c + d*x])/d - (5*a*b*Cos[c + d*x]^3)/(3*d) + ((2*a^2 - b^2)*Cot[c + d*x])/d - (a*b*Cos[c + d*x]^3*Cot[c + d*x]^2)/d - (a^2*Cot[c + d*x]^3)/(3*d) + ((4*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",12,9,29,0.3103,1,"{2911, 2592, 288, 302, 206, 456, 1259, 1261, 203}"
1248,1,174,0,0.2781321,"\int \cos (c+d x) \cot ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*Cot[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \cos (c+d x)}{d}-\frac{5 \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{\left(9 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{5 a b \cot ^3(c+d x)}{3 d}+\frac{5 a b \cot (c+d x)}{d}+\frac{a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+5 a b x-\frac{b^2 \cos ^3(c+d x)}{3 d}","\frac{\left(a^2-2 b^2\right) \cos (c+d x)}{d}-\frac{5 \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac{\left(9 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{5 a b \cot ^3(c+d x)}{3 d}+\frac{5 a b \cot (c+d x)}{d}+\frac{a b \cos ^2(c+d x) \cot ^3(c+d x)}{d}+5 a b x-\frac{b^2 \cos ^3(c+d x)}{3 d}",1,"5*a*b*x - (5*(3*a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) + ((a^2 - 2*b^2)*Cos[c + d*x])/d - (b^2*Cos[c + d*x]^3)/(3*d) + (5*a*b*Cot[c + d*x])/d - (5*a*b*Cot[c + d*x]^3)/(3*d) + (a*b*Cos[c + d*x]^2*Cot[c + d*x]^3)/d + ((9*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)","A",12,9,27,0.3333,1,"{2911, 2591, 288, 302, 203, 455, 1814, 1153, 206}"
1249,1,202,0,0.1881025,"\int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-a^2 x+\frac{15 a b \cos (c+d x)}{4 d}-\frac{a b \cos (c+d x) \cot ^4(c+d x)}{2 d}+\frac{5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}-\frac{15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{5 b^2 \cot ^3(c+d x)}{6 d}+\frac{5 b^2 \cot (c+d x)}{2 d}+\frac{b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac{5 b^2 x}{2}","-\frac{a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-a^2 x+\frac{15 a b \cos (c+d x)}{4 d}-\frac{a b \cos (c+d x) \cot ^4(c+d x)}{2 d}+\frac{5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}-\frac{15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{5 b^2 \cot ^3(c+d x)}{6 d}+\frac{5 b^2 \cot (c+d x)}{2 d}+\frac{b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac{5 b^2 x}{2}",1,"-(a^2*x) + (5*b^2*x)/2 - (15*a*b*ArcTanh[Cos[c + d*x]])/(4*d) + (15*a*b*Cos[c + d*x])/(4*d) - (a^2*Cot[c + d*x])/d + (5*b^2*Cot[c + d*x])/(2*d) + (5*a*b*Cos[c + d*x]*Cot[c + d*x]^2)/(4*d) + (a^2*Cot[c + d*x]^3)/(3*d) - (5*b^2*Cot[c + d*x]^3)/(6*d) + (b^2*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d) - (a*b*Cos[c + d*x]*Cot[c + d*x]^4)/(2*d) - (a^2*Cot[c + d*x]^5)/(5*d)","A",16,10,21,0.4762,1,"{2722, 2591, 288, 302, 203, 2592, 321, 206, 3473, 8}"
1250,1,175,0,0.26146,"\int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{5 \left(a^2-6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{\left(13 a^2-6 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{\left(11 a^2-18 b^2\right) \cot (c+d x) \csc (c+d x)}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{2 a b \cot ^5(c+d x)}{5 d}+\frac{2 a b \cot ^3(c+d x)}{3 d}-\frac{2 a b \cot (c+d x)}{d}-2 a b x+\frac{b^2 \cos (c+d x)}{d}","\frac{5 \left(a^2-6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{\left(13 a^2-6 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{\left(11 a^2-18 b^2\right) \cot (c+d x) \csc (c+d x)}{16 d}-\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{2 a b \cot ^5(c+d x)}{5 d}+\frac{2 a b \cot ^3(c+d x)}{3 d}-\frac{2 a b \cot (c+d x)}{d}-2 a b x+\frac{b^2 \cos (c+d x)}{d}",1,"-2*a*b*x + (5*(a^2 - 6*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) + (b^2*Cos[c + d*x])/d - (2*a*b*Cot[c + d*x])/d + (2*a*b*Cot[c + d*x]^3)/(3*d) - (2*a*b*Cot[c + d*x]^5)/(5*d) - ((11*a^2 - 18*b^2)*Cot[c + d*x]*Csc[c + d*x])/(16*d) + ((13*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)","A",11,9,27,0.3333,1,"{2911, 3473, 8, 4366, 455, 1814, 1157, 388, 206}"
1251,1,158,0,0.4143495,"\int \cot ^6(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","-\frac{a^2 \cot ^7(c+d x)}{7 d}+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a b \cot ^5(c+d x) \csc (c+d x)}{3 d}+\frac{5 a b \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac{5 a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b^2 \cot ^5(c+d x)}{5 d}+\frac{b^2 \cot ^3(c+d x)}{3 d}-\frac{b^2 \cot (c+d x)}{d}-b^2 x","-\frac{a^2 \cot ^7(c+d x)}{7 d}+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a b \cot ^5(c+d x) \csc (c+d x)}{3 d}+\frac{5 a b \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac{5 a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b^2 \cot ^5(c+d x)}{5 d}+\frac{b^2 \cot ^3(c+d x)}{3 d}-\frac{b^2 \cot (c+d x)}{d}-b^2 x",1,"-(b^2*x) + (5*a*b*ArcTanh[Cos[c + d*x]])/(8*d) - (b^2*Cot[c + d*x])/d + (b^2*Cot[c + d*x]^3)/(3*d) - (b^2*Cot[c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x]^7)/(7*d) - (5*a*b*Cot[c + d*x]*Csc[c + d*x])/(8*d) + (5*a*b*Cot[c + d*x]^3*Csc[c + d*x])/(12*d) - (a*b*Cot[c + d*x]^5*Csc[c + d*x])/(3*d)","A",9,5,29,0.1724,1,"{2911, 2611, 3770, 14, 203}"
1252,1,159,0,0.3192471,"\int \cot ^6(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{5 \left(a^2+8 b^2\right) \tanh ^{-1}(\cos (c+d x))}{128 d}+\frac{\left(17 a^2-8 b^2\right) \cot (c+d x) \csc ^5(c+d x)}{48 d}-\frac{\left(59 a^2-104 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{192 d}+\frac{\left(5 a^2-88 b^2\right) \cot (c+d x) \csc (c+d x)}{128 d}-\frac{a^2 \cot (c+d x) \csc ^7(c+d x)}{8 d}-\frac{2 a b \cot ^7(c+d x)}{7 d}","\frac{5 \left(a^2+8 b^2\right) \tanh ^{-1}(\cos (c+d x))}{128 d}+\frac{\left(17 a^2-8 b^2\right) \cot (c+d x) \csc ^5(c+d x)}{48 d}-\frac{\left(59 a^2-104 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{192 d}+\frac{\left(5 a^2-88 b^2\right) \cot (c+d x) \csc (c+d x)}{128 d}-\frac{a^2 \cot (c+d x) \csc ^7(c+d x)}{8 d}-\frac{2 a b \cot ^7(c+d x)}{7 d}",1,"(5*(a^2 + 8*b^2)*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a*b*Cot[c + d*x]^7)/(7*d) + ((5*a^2 - 88*b^2)*Cot[c + d*x]*Csc[c + d*x])/(128*d) - ((59*a^2 - 104*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(192*d) + ((17*a^2 - 8*b^2)*Cot[c + d*x]*Csc[c + d*x]^5)/(48*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^7)/(8*d)","A",9,9,29,0.3103,1,"{2911, 2607, 30, 4366, 455, 1814, 1157, 385, 206}"
1253,1,151,0,0.4054663,"\int \cot ^6(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2+b^2\right) \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^9(c+d x)}{9 d}+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac{a b \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}+\frac{5 a b \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a b \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac{5 a b \cot (c+d x) \csc (c+d x)}{64 d}","-\frac{\left(a^2+b^2\right) \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^9(c+d x)}{9 d}+\frac{5 a b \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac{a b \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}+\frac{5 a b \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a b \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac{5 a b \cot (c+d x) \csc (c+d x)}{64 d}",1,"(5*a*b*ArcTanh[Cos[c + d*x]])/(64*d) - ((a^2 + b^2)*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(9*d) + (5*a*b*Cot[c + d*x]*Csc[c + d*x])/(64*d) - (5*a*b*Cot[c + d*x]*Csc[c + d*x]^3)/(32*d) + (5*a*b*Cot[c + d*x]^3*Csc[c + d*x]^3)/(24*d) - (a*b*Cot[c + d*x]^5*Csc[c + d*x]^3)/(4*d)","A",9,5,29,0.1724,1,"{2911, 2611, 3768, 3770, 14}"
1254,1,210,0,0.3448883,"\int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{\left(3 a^2+10 b^2\right) \tanh ^{-1}(\cos (c+d x))}{256 d}+\frac{\left(21 a^2-10 b^2\right) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{\left(93 a^2-170 b^2\right) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac{\left(3 a^2-118 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{384 d}+\frac{\left(3 a^2+10 b^2\right) \cot (c+d x) \csc (c+d x)}{256 d}-\frac{a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}-\frac{2 a b \cot ^9(c+d x)}{9 d}-\frac{2 a b \cot ^7(c+d x)}{7 d}","\frac{\left(3 a^2+10 b^2\right) \tanh ^{-1}(\cos (c+d x))}{256 d}+\frac{\left(21 a^2-10 b^2\right) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac{\left(93 a^2-170 b^2\right) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac{\left(3 a^2-118 b^2\right) \cot (c+d x) \csc ^3(c+d x)}{384 d}+\frac{\left(3 a^2+10 b^2\right) \cot (c+d x) \csc (c+d x)}{256 d}-\frac{a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}-\frac{2 a b \cot ^9(c+d x)}{9 d}-\frac{2 a b \cot ^7(c+d x)}{7 d}",1,"((3*a^2 + 10*b^2)*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a*b*Cot[c + d*x]^7)/(7*d) - (2*a*b*Cot[c + d*x]^9)/(9*d) + ((3*a^2 + 10*b^2)*Cot[c + d*x]*Csc[c + d*x])/(256*d) + ((3*a^2 - 118*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(384*d) - ((93*a^2 - 170*b^2)*Cot[c + d*x]*Csc[c + d*x]^5)/(480*d) + ((21*a^2 - 10*b^2)*Cot[c + d*x]*Csc[c + d*x]^7)/(80*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^9)/(10*d)","A",11,10,29,0.3448,1,"{2911, 2607, 14, 4366, 455, 1814, 1157, 385, 199, 206}"
1255,1,198,0,0.456852,"\int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(2 a^2+b^2\right) \cot ^9(c+d x)}{9 d}-\frac{\left(a^2+b^2\right) \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 d}+\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac{a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a b \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{3 a b \cot (c+d x) \csc (c+d x)}{128 d}","-\frac{\left(2 a^2+b^2\right) \cot ^9(c+d x)}{9 d}-\frac{\left(a^2+b^2\right) \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 d}+\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac{a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a b \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{3 a b \cot (c+d x) \csc (c+d x)}{128 d}",1,"(3*a*b*ArcTanh[Cos[c + d*x]])/(128*d) - ((a^2 + b^2)*Cot[c + d*x]^7)/(7*d) - ((2*a^2 + b^2)*Cot[c + d*x]^9)/(9*d) - (a^2*Cot[c + d*x]^11)/(11*d) + (3*a*b*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (a*b*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) + (a*b*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) - (a*b*Cot[c + d*x]^5*Csc[c + d*x]^5)/(5*d)","A",10,5,29,0.1724,1,"{2911, 2611, 3768, 3770, 448}"
1256,1,525,0,1.9090977,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","\frac{\left(-1435 a^4 b^2+588 a^2 b^4+840 a^6-15 b^6\right) \cos (c+d x)}{105 b^8 d}-\frac{2 a^2 \left(8 a^2-3 b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^9 d}-\frac{\left(-30 a^2 b^2+20 a^4+9 b^4\right) \sin ^5(c+d x) \cos (c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}+\frac{\left(-340 a^2 b^2+224 a^4+105 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^4 d}-\frac{\left(-37 a^2 b^2+24 a^4+12 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{12 a b^5 d}+\frac{\left(-441 a^2 b^2+280 a^4+150 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}-\frac{a \left(-52 a^2 b^2+32 a^4+19 b^4\right) \sin (c+d x) \cos (c+d x)}{8 b^7 d}+\frac{a x \left(-120 a^4 b^2+60 a^2 b^4+64 a^6-5 b^6\right)}{8 b^9}-\frac{3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}","\frac{\left(-1435 a^4 b^2+588 a^2 b^4+840 a^6-15 b^6\right) \cos (c+d x)}{105 b^8 d}-\frac{2 a^2 \left(8 a^2-3 b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^9 d}-\frac{\left(-30 a^2 b^2+20 a^4+9 b^4\right) \sin ^5(c+d x) \cos (c+d x)}{15 a^2 b^3 d (a+b \sin (c+d x))}+\frac{\left(-340 a^2 b^2+224 a^4+105 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^4 d}-\frac{\left(-37 a^2 b^2+24 a^4+12 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{12 a b^5 d}+\frac{\left(-441 a^2 b^2+280 a^4+150 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}-\frac{a \left(-52 a^2 b^2+32 a^4+19 b^4\right) \sin (c+d x) \cos (c+d x)}{8 b^7 d}+\frac{a x \left(-120 a^4 b^2+60 a^2 b^4+64 a^6-5 b^6\right)}{8 b^9}-\frac{3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}-\frac{4 a \sin ^6(c+d x) \cos (c+d x)}{21 b^2 d (a+b \sin (c+d x))}+\frac{\sin ^7(c+d x) \cos (c+d x)}{7 b d (a+b \sin (c+d x))}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))}",1,"(a*(64*a^6 - 120*a^4*b^2 + 60*a^2*b^4 - 5*b^6)*x)/(8*b^9) - (2*a^2*(8*a^2 - 3*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^9*d) + ((840*a^6 - 1435*a^4*b^2 + 588*a^2*b^4 - 15*b^6)*Cos[c + d*x])/(105*b^8*d) - (a*(32*a^4 - 52*a^2*b^2 + 19*b^4)*Cos[c + d*x]*Sin[c + d*x])/(8*b^7*d) + ((280*a^4 - 441*a^2*b^2 + 150*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(105*b^6*d) - ((24*a^4 - 37*a^2*b^2 + 12*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(12*a*b^5*d) + ((224*a^4 - 340*a^2*b^2 + 105*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(140*a^2*b^4*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d*(a + b*Sin[c + d*x])) - (3*b*Cos[c + d*x]*Sin[c + d*x]^5)/(20*a^2*d*(a + b*Sin[c + d*x])) - ((20*a^4 - 30*a^2*b^2 + 9*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(15*a^2*b^3*d*(a + b*Sin[c + d*x])) - (4*a*Cos[c + d*x]*Sin[c + d*x]^6)/(21*b^2*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*Sin[c + d*x]^7)/(7*b*d*(a + b*Sin[c + d*x]))","A",11,8,29,0.2759,1,"{2896, 3047, 3049, 3023, 2735, 2660, 618, 204}"
1257,1,471,0,1.5497734,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","-\frac{a \left(-170 a^2 b^2+105 a^4+61 b^4\right) \cos (c+d x)}{15 b^7 d}+\frac{2 a \left(7 a^2-2 b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^8 d}-\frac{\left(-20 a^2 b^2+14 a^4+5 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}+\frac{\left(-61 a^2 b^2+42 a^4+16 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^4 d}-\frac{\left(-52 a^2 b^2+35 a^4+15 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{15 a b^5 d}+\frac{\left(-86 a^2 b^2+56 a^4+27 b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^6 d}-\frac{x \left(-200 a^4 b^2+90 a^2 b^4+112 a^6-5 b^6\right)}{16 b^8}-\frac{b \sin ^4(c+d x) \cos (c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\sin ^6(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))}+\frac{\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))}","-\frac{a \left(-170 a^2 b^2+105 a^4+61 b^4\right) \cos (c+d x)}{15 b^7 d}+\frac{2 a \left(7 a^2-2 b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^8 d}-\frac{\left(-20 a^2 b^2+14 a^4+5 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{10 a^2 b^3 d (a+b \sin (c+d x))}+\frac{\left(-61 a^2 b^2+42 a^4+16 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^4 d}-\frac{\left(-52 a^2 b^2+35 a^4+15 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{15 a b^5 d}+\frac{\left(-86 a^2 b^2+56 a^4+27 b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^6 d}-\frac{x \left(-200 a^4 b^2+90 a^2 b^4+112 a^6-5 b^6\right)}{16 b^8}-\frac{b \sin ^4(c+d x) \cos (c+d x)}{6 a^2 d (a+b \sin (c+d x))}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{30 b^2 d (a+b \sin (c+d x))}+\frac{\sin ^6(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))}+\frac{\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))}",1,"-((112*a^6 - 200*a^4*b^2 + 90*a^2*b^4 - 5*b^6)*x)/(16*b^8) + (2*a*(7*a^2 - 2*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^8*d) - (a*(105*a^4 - 170*a^2*b^2 + 61*b^4)*Cos[c + d*x])/(15*b^7*d) + ((56*a^4 - 86*a^2*b^2 + 27*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^6*d) - ((35*a^4 - 52*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(15*a*b^5*d) + ((42*a^4 - 61*a^2*b^2 + 16*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^2*b^4*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d*(a + b*Sin[c + d*x])) - (b*Cos[c + d*x]*Sin[c + d*x]^4)/(6*a^2*d*(a + b*Sin[c + d*x])) - ((14*a^4 - 20*a^2*b^2 + 5*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(10*a^2*b^3*d*(a + b*Sin[c + d*x])) - (7*a*Cos[c + d*x]*Sin[c + d*x]^5)/(30*b^2*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*Sin[c + d*x]^6)/(6*b*d*(a + b*Sin[c + d*x]))","A",10,8,29,0.2759,1,"{2896, 3047, 3049, 3023, 2735, 2660, 618, 204}"
1258,1,231,0,0.5126106,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(a^2-b^2\right)^{3/2} \left(6 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d}-\frac{\cos ^3(c+d x) \left(2 \left(6 a^2-b^2\right)-9 a b \sin (c+d x)\right)}{6 b^4 d}+\frac{\cos (c+d x) \left(4 \left(-7 a^2 b^2+6 a^4+b^4\right)-a b \left(12 a^2-11 b^2\right) \sin (c+d x)\right)}{4 b^6 d}+\frac{a x \left(-40 a^2 b^2+24 a^4+15 b^4\right)}{4 b^7}+\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}","-\frac{2 \left(a^2-b^2\right)^{3/2} \left(6 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d}-\frac{\cos ^3(c+d x) \left(2 \left(6 a^2-b^2\right)-9 a b \sin (c+d x)\right)}{6 b^4 d}+\frac{\cos (c+d x) \left(4 \left(-7 a^2 b^2+6 a^4+b^4\right)-a b \left(12 a^2-11 b^2\right) \sin (c+d x)\right)}{4 b^6 d}+\frac{a x \left(-40 a^2 b^2+24 a^4+15 b^4\right)}{4 b^7}+\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}",1,"(a*(24*a^4 - 40*a^2*b^2 + 15*b^4)*x)/(4*b^7) - (2*(a^2 - b^2)^(3/2)*(6*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^7*d) + (Cos[c + d*x]^5*(6*a + b*Sin[c + d*x]))/(5*b^2*d*(a + b*Sin[c + d*x])) - (Cos[c + d*x]^3*(2*(6*a^2 - b^2) - 9*a*b*Sin[c + d*x]))/(6*b^4*d) + (Cos[c + d*x]*(4*(6*a^4 - 7*a^2*b^2 + b^4) - a*b*(12*a^2 - 11*b^2)*Sin[c + d*x]))/(4*b^6*d)","A",7,6,27,0.2222,1,"{2863, 2865, 2735, 2660, 618, 204}"
1259,1,266,0,0.3505747,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{3 \left(a^2-b^2\right) \cos (c+d x)}{b^4 d}+\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d}-\frac{2 \left(a^2-b^2\right)^{3/2} \left(5 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^5 d}+\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a b^4 d (a+b \sin (c+d x))}+\frac{2 a x \left(2 a^2-3 b^2\right)}{b^5}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{a \sin (c+d x) \cos (c+d x)}{b^3 d}+\frac{a x}{b^3}-\frac{\cos ^3(c+d x)}{3 b^2 d}+\frac{\cos (c+d x)}{b^2 d}","\frac{3 \left(a^2-b^2\right) \cos (c+d x)}{b^4 d}+\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d}-\frac{2 \left(a^2-b^2\right)^{3/2} \left(5 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^5 d}+\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a b^4 d (a+b \sin (c+d x))}+\frac{2 a x \left(2 a^2-3 b^2\right)}{b^5}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{a \sin (c+d x) \cos (c+d x)}{b^3 d}+\frac{a x}{b^3}-\frac{\cos ^3(c+d x)}{3 b^2 d}+\frac{\cos (c+d x)}{b^2 d}",1,"(a*x)/b^3 + (2*a*(2*a^2 - 3*b^2)*x)/b^5 + (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^5*d) - (2*(a^2 - b^2)^(3/2)*(5*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^5*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(b^2*d) + (3*(a^2 - b^2)*Cos[c + d*x])/(b^4*d) - Cos[c + d*x]^3/(3*b^2*d) - (a*Cos[c + d*x]*Sin[c + d*x])/(b^3*d) + ((a^2 - b^2)^2*Cos[c + d*x])/(a*b^4*d*(a + b*Sin[c + d*x]))","A",16,11,27,0.4074,1,"{2897, 3770, 2638, 2635, 8, 2633, 2664, 12, 2660, 618, 204}"
1260,1,254,0,0.3418059,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b^4 d}+\frac{4 \left(-3 a^4 b^2+2 a^6+b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^4 d \sqrt{a^2-b^2}}-\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a^2 b^3 d (a+b \sin (c+d x))}-\frac{3 x \left(a^2-b^2\right)}{b^4}+\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{2 a \cos (c+d x)}{b^3 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 b^2 d}-\frac{x}{2 b^2}","-\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b^4 d}+\frac{4 \left(-3 a^4 b^2+2 a^6+b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^4 d \sqrt{a^2-b^2}}-\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a^2 b^3 d (a+b \sin (c+d x))}-\frac{3 x \left(a^2-b^2\right)}{b^4}+\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{2 a \cos (c+d x)}{b^3 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 b^2 d}-\frac{x}{2 b^2}",1,"-x/(2*b^2) - (3*(a^2 - b^2)*x)/b^4 - (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*b^4*d) + (4*(2*a^6 - 3*a^4*b^2 + b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b^4*Sqrt[a^2 - b^2]*d) + (2*b*ArcTanh[Cos[c + d*x]])/(a^3*d) - (2*a*Cos[c + d*x])/(b^3*d) - Cot[c + d*x]/(a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) - ((a^2 - b^2)^2*Cos[c + d*x])/(a^2*b^3*d*(a + b*Sin[c + d*x]))","A",16,11,29,0.3793,1,"{2897, 3770, 3767, 8, 2638, 2635, 2664, 12, 2660, 618, 204}"
1261,1,251,0,0.3358014,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","-\frac{6 \left(a^2+b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 b^3 d}+\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^3 d}+\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac{3 \left(a^2-b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{2 a x}{b^3}+\frac{\cos (c+d x)}{b^2 d}","-\frac{6 \left(a^2+b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 b^3 d}+\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^3 d}+\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac{3 \left(a^2-b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{2 a x}{b^3}+\frac{\cos (c+d x)}{b^2 d}",1,"(2*a*x)/b^3 + (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^3*d) - (6*(a^2 - b^2)^(3/2)*(a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*b^3*d) - ArcTanh[Cos[c + d*x]]/(2*a^2*d) + (3*(a^2 - b^2)*ArcTanh[Cos[c + d*x]])/(a^4*d) + Cos[c + d*x]/(b^2*d) + (2*b*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) + ((a^2 - b^2)^2*Cos[c + d*x])/(a^3*b^2*d*(a + b*Sin[c + d*x]))","A",16,11,29,0.3793,1,"{2897, 3770, 3767, 8, 3768, 2638, 2664, 12, 2660, 618, 204}"
1262,1,287,0,0.376761,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^2 d}+\frac{4 \left(-3 a^2 b^4+a^6+2 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 b^2 d \sqrt{a^2-b^2}}+\frac{3 \left(a^2-b^2\right) \cot (c+d x)}{a^4 d}-\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a^4 b d (a+b \sin (c+d x))}-\frac{2 b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{x}{b^2}","-\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^2 d}+\frac{4 \left(-3 a^2 b^4+a^6+2 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 b^2 d \sqrt{a^2-b^2}}+\frac{3 \left(a^2-b^2\right) \cot (c+d x)}{a^4 d}-\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a^4 b d (a+b \sin (c+d x))}-\frac{2 b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\cot (c+d x)}{a^2 d}-\frac{x}{b^2}",1,"-(x/b^2) - (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b^2*d) + (4*(a^6 - 3*a^2*b^4 + 2*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*b^2*Sqrt[a^2 - b^2]*d) + (b*ArcTanh[Cos[c + d*x]])/(a^3*d) - (2*b*(3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) - Cot[c + d*x]/(a^2*d) + (3*(a^2 - b^2)*Cot[c + d*x])/(a^4*d) - Cot[c + d*x]^3/(3*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(a^3*d) - ((a^2 - b^2)^2*Cos[c + d*x])/(a^4*b*d*(a + b*Sin[c + d*x]))","A",17,10,29,0.3448,1,"{2897, 3770, 3767, 8, 3768, 2664, 12, 2660, 618, 204}"
1263,1,303,0,1.2109917,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x])^2,x]","-\frac{10 b \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}+\frac{\left(-20 a^2 b^2+3 a^4+15 b^4\right) \cot (c+d x)}{3 a^5 b d}-\frac{5 \left(-12 a^2 b^2+3 a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}+\frac{5 \left(5 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac{\left(6 a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{3 a^3 d (a+b \sin (c+d x))}+\frac{5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac{\cot (c+d x)}{b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}","-\frac{10 b \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}+\frac{\left(-20 a^2 b^2+3 a^4+15 b^4\right) \cot (c+d x)}{3 a^5 b d}-\frac{5 \left(-12 a^2 b^2+3 a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}+\frac{5 \left(5 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 d}-\frac{\left(6 a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{3 a^3 d (a+b \sin (c+d x))}+\frac{5 b \cot (c+d x) \csc ^2(c+d x)}{12 a^2 d (a+b \sin (c+d x))}-\frac{\cot (c+d x)}{b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))}",1,"(-10*b*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*d) - (5*(3*a^4 - 12*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) + ((3*a^4 - 20*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(3*a^5*b*d) + (5*(5*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) - Cot[c + d*x]/(b*d*(a + b*Sin[c + d*x])) - ((6*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^3*d*(a + b*Sin[c + d*x])) + (5*b*Cot[c + d*x]*Csc[c + d*x]^2)/(12*a^2*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x]))","A",9,7,27,0.2593,1,"{2896, 3055, 3001, 3770, 2660, 618, 204}"
1264,1,424,0,1.5208764,"\int \frac{\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cot[c + d*x]^6/(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(a^2-6 b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^7 d}-\frac{\left(-135 a^2 b^2+38 a^4+90 b^4\right) \cot (c+d x)}{15 a^6 d}+\frac{b \left(-40 a^2 b^2+15 a^4+24 b^4\right) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac{\left(-82 a^2 b^2+15 a^4+60 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}+\frac{\left(-17 a^2 b^2+4 a^4+12 b^4\right) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}+\frac{\left(-12 a^2 b^2+2 a^4+9 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac{3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}","-\frac{2 \left(a^2-6 b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^7 d}-\frac{\left(-135 a^2 b^2+38 a^4+90 b^4\right) \cot (c+d x)}{15 a^6 d}+\frac{b \left(-40 a^2 b^2+15 a^4+24 b^4\right) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac{\left(-82 a^2 b^2+15 a^4+60 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}+\frac{\left(-17 a^2 b^2+4 a^4+12 b^4\right) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}+\frac{\left(-12 a^2 b^2+2 a^4+9 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac{3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}",1,"(-2*(a^2 - 6*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^7*d) + (b*(15*a^4 - 40*a^2*b^2 + 24*b^4)*ArcTanh[Cos[c + d*x]])/(4*a^7*d) - ((38*a^4 - 135*a^2*b^2 + 90*b^4)*Cot[c + d*x])/(15*a^6*d) + ((4*a^4 - 17*a^2*b^2 + 12*b^4)*Cot[c + d*x]*Csc[c + d*x])/(4*a^5*b*d) - ((15*a^4 - 82*a^2*b^2 + 60*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^4*b^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*b*d*(a + b*Sin[c + d*x])) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(6*b^2*d*(a + b*Sin[c + d*x])) + ((2*a^4 - 12*a^2*b^2 + 9*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(6*a^3*b^2*d*(a + b*Sin[c + d*x])) + (3*b*Cot[c + d*x]*Csc[c + d*x]^3)/(10*a^2*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d*(a + b*Sin[c + d*x]))","A",10,7,21,0.3333,1,"{2726, 3055, 3001, 3770, 2660, 618, 204}"
1265,1,480,0,1.9591966,"\int \frac{\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Cot[c + d*x]^6*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{2 b \left(2 a^2-7 b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^8 d}+\frac{b \left(-170 a^2 b^2+61 a^4+105 b^4\right) \cot (c+d x)}{15 a^7 d}+\frac{\left(-90 a^4 b^2+200 a^2 b^4+5 a^6-112 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^8 d}-\frac{\left(-61 a^2 b^2+16 a^4+42 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}+\frac{\left(-52 a^2 b^2+15 a^4+35 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}-\frac{\left(-86 a^2 b^2+27 a^4+56 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac{\left(-20 a^2 b^2+5 a^4+14 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac{7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}","\frac{2 b \left(2 a^2-7 b^2\right) \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^8 d}+\frac{b \left(-170 a^2 b^2+61 a^4+105 b^4\right) \cot (c+d x)}{15 a^7 d}+\frac{\left(-90 a^4 b^2+200 a^2 b^4+5 a^6-112 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^8 d}-\frac{\left(-61 a^2 b^2+16 a^4+42 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}+\frac{\left(-52 a^2 b^2+15 a^4+35 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}-\frac{\left(-86 a^2 b^2+27 a^4+56 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac{\left(-20 a^2 b^2+5 a^4+14 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac{7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}",1,"(2*b*(2*a^2 - 7*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^8*d) + ((5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^8*d) + (b*(61*a^4 - 170*a^2*b^2 + 105*b^4)*Cot[c + d*x])/(15*a^7*d) - ((27*a^4 - 86*a^2*b^2 + 56*b^4)*Cot[c + d*x]*Csc[c + d*x])/(16*a^6*d) + ((15*a^4 - 52*a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*a^5*b*d) - ((16*a^4 - 61*a^2*b^2 + 42*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^4*b^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*b*d*(a + b*Sin[c + d*x])) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(6*b^2*d*(a + b*Sin[c + d*x])) + ((5*a^4 - 20*a^2*b^2 + 14*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(10*a^3*b^2*d*(a + b*Sin[c + d*x])) + (7*b*Cot[c + d*x]*Csc[c + d*x]^4)/(30*a^2*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a*d*(a + b*Sin[c + d*x]))","A",11,7,27,0.2593,1,"{2896, 3055, 3001, 3770, 2660, 618, 204}"
1266,1,536,0,2.1993785,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]","-\frac{a \left(-985 a^2 b^2+840 a^4+213 b^4\right) \cos (c+d x)}{30 b^8 d}+\frac{a \sqrt{a^2-b^2} \left(-47 a^2 b^2+56 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^9 d}-\frac{\left(-60 a^2 b^2+56 a^4+9 b^4\right) \sin ^5(c+d x) \cos (c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{\left(-110 a^2 b^2+112 a^4+15 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\left(-169 a^2 b^2+168 a^4+24 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^5 d}-\frac{\left(-291 a^2 b^2+280 a^4+45 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{30 a b^6 d}+\frac{\left(-244 a^2 b^2+224 a^4+43 b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^7 d}-\frac{x \left(-600 a^4 b^2+180 a^2 b^4+448 a^6-5 b^6\right)}{16 b^9}-\frac{b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}","-\frac{a \left(-985 a^2 b^2+840 a^4+213 b^4\right) \cos (c+d x)}{30 b^8 d}+\frac{a \sqrt{a^2-b^2} \left(-47 a^2 b^2+56 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^9 d}-\frac{\left(-60 a^2 b^2+56 a^4+9 b^4\right) \sin ^5(c+d x) \cos (c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{\left(-110 a^2 b^2+112 a^4+15 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\left(-169 a^2 b^2+168 a^4+24 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^5 d}-\frac{\left(-291 a^2 b^2+280 a^4+45 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{30 a b^6 d}+\frac{\left(-244 a^2 b^2+224 a^4+43 b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^7 d}-\frac{x \left(-600 a^4 b^2+180 a^2 b^4+448 a^6-5 b^6\right)}{16 b^9}-\frac{b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac{\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}",1,"-((448*a^6 - 600*a^4*b^2 + 180*a^2*b^4 - 5*b^6)*x)/(16*b^9) + (a*Sqrt[a^2 - b^2]*(56*a^4 - 47*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^9*d) - (a*(840*a^4 - 985*a^2*b^2 + 213*b^4)*Cos[c + d*x])/(30*b^8*d) + ((224*a^4 - 244*a^2*b^2 + 43*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^7*d) - ((280*a^4 - 291*a^2*b^2 + 45*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(30*a*b^6*d) + ((168*a^4 - 169*a^2*b^2 + 24*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^2*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d*(a + b*Sin[c + d*x])^2) - (b*Cos[c + d*x]*Sin[c + d*x]^5)/(10*a^2*d*(a + b*Sin[c + d*x])^2) - ((56*a^4 - 60*a^2*b^2 + 9*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(60*a^2*b^3*d*(a + b*Sin[c + d*x])^2) - (4*a*Cos[c + d*x]*Sin[c + d*x]^6)/(15*b^2*d*(a + b*Sin[c + d*x])^2) + (Cos[c + d*x]*Sin[c + d*x]^7)/(6*b*d*(a + b*Sin[c + d*x])^2) - ((112*a^4 - 110*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(20*a^2*b^4*d*(a + b*Sin[c + d*x]))","A",11,8,29,0.2759,1,"{2896, 3047, 3049, 3023, 2735, 2660, 618, 204}"
1267,1,485,0,1.7169326,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{\left(-645 a^2 b^2+630 a^4+91 b^4\right) \cos (c+d x)}{30 b^7 d}-\frac{\sqrt{a^2-b^2} \left(-29 a^2 b^2+42 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^8 d}-\frac{\left(-60 a^2 b^2+63 a^4+5 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{\left(-54 a^2 b^2+63 a^4+4 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\left(-187 a^2 b^2+210 a^4+15 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{30 a^2 b^5 d}-\frac{\left(-79 a^2 b^2+84 a^4+8 b^4\right) \sin (c+d x) \cos (c+d x)}{8 a b^6 d}+\frac{a x \left(-200 a^2 b^2+168 a^4+45 b^4\right)}{8 b^8}-\frac{b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}","\frac{\left(-645 a^2 b^2+630 a^4+91 b^4\right) \cos (c+d x)}{30 b^7 d}-\frac{\sqrt{a^2-b^2} \left(-29 a^2 b^2+42 a^4+2 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^8 d}-\frac{\left(-60 a^2 b^2+63 a^4+5 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{\left(-54 a^2 b^2+63 a^4+4 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\left(-187 a^2 b^2+210 a^4+15 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{30 a^2 b^5 d}-\frac{\left(-79 a^2 b^2+84 a^4+8 b^4\right) \sin (c+d x) \cos (c+d x)}{8 a b^6 d}+\frac{a x \left(-200 a^2 b^2+168 a^4+45 b^4\right)}{8 b^8}-\frac{b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2}",1,"(a*(168*a^4 - 200*a^2*b^2 + 45*b^4)*x)/(8*b^8) - (Sqrt[a^2 - b^2]*(42*a^4 - 29*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^8*d) + ((630*a^4 - 645*a^2*b^2 + 91*b^4)*Cos[c + d*x])/(30*b^7*d) - ((84*a^4 - 79*a^2*b^2 + 8*b^4)*Cos[c + d*x]*Sin[c + d*x])/(8*a*b^6*d) + ((210*a^4 - 187*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(30*a^2*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d*(a + b*Sin[c + d*x])^2) - (b*Cos[c + d*x]*Sin[c + d*x]^4)/(12*a^2*d*(a + b*Sin[c + d*x])^2) - ((63*a^4 - 60*a^2*b^2 + 5*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(60*a^2*b^3*d*(a + b*Sin[c + d*x])^2) - (7*a*Cos[c + d*x]*Sin[c + d*x]^5)/(20*b^2*d*(a + b*Sin[c + d*x])^2) + (Cos[c + d*x]*Sin[c + d*x]^6)/(5*b*d*(a + b*Sin[c + d*x])^2) - ((63*a^4 - 54*a^2*b^2 + 4*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(12*a^2*b^4*d*(a + b*Sin[c + d*x]))","A",10,8,29,0.2759,1,"{2896, 3047, 3049, 3023, 2735, 2660, 618, 204}"
1268,1,237,0,0.4654552,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{15 a \left(-3 a^2 b^2+2 a^4+b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d \sqrt{a^2-b^2}}+\frac{5 \cos ^3(c+d x) \left(4 a^2+a b \sin (c+d x)-b^2\right)}{4 b^4 d (a+b \sin (c+d x))}-\frac{15 \cos (c+d x) \left(4 a \left(2 a^2-b^2\right)-b \left(4 a^2-b^2\right) \sin (c+d x)\right)}{8 b^6 d}-\frac{15 x \left(-8 a^2 b^2+8 a^4+b^4\right)}{8 b^7}+\frac{\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}","\frac{15 a \left(-3 a^2 b^2+2 a^4+b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d \sqrt{a^2-b^2}}+\frac{5 \cos ^3(c+d x) \left(4 a^2+a b \sin (c+d x)-b^2\right)}{4 b^4 d (a+b \sin (c+d x))}-\frac{15 \cos (c+d x) \left(4 a \left(2 a^2-b^2\right)-b \left(4 a^2-b^2\right) \sin (c+d x)\right)}{8 b^6 d}-\frac{15 x \left(-8 a^2 b^2+8 a^4+b^4\right)}{8 b^7}+\frac{\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}",1,"(-15*(8*a^4 - 8*a^2*b^2 + b^4)*x)/(8*b^7) + (15*a*(2*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) + (Cos[c + d*x]^5*(3*a + b*Sin[c + d*x]))/(4*b^2*d*(a + b*Sin[c + d*x])^2) + (5*Cos[c + d*x]^3*(4*a^2 - b^2 + a*b*Sin[c + d*x]))/(4*b^4*d*(a + b*Sin[c + d*x])) - (15*Cos[c + d*x]*(4*a*(2*a^2 - b^2) - b*(4*a^2 - b^2)*Sin[c + d*x]))/(8*b^6*d)","A",7,6,27,0.2222,1,"{2863, 2865, 2735, 2660, 618, 204}"
1269,1,399,0,0.517162,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^5*Cot[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{\left(2 a^2+b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b^5 d}-\frac{2 \left(5 a^2+b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b^5 d}+\frac{2 \left(-9 a^4 b^2+10 a^6-b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^5 d \sqrt{a^2-b^2}}+\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{2 a b^4 d (a+b \sin (c+d x))^2}-\frac{\left(5 a^2+b^2\right) \left(a^2-b^2\right) \cos (c+d x)}{a^2 b^4 d (a+b \sin (c+d x))}+\frac{3 \left(a^2-b^2\right) \cos (c+d x)}{2 b^4 d (a+b \sin (c+d x))}-\frac{3 x \left(2 a^2-b^2\right)}{b^5}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{3 a \cos (c+d x)}{b^4 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 b^3 d}-\frac{x}{2 b^3}","\frac{\left(2 a^2+b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b^5 d}-\frac{2 \left(5 a^2+b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b^5 d}+\frac{2 \left(-9 a^4 b^2+10 a^6-b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^5 d \sqrt{a^2-b^2}}+\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{2 a b^4 d (a+b \sin (c+d x))^2}-\frac{\left(5 a^2+b^2\right) \left(a^2-b^2\right) \cos (c+d x)}{a^2 b^4 d (a+b \sin (c+d x))}+\frac{3 \left(a^2-b^2\right) \cos (c+d x)}{2 b^4 d (a+b \sin (c+d x))}-\frac{3 x \left(2 a^2-b^2\right)}{b^5}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{3 a \cos (c+d x)}{b^4 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 b^3 d}-\frac{x}{2 b^3}",1,"-x/(2*b^3) - (3*(2*a^2 - b^2)*x)/b^5 + (Sqrt[a^2 - b^2]*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*b^5*d) - (2*Sqrt[a^2 - b^2]*(5*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*b^5*d) + (2*(10*a^6 - 9*a^4*b^2 - b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b^5*Sqrt[a^2 - b^2]*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) - (3*a*Cos[c + d*x])/(b^4*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b^3*d) + ((a^2 - b^2)^2*Cos[c + d*x])/(2*a*b^4*d*(a + b*Sin[c + d*x])^2) + (3*(a^2 - b^2)*Cos[c + d*x])/(2*b^4*d*(a + b*Sin[c + d*x])) - ((a^2 - b^2)*(5*a^2 + b^2)*Cos[c + d*x])/(a^2*b^4*d*(a + b*Sin[c + d*x]))","A",20,11,27,0.4074,1,"{2897, 3770, 2638, 2635, 8, 2664, 2754, 12, 2660, 618, 204}"
1270,1,314,0,0.4927996,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{3 \left(2 a^2+b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^4 d}-\frac{6 \left(-a^4 b^2+2 a^6-b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 b^4 d \sqrt{a^2-b^2}}-\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{2 a^2 b^3 d (a+b \sin (c+d x))^2}+\frac{2 \left(2 a^2+b^2\right) \left(a^2-b^2\right) \cos (c+d x)}{a^3 b^3 d (a+b \sin (c+d x))}-\frac{3 \left(a^2-b^2\right) \cos (c+d x)}{2 a b^3 d (a+b \sin (c+d x))}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a^3 d}+\frac{3 a x}{b^4}+\frac{\cos (c+d x)}{b^3 d}","\frac{3 \left(2 a^2+b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^4 d}-\frac{6 \left(-a^4 b^2+2 a^6-b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 b^4 d \sqrt{a^2-b^2}}-\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{2 a^2 b^3 d (a+b \sin (c+d x))^2}+\frac{2 \left(2 a^2+b^2\right) \left(a^2-b^2\right) \cos (c+d x)}{a^3 b^3 d (a+b \sin (c+d x))}-\frac{3 \left(a^2-b^2\right) \cos (c+d x)}{2 a b^3 d (a+b \sin (c+d x))}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a^3 d}+\frac{3 a x}{b^4}+\frac{\cos (c+d x)}{b^3 d}",1,"(3*a*x)/b^4 + (3*Sqrt[a^2 - b^2]*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^4*d) - (6*(2*a^6 - a^4*b^2 - b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*b^4*Sqrt[a^2 - b^2]*d) + (3*b*ArcTanh[Cos[c + d*x]])/(a^4*d) + Cos[c + d*x]/(b^3*d) - Cot[c + d*x]/(a^3*d) - ((a^2 - b^2)^2*Cos[c + d*x])/(2*a^2*b^3*d*(a + b*Sin[c + d*x])^2) - (3*(a^2 - b^2)*Cos[c + d*x])/(2*a*b^3*d*(a + b*Sin[c + d*x])) + (2*(a^2 - b^2)*(2*a^2 + b^2)*Cos[c + d*x])/(a^3*b^3*d*(a + b*Sin[c + d*x]))","A",20,11,29,0.3793,1,"{2897, 3770, 3767, 8, 2638, 2664, 2754, 12, 2660, 618, 204}"
1271,1,395,0,0.5234507,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]","-\frac{6 \left(a^2+b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^3 d}+\frac{\left(2 a^2+b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^3 d}+\frac{6 \left(a^2 b^4+a^6-2 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 b^3 d \sqrt{a^2-b^2}}+\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 \left(a^2-b^2\right) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{3 \left(a^4-b^4\right) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}+\frac{3 \left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{3 b \cot (c+d x)}{a^4 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{x}{b^3}","-\frac{6 \left(a^2+b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^3 d}+\frac{\left(2 a^2+b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^3 d}+\frac{6 \left(a^2 b^4+a^6-2 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 b^3 d \sqrt{a^2-b^2}}+\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 \left(a^2-b^2\right) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{3 \left(a^4-b^4\right) \cos (c+d x)}{a^4 b^2 d (a+b \sin (c+d x))}+\frac{3 \left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{3 b \cot (c+d x)}{a^4 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac{x}{b^3}",1,"-(x/b^3) - (6*Sqrt[a^2 - b^2]*(a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b^3*d) + (Sqrt[a^2 - b^2]*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b^3*d) + (6*(a^6 + a^2*b^4 - 2*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*b^3*Sqrt[a^2 - b^2]*d) - ArcTanh[Cos[c + d*x]]/(2*a^3*d) + (3*(a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + (3*b*Cot[c + d*x])/(a^4*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + ((a^2 - b^2)^2*Cos[c + d*x])/(2*a^3*b^2*d*(a + b*Sin[c + d*x])^2) + (3*(a^2 - b^2)*Cos[c + d*x])/(2*a^2*b^2*d*(a + b*Sin[c + d*x])) - (3*(a^4 - b^4)*Cos[c + d*x])/(a^4*b^2*d*(a + b*Sin[c + d*x]))","A",21,11,29,0.3793,1,"{2897, 3770, 3767, 8, 3768, 2664, 2754, 12, 2660, 618, 204}"
1272,1,329,0,1.3290387,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^3,x]","\frac{5 \left(a^2-4 b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}+\frac{\left(35 a^2 b^2+3 a^4-60 b^4\right) \cot (c+d x)}{6 a^5 b^2 d}-\frac{5 b \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}-\frac{5 \left(a^2-2 b^2\right) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac{\left(3 a^2-5 b^2\right) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac{5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac{a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac{\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}","\frac{5 \left(a^2-4 b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}+\frac{\left(35 a^2 b^2+3 a^4-60 b^4\right) \cot (c+d x)}{6 a^5 b^2 d}-\frac{5 b \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}-\frac{5 \left(a^2-2 b^2\right) \cot (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac{\left(3 a^2-5 b^2\right) \cot (c+d x)}{3 a^3 d (a+b \sin (c+d x))^2}+\frac{5 b \cot (c+d x) \csc (c+d x)}{6 a^2 d (a+b \sin (c+d x))^2}-\frac{a \cot (c+d x)}{2 b^2 d (a+b \sin (c+d x))^2}-\frac{\cos (c+d x)}{b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}",1,"(5*(a^2 - 4*b^2)*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*d) - (5*b*(3*a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^6*d) + ((3*a^4 + 35*a^2*b^2 - 60*b^4)*Cot[c + d*x])/(6*a^5*b^2*d) - Cos[c + d*x]/(b*d*(a + b*Sin[c + d*x])^2) - (a*Cot[c + d*x])/(2*b^2*d*(a + b*Sin[c + d*x])^2) - ((3*a^2 - 5*b^2)*Cot[c + d*x])/(3*a^3*d*(a + b*Sin[c + d*x])^2) + (5*b*Cot[c + d*x]*Csc[c + d*x])/(6*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^2) - (5*(a^2 - 2*b^2)*Cot[c + d*x])/(2*a^4*d*(a + b*Sin[c + d*x]))","A",9,7,29,0.2414,1,"{2896, 3055, 3001, 3770, 2660, 618, 204}"
1273,1,355,0,1.6816934,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x])^3,x]","-\frac{15 b \left(a^2-2 b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^7 d}+\frac{\left(-25 a^2 b^2+a^4+30 b^4\right) \cot (c+d x)}{2 a^6 b d}-\frac{15 \left(-8 a^2 b^2+a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}+\frac{15 \left(3 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\left(7 a^2-10 b^2\right) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac{\left(4 a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}","-\frac{15 b \left(a^2-2 b^2\right) \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^7 d}+\frac{\left(-25 a^2 b^2+a^4+30 b^4\right) \cot (c+d x)}{2 a^6 b d}-\frac{15 \left(-8 a^2 b^2+a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}+\frac{15 \left(3 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^5 d}-\frac{\left(7 a^2-10 b^2\right) \cot (c+d x) \csc (c+d x)}{2 a^4 d (a+b \sin (c+d x))}-\frac{\left(4 a^2-5 b^2\right) \cot (c+d x) \csc (c+d x)}{4 a^3 d (a+b \sin (c+d x))^2}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}",1,"(-15*b*(a^2 - 2*b^2)*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^7*d) - (15*(a^4 - 8*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^7*d) + ((a^4 - 25*a^2*b^2 + 30*b^4)*Cot[c + d*x])/(2*a^6*b*d) + (15*(3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^5*d) - Cot[c + d*x]/(2*b*d*(a + b*Sin[c + d*x])^2) - ((4*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(4*a^3*d*(a + b*Sin[c + d*x])^2) + (b*Cot[c + d*x]*Csc[c + d*x]^2)/(2*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x])^2) - ((7*a^2 - 10*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^4*d*(a + b*Sin[c + d*x]))","A",10,7,27,0.2593,1,"{2896, 3055, 3001, 3770, 2660, 618, 204}"
1274,1,492,0,2.1550686,"\int \frac{\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cot[c + d*x]^6/(a + b*Sin[c + d*x])^3,x]","-\frac{\sqrt{a^2-b^2} \left(-29 a^2 b^2+2 a^4+42 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^8 d}-\frac{\left(-645 a^2 b^2+91 a^4+630 b^4\right) \cot (c+d x)}{30 a^7 d}+\frac{b \left(-200 a^2 b^2+45 a^4+168 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac{\left(-187 a^2 b^2+15 a^4+210 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}+\frac{\left(-79 a^2 b^2+8 a^4+84 b^4\right) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}+\frac{\left(-54 a^2 b^2+4 a^4+63 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left(-60 a^2 b^2+5 a^4+63 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}","-\frac{\sqrt{a^2-b^2} \left(-29 a^2 b^2+2 a^4+42 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^8 d}-\frac{\left(-645 a^2 b^2+91 a^4+630 b^4\right) \cot (c+d x)}{30 a^7 d}+\frac{b \left(-200 a^2 b^2+45 a^4+168 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac{\left(-187 a^2 b^2+15 a^4+210 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}+\frac{\left(-79 a^2 b^2+8 a^4+84 b^4\right) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}+\frac{\left(-54 a^2 b^2+4 a^4+63 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left(-60 a^2 b^2+5 a^4+63 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}",1,"-((Sqrt[a^2 - b^2]*(2*a^4 - 29*a^2*b^2 + 42*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^8*d)) + (b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^8*d) - ((91*a^4 - 645*a^2*b^2 + 630*b^4)*Cot[c + d*x])/(30*a^7*d) + ((8*a^4 - 79*a^2*b^2 + 84*b^4)*Cot[c + d*x]*Csc[c + d*x])/(8*a^6*b*d) - ((15*a^4 - 187*a^2*b^2 + 210*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^5*b^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(3*b*d*(a + b*Sin[c + d*x])^2) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(12*b^2*d*(a + b*Sin[c + d*x])^2) + ((5*a^4 - 60*a^2*b^2 + 63*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(60*a^3*b^2*d*(a + b*Sin[c + d*x])^2) + (7*b*Cot[c + d*x]*Csc[c + d*x]^3)/(20*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d*(a + b*Sin[c + d*x])^2) + ((4*a^4 - 54*a^2*b^2 + 63*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(12*a^4*b^2*d*(a + b*Sin[c + d*x]))","A",11,7,21,0.3333,1,"{2726, 3055, 3001, 3770, 2660, 618, 204}"
1275,1,600,0,3.2202468,"\int \frac{\cot ^6(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Cot[c + d*x]^6*Csc[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","-\frac{3 b^2 \sqrt{a^2-b^2} \left(-23 a^2 b^2+4 a^4+24 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^{10} d}+\frac{\left(-889 a^4 b^2+3255 a^2 b^4+10 a^6-2520 b^6\right) \cot (c+d x)}{70 a^9 d}-\frac{3 b \left(-100 a^4 b^2+280 a^2 b^4+5 a^6-192 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^{10} d}-\frac{3 \left(-185 a^2 b^2+35 a^4+168 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}+\frac{\left(-81 a^2 b^2+16 a^4+72 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{\left(-973 a^2 b^2+205 a^4+840 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{3 b \left(-116 a^2 b^2+27 a^4+96 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^8 d}+\frac{\left(-65 a^2 b^2+12 a^4+60 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left(-35 a^2 b^2+7 a^4+30 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}","-\frac{3 b^2 \sqrt{a^2-b^2} \left(-23 a^2 b^2+4 a^4+24 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^{10} d}+\frac{\left(-889 a^4 b^2+3255 a^2 b^4+10 a^6-2520 b^6\right) \cot (c+d x)}{70 a^9 d}-\frac{3 b \left(-100 a^4 b^2+280 a^2 b^4+5 a^6-192 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^{10} d}-\frac{3 \left(-185 a^2 b^2+35 a^4+168 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}+\frac{\left(-81 a^2 b^2+16 a^4+72 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{\left(-973 a^2 b^2+205 a^4+840 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{3 b \left(-116 a^2 b^2+27 a^4+96 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^8 d}+\frac{\left(-65 a^2 b^2+12 a^4+60 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left(-35 a^2 b^2+7 a^4+30 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}",1,"(-3*b^2*Sqrt[a^2 - b^2]*(4*a^4 - 23*a^2*b^2 + 24*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^10*d) - (3*b*(5*a^6 - 100*a^4*b^2 + 280*a^2*b^4 - 192*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^10*d) + ((10*a^6 - 889*a^4*b^2 + 3255*a^2*b^4 - 2520*b^6)*Cot[c + d*x])/(70*a^9*d) + (3*b*(27*a^4 - 116*a^2*b^2 + 96*b^4)*Cot[c + d*x]*Csc[c + d*x])/(16*a^8*d) - ((205*a^4 - 973*a^2*b^2 + 840*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(70*a^7*d) + ((16*a^4 - 81*a^2*b^2 + 72*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^6*b*d) - (3*(35*a^4 - 185*a^2*b^2 + 168*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(70*a^5*b^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(5*b*d*(a + b*Sin[c + d*x])^2) + (a*Cot[c + d*x]*Csc[c + d*x]^4)/(10*b^2*d*(a + b*Sin[c + d*x])^2) + ((7*a^4 - 35*a^2*b^2 + 30*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(35*a^3*b^2*d*(a + b*Sin[c + d*x])^2) + (3*b*Cot[c + d*x]*Csc[c + d*x]^5)/(14*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^6)/(7*a*d*(a + b*Sin[c + d*x])^2) + ((12*a^4 - 65*a^2*b^2 + 60*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(10*a^4*b^2*d*(a + b*Sin[c + d*x]))","A",13,7,29,0.2414,1,"{2896, 3055, 3001, 3770, 2660, 618, 204}"
1276,1,712,0,2.6446162,"\int \frac{\cos ^6(e+f x)}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx","Int[Cos[e + f*x]^6/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(13/2)),x]","\frac{16 b \left(-93 a^2 b^2+93 a^4+32 b^4\right) \cos (e+f x)}{693 a^5 f \left(a^2-b^2\right)^3 \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{8 \left(-22 a^4 b^2+65 a^2 b^4+5 a^6-32 b^6\right) \cos (e+f x) \sqrt{d \sin (e+f x)}}{693 a^5 b^2 d f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))^{3/2}}-\frac{4 \left(-17 a^2 b^2+5 a^4+16 b^4\right) \cos (e+f x) \sqrt{d \sin (e+f x)}}{231 a^4 b^2 d f \left(a^2-b^2\right) (a+b \sin (e+f x))^{5/2}}+\frac{80 \left(3 a^2+2 b^2\right) \cos (e+f x) \sqrt{d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac{20 \left(a^2-b^2\right) \cos (e+f x) \sqrt{d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}-\frac{16 \left(-69 a^2 b^2-48 a^3 b+45 a^4+24 a b^3+32 b^4\right) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^6 \sqrt{d} f (a-b)^2 (a+b)^{5/2}}-\frac{16 b \left(-93 a^2 b^2+93 a^4+32 b^4\right) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^7 \sqrt{d} f (a-b)^2 (a+b)^{5/2}}+\frac{2 \cos ^5(e+f x) \sqrt{d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}","\frac{16 b \left(-93 a^2 b^2+93 a^4+32 b^4\right) \cos (e+f x)}{693 a^5 f \left(a^2-b^2\right)^3 \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{8 \left(-22 a^4 b^2+65 a^2 b^4+5 a^6-32 b^6\right) \cos (e+f x) \sqrt{d \sin (e+f x)}}{693 a^5 b^2 d f \left(a^2-b^2\right)^2 (a+b \sin (e+f x))^{3/2}}-\frac{4 \left(-17 a^2 b^2+5 a^4+16 b^4\right) \cos (e+f x) \sqrt{d \sin (e+f x)}}{231 a^4 b^2 d f \left(a^2-b^2\right) (a+b \sin (e+f x))^{5/2}}+\frac{80 \left(3 a^2+2 b^2\right) \cos (e+f x) \sqrt{d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac{20 \left(a^2-b^2\right) \cos (e+f x) \sqrt{d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}-\frac{16 \left(-69 a^2 b^2-48 a^3 b+45 a^4+24 a b^3+32 b^4\right) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^6 \sqrt{d} f (a-b)^2 (a+b)^{5/2}}-\frac{16 b \left(-93 a^2 b^2+93 a^4+32 b^4\right) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^7 \sqrt{d} f (a-b)^2 (a+b)^{5/2}}+\frac{2 \cos ^5(e+f x) \sqrt{d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}",1,"(2*Cos[e + f*x]^5*Sqrt[d*Sin[e + f*x]])/(11*a*d*f*(a + b*Sin[e + f*x])^(11/2)) - (20*(a^2 - b^2)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(99*a^2*b^2*d*f*(a + b*Sin[e + f*x])^(9/2)) + (80*(3*a^2 + 2*b^2)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(693*a^3*b^2*d*f*(a + b*Sin[e + f*x])^(7/2)) - (4*(5*a^4 - 17*a^2*b^2 + 16*b^4)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(231*a^4*b^2*(a^2 - b^2)*d*f*(a + b*Sin[e + f*x])^(5/2)) - (8*(5*a^6 - 22*a^4*b^2 + 65*a^2*b^4 - 32*b^6)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(693*a^5*b^2*(a^2 - b^2)^2*d*f*(a + b*Sin[e + f*x])^(3/2)) + (16*b*(93*a^4 - 93*a^2*b^2 + 32*b^4)*Cos[e + f*x])/(693*a^5*(a^2 - b^2)^3*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (16*b*(93*a^4 - 93*a^2*b^2 + 32*b^4)*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(693*a^7*(a - b)^2*(a + b)^(5/2)*Sqrt[d]*f) - (16*(45*a^4 - 48*a^3*b - 69*a^2*b^2 + 24*a*b^3 + 32*b^4)*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(693*a^6*(a - b)^2*(a + b)^(5/2)*Sqrt[d]*f)","A",8,7,35,0.2000,1,"{2887, 2891, 3055, 2993, 2998, 2816, 2994}"
1277,1,161,0,0.4278432,"\int \frac{(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt{d \sin (e+f x)}} \, dx","Int[(a + b*Sin[e + f*x])^2/((g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]]),x]","-\frac{2 \left(2 a^2-b^2\right) \left(1-\csc ^2(e+f x)\right)^{3/4} (d \sin (e+f x))^{3/2} F\left(\left.\frac{1}{2} \csc ^{-1}(\sin (e+f x))\right|2\right)}{3 d^2 f g (g \cos (e+f x))^{3/2}}+\frac{2 \left(a^2+b^2\right) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac{4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}","\frac{\left(2 a^2-b^2\right) \sqrt{\sin (2 e+2 f x)} F\left(\left.\frac{1}{4} (4 e-\pi )+f x\right|2\right)}{3 f g^2 \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 \left(a^2+b^2\right) \sqrt{d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac{4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}",1,"(2*(a^2 + b^2)*Sqrt[d*Sin[e + f*x]])/(3*d*f*g*(g*Cos[e + f*x])^(3/2)) + (4*a*b*(d*Sin[e + f*x])^(3/2))/(3*d^2*f*g*(g*Cos[e + f*x])^(3/2)) - (2*(2*a^2 - b^2)*(1 - Csc[e + f*x]^2)^(3/4)*EllipticF[ArcCsc[Sin[e + f*x]]/2, 2]*(d*Sin[e + f*x])^(3/2))/(3*d^2*f*g*(g*Cos[e + f*x])^(3/2))","A",9,9,37,0.2432,1,"{2911, 2563, 3202, 457, 329, 237, 335, 275, 232}"
1278,1,193,0,0.473507,"\int \frac{(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt{d \sin (e+f x)}} \, dx","Int[(a + b*Sin[e + f*x])^2/((g*Cos[e + f*x])^(7/2)*Sqrt[d*Sin[e + f*x]]),x]","\frac{8 a^2 \sqrt{d \sin (e+f x)}}{5 d f g^3 \sqrt{g \cos (e+f x)}}+\frac{8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt{g \cos (e+f x)}}-\frac{8 a b E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 d f g^4 \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}","\frac{8 a^2 \sqrt{d \sin (e+f x)}}{5 d f g^3 \sqrt{g \cos (e+f x)}}+\frac{8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt{g \cos (e+f x)}}-\frac{8 a b E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 d f g^4 \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}",1,"(8*a^2*Sqrt[d*Sin[e + f*x]])/(5*d*f*g^3*Sqrt[g*Cos[e + f*x]]) + (8*a*b*(d*Sin[e + f*x])^(3/2))/(5*d^2*f*g^3*Sqrt[g*Cos[e + f*x]]) + (2*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^2)/(5*d*f*g*(g*Cos[e + f*x])^(5/2)) - (8*a*b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*d*f*g^4*Sqrt[Sin[2*e + 2*f*x]])","A",6,6,37,0.1622,1,"{2888, 2838, 2563, 2571, 2572, 2639}"
1279,1,76,0,0.0922599,"\int \frac{\cos (c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{a^2 \sin (c+d x)}{b^3 d}-\frac{a^3 \log (a+b \sin (c+d x))}{b^4 d}-\frac{a \sin ^2(c+d x)}{2 b^2 d}+\frac{\sin ^3(c+d x)}{3 b d}","\frac{a^2 \sin (c+d x)}{b^3 d}-\frac{a^3 \log (a+b \sin (c+d x))}{b^4 d}-\frac{a \sin ^2(c+d x)}{2 b^2 d}+\frac{\sin ^3(c+d x)}{3 b d}",1,"-((a^3*Log[a + b*Sin[c + d*x]])/(b^4*d)) + (a^2*Sin[c + d*x])/(b^3*d) - (a*Sin[c + d*x]^2)/(2*b^2*d) + Sin[c + d*x]^3/(3*b*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
1280,1,55,0,0.0804525,"\int \frac{\cos (c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{a^2 \log (a+b \sin (c+d x))}{b^3 d}-\frac{a \sin (c+d x)}{b^2 d}+\frac{\sin ^2(c+d x)}{2 b d}","\frac{a^2 \log (a+b \sin (c+d x))}{b^3 d}-\frac{a \sin (c+d x)}{b^2 d}+\frac{\sin ^2(c+d x)}{2 b d}",1,"(a^2*Log[a + b*Sin[c + d*x]])/(b^3*d) - (a*Sin[c + d*x])/(b^2*d) + Sin[c + d*x]^2/(2*b*d)","A",4,3,27,0.1111,1,"{2833, 12, 43}"
1281,1,34,0,0.050181,"\int \frac{\cos (c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\sin (c+d x)}{b d}-\frac{a \log (a+b \sin (c+d x))}{b^2 d}","\frac{\sin (c+d x)}{b d}-\frac{a \log (a+b \sin (c+d x))}{b^2 d}",1,"-((a*Log[a + b*Sin[c + d*x]])/(b^2*d)) + Sin[c + d*x]/(b*d)","A",4,3,25,0.1200,1,"{2833, 12, 43}"
1282,1,34,0,0.0414976,"\int \frac{\cot (c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{\log (\sin (c+d x))}{a d}-\frac{\log (a+b \sin (c+d x))}{a d}","\frac{\log (\sin (c+d x))}{a d}-\frac{\log (a+b \sin (c+d x))}{a d}",1,"Log[Sin[c + d*x]]/(a*d) - Log[a + b*Sin[c + d*x]]/(a*d)","A",4,4,19,0.2105,1,"{2721, 36, 29, 31}"
1283,1,50,0,0.0724541,"\int \frac{\cot (c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{b \log (a+b \sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d}","-\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{b \log (a+b \sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d}",1,"-(Csc[c + d*x]/(a*d)) - (b*Log[Sin[c + d*x]])/(a^2*d) + (b*Log[a + b*Sin[c + d*x]])/(a^2*d)","A",4,3,25,0.1200,1,"{2833, 12, 44}"
1284,1,72,0,0.0913019,"\int \frac{\cot (c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{b^2 \log (\sin (c+d x))}{a^3 d}-\frac{b^2 \log (a+b \sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}","\frac{b^2 \log (\sin (c+d x))}{a^3 d}-\frac{b^2 \log (a+b \sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}",1,"(b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) + (b^2*Log[Sin[c + d*x]])/(a^3*d) - (b^2*Log[a + b*Sin[c + d*x]])/(a^3*d)","A",4,3,27,0.1111,1,"{2833, 12, 44}"
1285,1,235,0,0.9126404,"\int \frac{\cos ^2(c+d x) \sin ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","\frac{\left(-5 a^2 b^2+15 a^4-2 b^4\right) \cos (c+d x)}{15 b^5 d}-\frac{2 a^4 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d}+\frac{\left(5 a^2-b^2\right) \sin ^2(c+d x) \cos (c+d x)}{15 b^3 d}-\frac{a \left(4 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 d}+\frac{a x \left(-4 a^2 b^2+8 a^4-b^4\right)}{8 b^6}-\frac{a \sin ^3(c+d x) \cos (c+d x)}{4 b^2 d}+\frac{\sin ^4(c+d x) \cos (c+d x)}{5 b d}","\frac{\left(-5 a^2 b^2+15 a^4-2 b^4\right) \cos (c+d x)}{15 b^5 d}-\frac{2 a^4 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d}+\frac{\left(5 a^2-b^2\right) \sin ^2(c+d x) \cos (c+d x)}{15 b^3 d}-\frac{a \left(4 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 d}+\frac{a x \left(-4 a^2 b^2+8 a^4-b^4\right)}{8 b^6}-\frac{a \sin ^3(c+d x) \cos (c+d x)}{4 b^2 d}+\frac{\sin ^4(c+d x) \cos (c+d x)}{5 b d}",1,"(a*(8*a^4 - 4*a^2*b^2 - b^4)*x)/(8*b^6) - (2*a^4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^6*d) + ((15*a^4 - 5*a^2*b^2 - 2*b^4)*Cos[c + d*x])/(15*b^5*d) - (a*(4*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*d) + ((5*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(15*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^3)/(4*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(5*b*d)","A",10,8,29,0.2759,1,"{2889, 3050, 3049, 3023, 2735, 2660, 618, 204}"
1286,1,191,0,0.6636442,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","-\frac{a \left(3 a^2-b^2\right) \cos (c+d x)}{3 b^4 d}+\frac{2 a^3 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d}+\frac{\left(4 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}-\frac{x \left(-4 a^2 b^2+8 a^4-b^4\right)}{8 b^5}-\frac{a \sin ^2(c+d x) \cos (c+d x)}{3 b^2 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d}","-\frac{a \left(3 a^2-b^2\right) \cos (c+d x)}{3 b^4 d}+\frac{2 a^3 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d}+\frac{\left(4 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}-\frac{x \left(-4 a^2 b^2+8 a^4-b^4\right)}{8 b^5}-\frac{a \sin ^2(c+d x) \cos (c+d x)}{3 b^2 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d}",1,"-((8*a^4 - 4*a^2*b^2 - b^4)*x)/(8*b^5) + (2*a^3*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^5*d) - (a*(3*a^2 - b^2)*Cos[c + d*x])/(3*b^4*d) + ((4*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^2)/(3*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(4*b*d)","A",9,8,29,0.2759,1,"{2889, 3050, 3049, 3023, 2735, 2660, 618, 204}"
1287,1,148,0,0.4587288,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\left(3 a^2-b^2\right) \cos (c+d x)}{3 b^3 d}-\frac{2 a^2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d}+\frac{a x \left(2 a^2-b^2\right)}{2 b^4}-\frac{a \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{\sin ^2(c+d x) \cos (c+d x)}{3 b d}","\frac{\left(3 a^2-b^2\right) \cos (c+d x)}{3 b^3 d}-\frac{2 a^2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d}+\frac{a x \left(2 a^2-b^2\right)}{2 b^4}-\frac{a \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{\sin ^2(c+d x) \cos (c+d x)}{3 b d}",1,"(a*(2*a^2 - b^2)*x)/(2*b^4) - (2*a^2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*d) + ((3*a^2 - b^2)*Cos[c + d*x])/(3*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^2)/(3*b*d)","A",8,8,29,0.2759,1,"{2889, 3050, 3049, 3023, 2735, 2660, 618, 204}"
1288,1,100,0,0.1650189,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{2 a \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 d}-\frac{x \left(2 a^2-b^2\right)}{2 b^3}-\frac{\cos (c+d x) (2 a-b \sin (c+d x))}{2 b^2 d}","\frac{2 a \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 d}-\frac{x \left(2 a^2-b^2\right)}{2 b^3}-\frac{\cos (c+d x) (2 a-b \sin (c+d x))}{2 b^2 d}",1,"-((2*a^2 - b^2)*x)/(2*b^3) + (2*a*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^3*d) - (Cos[c + d*x]*(2*a - b*Sin[c + d*x]))/(2*b^2*d)","A",5,5,27,0.1852,1,"{2865, 2735, 2660, 618, 204}"
1289,1,75,0,0.1853805,"\int \frac{\cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{b}","\frac{2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{b}",1,"-(x/b) + (2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*b*d) - ArcTanh[Cos[c + d*x]]/(a*d)","A",6,6,25,0.2400,1,"{2889, 3058, 2660, 618, 204, 3770}"
1290,1,80,0,0.2528146,"\int \frac{\cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^2/(a + b*Sin[c + d*x]),x]","-\frac{2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}","-\frac{2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}",1,"(-2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a*d)","A",7,7,21,0.3333,1,"{2723, 3056, 3001, 3770, 2660, 618, 204}"
1291,1,114,0,0.456389,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{2 b \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d}+\frac{\left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{b \cot (c+d x)}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}","\frac{2 b \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d}+\frac{\left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{b \cot (c+d x)}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}",1,"(2*b*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*d) + ((a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (b*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)","A",8,8,27,0.2963,1,"{2889, 3056, 3055, 3001, 3770, 2660, 618, 204}"
1292,1,153,0,0.6739689,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{2 b^2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d}+\frac{\left(a^2-3 b^2\right) \cot (c+d x)}{3 a^3 d}-\frac{b \left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{b \cot (c+d x) \csc (c+d x)}{2 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d}","-\frac{2 b^2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d}+\frac{\left(a^2-3 b^2\right) \cot (c+d x)}{3 a^3 d}-\frac{b \left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{b \cot (c+d x) \csc (c+d x)}{2 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d}",1,"(-2*b^2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*d) - (b*(a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^4*d) + ((a^2 - 3*b^2)*Cot[c + d*x])/(3*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d)","A",9,8,29,0.2759,1,"{2889, 3056, 3055, 3001, 3770, 2660, 618, 204}"
1293,1,194,0,0.9532554,"\int \frac{\cot ^2(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{2 b^3 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d}-\frac{b \left(a^2-3 b^2\right) \cot (c+d x)}{3 a^4 d}+\frac{\left(4 a^2 b^2+a^4-8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}+\frac{\left(a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}","\frac{2 b^3 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d}-\frac{b \left(a^2-3 b^2\right) \cot (c+d x)}{3 a^4 d}+\frac{\left(4 a^2 b^2+a^4-8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}+\frac{\left(a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}",1,"(2*b^3*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*d) + ((a^4 + 4*a^2*b^2 - 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^5*d) - (b*(a^2 - 3*b^2)*Cot[c + d*x])/(3*a^4*d) + ((a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)","A",10,8,29,0.2759,1,"{2889, 3056, 3055, 3001, 3770, 2660, 618, 204}"
1294,1,238,0,1.2268252,"\int \frac{\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^2*Csc[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","-\frac{2 b^4 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}+\frac{\left(5 a^2 b^2+2 a^4-15 b^4\right) \cot (c+d x)}{15 a^5 d}-\frac{b \left(4 a^2 b^2+a^4-8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}+\frac{\left(a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}-\frac{b \left(a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}","-\frac{2 b^4 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}+\frac{\left(5 a^2 b^2+2 a^4-15 b^4\right) \cot (c+d x)}{15 a^5 d}-\frac{b \left(4 a^2 b^2+a^4-8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}+\frac{\left(a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}-\frac{b \left(a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}",1,"(-2*b^4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*d) - (b*(a^4 + 4*a^2*b^2 - 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) + ((2*a^4 + 5*a^2*b^2 - 15*b^4)*Cot[c + d*x])/(15*a^5*d) - (b*(a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) + ((a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)","A",11,8,29,0.2759,1,"{2889, 3056, 3055, 3001, 3770, 2660, 618, 204}"
1295,1,149,0,0.1988365,"\int \frac{\cos ^3(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","-\frac{\left(a^2-b^2\right) \sin ^3(c+d x)}{3 b^3 d}+\frac{a \left(a^2-b^2\right) \sin ^2(c+d x)}{2 b^4 d}-\frac{a^2 \left(a^2-b^2\right) \sin (c+d x)}{b^5 d}+\frac{a^3 \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^6 d}+\frac{a \sin ^4(c+d x)}{4 b^2 d}-\frac{\sin ^5(c+d x)}{5 b d}","-\frac{\left(a^2-b^2\right) \sin ^3(c+d x)}{3 b^3 d}+\frac{a \left(a^2-b^2\right) \sin ^2(c+d x)}{2 b^4 d}-\frac{a^2 \left(a^2-b^2\right) \sin (c+d x)}{b^5 d}+\frac{a^3 \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^6 d}+\frac{a \sin ^4(c+d x)}{4 b^2 d}-\frac{\sin ^5(c+d x)}{5 b d}",1,"(a^3*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^6*d) - (a^2*(a^2 - b^2)*Sin[c + d*x])/(b^5*d) + (a*(a^2 - b^2)*Sin[c + d*x]^2)/(2*b^4*d) - ((a^2 - b^2)*Sin[c + d*x]^3)/(3*b^3*d) + (a*Sin[c + d*x]^4)/(4*b^2*d) - Sin[c + d*x]^5/(5*b*d)","A",4,3,29,0.1034,1,"{2837, 12, 894}"
1296,1,119,0,0.1721538,"\int \frac{\cos ^3(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{\left(a^2-b^2\right) \sin ^2(c+d x)}{2 b^3 d}+\frac{a \left(a^2-b^2\right) \sin (c+d x)}{b^4 d}-\frac{a^2 \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^5 d}+\frac{a \sin ^3(c+d x)}{3 b^2 d}-\frac{\sin ^4(c+d x)}{4 b d}","-\frac{\left(a^2-b^2\right) \sin ^2(c+d x)}{2 b^3 d}+\frac{a \left(a^2-b^2\right) \sin (c+d x)}{b^4 d}-\frac{a^2 \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^5 d}+\frac{a \sin ^3(c+d x)}{3 b^2 d}-\frac{\sin ^4(c+d x)}{4 b d}",1,"-((a^2*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^5*d)) + (a*(a^2 - b^2)*Sin[c + d*x])/(b^4*d) - ((a^2 - b^2)*Sin[c + d*x]^2)/(2*b^3*d) + (a*Sin[c + d*x]^3)/(3*b^2*d) - Sin[c + d*x]^4/(4*b*d)","A",4,3,29,0.1034,1,"{2837, 12, 894}"
1297,1,89,0,0.1140422,"\int \frac{\cos ^3(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{\left(a^2-b^2\right) \sin (c+d x)}{b^3 d}+\frac{a \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^4 d}+\frac{a \sin ^2(c+d x)}{2 b^2 d}-\frac{\sin ^3(c+d x)}{3 b d}","-\frac{\left(a^2-b^2\right) \sin (c+d x)}{b^3 d}+\frac{a \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^4 d}+\frac{a \sin ^2(c+d x)}{2 b^2 d}-\frac{\sin ^3(c+d x)}{3 b d}",1,"(a*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^4*d) - ((a^2 - b^2)*Sin[c + d*x])/(b^3*d) + (a*Sin[c + d*x]^2)/(2*b^2*d) - Sin[c + d*x]^3/(3*b*d)","A",4,3,27,0.1111,1,"{2837, 12, 772}"
1298,1,59,0,0.117325,"\int \frac{\cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\left(a^2-b^2\right) \log (a+b \sin (c+d x))}{a b^2 d}+\frac{\log (\sin (c+d x))}{a d}-\frac{\sin (c+d x)}{b d}","\frac{\left(a^2-b^2\right) \log (a+b \sin (c+d x))}{a b^2 d}+\frac{\log (\sin (c+d x))}{a d}-\frac{\sin (c+d x)}{b d}",1,"Log[Sin[c + d*x]]/(a*d) + ((a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(a*b^2*d) - Sin[c + d*x]/(b*d)","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1299,1,60,0,0.1223802,"\int \frac{\cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{\left(1-\frac{b^2}{a^2}\right) \log (a+b \sin (c+d x))}{b d}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d}","-\frac{\left(1-\frac{b^2}{a^2}\right) \log (a+b \sin (c+d x))}{b d}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{\csc (c+d x)}{a d}",1,"-(Csc[c + d*x]/(a*d)) - (b*Log[Sin[c + d*x]])/(a^2*d) - ((1 - b^2/a^2)*Log[a + b*Sin[c + d*x]])/(b*d)","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1300,1,84,0,0.0918886,"\int \frac{\cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^3/(a + b*Sin[c + d*x]),x]","-\frac{\left(a^2-b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{\left(a^2-b^2\right) \log (a+b \sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}","-\frac{\left(a^2-b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{\left(a^2-b^2\right) \log (a+b \sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}",1,"(b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - ((a^2 - b^2)*Log[Sin[c + d*x]])/(a^3*d) + ((a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(a^3*d)","A",3,2,21,0.09524,1,"{2721, 894}"
1301,1,282,0,1.0012073,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{a \left(-20 a^2 b^2+15 a^4+3 b^4\right) \cos (c+d x)}{15 b^6 d}-\frac{2 a^3 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d}-\frac{\left(6 a^2-7 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{24 b^3 d}+\frac{a \left(5 a^2-6 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{15 b^4 d}-\frac{\left(-10 a^2 b^2+8 a^4+b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^5 d}+\frac{x \left(-24 a^4 b^2+6 a^2 b^4+16 a^6+b^6\right)}{16 b^7}+\frac{a \sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x)}{6 b d}","\frac{a \left(-20 a^2 b^2+15 a^4+3 b^4\right) \cos (c+d x)}{15 b^6 d}-\frac{2 a^3 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d}-\frac{\left(6 a^2-7 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{24 b^3 d}+\frac{a \left(5 a^2-6 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{15 b^4 d}-\frac{\left(-10 a^2 b^2+8 a^4+b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^5 d}+\frac{x \left(-24 a^4 b^2+6 a^2 b^4+16 a^6+b^6\right)}{16 b^7}+\frac{a \sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x)}{6 b d}",1,"((16*a^6 - 24*a^4*b^2 + 6*a^2*b^4 + b^6)*x)/(16*b^7) - (2*a^3*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^7*d) + (a*(15*a^4 - 20*a^2*b^2 + 3*b^4)*Cos[c + d*x])/(15*b^6*d) - ((8*a^4 - 10*a^2*b^2 + b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^5*d) + (a*(5*a^2 - 6*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(15*b^4*d) - ((6*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*b^3*d) + (a*Cos[c + d*x]*Sin[c + d*x]^4)/(5*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^5)/(6*b*d)","A",9,7,29,0.2414,1,"{2895, 3049, 3023, 2735, 2660, 618, 204}"
1302,1,235,0,0.7166394,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{\left(-20 a^2 b^2+15 a^4+3 b^4\right) \cos (c+d x)}{15 b^5 d}+\frac{2 a^2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d}-\frac{\left(5 a^2-6 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{15 b^3 d}+\frac{a \left(4 a^2-5 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 d}-\frac{a x \left(-12 a^2 b^2+8 a^4+3 b^4\right)}{8 b^6}+\frac{a \sin ^3(c+d x) \cos (c+d x)}{4 b^2 d}-\frac{\sin ^4(c+d x) \cos (c+d x)}{5 b d}","-\frac{\left(-20 a^2 b^2+15 a^4+3 b^4\right) \cos (c+d x)}{15 b^5 d}+\frac{2 a^2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d}-\frac{\left(5 a^2-6 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{15 b^3 d}+\frac{a \left(4 a^2-5 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 d}-\frac{a x \left(-12 a^2 b^2+8 a^4+3 b^4\right)}{8 b^6}+\frac{a \sin ^3(c+d x) \cos (c+d x)}{4 b^2 d}-\frac{\sin ^4(c+d x) \cos (c+d x)}{5 b d}",1,"-(a*(8*a^4 - 12*a^2*b^2 + 3*b^4)*x)/(8*b^6) + (2*a^2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^6*d) - ((15*a^4 - 20*a^2*b^2 + 3*b^4)*Cos[c + d*x])/(15*b^5*d) + (a*(4*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*d) - ((5*a^2 - 6*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(15*b^3*d) + (a*Cos[c + d*x]*Sin[c + d*x]^3)/(4*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^4)/(5*b*d)","A",8,7,29,0.2414,1,"{2895, 3049, 3023, 2735, 2660, 618, 204}"
1303,1,159,0,0.3247008,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{2 a \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d}+\frac{\cos (c+d x) \left(8 a \left(a^2-b^2\right)-b \left(4 a^2-3 b^2\right) \sin (c+d x)\right)}{8 b^4 d}+\frac{x \left(-12 a^2 b^2+8 a^4+3 b^4\right)}{8 b^5}-\frac{\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}","-\frac{2 a \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^5 d}+\frac{\cos (c+d x) \left(8 a \left(a^2-b^2\right)-b \left(4 a^2-3 b^2\right) \sin (c+d x)\right)}{8 b^4 d}+\frac{x \left(-12 a^2 b^2+8 a^4+3 b^4\right)}{8 b^5}-\frac{\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}",1,"((8*a^4 - 12*a^2*b^2 + 3*b^4)*x)/(8*b^5) - (2*a*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^5*d) - (Cos[c + d*x]^3*(4*a - 3*b*Sin[c + d*x]))/(12*b^2*d) + (Cos[c + d*x]*(8*a*(a^2 - b^2) - b*(4*a^2 - 3*b^2)*Sin[c + d*x]))/(8*b^4*d)","A",6,5,27,0.1852,1,"{2865, 2735, 2660, 618, 204}"
1304,1,124,0,0.2880462,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b^3 d}+\frac{x \left(2 a^2-3 b^2\right)}{2 b^3}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}","-\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b^3 d}+\frac{x \left(2 a^2-3 b^2\right)}{2 b^3}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}",1,"((2*a^2 - 3*b^2)*x)/(2*b^3) - (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*b^3*d) - ArcTanh[Cos[c + d*x]]/(a*d) + (a*Cos[c + d*x])/(b^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",6,6,27,0.2222,1,"{2895, 3057, 2660, 618, 204, 3770}"
1305,1,104,0,0.2695783,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^2 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{a x}{b^2}-\frac{\cot (c+d x)}{a d}-\frac{\cos (c+d x)}{b d}","\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^2 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{a x}{b^2}-\frac{\cot (c+d x)}{a d}-\frac{\cos (c+d x)}{b d}",1,"-((a*x)/b^2) + (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^2*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cos[c + d*x]/(b*d) - Cot[c + d*x]/(a*d)","A",6,6,29,0.2069,1,"{2894, 3057, 2660, 618, 204, 3770}"
1306,1,123,0,0.3013928,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","-\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b d}+\frac{\left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{b \cot (c+d x)}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}+\frac{x}{b}","-\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b d}+\frac{\left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{b \cot (c+d x)}{a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}+\frac{x}{b}",1,"x/b - (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b*d) + ((3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (b*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)","A",6,6,27,0.2222,1,"{2893, 3057, 2660, 618, 204, 3770}"
1307,1,154,0,0.4455492,"\int \frac{\cot ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d}+\frac{\left(4 a^2-3 b^2\right) \cot (c+d x)}{3 a^3 d}-\frac{b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{b \cot (c+d x) \csc (c+d x)}{2 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d}","\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d}+\frac{\left(4 a^2-3 b^2\right) \cot (c+d x)}{3 a^3 d}-\frac{b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{b \cot (c+d x) \csc (c+d x)}{2 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d}",1,"(2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*d) - (b*(3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^4*d) + ((4*a^2 - 3*b^2)*Cot[c + d*x])/(3*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d)","A",7,7,21,0.3333,1,"{2725, 3055, 3001, 3770, 2660, 618, 204}"
1308,1,198,0,0.7606781,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{2 b \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d}-\frac{b \left(4 a^2-3 b^2\right) \cot (c+d x)}{3 a^4 d}-\frac{\left(-12 a^2 b^2+3 a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}+\frac{\left(5 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}","-\frac{2 b \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d}-\frac{b \left(4 a^2-3 b^2\right) \cot (c+d x)}{3 a^4 d}-\frac{\left(-12 a^2 b^2+3 a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}+\frac{\left(5 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac{b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}",1,"(-2*b*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*d) - ((3*a^4 - 12*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^5*d) - (b*(4*a^2 - 3*b^2)*Cot[c + d*x])/(3*a^4*d) + ((5*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)","A",8,7,27,0.2593,1,"{2893, 3055, 3001, 3770, 2660, 618, 204}"
1309,1,244,0,1.049887,"\int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^4*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{2 b^2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}-\frac{\left(-20 a^2 b^2+3 a^4+15 b^4\right) \cot (c+d x)}{15 a^5 d}+\frac{b \left(-12 a^2 b^2+3 a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}+\frac{\left(6 a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}-\frac{b \left(5 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}","\frac{2 b^2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}-\frac{\left(-20 a^2 b^2+3 a^4+15 b^4\right) \cot (c+d x)}{15 a^5 d}+\frac{b \left(-12 a^2 b^2+3 a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}+\frac{\left(6 a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}-\frac{b \left(5 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}",1,"(2*b^2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*d) + (b*(3*a^4 - 12*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - ((3*a^4 - 20*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^5*d) - (b*(5*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) + ((6*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)","A",9,7,29,0.2414,1,"{2893, 3055, 3001, 3770, 2660, 618, 204}"
1310,1,212,0,0.2379863,"\int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{\left(a^2-2 b^2\right) \sin ^5(c+d x)}{5 b^3 d}-\frac{a \left(a^2-2 b^2\right) \sin ^4(c+d x)}{4 b^4 d}+\frac{\left(a^2-b^2\right)^2 \sin ^3(c+d x)}{3 b^5 d}-\frac{a \left(a^2-b^2\right)^2 \sin ^2(c+d x)}{2 b^6 d}+\frac{a^2 \left(a^2-b^2\right)^2 \sin (c+d x)}{b^7 d}-\frac{a^3 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^8 d}-\frac{a \sin ^6(c+d x)}{6 b^2 d}+\frac{\sin ^7(c+d x)}{7 b d}","\frac{\left(a^2-2 b^2\right) \sin ^5(c+d x)}{5 b^3 d}-\frac{a \left(a^2-2 b^2\right) \sin ^4(c+d x)}{4 b^4 d}+\frac{\left(a^2-b^2\right)^2 \sin ^3(c+d x)}{3 b^5 d}-\frac{a \left(a^2-b^2\right)^2 \sin ^2(c+d x)}{2 b^6 d}+\frac{a^2 \left(a^2-b^2\right)^2 \sin (c+d x)}{b^7 d}-\frac{a^3 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^8 d}-\frac{a \sin ^6(c+d x)}{6 b^2 d}+\frac{\sin ^7(c+d x)}{7 b d}",1,"-((a^3*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^8*d)) + (a^2*(a^2 - b^2)^2*Sin[c + d*x])/(b^7*d) - (a*(a^2 - b^2)^2*Sin[c + d*x]^2)/(2*b^6*d) + ((a^2 - b^2)^2*Sin[c + d*x]^3)/(3*b^5*d) - (a*(a^2 - 2*b^2)*Sin[c + d*x]^4)/(4*b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^5)/(5*b^3*d) - (a*Sin[c + d*x]^6)/(6*b^2*d) + Sin[c + d*x]^7/(7*b*d)","A",4,3,29,0.1034,1,"{2837, 12, 948}"
1311,1,180,0,0.2074086,"\int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\left(a^2-2 b^2\right) \sin ^4(c+d x)}{4 b^3 d}-\frac{a \left(a^2-2 b^2\right) \sin ^3(c+d x)}{3 b^4 d}+\frac{\left(a^2-b^2\right)^2 \sin ^2(c+d x)}{2 b^5 d}-\frac{a \left(a^2-b^2\right)^2 \sin (c+d x)}{b^6 d}+\frac{a^2 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^7 d}-\frac{a \sin ^5(c+d x)}{5 b^2 d}+\frac{\sin ^6(c+d x)}{6 b d}","\frac{\left(a^2-2 b^2\right) \sin ^4(c+d x)}{4 b^3 d}-\frac{a \left(a^2-2 b^2\right) \sin ^3(c+d x)}{3 b^4 d}+\frac{\left(a^2-b^2\right)^2 \sin ^2(c+d x)}{2 b^5 d}-\frac{a \left(a^2-b^2\right)^2 \sin (c+d x)}{b^6 d}+\frac{a^2 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^7 d}-\frac{a \sin ^5(c+d x)}{5 b^2 d}+\frac{\sin ^6(c+d x)}{6 b d}",1,"(a^2*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^7*d) - (a*(a^2 - b^2)^2*Sin[c + d*x])/(b^6*d) + ((a^2 - b^2)^2*Sin[c + d*x]^2)/(2*b^5*d) - (a*(a^2 - 2*b^2)*Sin[c + d*x]^3)/(3*b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^4)/(4*b^3*d) - (a*Sin[c + d*x]^5)/(5*b^2*d) + Sin[c + d*x]^6/(6*b*d)","A",4,3,29,0.1034,1,"{2837, 12, 948}"
1312,1,148,0,0.1407236,"\int \frac{\cos ^5(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\left(a^2-2 b^2\right) \sin ^3(c+d x)}{3 b^3 d}-\frac{a \left(a^2-2 b^2\right) \sin ^2(c+d x)}{2 b^4 d}+\frac{\left(a^2-b^2\right)^2 \sin (c+d x)}{b^5 d}-\frac{a \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^6 d}-\frac{a \sin ^4(c+d x)}{4 b^2 d}+\frac{\sin ^5(c+d x)}{5 b d}","\frac{\left(a^2-2 b^2\right) \sin ^3(c+d x)}{3 b^3 d}-\frac{a \left(a^2-2 b^2\right) \sin ^2(c+d x)}{2 b^4 d}+\frac{\left(a^2-b^2\right)^2 \sin (c+d x)}{b^5 d}-\frac{a \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^6 d}-\frac{a \sin ^4(c+d x)}{4 b^2 d}+\frac{\sin ^5(c+d x)}{5 b d}",1,"-((a*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^6*d)) + ((a^2 - b^2)^2*Sin[c + d*x])/(b^5*d) - (a*(a^2 - 2*b^2)*Sin[c + d*x]^2)/(2*b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^3)/(3*b^3*d) - (a*Sin[c + d*x]^4)/(4*b^2*d) + Sin[c + d*x]^5/(5*b*d)","A",4,3,27,0.1111,1,"{2837, 12, 772}"
1313,1,107,0,0.1403104,"\int \frac{\cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{b^3 d}-\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a b^4 d}-\frac{a \sin ^2(c+d x)}{2 b^2 d}+\frac{\log (\sin (c+d x))}{a d}+\frac{\sin ^3(c+d x)}{3 b d}","\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{b^3 d}-\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a b^4 d}-\frac{a \sin ^2(c+d x)}{2 b^2 d}+\frac{\log (\sin (c+d x))}{a d}+\frac{\sin ^3(c+d x)}{3 b d}",1,"Log[Sin[c + d*x]]/(a*d) - ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a*b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x])/(b^3*d) - (a*Sin[c + d*x]^2)/(2*b^2*d) + Sin[c + d*x]^3/(3*b*d)","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1314,1,96,0,0.1551761,"\int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^2 b^3 d}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{a \sin (c+d x)}{b^2 d}-\frac{\csc (c+d x)}{a d}+\frac{\sin ^2(c+d x)}{2 b d}","\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^2 b^3 d}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{a \sin (c+d x)}{b^2 d}-\frac{\csc (c+d x)}{a d}+\frac{\sin ^2(c+d x)}{2 b d}",1,"-(Csc[c + d*x]/(a*d)) - (b*Log[Sin[c + d*x]])/(a^2*d) + ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^2*b^3*d) - (a*Sin[c + d*x])/(b^2*d) + Sin[c + d*x]^2/(2*b*d)","A",4,3,29,0.1034,1,"{2837, 12, 894}"
1315,1,105,0,0.1798452,"\int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","-\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^3 b^2 d}-\frac{\left(2 a^2-b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\sin (c+d x)}{b d}","-\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^3 b^2 d}-\frac{\left(2 a^2-b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a d}+\frac{\sin (c+d x)}{b d}",1,"(b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - ((2*a^2 - b^2)*Log[Sin[c + d*x]])/(a^3*d) - ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^3*b^2*d) + Sin[c + d*x]/(b*d)","A",4,3,29,0.1034,1,"{2837, 12, 894}"
1316,1,120,0,0.1743879,"\int \frac{\cos (c+d x) \cot ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","\frac{\left(2 a^2-b^2\right) \csc (c+d x)}{a^3 d}+\frac{b \left(2 a^2-b^2\right) \log (\sin (c+d x))}{a^4 d}+\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^4 b d}+\frac{b \csc ^2(c+d x)}{2 a^2 d}-\frac{\csc ^3(c+d x)}{3 a d}","\frac{\left(2 a^2-b^2\right) \csc (c+d x)}{a^3 d}+\frac{b \left(2 a^2-b^2\right) \log (\sin (c+d x))}{a^4 d}+\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^4 b d}+\frac{b \csc ^2(c+d x)}{2 a^2 d}-\frac{\csc ^3(c+d x)}{3 a d}",1,"((2*a^2 - b^2)*Csc[c + d*x])/(a^3*d) + (b*Csc[c + d*x]^2)/(2*a^2*d) - Csc[c + d*x]^3/(3*a*d) + (b*(2*a^2 - b^2)*Log[Sin[c + d*x]])/(a^4*d) + ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^4*b*d)","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1317,1,148,0,0.1333274,"\int \frac{\cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^5/(a + b*Sin[c + d*x]),x]","\frac{\left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 a^3 d}-\frac{b \left(2 a^2-b^2\right) \csc (c+d x)}{a^4 d}+\frac{\left(a^2-b^2\right)^2 \log (\sin (c+d x))}{a^5 d}-\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^5 d}+\frac{b \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^4(c+d x)}{4 a d}","\frac{\left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 a^3 d}-\frac{b \left(2 a^2-b^2\right) \csc (c+d x)}{a^4 d}+\frac{\left(a^2-b^2\right)^2 \log (\sin (c+d x))}{a^5 d}-\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^5 d}+\frac{b \csc ^3(c+d x)}{3 a^2 d}-\frac{\csc ^4(c+d x)}{4 a d}",1,"-((b*(2*a^2 - b^2)*Csc[c + d*x])/(a^4*d)) + ((2*a^2 - b^2)*Csc[c + d*x]^2)/(2*a^3*d) + (b*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(4*a*d) + ((a^2 - b^2)^2*Log[Sin[c + d*x]])/(a^5*d) - ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^5*d)","A",3,2,21,0.09524,1,"{2721, 894}"
1318,1,179,0,0.2046359,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^5*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\left(2 a^2-b^2\right) \csc ^3(c+d x)}{3 a^3 d}-\frac{b \left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 a^4 d}-\frac{\left(a^2-b^2\right)^2 \csc (c+d x)}{a^5 d}-\frac{b \left(a^2-b^2\right)^2 \log (\sin (c+d x))}{a^6 d}+\frac{b \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^6 d}+\frac{b \csc ^4(c+d x)}{4 a^2 d}-\frac{\csc ^5(c+d x)}{5 a d}","\frac{\left(2 a^2-b^2\right) \csc ^3(c+d x)}{3 a^3 d}-\frac{b \left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 a^4 d}-\frac{\left(a^2-b^2\right)^2 \csc (c+d x)}{a^5 d}-\frac{b \left(a^2-b^2\right)^2 \log (\sin (c+d x))}{a^6 d}+\frac{b \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^6 d}+\frac{b \csc ^4(c+d x)}{4 a^2 d}-\frac{\csc ^5(c+d x)}{5 a d}",1,"-(((a^2 - b^2)^2*Csc[c + d*x])/(a^5*d)) - (b*(2*a^2 - b^2)*Csc[c + d*x]^2)/(2*a^4*d) + ((2*a^2 - b^2)*Csc[c + d*x]^3)/(3*a^3*d) + (b*Csc[c + d*x]^4)/(4*a^2*d) - Csc[c + d*x]^5/(5*a*d) - (b*(a^2 - b^2)^2*Log[Sin[c + d*x]])/(a^6*d) + (b*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^6*d)","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1319,1,212,0,0.2408169,"\int \frac{\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^5*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\left(2 a^2-b^2\right) \csc ^4(c+d x)}{4 a^3 d}-\frac{b \left(2 a^2-b^2\right) \csc ^3(c+d x)}{3 a^4 d}-\frac{\left(a^2-b^2\right)^2 \csc ^2(c+d x)}{2 a^5 d}+\frac{b \left(a^2-b^2\right)^2 \csc (c+d x)}{a^6 d}+\frac{b^2 \left(a^2-b^2\right)^2 \log (\sin (c+d x))}{a^7 d}-\frac{b^2 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^7 d}+\frac{b \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^6(c+d x)}{6 a d}","\frac{\left(2 a^2-b^2\right) \csc ^4(c+d x)}{4 a^3 d}-\frac{b \left(2 a^2-b^2\right) \csc ^3(c+d x)}{3 a^4 d}-\frac{\left(a^2-b^2\right)^2 \csc ^2(c+d x)}{2 a^5 d}+\frac{b \left(a^2-b^2\right)^2 \csc (c+d x)}{a^6 d}+\frac{b^2 \left(a^2-b^2\right)^2 \log (\sin (c+d x))}{a^7 d}-\frac{b^2 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^7 d}+\frac{b \csc ^5(c+d x)}{5 a^2 d}-\frac{\csc ^6(c+d x)}{6 a d}",1,"(b*(a^2 - b^2)^2*Csc[c + d*x])/(a^6*d) - ((a^2 - b^2)^2*Csc[c + d*x]^2)/(2*a^5*d) - (b*(2*a^2 - b^2)*Csc[c + d*x]^3)/(3*a^4*d) + ((2*a^2 - b^2)*Csc[c + d*x]^4)/(4*a^3*d) + (b*Csc[c + d*x]^5)/(5*a^2*d) - Csc[c + d*x]^6/(6*a*d) + (b^2*(a^2 - b^2)^2*Log[Sin[c + d*x]])/(a^7*d) - (b^2*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^7*d)","A",4,3,29,0.1034,1,"{2837, 12, 894}"
1320,1,467,0,1.814355,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","-\frac{a \left(-245 a^4 b^2+161 a^2 b^4+105 a^6-15 b^6\right) \cos (c+d x)}{105 b^8 d}+\frac{2 a^3 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^9 d}+\frac{\left(-85 a^2 b^2+40 a^4+48 b^4\right) \sin ^5(c+d x) \cos (c+d x)}{240 a^2 b^3 d}-\frac{\left(-60 a^2 b^2+28 a^4+35 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{140 a b^4 d}+\frac{\left(-104 a^2 b^2+48 a^4+59 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{192 b^5 d}-\frac{a \left(-77 a^2 b^2+35 a^4+45 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}+\frac{\left(-144 a^4 b^2+88 a^2 b^4+64 a^6-5 b^6\right) \sin (c+d x) \cos (c+d x)}{128 b^7 d}-\frac{x \left(-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6+128 a^8-5 b^8\right)}{128 b^9}-\frac{b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac{a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac{\sin ^7(c+d x) \cos (c+d x)}{8 b d}","-\frac{a \left(-245 a^4 b^2+161 a^2 b^4+105 a^6-15 b^6\right) \cos (c+d x)}{105 b^8 d}+\frac{2 a^3 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^9 d}+\frac{\left(-85 a^2 b^2+40 a^4+48 b^4\right) \sin ^5(c+d x) \cos (c+d x)}{240 a^2 b^3 d}-\frac{\left(-60 a^2 b^2+28 a^4+35 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{140 a b^4 d}+\frac{\left(-104 a^2 b^2+48 a^4+59 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{192 b^5 d}-\frac{a \left(-77 a^2 b^2+35 a^4+45 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}+\frac{\left(-144 a^4 b^2+88 a^2 b^4+64 a^6-5 b^6\right) \sin (c+d x) \cos (c+d x)}{128 b^7 d}-\frac{x \left(-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6+128 a^8-5 b^8\right)}{128 b^9}-\frac{b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac{a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac{\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac{\sin ^7(c+d x) \cos (c+d x)}{8 b d}",1,"-((128*a^8 - 320*a^6*b^2 + 240*a^4*b^4 - 40*a^2*b^6 - 5*b^8)*x)/(128*b^9) + (2*a^3*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^9*d) - (a*(105*a^6 - 245*a^4*b^2 + 161*a^2*b^4 - 15*b^6)*Cos[c + d*x])/(105*b^8*d) + ((64*a^6 - 144*a^4*b^2 + 88*a^2*b^4 - 5*b^6)*Cos[c + d*x]*Sin[c + d*x])/(128*b^7*d) - (a*(35*a^4 - 77*a^2*b^2 + 45*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(105*b^6*d) + ((48*a^4 - 104*a^2*b^2 + 59*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(192*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d) - ((28*a^4 - 60*a^2*b^2 + 35*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(140*a*b^4*d) - (b*Cos[c + d*x]*Sin[c + d*x]^5)/(5*a^2*d) + ((40*a^4 - 85*a^2*b^2 + 48*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(240*a^2*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^6)/(7*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^7)/(8*b*d)","A",11,7,29,0.2414,1,"{2896, 3049, 3023, 2735, 2660, 618, 204}"
1321,1,408,0,1.4578552,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\left(-245 a^4 b^2+161 a^2 b^4+105 a^6-15 b^6\right) \cos (c+d x)}{105 b^7 d}-\frac{2 a^2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^8 d}+\frac{\left(-60 a^2 b^2+28 a^4+35 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^3 d}-\frac{\left(-13 a^2 b^2+6 a^4+8 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{24 a b^4 d}+\frac{\left(-77 a^2 b^2+35 a^4+45 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{105 b^5 d}-\frac{a \left(-18 a^2 b^2+8 a^4+11 b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^6 d}+\frac{a x \left(-40 a^4 b^2+30 a^2 b^4+16 a^6-5 b^6\right)}{16 b^8}-\frac{b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac{a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac{\sin ^6(c+d x) \cos (c+d x)}{7 b d}","\frac{\left(-245 a^4 b^2+161 a^2 b^4+105 a^6-15 b^6\right) \cos (c+d x)}{105 b^7 d}-\frac{2 a^2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^8 d}+\frac{\left(-60 a^2 b^2+28 a^4+35 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^3 d}-\frac{\left(-13 a^2 b^2+6 a^4+8 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{24 a b^4 d}+\frac{\left(-77 a^2 b^2+35 a^4+45 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{105 b^5 d}-\frac{a \left(-18 a^2 b^2+8 a^4+11 b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^6 d}+\frac{a x \left(-40 a^4 b^2+30 a^2 b^4+16 a^6-5 b^6\right)}{16 b^8}-\frac{b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac{a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac{\sin ^6(c+d x) \cos (c+d x)}{7 b d}",1,"(a*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*x)/(16*b^8) - (2*a^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^8*d) + ((105*a^6 - 245*a^4*b^2 + 161*a^2*b^4 - 15*b^6)*Cos[c + d*x])/(105*b^7*d) - (a*(8*a^4 - 18*a^2*b^2 + 11*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^6*d) + ((35*a^4 - 77*a^2*b^2 + 45*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(105*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d) - ((6*a^4 - 13*a^2*b^2 + 8*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a*b^4*d) - (b*Cos[c + d*x]*Sin[c + d*x]^4)/(4*a^2*d) + ((28*a^4 - 60*a^2*b^2 + 35*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(140*a^2*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^5)/(6*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^6)/(7*b*d)","A",10,7,29,0.2414,1,"{2896, 3049, 3023, 2735, 2660, 618, 204}"
1322,1,228,0,0.5154057,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{2 a \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d}+\frac{\cos ^3(c+d x) \left(8 a \left(a^2-b^2\right)-b \left(6 a^2-5 b^2\right) \sin (c+d x)\right)}{24 b^4 d}-\frac{\cos (c+d x) \left(16 a \left(a^2-b^2\right)^2-b \left(-14 a^2 b^2+8 a^4+5 b^4\right) \sin (c+d x)\right)}{16 b^6 d}-\frac{x \left(-40 a^4 b^2+30 a^2 b^4+16 a^6-5 b^6\right)}{16 b^7}-\frac{\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}","\frac{2 a \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^7 d}+\frac{\cos ^3(c+d x) \left(8 a \left(a^2-b^2\right)-b \left(6 a^2-5 b^2\right) \sin (c+d x)\right)}{24 b^4 d}-\frac{\cos (c+d x) \left(16 a \left(a^2-b^2\right)^2-b \left(-14 a^2 b^2+8 a^4+5 b^4\right) \sin (c+d x)\right)}{16 b^6 d}-\frac{x \left(-40 a^4 b^2+30 a^2 b^4+16 a^6-5 b^6\right)}{16 b^7}-\frac{\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}",1,"-((16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*x)/(16*b^7) + (2*a*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^7*d) - (Cos[c + d*x]^5*(6*a - 5*b*Sin[c + d*x]))/(30*b^2*d) + (Cos[c + d*x]^3*(8*a*(a^2 - b^2) - b*(6*a^2 - 5*b^2)*Sin[c + d*x]))/(24*b^4*d) - (Cos[c + d*x]*(16*a*(a^2 - b^2)^2 - b*(8*a^4 - 14*a^2*b^2 + 5*b^4)*Sin[c + d*x]))/(16*b^6*d)","A",7,5,27,0.1852,1,"{2865, 2735, 2660, 618, 204}"
1323,1,252,0,0.285735,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{a \left(a^2-3 b^2\right) \cos (c+d x)}{b^4 d}+\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b^5 d}+\frac{\left(a^2-3 b^2\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d}-\frac{x \left(a^2-3 b^2\right)}{2 b^3}-\frac{x \left(-3 a^2 b^2+a^4+3 b^4\right)}{b^5}+\frac{a \cos ^3(c+d x)}{3 b^2 d}-\frac{a \cos (c+d x)}{b^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 b d}-\frac{3 x}{8 b}","-\frac{a \left(a^2-3 b^2\right) \cos (c+d x)}{b^4 d}+\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a b^5 d}+\frac{\left(a^2-3 b^2\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d}-\frac{x \left(a^2-3 b^2\right)}{2 b^3}-\frac{x \left(-3 a^2 b^2+a^4+3 b^4\right)}{b^5}+\frac{a \cos ^3(c+d x)}{3 b^2 d}-\frac{a \cos (c+d x)}{b^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 b d}-\frac{3 x}{8 b}",1,"(-3*x)/(8*b) - ((a^2 - 3*b^2)*x)/(2*b^3) - ((a^4 - 3*a^2*b^2 + 3*b^4)*x)/b^5 + (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*b^5*d) - ArcTanh[Cos[c + d*x]]/(a*d) - (a*Cos[c + d*x])/(b^2*d) - (a*(a^2 - 3*b^2)*Cos[c + d*x])/(b^4*d) + (a*Cos[c + d*x]^3)/(3*b^2*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) + ((a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(4*b*d)","A",14,9,27,0.3333,1,"{2897, 3770, 2638, 2635, 8, 2633, 2660, 618, 204}"
1324,1,183,0,0.2551592,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\left(a^2-3 b^2\right) \cos (c+d x)}{b^3 d}-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^4 d}+\frac{a x \left(a^2-3 b^2\right)}{b^4}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{a \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{a x}{2 b^2}-\frac{\cot (c+d x)}{a d}-\frac{\cos ^3(c+d x)}{3 b d}+\frac{\cos (c+d x)}{b d}","\frac{\left(a^2-3 b^2\right) \cos (c+d x)}{b^3 d}-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 b^4 d}+\frac{a x \left(a^2-3 b^2\right)}{b^4}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{a \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{a x}{2 b^2}-\frac{\cot (c+d x)}{a d}-\frac{\cos ^3(c+d x)}{3 b d}+\frac{\cos (c+d x)}{b d}",1,"(a*x)/(2*b^2) + (a*(a^2 - 3*b^2)*x)/b^4 - (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^4*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) + Cos[c + d*x]/(b*d) + ((a^2 - 3*b^2)*Cos[c + d*x])/(b^3*d) - Cos[c + d*x]^3/(3*b*d) - Cot[c + d*x]/(a*d) - (a*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d)","A",13,10,29,0.3448,1,"{2897, 3770, 3767, 8, 2638, 2635, 2633, 2660, 618, 204}"
1325,1,174,0,0.3846949,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^3 d}+\frac{\left(5 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{x \left(2 a^2-5 b^2\right)}{2 b^3}+\frac{b \cot (c+d x)}{a^2 d}-\frac{a \cos (c+d x)}{b^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}+\frac{\sin (c+d x) \cos (c+d x)}{2 b d}","\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 b^3 d}+\frac{\left(5 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{x \left(2 a^2-5 b^2\right)}{2 b^3}+\frac{b \cot (c+d x)}{a^2 d}-\frac{a \cos (c+d x)}{b^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}+\frac{\sin (c+d x) \cos (c+d x)}{2 b d}",1,"-((2*a^2 - 5*b^2)*x)/(2*b^3) + (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b^3*d) + ((5*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^3*d) - (a*Cos[c + d*x])/(b^2*d) + (b*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)","A",6,6,29,0.2069,1,"{2896, 3057, 2660, 618, 204, 3770}"
1326,1,197,0,0.2753579,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 b^2 d}+\frac{\left(3 a^2-b^2\right) \cot (c+d x)}{a^3 d}-\frac{b \left(3 a^2-b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}+\frac{b \cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{a x}{b^2}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}+\frac{\cos (c+d x)}{b d}","-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 b^2 d}+\frac{\left(3 a^2-b^2\right) \cot (c+d x)}{a^3 d}-\frac{b \left(3 a^2-b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{2 a^2 d}+\frac{b \cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{a x}{b^2}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}+\frac{\cos (c+d x)}{b d}",1,"(a*x)/b^2 - (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*b^2*d) + (b*ArcTanh[Cos[c + d*x]])/(2*a^2*d) - (b*(3*a^2 - b^2)*ArcTanh[Cos[c + d*x]])/(a^4*d) + Cos[c + d*x]/(b*d) - Cot[c + d*x]/(a*d) + ((3*a^2 - b^2)*Cot[c + d*x])/(a^3*d) - Cot[c + d*x]^3/(3*a*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d)","A",13,9,29,0.3103,1,"{2897, 3770, 3767, 8, 3768, 2638, 2660, 618, 204}"
1327,1,275,0,0.2895053,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 b d}-\frac{b \left(3 a^2-b^2\right) \cot (c+d x)}{a^4 d}+\frac{\left(3 a^2-b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\left(-3 a^2 b^2+3 a^4+b^4\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{\left(3 a^2-b^2\right) \cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{b \cot ^3(c+d x)}{3 a^2 d}+\frac{b \cot (c+d x)}{a^2 d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac{x}{b}","\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 b d}+\frac{b \left(b^2-2 a^2\right) \cot (c+d x)}{a^4 d}-\frac{\left(-20 a^2 b^2+15 a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}+\frac{\left(7 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac{b \cot ^3(c+d x)}{3 a^2 d}-\frac{\cot ^3(c+d x) \csc (c+d x)}{4 a d}-\frac{x}{b}",1,"-(x/b) + (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*b*d) - (3*ArcTanh[Cos[c + d*x]])/(8*a*d) + ((3*a^2 - b^2)*ArcTanh[Cos[c + d*x]])/(2*a^3*d) - ((3*a^4 - 3*a^2*b^2 + b^4)*ArcTanh[Cos[c + d*x]])/(a^5*d) + (b*Cot[c + d*x])/(a^2*d) - (b*(3*a^2 - b^2)*Cot[c + d*x])/(a^4*d) + (b*Cot[c + d*x]^3)/(3*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + ((3*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)","A",15,8,27,0.2963,1,"{2897, 3770, 3767, 8, 3768, 2660, 618, 204}"
1328,1,307,0,1.119807,"\int \frac{\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cot[c + d*x]^6/(a + b*Sin[c + d*x]),x]","-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}-\frac{\left(-35 a^2 b^2+23 a^4+15 b^4\right) \cot (c+d x)}{15 a^5 d}+\frac{b \left(-20 a^2 b^2+15 a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left(-22 a^2 b^2+15 a^4+10 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^3 b^2 d}+\frac{\left(-9 a^2 b^2+8 a^4+4 b^4\right) \cot (c+d x) \csc (c+d x)}{8 a^4 b d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{2 b^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac{\cot (c+d x) \csc (c+d x)}{b d}","-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^6 d}-\frac{\left(-35 a^2 b^2+23 a^4+15 b^4\right) \cot (c+d x)}{15 a^5 d}+\frac{b \left(-20 a^2 b^2+15 a^4+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}+\frac{\left(11 a^2-5 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac{b \left(4 b^2-9 a^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}",1,"(-2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*d) + (b*(15*a^4 - 20*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - ((23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^5*d) - (Cot[c + d*x]*Csc[c + d*x])/(b*d) + ((8*a^4 - 9*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(2*b^2*d) - ((15*a^4 - 22*a^2*b^2 + 10*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)","A",9,7,21,0.3333,1,"{2726, 3055, 3001, 3770, 2660, 618, 204}"
1329,1,363,0,1.4838907,"\int \frac{\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^6*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{2 b \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^7 d}+\frac{b \left(-35 a^2 b^2+23 a^4+15 b^4\right) \cot (c+d x)}{15 a^6 d}+\frac{\left(-30 a^4 b^2+40 a^2 b^4+5 a^6-16 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^7 d}-\frac{\left(-13 a^2 b^2+8 a^4+6 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac{\left(-22 a^2 b^2+15 a^4+10 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}-\frac{\left(-18 a^2 b^2+11 a^4+8 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^5 d}+\frac{b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{2 b d}","\frac{2 b \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^7 d}+\frac{b \left(-35 a^2 b^2+23 a^4+15 b^4\right) \cot (c+d x)}{15 a^6 d}+\frac{\left(-30 a^4 b^2+40 a^2 b^4+5 a^6-16 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^7 d}-\frac{\left(-13 a^2 b^2+8 a^4+6 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac{\left(-22 a^2 b^2+15 a^4+10 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}-\frac{\left(-18 a^2 b^2+11 a^4+8 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^5 d}+\frac{b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{2 b d}",1,"(2*b*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^7*d) + ((5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^7*d) + (b*(23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^6*d) - ((11*a^4 - 18*a^2*b^2 + 8*b^4)*Cot[c + d*x]*Csc[c + d*x])/(16*a^5*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(2*b*d) + ((15*a^4 - 22*a^2*b^2 + 10*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(3*b^2*d) - ((8*a^4 - 13*a^2*b^2 + 6*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^4)/(5*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a*d)","A",10,7,27,0.2593,1,"{2896, 3055, 3001, 3770, 2660, 618, 204}"
1330,1,417,0,1.8326131,"\int \frac{\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^6*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{2 b^2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^8 d}+\frac{\left(-161 a^4 b^2+245 a^2 b^4+15 a^6-105 b^6\right) \cot (c+d x)}{105 a^7 d}-\frac{b \left(-30 a^4 b^2+40 a^2 b^4+5 a^6-16 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^8 d}-\frac{\left(-60 a^2 b^2+35 a^4+28 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac{\left(-13 a^2 b^2+8 a^4+6 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}-\frac{\left(-77 a^2 b^2+45 a^4+35 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}+\frac{b \left(-18 a^2 b^2+11 a^4+8 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac{b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{3 b d}","-\frac{2 b^2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^8 d}+\frac{\left(-161 a^4 b^2+245 a^2 b^4+15 a^6-105 b^6\right) \cot (c+d x)}{105 a^7 d}-\frac{b \left(-30 a^4 b^2+40 a^2 b^4+5 a^6-16 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^8 d}-\frac{\left(-60 a^2 b^2+35 a^4+28 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac{\left(-13 a^2 b^2+8 a^4+6 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}-\frac{\left(-77 a^2 b^2+45 a^4+35 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}+\frac{b \left(-18 a^2 b^2+11 a^4+8 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac{b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{3 b d}",1,"(-2*b^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^8*d) - (b*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^8*d) + ((15*a^6 - 161*a^4*b^2 + 245*a^2*b^4 - 105*b^6)*Cot[c + d*x])/(105*a^7*d) + (b*(11*a^4 - 18*a^2*b^2 + 8*b^4)*Cot[c + d*x]*Csc[c + d*x])/(16*a^6*d) - ((45*a^4 - 77*a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(105*a^5*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(3*b*d) + ((8*a^4 - 13*a^2*b^2 + 6*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^4)/(4*b^2*d) - ((35*a^4 - 60*a^2*b^2 + 28*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(140*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^6)/(7*a*d)","A",11,7,29,0.2414,1,"{2896, 3055, 3001, 3770, 2660, 618, 204}"
1331,1,476,0,2.2145168,"\int \frac{\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Cot[c + d*x]^6*Csc[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{2 b^3 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^9 d}-\frac{b \left(-161 a^4 b^2+245 a^2 b^4+15 a^6-105 b^6\right) \cot (c+d x)}{105 a^8 d}+\frac{\left(40 a^6 b^2-240 a^4 b^4+320 a^2 b^6+5 a^8-128 b^8\right) \tanh ^{-1}(\cos (c+d x))}{128 a^9 d}-\frac{\left(-85 a^2 b^2+48 a^4+40 b^4\right) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{\left(-60 a^2 b^2+35 a^4+28 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}-\frac{\left(-104 a^2 b^2+59 a^4+48 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}+\frac{b \left(-77 a^2 b^2+45 a^4+35 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}+\frac{\left(-88 a^4 b^2+144 a^2 b^4+5 a^6-64 b^6\right) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}","\frac{2 b^3 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^9 d}-\frac{b \left(-161 a^4 b^2+245 a^2 b^4+15 a^6-105 b^6\right) \cot (c+d x)}{105 a^8 d}+\frac{\left(40 a^6 b^2-240 a^4 b^4+320 a^2 b^6+5 a^8-128 b^8\right) \tanh ^{-1}(\cos (c+d x))}{128 a^9 d}-\frac{\left(-85 a^2 b^2+48 a^4+40 b^4\right) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{\left(-60 a^2 b^2+35 a^4+28 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}-\frac{\left(-104 a^2 b^2+59 a^4+48 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}+\frac{b \left(-77 a^2 b^2+45 a^4+35 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}+\frac{\left(-88 a^4 b^2+144 a^2 b^4+5 a^6-64 b^6\right) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}",1,"(2*b^3*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^9*d) + ((5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*ArcTanh[Cos[c + d*x]])/(128*a^9*d) - (b*(15*a^6 - 161*a^4*b^2 + 245*a^2*b^4 - 105*b^6)*Cot[c + d*x])/(105*a^8*d) + ((5*a^6 - 88*a^4*b^2 + 144*a^2*b^4 - 64*b^6)*Cot[c + d*x]*Csc[c + d*x])/(128*a^7*d) + (b*(45*a^4 - 77*a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(105*a^6*d) - ((59*a^4 - 104*a^2*b^2 + 48*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(192*a^5*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(4*b*d) + ((35*a^4 - 60*a^2*b^2 + 28*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(140*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^5)/(5*b^2*d) - ((48*a^4 - 85*a^2*b^2 + 40*b^4)*Cot[c + d*x]*Csc[c + d*x]^5)/(240*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^6)/(7*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^7)/(8*a*d)","A",12,7,29,0.2414,1,"{2896, 3055, 3001, 3770, 2660, 618, 204}"
1332,1,93,0,0.1808539,"\int \frac{\sin ^2(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{a^3 \log (a+b \sin (c+d x))}{b^2 d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}-\frac{\sin (c+d x)}{b d}","\frac{a^3 \log (a+b \sin (c+d x))}{b^2 d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}-\frac{\sin (c+d x)}{b d}",1,"-Log[1 - Sin[c + d*x]]/(2*(a + b)*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)*d) + (a^3*Log[a + b*Sin[c + d*x]])/(b^2*(a^2 - b^2)*d) - Sin[c + d*x]/(b*d)","A",4,3,27,0.1111,1,"{2837, 12, 1629}"
1333,1,80,0,0.1557767,"\int \frac{\sin (c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{a^2 \log (a+b \sin (c+d x))}{b d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}","-\frac{a^2 \log (a+b \sin (c+d x))}{b 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,"-Log[1 - Sin[c + d*x]]/(2*(a + b)*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)*d) - (a^2*Log[a + b*Sin[c + d*x]])/(b*(a^2 - b^2)*d)","A",4,3,25,0.1200,1,"{2837, 12, 1629}"
1334,1,74,0,0.0643939,"\int \frac{\tan (c+d x)}{a+b \sin (c+d x)} \, dx","Int[Tan[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{a \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}","\frac{a \log (a+b \sin (c+d x))}{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,"-Log[1 - Sin[c + d*x]]/(2*(a + b)*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)*d) + (a*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)*d)","A",3,2,19,0.1053,1,"{2721, 801}"
1335,1,93,0,0.1458375,"\int \frac{\csc (c+d x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{b^2 \log (a+b \sin (c+d x))}{a d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}+\frac{\log (\sin (c+d x))}{a d}","\frac{b^2 \log (a+b \sin (c+d x))}{a d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}+\frac{\log (\sin (c+d x))}{a d}",1,"-Log[1 - Sin[c + d*x]]/(2*(a + b)*d) + Log[Sin[c + d*x]]/(a*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)*d) + (b^2*Log[a + b*Sin[c + d*x]])/(a*(a^2 - b^2)*d)","A",4,3,25,0.1200,1,"{2837, 12, 894}"
1336,1,110,0,0.1865485,"\int \frac{\csc ^2(c+d x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{b^3 \log (a+b \sin (c+d x))}{a^2 d \left(a^2-b^2\right)}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}-\frac{\csc (c+d x)}{a d}","-\frac{b^3 \log (a+b \sin (c+d x))}{a^2 d \left(a^2-b^2\right)}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}-\frac{\csc (c+d x)}{a d}",1,"-(Csc[c + d*x]/(a*d)) - Log[1 - Sin[c + d*x]]/(2*(a + b)*d) - (b*Log[Sin[c + d*x]])/(a^2*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)*d) - (b^3*Log[a + b*Sin[c + d*x]])/(a^2*(a^2 - b^2)*d)","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1337,1,132,0,0.2022669,"\int \frac{\csc ^3(c+d x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^3*Sec[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{b^4 \log (a+b \sin (c+d x))}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2+b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}-\frac{\csc ^2(c+d x)}{2 a d}","\frac{b^4 \log (a+b \sin (c+d x))}{a^3 d \left(a^2-b^2\right)}+\frac{\left(a^2+b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}-\frac{\csc ^2(c+d x)}{2 a d}",1,"(b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - Log[1 - Sin[c + d*x]]/(2*(a + b)*d) + ((a^2 + b^2)*Log[Sin[c + d*x]])/(a^3*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)*d) + (b^4*Log[a + b*Sin[c + d*x]])/(a^3*(a^2 - b^2)*d)","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1338,1,268,0,0.3959463,"\int \frac{\sin ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^3*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{a^3 \cos (c+d x)}{b^2 d \left(a^2-b^2\right)}+\frac{a \cos (c+d x)}{d \left(a^2-b^2\right)}+\frac{2 a^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 d \left(a^2-b^2\right)^{3/2}}-\frac{3 b \tan (c+d x)}{2 d \left(a^2-b^2\right)}+\frac{a \sec (c+d x)}{d \left(a^2-b^2\right)}+\frac{b \sin ^2(c+d x) \tan (c+d x)}{2 d \left(a^2-b^2\right)}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 b d \left(a^2-b^2\right)}-\frac{a^2 x \left(2 a^2+b^2\right)}{2 b^3 \left(a^2-b^2\right)}+\frac{3 b x}{2 \left(a^2-b^2\right)}","-\frac{a^3 \cos (c+d x)}{b^2 d \left(a^2-b^2\right)}+\frac{a \cos (c+d x)}{d \left(a^2-b^2\right)}+\frac{2 a^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^3 d \left(a^2-b^2\right)^{3/2}}-\frac{3 b \tan (c+d x)}{2 d \left(a^2-b^2\right)}+\frac{a \sec (c+d x)}{d \left(a^2-b^2\right)}+\frac{b \sin ^2(c+d x) \tan (c+d x)}{2 d \left(a^2-b^2\right)}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 b d \left(a^2-b^2\right)}-\frac{a^2 x \left(2 a^2+b^2\right)}{2 b^3 \left(a^2-b^2\right)}+\frac{3 b x}{2 \left(a^2-b^2\right)}",1,"(3*b*x)/(2*(a^2 - b^2)) - (a^2*(2*a^2 + b^2)*x)/(2*b^3*(a^2 - b^2)) + (2*a^5*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*d) + (a*Cos[c + d*x])/((a^2 - b^2)*d) - (a^3*Cos[c + d*x])/(b^2*(a^2 - b^2)*d) + (a*Sec[c + d*x])/((a^2 - b^2)*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d) - (3*b*Tan[c + d*x])/(2*(a^2 - b^2)*d) + (b*Sin[c + d*x]^2*Tan[c + d*x])/(2*(a^2 - b^2)*d)","A",14,13,29,0.4483,1,"{2902, 2590, 14, 2591, 288, 321, 203, 2793, 3023, 2735, 2660, 618, 204}"
1339,1,183,0,0.2734001,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{a^2 \cos (c+d x)}{b d \left(a^2-b^2\right)}-\frac{b \cos (c+d x)}{d \left(a^2-b^2\right)}-\frac{2 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{3/2}}+\frac{a \tan (c+d x)}{d \left(a^2-b^2\right)}-\frac{b \sec (c+d x)}{d \left(a^2-b^2\right)}+\frac{a^3 x}{b^2 \left(a^2-b^2\right)}-\frac{a x}{a^2-b^2}","\frac{a^2 \cos (c+d x)}{b d \left(a^2-b^2\right)}-\frac{b \cos (c+d x)}{d \left(a^2-b^2\right)}-\frac{2 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{3/2}}+\frac{a \tan (c+d x)}{d \left(a^2-b^2\right)}-\frac{b \sec (c+d x)}{d \left(a^2-b^2\right)}+\frac{a^3 x}{b^2 \left(a^2-b^2\right)}-\frac{a x}{a^2-b^2}",1,"-((a*x)/(a^2 - b^2)) + (a^3*x)/(b^2*(a^2 - b^2)) - (2*a^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)*d) + (a^2*Cos[c + d*x])/(b*(a^2 - b^2)*d) - (b*Cos[c + d*x])/((a^2 - b^2)*d) - (b*Sec[c + d*x])/((a^2 - b^2)*d) + (a*Tan[c + d*x])/((a^2 - b^2)*d)","A",12,11,29,0.3793,1,"{2902, 3473, 8, 2590, 14, 2746, 12, 2735, 2660, 618, 204}"
1340,1,133,0,0.1779637,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{2 a^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d \left(a^2-b^2\right)^{3/2}}-\frac{b \tan (c+d x)}{d \left(a^2-b^2\right)}+\frac{a \sec (c+d x)}{d \left(a^2-b^2\right)}-\frac{a^2 x}{b \left(a^2-b^2\right)}+\frac{b x}{a^2-b^2}","\frac{2 a^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d \left(a^2-b^2\right)^{3/2}}-\frac{b \tan (c+d x)}{d \left(a^2-b^2\right)}+\frac{a \sec (c+d x)}{d \left(a^2-b^2\right)}-\frac{a^2 x}{b \left(a^2-b^2\right)}+\frac{b x}{a^2-b^2}",1,"-((a^2*x)/(b*(a^2 - b^2))) + (b*x)/(a^2 - b^2) + (2*a^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(3/2)*d) + (a*Sec[c + d*x])/((a^2 - b^2)*d) - (b*Tan[c + d*x])/((a^2 - b^2)*d)","A",9,8,27,0.2963,1,"{2902, 2606, 8, 3473, 2735, 2660, 618, 204}"
1341,1,96,0,0.1071086,"\int \frac{\tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Tan[c + d*x]^2/(a + b*Sin[c + d*x]),x]","-\frac{2 a^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{a \tan (c+d x)}{d \left(a^2-b^2\right)}-\frac{b \sec (c+d x)}{d \left(a^2-b^2\right)}","-\frac{2 a^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{a \tan (c+d x)}{d \left(a^2-b^2\right)}-\frac{b \sec (c+d x)}{d \left(a^2-b^2\right)}",1,"(-2*a^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d) - (b*Sec[c + d*x])/((a^2 - b^2)*d) + (a*Tan[c + d*x])/((a^2 - b^2)*d)","A",8,7,21,0.3333,1,"{2727, 3767, 8, 2606, 2660, 618, 204}"
1342,1,82,0,0.0994956,"\int \frac{\sec (c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{2 a b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{\sec (c+d x) (a-b \sin (c+d x))}{d \left(a^2-b^2\right)}","\frac{2 a b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{\sec (c+d x) (a-b \sin (c+d x))}{d \left(a^2-b^2\right)}",1,"(2*a*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d) + (Sec[c + d*x]*(a - b*Sin[c + d*x]))/((a^2 - b^2)*d)","A",5,5,25,0.2000,1,"{2866, 12, 2660, 618, 204}"
1343,1,118,0,0.2301906,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{2 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d \left(a^2-b^2\right)^{3/2}}+\frac{b \sec (c+d x) (b-a \sin (c+d x))}{a d \left(a^2-b^2\right)}+\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}","\frac{2 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d \left(a^2-b^2\right)^{3/2}}+\frac{b \sec (c+d x) (b-a \sin (c+d x))}{a d \left(a^2-b^2\right)}+\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"(2*b^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(3/2)*d) - ArcTanh[Cos[c + d*x]]/(a*d) + Sec[c + d*x]/(a*d) + (b*Sec[c + d*x]*(b - a*Sin[c + d*x]))/(a*(a^2 - b^2)*d)","A",10,9,27,0.3333,1,"{2898, 2622, 321, 207, 2696, 12, 2660, 618, 204}"
1344,1,150,0,0.2883542,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{2 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{3/2}}-\frac{b^2 \sec (c+d x) (b-a \sin (c+d x))}{a^2 d \left(a^2-b^2\right)}-\frac{b \sec (c+d x)}{a^2 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\tan (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}","-\frac{2 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{3/2}}+\frac{a \tan (c+d x)}{d \left(a^2-b^2\right)}-\frac{b \sec (c+d x)}{d \left(a^2-b^2\right)}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}",1,"(-2*b^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2)*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a*d) - (b*Sec[c + d*x])/(a^2*d) - (b^2*Sec[c + d*x]*(b - a*Sin[c + d*x]))/(a^2*(a^2 - b^2)*d) + Tan[c + d*x]/(a*d)","A",13,11,29,0.3793,1,"{2898, 2622, 321, 207, 2620, 14, 2696, 12, 2660, 618, 204}"
1345,1,212,0,0.3597321,"\int \frac{\csc ^3(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^3*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{2 b^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{3/2}}+\frac{b^2 \sec (c+d x)}{a^3 d}-\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{b^3 \sec (c+d x) (b-a \sin (c+d x))}{a^3 d \left(a^2-b^2\right)}-\frac{b \tan (c+d x)}{a^2 d}+\frac{b \cot (c+d x)}{a^2 d}+\frac{3 \sec (c+d x)}{2 a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\csc ^2(c+d x) \sec (c+d x)}{2 a d}","\frac{2 b^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{3/2}}-\frac{b \tan (c+d x)}{d \left(a^2-b^2\right)}+\frac{\left(3 a^2-b^2\right) \sec (c+d x)}{2 a d \left(a^2-b^2\right)}-\frac{\left(3 a^2+2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{b \cot (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x) \sec (c+d x)}{2 a d}",1,"(2*b^5*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(3/2)*d) - (3*ArcTanh[Cos[c + d*x]])/(2*a*d) - (b^2*ArcTanh[Cos[c + d*x]])/(a^3*d) + (b*Cot[c + d*x])/(a^2*d) + (3*Sec[c + d*x])/(2*a*d) + (b^2*Sec[c + d*x])/(a^3*d) - (Csc[c + d*x]^2*Sec[c + d*x])/(2*a*d) + (b^3*Sec[c + d*x]*(b - a*Sin[c + d*x]))/(a^3*(a^2 - b^2)*d) - (b*Tan[c + d*x])/(a^2*d)","A",17,12,29,0.4138,1,"{2898, 2622, 321, 207, 2620, 14, 288, 2696, 12, 2660, 618, 204}"
1346,1,126,0,0.2024716,"\int \frac{\tan ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Tan[c + d*x]^3/(a + b*Sin[c + d*x]),x]","-\frac{a^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}+\frac{\sec ^2(c+d x) (a-b \sin (c+d x))}{2 d \left(a^2-b^2\right)}+\frac{(2 a+b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(2 a-b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}","-\frac{a^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}+\frac{\sec ^2(c+d x) (a-b \sin (c+d x))}{2 d \left(a^2-b^2\right)}+\frac{(2 a+b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(2 a-b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}",1,"((2*a + b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((2*a - b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (a^3*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + (Sec[c + d*x]^2*(a - b*Sin[c + d*x]))/(2*(a^2 - b^2)*d)","A",4,3,21,0.1429,1,"{2721, 1647, 801}"
1347,1,116,0,0.2307344,"\int \frac{\sec (c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{a^2 b \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right)}+\frac{a \log (1-\sin (c+d x))}{4 d (a+b)^2}-\frac{a \log (\sin (c+d x)+1)}{4 d (a-b)^2}","\frac{a^2 b \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right)}+\frac{a \log (1-\sin (c+d x))}{4 d (a+b)^2}-\frac{a \log (\sin (c+d x)+1)}{4 d (a-b)^2}",1,"(a*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) - (a*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) + (a^2*b*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d)","A",5,4,27,0.1481,1,"{2837, 12, 1647, 801}"
1348,1,117,0,0.1651215,"\int \frac{\sec ^2(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^2*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{a b^2 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}+\frac{\sec ^2(c+d x) (a-b \sin (c+d x))}{2 d \left(a^2-b^2\right)}-\frac{b \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{b \log (\sin (c+d x)+1)}{4 d (a-b)^2}","-\frac{a b^2 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}+\frac{\sec ^2(c+d x) (a-b \sin (c+d x))}{2 d \left(a^2-b^2\right)}-\frac{b \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{b \log (\sin (c+d x)+1)}{4 d (a-b)^2}",1,"-(b*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + (b*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (a*b^2*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + (Sec[c + d*x]^2*(a - b*Sin[c + d*x]))/(2*(a^2 - b^2)*d)","A",5,4,27,0.1481,1,"{2837, 12, 823, 801}"
1349,1,156,0,0.2415248,"\int \frac{\csc (c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","-\frac{b^4 \log (a+b \sin (c+d x))}{a 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{(2 a+3 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}-\frac{(2 a-3 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}+\frac{\log (\sin (c+d x))}{a d}","-\frac{b^4 \log (a+b \sin (c+d x))}{a 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{(2 a+3 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}-\frac{(2 a-3 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}+\frac{\log (\sin (c+d x))}{a d}",1,"-((2*a + 3*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + Log[Sin[c + d*x]]/(a*d) - ((2*a - 3*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (b^4*Log[a + b*Sin[c + d*x]])/(a*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Sin[c + d*x])) + 1/(4*(a - b)*d*(1 + Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1350,1,171,0,0.2696138,"\int \frac{\csc ^2(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{b^5 \log (a+b \sin (c+d x))}{a^2 d \left(a^2-b^2\right)^2}-\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{1}{4 d (a+b) (1-\sin (c+d x))}-\frac{1}{4 d (a-b) (\sin (c+d x)+1)}-\frac{(3 a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(3 a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}-\frac{\csc (c+d x)}{a d}","\frac{b^5 \log (a+b \sin (c+d x))}{a^2 d \left(a^2-b^2\right)^2}-\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{1}{4 d (a+b) (1-\sin (c+d x))}-\frac{1}{4 d (a-b) (\sin (c+d x)+1)}-\frac{(3 a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(3 a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}-\frac{\csc (c+d x)}{a d}",1,"-(Csc[c + d*x]/(a*d)) - ((3*a + 4*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) - (b*Log[Sin[c + d*x]])/(a^2*d) + ((3*a - 4*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) + (b^5*Log[a + b*Sin[c + d*x]])/(a^2*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Sin[c + d*x])) - 1/(4*(a - b)*d*(1 + Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2837, 12, 894}"
1351,1,197,0,0.3123789,"\int \frac{\csc ^3(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^3*Sec[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","-\frac{b^6 \log (a+b \sin (c+d x))}{a^3 d \left(a^2-b^2\right)^2}+\frac{\left(2 a^2+b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}+\frac{1}{4 d (a+b) (1-\sin (c+d x))}+\frac{1}{4 d (a-b) (\sin (c+d x)+1)}-\frac{(4 a+5 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}-\frac{(4 a-5 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}-\frac{\csc ^2(c+d x)}{2 a d}","-\frac{b^6 \log (a+b \sin (c+d x))}{a^3 d \left(a^2-b^2\right)^2}+\frac{\left(2 a^2+b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}+\frac{1}{4 d (a+b) (1-\sin (c+d x))}+\frac{1}{4 d (a-b) (\sin (c+d x)+1)}-\frac{(4 a+5 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}-\frac{(4 a-5 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}-\frac{\csc ^2(c+d x)}{2 a d}",1,"(b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - ((4*a + 5*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((2*a^2 + b^2)*Log[Sin[c + d*x]])/(a^3*d) - ((4*a - 5*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (b^6*Log[a + b*Sin[c + d*x]])/(a^3*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Sin[c + d*x])) + 1/(4*(a - b)*d*(1 + Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2837, 12, 894}"
1352,1,177,0,0.2286663,"\int \frac{\tan ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Tan[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{2 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{a \tan ^3(c+d x)}{3 d \left(a^2-b^2\right)}-\frac{a^3 \tan (c+d x)}{d \left(a^2-b^2\right)^2}-\frac{b \sec ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{a^2 b \sec (c+d x)}{d \left(a^2-b^2\right)^2}+\frac{b \sec (c+d x)}{d \left(a^2-b^2\right)}","\frac{2 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{a \tan ^3(c+d x)}{3 d \left(a^2-b^2\right)}-\frac{a^3 \tan (c+d x)}{d \left(a^2-b^2\right)^2}-\frac{b \sec ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{a^2 b \sec (c+d x)}{d \left(a^2-b^2\right)^2}+\frac{b \sec (c+d x)}{d \left(a^2-b^2\right)}",1,"(2*a^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (a^2*b*Sec[c + d*x])/((a^2 - b^2)^2*d) + (b*Sec[c + d*x])/((a^2 - b^2)*d) - (b*Sec[c + d*x]^3)/(3*(a^2 - b^2)*d) - (a^3*Tan[c + d*x])/((a^2 - b^2)^2*d) + (a*Tan[c + d*x]^3)/(3*(a^2 - b^2)*d)","A",13,9,21,0.4286,1,"{2727, 2607, 30, 2606, 3767, 8, 2660, 618, 204}"
1353,1,142,0,0.238903,"\int \frac{\sec (c+d x) \tan ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","-\frac{2 a^3 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{b \tan ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{a \sec ^3(c+d x)}{3 d \left(a^2-b^2\right)}-\frac{a^2 \sec (c+d x) (a-b \sin (c+d x))}{d \left(a^2-b^2\right)^2}","-\frac{2 a^3 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{b \tan ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{a \sec ^3(c+d x)}{3 d \left(a^2-b^2\right)}-\frac{a^2 \sec (c+d x) (a-b \sin (c+d x))}{d \left(a^2-b^2\right)^2}",1,"(-2*a^3*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (a*Sec[c + d*x]^3)/(3*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]*(a - b*Sin[c + d*x]))/((a^2 - b^2)^2*d) - (b*Tan[c + d*x]^3)/(3*(a^2 - b^2)*d)","A",10,9,27,0.3333,1,"{2902, 2606, 30, 2607, 2866, 12, 2660, 618, 204}"
1354,1,165,0,0.2377244,"\int \frac{\sec ^2(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^2*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{a \tan ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{2 a^2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{a \tan (c+d x)}{d \left(a^2-b^2\right)}-\frac{b \sec ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{a^2 \sec (c+d x) (b-a \sin (c+d x))}{d \left(a^2-b^2\right)^2}","\frac{a \tan ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{2 a^2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{a \tan (c+d x)}{d \left(a^2-b^2\right)}-\frac{b \sec ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\frac{a^2 \sec (c+d x) (b-a \sin (c+d x))}{d \left(a^2-b^2\right)^2}",1,"(2*a^2*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (b*Sec[c + d*x]^3)/(3*(a^2 - b^2)*d) + (a^2*Sec[c + d*x]*(b - a*Sin[c + d*x]))/((a^2 - b^2)^2*d) + (a*Tan[c + d*x])/((a^2 - b^2)*d) + (a*Tan[c + d*x]^3)/(3*(a^2 - b^2)*d)","A",10,9,29,0.3103,1,"{2902, 3767, 2606, 30, 2696, 12, 2660, 618, 204}"
1355,1,138,0,0.2231764,"\int \frac{\sec ^3(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^3*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{2 a b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{\sec ^3(c+d x) (a-b \sin (c+d x))}{3 d \left(a^2-b^2\right)}-\frac{\sec (c+d x) \left(3 a b^2-b \left(a^2+2 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}","-\frac{2 a b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{\sec ^3(c+d x) (a-b \sin (c+d x))}{3 d \left(a^2-b^2\right)}-\frac{\sec (c+d x) \left(3 a b^2-b \left(a^2+2 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}",1,"(-2*a*b^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (Sec[c + d*x]^3*(a - b*Sin[c + d*x]))/(3*(a^2 - b^2)*d) - (Sec[c + d*x]*(3*a*b^2 - b*(a^2 + 2*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^2*d)","A",6,5,27,0.1852,1,"{2866, 12, 2660, 618, 204}"
1356,1,194,0,0.4086958,"\int \frac{\csc (c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","-\frac{2 b^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d \left(a^2-b^2\right)^{5/2}}+\frac{b \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a d \left(a^2-b^2\right)}-\frac{b \sec (c+d x) \left(a \left(2 a^2-5 b^2\right) \sin (c+d x)+3 b^3\right)}{3 a d \left(a^2-b^2\right)^2}+\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}","-\frac{2 b^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d \left(a^2-b^2\right)^{5/2}}+\frac{b \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a d \left(a^2-b^2\right)}-\frac{b \sec (c+d x) \left(a \left(2 a^2-5 b^2\right) \sin (c+d x)+3 b^3\right)}{3 a d \left(a^2-b^2\right)^2}+\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"(-2*b^5*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(5/2)*d) - ArcTanh[Cos[c + d*x]]/(a*d) + Sec[c + d*x]/(a*d) + Sec[c + d*x]^3/(3*a*d) + (b*Sec[c + d*x]^3*(b - a*Sin[c + d*x]))/(3*a*(a^2 - b^2)*d) - (b*Sec[c + d*x]*(3*b^3 + a*(2*a^2 - 5*b^2)*Sin[c + d*x]))/(3*a*(a^2 - b^2)^2*d)","A",12,10,27,0.3704,1,"{2898, 2622, 302, 207, 2696, 2866, 12, 2660, 618, 204}"
1357,1,247,0,0.473599,"\int \frac{\csc ^2(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","\frac{2 b^6 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{5/2}}-\frac{b^2 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^2 d \left(a^2-b^2\right)}+\frac{b^2 \sec (c+d x) \left(a \left(2 a^2-5 b^2\right) \sin (c+d x)+3 b^3\right)}{3 a^2 d \left(a^2-b^2\right)^2}-\frac{b \sec ^3(c+d x)}{3 a^2 d}-\frac{b \sec (c+d x)}{a^2 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\tan ^3(c+d x)}{3 a d}+\frac{2 \tan (c+d x)}{a d}-\frac{\cot (c+d x)}{a d}","\frac{2 b^6 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{5/2}}+\frac{\left(-10 a^2 b^2+6 a^4+b^4\right) \tan (c+d x)}{3 a d \left(a^2-b^2\right)^2}+\frac{b \left(2 b^2-a^2\right) \sec (c+d x)}{d \left(a^2-b^2\right)^2}+\frac{b \sec ^3(c+d x) (b \sin (c+d x)-a)}{3 a d \left(a^2-b^2\right)}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\tan ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}",1,"(2*b^6*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a*d) - (b*Sec[c + d*x])/(a^2*d) - (b*Sec[c + d*x]^3)/(3*a^2*d) - (b^2*Sec[c + d*x]^3*(b - a*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*d) + (b^2*Sec[c + d*x]*(3*b^3 + a*(2*a^2 - 5*b^2)*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*d) + (2*Tan[c + d*x])/(a*d) + Tan[c + d*x]^3/(3*a*d)","A",15,12,29,0.4138,1,"{2898, 2622, 302, 207, 2620, 270, 2696, 2866, 12, 2660, 618, 204}"
1358,1,332,0,0.5479125,"\int \frac{\csc ^3(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^3*Sec[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","-\frac{2 b^7 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{5/2}}+\frac{b^2 \sec ^3(c+d x)}{3 a^3 d}+\frac{b^2 \sec (c+d x)}{a^3 d}-\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 d \left(a^2-b^2\right)}-\frac{b^3 \sec (c+d x) \left(a \left(2 a^2-5 b^2\right) \sin (c+d x)+3 b^3\right)}{3 a^3 d \left(a^2-b^2\right)^2}-\frac{b \tan ^3(c+d x)}{3 a^2 d}-\frac{2 b \tan (c+d x)}{a^2 d}+\frac{b \cot (c+d x)}{a^2 d}+\frac{5 \sec ^3(c+d x)}{6 a d}+\frac{5 \sec (c+d x)}{2 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\csc ^2(c+d x) \sec ^3(c+d x)}{2 a d}","-\frac{2 b^7 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{5/2}}+\frac{b^2 \sec ^3(c+d x)}{3 a^3 d}+\frac{b^2 \sec (c+d x)}{a^3 d}-\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{b^3 \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a^3 d \left(a^2-b^2\right)}-\frac{b^3 \sec (c+d x) \left(a \left(2 a^2-5 b^2\right) \sin (c+d x)+3 b^3\right)}{3 a^3 d \left(a^2-b^2\right)^2}-\frac{b \tan ^3(c+d x)}{3 a^2 d}-\frac{2 b \tan (c+d x)}{a^2 d}+\frac{b \cot (c+d x)}{a^2 d}+\frac{5 \sec ^3(c+d x)}{6 a d}+\frac{5 \sec (c+d x)}{2 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\csc ^2(c+d x) \sec ^3(c+d x)}{2 a d}",1,"(-2*b^7*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(5/2)*d) - (5*ArcTanh[Cos[c + d*x]])/(2*a*d) - (b^2*ArcTanh[Cos[c + d*x]])/(a^3*d) + (b*Cot[c + d*x])/(a^2*d) + (5*Sec[c + d*x])/(2*a*d) + (b^2*Sec[c + d*x])/(a^3*d) + (5*Sec[c + d*x]^3)/(6*a*d) + (b^2*Sec[c + d*x]^3)/(3*a^3*d) - (Csc[c + d*x]^2*Sec[c + d*x]^3)/(2*a*d) + (b^3*Sec[c + d*x]^3*(b - a*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*d) - (b^3*Sec[c + d*x]*(3*b^3 + a*(2*a^2 - 5*b^2)*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*d) - (2*b*Tan[c + d*x])/(a^2*d) - (b*Tan[c + d*x]^3)/(3*a^2*d)","A",20,13,29,0.4483,1,"{2898, 2622, 302, 207, 2620, 270, 288, 2696, 2866, 12, 2660, 618, 204}"
1359,1,240,0,0.617745,"\int \frac{\sin ^3(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^3*Tan[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","-\frac{a^8 \log (a+b \sin (c+d x))}{b^3 d \left(a^2-b^2\right)^3}-\frac{\left(35 a^2+57 a b+24 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(35 a^2-57 a b+24 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(4 b \left(4 a^2-3 b^2\right)-a \left(13 a^2-9 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}+\frac{a \sin (c+d x)}{b^2 d}-\frac{\sin ^2(c+d x)}{2 b d}","-\frac{a^8 \log (a+b \sin (c+d x))}{b^3 d \left(a^2-b^2\right)^3}-\frac{\left(35 a^2+57 a b+24 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(35 a^2-57 a b+24 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(4 b \left(4 a^2-3 b^2\right)-a \left(13 a^2-9 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}+\frac{a \sin (c+d x)}{b^2 d}-\frac{\sin ^2(c+d x)}{2 b d}",1,"-((35*a^2 + 57*a*b + 24*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + ((35*a^2 - 57*a*b + 24*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (a^8*Log[a + b*Sin[c + d*x]])/(b^3*(a^2 - b^2)^3*d) + (a*Sin[c + d*x])/(b^2*d) - Sin[c + d*x]^2/(2*b*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(4*b*(4*a^2 - 3*b^2) - a*(13*a^2 - 9*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",6,4,29,0.1379,1,"{2837, 12, 1647, 1629}"
1360,1,221,0,0.5391194,"\int \frac{\sin ^2(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","\frac{a^7 \log (a+b \sin (c+d x))}{b^2 d \left(a^2-b^2\right)^3}-\frac{\left(24 a^2+37 a b+15 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{\left(24 a^2-37 a b+15 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{\sec ^4(c+d x) (a-b \sin (c+d x))}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \left(4 a \left(3 a^2-2 b^2\right)-b \left(13 a^2-9 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x)}{b d}","\frac{a^7 \log (a+b \sin (c+d x))}{b^2 d \left(a^2-b^2\right)^3}-\frac{\left(24 a^2+37 a b+15 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{\left(24 a^2-37 a b+15 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{\sec ^4(c+d x) (a-b \sin (c+d x))}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \left(4 a \left(3 a^2-2 b^2\right)-b \left(13 a^2-9 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x)}{b d}",1,"-((24*a^2 + 37*a*b + 15*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) - ((24*a^2 - 37*a*b + 15*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (a^7*Log[a + b*Sin[c + d*x]])/(b^2*(a^2 - b^2)^3*d) - Sin[c + d*x]/(b*d) + (Sec[c + d*x]^4*(a - b*Sin[c + d*x]))/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*(4*a*(3*a^2 - 2*b^2) - b*(13*a^2 - 9*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",6,4,29,0.1379,1,"{2837, 12, 1647, 1629}"
1361,1,208,0,0.5121659,"\int \frac{\sin (c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","-\frac{a^6 \log (a+b \sin (c+d x))}{b d \left(a^2-b^2\right)^3}-\frac{\left(15 a^2+21 a b+8 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(15 a^2-21 a b+8 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(4 b \left(3 a^2-2 b^2\right)-a \left(9 a^2-5 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}","-\frac{a^6 \log (a+b \sin (c+d x))}{b d \left(a^2-b^2\right)^3}-\frac{\left(15 a^2+21 a b+8 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(15 a^2-21 a b+8 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(4 b \left(3 a^2-2 b^2\right)-a \left(9 a^2-5 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}",1,"-((15*a^2 + 21*a*b + 8*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + ((15*a^2 - 21*a*b + 8*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (a^6*Log[a + b*Sin[c + d*x]])/(b*(a^2 - b^2)^3*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(4*b*(3*a^2 - 2*b^2) - a*(9*a^2 - 5*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",6,4,27,0.1481,1,"{2837, 12, 1647, 1629}"
1362,1,204,0,0.3505262,"\int \frac{\tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Tan[c + d*x]^5/(a + b*Sin[c + d*x]),x]","\frac{a^5 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\left(8 a^2+9 a b+3 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{\left(8 a^2-9 a b+3 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{\sec ^4(c+d x) (a-b \sin (c+d x))}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \left(4 a \left(2 a^2-b^2\right)-b \left(9 a^2-5 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}","\frac{a^5 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\left(8 a^2+9 a b+3 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{\left(8 a^2-9 a b+3 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{\sec ^4(c+d x) (a-b \sin (c+d x))}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \left(4 a \left(2 a^2-b^2\right)-b \left(9 a^2-5 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}",1,"-((8*a^2 + 9*a*b + 3*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) - ((8*a^2 - 9*a*b + 3*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (a^5*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + (Sec[c + d*x]^4*(a - b*Sin[c + d*x]))/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*(4*a*(2*a^2 - b^2) - b*(9*a^2 - 5*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",5,3,21,0.1429,1,"{2721, 1647, 801}"
1363,1,190,0,0.4313292,"\int \frac{\sec (c+d x) \tan ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","-\frac{a^4 b \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(4 b \left(2 a^2-b^2\right)-a \left(5 a^2-b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}-\frac{a (3 a+b) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{a (3 a-b) \log (\sin (c+d x)+1)}{16 d (a-b)^3}","-\frac{a^4 b \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(4 b \left(2 a^2-b^2\right)-a \left(5 a^2-b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}-\frac{a (3 a+b) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{a (3 a-b) \log (\sin (c+d x)+1)}{16 d (a-b)^3}",1,"-(a*(3*a + b)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + (a*(3*a - b)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (a^4*b*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(4*b*(2*a^2 - b^2) - a*(5*a^2 - b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",6,4,27,0.1481,1,"{2837, 12, 1647, 801}"
1364,1,182,0,0.3727059,"\int \frac{\sec ^2(c+d x) \tan ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^2*Tan[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{a^3 b^2 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}+\frac{\sec ^4(c+d x) (a-b \sin (c+d x))}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \left(4 a^3-b \left(5 a^2-b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}+\frac{b (3 a+b) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{b (3 a-b) \log (\sin (c+d x)+1)}{16 d (a-b)^3}","\frac{a^3 b^2 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}+\frac{\sec ^4(c+d x) (a-b \sin (c+d x))}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \left(4 a^3-b \left(5 a^2-b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}+\frac{b (3 a+b) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{b (3 a-b) \log (\sin (c+d x)+1)}{16 d (a-b)^3}",1,"(b*(3*a + b)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) - ((3*a - b)*b*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (a^3*b^2*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + (Sec[c + d*x]^4*(a - b*Sin[c + d*x]))/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*(4*a^3 - b*(5*a^2 - b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",6,5,29,0.1724,1,"{2837, 12, 1647, 823, 801}"
1365,1,178,0,0.3796823,"\int \frac{\sec ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^3*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{a^2 b^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{a \sec ^2(c+d x) \left(4 a b-\left(a^2+3 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}+\frac{a (a+3 b) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{a (a-3 b) \log (\sin (c+d x)+1)}{16 d (a-b)^3}","-\frac{a^2 b^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{a \sec ^2(c+d x) \left(4 a b-\left(a^2+3 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}+\frac{a (a+3 b) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{a (a-3 b) \log (\sin (c+d x)+1)}{16 d (a-b)^3}",1,"(a*(a + 3*b)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) - (a*(a - 3*b)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (a^2*b^3*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (a*Sec[c + d*x]^2*(4*a*b - (a^2 + 3*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",6,5,29,0.1724,1,"{2837, 12, 1647, 823, 801}"
1366,1,177,0,0.2427199,"\int \frac{\sec ^4(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^4*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{a b^4 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}+\frac{\sec ^4(c+d x) (a-b \sin (c+d x))}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \left(4 a b^2-b \left(a^2+3 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}-\frac{b (a+3 b) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{b (a-3 b) \log (\sin (c+d x)+1)}{16 d (a-b)^3}","\frac{a b^4 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}+\frac{\sec ^4(c+d x) (a-b \sin (c+d x))}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \left(4 a b^2-b \left(a^2+3 b^2\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}-\frac{b (a+3 b) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{b (a-3 b) \log (\sin (c+d x)+1)}{16 d (a-b)^3}",1,"-(b*(a + 3*b)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + ((a - 3*b)*b*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (a*b^4*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + (Sec[c + d*x]^4*(a - b*Sin[c + d*x]))/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*(4*a*b^2 - b*(a^2 + 3*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",6,4,27,0.1481,1,"{2837, 12, 823, 801}"
1367,1,233,0,0.365385,"\int \frac{\csc (c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","\frac{b^6 \log (a+b \sin (c+d x))}{a d \left(a^2-b^2\right)^3}-\frac{\left(8 a^2+21 a b+15 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{\left(8 a^2-21 a b+15 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{5 a+7 b}{16 d (a+b)^2 (1-\sin (c+d x))}+\frac{5 a-7 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a+b) (1-\sin (c+d x))^2}+\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2}+\frac{\log (\sin (c+d x))}{a d}","\frac{b^6 \log (a+b \sin (c+d x))}{a d \left(a^2-b^2\right)^3}-\frac{\left(8 a^2+21 a b+15 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}-\frac{\left(8 a^2-21 a b+15 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{5 a+7 b}{16 d (a+b)^2 (1-\sin (c+d x))}+\frac{5 a-7 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a+b) (1-\sin (c+d x))^2}+\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2}+\frac{\log (\sin (c+d x))}{a d}",1,"-((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + Log[Sin[c + d*x]]/(a*d) - ((8*a^2 - 21*a*b + 15*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (b^6*Log[a + b*Sin[c + d*x]])/(a*(a^2 - b^2)^3*d) + 1/(16*(a + b)*d*(1 - Sin[c + d*x])^2) + (5*a + 7*b)/(16*(a + b)^2*d*(1 - Sin[c + d*x])) + 1/(16*(a - b)*d*(1 + Sin[c + d*x])^2) + (5*a - 7*b)/(16*(a - b)^2*d*(1 + Sin[c + d*x]))","A",4,3,27,0.1111,1,"{2837, 12, 894}"
1368,1,250,0,0.4045452,"\int \frac{\csc ^2(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","-\frac{b^7 \log (a+b \sin (c+d x))}{a^2 d \left(a^2-b^2\right)^3}-\frac{\left(15 a^2+37 a b+24 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(15 a^2-37 a b+24 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{7 a+9 b}{16 d (a+b)^2 (1-\sin (c+d x))}-\frac{7 a-9 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a+b) (1-\sin (c+d x))^2}-\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2}-\frac{\csc (c+d x)}{a d}","-\frac{b^7 \log (a+b \sin (c+d x))}{a^2 d \left(a^2-b^2\right)^3}-\frac{\left(15 a^2+37 a b+24 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(15 a^2-37 a b+24 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{b \log (\sin (c+d x))}{a^2 d}+\frac{7 a+9 b}{16 d (a+b)^2 (1-\sin (c+d x))}-\frac{7 a-9 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a+b) (1-\sin (c+d x))^2}-\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2}-\frac{\csc (c+d x)}{a d}",1,"-(Csc[c + d*x]/(a*d)) - ((15*a^2 + 37*a*b + 24*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) - (b*Log[Sin[c + d*x]])/(a^2*d) + ((15*a^2 - 37*a*b + 24*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (b^7*Log[a + b*Sin[c + d*x]])/(a^2*(a^2 - b^2)^3*d) + 1/(16*(a + b)*d*(1 - Sin[c + d*x])^2) + (7*a + 9*b)/(16*(a + b)^2*d*(1 - Sin[c + d*x])) - 1/(16*(a - b)*d*(1 + Sin[c + d*x])^2) - (7*a - 9*b)/(16*(a - b)^2*d*(1 + Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2837, 12, 894}"
1369,1,274,0,0.4738421,"\int \frac{\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Csc[c + d*x]^3*Sec[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","\frac{b^8 \log (a+b \sin (c+d x))}{a^3 d \left(a^2-b^2\right)^3}-\frac{\left(24 a^2+57 a b+35 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(3 a^2+b^2\right) \log (\sin (c+d x))}{a^3 d}-\frac{\left(24 a^2-57 a b+35 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{b \csc (c+d x)}{a^2 d}+\frac{9 a+11 b}{16 d (a+b)^2 (1-\sin (c+d x))}+\frac{9 a-11 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a+b) (1-\sin (c+d x))^2}+\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2}-\frac{\csc ^2(c+d x)}{2 a d}","\frac{b^8 \log (a+b \sin (c+d x))}{a^3 d \left(a^2-b^2\right)^3}-\frac{\left(24 a^2+57 a b+35 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(3 a^2+b^2\right) \log (\sin (c+d x))}{a^3 d}-\frac{\left(24 a^2-57 a b+35 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}+\frac{b \csc (c+d x)}{a^2 d}+\frac{9 a+11 b}{16 d (a+b)^2 (1-\sin (c+d x))}+\frac{9 a-11 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{16 d (a+b) (1-\sin (c+d x))^2}+\frac{1}{16 d (a-b) (\sin (c+d x)+1)^2}-\frac{\csc ^2(c+d x)}{2 a d}",1,"(b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - ((24*a^2 + 57*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + ((3*a^2 + b^2)*Log[Sin[c + d*x]])/(a^3*d) - ((24*a^2 - 57*a*b + 35*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (b^8*Log[a + b*Sin[c + d*x]])/(a^3*(a^2 - b^2)^3*d) + 1/(16*(a + b)*d*(1 - Sin[c + d*x])^2) + (9*a + 11*b)/(16*(a + b)^2*d*(1 - Sin[c + d*x])) + 1/(16*(a - b)*d*(1 + Sin[c + d*x])^2) + (9*a - 11*b)/(16*(a - b)^2*d*(1 + Sin[c + d*x]))","A",4,3,29,0.1034,1,"{2837, 12, 894}"
1370,1,500,0,1.237415,"\int \frac{\sqrt{g \cos (e+f x)} \sin ^4(e+f x)}{a+b \sin (e+f x)} \, dx","Int[(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^4)/(a + b*Sin[e + f*x]),x]","-\frac{2 a^2 (g \cos (e+f x))^{3/2}}{3 b^3 f g}+\frac{a^4 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f \sqrt[4]{b^2-a^2}}-\frac{a^4 \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f \sqrt[4]{b^2-a^2}}-\frac{2 a^3 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^4 f \sqrt{\cos (e+f x)}}+\frac{a^5 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^5 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x) (g \cos (e+f x))^{3/2}}{5 b^2 f g}-\frac{4 a E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 b^2 f \sqrt{\cos (e+f x)}}+\frac{2 (g \cos (e+f x))^{7/2}}{7 b f g^3}-\frac{2 (g \cos (e+f x))^{3/2}}{3 b f g}","-\frac{2 a^2 (g \cos (e+f x))^{3/2}}{3 b^3 f g}+\frac{a^4 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f \sqrt[4]{b^2-a^2}}-\frac{a^4 \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f \sqrt[4]{b^2-a^2}}-\frac{2 a^3 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^4 f \sqrt{\cos (e+f x)}}+\frac{a^5 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^5 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x) (g \cos (e+f x))^{3/2}}{5 b^2 f g}-\frac{4 a E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 b^2 f \sqrt{\cos (e+f x)}}+\frac{2 (g \cos (e+f x))^{7/2}}{7 b f g^3}-\frac{2 (g \cos (e+f x))^{3/2}}{3 b f g}",1,"(a^4*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*(-a^2 + b^2)^(1/4)*f) - (a^4*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*(-a^2 + b^2)^(1/4)*f) - (2*a^2*(g*Cos[e + f*x])^(3/2))/(3*b^3*f*g) - (2*(g*Cos[e + f*x])^(3/2))/(3*b*f*g) + (2*(g*Cos[e + f*x])^(7/2))/(7*b*f*g^3) - (2*a^3*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b^4*f*Sqrt[Cos[e + f*x]]) - (4*a*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*b^2*f*Sqrt[Cos[e + f*x]]) + (a^5*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^5*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (a^5*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^5*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (2*a*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(5*b^2*f*g)","A",21,14,33,0.4242,1,"{2898, 2640, 2639, 2565, 30, 2568, 14, 2701, 2807, 2805, 329, 298, 205, 208}"
1371,1,448,0,0.9602731,"\int \frac{\sqrt{g \cos (e+f x)} \sin ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Int[(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","-\frac{a^3 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f \sqrt[4]{b^2-a^2}}+\frac{a^3 \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f \sqrt[4]{b^2-a^2}}+\frac{2 a^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^3 f \sqrt{\cos (e+f x)}}-\frac{a^4 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{a^4 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{2 a (g \cos (e+f x))^{3/2}}{3 b^2 f g}-\frac{2 \sin (e+f x) (g \cos (e+f x))^{3/2}}{5 b f g}+\frac{4 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 b f \sqrt{\cos (e+f x)}}","-\frac{a^3 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f \sqrt[4]{b^2-a^2}}+\frac{a^3 \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f \sqrt[4]{b^2-a^2}}+\frac{2 a^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^3 f \sqrt{\cos (e+f x)}}-\frac{a^4 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{a^4 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{2 a (g \cos (e+f x))^{3/2}}{3 b^2 f g}-\frac{2 \sin (e+f x) (g \cos (e+f x))^{3/2}}{5 b f g}+\frac{4 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 b f \sqrt{\cos (e+f x)}}",1,"-((a^3*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*(-a^2 + b^2)^(1/4)*f)) + (a^3*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*(-a^2 + b^2)^(1/4)*f) + (2*a*(g*Cos[e + f*x])^(3/2))/(3*b^2*f*g) + (2*a^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b^3*f*Sqrt[Cos[e + f*x]]) + (4*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*b*f*Sqrt[Cos[e + f*x]]) - (a^4*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (a^4*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (2*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(5*b*f*g)","A",18,13,33,0.3939,1,"{2898, 2640, 2639, 2565, 30, 2568, 2701, 2807, 2805, 329, 298, 205, 208}"
1372,1,369,0,0.8720777,"\int \frac{\sqrt{g \cos (e+f x)} \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Int[(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","\frac{a^2 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f \sqrt[4]{b^2-a^2}}-\frac{a^2 \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f \sqrt[4]{b^2-a^2}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{2 a E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^2 f \sqrt{\cos (e+f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{3 b f g}","\frac{a^2 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f \sqrt[4]{b^2-a^2}}-\frac{a^2 \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f \sqrt[4]{b^2-a^2}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^3 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{2 a E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^2 f \sqrt{\cos (e+f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{3 b f g}",1,"(a^2*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(1/4)*f) - (a^2*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(1/4)*f) - (2*(g*Cos[e + f*x])^(3/2))/(3*b*f*g) - (2*a*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b^2*f*Sqrt[Cos[e + f*x]]) + (a^3*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (a^3*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])","A",15,12,33,0.3636,1,"{2898, 2640, 2639, 2565, 30, 2701, 2807, 2805, 329, 298, 205, 208}"
1373,1,341,0,0.7347098,"\int \frac{\sqrt{g \cos (e+f x)} \sin (e+f x)}{a+b \sin (e+f x)} \, dx","Int[(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(a + b*Sin[e + f*x]),x]","-\frac{a \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f \sqrt[4]{b^2-a^2}}+\frac{a \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f \sqrt[4]{b^2-a^2}}-\frac{a^2 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{a^2 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b f \sqrt{\cos (e+f x)}}","-\frac{a \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f \sqrt[4]{b^2-a^2}}+\frac{a \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f \sqrt[4]{b^2-a^2}}-\frac{a^2 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{a^2 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b f \sqrt{\cos (e+f x)}}",1,"-((a*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(1/4)*f)) + (a*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(1/4)*f) + (2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b*f*Sqrt[Cos[e + f*x]]) - (a^2*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (a^2*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])","A",12,10,31,0.3226,1,"{2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
1374,1,355,0,0.8325318,"\int \frac{\sqrt{g \cos (e+f x)} \csc (e+f x)}{a+b \sin (e+f x)} \, dx","Int[(Sqrt[g*Cos[e + f*x]]*Csc[e + f*x])/(a + b*Sin[e + f*x]),x]","-\frac{\sqrt{b} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f \sqrt[4]{b^2-a^2}}+\frac{\sqrt{b} \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f \sqrt[4]{b^2-a^2}}-\frac{g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{\sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}-\frac{\sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}","-\frac{\sqrt{b} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f \sqrt[4]{b^2-a^2}}+\frac{\sqrt{b} \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f \sqrt[4]{b^2-a^2}}-\frac{g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{\sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}-\frac{\sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}",1,"(Sqrt[g]*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f) - (Sqrt[b]*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(1/4)*f) - (Sqrt[g]*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f) + (Sqrt[b]*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(1/4)*f) - (g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])","A",16,11,31,0.3548,1,"{2898, 2565, 329, 298, 203, 206, 2701, 2807, 2805, 205, 208}"
1375,1,433,0,0.9386721,"\int \frac{\sqrt{g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Int[(Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","\frac{b^{3/2} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f \sqrt[4]{b^2-a^2}}-\frac{b^{3/2} \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f \sqrt[4]{b^2-a^2}}+\frac{b g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{b g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{b \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}+\frac{b \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}-\frac{\csc (e+f x) (g \cos (e+f x))^{3/2}}{a f g}-\frac{E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{\cos (e+f x)}}","\frac{b^{3/2} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f \sqrt[4]{b^2-a^2}}-\frac{b^{3/2} \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f \sqrt[4]{b^2-a^2}}+\frac{b g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{b g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{b \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}+\frac{b \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}-\frac{\csc (e+f x) (g \cos (e+f x))^{3/2}}{a f g}-\frac{E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{\cos (e+f x)}}",1,"-((b*Sqrt[g]*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f)) + (b^(3/2)*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(1/4)*f) + (b*Sqrt[g]*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f) - (b^(3/2)*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(1/4)*f) - ((g*Cos[e + f*x])^(3/2)*Csc[e + f*x])/(a*f*g) - (Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*f*Sqrt[Cos[e + f*x]]) + (b*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (b*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])","A",19,14,33,0.4242,1,"{2898, 2565, 329, 298, 203, 206, 2570, 2640, 2639, 2701, 2807, 2805, 205, 208}"
1376,1,544,0,1.0572614,"\int \frac{\sqrt{g \cos (e+f x)} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Int[(Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","-\frac{b^{5/2} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f \sqrt[4]{b^2-a^2}}+\frac{b^2 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}+\frac{b^{5/2} \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f \sqrt[4]{b^2-a^2}}-\frac{b^2 \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}-\frac{b^2 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{b^2 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{b \csc (e+f x) (g \cos (e+f x))^{3/2}}{a^2 f g}+\frac{b E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a^2 f \sqrt{\cos (e+f x)}}+\frac{\sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}-\frac{\csc ^2(e+f x) (g \cos (e+f x))^{3/2}}{2 a f g}-\frac{\sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}","-\frac{b^{5/2} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f \sqrt[4]{b^2-a^2}}+\frac{b^2 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}+\frac{b^{5/2} \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f \sqrt[4]{b^2-a^2}}-\frac{b^2 \sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}-\frac{b^2 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{b^2 g \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{b \csc (e+f x) (g \cos (e+f x))^{3/2}}{a^2 f g}+\frac{b E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a^2 f \sqrt{\cos (e+f x)}}+\frac{\sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}-\frac{\csc ^2(e+f x) (g \cos (e+f x))^{3/2}}{2 a f g}-\frac{\sqrt{g} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}",1,"(Sqrt[g]*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f) + (b^2*Sqrt[g]*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) - (b^(5/2)*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*(-a^2 + b^2)^(1/4)*f) - (Sqrt[g]*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f) - (b^2*Sqrt[g]*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) + (b^(5/2)*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*(-a^2 + b^2)^(1/4)*f) + (b*(g*Cos[e + f*x])^(3/2)*Csc[e + f*x])/(a^2*f*g) - ((g*Cos[e + f*x])^(3/2)*Csc[e + f*x]^2)/(2*a*f*g) + (b*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a^2*f*Sqrt[Cos[e + f*x]]) - (b^2*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (b^2*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])","A",25,15,33,0.4545,1,"{2898, 2565, 329, 298, 203, 206, 2570, 2640, 2639, 290, 2701, 2807, 2805, 205, 208}"
1377,1,621,0,1.5407208,"\int \frac{(g \cos (e+f x))^{3/2} \sin ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","\frac{a^3 g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f}+\frac{a^3 g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f}-\frac{2 a^4 g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \sqrt{g \cos (e+f x)}}+\frac{2 a^2 g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 b^3 f \sqrt{g \cos (e+f x)}}+\frac{a^4 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 a^3 g \sqrt{g \cos (e+f x)}}{b^4 f}+\frac{2 a^2 g \sin (e+f x) \sqrt{g \cos (e+f x)}}{3 b^3 f}+\frac{2 a (g \cos (e+f x))^{5/2}}{5 b^2 f g}+\frac{4 g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{21 b f \sqrt{g \cos (e+f x)}}-\frac{2 \sin (e+f x) (g \cos (e+f x))^{5/2}}{7 b f g}+\frac{4 g \sin (e+f x) \sqrt{g \cos (e+f x)}}{21 b f}","\frac{a^3 g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f}+\frac{a^3 g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f}-\frac{2 a^4 g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \sqrt{g \cos (e+f x)}}+\frac{2 a^2 g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 b^3 f \sqrt{g \cos (e+f x)}}+\frac{a^4 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 a^3 g \sqrt{g \cos (e+f x)}}{b^4 f}+\frac{2 a^2 g \sin (e+f x) \sqrt{g \cos (e+f x)}}{3 b^3 f}+\frac{2 a (g \cos (e+f x))^{5/2}}{5 b^2 f g}+\frac{4 g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{21 b f \sqrt{g \cos (e+f x)}}-\frac{2 \sin (e+f x) (g \cos (e+f x))^{5/2}}{7 b f g}+\frac{4 g \sin (e+f x) \sqrt{g \cos (e+f x)}}{21 b f}",1,"(a^3*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*f) + (a^3*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*f) - (2*a^3*g*Sqrt[g*Cos[e + f*x]])/(b^4*f) + (2*a*(g*Cos[e + f*x])^(5/2))/(5*b^2*f*g) - (2*a^4*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b^5*f*Sqrt[g*Cos[e + f*x]]) + (2*a^2*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*b^3*f*Sqrt[g*Cos[e + f*x]]) + (4*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(21*b*f*Sqrt[g*Cos[e + f*x]]) + (a^4*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^5*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (a^4*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^5*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (2*a^2*g*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(3*b^3*f) + (4*g*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(21*b*f) - (2*(g*Cos[e + f*x])^(5/2)*Sin[e + f*x])/(7*b*f*g)","A",24,16,33,0.4848,1,"{2898, 2635, 2642, 2641, 2565, 30, 2568, 2695, 2867, 2702, 2807, 2805, 329, 212, 208, 205}"
1378,1,514,0,1.2144841,"\int \frac{(g \cos (e+f x))^{3/2} \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","-\frac{a^2 g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f}-\frac{a^2 g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f}+\frac{2 a^3 g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \sqrt{g \cos (e+f x)}}-\frac{a^3 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 a^2 g \sqrt{g \cos (e+f x)}}{b^3 f}-\frac{2 a g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 b^2 f \sqrt{g \cos (e+f x)}}-\frac{2 a g \sin (e+f x) \sqrt{g \cos (e+f x)}}{3 b^2 f}-\frac{2 (g \cos (e+f x))^{5/2}}{5 b f g}","-\frac{a^2 g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f}-\frac{a^2 g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f}+\frac{2 a^3 g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \sqrt{g \cos (e+f x)}}-\frac{a^3 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 a^2 g \sqrt{g \cos (e+f x)}}{b^3 f}-\frac{2 a g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 b^2 f \sqrt{g \cos (e+f x)}}-\frac{2 a g \sin (e+f x) \sqrt{g \cos (e+f x)}}{3 b^2 f}-\frac{2 (g \cos (e+f x))^{5/2}}{5 b f g}",1,"-((a^2*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*f)) - (a^2*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*f) + (2*a^2*g*Sqrt[g*Cos[e + f*x]])/(b^3*f) - (2*(g*Cos[e + f*x])^(5/2))/(5*b*f*g) + (2*a^3*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b^4*f*Sqrt[g*Cos[e + f*x]]) - (2*a*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*b^2*f*Sqrt[g*Cos[e + f*x]]) - (a^3*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (a^3*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (2*a*g*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(3*b^2*f)","A",20,15,33,0.4545,1,"{2898, 2635, 2642, 2641, 2565, 30, 2695, 2867, 2702, 2807, 2805, 329, 212, 208, 205}"
1379,1,426,0,0.9831973,"\int \frac{(g \cos (e+f x))^{3/2} \sin (e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(a + b*Sin[e + f*x]),x]","\frac{a g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f}+\frac{a g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f}-\frac{2 g^2 \left(3 a^2-b^2\right) \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 b^3 f \sqrt{g \cos (e+f x)}}+\frac{a^2 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 g \sqrt{g \cos (e+f x)} (3 a-b \sin (e+f x))}{3 b^2 f}","\frac{a g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f}+\frac{a g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f}-\frac{2 g^2 \left(3 a^2-b^2\right) \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 b^3 f \sqrt{g \cos (e+f x)}}+\frac{a^2 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 g \sqrt{g \cos (e+f x)} (3 a-b \sin (e+f x))}{3 b^2 f}",1,"(a*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*f) + (a*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*f) - (2*(3*a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*b^3*f*Sqrt[g*Cos[e + f*x]]) + (a^2*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (a^2*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (2*g*Sqrt[g*Cos[e + f*x]]*(3*a - b*Sin[e + f*x]))/(3*b^2*f)","A",13,11,31,0.3548,1,"{2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
1380,1,439,0,1.1378211,"\int \frac{(g \cos (e+f x))^{3/2} \csc (e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Csc[e + f*x])/(a + b*Sin[e + f*x]),x]","\frac{g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a \sqrt{b} f}+\frac{g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a \sqrt{b} f}+\frac{g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}-\frac{g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}-\frac{2 g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \sqrt{g \cos (e+f x)}}","\frac{g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a \sqrt{b} f}+\frac{g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a \sqrt{b} f}+\frac{g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}-\frac{g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}-\frac{2 g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \sqrt{g \cos (e+f x)}}",1,"-((g^(3/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f)) + ((-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*Sqrt[b]*f) - (g^(3/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f) + ((-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*Sqrt[b]*f) - (2*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b*f*Sqrt[g*Cos[e + f*x]]) + ((a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + ((a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])","A",21,16,31,0.5161,1,"{2898, 2565, 321, 329, 212, 206, 203, 2695, 2867, 2642, 2641, 2702, 2807, 2805, 208, 205}"
1381,1,469,0,1.2572322,"\int \frac{(g \cos (e+f x))^{3/2} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Csc[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","-\frac{\sqrt{b} g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f}-\frac{\sqrt{b} g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f}-\frac{g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{b g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}+\frac{b g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}+\frac{g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \sqrt{g \cos (e+f x)}}-\frac{g \csc (e+f x) \sqrt{g \cos (e+f x)}}{a f}","-\frac{\sqrt{b} g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f}-\frac{\sqrt{b} g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f}-\frac{g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{b g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}+\frac{b g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}+\frac{g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \sqrt{g \cos (e+f x)}}-\frac{g \csc (e+f x) \sqrt{g \cos (e+f x)}}{a f}",1,"(b*g^(3/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f) - (Sqrt[b]*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*f) + (b*g^(3/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f) - (Sqrt[b]*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*f) - (g*Sqrt[g*Cos[e + f*x]]*Csc[e + f*x])/(a*f) + (g^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(a*f*Sqrt[g*Cos[e + f*x]]) - ((a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - ((a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])","A",24,17,33,0.5152,1,"{2898, 2565, 321, 329, 212, 206, 203, 2567, 2642, 2641, 2695, 2867, 2702, 2807, 2805, 208, 205}"
1382,1,574,0,1.3702924,"\int \frac{(g \cos (e+f x))^{3/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Csc[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","-\frac{b^2 g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}+\frac{b^{3/2} g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f}-\frac{b^2 g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}+\frac{b^{3/2} g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f}+\frac{b g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{b g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{b g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \sqrt{g \cos (e+f x)}}+\frac{b g \csc (e+f x) \sqrt{g \cos (e+f x)}}{a^2 f}+\frac{g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}+\frac{g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}-\frac{g \csc ^2(e+f x) \sqrt{g \cos (e+f x)}}{2 a f}","-\frac{b^2 g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}+\frac{b^{3/2} g^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f}-\frac{b^2 g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}+\frac{b^{3/2} g^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f}+\frac{b g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{b g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{b g^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \sqrt{g \cos (e+f x)}}+\frac{b g \csc (e+f x) \sqrt{g \cos (e+f x)}}{a^2 f}+\frac{g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}+\frac{g^{3/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}-\frac{g \csc ^2(e+f x) \sqrt{g \cos (e+f x)}}{2 a f}",1,"(g^(3/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f) - (b^2*g^(3/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) + (b^(3/2)*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*f) + (g^(3/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f) - (b^2*g^(3/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) + (b^(3/2)*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*f) + (b*g*Sqrt[g*Cos[e + f*x]]*Csc[e + f*x])/(a^2*f) - (g*Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^2)/(2*a*f) - (b*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(a^2*f*Sqrt[g*Cos[e + f*x]]) + (b*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (b*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])","A",30,18,33,0.5455,1,"{2898, 2565, 321, 329, 212, 206, 203, 2567, 2642, 2641, 288, 2695, 2867, 2702, 2807, 2805, 208, 205}"
1383,1,610,0,1.3591357,"\int \frac{(g \cos (e+f x))^{5/2} \sin ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","-\frac{a^3 g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} f}+\frac{a^3 g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} f}-\frac{2 a^4 g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^5 f \sqrt{\cos (e+f x)}}+\frac{6 a^2 g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 b^3 f \sqrt{\cos (e+f x)}}+\frac{a^4 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^6 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^6 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{2 a^3 g (g \cos (e+f x))^{3/2}}{3 b^4 f}+\frac{2 a^2 g \sin (e+f x) (g \cos (e+f x))^{3/2}}{5 b^3 f}+\frac{2 a (g \cos (e+f x))^{7/2}}{7 b^2 f g}+\frac{4 g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 b f \sqrt{\cos (e+f x)}}-\frac{2 \sin (e+f x) (g \cos (e+f x))^{7/2}}{9 b f g}+\frac{4 g \sin (e+f x) (g \cos (e+f x))^{3/2}}{45 b f}","-\frac{a^3 g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} f}+\frac{a^3 g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} f}-\frac{2 a^4 g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^5 f \sqrt{\cos (e+f x)}}+\frac{6 a^2 g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 b^3 f \sqrt{\cos (e+f x)}}+\frac{a^4 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^6 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^6 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{2 a^3 g (g \cos (e+f x))^{3/2}}{3 b^4 f}+\frac{2 a^2 g \sin (e+f x) (g \cos (e+f x))^{3/2}}{5 b^3 f}+\frac{2 a (g \cos (e+f x))^{7/2}}{7 b^2 f g}+\frac{4 g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{15 b f \sqrt{\cos (e+f x)}}-\frac{2 \sin (e+f x) (g \cos (e+f x))^{7/2}}{9 b f g}+\frac{4 g \sin (e+f x) (g \cos (e+f x))^{3/2}}{45 b f}",1,"-((a^3*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(11/2)*f)) + (a^3*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(11/2)*f) - (2*a^3*g*(g*Cos[e + f*x])^(3/2))/(3*b^4*f) + (2*a*(g*Cos[e + f*x])^(7/2))/(7*b^2*f*g) - (2*a^4*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b^5*f*Sqrt[Cos[e + f*x]]) + (6*a^2*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*b^3*f*Sqrt[Cos[e + f*x]]) + (4*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*b*f*Sqrt[Cos[e + f*x]]) + (a^4*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^6*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (a^4*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^6*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (2*a^2*g*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(5*b^3*f) + (4*g*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(45*b*f) - (2*(g*Cos[e + f*x])^(7/2)*Sin[e + f*x])/(9*b*f*g)","A",24,16,33,0.4848,1,"{2898, 2635, 2640, 2639, 2565, 30, 2568, 2695, 2867, 2701, 2807, 2805, 329, 298, 205, 208}"
1384,1,501,0,1.1936833,"\int \frac{(g \cos (e+f x))^{5/2} \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","\frac{a^2 g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f}-\frac{a^2 g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f}+\frac{2 a^3 g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^4 f \sqrt{\cos (e+f x)}}-\frac{a^3 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{2 a^2 g (g \cos (e+f x))^{3/2}}{3 b^3 f}-\frac{6 a g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 b^2 f \sqrt{\cos (e+f x)}}-\frac{2 a g \sin (e+f x) (g \cos (e+f x))^{3/2}}{5 b^2 f}-\frac{2 (g \cos (e+f x))^{7/2}}{7 b f g}","\frac{a^2 g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f}-\frac{a^2 g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} f}+\frac{2 a^3 g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^4 f \sqrt{\cos (e+f x)}}-\frac{a^3 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^5 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{2 a^2 g (g \cos (e+f x))^{3/2}}{3 b^3 f}-\frac{6 a g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 b^2 f \sqrt{\cos (e+f x)}}-\frac{2 a g \sin (e+f x) (g \cos (e+f x))^{3/2}}{5 b^2 f}-\frac{2 (g \cos (e+f x))^{7/2}}{7 b f g}",1,"(a^2*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*f) - (a^2*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*f) + (2*a^2*g*(g*Cos[e + f*x])^(3/2))/(3*b^3*f) - (2*(g*Cos[e + f*x])^(7/2))/(7*b*f*g) + (2*a^3*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b^4*f*Sqrt[Cos[e + f*x]]) - (6*a*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*b^2*f*Sqrt[Cos[e + f*x]]) - (a^3*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^5*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (a^3*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^5*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (2*a*g*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(5*b^2*f)","A",20,15,33,0.4545,1,"{2898, 2635, 2640, 2639, 2565, 30, 2695, 2867, 2701, 2807, 2805, 329, 298, 205, 208}"
1385,1,413,0,0.9722546,"\int \frac{(g \cos (e+f x))^{5/2} \sin (e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(5/2)*Sin[e + f*x])/(a + b*Sin[e + f*x]),x]","-\frac{a g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f}+\frac{a g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f}-\frac{2 g^2 \left(5 a^2-3 b^2\right) E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 b^3 f \sqrt{\cos (e+f x)}}+\frac{a^2 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{2 g (g \cos (e+f x))^{3/2} (5 a-3 b \sin (e+f x))}{15 b^2 f}","-\frac{a g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f}+\frac{a g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f}-\frac{2 g^2 \left(5 a^2-3 b^2\right) E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{5 b^3 f \sqrt{\cos (e+f x)}}+\frac{a^2 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{2 g (g \cos (e+f x))^{3/2} (5 a-3 b \sin (e+f x))}{15 b^2 f}",1,"-((a*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*f)) + (a*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*f) - (2*(5*a^2 - 3*b^2)*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*b^3*f*Sqrt[Cos[e + f*x]]) + (a^2*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (a^2*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (2*g*(g*Cos[e + f*x])^(3/2)*(5*a - 3*b*Sin[e + f*x]))/(15*b^2*f)","A",13,11,31,0.3548,1,"{2865, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
1386,1,425,0,1.1485433,"\int \frac{(g \cos (e+f x))^{5/2} \csc (e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(5/2)*Csc[e + f*x])/(a + b*Sin[e + f*x]),x]","-\frac{g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a b^{3/2} f}+\frac{g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a b^{3/2} f}+\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{g^{5/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}-\frac{g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}-\frac{2 g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b f \sqrt{\cos (e+f x)}}","-\frac{g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a b^{3/2} f}+\frac{g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a b^{3/2} f}+\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{g^{5/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}-\frac{g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f}-\frac{2 g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b f \sqrt{\cos (e+f x)}}",1,"(g^(5/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f) - ((-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*b^(3/2)*f) - (g^(5/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f) + ((-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*b^(3/2)*f) - (2*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b*f*Sqrt[Cos[e + f*x]]) + ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])","A",21,16,31,0.5161,1,"{2898, 2565, 321, 329, 298, 203, 206, 2695, 2867, 2640, 2639, 2701, 2807, 2805, 205, 208}"
1387,1,462,0,1.2614049,"\int \frac{(g \cos (e+f x))^{5/2} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(5/2)*Csc[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","\frac{g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 \sqrt{b} f}-\frac{g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 \sqrt{b} f}-\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a b f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a b f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{b g^{5/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}+\frac{b g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}-\frac{g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{\cos (e+f x)}}-\frac{g \csc (e+f x) (g \cos (e+f x))^{3/2}}{a f}","\frac{g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 \sqrt{b} f}-\frac{g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 \sqrt{b} f}-\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a b f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a b f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{b g^{5/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}+\frac{b g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f}-\frac{g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f \sqrt{\cos (e+f x)}}-\frac{g \csc (e+f x) (g \cos (e+f x))^{3/2}}{a f}",1,"-((b*g^(5/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f)) + ((-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*Sqrt[b]*f) + (b*g^(5/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f) - ((-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*Sqrt[b]*f) - (g*(g*Cos[e + f*x])^(3/2)*Csc[e + f*x])/(a*f) - (g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*f*Sqrt[Cos[e + f*x]]) - ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*b*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*b*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])","A",24,17,33,0.5152,1,"{2898, 2565, 321, 329, 298, 203, 206, 2567, 2640, 2639, 2695, 2867, 2701, 2807, 2805, 205, 208}"
1388,1,557,0,1.3434607,"\int \frac{(g \cos (e+f x))^{5/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(5/2)*Csc[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","\frac{b^2 g^{5/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}-\frac{\sqrt{b} g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f}-\frac{b^2 g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}+\frac{\sqrt{b} g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f}+\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{b g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a^2 f \sqrt{\cos (e+f x)}}+\frac{b g \csc (e+f x) (g \cos (e+f x))^{3/2}}{a^2 f}-\frac{3 g^{5/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}+\frac{3 g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}-\frac{g \csc ^2(e+f x) (g \cos (e+f x))^{3/2}}{2 a f}","\frac{b^2 g^{5/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}-\frac{\sqrt{b} g^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f}-\frac{b^2 g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}+\frac{\sqrt{b} g^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f}+\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{g^3 \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{b g^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a^2 f \sqrt{\cos (e+f x)}}+\frac{b g \csc (e+f x) (g \cos (e+f x))^{3/2}}{a^2 f}-\frac{3 g^{5/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}+\frac{3 g^{5/2} \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f}-\frac{g \csc ^2(e+f x) (g \cos (e+f x))^{3/2}}{2 a f}",1,"(-3*g^(5/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f) + (b^2*g^(5/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) - (Sqrt[b]*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*f) + (3*g^(5/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f) - (b^2*g^(5/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) + (Sqrt[b]*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*f) + (b*g*(g*Cos[e + f*x])^(3/2)*Csc[e + f*x])/(a^2*f) - (g*(g*Cos[e + f*x])^(3/2)*Csc[e + f*x]^2)/(2*a*f) + (b*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a^2*f*Sqrt[Cos[e + f*x]]) + ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])","A",30,18,33,0.5455,1,"{2898, 2565, 321, 329, 298, 203, 206, 2567, 2640, 2639, 288, 2695, 2867, 2701, 2807, 2805, 205, 208}"
1389,1,509,0,1.5061094,"\int \frac{\sin ^4(e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]^4/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","-\frac{2 a^2 \sqrt{g \cos (e+f x)}}{b^3 f g}-\frac{a^4 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{a^4 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{2 a^3 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \sqrt{g \cos (e+f x)}}+\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x) \sqrt{g \cos (e+f x)}}{3 b^2 f g}-\frac{4 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 b^2 f \sqrt{g \cos (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{5 b f g^3}-\frac{2 \sqrt{g \cos (e+f x)}}{b f g}","-\frac{2 a^2 \sqrt{g \cos (e+f x)}}{b^3 f g}-\frac{a^4 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{a^4 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{2 a^3 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \sqrt{g \cos (e+f x)}}+\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^4 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x) \sqrt{g \cos (e+f x)}}{3 b^2 f g}-\frac{4 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 b^2 f \sqrt{g \cos (e+f x)}}+\frac{2 (g \cos (e+f x))^{5/2}}{5 b f g^3}-\frac{2 \sqrt{g \cos (e+f x)}}{b f g}",1,"-((a^4*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g])) - (a^4*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - (2*a^2*Sqrt[g*Cos[e + f*x]])/(b^3*f*g) - (2*Sqrt[g*Cos[e + f*x]])/(b*f*g) + (2*(g*Cos[e + f*x])^(5/2))/(5*b*f*g^3) - (2*a^3*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b^4*f*Sqrt[g*Cos[e + f*x]]) - (4*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*b^2*f*Sqrt[g*Cos[e + f*x]]) + (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (2*a*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(3*b^2*f*g)","A",23,15,33,0.4545,1,"{2909, 2565, 14, 2568, 2642, 2641, 30, 2867, 2702, 2807, 2805, 329, 212, 208, 205}"
1390,1,457,0,1.1799071,"\int \frac{\sin ^3(e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]^3/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{a^3 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{2 a^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \sqrt{g \cos (e+f x)}}-\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 a \sqrt{g \cos (e+f x)}}{b^2 f g}-\frac{2 \sin (e+f x) \sqrt{g \cos (e+f x)}}{3 b f g}+\frac{4 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 b f \sqrt{g \cos (e+f x)}}","\frac{a^3 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{2 a^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \sqrt{g \cos (e+f x)}}-\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 a \sqrt{g \cos (e+f x)}}{b^2 f g}-\frac{2 \sin (e+f x) \sqrt{g \cos (e+f x)}}{3 b f g}+\frac{4 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 b f \sqrt{g \cos (e+f x)}}",1,"(a^3*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (a^3*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (2*a*Sqrt[g*Cos[e + f*x]])/(b^2*f*g) + (2*a^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b^3*f*Sqrt[g*Cos[e + f*x]]) + (4*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*b*f*Sqrt[g*Cos[e + f*x]]) - (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(3*b*f*g)","A",19,14,33,0.4242,1,"{2909, 2568, 2642, 2641, 2565, 30, 2867, 2702, 2807, 2805, 329, 212, 208, 205}"
1391,1,380,0,0.9309468,"\int \frac{\sin ^2(e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]^2/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{g \cos (e+f x)}}{b f g}","-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{g \cos (e+f x)}}{b f g}",1,"-((a^2*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g])) - (a^2*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - (2*Sqrt[g*Cos[e + f*x]])/(b*f*g) - (2*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b^2*f*Sqrt[g*Cos[e + f*x]]) + (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])","A",15,13,33,0.3939,1,"{2909, 2565, 30, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
1392,1,352,0,0.7254197,"\int \frac{\sin (e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{a \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{a^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \left(b \sqrt{b^2-a^2}+a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{a^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \sqrt{g \cos (e+f x)}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{a^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \left(b \sqrt{b^2-a^2}+a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{a^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b f \sqrt{g \cos (e+f x)}}",1,"(a*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (a*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b*f*Sqrt[g*Cos[e + f*x]]) - (a^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b^2 + b*Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (a^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])","A",12,10,31,0.3226,1,"{2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
1393,1,369,0,0.8180026,"\int \frac{\csc (e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Int[Csc[e + f*x]/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{b \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{b \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f \sqrt{g}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f \sqrt{g}}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{b \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{b \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f \sqrt{g}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f \sqrt{g}}",1,"-(ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*Sqrt[g])) + (b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*Sqrt[g]) + (b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - (b*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (b*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])","A",16,11,31,0.3548,1,"{2898, 2565, 329, 212, 206, 203, 2702, 2807, 2805, 208, 205}"
1394,1,448,0,0.975404,"\int \frac{\csc ^2(e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Int[Csc[e + f*x]^2/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{b^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(b \sqrt{b^2-a^2}+a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{b^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f \sqrt{g}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f \sqrt{g}}-\frac{\csc (e+f x) \sqrt{g \cos (e+f x)}}{a f g}+\frac{\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \sqrt{g \cos (e+f x)}}","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f \sqrt{g} \left(b^2-a^2\right)^{3/4}}+\frac{b^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(b \sqrt{b^2-a^2}+a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{b^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f \sqrt{g}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f \sqrt{g}}-\frac{\csc (e+f x) \sqrt{g \cos (e+f x)}}{a f g}+\frac{\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{a f \sqrt{g \cos (e+f x)}}",1,"(b*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*Sqrt[g]) - (b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (b*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*Sqrt[g]) - (b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - (Sqrt[g*Cos[e + f*x]]*Csc[e + f*x])/(a*f*g) + (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(a*f*Sqrt[g*Cos[e + f*x]]) + (b^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(a^2 - b^2 + b*Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (b^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])","A",19,14,33,0.4242,1,"{2898, 2565, 329, 212, 206, 203, 2570, 2642, 2641, 2702, 2807, 2805, 208, 205}"
1395,1,557,0,1.0316495,"\int \frac{\csc ^3(e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Int[Csc[e + f*x]^3/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f \sqrt{g}}+\frac{b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f \sqrt{g}}-\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{b \csc (e+f x) \sqrt{g \cos (e+f x)}}{a^2 f g}-\frac{b \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \sqrt{g \cos (e+f x)}}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f \sqrt{g}}-\frac{\csc ^2(e+f x) \sqrt{g \cos (e+f x)}}{2 a f g}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f \sqrt{g}}","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f \sqrt{g}}+\frac{b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^3 f \sqrt{g} \left(b^2-a^2\right)^{3/4}}-\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f \sqrt{g}}-\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{b \csc (e+f x) \sqrt{g \cos (e+f x)}}{a^2 f g}-\frac{b \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{a^2 f \sqrt{g \cos (e+f x)}}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f \sqrt{g}}-\frac{\csc ^2(e+f x) \sqrt{g \cos (e+f x)}}{2 a f g}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{4 a f \sqrt{g}}",1,"(-3*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f*Sqrt[g]) - (b^2*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f*Sqrt[g]) + (b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - (3*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f*Sqrt[g]) - (b^2*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f*Sqrt[g]) + (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (b*Sqrt[g*Cos[e + f*x]]*Csc[e + f*x])/(a^2*f*g) - (Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^2)/(2*a*f*g) - (b*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(a^2*f*Sqrt[g*Cos[e + f*x]]) - (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])","A",25,15,33,0.4545,1,"{2898, 2565, 329, 212, 206, 203, 2570, 2642, 2641, 290, 2702, 2807, 2805, 208, 205}"
1396,1,584,0,1.2758566,"\int \frac{\sin ^4(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]^4/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{2 a^2 (g \cos (e+f x))^{3/2}}{3 b f g^3 \left(a^2-b^2\right)}-\frac{2 b (g \cos (e+f x))^{3/2}}{3 f g^3 \left(a^2-b^2\right)}+\frac{a^4 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{a^4 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{2 a^3 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^2 f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}-\frac{4 a E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}-\frac{2 b}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x)}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}","\frac{2 a^2 (g \cos (e+f x))^{3/2}}{3 b f g^3 \left(a^2-b^2\right)}-\frac{2 b (g \cos (e+f x))^{3/2}}{3 f g^3 \left(a^2-b^2\right)}+\frac{a^4 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{a^4 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{2 a^3 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b^2 f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}-\frac{4 a E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}-\frac{2 b}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x)}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^3 f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}",1,"(a^4*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (a^4*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (2*b)/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) + (2*a^2*(g*Cos[e + f*x])^(3/2))/(3*b*(a^2 - b^2)*f*g^3) - (2*b*(g*Cos[e + f*x])^(3/2))/(3*(a^2 - b^2)*f*g^3) - (4*a*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/((a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) + (2*a^3*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b^2*(a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) - (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) - (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (2*a*Sin[e + f*x])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])","A",22,15,33,0.4545,1,"{2902, 2566, 2640, 2639, 2565, 14, 2898, 30, 2701, 2807, 2805, 329, 298, 205, 208}"
1397,1,509,0,1.0674571,"\int \frac{\sin ^3(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]^3/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","-\frac{a^3 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{2 a^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}+\frac{4 b E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}+\frac{2 a}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{2 b \sin (e+f x)}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}","-\frac{a^3 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{2 a^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{b f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}+\frac{4 b E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}+\frac{2 a}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{2 b \sin (e+f x)}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}",1,"-((a^3*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2))) + (a^3*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2)) + (2*a)/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) - (2*a^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b*(a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) + (4*b*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/((a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) + (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) - (2*b*Sin[e + f*x])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])","A",18,14,33,0.4242,1,"{2902, 2565, 30, 2566, 2640, 2639, 2867, 2701, 2807, 2805, 329, 298, 205, 208}"
1398,1,453,0,0.8806795,"\int \frac{\sin ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]^2/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{2 a E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}-\frac{2 b}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x)}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}","\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{2 a E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}-\frac{2 b}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 a \sin (e+f x)}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}",1,"(a^2*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (a^2*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (2*b)/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) - (2*a*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/((a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) - (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) - (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (2*a*Sin[e + f*x])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])","A",15,13,33,0.3939,1,"{2902, 2636, 2640, 2639, 2565, 30, 2701, 2807, 2805, 329, 298, 205, 208}"
1399,1,413,0,0.953311,"\int \frac{\sin (e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","-\frac{a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{2 b E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}+\frac{2 (a-b \sin (e+f x))}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}","-\frac{a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{2 b E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}+\frac{2 (a-b \sin (e+f x))}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}",1,"-((a*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(5/4)*f*g^(3/2))) + (a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(5/4)*f*g^(3/2)) + (2*b*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/((a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) + (a^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (a^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (2*(a - b*Sin[e + f*x]))/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])","A",13,11,31,0.3548,1,"{2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
1400,1,507,0,1.3613993,"\int \frac{\csc (e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[Csc[e + f*x]/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{2 b E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}+\frac{2 b (b-a \sin (e+f x))}{a f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{b^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{b^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f g^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f g^{3/2}}+\frac{2}{a f g \sqrt{g \cos (e+f x)}}","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f g^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{2 b E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}+\frac{2 b (b-a \sin (e+f x))}{a f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{b^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}+\frac{b^2 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f g^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f g^{3/2}}+\frac{2}{a f g \sqrt{g \cos (e+f x)}}",1,"ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*g^(3/2)) - (b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*g^(3/2)) + (b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(5/4)*f*g^(3/2)) + 2/(a*f*g*Sqrt[g*Cos[e + f*x]]) + (2*b*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/((a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) + (b^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (b^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (2*b*(b - a*Sin[e + f*x]))/(a*(a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])","A",21,16,31,0.5161,1,"{2898, 2565, 325, 329, 298, 203, 206, 2696, 2867, 2640, 2639, 2701, 2807, 2805, 205, 208}"
1401,1,627,0,1.4932768,"\int \frac{\csc ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[Csc[e + f*x]^2/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{2 b^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}-\frac{2 b^2 (b-a \sin (e+f x))}{a^2 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{b \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f g^{3/2}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f g^{3/2}}-\frac{2 b}{a^2 f g \sqrt{g \cos (e+f x)}}-\frac{3 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f g^2 \sqrt{\cos (e+f x)}}+\frac{3 \sin (e+f x)}{a f g \sqrt{g \cos (e+f x)}}-\frac{\csc (e+f x)}{a f g \sqrt{g \cos (e+f x)}}","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f g^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{2 b^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f g^2 \left(a^2-b^2\right) \sqrt{\cos (e+f x)}}-\frac{2 b^2 (b-a \sin (e+f x))}{a^2 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f g \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{g \cos (e+f x)}}-\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f g \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{g \cos (e+f x)}}-\frac{b \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f g^{3/2}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f g^{3/2}}-\frac{2 b}{a^2 f g \sqrt{g \cos (e+f x)}}-\frac{3 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{g \cos (e+f x)}}{a f g^2 \sqrt{\cos (e+f x)}}+\frac{3 \sin (e+f x)}{a f g \sqrt{g \cos (e+f x)}}-\frac{\csc (e+f x)}{a f g \sqrt{g \cos (e+f x)}}",1,"-((b*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*g^(3/2))) + (b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(5/4)*f*g^(3/2)) + (b*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*g^(3/2)) - (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (2*b)/(a^2*f*g*Sqrt[g*Cos[e + f*x]]) - Csc[e + f*x]/(a*f*g*Sqrt[g*Cos[e + f*x]]) - (3*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*f*g^2*Sqrt[Cos[e + f*x]]) - (2*b^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*(a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) - (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) - (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (3*Sin[e + f*x])/(a*f*g*Sqrt[g*Cos[e + f*x]]) - (2*b^2*(b - a*Sin[e + f*x]))/(a^2*(a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])","A",25,18,33,0.5455,1,"{2898, 2565, 325, 329, 298, 203, 206, 2570, 2636, 2640, 2639, 2696, 2867, 2701, 2807, 2805, 205, 208}"
1402,1,601,0,1.3980369,"\int \frac{\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]^4/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{2 a^2 \sqrt{g \cos (e+f x)}}{b f g^3 \left(a^2-b^2\right)}-\frac{2 b \sqrt{g \cos (e+f x)}}{f g^3 \left(a^2-b^2\right)}-\frac{a^4 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{a^4 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{2 a^3 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{4 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 b}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}+\frac{2 a \sin (e+f x)}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}","\frac{2 a^2 \sqrt{g \cos (e+f x)}}{b f g^3 \left(a^2-b^2\right)}-\frac{2 b \sqrt{g \cos (e+f x)}}{f g^3 \left(a^2-b^2\right)}-\frac{a^4 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{a^4 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{2 a^3 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{4 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{a^5 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 b}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}+\frac{2 a \sin (e+f x)}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}",1,"-((a^4*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(7/4)*f*g^(5/2))) - (a^4*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b)/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2)) + (2*a^2*Sqrt[g*Cos[e + f*x]])/(b*(a^2 - b^2)*f*g^3) - (2*b*Sqrt[g*Cos[e + f*x]])/((a^2 - b^2)*f*g^3) - (4*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*a^3*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b^2*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) - (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) - (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*a*Sin[e + f*x])/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))","A",22,16,33,0.4848,1,"{2902, 2566, 2642, 2641, 2565, 14, 2909, 30, 2867, 2702, 2807, 2805, 329, 212, 208, 205}"
1403,1,528,0,1.0864916,"\int \frac{\sin ^3(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]^3/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{a^3 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{2 a^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{4 b \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 a}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}-\frac{2 b \sin (e+f x)}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}","\frac{a^3 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{2 a^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{4 b \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{b f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 a}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}-\frac{2 b \sin (e+f x)}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}",1,"(a^3*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + (a^3*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + (2*a)/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2)) - (2*a^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) + (4*b*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) + (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) - (2*b*Sin[e + f*x])/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))","A",18,14,33,0.4242,1,"{2902, 2565, 30, 2566, 2642, 2641, 2867, 2702, 2807, 2805, 329, 212, 208, 205}"
1404,1,468,0,0.9007209,"\int \frac{\sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]^2/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","-\frac{a^2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{a^2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{2 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 b}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}+\frac{2 a \sin (e+f x)}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}","-\frac{a^2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{a^2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{2 a \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{a^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 b}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}+\frac{2 a \sin (e+f x)}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}",1,"-((a^2*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(7/4)*f*g^(5/2))) - (a^2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b)/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2)) + (2*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) - (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) - (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*a*Sin[e + f*x])/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))","A",15,13,33,0.3939,1,"{2902, 2636, 2642, 2641, 2565, 30, 2702, 2807, 2805, 329, 212, 208, 205}"
1405,1,432,0,0.9212833,"\int \frac{\sin (e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Int[Sin[e + f*x]/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{a b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{a b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{2 b \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 b \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 b \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 (a-b \sin (e+f x))}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}","\frac{a b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{a b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{2 b \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 b \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{a^2 b \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 (a-b \sin (e+f x))}{3 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}",1,"(a*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(7/4)*f*g^(5/2)) + (a*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) + (a^2*b*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (a^2*b*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*(a - b*Sin[e + f*x]))/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))","A",13,11,31,0.3548,1,"{2866, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
1406,1,527,0,1.313508,"\int \frac{\csc (e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Int[Csc[e + f*x]/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{2 b \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 b (b-a \sin (e+f x))}{3 a f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f g^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f g^{5/2}}+\frac{2}{3 a f g (g \cos (e+f x))^{3/2}}","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{2 b \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}+\frac{b^3 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}+\frac{2 b (b-a \sin (e+f x))}{3 a f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f g^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a f g^{5/2}}+\frac{2}{3 a f g (g \cos (e+f x))^{3/2}}",1,"-(ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*g^(5/2))) + (b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(7/4)*f*g^(5/2)) - ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*g^(5/2)) + (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + 2/(3*a*f*g*(g*Cos[e + f*x])^(3/2)) - (2*b*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) + (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*b*(b - a*Sin[e + f*x]))/(3*a*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))","A",21,16,31,0.5161,1,"{2898, 2565, 325, 329, 212, 206, 203, 2696, 2867, 2642, 2641, 2702, 2807, 2805, 208, 205}"
1407,1,651,0,1.4643253,"\int \frac{\csc ^2(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Int[Csc[e + f*x]^2/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","-\frac{b^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{b^{9/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{2 b^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 a f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{b^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{b^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 b^2 (b-a \sin (e+f x))}{3 a^2 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f g^{5/2}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f g^{5/2}}-\frac{2 b}{3 a^2 f g (g \cos (e+f x))^{3/2}}+\frac{5 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 a f g^2 \sqrt{g \cos (e+f x)}}+\frac{5 \sin (e+f x)}{3 a f g (g \cos (e+f x))^{3/2}}-\frac{\csc (e+f x)}{a f g (g \cos (e+f x))^{3/2}}","-\frac{b^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f g^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{b^{9/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt[4]{b^2-a^2}}\right)}{a^2 f g^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{2 b^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 a f g^2 \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{b^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f g^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{g \cos (e+f x)}}-\frac{b^4 \sqrt{\cos (e+f x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (e+f x)\right|2\right)}{a f g^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{g \cos (e+f x)}}-\frac{2 b^2 (b-a \sin (e+f x))}{3 a^2 f g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f g^{5/2}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^2 f g^{5/2}}-\frac{2 b}{3 a^2 f g (g \cos (e+f x))^{3/2}}+\frac{5 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 a f g^2 \sqrt{g \cos (e+f x)}}+\frac{5 \sin (e+f x)}{3 a f g (g \cos (e+f x))^{3/2}}-\frac{\csc (e+f x)}{a f g (g \cos (e+f x))^{3/2}}",1,"(b*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*g^(5/2)) - (b^(9/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + (b*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*g^(5/2)) - (b^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b)/(3*a^2*f*g*(g*Cos[e + f*x])^(3/2)) - Csc[e + f*x]/(a*f*g*(g*Cos[e + f*x])^(3/2)) + (5*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*a*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*b^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*a*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) - (b^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) - (b^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (5*Sin[e + f*x])/(3*a*f*g*(g*Cos[e + f*x])^(3/2)) - (2*b^2*(b - a*Sin[e + f*x]))/(3*a^2*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))","A",25,18,33,0.5455,1,"{2898, 2565, 325, 329, 212, 206, 203, 2570, 2636, 2642, 2641, 2696, 2867, 2702, 2807, 2805, 208, 205}"
1408,1,926,0,1.8422557,"\int \frac{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx","Int[(Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2))/(a + b*Sin[e + f*x]),x]","-\frac{2 \sqrt{2} a^3 \sqrt{g} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} d^3}{b^3 \sqrt{b-a} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a^3 \sqrt{g} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} d^3}{b^3 \sqrt{b-a} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}+\frac{\sqrt{g} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) d^{5/2}}{4 \sqrt{2} b f}+\frac{a^2 \sqrt{g} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) d^{5/2}}{\sqrt{2} b^3 f}-\frac{\sqrt{g} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) d^{5/2}}{4 \sqrt{2} b f}-\frac{a^2 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) d^{5/2}}{\sqrt{2} b^3 f}-\frac{\sqrt{g} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{8 \sqrt{2} b f}-\frac{a^2 \sqrt{g} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} b^3 f}+\frac{\sqrt{g} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{8 \sqrt{2} b f}+\frac{a^2 \sqrt{g} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} b^3 f}-\frac{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)} d^2}{2 b f g}-\frac{a \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} d^2}{b^2 f \sqrt{\sin (2 e+2 f x)}}","-\frac{2 \sqrt{2} a^3 \sqrt{g} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} d^3}{b^3 \sqrt{b-a} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a^3 \sqrt{g} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} d^3}{b^3 \sqrt{b-a} \sqrt{a+b} f \sqrt{d \sin (e+f x)}}+\frac{\sqrt{g} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) d^{5/2}}{4 \sqrt{2} b f}+\frac{a^2 \sqrt{g} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) d^{5/2}}{\sqrt{2} b^3 f}-\frac{\sqrt{g} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) d^{5/2}}{4 \sqrt{2} b f}-\frac{a^2 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) d^{5/2}}{\sqrt{2} b^3 f}-\frac{\sqrt{g} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{8 \sqrt{2} b f}-\frac{a^2 \sqrt{g} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} b^3 f}+\frac{\sqrt{g} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{8 \sqrt{2} b f}+\frac{a^2 \sqrt{g} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} b^3 f}-\frac{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)} d^2}{2 b f g}-\frac{a \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} d^2}{b^2 f \sqrt{\sin (2 e+2 f x)}}",1,"(a^2*d^(5/2)*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^3*f) + (d^(5/2)*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(4*Sqrt[2]*b*f) - (a^2*d^(5/2)*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^3*f) - (d^(5/2)*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(4*Sqrt[2]*b*f) - (a^2*d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^3*f) - (d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) + (a^2*d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^3*f) + (d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) - (2*Sqrt[2]*a^3*d^3*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*a^3*d^3*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) - (d^2*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])/(2*b*f*g) - (a*d^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(b^2*f*Sqrt[Sin[2*e + 2*f*x]])","A",31,15,37,0.4054,1,"{2909, 2568, 2575, 297, 1162, 617, 204, 1165, 628, 2572, 2639, 2906, 2905, 490, 1218}"
1409,1,578,0,1.1335842,"\int \frac{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx","Int[(Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(3/2))/(a + b*Sin[e + f*x]),x]","\frac{2 \sqrt{2} a^2 d^2 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} a^2 d^2 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{a d^{3/2} \sqrt{g} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right)}{\sqrt{2} b^2 f}+\frac{a d^{3/2} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right)}{\sqrt{2} b^2 f}+\frac{a d^{3/2} \sqrt{g} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b^2 f}-\frac{a d^{3/2} \sqrt{g} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b^2 f}+\frac{d E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b f \sqrt{\sin (2 e+2 f x)}}","\frac{2 \sqrt{2} a^2 d^2 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} a^2 d^2 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{a d^{3/2} \sqrt{g} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right)}{\sqrt{2} b^2 f}+\frac{a d^{3/2} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right)}{\sqrt{2} b^2 f}+\frac{a d^{3/2} \sqrt{g} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b^2 f}-\frac{a d^{3/2} \sqrt{g} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b^2 f}+\frac{d E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b f \sqrt{\sin (2 e+2 f x)}}",1,"-((a*d^(3/2)*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^2*f)) + (a*d^(3/2)*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^2*f) + (a*d^(3/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^2*f) - (a*d^(3/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^2*f) + (2*Sqrt[2]*a^2*d^2*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^2*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*a^2*d^2*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^2*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) + (d*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(b*f*Sqrt[Sin[2*e + 2*f*x]])","A",19,14,37,0.3784,1,"{2909, 2572, 2639, 2575, 297, 1162, 617, 204, 1165, 628, 2906, 2905, 490, 1218}"
1410,1,509,0,0.8345676,"\int \frac{\sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Int[(Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(a + b*Sin[e + f*x]),x]","-\frac{2 \sqrt{2} a d \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a d \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{\sqrt{d} \sqrt{g} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right)}{\sqrt{2} b f}-\frac{\sqrt{d} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right)}{\sqrt{2} b f}-\frac{\sqrt{d} \sqrt{g} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b f}+\frac{\sqrt{d} \sqrt{g} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b f}","-\frac{2 \sqrt{2} a d \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a d \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{\sqrt{d} \sqrt{g} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right)}{\sqrt{2} b f}-\frac{\sqrt{d} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right)}{\sqrt{2} b f}-\frac{\sqrt{d} \sqrt{g} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b f}+\frac{\sqrt{d} \sqrt{g} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b f}",1,"(Sqrt[d]*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*f) - (Sqrt[d]*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*f) - (Sqrt[d]*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*f) + (Sqrt[d]*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*f) - (2*Sqrt[2]*a*d*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*a*d*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]])","A",16,12,37,0.3243,1,"{2909, 2575, 297, 1162, 617, 204, 1165, 628, 2906, 2905, 490, 1218}"
1411,1,208,0,0.4148905,"\int \frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx","Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{2 \sqrt{2} \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}","\frac{2 \sqrt{2} \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}",1,"(2*Sqrt[2]*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]])","A",5,4,37,0.1081,1,"{2906, 2905, 490, 1218}"
1412,1,320,0,0.7477194,"\int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","-\frac{2 \sqrt{2} b \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a d f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} b \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a d f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a d^2 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{a d f g \sqrt{d \sin (e+f x)}}","-\frac{2 \sqrt{2} b \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a d f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} b \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a d f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a d^2 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{a d f g \sqrt{d \sin (e+f x)}}",1,"(-2*(g*Cos[e + f*x])^(3/2))/(a*d*f*g*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*Sqrt[-a + b]*Sqrt[a + b]*d*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*Sqrt[-a + b]*Sqrt[a + b]*d*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a*d^2*f*Sqrt[Sin[2*e + 2*f*x]])","A",9,8,37,0.2162,1,"{2910, 2570, 2572, 2639, 2906, 2905, 490, 1218}"
1413,1,366,0,1.0294386,"\int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Int[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{2 \sqrt{2} b^2 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^2 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{2 b (g \cos (e+f x))^{3/2}}{a^2 d^2 f g \sqrt{d \sin (e+f x)}}+\frac{2 b E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^2 d^3 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}","\frac{2 \sqrt{2} b^2 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^2 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{2 b (g \cos (e+f x))^{3/2}}{a^2 d^2 f g \sqrt{d \sin (e+f x)}}+\frac{2 b E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^2 d^3 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}",1,"(-2*(g*Cos[e + f*x])^(3/2))/(3*a*d*f*g*(d*Sin[e + f*x])^(3/2)) + (2*b*(g*Cos[e + f*x])^(3/2))/(a^2*d^2*f*g*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^2*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*Sqrt[-a + b]*Sqrt[a + b]*d^2*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^2*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*Sqrt[-a + b]*Sqrt[a + b]*d^2*f*Sqrt[d*Sin[e + f*x]]) + (2*b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^2*d^3*f*Sqrt[Sin[2*e + 2*f*x]])","A",11,9,37,0.2432,1,"{2910, 2563, 2570, 2572, 2639, 2906, 2905, 490, 1218}"
1414,1,513,0,1.4535461,"\int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx","Int[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(7/2)*(a + b*Sin[e + f*x])),x]","-\frac{2 b^2 (g \cos (e+f x))^{3/2}}{a^3 d^3 f g \sqrt{d \sin (e+f x)}}-\frac{2 b^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^3 d^4 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 \sqrt{2} b^3 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^3 d^3 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} b^3 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^3 d^3 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{2 b (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f g (d \sin (e+f x))^{3/2}}-\frac{4 (g \cos (e+f x))^{3/2}}{5 a d^3 f g \sqrt{d \sin (e+f x)}}-\frac{4 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 a d^4 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}}","-\frac{2 b^2 (g \cos (e+f x))^{3/2}}{a^3 d^3 f g \sqrt{d \sin (e+f x)}}-\frac{2 b^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^3 d^4 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 \sqrt{2} b^3 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^3 d^3 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} b^3 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^3 d^3 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{2 b (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f g (d \sin (e+f x))^{3/2}}-\frac{4 (g \cos (e+f x))^{3/2}}{5 a d^3 f g \sqrt{d \sin (e+f x)}}-\frac{4 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 a d^4 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}}",1,"(-2*(g*Cos[e + f*x])^(3/2))/(5*a*d*f*g*(d*Sin[e + f*x])^(5/2)) + (2*b*(g*Cos[e + f*x])^(3/2))/(3*a^2*d^2*f*g*(d*Sin[e + f*x])^(3/2)) - (4*(g*Cos[e + f*x])^(3/2))/(5*a*d^3*f*g*Sqrt[d*Sin[e + f*x]]) - (2*b^2*(g*Cos[e + f*x])^(3/2))/(a^3*d^3*f*g*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^3*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^3*Sqrt[-a + b]*Sqrt[a + b]*d^3*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^3*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^3*Sqrt[-a + b]*Sqrt[a + b]*d^3*f*Sqrt[d*Sin[e + f*x]]) - (4*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*a*d^4*f*Sqrt[Sin[2*e + 2*f*x]]) - (2*b^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^3*d^4*f*Sqrt[Sin[2*e + 2*f*x]])","A",16,9,37,0.2432,1,"{2910, 2570, 2572, 2639, 2563, 2906, 2905, 490, 1218}"
1415,1,598,0,1.8242226,"\int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx","Int[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(9/2)*(a + b*Sin[e + f*x])),x]","\frac{2 b^3 (g \cos (e+f x))^{3/2}}{a^4 d^4 f g \sqrt{d \sin (e+f x)}}-\frac{2 b^2 (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f g (d \sin (e+f x))^{3/2}}+\frac{2 b^3 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^4 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{2} b^4 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^4 d^4 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^4 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^4 d^4 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{4 b (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f g \sqrt{d \sin (e+f x)}}+\frac{2 b (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f g (d \sin (e+f x))^{5/2}}+\frac{4 b E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 a^2 d^5 f \sqrt{\sin (2 e+2 f x)}}-\frac{8 (g \cos (e+f x))^{3/2}}{21 a d^3 f g (d \sin (e+f x))^{3/2}}-\frac{2 (g \cos (e+f x))^{3/2}}{7 a d f g (d \sin (e+f x))^{7/2}}","\frac{2 b^3 (g \cos (e+f x))^{3/2}}{a^4 d^4 f g \sqrt{d \sin (e+f x)}}-\frac{2 b^2 (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f g (d \sin (e+f x))^{3/2}}+\frac{2 b^3 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^4 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{2} b^4 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^4 d^4 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^4 \sqrt{g} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^4 d^4 f \sqrt{b-a} \sqrt{a+b} \sqrt{d \sin (e+f x)}}+\frac{4 b (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f g \sqrt{d \sin (e+f x)}}+\frac{2 b (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f g (d \sin (e+f x))^{5/2}}+\frac{4 b E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 a^2 d^5 f \sqrt{\sin (2 e+2 f x)}}-\frac{8 (g \cos (e+f x))^{3/2}}{21 a d^3 f g (d \sin (e+f x))^{3/2}}-\frac{2 (g \cos (e+f x))^{3/2}}{7 a d f g (d \sin (e+f x))^{7/2}}",1,"(-2*(g*Cos[e + f*x])^(3/2))/(7*a*d*f*g*(d*Sin[e + f*x])^(7/2)) + (2*b*(g*Cos[e + f*x])^(3/2))/(5*a^2*d^2*f*g*(d*Sin[e + f*x])^(5/2)) - (8*(g*Cos[e + f*x])^(3/2))/(21*a*d^3*f*g*(d*Sin[e + f*x])^(3/2)) - (2*b^2*(g*Cos[e + f*x])^(3/2))/(3*a^3*d^3*f*g*(d*Sin[e + f*x])^(3/2)) + (4*b*(g*Cos[e + f*x])^(3/2))/(5*a^2*d^4*f*g*Sqrt[d*Sin[e + f*x]]) + (2*b^3*(g*Cos[e + f*x])^(3/2))/(a^4*d^4*f*g*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^4*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^4*Sqrt[-a + b]*Sqrt[a + b]*d^4*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^4*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^4*Sqrt[-a + b]*Sqrt[a + b]*d^4*f*Sqrt[d*Sin[e + f*x]]) + (4*b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*a^2*d^5*f*Sqrt[Sin[2*e + 2*f*x]]) + (2*b^3*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^4*d^5*f*Sqrt[Sin[2*e + 2*f*x]])","A",19,9,37,0.2432,1,"{2910, 2570, 2563, 2572, 2639, 2906, 2905, 490, 1218}"
1416,1,982,0,1.6913392,"\int \frac{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2))/(a + b*Sin[e + f*x]),x]","\frac{2 \sqrt{2} a \sqrt{b^2-a^2} d^{3/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right) g^2}{b^3 f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} a \sqrt{b^2-a^2} d^{3/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right) g^2}{b^3 f \sqrt{g \cos (e+f x)}}+\frac{a d^2 F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{\sin (2 e+2 f x)} g^2}{2 b^2 f \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}+\frac{\left(a^2-b^2\right) d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right) g^{3/2}}{\sqrt{2} b^3 f}+\frac{3 d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right) g^{3/2}}{4 \sqrt{2} b f}-\frac{\left(a^2-b^2\right) d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right) g^{3/2}}{\sqrt{2} b^3 f}-\frac{3 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right) g^{3/2}}{4 \sqrt{2} b f}-\frac{\left(a^2-b^2\right) d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{d}-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right) g^{3/2}}{2 \sqrt{2} b^3 f}-\frac{3 d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{d}-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right) g^{3/2}}{8 \sqrt{2} b f}+\frac{\left(a^2-b^2\right) d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{d}+\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right) g^{3/2}}{2 \sqrt{2} b^3 f}+\frac{3 d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{d}+\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right) g^{3/2}}{8 \sqrt{2} b f}+\frac{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{3/2} g}{2 b f}-\frac{a d \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)} g}{b^2 f}","\frac{2 \sqrt{2} a \sqrt{b^2-a^2} d^{3/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right) g^2}{b^3 f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} a \sqrt{b^2-a^2} d^{3/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right) g^2}{b^3 f \sqrt{g \cos (e+f x)}}+\frac{a d^2 F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{\sin (2 e+2 f x)} g^2}{2 b^2 f \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}+\frac{\left(a^2-b^2\right) d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right) g^{3/2}}{\sqrt{2} b^3 f}+\frac{3 d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right) g^{3/2}}{4 \sqrt{2} b f}-\frac{\left(a^2-b^2\right) d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right) g^{3/2}}{\sqrt{2} b^3 f}-\frac{3 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right) g^{3/2}}{4 \sqrt{2} b f}-\frac{\left(a^2-b^2\right) d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{d}-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right) g^{3/2}}{2 \sqrt{2} b^3 f}-\frac{3 d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{d}-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right) g^{3/2}}{8 \sqrt{2} b f}+\frac{\left(a^2-b^2\right) d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{d}+\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right) g^{3/2}}{2 \sqrt{2} b^3 f}+\frac{3 d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{d}+\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}\right) g^{3/2}}{8 \sqrt{2} b f}+\frac{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{3/2} g}{2 b f}-\frac{a d \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)} g}{b^2 f}",1,"(3*d^(3/2)*g^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(4*Sqrt[2]*b*f) + ((a^2 - b^2)*d^(3/2)*g^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^3*f) - (3*d^(3/2)*g^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(4*Sqrt[2]*b*f) - ((a^2 - b^2)*d^(3/2)*g^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^3*f) + (2*Sqrt[2]*a*Sqrt[-a^2 + b^2]*d^(3/2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^3*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*a*Sqrt[-a^2 + b^2]*d^(3/2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^3*f*Sqrt[g*Cos[e + f*x]]) - (3*d^(3/2)*g^(3/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(8*Sqrt[2]*b*f) - ((a^2 - b^2)*d^(3/2)*g^(3/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^3*f) + (3*d^(3/2)*g^(3/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(8*Sqrt[2]*b*f) + ((a^2 - b^2)*d^(3/2)*g^(3/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^3*f) - (a*d*g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(b^2*f) + (g*Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(3/2))/(2*b*f) + (a*d^2*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(2*b^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",31,16,37,0.4324,1,"{2901, 2838, 2568, 2573, 2641, 2574, 297, 1162, 617, 204, 1165, 628, 2909, 2908, 2907, 1218}"
1417,1,611,0,1.0299023,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])/(a + b*Sin[e + f*x]),x]","-\frac{2 \sqrt{2} \sqrt{d} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} \sqrt{d} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{g \cos (e+f x)}}-\frac{a \sqrt{d} g^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right)}{\sqrt{2} b^2 f}+\frac{a \sqrt{d} g^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right)}{\sqrt{2} b^2 f}+\frac{a \sqrt{d} g^{3/2} \log \left(-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b^2 f}-\frac{a \sqrt{d} g^{3/2} \log \left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b^2 f}-\frac{d g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{2 b f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{g \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b f}","-\frac{2 \sqrt{2} \sqrt{d} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} \sqrt{d} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{g \cos (e+f x)}}-\frac{a \sqrt{d} g^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right)}{\sqrt{2} b^2 f}+\frac{a \sqrt{d} g^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right)}{\sqrt{2} b^2 f}+\frac{a \sqrt{d} g^{3/2} \log \left(-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b^2 f}-\frac{a \sqrt{d} g^{3/2} \log \left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b^2 f}-\frac{d g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{2 b f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{g \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b f}",1,"-((a*Sqrt[d]*g^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^2*f)) + (a*Sqrt[d]*g^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^2*f) - (2*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[d]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^2*f*Sqrt[g*Cos[e + f*x]]) + (2*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[d]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^2*f*Sqrt[g*Cos[e + f*x]]) + (a*Sqrt[d]*g^(3/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^2*f) - (a*Sqrt[d]*g^(3/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^2*f) + (g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(b*f) - (d*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(2*b*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",19,15,37,0.4054,1,"{2901, 2838, 2574, 297, 1162, 617, 204, 1165, 628, 2568, 2573, 2641, 2908, 2907, 1218}"
1418,1,577,0,0.9723016,"\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^(3/2)/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a b \sqrt{d} f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a b \sqrt{d} f \sqrt{g \cos (e+f x)}}+\frac{g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{g^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right)}{\sqrt{2} b \sqrt{d} f}-\frac{g^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right)}{\sqrt{2} b \sqrt{d} f}-\frac{g^{3/2} \log \left(-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b \sqrt{d} f}+\frac{g^{3/2} \log \left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b \sqrt{d} f}","\frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a b \sqrt{d} f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a b \sqrt{d} f \sqrt{g \cos (e+f x)}}+\frac{g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{g^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right)}{\sqrt{2} b \sqrt{d} f}-\frac{g^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right)}{\sqrt{2} b \sqrt{d} f}-\frac{g^{3/2} \log \left(-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b \sqrt{d} f}+\frac{g^{3/2} \log \left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b \sqrt{d} f}",1,"(g^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*Sqrt[d]*f) - (g^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*Sqrt[d]*f) + (2*Sqrt[2]*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*b*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*b*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (g^(3/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b*Sqrt[d]*f) + (g^(3/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b*Sqrt[d]*f) + (g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",18,14,37,0.3784,1,"{2900, 2838, 2573, 2641, 2574, 297, 1162, 617, 204, 1165, 628, 2908, 2907, 1218}"
1419,1,321,0,0.7016256,"\int \frac{(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","-\frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 d^{3/2} f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 d^{3/2} f \sqrt{g \cos (e+f x)}}-\frac{b g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^2 d f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 g \sqrt{g \cos (e+f x)}}{a d f \sqrt{d \sin (e+f x)}}","-\frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 d^{3/2} f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 d^{3/2} f \sqrt{g \cos (e+f x)}}-\frac{b g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^2 d f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 g \sqrt{g \cos (e+f x)}}{a d f \sqrt{d \sin (e+f x)}}",1,"(-2*Sqrt[2]*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*d^(3/2)*f*Sqrt[g*Cos[e + f*x]]) + (2*Sqrt[2]*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*d^(3/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*g*Sqrt[g*Cos[e + f*x]])/(a*d*f*Sqrt[d*Sin[e + f*x]]) - (b*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^2*d*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",8,7,37,0.1892,1,"{2899, 2563, 2573, 2641, 2908, 2907, 1218}"
1420,1,435,0,1.0158305,"\int \frac{(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","-\frac{g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^3 d^2 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^3 d^{5/2} f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^3 d^{5/2} f \sqrt{g \cos (e+f x)}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{a^2 d^2 f \sqrt{d \sin (e+f x)}}+\frac{2 g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{3 a d^2 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 g \sqrt{g \cos (e+f x)}}{3 a d f (d \sin (e+f x))^{3/2}}","-\frac{g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^3 d^2 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^3 d^{5/2} f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^3 d^{5/2} f \sqrt{g \cos (e+f x)}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{a^2 d^2 f \sqrt{d \sin (e+f x)}}+\frac{2 g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{3 a d^2 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 g \sqrt{g \cos (e+f x)}}{3 a d f (d \sin (e+f x))^{3/2}}",1,"(2*Sqrt[2]*b*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^3*d^(5/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*b*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^3*d^(5/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*g*Sqrt[g*Cos[e + f*x]])/(3*a*d*f*(d*Sin[e + f*x])^(3/2)) + (2*b*g*Sqrt[g*Cos[e + f*x]])/(a^2*d^2*f*Sqrt[d*Sin[e + f*x]]) + (2*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3*a*d^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) - ((a^2 - b^2)*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^3*d^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",12,9,37,0.2432,1,"{2899, 2570, 2573, 2641, 2563, 2910, 2908, 2907, 1218}"
1421,1,525,0,1.352881,"\int \frac{(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((d*Sin[e + f*x])^(7/2)*(a + b*Sin[e + f*x])),x]","\frac{b g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^4 d^3 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b^2 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^4 d^{7/2} f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^2 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^4 d^{7/2} f \sqrt{g \cos (e+f x)}}+\frac{2 g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}{a^3 d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 b g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{3 a^2 d^3 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{3 a^2 d^2 f (d \sin (e+f x))^{3/2}}-\frac{8 g \sqrt{g \cos (e+f x)}}{5 a d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 g \sqrt{g \cos (e+f x)}}{5 a d f (d \sin (e+f x))^{5/2}}","\frac{b g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^4 d^3 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b^2 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^4 d^{7/2} f \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^2 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^4 d^{7/2} f \sqrt{g \cos (e+f x)}}+\frac{2 g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}{a^3 d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 b g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{3 a^2 d^3 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{3 a^2 d^2 f (d \sin (e+f x))^{3/2}}-\frac{8 g \sqrt{g \cos (e+f x)}}{5 a d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 g \sqrt{g \cos (e+f x)}}{5 a d f (d \sin (e+f x))^{5/2}}",1,"(-2*Sqrt[2]*b^2*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^4*d^(7/2)*f*Sqrt[g*Cos[e + f*x]]) + (2*Sqrt[2]*b^2*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^4*d^(7/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*g*Sqrt[g*Cos[e + f*x]])/(5*a*d*f*(d*Sin[e + f*x])^(5/2)) + (2*b*g*Sqrt[g*Cos[e + f*x]])/(3*a^2*d^2*f*(d*Sin[e + f*x])^(3/2)) - (8*g*Sqrt[g*Cos[e + f*x]])/(5*a*d^3*f*Sqrt[d*Sin[e + f*x]]) + (2*(a^2 - b^2)*g*Sqrt[g*Cos[e + f*x]])/(a^3*d^3*f*Sqrt[d*Sin[e + f*x]]) - (2*b*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3*a^2*d^3*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) + (b*(a^2 - b^2)*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^4*d^3*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",15,9,37,0.2432,1,"{2899, 2570, 2563, 2573, 2641, 2910, 2908, 2907, 1218}"
1422,1,688,0,1.7793705,"\int \frac{(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^(3/2)/((d*Sin[e + f*x])^(9/2)*(a + b*Sin[e + f*x])),x]","-\frac{b^2 g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^5 d^4 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{3 a^3 d^4 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^3 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^5 d^{9/2} f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b^3 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^5 d^{9/2} f \sqrt{g \cos (e+f x)}}-\frac{2 b g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac{8 b g \sqrt{g \cos (e+f x)}}{5 a^2 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}+\frac{4 g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{7 a d^4 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{4 g \sqrt{g \cos (e+f x)}}{7 a d^3 f (d \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{g \cos (e+f x)}}{7 a d f (d \sin (e+f x))^{7/2}}","-\frac{b^2 g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^5 d^4 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{3 a^3 d^4 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^3 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^5 d^{9/2} f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b^3 g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^5 d^{9/2} f \sqrt{g \cos (e+f x)}}-\frac{2 b g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac{8 b g \sqrt{g \cos (e+f x)}}{5 a^2 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 b g \sqrt{g \cos (e+f x)}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}+\frac{4 g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{7 a d^4 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{4 g \sqrt{g \cos (e+f x)}}{7 a d^3 f (d \sin (e+f x))^{3/2}}-\frac{2 g \sqrt{g \cos (e+f x)}}{7 a d f (d \sin (e+f x))^{7/2}}",1,"(2*Sqrt[2]*b^3*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^5*d^(9/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*b^3*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^5*d^(9/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*g*Sqrt[g*Cos[e + f*x]])/(7*a*d*f*(d*Sin[e + f*x])^(7/2)) + (2*b*g*Sqrt[g*Cos[e + f*x]])/(5*a^2*d^2*f*(d*Sin[e + f*x])^(5/2)) - (4*g*Sqrt[g*Cos[e + f*x]])/(7*a*d^3*f*(d*Sin[e + f*x])^(3/2)) + (2*(a^2 - b^2)*g*Sqrt[g*Cos[e + f*x]])/(3*a^3*d^3*f*(d*Sin[e + f*x])^(3/2)) + (8*b*g*Sqrt[g*Cos[e + f*x]])/(5*a^2*d^4*f*Sqrt[d*Sin[e + f*x]]) - (2*b*(a^2 - b^2)*g*Sqrt[g*Cos[e + f*x]])/(a^4*d^4*f*Sqrt[d*Sin[e + f*x]]) + (4*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(7*a*d^4*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) - (2*(a^2 - b^2)*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3*a^3*d^4*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) - (b^2*(a^2 - b^2)*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^5*d^4*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",20,9,37,0.2432,1,"{2899, 2570, 2573, 2641, 2563, 2910, 2908, 2907, 1218}"
1423,1,936,0,1.600188,"\int \frac{(g \cos (e+f x))^{5/2} \sqrt{d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Int[((g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]])/(a + b*Sin[e + f*x]),x]","-\frac{\left(a^2-b^2\right) \sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) g^{5/2}}{\sqrt{2} b^3 f}-\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) g^{5/2}}{4 \sqrt{2} b f}+\frac{\left(a^2-b^2\right) \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) g^{5/2}}{\sqrt{2} b^3 f}+\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) g^{5/2}}{4 \sqrt{2} b f}+\frac{\left(a^2-b^2\right) \sqrt{d} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) g^{5/2}}{2 \sqrt{2} b^3 f}+\frac{\sqrt{d} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) g^{5/2}}{8 \sqrt{2} b f}-\frac{\left(a^2-b^2\right) \sqrt{d} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) g^{5/2}}{2 \sqrt{2} b^3 f}-\frac{\sqrt{d} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) g^{5/2}}{8 \sqrt{2} b f}-\frac{2 \sqrt{2} a \sqrt{b-a} \sqrt{a+b} d \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} g^{5/2}}{b^3 f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a \sqrt{b-a} \sqrt{a+b} d \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} g^{5/2}}{b^3 f \sqrt{d \sin (e+f x)}}+\frac{a \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} g^2}{b^2 f \sqrt{\sin (2 e+2 f x)}}+\frac{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)} g}{2 b f}","-\frac{\left(a^2-b^2\right) \sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) g^{5/2}}{\sqrt{2} b^3 f}-\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) g^{5/2}}{4 \sqrt{2} b f}+\frac{\left(a^2-b^2\right) \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) g^{5/2}}{\sqrt{2} b^3 f}+\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) g^{5/2}}{4 \sqrt{2} b f}+\frac{\left(a^2-b^2\right) \sqrt{d} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) g^{5/2}}{2 \sqrt{2} b^3 f}+\frac{\sqrt{d} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) g^{5/2}}{8 \sqrt{2} b f}-\frac{\left(a^2-b^2\right) \sqrt{d} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) g^{5/2}}{2 \sqrt{2} b^3 f}-\frac{\sqrt{d} \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) g^{5/2}}{8 \sqrt{2} b f}-\frac{2 \sqrt{2} a \sqrt{b-a} \sqrt{a+b} d \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} g^{5/2}}{b^3 f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a \sqrt{b-a} \sqrt{a+b} d \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} g^{5/2}}{b^3 f \sqrt{d \sin (e+f x)}}+\frac{a \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} g^2}{b^2 f \sqrt{\sin (2 e+2 f x)}}+\frac{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)} g}{2 b f}",1,"-(Sqrt[d]*g^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(4*Sqrt[2]*b*f) - ((a^2 - b^2)*Sqrt[d]*g^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^3*f) + (Sqrt[d]*g^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(4*Sqrt[2]*b*f) + ((a^2 - b^2)*Sqrt[d]*g^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^3*f) + (Sqrt[d]*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) + ((a^2 - b^2)*Sqrt[d]*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^3*f) - (Sqrt[d]*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) - ((a^2 - b^2)*Sqrt[d]*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^3*f) - (2*Sqrt[2]*a*Sqrt[-a + b]*Sqrt[a + b]*d*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*a*Sqrt[-a + b]*Sqrt[a + b]*d*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*f*Sqrt[d*Sin[e + f*x]]) + (g*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])/(2*b*f) + (a*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(b^2*f*Sqrt[Sin[2*e + 2*f*x]])","A",31,17,37,0.4595,1,"{2901, 2838, 2572, 2639, 2568, 2575, 297, 1162, 617, 204, 1165, 628, 2909, 2906, 2905, 490, 1218}"
1424,1,572,0,1.0230972,"\int \frac{(g \cos (e+f x))^{5/2}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^(5/2)/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{a g^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right)}{\sqrt{2} b^2 \sqrt{d} f}-\frac{a g^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right)}{\sqrt{2} b^2 \sqrt{d} f}+\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{a g^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b^2 \sqrt{d} f}+\frac{a g^{5/2} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b^2 \sqrt{d} f}-\frac{g^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b d f \sqrt{\sin (2 e+2 f x)}}","\frac{a g^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right)}{\sqrt{2} b^2 \sqrt{d} f}-\frac{a g^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right)}{\sqrt{2} b^2 \sqrt{d} f}+\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{d \sin (e+f x)}}-\frac{a g^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b^2 \sqrt{d} f}+\frac{a g^{5/2} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b^2 \sqrt{d} f}-\frac{g^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b d f \sqrt{\sin (2 e+2 f x)}}",1,"(a*g^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^2*Sqrt[d]*f) - (a*g^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^2*Sqrt[d]*f) - (a*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^2*Sqrt[d]*f) + (a*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^2*Sqrt[d]*f) + (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^2*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^2*f*Sqrt[d*Sin[e + f*x]]) - (g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(b*d*f*Sqrt[Sin[2*e + 2*f*x]])","A",19,15,37,0.4054,1,"{2901, 2838, 2575, 297, 1162, 617, 204, 1165, 628, 2572, 2639, 2906, 2905, 490, 1218}"
1425,1,616,0,1.1360626,"\int \frac{(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","-\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a b d f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a b d f \sqrt{d \sin (e+f x)}}-\frac{2 g^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a d^2 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 g (g \cos (e+f x))^{3/2}}{a d f \sqrt{d \sin (e+f x)}}-\frac{g^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right)}{\sqrt{2} b d^{3/2} f}+\frac{g^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right)}{\sqrt{2} b d^{3/2} f}+\frac{g^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b d^{3/2} f}-\frac{g^{5/2} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b d^{3/2} f}","-\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a b d f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a b d f \sqrt{d \sin (e+f x)}}-\frac{2 g^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a d^2 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 g (g \cos (e+f x))^{3/2}}{a d f \sqrt{d \sin (e+f x)}}-\frac{g^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right)}{\sqrt{2} b d^{3/2} f}+\frac{g^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right)}{\sqrt{2} b d^{3/2} f}+\frac{g^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b d^{3/2} f}-\frac{g^{5/2} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}+\sqrt{g} \cot (e+f x)+\sqrt{g}\right)}{2 \sqrt{2} b d^{3/2} f}",1,"-((g^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*d^(3/2)*f)) + (g^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*d^(3/2)*f) + (g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*d^(3/2)*f) - (g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*d^(3/2)*f) - (2*g*(g*Cos[e + f*x])^(3/2))/(a*d*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*b*d*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*b*d*f*Sqrt[d*Sin[e + f*x]]) - (2*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a*d^2*f*Sqrt[Sin[2*e + 2*f*x]])","A",20,16,37,0.4324,1,"{2900, 2838, 2570, 2572, 2639, 2575, 297, 1162, 617, 204, 1165, 628, 2906, 2905, 490, 1218}"
1426,1,359,0,0.8121698,"\int \frac{(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{2 b g^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^2 d^3 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f \sqrt{d \sin (e+f x)}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{a^2 d^2 f \sqrt{d \sin (e+f x)}}-\frac{2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}","\frac{2 b g^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^2 d^3 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f \sqrt{d \sin (e+f x)}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{a^2 d^2 f \sqrt{d \sin (e+f x)}}-\frac{2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}",1,"(-2*g*(g*Cos[e + f*x])^(3/2))/(3*a*d*f*(d*Sin[e + f*x])^(3/2)) + (2*b*g*(g*Cos[e + f*x])^(3/2))/(a^2*d^2*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*d^2*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*d^2*f*Sqrt[d*Sin[e + f*x]]) + (2*b*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^2*d^3*f*Sqrt[Sin[2*e + 2*f*x]])","A",10,9,37,0.2432,1,"{2899, 2563, 2570, 2572, 2639, 2906, 2905, 490, 1218}"
1427,1,519,0,1.2098393,"\int \frac{(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(7/2)*(a + b*Sin[e + f*x])),x]","\frac{2 g^2 \left(a^2-b^2\right) E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^3 d^4 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}{a^3 d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^3 d^3 f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} b g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^3 d^3 f \sqrt{d \sin (e+f x)}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f (d \sin (e+f x))^{3/2}}-\frac{4 g^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 a d^4 f \sqrt{\sin (2 e+2 f x)}}-\frac{4 g (g \cos (e+f x))^{3/2}}{5 a d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 g (g \cos (e+f x))^{3/2}}{5 a d f (d \sin (e+f x))^{5/2}}","\frac{2 g^2 \left(a^2-b^2\right) E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^3 d^4 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}{a^3 d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^3 d^3 f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} b g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^3 d^3 f \sqrt{d \sin (e+f x)}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f (d \sin (e+f x))^{3/2}}-\frac{4 g^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 a d^4 f \sqrt{\sin (2 e+2 f x)}}-\frac{4 g (g \cos (e+f x))^{3/2}}{5 a d^3 f \sqrt{d \sin (e+f x)}}-\frac{2 g (g \cos (e+f x))^{3/2}}{5 a d f (d \sin (e+f x))^{5/2}}",1,"(-2*g*(g*Cos[e + f*x])^(3/2))/(5*a*d*f*(d*Sin[e + f*x])^(5/2)) + (2*b*g*(g*Cos[e + f*x])^(3/2))/(3*a^2*d^2*f*(d*Sin[e + f*x])^(3/2)) - (4*g*(g*Cos[e + f*x])^(3/2))/(5*a*d^3*f*Sqrt[d*Sin[e + f*x]]) + (2*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(a^3*d^3*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^3*d^3*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^3*d^3*f*Sqrt[d*Sin[e + f*x]]) - (4*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*a*d^4*f*Sqrt[Sin[2*e + 2*f*x]]) + (2*(a^2 - b^2)*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^3*d^4*f*Sqrt[Sin[2*e + 2*f*x]])","A",15,10,37,0.2703,1,"{2899, 2570, 2572, 2639, 2563, 2910, 2906, 2905, 490, 1218}"
1428,1,612,0,1.5839689,"\int \frac{(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(9/2)*(a + b*Sin[e + f*x])),x]","-\frac{2 b g^2 \left(a^2-b^2\right) E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^4 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{2} b^2 g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^4 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^2 g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^4 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 b g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac{4 b g^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 a^2 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{4 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}-\frac{8 g (g \cos (e+f x))^{3/2}}{21 a d^3 f (d \sin (e+f x))^{3/2}}-\frac{2 g (g \cos (e+f x))^{3/2}}{7 a d f (d \sin (e+f x))^{7/2}}","-\frac{2 b g^2 \left(a^2-b^2\right) E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^4 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{2} b^2 g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^4 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^2 g^{5/2} \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^4 d^4 f \sqrt{d \sin (e+f x)}}-\frac{2 b g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}{a^4 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 g \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}{3 a^3 d^3 f (d \sin (e+f x))^{3/2}}+\frac{4 b g^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 a^2 d^5 f \sqrt{\sin (2 e+2 f x)}}+\frac{4 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^4 f \sqrt{d \sin (e+f x)}}+\frac{2 b g (g \cos (e+f x))^{3/2}}{5 a^2 d^2 f (d \sin (e+f x))^{5/2}}-\frac{8 g (g \cos (e+f x))^{3/2}}{21 a d^3 f (d \sin (e+f x))^{3/2}}-\frac{2 g (g \cos (e+f x))^{3/2}}{7 a d f (d \sin (e+f x))^{7/2}}",1,"(-2*g*(g*Cos[e + f*x])^(3/2))/(7*a*d*f*(d*Sin[e + f*x])^(7/2)) + (2*b*g*(g*Cos[e + f*x])^(3/2))/(5*a^2*d^2*f*(d*Sin[e + f*x])^(5/2)) - (8*g*(g*Cos[e + f*x])^(3/2))/(21*a*d^3*f*(d*Sin[e + f*x])^(3/2)) + (2*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(3*a^3*d^3*f*(d*Sin[e + f*x])^(3/2)) + (4*b*g*(g*Cos[e + f*x])^(3/2))/(5*a^2*d^4*f*Sqrt[d*Sin[e + f*x]]) - (2*b*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(a^4*d^4*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^2*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^4*d^4*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^2*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^4*d^4*f*Sqrt[d*Sin[e + f*x]]) + (4*b*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*a^2*d^5*f*Sqrt[Sin[2*e + 2*f*x]]) - (2*b*(a^2 - b^2)*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^4*d^5*f*Sqrt[Sin[2*e + 2*f*x]])","A",18,10,37,0.2703,1,"{2899, 2570, 2563, 2572, 2639, 2910, 2906, 2905, 490, 1218}"
1429,1,822,0,2.1413023,"\int \frac{(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(11/2)*(a + b*Sin[e + f*x])),x]","-\frac{2 \sqrt{2} \sqrt{b-a} \sqrt{a+b} g^{5/2} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} b^3}{a^5 d^5 f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} \sqrt{b-a} \sqrt{a+b} g^{5/2} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} b^3}{a^5 d^5 f \sqrt{d \sin (e+f x)}}+\frac{2 \left(a^2-b^2\right) g (g \cos (e+f x))^{3/2} b^2}{a^5 d^5 f \sqrt{d \sin (e+f x)}}+\frac{2 \left(a^2-b^2\right) g^2 \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} b^2}{a^5 d^6 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 \left(a^2-b^2\right) g (g \cos (e+f x))^{3/2} b}{3 a^4 d^4 f (d \sin (e+f x))^{3/2}}+\frac{8 g (g \cos (e+f x))^{3/2} b}{21 a^2 d^4 f (d \sin (e+f x))^{3/2}}+\frac{2 g (g \cos (e+f x))^{3/2} b}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}+\frac{4 \left(a^2-b^2\right) g (g \cos (e+f x))^{3/2}}{5 a^3 d^5 f \sqrt{d \sin (e+f x)}}-\frac{8 g (g \cos (e+f x))^{3/2}}{15 a d^5 f \sqrt{d \sin (e+f x)}}+\frac{4 \left(a^2-b^2\right) g^2 \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)}}{5 a^3 d^6 f \sqrt{\sin (2 e+2 f x)}}-\frac{8 g^2 \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)}}{15 a d^6 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 \left(a^2-b^2\right) g (g \cos (e+f x))^{3/2}}{5 a^3 d^3 f (d \sin (e+f x))^{5/2}}-\frac{4 g (g \cos (e+f x))^{3/2}}{15 a d^3 f (d \sin (e+f x))^{5/2}}-\frac{2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}}","-\frac{2 \sqrt{2} \sqrt{b-a} \sqrt{a+b} g^{5/2} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} b^3}{a^5 d^5 f \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} \sqrt{b-a} \sqrt{a+b} g^{5/2} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} b^3}{a^5 d^5 f \sqrt{d \sin (e+f x)}}+\frac{2 \left(a^2-b^2\right) g (g \cos (e+f x))^{3/2} b^2}{a^5 d^5 f \sqrt{d \sin (e+f x)}}+\frac{2 \left(a^2-b^2\right) g^2 \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} b^2}{a^5 d^6 f \sqrt{\sin (2 e+2 f x)}}-\frac{2 \left(a^2-b^2\right) g (g \cos (e+f x))^{3/2} b}{3 a^4 d^4 f (d \sin (e+f x))^{3/2}}+\frac{8 g (g \cos (e+f x))^{3/2} b}{21 a^2 d^4 f (d \sin (e+f x))^{3/2}}+\frac{2 g (g \cos (e+f x))^{3/2} b}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}+\frac{4 \left(a^2-b^2\right) g (g \cos (e+f x))^{3/2}}{5 a^3 d^5 f \sqrt{d \sin (e+f x)}}-\frac{8 g (g \cos (e+f x))^{3/2}}{15 a d^5 f \sqrt{d \sin (e+f x)}}+\frac{4 \left(a^2-b^2\right) g^2 \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)}}{5 a^3 d^6 f \sqrt{\sin (2 e+2 f x)}}-\frac{8 g^2 \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)}}{15 a d^6 f \sqrt{\sin (2 e+2 f x)}}+\frac{2 \left(a^2-b^2\right) g (g \cos (e+f x))^{3/2}}{5 a^3 d^3 f (d \sin (e+f x))^{5/2}}-\frac{4 g (g \cos (e+f x))^{3/2}}{15 a d^3 f (d \sin (e+f x))^{5/2}}-\frac{2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}}",1,"(-2*g*(g*Cos[e + f*x])^(3/2))/(9*a*d*f*(d*Sin[e + f*x])^(9/2)) + (2*b*g*(g*Cos[e + f*x])^(3/2))/(7*a^2*d^2*f*(d*Sin[e + f*x])^(7/2)) - (4*g*(g*Cos[e + f*x])^(3/2))/(15*a*d^3*f*(d*Sin[e + f*x])^(5/2)) + (2*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(5*a^3*d^3*f*(d*Sin[e + f*x])^(5/2)) + (8*b*g*(g*Cos[e + f*x])^(3/2))/(21*a^2*d^4*f*(d*Sin[e + f*x])^(3/2)) - (2*b*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(3*a^4*d^4*f*(d*Sin[e + f*x])^(3/2)) - (8*g*(g*Cos[e + f*x])^(3/2))/(15*a*d^5*f*Sqrt[d*Sin[e + f*x]]) + (4*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(5*a^3*d^5*f*Sqrt[d*Sin[e + f*x]]) + (2*b^2*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(a^5*d^5*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^3*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^5*d^5*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^3*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^5*d^5*f*Sqrt[d*Sin[e + f*x]]) - (8*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(15*a*d^6*f*Sqrt[Sin[2*e + 2*f*x]]) + (4*(a^2 - b^2)*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*a^3*d^6*f*Sqrt[Sin[2*e + 2*f*x]]) + (2*b^2*(a^2 - b^2)*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^5*d^6*f*Sqrt[Sin[2*e + 2*f*x]])","A",24,10,37,0.2703,1,"{2899, 2570, 2572, 2639, 2563, 2910, 2906, 2905, 490, 1218}"
1430,1,616,0,1.1410449,"\int \frac{(d \sin (e+f x))^{5/2}}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Int[(d*Sin[e + f*x])^(5/2)/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","-\frac{2 \sqrt{2} a^2 d^{5/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} a^2 d^{5/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{a d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right)}{\sqrt{2} b^2 f \sqrt{g}}-\frac{a d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right)}{\sqrt{2} b^2 f \sqrt{g}}-\frac{a d^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b^2 f \sqrt{g}}+\frac{a d^{5/2} \log \left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b^2 f \sqrt{g}}-\frac{d^2 \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b f g}+\frac{d^3 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{2 b f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}","-\frac{2 \sqrt{2} a^2 d^{5/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} a^2 d^{5/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b^2 f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{a d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right)}{\sqrt{2} b^2 f \sqrt{g}}-\frac{a d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right)}{\sqrt{2} b^2 f \sqrt{g}}-\frac{a d^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b^2 f \sqrt{g}}+\frac{a d^{5/2} \log \left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b^2 f \sqrt{g}}-\frac{d^2 \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b f g}+\frac{d^3 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{2 b f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}",1,"(a*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^2*f*Sqrt[g]) - (a*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^2*f*Sqrt[g]) - (2*Sqrt[2]*a^2*d^(5/2)*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^2*Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]]) + (2*Sqrt[2]*a^2*d^(5/2)*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^2*Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]]) - (a*d^(5/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^2*f*Sqrt[g]) + (a*d^(5/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^2*f*Sqrt[g]) - (d^2*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(b*f*g) + (d^3*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(2*b*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",19,14,37,0.3784,1,"{2909, 2568, 2573, 2641, 2574, 297, 1162, 617, 204, 1165, 628, 2908, 2907, 1218}"
1431,1,508,0,0.7723788,"\int \frac{(d \sin (e+f x))^{3/2}}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Int[(d*Sin[e + f*x])^(3/2)/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{2 \sqrt{2} a d^{3/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} a d^{3/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right)}{\sqrt{2} b f \sqrt{g}}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right)}{\sqrt{2} b f \sqrt{g}}+\frac{d^{3/2} \log \left(-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b f \sqrt{g}}-\frac{d^{3/2} \log \left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b f \sqrt{g}}","\frac{2 \sqrt{2} a d^{3/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} a d^{3/2} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{b f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right)}{\sqrt{2} b f \sqrt{g}}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right)}{\sqrt{2} b f \sqrt{g}}+\frac{d^{3/2} \log \left(-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b f \sqrt{g}}-\frac{d^{3/2} \log \left(\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right)}{2 \sqrt{2} b f \sqrt{g}}",1,"-((d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*f*Sqrt[g])) + (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*f*Sqrt[g]) + (2*Sqrt[2]*a*d^(3/2)*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b*Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*a*d^(3/2)*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b*Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]]) + (d^(3/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b*f*Sqrt[g]) - (d^(3/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b*f*Sqrt[g])","A",15,11,37,0.2973,1,"{2909, 2574, 297, 1162, 617, 204, 1165, 628, 2908, 2907, 1218}"
1432,1,209,0,0.3541307,"\int \frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Int[Sqrt[d*Sin[e + f*x]]/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{2 \sqrt{2} \sqrt{d} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} \sqrt{d} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}","\frac{2 \sqrt{2} \sqrt{d} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} \sqrt{d} \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}",1,"(-2*Sqrt[2]*Sqrt[d]*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]]) + (2*Sqrt[2]*Sqrt[d]*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]])","A",4,3,37,0.08108,1,"{2908, 2907, 1218}"
1433,1,273,0,0.5860071,"\int \frac{1}{\sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx","Int[1/(Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{2 \sqrt{2} b \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{\sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}","\frac{2 \sqrt{2} b \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{\sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}",1,"(2*Sqrt[2]*b*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*b*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) + (EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",7,6,37,0.1622,1,"{2910, 2573, 2641, 2908, 2907, 1218}"
1434,1,320,0,0.8348402,"\int \frac{1}{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[1/(Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","-\frac{2 \sqrt{2} b^2 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 d^{3/2} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^2 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 d^{3/2} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{b \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^2 d f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{g \cos (e+f x)}}{a d f g \sqrt{d \sin (e+f x)}}","-\frac{2 \sqrt{2} b^2 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 d^{3/2} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^2 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 d^{3/2} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{b \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^2 d f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{g \cos (e+f x)}}{a d f g \sqrt{d \sin (e+f x)}}",1,"(-2*Sqrt[2]*b^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*Sqrt[-a^2 + b^2]*d^(3/2)*f*Sqrt[g*Cos[e + f*x]]) + (2*Sqrt[2]*b^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*Sqrt[-a^2 + b^2]*d^(3/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[g*Cos[e + f*x]])/(a*d*f*g*Sqrt[d*Sin[e + f*x]]) - (b*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^2*d*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",9,7,37,0.1892,1,"{2910, 2563, 2573, 2641, 2908, 2907, 1218}"
1435,1,424,0,1.1638377,"\int \frac{1}{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Int[1/(Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{b^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^3 d^2 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^3 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^3 d^{5/2} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b^3 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^3 d^{5/2} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{2 b \sqrt{g \cos (e+f x)}}{a^2 d^2 f g \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{3 a d^2 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{g \cos (e+f x)}}{3 a d f g (d \sin (e+f x))^{3/2}}","\frac{b^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{a^3 d^2 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^3 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^3 d^{5/2} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b^3 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^3 d^{5/2} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{2 b \sqrt{g \cos (e+f x)}}{a^2 d^2 f g \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{3 a d^2 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{g \cos (e+f x)}}{3 a d f g (d \sin (e+f x))^{3/2}}",1,"(2*Sqrt[2]*b^3*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^3*Sqrt[-a^2 + b^2]*d^(5/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*b^3*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^3*Sqrt[-a^2 + b^2]*d^(5/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[g*Cos[e + f*x]])/(3*a*d*f*g*(d*Sin[e + f*x])^(3/2)) + (2*b*Sqrt[g*Cos[e + f*x]])/(a^2*d^2*f*g*Sqrt[d*Sin[e + f*x]]) + (2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3*a*d^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) + (b^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^3*d^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",13,8,37,0.2162,1,"{2910, 2570, 2573, 2641, 2563, 2908, 2907, 1218}"
1436,1,1064,0,1.6241789,"\int \frac{(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[(d*Sin[e + f*x])^(5/2)/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","-\frac{2 \sqrt{2} a^3 \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} d^3}{b (b-a)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a^3 \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} d^3}{b (b-a)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt{d \sin (e+f x)}}+\frac{b \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) d^{5/2}}{\sqrt{2} \left(a^2-b^2\right) f g^{3/2}}-\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) d^{5/2}}{\sqrt{2} b \left(a^2-b^2\right) f g^{3/2}}-\frac{b \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) d^{5/2}}{\sqrt{2} \left(a^2-b^2\right) f g^{3/2}}+\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) d^{5/2}}{\sqrt{2} b \left(a^2-b^2\right) f g^{3/2}}-\frac{b \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} \left(a^2-b^2\right) f g^{3/2}}+\frac{a^2 \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} b \left(a^2-b^2\right) f g^{3/2}}+\frac{b \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} \left(a^2-b^2\right) f g^{3/2}}-\frac{a^2 \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} b \left(a^2-b^2\right) f g^{3/2}}-\frac{2 b \sqrt{d \sin (e+f x)} d^2}{\left(a^2-b^2\right) f g \sqrt{g \cos (e+f x)}}-\frac{2 a \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} d^2}{\left(a^2-b^2\right) f g^2 \sqrt{\sin (2 e+2 f x)}}+\frac{2 a (d \sin (e+f x))^{3/2} d}{\left(a^2-b^2\right) f g \sqrt{g \cos (e+f x)}}","-\frac{2 \sqrt{2} a^3 \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} d^3}{b (b-a)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a^3 \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right) \sqrt{\sin (e+f x)} d^3}{b (b-a)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt{d \sin (e+f x)}}+\frac{b \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) d^{5/2}}{\sqrt{2} \left(a^2-b^2\right) f g^{3/2}}-\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}\right) d^{5/2}}{\sqrt{2} b \left(a^2-b^2\right) f g^{3/2}}-\frac{b \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) d^{5/2}}{\sqrt{2} \left(a^2-b^2\right) f g^{3/2}}+\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{d \sin (e+f x)}}+1\right) d^{5/2}}{\sqrt{2} b \left(a^2-b^2\right) f g^{3/2}}-\frac{b \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} \left(a^2-b^2\right) f g^{3/2}}+\frac{a^2 \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}-\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} b \left(a^2-b^2\right) f g^{3/2}}+\frac{b \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} \left(a^2-b^2\right) f g^{3/2}}-\frac{a^2 \log \left(\sqrt{g} \cot (e+f x)+\sqrt{g}+\frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)}}\right) d^{5/2}}{2 \sqrt{2} b \left(a^2-b^2\right) f g^{3/2}}-\frac{2 b \sqrt{d \sin (e+f x)} d^2}{\left(a^2-b^2\right) f g \sqrt{g \cos (e+f x)}}-\frac{2 a \sqrt{g \cos (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} d^2}{\left(a^2-b^2\right) f g^2 \sqrt{\sin (2 e+2 f x)}}+\frac{2 a (d \sin (e+f x))^{3/2} d}{\left(a^2-b^2\right) f g \sqrt{g \cos (e+f x)}}",1,"-((a^2*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*(a^2 - b^2)*f*g^(3/2))) + (b*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*(a^2 - b^2)*f*g^(3/2)) + (a^2*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*(a^2 - b^2)*f*g^(3/2)) - (b*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*(a^2 - b^2)*f*g^(3/2)) + (a^2*d^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*(a^2 - b^2)*f*g^(3/2)) - (b*d^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*(a^2 - b^2)*f*g^(3/2)) - (a^2*d^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*(a^2 - b^2)*f*g^(3/2)) + (b*d^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*(a^2 - b^2)*f*g^(3/2)) - (2*Sqrt[2]*a^3*d^3*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b*(-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*a^3*d^3*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b*(-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*b*d^2*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) + (2*a*d*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) - (2*a*d^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g^2*Sqrt[Sin[2*e + 2*f*x]])","A",31,17,37,0.4595,1,"{2902, 2571, 2572, 2639, 2566, 2575, 297, 1162, 617, 204, 1165, 628, 2909, 2906, 2905, 490, 1218}"
1437,1,379,0,0.8272312,"\int \frac{(d \sin (e+f x))^{3/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[(d*Sin[e + f*x])^(3/2)/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{2 b d E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}+\frac{2 a d \sqrt{d \sin (e+f x)}}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{2 b (d \sin (e+f x))^{3/2}}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} a^2 d^2 \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} a^2 d^2 \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}","\frac{2 b d E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}+\frac{2 a d \sqrt{d \sin (e+f x)}}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{2 b (d \sin (e+f x))^{3/2}}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} a^2 d^2 \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} a^2 d^2 \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}",1,"(2*Sqrt[2]*a^2*d^2*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*a^2*d^2*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (2*a*d*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) - (2*b*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) + (2*b*d*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g^2*Sqrt[Sin[2*e + 2*f*x]])","A",10,9,37,0.2432,1,"{2902, 2563, 2571, 2572, 2639, 2906, 2905, 490, 1218}"
1438,1,374,0,0.9112557,"\int \frac{\sqrt{d \sin (e+f x)}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[Sqrt[d*Sin[e + f*x]]/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","-\frac{2 a E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}+\frac{2 a (d \sin (e+f x))^{3/2}}{d f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{2 b \sqrt{d \sin (e+f x)}}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} a b d \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a b d \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}","-\frac{2 a E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}+\frac{2 a (d \sin (e+f x))^{3/2}}{d f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{2 b \sqrt{d \sin (e+f x)}}{f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} a b d \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a b d \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}",1,"(-2*Sqrt[2]*a*b*d*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*a*b*d*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*b*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) + (2*a*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*d*f*g*Sqrt[g*Cos[e + f*x]]) - (2*a*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g^2*Sqrt[Sin[2*e + 2*f*x]])","A",11,10,37,0.2703,1,"{2903, 2838, 2563, 2571, 2572, 2639, 2906, 2905, 490, 1218}"
1439,1,380,0,0.920007,"\int \frac{1}{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx","Int[1/((g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]","-\frac{2 b (d \sin (e+f x))^{3/2}}{d^2 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 b E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{d f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}+\frac{2 a \sqrt{d \sin (e+f x)}}{d f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^2 \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^2 \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}","-\frac{2 b (d \sin (e+f x))^{3/2}}{d^2 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 b E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{d f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}+\frac{2 a \sqrt{d \sin (e+f x)}}{d f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{2} b^2 \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^2 \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}",1,"(2*Sqrt[2]*b^2*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^2*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (2*a*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d*f*g*Sqrt[g*Cos[e + f*x]]) - (2*b*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*d^2*f*g*Sqrt[g*Cos[e + f*x]]) + (2*b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d*f*g^2*Sqrt[Sin[2*e + 2*f*x]])","A",11,10,37,0.2703,1,"{2904, 2838, 2563, 2571, 2572, 2639, 2906, 2905, 490, 1218}"
1440,1,568,0,1.4129479,"\int \frac{1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Int[1/((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{2 b^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a d^2 f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}-\frac{4 a E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{d^2 f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}-\frac{2 b \sqrt{d \sin (e+f x)}}{d^2 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{4 a (d \sin (e+f x))^{3/2}}{d^3 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 b^2 (g \cos (e+f x))^{3/2}}{a d f g^3 \left(a^2-b^2\right) \sqrt{d \sin (e+f x)}}-\frac{2 a}{d f g \left(a^2-b^2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b^3 \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a d f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} b^3 \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a d f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}","\frac{2 b^2 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a d^2 f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}-\frac{4 a E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{d^2 f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}-\frac{2 b \sqrt{d \sin (e+f x)}}{d^2 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{4 a (d \sin (e+f x))^{3/2}}{d^3 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 b^2 (g \cos (e+f x))^{3/2}}{a d f g^3 \left(a^2-b^2\right) \sqrt{d \sin (e+f x)}}-\frac{2 a}{d f g \left(a^2-b^2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} b^3 \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a d f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} b^3 \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a d f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}",1,"(-2*a)/((a^2 - b^2)*d*f*g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) + (2*b^2*(g*Cos[e + f*x])^(3/2))/(a*(a^2 - b^2)*d*f*g^3*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^3*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*(-a + b)^(3/2)*(a + b)^(3/2)*d*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^3*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*(-a + b)^(3/2)*(a + b)^(3/2)*d*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*b*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d^2*f*g*Sqrt[g*Cos[e + f*x]]) + (4*a*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*d^3*f*g*Sqrt[g*Cos[e + f*x]]) - (4*a*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d^2*f*g^2*Sqrt[Sin[2*e + 2*f*x]]) + (2*b^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a*(a^2 - b^2)*d^2*f*g^2*Sqrt[Sin[2*e + 2*f*x]])","A",16,12,37,0.3243,1,"{2904, 2838, 2570, 2571, 2572, 2639, 2563, 2910, 2906, 2905, 490, 1218}"
1441,1,673,0,1.8109192,"\int \frac{1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Int[1/((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","-\frac{2 b^3 (g \cos (e+f x))^{3/2}}{a^2 d^2 f g^3 \left(a^2-b^2\right) \sqrt{d \sin (e+f x)}}-\frac{2 b^3 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^2 d^3 f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}+\frac{4 b E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{d^3 f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{2} b^4 \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^4 \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}-\frac{4 b (d \sin (e+f x))^{3/2}}{d^4 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 b}{d^2 f g \left(a^2-b^2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{8 a \sqrt{d \sin (e+f x)}}{3 d^3 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 b^2 (g \cos (e+f x))^{3/2}}{3 a d f g^3 \left(a^2-b^2\right) (d \sin (e+f x))^{3/2}}-\frac{2 a}{3 d f g \left(a^2-b^2\right) (d \sin (e+f x))^{3/2} \sqrt{g \cos (e+f x)}}","-\frac{2 b^3 (g \cos (e+f x))^{3/2}}{a^2 d^2 f g^3 \left(a^2-b^2\right) \sqrt{d \sin (e+f x)}}-\frac{2 b^3 E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a^2 d^3 f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}+\frac{4 b E\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{d^3 f g^2 \left(a^2-b^2\right) \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{2} b^4 \sqrt{\sin (e+f x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}-\frac{2 \sqrt{2} b^4 \sqrt{\sin (e+f x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right)\right|-1\right)}{a^2 d^2 f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}-\frac{4 b (d \sin (e+f x))^{3/2}}{d^4 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 b}{d^2 f g \left(a^2-b^2\right) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{8 a \sqrt{d \sin (e+f x)}}{3 d^3 f g \left(a^2-b^2\right) \sqrt{g \cos (e+f x)}}+\frac{2 b^2 (g \cos (e+f x))^{3/2}}{3 a d f g^3 \left(a^2-b^2\right) (d \sin (e+f x))^{3/2}}-\frac{2 a}{3 d f g \left(a^2-b^2\right) (d \sin (e+f x))^{3/2} \sqrt{g \cos (e+f x)}}",1,"(-2*a)/(3*(a^2 - b^2)*d*f*g*Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(3/2)) + (2*b^2*(g*Cos[e + f*x])^(3/2))/(3*a*(a^2 - b^2)*d*f*g^3*(d*Sin[e + f*x])^(3/2)) + (2*b)/((a^2 - b^2)*d^2*f*g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) - (2*b^3*(g*Cos[e + f*x])^(3/2))/(a^2*(a^2 - b^2)*d^2*f*g^3*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^4*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*(-a + b)^(3/2)*(a + b)^(3/2)*d^2*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^4*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*(-a + b)^(3/2)*(a + b)^(3/2)*d^2*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (8*a*Sqrt[d*Sin[e + f*x]])/(3*(a^2 - b^2)*d^3*f*g*Sqrt[g*Cos[e + f*x]]) - (4*b*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*d^4*f*g*Sqrt[g*Cos[e + f*x]]) + (4*b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d^3*f*g^2*Sqrt[Sin[2*e + 2*f*x]]) - (2*b^3*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^2*(a^2 - b^2)*d^3*f*g^2*Sqrt[Sin[2*e + 2*f*x]])","A",19,12,37,0.3243,1,"{2904, 2838, 2570, 2563, 2571, 2572, 2639, 2910, 2906, 2905, 490, 1218}"
1442,1,331,0,0.7933522,"\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^2} \, dx","Int[(g*Cos[e + f*x])^(3/2)/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^2),x]","\frac{\sqrt{2} b g^2 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{\sqrt{2} b g^2 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{2 a^2 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{g \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a d f (a+b \sin (e+f x))}","\frac{\sqrt{2} b g^2 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b-\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}-\frac{\sqrt{2} b g^2 \sqrt{\cos (e+f x)} \Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right)\right|-1\right)}{a^2 \sqrt{d} f \sqrt{b^2-a^2} \sqrt{g \cos (e+f x)}}+\frac{g^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{2 a^2 f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{g \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{a d f (a+b \sin (e+f x))}",1,"(Sqrt[2]*b*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (Sqrt[2]*b*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) + (g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(a*d*f*(a + b*Sin[e + f*x])) + (g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(2*a^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",8,7,37,0.1892,1,"{2887, 2910, 2573, 2641, 2908, 2907, 1218}"
1443,1,82,0,0.1366802,"\int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx","Int[Sin[c + d*x]^2*(a + b*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{3 a \tan (c+d x)}{2 d}-\frac{a \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 a x}{2}-\frac{b \cos ^3(c+d x)}{3 d}+\frac{2 b \cos (c+d x)}{d}+\frac{b \sec (c+d x)}{d}","\frac{3 a \tan (c+d x)}{2 d}-\frac{a \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 a x}{2}-\frac{b \cos ^3(c+d x)}{3 d}+\frac{2 b \cos (c+d x)}{d}+\frac{b \sec (c+d x)}{d}",1,"(-3*a*x)/2 + (2*b*Cos[c + d*x])/d - (b*Cos[c + d*x]^3)/(3*d) + (b*Sec[c + d*x])/d + (3*a*Tan[c + d*x])/(2*d) - (a*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)","A",8,7,27,0.2593,1,"{2838, 2591, 288, 321, 203, 2590, 270}"
1444,1,65,0,0.1074182,"\int \sin (c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx","Int[Sin[c + d*x]*(a + b*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{a \cos (c+d x)}{d}+\frac{a \sec (c+d x)}{d}+\frac{3 b \tan (c+d x)}{2 d}-\frac{b \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 b x}{2}","\frac{a \cos (c+d x)}{d}+\frac{a \sec (c+d x)}{d}+\frac{3 b \tan (c+d x)}{2 d}-\frac{b \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 b x}{2}",1,"(-3*b*x)/2 + (a*Cos[c + d*x])/d + (a*Sec[c + d*x])/d + (3*b*Tan[c + d*x])/(2*d) - (b*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)","A",8,7,25,0.2800,1,"{2838, 2590, 14, 2591, 288, 321, 203}"
1445,1,38,0,0.0640617,"\int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx","Int[(a + b*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{a \tan (c+d x)}{d}-a x+\frac{b \cos (c+d x)}{d}+\frac{b \sec (c+d x)}{d}","\frac{a \tan (c+d x)}{d}-a x+\frac{b \cos (c+d x)}{d}+\frac{b \sec (c+d x)}{d}",1,"-(a*x) + (b*Cos[c + d*x])/d + (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",7,5,19,0.2632,1,"{2722, 3473, 8, 2590, 14}"
1446,1,27,0,0.0491544,"\int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])*Tan[c + d*x],x]","\frac{a \sec (c+d x)}{d}+\frac{b \tan (c+d x)}{d}-b x","\frac{a \sec (c+d x)}{d}+\frac{b \tan (c+d x)}{d}-b x",1,"-(b*x) + (a*Sec[c + d*x])/d + (b*Tan[c + d*x])/d","A",5,4,23,0.1739,1,"{2838, 2606, 8, 3473}"
1447,1,36,0,0.0803195,"\int \csc (c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^2*(a + b*Sin[c + d*x]),x]","\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b \tan (c+d x)}{d}","\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b \tan (c+d x)}{d}",1,"-((a*ArcTanh[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d + (b*Tan[c + d*x])/d","A",6,6,25,0.2400,1,"{2838, 2622, 321, 207, 3767, 8}"
1448,1,48,0,0.1113079,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^2*(a + b*Sin[c + d*x]),x]","\frac{a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{b \sec (c+d x)}{d}-\frac{b \tanh ^{-1}(\cos (c+d x))}{d}","\frac{a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{b \sec (c+d x)}{d}-\frac{b \tanh ^{-1}(\cos (c+d x))}{d}",1,"-((b*ArcTanh[Cos[c + d*x]])/d) - (a*Cot[c + d*x])/d + (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",7,6,27,0.2222,1,"{2838, 2620, 14, 2622, 321, 207}"
1449,1,75,0,0.1367874,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + b*Sin[c + d*x]),x]","\frac{3 a \sec (c+d x)}{2 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{b \tan (c+d x)}{d}-\frac{b \cot (c+d x)}{d}","\frac{3 a \sec (c+d x)}{2 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{b \tan (c+d x)}{d}-\frac{b \cot (c+d x)}{d}",1,"(-3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (b*Cot[c + d*x])/d + (3*a*Sec[c + d*x])/(2*d) - (a*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (b*Tan[c + d*x])/d","A",8,7,27,0.2593,1,"{2838, 2622, 288, 321, 207, 2620, 14}"
1450,1,94,0,0.1834404,"\int \sin (c+d x) (a+b \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Int[Sin[c + d*x]*(a + b*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{\left(a^2+2 b^2\right) \cos (c+d x)}{d}+\frac{\left(a^2+b^2\right) \sec (c+d x)}{d}+\frac{3 a b \tan (c+d x)}{d}-\frac{a b \sin ^2(c+d x) \tan (c+d x)}{d}-3 a b x-\frac{b^2 \cos ^3(c+d x)}{3 d}","\frac{\left(a^2+2 b^2\right) \cos (c+d x)}{d}+\frac{\left(a^2+b^2\right) \sec (c+d x)}{d}+\frac{3 a b \tan (c+d x)}{d}-\frac{a b \sin ^2(c+d x) \tan (c+d x)}{d}-3 a b x-\frac{b^2 \cos ^3(c+d x)}{3 d}",1,"-3*a*b*x + ((a^2 + 2*b^2)*Cos[c + d*x])/d - (b^2*Cos[c + d*x]^3)/(3*d) + ((a^2 + b^2)*Sec[c + d*x])/d + (3*a*b*Tan[c + d*x])/d - (a*b*Sin[c + d*x]^2*Tan[c + d*x])/d","A",8,7,27,0.2593,1,"{2911, 2591, 288, 321, 203, 4357, 448}"
1451,1,94,0,0.1385158,"\int (a+b \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Int[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{a^2 \tan (c+d x)}{d}+a^2 (-x)+\frac{2 a b \cos (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}+\frac{3 b^2 \tan (c+d x)}{2 d}-\frac{b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 b^2 x}{2}","\frac{a^2 \tan (c+d x)}{d}+a^2 (-x)+\frac{2 a b \cos (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}+\frac{3 b^2 \tan (c+d x)}{2 d}-\frac{b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{3 b^2 x}{2}",1,"-(a^2*x) - (3*b^2*x)/2 + (2*a*b*Cos[c + d*x])/d + (2*a*b*Sec[c + d*x])/d + (a^2*Tan[c + d*x])/d + (3*b^2*Tan[c + d*x])/(2*d) - (b^2*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)","A",11,9,21,0.4286,1,"{2722, 3473, 8, 2590, 14, 2591, 288, 321, 203}"
1452,1,42,0,0.0636757,"\int \sec (c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^2*Tan[c + d*x],x]","\frac{\sec (c+d x) (a+b \sin (c+d x))^2}{d}-2 a b x+\frac{2 b^2 \cos (c+d x)}{d}","\frac{\sec (c+d x) (a+b \sin (c+d x))^2}{d}-2 a b x+\frac{2 b^2 \cos (c+d x)}{d}",1,"-2*a*b*x + (2*b^2*Cos[c + d*x])/d + (Sec[c + d*x]*(a + b*Sin[c + d*x])^2)/d","A",4,3,25,0.1200,1,"{2861, 12, 2638}"
1453,1,70,0,0.1681346,"\int \csc (c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2+b^2\right) \sec (c+d x)}{d}-\frac{a^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \tanh ^{-1}\left(\sqrt{\cos ^2(c+d x)}\right)}{d}+\frac{2 a b \tan (c+d x)}{d}","\frac{\left(a^2+b^2\right) \sec (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{2 a b \tan (c+d x)}{d}",1,"((a^2 + b^2)*Sec[c + d*x])/d - (a^2*ArcTanh[Sqrt[Cos[c + d*x]^2]]*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x])/d + (2*a*b*Tan[c + d*x])/d","A",8,8,27,0.2963,1,"{2911, 3767, 8, 3201, 446, 78, 63, 206}"
1454,1,59,0,0.2811593,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2+b^2\right) \tan (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}","\frac{\left(a^2+b^2\right) \tan (c+d x)}{d}-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}",1,"(-2*a*b*ArcTanh[Cos[c + d*x]])/d - (a^2*Cot[c + d*x])/d + (2*a*b*Sec[c + d*x])/d + ((a^2 + b^2)*Tan[c + d*x])/d","A",7,6,29,0.2069,1,"{2911, 2622, 321, 207, 3200, 14}"
1455,1,124,0,0.25142,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\left(3 a^2+2 b^2\right) \sec (c+d x)}{2 d}-\frac{\left(3 a^2+2 b^2\right) \sqrt{\cos ^2(c+d x)} \sec (c+d x) \tanh ^{-1}\left(\sqrt{\cos ^2(c+d x)}\right)}{2 d}-\frac{a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{2 a b \tan (c+d x)}{d}-\frac{2 a b \cot (c+d x)}{d}","\frac{\left(3 a^2+2 b^2\right) \sec (c+d x)}{2 d}-\frac{\left(3 a^2+2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{2 a b \tan (c+d x)}{d}-\frac{2 a b \cot (c+d x)}{d}",1,"(-2*a*b*Cot[c + d*x])/d + ((3*a^2 + 2*b^2)*Sec[c + d*x])/(2*d) - ((3*a^2 + 2*b^2)*ArcTanh[Sqrt[Cos[c + d*x]^2]]*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x])/(2*d) - (a^2*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (2*a*b*Tan[c + d*x])/d","A",10,9,29,0.3103,1,"{2911, 2620, 14, 3201, 446, 78, 51, 63, 206}"
1456,1,104,0,0.2309403,"\int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2+b^2\right) \tan (c+d x)}{d}-\frac{\left(2 a^2+b^2\right) \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{3 a b \sec (c+d x)}{d}-\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a b \csc ^2(c+d x) \sec (c+d x)}{d}","\frac{\left(a^2+b^2\right) \tan (c+d x)}{d}-\frac{\left(2 a^2+b^2\right) \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{3 a b \sec (c+d x)}{d}-\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a b \csc ^2(c+d x) \sec (c+d x)}{d}",1,"(-3*a*b*ArcTanh[Cos[c + d*x]])/d - ((2*a^2 + b^2)*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) + (3*a*b*Sec[c + d*x])/d - (a*b*Csc[c + d*x]^2*Sec[c + d*x])/d + ((a^2 + b^2)*Tan[c + d*x])/d","A",8,7,29,0.2414,1,"{2911, 2622, 288, 321, 207, 3200, 448}"
1457,1,197,0,0.2608165,"\int \sin (c+d x) (a+b \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Int[Sin[c + d*x]*(a + b*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{9 a^2 b \tan (c+d x)}{2 d}-\frac{3 a^2 b \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{9}{2} a^2 b x+\frac{a^3 \cos (c+d x)}{d}+\frac{a^3 \sec (c+d x)}{d}-\frac{a b^2 \cos ^3(c+d x)}{d}+\frac{6 a b^2 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{15 b^3 \tan (c+d x)}{8 d}-\frac{b^3 \sin ^4(c+d x) \tan (c+d x)}{4 d}-\frac{5 b^3 \sin ^2(c+d x) \tan (c+d x)}{8 d}-\frac{15 b^3 x}{8}","\frac{9 a^2 b \tan (c+d x)}{2 d}-\frac{3 a^2 b \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{9}{2} a^2 b x+\frac{a^3 \cos (c+d x)}{d}+\frac{a^3 \sec (c+d x)}{d}-\frac{a b^2 \cos ^3(c+d x)}{d}+\frac{6 a b^2 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{15 b^3 \tan (c+d x)}{8 d}-\frac{b^3 \sin ^4(c+d x) \tan (c+d x)}{4 d}-\frac{5 b^3 \sin ^2(c+d x) \tan (c+d x)}{8 d}-\frac{15 b^3 x}{8}",1,"(-9*a^2*b*x)/2 - (15*b^3*x)/8 + (a^3*Cos[c + d*x])/d + (6*a*b^2*Cos[c + d*x])/d - (a*b^2*Cos[c + d*x]^3)/d + (a^3*Sec[c + d*x])/d + (3*a*b^2*Sec[c + d*x])/d + (9*a^2*b*Tan[c + d*x])/(2*d) + (15*b^3*Tan[c + d*x])/(8*d) - (3*a^2*b*Sin[c + d*x]^2*Tan[c + d*x])/(2*d) - (5*b^3*Sin[c + d*x]^2*Tan[c + d*x])/(8*d) - (b^3*Sin[c + d*x]^4*Tan[c + d*x])/(4*d)","A",17,8,27,0.2963,1,"{2912, 2590, 14, 2591, 288, 321, 203, 270}"
1458,1,146,0,0.2090682,"\int (a+b \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Int[(a + b*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{3 a^2 b \cos (c+d x)}{d}+\frac{3 a^2 b \sec (c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}+a^3 (-x)+\frac{9 a b^2 \tan (c+d x)}{2 d}-\frac{3 a b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{9}{2} a b^2 x-\frac{b^3 \cos ^3(c+d x)}{3 d}+\frac{2 b^3 \cos (c+d x)}{d}+\frac{b^3 \sec (c+d x)}{d}","\frac{3 a^2 b \cos (c+d x)}{d}+\frac{3 a^2 b \sec (c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}+a^3 (-x)+\frac{9 a b^2 \tan (c+d x)}{2 d}-\frac{3 a b^2 \sin ^2(c+d x) \tan (c+d x)}{2 d}-\frac{9}{2} a b^2 x-\frac{b^3 \cos ^3(c+d x)}{3 d}+\frac{2 b^3 \cos (c+d x)}{d}+\frac{b^3 \sec (c+d x)}{d}",1,"-(a^3*x) - (9*a*b^2*x)/2 + (3*a^2*b*Cos[c + d*x])/d + (2*b^3*Cos[c + d*x])/d - (b^3*Cos[c + d*x]^3)/(3*d) + (3*a^2*b*Sec[c + d*x])/d + (b^3*Sec[c + d*x])/d + (a^3*Tan[c + d*x])/d + (9*a*b^2*Tan[c + d*x])/(2*d) - (3*a*b^2*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)","A",14,10,21,0.4762,1,"{2722, 3473, 8, 2590, 14, 2591, 288, 321, 203, 270}"
1459,1,75,0,0.0720454,"\int \sec (c+d x) (a+b \sin (c+d x))^3 \tan (c+d x) \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^3*Tan[c + d*x],x]","-\frac{3}{2} b x \left(2 a^2+b^2\right)+\frac{6 a b^2 \cos (c+d x)}{d}+\frac{\sec (c+d x) (a+b \sin (c+d x))^3}{d}+\frac{3 b^3 \sin (c+d x) \cos (c+d x)}{2 d}","-\frac{3}{2} b x \left(2 a^2+b^2\right)+\frac{6 a b^2 \cos (c+d x)}{d}+\frac{\sec (c+d x) (a+b \sin (c+d x))^3}{d}+\frac{3 b^3 \sin (c+d x) \cos (c+d x)}{2 d}",1,"(-3*b*(2*a^2 + b^2)*x)/2 + (6*a*b^2*Cos[c + d*x])/d + (3*b^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (Sec[c + d*x]*(a + b*Sin[c + d*x])^3)/d","A",3,3,25,0.1200,1,"{2861, 12, 2644}"
1460,1,78,0,0.1515674,"\int \csc (c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{3 a^2 b \tan (c+d x)}{d}+\frac{a^3 \sec (c+d x)}{d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \tan (c+d x)}{d}-b^3 x","\frac{3 a^2 b \tan (c+d x)}{d}+\frac{a^3 \sec (c+d x)}{d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \tan (c+d x)}{d}-b^3 x",1,"-(b^3*x) - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Sec[c + d*x])/d + (3*a*b^2*Sec[c + d*x])/d + (3*a^2*b*Tan[c + d*x])/d + (b^3*Tan[c + d*x])/d","A",11,8,27,0.2963,1,"{2912, 3767, 8, 2622, 321, 207, 2606, 3473}"
1461,1,87,0,0.1954146,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{3 a^2 b \sec (c+d x)}{d}-\frac{3 a^2 b \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 \tan (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \sec (c+d x)}{d}","\frac{3 a^2 b \sec (c+d x)}{d}-\frac{3 a^2 b \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 \tan (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \sec (c+d x)}{d}",1,"(-3*a^2*b*ArcTanh[Cos[c + d*x]])/d - (a^3*Cot[c + d*x])/d + (3*a^2*b*Sec[c + d*x])/d + (b^3*Sec[c + d*x])/d + (a^3*Tan[c + d*x])/d + (3*a*b^2*Tan[c + d*x])/d","A",12,9,29,0.3103,1,"{2912, 3767, 8, 2622, 321, 207, 2620, 14, 2606}"
1462,1,132,0,0.2278328,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{3 a^2 b \tan (c+d x)}{d}-\frac{3 a^2 b \cot (c+d x)}{d}+\frac{3 a^3 \sec (c+d x)}{2 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{3 a b^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^3 \tan (c+d x)}{d}","\frac{3 a^2 b \tan (c+d x)}{d}-\frac{3 a^2 b \cot (c+d x)}{d}+\frac{3 a^3 \sec (c+d x)}{2 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{3 a b^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^3 \tan (c+d x)}{d}",1,"(-3*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a*b^2*ArcTanh[Cos[c + d*x]])/d - (3*a^2*b*Cot[c + d*x])/d + (3*a^3*Sec[c + d*x])/(2*d) + (3*a*b^2*Sec[c + d*x])/d - (a^3*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (3*a^2*b*Tan[c + d*x])/d + (b^3*Tan[c + d*x])/d","A",14,9,29,0.3103,1,"{2912, 3767, 8, 2622, 321, 207, 2620, 14, 288}"
1463,1,164,0,0.2618615,"\int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{9 a^2 b \sec (c+d x)}{2 d}-\frac{9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^2 b \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{a^3 \tan (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{2 a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{3 a b^2 \cot (c+d x)}{d}+\frac{b^3 \sec (c+d x)}{d}-\frac{b^3 \tanh ^{-1}(\cos (c+d x))}{d}","\frac{9 a^2 b \sec (c+d x)}{2 d}-\frac{9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^2 b \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{a^3 \tan (c+d x)}{d}-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{2 a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{3 a b^2 \cot (c+d x)}{d}+\frac{b^3 \sec (c+d x)}{d}-\frac{b^3 \tanh ^{-1}(\cos (c+d x))}{d}",1,"(-9*a^2*b*ArcTanh[Cos[c + d*x]])/(2*d) - (b^3*ArcTanh[Cos[c + d*x]])/d - (2*a^3*Cot[c + d*x])/d - (3*a*b^2*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (9*a^2*b*Sec[c + d*x])/(2*d) + (b^3*Sec[c + d*x])/d - (3*a^2*b*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (a^3*Tan[c + d*x])/d + (3*a*b^2*Tan[c + d*x])/d","A",15,8,29,0.2759,1,"{2912, 2622, 321, 207, 2620, 14, 288, 270}"
1464,1,222,0,0.3644948,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","-\frac{2 a^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{5/2}}+\frac{4 a^3 \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{5/2}}-\frac{a^4 \cos (c+d x)}{b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}-\frac{x}{b^2}","-\frac{2 a^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{5/2}}+\frac{4 a^3 \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{5/2}}-\frac{a^4 \cos (c+d x)}{b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}-\frac{x}{b^2}",1,"-(x/b^2) - (2*a^5*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(5/2)*d) + (4*a^3*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(5/2)*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) - (a^4*Cos[c + d*x])/(b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",12,7,29,0.2414,1,"{2897, 2648, 2664, 12, 2660, 618, 204}"
1465,1,212,0,0.2978675,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{2 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d \left(a^2-b^2\right)^{5/2}}-\frac{2 a^2 \left(a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d \left(a^2-b^2\right)^{5/2}}+\frac{a^3 \cos (c+d x)}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}","\frac{2 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d \left(a^2-b^2\right)^{5/2}}-\frac{2 a^2 \left(a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d \left(a^2-b^2\right)^{5/2}}+\frac{a^3 \cos (c+d x)}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}",1,"(2*a^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(5/2)*d) - (2*a^2*(a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(5/2)*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) + (a^3*Cos[c + d*x])/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",12,7,27,0.2593,1,"{2897, 2648, 2664, 12, 2660, 618, 204}"
1466,1,200,0,0.3081525,"\int \frac{\tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Tan[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","-\frac{2 a^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{4 a b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{a^2 b \cos (c+d x)}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}","-\frac{2 a^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{4 a b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{a^2 b \cos (c+d x)}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}",1,"(-2*a^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (4*a*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) - (a^2*b*Cos[c + d*x])/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",12,7,21,0.3333,1,"{2731, 2648, 2664, 12, 2660, 618, 204}"
1467,1,133,0,0.2107719,"\int \frac{\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{2 b \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{\sec (c+d x) \left(2 a^2-3 a b \sin (c+d x)+b^2\right)}{d \left(a^2-b^2\right)^2}-\frac{a \sec (c+d x)}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}","\frac{2 b \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{\sec (c+d x) \left(2 a^2-3 a b \sin (c+d x)+b^2\right)}{d \left(a^2-b^2\right)^2}-\frac{a \sec (c+d x)}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}",1,"(2*b*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (a*Sec[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]*(2*a^2 + b^2 - 3*a*b*Sin[c + d*x]))/((a^2 - b^2)^2*d)","A",6,6,25,0.2400,1,"{2864, 2866, 12, 2660, 618, 204}"
1468,1,229,0,0.3104988,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{2 b^3 \left(3 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{5/2}}+\frac{2 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{b^4 \cos (c+d x)}{a d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}","\frac{2 b^3 \left(3 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{5/2}}+\frac{2 b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{b^4 \cos (c+d x)}{a d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}",1,"(2*b^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (2*b^3*(3*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) + (b^4*Cos[c + d*x])/(a*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",13,8,27,0.2963,1,"{2897, 3770, 2648, 2664, 12, 2660, 618, 204}"
1469,1,248,0,0.3722254,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","-\frac{4 b^4 \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{5/2}}-\frac{2 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d \left(a^2-b^2\right)^{5/2}}-\frac{b^5 \cos (c+d x)}{a^2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{\cot (c+d x)}{a^2 d}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}","-\frac{4 b^4 \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{5/2}}-\frac{2 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d \left(a^2-b^2\right)^{5/2}}-\frac{b^5 \cos (c+d x)}{a^2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{\cot (c+d x)}{a^2 d}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}",1,"(-2*b^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(5/2)*d) - (4*b^4*(2*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(5/2)*d) + (2*b*ArcTanh[Cos[c + d*x]])/(a^3*d) - Cot[c + d*x]/(a^2*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) - (b^5*Cos[c + d*x])/(a^2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",15,10,29,0.3448,1,"{2897, 3770, 3767, 8, 2648, 2664, 12, 2660, 618, 204}"
1470,1,295,0,0.3949895,"\int \frac{\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[(Csc[c + d*x]^3*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{2 b^5 \left(5 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \left(a^2-b^2\right)^{5/2}}+\frac{2 b^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{5/2}}+\frac{b^6 \cos (c+d x)}{a^3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{\left(a^2+3 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}","\frac{2 b^5 \left(5 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \left(a^2-b^2\right)^{5/2}}+\frac{2 b^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{5/2}}+\frac{b^6 \cos (c+d x)}{a^3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{\left(a^2+3 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}",1,"(2*b^5*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)*d) + (2*b^5*(5*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(5/2)*d) - ArcTanh[Cos[c + d*x]]/(2*a^2*d) - ((a^2 + 3*b^2)*ArcTanh[Cos[c + d*x]])/(a^4*d) + (2*b*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) + (b^6*Cos[c + d*x])/(a^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",17,11,29,0.3793,1,"{2897, 3770, 3767, 8, 3768, 2648, 2664, 12, 2660, 618, 204}"
1471,1,388,0,0.5800886,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","-\frac{a^4 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{7/2}}+\frac{4 a^4 \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{7/2}}-\frac{2 a^2 \left(-3 a^2 b^2+a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{7/2}}-\frac{3 a^5 \cos (c+d x)}{2 b d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{a^4 \cos (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{2 a^3 \left(a^2-2 b^2\right) \cos (c+d x)}{b d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}","-\frac{a^4 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{7/2}}+\frac{4 a^4 \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{7/2}}-\frac{2 a^2 \left(-3 a^2 b^2+a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \left(a^2-b^2\right)^{7/2}}-\frac{3 a^5 \cos (c+d x)}{2 b d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{a^4 \cos (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{2 a^3 \left(a^2-2 b^2\right) \cos (c+d x)}{b d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}",1,"(4*a^4*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(7/2)*d) - (a^4*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(7/2)*d) - (2*a^2*(a^4 - 3*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(7/2)*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) - (a^4*Cos[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) - (3*a^5*Cos[c + d*x])/(2*b*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (2*a^3*(a^2 - 2*b^2)*Cos[c + d*x])/(b*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",18,8,29,0.2759,1,"{2897, 2648, 2664, 2754, 12, 2660, 618, 204}"
1472,1,366,0,0.4906593,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{a^3 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d \left(a^2-b^2\right)^{7/2}}-\frac{2 a^3 \left(a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d \left(a^2-b^2\right)^{7/2}}+\frac{2 a b \left(a^2+3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{3 a^4 \cos (c+d x)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{a^3 \cos (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{a^2 \left(a^2-3 b^2\right) \cos (c+d x)}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}","\frac{a^3 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d \left(a^2-b^2\right)^{7/2}}-\frac{2 a^3 \left(a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d \left(a^2-b^2\right)^{7/2}}+\frac{2 a b \left(a^2+3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{3 a^4 \cos (c+d x)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{a^3 \cos (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{a^2 \left(a^2-3 b^2\right) \cos (c+d x)}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}",1,"(-2*a^3*(a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(7/2)*d) + (a^3*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(7/2)*d) + (2*a*b*(a^2 + 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) + (a^3*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (3*a^4*Cos[c + d*x])/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (a^2*(a^2 - 3*b^2)*Cos[c + d*x])/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",18,8,27,0.2963,1,"{2897, 2648, 2664, 2754, 12, 2660, 618, 204}"
1473,1,350,0,0.5610496,"\int \frac{\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Tan[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","-\frac{a^2 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{4 a^2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{2 b^2 \left(3 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{3 a^3 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{a^2 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{2 a b^3 \cos (c+d x)}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}","-\frac{a^2 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{4 a^2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{2 b^2 \left(3 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{3 a^3 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{a^2 b \cos (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{2 a b^3 \cos (c+d x)}{d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}",1,"(-4*a^2*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - (a^2*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - (2*b^2*(3*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) - (a^2*b*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) - (3*a^3*b*Cos[c + d*x])/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (2*a*b^3*Cos[c + d*x])/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",18,8,21,0.3810,1,"{2731, 2648, 2664, 2754, 12, 2660, 618, 204}"
1474,1,204,0,0.363555,"\int \frac{\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Sec[c + d*x]*Tan[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{3 a b \left(2 a^2+3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{\sec (c+d x) \left(3 a \left(2 a^2+3 b^2\right)-b \left(11 a^2+4 b^2\right) \sin (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}-\frac{\left(3 a^2+2 b^2\right) \sec (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{a \sec (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}","\frac{3 a b \left(2 a^2+3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{\sec (c+d x) \left(3 a \left(2 a^2+3 b^2\right)-b \left(11 a^2+4 b^2\right) \sin (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}-\frac{\left(3 a^2+2 b^2\right) \sec (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{a \sec (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}",1,"(3*a*b*(2*a^2 + 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - (a*Sec[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - ((3*a^2 + 2*b^2)*Sec[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]*(3*a*(2*a^2 + 3*b^2) - b*(11*a^2 + 4*b^2)*Sin[c + d*x]))/(2*(a^2 - b^2)^3*d)","A",7,6,25,0.2400,1,"{2864, 2866, 12, 2660, 618, 204}"
1475,1,402,0,0.509691,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{2 b^3 \left(3 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d \left(a^2-b^2\right)^{7/2}}+\frac{b^3 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d \left(a^2-b^2\right)^{7/2}}+\frac{2 b^3 \left(-3 a^2 b^2+6 a^4+b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{7/2}}+\frac{b^4 \left(3 a^2-b^2\right) \cos (c+d x)}{a^2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{3 b^4 \cos (c+d x)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{b^4 \cos (c+d x)}{2 a d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}","\frac{2 b^3 \left(3 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d \left(a^2-b^2\right)^{7/2}}+\frac{b^3 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d \left(a^2-b^2\right)^{7/2}}+\frac{2 b^3 \left(-3 a^2 b^2+6 a^4+b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{7/2}}+\frac{b^4 \left(3 a^2-b^2\right) \cos (c+d x)}{a^2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{3 b^4 \cos (c+d x)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{b^4 \cos (c+d x)}{2 a d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}",1,"(2*b^3*(3*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(7/2)*d) + (b^3*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(7/2)*d) + (2*b^3*(6*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) + (b^4*Cos[c + d*x])/(2*a*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (3*b^4*Cos[c + d*x])/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (b^4*(3*a^2 - b^2)*Cos[c + d*x])/(a^2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",19,9,27,0.3333,1,"{2897, 3770, 2648, 2664, 2754, 12, 2660, 618, 204}"
1476,1,424,0,0.5897521,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","-\frac{4 b^4 \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{7/2}}-\frac{b^4 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{7/2}}-\frac{2 b^4 \left(-9 a^2 b^2+10 a^4+3 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \left(a^2-b^2\right)^{7/2}}-\frac{2 b^5 \left(2 a^2-b^2\right) \cos (c+d x)}{a^3 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{3 b^5 \cos (c+d x)}{2 a d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{b^5 \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a^3 d}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}","-\frac{4 b^4 \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{7/2}}-\frac{b^4 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{7/2}}-\frac{2 b^4 \left(-9 a^2 b^2+10 a^4+3 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^4 d \left(a^2-b^2\right)^{7/2}}-\frac{2 b^5 \left(2 a^2-b^2\right) \cos (c+d x)}{a^3 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{3 b^5 \cos (c+d x)}{2 a d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{b^5 \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a^3 d}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}",1,"(-4*b^4*(2*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(7/2)*d) - (b^4*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(7/2)*d) - (2*b^4*(10*a^4 - 9*a^2*b^2 + 3*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(7/2)*d) + (3*b*ArcTanh[Cos[c + d*x]])/(a^4*d) - Cot[c + d*x]/(a^3*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) - (b^5*Cos[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) - (3*b^5*Cos[c + d*x])/(2*a*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (2*b^5*(2*a^2 - b^2)*Cos[c + d*x])/(a^3*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",21,11,29,0.3793,1,"{2897, 3770, 3767, 8, 2648, 2664, 2754, 12, 2660, 618, 204}"
1477,1,470,0,0.6082766,"\int \frac{\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[(Csc[c + d*x]^3*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{b^5 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{7/2}}+\frac{2 b^5 \left(-17 a^2 b^2+15 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \left(a^2-b^2\right)^{7/2}}+\frac{2 b^5 \left(5 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{7/2}}+\frac{b^6 \left(5 a^2-3 b^2\right) \cos (c+d x)}{a^4 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{3 b^6 \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{b^6 \cos (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{\left(a^2+6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{3 b \cot (c+d x)}{a^4 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}","\frac{b^5 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{7/2}}+\frac{2 b^5 \left(-17 a^2 b^2+15 a^4+6 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^5 d \left(a^2-b^2\right)^{7/2}}+\frac{2 b^5 \left(5 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{7/2}}+\frac{b^6 \left(5 a^2-3 b^2\right) \cos (c+d x)}{a^4 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{3 b^6 \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{b^6 \cos (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{\left(a^2+6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{3 b \cot (c+d x)}{a^4 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)}",1,"(2*b^5*(5*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) + (b^5*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) + (2*b^5*(15*a^4 - 17*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(7/2)*d) - ArcTanh[Cos[c + d*x]]/(2*a^3*d) - ((a^2 + 6*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + (3*b*Cot[c + d*x])/(a^4*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) + (b^6*Cos[c + d*x])/(2*a^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (3*b^6*Cos[c + d*x])/(2*a^2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (b^6*(5*a^2 - 3*b^2)*Cos[c + d*x])/(a^4*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",23,12,29,0.4138,1,"{2897, 3770, 3767, 8, 3768, 2648, 2664, 2754, 12, 2660, 618, 204}"
1478,1,158,0,0.2654906,"\int \frac{\sec ^2(e+f x) \sqrt{a+b \sin (e+f x)}}{\sqrt{d \sin (e+f x)}} \, dx","Int[(Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]])/Sqrt[d*Sin[e + f*x]],x]","\frac{\sec (e+f x) \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{d f}-\frac{\sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{\sqrt{d} f}","\frac{\sec (e+f x) \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{d f}-\frac{\sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{\sqrt{d} f}",1,"(Sec[e + f*x]*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(d*f) - (Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(Sqrt[d]*f)","A",2,2,35,0.05714,1,"{2888, 2816}"
1479,0,0,0,0.1899602,"\int \frac{\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)}} \, dx","Int[(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]],x]","\int \frac{\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)}} \, dx","\frac{\sec (e+f x) (a \sin (e+f x)+b) \sqrt{a+b \sin (e+f x)}}{f \sqrt{d \sin (e+f x)}}-\frac{(a+b)^{3/2} \tan (e+f x) \sqrt{-\frac{a (\csc (e+f x)-1)}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{\sqrt{d} f}-\frac{b (a+b) (\sin (e+f x)+1) \tan (e+f x) \sqrt{-\frac{a (\csc (e+f x)-1)}{a+b}} \sqrt{\frac{a \csc (e+f x)+b}{b-a}} E\left(\sin ^{-1}\left(\sqrt{-\frac{b+a \csc (e+f x)}{a-b}}\right)|\frac{b-a}{a+b}\right)}{f \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}}}",1,"Defer[Int][(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]], x]","F",0,0,0,0,-1,"{}"
1480,0,0,0,0.3910585,"\int \frac{\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt{d \sin (e+f x)}} \, dx","Int[(Sec[e + f*x]^4*(a + b*Sin[e + f*x])^(5/2))/Sqrt[d*Sin[e + f*x]],x]","\int \frac{\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt{d \sin (e+f x)}} \, dx","\frac{\sec ^3(e+f x) \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{3 d f}+\frac{5 a \sec (e+f x) (a \sin (e+f x)+b) \sqrt{a+b \sin (e+f x)}}{6 f \sqrt{d \sin (e+f x)}}-\frac{5 a (a+b)^{3/2} \tan (e+f x) \sqrt{-\frac{a (\csc (e+f x)-1)}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{6 \sqrt{d} f}-\frac{5 a b (a+b) (\sin (e+f x)+1) \tan (e+f x) \sqrt{-\frac{a (\csc (e+f x)-1)}{a+b}} \sqrt{\frac{a \csc (e+f x)+b}{b-a}} E\left(\sin ^{-1}\left(\sqrt{-\frac{b+a \csc (e+f x)}{a-b}}\right)|\frac{b-a}{a+b}\right)}{6 f \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}}}",1,"(Sec[e + f*x]^3*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(5/2))/(3*d*f) + (5*a*Defer[Int][(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]], x])/6","F",0,0,0,0,-1,"{}"
1481,1,155,0,0.1686418,"\int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx","Int[Sin[c + d*x]^2*(a + b*Sin[c + d*x])*Tan[c + d*x]^5,x]","\frac{a \cos ^2(c+d x)}{2 d}+\frac{a \sec ^4(c+d x)}{4 d}-\frac{3 a \sec ^2(c+d x)}{2 d}-\frac{3 a \log (\cos (c+d x))}{d}-\frac{35 b \sin ^3(c+d x)}{24 d}-\frac{35 b \sin (c+d x)}{8 d}+\frac{b \sin ^3(c+d x) \tan ^4(c+d x)}{4 d}-\frac{7 b \sin ^3(c+d x) \tan ^2(c+d x)}{8 d}+\frac{35 b \tanh ^{-1}(\sin (c+d x))}{8 d}","\frac{a \cos ^2(c+d x)}{2 d}+\frac{a \sec ^4(c+d x)}{4 d}-\frac{3 a \sec ^2(c+d x)}{2 d}-\frac{3 a \log (\cos (c+d x))}{d}-\frac{35 b \sin ^3(c+d x)}{24 d}-\frac{35 b \sin (c+d x)}{8 d}+\frac{b \sin ^3(c+d x) \tan ^4(c+d x)}{4 d}-\frac{7 b \sin ^3(c+d x) \tan ^2(c+d x)}{8 d}+\frac{35 b \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"(35*b*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Cos[c + d*x]^2)/(2*d) - (3*a*Log[Cos[c + d*x]])/d - (3*a*Sec[c + d*x]^2)/(2*d) + (a*Sec[c + d*x]^4)/(4*d) - (35*b*Sin[c + d*x])/(8*d) - (35*b*Sin[c + d*x]^3)/(24*d) - (7*b*Sin[c + d*x]^3*Tan[c + d*x]^2)/(8*d) + (b*Sin[c + d*x]^3*Tan[c + d*x]^4)/(4*d)","A",11,8,27,0.2963,1,"{2834, 2590, 266, 43, 2592, 288, 302, 206}"
1482,1,135,0,0.1333456,"\int \sin (c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx","Int[Sin[c + d*x]*(a + b*Sin[c + d*x])*Tan[c + d*x]^5,x]","-\frac{15 a \sin (c+d x)}{8 d}+\frac{a \sin (c+d x) \tan ^4(c+d x)}{4 d}-\frac{5 a \sin (c+d x) \tan ^2(c+d x)}{8 d}+\frac{15 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \cos ^2(c+d x)}{2 d}+\frac{b \sec ^4(c+d x)}{4 d}-\frac{3 b \sec ^2(c+d x)}{2 d}-\frac{3 b \log (\cos (c+d x))}{d}","-\frac{15 a \sin (c+d x)}{8 d}+\frac{a \sin (c+d x) \tan ^4(c+d x)}{4 d}-\frac{5 a \sin (c+d x) \tan ^2(c+d x)}{8 d}+\frac{15 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \cos ^2(c+d x)}{2 d}+\frac{b \sec ^4(c+d x)}{4 d}-\frac{3 b \sec ^2(c+d x)}{2 d}-\frac{3 b \log (\cos (c+d x))}{d}",1,"(15*a*ArcTanh[Sin[c + d*x]])/(8*d) + (b*Cos[c + d*x]^2)/(2*d) - (3*b*Log[Cos[c + d*x]])/d - (3*b*Sec[c + d*x]^2)/(2*d) + (b*Sec[c + d*x]^4)/(4*d) - (15*a*Sin[c + d*x])/(8*d) - (5*a*Sin[c + d*x]*Tan[c + d*x]^2)/(8*d) + (a*Sin[c + d*x]*Tan[c + d*x]^4)/(4*d)","A",10,8,25,0.3200,1,"{2834, 2592, 288, 321, 206, 2590, 266, 43}"
1483,1,116,0,0.1086346,"\int (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx","Int[(a + b*Sin[c + d*x])*Tan[c + d*x]^5,x]","-\frac{(8 a+15 b) \log (1-\sin (c+d x))}{16 d}-\frac{(8 a-15 b) \log (\sin (c+d x)+1)}{16 d}+\frac{\tan ^4(c+d x) (a+b \sin (c+d x))}{4 d}-\frac{\tan ^2(c+d x) (4 a+5 b \sin (c+d x))}{8 d}-\frac{15 b \sin (c+d x)}{8 d}","-\frac{(8 a+15 b) \log (1-\sin (c+d x))}{16 d}-\frac{(8 a-15 b) \log (\sin (c+d x)+1)}{16 d}+\frac{\tan ^4(c+d x) (a+b \sin (c+d x))}{4 d}-\frac{\tan ^2(c+d x) (4 a+5 b \sin (c+d x))}{8 d}-\frac{15 b \sin (c+d x)}{8 d}",1,"-((8*a + 15*b)*Log[1 - Sin[c + d*x]])/(16*d) - ((8*a - 15*b)*Log[1 + Sin[c + d*x]])/(16*d) - (15*b*Sin[c + d*x])/(8*d) - ((4*a + 5*b*Sin[c + d*x])*Tan[c + d*x]^2)/(8*d) + ((a + b*Sin[c + d*x])*Tan[c + d*x]^4)/(4*d)","A",7,5,19,0.2632,1,"{2721, 819, 774, 633, 31}"
1484,1,103,0,0.1204091,"\int \sec (c+d x) (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])*Tan[c + d*x]^4,x]","\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan ^4(c+d x)}{4 d}-\frac{b \tan ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d}","\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan ^4(c+d x)}{4 d}-\frac{b \tan ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d}",1,"(3*a*ArcTanh[Sin[c + d*x]])/(8*d) - (b*Log[Cos[c + d*x]])/d - (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (b*Tan[c + d*x]^2)/(2*d) + (a*Sec[c + d*x]*Tan[c + d*x]^3)/(4*d) + (b*Tan[c + d*x]^4)/(4*d)","A",7,5,25,0.2000,1,"{2834, 2611, 3770, 3473, 3475}"
1485,1,74,0,0.1329364,"\int \sec ^2(c+d x) (a+b \sin (c+d x)) \tan ^3(c+d x) \, dx","Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x])*Tan[c + d*x]^3,x]","\frac{a \tan ^4(c+d x)}{4 d}+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac{3 b \tan (c+d x) \sec (c+d x)}{8 d}","\frac{a \tan ^4(c+d x)}{4 d}+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac{3 b \tan (c+d x) \sec (c+d x)}{8 d}",1,"(3*b*ArcTanh[Sin[c + d*x]])/(8*d) - (3*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]*Tan[c + d*x]^3)/(4*d) + (a*Tan[c + d*x]^4)/(4*d)","A",6,5,27,0.1852,1,"{2834, 2607, 30, 2611, 3770}"
1486,1,74,0,0.1391926,"\int \sec ^3(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])*Tan[c + d*x]^2,x]","-\frac{a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan ^4(c+d x)}{4 d}","-\frac{a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan ^4(c+d x)}{4 d}",1,"-(a*ArcTanh[Sin[c + d*x]])/(8*d) - (a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b*Tan[c + d*x]^4)/(4*d)","A",6,6,27,0.2222,1,"{2834, 2611, 3768, 3770, 2607, 30}"
1487,1,74,0,0.105266,"\int \sec ^4(c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx","Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])*Tan[c + d*x],x]","\frac{a \sec ^4(c+d x)}{4 d}-\frac{b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{b \tan (c+d x) \sec (c+d x)}{8 d}","\frac{a \sec ^4(c+d x)}{4 d}-\frac{b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{b \tan (c+d x) \sec (c+d x)}{8 d}",1,"-(b*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Sec[c + d*x]^4)/(4*d) - (b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",6,6,25,0.2400,1,"{2834, 2606, 30, 2611, 3768, 3770}"
1488,1,99,0,0.1052854,"\int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^5*(a + b*Sin[c + d*x]),x]","\frac{a \tan ^4(c+d x)}{4 d}+\frac{a \tan ^2(c+d x)}{d}+\frac{a \log (\tan (c+d x))}{d}+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \tan (c+d x) \sec (c+d x)}{8 d}","\frac{a \tan ^4(c+d x)}{4 d}+\frac{a \tan ^2(c+d x)}{d}+\frac{a \log (\tan (c+d x))}{d}+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \tan (c+d x) \sec (c+d x)}{8 d}",1,"(3*b*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Log[Tan[c + d*x]])/d + (3*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*Tan[c + d*x]^2)/d + (a*Tan[c + d*x]^4)/(4*d)","A",8,6,25,0.2400,1,"{2834, 2620, 266, 43, 3768, 3770}"
1489,1,115,0,0.1427618,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^5*(a + b*Sin[c + d*x]),x]","-\frac{15 a \csc (c+d x)}{8 d}+\frac{15 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \csc (c+d x) \sec ^4(c+d x)}{4 d}+\frac{5 a \csc (c+d x) \sec ^2(c+d x)}{8 d}+\frac{b \tan ^4(c+d x)}{4 d}+\frac{b \tan ^2(c+d x)}{d}+\frac{b \log (\tan (c+d x))}{d}","-\frac{15 a \csc (c+d x)}{8 d}+\frac{15 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \csc (c+d x) \sec ^4(c+d x)}{4 d}+\frac{5 a \csc (c+d x) \sec ^2(c+d x)}{8 d}+\frac{b \tan ^4(c+d x)}{4 d}+\frac{b \tan ^2(c+d x)}{d}+\frac{b \log (\tan (c+d x))}{d}",1,"(15*a*ArcTanh[Sin[c + d*x]])/(8*d) - (15*a*Csc[c + d*x])/(8*d) + (b*Log[Tan[c + d*x]])/d + (5*a*Csc[c + d*x]*Sec[c + d*x]^2)/(8*d) + (a*Csc[c + d*x]*Sec[c + d*x]^4)/(4*d) + (b*Tan[c + d*x]^2)/d + (b*Tan[c + d*x]^4)/(4*d)","A",10,8,27,0.2963,1,"{2834, 2621, 288, 321, 207, 2620, 266, 43}"
1490,1,135,0,0.1565851,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + b*Sin[c + d*x]),x]","\frac{a \tan ^4(c+d x)}{4 d}+\frac{3 a \tan ^2(c+d x)}{2 d}-\frac{a \cot ^2(c+d x)}{2 d}+\frac{3 a \log (\tan (c+d x))}{d}-\frac{15 b \csc (c+d x)}{8 d}+\frac{15 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \csc (c+d x) \sec ^4(c+d x)}{4 d}+\frac{5 b \csc (c+d x) \sec ^2(c+d x)}{8 d}","\frac{a \tan ^4(c+d x)}{4 d}+\frac{3 a \tan ^2(c+d x)}{2 d}-\frac{a \cot ^2(c+d x)}{2 d}+\frac{3 a \log (\tan (c+d x))}{d}-\frac{15 b \csc (c+d x)}{8 d}+\frac{15 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \csc (c+d x) \sec ^4(c+d x)}{4 d}+\frac{5 b \csc (c+d x) \sec ^2(c+d x)}{8 d}",1,"(15*b*ArcTanh[Sin[c + d*x]])/(8*d) - (a*Cot[c + d*x]^2)/(2*d) - (15*b*Csc[c + d*x])/(8*d) + (3*a*Log[Tan[c + d*x]])/d + (5*b*Csc[c + d*x]*Sec[c + d*x]^2)/(8*d) + (b*Csc[c + d*x]*Sec[c + d*x]^4)/(4*d) + (3*a*Tan[c + d*x]^2)/(2*d) + (a*Tan[c + d*x]^4)/(4*d)","A",10,8,27,0.2963,1,"{2834, 2620, 266, 43, 2621, 288, 321, 207}"
1491,1,155,0,0.1605651,"\int \csc ^4(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx","Int[Csc[c + d*x]^4*Sec[c + d*x]^5*(a + b*Sin[c + d*x]),x]","-\frac{35 a \csc ^3(c+d x)}{24 d}-\frac{35 a \csc (c+d x)}{8 d}+\frac{35 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \csc ^3(c+d x) \sec ^4(c+d x)}{4 d}+\frac{7 a \csc ^3(c+d x) \sec ^2(c+d x)}{8 d}+\frac{b \tan ^4(c+d x)}{4 d}+\frac{3 b \tan ^2(c+d x)}{2 d}-\frac{b \cot ^2(c+d x)}{2 d}+\frac{3 b \log (\tan (c+d x))}{d}","-\frac{35 a \csc ^3(c+d x)}{24 d}-\frac{35 a \csc (c+d x)}{8 d}+\frac{35 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \csc ^3(c+d x) \sec ^4(c+d x)}{4 d}+\frac{7 a \csc ^3(c+d x) \sec ^2(c+d x)}{8 d}+\frac{b \tan ^4(c+d x)}{4 d}+\frac{3 b \tan ^2(c+d x)}{2 d}-\frac{b \cot ^2(c+d x)}{2 d}+\frac{3 b \log (\tan (c+d x))}{d}",1,"(35*a*ArcTanh[Sin[c + d*x]])/(8*d) - (b*Cot[c + d*x]^2)/(2*d) - (35*a*Csc[c + d*x])/(8*d) - (35*a*Csc[c + d*x]^3)/(24*d) + (3*b*Log[Tan[c + d*x]])/d + (7*a*Csc[c + d*x]^3*Sec[c + d*x]^2)/(8*d) + (a*Csc[c + d*x]^3*Sec[c + d*x]^4)/(4*d) + (3*b*Tan[c + d*x]^2)/(2*d) + (b*Tan[c + d*x]^4)/(4*d)","A",11,8,27,0.2963,1,"{2834, 2621, 288, 302, 207, 2620, 266, 43}"
1492,1,189,0,0.358873,"\int \sin (c+d x) (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx","Int[Sin[c + d*x]*(a + b*Sin[c + d*x])^2*Tan[c + d*x]^5,x]","-\frac{\left(a^2+3 b^2\right) \sin (c+d x)}{d}-\frac{\left(15 a^2+48 a b+35 b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{\left(15 a^2-48 a b+35 b^2\right) \log (\sin (c+d x)+1)}{16 d}-\frac{a b \sin ^2(c+d x)}{d}-\frac{\sec ^2(c+d x) (9 a \sin (c+d x)+11 b) (a+b \sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{b^2 \sin ^3(c+d x)}{3 d}","-\frac{\left(a^2+3 b^2\right) \sin (c+d x)}{d}-\frac{\left(15 a^2+48 a b+35 b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{\left(15 a^2-48 a b+35 b^2\right) \log (\sin (c+d x)+1)}{16 d}-\frac{a b \sin ^2(c+d x)}{d}-\frac{\sec ^2(c+d x) (9 a \sin (c+d x)+11 b) (a+b \sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{b^2 \sin ^3(c+d x)}{3 d}",1,"-((15*a^2 + 48*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*d) + ((15*a^2 - 48*a*b + 35*b^2)*Log[1 + Sin[c + d*x]])/(16*d) - ((a^2 + 3*b^2)*Sin[c + d*x])/d - (a*b*Sin[c + d*x]^2)/d - (b^2*Sin[c + d*x]^3)/(3*d) - (Sec[c + d*x]^2*(11*b + 9*a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*Tan[c + d*x])/(4*d)","A",9,6,27,0.2222,1,"{2837, 12, 1645, 1810, 633, 31}"
1493,1,162,0,0.2702699,"\int (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx","Int[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^5,x]","-\frac{\left(4 a^2+15 a b+12 b^2\right) \log (1-\sin (c+d x))}{8 d}+\frac{\left(15 a b-4 \left(a^2+3 b^2\right)\right) \log (\sin (c+d x)+1)}{8 d}-\frac{2 a b \sin (c+d x)}{d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x)) (4 a+5 b \sin (c+d x))}{4 d}-\frac{b^2 \sin ^2(c+d x)}{2 d}","-\frac{\left(4 a^2+15 a b+12 b^2\right) \log (1-\sin (c+d x))}{8 d}+\frac{\left(15 a b-4 \left(a^2+3 b^2\right)\right) \log (\sin (c+d x)+1)}{8 d}-\frac{2 a b \sin (c+d x)}{d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x)) (4 a+5 b \sin (c+d x))}{4 d}-\frac{b^2 \sin ^2(c+d x)}{2 d}",1,"-((4*a^2 + 15*a*b + 12*b^2)*Log[1 - Sin[c + d*x]])/(8*d) + ((15*a*b - 4*(a^2 + 3*b^2))*Log[1 + Sin[c + d*x]])/(8*d) - (2*a*b*Sin[c + d*x])/d - (b^2*Sin[c + d*x]^2)/(2*d) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])*(4*a + 5*b*Sin[c + d*x]))/(4*d)","A",8,5,21,0.2381,1,"{2721, 1645, 1810, 633, 31}"
1494,1,150,0,0.2849445,"\int \sec (c+d x) (a+b \sin (c+d x))^2 \tan ^4(c+d x) \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^2*Tan[c + d*x]^4,x]","-\frac{\left(3 a^2+16 a b+15 b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{\left(3 a^2-16 a b+15 b^2\right) \log (\sin (c+d x)+1)}{16 d}-\frac{\sec ^2(c+d x) (5 a \sin (c+d x)+7 b) (a+b \sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{b^2 \sin (c+d x)}{d}","-\frac{\left(3 a^2+16 a b+15 b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{\left(3 a^2-16 a b+15 b^2\right) \log (\sin (c+d x)+1)}{16 d}-\frac{\sec ^2(c+d x) (5 a \sin (c+d x)+7 b) (a+b \sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{b^2 \sin (c+d x)}{d}",1,"-((3*a^2 + 16*a*b + 15*b^2)*Log[1 - Sin[c + d*x]])/(16*d) + ((3*a^2 - 16*a*b + 15*b^2)*Log[1 + Sin[c + d*x]])/(16*d) - (b^2*Sin[c + d*x])/d - (Sec[c + d*x]^2*(7*b + 5*a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*Tan[c + d*x])/(4*d)","A",9,6,27,0.2222,1,"{2837, 12, 1645, 1810, 633, 31}"
1495,1,116,0,0.2230905,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","-\frac{b (3 a+4 b) \log (1-\sin (c+d x))}{8 d}+\frac{b (3 a-4 b) \log (\sin (c+d x)+1)}{8 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x)) (2 a+3 b \sin (c+d x))}{4 d}","-\frac{b (3 a+4 b) \log (1-\sin (c+d x))}{8 d}+\frac{b (3 a-4 b) \log (\sin (c+d x)+1)}{8 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x)) (2 a+3 b \sin (c+d x))}{4 d}",1,"-(b*(3*a + 4*b)*Log[1 - Sin[c + d*x]])/(8*d) + ((3*a - 4*b)*b*Log[1 + Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])*(2*a + 3*b*Sin[c + d*x]))/(4*d)","A",7,5,29,0.1724,1,"{2837, 12, 1645, 633, 31}"
1496,1,93,0,0.194086,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","-\frac{\left(a^2-3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{\sec ^2(c+d x) \left(\left(a^2+3 b^2\right) \sin (c+d x)+4 a b\right)}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}","-\frac{\left(a^2-3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{\sec ^2(c+d x) \left(\left(a^2+3 b^2\right) \sin (c+d x)+4 a b\right)}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}",1,"-((a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) - (Sec[c + d*x]^2*(4*a*b + (a^2 + 3*b^2)*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*Tan[c + d*x])/(4*d)","A",5,5,29,0.1724,1,"{2837, 12, 1645, 778, 206}"
1497,1,72,0,0.0988589,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx","Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^2*Tan[c + d*x],x]","-\frac{\sec ^2(c+d x) \left(a b \sin (c+d x)+b^2\right)}{4 d}-\frac{a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}","-\frac{\sec ^2(c+d x) \left(a b \sin (c+d x)+b^2\right)}{4 d}-\frac{a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}",1,"-(a*b*ArcTanh[Sin[c + d*x]])/(4*d) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(4*d) - (Sec[c + d*x]^2*(b^2 + a*b*Sin[c + d*x]))/(4*d)","A",6,5,27,0.1852,1,"{2837, 12, 821, 639, 206}"
1498,1,126,0,0.2170927,"\int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{\sec ^4(c+d x) \left(a^2+2 a b \sin (c+d x)+b^2\right)}{4 d}+\frac{a^2 \log (\sin (c+d x))}{d}-\frac{a (4 a+3 b) \log (1-\sin (c+d x))}{8 d}-\frac{a (4 a-3 b) \log (\sin (c+d x)+1)}{8 d}+\frac{a \sec ^2(c+d x) (2 a+3 b \sin (c+d x))}{4 d}","\frac{\sec ^4(c+d x) \left(a^2+2 a b \sin (c+d x)+b^2\right)}{4 d}+\frac{a^2 \log (\sin (c+d x))}{d}-\frac{a (4 a+3 b) \log (1-\sin (c+d x))}{8 d}-\frac{a (4 a-3 b) \log (\sin (c+d x)+1)}{8 d}+\frac{a \sec ^2(c+d x) (2 a+3 b \sin (c+d x))}{4 d}",1,"-(a*(4*a + 3*b)*Log[1 - Sin[c + d*x]])/(8*d) + (a^2*Log[Sin[c + d*x]])/d - (a*(4*a - 3*b)*Log[1 + Sin[c + d*x]])/(8*d) + (a*Sec[c + d*x]^2*(2*a + 3*b*Sin[c + d*x]))/(4*d) + (Sec[c + d*x]^4*(a^2 + b^2 + 2*a*b*Sin[c + d*x]))/(4*d)","A",6,5,27,0.1852,1,"{2837, 12, 1805, 823, 801}"
1499,1,168,0,0.3539476,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(15 a^2+16 a b+3 b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{\left(15 a^2-16 a b+3 b^2\right) \log (\sin (c+d x)+1)}{16 d}+\frac{b \sec ^4(c+d x) \left(\frac{\left(a^2+b^2\right) \sin (c+d x)}{b}+2 a\right)}{4 d}+\frac{b \sec ^2(c+d x) \left(b \left(\frac{7 a^2}{b^2}+3\right) \sin (c+d x)+8 a\right)}{8 d}-\frac{a^2 \csc (c+d x)}{d}+\frac{2 a b \log (\sin (c+d x))}{d}","-\frac{\left(15 a^2+16 a b+3 b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{\left(15 a^2-16 a b+3 b^2\right) \log (\sin (c+d x)+1)}{16 d}+\frac{b \sec ^4(c+d x) \left(\frac{\left(a^2+b^2\right) \sin (c+d x)}{b}+2 a\right)}{4 d}+\frac{b \sec ^2(c+d x) \left(b \left(\frac{7 a^2}{b^2}+3\right) \sin (c+d x)+8 a\right)}{8 d}-\frac{a^2 \csc (c+d x)}{d}+\frac{2 a b \log (\sin (c+d x))}{d}",1,"-((a^2*Csc[c + d*x])/d) - ((15*a^2 + 16*a*b + 3*b^2)*Log[1 - Sin[c + d*x]])/(16*d) + (2*a*b*Log[Sin[c + d*x]])/d + ((15*a^2 - 16*a*b + 3*b^2)*Log[1 + Sin[c + d*x]])/(16*d) + (b*Sec[c + d*x]^2*(8*a + (3 + (7*a^2)/b^2)*b*Sin[c + d*x]))/(8*d) + (b*Sec[c + d*x]^4*(2*a + ((a^2 + b^2)*Sin[c + d*x])/b))/(4*d)","A",6,4,29,0.1379,1,"{2837, 12, 1805, 1802}"
1500,1,185,0,0.3837804,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(12 a^2+15 a b+4 b^2\right) \log (1-\sin (c+d x))}{8 d}+\frac{\left(3 a^2+b^2\right) \log (\sin (c+d x))}{d}-\frac{\left(12 a^2-15 a b+4 b^2\right) \log (\sin (c+d x)+1)}{8 d}+\frac{\sec ^4(c+d x) \left(a^2+2 a b \sin (c+d x)+b^2\right)}{4 d}+\frac{\sec ^2(c+d x) \left(2 \left(2 a^2+b^2\right)+7 a b \sin (c+d x)\right)}{4 d}-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a b \csc (c+d x)}{d}","-\frac{\left(12 a^2+15 a b+4 b^2\right) \log (1-\sin (c+d x))}{8 d}+\frac{\left(3 a^2+b^2\right) \log (\sin (c+d x))}{d}-\frac{\left(12 a^2-15 a b+4 b^2\right) \log (\sin (c+d x)+1)}{8 d}+\frac{\sec ^4(c+d x) \left(a^2+2 a b \sin (c+d x)+b^2\right)}{4 d}+\frac{\sec ^2(c+d x) \left(2 \left(2 a^2+b^2\right)+7 a b \sin (c+d x)\right)}{4 d}-\frac{a^2 \csc ^2(c+d x)}{2 d}-\frac{2 a b \csc (c+d x)}{d}",1,"(-2*a*b*Csc[c + d*x])/d - (a^2*Csc[c + d*x]^2)/(2*d) - ((12*a^2 + 15*a*b + 4*b^2)*Log[1 - Sin[c + d*x]])/(8*d) + ((3*a^2 + b^2)*Log[Sin[c + d*x]])/d - ((12*a^2 - 15*a*b + 4*b^2)*Log[1 + Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(a^2 + b^2 + 2*a*b*Sin[c + d*x]))/(4*d) + (Sec[c + d*x]^2*(2*(2*a^2 + b^2) + 7*a*b*Sin[c + d*x]))/(4*d)","A",6,4,29,0.1379,1,"{2837, 12, 1805, 1802}"
1501,1,202,0,0.3408399,"\int (a+b \sin (c+d x))^3 \tan ^5(c+d x) \, dx","Int[(a + b*Sin[c + d*x])^3*Tan[c + d*x]^5,x]","-\frac{b \left(24 a^2+35 b^2\right) \sin (c+d x)}{8 d}-\frac{(a+b) \left(8 a^2+37 a b+35 b^2\right) \log (1-\sin (c+d x))}{16 d}-\frac{(a-b) \left(8 a^2-37 a b+35 b^2\right) \log (\sin (c+d x)+1)}{16 d}-\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^2 (8 a+11 b \sin (c+d x))}{8 d}-\frac{b^3 \sin ^3(c+d x)}{3 d}","-\frac{b \left(24 a^2+35 b^2\right) \sin (c+d x)}{8 d}-\frac{(a+b) \left(8 a^2+37 a b+35 b^2\right) \log (1-\sin (c+d x))}{16 d}-\frac{(a-b) \left(8 a^2-37 a b+35 b^2\right) \log (\sin (c+d x)+1)}{16 d}-\frac{3 a b^2 \sin ^2(c+d x)}{2 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^2 (8 a+11 b \sin (c+d x))}{8 d}-\frac{b^3 \sin ^3(c+d x)}{3 d}",1,"-((a + b)*(8*a^2 + 37*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*d) - ((a - b)*(8*a^2 - 37*a*b + 35*b^2)*Log[1 + Sin[c + d*x]])/(16*d) - (b*(24*a^2 + 35*b^2)*Sin[c + d*x])/(8*d) - (3*a*b^2*Sin[c + d*x]^2)/(2*d) - (b^3*Sin[c + d*x]^3)/(3*d) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2*(8*a + 11*b*Sin[c + d*x]))/(8*d)","A",8,5,21,0.2381,1,"{2721, 1645, 1629, 633, 31}"
1502,1,177,0,0.3678326,"\int \sec (c+d x) (a+b \sin (c+d x))^3 \tan ^4(c+d x) \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^3*Tan[c + d*x]^4,x]","-\frac{3 (a+b) \left(a^2+7 a b+8 b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{3 (a-b) \left(a^2-7 a b+8 b^2\right) \log (\sin (c+d x)+1)}{16 d}-\frac{29 a b^2 \sin (c+d x)}{8 d}-\frac{\sec ^2(c+d x) (5 a \sin (c+d x)+8 b) (a+b \sin (c+d x))^2}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac{b^3 \sin ^2(c+d x)}{2 d}","-\frac{3 (a+b) \left(a^2+7 a b+8 b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{3 (a-b) \left(a^2-7 a b+8 b^2\right) \log (\sin (c+d x)+1)}{16 d}-\frac{29 a b^2 \sin (c+d x)}{8 d}-\frac{\sec ^2(c+d x) (5 a \sin (c+d x)+8 b) (a+b \sin (c+d x))^2}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac{b^3 \sin ^2(c+d x)}{2 d}",1,"(-3*(a + b)*(a^2 + 7*a*b + 8*b^2)*Log[1 - Sin[c + d*x]])/(16*d) + (3*(a - b)*(a^2 - 7*a*b + 8*b^2)*Log[1 + Sin[c + d*x]])/(16*d) - (29*a*b^2*Sin[c + d*x])/(8*d) - (b^3*Sin[c + d*x]^2)/(2*d) - (Sec[c + d*x]^2*(8*b + 5*a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^3*Tan[c + d*x])/(4*d)","A",9,6,27,0.2222,1,"{2837, 12, 1645, 1629, 633, 31}"
1503,1,142,0,0.3036079,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^3 \tan ^3(c+d x) \, dx","Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3*Tan[c + d*x]^3,x]","-\frac{3 b (a+b) (3 a+5 b) \log (1-\sin (c+d x))}{16 d}+\frac{3 b (3 a-5 b) (a-b) \log (\sin (c+d x)+1)}{16 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^2 (4 a+7 b \sin (c+d x))}{8 d}-\frac{15 b^3 \sin (c+d x)}{8 d}","-\frac{3 b (a+b) (3 a+5 b) \log (1-\sin (c+d x))}{16 d}+\frac{3 b (3 a-5 b) (a-b) \log (\sin (c+d x)+1)}{16 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^2 (4 a+7 b \sin (c+d x))}{8 d}-\frac{15 b^3 \sin (c+d x)}{8 d}",1,"(-3*b*(a + b)*(3*a + 5*b)*Log[1 - Sin[c + d*x]])/(16*d) + (3*(3*a - 5*b)*(a - b)*b*Log[1 + Sin[c + d*x]])/(16*d) - (15*b^3*Sin[c + d*x])/(8*d) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2*(4*a + 7*b*Sin[c + d*x]))/(8*d)","A",8,6,29,0.2069,1,"{2837, 12, 1645, 774, 633, 31}"
1504,1,144,0,0.2527686,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{\left(a^3-9 a b^2-8 b^3\right) \log (1-\sin (c+d x))}{16 d}-\frac{\left(a^3-9 a b^2+8 b^3\right) \log (\sin (c+d x)+1)}{16 d}-\frac{\sec ^2(c+d x) \left(\left(a^2+4 b^2\right) \sin (c+d x)+5 a b\right) (a+b \sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}","\frac{\left(a^3-9 a b^2-8 b^3\right) \log (1-\sin (c+d x))}{16 d}-\frac{\left(a^3-9 a b^2+8 b^3\right) \log (\sin (c+d x)+1)}{16 d}-\frac{\sec ^2(c+d x) \left(\left(a^2+4 b^2\right) \sin (c+d x)+5 a b\right) (a+b \sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}",1,"((a^3 - 9*a*b^2 - 8*b^3)*Log[1 - Sin[c + d*x]])/(16*d) - ((a^3 - 9*a*b^2 + 8*b^3)*Log[1 + Sin[c + d*x]])/(16*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])*(5*a*b + (a^2 + 4*b^2)*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^3*Tan[c + d*x])/(4*d)","A",7,6,29,0.2069,1,"{2837, 12, 1645, 819, 633, 31}"
1505,1,90,0,0.1110328,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^3 \tan (c+d x) \, dx","Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3*Tan[c + d*x],x]","-\frac{3 b \left(a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{3 \sec ^2(c+d x) (a+b \sin (c+d x)) \left(a b \sin (c+d x)+b^2\right)}{8 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^3}{4 d}","-\frac{3 b \left(a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{3 \sec ^2(c+d x) (a+b \sin (c+d x)) \left(a b \sin (c+d x)+b^2\right)}{8 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^3}{4 d}",1,"(-3*b*(a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(4*d) - (3*Sec[c + d*x]^2*(a + b*Sin[c + d*x])*(b^2 + a*b*Sin[c + d*x]))/(8*d)","A",5,5,27,0.1852,1,"{2837, 12, 805, 723, 206}"
1506,1,165,0,0.2473414,"\int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","-\frac{\left(9 a^2 b+8 a^3-b^3\right) \log (1-\sin (c+d x))}{16 d}-\frac{\left(-9 a^2 b+8 a^3+b^3\right) \log (\sin (c+d x)+1)}{16 d}+\frac{\sec ^4(c+d x) \left(b \left(3 a^2+b^2\right) \sin (c+d x)+a \left(a^2+3 b^2\right)\right)}{4 d}+\frac{\sec ^2(c+d x) \left(b \left(9 a^2-b^2\right) \sin (c+d x)+4 a^3\right)}{8 d}+\frac{a^3 \log (\sin (c+d x))}{d}","-\frac{\left(9 a^2 b+8 a^3-b^3\right) \log (1-\sin (c+d x))}{16 d}-\frac{\left(-9 a^2 b+8 a^3+b^3\right) \log (\sin (c+d x)+1)}{16 d}+\frac{\sec ^4(c+d x) \left(b \left(3 a^2+b^2\right) \sin (c+d x)+a \left(a^2+3 b^2\right)\right)}{4 d}+\frac{\sec ^2(c+d x) \left(b \left(9 a^2-b^2\right) \sin (c+d x)+4 a^3\right)}{8 d}+\frac{a^3 \log (\sin (c+d x))}{d}",1,"-((8*a^3 + 9*a^2*b - b^3)*Log[1 - Sin[c + d*x]])/(16*d) + (a^3*Log[Sin[c + d*x]])/d - ((8*a^3 - 9*a^2*b + b^3)*Log[1 + Sin[c + d*x]])/(16*d) + (Sec[c + d*x]^2*(4*a^3 + b*(9*a^2 - b^2)*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^4*(a*(a^2 + 3*b^2) + b*(3*a^2 + b^2)*Sin[c + d*x]))/(4*d)","A",6,5,27,0.1852,1,"{2837, 12, 1805, 823, 801}"
1507,1,171,0,0.3706692,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^2*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","\frac{b \sec ^4(c+d x) \left(a b \left(\frac{a^2}{b^2}+3\right) \sin (c+d x)+3 a^2+b^2\right)}{4 d}+\frac{a b \sec ^2(c+d x) \left(b \left(\frac{7 a^2}{b^2}+9\right) \sin (c+d x)+12 a\right)}{8 d}+\frac{3 a^2 b \log (\sin (c+d x))}{d}-\frac{a^3 \csc (c+d x)}{d}-\frac{3 a (a+b) (5 a+3 b) \log (1-\sin (c+d x))}{16 d}+\frac{3 a (5 a-3 b) (a-b) \log (\sin (c+d x)+1)}{16 d}","\frac{b \sec ^4(c+d x) \left(a b \left(\frac{a^2}{b^2}+3\right) \sin (c+d x)+3 a^2+b^2\right)}{4 d}+\frac{a b \sec ^2(c+d x) \left(b \left(\frac{7 a^2}{b^2}+9\right) \sin (c+d x)+12 a\right)}{8 d}+\frac{3 a^2 b \log (\sin (c+d x))}{d}-\frac{a^3 \csc (c+d x)}{d}-\frac{3 a (a+b) (5 a+3 b) \log (1-\sin (c+d x))}{16 d}+\frac{3 a (5 a-3 b) (a-b) \log (\sin (c+d x)+1)}{16 d}",1,"-((a^3*Csc[c + d*x])/d) - (3*a*(a + b)*(5*a + 3*b)*Log[1 - Sin[c + d*x]])/(16*d) + (3*a^2*b*Log[Sin[c + d*x]])/d + (3*a*(5*a - 3*b)*(a - b)*Log[1 + Sin[c + d*x]])/(16*d) + (b*Sec[c + d*x]^4*(3*a^2 + b^2 + a*(3 + a^2/b^2)*b*Sin[c + d*x]))/(4*d) + (a*b*Sec[c + d*x]^2*(12*a + (9 + (7*a^2)/b^2)*b*Sin[c + d*x]))/(8*d)","A",6,4,29,0.1379,1,"{2837, 12, 1805, 1802}"
1508,1,221,0,0.4344523,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","-\frac{3 (a+b) \left(8 a^2+7 a b+b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{3 a \left(a^2+b^2\right) \log (\sin (c+d x))}{d}-\frac{3 (a-b) \left(8 a^2-7 a b+b^2\right) \log (\sin (c+d x)+1)}{16 d}+\frac{b^2 \sec ^4(c+d x) \left(b \left(\frac{3 a^2}{b^2}+1\right) \sin (c+d x)+a \left(\frac{a^2}{b^2}+3\right)\right)}{4 d}+\frac{b^2 \sec ^2(c+d x) \left(3 b \left(\frac{7 a^2}{b^2}+1\right) \sin (c+d x)+4 a \left(\frac{2 a^2}{b^2}+3\right)\right)}{8 d}-\frac{3 a^2 b \csc (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}","-\frac{3 (a+b) \left(8 a^2+7 a b+b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{3 a \left(a^2+b^2\right) \log (\sin (c+d x))}{d}-\frac{3 (a-b) \left(8 a^2-7 a b+b^2\right) \log (\sin (c+d x)+1)}{16 d}+\frac{b^2 \sec ^4(c+d x) \left(b \left(\frac{3 a^2}{b^2}+1\right) \sin (c+d x)+a \left(\frac{a^2}{b^2}+3\right)\right)}{4 d}+\frac{b^2 \sec ^2(c+d x) \left(3 b \left(\frac{7 a^2}{b^2}+1\right) \sin (c+d x)+4 a \left(\frac{2 a^2}{b^2}+3\right)\right)}{8 d}-\frac{3 a^2 b \csc (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}",1,"(-3*a^2*b*Csc[c + d*x])/d - (a^3*Csc[c + d*x]^2)/(2*d) - (3*(a + b)*(8*a^2 + 7*a*b + b^2)*Log[1 - Sin[c + d*x]])/(16*d) + (3*a*(a^2 + b^2)*Log[Sin[c + d*x]])/d - (3*(a - b)*(8*a^2 - 7*a*b + b^2)*Log[1 + Sin[c + d*x]])/(16*d) + (b^2*Sec[c + d*x]^4*(a*(3 + a^2/b^2) + (1 + (3*a^2)/b^2)*b*Sin[c + d*x]))/(4*d) + (b^2*Sec[c + d*x]^2*(4*a*(3 + (2*a^2)/b^2) + 3*(1 + (7*a^2)/b^2)*b*Sin[c + d*x]))/(8*d)","A",6,4,29,0.1379,1,"{2837, 12, 1805, 1802}"
1509,1,295,0,0.5304052,"\int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^4,x]","-\frac{\left(6 a^2 b^2 \left(1-n^2\right)+a^4 \left(-\left(n^2-4 n+3\right)\right)-b^4 \left(n^2+4 n+3\right)\right) \sin ^{n+1}(c+d x) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{8 d (n+1)}-\frac{a b n \left(a^2 (2-n)-b^2 (n+2)\right) \sin ^{n+2}(c+d x) \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{2 d (n+2)}+\frac{\sec ^4(c+d x) \sin ^{n+1}(c+d x) \left(4 a b \left(a^2+b^2\right) \sin (c+d x)+6 a^2 b^2+a^4+b^4\right)}{4 d}+\frac{\sec ^2(c+d x) \sin ^{n+1}(c+d x) \left(4 a b \left(a^2 (2-n)-b^2 (n+2)\right) \sin (c+d x)-6 a^2 b^2 (n+1)+a^4 (3-n)-b^4 (n+5)\right)}{8 d}","-\frac{\left(6 a^2 b^2 \left(1-n^2\right)+a^4 \left(-\left(n^2-4 n+3\right)\right)-b^4 \left(n^2+4 n+3\right)\right) \sin ^{n+1}(c+d x) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{8 d (n+1)}-\frac{a b n \left(a^2 (2-n)-b^2 (n+2)\right) \sin ^{n+2}(c+d x) \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{2 d (n+2)}+\frac{\sec ^4(c+d x) \sin ^{n+1}(c+d x) \left(4 a b \left(a^2+b^2\right) \sin (c+d x)+6 a^2 b^2+a^4+b^4\right)}{4 d}+\frac{\sec ^2(c+d x) \sin ^{n+1}(c+d x) \left(4 a b \left(a^2 (2-n)-b^2 (n+2)\right) \sin (c+d x)-6 a^2 b^2 (n+1)+a^4 (3-n)-b^4 (n+5)\right)}{8 d}",1,"-((6*a^2*b^2*(1 - n^2) - a^4*(3 - 4*n + n^2) - b^4*(3 + 4*n + n^2))*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(8*d*(1 + n)) - (a*b*n*(a^2*(2 - n) - b^2*(2 + n))*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(2*d*(2 + n)) + (Sec[c + d*x]^4*Sin[c + d*x]^(1 + n)*(a^4 + 6*a^2*b^2 + b^4 + 4*a*b*(a^2 + b^2)*Sin[c + d*x]))/(4*d) + (Sec[c + d*x]^2*Sin[c + d*x]^(1 + n)*(a^4*(3 - n) - 6*a^2*b^2*(1 + n) - b^4*(5 + n) + 4*a*b*(a^2*(2 - n) - b^2*(2 + n))*Sin[c + d*x]))/(8*d)","A",6,4,29,0.1379,1,"{2837, 1806, 808, 364}"
1510,1,186,0,0.2961522,"\int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^3,x]","\frac{a \left(a^2 (3-n)-3 b^2 (n+1)\right) \sin ^{n+1}(c+d x) \, _2F_1\left(2,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{4 d (n+1)}+\frac{b \left(3 a^2 (2-n)-b^2 (n+2)\right) \sin ^{n+2}(c+d x) \, _2F_1\left(2,\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{4 d (n+2)}+\frac{\sec ^4(c+d x) \sin ^{n+1}(c+d x) \left(b \left(3 a^2+b^2\right) \sin (c+d x)+a \left(a^2+3 b^2\right)\right)}{4 d}","\frac{a \left(a^2 (3-n)-3 b^2 (n+1)\right) \sin ^{n+1}(c+d x) \, _2F_1\left(2,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{4 d (n+1)}+\frac{b \left(3 a^2 (2-n)-b^2 (n+2)\right) \sin ^{n+2}(c+d x) \, _2F_1\left(2,\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{4 d (n+2)}+\frac{\sec ^4(c+d x) \sin ^{n+1}(c+d x) \left(b \left(3 a^2+b^2\right) \sin (c+d x)+a \left(a^2+3 b^2\right)\right)}{4 d}",1,"(a*(a^2*(3 - n) - 3*b^2*(1 + n))*Hypergeometric2F1[2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(4*d*(1 + n)) + (b*(3*a^2*(2 - n) - b^2*(2 + n))*Hypergeometric2F1[2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(4*d*(2 + n)) + (Sec[c + d*x]^4*Sin[c + d*x]^(1 + n)*(a*(a^2 + 3*b^2) + b*(3*a^2 + b^2)*Sin[c + d*x]))/(4*d)","A",5,4,29,0.1379,1,"{2837, 1806, 808, 364}"
1511,1,160,0,0.258059,"\int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2 (3-n)-b^2 (n+1)\right) \sin ^{n+1}(c+d x) \, _2F_1\left(2,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{4 d (n+1)}+\frac{\sec ^4(c+d x) \sin ^{n+1}(c+d x) \left(a^2+2 a b \sin (c+d x)+b^2\right)}{4 d}+\frac{a b (2-n) \sin ^{n+2}(c+d x) \, _2F_1\left(2,\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{2 d (n+2)}","\frac{\left(a^2 (3-n)-b^2 (n+1)\right) \sin ^{n+1}(c+d x) \, _2F_1\left(2,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{4 d (n+1)}+\frac{\sec ^4(c+d x) \sin ^{n+1}(c+d x) \left(a^2+2 a b \sin (c+d x)+b^2\right)}{4 d}+\frac{a b (2-n) \sin ^{n+2}(c+d x) \, _2F_1\left(2,\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{2 d (n+2)}",1,"((a^2*(3 - n) - b^2*(1 + n))*Hypergeometric2F1[2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(4*d*(1 + n)) + (a*b*(2 - n)*Hypergeometric2F1[2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(2*d*(2 + n)) + (Sec[c + d*x]^4*Sin[c + d*x]^(1 + n)*(a^2 + b^2 + 2*a*b*Sin[c + d*x]))/(4*d)","A",5,4,29,0.1379,1,"{2837, 1806, 808, 364}"
1512,1,89,0,0.1136084,"\int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x]),x]","\frac{a \sin ^{n+1}(c+d x) \, _2F_1\left(3,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1)}+\frac{b \sin ^{n+2}(c+d x) \, _2F_1\left(3,\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2)}","\frac{a \sin ^{n+1}(c+d x) \, _2F_1\left(3,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{d (n+1)}+\frac{b \sin ^{n+2}(c+d x) \, _2F_1\left(3,\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{d (n+2)}",1,"(a*Hypergeometric2F1[3, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (b*Hypergeometric2F1[3, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n))","A",4,3,27,0.1111,1,"{2837, 808, 364}"
1513,1,360,0,0.5412209,"\int \frac{\sec ^5(c+d x) \sin ^n(c+d x)}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^5*Sin[c + d*x]^n)/(a + b*Sin[c + d*x]),x]","\frac{\left(3 a^2-9 a b+8 b^2\right) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{16 d (n+1) (a-b)^3}+\frac{\left(3 a^2+9 a b+8 b^2\right) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;\sin (c+d x))}{16 d (n+1) (a+b)^3}-\frac{b^6 \sin ^{n+1}(c+d x) \, _2F_1\left(1,n+1;n+2;-\frac{b \sin (c+d x)}{a}\right)}{a d (n+1) \left(a^2-b^2\right)^3}+\frac{(3 a-5 b) \sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;-\sin (c+d x))}{16 d (n+1) (a-b)^2}+\frac{(3 a+5 b) \sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;\sin (c+d x))}{16 d (n+1) (a+b)^2}+\frac{\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{8 d (n+1) (a-b)}+\frac{\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;\sin (c+d x))}{8 d (n+1) (a+b)}","\frac{\left(3 a^2-9 a b+8 b^2\right) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{16 d (n+1) (a-b)^3}+\frac{\left(3 a^2+9 a b+8 b^2\right) \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;\sin (c+d x))}{16 d (n+1) (a+b)^3}-\frac{b^6 \sin ^{n+1}(c+d x) \, _2F_1\left(1,n+1;n+2;-\frac{b \sin (c+d x)}{a}\right)}{a d (n+1) \left(a^2-b^2\right)^3}+\frac{(3 a-5 b) \sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;-\sin (c+d x))}{16 d (n+1) (a-b)^2}+\frac{(3 a+5 b) \sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;\sin (c+d x))}{16 d (n+1) (a+b)^2}+\frac{\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{8 d (n+1) (a-b)}+\frac{\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;\sin (c+d x))}{8 d (n+1) (a+b)}",1,"((3*a^2 - 9*a*b + 8*b^2)*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(16*(a - b)^3*d*(1 + n)) + ((3*a^2 + 9*a*b + 8*b^2)*Hypergeometric2F1[1, 1 + n, 2 + n, Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(16*(a + b)^3*d*(1 + n)) - (b^6*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Sin[c + d*x])/a)]*Sin[c + d*x]^(1 + n))/(a*(a^2 - b^2)^3*d*(1 + n)) + ((3*a - 5*b)*Hypergeometric2F1[2, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(16*(a - b)^2*d*(1 + n)) + ((3*a + 5*b)*Hypergeometric2F1[2, 1 + n, 2 + n, Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(16*(a + b)^2*d*(1 + n)) + (Hypergeometric2F1[3, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(8*(a - b)*d*(1 + n)) + (Hypergeometric2F1[3, 1 + n, 2 + n, Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(8*(a + b)*d*(1 + n))","A",10,3,29,0.1034,1,"{2837, 961, 64}"
1514,1,487,0,0.5662356,"\int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx","Int[Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p,x]","\frac{3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,1;n+2;-\frac{b \sin (c+d x)}{a},-\sin (c+d x)\right)}{16 d (n+1)}+\frac{3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,1;n+2;-\frac{b \sin (c+d x)}{a},\sin (c+d x)\right)}{16 d (n+1)}+\frac{3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,2;n+2;-\frac{b \sin (c+d x)}{a},-\sin (c+d x)\right)}{16 d (n+1)}+\frac{3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,2;n+2;-\frac{b \sin (c+d x)}{a},\sin (c+d x)\right)}{16 d (n+1)}+\frac{\sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,3;n+2;-\frac{b \sin (c+d x)}{a},-\sin (c+d x)\right)}{8 d (n+1)}+\frac{\sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,3;n+2;-\frac{b \sin (c+d x)}{a},\sin (c+d x)\right)}{8 d (n+1)}","\frac{3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,1;n+2;-\frac{b \sin (c+d x)}{a},-\sin (c+d x)\right)}{16 d (n+1)}+\frac{3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,1;n+2;-\frac{b \sin (c+d x)}{a},\sin (c+d x)\right)}{16 d (n+1)}+\frac{3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,2;n+2;-\frac{b \sin (c+d x)}{a},-\sin (c+d x)\right)}{16 d (n+1)}+\frac{3 \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,2;n+2;-\frac{b \sin (c+d x)}{a},\sin (c+d x)\right)}{16 d (n+1)}+\frac{\sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,3;n+2;-\frac{b \sin (c+d x)}{a},-\sin (c+d x)\right)}{8 d (n+1)}+\frac{\sin ^{n+1}(c+d x) (a+b \sin (c+d x))^p \left(\frac{b \sin (c+d x)}{a}+1\right)^{-p} F_1\left(n+1;-p,3;n+2;-\frac{b \sin (c+d x)}{a},\sin (c+d x)\right)}{8 d (n+1)}",1,"(3*AppellF1[1 + n, -p, 1, 2 + n, -((b*Sin[c + d*x])/a), -Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(16*d*(1 + n)*(1 + (b*Sin[c + d*x])/a)^p) + (3*AppellF1[1 + n, -p, 1, 2 + n, -((b*Sin[c + d*x])/a), Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(16*d*(1 + n)*(1 + (b*Sin[c + d*x])/a)^p) + (3*AppellF1[1 + n, -p, 2, 2 + n, -((b*Sin[c + d*x])/a), -Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(16*d*(1 + n)*(1 + (b*Sin[c + d*x])/a)^p) + (3*AppellF1[1 + n, -p, 2, 2 + n, -((b*Sin[c + d*x])/a), Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(16*d*(1 + n)*(1 + (b*Sin[c + d*x])/a)^p) + (AppellF1[1 + n, -p, 3, 2 + n, -((b*Sin[c + d*x])/a), -Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(8*d*(1 + n)*(1 + (b*Sin[c + d*x])/a)^p) + (AppellF1[1 + n, -p, 3, 2 + n, -((b*Sin[c + d*x])/a), Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(8*d*(1 + n)*(1 + (b*Sin[c + d*x])/a)^p)","A",17,5,29,0.1724,1,"{2837, 961, 135, 133, 912}"
1515,0,0,0,0.3702133,"\int \frac{\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt{d \sin (e+f x)}} \, dx","Int[(Sec[e + f*x]^6*(a + b*Sin[e + f*x])^(9/2))/Sqrt[d*Sin[e + f*x]],x]","\int \frac{\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt{d \sin (e+f x)}} \, dx","-\frac{3 a b \left(b^2-2 a^2\right) \cos (e+f x) \sqrt{a+b \sin (e+f x)}}{5 f \sqrt{d \sin (e+f x)}}-\frac{3 a \sec ^3(e+f x) \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)} \left(\left(8 a^2 b-4 b^3\right) \sin ^3(e+f x)+5 a \left(a^2-b^2\right) \sin ^2(e+f x)+2 b \left(b^2-7 a^2\right) \sin (e+f x)-a \left(7 a^2+b^2\right)\right)}{20 d f}-\frac{3 b \left(-3 a^2 b^2+2 a^4+b^4\right) \tan (e+f x) \sqrt{d \sin (e+f x)} \sqrt{-\frac{a (\csc (e+f x)-1)}{a+b}} \sqrt{-\frac{a (\sin (e+f x)+1) \csc ^2(e+f x) (a+b \sin (e+f x))}{(a-b)^2}} E\left(\sin ^{-1}\left(\sqrt{-\frac{b+a \csc (e+f x)}{a-b}}\right)|1-\frac{2 a}{a+b}\right)}{5 d f \sqrt{a+b \sin (e+f x)}}-\frac{3 a (a+b)^{3/2} \left(5 a^2+3 a b-4 b^2\right) \tan (e+f x) \sqrt{-\frac{a (\csc (e+f x)-1)}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right)|-\frac{a+b}{a-b}\right)}{20 \sqrt{d} f}+\frac{\sec ^5(e+f x) \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f}",1,"(Sec[e + f*x]^5*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(9/2))/(5*d*f) + (9*a*Defer[Int][(Sec[e + f*x]^4*(a + b*Sin[e + f*x])^(7/2))/Sqrt[d*Sin[e + f*x]], x])/10","F",0,0,0,0,-1,"{}"
1516,1,458,0,1.2422472,"\int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(4/3),x]","-\frac{3 (c+d)^2 \left(208 a^2 c d^2-64 a b d \left(3 c^2-5 d^2\right)+b^2 c \left(54 c^2+d^2\right)\right) \cos (e+f x) \sqrt[3]{c+d \sin (e+f x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{7}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{1040 \sqrt{2} d^4 f \sqrt{\sin (e+f x)+1} \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{3 (c-d) (c+d)^2 \left(-208 a^2 d^2+192 a b c d+b^2 \left(-\left(54 c^2+91 d^2\right)\right)\right) \cos (e+f x) \sqrt[3]{c+d \sin (e+f x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{1040 \sqrt{2} d^4 f \sqrt{\sin (e+f x)+1} \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{9 \left(-26 a^2 d^2+64 a b c d+b^2 \left(-\left(18 c^2-13 d^2\right)\right)\right) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{2080 d^3 f}-\frac{9 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{208 d^2 f}+\frac{3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}","-\frac{3 (c+d)^2 \left(208 a^2 c d^2-64 a b d \left(3 c^2-5 d^2\right)+b^2 c \left(54 c^2+d^2\right)\right) \cos (e+f x) \sqrt[3]{c+d \sin (e+f x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{7}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{1040 \sqrt{2} d^4 f \sqrt{\sin (e+f x)+1} \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{3 (c-d) (c+d)^2 \left(-208 a^2 d^2+192 a b c d+b^2 \left(-\left(54 c^2+91 d^2\right)\right)\right) \cos (e+f x) \sqrt[3]{c+d \sin (e+f x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{1040 \sqrt{2} d^4 f \sqrt{\sin (e+f x)+1} \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{9 \left(-26 a^2 d^2+64 a b c d+b^2 \left(-\left(18 c^2-13 d^2\right)\right)\right) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{2080 d^3 f}-\frac{9 b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{208 d^2 f}+\frac{3 \cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{7/3}}{16 d f}",1,"(-9*(64*a*b*c*d - 26*a^2*d^2 - b^2*(18*c^2 - 13*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(2080*d^3*f) - (9*b*(3*b*c - 2*a*d)*Cos[e + f*x]*Sin[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(208*d^2*f) + (3*Cos[e + f*x]*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(7/3))/(16*d*f) - (3*(c + d)^2*(208*a^2*c*d^2 - 64*a*b*d*(3*c^2 - 5*d^2) + b^2*c*(54*c^2 + d^2))*AppellF1[1/2, 1/2, -7/3, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(1040*Sqrt[2]*d^4*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3)) - (3*(c - d)*(c + d)^2*(192*a*b*c*d - 208*a^2*d^2 - b^2*(54*c^2 + 91*d^2))*AppellF1[1/2, 1/2, -4/3, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(1040*Sqrt[2]*d^4*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3))","A",11,8,35,0.2286,1,"{2922, 3050, 3033, 3023, 2756, 2665, 139, 138}"
1517,1,341,0,0.6513506,"\int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(4/3),x]","\frac{3 (c+d)^2 \left(-13 a c d+6 b c^2-10 b d^2\right) \cos (e+f x) \sqrt[3]{c+d \sin (e+f x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{7}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{65 \sqrt{2} d^3 f \sqrt{\sin (e+f x)+1} \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{3 (c-d) (c+d)^2 (6 b c-13 a d) \cos (e+f x) \sqrt[3]{c+d \sin (e+f x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{65 \sqrt{2} d^3 f \sqrt{\sin (e+f x)+1} \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{3 (6 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{130 d^2 f}+\frac{3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}","\frac{3 (c+d)^2 \left(-13 a c d+6 b c^2-10 b d^2\right) \cos (e+f x) \sqrt[3]{c+d \sin (e+f x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{7}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{65 \sqrt{2} d^3 f \sqrt{\sin (e+f x)+1} \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{3 (c-d) (c+d)^2 (6 b c-13 a d) \cos (e+f x) \sqrt[3]{c+d \sin (e+f x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{65 \sqrt{2} d^3 f \sqrt{\sin (e+f x)+1} \sqrt[3]{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{3 (6 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{130 d^2 f}+\frac{3 b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{13 d f}",1,"(-3*(6*b*c - 13*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(130*d^2*f) + (3*b*Cos[e + f*x]*Sin[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(13*d*f) + (3*(c + d)^2*(6*b*c^2 - 13*a*c*d - 10*b*d^2)*AppellF1[1/2, 1/2, -7/3, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(65*Sqrt[2]*d^3*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3)) - (3*(c - d)*(c + d)^2*(6*b*c - 13*a*d)*AppellF1[1/2, 1/2, -4/3, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(65*Sqrt[2]*d^3*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3))","A",10,7,33,0.2121,1,"{2922, 3034, 3023, 2756, 2665, 139, 138}"
1518,1,125,0,0.1297025,"\int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx","Int[Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3),x]","\frac{3 \cos (e+f x) (c+d \sin (e+f x))^{7/3} F_1\left(\frac{7}{3};-\frac{1}{2},-\frac{1}{2};\frac{10}{3};\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right)}{7 d f \sqrt{1-\frac{c+d \sin (e+f x)}{c-d}} \sqrt{1-\frac{c+d \sin (e+f x)}{c+d}}}","\frac{3 \cos (e+f x) (c+d \sin (e+f x))^{7/3} F_1\left(\frac{7}{3};-\frac{1}{2},-\frac{1}{2};\frac{10}{3};\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right)}{7 d f \sqrt{1-\frac{c+d \sin (e+f x)}{c-d}} \sqrt{1-\frac{c+d \sin (e+f x)}{c+d}}}",1,"(3*AppellF1[7/3, -1/2, -1/2, 10/3, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(7*d*f*Sqrt[1 - (c + d*Sin[e + f*x])/(c - d)]*Sqrt[1 - (c + d*Sin[e + f*x])/(c + d)])","A",2,2,23,0.08696,1,"{2704, 138}"
1519,0,0,0,0.2108106,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{a+b \sin (e+f x)} \, dx","Int[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3))/(a + b*Sin[e + f*x]),x]","\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{a+b \sin (e+f x)} \, dx","\text{Int}\left(\frac{\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{a+b \sin (e+f x)},x\right)",0,"Defer[Int][(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3))/(a + b*Sin[e + f*x]), x]","A",0,0,0,0,-1,"{}"
1520,0,0,0,0.2077385,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx","Int[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3))/(a + b*Sin[e + f*x])^2,x]","\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx","\text{Int}\left(\frac{\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2},x\right)",0,"Defer[Int][(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3))/(a + b*Sin[e + f*x])^2, x]","A",0,0,0,0,-1,"{}"
1521,0,0,0,0.1309238,"\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","\text{Int}\left(\cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n,x\right)",0,"Defer[Int][Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x]","A",0,0,0,0,-1,"{}"
1522,0,0,0,0.1956108,"\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(4/3),x]","\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx","\text{Int}\left(\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} (a+b \sin (e+f x))^m,x\right)",0,"Defer[Int][Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(4/3), x]","A",0,0,0,0,-1,"{}"
1523,1,552,0,1.5122766,"\int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","-\frac{\sqrt{2} (c+d) \cos (e+f x) \left(a^2 c d^2 \left(n^2+7 n+12\right)-2 a b d (n+4) \left(2 c^2-d^2 (n+2)\right)+b^2 \left(6 c^3-c d^2 \left(-n^2-n+3\right)\right)\right) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{d^4 f (n+2) (n+3) (n+4) \sqrt{\sin (e+f x)+1}}-\frac{\sqrt{2} \left(c^2-d^2\right) \cos (e+f x) \left(-a^2 d^2 \left(n^2+7 n+12\right)+4 a b c d (n+4)+b^2 \left(-\left(6 c^2+d^2 \left(n^2+4 n+3\right)\right)\right)\right) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{d^4 f (n+2) (n+3) (n+4) \sqrt{\sin (e+f x)+1}}+\frac{\cos (e+f x) \left(2 a^2 d^2 (n+3)-4 a b c d (n+4)+b^2 \left(6 c^2-d^2 (n+3)\right)\right) (c+d \sin (e+f x))^{n+1}}{d^3 f (n+2) (n+3) (n+4)}-\frac{b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d^2 f (n+3) (n+4)}+\frac{\cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{n+1}}{d f (n+4)}","-\frac{\sqrt{2} (c+d) \cos (e+f x) \left(a^2 c d^2 \left(n^2+7 n+12\right)-2 a b d (n+4) \left(2 c^2-d^2 (n+2)\right)+b^2 c \left(6 c^2-d^2 \left(-n^2-n+3\right)\right)\right) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{d^4 f (n+2) (n+3) (n+4) \sqrt{\sin (e+f x)+1}}-\frac{\sqrt{2} \left(c^2-d^2\right) \cos (e+f x) \left(-a^2 d^2 \left(n^2+7 n+12\right)+4 a b c d (n+4)+b^2 \left(-\left(6 c^2+d^2 \left(n^2+4 n+3\right)\right)\right)\right) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{d^4 f (n+2) (n+3) (n+4) \sqrt{\sin (e+f x)+1}}+\frac{\cos (e+f x) \left(2 a^2 d^2 (n+3)-4 a b c d (n+4)+b^2 \left(6 c^2-d^2 (n+3)\right)\right) (c+d \sin (e+f x))^{n+1}}{d^3 f (n+2) (n+3) (n+4)}-\frac{b (3 b c-2 a d) \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d^2 f (n+3) (n+4)}+\frac{\cos (e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{n+1}}{d f (n+4)}",1,"((2*a^2*d^2*(3 + n) - 4*a*b*c*d*(4 + n) + b^2*(6*c^2 - d^2*(3 + n)))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^3*f*(2 + n)*(3 + n)*(4 + n)) - (b*(3*b*c - 2*a*d)*Cos[e + f*x]*Sin[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(3 + n)*(4 + n)) + (Cos[e + f*x]*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(4 + n)) - (Sqrt[2]*(c + d)*(a^2*c*d^2*(12 + 7*n + n^2) - 2*a*b*d*(4 + n)*(2*c^2 - d^2*(2 + n)) + b^2*(6*c^3 - c*d^2*(3 - n - n^2)))*AppellF1[1/2, 1/2, -1 - n, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(d^4*f*(2 + n)*(3 + n)*(4 + n)*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n) - (Sqrt[2]*(c^2 - d^2)*(4*a*b*c*d*(4 + n) - a^2*d^2*(12 + 7*n + n^2) - b^2*(6*c^2 + d^2*(3 + 4*n + n^2)))*AppellF1[1/2, 1/2, -n, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(d^4*f*(2 + n)*(3 + n)*(4 + n)*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",11,8,33,0.2424,1,"{2922, 3050, 3033, 3023, 2756, 2665, 139, 138}"
1524,1,373,0,0.6351508,"\int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^2*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\frac{\sqrt{2} (c+d) \cos (e+f x) \left(-a c d (n+3)+2 b c^2-b d^2 (n+2)\right) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{d^3 f (n+2) (n+3) \sqrt{\sin (e+f x)+1}}-\frac{\sqrt{2} \left(c^2-d^2\right) \cos (e+f x) (2 b c-a d (n+3)) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{d^3 f (n+2) (n+3) \sqrt{\sin (e+f x)+1}}-\frac{\cos (e+f x) (2 b c-a d (n+3)) (c+d \sin (e+f x))^{n+1}}{d^2 f (n+2) (n+3)}+\frac{b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (n+3)}","-\frac{\sqrt{2} (c+d) \cos (e+f x) \left(a c d (n+3)-b \left(2 c^2-d^2 (n+2)\right)\right) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n-1;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{d^3 f (n+2) (n+3) \sqrt{\sin (e+f x)+1}}-\frac{\sqrt{2} \left(c^2-d^2\right) \cos (e+f x) (2 b c-a d (n+3)) (c+d \sin (e+f x))^n \left(\frac{c+d \sin (e+f x)}{c+d}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right)}{d^3 f (n+2) (n+3) \sqrt{\sin (e+f x)+1}}-\frac{\cos (e+f x) (2 b c-a d (n+3)) (c+d \sin (e+f x))^{n+1}}{d^2 f (n+2) (n+3)}+\frac{b \sin (e+f x) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (n+3)}",1,"-(((2*b*c - a*d*(3 + n))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(2 + n)*(3 + n))) + (b*Cos[e + f*x]*Sin[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(3 + n)) + (Sqrt[2]*(c + d)*(2*b*c^2 - b*d^2*(2 + n) - a*c*d*(3 + n))*AppellF1[1/2, 1/2, -1 - n, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(d^3*f*(2 + n)*(3 + n)*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n) - (Sqrt[2]*(c^2 - d^2)*(2*b*c - a*d*(3 + n))*AppellF1[1/2, 1/2, -n, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(d^3*f*(2 + n)*(3 + n)*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n)","A",10,7,31,0.2258,1,"{2922, 3034, 3023, 2756, 2665, 139, 138}"
1525,1,127,0,0.0930213,"\int \cos ^2(e+f x) (c+d \sin (e+f x))^n \, dx","Int[Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n,x]","\frac{\cos (e+f x) (c+d \sin (e+f x))^{n+1} F_1\left(n+1;-\frac{1}{2},-\frac{1}{2};n+2;\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right)}{d f (n+1) \sqrt{1-\frac{c+d \sin (e+f x)}{c-d}} \sqrt{1-\frac{c+d \sin (e+f x)}{c+d}}}","\frac{\cos (e+f x) (c+d \sin (e+f x))^{n+1} F_1\left(n+1;-\frac{1}{2},-\frac{1}{2};n+2;\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right)}{d f (n+1) \sqrt{1-\frac{c+d \sin (e+f x)}{c-d}} \sqrt{1-\frac{c+d \sin (e+f x)}{c+d}}}",1,"(AppellF1[1 + n, -1/2, -1/2, 2 + n, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*Sqrt[1 - (c + d*Sin[e + f*x])/(c - d)]*Sqrt[1 - (c + d*Sin[e + f*x])/(c + d)])","A",2,2,21,0.09524,1,"{2704, 138}"
1526,0,0,0,0.1405708,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx","Int[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x]),x]","\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx","\text{Int}\left(\frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)},x\right)",0,"Defer[Int][(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x]), x]","A",0,0,0,0,-1,"{}"
1527,0,0,0,0.1333499,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx","Int[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x])^2,x]","\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx","\text{Int}\left(\frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2},x\right)",0,"Defer[Int][(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x])^2, x]","A",0,0,0,0,-1,"{}"
1528,1,188,0,0.2415268,"\int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{(a B+A b) \sin ^8(c+d x)}{8 d}-\frac{(a A-3 b B) \sin ^7(c+d x)}{7 d}+\frac{(a B+A b) \sin ^6(c+d x)}{2 d}+\frac{3 (a A-b B) \sin ^5(c+d x)}{5 d}-\frac{3 (a B+A b) \sin ^4(c+d x)}{4 d}-\frac{(3 a A-b B) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}-\frac{b B \sin ^9(c+d x)}{9 d}","-\frac{(a B+A b) \sin ^8(c+d x)}{8 d}-\frac{(a A-3 b B) \sin ^7(c+d x)}{7 d}+\frac{(a B+A b) \sin ^6(c+d x)}{2 d}+\frac{3 (a A-b B) \sin ^5(c+d x)}{5 d}-\frac{3 (a B+A b) \sin ^4(c+d x)}{4 d}-\frac{(3 a A-b B) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}-\frac{b B \sin ^9(c+d x)}{9 d}",1,"(a*A*Sin[c + d*x])/d + ((A*b + a*B)*Sin[c + d*x]^2)/(2*d) - ((3*a*A - b*B)*Sin[c + d*x]^3)/(3*d) - (3*(A*b + a*B)*Sin[c + d*x]^4)/(4*d) + (3*(a*A - b*B)*Sin[c + d*x]^5)/(5*d) + ((A*b + a*B)*Sin[c + d*x]^6)/(2*d) - ((a*A - 3*b*B)*Sin[c + d*x]^7)/(7*d) - ((A*b + a*B)*Sin[c + d*x]^8)/(8*d) - (b*B*Sin[c + d*x]^9)/(9*d)","A",3,2,29,0.06897,1,"{2837, 772}"
1529,1,143,0,0.1707956,"\int \cos ^5(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{(a B+A b) \sin ^6(c+d x)}{6 d}+\frac{(a A-2 b B) \sin ^5(c+d x)}{5 d}-\frac{(a B+A b) \sin ^4(c+d x)}{2 d}-\frac{(2 a A-b B) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}+\frac{b B \sin ^7(c+d x)}{7 d}","\frac{(a B+A b) \sin ^6(c+d x)}{6 d}+\frac{(a A-2 b B) \sin ^5(c+d x)}{5 d}-\frac{(a B+A b) \sin ^4(c+d x)}{2 d}-\frac{(2 a A-b B) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}+\frac{b B \sin ^7(c+d x)}{7 d}",1,"(a*A*Sin[c + d*x])/d + ((A*b + a*B)*Sin[c + d*x]^2)/(2*d) - ((2*a*A - b*B)*Sin[c + d*x]^3)/(3*d) - ((A*b + a*B)*Sin[c + d*x]^4)/(2*d) + ((a*A - 2*b*B)*Sin[c + d*x]^5)/(5*d) + ((A*b + a*B)*Sin[c + d*x]^6)/(6*d) + (b*B*Sin[c + d*x]^7)/(7*d)","A",3,2,29,0.06897,1,"{2837, 772}"
1530,1,97,0,0.1152968,"\int \cos ^3(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{(a B+A b) \sin ^4(c+d x)}{4 d}-\frac{(a A-b B) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}-\frac{b B \sin ^5(c+d x)}{5 d}","-\frac{(a B+A b) \sin ^4(c+d x)}{4 d}-\frac{(a A-b B) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}-\frac{b B \sin ^5(c+d x)}{5 d}",1,"(a*A*Sin[c + d*x])/d + ((A*b + a*B)*Sin[c + d*x]^2)/(2*d) - ((a*A - b*B)*Sin[c + d*x]^3)/(3*d) - ((A*b + a*B)*Sin[c + d*x]^4)/(4*d) - (b*B*Sin[c + d*x]^5)/(5*d)","A",3,2,29,0.06897,1,"{2837, 772}"
1531,1,52,0,0.0537233,"\int \cos (c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{(a B+A b) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}+\frac{b B \sin ^3(c+d x)}{3 d}","\frac{(a B+A b) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}+\frac{b B \sin ^3(c+d x)}{3 d}",1,"(a*A*Sin[c + d*x])/d + ((A*b + a*B)*Sin[c + d*x]^2)/(2*d) + (b*B*Sin[c + d*x]^3)/(3*d)","A",3,2,27,0.07407,1,"{2833, 43}"
1532,1,64,0,0.1082338,"\int \sec (c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{(a+b) (A+B) \log (1-\sin (c+d x))}{2 d}+\frac{(a-b) (A-B) \log (\sin (c+d x)+1)}{2 d}-\frac{b B \sin (c+d x)}{d}","-\frac{(a+b) (A+B) \log (1-\sin (c+d x))}{2 d}+\frac{(a-b) (A-B) \log (\sin (c+d x)+1)}{2 d}-\frac{b B \sin (c+d x)}{d}",1,"-((a + b)*(A + B)*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)*(A - B)*Log[1 + Sin[c + d*x]])/(2*d) - (b*B*Sin[c + d*x])/d","A",5,4,27,0.1481,1,"{2837, 774, 633, 31}"
1533,1,59,0,0.0757744,"\int \sec ^3(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{(a A-b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\sec ^2(c+d x) ((a A+b B) \sin (c+d x)+a B+A b)}{2 d}","\frac{(a A-b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\sec ^2(c+d x) ((a A+b B) \sin (c+d x)+a B+A b)}{2 d}",1,"((a*A - b*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (Sec[c + d*x]^2*(A*b + a*B + (a*A + b*B)*Sin[c + d*x]))/(2*d)","A",3,3,29,0.1034,1,"{2837, 778, 206}"
1534,1,88,0,0.0897351,"\int \sec ^5(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^5*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{(3 a A-b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\sec ^4(c+d x) ((a A+b B) \sin (c+d x)+a B+A b)}{4 d}+\frac{(3 a A-b B) \tan (c+d x) \sec (c+d x)}{8 d}","\frac{(3 a A-b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\sec ^4(c+d x) ((a A+b B) \sin (c+d x)+a B+A b)}{4 d}+\frac{(3 a A-b B) \tan (c+d x) \sec (c+d x)}{8 d}",1,"((3*a*A - b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(A*b + a*B + (a*A + b*B)*Sin[c + d*x]))/(4*d) + ((3*a*A - b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d)","A",4,4,29,0.1379,1,"{2837, 778, 199, 206}"
1535,1,118,0,0.1053686,"\int \sec ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^7*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{(5 a A-b B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\sec ^6(c+d x) ((a A+b B) \sin (c+d x)+a B+A b)}{6 d}+\frac{(5 a A-b B) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{(5 a A-b B) \tan (c+d x) \sec (c+d x)}{16 d}","\frac{(5 a A-b B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\sec ^6(c+d x) ((a A+b B) \sin (c+d x)+a B+A b)}{6 d}+\frac{(5 a A-b B) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{(5 a A-b B) \tan (c+d x) \sec (c+d x)}{16 d}",1,"((5*a*A - b*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (Sec[c + d*x]^6*(A*b + a*B + (a*A + b*B)*Sin[c + d*x]))/(6*d) + ((5*a*A - b*B)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((5*a*A - b*B)*Sec[c + d*x]^3*Tan[c + d*x])/(24*d)","A",5,4,29,0.1379,1,"{2837, 778, 199, 206}"
1536,1,349,0,0.3921808,"\int \cos ^7(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{3 \left(-7 a^2 B+2 a A b+b^2 B\right) (a+b \sin (c+d x))^8}{8 b^8 d}-\frac{\left(15 a^2 A b-35 a^3 B+15 a b^2 B-3 A b^3\right) (a+b \sin (c+d x))^7}{7 b^8 d}+\frac{\left(20 a^3 A b+30 a^2 b^2 B-35 a^4 B-12 a A b^3-3 b^4 B\right) (a+b \sin (c+d x))^6}{6 b^8 d}-\frac{3 \left(a^2-b^2\right) \left(5 a^2 A b-7 a^3 B+3 a b^2 B-A b^3\right) (a+b \sin (c+d x))^5}{5 b^8 d}+\frac{\left(a^2-b^2\right)^2 \left(-7 a^2 B+6 a A b+b^2 B\right) (a+b \sin (c+d x))^4}{4 b^8 d}-\frac{\left(a^2-b^2\right)^3 (A b-a B) (a+b \sin (c+d x))^3}{3 b^8 d}-\frac{(A b-7 a B) (a+b \sin (c+d x))^9}{9 b^8 d}-\frac{B (a+b \sin (c+d x))^{10}}{10 b^8 d}","\frac{3 \left(-7 a^2 B+2 a A b+b^2 B\right) (a+b \sin (c+d x))^8}{8 b^8 d}-\frac{\left(15 a^2 A b-35 a^3 B+15 a b^2 B-3 A b^3\right) (a+b \sin (c+d x))^7}{7 b^8 d}+\frac{\left(20 a^3 A b+30 a^2 b^2 B-35 a^4 B-12 a A b^3-3 b^4 B\right) (a+b \sin (c+d x))^6}{6 b^8 d}-\frac{3 \left(a^2-b^2\right) \left(5 a^2 A b-7 a^3 B+3 a b^2 B-A b^3\right) (a+b \sin (c+d x))^5}{5 b^8 d}+\frac{\left(a^2-b^2\right)^2 \left(-7 a^2 B+6 a A b+b^2 B\right) (a+b \sin (c+d x))^4}{4 b^8 d}-\frac{\left(a^2-b^2\right)^3 (A b-a B) (a+b \sin (c+d x))^3}{3 b^8 d}-\frac{(A b-7 a B) (a+b \sin (c+d x))^9}{9 b^8 d}-\frac{B (a+b \sin (c+d x))^{10}}{10 b^8 d}",1,"-((a^2 - b^2)^3*(A*b - a*B)*(a + b*Sin[c + d*x])^3)/(3*b^8*d) + ((a^2 - b^2)^2*(6*a*A*b - 7*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^4)/(4*b^8*d) - (3*(a^2 - b^2)*(5*a^2*A*b - A*b^3 - 7*a^3*B + 3*a*b^2*B)*(a + b*Sin[c + d*x])^5)/(5*b^8*d) + ((20*a^3*A*b - 12*a*A*b^3 - 35*a^4*B + 30*a^2*b^2*B - 3*b^4*B)*(a + b*Sin[c + d*x])^6)/(6*b^8*d) - ((15*a^2*A*b - 3*A*b^3 - 35*a^3*B + 15*a*b^2*B)*(a + b*Sin[c + d*x])^7)/(7*b^8*d) + (3*(2*a*A*b - 7*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^8)/(8*b^8*d) - ((A*b - 7*a*B)*(a + b*Sin[c + d*x])^9)/(9*b^8*d) - (B*(a + b*Sin[c + d*x])^10)/(10*b^8*d)","A",3,2,31,0.06452,1,"{2837, 772}"
1537,1,231,0,0.2590031,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{\left(-5 a^2 B+2 a A b+b^2 B\right) (a+b \sin (c+d x))^6}{3 b^6 d}+\frac{2 \left(3 a^2 A b-5 a^3 B+3 a b^2 B-A b^3\right) (a+b \sin (c+d x))^5}{5 b^6 d}-\frac{\left(a^2-b^2\right) \left(-5 a^2 B+4 a A b+b^2 B\right) (a+b \sin (c+d x))^4}{4 b^6 d}+\frac{\left(a^2-b^2\right)^2 (A b-a B) (a+b \sin (c+d x))^3}{3 b^6 d}+\frac{(A b-5 a B) (a+b \sin (c+d x))^7}{7 b^6 d}+\frac{B (a+b \sin (c+d x))^8}{8 b^6 d}","-\frac{\left(-5 a^2 B+2 a A b+b^2 B\right) (a+b \sin (c+d x))^6}{3 b^6 d}+\frac{2 \left(3 a^2 A b-5 a^3 B+3 a b^2 B-A b^3\right) (a+b \sin (c+d x))^5}{5 b^6 d}-\frac{\left(a^2-b^2\right) \left(-5 a^2 B+4 a A b+b^2 B\right) (a+b \sin (c+d x))^4}{4 b^6 d}+\frac{\left(a^2-b^2\right)^2 (A b-a B) (a+b \sin (c+d x))^3}{3 b^6 d}+\frac{(A b-5 a B) (a+b \sin (c+d x))^7}{7 b^6 d}+\frac{B (a+b \sin (c+d x))^8}{8 b^6 d}",1,"((a^2 - b^2)^2*(A*b - a*B)*(a + b*Sin[c + d*x])^3)/(3*b^6*d) - ((a^2 - b^2)*(4*a*A*b - 5*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^4)/(4*b^6*d) + (2*(3*a^2*A*b - A*b^3 - 5*a^3*B + 3*a*b^2*B)*(a + b*Sin[c + d*x])^5)/(5*b^6*d) - ((2*a*A*b - 5*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^6)/(3*b^6*d) + ((A*b - 5*a*B)*(a + b*Sin[c + d*x])^7)/(7*b^6*d) + (B*(a + b*Sin[c + d*x])^8)/(8*b^6*d)","A",3,2,31,0.06452,1,"{2837, 772}"
1538,1,132,0,0.1702266,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{\left(-3 a^2 B+2 a A b+b^2 B\right) (a+b \sin (c+d x))^4}{4 b^4 d}-\frac{\left(a^2-b^2\right) (A b-a B) (a+b \sin (c+d x))^3}{3 b^4 d}-\frac{(A b-3 a B) (a+b \sin (c+d x))^5}{5 b^4 d}-\frac{B (a+b \sin (c+d x))^6}{6 b^4 d}","\frac{\left(-3 a^2 B+2 a A b+b^2 B\right) (a+b \sin (c+d x))^4}{4 b^4 d}-\frac{\left(a^2-b^2\right) (A b-a B) (a+b \sin (c+d x))^3}{3 b^4 d}-\frac{(A b-3 a B) (a+b \sin (c+d x))^5}{5 b^4 d}-\frac{B (a+b \sin (c+d x))^6}{6 b^4 d}",1,"-((a^2 - b^2)*(A*b - a*B)*(a + b*Sin[c + d*x])^3)/(3*b^4*d) + ((2*a*A*b - 3*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^4)/(4*b^4*d) - ((A*b - 3*a*B)*(a + b*Sin[c + d*x])^5)/(5*b^4*d) - (B*(a + b*Sin[c + d*x])^6)/(6*b^4*d)","A",3,2,31,0.06452,1,"{2837, 772}"
1539,1,54,0,0.07638,"\int \cos (c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{(A b-a B) (a+b \sin (c+d x))^3}{3 b^2 d}+\frac{B (a+b \sin (c+d x))^4}{4 b^2 d}","\frac{(A b-a B) (a+b \sin (c+d x))^3}{3 b^2 d}+\frac{B (a+b \sin (c+d x))^4}{4 b^2 d}",1,"((A*b - a*B)*(a + b*Sin[c + d*x])^3)/(3*b^2*d) + (B*(a + b*Sin[c + d*x])^4)/(4*b^2*d)","A",3,2,29,0.06897,1,"{2833, 43}"
1540,1,94,0,0.1736672,"\int \sec (c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{b (2 a B+A b) \sin (c+d x)}{d}+\frac{(a-b)^2 (A-B) \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^2 (A+B) \log (1-\sin (c+d x))}{2 d}-\frac{b^2 B \sin ^2(c+d x)}{2 d}","-\frac{b (2 a B+A b) \sin (c+d x)}{d}+\frac{(a-b)^2 (A-B) \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^2 (A+B) \log (1-\sin (c+d x))}{2 d}-\frac{b^2 B \sin ^2(c+d x)}{2 d}",1,"-((a + b)^2*(A + B)*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)^2*(A - B)*Log[1 + Sin[c + d*x]])/(2*d) - (b*(A*b + 2*a*B)*Sin[c + d*x])/d - (b^2*B*Sin[c + d*x]^2)/(2*d)","A",6,4,29,0.1379,1,"{2837, 801, 633, 31}"
1541,1,112,0,0.1799819,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{(a+b) (a A-b (A+2 B)) \log (1-\sin (c+d x))}{4 d}+\frac{(a-b) (a A+b (A-2 B)) \log (\sin (c+d x)+1)}{4 d}+\frac{\sec ^2(c+d x) (a+b \sin (c+d x)) ((a A+b B) \sin (c+d x)+a B+A b)}{2 d}","-\frac{(a+b) (a A-b (A+2 B)) \log (1-\sin (c+d x))}{4 d}+\frac{(a-b) (a A+b (A-2 B)) \log (\sin (c+d x)+1)}{4 d}+\frac{\sec ^2(c+d x) (a+b \sin (c+d x)) ((a A+b B) \sin (c+d x)+a B+A b)}{2 d}",1,"-((a + b)*(a*A - b*(A + 2*B))*Log[1 - Sin[c + d*x]])/(4*d) + ((a - b)*(a*A + b*(A - 2*B))*Log[1 + Sin[c + d*x]])/(4*d) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x])*(A*b + a*B + (a*A + b*B)*Sin[c + d*x]))/(2*d)","A",5,4,31,0.1290,1,"{2837, 819, 633, 31}"
1542,1,122,0,0.155705,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{\left(3 a^2 A-2 a b B-A b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\sec ^2(c+d x) \left(\left(3 a^2 A-2 a b B+A b^2\right) \sin (c+d x)+2 b (2 a A-b B)\right)}{8 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^2 (A \sin (c+d x)+B)}{4 d}","\frac{\left(3 a^2 A-2 a b B-A b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\sec ^2(c+d x) \left(\left(3 a^2 A-2 a b B+A b^2\right) \sin (c+d x)+2 b (2 a A-b B)\right)}{8 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^2 (A \sin (c+d x)+B)}{4 d}",1,"((3*a^2*A - A*b^2 - 2*a*b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(B + A*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(4*d) + (Sec[c + d*x]^2*(2*b*(2*a*A - b*B) + (3*a^2*A + A*b^2 - 2*a*b*B)*Sin[c + d*x]))/(8*d)","A",4,4,31,0.1290,1,"{2837, 821, 778, 206}"
1543,1,160,0,0.2051963,"\int \sec ^7(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Int[Sec[c + d*x]^7*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{\left(5 a^2 A-2 a b B-A b^2\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\sec ^4(c+d x) \left(\left(5 a^2 A-2 a b B+3 A b^2\right) \sin (c+d x)+2 b (4 a A-b B)\right)}{24 d}+\frac{\left(5 a^2 A-2 a b B-A b^2\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{\sec ^6(c+d x) (a+b \sin (c+d x))^2 (A \sin (c+d x)+B)}{6 d}","\frac{\left(5 a^2 A-2 a b B-A b^2\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\sec ^4(c+d x) \left(\left(5 a^2 A-2 a b B+3 A b^2\right) \sin (c+d x)+2 b (4 a A-b B)\right)}{24 d}+\frac{\left(5 a^2 A-2 a b B-A b^2\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{\sec ^6(c+d x) (a+b \sin (c+d x))^2 (A \sin (c+d x)+B)}{6 d}",1,"((5*a^2*A - A*b^2 - 2*a*b*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (Sec[c + d*x]^6*(B + A*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(6*d) + (Sec[c + d*x]^4*(2*b*(4*a*A - b*B) + (5*a^2*A + 3*A*b^2 - 2*a*b*B)*Sin[c + d*x]))/(24*d) + ((5*a^2*A - A*b^2 - 2*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(16*d)","A",5,5,31,0.1613,1,"{2837, 821, 778, 199, 206}"
1544,1,315,0,0.3560665,"\int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{\left(a^2 (-B)+a A b+3 b^2 B\right) \sin ^5(c+d x)}{5 b^3 d}-\frac{\left(a^2-3 b^2\right) (A b-a B) \sin ^4(c+d x)}{4 b^4 d}+\frac{\left(a^3 A b+3 a^2 b^2 B+a^4 (-B)-3 a A b^3-3 b^4 B\right) \sin ^3(c+d x)}{3 b^5 d}-\frac{\left(-3 a^2 b^2+a^4+3 b^4\right) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac{\left(-3 a^3 A b^3+a^5 A b+3 a^4 b^2 B-3 a^2 b^4 B+a^6 (-B)+3 a A b^5+b^6 B\right) \sin (c+d x)}{b^7 d}-\frac{\left(a^2-b^2\right)^3 (A b-a B) \log (a+b \sin (c+d x))}{b^8 d}-\frac{(A b-a B) \sin ^6(c+d x)}{6 b^2 d}-\frac{B \sin ^7(c+d x)}{7 b d}","\frac{\left(a^2 (-B)+a A b+3 b^2 B\right) \sin ^5(c+d x)}{5 b^3 d}-\frac{\left(a^2-3 b^2\right) (A b-a B) \sin ^4(c+d x)}{4 b^4 d}+\frac{\left(a^3 A b+3 a^2 b^2 B+a^4 (-B)-3 a A b^3-3 b^4 B\right) \sin ^3(c+d x)}{3 b^5 d}-\frac{\left(-3 a^2 b^2+a^4+3 b^4\right) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac{\left(-3 a^3 A b^3+a^5 A b+3 a^4 b^2 B-3 a^2 b^4 B+a^6 (-B)+3 a A b^5+b^6 B\right) \sin (c+d x)}{b^7 d}-\frac{\left(a^2-b^2\right)^3 (A b-a B) \log (a+b \sin (c+d x))}{b^8 d}-\frac{(A b-a B) \sin ^6(c+d x)}{6 b^2 d}-\frac{B \sin ^7(c+d x)}{7 b d}",1,"-(((a^2 - b^2)^3*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^8*d)) + ((a^5*A*b - 3*a^3*A*b^3 + 3*a*A*b^5 - a^6*B + 3*a^4*b^2*B - 3*a^2*b^4*B + b^6*B)*Sin[c + d*x])/(b^7*d) - ((a^4 - 3*a^2*b^2 + 3*b^4)*(A*b - a*B)*Sin[c + d*x]^2)/(2*b^6*d) + ((a^3*A*b - 3*a*A*b^3 - a^4*B + 3*a^2*b^2*B - 3*b^4*B)*Sin[c + d*x]^3)/(3*b^5*d) - ((a^2 - 3*b^2)*(A*b - a*B)*Sin[c + d*x]^4)/(4*b^4*d) + ((a*A*b - a^2*B + 3*b^2*B)*Sin[c + d*x]^5)/(5*b^3*d) - ((A*b - a*B)*Sin[c + d*x]^6)/(6*b^2*d) - (B*Sin[c + d*x]^7)/(7*b*d)","A",3,2,31,0.06452,1,"{2837, 772}"
1545,1,202,0,0.2478784,"\int \frac{\cos ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","-\frac{\left(a^2 (-B)+a A b+2 b^2 B\right) \sin ^3(c+d x)}{3 b^3 d}+\frac{\left(a^2-2 b^2\right) (A b-a B) \sin ^2(c+d x)}{2 b^4 d}-\frac{\left(a^3 A b+2 a^2 b^2 B+a^4 (-B)-2 a A b^3-b^4 B\right) \sin (c+d x)}{b^5 d}+\frac{\left(a^2-b^2\right)^2 (A b-a B) \log (a+b \sin (c+d x))}{b^6 d}+\frac{(A b-a B) \sin ^4(c+d x)}{4 b^2 d}+\frac{B \sin ^5(c+d x)}{5 b d}","-\frac{\left(a^2 (-B)+a A b+2 b^2 B\right) \sin ^3(c+d x)}{3 b^3 d}+\frac{\left(a^2-2 b^2\right) (A b-a B) \sin ^2(c+d x)}{2 b^4 d}-\frac{\left(a^3 A b+2 a^2 b^2 B+a^4 (-B)-2 a A b^3-b^4 B\right) \sin (c+d x)}{b^5 d}+\frac{\left(a^2-b^2\right)^2 (A b-a B) \log (a+b \sin (c+d x))}{b^6 d}+\frac{(A b-a B) \sin ^4(c+d x)}{4 b^2 d}+\frac{B \sin ^5(c+d x)}{5 b d}",1,"((a^2 - b^2)^2*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^6*d) - ((a^3*A*b - 2*a*A*b^3 - a^4*B + 2*a^2*b^2*B - b^4*B)*Sin[c + d*x])/(b^5*d) + ((a^2 - 2*b^2)*(A*b - a*B)*Sin[c + d*x]^2)/(2*b^4*d) - ((a*A*b - a^2*B + 2*b^2*B)*Sin[c + d*x]^3)/(3*b^3*d) + ((A*b - a*B)*Sin[c + d*x]^4)/(4*b^2*d) + (B*Sin[c + d*x]^5)/(5*b*d)","A",3,2,31,0.06452,1,"{2837, 772}"
1546,1,111,0,0.1627421,"\int \frac{\cos ^3(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{\left(a^2 (-B)+a A b+b^2 B\right) \sin (c+d x)}{b^3 d}-\frac{\left(a^2-b^2\right) (A b-a B) \log (a+b \sin (c+d x))}{b^4 d}-\frac{(A b-a B) \sin ^2(c+d x)}{2 b^2 d}-\frac{B \sin ^3(c+d x)}{3 b d}","\frac{\left(a^2 (-B)+a A b+b^2 B\right) \sin (c+d x)}{b^3 d}-\frac{\left(a^2-b^2\right) (A b-a B) \log (a+b \sin (c+d x))}{b^4 d}-\frac{(A b-a B) \sin ^2(c+d x)}{2 b^2 d}-\frac{B \sin ^3(c+d x)}{3 b d}",1,"-(((a^2 - b^2)*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^4*d)) + ((a*A*b - a^2*B + b^2*B)*Sin[c + d*x])/(b^3*d) - ((A*b - a*B)*Sin[c + d*x]^2)/(2*b^2*d) - (B*Sin[c + d*x]^3)/(3*b*d)","A",3,2,31,0.06452,1,"{2837, 772}"
1547,1,41,0,0.0701739,"\int \frac{\cos (c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{(A b-a B) \log (a+b \sin (c+d x))}{b^2 d}+\frac{B \sin (c+d x)}{b d}","\frac{(A b-a B) \log (a+b \sin (c+d x))}{b^2 d}+\frac{B \sin (c+d x)}{b d}",1,"((A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^2*d) + (B*Sin[c + d*x])/(b*d)","A",3,2,29,0.06897,1,"{2833, 43}"
1548,1,90,0,0.1483704,"\int \frac{\sec (c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","-\frac{(A b-a B) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)}-\frac{(A+B) \log (1-\sin (c+d x))}{2 d (a+b)}+\frac{(A-B) \log (\sin (c+d x)+1)}{2 d (a-b)}","-\frac{(A b-a B) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)}-\frac{(A+B) \log (1-\sin (c+d x))}{2 d (a+b)}+\frac{(A-B) \log (\sin (c+d x)+1)}{2 d (a-b)}",1,"-((A + B)*Log[1 - Sin[c + d*x]])/(2*(a + b)*d) + ((A - B)*Log[1 + Sin[c + d*x]])/(2*(a - b)*d) - ((A*b - a*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)*d)","A",3,2,29,0.06897,1,"{2837, 801}"
1549,1,159,0,0.2897023,"\int \frac{\sec ^3(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{b^2 (A b-a B) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{\sec ^2(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{2 d \left(a^2-b^2\right)}-\frac{(a A+b (2 A+B)) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(a A-b (2 A-B)) \log (\sin (c+d x)+1)}{4 d (a-b)^2}","\frac{b^2 (A b-a B) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{\sec ^2(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{2 d \left(a^2-b^2\right)}-\frac{(a A+b (2 A+B)) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(a A-b (2 A-B)) \log (\sin (c+d x)+1)}{4 d (a-b)^2}",1,"-((a*A + b*(2*A + B))*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((a*A - b*(2*A - B))*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) + (b^2*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) - (Sec[c + d*x]^2*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(2*(a^2 - b^2)*d)","A",4,3,31,0.09677,1,"{2837, 823, 801}"
1550,1,263,0,0.4477821,"\int \frac{\sec ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","-\frac{b^4 (A b-a B) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\left(3 a^2 A+a b (9 A+B)+b^2 (8 A+3 B)\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(3 a^2 A-a b (9 A-B)+b^2 (8 A-3 B)\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{\sec ^4(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(\left(3 a^3 A+a^2 b B-7 a A b^2+3 b^3 B\right) \sin (c+d x)+4 b^2 (A b-a B)\right)}{8 d \left(a^2-b^2\right)^2}","-\frac{b^4 (A b-a B) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\left(3 a^2 A+a b (9 A+B)+b^2 (8 A+3 B)\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(3 a^2 A-a b (9 A-B)+b^2 (8 A-3 B)\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{\sec ^4(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(\left(3 a^3 A+a^2 b B-7 a A b^2+3 b^3 B\right) \sin (c+d x)+4 b^2 (A b-a B)\right)}{8 d \left(a^2-b^2\right)^2}",1,"-((3*a^2*A + a*b*(9*A + B) + b^2*(8*A + 3*B))*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + ((3*a^2*A + b^2*(8*A - 3*B) - a*b*(9*A - B))*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (b^4*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (Sec[c + d*x]^4*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(4*b^2*(A*b - a*B) + (3*a^3*A - 7*a*A*b^2 + a^2*b*B + 3*b^3*B)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",5,3,31,0.09677,1,"{2837, 823, 801}"
1551,1,383,0,0.6808531,"\int \frac{\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Int[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{b^6 (A b-a B) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}-\frac{\left(a^2 b (20 A+B)+5 a^3 A+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right) \log (1-\sin (c+d x))}{32 d (a+b)^4}+\frac{\left(-a^2 b (20 A-B)+5 a^3 A+a b^2 (29 A-4 B)-b^3 (16 A-5 B)\right) \log (\sin (c+d x)+1)}{32 d (a-b)^4}-\frac{\sec ^6(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{6 d \left(a^2-b^2\right)}+\frac{\sec ^4(c+d x) \left(\left(5 a^3 A+a^2 b B-11 a A b^2+5 b^3 B\right) \sin (c+d x)+6 b^2 (A b-a B)\right)}{24 d \left(a^2-b^2\right)^2}-\frac{\sec ^2(c+d x) \left(8 b^4 (A b-a B)-\left(-16 a^3 A b^2+5 a^5 A-4 a^2 b^3 B+a^4 b B+19 a A b^4-5 b^5 B\right) \sin (c+d x)\right)}{16 d \left(a^2-b^2\right)^3}","\frac{b^6 (A b-a B) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}-\frac{\left(a^2 b (20 A+B)+5 a^3 A+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right) \log (1-\sin (c+d x))}{32 d (a+b)^4}+\frac{\left(-a^2 b (20 A-B)+5 a^3 A+a b^2 (29 A-4 B)-b^3 (16 A-5 B)\right) \log (\sin (c+d x)+1)}{32 d (a-b)^4}-\frac{\sec ^6(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{6 d \left(a^2-b^2\right)}+\frac{\sec ^4(c+d x) \left(\left(5 a^3 A+a^2 b B-11 a A b^2+5 b^3 B\right) \sin (c+d x)+6 b^2 (A b-a B)\right)}{24 d \left(a^2-b^2\right)^2}-\frac{\sec ^2(c+d x) \left(8 b^4 (A b-a B)-\left(-16 a^3 A b^2+5 a^5 A-4 a^2 b^3 B+a^4 b B+19 a A b^4-5 b^5 B\right) \sin (c+d x)\right)}{16 d \left(a^2-b^2\right)^3}",1,"-((5*a^3*A + a^2*b*(20*A + B) + a*b^2*(29*A + 4*B) + b^3*(16*A + 5*B))*Log[1 - Sin[c + d*x]])/(32*(a + b)^4*d) + ((5*a^3*A - b^3*(16*A - 5*B) + a*b^2*(29*A - 4*B) - a^2*b*(20*A - B))*Log[1 + Sin[c + d*x]])/(32*(a - b)^4*d) + (b^6*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) - (Sec[c + d*x]^6*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(6*(a^2 - b^2)*d) + (Sec[c + d*x]^4*(6*b^2*(A*b - a*B) + (5*a^3*A - 11*a*A*b^2 + a^2*b*B + 5*b^3*B)*Sin[c + d*x]))/(24*(a^2 - b^2)^2*d) - (Sec[c + d*x]^2*(8*b^4*(A*b - a*B) - (5*a^5*A - 16*a^3*A*b^2 + 19*a*A*b^4 + a^4*b*B - 4*a^2*b^3*B - 5*b^5*B)*Sin[c + d*x]))/(16*(a^2 - b^2)^3*d)","A",6,3,31,0.09677,1,"{2837, 823, 801}"
1552,1,324,0,0.3992743,"\int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","\frac{\left(-3 a^2 B+2 a A b+3 b^2 B\right) \sin ^4(c+d x)}{4 b^4 d}-\frac{\left(3 a^2 A b-4 a^3 B+6 a b^2 B-3 A b^3\right) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left(4 a^3 A b+9 a^2 b^2 B-5 a^4 B-6 a A b^3-3 b^4 B\right) \sin ^2(c+d x)}{2 b^6 d}-\frac{\left(-9 a^2 A b^3+5 a^4 A b+12 a^3 b^2 B-6 a^5 B-6 a b^4 B+3 A b^5\right) \sin (c+d x)}{b^7 d}+\frac{\left(a^2-b^2\right)^3 (A b-a B)}{b^8 d (a+b \sin (c+d x))}+\frac{\left(a^2-b^2\right)^2 \left(-7 a^2 B+6 a A b+b^2 B\right) \log (a+b \sin (c+d x))}{b^8 d}-\frac{(A b-2 a B) \sin ^5(c+d x)}{5 b^3 d}-\frac{B \sin ^6(c+d x)}{6 b^2 d}","\frac{\left(-3 a^2 B+2 a A b+3 b^2 B\right) \sin ^4(c+d x)}{4 b^4 d}-\frac{\left(3 a^2 A b-4 a^3 B+6 a b^2 B-3 A b^3\right) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left(4 a^3 A b+9 a^2 b^2 B-5 a^4 B-6 a A b^3-3 b^4 B\right) \sin ^2(c+d x)}{2 b^6 d}-\frac{\left(-9 a^2 A b^3+5 a^4 A b+12 a^3 b^2 B-6 a^5 B-6 a b^4 B+3 A b^5\right) \sin (c+d x)}{b^7 d}+\frac{\left(a^2-b^2\right)^3 (A b-a B)}{b^8 d (a+b \sin (c+d x))}+\frac{\left(a^2-b^2\right)^2 \left(-7 a^2 B+6 a A b+b^2 B\right) \log (a+b \sin (c+d x))}{b^8 d}-\frac{(A b-2 a B) \sin ^5(c+d x)}{5 b^3 d}-\frac{B \sin ^6(c+d x)}{6 b^2 d}",1,"((a^2 - b^2)^2*(6*a*A*b - 7*a^2*B + b^2*B)*Log[a + b*Sin[c + d*x]])/(b^8*d) - ((5*a^4*A*b - 9*a^2*A*b^3 + 3*A*b^5 - 6*a^5*B + 12*a^3*b^2*B - 6*a*b^4*B)*Sin[c + d*x])/(b^7*d) + ((4*a^3*A*b - 6*a*A*b^3 - 5*a^4*B + 9*a^2*b^2*B - 3*b^4*B)*Sin[c + d*x]^2)/(2*b^6*d) - ((3*a^2*A*b - 3*A*b^3 - 4*a^3*B + 6*a*b^2*B)*Sin[c + d*x]^3)/(3*b^5*d) + ((2*a*A*b - 3*a^2*B + 3*b^2*B)*Sin[c + d*x]^4)/(4*b^4*d) - ((A*b - 2*a*B)*Sin[c + d*x]^5)/(5*b^3*d) - (B*Sin[c + d*x]^6)/(6*b^2*d) + ((a^2 - b^2)^3*(A*b - a*B))/(b^8*d*(a + b*Sin[c + d*x]))","A",3,2,31,0.06452,1,"{2837, 772}"
1553,1,206,0,0.2719051,"\int \frac{\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","-\frac{\left(-3 a^2 B+2 a A b+2 b^2 B\right) \sin ^2(c+d x)}{2 b^4 d}+\frac{\left(3 a^2 A b-4 a^3 B+4 a b^2 B-2 A b^3\right) \sin (c+d x)}{b^5 d}-\frac{\left(a^2-b^2\right)^2 (A b-a B)}{b^6 d (a+b \sin (c+d x))}-\frac{\left(a^2-b^2\right) \left(-5 a^2 B+4 a A b+b^2 B\right) \log (a+b \sin (c+d x))}{b^6 d}+\frac{(A b-2 a B) \sin ^3(c+d x)}{3 b^3 d}+\frac{B \sin ^4(c+d x)}{4 b^2 d}","-\frac{\left(-3 a^2 B+2 a A b+2 b^2 B\right) \sin ^2(c+d x)}{2 b^4 d}+\frac{\left(3 a^2 A b-4 a^3 B+4 a b^2 B-2 A b^3\right) \sin (c+d x)}{b^5 d}-\frac{\left(a^2-b^2\right)^2 (A b-a B)}{b^6 d (a+b \sin (c+d x))}-\frac{\left(a^2-b^2\right) \left(-5 a^2 B+4 a A b+b^2 B\right) \log (a+b \sin (c+d x))}{b^6 d}+\frac{(A b-2 a B) \sin ^3(c+d x)}{3 b^3 d}+\frac{B \sin ^4(c+d x)}{4 b^2 d}",1,"-(((a^2 - b^2)*(4*a*A*b - 5*a^2*B + b^2*B)*Log[a + b*Sin[c + d*x]])/(b^6*d)) + ((3*a^2*A*b - 2*A*b^3 - 4*a^3*B + 4*a*b^2*B)*Sin[c + d*x])/(b^5*d) - ((2*a*A*b - 3*a^2*B + 2*b^2*B)*Sin[c + d*x]^2)/(2*b^4*d) + ((A*b - 2*a*B)*Sin[c + d*x]^3)/(3*b^3*d) + (B*Sin[c + d*x]^4)/(4*b^2*d) - ((a^2 - b^2)^2*(A*b - a*B))/(b^6*d*(a + b*Sin[c + d*x]))","A",3,2,31,0.06452,1,"{2837, 772}"
1554,1,113,0,0.168508,"\int \frac{\cos ^3(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-b^2\right) (A b-a B)}{b^4 d (a+b \sin (c+d x))}+\frac{\left(-3 a^2 B+2 a A b+b^2 B\right) \log (a+b \sin (c+d x))}{b^4 d}-\frac{(A b-2 a B) \sin (c+d x)}{b^3 d}-\frac{B \sin ^2(c+d x)}{2 b^2 d}","\frac{\left(a^2-b^2\right) (A b-a B)}{b^4 d (a+b \sin (c+d x))}+\frac{\left(-3 a^2 B+2 a A b+b^2 B\right) \log (a+b \sin (c+d x))}{b^4 d}-\frac{(A b-2 a B) \sin (c+d x)}{b^3 d}-\frac{B \sin ^2(c+d x)}{2 b^2 d}",1,"((2*a*A*b - 3*a^2*B + b^2*B)*Log[a + b*Sin[c + d*x]])/(b^4*d) - ((A*b - 2*a*B)*Sin[c + d*x])/(b^3*d) - (B*Sin[c + d*x]^2)/(2*b^2*d) + ((a^2 - b^2)*(A*b - a*B))/(b^4*d*(a + b*Sin[c + d*x]))","A",3,2,31,0.06452,1,"{2837, 772}"
1555,1,48,0,0.0767375,"\int \frac{\cos (c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","\frac{B \log (a+b \sin (c+d x))}{b^2 d}-\frac{A b-a B}{b^2 d (a+b \sin (c+d x))}","\frac{B \log (a+b \sin (c+d x))}{b^2 d}-\frac{A b-a B}{b^2 d (a+b \sin (c+d x))}",1,"(B*Log[a + b*Sin[c + d*x]])/(b^2*d) - (A*b - a*B)/(b^2*d*(a + b*Sin[c + d*x]))","A",3,2,29,0.06897,1,"{2833, 43}"
1556,1,135,0,0.1937157,"\int \frac{\sec (c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","\frac{A b-a B}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\left(a^2 (-B)+2 a A b-b^2 B\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{(A+B) \log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{(A-B) \log (\sin (c+d x)+1)}{2 d (a-b)^2}","\frac{A b-a B}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\left(a^2 (-B)+2 a A b-b^2 B\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{(A+B) \log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{(A-B) \log (\sin (c+d x)+1)}{2 d (a-b)^2}",1,"-((A + B)*Log[1 - Sin[c + d*x]])/(2*(a + b)^2*d) + ((A - B)*Log[1 + Sin[c + d*x]])/(2*(a - b)^2*d) - ((2*a*A*b - a^2*B - b^2*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + (A*b - a*B)/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",3,2,29,0.06897,1,"{2837, 801}"
1557,1,228,0,0.3262916,"\int \frac{\sec ^3(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","-\frac{b \left(a^2 A-4 a b B+3 A b^2\right)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b^2 \left(-3 a^2 B+4 a A b-b^2 B\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\sec ^2(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{(a A+3 A b+2 b B) \log (1-\sin (c+d x))}{4 d (a+b)^3}+\frac{(a A-3 A b+2 b B) \log (\sin (c+d x)+1)}{4 d (a-b)^3}","-\frac{b \left(a^2 A-4 a b B+3 A b^2\right)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b^2 \left(-3 a^2 B+4 a A b-b^2 B\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\sec ^2(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{(a A+3 A b+2 b B) \log (1-\sin (c+d x))}{4 d (a+b)^3}+\frac{(a A-3 A b+2 b B) \log (\sin (c+d x)+1)}{4 d (a-b)^3}",1,"-((a*A + 3*A*b + 2*b*B)*Log[1 - Sin[c + d*x]])/(4*(a + b)^3*d) + ((a*A - 3*A*b + 2*b*B)*Log[1 + Sin[c + d*x]])/(4*(a - b)^3*d) + (b^2*(4*a*A*b - 3*a^2*B - b^2*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (b*(a^2*A + 3*A*b^2 - 4*a*b*B))/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^2*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",4,3,31,0.09677,1,"{2837, 823, 801}"
1558,1,372,0,0.5641633,"\int \frac{\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","-\frac{b \left(-12 a^2 A b^2+3 a^4 A+2 a^3 b B+22 a b^3 B-15 A b^4\right)}{8 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{b^4 \left(-5 a^2 B+6 a A b-b^2 B\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}-\frac{\left(3 a^2 A+2 a b (6 A+B)+b^2 (15 A+8 B)\right) \log (1-\sin (c+d x))}{16 d (a+b)^4}+\frac{\left(3 a^2 A-2 a b (6 A-B)+b^2 (15 A-8 B)\right) \log (\sin (c+d x)+1)}{16 d (a-b)^4}-\frac{\sec ^4(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{4 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left(\left(3 a^3 A+2 a^2 b B-9 a A b^2+4 b^3 B\right) \sin (c+d x)+b \left(a^2 A-6 a b B+5 A b^2\right)\right)}{8 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}","-\frac{b \left(-12 a^2 A b^2+3 a^4 A+2 a^3 b B+22 a b^3 B-15 A b^4\right)}{8 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{b^4 \left(-5 a^2 B+6 a A b-b^2 B\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}-\frac{\left(3 a^2 A+2 a b (6 A+B)+b^2 (15 A+8 B)\right) \log (1-\sin (c+d x))}{16 d (a+b)^4}+\frac{\left(3 a^2 A-2 a b (6 A-B)+b^2 (15 A-8 B)\right) \log (\sin (c+d x)+1)}{16 d (a-b)^4}-\frac{\sec ^4(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{4 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left(\left(3 a^3 A+2 a^2 b B-9 a A b^2+4 b^3 B\right) \sin (c+d x)+b \left(a^2 A-6 a b B+5 A b^2\right)\right)}{8 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}",1,"-((3*a^2*A + 2*a*b*(6*A + B) + b^2*(15*A + 8*B))*Log[1 - Sin[c + d*x]])/(16*(a + b)^4*d) + ((3*a^2*A + b^2*(15*A - 8*B) - 2*a*b*(6*A - B))*Log[1 + Sin[c + d*x]])/(16*(a - b)^4*d) - (b^4*(6*a*A*b - 5*a^2*B - b^2*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) - (b*(3*a^4*A - 12*a^2*A*b^2 - 15*A*b^4 + 2*a^3*b*B + 22*a*b^3*B))/(8*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^4*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(4*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(b*(a^2*A + 5*A*b^2 - 6*a*b*B) + (3*a^3*A - 9*a*A*b^2 + 2*a^2*b*B + 4*b^3*B)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",5,3,31,0.09677,1,"{2837, 823, 801}"
1559,1,550,0,0.9281595,"\int \frac{\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Int[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","-\frac{b \left(-23 a^4 A b^2+47 a^2 A b^4+5 a^6 A-12 a^3 b^3 B+2 a^5 b B-54 a b^5 B+35 A b^6\right)}{16 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}+\frac{b^6 \left(-7 a^2 B+8 a A b-b^2 B\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^5}-\frac{\left(a^2 b (25 A+2 B)+5 a^3 A+a b^2 (47 A+10 B)+b^3 (35 A+16 B)\right) \log (1-\sin (c+d x))}{32 d (a+b)^5}+\frac{\left(-a^2 (25 A b-2 b B)+5 a^3 A+a b^2 (47 A-10 B)-b^3 (35 A-16 B)\right) \log (\sin (c+d x)+1)}{32 d (a-b)^5}-\frac{\sec ^6(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{6 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{\sec ^4(c+d x) \left(\left(5 a^3 A+2 a^2 b B-13 a A b^2+6 b^3 B\right) \sin (c+d x)+b \left(a^2 A-8 a b B+7 A b^2\right)\right)}{24 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left(3 \left(-18 a^3 A b^2+5 a^5 A-10 a^2 b^3 B+2 a^4 b B+29 a A b^4-8 b^5 B\right) \sin (c+d x)+b \left(-18 a^2 A b^2+5 a^4 A+2 a^3 b B+46 a b^3 B-35 A b^4\right)\right)}{48 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}","-\frac{b \left(-23 a^4 A b^2+47 a^2 A b^4+5 a^6 A-12 a^3 b^3 B+2 a^5 b B-54 a b^5 B+35 A b^6\right)}{16 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}+\frac{b^6 \left(-7 a^2 B+8 a A b-b^2 B\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^5}-\frac{\left(a^2 b (25 A+2 B)+5 a^3 A+a b^2 (47 A+10 B)+b^3 (35 A+16 B)\right) \log (1-\sin (c+d x))}{32 d (a+b)^5}+\frac{\left(-a^2 (25 A b-2 b B)+5 a^3 A+a b^2 (47 A-10 B)-b^3 (35 A-16 B)\right) \log (\sin (c+d x)+1)}{32 d (a-b)^5}-\frac{\sec ^6(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{6 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{\sec ^4(c+d x) \left(\left(5 a^3 A+2 a^2 b B-13 a A b^2+6 b^3 B\right) \sin (c+d x)+b \left(a^2 A-8 a b B+7 A b^2\right)\right)}{24 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left(3 \left(-18 a^3 A b^2+5 a^5 A-10 a^2 b^3 B+2 a^4 b B+29 a A b^4-8 b^5 B\right) \sin (c+d x)+b \left(-18 a^2 A b^2+5 a^4 A+2 a^3 b B+46 a b^3 B-35 A b^4\right)\right)}{48 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}",1,"-((5*a^3*A + a^2*b*(25*A + 2*B) + a*b^2*(47*A + 10*B) + b^3*(35*A + 16*B))*Log[1 - Sin[c + d*x]])/(32*(a + b)^5*d) + ((5*a^3*A - b^3*(35*A - 16*B) + a*b^2*(47*A - 10*B) - a^2*(25*A*b - 2*b*B))*Log[1 + Sin[c + d*x]])/(32*(a - b)^5*d) + (b^6*(8*a*A*b - 7*a^2*B - b^2*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^5*d) - (b*(5*a^6*A - 23*a^4*A*b^2 + 47*a^2*A*b^4 + 35*A*b^6 + 2*a^5*b*B - 12*a^3*b^3*B - 54*a*b^5*B))/(16*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^6*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(6*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^4*(b*(a^2*A + 7*A*b^2 - 8*a*b*B) + (5*a^3*A - 13*a*A*b^2 + 2*a^2*b*B + 6*b^3*B)*Sin[c + d*x]))/(24*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(b*(5*a^4*A - 18*a^2*A*b^2 - 35*A*b^4 + 2*a^3*b*B + 46*a*b^3*B) + 3*(5*a^5*A - 18*a^3*A*b^2 + 29*a*A*b^4 + 2*a^4*b*B - 10*a^2*b^3*B - 8*b^5*B)*Sin[c + d*x]))/(48*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",6,3,31,0.09677,1,"{2837, 823, 801}"
1560,0,0,0,0.0973966,"\int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Int[(g*Cos[e + f*x])^(-1 - m)*(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","\text{Int}\left((A+B \sin (e+f x)) (g \cos (e+f x))^{-m-1} (a+b \sin (e+f x))^m,x\right)",0,"Defer[Int][(g*Cos[e + f*x])^(-1 - m)*(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x]","A",0,0,0,0,-1,"{}"
1561,1,330,0,0.4158483,"\int \frac{(g \cos (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx","Int[(g*Cos[e + f*x])^p/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])),x]","\frac{g (g \cos (e+f x))^{p-1} \left(-\frac{d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} \left(\frac{d (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} F_1\left(1-p;\frac{1-p}{2},\frac{1-p}{2};2-p;\frac{c+d}{c+d \sin (e+f x)},\frac{c-d}{c+d \sin (e+f x)}\right)}{f (1-p) (b c-a d)}-\frac{g (g \cos (e+f x))^{p-1} \left(-\frac{b (1-\sin (e+f x))}{a+b \sin (e+f x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (e+f x)+1)}{a+b \sin (e+f x)}\right)^{\frac{1-p}{2}} F_1\left(1-p;\frac{1-p}{2},\frac{1-p}{2};2-p;\frac{a+b}{a+b \sin (e+f x)},\frac{a-b}{a+b \sin (e+f x)}\right)}{f (1-p) (b c-a d)}","\frac{g (g \cos (e+f x))^{p-1} \left(-\frac{d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} \left(\frac{d (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} F_1\left(1-p;\frac{1-p}{2},\frac{1-p}{2};2-p;\frac{c+d}{c+d \sin (e+f x)},\frac{c-d}{c+d \sin (e+f x)}\right)}{f (1-p) (b c-a d)}-\frac{g (g \cos (e+f x))^{p-1} \left(-\frac{b (1-\sin (e+f x))}{a+b \sin (e+f x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (e+f x)+1)}{a+b \sin (e+f x)}\right)^{\frac{1-p}{2}} F_1\left(1-p;\frac{1-p}{2},\frac{1-p}{2};2-p;\frac{a+b}{a+b \sin (e+f x)},\frac{a-b}{a+b \sin (e+f x)}\right)}{f (1-p) (b c-a d)}",1,"-((g*AppellF1[1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (a + b)/(a + b*Sin[e + f*x]), (a - b)/(a + b*Sin[e + f*x])]*(g*Cos[e + f*x])^(-1 + p)*(-((b*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])))^((1 - p)/2)*((b*(1 + Sin[e + f*x]))/(a + b*Sin[e + f*x]))^((1 - p)/2))/((b*c - a*d)*f*(1 - p))) + (g*AppellF1[1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (c + d)/(c + d*Sin[e + f*x]), (c - d)/(c + d*Sin[e + f*x])]*(g*Cos[e + f*x])^(-1 + p)*(-((d*(1 - Sin[e + f*x]))/(c + d*Sin[e + f*x])))^((1 - p)/2)*((d*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^((1 - p)/2))/((b*c - a*d)*f*(1 - p))","A",4,2,35,0.05714,1,"{2924, 2703}"
1562,1,508,0,0.5244934,"\int \frac{(g \cos (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx","Int[(g*Cos[e + f*x])^p/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2),x]","-\frac{b g (g \cos (e+f x))^{p-1} \left(-\frac{b (1-\sin (e+f x))}{a+b \sin (e+f x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (e+f x)+1)}{a+b \sin (e+f x)}\right)^{\frac{1-p}{2}} F_1\left(1-p;\frac{1-p}{2},\frac{1-p}{2};2-p;\frac{a+b}{a+b \sin (e+f x)},\frac{a-b}{a+b \sin (e+f x)}\right)}{f (1-p) (b c-a d)^2}+\frac{b g (g \cos (e+f x))^{p-1} \left(-\frac{d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} \left(\frac{d (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} F_1\left(1-p;\frac{1-p}{2},\frac{1-p}{2};2-p;\frac{c+d}{c+d \sin (e+f x)},\frac{c-d}{c+d \sin (e+f x)}\right)}{f (1-p) (b c-a d)^2}+\frac{g (g \cos (e+f x))^{p-1} \left(-\frac{d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} \left(\frac{d (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} F_1\left(2-p;\frac{1-p}{2},\frac{1-p}{2};3-p;\frac{c+d}{c+d \sin (e+f x)},\frac{c-d}{c+d \sin (e+f x)}\right)}{f (2-p) (b c-a d) (c+d \sin (e+f x))}","-\frac{b g (g \cos (e+f x))^{p-1} \left(-\frac{b (1-\sin (e+f x))}{a+b \sin (e+f x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (e+f x)+1)}{a+b \sin (e+f x)}\right)^{\frac{1-p}{2}} F_1\left(1-p;\frac{1-p}{2},\frac{1-p}{2};2-p;\frac{a+b}{a+b \sin (e+f x)},\frac{a-b}{a+b \sin (e+f x)}\right)}{f (1-p) (b c-a d)^2}+\frac{b g (g \cos (e+f x))^{p-1} \left(-\frac{d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} \left(\frac{d (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} F_1\left(1-p;\frac{1-p}{2},\frac{1-p}{2};2-p;\frac{c+d}{c+d \sin (e+f x)},\frac{c-d}{c+d \sin (e+f x)}\right)}{f (1-p) (b c-a d)^2}+\frac{g (g \cos (e+f x))^{p-1} \left(-\frac{d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} \left(\frac{d (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right)^{\frac{1-p}{2}} F_1\left(2-p;\frac{1-p}{2},\frac{1-p}{2};3-p;\frac{c+d}{c+d \sin (e+f x)},\frac{c-d}{c+d \sin (e+f x)}\right)}{f (2-p) (b c-a d) (c+d \sin (e+f x))}",1,"-((b*g*AppellF1[1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (a + b)/(a + b*Sin[e + f*x]), (a - b)/(a + b*Sin[e + f*x])]*(g*Cos[e + f*x])^(-1 + p)*(-((b*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])))^((1 - p)/2)*((b*(1 + Sin[e + f*x]))/(a + b*Sin[e + f*x]))^((1 - p)/2))/((b*c - a*d)^2*f*(1 - p))) + (b*g*AppellF1[1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (c + d)/(c + d*Sin[e + f*x]), (c - d)/(c + d*Sin[e + f*x])]*(g*Cos[e + f*x])^(-1 + p)*(-((d*(1 - Sin[e + f*x]))/(c + d*Sin[e + f*x])))^((1 - p)/2)*((d*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^((1 - p)/2))/((b*c - a*d)^2*f*(1 - p)) + (g*AppellF1[2 - p, (1 - p)/2, (1 - p)/2, 3 - p, (c + d)/(c + d*Sin[e + f*x]), (c - d)/(c + d*Sin[e + f*x])]*(g*Cos[e + f*x])^(-1 + p)*(-((d*(1 - Sin[e + f*x]))/(c + d*Sin[e + f*x])))^((1 - p)/2)*((d*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^((1 - p)/2))/((b*c - a*d)*f*(2 - p)*(c + d*Sin[e + f*x]))","A",5,2,35,0.05714,1,"{2924, 2703}"
1563,1,308,0,0.6157346,"\int \frac{(g \sec (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx","Int[(g*Sec[e + f*x])^p/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])),x]","\frac{\sec (e+f x) (g \sec (e+f x))^p \left(-\frac{d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right)^{\frac{p+1}{2}} \left(\frac{d (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right)^{\frac{p+1}{2}} F_1\left(p+1;\frac{p+1}{2},\frac{p+1}{2};p+2;\frac{c+d}{c+d \sin (e+f x)},\frac{c-d}{c+d \sin (e+f x)}\right)}{f (p+1) (b c-a d)}-\frac{\sec (e+f x) (g \sec (e+f x))^p \left(-\frac{b (1-\sin (e+f x))}{a+b \sin (e+f x)}\right)^{\frac{p+1}{2}} \left(\frac{b (\sin (e+f x)+1)}{a+b \sin (e+f x)}\right)^{\frac{p+1}{2}} F_1\left(p+1;\frac{p+1}{2},\frac{p+1}{2};p+2;\frac{a+b}{a+b \sin (e+f x)},\frac{a-b}{a+b \sin (e+f x)}\right)}{f (p+1) (b c-a d)}","\frac{\sec (e+f x) (g \sec (e+f x))^p \left(-\frac{d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right)^{\frac{p+1}{2}} \left(\frac{d (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right)^{\frac{p+1}{2}} F_1\left(p+1;\frac{p+1}{2},\frac{p+1}{2};p+2;\frac{c+d}{c+d \sin (e+f x)},\frac{c-d}{c+d \sin (e+f x)}\right)}{f (p+1) (b c-a d)}-\frac{\sec (e+f x) (g \sec (e+f x))^p \left(-\frac{b (1-\sin (e+f x))}{a+b \sin (e+f x)}\right)^{\frac{p+1}{2}} \left(\frac{b (\sin (e+f x)+1)}{a+b \sin (e+f x)}\right)^{\frac{p+1}{2}} F_1\left(p+1;\frac{p+1}{2},\frac{p+1}{2};p+2;\frac{a+b}{a+b \sin (e+f x)},\frac{a-b}{a+b \sin (e+f x)}\right)}{f (p+1) (b c-a d)}",1,"-((AppellF1[1 + p, (1 + p)/2, (1 + p)/2, 2 + p, (a + b)/(a + b*Sin[e + f*x]), (a - b)/(a + b*Sin[e + f*x])]*Sec[e + f*x]*(g*Sec[e + f*x])^p*(-((b*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])))^((1 + p)/2)*((b*(1 + Sin[e + f*x]))/(a + b*Sin[e + f*x]))^((1 + p)/2))/((b*c - a*d)*f*(1 + p))) + (AppellF1[1 + p, (1 + p)/2, (1 + p)/2, 2 + p, (c + d)/(c + d*Sin[e + f*x]), (c - d)/(c + d*Sin[e + f*x])]*Sec[e + f*x]*(g*Sec[e + f*x])^p*(-((d*(1 - Sin[e + f*x]))/(c + d*Sin[e + f*x])))^((1 + p)/2)*((d*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^((1 + p)/2))/((b*c - a*d)*f*(1 + p))","A",5,3,35,0.08571,1,"{2926, 2924, 2703}"