1,1,104,92,0.5681656,"\int \cos ^2(e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx","Integrate[Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2),x]","\frac{c^3 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (1080 \sin (e+f x)+20 \sin (3 (e+f x))-36 \sin (5 (e+f x))+405 \cos (2 (e+f x))+90 \cos (4 (e+f x))-5 \cos (6 (e+f x)))}{960 f}","-\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,"(c^3*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(405*Cos[2*(e + f*x)] + 90*Cos[4*(e + f*x)] - 5*Cos[6*(e + f*x)] + 1080*Sin[e + f*x] + 20*Sin[3*(e + f*x)] - 36*Sin[5*(e + f*x)]))/(960*f)","A",1
2,1,94,92,0.4760161,"\int \cos ^2(e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx","Integrate[Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2),x]","\frac{c^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (70 \sin (e+f x)+5 \sin (3 (e+f x))-\sin (5 (e+f x))+20 \cos (2 (e+f x))+5 \cos (4 (e+f x)))}{80 f}","-\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,"(c^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(20*Cos[2*(e + f*x)] + 5*Cos[4*(e + f*x)] + 70*Sin[e + f*x] + 5*Sin[3*(e + f*x)] - Sin[5*(e + f*x)]))/(80*f)","A",1
3,1,83,92,0.3988941,"\int \cos ^2(e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx","Integrate[Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2),x]","\frac{c \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (8 (9 \sin (e+f x)+\sin (3 (e+f x)))+12 \cos (2 (e+f x))+3 \cos (4 (e+f x)))}{96 f}","-\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,"(c*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(12*Cos[2*(e + f*x)] + 3*Cos[4*(e + f*x)] + 8*(9*Sin[e + f*x] + Sin[3*(e + f*x)])))/(96*f)","A",1
4,1,59,92,0.1701798,"\int \cos ^2(e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]],x]","\frac{(9 \sin (e+f x)+\sin (3 (e+f x))) \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{12 f}","-\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,"(Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(9*Sin[e + f*x] + Sin[3*(e + f*x)]))/(12*f)","A",1
5,1,62,45,0.3026839,"\int \frac{\cos ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{\sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (\cos (2 (e+f x))-4 \sin (e+f x))}{4 c f}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 a f \sqrt{c-c \sin (e+f x)}}",1,"-1/4*(Sec[e + f*x]*(Cos[2*(e + f*x)] - 4*Sin[e + f*x])*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/(c*f)","A",1
6,1,115,99,1.087841,"\int \frac{\cos ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{\sqrt{a (\sin (e+f x)+1)} \left(4 \log \left(i-e^{i (e+f x)}\right)+\sin (e+f x)-2 i f x\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{f (c-c \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\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,"-(((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*((-2*I)*f*x + 4*Log[I - E^(I*(e + f*x))] + Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^(3/2)))","C",1
7,1,104,97,0.8344414,"\int \frac{\cos ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(5/2),x]","\frac{\sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \left(2 \log \left(i-e^{i (e+f x)}\right)+\left(i f x-2 \log \left(i-e^{i (e+f x)}\right)\right) \sin (e+f x)-i f x+2\right)}{c^2 f \sqrt{c-c \sin (e+f 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}}",1,"(Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*(2 - I*f*x + 2*Log[I - E^(I*(e + f*x))] + (I*f*x - 2*Log[I - E^(I*(e + f*x))])*Sin[e + f*x]))/(c^2*f*Sqrt[c - c*Sin[e + f*x]])","C",1
8,1,90,48,0.394475,"\int \frac{\cos ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(7/2),x]","\frac{\sin (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{c^4 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"(Sin[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/(c^4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","A",1
9,1,166,140,1.2119248,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{c^3 (\sin (e+f x)-1)^3 (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)} (4725 \sin (e+f x)+665 \sin (3 (e+f x))+21 \sin (5 (e+f x))-15 \sin (7 (e+f x))+1050 \cos (2 (e+f x))+420 \cos (4 (e+f x))+70 \cos (6 (e+f x)))}{6720 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\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,"-1/6720*(c^3*(-1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(1050*Cos[2*(e + f*x)] + 420*Cos[4*(e + f*x)] + 70*Cos[6*(e + f*x)] + 4725*Sin[e + f*x] + 665*Sin[3*(e + f*x)] + 21*Sin[5*(e + f*x)] - 15*Sin[7*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
10,1,156,140,0.967441,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2),x]","\frac{c^2 (\sin (e+f x)-1)^2 (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)} (600 \sin (e+f x)+100 \sin (3 (e+f x))+12 \sin (5 (e+f x))+75 \cos (2 (e+f x))+30 \cos (4 (e+f x))+5 \cos (6 (e+f x)))}{960 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\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,"(c^2*(-1 + Sin[e + f*x])^2*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(75*Cos[2*(e + f*x)] + 30*Cos[4*(e + f*x)] + 5*Cos[6*(e + f*x)] + 600*Sin[e + f*x] + 100*Sin[3*(e + f*x)] + 12*Sin[5*(e + f*x)]))/(960*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
11,1,82,140,0.6161241,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{c (\sin (e+f x)-1) (150 \sin (e+f x)+25 \sin (3 (e+f x))+3 \sin (5 (e+f x))) \sec ^3(e+f x) (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)}}{240 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,"-1/240*(c*Sec[e + f*x]^3*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(150*Sin[e + f*x] + 25*Sin[3*(e + f*x)] + 3*Sin[5*(e + f*x)]))/f","A",1
12,1,83,92,0.3916552,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]],x]","\frac{a \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (8 (9 \sin (e+f x)+\sin (3 (e+f x)))-12 \cos (2 (e+f x))-3 \cos (4 (e+f x)))}{96 f}","\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,"(a*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-12*Cos[2*(e + f*x)] - 3*Cos[4*(e + f*x)] + 8*(9*Sin[e + f*x] + Sin[3*(e + f*x)])))/(96*f)","A",1
13,1,111,45,0.5463534,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/Sqrt[c - c*Sin[e + f*x]],x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (15 \sin (e+f x)-\sin (3 (e+f x))-6 \cos (2 (e+f x)))}{12 f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","\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)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2)*(-6*Cos[2*(e + f*x)] + 15*Sin[e + f*x] - Sin[3*(e + f*x)]))/(12*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]])","B",1
14,1,130,147,1.063525,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{(a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(12 \sin (e+f x)-\cos (2 (e+f x))+32 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{4 c f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\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,"-1/4*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2)*(-Cos[2*(e + f*x)] + 32*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 12*Sin[e + f*x]))/(c*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]])","A",1
15,1,169,144,1.0590172,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos (2 (e+f x))+16 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+\sin (e+f x) \left(2-16 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+7\right)}{2 c^2 f (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(7 + Cos[2*(e + f*x)] + 16*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + (2 - 16*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x]))/(2*c^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","A",1
16,1,191,147,1.3635613,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(7/2),x]","-\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos (2 (e+f x)) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-3 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+4 \sin (e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+1\right)-2\right)}{c^3 f (\sin (e+f x)-1)^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\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,"-((a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(-2 - 3*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Cos[2*(e + f*x)]*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 4*(1 + Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x]))/(c^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]]))","A",1
17,1,110,48,1.4240313,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(9/2),x]","-\frac{a (3 \cos (2 (e+f x))-5) \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{6 c^4 f (\sin (e+f x)-1)^4 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"-1/6*(a*(-5 + 3*Cos[2*(e + f*x)])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])])/(c^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^4*Sqrt[c - c*Sin[e + f*x]])","B",1
18,1,118,97,1.9527723,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(11/2),x]","-\frac{a \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 (4 \sin (e+f x)-3 \cos (2 (e+f x))+5)}{12 c^5 f (\sin (e+f x)-1)^5 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"-1/12*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(5 - 3*Cos[2*(e + f*x)] + 4*Sin[e + f*x]))/(c^5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^5*Sqrt[c - c*Sin[e + f*x]])","A",1
19,1,176,188,3.1095862,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2),x]","-\frac{c^3 (\sin (e+f x)-1)^3 (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)} (19600 \sin (e+f x)+3920 \sin (3 (e+f x))+784 \sin (5 (e+f x))+80 \sin (7 (e+f x))+1960 \cos (2 (e+f x))+980 \cos (4 (e+f x))+280 \cos (6 (e+f x))+35 \cos (8 (e+f x)))}{35840 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\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{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{\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,"-1/35840*(c^3*(-1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]]*(1960*Cos[2*(e + f*x)] + 980*Cos[4*(e + f*x)] + 280*Cos[6*(e + f*x)] + 35*Cos[8*(e + f*x)] + 19600*Sin[e + f*x] + 3920*Sin[3*(e + f*x)] + 784*Sin[5*(e + f*x)] + 80*Sin[7*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
20,1,87,188,0.6479096,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^2 c^2 (1225 \sin (e+f x)+245 \sin (3 (e+f x))+49 \sin (5 (e+f x))+5 \sin (7 (e+f x))) \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{2240 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{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{\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,"(a^2*c^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(1225*Sin[e + f*x] + 245*Sin[3*(e + f*x)] + 49*Sin[5*(e + f*x)] + 5*Sin[7*(e + f*x)]))/(2240*f)","A",1
21,1,152,140,0.7671999,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{c (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)} (600 \sin (e+f x)+100 \sin (3 (e+f x))+12 \sin (5 (e+f x))-75 \cos (2 (e+f x))-30 \cos (4 (e+f x))-5 \cos (6 (e+f x)))}{960 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\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,"-1/960*(c*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]]*(-75*Cos[2*(e + f*x)] - 30*Cos[4*(e + f*x)] - 5*Cos[6*(e + f*x)] + 600*Sin[e + f*x] + 100*Sin[3*(e + f*x)] + 12*Sin[5*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
22,1,92,92,0.5046525,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (-70 \sin (e+f x)-5 \sin (3 (e+f x))+\sin (5 (e+f x))+20 \cos (2 (e+f x))+5 \cos (4 (e+f x)))}{80 f}","\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,"-1/80*(a^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(20*Cos[2*(e + f*x)] + 5*Cos[4*(e + f*x)] - 70*Sin[e + f*x] - 5*Sin[3*(e + f*x)] + Sin[5*(e + f*x)]))/f","A",1
23,1,119,45,0.9221456,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/Sqrt[c - c*Sin[e + f*x]],x]","\frac{(a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (56 \sin (e+f x)-8 \sin (3 (e+f x))-28 \cos (2 (e+f x))+\cos (4 (e+f x)))}{32 f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\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)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2)*(-28*Cos[2*(e + f*x)] + Cos[4*(e + f*x)] + 56*Sin[e + f*x] - 8*Sin[3*(e + f*x)]))/(32*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sqrt[c - c*Sin[e + f*x]])","B",1
24,1,140,193,2.6089162,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{(a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(87 \sin (e+f x)-\sin (3 (e+f x))-12 \cos (2 (e+f x))+192 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{12 c f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\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{4 a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \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,"-1/12*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2)*(-12*Cos[2*(e + f*x)] + 192*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 87*Sin[e + f*x] - Sin[3*(e + f*x)]))/(c*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sqrt[c - c*Sin[e + f*x]])","A",1
25,1,181,192,2.3760816,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin (3 (e+f x))+18 \cos (2 (e+f x))+192 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+\sin (e+f x) \left(39-192 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+44\right)}{8 c^2 f (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\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{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{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,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(44 + 18*Cos[2*(e + f*x)] + 192*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + (39 - 192*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x] + Sin[3*(e + f*x)]))/(8*c^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","A",1
26,1,209,195,2.7442865,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[(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 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin (3 (e+f x))-72 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+4 \cos (2 (e+f x)) \left(6 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-1\right)+\sin (e+f x) \left(96 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+41\right)-28\right)}{4 c^3 f (\sin (e+f x)-1)^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\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^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^3 f \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,"-1/4*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(-28 - 72*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 4*Cos[2*(e + f*x)]*(-1 + 6*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]) + (41 + 96*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x] + Sin[3*(e + f*x)]))/(c^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]])","A",1
27,1,234,193,4.1348132,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[(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 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(3 \sin (3 (e+f x)) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+30 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-18 \cos (2 (e+f x)) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+1\right)-9 \sin (e+f x) \left(5 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+4\right)+34\right)}{6 c^4 f (\sin (e+f x)-1)^4 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\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^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 \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,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(34 + 30*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 18*Cos[2*(e + f*x)]*(1 + Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]) - 9*(4 + 5*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x] + 3*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)]))/(6*c^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^4*Sqrt[c - c*Sin[e + f*x]])","A",1
28,1,117,48,4.287604,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(11/2),x]","\frac{a^2 (\sin (3 (e+f x))-7 \sin (e+f x)) \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{4 c^5 f (\sin (e+f x)-1)^5 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(-7*Sin[e + f*x] + Sin[3*(e + f*x)]))/(4*c^5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^5*Sqrt[c - c*Sin[e + f*x]])","B",1
29,1,130,97,6.2748638,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(13/2),x]","\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 (35 \sin (e+f x)-5 \sin (3 (e+f x))-10 \cos (2 (e+f x))+14)}{40 c^6 f (\sin (e+f x)-1)^6 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(14 - 10*Cos[2*(e + f*x)] + 35*Sin[e + f*x] - 5*Sin[3*(e + f*x)]))/(40*c^6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^6*Sqrt[c - c*Sin[e + f*x]])","A",1
30,1,209,236,5.6258214,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2),x]","\frac{a^3 c^4 (\sin (e+f x)-1)^4 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (158760 \sin (e+f x)+35280 \sin (3 (e+f x))+9072 \sin (5 (e+f x))+1620 \sin (7 (e+f x))+140 \sin (9 (e+f x))+13230 \cos (2 (e+f x))+7560 \cos (4 (e+f x))+2835 \cos (6 (e+f x))+630 \cos (8 (e+f x))+63 \cos (10 (e+f x)))}{322560 f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\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{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{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{\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,"(a^3*c^4*(-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(13230*Cos[2*(e + f*x)] + 7560*Cos[4*(e + f*x)] + 2835*Cos[6*(e + f*x)] + 630*Cos[8*(e + f*x)] + 63*Cos[10*(e + f*x)] + 158760*Sin[e + f*x] + 35280*Sin[3*(e + f*x)] + 9072*Sin[5*(e + f*x)] + 1620*Sin[7*(e + f*x)] + 140*Sin[9*(e + f*x)]))/(322560*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7)","A",1
31,1,97,236,1.2632904,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(7/2),x]","\frac{a^3 c^3 (39690 \sin (e+f x)+8820 \sin (3 (e+f x))+2268 \sin (5 (e+f x))+405 \sin (7 (e+f x))+35 \sin (9 (e+f x))) \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{80640 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{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{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{\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,"(a^3*c^3*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(39690*Sin[e + f*x] + 8820*Sin[3*(e + f*x)] + 2268*Sin[5*(e + f*x)] + 405*Sin[7*(e + f*x)] + 35*Sin[9*(e + f*x)]))/(80640*f)","A",1
32,1,127,188,2.4144175,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2),x]","\frac{a^3 c^2 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (19600 \sin (e+f x)+3920 \sin (3 (e+f x))+784 \sin (5 (e+f x))+80 \sin (7 (e+f x))-1960 \cos (2 (e+f x))-980 \cos (4 (e+f x))-280 \cos (6 (e+f x))-35 \cos (8 (e+f x)))}{35840 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{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{\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,"(a^3*c^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-1960*Cos[2*(e + f*x)] - 980*Cos[4*(e + f*x)] - 280*Cos[6*(e + f*x)] - 35*Cos[8*(e + f*x)] + 19600*Sin[e + f*x] + 3920*Sin[3*(e + f*x)] + 784*Sin[5*(e + f*x)] + 80*Sin[7*(e + f*x)]))/(35840*f)","A",1
33,1,115,140,1.2787579,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2),x]","\frac{a^3 c \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (4725 \sin (e+f x)+665 \sin (3 (e+f x))+21 \sin (5 (e+f x))-15 \sin (7 (e+f x))-1050 \cos (2 (e+f x))-420 \cos (4 (e+f x))-70 \cos (6 (e+f x)))}{6720 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,"(a^3*c*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-1050*Cos[2*(e + f*x)] - 420*Cos[4*(e + f*x)] - 70*Cos[6*(e + f*x)] + 4725*Sin[e + f*x] + 665*Sin[3*(e + f*x)] + 21*Sin[5*(e + f*x)] - 15*Sin[7*(e + f*x)]))/(6720*f)","A",1
34,1,104,92,0.5380229,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]],x]","\frac{a^3 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (1080 \sin (e+f x)+20 \sin (3 (e+f x))-36 \sin (5 (e+f x))-405 \cos (2 (e+f x))-90 \cos (4 (e+f x))+5 \cos (6 (e+f x)))}{960 f}","\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,"(a^3*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-405*Cos[2*(e + f*x)] - 90*Cos[4*(e + f*x)] + 5*Cos[6*(e + f*x)] + 1080*Sin[e + f*x] + 20*Sin[3*(e + f*x)] - 36*Sin[5*(e + f*x)]))/(960*f)","A",1
35,1,142,45,1.4424778,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/Sqrt[c - c*Sin[e + f*x]],x]","\frac{a^3 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (210 \sin (e+f x)-45 \sin (3 (e+f x))+\sin (5 (e+f x))-120 \cos (2 (e+f x))+10 \cos (4 (e+f x)))}{80 f \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{5 a f \sqrt{c-c \sin (e+f x)}}",1,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*(-120*Cos[2*(e + f*x)] + 10*Cos[4*(e + f*x)] + 210*Sin[e + f*x] - 45*Sin[3*(e + f*x)] + Sin[5*(e + f*x)]))/(80*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*Sqrt[c - c*Sin[e + f*x]])","B",1
36,1,473,241,6.4730522,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(3/2),x]","\frac{5 \sin (3 (e+f x)) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{12 f (c-c \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{65 \sin (e+f x) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{4 f (c-c \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{23 \cos (2 (e+f x)) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{8 f (c-c \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{\cos (4 (e+f x)) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{32 f (c-c \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{32 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f (c-c \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\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{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{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,"(23*Cos[2*(e + f*x)]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(8*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(3/2)) - (Cos[4*(e + f*x)]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(32*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(3/2)) - (32*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(3/2)) - (65*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sin[e + f*x]*(a*(1 + Sin[e + f*x]))^(7/2))/(4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(3/2)) + (5*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2)*Sin[3*(e + f*x)])/(12*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(3/2))","A",1
37,1,196,238,4.8092634,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(5/2),x]","-\frac{a^3 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(-396 \sin (e+f x)-16 \sin (3 (e+f x))-172 \cos (2 (e+f x))+\cos (4 (e+f x))-1536 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+1536 \sin (e+f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-177\right)}{24 c^2 f (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\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{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{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,"-1/24*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(-177 - 172*Cos[2*(e + f*x)] + Cos[4*(e + f*x)] - 1536*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 396*Sin[e + f*x] + 1536*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] - 16*Sin[3*(e + f*x)]))/(c^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","A",1
38,1,223,239,6.5534101,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(7/2),x]","\frac{a^3 \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(-320 \sin (e+f x)-24 \sin (3 (e+f x))+\cos (4 (e+f x))+\cos (2 (e+f x)) \left(106-384 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+1152 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-1536 \sin (e+f x) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+273\right)}{16 c^3 f (\sin (e+f x)-1)^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\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{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{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,"(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(273 + Cos[4*(e + f*x)] + Cos[2*(e + f*x)]*(106 - 384*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]) + 1152*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 320*Sin[e + f*x] - 1536*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] - 24*Sin[3*(e + f*x)]))/(16*c^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]])","A",1
39,1,442,241,6.6115537,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(9/2),x]","\frac{\sin (e+f x) (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{f (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{24 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}{f (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{16 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{f (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{16 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{3 f (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{16 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\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{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{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,"(16*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(9/2)) - (16*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(9/2)) + (24*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(9/2)) + (16*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(9/2)) + ((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*Sin[e + f*x]*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(9/2))","A",1
40,1,437,243,6.6493501,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(11/2),x]","-\frac{8 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{12 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}{f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{32 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{3 f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{4 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{2 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11} \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\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^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 \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,"(4*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2)) - (32*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2)) + (12*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2)) - (8*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2)) - (2*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2))","A",1
41,1,412,48,6.7410543,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(13/2),x]","\frac{(a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}{f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{4 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{8 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}{f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{8 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{16 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{5 f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\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,"(16*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2)) - (8*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2)) + (8*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2)) - (4*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2)) + ((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2))","B",1
42,1,419,97,6.7905711,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(15/2),x]","\frac{(a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}{2 f (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{8 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{3 f (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{6 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}{f (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{32 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{5 f (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{8 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{3 f (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\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,"(8*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) - (32*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) + (6*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) - (8*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) + ((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(a*(1 + Sin[e + f*x]))^(7/2))/(2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2))","B",1
43,1,419,145,6.8675411,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(17/2),x]","\frac{(a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}{3 f (c-c \sin (e+f x))^{17/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{2 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{f (c-c \sin (e+f x))^{17/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{24 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}{5 f (c-c \sin (e+f x))^{17/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{16 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}{3 f (c-c \sin (e+f x))^{17/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{16 (a (\sin (e+f x)+1))^{7/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{7 f (c-c \sin (e+f x))^{17/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\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,"(16*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) - (16*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + (24*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) - (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + ((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2))","B",1
44,1,134,45,0.8970999,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(5/2))/Sqrt[a + a*Sin[e + f*x]],x]","\frac{c^2 (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (56 \sin (e+f x)-8 \sin (3 (e+f x))+28 \cos (2 (e+f x))-\cos (4 (e+f x)))}{32 f \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 c f \sqrt{a \sin (e+f x)+a}}",1,"(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]]*(28*Cos[2*(e + f*x)] - Cos[4*(e + f*x)] + 56*Sin[e + f*x] - 8*Sin[3*(e + f*x)]))/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*Sqrt[a*(1 + Sin[e + f*x])])","B",1
45,1,120,45,0.511911,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(3/2))/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{c (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) (15 \sin (e+f x)-\sin (3 (e+f x))+6 \cos (2 (e+f x)))}{12 f \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt{a \sin (e+f x)+a}}",1,"-1/12*(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]]*(6*Cos[2*(e + f*x)] + 15*Sin[e + f*x] - Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])])","B",1
46,1,62,45,0.2850954,"\int \frac{\cos ^2(e+f x) \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[(Cos[e + f*x]^2*Sqrt[c - c*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (4 \sin (e+f x)+\cos (2 (e+f x)))}{4 a f}","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt{a \sin (e+f x)+a}}",1,"(Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*(Cos[2*(e + f*x)] + 4*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(4*a*f)","A",1
47,1,44,43,0.2988559,"\int \frac{\cos ^2(e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[Cos[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{\sin (2 (e+f x))}{2 f \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}","-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{c f \sqrt{a \sin (e+f x)+a}}",1,"Sin[2*(e + f*x)]/(2*f*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])","A",1
48,1,104,54,0.4143262,"\int \frac{\cos ^2(e+f x)}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[Cos[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)),x]","-\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}{f \sqrt{a (\sin (e+f x)+1)} (c-c \sin (e+f x))^{3/2}}","-\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,"(-2*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(f*Sqrt[a*(1 + Sin[e + f*x])]*(c - c*Sin[e + f*x])^(3/2))","A",1
49,1,79,42,0.4936522,"\int \frac{\cos ^2(e+f x)}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[Cos[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f \sqrt{a (\sin (e+f x)+1)} (c-c \sin (e+f x))^{5/2}}","\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)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(f*Sqrt[a*(1 + Sin[e + f*x])]*(c - c*Sin[e + f*x])^(5/2))","A",1
50,1,471,239,6.4688472,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(3/2),x]","\frac{5 \sin (3 (e+f x)) (c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{12 f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{23 \cos (2 (e+f x)) (c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{8 f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{\cos (4 (e+f x)) (c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{32 f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{65 \sin (e+f x) (c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{4 f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{32 (c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}","\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{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{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,"(-23*Cos[2*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(7/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(3/2)) + (Cos[4*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(7/2))/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(3/2)) + (32*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(7/2))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(3/2)) - (65*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(3/2)) + (5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[3*(e + f*x)])/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(3/2))","A",1
51,1,138,190,2.3704104,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(3/2),x]","\frac{c^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(-87 \sin (e+f x)+\sin (3 (e+f x))-12 \cos (2 (e+f x))+192 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{12 f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\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{4 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f \sqrt{a \sin (e+f x)+a}}+\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,"(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]]*(-12*Cos[2*(e + f*x)] + 192*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] - 87*Sin[e + f*x] + Sin[3*(e + f*x)]))/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2))","A",1
52,1,134,145,1.1152886,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(3/2),x]","\frac{c (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(12 \sin (e+f x)+\cos (2 (e+f x))-32 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{4 f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}","\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,"(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(-1 + Sin[e + f*x])*(Cos[2*(e + f*x)] - 32*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 12*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(3/2))","A",1
53,1,113,96,1.0568924,"\int \frac{\cos ^2(e+f x) \sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[(Cos[e + f*x]^2*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{\sqrt{c-c \sin (e+f x)} \left(-4 \log \left(e^{i (e+f x)}+i\right)+\sin (e+f x)+2 i f x\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"-(((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*((2*I)*f*x - 4*Log[I + E^(I*(e + f*x))] + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2)))","C",1
54,1,102,51,0.4300718,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f (a (\sin (e+f x)+1))^{3/2} \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,"(2*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/(f*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
55,1,103,52,0.5344719,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\cos ^3(e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{c f (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{3/2} \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,"(Cos[e + f*x]^3*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]))/(c*f*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
56,1,163,104,0.7746462,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\cos ^3(e+f x) \left(-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\sin (e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+1\right)}{2 c^2 f (\sin (e+f x)-1)^2 (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f 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}}",1,"(Cos[e + f*x]^3*(1 - Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x]))/(2*c^2*f*(-1 + Sin[e + f*x])^2*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
57,1,553,285,6.6418155,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^(5/2),x]","\frac{203 \sin (e+f x) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{4 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}+\frac{47 \cos (2 (e+f x)) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{8 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}-\frac{\cos (4 (e+f x)) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{32 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}-\frac{7 \sin (3 (e+f x)) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{12 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}-\frac{32 (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}-\frac{160 (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}","-\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{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{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,"(-32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(9/2))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) + (47*Cos[2*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) - (Cos[4*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2))/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) - (160*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) + (203*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sin[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) - (7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2)*Sin[3*(e + f*x)])/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
58,1,179,237,5.3086742,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(5/2),x]","\frac{c^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(396 \sin (e+f x)+16 \sin (3 (e+f x))-172 \cos (2 (e+f x))+\cos (4 (e+f x))-1536 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-1536 \sin (e+f x) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-177\right)}{24 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\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{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{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,"(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]]*(-177 - 172*Cos[2*(e + f*x)] + Cos[4*(e + f*x)] - 1536*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 396*Sin[e + f*x] - 1536*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] + 16*Sin[3*(e + f*x)]))/(24*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
59,1,164,191,2.4944001,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(5/2),x]","\frac{c^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin (3 (e+f x))-18 \cos (2 (e+f x))-192 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\sin (e+f x) \left(39-192 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-44\right)}{8 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\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{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{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,"(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]]*(-44 - 18*Cos[2*(e + f*x)] - 192*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (39 - 192*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x] + Sin[3*(e + f*x)]))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
60,1,153,143,1.1329962,"\int \frac{\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(5/2),x]","-\frac{c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\cos (2 (e+f x))+16 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+2 \sin (e+f x) \left(8 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-1\right)+7\right)}{2 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\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,"-1/2*(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]]*(7 + Cos[2*(e + f*x)] + 16*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + 2*(-1 + 8*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
61,1,100,97,0.9847945,"\int \frac{\cos ^2(e+f x) \sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{5/2}} \, dx","Integrate[(Cos[e + f*x]^2*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(5/2),x]","\frac{\sec (e+f x) \sqrt{c-c \sin (e+f x)} \left(-2 \log \left(e^{i (e+f x)}+i\right)+\left(i f x-2 \log \left(e^{i (e+f x)}+i\right)\right) \sin (e+f x)+i f x-2\right)}{a^2 f \sqrt{a (\sin (e+f x)+1)}}","-\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,"(Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]]*(-2 + I*f*x - 2*Log[I + E^(I*(e + f*x))] + (I*f*x - 2*Log[I + E^(I*(e + f*x))])*Sin[e + f*x]))/(a^2*f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
62,1,80,43,0.5208762,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{f (a (\sin (e+f x)+1))^{5/2} \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)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/(f*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]]))","A",1
63,1,163,104,0.8033261,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx","Integrate[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)),x]","\frac{\cos ^3(e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\sin (e+f x) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+1\right)}{2 c f (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{5/2} \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]^3*(1 + Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[e + f*x]))/(2*c*f*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
64,1,163,152,0.8780455,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}} \, dx","Integrate[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{\sec (e+f x) \left(2 \sin (e+f x)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\cos (2 (e+f x)) \left(\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{4 a^2 c^2 f \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f 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}}",1,"(Sec[e + f*x]*(-Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + Cos[2*(e + f*x)]*(-Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 2*Sin[e + f*x]))/(4*a^2*c^2*f*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])","A",1
65,1,3426,114,14.2582583,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\text{Result too large to show}","\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,"(-64*(AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 5*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2 + 2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2 + 2*n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*Tan[(-e + Pi/2 - f*x)/4])/(f*(1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)*((16*m*(AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 5*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n)*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 - 2*m))/(1 + 2*n) + (4*(AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 5*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n))/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (16*n*(AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 5*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(-1 + 2*n)*Tan[(-e + Pi/2 - f*x)/4])/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (16*m*(AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 5*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(1 + 2*n)*Tan[(-e + Pi/2 - f*x)/4])/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (16*(m + n)*(AppellF1[1/2 + n, -2*m, 2*(1 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*AppellF1[1/2 + n, -2*m, 2*(2 + m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 5*AppellF1[1/2 + n, -2*m, 3 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*AppellF1[1/2 + n, -2*m, 5 + 2*(m + n), 3/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n)*Tan[(-e + Pi/2 - f*x)/4]^2)/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (16*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*(m + n))*Sin[(-e + Pi/2 - f*x)/2]^(2*n)*Tan[(-e + Pi/2 - f*x)/4]*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 2*(1 + m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(1 + m + n)*AppellF1[3/2 + n, -2*m, 1 + 2*(1 + m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n) - 5*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 3 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(3 + 2*(m + n))*AppellF1[3/2 + n, -2*m, 4 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 + n))) - 4*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 5 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(5 + 2*(m + n))*AppellF1[3/2 + n, -2*m, 6 + 2*(m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 + n))) + 8*(-((m*(1/2 + n)*AppellF1[3/2 + n, 1 - 2*m, 2*(2 + m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n)) - ((1/2 + n)*(2 + m + n)*AppellF1[3/2 + n, -2*m, 1 + 2*(2 + m + n), 5/2 + n, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 + n))))/((1 + 2*n)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))))","C",0
66,-1,0,86,180.013457,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3,x]","\text{\$Aborted}","-\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,"$Aborted","F",-1
67,-1,0,86,180.0011961,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^2 \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^2,x]","\text{\$Aborted}","-\frac{a^3 c^2 2^{m+\frac{3}{2}} \cos ^7(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left(\frac{7}{2},-m-\frac{1}{2};\frac{9}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{7 f}",1,"$Aborted","F",-1
68,-1,0,84,180.0030836,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x)) \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x]),x]","\text{\$Aborted}","-\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,"$Aborted","F",-1
69,1,78,81,0.1085822,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m,x]","-\frac{2^{m+\frac{3}{2}} \cos ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac{3}{2}} (a (\sin (e+f x)+1))^m \, _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,"-1/3*(2^(3/2 + 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])^(-3/2 - m)*(a*(1 + Sin[e + f*x]))^m)/f","A",1
70,1,6442,77,19.6840209,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x]),x]","\text{Result too large to show}","-\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,"Result too large to show","C",0
71,1,88,81,0.2066904,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^2} \, dx","Integrate[(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}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a (\sin (e+f x)+1))^m \, _2F_1\left(-\frac{1}{2},-m-\frac{1}{2};\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{c^2 f (1-\sin (e+f 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}",1,"(2^(3/2 + m)*Cos[e + f*x]*Hypergeometric2F1[-1/2, -1/2 - m, 1/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a*(1 + Sin[e + f*x]))^m)/(c^2*f*(1 - Sin[e + f*x]))","A",1
72,1,91,86,0.1473711,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx","Integrate[(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}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a (\sin (e+f x)+1))^m \, _2F_1\left(-\frac{3}{2},-m-\frac{1}{2};-\frac{1}{2};\frac{1}{2} (1-\sin (e+f x))\right)}{3 c^3 f (1-\sin (e+f x))^2}","\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)*Cos[e + f*x]*Hypergeometric2F1[-3/2, -1/2 - m, -1/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a*(1 + Sin[e + f*x]))^m)/(3*c^3*f*(1 - Sin[e + f*x])^2)","A",1
73,1,695,244,6.5585008,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2),x]","\frac{(c-c \sin (e+f x))^{5/2} (a (\sin (e+f x)+1))^m \left(\frac{\left(8 m^3+108 m^2+590 m+2205\right) \left(\left(\frac{3}{8}-\frac{3 i}{8}\right) \sin \left(\frac{1}{2} (e+f x)\right)+\left(\frac{3}{8}+\frac{3 i}{8}\right) \cos \left(\frac{1}{2} (e+f x)\right)\right)}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left(8 m^3+108 m^2+590 m+2205\right) \left(\left(\frac{3}{8}+\frac{3 i}{8}\right) \sin \left(\frac{1}{2} (e+f x)\right)+\left(\frac{3}{8}-\frac{3 i}{8}\right) \cos \left(\frac{1}{2} (e+f x)\right)\right)}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left(4 m^3+48 m^2+191 m\right) \left((1-i) \cos \left(\frac{3}{2} (e+f x)\right)-(1+i) \sin \left(\frac{3}{2} (e+f x)\right)\right)}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left(4 m^3+48 m^2+191 m\right) \left((1+i) \cos \left(\frac{3}{2} (e+f x)\right)-(1-i) \sin \left(\frac{3}{2} (e+f x)\right)\right)}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{(2 m+21) \left(\left(\frac{3}{2}-\frac{3 i}{2}\right) \sin \left(\frac{5}{2} (e+f x)\right)+\left(\frac{3}{2}+\frac{3 i}{2}\right) \cos \left(\frac{5}{2} (e+f x)\right)\right)}{(2 m+5) (2 m+7) (2 m+9)}+\frac{(2 m+21) \left(\left(\frac{3}{2}+\frac{3 i}{2}\right) \sin \left(\frac{5}{2} (e+f x)\right)+\left(\frac{3}{2}-\frac{3 i}{2}\right) \cos \left(\frac{5}{2} (e+f x)\right)\right)}{(2 m+5) (2 m+7) (2 m+9)}+\frac{(2 m+15) \left(\left(\frac{3}{16}-\frac{3 i}{16}\right) \cos \left(\frac{7}{2} (e+f x)\right)-\left(\frac{3}{16}+\frac{3 i}{16}\right) \sin \left(\frac{7}{2} (e+f x)\right)\right)}{(2 m+7) (2 m+9)}+\frac{(2 m+15) \left(\left(\frac{3}{16}+\frac{3 i}{16}\right) \cos \left(\frac{7}{2} (e+f x)\right)-\left(\frac{3}{16}-\frac{3 i}{16}\right) \sin \left(\frac{7}{2} (e+f x)\right)\right)}{(2 m+7) (2 m+9)}+\frac{\left(-\frac{1}{16}+\frac{i}{16}\right) \cos \left(\frac{9}{2} (e+f x)\right)-\left(\frac{1}{16}+\frac{i}{16}\right) \sin \left(\frac{9}{2} (e+f x)\right)}{2 m+9}+\frac{\left(-\frac{1}{16}-\frac{i}{16}\right) \cos \left(\frac{9}{2} (e+f x)\right)-\left(\frac{1}{16}-\frac{i}{16}\right) \sin \left(\frac{9}{2} (e+f x)\right)}{2 m+9}\right)}{f \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","\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{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{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,"((a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^(5/2)*(((2205 + 590*m + 108*m^2 + 8*m^3)*((3/8 + (3*I)/8)*Cos[(e + f*x)/2] + (3/8 - (3*I)/8)*Sin[(e + f*x)/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((2205 + 590*m + 108*m^2 + 8*m^3)*((3/8 - (3*I)/8)*Cos[(e + f*x)/2] + (3/8 + (3*I)/8)*Sin[(e + f*x)/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((191*m + 48*m^2 + 4*m^3)*((1 - I)*Cos[(3*(e + f*x))/2] - (1 + I)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((191*m + 48*m^2 + 4*m^3)*((1 + I)*Cos[(3*(e + f*x))/2] - (1 - I)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((21 + 2*m)*((3/2 + (3*I)/2)*Cos[(5*(e + f*x))/2] + (3/2 - (3*I)/2)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((21 + 2*m)*((3/2 - (3*I)/2)*Cos[(5*(e + f*x))/2] + (3/2 + (3*I)/2)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((15 + 2*m)*((3/16 - (3*I)/16)*Cos[(7*(e + f*x))/2] - (3/16 + (3*I)/16)*Sin[(7*(e + f*x))/2]))/((7 + 2*m)*(9 + 2*m)) + ((15 + 2*m)*((3/16 + (3*I)/16)*Cos[(7*(e + f*x))/2] - (3/16 - (3*I)/16)*Sin[(7*(e + f*x))/2]))/((7 + 2*m)*(9 + 2*m)) + ((-1/16 + I/16)*Cos[(9*(e + f*x))/2] - (1/16 + I/16)*Sin[(9*(e + f*x))/2])/(9 + 2*m) + ((-1/16 - I/16)*Cos[(9*(e + f*x))/2] - (1/16 - I/16)*Sin[(9*(e + f*x))/2])/(9 + 2*m)))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5)","C",1
74,1,149,172,3.1043103,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2),x]","-\frac{c \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 (a (\sin (e+f x)+1))^m \left(4 \left(4 m^2+24 m+27\right) \sin (e+f x)+\left(4 m^2+16 m+15\right) \cos (2 (e+f x))-12 m^2-80 m-157\right)}{f (2 m+3) (2 m+5) (2 m+7) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"-((c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^m*Sqrt[c - c*Sin[e + f*x]]*(-157 - 80*m - 12*m^2 + (15 + 16*m + 4*m^2)*Cos[2*(e + f*x)] + 4*(27 + 24*m + 4*m^2)*Sin[e + f*x]))/(f*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))","A",1
75,1,111,107,0.5699285,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]],x]","-\frac{2 \sqrt{c-c \sin (e+f x)} ((2 m+3) \sin (e+f x)-2 m-7) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 (a (\sin (e+f x)+1))^m}{f (2 m+3) (2 m+5) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"(-2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^m*Sqrt[c - c*Sin[e + f*x]]*(-7 - 2*m + (3 + 2*m)*Sin[e + f*x]))/(f*(3 + 2*m)*(5 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
76,1,85,50,0.373736,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 (a (\sin (e+f x)+1))^m}{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)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^m)/(f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",1
77,1,218,76,6.655337,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(3/2),x]","-\frac{2^{-2 m-\frac{5}{2}} \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 (a \sin (e+f x)+a)^m \left(\sec ^4\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \sec ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)^{2 m} \, _2F_1\left(2 m+2,2 m+2;2 m+3;\frac{1}{2} \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right)-4^{m+1} \, _2F_1\left(1,2 m+2;2 m+3;\cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right)}{f (m+1) (c-c \sin (e+f x))^{3/2}}","\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,"-((2^(-5/2 - 2*m)*Cos[(-e + Pi/2 - f*x)/2]^2*(-(4^(1 + m)*Hypergeometric2F1[1, 2 + 2*m, 3 + 2*m, Cos[(-e + Pi/2 - f*x)/2]]) + Hypergeometric2F1[2 + 2*m, 2 + 2*m, 3 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2]*Sec[(-e + Pi/2 - f*x)/4]^4*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(c - c*Sin[e + f*x])^(3/2)))","B",0
78,1,3174,79,6.6694336,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","\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,"(2^(-3/2 - 2*m)*(-(4^m*Hypergeometric2F1[1, 2*m, 1 + 2*m, Cos[(-e + Pi/2 - f*x)/2]]) + Hypergeometric2F1[2*m, 2*m, 1 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a + a*Sin[e + f*x])^m)/(f*m*(c - c*Sin[e + f*x])^(5/2)) - ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a + a*Sin[e + f*x])^m*(AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2 - (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (2^(1 - 2*m)*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m)))/(4*Sqrt[2]*f*(c - c*Sin[e + f*x])^(5/2)*(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^3*(-1/8*(m*Cos[(-e + Pi/2 - f*x)/4]*(Cos[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/4]*(AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2 - (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (2^(1 - 2*m)*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m)))/Sqrt[2] + ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*((AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*Tan[(-e + Pi/2 - f*x)/4])/2 + m*AppellF1[1, -2*m, 2*m, 2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^3 + (Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]^2*(-1/2*(m*AppellF1[2, 1 - 2*m, 2*m, 3, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) - (m*AppellF1[2, -2*m, 1 + 2*m, 3, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/2) + (m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^3*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^3*(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(-1 - 2*m)*(Csc[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*(Csc[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(2*(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (Cot[(-e + Pi/2 - f*x)/4]^2*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*((m*AppellF1[2, 1 - 2*m, 2*m, 3, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*Csc[(-e + Pi/2 - f*x)/4]^2)/2 + (m*AppellF1[2, -2*m, 1 + 2*m, 3, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]*Csc[(-e + Pi/2 - f*x)/4]^2)/2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (m*AppellF1[1, -2*m, 2*m, 2, Cot[(-e + Pi/2 - f*x)/4]^2, -Cot[(-e + Pi/2 - f*x)/4]^2]*Csc[(-e + Pi/2 - f*x)/4]*(Csc[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Sec[(-e + Pi/2 - f*x)/4]*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m))/(1 - Cot[(-e + Pi/2 - f*x)/4]^2)^(2*m) + (AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(2^(2*m)*(1 + 2*m)) + (2^(1 - 2*m)*(-1/2*((1 + 2*m)*AppellF1[2 + 2*m, 2*m, 2, 3 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2 + 2*m) - (m*(1 + 2*m)*AppellF1[2 + 2*m, 1 + 2*m, 1, 3 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(2 + 2*m)))*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(2*m))/(1 + 2*m) - (2^(2 - 2*m)*m*AppellF1[1 + 2*m, 2*m, 1, 2 + 2*m, (1 - Tan[(-e + Pi/2 - f*x)/4]^2)/2, 1 - Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]^3*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(1 - Tan[(-e + Pi/2 - f*x)/4]^4)^(-1 + 2*m))/(1 + 2*m)))/(8*Sqrt[2])))","C",0
79,1,85,50,0.3359156,"\int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/Sqrt[c - c*Sin[e + f*x]],x]","\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 (a (\sin (e+f x)+1))^m}{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)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^m)/(f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])","A",1
80,1,85,50,0.3563752,"\int \frac{\cos ^2(e+f x) (c+c \sin (e+f x))^m}{\sqrt{a-a \sin (e+f x)}} \, dx","Integrate[(Cos[e + f*x]^2*(c + c*Sin[e + f*x])^m)/Sqrt[a - a*Sin[e + f*x]],x]","\frac{2 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 (c (\sin (e+f x)+1))^m}{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)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c*(1 + Sin[e + f*x]))^m)/(f*(3 + 2*m)*Sqrt[a - a*Sin[e + f*x]])","A",1
81,1,176,182,17.0326704,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-5 - m),x]","\frac{2^{-m-2} \cos ^3\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m-7}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-5} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (-m-5)} \left(-2 (2 m+5) \sin (e+f x)+\cos \left(2 \left(-e-f x+\frac{\pi }{2}\right)\right)+4 \left(m^2+5 m+6\right)\right)}{f (2 m+3) (2 m+5) (2 m+7)}","\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{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{\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,"(2^(-2 - m)*Cos[(-e + Pi/2 - f*x)/2]^3*Sin[(-e + Pi/2 - f*x)/2]^(-7 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-5 - m)*(4*(6 + 5*m + m^2) + Cos[2*(-e + Pi/2 - f*x)] - 2*(5 + 2*m)*Sin[e + f*x]))/(f*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-5 - m)))","A",0
82,1,142,114,12.609172,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-4-m} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-4 - m),x]","-\frac{2^{-m-1} \cos ^3\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m-5}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (\sin (e+f x)-2 (m+2)) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-4} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (-m-4)}}{f (2 m+3) (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,"-((2^(-1 - m)*Cos[(-e + Pi/2 - f*x)/2]^3*Sin[(-e + Pi/2 - f*x)/2]^(-5 - 2*m)*(-2*(2 + m) + Sin[e + f*x])*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-4 - m))/(f*(3 + 2*m)*(5 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-4 - m))))","A",1
83,1,109,54,5.5187288,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m),x]","\frac{2^{-m} \sin ^3\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \cos ^{-2 m-3}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^{-m} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{2 m}}{c^3 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[(2*e + Pi + 2*f*x)/4]^(-3 - 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*m)*(a*(1 + Sin[e + f*x]))^m*Sin[(2*e + Pi + 2*f*x)/4]^3)/(2^m*c^3*f*(3 + 2*m)*(c - c*Sin[e + f*x])^m)","B",1
84,1,589,113,21.555262,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m),x]","-\frac{2^{1-m} (2 m-3) \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \cot \left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (-m-2)} \left((2 m-1) \cot ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \, _2F_1\left(-m-\frac{1}{2},-2 (m+1);\frac{1}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)-(2 m+1) F_1\left(\frac{1}{2}-m;-2 (m+1),1;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right)}{f \left(4 m^2-1\right) \left(8 (m+1) F_1\left(\frac{3}{2}-m;-2 m-1,1;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)+4 F_1\left(\frac{3}{2}-m;-2 (m+1),2;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)+(2 m-3) \left(2 \cot ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) F_1\left(\frac{1}{2}-m;-2 (m+1),1;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)-(\sin (e+f x)+1) \csc ^4\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{2 m}\right)\right)}","\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 - m)*(-3 + 2*m)*Cos[(-e + Pi/2 - f*x)/2]^2*Cot[(-e + Pi/2 - f*x)/4]*Csc[(-e + Pi/2 - f*x)/4]^2*(-((1 + 2*m)*AppellF1[1/2 - m, -2*(1 + m), 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]) + (-1 + 2*m)*Cot[(-e + Pi/2 - f*x)/4]^2*Hypergeometric2F1[-1/2 - m, -2*(1 + m), 1/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2])*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(-1 + 4*m^2)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-2 - m))*(8*(1 + m)*AppellF1[3/2 - m, -1 - 2*m, 1, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 4*AppellF1[3/2 - m, -2*(1 + m), 2, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + (-3 + 2*m)*(2*AppellF1[1/2 - m, -2*(1 + m), 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^2 - Csc[(-e + Pi/2 - f*x)/4]^4*(1 + Sin[e + f*x])*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)))))","C",0
85,1,1045,114,29.6453833,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m),x]","\frac{2^{3-m} (2 m-3) \left(F_1\left(\frac{1}{2}-m;-2 m,1;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)-4 F_1\left(\frac{1}{2}-m;-2 m,2;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)+4 F_1\left(\frac{1}{2}-m;-2 m,3;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right) \cos ^3\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin \left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (-m-1)} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{-m-1}}{f (2 m-1) \left((2 m-3) F_1\left(\frac{1}{2}-m;-2 m,1;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-4 (2 m-3) F_1\left(\frac{1}{2}-m;-2 m,2;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+4 (2 m-3) F_1\left(\frac{1}{2}-m;-2 m,3;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+2 \left(2 m F_1\left(\frac{3}{2}-m;1-2 m,1;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)-8 m F_1\left(\frac{3}{2}-m;1-2 m,2;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)+8 m F_1\left(\frac{3}{2}-m;1-2 m,3;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)+F_1\left(\frac{3}{2}-m;-2 m,2;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)+12 F_1\left(\frac{3}{2}-m;-2 m,4;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right) \sin ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+8 F_1\left(\frac{3}{2}-m;-2 m,3;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(\cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)-1\right)\right)}","\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^(3 - m)*(-3 + 2*m)*(AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 4*AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/4]^3*Cos[(-e + Pi/2 - f*x)/2]^2*Sin[(-e + Pi/2 - f*x)/4]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(-1 + 2*m)*((-3 + 2*m)*AppellF1[1/2 - m, -2*m, 1, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^2 - 4*(-3 + 2*m)*AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^2 + 4*(-3 + 2*m)*AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^2 + 8*AppellF1[3/2 - m, -2*m, 3, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(-1 + Cos[(-e + Pi/2 - f*x)/2]) + 2*(2*m*AppellF1[3/2 - m, 1 - 2*m, 1, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 8*m*AppellF1[3/2 - m, 1 - 2*m, 2, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*m*AppellF1[3/2 - m, 1 - 2*m, 3, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2 - m, -2*m, 2, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 12*AppellF1[3/2 - m, -2*m, 4, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Sin[(-e + Pi/2 - f*x)/4]^2)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-1 - m)))","C",0
86,1,1519,116,21.7083184,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-m} \, dx","Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^m,x]","\frac{2^{6-m} (2 m-3) \left(F_1\left(\frac{1}{2}-m;-2 m,2;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)-5 F_1\left(\frac{1}{2}-m;-2 m,3;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)+8 F_1\left(\frac{1}{2}-m;-2 m,4;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)-4 F_1\left(\frac{1}{2}-m;-2 m,5;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right) \cos ^5\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^3\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin ^{-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{2 m} (\sin (e+f x) a+a)^m (c-c \sin (e+f x))^{-m}}{f (2 m-1) \left((2 m-3) F_1\left(\frac{1}{2}-m;-2 m,2;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-5 (2 m-3) F_1\left(\frac{1}{2}-m;-2 m,3;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+8 (2 m-3) F_1\left(\frac{1}{2}-m;-2 m,4;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-8 m F_1\left(\frac{1}{2}-m;-2 m,5;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+12 F_1\left(\frac{1}{2}-m;-2 m,5;\frac{3}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+4 m F_1\left(\frac{3}{2}-m;1-2 m,2;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \sin ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-20 m F_1\left(\frac{3}{2}-m;1-2 m,3;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \sin ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+32 m F_1\left(\frac{3}{2}-m;1-2 m,4;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \sin ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-16 m F_1\left(\frac{3}{2}-m;1-2 m,5;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \sin ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+4 F_1\left(\frac{3}{2}-m;-2 m,3;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \sin ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-30 F_1\left(\frac{3}{2}-m;-2 m,4;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \sin ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+64 F_1\left(\frac{3}{2}-m;-2 m,5;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \sin ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-40 F_1\left(\frac{3}{2}-m;-2 m,6;\frac{5}{2}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \sin ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)}","\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^(6 - m)*(-3 + 2*m)*(AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 5*AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 8*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/4]^5*Cos[(-e + Pi/2 - f*x)/2]^2*Sin[(-e + Pi/2 - f*x)/4]^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*m)*(a + a*Sin[e + f*x])^m)/(f*(-1 + 2*m)*((-3 + 2*m)*AppellF1[1/2 - m, -2*m, 2, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^2 - 5*(-3 + 2*m)*AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^2 + 8*(-3 + 2*m)*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^2 + 12*AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^2 - 8*m*AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^2 + 4*m*AppellF1[3/2 - m, 1 - 2*m, 2, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^2 - 20*m*AppellF1[3/2 - m, 1 - 2*m, 3, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^2 + 32*m*AppellF1[3/2 - m, 1 - 2*m, 4, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^2 - 16*m*AppellF1[3/2 - m, 1 - 2*m, 5, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^2 + 4*AppellF1[3/2 - m, -2*m, 3, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^2 - 30*AppellF1[3/2 - m, -2*m, 4, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^2 + 64*AppellF1[3/2 - m, -2*m, 5, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^2 - 40*AppellF1[3/2 - m, -2*m, 6, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sin[(-e + Pi/2 - f*x)/4]^2)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(c - c*Sin[e + f*x])^m)","C",0
87,1,4270,116,25.9005793,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 - m),x]","\text{Result too large to show}","\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^(9 - m)*(AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 6*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 13*AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 12*AppellF1[1/2 - m, -2*m, 6, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 4*AppellF1[1/2 - m, -2*m, 7, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 - m)*((Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Cos[Pi/4 + (e - Pi/2 + f*x)/2]^6)/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m) - (2*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Cos[Pi/4 + (e - Pi/2 + f*x)/2]^5*Sin[Pi/4 + (e - Pi/2 + f*x)/2])/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m) - (Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Cos[Pi/4 + (e - Pi/2 + f*x)/2]^4*Sin[Pi/4 + (e - Pi/2 + f*x)/2]^2)/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m) + (4*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Cos[Pi/4 + (e - Pi/2 + f*x)/2]^3*Sin[Pi/4 + (e - Pi/2 + f*x)/2]^3)/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m) - (Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Cos[Pi/4 + (e - Pi/2 + f*x)/2]^2*Sin[Pi/4 + (e - Pi/2 + f*x)/2]^4)/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m) - (2*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Cos[Pi/4 + (e - Pi/2 + f*x)/2]*Sin[Pi/4 + (e - Pi/2 + f*x)/2]^5)/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m) + (Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sin[Pi/4 + (e - Pi/2 + f*x)/2]^6)/(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(2*m))*Tan[(-e + Pi/2 - f*x)/4])/(f*(-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(1 - m))*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(-((2^(9 - m)*m*(AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 6*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 13*AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 12*AppellF1[1/2 - m, -2*m, 6, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 4*AppellF1[1/2 - m, -2*m, 7, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 - 2*m))/((-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m))) - (2^(7 - m)*(AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 6*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 13*AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 12*AppellF1[1/2 - m, -2*m, 6, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 4*AppellF1[1/2 - m, -2*m, 7, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Sec[(-e + Pi/2 - f*x)/4]^2)/((-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (2^(9 - m)*m*(AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 6*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 13*AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 12*AppellF1[1/2 - m, -2*m, 6, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 4*AppellF1[1/2 - m, -2*m, 7, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(-1 - 2*m)*Tan[(-e + Pi/2 - f*x)/4])/((-1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (2^(9 - m)*m*(AppellF1[1/2 - m, -2*m, 3, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 6*AppellF1[1/2 - m, -2*m, 4, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 13*AppellF1[1/2 - m, -2*m, 5, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 12*AppellF1[1/2 - m, -2*m, 6, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 4*AppellF1[1/2 - m, -2*m, 7, 3/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(1 - 2*m)*Tan[(-e + Pi/2 - f*x)/4])/((-1 + 2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)) - (2^(9 - m)*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Tan[(-e + Pi/2 - f*x)/4]*(-(((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 3, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) - (3*(1/2 - m)*AppellF1[3/2 - m, -2*m, 4, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 - m)) - 6*(-(((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 4, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) - (2*(1/2 - m)*AppellF1[3/2 - m, -2*m, 5, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) + 13*(-(((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 5, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) - (5*(1/2 - m)*AppellF1[3/2 - m, -2*m, 6, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 - m))) - 12*(-(((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 6, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) - (3*(1/2 - m)*AppellF1[3/2 - m, -2*m, 7, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) + 4*(-(((1/2 - m)*m*AppellF1[3/2 - m, 1 - 2*m, 7, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(3/2 - m)) - (7*(1/2 - m)*AppellF1[3/2 - m, -2*m, 8, 5/2 - m, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/(2*(3/2 - m)))))/((-1 + 2*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))))","C",0
88,1,311,343,8.0153192,"\int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2),x]","\frac{c^4 g e^{-5 i (e+f x)} \left(e^{i (e+f x)}-i\right) \left(4928 e^{7 i (e+f x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)+\sqrt{1+e^{2 i (e+f x)}} \left(154 e^{i (e+f x)}+423 i e^{2 i (e+f x)}-308 e^{3 i (e+f x)}+1374 i e^{4 i (e+f x)}-7392 e^{5 i (e+f x)}+1374 i e^{6 i (e+f x)}+308 e^{7 i (e+f x)}+423 i e^{8 i (e+f x)}-154 e^{9 i (e+f x)}-21 i e^{10 i (e+f x)}-21 i\right)\right) \sqrt{a (\sin (e+f x)+1)} \sqrt{g \cos (e+f x)}}{3696 f \left(e^{i (e+f x)}+i\right) \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-c \sin (e+f 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^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^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 (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,"(c^4*(-I + E^(I*(e + f*x)))*g*Sqrt[g*Cos[e + f*x]]*(Sqrt[1 + E^((2*I)*(e + f*x))]*(-21*I + 154*E^(I*(e + f*x)) + (423*I)*E^((2*I)*(e + f*x)) - 308*E^((3*I)*(e + f*x)) + (1374*I)*E^((4*I)*(e + f*x)) - 7392*E^((5*I)*(e + f*x)) + (1374*I)*E^((6*I)*(e + f*x)) + 308*E^((7*I)*(e + f*x)) + (423*I)*E^((8*I)*(e + f*x)) - 154*E^((9*I)*(e + f*x)) - (21*I)*E^((10*I)*(e + f*x))) + 4928*E^((7*I)*(e + f*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(3696*E^((5*I)*(e + f*x))*(I + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[c - c*Sin[e + f*x]])","C",1
89,1,281,290,1.9323036,"\int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2),x]","-\frac{c^3 g e^{-4 i (e+f x)} \left(e^{i (e+f x)}-i\right) \left(\sqrt{1+e^{2 i (e+f x)}} \left(-180 i e^{i (e+f x)}+238 e^{2 i (e+f x)}-540 i e^{3 i (e+f x)}+3696 e^{4 i (e+f x)}-540 i e^{5 i (e+f x)}-238 e^{6 i (e+f x)}-180 i e^{7 i (e+f x)}+35 e^{8 i (e+f x)}-35\right)-2464 e^{6 i (e+f x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)\right) \sqrt{a (\sin (e+f x)+1)} \sqrt{g \cos (e+f x)}}{2520 f \left(e^{i (e+f x)}+i\right) \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-c \sin (e+f 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^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{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{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,"-1/2520*(c^3*(-I + E^(I*(e + f*x)))*g*Sqrt[g*Cos[e + f*x]]*(Sqrt[1 + E^((2*I)*(e + f*x))]*(-35 - (180*I)*E^(I*(e + f*x)) + 238*E^((2*I)*(e + f*x)) - (540*I)*E^((3*I)*(e + f*x)) + 3696*E^((4*I)*(e + f*x)) - (540*I)*E^((5*I)*(e + f*x)) - 238*E^((6*I)*(e + f*x)) - (180*I)*E^((7*I)*(e + f*x)) + 35*E^((8*I)*(e + f*x))) - 2464*E^((6*I)*(e + f*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(E^((4*I)*(e + f*x))*(I + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[c - c*Sin[e + f*x]])","C",1
90,1,255,235,1.7169701,"\int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2),x]","\frac{c^2 g e^{-3 i (e+f x)} \left(e^{i (e+f x)}-i\right) \left(112 e^{5 i (e+f x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)+\sqrt{1+e^{2 i (e+f x)}} \left(-14 e^{i (e+f x)}+15 i e^{2 i (e+f x)}-168 e^{3 i (e+f x)}+15 i e^{4 i (e+f x)}+14 e^{5 i (e+f x)}+5 i e^{6 i (e+f x)}+5 i\right)\right) \sqrt{a (\sin (e+f x)+1)} \sqrt{g \cos (e+f x)}}{140 f \left(e^{i (e+f x)}+i\right) \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-c \sin (e+f 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}}",1,"(c^2*(-I + E^(I*(e + f*x)))*g*Sqrt[g*Cos[e + f*x]]*(Sqrt[1 + E^((2*I)*(e + f*x))]*(5*I - 14*E^(I*(e + f*x)) + (15*I)*E^((2*I)*(e + f*x)) - 168*E^((3*I)*(e + f*x)) + (15*I)*E^((4*I)*(e + f*x)) + 14*E^((5*I)*(e + f*x)) + (5*I)*E^((6*I)*(e + f*x))) + 112*E^((5*I)*(e + f*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(140*E^((3*I)*(e + f*x))*(I + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[c - c*Sin[e + f*x]])","C",1
91,1,249,178,2.3913748,"\int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]],x]","\frac{\csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) \sec ^3(e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(12 (\cos (f x)-i \sin (f x)) \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i f x} (\cos (e)+i \sin (e))^2\right)+4 (\cos (f x)+i \sin (f x)) \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i f x} (\cos (e)+i \sin (e))^2\right)-13 \cos (2 e+f x)+\cos (2 e+3 f x)-\cos (4 e+3 f x)-11 \cos (f x)\right)}{40 f}","-\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,"((g*Cos[e + f*x])^(3/2)*Csc[e/2]*Sec[e/2]*Sec[e + f*x]^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-11*Cos[f*x] - 13*Cos[2*e + f*x] + Cos[2*e + 3*f*x] - Cos[4*e + 3*f*x] + 12*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*f*x)*(Cos[e] + I*Sin[e])^2)]*(Cos[f*x] - I*Sin[f*x])*Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]] + 4*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*f*x)*(Cos[e] + I*Sin[e])^2)]*(Cos[f*x] + I*Sin[f*x])*Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]))/(40*f)","C",1
92,1,197,122,1.9211557,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{i g \sqrt{i c e^{-i (e+f x)} \left(e^{i (e+f x)}-i\right)^2} \sqrt{g e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)} \left(4 e^{3 i (e+f x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)-i \sqrt{1+e^{2 i (e+f x)}} \left(-6 i e^{i (e+f x)}+e^{2 i (e+f x)}+1\right)\right) \sqrt{a (\sin (e+f x)+1)}}{3 c f \left(1+e^{2 i (e+f x)}\right)^{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)}}{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,"((-1/3*I)*Sqrt[(I*c*(-I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*g*Sqrt[((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x))]*((-I)*Sqrt[1 + E^((2*I)*(e + f*x))]*(1 - (6*I)*E^(I*(e + f*x)) + E^((2*I)*(e + f*x))) + 4*E^((3*I)*(e + f*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(c*(1 + E^((2*I)*(e + f*x)))^(3/2)*f)","C",1
93,1,211,123,1.6909653,"\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","Integrate[((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 \sqrt{g e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)} \left(2 e^{2 i (e+f x)} \left(e^{i (e+f x)}-i\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)+\sqrt{1+e^{2 i (e+f x)}} \left(i-5 e^{i (e+f x)}\right)\right) \sqrt{a (\sin (e+f x)+1)}}{c f \left(e^{i (e+f x)}+i\right) \sqrt{1+e^{2 i (e+f x)}} \sqrt{i c e^{-i (e+f x)} \left(e^{i (e+f x)}-i\right)^2}}","\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,"(-2*g*Sqrt[((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x))]*((I - 5*E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))] + 2*E^((2*I)*(e + f*x))*(-I + E^(I*(e + f*x)))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(c*Sqrt[(I*c*(-I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*(I + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*f)","C",1
94,1,229,182,2.0770985,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(5/2),x]","\frac{4 i g \sqrt{g e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)} \left(e^{i (e+f x)} \left(e^{i (e+f x)}-i\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)+\left(4 i e^{i (e+f x)}-3 e^{2 i (e+f x)}+5\right) \sqrt{1+e^{2 i (e+f x)}}\right) \sqrt{a (\sin (e+f x)+1)}}{5 c f \left(e^{i (e+f x)}+i\right) \sqrt{1+e^{2 i (e+f x)}} \left(i c e^{-i (e+f x)} \left(e^{i (e+f x)}-i\right)^2\right)^{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)}}{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*I)/5)*g*Sqrt[((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x))]*((5 + (4*I)*E^(I*(e + f*x)) - 3*E^((2*I)*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))] + E^(I*(e + f*x))*(-I + E^(I*(e + f*x)))^3*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(c*((I*c*(-I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x)))^(3/2)*(I + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*f)","C",1
95,1,256,237,2.1801323,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(7/2),x]","\frac{4 e^{3 i (e+f x)} \left(g e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)\right)^{3/2} \left(\left(e^{i (e+f x)}-i\right)^5 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)+\sqrt{1+e^{2 i (e+f x)}} \left(e^{i (e+f x)}+15 i e^{2 i (e+f x)}-3 e^{3 i (e+f x)}-29 i\right)\right) \sqrt{a (\sin (e+f x)+1)}}{45 c^3 f \left(e^{i (e+f x)}-i\right)^4 \left(e^{i (e+f x)}+i\right) \left(1+e^{2 i (e+f x)}\right)^{3/2} \sqrt{i c e^{-i (e+f x)} \left(e^{i (e+f x)}-i\right)^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^2 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}}{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*E^((3*I)*(e + f*x))*(((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(e + f*x))]*(-29*I + E^(I*(e + f*x)) + (15*I)*E^((2*I)*(e + f*x)) - 3*E^((3*I)*(e + f*x))) + (-I + E^(I*(e + f*x)))^5*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(45*c^3*(-I + E^(I*(e + f*x)))^4*Sqrt[(I*c*(-I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*(I + E^(I*(e + f*x)))*(1 + E^((2*I)*(e + f*x)))^(3/2)*f)","C",1
96,1,291,292,2.6343631,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(9/2),x]","\frac{4 e^{3 i (e+f x)} \left(g e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)\right)^{3/2} \left(\sqrt{1+e^{2 i (e+f x)}} \left(149 i e^{i (e+f x)}+44 e^{2 i (e+f x)}-64 i e^{3 i (e+f x)}+21 e^{4 i (e+f x)}+3 i e^{5 i (e+f x)}-1\right)-i \left(e^{i (e+f x)}-i\right)^7 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)\right) \sqrt{a (\sin (e+f x)+1)}}{195 c^4 f \left(1-i e^{i (e+f x)}\right) \left(e^{i (e+f x)}-i\right)^6 \left(1+e^{2 i (e+f x)}\right)^{3/2} \sqrt{i c e^{-i (e+f x)} \left(e^{i (e+f x)}-i\right)^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}}{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 \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*E^((3*I)*(e + f*x))*(((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(e + f*x))]*(-1 + (149*I)*E^(I*(e + f*x)) + 44*E^((2*I)*(e + f*x)) - (64*I)*E^((3*I)*(e + f*x)) + 21*E^((4*I)*(e + f*x)) + (3*I)*E^((5*I)*(e + f*x))) - I*(-I + E^(I*(e + f*x)))^7*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(195*c^4*(1 - I*E^(I*(e + f*x)))*(-I + E^(I*(e + f*x)))^6*Sqrt[(I*c*(-I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*(1 + E^((2*I)*(e + f*x)))^(3/2)*f)","C",1
97,1,193,352,1.3084453,"\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","Integrate[(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{c^2 (\sin (e+f x)-1)^2 (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(3696 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)+\sqrt{\cos (e+f x)} (836 \sin (2 (e+f x))+110 \sin (4 (e+f x))+450 \cos (e+f x)+225 \cos (3 (e+f x))+45 \cos (5 (e+f x)))\right)}{3960 f \cos ^{\frac{3}{2}}(e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","\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{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^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{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,"(c^2*(g*Cos[e + f*x])^(3/2)*(-1 + Sin[e + f*x])^2*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(3696*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(450*Cos[e + f*x] + 225*Cos[3*(e + f*x)] + 45*Cos[5*(e + f*x)] + 836*Sin[2*(e + f*x)] + 110*Sin[4*(e + f*x)])))/(3960*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
98,1,113,295,0.694379,"\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","Integrate[(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{c (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(168 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)+(38 \sin (2 (e+f x))+5 \sin (4 (e+f x))) \sqrt{\cos (e+f x)}\right)}{180 f \cos ^{\frac{9}{2}}(e+f 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}",1,"-1/180*(c*(g*Cos[e + f*x])^(3/2)*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(168*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(38*Sin[2*(e + f*x)] + 5*Sin[4*(e + f*x)])))/(f*Cos[e + f*x]^(9/2))","A",1
99,1,257,235,7.5987333,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]],x]","\frac{i a^2 g e^{-3 i (e+f x)} \left(e^{i (e+f x)}+i\right) \left(112 i e^{5 i (e+f x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)+\sqrt{1+e^{2 i (e+f x)}} \left(-14 i e^{i (e+f x)}+15 e^{2 i (e+f x)}-168 i e^{3 i (e+f x)}+15 e^{4 i (e+f x)}+14 i e^{5 i (e+f x)}+5 e^{6 i (e+f x)}+5\right)\right) \sqrt{c-c \sin (e+f x)} \sqrt{g \cos (e+f x)}}{140 f \left(e^{i (e+f x)}-i\right) \sqrt{1+e^{2 i (e+f x)}} \sqrt{a (\sin (e+f x)+1)}}","-\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,"((I/140)*a^2*(I + E^(I*(e + f*x)))*g*Sqrt[g*Cos[e + f*x]]*(Sqrt[1 + E^((2*I)*(e + f*x))]*(5 - (14*I)*E^(I*(e + f*x)) + 15*E^((2*I)*(e + f*x)) - (168*I)*E^((3*I)*(e + f*x)) + 15*E^((4*I)*(e + f*x)) + (14*I)*E^((5*I)*(e + f*x)) + 5*E^((6*I)*(e + f*x))) + (112*I)*E^((5*I)*(e + f*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[c - c*Sin[e + f*x]])/(E^((3*I)*(e + f*x))*(-I + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
100,1,148,180,0.7728298,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{(a (\sin (e+f x)+1))^{3/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sqrt{\cos (e+f x)} (3 \sin (2 (e+f x))+20 \cos (e+f x))-42 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{15 f \cos ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\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,"-1/15*((g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2)*(-42*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(20*Cos[e + f*x] + 3*Sin[2*(e + f*x)])))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c - c*Sin[e + f*x]])","A",1
101,1,207,182,1.7461107,"\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","Integrate[((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 (a (\sin (e+f x)+1))^{3/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sqrt{\cos (e+f x)} \left(\cos \left(\frac{1}{2} (e+f x)\right) (\cos (e+f x)+12)-\sin \left(\frac{1}{2} (e+f x)\right) (\cos (e+f x)-12)\right)-21 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{3 c f (\sin (e+f x)-1) \cos ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","\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,"(-2*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(-21*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + Sqrt[Cos[e + f*x]]*(Cos[(e + f*x)/2]*(12 + Cos[e + f*x]) - (-12 + Cos[e + f*x])*Sin[(e + f*x)/2]))*(a*(1 + Sin[e + f*x]))^(3/2))/(3*c*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])","A",1
102,1,191,186,1.393458,"\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","Integrate[((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{a \sqrt{\cos (e+f x)} \sqrt{a (\sin (e+f x)+1)} (g \cos (e+f x))^{3/2} \left(8 \sqrt{\cos (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)-2 \sin \left(\frac{3}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+2 \cos \left(\frac{3}{2} (e+f x)\right)\right)-42 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{5 c^2 f (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","\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,"-1/5*(a*Sqrt[Cos[e + f*x]]*(g*Cos[e + f*x])^(3/2)*Sqrt[a*(1 + Sin[e + f*x])]*(-42*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + 8*Sqrt[Cos[e + f*x]]*(Cos[(e + f*x)/2] + 2*Cos[(3*(e + f*x))/2] + Sin[(e + f*x)/2] - 2*Sin[(3*(e + f*x))/2])))/(c^2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","A",1
103,1,218,243,2.3368554,"\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","Integrate[((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{a \sqrt{\cos (e+f x)} \sqrt{a (\sin (e+f x)+1)} (g \cos (e+f x))^{3/2} \left(\sqrt{\cos (e+f x)} \left(-74 \sin \left(\frac{1}{2} (e+f x)\right)+15 \sin \left(\frac{3}{2} (e+f x)\right)+21 \sin \left(\frac{5}{2} (e+f x)\right)-74 \cos \left(\frac{1}{2} (e+f x)\right)-15 \cos \left(\frac{3}{2} (e+f x)\right)+21 \cos \left(\frac{5}{2} (e+f x)\right)\right)+84 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5\right)}{90 c^3 f (\sin (e+f x)-1)^3 \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\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{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{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,"(a*Sqrt[Cos[e + f*x]]*(g*Cos[e + f*x])^(3/2)*Sqrt[a*(1 + Sin[e + f*x])]*(84*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + Sqrt[Cos[e + f*x]]*(-74*Cos[(e + f*x)/2] - 15*Cos[(3*(e + f*x))/2] + 21*Cos[(5*(e + f*x))/2] - 74*Sin[(e + f*x)/2] + 15*Sin[(3*(e + f*x))/2] + 21*Sin[(5*(e + f*x))/2])))/(90*c^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(-1 + Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]])","A",1
104,1,464,300,6.4719123,"\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","Integrate[((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{\sec (e+f x) (a (\sin (e+f x)+1))^{3/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9 \left(\frac{28 \sin \left(\frac{1}{2} (e+f x)\right)}{195 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{28 \sin \left(\frac{1}{2} (e+f x)\right)}{195 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}-\frac{128 \sin \left(\frac{1}{2} (e+f x)\right)}{117 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{16 \sin \left(\frac{1}{2} (e+f x)\right)}{13 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{14}{195 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{64}{117 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{8}{13 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{14}{195}\right)}{f (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}-\frac{14 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a (\sin (e+f x)+1))^{3/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{195 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\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{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{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,"(-14*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(3/2))/(195*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(9/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(14/195 + 8/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) - 64/(117*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) + 14/(195*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (16*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) - (128*Sin[(e + f*x)/2])/(117*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (28*Sin[(e + f*x)/2])/(195*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (28*Sin[(e + f*x)/2])/(195*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(3/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(9/2))","A",1
105,1,532,357,6.4941948,"\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","Integrate[((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{\sec (e+f x) (a (\sin (e+f x)+1))^{3/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11} \left(\frac{28 \sin \left(\frac{1}{2} (e+f x)\right)}{1105 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{28 \sin \left(\frac{1}{2} (e+f x)\right)}{1105 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{28 \sin \left(\frac{1}{2} (e+f x)\right)}{663 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}-\frac{160 \sin \left(\frac{1}{2} (e+f x)\right)}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{16 \sin \left(\frac{1}{2} (e+f x)\right)}{17 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}+\frac{14}{1105 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{14}{663 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}-\frac{80}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{8}{17 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8}+\frac{14}{1105}\right)}{f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}-\frac{14 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a (\sin (e+f x)+1))^{3/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}{1105 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\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{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{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,"(-14*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(a*(1 + Sin[e + f*x]))^(3/2))/(1105*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(11/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(14/1105 + 8/(17*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) - 80/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) + 14/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) + 14/(1105*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (16*Sin[(e + f*x)/2])/(17*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) - (160*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) + (28*Sin[(e + f*x)/2])/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (28*Sin[(e + f*x)/2])/(1105*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (28*Sin[(e + f*x)/2])/(1105*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(3/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(11/2))","A",1
106,1,120,406,1.1642366,"\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","Integrate[(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{a^2 c^2 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(7392 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)+(1897 \sin (2 (e+f x))+400 \sin (4 (e+f x))+45 \sin (6 (e+f x))) \sqrt{\cos (e+f x)}\right)}{9360 f \cos ^{\frac{5}{2}}(e+f x)}","\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{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{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,"(a^2*c^2*(g*Cos[e + f*x])^(3/2)*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(7392*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(1897*Sin[2*(e + f*x)] + 400*Sin[4*(e + f*x)] + 45*Sin[6*(e + f*x)])))/(9360*f*Cos[e + f*x]^(5/2))","A",1
107,1,189,352,1.280112,"\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","Integrate[(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{c (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(\sqrt{\cos (e+f x)} (-836 \sin (2 (e+f x))-110 \sin (4 (e+f x))+450 \cos (e+f x)+225 \cos (3 (e+f x))+45 \cos (5 (e+f x)))-3696 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{3960 f \cos ^{\frac{3}{2}}(e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\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{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^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{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,"(c*(g*Cos[e + f*x])^(3/2)*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]]*(-3696*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(450*Cos[e + f*x] + 225*Cos[3*(e + f*x)] + 45*Cos[5*(e + f*x)] - 836*Sin[2*(e + f*x)] - 110*Sin[4*(e + f*x)])))/(3960*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)","A",1
108,1,281,290,2.0443525,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]],x]","\frac{a^3 g e^{-4 i (e+f x)} \left(e^{i (e+f x)}+i\right) \left(\sqrt{1+e^{2 i (e+f x)}} \left(180 i e^{i (e+f x)}+238 e^{2 i (e+f x)}+540 i e^{3 i (e+f x)}+3696 e^{4 i (e+f x)}+540 i e^{5 i (e+f x)}-238 e^{6 i (e+f x)}+180 i e^{7 i (e+f x)}+35 e^{8 i (e+f x)}-35\right)-2464 e^{6 i (e+f x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)\right) \sqrt{c-c \sin (e+f x)} \sqrt{g \cos (e+f x)}}{2520 f \left(e^{i (e+f x)}-i\right) \sqrt{1+e^{2 i (e+f x)}} \sqrt{a (\sin (e+f x)+1)}}","-\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^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{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{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,"(a^3*(I + E^(I*(e + f*x)))*g*Sqrt[g*Cos[e + f*x]]*(Sqrt[1 + E^((2*I)*(e + f*x))]*(-35 + (180*I)*E^(I*(e + f*x)) + 238*E^((2*I)*(e + f*x)) + (540*I)*E^((3*I)*(e + f*x)) + 3696*E^((4*I)*(e + f*x)) + (540*I)*E^((5*I)*(e + f*x)) - 238*E^((6*I)*(e + f*x)) + (180*I)*E^((7*I)*(e + f*x)) + 35*E^((8*I)*(e + f*x))) - 2464*E^((6*I)*(e + f*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[c - c*Sin[e + f*x]])/(2520*E^((4*I)*(e + f*x))*(-I + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[a*(1 + Sin[e + f*x])])","C",1
109,1,158,234,1.7708352,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{(a (\sin (e+f x)+1))^{5/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sqrt{\cos (e+f x)} (126 \sin (2 (e+f x))+515 \cos (e+f x)-15 \cos (3 (e+f x)))-924 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{210 f \cos ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\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^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{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{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,"-1/210*((g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2)*(-924*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(515*Cos[e + f*x] - 15*Cos[3*(e + f*x)] + 126*Sin[2*(e + f*x)])))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sqrt[c - c*Sin[e + f*x]])","A",1
110,1,240,241,6.3407987,"\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","Integrate[((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{(a (\sin (e+f x)+1))^{5/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sqrt{\cos (e+f x)} \left(520 \sin \left(\frac{1}{2} (e+f x)\right)-37 \sin \left(\frac{3}{2} (e+f x)\right)+3 \sin \left(\frac{5}{2} (e+f x)\right)+520 \cos \left(\frac{1}{2} (e+f x)\right)+37 \cos \left(\frac{3}{2} (e+f x)\right)+3 \cos \left(\frac{5}{2} (e+f x)\right)\right)-924 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{30 c f (\sin (e+f x)-1) \cos ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\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{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{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{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,"-1/30*((g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(a*(1 + Sin[e + f*x]))^(5/2)*(-924*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + Sqrt[Cos[e + f*x]]*(520*Cos[(e + f*x)/2] + 37*Cos[(3*(e + f*x))/2] + 3*Cos[(5*(e + f*x))/2] + 520*Sin[(e + f*x)/2] - 37*Sin[(3*(e + f*x))/2] + 3*Sin[(5*(e + f*x))/2])))/(c*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])","A",1
111,1,245,243,2.4350801,"\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","Integrate[((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{a^2 \sqrt{a (\sin (e+f x)+1)} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sqrt{\cos (e+f x)} \left(226 \sin \left(\frac{1}{2} (e+f x)\right)-327 \sin \left(\frac{3}{2} (e+f x)\right)-5 \sin \left(\frac{5}{2} (e+f x)\right)+226 \cos \left(\frac{1}{2} (e+f x)\right)+327 \cos \left(\frac{3}{2} (e+f x)\right)-5 \cos \left(\frac{5}{2} (e+f x)\right)\right)-924 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{30 c^2 f (\sin (e+f x)-1)^2 \cos ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\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,"-1/30*(a^2*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sqrt[a*(1 + Sin[e + f*x])]*(-924*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + Sqrt[Cos[e + f*x]]*(226*Cos[(e + f*x)/2] + 327*Cos[(3*(e + f*x))/2] - 5*Cos[(5*(e + f*x))/2] + 226*Sin[(e + f*x)/2] - 327*Sin[(3*(e + f*x))/2] - 5*Sin[(5*(e + f*x))/2])))/(c^2*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","A",1
112,1,246,243,2.6393955,"\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","Integrate[((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{a^2 \sqrt{a (\sin (e+f x)+1)} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(2 \sqrt{\cos (e+f x)} \left(182 \sin \left(\frac{1}{2} (e+f x)\right)-195 \sin \left(\frac{3}{2} (e+f x)\right)-93 \sin \left(\frac{5}{2} (e+f x)\right)+182 \cos \left(\frac{1}{2} (e+f x)\right)+195 \cos \left(\frac{3}{2} (e+f x)\right)-93 \cos \left(\frac{5}{2} (e+f x)\right)\right)-924 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5\right)}{90 c^3 f (\sin (e+f x)-1)^3 \cos ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\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{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{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,"-1/90*(a^2*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sqrt[a*(1 + Sin[e + f*x])]*(-924*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + 2*Sqrt[Cos[e + f*x]]*(182*Cos[(e + f*x)/2] + 195*Cos[(3*(e + f*x))/2] - 93*Cos[(5*(e + f*x))/2] + 182*Sin[(e + f*x)/2] - 195*Sin[(3*(e + f*x))/2] - 93*Sin[(5*(e + f*x))/2])))/(c^3*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]])","A",1
113,1,464,300,6.5384239,"\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","Integrate[((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 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a (\sin (e+f x)+1))^{5/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{195 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\sec (e+f x) (a (\sin (e+f x)+1))^{5/2} (g \cos (e+f x))^{3/2} \left(-\frac{308 \sin \left(\frac{1}{2} (e+f x)\right)}{195 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{472 \sin \left(\frac{1}{2} (e+f x)\right)}{195 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}-\frac{464 \sin \left(\frac{1}{2} (e+f x)\right)}{117 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{32 \sin \left(\frac{1}{2} (e+f x)\right)}{13 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{236}{195 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{232}{117 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{16}{13 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{154}{195}\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{f (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\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{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{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,"(154*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2))/(195*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(-154/195 + 16/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) - 232/(117*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) + 236/(195*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (32*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) - (464*Sin[(e + f*x)/2])/(117*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (472*Sin[(e + f*x)/2])/(195*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) - (308*Sin[(e + f*x)/2])/(195*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(5/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2))","A",1
114,1,532,357,6.5695395,"\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","Integrate[((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 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a (\sin (e+f x)+1))^{5/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}{3315 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\sec (e+f x) (a (\sin (e+f x)+1))^{5/2} (g \cos (e+f x))^{3/2} \left(-\frac{308 \sin \left(\frac{1}{2} (e+f x)\right)}{3315 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}-\frac{308 \sin \left(\frac{1}{2} (e+f x)\right)}{3315 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{2344 \sin \left(\frac{1}{2} (e+f x)\right)}{1989 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}-\frac{592 \sin \left(\frac{1}{2} (e+f x)\right)}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{32 \sin \left(\frac{1}{2} (e+f x)\right)}{17 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}-\frac{154}{3315 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{1172}{1989 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}-\frac{296}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{16}{17 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8}-\frac{154}{3315}\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}{f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\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{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{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,"(154*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(a*(1 + Sin[e + f*x]))^(5/2))/(3315*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(11/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(-154/3315 + 16/(17*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) - 296/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) + 1172/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) - 154/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (32*Sin[(e + f*x)/2])/(17*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) - (592*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) + (2344*Sin[(e + f*x)/2])/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (308*Sin[(e + f*x)/2])/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) - (308*Sin[(e + f*x)/2])/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(5/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(11/2))","A",1
115,1,600,414,6.6792752,"\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","Integrate[((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 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a (\sin (e+f x)+1))^{5/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{13}}{3315 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{\sec (e+f x) (a (\sin (e+f x)+1))^{5/2} (g \cos (e+f x))^{3/2} \left(-\frac{44 \sin \left(\frac{1}{2} (e+f x)\right)}{3315 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}-\frac{44 \sin \left(\frac{1}{2} (e+f x)\right)}{3315 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}-\frac{44 \sin \left(\frac{1}{2} (e+f x)\right)}{1989 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{168 \sin \left(\frac{1}{2} (e+f x)\right)}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{240 \sin \left(\frac{1}{2} (e+f x)\right)}{119 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}+\frac{32 \sin \left(\frac{1}{2} (e+f x)\right)}{21 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}-\frac{22}{3315 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{22}{1989 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{84}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{120}{119 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8}+\frac{16}{21 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{10}}-\frac{22}{3315}\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{13}}{f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}","\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{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{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,"(22*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^13*(a*(1 + Sin[e + f*x]))^(5/2))/(3315*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(13/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^13*(-22/3315 + 16/(21*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10) - 120/(119*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) + 84/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) - 22/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) - 22/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (32*Sin[(e + f*x)/2])/(21*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11) - (240*Sin[(e + f*x)/2])/(119*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) + (168*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) - (44*Sin[(e + f*x)/2])/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (44*Sin[(e + f*x)/2])/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) - (44*Sin[(e + f*x)/2])/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(5/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(13/2))","A",1
116,1,226,463,3.5233649,"\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","Integrate[(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{a^3 c^2 (\sin (e+f x)-1)^2 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(\sqrt{\cos (e+f x)} (-3794 \sin (2 (e+f x))-800 \sin (4 (e+f x))-90 \sin (6 (e+f x))+1365 \cos (e+f x)+819 \cos (3 (e+f x))+273 \cos (5 (e+f x))+39 \cos (7 (e+f x)))-14784 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{18720 f \cos ^{\frac{3}{2}}(e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\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{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{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{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{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{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{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,"-1/18720*(a^3*c^2*(g*Cos[e + f*x])^(3/2)*(-1 + Sin[e + f*x])^2*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-14784*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(1365*Cos[e + f*x] + 819*Cos[3*(e + f*x)] + 273*Cos[5*(e + f*x)] + 39*Cos[7*(e + f*x)] - 3794*Sin[2*(e + f*x)] - 800*Sin[4*(e + f*x)] - 90*Sin[6*(e + f*x)])))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7)","A",1
117,1,212,409,2.9212758,"\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","Integrate[(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{a^3 c (\sin (e+f x)-1) (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(\sqrt{\cos (e+f x)} (-1507 \sin (2 (e+f x))-88 \sin (4 (e+f x))+33 \sin (6 (e+f x))+1560 \cos (e+f x)+780 \cos (3 (e+f x))+156 \cos (5 (e+f x)))-7392 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{6864 f \cos ^{\frac{3}{2}}(e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\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{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{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 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,"(a^3*c*(g*Cos[e + f*x])^(3/2)*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-7392*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(1560*Cos[e + f*x] + 780*Cos[3*(e + f*x)] + 156*Cos[5*(e + f*x)] - 1507*Sin[2*(e + f*x)] - 88*Sin[4*(e + f*x)] + 33*Sin[6*(e + f*x)])))/(6864*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7)","A",1
118,1,360,343,4.3217207,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]],x]","\frac{\sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(-\frac{a^3 \sqrt{\cos (e+f x)} \sqrt{a (\sin (e+f x)+1)} (1374 \cos (e+f x)+423 \cos (3 (e+f x))-7 (44 \sin (2 (e+f x))-22 \sin (4 (e+f x))+3 \cos (5 (e+f x))-528 \cot (e)))}{1848 f \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{(2+2 i) a^4 e^{\frac{1}{2} i (e+f x)} \left(e^{i (e+f x)}+i\right) \left(e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)\right)^{3/2} \left(\left(-1+e^{2 i e}\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (e+f x)}\right)+\sqrt{1+e^{2 i (e+f x)}}\right)}{\left(-1+e^{2 i e}\right) f \left(1+e^{2 i (e+f x)}\right)^{3/2} \sqrt{-i a e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2}}\right)}{\cos ^{\frac{3}{2}}(e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\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^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^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 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,"((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(((2 + 2*I)*a^4*E^((I/2)*(e + f*x))*(I + E^(I*(e + f*x)))*((1 + E^((2*I)*(e + f*x)))/E^(I*(e + f*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(e + f*x))] + (-1 + E^((2*I)*e))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(e + f*x))]))/((-1 + E^((2*I)*e))*Sqrt[((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*(1 + E^((2*I)*(e + f*x)))^(3/2)*f) - (a^3*Sqrt[Cos[e + f*x]]*Sqrt[a*(1 + Sin[e + f*x])]*(1374*Cos[e + f*x] + 423*Cos[3*(e + f*x)] - 7*(3*Cos[5*(e + f*x)] - 528*Cot[e] + 44*Sin[2*(e + f*x)] - 22*Sin[4*(e + f*x)])))/(1848*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))))/(Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","C",1
119,1,181,288,3.7142643,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{a^3 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sqrt{\cos (e+f x)} (350 \sin (2 (e+f x))-7 \sin (4 (e+f x))+1128 \cos (e+f x)-72 \cos (3 (e+f x)))-1848 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{252 f \cos ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\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^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{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{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,"-1/252*(a^3*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*(-1848*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(1128*Cos[e + f*x] - 72*Cos[3*(e + f*x)] + 350*Sin[2*(e + f*x)] - 7*Sin[4*(e + f*x)])))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*Sqrt[c - c*Sin[e + f*x]])","A",1
120,1,284,294,6.5212318,"\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","Integrate[((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{\sec (e+f x) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin (2 (e+f x))+\frac{109}{14} \cos (e+f x)-\frac{1}{14} \cos (3 (e+f x))+\frac{64 \sin \left(\frac{1}{2} (e+f x)\right)}{\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)}+32\right)}{f (c-c \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{66 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}{f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\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^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{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{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,"(-66*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(3/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2)*(32 + (109*Cos[e + f*x])/14 - Cos[3*(e + f*x)]/14 + (64*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + Sin[2*(e + f*x)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(3/2))","A",1
121,1,267,298,4.6379204,"\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","Integrate[((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{a^3 \sqrt{a (\sin (e+f x)+1)} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sqrt{\cos (e+f x)} \left(487 \sin \left(\frac{1}{2} (e+f x)\right)-633 \sin \left(\frac{3}{2} (e+f x)\right)-17 \sin \left(\frac{5}{2} (e+f x)\right)-\sin \left(\frac{7}{2} (e+f x)\right)+487 \cos \left(\frac{1}{2} (e+f x)\right)+633 \cos \left(\frac{3}{2} (e+f x)\right)-17 \cos \left(\frac{5}{2} (e+f x)\right)+\cos \left(\frac{7}{2} (e+f x)\right)\right)-1848 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{20 c^2 f (\sin (e+f x)-1)^2 \cos ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","-\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{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{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{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,"-1/20*(a^3*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sqrt[a*(1 + Sin[e + f*x])]*(-1848*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + Sqrt[Cos[e + f*x]]*(487*Cos[(e + f*x)/2] + 633*Cos[(3*(e + f*x))/2] - 17*Cos[(5*(e + f*x))/2] + Cos[(7*(e + f*x))/2] + 487*Sin[(e + f*x)/2] - 633*Sin[(3*(e + f*x))/2] - 17*Sin[(5*(e + f*x))/2] - Sin[(7*(e + f*x))/2])))/(c^2*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]])","A",1
122,1,406,300,6.5820622,"\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","Integrate[((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{\sec (e+f x) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7 \left(\frac{2}{3} \cos (e+f x)+\frac{224 \sin \left(\frac{1}{2} (e+f x)\right)}{3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}-\frac{64 \sin \left(\frac{1}{2} (e+f x)\right)}{3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{64 \sin \left(\frac{1}{2} (e+f x)\right)}{9 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}-\frac{32}{3 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{32}{9 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{112}{3}\right)}{f (c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{154 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}{3 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{7/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\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{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{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{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,"(-154*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(7/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(112/3 + (2*Cos[e + f*x])/3 + 32/(9*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) - 32/(3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(9*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (64*Sin[(e + f*x)/2])/(3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (224*Sin[(e + f*x)/2])/(3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(7/2))","A",1
123,1,464,300,6.632315,"\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","Integrate[((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{154 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{13 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{\sec (e+f x) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(-\frac{256 \sin \left(\frac{1}{2} (e+f x)\right)}{13 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{160 \sin \left(\frac{1}{2} (e+f x)\right)}{13 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}-\frac{448 \sin \left(\frac{1}{2} (e+f x)\right)}{39 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{64 \sin \left(\frac{1}{2} (e+f x)\right)}{13 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{80}{13 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{224}{39 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{32}{13 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{128}{13}\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}{f (c-c \sin (e+f x))^{9/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\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{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{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,"(154*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(13*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(9/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(-128/13 + 32/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) - 224/(39*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) + 80/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) - (448*Sin[(e + f*x)/2])/(39*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (160*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) - (256*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(9/2))","A",1
124,1,532,357,6.6836798,"\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","Integrate[((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{\sec (e+f x) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11} \left(\frac{308 \sin \left(\frac{1}{2} (e+f x)\right)}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}-\frac{576 \sin \left(\frac{1}{2} (e+f x)\right)}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{4192 \sin \left(\frac{1}{2} (e+f x)\right)}{663 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}-\frac{1728 \sin \left(\frac{1}{2} (e+f x)\right)}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{64 \sin \left(\frac{1}{2} (e+f x)\right)}{17 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}-\frac{288}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{2096}{663 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}-\frac{864}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{32}{17 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8}+\frac{154}{221}\right)}{f (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{154 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}{221 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{11/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\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{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{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,"(-154*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(a*(1 + Sin[e + f*x]))^(7/2))/(221*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(154/221 + 32/(17*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) - 864/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) + 2096/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) - 288/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(17*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) - (1728*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) + (4192*Sin[(e + f*x)/2])/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (576*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (308*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2))","A",1
125,1,600,414,6.7860706,"\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","Integrate[((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{\sec (e+f x) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{13} \left(\frac{44 \sin \left(\frac{1}{2} (e+f x)\right)}{663 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{44 \sin \left(\frac{1}{2} (e+f x)\right)}{663 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}-\frac{2432 \sin \left(\frac{1}{2} (e+f x)\right)}{1989 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}+\frac{928 \sin \left(\frac{1}{2} (e+f x)\right)}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{704 \sin \left(\frac{1}{2} (e+f x)\right)}{119 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}+\frac{64 \sin \left(\frac{1}{2} (e+f x)\right)}{21 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}+\frac{22}{663 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{1216}{1989 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}+\frac{464}{221 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}-\frac{352}{119 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8}+\frac{32}{21 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{10}}+\frac{22}{663}\right)}{f (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{22 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{13}}{663 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{13/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\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{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{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,"(-22*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^13*(a*(1 + Sin[e + f*x]))^(7/2))/(663*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^13*(22/663 + 32/(21*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10) - 352/(119*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) + 464/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) - 1216/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) + 22/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(21*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11) - (704*Sin[(e + f*x)/2])/(119*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) + (928*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) - (2432*Sin[(e + f*x)/2])/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (44*Sin[(e + f*x)/2])/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (44*Sin[(e + f*x)/2])/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(13/2))","A",1
126,1,668,471,6.8654295,"\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","Integrate[((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{\sec (e+f x) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{15} \left(\frac{44 \sin \left(\frac{1}{2} (e+f x)\right)}{5525 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{44 \sin \left(\frac{1}{2} (e+f x)\right)}{5525 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{44 \sin \left(\frac{1}{2} (e+f x)\right)}{3315 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}-\frac{4288 \sin \left(\frac{1}{2} (e+f x)\right)}{5525 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}+\frac{9312 \sin \left(\frac{1}{2} (e+f x)\right)}{2975 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}-\frac{832 \sin \left(\frac{1}{2} (e+f x)\right)}{175 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{11}}+\frac{64 \sin \left(\frac{1}{2} (e+f x)\right)}{25 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{13}}+\frac{22}{5525 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{22}{3315 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4}-\frac{2144}{5525 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^6}+\frac{4656}{2975 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^8}-\frac{416}{175 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{10}}+\frac{32}{25 \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{12}}+\frac{22}{5525}\right)}{f (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{22 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{15}}{5525 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{15/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\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{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{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,"(-22*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^15*(a*(1 + Sin[e + f*x]))^(7/2))/(5525*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^15*(22/5525 + 32/(25*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^12) - 416/(175*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10) + 4656/(2975*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) - 2144/(5525*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) + 22/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) + 22/(5525*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(25*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^13) - (832*Sin[(e + f*x)/2])/(175*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11) + (9312*Sin[(e + f*x)/2])/(2975*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) - (4288*Sin[(e + f*x)/2])/(5525*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) + (44*Sin[(e + f*x)/2])/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (44*Sin[(e + f*x)/2])/(5525*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (44*Sin[(e + f*x)/2])/(5525*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2))","A",1
127,1,174,234,1.4594515,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/Sqrt[a + a*Sin[e + f*x]],x]","\frac{c^2 (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(924 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)+\sqrt{\cos (e+f x)} (515 \cos (e+f x)-3 (42 \sin (2 (e+f x))+5 \cos (3 (e+f x))))\right)}{210 f \cos ^{\frac{3}{2}}(e+f x) \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","\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^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{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{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,"(c^2*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]]*(924*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(515*Cos[e + f*x] - 3*(5*Cos[3*(e + f*x)] + 42*Sin[2*(e + f*x)]))))/(210*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*Sqrt[a*(1 + Sin[e + f*x])])","A",1
128,1,157,180,0.6770864,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2))/Sqrt[a + a*Sin[e + f*x]],x]","-\frac{c (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(42 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)+\sqrt{\cos (e+f x)} (20 \cos (e+f x)-3 \sin (2 (e+f x)))\right)}{15 f \cos ^{\frac{3}{2}}(e+f x) \sqrt{a (\sin (e+f x)+1)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}","\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,"-1/15*(c*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]]*(42*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(20*Cos[e + f*x] - 3*Sin[2*(e + f*x)])))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])])","A",1
129,1,215,122,2.4980715,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]],x]","\frac{\sqrt{e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)} \sqrt{-i a e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2} \left(12 i e^{i (e+f x)} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (e+f x)}\right)+\sqrt{1+e^{2 i (e+f x)}} \left(-6 i e^{i (e+f x)}+e^{2 i (e+f x)}+1\right)\right) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2}}{3 a f \left(1+e^{2 i (e+f x)}\right)^{3/2} \cos ^{\frac{3}{2}}(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,"(Sqrt[((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*Sqrt[(1 + E^((2*I)*(e + f*x)))/E^(I*(e + f*x))]*(g*Cos[e + f*x])^(3/2)*(Sqrt[1 + E^((2*I)*(e + f*x))]*(1 - (6*I)*E^(I*(e + f*x)) + E^((2*I)*(e + f*x))) + (12*I)*E^(I*(e + f*x))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(e + f*x))])*Sqrt[c - c*Sin[e + f*x]])/(3*a*(1 + E^((2*I)*(e + f*x)))^(3/2)*f*Cos[e + f*x]^(3/2))","C",1
130,1,111,68,0.3601339,"\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]),x]","\frac{2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{f \cos ^{\frac{3}{2}}(e+f x) \sqrt{a (\sin (e+f x)+1)} \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*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(f*Cos[e + f*x]^(3/2)*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])","A",1
131,1,148,121,0.7260959,"\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","Integrate[(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))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\sqrt{\cos (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)-\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{c f \cos ^{\frac{3}{2}}(e+f x) \sqrt{a (\sin (e+f x)+1)} \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])^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(EllipticE[(e + f*x)/2, 2]*(-Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + Sqrt[Cos[e + f*x]]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(c*f*Cos[e + f*x]^(3/2)*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])","A",1
132,1,204,179,1.523216,"\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","Integrate[(g*Cos[e + f*x])^(3/2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)),x]","\frac{(g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\sqrt{\cos (e+f x)} \left(4 \sin ^3\left(\frac{1}{2} (e+f x)\right)+3 \cos \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{3}{2} (e+f x)\right)\right)-2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{5 c^2 f (\sin (e+f x)-1)^2 \cos ^{\frac{3}{2}}(e+f x) \sqrt{a (\sin (e+f x)+1)} \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 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,"((g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-2*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + Sqrt[Cos[e + f*x]]*(3*Cos[(e + f*x)/2] + Cos[(3*(e + f*x))/2] + 4*Sin[(e + f*x)/2]^3)))/(5*c^2*f*Cos[e + f*x]^(3/2)*(-1 + Sin[e + f*x])^2*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])","A",1
133,1,240,233,2.1492453,"\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","Integrate[(g*Cos[e + f*x])^(3/2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)),x]","\frac{(g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\sqrt{\cos (e+f x)} \left(-32 \sin \left(\frac{1}{2} (e+f x)\right)+15 \sin \left(\frac{3}{2} (e+f x)\right)+3 \sin \left(\frac{5}{2} (e+f x)\right)-32 \cos \left(\frac{1}{2} (e+f x)\right)-15 \cos \left(\frac{3}{2} (e+f x)\right)+3 \cos \left(\frac{5}{2} (e+f x)\right)\right)+12 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5\right)}{90 c^3 f (\sin (e+f x)-1)^3 \cos ^{\frac{3}{2}}(e+f x) \sqrt{a (\sin (e+f x)+1)} \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)}}{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^2 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}}{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,"((g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(12*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + Sqrt[Cos[e + f*x]]*(-32*Cos[(e + f*x)/2] - 15*Cos[(3*(e + f*x))/2] + 3*Cos[(5*(e + f*x))/2] - 32*Sin[(e + f*x)/2] + 15*Sin[(3*(e + f*x))/2] + 3*Sin[(5*(e + f*x))/2])))/(90*c^3*f*Cos[e + f*x]^(3/2)*(-1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])","A",1
134,1,282,294,6.531808,"\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","Integrate[((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{\sec (e+f x) (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sin (2 (e+f x))-\frac{109}{14} \cos (e+f x)+\frac{1}{14} \cos (3 (e+f x))+\frac{64 \sin \left(\frac{1}{2} (e+f x)\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}-32\right)}{f (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}-\frac{66 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}{f \cos ^{\frac{3}{2}}(e+f x) (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^7}","-\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^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{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{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,"(-66*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(7/2))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(3/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(7/2)*(-32 - (109*Cos[e + f*x])/14 + Cos[3*(e + f*x)]/14 + (64*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + Sin[2*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(3/2))","A",1
135,1,238,241,4.8060688,"\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","Integrate[((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{c^2 (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sqrt{\cos (e+f x)} \left(-520 \sin \left(\frac{1}{2} (e+f x)\right)+37 \sin \left(\frac{3}{2} (e+f x)\right)-3 \sin \left(\frac{5}{2} (e+f x)\right)+520 \cos \left(\frac{1}{2} (e+f x)\right)+37 \cos \left(\frac{3}{2} (e+f x)\right)+3 \cos \left(\frac{5}{2} (e+f x)\right)\right)+924 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{30 f \cos ^{\frac{3}{2}}(e+f x) (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^5}","-\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{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{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{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,"-1/30*(c^2*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]]*(924*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + Sqrt[Cos[e + f*x]]*(520*Cos[(e + f*x)/2] + 37*Cos[(3*(e + f*x))/2] + 3*Cos[(5*(e + f*x))/2] - 520*Sin[(e + f*x)/2] + 37*Sin[(3*(e + f*x))/2] - 3*Sin[(5*(e + f*x))/2])))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(3/2))","A",1
136,1,200,182,1.5515017,"\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","Integrate[((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{2 c (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sqrt{\cos (e+f x)} \left(\cos \left(\frac{1}{2} (e+f x)\right) (\cos (e+f x)+12)+\sin \left(\frac{1}{2} (e+f x)\right) (\cos (e+f x)-12)\right)+21 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{3 f \cos ^{\frac{3}{2}}(e+f x) (a (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}","-\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,"(2*c*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(21*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + Sqrt[Cos[e + f*x]]*(Cos[(e + f*x)/2]*(12 + Cos[e + f*x]) + (-12 + Cos[e + f*x])*Sin[(e + f*x)/2]))*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(3*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(3/2))","A",1
137,1,213,123,1.8699828,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(3/2),x]","\frac{2 g \sqrt{g e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)} \left(2 e^{2 i (e+f x)} \left(e^{i (e+f x)}+i\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)-\left(5 e^{i (e+f x)}+i\right) \sqrt{1+e^{2 i (e+f x)}}\right) \sqrt{c-c \sin (e+f x)}}{a f \left(e^{i (e+f x)}-i\right) \sqrt{1+e^{2 i (e+f x)}} \sqrt{-i a e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2}}","-\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,"(2*g*Sqrt[((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x))]*(-((I + 5*E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]) + 2*E^((2*I)*(e + f*x))*(I + E^(I*(e + f*x)))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[c - c*Sin[e + f*x]])/(a*(-I + E^(I*(e + f*x)))*Sqrt[((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*Sqrt[1 + E^((2*I)*(e + f*x))]*f)","C",1
138,1,170,121,0.6724369,"\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","Integrate[(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))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sqrt{\cos (e+f x)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{f \cos ^{\frac{3}{2}}(e+f x) (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)}}","-\frac{2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{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])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(Sqrt[Cos[e + f*x]]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(f*Cos[e + f*x]^(3/2)*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
139,1,92,176,0.7256088,"\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","Integrate[(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} \left(\sin (e+f x)-\sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{c f g (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{3/2} \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)*(-(Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2]) + Sin[e + f*x]))/(c*f*g*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
140,1,134,237,1.1078942,"\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","Integrate[(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{\sqrt{\cos (e+f x)} (g \cos (e+f x))^{3/2} \left(\sqrt{\cos (e+f x)} (-6 \sin (e+f x)-3 \cos (2 (e+f x))+1)+E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (6 \cos (e+f x)-3 \sin (2 (e+f x)))\right)}{5 c^2 f (\sin (e+f x)-1)^2 (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f 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}}",1,"-1/5*(Sqrt[Cos[e + f*x]]*(g*Cos[e + f*x])^(3/2)*(Sqrt[Cos[e + f*x]]*(1 - 3*Cos[2*(e + f*x)] - 6*Sin[e + f*x]) + EllipticE[(e + f*x)/2, 2]*(6*Cos[e + f*x] - 3*Sin[2*(e + f*x)])))/(c^2*f*(-1 + Sin[e + f*x])^2*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
141,1,155,294,1.4488412,"\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","Integrate[(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{\sqrt{\cos (e+f x)} (g \cos (e+f x))^{3/2} \left(\sqrt{\cos (e+f x)} (-17 \sin (e+f x)+3 \sin (3 (e+f x))-12 \cos (2 (e+f x))+4)+3 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (-4 \sin (2 (e+f x))+5 \cos (e+f x)-\cos (3 (e+f x)))\right)}{18 c^3 f (\sin (e+f x)-1)^3 (a (\sin (e+f x)+1))^{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)}}{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^2 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}}{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,"(Sqrt[Cos[e + f*x]]*(g*Cos[e + f*x])^(3/2)*(3*EllipticE[(e + f*x)/2, 2]*(5*Cos[e + f*x] - Cos[3*(e + f*x)] - 4*Sin[2*(e + f*x)]) + Sqrt[Cos[e + f*x]]*(4 - 12*Cos[2*(e + f*x)] - 17*Sin[e + f*x] + 3*Sin[3*(e + f*x)])))/(18*c^3*f*(-1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
142,1,356,357,6.7123412,"\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","Integrate[((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{1254 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (c-c \sin (e+f x))^{9/2} (g \cos (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{5 f \cos ^{\frac{3}{2}}(e+f x) (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}+\frac{\sec (e+f x) (c-c \sin (e+f x))^{9/2} (g \cos (e+f x))^{3/2} \left(-\frac{7}{5} \sin (2 (e+f x))+\frac{221}{14} \cos (e+f x)-\frac{1}{14} \cos (3 (e+f x))-\frac{1472 \sin \left(\frac{1}{2} (e+f x)\right)}{5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}-\frac{64}{5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{128 \sin \left(\frac{1}{2} (e+f x)\right)}{5 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{736}{5}\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^5}{f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^9}","\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^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{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{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,"(1254*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2))/(5*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2)*(736/5 + (221*Cos[e + f*x])/14 - Cos[3*(e + f*x)]/14 + (128*Sin[(e + f*x)/2])/(5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) - 64/(5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) - (1472*Sin[(e + f*x)/2])/(5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - (7*Sin[2*(e + f*x)])/5))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
143,1,250,298,4.6805139,"\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","Integrate[((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{c^3 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sqrt{\cos (e+f x)} \left(-487 \sin \left(\frac{1}{2} (e+f x)\right)+633 \sin \left(\frac{3}{2} (e+f x)\right)+17 \sin \left(\frac{5}{2} (e+f x)\right)+\sin \left(\frac{7}{2} (e+f x)\right)+487 \cos \left(\frac{1}{2} (e+f x)\right)+633 \cos \left(\frac{3}{2} (e+f x)\right)-17 \cos \left(\frac{5}{2} (e+f x)\right)+\cos \left(\frac{7}{2} (e+f x)\right)\right)+1848 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{20 f \cos ^{\frac{3}{2}}(e+f x) (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\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{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{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{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,"(c^3*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[c - c*Sin[e + f*x]]*(1848*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + Sqrt[Cos[e + f*x]]*(487*Cos[(e + f*x)/2] + 633*Cos[(3*(e + f*x))/2] - 17*Cos[(5*(e + f*x))/2] + Cos[(7*(e + f*x))/2] - 487*Sin[(e + f*x)/2] + 633*Sin[(3*(e + f*x))/2] + 17*Sin[(5*(e + f*x))/2] + Sin[(7*(e + f*x))/2])))/(20*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
144,1,230,243,2.3178504,"\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","Integrate[((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{c^2 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sqrt{\cos (e+f x)} \left(-226 \sin \left(\frac{1}{2} (e+f x)\right)+327 \sin \left(\frac{3}{2} (e+f x)\right)+5 \sin \left(\frac{5}{2} (e+f x)\right)+226 \cos \left(\frac{1}{2} (e+f x)\right)+327 \cos \left(\frac{3}{2} (e+f x)\right)-5 \cos \left(\frac{5}{2} (e+f x)\right)\right)+924 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{30 f \cos ^{\frac{3}{2}}(e+f x) (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"(c^2*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[c - c*Sin[e + f*x]]*(924*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + Sqrt[Cos[e + f*x]]*(226*Cos[(e + f*x)/2] + 327*Cos[(3*(e + f*x))/2] - 5*Cos[(5*(e + f*x))/2] - 226*Sin[(e + f*x)/2] + 327*Sin[(3*(e + f*x))/2] + 5*Sin[(5*(e + f*x))/2])))/(30*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
145,1,180,186,1.159611,"\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","Integrate[((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{c \sqrt{\cos (e+f x)} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(8 \sqrt{\cos (e+f x)} \left(-\sin \left(\frac{1}{2} (e+f x)\right)+2 \sin \left(\frac{3}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)+2 \cos \left(\frac{3}{2} (e+f x)\right)\right)+42 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{5 f (a (\sin (e+f x)+1))^{5/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}","\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,"(c*Sqrt[Cos[e + f*x]]*(g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(42*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 8*Sqrt[Cos[e + f*x]]*(Cos[(e + f*x)/2] + 2*Cos[(3*(e + f*x))/2] - Sin[(e + f*x)/2] + 2*Sin[(3*(e + f*x))/2])))/(5*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(5/2))","A",1
146,1,230,182,2.099474,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(5/2),x]","\frac{4 i g \sqrt{g e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)} \left(e^{i (e+f x)} \left(e^{i (e+f x)}+i\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right)+\left(-4 i e^{i (e+f x)}-3 e^{2 i (e+f x)}+5\right) \sqrt{1+e^{2 i (e+f x)}}\right) \sqrt{c-c \sin (e+f x)}}{5 a f \left(e^{i (e+f x)}-i\right) \sqrt{1+e^{2 i (e+f x)}} \left(-i a e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2\right)^{3/2}}","\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*I)/5)*g*Sqrt[((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x))]*((5 - (4*I)*E^(I*(e + f*x)) - 3*E^((2*I)*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))] + E^(I*(e + f*x))*(I + E^(I*(e + f*x)))^3*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[c - c*Sin[e + f*x]])/(a*(-I + E^(I*(e + f*x)))*(((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x)))^(3/2)*Sqrt[1 + E^((2*I)*(e + f*x))]*f)","C",1
147,1,189,179,1.5183713,"\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","Integrate[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]),x]","-\frac{(g \cos (e+f x))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \left(\sqrt{\cos (e+f x)} \left(-4 \sin ^3\left(\frac{1}{2} (e+f x)\right)+3 \cos \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{3}{2} (e+f x)\right)\right)+2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3\right)}{5 f \cos ^{\frac{3}{2}}(e+f x) (a (\sin (e+f x)+1))^{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,"-1/5*((g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(2*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + Sqrt[Cos[e + f*x]]*(3*Cos[(e + f*x)/2] + Cos[(3*(e + f*x))/2] - 4*Sin[(e + f*x)/2]^3)))/(f*Cos[e + f*x]^(3/2)*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
148,1,133,237,1.0589485,"\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","Integrate[(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{\sqrt{\cos (e+f x)} (g \cos (e+f x))^{3/2} \left(\sqrt{\cos (e+f x)} (-6 \sin (e+f x)+3 \cos (2 (e+f x))-1)+3 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) (\sin (2 (e+f x))+2 \cos (e+f x))\right)}{5 c f (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f 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}}",1,"(Sqrt[Cos[e + f*x]]*(g*Cos[e + f*x])^(3/2)*(Sqrt[Cos[e + f*x]]*(-1 + 3*Cos[2*(e + f*x)] - 6*Sin[e + f*x]) + 3*EllipticE[(e + f*x)/2, 2]*(2*Cos[e + f*x] + Sin[2*(e + f*x)])))/(5*c*f*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
149,1,104,291,1.1073056,"\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","Integrate[(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{\sec ^3(e+f x) (g \cos (e+f x))^{3/2} \left(7 \sin (e+f x)+3 \sin (3 (e+f x))-12 \cos ^{\frac{5}{2}}(e+f x) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{10 a^2 c^2 f \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f 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}}",1,"((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]^3*(-12*Cos[e + f*x]^(5/2)*EllipticE[(e + f*x)/2, 2] + 7*Sin[e + f*x] + 3*Sin[3*(e + f*x)]))/(10*a^2*c^2*f*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])","A",1
150,1,171,350,6.0490707,"\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","Integrate[(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{\sqrt{\cos (e+f x)} (g \cos (e+f x))^{3/2} \left(\sqrt{\cos (e+f x)} (98 \sin (e+f x)+42 \sin (3 (e+f x))+28 \cos (2 (e+f x))+21 \cos (4 (e+f x))-9)+42 E\left(\left.\frac{1}{2} (e+f x)\right|2\right) \left(-3 \cos (e+f x)-\cos (3 (e+f x))+4 \sin (e+f x) \cos ^3(e+f x)\right)\right)}{180 c^3 f (\sin (e+f x)-1)^3 (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)}}","-\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^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\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,"-1/180*(Sqrt[Cos[e + f*x]]*(g*Cos[e + f*x])^(3/2)*(42*EllipticE[(e + f*x)/2, 2]*(-3*Cos[e + f*x] - Cos[3*(e + f*x)] + 4*Cos[e + f*x]^3*Sin[e + f*x]) + Sqrt[Cos[e + f*x]]*(-9 + 28*Cos[2*(e + f*x)] + 21*Cos[4*(e + f*x)] + 98*Sin[e + f*x] + 42*Sin[3*(e + f*x)])))/(c^3*f*(-1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]])","A",1
151,1,126,119,2.15348,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","-\frac{8 g \cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \sqrt{g \cos (e+f x)} (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n \csc ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{m+n+\frac{3}{2}} \, _2F_1\left(n+\frac{5}{4},m+n+\frac{5}{2};n+\frac{9}{4};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f (4 n+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,"(-8*g*Sqrt[g*Cos[e + f*x]]*Cos[(2*e + Pi + 2*f*x)/4]^2*(Csc[(2*e + Pi + 2*f*x)/4]^2)^(3/2 + m + n)*Hypergeometric2F1[5/4 + n, 5/2 + m + n, 9/4 + n, -Tan[(2*e - Pi + 2*f*x)/4]^2]*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n)/(f*(5 + 4*n))","A",1
152,-1,0,93,180.0069097,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3,x]","\text{\$Aborted}","-\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,"$Aborted","F",-1
153,-1,0,93,180.0102496,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^2 \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^2,x]","\text{\$Aborted}","-\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,"$Aborted","F",-1
154,0,0,91,169.6010033,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x)) \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x]),x]","\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x)) \, dx","-\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,"Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x]), x]","F",-1
155,1,85,88,0.1531748,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m,x]","-\frac{2^{m+\frac{9}{4}} (g \cos (e+f x))^{5/2} (\sin (e+f x)+1)^{-m-\frac{5}{4}} (a (\sin (e+f x)+1))^m \, _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,"-1/5*(2^(9/4 + m)*(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])^(-5/4 - m)*(a*(1 + Sin[e + f*x]))^m)/(f*g)","A",1
156,1,84,84,0.1193646,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx","Integrate[((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)+1))^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*(1 + Sin[e + f*x]))^m)/(c*f))","A",1
157,1,96,93,0.2036717,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^2} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^2,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)+1))^m \, _2F_1\left(-\frac{3}{4},-m-\frac{1}{4};\frac{1}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{3 c^2 f (\sin (e+f x)-1)}","\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,"-1/3*(2^(9/4 + m)*g*Sqrt[g*Cos[e + f*x]]*Hypergeometric2F1[-3/4, -1/4 - m, 1/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a*(1 + Sin[e + f*x]))^m)/(c^2*f*(-1 + Sin[e + f*x]))","A",1
158,1,96,93,0.2096399,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^3,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)+1))^m \, _2F_1\left(-\frac{7}{4},-m-\frac{1}{4};-\frac{3}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{7 c^3 f (\sin (e+f x)-1)^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*Sqrt[g*Cos[e + f*x]]*Hypergeometric2F1[-7/4, -1/4 - m, -3/4, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/4 - m)*(a*(1 + Sin[e + f*x]))^m)/(7*c^3*f*(-1 + Sin[e + f*x])^2)","A",1
159,-1,0,114,180.0028011,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2),x]","\text{\$Aborted}","-\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,"$Aborted","F",-1
160,-1,0,112,180.0038528,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2),x]","\text{\$Aborted}","-\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,"$Aborted","F",-1
161,-1,0,109,180.0005023,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]],x]","\text{\$Aborted}","-\frac{a 2^{m+\frac{9}{4}} \sec (e+f x) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{7/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-1} \, _2F_1\left(\frac{7}{4},-m-\frac{1}{4};\frac{11}{4};\frac{1}{2} (1-\sin (e+f x))\right)}{7 f g^2}",1,"$Aborted","F",-1
162,-1,0,106,180.010585,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/Sqrt[c - c*Sin[e + f*x]],x]","\text{\$Aborted}","-\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,"$Aborted","F",-1
163,0,0,106,81.6798036,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(3/2),x]","\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","\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,"Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(3/2), x]","F",-1
164,1,2320,114,53.6582215,"\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","Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(5/2),x]","\text{Result too large to show}","\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,"((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Cos[(-e + Pi/2 - f*x)/2]*(g*Cos[e + f*x])^(3/2)*(5*AppellF1[3/4, -1/2 - 2*m, 2*m, 7/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 3*AppellF1[-5/4, -1/2 - 2*m, 2*m, -1/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^4)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a + a*Sin[e + f*x])^m*Tan[(-e + Pi/2 - f*x)/4])/(60*Sqrt[2]*f*Sqrt[Cos[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)*(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^4*Sqrt[2 - 2*Tan[(-e + Pi/2 - f*x)/4]^2]*(-1/240*((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Sqrt[Cos[e + f*x]]*(5*AppellF1[3/4, -1/2 - 2*m, 2*m, 7/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 3*AppellF1[-5/4, -1/2 - 2*m, 2*m, -1/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^4)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2 + 2*m)*Tan[(-e + Pi/2 - f*x)/4]^2)/(2 - 2*Tan[(-e + Pi/2 - f*x)/4]^2)^(3/2) - ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Sqrt[Cos[e + f*x]]*(5*AppellF1[3/4, -1/2 - 2*m, 2*m, 7/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 3*AppellF1[-5/4, -1/2 - 2*m, 2*m, -1/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^4)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2 + 2*m))/(480*Sqrt[2 - 2*Tan[(-e + Pi/2 - f*x)/4]^2]) + (m*(Cos[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m)*Sqrt[Cos[e + f*x]]*(5*AppellF1[3/4, -1/2 - 2*m, 2*m, 7/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 3*AppellF1[-5/4, -1/2 - 2*m, 2*m, -1/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^4)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*Sin[(-e + Pi/2 - f*x)/4]^2)/(120*Sqrt[2 - 2*Tan[(-e + Pi/2 - f*x)/4]^2]) - ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(5*AppellF1[3/4, -1/2 - 2*m, 2*m, 7/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 3*AppellF1[-5/4, -1/2 - 2*m, 2*m, -1/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^4)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*Sin[e + f*x]*Tan[(-e + Pi/2 - f*x)/4])/(240*Sqrt[Cos[e + f*x]]*Sqrt[2 - 2*Tan[(-e + Pi/2 - f*x)/4]^2]) - ((1 + 2*m)*(Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Sqrt[Cos[e + f*x]]*(5*AppellF1[3/4, -1/2 - 2*m, 2*m, 7/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 3*AppellF1[-5/4, -1/2 - 2*m, 2*m, -1/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^4)*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*Tan[(-e + Pi/2 - f*x)/4]^2)/(240*Sqrt[2 - 2*Tan[(-e + Pi/2 - f*x)/4]^2]) - ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Sqrt[Cos[e + f*x]]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(1 + 2*m)*Tan[(-e + Pi/2 - f*x)/4]*(-3*AppellF1[-5/4, -1/2 - 2*m, 2*m, -1/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cot[(-e + Pi/2 - f*x)/4]^3*Csc[(-e + Pi/2 - f*x)/4]^2 + 3*Cot[(-e + Pi/2 - f*x)/4]^4*(-5*m*AppellF1[-1/4, -1/2 - 2*m, 1 + 2*m, 3/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4] + (5*(-1/2 - 2*m)*AppellF1[-1/4, 1/2 - 2*m, 2*m, 3/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/2) + 5*((-3*m*AppellF1[7/4, -1/2 - 2*m, 1 + 2*m, 11/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/7 + (3*(-1/2 - 2*m)*AppellF1[7/4, 1/2 - 2*m, 2*m, 11/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/14)))/(120*Sqrt[2 - 2*Tan[(-e + Pi/2 - f*x)/4]^2])))","C",0
165,1,832,106,13.5918633,"\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/Sqrt[c - c*Sin[e + f*x]],x]","-\frac{88 \sqrt{2} \cos ^6\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \cos (e+f x) (g \cos (e+f x))^{3/2} \csc \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sec ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) (\sin (e+f x) a+a)^m \left(\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-1\right) \left(3 F_1\left(\frac{7}{4};-2 m-\frac{1}{2},2 m+3;\frac{11}{4};\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-7 F_1\left(\frac{3}{4};-2 m-\frac{1}{2},2 m+3;\frac{7}{4};\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right)}{3 f \left(616 F_1\left(\frac{3}{4};-2 m-\frac{1}{2},2 m+3;\frac{7}{4};\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^4\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-4 \sin ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(88 (2 m+3) F_1\left(\frac{7}{4};-2 m-\frac{1}{2},2 m+4;\frac{11}{4};\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+44 (4 m+1) F_1\left(\frac{7}{4};\frac{1}{2}-2 m,2 m+3;\frac{11}{4};\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \cos ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)-28 \left((4 m+6) F_1\left(\frac{11}{4};-2 m-\frac{1}{2},2 m+4;\frac{15}{4};\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)+(4 m+1) F_1\left(\frac{11}{4};\frac{1}{2}-2 m,2 m+3;\frac{15}{4};\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right) \sin ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)+77 F_1\left(\frac{7}{4};-2 m-\frac{1}{2},2 m+3;\frac{11}{4};\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right),-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right) \left(\cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+1\right)\right)\right) \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,"(-88*Sqrt[2]*Cos[(-e + Pi/2 - f*x)/4]^6*Cos[e + f*x]*(g*Cos[e + f*x])^(3/2)*Csc[(-e + Pi/2 - f*x)/2]*Sec[(-e + Pi/2 - f*x)/2]^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a + a*Sin[e + f*x])^m*(-1 + Tan[(-e + Pi/2 - f*x)/4]^2)*(-7*AppellF1[3/4, -1/2 - 2*m, 3 + 2*m, 7/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + 3*AppellF1[7/4, -1/2 - 2*m, 3 + 2*m, 11/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2))/(3*f*(616*AppellF1[3/4, -1/2 - 2*m, 3 + 2*m, 7/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^4 - 4*Sin[(-e + Pi/2 - f*x)/4]^2*(88*(3 + 2*m)*AppellF1[7/4, -1/2 - 2*m, 4 + 2*m, 11/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^2 + 44*(1 + 4*m)*AppellF1[7/4, 1/2 - 2*m, 3 + 2*m, 11/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Cos[(-e + Pi/2 - f*x)/4]^2 + 77*AppellF1[7/4, -1/2 - 2*m, 3 + 2*m, 11/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(1 + Cos[(-e + Pi/2 - f*x)/2]) - 28*((6 + 4*m)*AppellF1[11/4, -1/2 - 2*m, 4 + 2*m, 15/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + (1 + 4*m)*AppellF1[11/4, 1/2 - 2*m, 3 + 2*m, 15/4, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Sin[(-e + Pi/2 - f*x)/4]^2))*Sqrt[c - c*Sin[e + f*x]])","C",0
166,-1,0,106,180.0134636,"\int \frac{(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt{a-a \sin (e+f x)}} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*(c + c*Sin[e + f*x])^m)/Sqrt[a - a*Sin[e + f*x]],x]","\text{\$Aborted}","-\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,"$Aborted","F",-1
167,1,382,123,18.0028569,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-3-m} \, dx","Integrate[(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-4} \sec ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \sec (e+f x) (g \cos (e+f x))^{3/2} \sin ^{-2 m}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \left(1-\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)^{-2 m-\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (-m-3)} \left(\left(16 m^2+8 m-3\right) \cot ^4\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \, _2F_1\left(-2 m-\frac{3}{2},-m-\frac{7}{4};-m-\frac{3}{4};\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)+(4 m+7) \left((4 m+3) \, _2F_1\left(-2 m-\frac{3}{2},\frac{1}{4}-m;\frac{5}{4}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)+2 (4 m-1) \cot ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right) \, _2F_1\left(-2 m-\frac{3}{2},-m-\frac{3}{4};\frac{1}{4}-m;\tan ^2\left(\frac{1}{4} \left(-e-f x+\frac{\pi }{2}\right)\right)\right)\right)\right)}{f (4 m-1) (4 m+3) (4 m+7)}","\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^(-4 - m)*(g*Cos[e + f*x])^(3/2)*((-3 + 8*m + 16*m^2)*Cot[(-e + Pi/2 - f*x)/4]^4*Hypergeometric2F1[-3/2 - 2*m, -7/4 - m, -3/4 - m, Tan[(-e + Pi/2 - f*x)/4]^2] + (7 + 4*m)*(2*(-1 + 4*m)*Cot[(-e + Pi/2 - f*x)/4]^2*Hypergeometric2F1[-3/2 - 2*m, -3/4 - m, 1/4 - m, Tan[(-e + Pi/2 - f*x)/4]^2] + (3 + 4*m)*Hypergeometric2F1[-3/2 - 2*m, 1/4 - m, 5/4 - m, Tan[(-e + Pi/2 - f*x)/4]^2]))*Sec[(-e + Pi/2 - f*x)/4]^2*Sec[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1/2 - 2*m))/(f*(-1 + 4*m)*(3 + 4*m)*(7 + 4*m)*Sin[(-e + Pi/2 - f*x)/2]^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-3 - m)))","B",0
168,1,202,123,6.1958074,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-2-m} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m),x]","\frac{g 2^{-m-1} \csc ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right) \sqrt{g \cos (e+f x)} \cos ^{-2 m}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) \left(1-\tan ^2\left(\frac{1}{8} (2 e+2 f x-\pi )\right)\right)^{-2 m-\frac{1}{2}} (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^{-m} \, _2F_1\left(-2 m-\frac{3}{2},-m-\frac{3}{4};\frac{1}{4}-m;\tan ^2\left(\frac{1}{8} (-2 e-2 f x+\pi )\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{2 (m+2)}}{c^2 f (4 m+3) (\sin (e+f x)-1)^2}","\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 - m)*g*Sqrt[g*Cos[e + f*x]]*Csc[(-2*e + Pi - 2*f*x)/8]^2*Hypergeometric2F1[-3/2 - 2*m, -3/4 - m, 1/4 - m, Tan[(-2*e + Pi - 2*f*x)/8]^2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(2 + m))*(a*(1 + Sin[e + f*x]))^m*(1 - Tan[(2*e - Pi + 2*f*x)/8]^2)^(-1/2 - 2*m))/(c^2*f*(3 + 4*m)*Cos[(2*e + Pi + 2*f*x)/4]^(2*m)*(-1 + Sin[e + f*x])^2*(c - c*Sin[e + f*x])^m)","A",0
169,0,0,120,104.9756944,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m),x]","\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx","\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,"Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m), x]","F",-1
170,0,0,121,16.2961889,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-m} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^m,x]","\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-m} \, dx","\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,"Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^m, x]","F",-1
171,0,0,123,176.6402888,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 - m),x]","\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{1-m} \, dx","\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,"Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 - m), x]","F",-1
172,0,0,123,84.4954551,"\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2-m} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(2 - m),x]","\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{2-m} \, dx","\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,"Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(2 - m), x]","F",-1
173,1,133,135,41.7319541,"\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{2 \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n (g \cos (e+f x))^p \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{m+n+p} \, _2F_1\left(m+n+p+1,\frac{1}{2} (2 n+p+1);\frac{1}{2} (2 n+p+3);-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f (2 n+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*(g*Cos[e + f*x])^p*Hypergeometric2F1[1 + m + n + p, (1 + 2*n + p)/2, (3 + 2*n + p)/2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*(Sec[(2*e - Pi + 2*f*x)/4]^2)^(m + n + p)*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n*Tan[(2*e - Pi + 2*f*x)/4])/(f*(1 + 2*n + p))","A",1
174,1,132,57,84.6349475,"\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","Integrate[(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 2^{m+1} \cos ^{2 m}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 m} \left(\log \left(\csc ^2\left(\frac{1}{8} (2 e+2 f x+3 \pi )\right)\right)-\log \left(\tan \left(\frac{1}{8} (-2 e-2 f x+\pi )\right)\right)\right)}{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,"(2^(1 + m)*g*Cos[(2*e + Pi + 2*f*x)/4]^(2*m)*(Log[Csc[(2*e + 3*Pi + 2*f*x)/8]^2] - Log[Tan[(-2*e + Pi - 2*f*x)/8]])*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^m)/(c*f*(g*Cos[e + f*x])^(2*m)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*m))","B",1
175,1,1513,203,6.8088207,"\int (g \cos (e+f x))^{5-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[(g*Cos[e + f*x])^(5 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{\cos ^{2 n-5}(e+f x) (g \cos (e+f x))^{5-2 m} (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^{n-\frac{n (\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))}{\log (c-c \sin (e+f x))}} \left(\frac{e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \left(3 m^2-6 n m-41 m+3 n^2+41 n+256\right)}{8 (m-n-5) (m-n-4) (m-n-3)}+\frac{\left(m^2-2 n m-23 m+n^2+23 n+300\right) \left(-\frac{1}{16} i e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \cos (e+f x)-\frac{1}{16} e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \sin (e+f x)\right)}{(m-n-5) (m-n-4) (m-n-3)}+\frac{\left(m^2-2 n m-23 m+n^2+23 n+300\right) \left(\frac{1}{16} i e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \cos (e+f x)-\frac{1}{16} e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \sin (e+f x)\right)}{(m-n-5) (m-n-4) (m-n-3)}+\frac{\left(m^2-2 n m-11 m+n^2+11 n\right) \left(\frac{1}{4} e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \cos (2 (e+f x))-\frac{1}{4} i e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \sin (2 (e+f x))\right)}{(m-n-5) (m-n-4) (m-n-3)}+\frac{\left(m^2-2 n m-11 m+n^2+11 n\right) \left(\frac{1}{4} e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \cos (2 (e+f x))+\frac{1}{4} i e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \sin (2 (e+f x))\right)}{(m-n-5) (m-n-4) (m-n-3)}+\frac{\left(3 m^2-6 n m-53 m+3 n^2+53 n+100\right) \left(-\frac{1}{32} i e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \cos (3 (e+f x))-\frac{1}{32} e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \sin (3 (e+f x))\right)}{(m-n-5) (m-n-4) (m-n-3)}+\frac{\left(3 m^2-6 n m-53 m+3 n^2+53 n+100\right) \left(\frac{1}{32} i e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \cos (3 (e+f x))-\frac{1}{32} e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \sin (3 (e+f x))\right)}{(m-n-5) (m-n-4) (m-n-3)}+\frac{(m-n) \left(\frac{1}{16} e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \cos (4 (e+f x))-\frac{1}{16} i e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \sin (4 (e+f x))\right)}{(m-n-5) (m-n-4)}+\frac{(m-n) \left(\frac{1}{16} e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \cos (4 (e+f x))+\frac{1}{16} i e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \sin (4 (e+f x))\right)}{(m-n-5) (m-n-4)}+\frac{-\frac{1}{32} i e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \cos (5 (e+f x))-\frac{1}{32} e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \sin (5 (e+f x))}{m-n-5}+\frac{\frac{1}{32} i e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \cos (5 (e+f x))-\frac{1}{32} e^{n (-2 \log (\cos (e+f x))+\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x)))} \sin (5 (e+f x))}{m-n-5}\right)}{f}","-\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,"(Cos[e + f*x]^(-5 + 2*n)*(g*Cos[e + f*x])^(5 - 2*m)*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^(n - (n*(Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))/Log[c - c*Sin[e + f*x]])*((E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*(256 - 41*m + 3*m^2 + 41*n - 6*m*n + 3*n^2))/(8*(-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((300 - 23*m + m^2 + 23*n - 2*m*n + n^2)*((-1/16*I)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[e + f*x] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[e + f*x])/16))/((-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((300 - 23*m + m^2 + 23*n - 2*m*n + n^2)*((I/16)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[e + f*x] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[e + f*x])/16))/((-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((-11*m + m^2 + 11*n - 2*m*n + n^2)*((E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[2*(e + f*x)])/4 - (I/4)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[2*(e + f*x)]))/((-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((-11*m + m^2 + 11*n - 2*m*n + n^2)*((E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[2*(e + f*x)])/4 + (I/4)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[2*(e + f*x)]))/((-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((100 - 53*m + 3*m^2 + 53*n - 6*m*n + 3*n^2)*((-1/32*I)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[3*(e + f*x)] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[3*(e + f*x)])/32))/((-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((100 - 53*m + 3*m^2 + 53*n - 6*m*n + 3*n^2)*((I/32)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[3*(e + f*x)] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[3*(e + f*x)])/32))/((-5 + m - n)*(-4 + m - n)*(-3 + m - n)) + ((m - n)*((E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[4*(e + f*x)])/16 - (I/16)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[4*(e + f*x)]))/((-5 + m - n)*(-4 + m - n)) + ((m - n)*((E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[4*(e + f*x)])/16 + (I/16)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[4*(e + f*x)]))/((-5 + m - n)*(-4 + m - n)) + ((-1/32*I)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[5*(e + f*x)] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[5*(e + f*x)])/32)/(-5 + m - n) + ((I/32)*E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Cos[5*(e + f*x)] - (E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*Sin[5*(e + f*x)])/32)/(-5 + m - n)))/f","C",1
176,1,143,127,1.3762529,"\int (g \cos (e+f x))^{3-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[(g*Cos[e + f*x])^(3 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","-\frac{g^3 \cos ^{2 n}(e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^4 (g \cos (e+f x))^{-2 m} ((-m+n+2) \sin (e+f x)-m+n+4) (a (\sin (e+f x)+1))^{m-n} \exp (n (\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x))-2 \log (\cos (e+f x))))}{f (-m+n+2) (-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,"-((E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*g^3*Cos[e + f*x]^(2*n)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*(a*(1 + Sin[e + f*x]))^(m - n)*(4 - m + n + (2 - m + n)*Sin[e + f*x]))/(f*(2 - m + n)*(3 - m + n)*(g*Cos[e + f*x])^(2*m)))","A",1
177,1,96,58,0.7205508,"\int (g \cos (e+f x))^{1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[(g*Cos[e + f*x])^(1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{g (\sin (e+f x)-1) \cos ^{2 n}(e+f x) (g \cos (e+f x))^{-2 m} (a (\sin (e+f x)+1))^{m-n} \exp (n (\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x))-2 \log (\cos (e+f x))))}{f (-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,"(E^(n*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*g*Cos[e + f*x]^(2*n)*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(m - n))/(f*(1 - m + n)*(g*Cos[e + f*x])^(2*m))","A",1
178,1,115,81,71.4449437,"\int (g \cos (e+f x))^{-1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[(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)+1))^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-2 m} \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{n-m} \, _2F_1\left(n-m,n-m;-m+n+1;-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\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[-m + n, -m + n, 1 - m + n, -Tan[(2*e - Pi + 2*f*x)/4]^2]*(Sec[(2*e - Pi + 2*f*x)/4]^2)^(-m + n)*(a*(1 + 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",1
179,1,135,85,49.7003925,"\int (g \cos (e+f x))^{-3-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[(g*Cos[e + f*x])^(-3 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{\cot ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-2 m} \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{n-m} \, _2F_1\left(-m+n-2,-m+n-1;n-m;-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{8 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,"(Cot[(2*e - Pi + 2*f*x)/4]^2*Hypergeometric2F1[-2 - m + n, -1 - m + n, -m + n, -Tan[(2*e - Pi + 2*f*x)/4]^2]*(Sec[(2*e - Pi + 2*f*x)/4]^2)^(-m + n)*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n)/(8*f*g^3*(1 + m - n)*(g*Cos[e + f*x])^(2*m))","A",1
180,1,136,88,41.1430914,"\int (g \cos (e+f x))^{-5-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[(g*Cos[e + f*x])^(-5 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{\cot ^4\left(\frac{1}{4} (2 e+2 f x-\pi )\right) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-2 m} \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{n-m} \, _2F_1\left(-m+n-4,-m+n-2;-m+n-1;-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{32 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,"(Cot[(2*e - Pi + 2*f*x)/4]^4*Hypergeometric2F1[-4 - m + n, -2 - m + n, -1 - m + n, -Tan[(2*e - Pi + 2*f*x)/4]^2]*(Sec[(2*e - Pi + 2*f*x)/4]^2)^(-m + n)*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n)/(32*f*g^5*(2 + m - n)*(g*Cos[e + f*x])^(2*m))","A",1
181,1,94,51,1.0027965,"\int (g \cos (e+f x))^{-1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m \, dx","Integrate[(g*Cos[e + f*x])^(-1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^m,x]","\frac{\sqrt{-\tan ^2(e+f x)} \csc (e+f x) \cos ^{2 (m+1)}(e+f x) \sin ^{-1}(\sec (e+f x)) (g \cos (e+f x))^{-2 m-1} \exp (m (\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x))-2 \log (\cos (e+f x))))}{f}","\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,"(E^(m*(-2*Log[Cos[e + f*x]] + Log[a*(1 + Sin[e + f*x])] + Log[c - c*Sin[e + f*x]]))*ArcSin[Sec[e + f*x]]*Cos[e + f*x]^(2*(1 + m))*(g*Cos[e + f*x])^(-1 - 2*m)*Csc[e + f*x]*Sqrt[-Tan[e + f*x]^2])/f","A",1
182,0,0,134,149.7124772,"\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","Integrate[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3 + n),x]","\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","\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,"Integrate[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3 + n), x]","F",-1
183,0,0,134,142.8414965,"\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","Integrate[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(2 + n),x]","\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","\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,"Integrate[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(2 + n), x]","F",-1
184,1,207,131,27.10247,"\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","Integrate[(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{i c (\sin (e+f x)-1) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n (g \cos (e+f x))^{-m-n} \left(\, _2F_1\left(1,-m+n+1;-m+n+2;-\frac{i \left(\tan \left(\frac{1}{2} (e+f x)\right)-1\right)}{\tan \left(\frac{1}{2} (e+f x)\right)+1}\right)-\, _2F_1\left(1,-m+n+1;-m+n+2;\frac{i \left(\tan \left(\frac{1}{2} (e+f x)\right)-1\right)}{\tan \left(\frac{1}{2} (e+f x)\right)+1}\right)\right)}{f g (m-n-1) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}","\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,"(I*c*(g*Cos[e + f*x])^(-m - n)*(Hypergeometric2F1[1, 1 - m + n, 2 - m + n, ((-I)*(-1 + Tan[(e + f*x)/2]))/(1 + Tan[(e + f*x)/2])] - Hypergeometric2F1[1, 1 - m + n, 2 - m + n, (I*(-1 + Tan[(e + f*x)/2]))/(1 + Tan[(e + f*x)/2])])*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n)/(f*g*(-1 + m - n)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))","C",1
185,1,55,55,0.7922242,"\int (g \cos (e+f x))^{-1-m-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[(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)+1))^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*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n)/(f*g*(m - n))","A",1
186,1,132,125,27.8219648,"\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","Integrate[(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{2^{n-1} \cos ^{2 (n-1)}\left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^{n-1} (\sin (e+f x)-m+n-1) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{2-2 n} (g \cos (e+f x))^{-m-n}}{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,"-((2^(-1 + n)*(g*Cos[e + f*x])^(-m - n)*Cos[(2*e + Pi + 2*f*x)/4]^(2*(-1 + n))*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2 - 2*n)*(a*(1 + Sin[e + f*x]))^m*(-1 - m + n + Sin[e + f*x])*(c - c*Sin[e + f*x])^(-1 + n))/(f*g*(m - n)*(2 + m - n)))","A",1
187,1,183,204,32.5630148,"\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","Integrate[(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^{n-2} \cos (e+f x) \sin ^{2 n-4}\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (n-2)} (g \cos (e+f x))^{-m-n-1} \left(-2 (m-n+2) \sin (e+f x)+\cos \left(2 \left(-e-f x+\frac{\pi }{2}\right)\right)+m^2-2 m n+4 m+n^2-4 n+3\right)}{f (m-n) (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,"(2^(-2 + n)*Cos[e + f*x]*(g*Cos[e + f*x])^(-1 - m - n)*Sin[(-e + Pi/2 - f*x)/2]^(-4 + 2*n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 + n)*(3 + 4*m + m^2 - 4*n - 2*m*n + n^2 + Cos[2*(-e + Pi/2 - f*x)] - 2*(2 + m - n)*Sin[e + f*x]))/(f*(m - n)*(2 + m - n)*(4 + m - n)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-2 + n)))","A",1
188,1,2681,290,38.7005397,"\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","Integrate[(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 + n),x]","\text{Result too large to show}","\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{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{(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,"-((2^(-4 - m + 2*n)*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Cos[e + f*x]*(g*Cos[e + f*x])^(-1 - m - n)*Csc[(-e + Pi/2 - f*x)/2]^6*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))^(2*n)*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] + Sin[(7*(-e + Pi/2 - f*x))/8]))^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 + n)*(-30 - 46*m - 18*m^2 - 2*m^3 + 46*n + 36*m*n + 6*m^2*n - 18*n^2 - 6*m*n^2 + 2*n^3 - 6*(3 + m - n)*Cos[2*(-e + Pi/2 - f*x)] + 3*Cos[3*(-e + Pi/2 - f*x)] + 3*(15 + 2*m^2 - 4*m*(-3 + n) - 12*n + 2*n^2)*Sin[e + f*x])*(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^(-7 + 2*n))/(f*(m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(2*(-3 + n))*((2^(-4 - m + 2*n)*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Csc[(-e + Pi/2 - f*x)/2]^6*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))^(2*n)*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] + Sin[(7*(-e + Pi/2 - f*x))/8]))^(-m - n)*(-3*(15 + 2*m^2 - 4*m*(-3 + n) - 12*n + 2*n^2)*Cos[e + f*x] + 12*(3 + m - n)*Sin[2*(-e + Pi/2 - f*x)] - 9*Sin[3*(-e + Pi/2 - f*x)]))/((m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n)) - (2^(-4 - m + 2*n)*m*Cos[(-e + Pi/2 - f*x)/2]^(-1 + 2*m)*Csc[(-e + Pi/2 - f*x)/2]^5*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))^(2*n)*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] + Sin[(7*(-e + Pi/2 - f*x))/8]))^(-m - n)*(-30 - 46*m - 18*m^2 - 2*m^3 + 46*n + 36*m*n + 6*m^2*n - 18*n^2 - 6*m*n^2 + 2*n^3 - 6*(3 + m - n)*Cos[2*(-e + Pi/2 - f*x)] + 3*Cos[3*(-e + Pi/2 - f*x)] + 3*(15 + 2*m^2 - 4*m*(-3 + n) - 12*n + 2*n^2)*Sin[e + f*x]))/((m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n)) - (3*2^(-4 - m + 2*n)*Cos[(-e + Pi/2 - f*x)/2]^(1 + 2*m)*Csc[(-e + Pi/2 - f*x)/2]^7*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))^(2*n)*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] + Sin[(7*(-e + Pi/2 - f*x))/8]))^(-m - n)*(-30 - 46*m - 18*m^2 - 2*m^3 + 46*n + 36*m*n + 6*m^2*n - 18*n^2 - 6*m*n^2 + 2*n^3 - 6*(3 + m - n)*Cos[2*(-e + Pi/2 - f*x)] + 3*Cos[3*(-e + Pi/2 - f*x)] + 3*(15 + 2*m^2 - 4*m*(-3 + n) - 12*n + 2*n^2)*Sin[e + f*x]))/((m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n)) + (2^(-3 - m + 2*n)*n*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Csc[(-e + Pi/2 - f*x)/2]^6*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))^(-1 + 2*n)*(Cos[(-e + Pi/2 - f*x)/8]*(-1/8*Cos[(-e + Pi/2 - f*x)/8] + (3*Cos[(3*(-e + Pi/2 - f*x))/8])/8) - (Sin[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))/8)*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] + Sin[(7*(-e + Pi/2 - f*x))/8]))^(-m - n)*(-30 - 46*m - 18*m^2 - 2*m^3 + 46*n + 36*m*n + 6*m^2*n - 18*n^2 - 6*m*n^2 + 2*n^3 - 6*(3 + m - n)*Cos[2*(-e + Pi/2 - f*x)] + 3*Cos[3*(-e + Pi/2 - f*x)] + 3*(15 + 2*m^2 - 4*m*(-3 + n) - 12*n + 2*n^2)*Sin[e + f*x]))/((m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n)) + (2^(-4 - m + 2*n)*(-m - n)*Cos[(-e + Pi/2 - f*x)/2]^(2*m)*Csc[(-e + Pi/2 - f*x)/2]^6*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8]))^(2*n)*(Cos[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] + Sin[(7*(-e + Pi/2 - f*x))/8]))^(-1 - m - n)*(Cos[(-e + Pi/2 - f*x)/8]*(-1/8*Cos[(-e + Pi/2 - f*x)/8] + (3*Cos[(3*(-e + Pi/2 - f*x))/8])/8 - (5*Cos[(5*(-e + Pi/2 - f*x))/8])/8 + (7*Cos[(7*(-e + Pi/2 - f*x))/8])/8) - (Sin[(-e + Pi/2 - f*x)/8]*(-Sin[(-e + Pi/2 - f*x)/8] + Sin[(3*(-e + Pi/2 - f*x))/8] - Sin[(5*(-e + Pi/2 - f*x))/8] + Sin[(7*(-e + Pi/2 - f*x))/8]))/8)*(-30 - 46*m - 18*m^2 - 2*m^3 + 46*n + 36*m*n + 6*m^2*n - 18*n^2 - 6*m*n^2 + 2*n^3 - 6*(3 + m - n)*Cos[2*(-e + Pi/2 - f*x)] + 3*Cos[3*(-e + Pi/2 - f*x)] + 3*(15 + 2*m^2 - 4*m*(-3 + n) - 12*n + 2*n^2)*Sin[e + f*x]))/((m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n)))*(Cos[Pi/4 + (e - Pi/2 + f*x)/2] + Sin[Pi/4 + (e - Pi/2 + f*x)/2])))","B",1
189,1,139,138,42.5354935,"\int (g \sec (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, dx","Integrate[(g*Sec[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n,x]","\frac{2 \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n (g \sec (e+f x))^p \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)^{m+n-p} \, _2F_1\left(m+n-p+1,n-\frac{p}{2}+\frac{1}{2};n-\frac{p}{2}+\frac{3}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f (2 n-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*Hypergeometric2F1[1 + m + n - p, 1/2 + n - p/2, 3/2 + n - p/2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*(g*Sec[e + f*x])^p*(Sec[(2*e - Pi + 2*f*x)/4]^2)^(m + n - p)*(a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^n*Tan[(2*e - Pi + 2*f*x)/4])/(f*(1 + 2*n - p))","A",1
190,1,33,33,0.0108742,"\int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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",1
191,1,30,33,0.0902642,"\int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{4 a \sin ^3(c+d x)-3 a \cos (2 (c+d x))}{12 d}","\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^2(c+d x)}{2 d}",1,"(-3*a*Cos[2*(c + d*x)] + 4*a*Sin[c + d*x]^3)/(12*d)","A",1
192,1,26,24,0.0379058,"\int \cot (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a (\sin (c+d x)+\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}","\frac{a \sin (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}",1,"(a*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]] + Sin[c + d*x]))/d","A",1
193,1,33,25,0.0437882,"\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a (\log (\tan (c+d x))+\log (\cos (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[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","A",1
194,1,29,30,0.0196688,"\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^2(c+d x)}{2 d}-\frac{a \csc (c+d x)}{d}","-\frac{\csc ^2(c+d x) (a \sin (c+d x)+a)^2}{2 a d}",1,"-((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d)","A",1
195,1,33,33,0.0246189,"\int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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,"-1/2*(a*Csc[c + d*x]^2)/d - (a*Csc[c + d*x]^3)/(3*d)","A",1
196,1,33,33,0.0193142,"\int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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,"-1/3*(a*Csc[c + d*x]^3)/d - (a*Csc[c + d*x]^4)/(4*d)","A",1
197,1,53,55,0.3366322,"\int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(104 \sin ^3(c+d x)+15 \cos (4 (c+d x))-12 \left(2 \sin ^3(c+d x)+5\right) \cos (2 (c+d x))\right)}{240 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*(15*Cos[4*(c + d*x)] + 104*Sin[c + d*x]^3 - 12*Cos[2*(c + d*x)]*(5 + 2*Sin[c + d*x]^3)))/(240*d)","A",1
198,1,38,55,0.0529971,"\int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^2(c+d x) \left(3 \sin ^2(c+d x)+8 \sin (c+d x)+6\right)}{12 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*(6 + 8*Sin[c + d*x] + 3*Sin[c + d*x]^2))/(12*d)","A",1
199,1,47,47,0.0254018,"\int \cot (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[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",1
200,1,38,43,0.0223318,"\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]","a^2 \left(\frac{\sin (c+d x)}{d}-\frac{\csc (c+d x)}{d}+\frac{2 \log (\sin (c+d x))}{d}\right)","\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*Log[Sin[c + d*x]])/d + Sin[c + d*x]/d)","A",1
201,1,42,47,0.0177826,"\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","a^2 \left(-\frac{\csc ^2(c+d x)}{2 d}-\frac{2 \csc (c+d x)}{d}+\frac{\log (\sin (c+d x))}{d}\right)","-\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,"a^2*((-2*Csc[c + d*x])/d - Csc[c + d*x]^2/(2*d) + Log[Sin[c + d*x]]/d)","A",1
202,1,20,30,0.0266786,"\int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 (\csc (c+d x)+1)^3}{3 d}","-\frac{\csc ^3(c+d x) (a \sin (c+d x)+a)^3}{3 a d}",1,"-1/3*(a^2*(1 + Csc[c + d*x])^3)/d","A",1
203,1,55,55,0.0289833,"\int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[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,"-1/2*(a^2*Csc[c + d*x]^2)/d - (2*a^2*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*d)","A",1
204,1,55,55,0.0263583,"\int \cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[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,"-1/3*(a^2*Csc[c + d*x]^3)/d - (a^2*Csc[c + d*x]^4)/(2*d) - (a^2*Csc[c + d*x]^5)/(5*d)","A",1
205,1,55,55,0.0329839,"\int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[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,"-1/4*(a^2*Csc[c + d*x]^4)/d - (2*a^2*Csc[c + d*x]^5)/(5*d) - (a^2*Csc[c + d*x]^6)/(6*d)","A",1
206,1,80,73,0.3363946,"\int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 (-1015 \sin (c+d x)+525 \sin (3 (c+d x))-119 \sin (5 (c+d x))+5 \sin (7 (c+d x))+805 \cos (2 (c+d x))-280 \cos (4 (c+d x))+35 \cos (6 (c+d x))-350)}{2240 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,"-1/2240*(a^3*(-350 + 805*Cos[2*(c + d*x)] - 280*Cos[4*(c + d*x)] + 35*Cos[6*(c + d*x)] - 1015*Sin[c + d*x] + 525*Sin[3*(c + d*x)] - 119*Sin[5*(c + d*x)] + 5*Sin[7*(c + d*x)]))/d","A",1
207,1,70,73,0.2819206,"\int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 (-1200 \sin (c+d x)+520 \sin (3 (c+d x))-72 \sin (5 (c+d x))+870 \cos (2 (c+d x))-240 \cos (4 (c+d x))+10 \cos (6 (c+d x))-45)}{1920 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,"-1/1920*(a^3*(-45 + 870*Cos[2*(c + d*x)] - 240*Cos[4*(c + d*x)] + 10*Cos[6*(c + d*x)] - 1200*Sin[c + d*x] + 520*Sin[3*(c + d*x)] - 72*Sin[5*(c + d*x)]))/d","A",1
208,1,30,45,0.12353,"\int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^4 (4 \sin (c+d x)-1)}{20 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^3*(1 + Sin[c + d*x])^4*(-1 + 4*Sin[c + d*x]))/(20*d)","A",1
209,1,65,65,0.0284888,"\int \cot (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[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",1
210,1,62,62,0.0296658,"\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[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",1
211,1,53,61,0.0222852,"\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","a^3 \left(\frac{\sin (c+d x)}{d}-\frac{\csc ^2(c+d x)}{2 d}-\frac{3 \csc (c+d x)}{d}+\frac{3 \log (\sin (c+d x))}{d}\right)","\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,"a^3*((-3*Csc[c + d*x])/d - Csc[c + d*x]^2/(2*d) + (3*Log[Sin[c + d*x]])/d + Sin[c + d*x]/d)","A",1
212,1,57,65,0.0195565,"\int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","a^3 \left(-\frac{\csc ^3(c+d x)}{3 d}-\frac{3 \csc ^2(c+d x)}{2 d}-\frac{3 \csc (c+d x)}{d}+\frac{\log (\sin (c+d x))}{d}\right)","-\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,"a^3*((-3*Csc[c + d*x])/d - (3*Csc[c + d*x]^2)/(2*d) - Csc[c + d*x]^3/(3*d) + Log[Sin[c + d*x]]/d)","A",1
213,1,20,30,0.0231767,"\int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 (\csc (c+d x)+1)^4}{4 d}","-\frac{\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{4 a d}",1,"-1/4*(a^3*(1 + Csc[c + d*x])^4)/d","A",1
214,1,71,61,0.0307919,"\int \cot (c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","-\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)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 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,"-1/2*(a^3*Csc[c + d*x]^2)/d - (a^3*Csc[c + d*x]^3)/d - (3*a^3*Csc[c + d*x]^4)/(4*d) - (a^3*Csc[c + d*x]^5)/(5*d)","A",1
215,1,73,73,0.0303715,"\int \cot (c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[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,"-1/3*(a^3*Csc[c + d*x]^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",1
216,1,73,73,0.0310635,"\int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[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,"-1/4*(a^3*Csc[c + d*x]^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",1
217,1,100,91,0.8442312,"\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 (52290 \sin (c+d x)-30660 \sin (3 (c+d x))+9828 \sin (5 (c+d x))-1395 \sin (7 (c+d x))+35 \sin (9 (c+d x))-42840 \cos (2 (c+d x))+18900 \cos (4 (c+d x))-4200 \cos (6 (c+d x))+315 \cos (8 (c+d x))+4095)}{80640 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*(4095 - 42840*Cos[2*(c + d*x)] + 18900*Cos[4*(c + d*x)] - 4200*Cos[6*(c + d*x)] + 315*Cos[8*(c + d*x)] + 52290*Sin[c + d*x] - 30660*Sin[3*(c + d*x)] + 9828*Sin[5*(c + d*x)] - 1395*Sin[7*(c + d*x)] + 35*Sin[9*(c + d*x)]))/(80640*d)","A",1
218,1,90,88,0.5340577,"\int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 (87360 \sin (c+d x)-47040 \sin (3 (c+d x))+12096 \sin (5 (c+d x))-960 \sin (7 (c+d x))-69720 \cos (2 (c+d x))+26460 \cos (4 (c+d x))-4200 \cos (6 (c+d x))+105 \cos (8 (c+d x))+36400)}{107520 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*(36400 - 69720*Cos[2*(c + d*x)] + 26460*Cos[4*(c + d*x)] - 4200*Cos[6*(c + d*x)] + 105*Cos[8*(c + d*x)] + 87360*Sin[c + d*x] - 47040*Sin[3*(c + d*x)] + 12096*Sin[5*(c + d*x)] - 960*Sin[7*(c + d*x)]))/(107520*d)","A",1
219,1,80,67,0.3370764,"\int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","-\frac{a^4 (-7245 \sin (c+d x)+3395 \sin (3 (c+d x))-609 \sin (5 (c+d x))+15 \sin (7 (c+d x))+5460 \cos (2 (c+d x))-1680 \cos (4 (c+d x))+140 \cos (6 (c+d x))-630)}{6720 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,"-1/6720*(a^4*(-630 + 5460*Cos[2*(c + d*x)] - 1680*Cos[4*(c + d*x)] + 140*Cos[6*(c + d*x)] - 7245*Sin[c + d*x] + 3395*Sin[3*(c + d*x)] - 609*Sin[5*(c + d*x)] + 15*Sin[7*(c + d*x)]))/d","A",1
220,1,30,45,0.0985354,"\int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 (\sin (c+d x)+1)^5 (5 \sin (c+d x)-1)}{30 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^4*(1 + Sin[c + d*x])^5*(-1 + 5*Sin[c + d*x]))/(30*d)","A",1
221,1,81,81,0.03826,"\int \cot (c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[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",1
222,1,78,78,0.0373108,"\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[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",1
223,1,54,80,0.0771437,"\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","-\frac{a^4 \left(-\sin ^2(c+d x)-8 \sin (c+d x)+\csc ^2(c+d x)+8 \csc (c+d x)-12 \log (\sin (c+d x))\right)}{2 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,"-1/2*(a^4*(8*Csc[c + d*x] + Csc[c + d*x]^2 - 12*Log[Sin[c + d*x]] - 8*Sin[c + d*x] - Sin[c + d*x]^2))/d","A",1
224,1,60,85,0.1318958,"\int \frac{\cos (c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{3 \sin ^4(c+d x)-4 \sin ^3(c+d x)+6 \sin ^2(c+d x)-12 \sin (c+d x)+12 \log (\sin (c+d x)+1)}{12 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,"(12*Log[1 + Sin[c + d*x]] - 12*Sin[c + d*x] + 6*Sin[c + d*x]^2 - 4*Sin[c + d*x]^3 + 3*Sin[c + d*x]^4)/(12*a*d)","A",1
225,1,50,67,0.1086549,"\int \frac{\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{2 \sin ^3(c+d x)-3 \sin ^2(c+d x)+6 \sin (c+d x)-6 \log (\sin (c+d x)+1)}{6 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,"(-6*Log[1 + Sin[c + d*x]] + 6*Sin[c + d*x] - 3*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3)/(6*a*d)","A",1
226,1,38,49,0.0606178,"\int \frac{\cos (c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^2(c+d x)-2 \sin (c+d x)+2 \log (\sin (c+d x)+1)}{2 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,"(2*Log[1 + Sin[c + d*x]] - 2*Sin[c + d*x] + Sin[c + d*x]^2)/(2*a*d)","A",1
227,1,25,31,0.0205616,"\int \frac{\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\sin (c+d x)-\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]] + Sin[c + d*x])/(a*d)","A",1
228,1,32,32,0.0165568,"\int \frac{\cot (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[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",1
229,1,46,46,0.0345626,"\int \frac{\cot (c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(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",1
230,1,63,63,0.0386237,"\int \frac{\cot (c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(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",1
231,1,82,82,0.0471201,"\int \frac{\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(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",1
232,1,73,87,0.582242,"\int \frac{\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^3(c+d x)-3 \sin ^2(c+d x)+9 \sin (c+d x)-12 \log (\sin (c+d x)+1)-\frac{3}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{3 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,"(-12*Log[1 + Sin[c + d*x]] - 3/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 9*Sin[c + d*x] - 3*Sin[c + d*x]^2 + Sin[c + d*x]^3)/(3*a^2*d)","A",1
233,1,71,70,0.183631,"\int \frac{\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^3(c+d x)-3 \sin ^2(c+d x)+\sin (c+d x) (6 \log (\sin (c+d x)+1)-4)+6 \log (\sin (c+d x)+1)+2}{2 a^2 d (\sin (c+d x)+1)}","\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,"(2 + 6*Log[1 + Sin[c + d*x]] + (-4 + 6*Log[1 + Sin[c + d*x]])*Sin[c + d*x] - 3*Sin[c + d*x]^2 + Sin[c + d*x]^3)/(2*a^2*d*(1 + Sin[c + d*x]))","A",1
234,1,55,52,0.2038878,"\int \frac{\cos (c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{4 \sin (c+d x)-8 \log (\sin (c+d x)+1)-\frac{4}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{4 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,"(-8*Log[1 + Sin[c + d*x]] - 4/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 4*Sin[c + d*x])/(4*a^2*d)","A",1
235,1,27,37,0.0288209,"\int \frac{\cos (c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{\frac{1}{\sin (c+d x)+1}+\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]] + (1 + Sin[c + d*x])^(-1))/(a^2*d)","A",1
236,1,36,52,0.0553881,"\int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]/(a + a*Sin[c + d*x])^2,x]","\frac{\frac{1}{\sin (c+d x)+1}+\log (\sin (c+d x))-\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]] - Log[1 + Sin[c + d*x]] + (1 + Sin[c + d*x])^(-1))/(a^2*d)","A",1
237,1,45,68,0.1600946,"\int \frac{\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\frac{1}{\sin (c+d x)+1}+\csc (c+d x)+2 \log (\sin (c+d x))-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] + 2*Log[Sin[c + d*x]] - 2*Log[1 + Sin[c + d*x]] + (1 + Sin[c + d*x])^(-1))/(a^2*d))","A",1
238,1,61,85,0.2043582,"\int \frac{\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\frac{2}{\sin (c+d x)+1}-\csc ^2(c+d x)+4 \csc (c+d x)+6 \log (\sin (c+d x))-6 \log (\sin (c+d x)+1)}{2 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,"(4*Csc[c + d*x] - Csc[c + d*x]^2 + 6*Log[Sin[c + d*x]] - 6*Log[1 + Sin[c + d*x]] + 2/(1 + Sin[c + d*x]))/(2*a^2*d)","A",1
239,1,98,101,2.4584872,"\int \frac{\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{1}{a^2 d (\sin (c+d x)+1)}-\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/(a^2*d*(1 + Sin[c + d*x]))","A",1
240,1,106,111,0.6962241,"\int \frac{\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]","\frac{32 \sin ^5(c+d x)-80 \sin ^4(c+d x)+320 \sin ^3(c+d x)+\sin ^2(c+d x) (1023-960 \log (\sin (c+d x)+1))-6 \sin (c+d x) (320 \log (\sin (c+d x)+1)-21)-960 \log (\sin (c+d x)+1)-417}{96 a^3 d (\sin (c+d x)+1)^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,"(-417 - 960*Log[1 + Sin[c + d*x]] - 6*(-21 + 320*Log[1 + Sin[c + d*x]])*Sin[c + d*x] + (1023 - 960*Log[1 + Sin[c + d*x]])*Sin[c + d*x]^2 + 320*Sin[c + d*x]^3 - 80*Sin[c + d*x]^4 + 32*Sin[c + d*x]^5)/(96*a^3*d*(1 + Sin[c + d*x])^2)","A",1
241,1,78,93,2.0827681,"\int \frac{\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","\frac{8 \sin ^2(c+d x)+\left(\frac{64}{(\sin (c+d x)+1)^2}-48\right) \sin (c+d x)+96 \log (\sin (c+d x)+1)+\frac{56}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}}{16 a^3 d}","\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,"(96*Log[1 + Sin[c + d*x]] + 56/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + 8*Sin[c + d*x]^2 + Sin[c + d*x]*(-48 + 64/(1 + Sin[c + d*x])^2))/(16*a^3*d)","A",1
242,1,70,74,0.4025726,"\int \frac{\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{\frac{\sin ^2(c+d x)}{(\sin (c+d x)+1)^2}+4 \sin (c+d x)+\frac{-10 \sin (c+d x)-9}{(\sin (c+d x)+1)^2}-12 \log (\sin (c+d x)+1)}{4 a^3 d}","\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,"(-12*Log[1 + Sin[c + d*x]] + 4*Sin[c + d*x] + (-9 - 10*Sin[c + d*x])/(1 + Sin[c + d*x])^2 + Sin[c + d*x]^2/(1 + Sin[c + d*x])^2)/(4*a^3*d)","A",1
243,1,65,60,0.6497399,"\int \frac{\cos (c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\frac{16 \sin (c+d x)}{(\sin (c+d x)+1)^2}+8 \log (\sin (c+d x)+1)+\frac{12}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}}{8 a^3 d}","\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,"(8*Log[1 + Sin[c + d*x]] + 12/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + (16*Sin[c + d*x])/(1 + Sin[c + d*x])^2)/(8*a^3*d)","A",1
244,1,30,30,0.0299011,"\int \frac{\cos (c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(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",1
245,1,52,74,0.1795919,"\int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]/(a + a*Sin[c + d*x])^3,x]","\frac{\frac{2 \sin (c+d x)+3}{(\sin (c+d x)+1)^2}+2 \log (\sin (c+d x))-2 \log (\sin (c+d x)+1)}{2 a^3 d}","\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,"(2*Log[Sin[c + d*x]] - 2*Log[1 + Sin[c + d*x]] + (3 + 2*Sin[c + d*x])/(1 + Sin[c + d*x])^2)/(2*a^3*d)","A",1
246,1,61,90,0.3142651,"\int \frac{\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{\frac{4}{\sin (c+d x)+1}+\frac{1}{(\sin (c+d x)+1)^2}+2 \csc (c+d x)+6 \log (\sin (c+d x))-6 \log (\sin (c+d x)+1)}{2 a^3 d}","-\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,"-1/2*(2*Csc[c + d*x] + 6*Log[Sin[c + d*x]] - 6*Log[1 + Sin[c + d*x]] + (1 + Sin[c + d*x])^(-2) + 4/(1 + Sin[c + d*x]))/(a^3*d)","A",1
247,1,71,108,0.5739199,"\int \frac{\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\frac{6}{\sin (c+d x)+1}+\frac{1}{(\sin (c+d x)+1)^2}-\csc ^2(c+d x)+6 \csc (c+d x)+12 \log (\sin (c+d x))-12 \log (\sin (c+d x)+1)}{2 a^3 d}","\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,"(6*Csc[c + d*x] - Csc[c + d*x]^2 + 12*Log[Sin[c + d*x]] - 12*Log[1 + Sin[c + d*x]] + (1 + Sin[c + d*x])^(-2) + 6/(1 + Sin[c + d*x]))/(2*a^3*d)","A",1
248,1,81,126,5.4325839,"\int \frac{\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","-\frac{\frac{3 (8 \sin (c+d x)+9)}{(\sin (c+d x)+1)^2}+2 \csc ^3(c+d x)-9 \csc ^2(c+d x)+36 \csc (c+d x)+60 \log (\sin (c+d x))-60 \log (\sin (c+d x)+1)}{6 a^3 d}","-\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,"-1/6*(36*Csc[c + d*x] - 9*Csc[c + d*x]^2 + 2*Csc[c + d*x]^3 + 60*Log[Sin[c + d*x]] - 60*Log[1 + Sin[c + d*x]] + (3*(9 + 8*Sin[c + d*x]))/(1 + Sin[c + d*x])^2)/(a^3*d)","A",1
249,1,119,116,0.8830325,"\int \frac{\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^4,x]","\frac{3 \sin ^5(c+d x)-15 \sin ^4(c+d x)+\sin ^3(c+d x) (60 \log (\sin (c+d x)+1)-63)+9 \sin ^2(c+d x) (20 \log (\sin (c+d x)+1)-1)+9 \sin (c+d x) (20 \log (\sin (c+d x)+1)+9)+60 \log (\sin (c+d x)+1)+47}{6 a^4 d (\sin (c+d x)+1)^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{10 \log (\sin (c+d x)+1)}{a^4 d}-\frac{5}{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}",1,"(47 + 60*Log[1 + Sin[c + d*x]] + 9*(9 + 20*Log[1 + Sin[c + d*x]])*Sin[c + d*x] + 9*(-1 + 20*Log[1 + Sin[c + d*x]])*Sin[c + d*x]^2 + (-63 + 60*Log[1 + Sin[c + d*x]])*Sin[c + d*x]^3 - 15*Sin[c + d*x]^4 + 3*Sin[c + d*x]^5)/(6*a^4*d*(1 + Sin[c + d*x])^3)","A",1
250,1,127,95,6.5606178,"\int \frac{\cos (c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^4,x]","-\frac{3 (2 \sin (c+d x)+1)^2}{16 a^4 d (\sin (c+d x)+1)^3}-\frac{\frac{252 \sin ^2(c+d x)+444 \sin (c+d x)+197}{(\sin (c+d x)+1)^3}-48 \sin (c+d x)+192 \log (\sin (c+d x)+1)}{48 a^4 d}-\frac{1}{24 a^4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\frac{\sin (c+d x)}{a^4 d}-\frac{6}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{4 \log (\sin (c+d x)+1)}{a^4 d}+\frac{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}",1,"-1/24*1/(a^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (3*(1 + 2*Sin[c + d*x])^2)/(16*a^4*d*(1 + Sin[c + d*x])^3) - (192*Log[1 + Sin[c + d*x]] - 48*Sin[c + d*x] + (197 + 444*Sin[c + d*x] + 252*Sin[c + d*x]^2)/(1 + Sin[c + d*x])^3)/(48*a^4*d)","A",1
251,1,61,83,0.3677174,"\int \frac{\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^4,x]","\frac{18 \sin ^2(c+d x)+27 \sin (c+d x)+6 (\sin (c+d x)+1)^3 \log (\sin (c+d x)+1)+11}{6 a^4 d (\sin (c+d x)+1)^3}","\frac{3}{d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{\log (\sin (c+d x)+1)}{a^4 d}-\frac{3}{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}",1,"(11 + 27*Sin[c + d*x] + 18*Sin[c + d*x]^2 + 6*Log[1 + Sin[c + d*x]]*(1 + Sin[c + d*x])^3)/(6*a^4*d*(1 + Sin[c + d*x])^3)","A",1
252,1,53,30,0.1802139,"\int \frac{\cos (c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^4,x]","\frac{-6 \sin (c+d x)+3 \cos (2 (c+d x))-5}{6 a^4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\frac{\sin ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^3}",1,"(-5 + 3*Cos[2*(c + d*x)] - 6*Sin[c + d*x])/(6*a^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
253,1,30,46,0.0330282,"\int \frac{\cos (c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x])/(a + a*Sin[c + d*x])^4,x]","-\frac{3 \sin (c+d x)+1}{6 a^4 d (\sin (c+d x)+1)^3}","\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,"-1/6*(1 + 3*Sin[c + d*x])/(a^4*d*(1 + Sin[c + d*x])^3)","A",1
254,1,62,97,0.3620299,"\int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Cot[c + d*x]/(a + a*Sin[c + d*x])^4,x]","\frac{\frac{6 \sin ^2(c+d x)+15 \sin (c+d x)+11}{(\sin (c+d x)+1)^3}+6 \log (\sin (c+d x))-6 \log (\sin (c+d x)+1)}{6 a^4 d}","\frac{1}{d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{\log (\sin (c+d x))}{a^4 d}-\frac{\log (\sin (c+d x)+1)}{a^4 d}+\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}",1,"(6*Log[Sin[c + d*x]] - 6*Log[1 + Sin[c + d*x]] + (11 + 15*Sin[c + d*x] + 6*Sin[c + d*x]^2)/(1 + Sin[c + d*x])^3)/(6*a^4*d)","A",1
255,1,73,111,1.0056807,"\int \frac{\cot (c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Cot[c + d*x]*Csc[c + d*x])/(a + a*Sin[c + d*x])^4,x]","-\frac{\frac{9}{\sin (c+d x)+1}+\frac{3}{(\sin (c+d x)+1)^2}+\frac{1}{(\sin (c+d x)+1)^3}+3 \csc (c+d x)+12 \log (\sin (c+d x))-12 \log (\sin (c+d x)+1)}{3 a^4 d}","-\frac{3}{d \left(a^4 \sin (c+d x)+a^4\right)}-\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}{d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{1}{3 a d (a \sin (c+d x)+a)^3}",1,"-1/3*(3*Csc[c + d*x] + 12*Log[Sin[c + d*x]] - 12*Log[1 + Sin[c + d*x]] + (1 + Sin[c + d*x])^(-3) + 3/(1 + Sin[c + d*x])^2 + 9/(1 + Sin[c + d*x]))/(a^4*d)","A",1
256,1,85,131,3.4179253,"\int \frac{\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^4,x]","\frac{\frac{36}{\sin (c+d x)+1}+\frac{9}{(\sin (c+d x)+1)^2}+\frac{2}{(\sin (c+d x)+1)^3}-3 \csc ^2(c+d x)+24 \csc (c+d x)+60 \log (\sin (c+d x))-60 \log (\sin (c+d x)+1)}{6 a^4 d}","\frac{6}{d \left(a^4 \sin (c+d x)+a^4\right)}-\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{3}{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}",1,"(24*Csc[c + d*x] - 3*Csc[c + d*x]^2 + 60*Log[Sin[c + d*x]] - 60*Log[1 + Sin[c + d*x]] + 2/(1 + Sin[c + d*x])^3 + 9/(1 + Sin[c + d*x])^2 + 36/(1 + Sin[c + d*x]))/(6*a^4*d)","A",1
257,1,118,51,0.1458961,"\int \cot (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)+\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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*Cos[(c + d*x)/2] + Log[1 - Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[1 + Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*Sin[(c + d*x)/2])*Sqrt[a*(1 + Sin[c + d*x])])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
258,1,80,114,0.2656331,"\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[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) \left(\frac{\sin ^4(c+d x)}{n+5}+\frac{4 \sin ^3(c+d x)}{n+4}+\frac{6 \sin ^2(c+d x)}{n+3}+\frac{4 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right)}{d}","\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)*((1 + n)^(-1) + (4*Sin[c + d*x])/(2 + n) + (6*Sin[c + d*x]^2)/(3 + n) + (4*Sin[c + d*x]^3)/(4 + n) + Sin[c + d*x]^4/(5 + n)))/d","A",1
259,1,65,91,0.1744652,"\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[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) \left(\frac{\sin ^3(c+d x)}{n+4}+\frac{3 \sin ^2(c+d x)}{n+3}+\frac{3 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right)}{d}","\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)*((1 + n)^(-1) + (3*Sin[c + d*x])/(2 + n) + (3*Sin[c + d*x]^2)/(3 + n) + Sin[c + d*x]^3/(4 + n)))/d","A",1
260,1,50,68,0.2179196,"\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[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) \left(\frac{\sin ^2(c+d x)}{n+3}+\frac{2 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right)}{d}","\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)*((1 + n)^(-1) + (2*Sin[c + d*x])/(2 + n) + Sin[c + d*x]^2/(3 + n)))/d","A",1
261,1,38,41,0.2963511,"\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^{n+1}(c+d x) ((n+1) \sin (c+d x)+n+2)}{d (n+1) (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)*(2 + n + (1 + n)*Sin[c + d*x]))/(d*(1 + n)*(2 + n))","A",1
262,1,38,38,0.0373731,"\int \frac{\cos (c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(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",1
263,1,38,38,0.0375781,"\int \frac{\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(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",1
264,1,38,38,0.0578591,"\int \frac{\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(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",1
265,1,38,38,0.0309856,"\int \frac{\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(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",1
266,1,71,105,0.2004623,"\int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a (-15 \sin (2 (c+d x))-15 \sin (4 (c+d x))+5 \sin (6 (c+d x))-120 \cos (c+d x)-20 \cos (3 (c+d x))+12 \cos (5 (c+d x))+60 d x)}{960 d}","\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*(60*d*x - 120*Cos[c + d*x] - 20*Cos[3*(c + d*x)] + 12*Cos[5*(c + d*x)] - 15*Sin[2*(c + d*x)] - 15*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)","A",1
267,1,54,81,0.1096063,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a (-15 \sin (4 (c+d x))-60 \cos (c+d x)-10 \cos (3 (c+d x))+6 \cos (5 (c+d x))+60 c+60 d x)}{480 d}","\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*(60*c + 60*d*x - 60*Cos[c + d*x] - 10*Cos[3*(c + d*x)] + 6*Cos[5*(c + d*x)] - 15*Sin[4*(c + d*x)]))/(480*d)","A",1
268,1,42,65,0.102778,"\int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a (3 (\sin (4 (c+d x))-4 d x)+24 \cos (c+d x)+8 \cos (3 (c+d x)))}{96 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,"-1/96*(a*(24*Cos[c + d*x] + 8*Cos[3*(c + d*x)] + 3*(-4*d*x + Sin[4*(c + d*x)])))/d","A",1
269,1,74,51,0.073333,"\int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \cos (c+d x)}{d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}","\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*(c + d*x))/(2*d) + (a*Cos[c + d*x])/d - (a*Log[Cos[(c + d*x)/2]])/d + (a*Log[Sin[(c + d*x)/2]])/d + (a*Sin[2*(c + d*x)])/(4*d)","A",1
270,1,75,41,0.0429008,"\int \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}+\frac{a \cos (c+d x)}{d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}","\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*Cos[c + d*x])/d - (a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d - (a*Log[Cos[(c + d*x)/2]])/d + (a*Log[Sin[(c + d*x)/2]])/d","C",1
271,1,109,52,0.0441831,"\int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","-\frac{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,"-1/8*(a*Csc[(c + d*x)/2]^2)/d - (a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d + (a*Log[Cos[(c + d*x)/2]])/(2*d) - (a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)","C",1
272,1,95,52,0.0380341,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","-\frac{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,"-1/3*(a*Cot[c + d*x]^3)/d - (a*Csc[(c + d*x)/2]^2)/(8*d) + (a*Log[Cos[(c + d*x)/2]])/(2*d) - (a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)","A",1
273,1,135,74,0.0536802,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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 \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-1/3*(a*Cot[c + d*x]^3)/d + (a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) + (a*Log[Cos[(c + d*x)/2]])/(8*d) - (a*Log[Sin[(c + d*x)/2]])/(8*d) - (a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)","A",1
274,1,177,90,0.0728289,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{2 a \cot (c+d x)}{15 d}-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{15 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,"(2*a*Cot[c + d*x])/(15*d) + (a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d) + (a*Log[Cos[(c + d*x)/2]])/(8*d) - (a*Log[Sin[(c + d*x)/2]])/(8*d) - (a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)","A",1
275,1,86,135,0.5017637,"\int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (-210 \sin (2 (c+d x))-210 \sin (4 (c+d x))+70 \sin (6 (c+d x))-1365 \cos (c+d x)-175 \cos (3 (c+d x))+147 \cos (5 (c+d x))-15 \cos (7 (c+d x))+840 c+840 d x)}{6720 d}","-\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*(840*c + 840*d*x - 1365*Cos[c + d*x] - 175*Cos[3*(c + d*x)] + 147*Cos[5*(c + d*x)] - 15*Cos[7*(c + d*x)] - 210*Sin[2*(c + d*x)] - 210*Sin[4*(c + d*x)] + 70*Sin[6*(c + d*x)]))/(6720*d)","A",1
276,1,76,103,0.4481822,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (-15 \sin (2 (c+d x))-45 \sin (4 (c+d x))+5 \sin (6 (c+d x))-240 \cos (c+d x)-40 \cos (3 (c+d x))+24 \cos (5 (c+d x))+180 c+180 d x)}{960 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,"(a^2*(180*c + 180*d*x - 240*Cos[c + d*x] - 40*Cos[3*(c + d*x)] + 24*Cos[5*(c + d*x)] - 15*Sin[2*(c + d*x)] - 45*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)","A",1
277,1,57,91,0.1989437,"\int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (-90 \cos (c+d x)-25 \cos (3 (c+d x))+3 (-5 \sin (4 (c+d x))+\cos (5 (c+d x))+20 c+20 d x))}{240 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*(-90*Cos[c + d*x] - 25*Cos[3*(c + d*x)] + 3*(20*c + 20*d*x + Cos[5*(c + d*x)] - 5*Sin[4*(c + d*x)])))/(240*d)","A",1
278,1,71,71,0.3810023,"\int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(9 \cos (c+d x)-\cos (3 (c+d x))+6 \left(\sin (2 (c+d x))+2 \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)\right)\right)}{12 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 (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+a^2 x",1,"(a^2*(9*Cos[c + d*x] - Cos[3*(c + d*x)] + 6*(2*(c + d*x - Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]) + Sin[2*(c + d*x)])))/(12*d)","A",1
279,1,94,74,0.5906772,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(7 \cos (c+d x)+\cos (3 (c+d x))+4 \sin (c+d x) \left(-4 \cos (c+d x)-4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)\right)}{16 d}","\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,"-1/16*(a^2*Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(7*Cos[c + d*x] + Cos[3*(c + d*x)] + 4*(c + d*x - 4*Cos[c + d*x] + 4*Log[Cos[(c + d*x)/2]] - 4*Log[Sin[(c + d*x)/2]])*Sin[c + d*x]))/d","A",1
280,1,102,73,0.6734332,"\int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(8 \cos (c+d x)+8 \tan \left(\frac{1}{2} (c+d x)\right)-8 \cot \left(\frac{1}{2} (c+d x)\right)-\csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^2\left(\frac{1}{2} (c+d x)\right)+4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-16 c-16 d x\right)}{8 d}","\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,"(a^2*(-16*c - 16*d*x + 8*Cos[c + d*x] - 8*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 - 4*Log[Cos[(c + d*x)/2]] + 4*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 + 8*Tan[(c + d*x)/2]))/(8*d)","A",1
281,1,140,73,0.5801929,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \left(-8 \tan \left(\frac{1}{2} (c+d x)\right)+8 \cot \left(\frac{1}{2} (c+d x)\right)+6 \csc ^2\left(\frac{1}{2} (c+d x)\right)-6 \sec ^2\left(\frac{1}{2} (c+d x)\right)+24 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-24 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{1}{2} \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-8 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+24 c+24 d x\right)}{24 d}","-\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,"-1/24*(a^2*(24*c + 24*d*x + 8*Cot[(c + d*x)/2] + 6*Csc[(c + d*x)/2]^2 - 24*Log[Cos[(c + d*x)/2]] + 24*Log[Sin[(c + d*x)/2]] - 6*Sec[(c + d*x)/2]^2 - 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + (Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 - 8*Tan[(c + d*x)/2]))/d","A",1
282,1,209,82,0.1120109,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","a^2 \left(-\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{3 d}+\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{3 d}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{3 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{12 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{12 d}\right)","-\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,"a^2*(Cot[(c + d*x)/2]/(3*d) - (3*Csc[(c + d*x)/2]^2)/(32*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(12*d) - Csc[(c + d*x)/2]^4/(64*d) + (5*Log[Cos[(c + d*x)/2]])/(8*d) - (5*Log[Sin[(c + d*x)/2]])/(8*d) + (3*Sec[(c + d*x)/2]^2)/(32*d) + Sec[(c + d*x)/2]^4/(64*d) - Tan[(c + d*x)/2]/(3*d) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(12*d))","B",1
283,1,189,100,0.7545693,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc ^5(c+d x) \left(180 \sin (2 (c+d x))+30 \sin (4 (c+d x))+200 \cos (c+d x)+20 \cos (3 (c+d x))-28 \cos (5 (c+d x))+150 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-75 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-150 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+75 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-15 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{960 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,"-1/960*(a^2*Csc[c + d*x]^5*(200*Cos[c + d*x] + 20*Cos[3*(c + d*x)] - 28*Cos[5*(c + d*x)] - 150*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 150*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 180*Sin[2*(c + d*x)] + 75*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 75*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 30*Sin[4*(c + d*x)] - 15*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 15*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/d","A",1
284,1,229,124,0.7164338,"\int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \csc ^6(c+d x) \left(-960 \sin (2 (c+d x))-384 \sin (4 (c+d x))+64 \sin (6 (c+d x))-1500 \cos (c+d x)+130 \cos (3 (c+d x))+90 \cos (5 (c+d x))-450 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-675 \cos (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+270 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-45 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+450 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+675 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-270 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+45 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{7680 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,"(a^2*Csc[c + d*x]^6*(-1500*Cos[c + d*x] + 130*Cos[3*(c + d*x)] + 90*Cos[5*(c + d*x)] + 450*Log[Cos[(c + d*x)/2]] - 675*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 270*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 45*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 450*Log[Sin[(c + d*x)/2]] + 675*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 270*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 45*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 960*Sin[2*(c + d*x)] - 384*Sin[4*(c + d*x)] + 64*Sin[6*(c + d*x)]))/(7680*d)","A",1
285,1,86,132,0.6700901,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (-63 \sin (2 (c+d x))-105 \sin (4 (c+d x))+21 \sin (6 (c+d x))-609 \cos (c+d x)-91 \cos (3 (c+d x))+63 \cos (5 (c+d x))-3 \cos (7 (c+d x))+420 c+420 d x)}{1344 d}","-\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,"(a^3*(420*c + 420*d*x - 609*Cos[c + d*x] - 91*Cos[3*(c + d*x)] + 63*Cos[5*(c + d*x)] - 3*Cos[7*(c + d*x)] - 63*Sin[2*(c + d*x)] - 105*Sin[4*(c + d*x)] + 21*Sin[6*(c + d*x)]))/(1344*d)","A",1
286,1,76,117,0.3791609,"\int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (-15 \sin (2 (c+d x))-105 \sin (4 (c+d x))+5 \sin (6 (c+d x))-600 \cos (c+d x)-140 \cos (3 (c+d x))+36 \cos (5 (c+d x))+450 c+420 d x)}{960 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,"(a^3*(450*c + 420*d*x - 600*Cos[c + d*x] - 140*Cos[3*(c + d*x)] + 36*Cos[5*(c + d*x)] - 15*Sin[2*(c + d*x)] - 105*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)","A",1
287,1,82,99,0.6575816,"\int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(24 \sin (2 (c+d x))-\sin (4 (c+d x))+8 \cos (c+d x)-8 \cos (3 (c+d x))+32 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-32 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+52 c+52 d x\right)}{32 d}","-\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,"(a^3*(52*c + 52*d*x + 8*Cos[c + d*x] - 8*Cos[3*(c + d*x)] - 32*Log[Cos[(c + d*x)/2]] + 32*Log[Sin[(c + d*x)/2]] + 24*Sin[2*(c + d*x)] - Sin[4*(c + d*x)]))/(32*d)","A",1
288,1,106,92,1.1295704,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left((15-66 \sin (c+d x)) \cos (c+d x)+(2 \sin (c+d x)+9) \cos (3 (c+d x))-12 \sin (c+d x) \left(6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)\right)}{48 d}","-\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,"-1/48*(a^3*Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Cos[c + d*x]*(15 - 66*Sin[c + d*x]) - 12*(c + d*x - 6*Log[Cos[(c + d*x)/2]] + 6*Log[Sin[(c + d*x)/2]])*Sin[c + d*x] + Cos[3*(c + d*x)]*(9 + 2*Sin[c + d*x])))/d","A",1
289,1,112,98,1.0403909,"\int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(2 \sin (2 (c+d x))+24 \cos (c+d x)+12 \tan \left(\frac{1}{2} (c+d x)\right)-12 \cot \left(\frac{1}{2} (c+d x)\right)-\csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^2\left(\frac{1}{2} (c+d x)\right)+20 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-20 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-20 c-20 d x\right)}{8 d}","\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,"(a^3*(-20*c - 20*d*x + 24*Cos[c + d*x] - 12*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 - 20*Log[Cos[(c + d*x)/2]] + 20*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 + 2*Sin[2*(c + d*x)] + 12*Tan[(c + d*x)/2]))/(8*d)","A",1
290,1,148,91,0.42732,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(24 \cos (c+d x)+32 \tan \left(\frac{1}{2} (c+d x)\right)-32 \cot \left(\frac{1}{2} (c+d x)\right)-9 \csc ^2\left(\frac{1}{2} (c+d x)\right)+9 \sec ^2\left(\frac{1}{2} (c+d x)\right)-12 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{1}{2} \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+8 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-72 c-72 d x\right)}{24 d}","\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,"(a^3*(-72*c - 72*d*x + 24*Cos[c + d*x] - 32*Cot[(c + d*x)/2] - 9*Csc[(c + d*x)/2]^2 + 12*Log[Cos[(c + d*x)/2]] - 12*Log[Sin[(c + d*x)/2]] + 9*Sec[(c + d*x)/2]^2 + 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - (Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 + 32*Tan[(c + d*x)/2]))/(24*d)","A",1
291,1,133,100,0.5580259,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(-22 \csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^4\left(\frac{1}{2} (c+d x)\right)+22 \sec ^2\left(\frac{1}{2} (c+d x)\right)-\left((4 \sin (c+d x)+1) \csc ^4\left(\frac{1}{2} (c+d x)\right)\right)-8 \left(13 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-13 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-8 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+8 c+8 d x\right)\right)}{64 d}","-\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*(-22*Csc[(c + d*x)/2]^2 + 22*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^4 - 8*(8*c + 8*d*x - 13*Log[Cos[(c + d*x)/2]] + 13*Log[Sin[(c + d*x)/2]] - 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4) - Csc[(c + d*x)/2]^4*(1 + 4*Sin[c + d*x])))/(64*d)","A",1
292,1,267,100,0.1271415,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","a^3 \left(-\frac{17 \tan \left(\frac{1}{2} (c+d x)\right)}{30 d}+\frac{17 \cot \left(\frac{1}{2} (c+d x)\right)}{30 d}-\frac{3 \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{7 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{7 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^4\left(\frac{1}{2} (c+d x)\right)}{160 d}-\frac{59 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{480 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)}{160 d}+\frac{59 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{480 d}\right)","-\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,"a^3*((17*Cot[(c + d*x)/2])/(30*d) - Csc[(c + d*x)/2]^2/(32*d) - (59*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(480*d) - (3*Csc[(c + d*x)/2]^4)/(64*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(160*d) + (7*Log[Cos[(c + d*x)/2]])/(8*d) - (7*Log[Sin[(c + d*x)/2]])/(8*d) + Sec[(c + d*x)/2]^2/(32*d) + (3*Sec[(c + d*x)/2]^4)/(64*d) - (17*Tan[(c + d*x)/2])/(30*d) + (59*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(480*d) + (Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(160*d))","B",1
293,1,252,124,3.5900017,"\int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \sin (c+d x) (\sin (c+d x)+1)^3 \left(\csc ^6\left(\frac{1}{2} (c+d x)\right) (5 \csc (c+d x)+18)+\csc ^4\left(\frac{1}{2} (c+d x)\right) (90 \csc (c+d x)+34)-2 \csc ^2\left(\frac{1}{2} (c+d x)\right) (105 \csc (c+d x)+176)+(159 \cos (c+d x)+44 \cos (2 (c+d x))+97) \sec ^6\left(\frac{1}{2} (c+d x)\right)-320 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^7(c+d x)-1440 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+840 \sin ^2\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-840 \csc (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{1920 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"-1/1920*(a^3*(Csc[(c + d*x)/2]^6*(18 + 5*Csc[c + d*x]) + Csc[(c + d*x)/2]^4*(34 + 90*Csc[c + d*x]) - 2*Csc[(c + d*x)/2]^2*(176 + 105*Csc[c + d*x]) - 840*Csc[c + d*x]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + (97 + 159*Cos[c + d*x] + 44*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^6 + 840*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 - 1440*Csc[c + d*x]^5*Sin[(c + d*x)/2]^4 - 320*Csc[c + d*x]^7*Sin[(c + d*x)/2]^6)*Sin[c + d*x]*(1 + Sin[c + d*x])^3)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","B",1
294,1,151,137,0.4425778,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","-\frac{a^4 \left(630 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+\sqrt{\sin (c+d x)+1} \left(40 \sin ^6(c+d x)+152 \sin ^5(c+d x)+158 \sin ^4(c+d x)-94 \sin ^3(c+d x)-331 \sin ^2(c+d x)-373 \sin (c+d x)+448\right)\right) \cos ^3(c+d x)}{240 d (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^{3/2}}","-\frac{7 a^4 \cos ^3(c+d x)}{8 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{3 \cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^2}{10 d}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d}",1,"-1/240*(a^4*Cos[c + d*x]^3*(630*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(448 - 373*Sin[c + d*x] - 331*Sin[c + d*x]^2 - 94*Sin[c + d*x]^3 + 158*Sin[c + d*x]^4 + 152*Sin[c + d*x]^5 + 40*Sin[c + d*x]^6)))/(d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(3/2))","A",1
295,1,95,117,0.8779586,"\int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \left(-150 \cos (c+d x)-125 \cos (3 (c+d x))+3 \cos (5 (c+d x))+30 \left(8 \sin (2 (c+d x))-\sin (4 (c+d x))+8 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+20 c+20 d x\right)\right)}{240 d}","\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,"(a^4*(-150*Cos[c + d*x] - 125*Cos[3*(c + d*x)] + 3*Cos[5*(c + d*x)] + 30*(20*c + 20*d*x - 8*Log[Cos[(c + d*x)/2]] + 8*Log[Sin[(c + d*x)/2]] + 8*Sin[2*(c + d*x)] - Sin[4*(c + d*x)])))/(240*d)","A",1
296,1,136,116,1.6256377,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(408 c \sin (c+d x)+408 d x \sin (c+d x)+320 \sin (2 (c+d x))-32 \sin (4 (c+d x))-48 \cos (c+d x)-147 \cos (3 (c+d x))+3 \cos (5 (c+d x))+768 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-768 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{384 d}","-\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,"(a^4*Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(-48*Cos[c + d*x] - 147*Cos[3*(c + d*x)] + 3*Cos[5*(c + d*x)] + 408*c*Sin[c + d*x] + 408*d*x*Sin[c + d*x] - 768*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 768*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 320*Sin[2*(c + d*x)] - 32*Sin[4*(c + d*x)]))/(384*d)","A",1
297,1,281,104,5.2380084,"\int \frac{\cos ^2(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{1}{480} \left(\frac{60 \sin ^2\left(\frac{1}{2} (c+d x)\right)}{d (a \sin (c+d x)+a)}-\frac{300 \sin (c) \sin (d x)}{a d}+\frac{50 \sin (3 c) \sin (3 d x)}{a d}-\frac{6 \sin (5 c) \sin (5 d x)}{a d}+\frac{30 \sin (c+d x)}{a d (\sin (c+d x)+1)}+\frac{300 \cos (c) \cos (d x)}{a d}-\frac{50 \cos (3 c) \cos (3 d x)}{a d}+\frac{6 \cos (5 c) \cos (5 d x)}{a d}-\frac{120 \sin (2 c) \cos (2 d x)}{a d}+\frac{15 \sin (4 c) \cos (4 d x)}{a d}-\frac{120 \cos (2 c) \sin (2 d x)}{a d}+\frac{15 \cos (4 c) \sin (4 d x)}{a d}-\frac{60 \sin \left(\frac{d x}{2}\right)}{a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{180 x}{a}\right)","\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,"((180*x)/a + (300*Cos[c]*Cos[d*x])/(a*d) - (50*Cos[3*c]*Cos[3*d*x])/(a*d) + (6*Cos[5*c]*Cos[5*d*x])/(a*d) - (120*Cos[2*d*x]*Sin[2*c])/(a*d) + (15*Cos[4*d*x]*Sin[4*c])/(a*d) - (300*Sin[c]*Sin[d*x])/(a*d) - (120*Cos[2*c]*Sin[2*d*x])/(a*d) + (50*Sin[3*c]*Sin[3*d*x])/(a*d) + (15*Cos[4*c]*Sin[4*d*x])/(a*d) - (6*Sin[5*c]*Sin[5*d*x])/(a*d) - (60*Sin[(d*x)/2])/(a*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (30*Sin[c + d*x])/(a*d*(1 + Sin[c + d*x])) + (60*Sin[(c + d*x)/2]^2)/(d*(a + a*Sin[c + d*x])))/480","B",1
298,1,271,87,1.7224735,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{-72 d x \sin \left(\frac{c}{2}\right)+72 \sin \left(\frac{c}{2}+d x\right)-72 \sin \left(\frac{3 c}{2}+d x\right)+24 \sin \left(\frac{3 c}{2}+2 d x\right)+24 \sin \left(\frac{5 c}{2}+2 d x\right)-8 \sin \left(\frac{5 c}{2}+3 d x\right)+8 \sin \left(\frac{7 c}{2}+3 d x\right)-3 \sin \left(\frac{7 c}{2}+4 d x\right)-3 \sin \left(\frac{9 c}{2}+4 d x\right)+24 \cos \left(\frac{c}{2}\right) (c-3 d x)-72 \cos \left(\frac{c}{2}+d x\right)-72 \cos \left(\frac{3 c}{2}+d x\right)+24 \cos \left(\frac{3 c}{2}+2 d x\right)-24 \cos \left(\frac{5 c}{2}+2 d x\right)+8 \cos \left(\frac{5 c}{2}+3 d x\right)+8 \cos \left(\frac{7 c}{2}+3 d x\right)-3 \cos \left(\frac{7 c}{2}+4 d x\right)+3 \cos \left(\frac{9 c}{2}+4 d x\right)+24 c \sin \left(\frac{c}{2}\right)-48 \sin \left(\frac{c}{2}\right)}{192 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","\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,"(24*(c - 3*d*x)*Cos[c/2] - 72*Cos[c/2 + d*x] - 72*Cos[(3*c)/2 + d*x] + 24*Cos[(3*c)/2 + 2*d*x] - 24*Cos[(5*c)/2 + 2*d*x] + 8*Cos[(5*c)/2 + 3*d*x] + 8*Cos[(7*c)/2 + 3*d*x] - 3*Cos[(7*c)/2 + 4*d*x] + 3*Cos[(9*c)/2 + 4*d*x] - 48*Sin[c/2] + 24*c*Sin[c/2] - 72*d*x*Sin[c/2] + 72*Sin[c/2 + d*x] - 72*Sin[(3*c)/2 + d*x] + 24*Sin[(3*c)/2 + 2*d*x] + 24*Sin[(5*c)/2 + 2*d*x] - 8*Sin[(5*c)/2 + 3*d*x] + 8*Sin[(7*c)/2 + 3*d*x] - 3*Sin[(7*c)/2 + 4*d*x] - 3*Sin[(9*c)/2 + 4*d*x])/(192*a*d*(Cos[c/2] + Sin[c/2]))","B",1
299,1,46,62,0.086835,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{-3 \sin (2 (c+d x))+9 \cos (c+d x)-\cos (3 (c+d x))+6 c+6 d x}{12 a d}","-\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,"(6*c + 6*d*x + 9*Cos[c + d*x] - Cos[3*(c + d*x)] - 3*Sin[2*(c + d*x)])/(12*a*d)","A",1
300,1,161,45,0.5883968,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{-4 d x \sin \left(\frac{c}{2}\right)+4 \sin \left(\frac{c}{2}+d x\right)-4 \sin \left(\frac{3 c}{2}+d x\right)+\sin \left(\frac{3 c}{2}+2 d x\right)+\sin \left(\frac{5 c}{2}+2 d x\right)+2 \cos \left(\frac{c}{2}\right) (c-2 d x)-4 \cos \left(\frac{c}{2}+d x\right)-4 \cos \left(\frac{3 c}{2}+d x\right)+\cos \left(\frac{3 c}{2}+2 d x\right)-\cos \left(\frac{5 c}{2}+2 d x\right)+2 c \sin \left(\frac{c}{2}\right)-4 \sin \left(\frac{c}{2}\right)}{8 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\frac{\cos (c+d x)}{a d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x}{2 a}",1,"(2*(c - 2*d*x)*Cos[c/2] - 4*Cos[c/2 + d*x] - 4*Cos[(3*c)/2 + d*x] + Cos[(3*c)/2 + 2*d*x] - Cos[(5*c)/2 + 2*d*x] - 4*Sin[c/2] + 2*c*Sin[c/2] - 4*d*x*Sin[c/2] + 4*Sin[c/2 + d*x] - 4*Sin[(3*c)/2 + d*x] + Sin[(3*c)/2 + 2*d*x] + Sin[(5*c)/2 + 2*d*x])/(8*a*d*(Cos[c/2] + Sin[c/2]))","B",1
301,1,37,22,0.0919119,"\int \frac{\cos (c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x}{a d}","-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{a}",1,"-((c + d*x + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])/(a*d))","A",1
302,1,69,29,0.2388432,"\int \frac{\cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^2/(a + a*Sin[c + d*x]),x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\cos (c+d x)+\sin (c+d x) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{2 a d}","\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\cot (c+d x)}{a d}",1,"-1/2*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Cos[c + d*x] + (-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]])*Sin[c + d*x]))/(a*d)","B",1
303,1,94,53,0.4150946,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(\sin (2 (c+d x))-\cos (c+d x)+\sin ^2(c+d x) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{8 a d (\sin (c+d x)+1)}","\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,"((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(-Cos[c + d*x] + (-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^2 + Sin[2*(c + d*x)]))/(8*a*d*(1 + Sin[c + d*x]))","A",1
304,1,126,72,0.5978836,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-12 (\sin (c+d x)-1) \cos (c+d x)-4 \left(\cos (3 (c+d x))+3 \sin ^3(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}{192 a d (\sin (c+d x)+1)}","-\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,"-1/192*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(-12*Cos[c + d*x]*(-1 + Sin[c + d*x]) - 4*(Cos[3*(c + d*x)] + 3*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^3)))/(a*d*(1 + Sin[c + d*x]))","A",1
305,1,125,95,1.1288103,"\int \frac{\cot ^2(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^4(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-48 \sin (2 (c+d x))+66 \cos (c+d x)+2 (16 \sin (c+d x)-9) \cos (3 (c+d x))+72 \sin ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{192 a d (\sin (c+d x)+1)}","\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,"-1/192*(Csc[c + d*x]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(66*Cos[c + d*x] + 72*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^4 + 2*Cos[3*(c + d*x)]*(-9 + 16*Sin[c + d*x]) - 48*Sin[2*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","A",1
306,1,189,114,0.6581225,"\int \frac{\cot ^2(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\csc ^5(c+d x) \left(420 \sin (2 (c+d x))-90 \sin (4 (c+d x))-640 \cos (c+d x)+320 \cos (3 (c+d x))-64 \cos (5 (c+d x))-450 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+225 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-45 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+450 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-225 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+45 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1920 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,"(Csc[c + d*x]^5*(-640*Cos[c + d*x] + 320*Cos[3*(c + d*x)] - 64*Cos[5*(c + d*x)] + 450*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 450*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 420*Sin[2*(c + d*x)] - 225*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 225*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 90*Sin[4*(c + d*x)] + 45*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 45*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(1920*a*d)","A",1
307,1,209,111,1.489209,"\int \frac{\cos ^2(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{-648 d x \sin \left(c+\frac{d x}{2}\right)+4 \sin \left(c+\frac{d x}{2}\right)-264 \sin \left(2 c+\frac{3 d x}{2}\right)+56 \sin \left(2 c+\frac{5 d x}{2}\right)+13 \sin \left(4 c+\frac{7 d x}{2}\right)-3 \sin \left(4 c+\frac{9 d x}{2}\right)-340 \cos \left(c+\frac{d x}{2}\right)-264 \cos \left(c+\frac{3 d x}{2}\right)-56 \cos \left(3 c+\frac{5 d x}{2}\right)+13 \cos \left(3 c+\frac{7 d x}{2}\right)+3 \cos \left(5 c+\frac{9 d x}{2}\right)+1100 \sin \left(\frac{d x}{2}\right)+(4-648 d x) \cos \left(\frac{d x}{2}\right)}{192 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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,"((4 - 648*d*x)*Cos[(d*x)/2] - 340*Cos[c + (d*x)/2] - 264*Cos[c + (3*d*x)/2] - 56*Cos[3*c + (5*d*x)/2] + 13*Cos[3*c + (7*d*x)/2] + 3*Cos[5*c + (9*d*x)/2] + 1100*Sin[(d*x)/2] + 4*Sin[c + (d*x)/2] - 648*d*x*Sin[c + (d*x)/2] - 264*Sin[2*c + (3*d*x)/2] + 56*Sin[2*c + (5*d*x)/2] + 13*Sin[4*c + (7*d*x)/2] - 3*Sin[4*c + (9*d*x)/2])/(192*a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
308,1,165,83,0.9699704,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{-72 d x \sin \left(c+\frac{d x}{2}\right)-27 \sin \left(2 c+\frac{3 d x}{2}\right)+5 \sin \left(2 c+\frac{5 d x}{2}\right)+\sin \left(4 c+\frac{7 d x}{2}\right)-31 \cos \left(c+\frac{d x}{2}\right)-27 \cos \left(c+\frac{3 d x}{2}\right)-5 \cos \left(3 c+\frac{5 d x}{2}\right)+\cos \left(3 c+\frac{7 d x}{2}\right)+131 \sin \left(\frac{d x}{2}\right)-72 d x \cos \left(\frac{d x}{2}\right)}{24 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\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,"-1/24*(-72*d*x*Cos[(d*x)/2] - 31*Cos[c + (d*x)/2] - 27*Cos[c + (3*d*x)/2] - 5*Cos[3*c + (5*d*x)/2] + Cos[3*c + (7*d*x)/2] + 131*Sin[(d*x)/2] - 72*d*x*Sin[c + (d*x)/2] - 27*Sin[2*c + (3*d*x)/2] + 5*Sin[2*c + (5*d*x)/2] + Sin[4*c + (7*d*x)/2])/(a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
309,1,69,69,0.152475,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{-10 (c+d x)+\sin (2 (c+d x))-8 \cos (c+d x)+\frac{16 \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}}{4 a^2 d}","-\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,"(-10*(c + d*x) - 8*Cos[c + d*x] + (16*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + Sin[2*(c + d*x)])/(4*a^2*d)","A",1
310,1,117,47,0.3270721,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{12 d x \sin \left(c+\frac{d x}{2}\right)+3 \sin \left(2 c+\frac{3 d x}{2}\right)+2 \cos \left(c+\frac{d x}{2}\right)+3 \cos \left(c+\frac{3 d x}{2}\right)-28 \sin \left(\frac{d x}{2}\right)+12 d x \cos \left(\frac{d x}{2}\right)}{6 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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,"(12*d*x*Cos[(d*x)/2] + 2*Cos[c + (d*x)/2] + 3*Cos[c + (3*d*x)/2] - 28*Sin[(d*x)/2] + 12*d*x*Sin[c + (d*x)/2] + 3*Sin[2*c + (3*d*x)/2])/(6*a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
311,1,115,40,0.1579216,"\int \frac{\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sin \left(\frac{1}{2} (c+d x)\right) \left(-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+4\right)\right)}{a^2 d (\sin (c+d x)+1)^2}","\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,"-(((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + (4 + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[(c + d*x)/2]))/(a^2*d*(1 + Sin[c + d*x])^2))","B",1
312,1,216,54,0.7981758,"\int \frac{\cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^2/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\cos \left(\frac{3}{2} (c+d x)\right) \left(-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+5\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left(2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-3\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \left(2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\cos (c+d x) \left(2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+1\right)\right)\right)}{4 a^2 d (\sin (c+d x)+1)^2}","-\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,"-1/4*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(Cos[(3*(c + d*x))/2]*(5 + 2*Log[Cos[(c + d*x)/2]] - 2*Log[Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*(-3 - 2*Log[Cos[(c + d*x)/2]] + 2*Log[Sin[(c + d*x)/2]]) + 2*(-2*Log[Cos[(c + d*x)/2]] + 2*Log[Sin[(c + d*x)/2]] + Cos[c + d*x]*(1 - 2*Log[Cos[(c + d*x)/2]] + 2*Log[Sin[(c + d*x)/2]]))*Sin[(c + d*x)/2]))/(a^2*d*(1 + Sin[c + d*x])^2)","B",1
313,1,364,78,0.784655,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{4 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}{d (a \sin (c+d x)+a)^2}-\frac{5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{2 d (a \sin (c+d x)+a)^2}+\frac{5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{2 d (a \sin (c+d x)+a)^2}-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{d (a \sin (c+d x)+a)^2}+\frac{\cot \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{d (a \sin (c+d x)+a)^2}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{8 d (a \sin (c+d x)+a)^2}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{8 d (a \sin (c+d x)+a)^2}","\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,"(-4*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)/(d*(a + a*Sin[c + d*x])^2) + (Cot[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(d*(a + a*Sin[c + d*x])^2) - (Csc[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(8*d*(a + a*Sin[c + d*x])^2) - (5*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(2*d*(a + a*Sin[c + d*x])^2) + (5*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(2*d*(a + a*Sin[c + d*x])^2) + (Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(8*d*(a + a*Sin[c + d*x])^2) - ((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*Tan[(c + d*x)/2])/(d*(a + a*Sin[c + d*x])^2)","B",1
314,1,472,91,1.2920504,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(12 \sin \left(\frac{1}{2} (c+d x)\right)-6 \sin \left(\frac{3}{2} (c+d x)\right)-2 \sin \left(\frac{5}{2} (c+d x)\right)+8 \sin \left(\frac{7}{2} (c+d x)\right)-10 \cos \left(\frac{5}{2} (c+d x)\right)+20 \cos \left(\frac{7}{2} (c+d x)\right)-27 \sin \left(\frac{1}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-27 \sin \left(\frac{3}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 \sin \left(\frac{5}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 \sin \left(\frac{7}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-9 \cos \left(\frac{5}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+9 \cos \left(\frac{7}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+9 \cos \left(\frac{5}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right) \left(-9 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+8\right)-3 \cos \left(\frac{3}{2} (c+d x)\right) \left(-9 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+14\right)-9 \cos \left(\frac{7}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+27 \sin \left(\frac{1}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+27 \sin \left(\frac{3}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-9 \sin \left(\frac{5}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-9 \sin \left(\frac{7}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 a^2 d (\sin (c+d x)+1)^2}","-\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,"((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^3*(-10*Cos[(5*(c + d*x))/2] + 20*Cos[(7*(c + d*x))/2] - 9*Cos[(5*(c + d*x))/2]*Log[Cos[(c + d*x)/2]] + 9*Cos[(7*(c + d*x))/2]*Log[Cos[(c + d*x)/2]] + 3*Cos[(c + d*x)/2]*(8 + 9*Log[Cos[(c + d*x)/2]] - 9*Log[Sin[(c + d*x)/2]]) - 3*Cos[(3*(c + d*x))/2]*(14 + 9*Log[Cos[(c + d*x)/2]] - 9*Log[Sin[(c + d*x)/2]]) + 9*Cos[(5*(c + d*x))/2]*Log[Sin[(c + d*x)/2]] - 9*Cos[(7*(c + d*x))/2]*Log[Sin[(c + d*x)/2]] + 12*Sin[(c + d*x)/2] + 27*Log[Cos[(c + d*x)/2]]*Sin[(c + d*x)/2] - 27*Log[Sin[(c + d*x)/2]]*Sin[(c + d*x)/2] - 6*Sin[(3*(c + d*x))/2] + 27*Log[Cos[(c + d*x)/2]]*Sin[(3*(c + d*x))/2] - 27*Log[Sin[(c + d*x)/2]]*Sin[(3*(c + d*x))/2] - 2*Sin[(5*(c + d*x))/2] - 9*Log[Cos[(c + d*x)/2]]*Sin[(5*(c + d*x))/2] + 9*Log[Sin[(c + d*x)/2]]*Sin[(5*(c + d*x))/2] + 8*Sin[(7*(c + d*x))/2] - 9*Log[Cos[(c + d*x)/2]]*Sin[(7*(c + d*x))/2] + 9*Log[Sin[(c + d*x)/2]]*Sin[(7*(c + d*x))/2]))/(192*a^2*d*(1 + Sin[c + d*x])^2)","B",1
315,1,197,97,1.0892262,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","-\frac{1980 d x \sin \left(c+\frac{d x}{2}\right)+660 d x \sin \left(c+\frac{3 d x}{2}\right)+498 \sin \left(2 c+\frac{3 d x}{2}\right)+135 \sin \left(2 c+\frac{5 d x}{2}\right)+15 \sin \left(4 c+\frac{7 d x}{2}\right)-1326 \cos \left(c+\frac{d x}{2}\right)+2012 \cos \left(c+\frac{3 d x}{2}\right)-660 d x \cos \left(2 c+\frac{3 d x}{2}\right)-135 \cos \left(3 c+\frac{5 d x}{2}\right)+15 \cos \left(3 c+\frac{7 d x}{2}\right)-3216 \sin \left(\frac{d x}{2}\right)+1980 d x \cos \left(\frac{d x}{2}\right)}{240 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^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,"-1/240*(1980*d*x*Cos[(d*x)/2] - 1326*Cos[c + (d*x)/2] + 2012*Cos[c + (3*d*x)/2] - 660*d*x*Cos[2*c + (3*d*x)/2] - 135*Cos[3*c + (5*d*x)/2] + 15*Cos[3*c + (7*d*x)/2] - 3216*Sin[(d*x)/2] + 1980*d*x*Sin[c + (d*x)/2] + 660*d*x*Sin[c + (3*d*x)/2] + 498*Sin[2*c + (3*d*x)/2] + 135*Sin[2*c + (5*d*x)/2] + 15*Sin[4*c + (7*d*x)/2])/(a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","B",1
316,1,96,76,0.6711273,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{3 \cos (c+d x)-\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) (13 \sin (c+d x)+11)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+9 c+9 d x}{3 a^3 d}","\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,"(9*c + 9*d*x + 3*Cos[c + d*x] - 2/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (2*Sin[(c + d*x)/2]*(11 + 13*Sin[c + d*x]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)/(3*a^3*d)","A",1
317,1,145,61,0.4173929,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{180 d x \sin \left(c+\frac{d x}{2}\right)+60 d x \sin \left(c+\frac{3 d x}{2}\right)+3 \sin \left(2 c+\frac{3 d x}{2}\right)-351 \cos \left(c+\frac{d x}{2}\right)+277 \cos \left(c+\frac{3 d x}{2}\right)-60 d x \cos \left(2 c+\frac{3 d x}{2}\right)-471 \sin \left(\frac{d x}{2}\right)+180 d x \cos \left(\frac{d x}{2}\right)}{120 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\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,"-1/120*(180*d*x*Cos[(d*x)/2] - 351*Cos[c + (d*x)/2] + 277*Cos[c + (3*d*x)/2] - 60*d*x*Cos[2*c + (3*d*x)/2] - 471*Sin[(d*x)/2] + 180*d*x*Sin[c + (d*x)/2] + 60*d*x*Sin[c + (3*d*x)/2] + 3*Sin[2*c + (3*d*x)/2])/(a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","B",1
318,1,185,68,0.3929226,"\int \frac{\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(-4 \sin \left(\frac{1}{2} (c+d x)\right)-10 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3+3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3\right)}{3 d (a \sin (c+d x)+a)^3}","\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(-4*Sin[(c + d*x)/2] + 2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 10*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 3*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 3*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3))/(3*d*(a + a*Sin[c + d*x])^3)","B",1
319,1,255,82,1.5639016,"\int \frac{\cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^2/(a + a*Sin[c + d*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(8 \sin \left(\frac{1}{2} (c+d x)\right)+44 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2-4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+18 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3-18 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3+3 \tan \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3-3 \cot \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3\right)}{6 d (a \sin (c+d x)+a)^3}","-\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(8*Sin[(c + d*x)/2] - 4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 44*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 3*Cot[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 18*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 18*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*Tan[(c + d*x)/2]))/(6*d*(a + a*Sin[c + d*x])^3)","B",1
320,1,308,106,5.9461164,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(-32 \sin \left(\frac{1}{2} (c+d x)\right)-272 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+16 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right)+1\right)^3-3 \sin \left(\frac{1}{2} (c+d x)\right) \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right)^3-132 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3+132 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3-36 \tan \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3+36 \cot \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3\right)}{24 a^3 d (\sin (c+d x)+1)^3}","\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(-32*Sin[(c + d*x)/2] - 3*(1 + Cot[(c + d*x)/2])^3*Sin[(c + d*x)/2] + 16*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 272*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 36*Cot[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 132*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 132*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 36*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*Tan[(c + d*x)/2] + 3*Cos[(c + d*x)/2]*(1 + Tan[(c + d*x)/2])^3))/(24*a^3*d*(1 + Sin[c + d*x])^3)","B",1
321,1,171,144,0.9643312,"\int \frac{\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx","Integrate[(Cos[e + f*x]^2*Sin[e + f*x])/(a + a*Sin[e + f*x])^6,x]","-\frac{2562 \sin \left(2 e+\frac{3 f x}{2}\right)-900 \sin \left(2 e+\frac{5 f x}{2}\right)-27 \sin \left(4 e+\frac{7 f x}{2}\right)+25 \sin \left(4 e+\frac{9 f x}{2}\right)+378 \cos \left(e+\frac{f x}{2}\right)+210 \cos \left(e+\frac{3 f x}{2}\right)-108 \cos \left(3 e+\frac{5 f x}{2}\right)+225 \cos \left(3 e+\frac{7 f x}{2}\right)+3 \cos \left(5 e+\frac{9 f x}{2}\right)+3150 \sin \left(\frac{f x}{2}\right)}{13860 a^6 f \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^9}","\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,"-1/13860*(378*Cos[e + (f*x)/2] + 210*Cos[e + (3*f*x)/2] - 108*Cos[3*e + (5*f*x)/2] + 225*Cos[3*e + (7*f*x)/2] + 3*Cos[5*e + (9*f*x)/2] + 3150*Sin[(f*x)/2] + 2562*Sin[2*e + (3*f*x)/2] - 900*Sin[2*e + (5*f*x)/2] - 27*Sin[4*e + (7*f*x)/2] + 25*Sin[4*e + (9*f*x)/2])/(a^6*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^9)","A",1
322,1,109,193,1.2304466,"\int \cos ^2(c+d x) \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (7638 \sin (c+d x)-1330 \sin (3 (c+d x))-3540 \cos (2 (c+d x))+315 \cos (4 (c+d x))+5657)}{13860 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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,"-1/13860*((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*Sqrt[a*(1 + Sin[c + d*x])]*(5657 - 3540*Cos[2*(c + d*x)] + 315*Cos[4*(c + d*x)] + 7638*Sin[c + d*x] - 1330*Sin[3*(c + d*x)]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
323,1,99,124,0.5988168,"\int \cos ^2(c+d x) \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (-69 \sin (c+d x)+7 \sin (3 (c+d x))+30 \cos (2 (c+d x))-62)}{126 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\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,"((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*Sqrt[a*(1 + Sin[c + d*x])]*(-62 + 30*Cos[2*(c + d*x)] - 69*Sin[c + d*x] + 7*Sin[3*(c + d*x)]))/(126*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
324,1,89,92,0.4035839,"\int \cos ^2(c+d x) \sin (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (66 \sin (c+d x)-15 \cos (2 (c+d x))+59)}{105 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\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,"-1/105*((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*Sqrt[a*(1 + Sin[c + d*x])]*(59 - 15*Cos[2*(c + d*x)] + 66*Sin[c + d*x]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
325,1,143,93,0.2111434,"\int \cos (c+d x) \cot (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(-3 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)-3 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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,"(Sqrt[a*(1 + Sin[c + d*x])]*(3*Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2] - 3*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 3*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
326,1,206,89,1.0026115,"\int \cot ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right)+2 \sin \left(\frac{3}{2} (c+d x)\right)-4 \cos \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{3}{2} (c+d x)\right)-\sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+\sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)-\sec \left(\frac{1}{4} (c+d x)\right)\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)+\sec \left(\frac{1}{4} (c+d x)\right)\right)}","\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,"(Csc[(c + d*x)/2]^4*Sqrt[a*(1 + Sin[c + d*x])]*(-4*Cos[(c + d*x)/2] + 2*Cos[(3*(c + d*x))/2] + 4*Sin[(c + d*x)/2] - Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 2*Sin[(3*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4] - Sec[(c + d*x)/4])*(Csc[(c + d*x)/4] + Sec[(c + d*x)/4]))","B",1
327,1,249,101,0.8057299,"\int \cot ^2(c+d x) \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\csc ^7\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-2 \sin \left(\frac{1}{2} (c+d x)\right)+6 \sin \left(\frac{3}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)+6 \cos \left(\frac{3}{2} (c+d x)\right)+5 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-5 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-5 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{4 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^2}","-\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,"-1/4*(Csc[(c + d*x)/2]^7*Sqrt[a*(1 + Sin[c + d*x])]*(2*Cos[(c + d*x)/2] + 6*Cos[(3*(c + d*x))/2] - 5*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 5*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 5*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 5*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2*Sin[(c + d*x)/2] + 6*Sin[(3*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^2)","B",1
328,1,285,137,1.4191203,"\int \cot ^2(c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\csc ^{10}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(12 \sin \left(\frac{1}{2} (c+d x)\right)+58 \sin \left(\frac{3}{2} (c+d x)\right)-18 \sin \left(\frac{5}{2} (c+d x)\right)-12 \cos \left(\frac{1}{2} (c+d x)\right)+58 \cos \left(\frac{3}{2} (c+d x)\right)+18 \cos \left(\frac{5}{2} (c+d x)\right)-27 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+27 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+9 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-9 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{24 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}","\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,"-1/24*(Csc[(c + d*x)/2]^10*Sqrt[a*(1 + Sin[c + d*x])]*(-12*Cos[(c + d*x)/2] + 58*Cos[(3*(c + d*x))/2] + 18*Cos[(5*(c + d*x))/2] + 12*Sin[(c + d*x)/2] - 27*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 27*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 58*Sin[(3*(c + d*x))/2] - 18*Sin[(5*(c + d*x))/2] + 9*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 9*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3)","B",1
329,1,120,233,3.8735045,"\int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (381174 \sin (c+d x)-77665 \sin (3 (c+d x))+3465 \sin (5 (c+d x))-194160 \cos (2 (c+d x))+22680 \cos (4 (c+d x))+281816)}{360360 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\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,"-1/360360*(a*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*Sqrt[a*(1 + Sin[c + d*x])]*(281816 - 194160*Cos[2*(c + d*x)] + 22680*Cos[4*(c + d*x)] + 381174*Sin[c + d*x] - 77665*Sin[3*(c + d*x)] + 3465*Sin[5*(c + d*x)]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
330,1,110,156,1.9192917,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (5076 \sin (c+d x)-700 \sin (3 (c+d x))-2280 \cos (2 (c+d x))+105 \cos (4 (c+d x))+4159)}{4620 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{64 a^3 \cos ^3(c+d x)}{385 d (a \sin (c+d x)+a)^{3/2}}-\frac{48 a^2 \cos ^3(c+d x)}{385 d \sqrt{a \sin (c+d x)+a}}-\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,"-1/4620*(a*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*Sqrt[a*(1 + Sin[c + d*x])]*(4159 - 2280*Cos[2*(c + d*x)] + 105*Cos[4*(c + d*x)] + 5076*Sin[c + d*x] - 700*Sin[3*(c + d*x)]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
331,1,100,124,1.4339231,"\int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","\frac{a \sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (-741 \sin (c+d x)+35 \sin (3 (c+d x))+240 \cos (2 (c+d x))-664)}{630 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{64 a^3 \cos ^3(c+d x)}{315 d (a \sin (c+d x)+a)^{3/2}}-\frac{16 a^2 \cos ^3(c+d x)}{105 d \sqrt{a \sin (c+d x)+a}}-\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,"(a*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*Sqrt[a*(1 + Sin[c + d*x])]*(-664 + 240*Cos[2*(c + d*x)] - 741*Sin[c + d*x] + 35*Sin[3*(c + d*x)]))/(630*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
332,1,145,123,0.2620241,"\int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","\frac{(a (\sin (c+d x)+1))^{3/2} \left(5 \sin \left(\frac{3}{2} (c+d x)\right)+\sin \left(\frac{5}{2} (c+d x)\right)+5 \cos \left(\frac{3}{2} (c+d x)\right)-\cos \left(\frac{5}{2} (c+d x)\right)-10 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+10 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{10 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\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^2 \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\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,"((a*(1 + Sin[c + d*x]))^(3/2)*(5*Cos[(3*(c + d*x))/2] - Cos[(5*(c + d*x))/2] - 10*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 10*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 5*Sin[(3*(c + d*x))/2] + Sin[(5*(c + d*x))/2]))/(10*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","A",1
333,1,233,121,0.7923085,"\int \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-14 \sin \left(\frac{1}{2} (c+d x)\right)-9 \sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{5}{2} (c+d x)\right)+14 \cos \left(\frac{1}{2} (c+d x)\right)-9 \cos \left(\frac{3}{2} (c+d x)\right)+\cos \left(\frac{5}{2} (c+d x)\right)+9 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-9 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{3 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)-\sec \left(\frac{1}{4} (c+d x)\right)\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)+\sec \left(\frac{1}{4} (c+d x)\right)\right)}","-\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{11 a^2 \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\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,"-1/3*(a*Csc[(c + d*x)/2]^4*Sqrt[a*(1 + Sin[c + d*x])]*(14*Cos[(c + d*x)/2] - 9*Cos[(3*(c + d*x))/2] + Cos[(5*(c + d*x))/2] - 14*Sin[(c + d*x)/2] + 9*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 9*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 9*Sin[(3*(c + d*x))/2] - Sin[(5*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4] - Sec[(c + d*x)/4])*(Csc[(c + d*x)/4] + Sec[(c + d*x)/4]))","A",1
334,1,271,131,0.6787559,"\int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \csc ^7\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(22 \sin \left(\frac{1}{2} (c+d x)\right)+22 \sin \left(\frac{3}{2} (c+d x)\right)-8 \sin \left(\frac{5}{2} (c+d x)\right)-22 \cos \left(\frac{1}{2} (c+d x)\right)+22 \cos \left(\frac{3}{2} (c+d x)\right)+8 \cos \left(\frac{5}{2} (c+d x)\right)+\cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{4 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^2}","\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{13 a^2 \cos (c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}-\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,"-1/4*(a*Csc[(c + d*x)/2]^7*Sqrt[a*(1 + Sin[c + d*x])]*(-22*Cos[(c + d*x)/2] + 22*Cos[(3*(c + d*x))/2] + 8*Cos[(5*(c + d*x))/2] - Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 22*Sin[(c + d*x)/2] + 22*Sin[(3*(c + d*x))/2] - 8*Sin[(5*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^2)","B",1
335,1,286,139,0.9199646,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \csc ^{10}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-12 \sin \left(\frac{1}{2} (c+d x)\right)+70 \sin \left(\frac{3}{2} (c+d x)\right)+18 \sin \left(\frac{5}{2} (c+d x)\right)+12 \cos \left(\frac{1}{2} (c+d x)\right)+70 \cos \left(\frac{3}{2} (c+d x)\right)-18 \cos \left(\frac{5}{2} (c+d x)\right)-117 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+117 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+39 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-39 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{24 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}","\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{5 a^2 \cot (c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}-\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,"-1/24*(a*Csc[(c + d*x)/2]^10*Sqrt[a*(1 + Sin[c + d*x])]*(12*Cos[(c + d*x)/2] + 70*Cos[(3*(c + d*x))/2] - 18*Cos[(5*(c + d*x))/2] - 12*Sin[(c + d*x)/2] - 117*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 117*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 70*Sin[(3*(c + d*x))/2] + 18*Sin[(5*(c + d*x))/2] + 39*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 39*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3)","B",1
336,1,97,158,1.2728711,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^3)/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (-201 \sin (c+d x)+35 \sin (3 (c+d x))+60 \cos (2 (c+d x))-124)}{630 d \sqrt{a (\sin (c+d x)+1)}}","-\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,"((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-124 + 60*Cos[2*(c + d*x)] - 201*Sin[c + d*x] + 35*Sin[3*(c + d*x)]))/(630*d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
337,1,87,92,0.3522095,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (24 \sin (c+d x)-15 \cos (2 (c+d x))+31)}{105 d \sqrt{a (\sin (c+d x)+1)}}","-\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,"-1/105*((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(31 - 15*Cos[2*(c + d*x)] + 24*Sin[c + d*x]))/(d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
338,1,77,60,0.4324619,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 (3 \sin (c+d x)+2) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{15 d \sqrt{a (\sin (c+d x)+1)}}","\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*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(2 + 3*Sin[c + d*x]))/(15*d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
339,1,116,63,0.1310373,"\int \frac{\cos (c+d x) \cot (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-2 \sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)-\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{d \sqrt{a (\sin (c+d x)+1)}}","\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*Cos[(c + d*x)/2] - Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2*Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
340,1,138,62,0.3049499,"\int \frac{\cot ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Cot[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\left(\tan \left(\frac{1}{2} (c+d x)\right)+1\right) \csc \left(\frac{1}{4} (c+d x)\right) \sec \left(\frac{1}{4} (c+d x)\right) \left(2 \sin \left(\frac{1}{2} (c+d x)\right)-2 \cos \left(\frac{1}{2} (c+d x)\right)+\sin (c+d x) \left(\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)\right)}{8 d \sqrt{a (\sin (c+d x)+1)}}","\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,"(Csc[(c + d*x)/4]*Sec[(c + d*x)/4]*(-2*Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2] + (Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sin[c + d*x])*(1 + Tan[(c + d*x)/2]))/(8*d*Sqrt[a*(1 + Sin[c + d*x])])","B",1
341,1,272,100,1.8279227,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(4 \tan \left(\frac{1}{4} (c+d x)\right)+4 \cot \left(\frac{1}{4} (c+d x)\right)-\csc ^2\left(\frac{1}{4} (c+d x)\right)+\sec ^2\left(\frac{1}{4} (c+d x)\right)-\frac{8 \sin \left(\frac{1}{4} (c+d x)\right)}{\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}+\frac{8 \sin \left(\frac{1}{4} (c+d x)\right)}{\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)}+\frac{2}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{2}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}+4 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-8\right)}{32 d \sqrt{a (\sin (c+d x)+1)}}","\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-8 + 4*Cot[(c + d*x)/4] - Csc[(c + d*x)/4]^2 + 4*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 4*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sec[(c + d*x)/4]^2 + 2/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 - (8*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - 2/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2 + (8*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) + 4*Tan[(c + d*x)/4]))/(32*d*Sqrt[a*(1 + Sin[c + d*x])])","B",1
342,1,292,135,0.7353916,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\csc ^9\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(60 \sin \left(\frac{1}{2} (c+d x)\right)+2 \sin \left(\frac{3}{2} (c+d x)\right)+6 \sin \left(\frac{5}{2} (c+d x)\right)-60 \cos \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{3}{2} (c+d x)\right)-6 \cos \left(\frac{5}{2} (c+d x)\right)+9 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-9 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-3 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+3 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{24 d \sqrt{a (\sin (c+d x)+1)} \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}","\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,"(Csc[(c + d*x)/2]^9*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-60*Cos[(c + d*x)/2] + 2*Cos[(3*(c + d*x))/2] - 6*Cos[(5*(c + d*x))/2] + 60*Sin[(c + d*x)/2] + 9*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 9*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 2*Sin[(3*(c + d*x))/2] + 6*Sin[(5*(c + d*x))/2] - 3*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 3*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(24*d*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3*Sqrt[a*(1 + Sin[c + d*x])])","B",1
343,1,201,184,1.8777094,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(1365 \sin \left(\frac{1}{2} (c+d x)\right)+245 \sin \left(\frac{3}{2} (c+d x)\right)-63 \sin \left(\frac{5}{2} (c+d x)\right)-15 \sin \left(\frac{7}{2} (c+d x)\right)-1365 \cos \left(\frac{1}{2} (c+d x)\right)+245 \cos \left(\frac{3}{2} (c+d x)\right)+63 \cos \left(\frac{5}{2} (c+d x)\right)-15 \cos \left(\frac{7}{2} (c+d x)\right)+(1680+1680 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{d x}{4}\right) \left(\cos \left(\frac{1}{4} (2 c+d x)\right)-\sin \left(\frac{1}{4} (2 c+d x)\right)\right)\right)\right)}{420 a^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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{76 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{105 a^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,"(Sqrt[a*(1 + Sin[c + d*x])]*((1680 + 1680*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] - Sin[(2*c + d*x)/4])] - 1365*Cos[(c + d*x)/2] + 245*Cos[(3*(c + d*x))/2] + 63*Cos[(5*(c + d*x))/2] - 15*Cos[(7*(c + d*x))/2] + 1365*Sin[(c + d*x)/2] + 245*Sin[(3*(c + d*x))/2] - 63*Sin[(5*(c + d*x))/2] - 15*Sin[(7*(c + d*x))/2]))/(420*a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
344,1,150,140,0.2867314,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(-30 \sin \left(\frac{1}{2} (c+d x)\right)-5 \sin \left(\frac{3}{2} (c+d x)\right)+\sin \left(\frac{5}{2} (c+d x)\right)+30 \cos \left(\frac{1}{2} (c+d x)\right)-5 \cos \left(\frac{3}{2} (c+d x)\right)-\cos \left(\frac{5}{2} (c+d x)\right)+(40+40 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{10 d (a (\sin (c+d x)+1))^{3/2}}","-\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{4 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{5 a^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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*((40 + 40*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + 30*Cos[(c + d*x)/2] - 5*Cos[(3*(c + d*x))/2] - Cos[(5*(c + d*x))/2] - 30*Sin[(c + d*x)/2] - 5*Sin[(3*(c + d*x))/2] + Sin[(5*(c + d*x))/2]))/(10*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
345,1,149,108,0.8954792,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(9 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)-9 \cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)+(12+12 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{d x}{4}\right) \left(\cos \left(\frac{1}{4} (2 c+d x)\right)-\sin \left(\frac{1}{4} (2 c+d x)\right)\right)\right)\right)}{3 a^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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 (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a^2 d}-\frac{10 \cos (c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}",1,"(Sqrt[a*(1 + Sin[c + d*x])]*((12 + 12*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] - Sin[(2*c + d*x)/4])] - 9*Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2] + 9*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(3*a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
346,1,130,85,0.2087022,"\int \frac{\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left((4+4 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)+\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{d (a (\sin (c+d x)+1))^{3/2}}","\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,"-((((4 + 4*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)/(d*(a*(1 + Sin[c + d*x]))^(3/2)))","C",1
347,1,206,113,2.1480165,"\int \frac{\cot ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(-\cot \left(\frac{1}{4} (c+d x)\right)+(16+16 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)+2 \left(\sec \left(\frac{1}{2} (c+d x)\right)+\sin ^2\left(\frac{1}{4} (c+d x)\right) \csc (c+d x)-\sin \left(\frac{3}{4} (c+d x)\right) \sin \left(\frac{1}{4} (c+d x)\right) \csc (c+d x)+3 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)\right)}{4 d (a (\sin (c+d x)+1))^{3/2}}","\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*((16 + 16*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] - Cot[(c + d*x)/4] + 2*(3*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 3*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sec[(c + d*x)/2] + Csc[c + d*x]*Sin[(c + d*x)/4]^2 - Csc[c + d*x]*Sin[(c + d*x)/4]*Sin[(3*(c + d*x))/4])))/(4*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
348,1,309,153,3.5564985,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(12 \tan \left(\frac{1}{4} (c+d x)\right)+12 \cot \left(\frac{1}{4} (c+d x)\right)-\csc ^2\left(\frac{1}{4} (c+d x)\right)+\sec ^2\left(\frac{1}{4} (c+d x)\right)-\frac{24 \sin \left(\frac{1}{4} (c+d x)\right)}{\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}+\frac{24 \sin \left(\frac{1}{4} (c+d x)\right)}{\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)}+\frac{2}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{2}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-(128+128 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)-44 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+44 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-24\right)}{32 d (a (\sin (c+d x)+1))^{3/2}}","-\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(-24 - (128 + 128*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + 12*Cot[(c + d*x)/4] - Csc[(c + d*x)/4]^2 - 44*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 44*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sec[(c + d*x)/4]^2 + 2/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 - (24*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - 2/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2 + (24*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) + 12*Tan[(c + d*x)/4]))/(32*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
349,1,332,191,2.3522325,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(-\frac{8 \csc ^9\left(\frac{1}{2} (c+d x)\right) \left(-228 \sin \left(\frac{1}{2} (c+d x)\right)-110 \sin \left(\frac{3}{2} (c+d x)\right)+54 \sin \left(\frac{5}{2} (c+d x)\right)+228 \cos \left(\frac{1}{2} (c+d x)\right)-110 \cos \left(\frac{3}{2} (c+d x)\right)-54 \cos \left(\frac{5}{2} (c+d x)\right)-207 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+207 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+69 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-69 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{\left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}+(768+768 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{192 d (a (\sin (c+d x)+1))^{3/2}}","\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*((768 + 768*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] - (8*Csc[(c + d*x)/2]^9*(228*Cos[(c + d*x)/2] - 110*Cos[(3*(c + d*x))/2] - 54*Cos[(5*(c + d*x))/2] - 228*Sin[(c + d*x)/2] - 207*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 207*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 110*Sin[(3*(c + d*x))/2] + 54*Sin[(5*(c + d*x))/2] + 69*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 69*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3))/(192*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",0
350,1,51,65,0.3009549,"\int \cos ^3(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a \left(-315 \cos (2 (c+d x))+35 \cos (6 (c+d x))+96 \sin ^5(c+d x) (5 \cos (2 (c+d x))+9)\right)}{6720 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*(-315*Cos[2*(c + d*x)] + 35*Cos[6*(c + d*x)] + 96*(9 + 5*Cos[2*(c + d*x)])*Sin[c + d*x]^5))/(6720*d)","A",1
351,1,51,65,0.2212765,"\int \cos ^3(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \left(-45 \cos (2 (c+d x))+5 \cos (6 (c+d x))+32 \sin ^3(c+d x) (3 \cos (2 (c+d x))+7)\right)}{960 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*(-45*Cos[2*(c + d*x)] + 5*Cos[6*(c + d*x)] + 32*(7 + 3*Cos[2*(c + d*x)])*Sin[c + d*x]^3))/(960*d)","A",1
352,1,58,49,0.100184,"\int \cos ^3(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a (-60 \sin (c+d x)+10 \sin (3 (c+d x))+6 \sin (5 (c+d x))+60 \cos (2 (c+d x))+15 \cos (4 (c+d x))+45)}{480 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,"-1/480*(a*(45 + 60*Cos[2*(c + d*x)] + 15*Cos[4*(c + d*x)] - 60*Sin[c + d*x] + 10*Sin[3*(c + d*x)] + 6*Sin[5*(c + d*x)]))/d","A",1
353,1,44,45,0.0157647,"\int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{a \cos ^4(c+d x)}{4 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,"-1/4*(a*Cos[c + d*x]^4)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)","A",1
354,1,56,56,0.0318388,"\int \cos ^2(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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",1
355,1,53,53,0.034722,"\int \cos (c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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",1
356,1,60,54,0.113844,"\int \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}-\frac{a \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{2 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*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/(2*d) - (a*Sin[c + d*x])/d","A",1
357,1,28,37,0.1326754,"\int \frac{\cos ^3(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{(4-3 \sin (c+d x)) \sin ^3(c+d x)}{12 a d}","\frac{\sin ^3(c+d x)}{3 a d}-\frac{\sin ^4(c+d x)}{4 a d}",1,"((4 - 3*Sin[c + d*x])*Sin[c + d*x]^3)/(12*a*d)","A",1
358,1,28,37,0.0986587,"\int \frac{\cos ^3(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{(3-2 \sin (c+d x)) \sin ^2(c+d x)}{6 a d}","\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin ^3(c+d x)}{3 a d}",1,"((3 - 2*Sin[c + d*x])*Sin[c + d*x]^2)/(6*a*d)","A",1
359,1,24,32,0.041745,"\int \frac{\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Sin[c + d*x]),x]","-\frac{(\sin (c+d x)-2) \sin (c+d x)}{2 a d}","\frac{\sin (c+d x)}{a d}-\frac{\sin ^2(c+d x)}{2 a d}",1,"-1/2*((-2 + Sin[c + d*x])*Sin[c + d*x])/(a*d)","A",1
360,1,23,29,0.0328794,"\int \frac{\cos ^2(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\log (\sin (c+d x))-\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]] - Sin[c + d*x])/(a*d)","A",1
361,1,22,30,0.0379138,"\int \frac{\cos (c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\csc (c+d x)+\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] + Log[Sin[c + d*x]])/(a*d))","A",1
362,1,24,32,0.0317162,"\int \frac{\cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^3/(a + a*Sin[c + d*x]),x]","-\frac{(\csc (c+d x)-2) \csc (c+d x)}{2 a d}","\frac{\csc (c+d x)}{a d}-\frac{\csc ^2(c+d x)}{2 a d}",1,"-1/2*((-2 + Csc[c + d*x])*Csc[c + d*x])/(a*d)","A",1
363,1,28,37,0.0559228,"\int \frac{\cot ^3(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^3*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{(3 \sin (c+d x)-2) \csc ^3(c+d x)}{6 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]^3*(-2 + 3*Sin[c + d*x]))/(6*a*d)","A",1
364,1,28,37,0.044678,"\int \frac{\cot ^3(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^3*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{(4 \sin (c+d x)-3) \csc ^4(c+d x)}{12 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]^4*(-3 + 4*Sin[c + d*x]))/(12*a*d)","A",1
365,1,84,143,0.2780093,"\int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{a (-2520 \sin (4 (c+d x))+315 \sin (8 (c+d x))-7560 \cos (c+d x)-1680 \cos (3 (c+d x))+1008 \cos (5 (c+d x))+180 \cos (7 (c+d x))-140 \cos (9 (c+d x))+7560 c+7560 d x)}{322560 d}","-\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,"(a*(7560*c + 7560*d*x - 7560*Cos[c + d*x] - 1680*Cos[3*(c + d*x)] + 1008*Cos[5*(c + d*x)] + 180*Cos[7*(c + d*x)] - 140*Cos[9*(c + d*x)] - 2520*Sin[4*(c + d*x)] + 315*Sin[8*(c + d*x)]))/(322560*d)","A",1
366,1,71,127,0.217064,"\int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a (-280 \sin (4 (c+d x))+35 \sin (8 (c+d x))-1680 \cos (c+d x)-560 \cos (3 (c+d x))+112 \cos (5 (c+d x))+80 \cos (7 (c+d x))+840 d x)}{35840 d}","\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,"(a*(840*d*x - 1680*Cos[c + d*x] - 560*Cos[3*(c + d*x)] + 112*Cos[5*(c + d*x)] + 80*Cos[7*(c + d*x)] - 280*Sin[4*(c + d*x)] + 35*Sin[8*(c + d*x)]))/(35840*d)","A",1
367,1,81,103,0.2079673,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a (105 \sin (2 (c+d x))-105 \sin (4 (c+d x))-35 \sin (6 (c+d x))-315 \cos (c+d x)-105 \cos (3 (c+d x))+21 \cos (5 (c+d x))+15 \cos (7 (c+d x))+420 d x)}{6720 d}","\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*(420*d*x - 315*Cos[c + d*x] - 105*Cos[3*(c + d*x)] + 21*Cos[5*(c + d*x)] + 15*Cos[7*(c + d*x)] + 105*Sin[2*(c + d*x)] - 105*Sin[4*(c + d*x)] - 35*Sin[6*(c + d*x)]))/(6720*d)","A",1
368,1,71,87,0.1590416,"\int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a (-15 \sin (2 (c+d x))+15 \sin (4 (c+d x))+5 \sin (6 (c+d x))+120 \cos (c+d x)+60 \cos (3 (c+d x))+12 \cos (5 (c+d x))-60 d x)}{960 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,"-1/960*(a*(-60*d*x + 120*Cos[c + d*x] + 60*Cos[3*(c + d*x)] + 12*Cos[5*(c + d*x)] - 15*Sin[2*(c + d*x)] + 15*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)]))/d","A",1
369,1,81,89,0.1491328,"\int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \left(120 \cos (c+d x)+8 \cos (3 (c+d x))+3 \left(8 \sin (2 (c+d x))+\sin (4 (c+d x))+32 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-32 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 c+12 d x\right)\right)}{96 d}","\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,"(a*(120*Cos[c + d*x] + 8*Cos[3*(c + d*x)] + 3*(12*c + 12*d*x - 32*Log[Cos[(c + d*x)/2]] + 32*Log[Sin[(c + d*x)/2]] + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])))/(96*d)","A",1
370,1,77,83,0.4305907,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \left(15 \cos (c+d x)+\cos (3 (c+d x))-3 \left(\sin (2 (c+d x))+4 \cot (c+d x)-4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+6 c+6 d x\right)\right)}{12 d}","\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,"(a*(15*Cos[c + d*x] + Cos[3*(c + d*x)] - 3*(6*c + 6*d*x + 4*Cot[c + d*x] + 4*Log[Cos[(c + d*x)/2]] - 4*Log[Sin[(c + d*x)/2]] + Sin[2*(c + d*x)])))/(12*d)","A",1
371,1,94,94,0.7976311,"\int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \left(2 \sin (2 (c+d x))+8 \cos (c+d x)+8 \cot (c+d x)+\csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^2\left(\frac{1}{2} (c+d x)\right)+12 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-12 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 c+12 d x\right)}{8 d}","-\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,"-1/8*(a*(12*c + 12*d*x + 8*Cos[c + d*x] + 8*Cot[c + d*x] + Csc[(c + d*x)/2]^2 - 12*Log[Cos[(c + d*x)/2]] + 12*Log[Sin[(c + d*x)/2]] - Sec[(c + d*x)/2]^2 + 2*Sin[2*(c + d*x)]))/d","A",1
372,1,125,82,0.0509123,"\int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","-\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*Cos[c + d*x])/d) - (a*Csc[(c + d*x)/2]^2)/(8*d) - (a*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + (3*a*Log[Cos[(c + d*x)/2]])/(2*d) - (3*a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)","C",1
373,1,153,88,0.0491484,"\int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 d}-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{5 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}","-\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,"(5*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (a*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) - (3*a*Log[Cos[(c + d*x)/2]])/(8*d) + (3*a*Log[Sin[(c + d*x)/2]])/(8*d) - (5*a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)","C",1
374,1,135,74,0.0353305,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{5 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-1/5*(a*Cot[c + d*x]^5)/d + (5*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (3*a*Log[Cos[(c + d*x)/2]])/(8*d) + (3*a*Log[Sin[(c + d*x)/2]])/(8*d) - (5*a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)","A",1
375,1,175,98,0.0442826,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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 \csc ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-1/5*(a*Cot[c + d*x]^5)/d - (a*Csc[(c + d*x)/2]^2)/(64*d) + (a*Csc[(c + d*x)/2]^4)/(64*d) - (a*Csc[(c + d*x)/2]^6)/(384*d) - (a*Log[Cos[(c + d*x)/2]])/(16*d) + (a*Log[Sin[(c + d*x)/2]])/(16*d) + (a*Sec[(c + d*x)/2]^2)/(64*d) - (a*Sec[(c + d*x)/2]^4)/(64*d) + (a*Sec[(c + d*x)/2]^6)/(384*d)","A",1
376,1,239,114,0.079216,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{2 a \cot (c+d x)}{35 d}-\frac{a \csc ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{a \cot (c+d x) \csc ^6(c+d x)}{7 d}+\frac{8 a \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{35 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,"(-2*a*Cot[c + d*x])/(35*d) - (a*Csc[(c + d*x)/2]^2)/(64*d) + (a*Csc[(c + d*x)/2]^4)/(64*d) - (a*Csc[(c + d*x)/2]^6)/(384*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(35*d) + (8*a*Cot[c + d*x]*Csc[c + d*x]^4)/(35*d) - (a*Cot[c + d*x]*Csc[c + d*x]^6)/(7*d) - (a*Log[Cos[(c + d*x)/2]])/(16*d) + (a*Log[Sin[(c + d*x)/2]])/(16*d) + (a*Sec[(c + d*x)/2]^2)/(64*d) - (a*Sec[(c + d*x)/2]^4)/(64*d) + (a*Sec[(c + d*x)/2]^6)/(384*d)","B",1
377,1,279,136,0.0810234,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{2 a \cot (c+d x)}{35 d}-\frac{a \csc ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}+\frac{a \csc ^6\left(\frac{1}{2} (c+d x)\right)}{512 d}+\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}-\frac{3 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}+\frac{a \sec ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}-\frac{a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{512 d}-\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}+\frac{3 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{128 d}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{128 d}-\frac{a \cot (c+d x) \csc ^6(c+d x)}{7 d}+\frac{8 a \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{35 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,"(-2*a*Cot[c + d*x])/(35*d) - (3*a*Csc[(c + d*x)/2]^2)/(512*d) + (a*Csc[(c + d*x)/2]^4)/(1024*d) + (a*Csc[(c + d*x)/2]^6)/(512*d) - (a*Csc[(c + d*x)/2]^8)/(2048*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(35*d) + (8*a*Cot[c + d*x]*Csc[c + d*x]^4)/(35*d) - (a*Cot[c + d*x]*Csc[c + d*x]^6)/(7*d) - (3*a*Log[Cos[(c + d*x)/2]])/(128*d) + (3*a*Log[Sin[(c + d*x)/2]])/(128*d) + (3*a*Sec[(c + d*x)/2]^2)/(512*d) - (a*Sec[(c + d*x)/2]^4)/(1024*d) - (a*Sec[(c + d*x)/2]^6)/(512*d) + (a*Sec[(c + d*x)/2]^8)/(2048*d)","B",1
378,1,116,185,0.7275285,"\int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (-1260 \sin (2 (c+d x))-7560 \sin (4 (c+d x))+630 \sin (6 (c+d x))+945 \sin (8 (c+d x))-126 \sin (10 (c+d x))-30240 \cos (c+d x)-6720 \cos (3 (c+d x))+4032 \cos (5 (c+d x))+720 \cos (7 (c+d x))-560 \cos (9 (c+d x))+22680 c+22680 d x)}{645120 d}","-\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,"(a^2*(22680*c + 22680*d*x - 30240*Cos[c + d*x] - 6720*Cos[3*(c + d*x)] + 4032*Cos[5*(c + d*x)] + 720*Cos[7*(c + d*x)] - 560*Cos[9*(c + d*x)] - 1260*Sin[2*(c + d*x)] - 7560*Sin[4*(c + d*x)] + 630*Sin[6*(c + d*x)] + 945*Sin[8*(c + d*x)] - 126*Sin[10*(c + d*x)]))/(645120*d)","A",1
379,1,86,159,0.7079152,"\int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (-2520 \sin (4 (c+d x))+315 \sin (8 (c+d x))-11340 \cos (c+d x)-3360 \cos (3 (c+d x))+1008 \cos (5 (c+d x))+450 \cos (7 (c+d x))-70 \cos (9 (c+d x))+7560 c+7560 d x)}{161280 d}","-\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,"(a^2*(7560*c + 7560*d*x - 11340*Cos[c + d*x] - 3360*Cos[3*(c + d*x)] + 1008*Cos[5*(c + d*x)] + 450*Cos[7*(c + d*x)] - 70*Cos[9*(c + d*x)] - 2520*Sin[4*(c + d*x)] + 315*Sin[8*(c + d*x)]))/(161280*d)","A",1
380,1,96,141,0.5143352,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (1680 \sin (2 (c+d x))-2520 \sin (4 (c+d x))-560 \sin (6 (c+d x))+105 \sin (8 (c+d x))-10080 \cos (c+d x)-3360 \cos (3 (c+d x))+672 \cos (5 (c+d x))+480 \cos (7 (c+d x))+3360 c+9240 d x)}{107520 d}","\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,"(a^2*(3360*c + 9240*d*x - 10080*Cos[c + d*x] - 3360*Cos[3*(c + d*x)] + 672*Cos[5*(c + d*x)] + 480*Cos[7*(c + d*x)] + 1680*Sin[2*(c + d*x)] - 2520*Sin[4*(c + d*x)] - 560*Sin[6*(c + d*x)] + 105*Sin[8*(c + d*x)]))/(107520*d)","A",1
381,1,86,129,0.3254179,"\int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (210 \sin (2 (c+d x))-210 \sin (4 (c+d x))-70 \sin (6 (c+d x))-1155 \cos (c+d x)-525 \cos (3 (c+d x))-63 \cos (5 (c+d x))+15 \cos (7 (c+d x))+840 c+840 d x)}{6720 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*(840*c + 840*d*x - 1155*Cos[c + d*x] - 525*Cos[3*(c + d*x)] - 63*Cos[5*(c + d*x)] + 15*Cos[7*(c + d*x)] + 210*Sin[2*(c + d*x)] - 210*Sin[4*(c + d*x)] - 70*Sin[6*(c + d*x)]))/(6720*d)","A",1
382,1,96,119,0.8489951,"\int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(270 \cos (c+d x)+5 \cos (3 (c+d x))-3 \cos (5 (c+d x))+15 \left(8 \sin (2 (c+d x))+\sin (4 (c+d x))+4 \left(4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 c+3 d x\right)\right)\right)}{240 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 ^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,"(a^2*(270*Cos[c + d*x] + 5*Cos[3*(c + d*x)] - 3*Cos[5*(c + d*x)] + 15*(4*(3*c + 3*d*x - 4*Log[Cos[(c + d*x)/2]] + 4*Log[Sin[(c + d*x)/2]]) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])))/(240*d)","A",1
383,1,83,116,0.4908496,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(240 \cos (c+d x)+16 \cos (3 (c+d x))-3 \left(-\sin (4 (c+d x))+32 \cot (c+d x)-64 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+64 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+36 c+36 d x\right)\right)}{96 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 ^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,"(a^2*(240*Cos[c + d*x] + 16*Cos[3*(c + d*x)] - 3*(36*c + 36*d*x + 32*Cot[c + d*x] + 64*Log[Cos[(c + d*x)/2]] - 64*Log[Sin[(c + d*x)/2]] - Sin[4*(c + d*x)])))/(96*d)","A",1
384,1,158,98,2.1212162,"\int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (\sin (c+d x)+1)^2 \left(6 \cos (c+d x)+2 \cos (3 (c+d x))+3 \left(-4 \sin (2 (c+d x))+8 \tan \left(\frac{1}{2} (c+d x)\right)-8 \cot \left(\frac{1}{2} (c+d x)\right)-\csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^2\left(\frac{1}{2} (c+d x)\right)-4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-24 c-24 d x\right)\right)}{24 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","\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,"(a^2*(1 + Sin[c + d*x])^2*(6*Cos[c + d*x] + 2*Cos[3*(c + d*x)] + 3*(-24*c - 24*d*x - 8*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 + 4*Log[Cos[(c + d*x)/2]] - 4*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 - 4*Sin[2*(c + d*x)] + 8*Tan[(c + d*x)/2])))/(24*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
385,1,191,98,5.3224496,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (\sin (c+d x)+1)^2 \left(-12 (c+d x)-6 \sin (2 (c+d x))-48 \cos (c+d x)-4 \tan \left(\frac{1}{2} (c+d x)\right)+4 \cot \left(\frac{1}{2} (c+d x)\right)-6 \csc ^2\left(\frac{1}{2} (c+d x)\right)+6 \sec ^2\left(\frac{1}{2} (c+d x)\right)-72 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+72 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{1}{2} \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+8 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{24 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\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*(1 + Sin[c + d*x])^2*(-12*(c + d*x) - 48*Cos[c + d*x] + 4*Cot[(c + d*x)/2] - 6*Csc[(c + d*x)/2]^2 + 72*Log[Cos[(c + d*x)/2]] - 72*Log[Sin[(c + d*x)/2]] + 6*Sec[(c + d*x)/2]^2 + 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - (Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 - 6*Sin[2*(c + d*x)] - 4*Tan[(c + d*x)/2]))/(24*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
386,1,215,116,1.2346575,"\int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \sin (c+d x) (\sin (c+d x)+1)^2 \left(192 \cot (c+d x)+\csc ^4\left(\frac{1}{2} (c+d x)\right) (3 \csc (c+d x)+8)-2 \csc ^2\left(\frac{1}{2} (c+d x)\right) (3 \csc (c+d x)+64)+8 (8 \cos (c+d x)+7) \sec ^4\left(\frac{1}{2} (c+d x)\right)-48 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+24 \sin ^2\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-24 \csc (c+d x) \left(16 (c+d x)-9 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{192 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\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,"-1/192*(a^2*(192*Cot[c + d*x] + Csc[(c + d*x)/2]^4*(8 + 3*Csc[c + d*x]) - 2*Csc[(c + d*x)/2]^2*(64 + 3*Csc[c + d*x]) - 24*Csc[c + d*x]*(16*(c + d*x) + 9*Log[Cos[(c + d*x)/2]] - 9*Log[Sin[(c + d*x)/2]]) + 8*(7 + 8*Cos[c + d*x])*Sec[(c + d*x)/2]^4 + 24*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 - 48*Csc[c + d*x]^5*Sin[(c + d*x)/2]^4)*Sin[c + d*x]*(1 + Sin[c + d*x])^2)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
387,1,200,118,0.5328226,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(-272 \tan \left(\frac{1}{2} (c+d x)\right)+272 \cot \left(\frac{1}{2} (c+d x)\right)+150 \csc ^2\left(\frac{1}{2} (c+d x)\right)+15 \sec ^4\left(\frac{1}{2} (c+d x)\right)-150 \sec ^2\left(\frac{1}{2} (c+d x)\right)+360 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-360 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{3}{2} \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+96 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+\frac{1}{2} (\sin (c+d x)-30) \csc ^4\left(\frac{1}{2} (c+d x)\right)-8 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+480 c+480 d x\right)}{480 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{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*(480*c + 480*d*x + 272*Cot[(c + d*x)/2] + 150*Csc[(c + d*x)/2]^2 - 360*Log[Cos[(c + d*x)/2]] + 360*Log[Sin[(c + d*x)/2]] - 150*Sec[(c + d*x)/2]^2 + 15*Sec[(c + d*x)/2]^4 - 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 96*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 + (Csc[(c + d*x)/2]^4*(-30 + Sin[c + d*x]))/2 - (3*Csc[(c + d*x)/2]^6*Sin[c + d*x])/2 - 272*Tan[(c + d*x)/2]))/(480*d)","A",1
388,1,267,132,0.1058212,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","a^2 \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{5 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{5 d}-\frac{\csc ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{9 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{\sec ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{9 \sec ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{7 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{7 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^4\left(\frac{1}{2} (c+d x)\right)}{80 d}+\frac{7 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{80 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)}{80 d}-\frac{7 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{80 d}\right)","-\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,"a^2*(-1/5*Cot[(c + d*x)/2]/d + (9*Csc[(c + d*x)/2]^2)/(64*d) + (7*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(80*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(80*d) - Csc[(c + d*x)/2]^6/(384*d) - (7*Log[Cos[(c + d*x)/2]])/(16*d) + (7*Log[Sin[(c + d*x)/2]])/(16*d) - (9*Sec[(c + d*x)/2]^2)/(64*d) + Sec[(c + d*x)/2]^6/(384*d) + Tan[(c + d*x)/2]/(5*d) - (7*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(80*d) + (Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(80*d))","B",1
389,1,291,176,0.9551714,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc ^8(c+d x) \left(86016 \sin (2 (c+d x))+64512 \sin (4 (c+d x))+12288 \sin (6 (c+d x))-1536 \sin (8 (c+d x))+158270 \cos (c+d x)+77210 \cos (3 (c+d x))-18130 \cos (5 (c+d x))-2310 \cos (7 (c+d x))-40425 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-64680 \cos (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+32340 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-9240 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+1155 \cos (8 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+40425 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+64680 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-32340 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+9240 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-1155 \cos (8 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1720320 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,"-1/1720320*(a^2*Csc[c + d*x]^8*(158270*Cos[c + d*x] + 77210*Cos[3*(c + d*x)] - 18130*Cos[5*(c + d*x)] - 2310*Cos[7*(c + d*x)] + 40425*Log[Cos[(c + d*x)/2]] - 64680*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 32340*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 9240*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 1155*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 40425*Log[Sin[(c + d*x)/2]] + 64680*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 32340*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 9240*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 1155*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 86016*Sin[2*(c + d*x)] + 64512*Sin[4*(c + d*x)] + 12288*Sin[6*(c + d*x)] - 1536*Sin[8*(c + d*x)]))/d","A",1
390,1,313,168,1.3706895,"\int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc ^9(c+d x) \left(212940 \sin (2 (c+d x))+195300 \sin (4 (c+d x))+16380 \sin (6 (c+d x))-1890 \sin (8 (c+d x))+451584 \cos (c+d x)+155904 \cos (3 (c+d x))-20736 \cos (5 (c+d x))-14976 \cos (7 (c+d x))+1664 \cos (9 (c+d x))-119070 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+79380 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-34020 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+8505 \sin (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-945 \sin (9 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+119070 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-79380 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+34020 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-8505 \sin (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+945 \sin (9 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{5160960 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,"-1/5160960*(a^2*Csc[c + d*x]^9*(451584*Cos[c + d*x] + 155904*Cos[3*(c + d*x)] - 20736*Cos[5*(c + d*x)] - 14976*Cos[7*(c + d*x)] + 1664*Cos[9*(c + d*x)] + 119070*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 119070*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 212940*Sin[2*(c + d*x)] - 79380*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 79380*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 195300*Sin[4*(c + d*x)] + 34020*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 34020*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 16380*Sin[6*(c + d*x)] - 8505*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] + 8505*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 1890*Sin[8*(c + d*x)] + 945*Log[Cos[(c + d*x)/2]]*Sin[9*(c + d*x)] - 945*Log[Sin[(c + d*x)/2]]*Sin[9*(c + d*x)]))/d","A",1
391,1,353,218,1.2228382,"\int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc ^{10}(c+d x) \left(1720320 \sin (2 (c+d x))+1228800 \sin (4 (c+d x))+184320 \sin (6 (c+d x))-40960 \sin (8 (c+d x))+4096 \sin (10 (c+d x))+3219300 \cos (c+d x)+1237320 \cos (3 (c+d x))-278712 \cos (5 (c+d x))-54810 \cos (7 (c+d x))+5670 \cos (9 (c+d x))-357210 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-595350 \cos (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+340200 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-127575 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+28350 \cos (8 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2835 \cos (10 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+357210 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+595350 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-340200 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+127575 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-28350 \cos (8 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2835 \cos (10 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{41287680 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,"-1/41287680*(a^2*Csc[c + d*x]^10*(3219300*Cos[c + d*x] + 1237320*Cos[3*(c + d*x)] - 278712*Cos[5*(c + d*x)] - 54810*Cos[7*(c + d*x)] + 5670*Cos[9*(c + d*x)] + 357210*Log[Cos[(c + d*x)/2]] - 595350*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 340200*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 127575*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 28350*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 2835*Cos[10*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 357210*Log[Sin[(c + d*x)/2]] + 595350*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 340200*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 127575*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 28350*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 2835*Cos[10*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 1720320*Sin[2*(c + d*x)] + 1228800*Sin[4*(c + d*x)] + 184320*Sin[6*(c + d*x)] - 40960*Sin[8*(c + d*x)] + 4096*Sin[10*(c + d*x)]))/d","A",1
392,1,126,203,1.0401557,"\int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (-13860 \sin (2 (c+d x))-46200 \sin (4 (c+d x))+6930 \sin (6 (c+d x))+5775 \sin (8 (c+d x))-1386 \sin (10 (c+d x))-198660 \cos (c+d x)-41580 \cos (3 (c+d x))+27258 \cos (5 (c+d x))+3630 \cos (7 (c+d x))-3850 \cos (9 (c+d x))+210 \cos (11 (c+d x))+138600 c+138600 d x)}{2365440 d}","\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,"(a^3*(138600*c + 138600*d*x - 198660*Cos[c + d*x] - 41580*Cos[3*(c + d*x)] + 27258*Cos[5*(c + d*x)] + 3630*Cos[7*(c + d*x)] - 3850*Cos[9*(c + d*x)] + 210*Cos[11*(c + d*x)] - 13860*Sin[2*(c + d*x)] - 46200*Sin[4*(c + d*x)] + 6930*Sin[6*(c + d*x)] + 5775*Sin[8*(c + d*x)] - 1386*Sin[10*(c + d*x)]))/(2365440*d)","A",1
393,1,116,182,1.0815708,"\int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (-60 \sin (2 (c+d x))-840 \sin (4 (c+d x))+30 \sin (6 (c+d x))+105 \sin (8 (c+d x))-6 \sin (10 (c+d x))-3600 \cos (c+d x)-960 \cos (3 (c+d x))+384 \cos (5 (c+d x))+120 \cos (7 (c+d x))-40 \cos (9 (c+d x))+2700 c+2520 d x)}{30720 d}","-\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,"(a^3*(2700*c + 2520*d*x - 3600*Cos[c + d*x] - 960*Cos[3*(c + d*x)] + 384*Cos[5*(c + d*x)] + 120*Cos[7*(c + d*x)] - 40*Cos[9*(c + d*x)] - 60*Sin[2*(c + d*x)] - 840*Sin[4*(c + d*x)] + 30*Sin[6*(c + d*x)] + 105*Sin[8*(c + d*x)] - 6*Sin[10*(c + d*x)]))/(30720*d)","A",1
394,1,106,159,0.848825,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (5040 \sin (2 (c+d x))-12600 \sin (4 (c+d x))-1680 \sin (6 (c+d x))+945 \sin (8 (c+d x))-52920 \cos (c+d x)-16800 \cos (3 (c+d x))+4032 \cos (5 (c+d x))+2340 \cos (7 (c+d x))-140 \cos (9 (c+d x))+30240 c+42840 d x)}{322560 d}","-\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,"(a^3*(30240*c + 42840*d*x - 52920*Cos[c + d*x] - 16800*Cos[3*(c + d*x)] + 4032*Cos[5*(c + d*x)] + 2340*Cos[7*(c + d*x)] - 140*Cos[9*(c + d*x)] + 5040*Sin[2*(c + d*x)] - 12600*Sin[4*(c + d*x)] - 1680*Sin[6*(c + d*x)] + 945*Sin[8*(c + d*x)]))/(322560*d)","A",1
395,1,96,157,0.4524824,"\int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (1680 \sin (2 (c+d x))-1960 \sin (4 (c+d x))-560 \sin (6 (c+d x))+35 \sin (8 (c+d x))-9520 \cos (c+d x)-3920 \cos (3 (c+d x))-112 \cos (5 (c+d x))+240 \cos (7 (c+d x))+8400 c+7560 d x)}{35840 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,"(a^3*(8400*c + 7560*d*x - 9520*Cos[c + d*x] - 3920*Cos[3*(c + d*x)] - 112*Cos[5*(c + d*x)] + 240*Cos[7*(c + d*x)] + 1680*Sin[2*(c + d*x)] - 1960*Sin[4*(c + d*x)] - 560*Sin[6*(c + d*x)] + 35*Sin[8*(c + d*x)]))/(35840*d)","A",1
396,1,102,143,1.0209031,"\int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(735 \sin (2 (c+d x))+75 \sin (4 (c+d x))-5 \sin (6 (c+d x))+840 \cos (c+d x)-100 \cos (3 (c+d x))-36 \cos (5 (c+d x))+960 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-960 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+1140 c+1140 d x\right)}{960 d}","-\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,"(a^3*(1140*c + 1140*d*x + 840*Cos[c + d*x] - 100*Cos[3*(c + d*x)] - 36*Cos[5*(c + d*x)] - 960*Log[Cos[(c + d*x)/2]] + 960*Log[Sin[(c + d*x)/2]] + 735*Sin[2*(c + d*x)] + 75*Sin[4*(c + d*x)] - 5*Sin[6*(c + d*x)]))/(960*d)","A",1
397,1,148,131,2.0541768,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{(a \sin (c+d x)+a)^3 \left(-60 (c+d x)+80 \sin (2 (c+d x))+15 \sin (4 (c+d x))+580 \cos (c+d x)+30 \cos (3 (c+d x))-2 \cos (5 (c+d x))+80 \tan \left(\frac{1}{2} (c+d x)\right)-80 \cot \left(\frac{1}{2} (c+d x)\right)+480 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-480 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{160 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"((a + a*Sin[c + d*x])^3*(-60*(c + d*x) + 580*Cos[c + d*x] + 30*Cos[3*(c + d*x)] - 2*Cos[5*(c + d*x)] - 80*Cot[(c + d*x)/2] - 480*Log[Cos[(c + d*x)/2]] + 480*Log[Sin[(c + d*x)/2]] + 80*Sin[2*(c + d*x)] + 15*Sin[4*(c + d*x)] + 80*Tan[(c + d*x)/2]))/(160*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
398,1,164,137,2.8592705,"\int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{(a \sin (c+d x)+a)^3 \left(-132 (c+d x)-16 \sin (2 (c+d x))+\sin (4 (c+d x))+88 \cos (c+d x)+8 \cos (3 (c+d x))+48 \tan \left(\frac{1}{2} (c+d x)\right)-48 \cot \left(\frac{1}{2} (c+d x)\right)-4 \csc ^2\left(\frac{1}{2} (c+d x)\right)+4 \sec ^2\left(\frac{1}{2} (c+d x)\right)+48 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-48 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{32 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\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,"((a + a*Sin[c + d*x])^3*(-132*(c + d*x) + 88*Cos[c + d*x] + 8*Cos[3*(c + d*x)] - 48*Cot[(c + d*x)/2] - 4*Csc[(c + d*x)/2]^2 - 48*Log[Cos[(c + d*x)/2]] + 48*Log[Sin[(c + d*x)/2]] + 4*Sec[(c + d*x)/2]^2 - 16*Sin[2*(c + d*x)] + Sin[4*(c + d*x)] + 48*Tan[(c + d*x)/2]))/(32*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
399,1,201,134,6.1321647,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(-84 (c+d x)-18 \sin (2 (c+d x))-42 \cos (c+d x)+2 \cos (3 (c+d x))+20 \tan \left(\frac{1}{2} (c+d x)\right)-20 \cot \left(\frac{1}{2} (c+d x)\right)-9 \csc ^2\left(\frac{1}{2} (c+d x)\right)+9 \sec ^2\left(\frac{1}{2} (c+d x)\right)-84 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+84 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{1}{2} \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+8 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{24 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\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,"(a^3*(1 + Sin[c + d*x])^3*(-84*(c + d*x) - 42*Cos[c + d*x] + 2*Cos[3*(c + d*x)] - 20*Cot[(c + d*x)/2] - 9*Csc[(c + d*x)/2]^2 + 84*Log[Cos[(c + d*x)/2]] - 84*Log[Sin[(c + d*x)/2]] + 9*Sec[(c + d*x)/2]^2 + 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - (Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 - 18*Sin[2*(c + d*x)] + 20*Tan[(c + d*x)/2]))/(24*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
400,1,215,138,1.218889,"\int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(96 (c+d x)-16 \sin (2 (c+d x))-192 \cos (c+d x)-96 \tan \left(\frac{1}{2} (c+d x)\right)+96 \cot \left(\frac{1}{2} (c+d x)\right)-\csc ^4\left(\frac{1}{2} (c+d x)\right)-14 \csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^4\left(\frac{1}{2} (c+d x)\right)+14 \sec ^2\left(\frac{1}{2} (c+d x)\right)-264 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+264 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+64 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{64 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"(a^3*(1 + Sin[c + d*x])^3*(96*(c + d*x) - 192*Cos[c + d*x] + 96*Cot[(c + d*x)/2] - 14*Csc[(c + d*x)/2]^2 - Csc[(c + d*x)/2]^4 + 264*Log[Cos[(c + d*x)/2]] - 264*Log[Sin[(c + d*x)/2]] + 14*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^4 + 64*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 4*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 16*Sin[2*(c + d*x)] - 96*Tan[(c + d*x)/2]))/(64*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
401,1,216,132,0.5114398,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(-320 \cos (c+d x)-608 \tan \left(\frac{1}{2} (c+d x)\right)+608 \cot \left(\frac{1}{2} (c+d x)\right)-15 \csc ^4\left(\frac{1}{2} (c+d x)\right)+110 \csc ^2\left(\frac{1}{2} (c+d x)\right)+15 \sec ^4\left(\frac{1}{2} (c+d x)\right)-110 \sec ^2\left(\frac{1}{2} (c+d x)\right)-120 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+120 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+64 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)-13 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+208 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+960 c+960 d x\right)}{320 d}","-\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,"(a^3*(960*c + 960*d*x - 320*Cos[c + d*x] + 608*Cot[(c + d*x)/2] + 110*Csc[(c + d*x)/2]^2 - 15*Csc[(c + d*x)/2]^4 + 120*Log[Cos[(c + d*x)/2]] - 120*Log[Sin[(c + d*x)/2]] - 110*Sec[(c + d*x)/2]^2 + 15*Sec[(c + d*x)/2]^4 + 208*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 64*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 - 13*Csc[(c + d*x)/2]^4*Sin[c + d*x] - Csc[(c + d*x)/2]^6*Sin[c + d*x] - 608*Tan[(c + d*x)/2]))/(320*d)","A",1
402,1,217,168,0.7814623,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(-704 \tan \left(\frac{1}{2} (c+d x)\right)+704 \cot \left(\frac{1}{2} (c+d x)\right)+870 \csc ^2\left(\frac{1}{2} (c+d x)\right)+5 \sec ^6\left(\frac{1}{2} (c+d x)\right)+60 \sec ^4\left(\frac{1}{2} (c+d x)\right)-870 \sec ^2\left(\frac{1}{2} (c+d x)\right)+2280 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2280 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\left((18 \sin (c+d x)+5) \csc ^6\left(\frac{1}{2} (c+d x)\right)\right)+(86 \sin (c+d x)-60) \csc ^4\left(\frac{1}{2} (c+d x)\right)-1376 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+36 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)+1920 c+1920 d x\right)}{1920 d}","-\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*(1920*c + 1920*d*x + 704*Cot[(c + d*x)/2] + 870*Csc[(c + d*x)/2]^2 - 2280*Log[Cos[(c + d*x)/2]] + 2280*Log[Sin[(c + d*x)/2]] - 870*Sec[(c + d*x)/2]^2 + 60*Sec[(c + d*x)/2]^4 + 5*Sec[(c + d*x)/2]^6 - 1376*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - Csc[(c + d*x)/2]^6*(5 + 18*Sin[c + d*x]) + Csc[(c + d*x)/2]^4*(-60 + 86*Sin[c + d*x]) - 704*Tan[(c + d*x)/2] + 36*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(1920*d)","A",1
403,1,363,150,0.1283676,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","a^3 \left(\frac{23 \tan \left(\frac{1}{2} (c+d x)\right)}{70 d}-\frac{23 \cot \left(\frac{1}{2} (c+d x)\right)}{70 d}-\frac{\csc ^6\left(\frac{1}{2} (c+d x)\right)}{128 d}+\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{7 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{\sec ^6\left(\frac{1}{2} (c+d x)\right)}{128 d}-\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{7 \sec ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{9 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{9 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^6\left(\frac{1}{2} (c+d x)\right)}{896 d}-\frac{31 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^4\left(\frac{1}{2} (c+d x)\right)}{2240 d}+\frac{297 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{2240 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right)}{896 d}+\frac{31 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)}{2240 d}-\frac{297 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{2240 d}\right)","-\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,"a^3*((-23*Cot[(c + d*x)/2])/(70*d) + (7*Csc[(c + d*x)/2]^2)/(64*d) + (297*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(2240*d) + Csc[(c + d*x)/2]^4/(32*d) - (31*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(2240*d) - Csc[(c + d*x)/2]^6/(128*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^6)/(896*d) - (9*Log[Cos[(c + d*x)/2]])/(16*d) + (9*Log[Sin[(c + d*x)/2]])/(16*d) - (7*Sec[(c + d*x)/2]^2)/(64*d) - Sec[(c + d*x)/2]^4/(32*d) + Sec[(c + d*x)/2]^6/(128*d) + (23*Tan[(c + d*x)/2])/(70*d) - (297*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(2240*d) + (31*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(2240*d) + (Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2])/(896*d))","B",1
404,1,313,176,5.092008,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \sin (c+d x) (\sin (c+d x)+1)^3 \left(10 (7 \csc (c+d x)+24) \csc ^8\left(\frac{1}{2} (c+d x)\right)+8 (105 \csc (c+d x)-76) \csc ^6\left(\frac{1}{2} (c+d x)\right)-4 (1715 \csc (c+d x)+856) \csc ^4\left(\frac{1}{2} (c+d x)\right)+8 (945 \csc (c+d x)+1664) \csc ^2\left(\frac{1}{2} (c+d x)\right)-4 \left((1056 \cos (c+d x)+517 \cos (2 (c+d x))+104 \cos (3 (c+d x))+703) \sec ^8\left(\frac{1}{2} (c+d x)\right)+4480 \sin ^8\left(\frac{1}{2} (c+d x)\right) \csc ^9(c+d x)+13440 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^7(c+d x)-27440 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+7560 \sin ^2\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-7560 \csc (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}{143360 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"-1/143360*(a^3*(10*Csc[(c + d*x)/2]^8*(24 + 7*Csc[c + d*x]) + 8*Csc[(c + d*x)/2]^6*(-76 + 105*Csc[c + d*x]) + 8*Csc[(c + d*x)/2]^2*(1664 + 945*Csc[c + d*x]) - 4*Csc[(c + d*x)/2]^4*(856 + 1715*Csc[c + d*x]) - 4*(-7560*Csc[c + d*x]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + (703 + 1056*Cos[c + d*x] + 517*Cos[2*(c + d*x)] + 104*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^8 + 7560*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 - 27440*Csc[c + d*x]^5*Sin[(c + d*x)/2]^4 + 13440*Csc[c + d*x]^7*Sin[(c + d*x)/2]^6 + 4480*Csc[c + d*x]^9*Sin[(c + d*x)/2]^8))*Sin[c + d*x]*(1 + Sin[c + d*x])^3)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
405,1,313,194,1.3331033,"\int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \csc ^9(c+d x) \left(669060 \sin (2 (c+d x))+676620 \sin (4 (c+d x))-14700 \sin (6 (c+d x))-10710 \sin (8 (c+d x))+1161216 \cos (c+d x)+247296 \cos (3 (c+d x))-198144 \cos (5 (c+d x))-71424 \cos (7 (c+d x))+7936 \cos (9 (c+d x))-674730 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+449820 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-192780 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+48195 \sin (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-5355 \sin (9 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+674730 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-449820 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+192780 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-48195 \sin (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+5355 \sin (9 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{10321920 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,"-1/10321920*(a^3*Csc[c + d*x]^9*(1161216*Cos[c + d*x] + 247296*Cos[3*(c + d*x)] - 198144*Cos[5*(c + d*x)] - 71424*Cos[7*(c + d*x)] + 7936*Cos[9*(c + d*x)] + 674730*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 674730*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 669060*Sin[2*(c + d*x)] - 449820*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 449820*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 676620*Sin[4*(c + d*x)] + 192780*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 192780*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 14700*Sin[6*(c + d*x)] - 48195*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] + 48195*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 10710*Sin[8*(c + d*x)] + 5355*Log[Cos[(c + d*x)/2]]*Sin[9*(c + d*x)] - 5355*Log[Sin[(c + d*x)/2]]*Sin[9*(c + d*x)]))/d","A",1
406,1,366,216,2.1713588,"\int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(4096 \tan \left(\frac{1}{2} (c+d x)\right)-4096 \cot \left(\frac{1}{2} (c+d x)\right)-1260 \csc ^2\left(\frac{1}{2} (c+d x)\right)+6 \sec ^{10}\left(\frac{1}{2} (c+d x)\right)+75 \sec ^8\left(\frac{1}{2} (c+d x)\right)-390 \sec ^6\left(\frac{1}{2} (c+d x)\right)-180 \sec ^4\left(\frac{1}{2} (c+d x)\right)+1260 \sec ^2\left(\frac{1}{2} (c+d x)\right)+5040 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-5040 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 (10 \sin (c+d x)+3) \csc ^{10}\left(\frac{1}{2} (c+d x)\right)+5 (4 \sin (c+d x)-15) \csc ^8\left(\frac{1}{2} (c+d x)\right)+6 (42 \sin (c+d x)+65) \csc ^6\left(\frac{1}{2} (c+d x)\right)-4 (\sin (c+d x)-45) \csc ^4\left(\frac{1}{2} (c+d x)\right)+64 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+40 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^8\left(\frac{1}{2} (c+d x)\right)-40 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right)-504 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)\right)}{61440 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"(a^3*(1 + Sin[c + d*x])^3*(-4096*Cot[(c + d*x)/2] - 1260*Csc[(c + d*x)/2]^2 - 5040*Log[Cos[(c + d*x)/2]] + 5040*Log[Sin[(c + d*x)/2]] + 1260*Sec[(c + d*x)/2]^2 - 180*Sec[(c + d*x)/2]^4 - 390*Sec[(c + d*x)/2]^6 + 75*Sec[(c + d*x)/2]^8 + 6*Sec[(c + d*x)/2]^10 + 64*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 4*Csc[(c + d*x)/2]^4*(-45 + Sin[c + d*x]) + 5*Csc[(c + d*x)/2]^8*(-15 + 4*Sin[c + d*x]) - 2*Csc[(c + d*x)/2]^10*(3 + 10*Sin[c + d*x]) + 6*Csc[(c + d*x)/2]^6*(65 + 42*Sin[c + d*x]) + 4096*Tan[(c + d*x)/2] - 504*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] - 40*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2] + 40*Sec[(c + d*x)/2]^8*Tan[(c + d*x)/2]))/(61440*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
407,1,116,187,1.1882221,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 (8820 \sin (2 (c+d x))-42840 \sin (4 (c+d x))-2730 \sin (6 (c+d x))+4095 \sin (8 (c+d x))-126 \sin (10 (c+d x))-181440 \cos (c+d x)-53760 \cos (3 (c+d x))+16128 \cos (5 (c+d x))+7200 \cos (7 (c+d x))-1120 \cos (9 (c+d x))+136080 c+138600 d x)}{645120 d}","-\frac{11 a^4 \cos ^7(c+d x)}{112 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 ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^2}{18 d}-\frac{\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}",1,"(a^4*(136080*c + 138600*d*x - 181440*Cos[c + d*x] - 53760*Cos[3*(c + d*x)] + 16128*Cos[5*(c + d*x)] + 7200*Cos[7*(c + d*x)] - 1120*Cos[9*(c + d*x)] + 8820*Sin[2*(c + d*x)] - 42840*Sin[4*(c + d*x)] - 2730*Sin[6*(c + d*x)] + 4095*Sin[8*(c + d*x)] - 126*Sin[10*(c + d*x)]))/(645120*d)","A",1
408,1,209,140,5.2585016,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 (\sin (c+d x)+1)^4 \left(-732 (c+d x)-120 \sin (2 (c+d x))+3 \sin (4 (c+d x))+96 \cos (c+d x)+32 \cos (3 (c+d x))+224 \tan \left(\frac{1}{2} (c+d x)\right)-224 \cot \left(\frac{1}{2} (c+d x)\right)-48 \csc ^2\left(\frac{1}{2} (c+d x)\right)+48 \sec ^2\left(\frac{1}{2} (c+d x)\right)-192 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+192 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+32 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{96 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^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,"(a^4*(1 + Sin[c + d*x])^4*(-732*(c + d*x) + 96*Cos[c + d*x] + 32*Cos[3*(c + d*x)] - 224*Cot[(c + d*x)/2] - 48*Csc[(c + d*x)/2]^2 + 192*Log[Cos[(c + d*x)/2]] - 192*Log[Sin[(c + d*x)/2]] + 48*Sec[(c + d*x)/2]^2 + 32*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 120*Sin[2*(c + d*x)] + 3*Sin[4*(c + d*x)] + 224*Tan[(c + d*x)/2]))/(96*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)","A",1
409,1,86,135,0.258173,"\int \frac{\cos ^4(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{-105 \sin (2 (c+d x))-105 \sin (4 (c+d x))+35 \sin (6 (c+d x))+525 \cos (c+d x)+35 \cos (3 (c+d x))-63 \cos (5 (c+d x))+15 \cos (7 (c+d x))+420 c+420 d x}{6720 a d}","\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,"(420*c + 420*d*x + 525*Cos[c + d*x] + 35*Cos[3*(c + d*x)] - 63*Cos[5*(c + d*x)] + 15*Cos[7*(c + d*x)] - 105*Sin[2*(c + d*x)] - 105*Sin[4*(c + d*x)] + 35*Sin[6*(c + d*x)])/(6720*a*d)","A",1
410,1,377,117,4.9539476,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{-120 d x \sin \left(\frac{c}{2}\right)+120 \sin \left(\frac{c}{2}+d x\right)-120 \sin \left(\frac{3 c}{2}+d x\right)+15 \sin \left(\frac{3 c}{2}+2 d x\right)+15 \sin \left(\frac{5 c}{2}+2 d x\right)+20 \sin \left(\frac{5 c}{2}+3 d x\right)-20 \sin \left(\frac{7 c}{2}+3 d x\right)+15 \sin \left(\frac{7 c}{2}+4 d x\right)+15 \sin \left(\frac{9 c}{2}+4 d x\right)-12 \sin \left(\frac{9 c}{2}+5 d x\right)+12 \sin \left(\frac{11 c}{2}+5 d x\right)-5 \sin \left(\frac{11 c}{2}+6 d x\right)-5 \sin \left(\frac{13 c}{2}+6 d x\right)+30 \cos \left(\frac{c}{2}\right) (3 c-4 d x)-120 \cos \left(\frac{c}{2}+d x\right)-120 \cos \left(\frac{3 c}{2}+d x\right)+15 \cos \left(\frac{3 c}{2}+2 d x\right)-15 \cos \left(\frac{5 c}{2}+2 d x\right)-20 \cos \left(\frac{5 c}{2}+3 d x\right)-20 \cos \left(\frac{7 c}{2}+3 d x\right)+15 \cos \left(\frac{7 c}{2}+4 d x\right)-15 \cos \left(\frac{9 c}{2}+4 d x\right)+12 \cos \left(\frac{9 c}{2}+5 d x\right)+12 \cos \left(\frac{11 c}{2}+5 d x\right)-5 \cos \left(\frac{11 c}{2}+6 d x\right)+5 \cos \left(\frac{13 c}{2}+6 d x\right)+90 c \sin \left(\frac{c}{2}\right)-180 \sin \left(\frac{c}{2}\right)}{1920 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","\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,"(30*(3*c - 4*d*x)*Cos[c/2] - 120*Cos[c/2 + d*x] - 120*Cos[(3*c)/2 + d*x] + 15*Cos[(3*c)/2 + 2*d*x] - 15*Cos[(5*c)/2 + 2*d*x] - 20*Cos[(5*c)/2 + 3*d*x] - 20*Cos[(7*c)/2 + 3*d*x] + 15*Cos[(7*c)/2 + 4*d*x] - 15*Cos[(9*c)/2 + 4*d*x] + 12*Cos[(9*c)/2 + 5*d*x] + 12*Cos[(11*c)/2 + 5*d*x] - 5*Cos[(11*c)/2 + 6*d*x] + 5*Cos[(13*c)/2 + 6*d*x] - 180*Sin[c/2] + 90*c*Sin[c/2] - 120*d*x*Sin[c/2] + 120*Sin[c/2 + d*x] - 120*Sin[(3*c)/2 + d*x] + 15*Sin[(3*c)/2 + 2*d*x] + 15*Sin[(5*c)/2 + 2*d*x] + 20*Sin[(5*c)/2 + 3*d*x] - 20*Sin[(7*c)/2 + 3*d*x] + 15*Sin[(7*c)/2 + 4*d*x] + 15*Sin[(9*c)/2 + 4*d*x] - 12*Sin[(9*c)/2 + 5*d*x] + 12*Sin[(11*c)/2 + 5*d*x] - 5*Sin[(11*c)/2 + 6*d*x] - 5*Sin[(13*c)/2 + 6*d*x])/(1920*a*d*(Cos[c/2] + Sin[c/2]))","B",1
411,1,258,91,2.4351839,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{120 d x \sin \left(\frac{c}{2}\right)-60 \sin \left(\frac{c}{2}+d x\right)+60 \sin \left(\frac{3 c}{2}+d x\right)-10 \sin \left(\frac{5 c}{2}+3 d x\right)+10 \sin \left(\frac{7 c}{2}+3 d x\right)-15 \sin \left(\frac{7 c}{2}+4 d x\right)-15 \sin \left(\frac{9 c}{2}+4 d x\right)+6 \sin \left(\frac{9 c}{2}+5 d x\right)-6 \sin \left(\frac{11 c}{2}+5 d x\right)+120 d x \cos \left(\frac{c}{2}\right)+60 \cos \left(\frac{c}{2}+d x\right)+60 \cos \left(\frac{3 c}{2}+d x\right)+10 \cos \left(\frac{5 c}{2}+3 d x\right)+10 \cos \left(\frac{7 c}{2}+3 d x\right)-15 \cos \left(\frac{7 c}{2}+4 d x\right)+15 \cos \left(\frac{9 c}{2}+4 d x\right)-6 \cos \left(\frac{9 c}{2}+5 d x\right)-6 \cos \left(\frac{11 c}{2}+5 d x\right)+120 \sin \left(\frac{c}{2}\right)}{960 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"(120*d*x*Cos[c/2] + 60*Cos[c/2 + d*x] + 60*Cos[(3*c)/2 + d*x] + 10*Cos[(5*c)/2 + 3*d*x] + 10*Cos[(7*c)/2 + 3*d*x] - 15*Cos[(7*c)/2 + 4*d*x] + 15*Cos[(9*c)/2 + 4*d*x] - 6*Cos[(9*c)/2 + 5*d*x] - 6*Cos[(11*c)/2 + 5*d*x] + 120*Sin[c/2] + 120*d*x*Sin[c/2] - 60*Sin[c/2 + d*x] + 60*Sin[(3*c)/2 + d*x] - 10*Sin[(5*c)/2 + 3*d*x] + 10*Sin[(7*c)/2 + 3*d*x] - 15*Sin[(7*c)/2 + 4*d*x] - 15*Sin[(9*c)/2 + 4*d*x] + 6*Sin[(9*c)/2 + 5*d*x] - 6*Sin[(11*c)/2 + 5*d*x])/(960*a*d*(Cos[c/2] + Sin[c/2]))","B",1
412,1,219,73,1.6819731,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{24 d x \sin \left(\frac{c}{2}\right)-24 \sin \left(\frac{c}{2}+d x\right)+24 \sin \left(\frac{3 c}{2}+d x\right)-8 \sin \left(\frac{5 c}{2}+3 d x\right)+8 \sin \left(\frac{7 c}{2}+3 d x\right)-3 \sin \left(\frac{7 c}{2}+4 d x\right)-3 \sin \left(\frac{9 c}{2}+4 d x\right)-24 \cos \left(\frac{c}{2}\right) (c-d x)+24 \cos \left(\frac{c}{2}+d x\right)+24 \cos \left(\frac{3 c}{2}+d x\right)+8 \cos \left(\frac{5 c}{2}+3 d x\right)+8 \cos \left(\frac{7 c}{2}+3 d x\right)-3 \cos \left(\frac{7 c}{2}+4 d x\right)+3 \cos \left(\frac{9 c}{2}+4 d x\right)-24 c \sin \left(\frac{c}{2}\right)+48 \sin \left(\frac{c}{2}\right)}{192 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"-1/192*(-24*(c - d*x)*Cos[c/2] + 24*Cos[c/2 + d*x] + 24*Cos[(3*c)/2 + d*x] + 8*Cos[(5*c)/2 + 3*d*x] + 8*Cos[(7*c)/2 + 3*d*x] - 3*Cos[(7*c)/2 + 4*d*x] + 3*Cos[(9*c)/2 + 4*d*x] + 48*Sin[c/2] - 24*c*Sin[c/2] + 24*d*x*Sin[c/2] - 24*Sin[c/2 + d*x] + 24*Sin[(3*c)/2 + d*x] - 8*Sin[(5*c)/2 + 3*d*x] + 8*Sin[(7*c)/2 + 3*d*x] - 3*Sin[(7*c)/2 + 4*d*x] - 3*Sin[(9*c)/2 + 4*d*x])/(a*d*(Cos[c/2] + Sin[c/2]))","B",1
413,1,60,59,0.1811198,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\sin (2 (c+d x))-4 \cos (c+d x)+2 \left(-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)}{4 a d}","\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,"-1/4*(-4*Cos[c + d*x] + 2*(c + d*x + 2*Log[Cos[(c + d*x)/2]] - 2*Log[Sin[(c + d*x)/2]]) + Sin[2*(c + d*x)])/(a*d)","A",1
414,1,93,49,0.4249889,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right)^2 \left(\cos (c+d x)+\sin (c+d x) \left(\cos (c+d x)+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)\right)}{2 a d (\sin (c+d x)+1)}","-\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,"-1/2*((1 + Cot[(c + d*x)/2])^2*(Cos[c + d*x] + (c + d*x + Cos[c + d*x] - Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]])*Sin[c + d*x])*Tan[(c + d*x)/2])/(a*d*(1 + Sin[c + d*x]))","A",1
415,1,102,58,0.4640607,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^2 \left((2 \sin (c+d x)-1) \cos (c+d x)+\sin ^2(c+d x) \left(-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 c+2 d x\right)\right)}{8 a d (\sin (c+d x)+1)}","\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,"((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*((2*c + 2*d*x + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^2 + Cos[c + d*x]*(-1 + 2*Sin[c + d*x])))/(8*a*d*(1 + Sin[c + d*x]))","A",1
416,1,124,58,0.4879658,"\int \frac{\cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sin[c + d*x]),x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(\cos (3 (c+d x))+(3-6 \sin (c+d x)) \cos (c+d x)+6 \sin ^3(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{96 a d (\sin (c+d x)+1)}","-\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,"-1/96*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(Cos[3*(c + d*x)] + Cos[c + d*x]*(3 - 6*Sin[c + d*x]) + 6*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^3))/(a*d*(1 + Sin[c + d*x]))","B",1
417,1,125,82,0.9711372,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\csc ^4(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-42 \cos (c+d x)+2 (8 \sin (c+d x)-3) \cos (3 (c+d x))+24 \left(\sin (2 (c+d x))+\sin ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}{192 a d (\sin (c+d x)+1)}","\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,"(Csc[c + d*x]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(-42*Cos[c + d*x] + 2*Cos[3*(c + d*x)]*(-3 + 8*Sin[c + d*x]) + 24*((Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^4 + Sin[2*(c + d*x)])))/(192*a*d*(1 + Sin[c + d*x]))","A",1
418,1,189,100,0.5935774,"\int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^5(c+d x) \left(-180 \sin (2 (c+d x))-30 \sin (4 (c+d x))+320 \cos (c+d x)+80 \cos (3 (c+d x))-16 \cos (5 (c+d x))-150 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+75 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-15 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+150 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-75 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+15 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1920 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,"-1/1920*(Csc[c + d*x]^5*(320*Cos[c + d*x] + 80*Cos[3*(c + d*x)] - 16*Cos[5*(c + d*x)] + 150*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 150*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 180*Sin[2*(c + d*x)] - 75*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 75*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 30*Sin[4*(c + d*x)] + 15*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 15*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(a*d)","A",1
419,1,229,124,0.5643719,"\int \frac{\cot ^4(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^6(c+d x) \left(-480 \sin (2 (c+d x))-192 \sin (4 (c+d x))+32 \sin (6 (c+d x))+1140 \cos (c+d x)+170 \cos (3 (c+d x))-30 \cos (5 (c+d x))+150 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+225 \cos (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-90 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+15 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-150 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-225 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+90 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-15 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{7680 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,"-1/7680*(Csc[c + d*x]^6*(1140*Cos[c + d*x] + 170*Cos[3*(c + d*x)] - 30*Cos[5*(c + d*x)] - 150*Log[Cos[(c + d*x)/2]] + 225*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 90*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 15*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 150*Log[Sin[(c + d*x)/2]] - 225*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 90*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 15*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 480*Sin[2*(c + d*x)] - 192*Sin[4*(c + d*x)] + 32*Sin[6*(c + d*x)]))/(a*d)","A",1
420,1,418,147,4.8312989,"\int \frac{\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]","\frac{-8400 d x \sin \left(\frac{c}{2}\right)+7875 \sin \left(\frac{c}{2}+d x\right)-7875 \sin \left(\frac{3 c}{2}+d x\right)+3150 \sin \left(\frac{3 c}{2}+2 d x\right)+3150 \sin \left(\frac{5 c}{2}+2 d x\right)-1435 \sin \left(\frac{5 c}{2}+3 d x\right)+1435 \sin \left(\frac{7 c}{2}+3 d x\right)-630 \sin \left(\frac{7 c}{2}+4 d x\right)-630 \sin \left(\frac{9 c}{2}+4 d x\right)+231 \sin \left(\frac{9 c}{2}+5 d x\right)-231 \sin \left(\frac{11 c}{2}+5 d x\right)+70 \sin \left(\frac{11 c}{2}+6 d x\right)+70 \sin \left(\frac{13 c}{2}+6 d x\right)-15 \sin \left(\frac{13 c}{2}+7 d x\right)+15 \sin \left(\frac{15 c}{2}+7 d x\right)-210 \cos \left(\frac{c}{2}\right) (40 d x+1)-7875 \cos \left(\frac{c}{2}+d x\right)-7875 \cos \left(\frac{3 c}{2}+d x\right)+3150 \cos \left(\frac{3 c}{2}+2 d x\right)-3150 \cos \left(\frac{5 c}{2}+2 d x\right)+1435 \cos \left(\frac{5 c}{2}+3 d x\right)+1435 \cos \left(\frac{7 c}{2}+3 d x\right)-630 \cos \left(\frac{7 c}{2}+4 d x\right)+630 \cos \left(\frac{9 c}{2}+4 d x\right)-231 \cos \left(\frac{9 c}{2}+5 d x\right)-231 \cos \left(\frac{11 c}{2}+5 d x\right)+70 \cos \left(\frac{11 c}{2}+6 d x\right)-70 \cos \left(\frac{13 c}{2}+6 d x\right)+15 \cos \left(\frac{13 c}{2}+7 d x\right)+15 \cos \left(\frac{15 c}{2}+7 d x\right)+210 \sin \left(\frac{c}{2}\right)}{13440 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","\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,"(-210*(1 + 40*d*x)*Cos[c/2] - 7875*Cos[c/2 + d*x] - 7875*Cos[(3*c)/2 + d*x] + 3150*Cos[(3*c)/2 + 2*d*x] - 3150*Cos[(5*c)/2 + 2*d*x] + 1435*Cos[(5*c)/2 + 3*d*x] + 1435*Cos[(7*c)/2 + 3*d*x] - 630*Cos[(7*c)/2 + 4*d*x] + 630*Cos[(9*c)/2 + 4*d*x] - 231*Cos[(9*c)/2 + 5*d*x] - 231*Cos[(11*c)/2 + 5*d*x] + 70*Cos[(11*c)/2 + 6*d*x] - 70*Cos[(13*c)/2 + 6*d*x] + 15*Cos[(13*c)/2 + 7*d*x] + 15*Cos[(15*c)/2 + 7*d*x] + 210*Sin[c/2] - 8400*d*x*Sin[c/2] + 7875*Sin[c/2 + d*x] - 7875*Sin[(3*c)/2 + d*x] + 3150*Sin[(3*c)/2 + 2*d*x] + 3150*Sin[(5*c)/2 + 2*d*x] - 1435*Sin[(5*c)/2 + 3*d*x] + 1435*Sin[(7*c)/2 + 3*d*x] - 630*Sin[(7*c)/2 + 4*d*x] - 630*Sin[(9*c)/2 + 4*d*x] + 231*Sin[(9*c)/2 + 5*d*x] - 231*Sin[(11*c)/2 + 5*d*x] + 70*Sin[(11*c)/2 + 6*d*x] + 70*Sin[(13*c)/2 + 6*d*x] - 15*Sin[(13*c)/2 + 7*d*x] + 15*Sin[(15*c)/2 + 7*d*x])/(13440*a^2*d*(Cos[c/2] + Sin[c/2]))","B",1
421,1,76,129,0.2393776,"\int \frac{\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{-465 \sin (2 (c+d x))+75 \sin (4 (c+d x))-5 \sin (6 (c+d x))+1200 \cos (c+d x)-200 \cos (3 (c+d x))+24 \cos (5 (c+d x))+660 c+660 d x}{960 a^2 d}","\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,"(660*c + 660*d*x + 1200*Cos[c + d*x] - 200*Cos[3*(c + d*x)] + 24*Cos[5*(c + d*x)] - 465*Sin[2*(c + d*x)] + 75*Sin[4*(c + d*x)] - 5*Sin[6*(c + d*x)])/(960*a^2*d)","A",1
422,1,308,102,1.3219954,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{120 d x \sin \left(\frac{c}{2}\right)-110 \sin \left(\frac{c}{2}+d x\right)+110 \sin \left(\frac{3 c}{2}+d x\right)-40 \sin \left(\frac{3 c}{2}+2 d x\right)-40 \sin \left(\frac{5 c}{2}+2 d x\right)+15 \sin \left(\frac{5 c}{2}+3 d x\right)-15 \sin \left(\frac{7 c}{2}+3 d x\right)+5 \sin \left(\frac{7 c}{2}+4 d x\right)+5 \sin \left(\frac{9 c}{2}+4 d x\right)-\sin \left(\frac{9 c}{2}+5 d x\right)+\sin \left(\frac{11 c}{2}+5 d x\right)+5 \cos \left(\frac{c}{2}\right) (24 d x+1)+110 \cos \left(\frac{c}{2}+d x\right)+110 \cos \left(\frac{3 c}{2}+d x\right)-40 \cos \left(\frac{3 c}{2}+2 d x\right)+40 \cos \left(\frac{5 c}{2}+2 d x\right)-15 \cos \left(\frac{5 c}{2}+3 d x\right)-15 \cos \left(\frac{7 c}{2}+3 d x\right)+5 \cos \left(\frac{7 c}{2}+4 d x\right)-5 \cos \left(\frac{9 c}{2}+4 d x\right)+\cos \left(\frac{9 c}{2}+5 d x\right)+\cos \left(\frac{11 c}{2}+5 d x\right)-5 \sin \left(\frac{c}{2}\right)}{160 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"-1/160*(5*(1 + 24*d*x)*Cos[c/2] + 110*Cos[c/2 + d*x] + 110*Cos[(3*c)/2 + d*x] - 40*Cos[(3*c)/2 + 2*d*x] + 40*Cos[(5*c)/2 + 2*d*x] - 15*Cos[(5*c)/2 + 3*d*x] - 15*Cos[(7*c)/2 + 3*d*x] + 5*Cos[(7*c)/2 + 4*d*x] - 5*Cos[(9*c)/2 + 4*d*x] + Cos[(9*c)/2 + 5*d*x] + Cos[(11*c)/2 + 5*d*x] - 5*Sin[c/2] + 120*d*x*Sin[c/2] - 110*Sin[c/2 + d*x] + 110*Sin[(3*c)/2 + d*x] - 40*Sin[(3*c)/2 + 2*d*x] - 40*Sin[(5*c)/2 + 2*d*x] + 15*Sin[(5*c)/2 + 3*d*x] - 15*Sin[(7*c)/2 + 3*d*x] + 5*Sin[(7*c)/2 + 4*d*x] + 5*Sin[(9*c)/2 + 4*d*x] - Sin[(9*c)/2 + 5*d*x] + Sin[(11*c)/2 + 5*d*x])/(a^2*d*(Cos[c/2] + Sin[c/2]))","B",1
423,1,258,87,1.3548877,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{168 d x \sin \left(\frac{c}{2}\right)-144 \sin \left(\frac{c}{2}+d x\right)+144 \sin \left(\frac{3 c}{2}+d x\right)-48 \sin \left(\frac{3 c}{2}+2 d x\right)-48 \sin \left(\frac{5 c}{2}+2 d x\right)+16 \sin \left(\frac{5 c}{2}+3 d x\right)-16 \sin \left(\frac{7 c}{2}+3 d x\right)+3 \sin \left(\frac{7 c}{2}+4 d x\right)+3 \sin \left(\frac{9 c}{2}+4 d x\right)+168 d x \cos \left(\frac{c}{2}\right)+144 \cos \left(\frac{c}{2}+d x\right)+144 \cos \left(\frac{3 c}{2}+d x\right)-48 \cos \left(\frac{3 c}{2}+2 d x\right)+48 \cos \left(\frac{5 c}{2}+2 d x\right)-16 \cos \left(\frac{5 c}{2}+3 d x\right)-16 \cos \left(\frac{7 c}{2}+3 d x\right)+3 \cos \left(\frac{7 c}{2}+4 d x\right)-3 \cos \left(\frac{9 c}{2}+4 d x\right)+8 \sin \left(\frac{c}{2}\right)}{192 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"(168*d*x*Cos[c/2] + 144*Cos[c/2 + d*x] + 144*Cos[(3*c)/2 + d*x] - 48*Cos[(3*c)/2 + 2*d*x] + 48*Cos[(5*c)/2 + 2*d*x] - 16*Cos[(5*c)/2 + 3*d*x] - 16*Cos[(7*c)/2 + 3*d*x] + 3*Cos[(7*c)/2 + 4*d*x] - 3*Cos[(9*c)/2 + 4*d*x] + 8*Sin[c/2] + 168*d*x*Sin[c/2] - 144*Sin[c/2 + d*x] + 144*Sin[(3*c)/2 + d*x] - 48*Sin[(3*c)/2 + 2*d*x] - 48*Sin[(5*c)/2 + 2*d*x] + 16*Sin[(5*c)/2 + 3*d*x] - 16*Sin[(7*c)/2 + 3*d*x] + 3*Sin[(7*c)/2 + 4*d*x] + 3*Sin[(9*c)/2 + 4*d*x])/(192*a^2*d*(Cos[c/2] + Sin[c/2]))","B",1
424,1,204,70,0.8076483,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{-24 d x \sin \left(\frac{c}{2}\right)+21 \sin \left(\frac{c}{2}+d x\right)-21 \sin \left(\frac{3 c}{2}+d x\right)+6 \sin \left(\frac{3 c}{2}+2 d x\right)+6 \sin \left(\frac{5 c}{2}+2 d x\right)-\sin \left(\frac{5 c}{2}+3 d x\right)+\sin \left(\frac{7 c}{2}+3 d x\right)-2 \cos \left(\frac{c}{2}\right) (12 d x+1)-21 \cos \left(\frac{c}{2}+d x\right)-21 \cos \left(\frac{3 c}{2}+d x\right)+6 \cos \left(\frac{3 c}{2}+2 d x\right)-6 \cos \left(\frac{5 c}{2}+2 d x\right)+\cos \left(\frac{5 c}{2}+3 d x\right)+\cos \left(\frac{7 c}{2}+3 d x\right)+2 \sin \left(\frac{c}{2}\right)}{24 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"(-2*(1 + 12*d*x)*Cos[c/2] - 21*Cos[c/2 + d*x] - 21*Cos[(3*c)/2 + d*x] + 6*Cos[(3*c)/2 + 2*d*x] - 6*Cos[(5*c)/2 + 2*d*x] + Cos[(5*c)/2 + 3*d*x] + Cos[(7*c)/2 + 3*d*x] + 2*Sin[c/2] - 24*d*x*Sin[c/2] + 21*Sin[c/2 + d*x] - 21*Sin[(3*c)/2 + d*x] + 6*Sin[(3*c)/2 + 2*d*x] + 6*Sin[(5*c)/2 + 2*d*x] - Sin[(5*c)/2 + 3*d*x] + Sin[(7*c)/2 + 3*d*x])/(24*a^2*d*(Cos[c/2] + Sin[c/2]))","B",1
425,1,46,36,0.1335936,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\cos (c+d x)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 c+2 d x}{a^2 d}","-\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*c + 2*d*x + Cos[c + d*x] + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])/(a^2*d))","A",1
426,1,98,35,0.3668693,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(2 (c+d x)+\tan \left(\frac{1}{2} (c+d x)\right)-\cot \left(\frac{1}{2} (c+d x)\right)-4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 d (a \sin (c+d 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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(2*(c + d*x) - Cot[(c + d*x)/2] + 4*Log[Cos[(c + d*x)/2]] - 4*Log[Sin[(c + d*x)/2]] + Tan[(c + d*x)/2]))/(2*d*(a + a*Sin[c + d*x])^2)","B",1
427,1,86,54,0.5291366,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(\cot (c+d x) (\csc (c+d x)-4)+3 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{2 a^2 d (\sin (c+d x)+1)^2}","\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,"-1/2*((Cot[c + d*x]*(-4 + Csc[c + d*x]) + 3*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
428,1,121,66,0.9124064,"\int \frac{\cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^2,x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right)^4 \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-9 \cos (c+d x)+5 \cos (3 (c+d x))+6 \left(\sin (2 (c+d x))+2 \sin ^3(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}{96 a^2 d (\sin (c+d x)+1)^2}","-\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,"((1 + Cot[(c + d*x)/2])^4*Sec[(c + d*x)/2]^2*(-9*Cos[c + d*x] + 5*Cos[3*(c + d*x)] + 6*(2*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^3 + Sin[2*(c + d*x)]))*Tan[(c + d*x)/2])/(96*a^2*d*(1 + Sin[c + d*x])^2)","A",1
429,1,116,96,1.5167149,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(-48 \sin (2 (c+d x))+45 \cos (c+d x)+(32 \sin (c+d x)-21) \cos (3 (c+d x))+84 \sin ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{1536 a^2 d (\sin (c+d x)+1)^2}","\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,"-1/1536*((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^4*(45*Cos[c + d*x] + 84*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^4 + Cos[3*(c + d*x)]*(-21 + 32*Sin[c + d*x]) - 48*Sin[2*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
430,1,189,112,0.7470252,"\int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\csc ^5(c+d x) \left(140 \sin (2 (c+d x))-30 \sin (4 (c+d x))-160 \cos (c+d x)+120 \cos (3 (c+d x))-24 \cos (5 (c+d x))-150 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+75 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-15 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+150 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-75 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+15 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{320 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,"(Csc[c + d*x]^5*(-160*Cos[c + d*x] + 120*Cos[3*(c + d*x)] - 24*Cos[5*(c + d*x)] + 150*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 150*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 140*Sin[2*(c + d*x)] - 75*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 75*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 30*Sin[4*(c + d*x)] + 15*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 15*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(320*a^2*d)","A",1
431,1,229,138,0.8331456,"\int \frac{\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{\csc ^6(c+d x) \left(3840 \sin (2 (c+d x))-1536 \sin (4 (c+d x))+256 \sin (6 (c+d x))-2820 \cos (c+d x)+1870 \cos (3 (c+d x))-330 \cos (5 (c+d x))+1650 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2475 \cos (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-990 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+165 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-1650 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2475 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+990 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-165 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{7680 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,"(Csc[c + d*x]^6*(-2820*Cos[c + d*x] + 1870*Cos[3*(c + d*x)] - 330*Cos[5*(c + d*x)] - 1650*Log[Cos[(c + d*x)/2]] + 2475*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 990*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 165*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 1650*Log[Sin[(c + d*x)/2]] - 2475*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 990*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 165*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 3840*Sin[2*(c + d*x)] - 1536*Sin[4*(c + d*x)] + 256*Sin[6*(c + d*x)]))/(7680*a^2*d)","A",1
432,1,195,109,1.3366612,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{2040 d x \sin \left(c+\frac{d x}{2}\right)+800 \sin \left(2 c+\frac{3 d x}{2}\right)-160 \sin \left(2 c+\frac{5 d x}{2}\right)-35 \sin \left(4 c+\frac{7 d x}{2}\right)+5 \sin \left(4 c+\frac{9 d x}{2}\right)+997 \cos \left(c+\frac{d x}{2}\right)+800 \cos \left(c+\frac{3 d x}{2}\right)+160 \cos \left(3 c+\frac{5 d x}{2}\right)-35 \cos \left(3 c+\frac{7 d x}{2}\right)-5 \cos \left(5 c+\frac{9 d x}{2}\right)-3563 \sin \left(\frac{d x}{2}\right)+2040 d x \cos \left(\frac{d x}{2}\right)}{320 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\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,"(2040*d*x*Cos[(d*x)/2] + 997*Cos[c + (d*x)/2] + 800*Cos[c + (3*d*x)/2] + 160*Cos[3*c + (5*d*x)/2] - 35*Cos[3*c + (7*d*x)/2] - 5*Cos[5*c + (9*d*x)/2] - 3563*Sin[(d*x)/2] + 2040*d*x*Sin[c + (d*x)/2] + 800*Sin[2*c + (3*d*x)/2] - 160*Sin[2*c + (5*d*x)/2] - 35*Sin[4*c + (7*d*x)/2] + 5*Sin[4*c + (9*d*x)/2])/(320*a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
433,1,181,87,1.1358716,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{-660 d x \sin \left(c+\frac{d x}{2}\right)+\sin \left(c+\frac{d x}{2}\right)-240 \sin \left(2 c+\frac{3 d x}{2}\right)+40 \sin \left(2 c+\frac{5 d x}{2}\right)+5 \sin \left(4 c+\frac{7 d x}{2}\right)-286 \cos \left(c+\frac{d x}{2}\right)-240 \cos \left(c+\frac{3 d x}{2}\right)-40 \cos \left(3 c+\frac{5 d x}{2}\right)+5 \cos \left(3 c+\frac{7 d x}{2}\right)+1244 \sin \left(\frac{d x}{2}\right)+(1-660 d x) \cos \left(\frac{d x}{2}\right)}{120 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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,"((1 - 660*d*x)*Cos[(d*x)/2] - 286*Cos[c + (d*x)/2] - 240*Cos[c + (3*d*x)/2] - 40*Cos[3*c + (5*d*x)/2] + 5*Cos[3*c + (7*d*x)/2] + 1244*Sin[(d*x)/2] + Sin[c + (d*x)/2] - 660*d*x*Sin[c + (d*x)/2] - 240*Sin[2*c + (3*d*x)/2] + 40*Sin[2*c + (5*d*x)/2] + 5*Sin[4*c + (7*d*x)/2])/(120*a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
434,1,143,80,0.6446926,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{180 d x \sin \left(c+\frac{d x}{2}\right)+55 \sin \left(2 c+\frac{3 d x}{2}\right)-5 \sin \left(2 c+\frac{5 d x}{2}\right)+59 \cos \left(c+\frac{d x}{2}\right)+55 \cos \left(c+\frac{3 d x}{2}\right)+5 \cos \left(3 c+\frac{5 d x}{2}\right)-381 \sin \left(\frac{d x}{2}\right)+180 d x \cos \left(\frac{d x}{2}\right)}{40 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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,"(180*d*x*Cos[(d*x)/2] + 59*Cos[c + (d*x)/2] + 55*Cos[c + (3*d*x)/2] + 5*Cos[3*c + (5*d*x)/2] - 381*Sin[(d*x)/2] + 180*d*x*Sin[c + (d*x)/2] + 55*Sin[2*c + (3*d*x)/2] - 5*Sin[2*c + (5*d*x)/2])/(40*a^3*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
435,1,122,45,0.284098,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)+\sin \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x-8\right)\right)}{a^3 d (\sin (c+d x)+1)^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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*(Cos[(c + d*x)/2]*(c + d*x - Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]) + (-8 + c + d*x - Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]])*Sin[(c + d*x)/2]))/(a^3*d*(1 + Sin[c + d*x])^3)","B",1
436,1,156,54,0.6438135,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(\cot ^2\left(\frac{1}{2} (c+d x)\right)+6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\cot \left(\frac{1}{2} (c+d x)\right) \left(6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+1\right)-17\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^3 d (\sin (c+d x)+1)^3}","-\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,"-1/2*((Cos[(c + d*x)/2]*(-17 + Cot[(c + d*x)/2]^2 - 6*Log[Cos[(c + d*x)/2]] + 6*Log[Sin[(c + d*x)/2]] + Cot[(c + d*x)/2]*(1 - 6*Log[Cos[(c + d*x)/2]] + 6*Log[Sin[(c + d*x)/2]])) - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*Tan[(c + d*x)/2])/(a^3*d*(1 + Sin[c + d*x])^3)","B",1
437,1,213,78,5.9014112,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","-\frac{\sin ^8\left(\frac{1}{2} (c+d x)\right) \sin ^7(c+d x) \left(\csc ^2\left(\frac{1}{2} (c+d x)\right)+2 \csc (c+d x)\right)^5 \left((\csc (c+d x)-6) \csc ^6\left(\frac{1}{2} (c+d x)\right)-8 (\csc (c+d x)-6) \csc ^3(c+d x)+2 \csc (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right) \left(\csc (c+d x)-18 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+18 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-6\right)-4 \csc ^2(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) \left(\csc (c+d x)+18 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-18 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-38\right)\right)}{512 a^3 d (\sin (c+d x)+1)^3}","\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,"-1/512*((Csc[(c + d*x)/2]^2 + 2*Csc[c + d*x])^5*(Csc[(c + d*x)/2]^6*(-6 + Csc[c + d*x]) - 8*(-6 + Csc[c + d*x])*Csc[c + d*x]^3 + 2*Csc[(c + d*x)/2]^4*Csc[c + d*x]*(-6 + Csc[c + d*x] + 18*Log[Cos[(c + d*x)/2]] - 18*Log[Sin[(c + d*x)/2]]) - 4*Csc[(c + d*x)/2]^2*Csc[c + d*x]^2*(-38 + Csc[c + d*x] - 18*Log[Cos[(c + d*x)/2]] + 18*Log[Sin[(c + d*x)/2]]))*Sin[(c + d*x)/2]^8*Sin[c + d*x]^7)/(a^3*d*(1 + Sin[c + d*x])^3)","B",1
438,1,251,96,5.0330272,"\int \frac{\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]","-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right) \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right)^5 \csc ^3(c+d x) \left(-4 \sin ^8\left(\frac{1}{2} (c+d x)\right)-8 \sin (c+d x) (7 \sin (c+d x)-2) \sin ^6\left(\frac{1}{2} (c+d x)\right)+\frac{1}{4} \sin ^4(c+d x) \left(28 \sin (c+d x)+\cot \left(\frac{1}{2} (c+d x)\right)-8\right)+\sin ^2(c+d x) \sin ^4\left(\frac{1}{2} (c+d x)\right) \left(9-2 \sin (c+d x) \left(-33 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+33 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+62\right)\right)-\frac{1}{2} \sin ^3(c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin (c+d x) \left(-66 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+66 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-28\right)+9\right)\right)}{12 a^3 d (\sin (c+d x)+1)^3}","-\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,"-1/12*((1 + Cot[(c + d*x)/2])^5*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2*(-4*Sin[(c + d*x)/2]^8 - 8*Sin[(c + d*x)/2]^6*Sin[c + d*x]*(-2 + 7*Sin[c + d*x]) + (Sin[c + d*x]^4*(-8 + Cot[(c + d*x)/2] + 28*Sin[c + d*x]))/4 - (Sin[(c + d*x)/2]^2*Sin[c + d*x]^3*(9 + (-28 + 66*Log[Cos[(c + d*x)/2]] - 66*Log[Sin[(c + d*x)/2]])*Sin[c + d*x]))/2 + Sin[(c + d*x)/2]^4*Sin[c + d*x]^2*(9 - 2*(62 + 33*Log[Cos[(c + d*x)/2]] - 33*Log[Sin[(c + d*x)/2]])*Sin[c + d*x])))/(a^3*d*(1 + Sin[c + d*x])^3)","B",1
439,1,601,117,6.1630728,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{8 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}{d (a \sin (c+d x)+a)^3}-\frac{51 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{8 d (a \sin (c+d x)+a)^3}+\frac{51 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{8 d (a \sin (c+d x)+a)^3}-\frac{3 \tan \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{d (a \sin (c+d x)+a)^3}+\frac{3 \cot \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{d (a \sin (c+d x)+a)^3}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{64 d (a \sin (c+d x)+a)^3}-\frac{19 \csc ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{32 d (a \sin (c+d x)+a)^3}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{64 d (a \sin (c+d x)+a)^3}+\frac{19 \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{32 d (a \sin (c+d x)+a)^3}+\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{8 d (a \sin (c+d x)+a)^3}-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{8 d (a \sin (c+d x)+a)^3}","\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,"(-8*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)/(d*(a + a*Sin[c + d*x])^3) + (3*Cot[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)/(d*(a + a*Sin[c + d*x])^3) - (19*Csc[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)/(32*d*(a + a*Sin[c + d*x])^3) + (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)/(8*d*(a + a*Sin[c + d*x])^3) - (Csc[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)/(64*d*(a + a*Sin[c + d*x])^3) - (51*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)/(8*d*(a + a*Sin[c + d*x])^3) + (51*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)/(8*d*(a + a*Sin[c + d*x])^3) + (19*Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)/(32*d*(a + a*Sin[c + d*x])^3) + (Sec[(c + d*x)/2]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)/(64*d*(a + a*Sin[c + d*x])^3) - (3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*Tan[(c + d*x)/2])/(d*(a + a*Sin[c + d*x])^3) - (Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*Tan[(c + d*x)/2])/(8*d*(a + a*Sin[c + d*x])^3)","B",1
440,1,143,58,1.2613051,"\int \frac{\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx","Integrate[(Cos[e + f*x]^4*Sin[e + f*x])/(a + a*Sin[e + f*x])^6,x]","\frac{1134 \sin \left(2 e+\frac{3 f x}{2}\right)-224 \sin \left(2 e+\frac{5 f x}{2}\right)+\sin \left(4 e+\frac{7 f x}{2}\right)+4585 \cos \left(e+\frac{f x}{2}\right)-2982 \cos \left(e+\frac{3 f x}{2}\right)-1148 \cos \left(3 e+\frac{5 f x}{2}\right)+197 \cos \left(3 e+\frac{7 f x}{2}\right)+2275 \sin \left(\frac{f x}{2}\right)}{4620 a^6 f \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^7}","\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,"(4585*Cos[e + (f*x)/2] - 2982*Cos[e + (3*f*x)/2] - 1148*Cos[3*e + (5*f*x)/2] + 197*Cos[3*e + (7*f*x)/2] + 2275*Sin[(f*x)/2] + 1134*Sin[2*e + (3*f*x)/2] - 224*Sin[2*e + (5*f*x)/2] + Sin[4*e + (7*f*x)/2])/(4620*a^6*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7)","B",1
441,1,293,89,2.6094843,"\int \frac{\cos ^4(e+f x) \sin ^2(e+f x)}{(a+a \sin (e+f x))^7} \, dx","Integrate[(Cos[e + f*x]^4*Sin[e + f*x]^2)/(a + a*Sin[e + f*x])^7,x]","\frac{1890 \sin \left(e+\frac{f x}{2}\right)+1260 \sin \left(e+\frac{3 f x}{2}\right)+659400 \sin \left(2 e+\frac{3 f x}{2}\right)-303192 \sin \left(2 e+\frac{5 f x}{2}\right)-540 \sin \left(3 e+\frac{5 f x}{2}\right)-135 \sin \left(3 e+\frac{7 f x}{2}\right)-89955 \sin \left(4 e+\frac{7 f x}{2}\right)+13427 \sin \left(4 e+\frac{9 f x}{2}\right)+15 \sin \left(5 e+\frac{9 f x}{2}\right)+718830 \cos \left(e+\frac{f x}{2}\right)-467208 \cos \left(e+\frac{3 f x}{2}\right)-1260 \cos \left(2 e+\frac{3 f x}{2}\right)-540 \cos \left(2 e+\frac{5 f x}{2}\right)-179640 \cos \left(3 e+\frac{5 f x}{2}\right)+30753 \cos \left(3 e+\frac{7 f x}{2}\right)+135 \cos \left(4 e+\frac{7 f x}{2}\right)+15 \cos \left(4 e+\frac{9 f x}{2}\right)-15 \cos \left(5 e+\frac{9 f x}{2}\right)+971082 \sin \left(\frac{f x}{2}\right)+1890 \cos \left(\frac{f x}{2}\right)}{720720 a^7 f \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^9}","-\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,"(1890*Cos[(f*x)/2] + 718830*Cos[e + (f*x)/2] - 467208*Cos[e + (3*f*x)/2] - 1260*Cos[2*e + (3*f*x)/2] - 540*Cos[2*e + (5*f*x)/2] - 179640*Cos[3*e + (5*f*x)/2] + 30753*Cos[3*e + (7*f*x)/2] + 135*Cos[4*e + (7*f*x)/2] + 15*Cos[4*e + (9*f*x)/2] - 15*Cos[5*e + (9*f*x)/2] + 971082*Sin[(f*x)/2] + 1890*Sin[e + (f*x)/2] + 1260*Sin[e + (3*f*x)/2] + 659400*Sin[2*e + (3*f*x)/2] - 303192*Sin[2*e + (5*f*x)/2] - 540*Sin[3*e + (5*f*x)/2] - 135*Sin[3*e + (7*f*x)/2] - 89955*Sin[4*e + (7*f*x)/2] + 13427*Sin[4*e + (9*f*x)/2] + 15*Sin[5*e + (9*f*x)/2])/(720720*a^7*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^9)","B",1
442,1,195,157,3.1761003,"\int \frac{\cos ^4(e+f x) \sin ^3(e+f x)}{(a+a \sin (e+f x))^8} \, dx","Integrate[(Cos[e + f*x]^4*Sin[e + f*x]^3)/(a + a*Sin[e + f*x])^8,x]","-\frac{-299970 \sin \left(2 e+\frac{3 f x}{2}\right)+145695 \sin \left(2 e+\frac{5 f x}{2}\right)+44990 \sin \left(4 e+\frac{7 f x}{2}\right)-6710 \sin \left(4 e+\frac{9 f x}{2}\right)+\sin \left(6 e+\frac{11 f x}{2}\right)-486024 \cos \left(e+\frac{f x}{2}\right)+351450 \cos \left(e+\frac{3 f x}{2}\right)+180015 \cos \left(3 e+\frac{5 f x}{2}\right)-63580 \cos \left(3 e+\frac{7 f x}{2}\right)-15004 \cos \left(5 e+\frac{9 f x}{2}\right)+1975 \cos \left(5 e+\frac{11 f x}{2}\right)-425964 \sin \left(\frac{f x}{2}\right)}{240240 a^8 f \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^{11}}","-\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,"-1/240240*(-486024*Cos[e + (f*x)/2] + 351450*Cos[e + (3*f*x)/2] + 180015*Cos[3*e + (5*f*x)/2] - 63580*Cos[3*e + (7*f*x)/2] - 15004*Cos[5*e + (9*f*x)/2] + 1975*Cos[5*e + (11*f*x)/2] - 425964*Sin[(f*x)/2] - 299970*Sin[2*e + (3*f*x)/2] + 145695*Sin[2*e + (5*f*x)/2] + 44990*Sin[4*e + (7*f*x)/2] - 6710*Sin[4*e + (9*f*x)/2] + Sin[6*e + (11*f*x)/2])/(a^8*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^11)","A",1
443,1,109,156,3.9071812,"\int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (119780 \sin (c+d x)-21420 \sin (3 (c+d x))-62440 \cos (2 (c+d x))+3465 \cos (4 (c+d x))+81183)}{180180 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{1472 a^3 \cos ^5(c+d x)}{45045 d (a \sin (c+d x)+a)^{5/2}}-\frac{368 a^2 \cos ^5(c+d x)}{9009 d (a \sin (c+d x)+a)^{3/2}}-\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,"-1/180180*((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*Sqrt[a*(1 + Sin[c + d*x])]*(81183 - 62440*Cos[2*(c + d*x)] + 3465*Cos[4*(c + d*x)] + 119780*Sin[c + d*x] - 21420*Sin[3*(c + d*x)]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
444,1,99,124,2.0200861,"\int \cos ^4(c+d x) \sin (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (-5165 \sin (c+d x)+315 \sin (3 (c+d x))+1960 \cos (2 (c+d x))-3648)}{6930 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{64 a^3 \cos ^5(c+d x)}{3465 d (a \sin (c+d x)+a)^{5/2}}-\frac{16 a^2 \cos ^5(c+d x)}{693 d (a \sin (c+d x)+a)^{3/2}}-\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,"((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*Sqrt[a*(1 + Sin[c + d*x])]*(-3648 + 1960*Cos[2*(c + d*x)] - 5165*Sin[c + d*x] + 315*Sin[3*(c + d*x)]))/(6930*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
445,1,195,159,0.3284188,"\int \cos ^3(c+d x) \cot (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(-525 \sin \left(\frac{1}{2} (c+d x)\right)+175 \sin \left(\frac{3}{2} (c+d x)\right)-21 \sin \left(\frac{5}{2} (c+d x)\right)+15 \sin \left(\frac{7}{2} (c+d x)\right)+525 \cos \left(\frac{1}{2} (c+d x)\right)+175 \cos \left(\frac{3}{2} (c+d x)\right)+21 \cos \left(\frac{5}{2} (c+d x)\right)+15 \cos \left(\frac{7}{2} (c+d x)\right)-420 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+420 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{420 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\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,"(Sqrt[a*(1 + Sin[c + d*x])]*(525*Cos[(c + d*x)/2] + 175*Cos[(3*(c + d*x))/2] + 21*Cos[(5*(c + d*x))/2] + 15*Cos[(7*(c + d*x))/2] - 420*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 420*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 525*Sin[(c + d*x)/2] + 175*Sin[(3*(c + d*x))/2] - 21*Sin[(5*(c + d*x))/2] + 15*Sin[(7*(c + d*x))/2]))/(420*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
446,1,258,148,0.7406211,"\int \cos ^2(c+d x) \cot ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(155 \sin \left(\frac{1}{2} (c+d x)\right)+87 \sin \left(\frac{3}{2} (c+d x)\right)-5 \sin \left(\frac{5}{2} (c+d x)\right)+3 \sin \left(\frac{7}{2} (c+d x)\right)-155 \cos \left(\frac{1}{2} (c+d x)\right)+87 \cos \left(\frac{3}{2} (c+d x)\right)+5 \cos \left(\frac{5}{2} (c+d x)\right)+3 \cos \left(\frac{7}{2} (c+d x)\right)-30 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+30 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{30 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)-\sec \left(\frac{1}{4} (c+d x)\right)\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)+\sec \left(\frac{1}{4} (c+d x)\right)\right)}","-\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,"(Csc[(c + d*x)/2]^4*Sqrt[a*(1 + Sin[c + d*x])]*(-155*Cos[(c + d*x)/2] + 87*Cos[(3*(c + d*x))/2] + 5*Cos[(5*(c + d*x))/2] + 3*Cos[(7*(c + d*x))/2] + 155*Sin[(c + d*x)/2] - 30*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 30*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 87*Sin[(3*(c + d*x))/2] - 5*Sin[(5*(c + d*x))/2] + 3*Sin[(7*(c + d*x))/2]))/(30*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4] - Sec[(c + d*x)/4])*(Csc[(c + d*x)/4] + Sec[(c + d*x)/4]))","A",1
447,1,297,156,0.903315,"\int \cos (c+d x) \cot ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\csc ^7\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(26 \sin \left(\frac{1}{2} (c+d x)\right)-14 \sin \left(\frac{3}{2} (c+d x)\right)-12 \sin \left(\frac{5}{2} (c+d x)\right)+4 \sin \left(\frac{7}{2} (c+d x)\right)-26 \cos \left(\frac{1}{2} (c+d x)\right)-14 \cos \left(\frac{3}{2} (c+d x)\right)+12 \cos \left(\frac{5}{2} (c+d x)\right)+4 \cos \left(\frac{7}{2} (c+d x)\right)-39 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+39 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+39 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-39 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{12 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^2}","-\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,"(Csc[(c + d*x)/2]^7*Sqrt[a*(1 + Sin[c + d*x])]*(-26*Cos[(c + d*x)/2] - 14*Cos[(3*(c + d*x))/2] + 12*Cos[(5*(c + d*x))/2] + 4*Cos[(7*(c + d*x))/2] + 39*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 39*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 39*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 39*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 26*Sin[(c + d*x)/2] - 14*Sin[(3*(c + d*x))/2] - 12*Sin[(5*(c + d*x))/2] + 4*Sin[(7*(c + d*x))/2]))/(12*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^2)","A",1
448,1,309,163,1.4168455,"\int \cot ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\csc ^{10}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-252 \sin \left(\frac{1}{2} (c+d x)\right)-250 \sin \left(\frac{3}{2} (c+d x)\right)+114 \sin \left(\frac{5}{2} (c+d x)\right)+48 \sin \left(\frac{7}{2} (c+d x)\right)+252 \cos \left(\frac{1}{2} (c+d x)\right)-250 \cos \left(\frac{3}{2} (c+d x)\right)-114 \cos \left(\frac{5}{2} (c+d x)\right)+48 \cos \left(\frac{7}{2} (c+d x)\right)+99 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-99 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-33 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+33 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{24 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}","-\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,"(Csc[(c + d*x)/2]^10*Sqrt[a*(1 + Sin[c + d*x])]*(252*Cos[(c + d*x)/2] - 250*Cos[(3*(c + d*x))/2] - 114*Cos[(5*(c + d*x))/2] + 48*Cos[(7*(c + d*x))/2] - 252*Sin[(c + d*x)/2] + 99*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 99*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 250*Sin[(3*(c + d*x))/2] + 114*Sin[(5*(c + d*x))/2] - 33*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 33*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 48*Sin[(7*(c + d*x))/2]))/(24*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3)","A",1
449,1,367,173,2.7369137,"\int \cot ^4(c+d x) \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\csc ^{13}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-442 \sin \left(\frac{1}{2} (c+d x)\right)-162 \sin \left(\frac{3}{2} (c+d x)\right)-122 \sin \left(\frac{5}{2} (c+d x)\right)+366 \sin \left(\frac{7}{2} (c+d x)\right)+442 \cos \left(\frac{1}{2} (c+d x)\right)-162 \cos \left(\frac{3}{2} (c+d x)\right)+122 \cos \left(\frac{5}{2} (c+d x)\right)+366 \cos \left(\frac{7}{2} (c+d x)\right)-804 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+201 \cos (4 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+603 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+804 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-201 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-603 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{192 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^4}","\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,"-1/192*(Csc[(c + d*x)/2]^13*Sqrt[a*(1 + Sin[c + d*x])]*(442*Cos[(c + d*x)/2] - 162*Cos[(3*(c + d*x))/2] + 122*Cos[(5*(c + d*x))/2] + 366*Cos[(7*(c + d*x))/2] + 603*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 804*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 201*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 603*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 804*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 201*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 442*Sin[(c + d*x)/2] - 162*Sin[(3*(c + d*x))/2] - 122*Sin[(5*(c + d*x))/2] + 366*Sin[(7*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^4)","B",1
450,1,403,209,4.4377812,"\int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\csc ^{16}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-10180 \sin \left(\frac{1}{2} (c+d x)\right)-2240 \sin \left(\frac{3}{2} (c+d x)\right)+1392 \sin \left(\frac{5}{2} (c+d x)\right)+4810 \sin \left(\frac{7}{2} (c+d x)\right)-930 \sin \left(\frac{9}{2} (c+d x)\right)+10180 \cos \left(\frac{1}{2} (c+d x)\right)-2240 \cos \left(\frac{3}{2} (c+d x)\right)-1392 \cos \left(\frac{5}{2} (c+d x)\right)+4810 \cos \left(\frac{7}{2} (c+d x)\right)+930 \cos \left(\frac{9}{2} (c+d x)\right)+4650 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-4650 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-2325 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+2325 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+465 \sin (5 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-465 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{1920 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^5}","-\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,"-1/1920*(Csc[(c + d*x)/2]^16*Sqrt[a*(1 + Sin[c + d*x])]*(10180*Cos[(c + d*x)/2] - 2240*Cos[(3*(c + d*x))/2] - 1392*Cos[(5*(c + d*x))/2] + 4810*Cos[(7*(c + d*x))/2] + 930*Cos[(9*(c + d*x))/2] - 10180*Sin[(c + d*x)/2] + 4650*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 4650*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 2240*Sin[(3*(c + d*x))/2] + 1392*Sin[(5*(c + d*x))/2] - 2325*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 2325*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 4810*Sin[(7*(c + d*x))/2] - 930*Sin[(9*(c + d*x))/2] + 465*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 465*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^5)","A",1
451,1,485,245,7.685315,"\int \cot ^4(c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\csc ^{19}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(24540 \sin \left(\frac{1}{2} (c+d x)\right)-25684 \sin \left(\frac{3}{2} (c+d x)\right)+14490 \sin \left(\frac{5}{2} (c+d x)\right)-15006 \sin \left(\frac{7}{2} (c+d x)\right)+550 \sin \left(\frac{9}{2} (c+d x)\right)-1650 \sin \left(\frac{11}{2} (c+d x)\right)-24540 \cos \left(\frac{1}{2} (c+d x)\right)-25684 \cos \left(\frac{3}{2} (c+d x)\right)-14490 \cos \left(\frac{5}{2} (c+d x)\right)-15006 \cos \left(\frac{7}{2} (c+d x)\right)-550 \cos \left(\frac{9}{2} (c+d x)\right)-1650 \cos \left(\frac{11}{2} (c+d x)\right)+12375 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-4950 \cos (4 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+825 \cos (6 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-8250 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-12375 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+4950 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-825 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+8250 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{7680 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^6}","-\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,"(Csc[(c + d*x)/2]^19*Sqrt[a*(1 + Sin[c + d*x])]*(-24540*Cos[(c + d*x)/2] - 25684*Cos[(3*(c + d*x))/2] - 14490*Cos[(5*(c + d*x))/2] - 15006*Cos[(7*(c + d*x))/2] - 550*Cos[(9*(c + d*x))/2] - 1650*Cos[(11*(c + d*x))/2] - 8250*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12375*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 4950*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 825*Cos[6*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8250*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 12375*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4950*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 825*Cos[6*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24540*Sin[(c + d*x)/2] - 25684*Sin[(3*(c + d*x))/2] + 14490*Sin[(5*(c + d*x))/2] - 15006*Sin[(7*(c + d*x))/2] + 550*Sin[(9*(c + d*x))/2] - 1650*Sin[(11*(c + d*x))/2]))/(7680*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^6)","A",1
452,1,191,281,2.0304308,"\int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(-102480 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+102480 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+\csc ^7(c+d x) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (49128 \sin (c+d x)-179636 \sin (3 (c+d x))-8540 \sin (5 (c+d x))-244533 \cos (2 (c+d x))-52094 \cos (4 (c+d x))+6405 \cos (6 (c+d x))-201298)\right)}{3440640 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\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,"(Sqrt[a*(1 + Sin[c + d*x])]*(-102480*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 102480*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Csc[c + d*x]^7*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(-201298 - 244533*Cos[2*(c + d*x)] - 52094*Cos[4*(c + d*x)] + 6405*Cos[6*(c + d*x)] + 49128*Sin[c + d*x] - 179636*Sin[3*(c + d*x)] - 8540*Sin[5*(c + d*x)])))/(3440640*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
453,1,120,188,9.0797685,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (66470 \sin (c+d x)-14445 \sin (3 (c+d x))+429 \sin (5 (c+d x))-36640 \cos (2 (c+d x))+3630 \cos (4 (c+d x))+43122)}{51480 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{256 a^4 \cos ^5(c+d x)}{6435 d (a \sin (c+d x)+a)^{5/2}}-\frac{64 a^3 \cos ^5(c+d x)}{1287 d (a \sin (c+d x)+a)^{3/2}}-\frac{56 a^2 \cos ^5(c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}-\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,"-1/51480*(a*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*Sqrt[a*(1 + Sin[c + d*x])]*(43122 - 36640*Cos[2*(c + d*x)] + 3630*Cos[4*(c + d*x)] + 66470*Sin[c + d*x] - 14445*Sin[3*(c + d*x)] + 429*Sin[5*(c + d*x)]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
454,1,110,156,5.181496,"\int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (28230 \sin (c+d x)-3290 \sin (3 (c+d x))-12600 \cos (2 (c+d x))+385 \cos (4 (c+d x))+19559)}{20020 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{256 a^4 \cos ^5(c+d x)}{5005 d (a \sin (c+d x)+a)^{5/2}}-\frac{64 a^3 \cos ^5(c+d x)}{1001 d (a \sin (c+d x)+a)^{3/2}}-\frac{8 a^2 \cos ^5(c+d x)}{143 d \sqrt{a \sin (c+d x)+a}}-\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,"-1/20020*(a*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*Sqrt[a*(1 + Sin[c + d*x])]*(19559 - 12600*Cos[2*(c + d*x)] + 385*Cos[4*(c + d*x)] + 28230*Sin[c + d*x] - 3290*Sin[3*(c + d*x)]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
455,1,219,199,0.4765655,"\int \cos ^3(c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","\frac{(a (\sin (c+d x)+1))^{3/2} \left(-1260 \sin \left(\frac{1}{2} (c+d x)\right)+1470 \sin \left(\frac{3}{2} (c+d x)\right)+126 \sin \left(\frac{5}{2} (c+d x)\right)+135 \sin \left(\frac{7}{2} (c+d x)\right)+35 \sin \left(\frac{9}{2} (c+d x)\right)+1260 \cos \left(\frac{1}{2} (c+d x)\right)+1470 \cos \left(\frac{3}{2} (c+d x)\right)-126 \cos \left(\frac{5}{2} (c+d x)\right)+135 \cos \left(\frac{7}{2} (c+d x)\right)-35 \cos \left(\frac{9}{2} (c+d x)\right)-2520 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+2520 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{2520 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\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^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{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,"((a*(1 + Sin[c + d*x]))^(3/2)*(1260*Cos[(c + d*x)/2] + 1470*Cos[(3*(c + d*x))/2] - 126*Cos[(5*(c + d*x))/2] + 135*Cos[(7*(c + d*x))/2] - 35*Cos[(9*(c + d*x))/2] - 2520*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2520*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 1260*Sin[(c + d*x)/2] + 1470*Sin[(3*(c + d*x))/2] + 126*Sin[(5*(c + d*x))/2] + 135*Sin[(7*(c + d*x))/2] + 35*Sin[(9*(c + d*x))/2]))/(2520*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","A",1
456,1,283,178,1.3229915,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-840 \sin \left(\frac{1}{2} (c+d x)\right)-574 \sin \left(\frac{3}{2} (c+d x)\right)-30 \sin \left(\frac{5}{2} (c+d x)\right)-21 \sin \left(\frac{7}{2} (c+d x)\right)-5 \sin \left(\frac{9}{2} (c+d x)\right)+840 \cos \left(\frac{1}{2} (c+d x)\right)-574 \cos \left(\frac{3}{2} (c+d x)\right)+30 \cos \left(\frac{5}{2} (c+d x)\right)-21 \cos \left(\frac{7}{2} (c+d x)\right)+5 \cos \left(\frac{9}{2} (c+d x)\right)+420 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-420 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{140 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)-\sec \left(\frac{1}{4} (c+d x)\right)\right) \left(\csc \left(\frac{1}{4} (c+d x)\right)+\sec \left(\frac{1}{4} (c+d x)\right)\right)}","-\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{171 a^2 \cos (c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\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,"-1/140*(a*Csc[(c + d*x)/2]^4*Sqrt[a*(1 + Sin[c + d*x])]*(840*Cos[(c + d*x)/2] - 574*Cos[(3*(c + d*x))/2] + 30*Cos[(5*(c + d*x))/2] - 21*Cos[(7*(c + d*x))/2] + 5*Cos[(9*(c + d*x))/2] - 840*Sin[(c + d*x)/2] + 420*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 420*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 574*Sin[(3*(c + d*x))/2] - 30*Sin[(5*(c + d*x))/2] - 21*Sin[(7*(c + d*x))/2] - 5*Sin[(9*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4] - Sec[(c + d*x)/4])*(Csc[(c + d*x)/4] + Sec[(c + d*x)/4]))","A",1
457,1,322,186,1.0726344,"\int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \csc ^7\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(118 \sin \left(\frac{1}{2} (c+d x)\right)+130 \sin \left(\frac{3}{2} (c+d x)\right)-36 \sin \left(\frac{5}{2} (c+d x)\right)-10 \sin \left(\frac{7}{2} (c+d x)\right)-2 \sin \left(\frac{9}{2} (c+d x)\right)-118 \cos \left(\frac{1}{2} (c+d x)\right)+130 \cos \left(\frac{3}{2} (c+d x)\right)+36 \cos \left(\frac{5}{2} (c+d x)\right)-10 \cos \left(\frac{7}{2} (c+d x)\right)+2 \cos \left(\frac{9}{2} (c+d x)\right)+45 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-45 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-45 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+45 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{20 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^2}","\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{73 a^2 \cos (c+d x)}{20 d \sqrt{a \sin (c+d x)+a}}-\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,"-1/20*(a*Csc[(c + d*x)/2]^7*Sqrt[a*(1 + Sin[c + d*x])]*(-118*Cos[(c + d*x)/2] + 130*Cos[(3*(c + d*x))/2] + 36*Cos[(5*(c + d*x))/2] - 10*Cos[(7*(c + d*x))/2] + 2*Cos[(9*(c + d*x))/2] - 45*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 45*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 45*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 45*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 118*Sin[(c + d*x)/2] + 130*Sin[(3*(c + d*x))/2] - 36*Sin[(5*(c + d*x))/2] - 10*Sin[(7*(c + d*x))/2] - 2*Sin[(9*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^2)","A",1
458,1,334,197,1.2760835,"\int \cot ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \csc ^{10}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(276 \sin \left(\frac{1}{2} (c+d x)\right)+326 \sin \left(\frac{3}{2} (c+d x)\right)-78 \sin \left(\frac{5}{2} (c+d x)\right)-72 \sin \left(\frac{7}{2} (c+d x)\right)-8 \sin \left(\frac{9}{2} (c+d x)\right)-276 \cos \left(\frac{1}{2} (c+d x)\right)+326 \cos \left(\frac{3}{2} (c+d x)\right)+78 \cos \left(\frac{5}{2} (c+d x)\right)-72 \cos \left(\frac{7}{2} (c+d x)\right)+8 \cos \left(\frac{9}{2} (c+d x)\right)-333 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+333 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+111 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-111 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{24 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}","\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{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{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,"-1/24*(a*Csc[(c + d*x)/2]^10*Sqrt[a*(1 + Sin[c + d*x])]*(-276*Cos[(c + d*x)/2] + 326*Cos[(3*(c + d*x))/2] + 78*Cos[(5*(c + d*x))/2] - 72*Cos[(7*(c + d*x))/2] + 8*Cos[(9*(c + d*x))/2] + 276*Sin[(c + d*x)/2] - 333*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 333*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 326*Sin[(3*(c + d*x))/2] - 78*Sin[(5*(c + d*x))/2] + 111*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 111*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 72*Sin[(7*(c + d*x))/2] - 8*Sin[(9*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3)","A",1
459,1,392,205,1.4595908,"\int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \csc ^{13}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-1486 \sin \left(\frac{1}{2} (c+d x)\right)-1030 \sin \left(\frac{3}{2} (c+d x)\right)+754 \sin \left(\frac{5}{2} (c+d x)\right)+426 \sin \left(\frac{7}{2} (c+d x)\right)-128 \sin \left(\frac{9}{2} (c+d x)\right)+1486 \cos \left(\frac{1}{2} (c+d x)\right)-1030 \cos \left(\frac{3}{2} (c+d x)\right)-754 \cos \left(\frac{5}{2} (c+d x)\right)+426 \cos \left(\frac{7}{2} (c+d x)\right)+128 \cos \left(\frac{9}{2} (c+d x)\right)+84 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-21 \cos (4 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-63 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-84 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+21 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+63 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{64 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^4}","\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{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{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,"-1/64*(a*Csc[(c + d*x)/2]^13*Sqrt[a*(1 + Sin[c + d*x])]*(1486*Cos[(c + d*x)/2] - 1030*Cos[(3*(c + d*x))/2] - 754*Cos[(5*(c + d*x))/2] + 426*Cos[(7*(c + d*x))/2] + 128*Cos[(9*(c + d*x))/2] - 63*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 84*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 21*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 63*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 84*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 21*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 1486*Sin[(c + d*x)/2] - 1030*Sin[(3*(c + d*x))/2] + 754*Sin[(5*(c + d*x))/2] + 426*Sin[(7*(c + d*x))/2] - 128*Sin[(9*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^4)","A",1
460,1,404,215,1.6406814,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \csc ^{16}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-1380 \sin \left(\frac{1}{2} (c+d x)\right)+320 \sin \left(\frac{3}{2} (c+d x)\right)-1296 \sin \left(\frac{5}{2} (c+d x)\right)+2010 \sin \left(\frac{7}{2} (c+d x)\right)+910 \sin \left(\frac{9}{2} (c+d x)\right)+1380 \cos \left(\frac{1}{2} (c+d x)\right)+320 \cos \left(\frac{3}{2} (c+d x)\right)+1296 \cos \left(\frac{5}{2} (c+d x)\right)+2010 \cos \left(\frac{7}{2} (c+d x)\right)-910 \cos \left(\frac{9}{2} (c+d x)\right)+8250 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-8250 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-4125 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+4125 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+825 \sin (5 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-825 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{640 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^5}","-\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{91 a^2 \cot (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}+\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,"-1/640*(a*Csc[(c + d*x)/2]^16*Sqrt[a*(1 + Sin[c + d*x])]*(1380*Cos[(c + d*x)/2] + 320*Cos[(3*(c + d*x))/2] + 1296*Cos[(5*(c + d*x))/2] + 2010*Cos[(7*(c + d*x))/2] - 910*Cos[(9*(c + d*x))/2] - 1380*Sin[(c + d*x)/2] + 8250*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 8250*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 320*Sin[(3*(c + d*x))/2] - 1296*Sin[(5*(c + d*x))/2] - 4125*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 4125*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 2010*Sin[(7*(c + d*x))/2] + 910*Sin[(9*(c + d*x))/2] + 825*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 825*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^5)","A",1
461,1,486,253,2.507763,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","\frac{a \csc ^{19}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(-25140 \sin \left(\frac{1}{2} (c+d x)\right)-71972 \sin \left(\frac{3}{2} (c+d x)\right)+42690 \sin \left(\frac{5}{2} (c+d x)\right)-5718 \sin \left(\frac{7}{2} (c+d x)\right)-18690 \sin \left(\frac{9}{2} (c+d x)\right)-5370 \sin \left(\frac{11}{2} (c+d x)\right)+25140 \cos \left(\frac{1}{2} (c+d x)\right)-71972 \cos \left(\frac{3}{2} (c+d x)\right)-42690 \cos \left(\frac{5}{2} (c+d x)\right)-5718 \cos \left(\frac{7}{2} (c+d x)\right)+18690 \cos \left(\frac{9}{2} (c+d x)\right)-5370 \cos \left(\frac{11}{2} (c+d x)\right)+40275 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-16110 \cos (4 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+2685 \cos (6 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-26850 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-40275 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+16110 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-2685 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+26850 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{7680 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^6}","-\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{179 a^2 \cot (c+d x)}{512 d \sqrt{a \sin (c+d x)+a}}+\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,"(a*Csc[(c + d*x)/2]^19*Sqrt[a*(1 + Sin[c + d*x])]*(25140*Cos[(c + d*x)/2] - 71972*Cos[(3*(c + d*x))/2] - 42690*Cos[(5*(c + d*x))/2] - 5718*Cos[(7*(c + d*x))/2] + 18690*Cos[(9*(c + d*x))/2] - 5370*Cos[(11*(c + d*x))/2] - 26850*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 40275*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 16110*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2685*Cos[6*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 26850*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 40275*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 16110*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2685*Cos[6*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 25140*Sin[(c + d*x)/2] - 71972*Sin[(3*(c + d*x))/2] + 42690*Sin[(5*(c + d*x))/2] - 5718*Sin[(7*(c + d*x))/2] - 18690*Sin[(9*(c + d*x))/2] - 5370*Sin[(11*(c + d*x))/2]))/(7680*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^6)","A",1
462,1,522,291,4.9382719,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]","\frac{a \csc ^{22}\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sin (c+d x)+1)} \left(306488 \sin \left(\frac{1}{2} (c+d x)\right)-177170 \sin \left(\frac{3}{2} (c+d x)\right)-6566 \sin \left(\frac{5}{2} (c+d x)\right)-219540 \sin \left(\frac{7}{2} (c+d x)\right)-33292 \sin \left(\frac{9}{2} (c+d x)\right)-3990 \sin \left(\frac{11}{2} (c+d x)\right)-11970 \sin \left(\frac{13}{2} (c+d x)\right)-306488 \cos \left(\frac{1}{2} (c+d x)\right)-177170 \cos \left(\frac{3}{2} (c+d x)\right)+6566 \cos \left(\frac{5}{2} (c+d x)\right)-219540 \cos \left(\frac{7}{2} (c+d x)\right)+33292 \cos \left(\frac{9}{2} (c+d x)\right)-3990 \cos \left(\frac{11}{2} (c+d x)\right)+11970 \cos \left(\frac{13}{2} (c+d x)\right)-209475 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+209475 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+125685 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-125685 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-41895 \sin (5 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+41895 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+5985 \sin (7 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-5985 \sin (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{35840 d \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^7}","-\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{171 a^2 \cot (c+d x)}{1024 d \sqrt{a \sin (c+d x)+a}}+\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,"(a*Csc[(c + d*x)/2]^22*Sqrt[a*(1 + Sin[c + d*x])]*(-306488*Cos[(c + d*x)/2] - 177170*Cos[(3*(c + d*x))/2] + 6566*Cos[(5*(c + d*x))/2] - 219540*Cos[(7*(c + d*x))/2] + 33292*Cos[(9*(c + d*x))/2] - 3990*Cos[(11*(c + d*x))/2] + 11970*Cos[(13*(c + d*x))/2] + 306488*Sin[(c + d*x)/2] - 209475*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 209475*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 177170*Sin[(3*(c + d*x))/2] - 6566*Sin[(5*(c + d*x))/2] + 125685*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 125685*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 219540*Sin[(7*(c + d*x))/2] - 33292*Sin[(9*(c + d*x))/2] - 41895*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 41895*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 3990*Sin[(11*(c + d*x))/2] - 11970*Sin[(13*(c + d*x))/2] + 5985*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 5985*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/(35840*d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^7)","A",1
463,1,2303,329,6.2584923,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),x]","\text{Result too large to show}","-\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{1587 a^2 \cot (c+d x)}{16384 d \sqrt{a \sin (c+d x)+a}}+\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,"(6053*(a*(1 + Sin[c + d*x]))^(3/2))/(143360*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (6053*Cot[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(286720*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (179*Csc[(c + d*x)/4]^2*(a*(1 + Sin[c + d*x]))^(3/2))/(131072*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (107*Cot[(c + d*x)/4]*Csc[(c + d*x)/4]^2*(a*(1 + Sin[c + d*x]))^(3/2))/(573440*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (113*Csc[(c + d*x)/4]^4*(a*(1 + Sin[c + d*x]))^(3/2))/(262144*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (31*Cot[(c + d*x)/4]*Csc[(c + d*x)/4]^4*(a*(1 + Sin[c + d*x]))^(3/2))/(143360*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (Csc[(c + d*x)/4]^6*(a*(1 + Sin[c + d*x]))^(3/2))/(131072*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (3*Cot[(c + d*x)/4]*Csc[(c + d*x)/4]^6*(a*(1 + Sin[c + d*x]))^(3/2))/(229376*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (Csc[(c + d*x)/4]^8*(a*(1 + Sin[c + d*x]))^(3/2))/(524288*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (1587*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a*(1 + Sin[c + d*x]))^(3/2))/(32768*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (1587*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a*(1 + Sin[c + d*x]))^(3/2))/(32768*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (179*Sec[(c + d*x)/4]^2*(a*(1 + Sin[c + d*x]))^(3/2))/(131072*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (113*Sec[(c + d*x)/4]^4*(a*(1 + Sin[c + d*x]))^(3/2))/(262144*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (Sec[(c + d*x)/4]^6*(a*(1 + Sin[c + d*x]))^(3/2))/(131072*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (Sec[(c + d*x)/4]^8*(a*(1 + Sin[c + d*x]))^(3/2))/(524288*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (a*(1 + Sin[c + d*x]))^(3/2)/(32768*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^8*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (5*(a*(1 + Sin[c + d*x]))^(3/2))/(114688*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^6*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (5939*(a*(1 + Sin[c + d*x]))^(3/2))/(2293760*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (5409*(a*(1 + Sin[c + d*x]))^(3/2))/(2293760*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (3*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(14336*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^7*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (31*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(17920*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^5*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (107*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(143360*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (6053*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(143360*d*(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (a*(1 + Sin[c + d*x]))^(3/2)/(32768*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^8*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (3*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(14336*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^7*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (19*(a*(1 + Sin[c + d*x]))^(3/2))/(114688*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^6*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (31*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(17920*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^5*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (1971*(a*(1 + Sin[c + d*x]))^(3/2))/(2293760*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (107*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(143360*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (7121*(a*(1 + Sin[c + d*x]))^(3/2))/(2293760*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (6053*Sin[(c + d*x)/4]*(a*(1 + Sin[c + d*x]))^(3/2))/(143360*d*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (6053*(a*(1 + Sin[c + d*x]))^(3/2)*Tan[(c + d*x)/4])/(286720*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (107*Sec[(c + d*x)/4]^2*(a*(1 + Sin[c + d*x]))^(3/2)*Tan[(c + d*x)/4])/(573440*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (31*Sec[(c + d*x)/4]^4*(a*(1 + Sin[c + d*x]))^(3/2)*Tan[(c + d*x)/4])/(143360*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (3*Sec[(c + d*x)/4]^6*(a*(1 + Sin[c + d*x]))^(3/2)*Tan[(c + d*x)/4])/(229376*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","B",1
464,1,143,124,1.6966248,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(5773 \sin \left(\frac{1}{2} (c+d x)\right)+3495 \sin \left(\frac{3}{2} (c+d x)\right)-1505 \sin \left(\frac{5}{2} (c+d x)\right)-315 \sin \left(\frac{7}{2} (c+d x)\right)+5773 \cos \left(\frac{1}{2} (c+d x)\right)-3495 \cos \left(\frac{3}{2} (c+d x)\right)-1505 \cos \left(\frac{5}{2} (c+d x)\right)+315 \cos \left(\frac{7}{2} (c+d x)\right)\right)}{13860 d \sqrt{a (\sin (c+d x)+1)}}","-\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,"-1/13860*((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*(5773*Cos[(c + d*x)/2] - 3495*Cos[(3*(c + d*x))/2] - 1505*Cos[(5*(c + d*x))/2] + 315*Cos[(7*(c + d*x))/2] + 5773*Sin[(c + d*x)/2] + 3495*Sin[(3*(c + d*x))/2] - 1505*Sin[(5*(c + d*x))/2] - 315*Sin[(7*(c + d*x))/2]))/(d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
465,1,87,92,1.5948948,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (130 \sin (c+d x)-35 \cos (2 (c+d x))+87)}{315 d \sqrt{a (\sin (c+d x)+1)}}","\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,"-1/315*((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(87 - 35*Cos[2*(c + d*x)] + 130*Sin[c + d*x]))/(d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
466,1,169,130,0.250517,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-60 \sin \left(\frac{1}{2} (c+d x)\right)+5 \sin \left(\frac{3}{2} (c+d x)\right)-3 \sin \left(\frac{5}{2} (c+d x)\right)+60 \cos \left(\frac{1}{2} (c+d x)\right)+5 \cos \left(\frac{3}{2} (c+d x)\right)+3 \cos \left(\frac{5}{2} (c+d x)\right)-30 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+30 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{30 d \sqrt{a (\sin (c+d x)+1)}}","-\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(60*Cos[(c + d*x)/2] + 5*Cos[(3*(c + d*x))/2] + 3*Cos[(5*(c + d*x))/2] - 30*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 30*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 60*Sin[(c + d*x)/2] + 5*Sin[(3*(c + d*x))/2] - 3*Sin[(5*(c + d*x))/2]))/(30*d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
467,1,190,119,0.3810228,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\left(\tan \left(\frac{1}{2} (c+d x)\right)+1\right) \csc \left(\frac{1}{4} (c+d x)\right) \sec \left(\frac{1}{4} (c+d x)\right) \left(10 \sin \left(\frac{1}{2} (c+d x)\right)+3 \sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{5}{2} (c+d x)\right)-10 \cos \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{3}{2} (c+d x)\right)+\cos \left(\frac{5}{2} (c+d x)\right)+3 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-3 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{24 d \sqrt{a (\sin (c+d x)+1)}}","-\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,"(Csc[(c + d*x)/4]*Sec[(c + d*x)/4]*(-10*Cos[(c + d*x)/2] + 3*Cos[(3*(c + d*x))/2] + Cos[(5*(c + d*x))/2] + 10*Sin[(c + d*x)/2] + 3*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 3*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 3*Sin[(3*(c + d*x))/2] - Sin[(5*(c + d*x))/2])*(1 + Tan[(c + d*x)/2]))/(24*d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
468,1,296,125,3.7359852,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(64 \sin \left(\frac{1}{2} (c+d x)\right)-64 \cos \left(\frac{1}{2} (c+d x)\right)+4 \tan \left(\frac{1}{4} (c+d x)\right)+4 \cot \left(\frac{1}{4} (c+d x)\right)-\csc ^2\left(\frac{1}{4} (c+d x)\right)+\sec ^2\left(\frac{1}{4} (c+d x)\right)-\frac{8 \sin \left(\frac{1}{4} (c+d x)\right)}{\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}+\frac{8 \sin \left(\frac{1}{4} (c+d x)\right)}{\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)}+\frac{2}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{2}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}+36 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-36 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-8\right)}{32 d \sqrt{a (\sin (c+d x)+1)}}","-\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-8 - 64*Cos[(c + d*x)/2] + 4*Cot[(c + d*x)/4] - Csc[(c + d*x)/4]^2 + 36*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 36*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sec[(c + d*x)/4]^2 + 2/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 - (8*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - 2/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2 + (8*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) + 64*Sin[(c + d*x)/2] + 4*Tan[(c + d*x)/4]))/(32*d*Sqrt[a*(1 + Sin[c + d*x])])","B",1
469,1,292,135,0.6039855,"\int \frac{\cot ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Cot[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\csc ^9\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-36 \sin \left(\frac{1}{2} (c+d x)\right)-46 \sin \left(\frac{3}{2} (c+d x)\right)+54 \sin \left(\frac{5}{2} (c+d x)\right)+36 \cos \left(\frac{1}{2} (c+d x)\right)-46 \cos \left(\frac{3}{2} (c+d x)\right)-54 \cos \left(\frac{5}{2} (c+d x)\right)-63 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+63 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+21 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-21 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{24 d \sqrt{a (\sin (c+d x)+1)} \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}","\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,"(Csc[(c + d*x)/2]^9*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(36*Cos[(c + d*x)/2] - 46*Cos[(3*(c + d*x))/2] - 54*Cos[(5*(c + d*x))/2] - 36*Sin[(c + d*x)/2] - 63*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 63*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 46*Sin[(3*(c + d*x))/2] + 54*Sin[(5*(c + d*x))/2] + 21*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 21*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(24*d*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3*Sqrt[a*(1 + Sin[c + d*x])])","B",1
470,1,374,170,0.9003446,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\csc ^{12}\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-214 \sin \left(\frac{1}{2} (c+d x)\right)-558 \sin \left(\frac{3}{2} (c+d x)\right)+490 \sin \left(\frac{5}{2} (c+d x)\right)+66 \sin \left(\frac{7}{2} (c+d x)\right)+214 \cos \left(\frac{1}{2} (c+d x)\right)-558 \cos \left(\frac{3}{2} (c+d x)\right)-490 \cos \left(\frac{5}{2} (c+d x)\right)+66 \cos \left(\frac{7}{2} (c+d x)\right)+132 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-33 \cos (4 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-99 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-132 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+33 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+99 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{192 d \sqrt{a (\sin (c+d x)+1)} \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^4}","-\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,"(Csc[(c + d*x)/2]^12*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(214*Cos[(c + d*x)/2] - 558*Cos[(3*(c + d*x))/2] - 490*Cos[(5*(c + d*x))/2] + 66*Cos[(7*(c + d*x))/2] - 99*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 132*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 33*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 99*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 132*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 33*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 214*Sin[(c + d*x)/2] - 558*Sin[(3*(c + d*x))/2] + 490*Sin[(5*(c + d*x))/2] + 66*Sin[(7*(c + d*x))/2]))/(192*d*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^4*Sqrt[a*(1 + Sin[c + d*x])])","B",1
471,1,410,205,1.005721,"\int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x]^2)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\csc ^{15}\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(-820 \sin \left(\frac{1}{2} (c+d x)\right)+1600 \sin \left(\frac{3}{2} (c+d x)\right)-1616 \sin \left(\frac{5}{2} (c+d x)\right)-30 \sin \left(\frac{7}{2} (c+d x)\right)-90 \sin \left(\frac{9}{2} (c+d x)\right)+820 \cos \left(\frac{1}{2} (c+d x)\right)+1600 \cos \left(\frac{3}{2} (c+d x)\right)+1616 \cos \left(\frac{5}{2} (c+d x)\right)-30 \cos \left(\frac{7}{2} (c+d x)\right)+90 \cos \left(\frac{9}{2} (c+d x)\right)+450 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-450 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-225 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+225 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+45 \sin (5 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-45 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{640 d \sqrt{a (\sin (c+d x)+1)} \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^5}","-\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,"-1/640*(Csc[(c + d*x)/2]^15*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(820*Cos[(c + d*x)/2] + 1600*Cos[(3*(c + d*x))/2] + 1616*Cos[(5*(c + d*x))/2] - 30*Cos[(7*(c + d*x))/2] + 90*Cos[(9*(c + d*x))/2] - 820*Sin[(c + d*x)/2] + 450*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 450*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 1600*Sin[(3*(c + d*x))/2] - 1616*Sin[(5*(c + d*x))/2] - 225*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 225*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 30*Sin[(7*(c + d*x))/2] - 90*Sin[(9*(c + d*x))/2] + 45*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 45*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(d*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^5*Sqrt[a*(1 + Sin[c + d*x])])","A",1
472,1,102,205,5.585568,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (-475 \sin (c+d x)+105 \sin (3 (c+d x))+140 \cos (2 (c+d x))-204)}{2310 a^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{385 a^3 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{11 a^2 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,"((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*Sqrt[a*(1 + Sin[c + d*x])]*(-204 + 140*Cos[2*(c + d*x)] - 475*Sin[c + d*x] + 105*Sin[3*(c + d*x)]))/(2310*a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
473,1,92,92,3.7173818,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{\sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (40 \sin (c+d x)-35 \cos (2 (c+d x))+51)}{315 a^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\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,"-1/315*((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*Sqrt[a*(1 + Sin[c + d*x])]*(51 - 35*Cos[2*(c + d*x)] + 40*Sin[c + d*x]))/(a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
474,1,82,60,1.9011842,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 (5 \sin (c+d x)+2) \sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}{35 a^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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,"(-2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*Sqrt[a*(1 + Sin[c + d*x])]*(2 + 5*Sin[c + d*x]))/(35*a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
475,1,147,98,0.2676803,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(-9 \sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)+9 \cos \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{3}{2} (c+d x)\right)-3 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{3 d (a (\sin (c+d x)+1))^{3/2}}","-\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 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 a^2 d}+\frac{10 \cos (c+d x)}{3 a d \sqrt{a \sin (c+d x)+a}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(9*Cos[(c + d*x)/2] - Cos[(3*(c + d*x))/2] - 3*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 9*Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/(3*d*(a*(1 + Sin[c + d*x]))^(3/2))","A",1
476,1,220,94,0.7011174,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(8 \sin \left(\frac{1}{2} (c+d x)\right)-8 \cos \left(\frac{1}{2} (c+d x)\right)-\tan \left(\frac{1}{4} (c+d x)\right)-\cot \left(\frac{1}{4} (c+d x)\right)+\frac{2 \sin \left(\frac{1}{4} (c+d x)\right)}{\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}-\frac{2 \sin \left(\frac{1}{4} (c+d x)\right)}{\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)}+6 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+2\right)}{4 d (a (\sin (c+d x)+1))^{3/2}}","\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{\cot (c+d x) \sqrt{a \sin (c+d x)+a}}{a^2 d}-\frac{\cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(2 - 8*Cos[(c + d*x)/2] - Cot[(c + d*x)/4] + 6*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 6*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - (2*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) + 8*Sin[(c + d*x)/2] - Tan[(c + d*x)/4]))/(4*d*(a*(1 + Sin[c + d*x]))^(3/2))","B",1
477,1,274,106,1.818708,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(12 \tan \left(\frac{1}{4} (c+d x)\right)+12 \cot \left(\frac{1}{4} (c+d x)\right)-\csc ^2\left(\frac{1}{4} (c+d x)\right)+\sec ^2\left(\frac{1}{4} (c+d x)\right)-\frac{24 \sin \left(\frac{1}{4} (c+d x)\right)}{\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}+\frac{24 \sin \left(\frac{1}{4} (c+d x)\right)}{\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)}+\frac{2}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{2}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-12 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+12 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-24\right)}{32 d (a (\sin (c+d x)+1))^{3/2}}","-\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(-24 + 12*Cot[(c + d*x)/4] - Csc[(c + d*x)/4]^2 - 12*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sec[(c + d*x)/4]^2 + 2/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 - (24*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - 2/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2 + (24*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) + 12*Tan[(c + d*x)/4]))/(32*d*(a*(1 + Sin[c + d*x]))^(3/2))","B",1
478,1,294,144,0.7895568,"\int \frac{\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\csc ^9\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(132 \sin \left(\frac{1}{2} (c+d x)\right)+62 \sin \left(\frac{3}{2} (c+d x)\right)-6 \sin \left(\frac{5}{2} (c+d x)\right)-132 \cos \left(\frac{1}{2} (c+d x)\right)+62 \cos \left(\frac{3}{2} (c+d x)\right)+6 \cos \left(\frac{5}{2} (c+d x)\right)-9 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+9 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+3 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-3 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{24 d (a (\sin (c+d x)+1))^{3/2} \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}","-\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,"(Csc[(c + d*x)/2]^9*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(-132*Cos[(c + d*x)/2] + 62*Cos[(3*(c + d*x))/2] + 6*Cos[(5*(c + d*x))/2] + 132*Sin[(c + d*x)/2] - 9*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 9*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 62*Sin[(3*(c + d*x))/2] - 6*Sin[(5*(c + d*x))/2] + 3*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 3*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(24*d*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3*(a*(1 + Sin[c + d*x]))^(3/2))","B",0
479,1,376,182,0.9583528,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{\csc ^{12}\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(-446 \sin \left(\frac{1}{2} (c+d x)\right)-182 \sin \left(\frac{3}{2} (c+d x)\right)+2 \sin \left(\frac{5}{2} (c+d x)\right)-6 \sin \left(\frac{7}{2} (c+d x)\right)+446 \cos \left(\frac{1}{2} (c+d x)\right)-182 \cos \left(\frac{3}{2} (c+d x)\right)-2 \cos \left(\frac{5}{2} (c+d x)\right)-6 \cos \left(\frac{7}{2} (c+d x)\right)-12 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+3 \cos (4 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+9 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+12 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-3 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-9 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{64 d (a (\sin (c+d x)+1))^{3/2} \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^4}","-\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,"-1/64*(Csc[(c + d*x)/2]^12*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(446*Cos[(c + d*x)/2] - 182*Cos[(3*(c + d*x))/2] - 2*Cos[(5*(c + d*x))/2] - 6*Cos[(7*(c + d*x))/2] + 9*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 12*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 9*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 12*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 3*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 446*Sin[(c + d*x)/2] - 182*Sin[(3*(c + d*x))/2] + 2*Sin[(5*(c + d*x))/2] - 6*Sin[(7*(c + d*x))/2]))/(d*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^4*(a*(1 + Sin[c + d*x]))^(3/2))","B",1
480,1,412,220,1.3846136,"\int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{\csc ^{15}\left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(-7100 \sin \left(\frac{1}{2} (c+d x)\right)-2880 \sin \left(\frac{3}{2} (c+d x)\right)+144 \sin \left(\frac{5}{2} (c+d x)\right)-10 \sin \left(\frac{7}{2} (c+d x)\right)-30 \sin \left(\frac{9}{2} (c+d x)\right)+7100 \cos \left(\frac{1}{2} (c+d x)\right)-2880 \cos \left(\frac{3}{2} (c+d x)\right)-144 \cos \left(\frac{5}{2} (c+d x)\right)-10 \cos \left(\frac{7}{2} (c+d x)\right)+30 \cos \left(\frac{9}{2} (c+d x)\right)+150 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-150 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-75 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+75 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+15 \sin (5 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-15 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{640 d (a (\sin (c+d x)+1))^{3/2} \left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^5}","-\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,"-1/640*(Csc[(c + d*x)/2]^15*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(7100*Cos[(c + d*x)/2] - 2880*Cos[(3*(c + d*x))/2] - 144*Cos[(5*(c + d*x))/2] - 10*Cos[(7*(c + d*x))/2] + 30*Cos[(9*(c + d*x))/2] - 7100*Sin[(c + d*x)/2] + 150*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 150*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 2880*Sin[(3*(c + d*x))/2] + 144*Sin[(5*(c + d*x))/2] - 75*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 75*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 10*Sin[(7*(c + d*x))/2] - 30*Sin[(9*(c + d*x))/2] + 15*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 15*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(d*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^5*(a*(1 + Sin[c + d*x]))^(3/2))","A",1
481,1,224,260,1.0751902,"\int \frac{\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(-73458 \sin \left(\frac{1}{2} (c+d x)\right)-15246 \sin \left(\frac{3}{2} (c+d x)\right)+4851 \sin \left(\frac{5}{2} (c+d x)\right)+1485 \sin \left(\frac{7}{2} (c+d x)\right)-385 \sin \left(\frac{9}{2} (c+d x)\right)-63 \sin \left(\frac{11}{2} (c+d x)\right)+73458 \cos \left(\frac{1}{2} (c+d x)\right)-15246 \cos \left(\frac{3}{2} (c+d x)\right)-4851 \cos \left(\frac{5}{2} (c+d x)\right)+1485 \cos \left(\frac{7}{2} (c+d x)\right)+385 \cos \left(\frac{9}{2} (c+d x)\right)-63 \cos \left(\frac{11}{2} (c+d x)\right)+(88704+88704 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{11088 d (a (\sin (c+d x)+1))^{5/2}}","-\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{1048 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{693 a^3 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{4496 \cos (c+d x)}{693 a^2 d \sqrt{a \sin (c+d x)+a}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*((88704 + 88704*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + 73458*Cos[(c + d*x)/2] - 15246*Cos[(3*(c + d*x))/2] - 4851*Cos[(5*(c + d*x))/2] + 1485*Cos[(7*(c + d*x))/2] + 385*Cos[(9*(c + d*x))/2] - 63*Cos[(11*(c + d*x))/2] - 73458*Sin[(c + d*x)/2] - 15246*Sin[(3*(c + d*x))/2] + 4851*Sin[(5*(c + d*x))/2] + 1485*Sin[(7*(c + d*x))/2] - 385*Sin[(9*(c + d*x))/2] - 63*Sin[(11*(c + d*x))/2]))/(11088*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
482,1,225,222,3.2801312,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(16380 \sin \left(\frac{1}{2} (c+d x)\right)+3150 \sin \left(\frac{3}{2} (c+d x)\right)-882 \sin \left(\frac{5}{2} (c+d x)\right)-225 \sin \left(\frac{7}{2} (c+d x)\right)+35 \sin \left(\frac{9}{2} (c+d x)\right)-16380 \cos \left(\frac{1}{2} (c+d x)\right)+3150 \cos \left(\frac{3}{2} (c+d x)\right)+882 \cos \left(\frac{5}{2} (c+d x)\right)-225 \cos \left(\frac{7}{2} (c+d x)\right)-35 \cos \left(\frac{9}{2} (c+d x)\right)+(20160+20160 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{d x}{4}\right) \left(\cos \left(\frac{1}{4} (2 c+d x)\right)-\sin \left(\frac{1}{4} (2 c+d x)\right)\right)\right)\right)}{2520 a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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{472 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 a^3 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{2048 \cos (c+d x)}{315 a^2 d \sqrt{a \sin (c+d x)+a}}",1,"(Sqrt[a*(1 + Sin[c + d*x])]*((20160 + 20160*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] - Sin[(2*c + d*x)/4])] - 16380*Cos[(c + d*x)/2] + 3150*Cos[(3*(c + d*x))/2] + 882*Cos[(5*(c + d*x))/2] - 225*Cos[(7*(c + d*x))/2] - 35*Cos[(9*(c + d*x))/2] + 16380*Sin[(c + d*x)/2] + 3150*Sin[(3*(c + d*x))/2] - 882*Sin[(5*(c + d*x))/2] - 225*Sin[(7*(c + d*x))/2] + 35*Sin[(9*(c + d*x))/2]))/(2520*a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
483,1,201,169,2.5073773,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{\sqrt{a (\sin (c+d x)+1)} \left(525 \sin \left(\frac{1}{2} (c+d x)\right)+91 \sin \left(\frac{3}{2} (c+d x)\right)-21 \sin \left(\frac{5}{2} (c+d x)\right)-3 \sin \left(\frac{7}{2} (c+d x)\right)-525 \cos \left(\frac{1}{2} (c+d x)\right)+91 \cos \left(\frac{3}{2} (c+d x)\right)+21 \cos \left(\frac{5}{2} (c+d x)\right)-3 \cos \left(\frac{7}{2} (c+d x)\right)+(672+672 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{d x}{4}\right) \left(\cos \left(\frac{1}{4} (2 c+d x)\right)-\sin \left(\frac{1}{4} (2 c+d x)\right)\right)\right)\right)}{84 a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\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{4 \cos (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}-\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,"-1/84*(Sqrt[a*(1 + Sin[c + d*x])]*((672 + 672*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] - Sin[(2*c + d*x)/4])] - 525*Cos[(c + d*x)/2] + 91*Cos[(3*(c + d*x))/2] + 21*Cos[(5*(c + d*x))/2] - 3*Cos[(7*(c + d*x))/2] + 525*Sin[(c + d*x)/2] + 91*Sin[(3*(c + d*x))/2] - 21*Sin[(5*(c + d*x))/2] - 3*Sin[(7*(c + d*x))/2]))/(a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
484,1,177,137,1.619666,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(180 \sin \left(\frac{1}{2} (c+d x)\right)+25 \sin \left(\frac{3}{2} (c+d x)\right)-3 \sin \left(\frac{5}{2} (c+d x)\right)-180 \cos \left(\frac{1}{2} (c+d x)\right)+25 \cos \left(\frac{3}{2} (c+d x)\right)+3 \cos \left(\frac{5}{2} (c+d x)\right)+(240+240 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{d x}{4}\right) \left(\cos \left(\frac{1}{4} (2 c+d x)\right)-\sin \left(\frac{1}{4} (2 c+d x)\right)\right)\right)\right)}{30 a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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{4 \cos (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}-\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,"(Sqrt[a*(1 + Sin[c + d*x])]*((240 + 240*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] - Sin[(2*c + d*x)/4])] - 180*Cos[(c + d*x)/2] + 25*Cos[(3*(c + d*x))/2] + 3*Cos[(5*(c + d*x))/2] + 180*Sin[(c + d*x)/2] + 25*Sin[(3*(c + d*x))/2] - 3*Sin[(5*(c + d*x))/2]))/(30*a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
485,1,154,113,0.3998537,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(-2 \sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)+(8+8 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)+\log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{d (a (\sin (c+d x)+1))^{5/2}}","-\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}}",1,"-((((8 + 8*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + 2*Cos[(c + d*x)/2] + Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2*Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)/(d*(a*(1 + Sin[c + d*x]))^(5/2)))","C",1
486,1,170,113,2.6968285,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(-\tan \left(\frac{1}{4} (c+d x)\right)-\cot \left(\frac{1}{4} (c+d x)\right)+2 \sec \left(\frac{1}{2} (c+d x)\right)+(32+32 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)+10 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-10 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{4 d (a (\sin (c+d x)+1))^{5/2}}","\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}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*((32 + 32*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] - Cot[(c + d*x)/4] + 10*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 10*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*Sec[(c + d*x)/2] - Tan[(c + d*x)/4]))/(4*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
487,1,309,153,3.5583541,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(20 \tan \left(\frac{1}{4} (c+d x)\right)+20 \cot \left(\frac{1}{4} (c+d x)\right)-\csc ^2\left(\frac{1}{4} (c+d x)\right)+\sec ^2\left(\frac{1}{4} (c+d x)\right)-\frac{40 \sin \left(\frac{1}{4} (c+d x)\right)}{\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}+\frac{40 \sin \left(\frac{1}{4} (c+d x)\right)}{\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)}+\frac{2}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{2}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-(256+256 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)-92 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+92 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-40\right)}{32 d (a (\sin (c+d x)+1))^{5/2}}","-\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{9 \cot (c+d x)}{4 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d \sqrt{a \sin (c+d x)+a}}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*(-40 - (256 + 256*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + 20*Cot[(c + d*x)/4] - Csc[(c + d*x)/4]^2 - 92*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 92*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sec[(c + d*x)/4]^2 + 2/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 - (40*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4]) - 2/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2 + (40*Sin[(c + d*x)/4])/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4]) + 20*Tan[(c + d*x)/4]))/(32*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
488,1,332,191,2.4110718,"\int \frac{\cot ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(-\frac{8 \csc ^9\left(\frac{1}{2} (c+d x)\right) \left(-396 \sin \left(\frac{1}{2} (c+d x)\right)-218 \sin \left(\frac{3}{2} (c+d x)\right)+114 \sin \left(\frac{5}{2} (c+d x)\right)+396 \cos \left(\frac{1}{2} (c+d x)\right)-218 \cos \left(\frac{3}{2} (c+d x)\right)-114 \cos \left(\frac{5}{2} (c+d x)\right)-405 \sin (c+d x) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+405 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+135 \sin (3 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-135 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{\left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^3}+(1536+1536 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{192 d (a (\sin (c+d x)+1))^{5/2}}","\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{19 \cot (c+d x)}{8 a^2 d \sqrt{a \sin (c+d x)+a}}-\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*((1536 + 1536*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] - (8*Csc[(c + d*x)/2]^9*(396*Cos[(c + d*x)/2] - 218*Cos[(3*(c + d*x))/2] - 114*Cos[(5*(c + d*x))/2] - 396*Sin[(c + d*x)/2] - 405*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 405*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 218*Sin[(3*(c + d*x))/2] + 114*Sin[(5*(c + d*x))/2] + 135*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 135*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3))/(192*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",0
489,1,414,229,5.120652,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(-\frac{16 \csc ^{12}\left(\frac{1}{2} (c+d x)\right) \left(-6250 \sin \left(\frac{1}{2} (c+d x)\right)-4626 \sin \left(\frac{3}{2} (c+d x)\right)+1750 \sin \left(\frac{5}{2} (c+d x)\right)+894 \sin \left(\frac{7}{2} (c+d x)\right)+6250 \cos \left(\frac{1}{2} (c+d x)\right)-4626 \cos \left(\frac{3}{2} (c+d x)\right)-1750 \cos \left(\frac{5}{2} (c+d x)\right)+894 \cos \left(\frac{7}{2} (c+d x)\right)-4356 \cos (2 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+1089 \cos (4 (c+d x)) \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+3267 \log \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)+1\right)+4356 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-1089 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)-3267 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{\left(\csc ^2\left(\frac{1}{4} (c+d x)\right)-\sec ^2\left(\frac{1}{4} (c+d x)\right)\right)^4}-(24576+24576 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{3072 d (a (\sin (c+d x)+1))^{5/2}}","-\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{149 \cot (c+d x)}{64 a^2 d \sqrt{a \sin (c+d x)+a}}-\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*((-24576 - 24576*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] - (16*Csc[(c + d*x)/2]^12*(6250*Cos[(c + d*x)/2] - 4626*Cos[(3*(c + d*x))/2] - 1750*Cos[(5*(c + d*x))/2] + 894*Cos[(7*(c + d*x))/2] + 3267*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 4356*Cos[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 1089*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 3267*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4356*Cos[2*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 1089*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 6250*Sin[(c + d*x)/2] - 4626*Sin[(3*(c + d*x))/2] + 1750*Sin[(5*(c + d*x))/2] + 894*Sin[(7*(c + d*x))/2]))/(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^4))/(3072*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
490,1,164,200,0.2923509,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{n+1}(c+d x) \left(\left(n^2+5 n+6\right) \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)+(n+1) \sin (c+d x) \left(2 (n+3) \, _2F_1\left(-\frac{3}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)+(n+2) \sin (c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right)\right)\right)}{d (n+1) (n+2) (n+3)}","\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*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + n)*((6 + 5*n + n^2)*Hypergeometric2F1[-3/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2] + (1 + n)*Sin[c + d*x]*(2*(3 + n)*Hypergeometric2F1[-3/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2] + (2 + n)*Hypergeometric2F1[-3/2, (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x])))/(d*(1 + n)*(2 + n)*(3 + n))","A",1
491,0,0,129,0.2684213,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]","\int \cos ^4(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","\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,"Integrate[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + a*Sin[c + d*x]), x]","F",-1
492,1,441,134,11.479452,"\int \frac{\cos ^4(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^n)/(a + a*Sin[c + d*x]),x]","\frac{2^{n+1} \tan \left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^n \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^n \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(\frac{\, _2F_1\left(\frac{n+1}{2},n+4;\frac{n+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+1}+\tan \left(\frac{1}{2} (c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \left(-\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(n+4,\frac{n+5}{2};\frac{n+7}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+5}+\frac{4 \tan \left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(\frac{n+4}{2},n+4;\frac{n+6}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+4}-\frac{\, _2F_1\left(\frac{n+3}{2},n+4;\frac{n+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+3}+\frac{\tan ^4\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(n+4,\frac{n+7}{2};\frac{n+9}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+7}-\frac{2 \tan ^3\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(n+4,\frac{n+6}{2};\frac{n+8}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+6}\right)-\frac{2 \, _2F_1\left(\frac{n+2}{2},n+4;\frac{n+4}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+2}\right)\right)}{d (a \sin (c+d x)+a)}","\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,"(2^(1 + n)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*Tan[(c + d*x)/2]*(Tan[(c + d*x)/2]/(1 + Tan[(c + d*x)/2]^2))^n*(1 + Tan[(c + d*x)/2]^2)^n*(Hypergeometric2F1[(1 + n)/2, 4 + n, (3 + n)/2, -Tan[(c + d*x)/2]^2]/(1 + n) + Tan[(c + d*x)/2]*((-2*Hypergeometric2F1[(2 + n)/2, 4 + n, (4 + n)/2, -Tan[(c + d*x)/2]^2])/(2 + n) + Tan[(c + d*x)/2]*(-(Hypergeometric2F1[(3 + n)/2, 4 + n, (5 + n)/2, -Tan[(c + d*x)/2]^2]/(3 + n)) + (4*Hypergeometric2F1[(4 + n)/2, 4 + n, (6 + n)/2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2])/(4 + n) - (Hypergeometric2F1[4 + n, (5 + n)/2, (7 + n)/2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(5 + n) - (2*Hypergeometric2F1[4 + n, (6 + n)/2, (8 + n)/2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^3)/(6 + n) + (Hypergeometric2F1[4 + n, (7 + n)/2, (9 + n)/2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^4)/(7 + n)))))/(d*(a + a*Sin[c + d*x]))","B",1
493,1,312,173,4.9206655,"\int \frac{\cos ^4(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^2,x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sin ^n(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)^n \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(\frac{\, _2F_1\left(\frac{n+1}{2},n+3;\frac{n+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+1}+\tan \left(\frac{1}{2} (c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(n+3,\frac{n+5}{2};\frac{n+7}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+5}-\frac{4 \tan \left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(n+3,\frac{n+4}{2};\frac{n+6}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+4}+\frac{6 \, _2F_1\left(\frac{n+3}{2},n+3;\frac{n+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+3}\right)-\frac{4 \, _2F_1\left(\frac{n+2}{2},n+3;\frac{n+4}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n+2}\right)\right)}{d (a \sin (c+d x)+a)^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,"(2*(Sec[(c + d*x)/2]^2)^n*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*Sin[c + d*x]^n*Tan[(c + d*x)/2]*(Hypergeometric2F1[(1 + n)/2, 3 + n, (3 + n)/2, -Tan[(c + d*x)/2]^2]/(1 + n) + Tan[(c + d*x)/2]*((-4*Hypergeometric2F1[(2 + n)/2, 3 + n, (4 + n)/2, -Tan[(c + d*x)/2]^2])/(2 + n) + Tan[(c + d*x)/2]*((6*Hypergeometric2F1[(3 + n)/2, 3 + n, (5 + n)/2, -Tan[(c + d*x)/2]^2])/(3 + n) - (4*Hypergeometric2F1[3 + n, (4 + n)/2, (6 + n)/2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2])/(4 + n) + (Hypergeometric2F1[3 + n, (5 + n)/2, (7 + n)/2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(5 + n)))))/(d*(a + a*Sin[c + d*x])^2)","A",1
494,1,97,97,0.446744,"\int \cos ^5(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a (-34650 \sin (c+d x)+11550 \sin (3 (c+d x))+3465 \sin (5 (c+d x))-2475 \sin (7 (c+d x))-385 \sin (9 (c+d x))+315 \sin (11 (c+d x))+34650 \cos (2 (c+d x))-5775 \cos (6 (c+d x))+693 \cos (10 (c+d x)))}{3548160 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,"-1/3548160*(a*(34650*Cos[2*(c + d*x)] - 5775*Cos[6*(c + d*x)] + 693*Cos[10*(c + d*x)] - 34650*Sin[c + d*x] + 11550*Sin[3*(c + d*x)] + 3465*Sin[5*(c + d*x)] - 2475*Sin[7*(c + d*x)] - 385*Sin[9*(c + d*x)] + 315*Sin[11*(c + d*x)]))/d","A",1
495,1,87,97,0.3263656,"\int \cos ^5(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a (-7560 \sin (c+d x)+1680 \sin (3 (c+d x))+1008 \sin (5 (c+d x))-180 \sin (7 (c+d x))-140 \sin (9 (c+d x))+3150 \cos (2 (c+d x))-525 \cos (6 (c+d x))+63 \cos (10 (c+d x)))}{322560 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,"-1/322560*(a*(3150*Cos[2*(c + d*x)] - 525*Cos[6*(c + d*x)] + 63*Cos[10*(c + d*x)] - 7560*Sin[c + d*x] + 1680*Sin[3*(c + d*x)] + 1008*Sin[5*(c + d*x)] - 180*Sin[7*(c + d*x)] - 140*Sin[9*(c + d*x)]))/d","A",1
496,1,97,81,0.3614514,"\int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a (7560 \sin (c+d x)-1680 \sin (3 (c+d x))-1008 \sin (5 (c+d x))+180 \sin (7 (c+d x))+140 \sin (9 (c+d x))-7560 \cos (2 (c+d x))-1260 \cos (4 (c+d x))+840 \cos (6 (c+d x))+315 \cos (8 (c+d x)))}{322560 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*(-7560*Cos[2*(c + d*x)] - 1260*Cos[4*(c + d*x)] + 840*Cos[6*(c + d*x)] + 315*Cos[8*(c + d*x)] + 7560*Sin[c + d*x] - 1680*Sin[3*(c + d*x)] - 1008*Sin[5*(c + d*x)] + 180*Sin[7*(c + d*x)] + 140*Sin[9*(c + d*x)]))/(322560*d)","A",1
497,1,87,81,0.2847249,"\int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a (-8400 \sin (c+d x)+560 \sin (3 (c+d x))+1008 \sin (5 (c+d x))+240 \sin (7 (c+d x))+2520 \cos (2 (c+d x))+420 \cos (4 (c+d x))-280 \cos (6 (c+d x))-105 \cos (8 (c+d x)))}{107520 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,"-1/107520*(a*(2520*Cos[2*(c + d*x)] + 420*Cos[4*(c + d*x)] - 280*Cos[6*(c + d*x)] - 105*Cos[8*(c + d*x)] - 8400*Sin[c + d*x] + 560*Sin[3*(c + d*x)] + 1008*Sin[5*(c + d*x)] + 240*Sin[7*(c + d*x)]))/d","A",1
498,1,78,65,0.1607737,"\int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a (-525 \sin (c+d x)+35 \sin (3 (c+d x))+63 \sin (5 (c+d x))+15 \sin (7 (c+d x))+525 \cos (2 (c+d x))+210 \cos (4 (c+d x))+35 \cos (6 (c+d x))+350)}{6720 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,"-1/6720*(a*(350 + 525*Cos[2*(c + d*x)] + 210*Cos[4*(c + d*x)] + 35*Cos[6*(c + d*x)] - 525*Sin[c + d*x] + 35*Sin[3*(c + d*x)] + 63*Sin[5*(c + d*x)] + 15*Sin[7*(c + d*x)]))/d","A",1
499,1,86,86,0.037107,"\int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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",1
500,1,83,83,0.0386374,"\int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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",1
501,1,77,86,0.1124059,"\int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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{2 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}-\frac{a \left(-\sin ^2(c+d x)+\csc ^2(c+d x)+4 \log (\sin (c+d x))\right)}{2 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) - (2*a*Sin[c + d*x])/d + (a*Sin[c + d*x]^3)/(3*d) - (a*(Csc[c + d*x]^2 + 4*Log[Sin[c + d*x]] - Sin[c + d*x]^2))/(2*d)","A",1
502,1,76,85,0.149691,"\int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*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{a \left(-\sin ^2(c+d x)+\csc ^2(c+d x)+4 \log (\sin (c+d x))\right)}{2 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]^3)/(3*d) + (a*Sin[c + d*x])/d - (a*(Csc[c + d*x]^2 + 4*Log[Sin[c + d*x]] - Sin[c + d*x]^2))/(2*d)","A",1
503,1,87,81,0.2009221,"\int \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + a*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{a \left(-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right)}{4 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]^3)/(3*d) + (a*(2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/(4*d) + (a*Sin[c + d*x])/d","A",1
504,1,92,86,0.1687591,"\int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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{2 a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \left(-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right)}{4 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) + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d) + (a*(2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/(4*d)","A",1
505,1,61,61,0.0252673,"\int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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,"-1/6*(a*Cot[c + d*x]^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",1
506,1,65,65,0.0257627,"\int \cot ^5(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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,"-1/6*(a*Cot[c + d*x]^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",1
507,1,88,81,0.1599495,"\int \cot ^5(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + a*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{a \left(3 \csc ^8(c+d x)-8 \csc ^6(c+d x)+6 \csc ^4(c+d x)\right)}{24 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,"-1/3*(a*Csc[c + d*x]^3)/d + (2*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*(6*Csc[c + d*x]^4 - 8*Csc[c + d*x]^6 + 3*Csc[c + d*x]^8))/(24*d)","A",1
508,1,88,81,0.112984,"\int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]^5*(a + a*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{a \left(3 \csc ^8(c+d x)-8 \csc ^6(c+d x)+6 \csc ^4(c+d x)\right)}{24 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,"-1/5*(a*Csc[c + d*x]^5)/d + (2*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d) - (a*(6*Csc[c + d*x]^4 - 8*Csc[c + d*x]^6 + 3*Csc[c + d*x]^8))/(24*d)","A",1
509,1,88,97,0.1735807,"\int \cot ^5(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]^6*(a + a*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{a \left(6 \csc ^{10}(c+d x)-15 \csc ^8(c+d x)+10 \csc ^6(c+d x)\right)}{60 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,"-1/5*(a*Csc[c + d*x]^5)/d + (2*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d) - (a*(10*Csc[c + d*x]^6 - 15*Csc[c + d*x]^8 + 6*Csc[c + d*x]^10))/(60*d)","A",1
510,1,88,97,0.1935304,"\int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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{2 a \csc ^9(c+d x)}{9 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \left(6 \csc ^{10}(c+d x)-15 \csc ^8(c+d x)+10 \csc ^6(c+d x)\right)}{60 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,"-1/7*(a*Csc[c + d*x]^7)/d + (2*a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^11)/(11*d) - (a*(10*Csc[c + d*x]^6 - 15*Csc[c + d*x]^8 + 6*Csc[c + d*x]^10))/(60*d)","A",1
511,1,110,127,0.79147,"\int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 (-15120 \sin (c+d x)+3360 \sin (3 (c+d x))+2016 \sin (5 (c+d x))-360 \sin (7 (c+d x))-280 \sin (9 (c+d x))+10710 \cos (2 (c+d x))+1260 \cos (4 (c+d x))-1365 \cos (6 (c+d x))-315 \cos (8 (c+d x))+63 \cos (10 (c+d x))-2625)}{322560 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,"-1/322560*(a^2*(-2625 + 10710*Cos[2*(c + d*x)] + 1260*Cos[4*(c + d*x)] - 1365*Cos[6*(c + d*x)] - 315*Cos[8*(c + d*x)] + 63*Cos[10*(c + d*x)] - 15120*Sin[c + d*x] + 3360*Sin[3*(c + d*x)] + 2016*Sin[5*(c + d*x)] - 360*Sin[7*(c + d*x)] - 280*Sin[9*(c + d*x)]))/d","A",1
512,1,99,109,0.7245707,"\int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 (-16380 \sin (c+d x)+1680 \sin (3 (c+d x))+2016 \sin (5 (c+d x))+270 \sin (7 (c+d x))-70 \sin (9 (c+d x))+7560 \cos (2 (c+d x))+1260 \cos (4 (c+d x))-840 \cos (6 (c+d x))-315 \cos (8 (c+d x)))}{161280 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,"-1/161280*(a^2*(7560*Cos[2*(c + d*x)] + 1260*Cos[4*(c + d*x)] - 840*Cos[6*(c + d*x)] - 315*Cos[8*(c + d*x)] - 16380*Sin[c + d*x] + 1680*Sin[3*(c + d*x)] + 2016*Sin[5*(c + d*x)] + 270*Sin[7*(c + d*x)] - 70*Sin[9*(c + d*x)]))/d","A",1
513,1,90,89,0.3188721,"\int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 (-16800 \sin (c+d x)+1120 \sin (3 (c+d x))+2016 \sin (5 (c+d x))+480 \sin (7 (c+d x))+10920 \cos (2 (c+d x))+3780 \cos (4 (c+d x))+280 \cos (6 (c+d x))-105 \cos (8 (c+d x))-2590)}{107520 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,"-1/107520*(a^2*(-2590 + 10920*Cos[2*(c + d*x)] + 3780*Cos[4*(c + d*x)] + 280*Cos[6*(c + d*x)] - 105*Cos[8*(c + d*x)] - 16800*Sin[c + d*x] + 1120*Sin[3*(c + d*x)] + 2016*Sin[5*(c + d*x)] + 480*Sin[7*(c + d*x)]))/d","A",1
514,1,78,119,0.0830719,"\int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Cot[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(10 \sin ^6(c+d x)+24 \sin ^5(c+d x)-15 \sin ^4(c+d x)-80 \sin ^3(c+d x)-30 \sin ^2(c+d x)+120 \sin (c+d x)+60 \log (\sin (c+d x))\right)}{60 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*(60*Log[Sin[c + d*x]] + 120*Sin[c + d*x] - 30*Sin[c + d*x]^2 - 80*Sin[c + d*x]^3 - 15*Sin[c + d*x]^4 + 24*Sin[c + d*x]^5 + 10*Sin[c + d*x]^6))/(60*d)","A",1
515,1,114,114,0.0528327,"\int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[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",1
516,1,76,116,0.1714958,"\int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \left(-3 \sin ^4(c+d x)-8 \sin ^3(c+d x)+6 \sin ^2(c+d x)+48 \sin (c+d x)+6 \csc ^2(c+d x)+24 \csc (c+d x)+12 \log (\sin (c+d x))\right)}{12 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,"-1/12*(a^2*(24*Csc[c + d*x] + 6*Csc[c + d*x]^2 + 12*Log[Sin[c + d*x]] + 48*Sin[c + d*x] + 6*Sin[c + d*x]^2 - 8*Sin[c + d*x]^3 - 3*Sin[c + d*x]^4))/d","A",1
517,1,74,110,0.1822545,"\int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(\sin ^3(c+d x)+3 \sin ^2(c+d x)-3 \sin (c+d x)-\csc ^3(c+d x)-3 \csc ^2(c+d x)+3 \csc (c+d x)-12 \log (\sin (c+d x))\right)}{3 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*(3*Csc[c + d*x] - 3*Csc[c + d*x]^2 - Csc[c + d*x]^3 - 12*Log[Sin[c + d*x]] - 3*Sin[c + d*x] + 3*Sin[c + d*x]^2 + Sin[c + d*x]^3))/(3*d)","A",1
518,1,76,116,0.4535648,"\int \cot ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(6 \sin ^2(c+d x)+24 \sin (c+d x)-3 \csc ^4(c+d x)-8 \csc ^3(c+d x)+6 \csc ^2(c+d x)+48 \csc (c+d x)-12 \log (\sin (c+d x))\right)}{12 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,"(a^2*(48*Csc[c + d*x] + 6*Csc[c + d*x]^2 - 8*Csc[c + d*x]^3 - 3*Csc[c + d*x]^4 - 12*Log[Sin[c + d*x]] + 24*Sin[c + d*x] + 6*Sin[c + d*x]^2))/(12*d)","A",1
519,1,76,112,0.1289062,"\int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(30 \sin (c+d x)-6 \csc ^5(c+d x)-15 \csc ^4(c+d x)+10 \csc ^3(c+d x)+60 \csc ^2(c+d x)+30 \csc (c+d x)+60 \log (\sin (c+d x))\right)}{30 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*(30*Csc[c + d*x] + 60*Csc[c + d*x]^2 + 10*Csc[c + d*x]^3 - 15*Csc[c + d*x]^4 - 6*Csc[c + d*x]^5 + 60*Log[Sin[c + d*x]] + 30*Sin[c + d*x]))/(30*d)","A",1
520,1,102,119,0.0418361,"\int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","a^2 \left(-\frac{\csc ^6(c+d x)}{6 d}-\frac{2 \csc ^5(c+d x)}{5 d}+\frac{\csc ^4(c+d x)}{4 d}+\frac{4 \csc ^3(c+d x)}{3 d}+\frac{\csc ^2(c+d x)}{2 d}-\frac{2 \csc (c+d x)}{d}+\frac{\log (\sin (c+d x))}{d}\right)","-\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,"a^2*((-2*Csc[c + d*x])/d + Csc[c + d*x]^2/(2*d) + (4*Csc[c + d*x]^3)/(3*d) + Csc[c + d*x]^4/(4*d) - (2*Csc[c + d*x]^5)/(5*d) - Csc[c + d*x]^6/(6*d) + Log[Sin[c + d*x]]/d)","A",1
521,1,110,111,0.8564557,"\int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 (-63840 \sin (c+d x)+8960 \sin (3 (c+d x))+8064 \sin (5 (c+d x))+240 \sin (7 (c+d x))-560 \sin (9 (c+d x))+34440 \cos (2 (c+d x))+5040 \cos (4 (c+d x))-4060 \cos (6 (c+d x))-1260 \cos (8 (c+d x))+84 \cos (10 (c+d x))-2835)}{430080 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,"-1/430080*(a^3*(-2835 + 34440*Cos[2*(c + d*x)] + 5040*Cos[4*(c + d*x)] - 4060*Cos[6*(c + d*x)] - 1260*Cos[8*(c + d*x)] + 84*Cos[10*(c + d*x)] - 63840*Sin[c + d*x] + 8960*Sin[3*(c + d*x)] + 8064*Sin[5*(c + d*x)] + 240*Sin[7*(c + d*x)] - 560*Sin[9*(c + d*x)]))/d","A",1
522,1,100,89,0.6781049,"\int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (16632 \sin (c+d x)-1344 \sin (3 (c+d x))-2016 \sin (5 (c+d x))-396 \sin (7 (c+d x))+28 \sin (9 (c+d x))-9576 \cos (2 (c+d x))-2772 \cos (4 (c+d x))+168 \cos (6 (c+d x))+189 \cos (8 (c+d x))+4662)}{64512 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,"(a^3*(4662 - 9576*Cos[2*(c + d*x)] - 2772*Cos[4*(c + d*x)] + 168*Cos[6*(c + d*x)] + 189*Cos[8*(c + d*x)] + 16632*Sin[c + d*x] - 1344*Sin[3*(c + d*x)] - 2016*Sin[5*(c + d*x)] - 396*Sin[7*(c + d*x)] + 28*Sin[9*(c + d*x)]))/(64512*d)","A",1
523,1,88,137,0.1084493,"\int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(60 \sin ^7(c+d x)+210 \sin ^6(c+d x)+84 \sin ^5(c+d x)-525 \sin ^4(c+d x)-700 \sin ^3(c+d x)+210 \sin ^2(c+d x)+1260 \sin (c+d x)+420 \log (\sin (c+d x))\right)}{420 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*(420*Log[Sin[c + d*x]] + 1260*Sin[c + d*x] + 210*Sin[c + d*x]^2 - 700*Sin[c + d*x]^3 - 525*Sin[c + d*x]^4 + 84*Sin[c + d*x]^5 + 210*Sin[c + d*x]^6 + 60*Sin[c + d*x]^7))/(420*d)","A",1
524,1,86,133,0.1705133,"\int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \left(-10 \sin ^6(c+d x)-36 \sin ^5(c+d x)-15 \sin ^4(c+d x)+100 \sin ^3(c+d x)+150 \sin ^2(c+d x)-60 \sin (c+d x)+60 \csc (c+d x)-180 \log (\sin (c+d x))\right)}{60 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,"-1/60*(a^3*(60*Csc[c + d*x] - 180*Log[Sin[c + d*x]] - 60*Sin[c + d*x] + 150*Sin[c + d*x]^2 + 100*Sin[c + d*x]^3 - 15*Sin[c + d*x]^4 - 36*Sin[c + d*x]^5 - 10*Sin[c + d*x]^6))/d","A",1
525,1,86,133,0.1522272,"\int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \left(-12 \sin ^5(c+d x)-45 \sin ^4(c+d x)-20 \sin ^3(c+d x)+150 \sin ^2(c+d x)+300 \sin (c+d x)+30 \csc ^2(c+d x)+180 \csc (c+d x)-60 \log (\sin (c+d x))\right)}{60 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,"-1/60*(a^3*(180*Csc[c + d*x] + 30*Csc[c + d*x]^2 - 60*Log[Sin[c + d*x]] + 300*Sin[c + d*x] + 150*Sin[c + d*x]^2 - 20*Sin[c + d*x]^3 - 45*Sin[c + d*x]^4 - 12*Sin[c + d*x]^5))/d","A",1
526,1,86,131,0.2559436,"\int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \left(-3 \sin ^4(c+d x)-12 \sin ^3(c+d x)-6 \sin ^2(c+d x)+60 \sin (c+d x)+4 \csc ^3(c+d x)+18 \csc ^2(c+d x)+12 \csc (c+d x)+60 \log (\sin (c+d x))\right)}{12 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,"-1/12*(a^3*(12*Csc[c + d*x] + 18*Csc[c + d*x]^2 + 4*Csc[c + d*x]^3 + 60*Log[Sin[c + d*x]] + 60*Sin[c + d*x] - 6*Sin[c + d*x]^2 - 12*Sin[c + d*x]^3 - 3*Sin[c + d*x]^4))/d","A",1
527,1,86,131,0.4980731,"\int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(4 \sin ^3(c+d x)+18 \sin ^2(c+d x)+12 \sin (c+d x)-3 \csc ^4(c+d x)-12 \csc ^3(c+d x)-6 \csc ^2(c+d x)+60 \csc (c+d x)-60 \log (\sin (c+d x))\right)}{12 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,"(a^3*(60*Csc[c + d*x] - 6*Csc[c + d*x]^2 - 12*Csc[c + d*x]^3 - 3*Csc[c + d*x]^4 - 60*Log[Sin[c + d*x]] + 12*Sin[c + d*x] + 18*Sin[c + d*x]^2 + 4*Sin[c + d*x]^3))/(12*d)","A",1
528,1,86,133,0.1733459,"\int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(30 \sin ^2(c+d x)+180 \sin (c+d x)-12 \csc ^5(c+d x)-45 \csc ^4(c+d x)-20 \csc ^3(c+d x)+150 \csc ^2(c+d x)+300 \csc (c+d x)+60 \log (\sin (c+d x))\right)}{60 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,"(a^3*(300*Csc[c + d*x] + 150*Csc[c + d*x]^2 - 20*Csc[c + d*x]^3 - 45*Csc[c + d*x]^4 - 12*Csc[c + d*x]^5 + 60*Log[Sin[c + d*x]] + 180*Sin[c + d*x] + 30*Sin[c + d*x]^2))/(60*d)","A",1
529,1,113,133,0.0429106,"\int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","a^3 \left(\frac{\sin (c+d x)}{d}-\frac{\csc ^6(c+d x)}{6 d}-\frac{3 \csc ^5(c+d x)}{5 d}-\frac{\csc ^4(c+d x)}{4 d}+\frac{5 \csc ^3(c+d x)}{3 d}+\frac{5 \csc ^2(c+d x)}{2 d}-\frac{\csc (c+d x)}{d}+\frac{3 \log (\sin (c+d x))}{d}\right)","\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*Csc[c + d*x]^2)/(2*d) + (5*Csc[c + d*x]^3)/(3*d) - Csc[c + d*x]^4/(4*d) - (3*Csc[c + d*x]^5)/(5*d) - Csc[c + d*x]^6/(6*d) + (3*Log[Sin[c + d*x]])/d + Sin[c + d*x]/d)","A",1
530,1,96,145,0.1580924,"\int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]","-\frac{a^4 \left(-3 \sin ^5(c+d x)-15 \sin ^4(c+d x)-20 \sin ^3(c+d x)+30 \sin ^2(c+d x)+150 \sin (c+d x)+5 \csc ^3(c+d x)+30 \csc ^2(c+d x)+60 \csc (c+d x)+60 \log (\sin (c+d x))\right)}{15 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,"-1/15*(a^4*(60*Csc[c + d*x] + 30*Csc[c + d*x]^2 + 5*Csc[c + d*x]^3 + 60*Log[Sin[c + d*x]] + 150*Sin[c + d*x] + 30*Sin[c + d*x]^2 - 20*Sin[c + d*x]^3 - 15*Sin[c + d*x]^4 - 3*Sin[c + d*x]^5))/d","A",1
531,1,96,148,0.1535812,"\int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \left(3 \sin ^4(c+d x)+16 \sin ^3(c+d x)+24 \sin ^2(c+d x)-48 \sin (c+d x)-3 \csc ^4(c+d x)-16 \csc ^3(c+d x)-24 \csc ^2(c+d x)+48 \csc (c+d x)-120 \log (\sin (c+d x))\right)}{12 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,"(a^4*(48*Csc[c + d*x] - 24*Csc[c + d*x]^2 - 16*Csc[c + d*x]^3 - 3*Csc[c + d*x]^4 - 120*Log[Sin[c + d*x]] - 48*Sin[c + d*x] + 24*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3 + 3*Sin[c + d*x]^4))/(12*d)","A",1
532,1,96,146,0.1834138,"\int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 \left(5 \sin ^3(c+d x)+30 \sin ^2(c+d x)+60 \sin (c+d x)-3 \csc ^5(c+d x)-15 \csc ^4(c+d x)-20 \csc ^3(c+d x)+30 \csc ^2(c+d x)+150 \csc (c+d x)-60 \log (\sin (c+d x))\right)}{15 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,"(a^4*(150*Csc[c + d*x] + 30*Csc[c + d*x]^2 - 20*Csc[c + d*x]^3 - 15*Csc[c + d*x]^4 - 3*Csc[c + d*x]^5 - 60*Log[Sin[c + d*x]] + 60*Sin[c + d*x] + 30*Sin[c + d*x]^2 + 5*Sin[c + d*x]^3))/(15*d)","A",1
533,1,48,73,0.3545579,"\int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^4(c+d x) \left(60 \sin ^3(c+d x)-70 \sin ^2(c+d x)-84 \sin (c+d x)+105\right)}{420 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*(105 - 84*Sin[c + d*x] - 70*Sin[c + d*x]^2 + 60*Sin[c + d*x]^3))/(420*a*d)","A",1
534,1,48,73,0.2075054,"\int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^3(c+d x) \left(10 \sin ^3(c+d x)-12 \sin ^2(c+d x)-15 \sin (c+d x)+20\right)}{60 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*(20 - 15*Sin[c + d*x] - 12*Sin[c + d*x]^2 + 10*Sin[c + d*x]^3))/(60*a*d)","A",1
535,1,48,55,0.1555402,"\int \frac{\cos ^5(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\sin ^2(c+d x) \left(12 \sin ^3(c+d x)-15 \sin ^2(c+d x)-20 \sin (c+d x)+30\right)}{60 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,"(Sin[c + d*x]^2*(30 - 20*Sin[c + d*x] - 15*Sin[c + d*x]^2 + 12*Sin[c + d*x]^3))/(60*a*d)","A",1
536,1,49,65,0.0523841,"\int \frac{\cos ^4(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{2 \sin ^3(c+d x)-3 \sin ^2(c+d x)-6 \sin (c+d x)+6 \log (\sin (c+d x))-2}{6 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,"(-2 + 6*Log[Sin[c + d*x]] - 6*Sin[c + d*x] - 3*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3)/(6*a*d)","A",1
537,1,45,62,0.0612197,"\int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^2(c+d x)-2 \sin (c+d x)-2 \csc (c+d x)-2 \log (\sin (c+d x))+6}{2 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,"(6 - 2*Csc[c + d*x] - 2*Log[Sin[c + d*x]] - 2*Sin[c + d*x] + Sin[c + d*x]^2)/(2*a*d)","A",1
538,1,45,60,0.0863416,"\int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{-2 \sin (c+d x)+\csc ^2(c+d x)-2 \csc (c+d x)+2 \log (\sin (c+d x))+3}{2 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,"-1/2*(3 - 2*Csc[c + d*x] + Csc[c + d*x]^2 + 2*Log[Sin[c + d*x]] - 2*Sin[c + d*x])/(a*d)","A",1
539,1,48,64,0.0788957,"\int \frac{\cos (c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{-2 \csc ^3(c+d x)+3 \csc ^2(c+d x)+6 \csc (c+d x)+6 \log (\sin (c+d x))}{6 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,"(6*Csc[c + d*x] + 3*Csc[c + d*x]^2 - 2*Csc[c + d*x]^3 + 6*Log[Sin[c + d*x]])/(6*a*d)","A",1
540,1,30,51,0.0467833,"\int \frac{\cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sin[c + d*x]),x]","-\frac{(\csc (c+d x)-1)^3 (3 \csc (c+d x)+5)}{12 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,"-1/12*((-1 + Csc[c + d*x])^3*(5 + 3*Csc[c + d*x]))/(a*d)","A",1
541,1,48,55,0.1125025,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^5*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\csc ^2(c+d x) \left(-12 \csc ^3(c+d x)+15 \csc ^2(c+d x)+20 \csc (c+d x)-30\right)}{60 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,"(Csc[c + d*x]^2*(-30 + 20*Csc[c + d*x] + 15*Csc[c + d*x]^2 - 12*Csc[c + d*x]^3))/(60*a*d)","A",1
542,1,48,73,0.1043611,"\int \frac{\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^5*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\csc ^3(c+d x) \left(-10 \csc ^3(c+d x)+12 \csc ^2(c+d x)+15 \csc (c+d x)-20\right)}{60 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*(-20 + 15*Csc[c + d*x] + 12*Csc[c + d*x]^2 - 10*Csc[c + d*x]^3))/(60*a*d)","A",1
543,1,48,73,0.1060442,"\int \frac{\cot ^5(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^5*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\csc ^4(c+d x) \left(-60 \csc ^3(c+d x)+70 \csc ^2(c+d x)+84 \csc (c+d x)-105\right)}{420 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*(-105 + 84*Csc[c + d*x] + 70*Csc[c + d*x]^2 - 60*Csc[c + d*x]^3))/(420*a*d)","A",1
544,1,38,55,0.5358519,"\int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^4(c+d x) \left(10 \sin ^2(c+d x)-24 \sin (c+d x)+15\right)}{60 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*(15 - 24*Sin[c + d*x] + 10*Sin[c + d*x]^2))/(60*a^2*d)","A",1
545,1,38,55,0.6629036,"\int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{\sin ^3(c+d x) (15 \sin (c+d x)+3 \cos (2 (c+d x))-13)}{30 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,"-1/30*(Sin[c + d*x]^3*(-13 + 3*Cos[2*(c + d*x)] + 15*Sin[c + d*x]))/(a^2*d)","A",1
546,1,38,55,0.2144196,"\int \frac{\cos ^5(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^2(c+d x) \left(3 \sin ^2(c+d x)-8 \sin (c+d x)+6\right)}{12 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*(6 - 8*Sin[c + d*x] + 3*Sin[c + d*x]^2))/(12*a^2*d)","A",1
547,1,36,47,0.0393967,"\int \frac{\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{\sin ^2(c+d x)-4 \sin (c+d x)+2 \log (\sin (c+d x))}{2 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,"(2*Log[Sin[c + d*x]] - 4*Sin[c + d*x] + Sin[c + d*x]^2)/(2*a^2*d)","A",1
548,1,32,43,0.0428243,"\int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{-\sin (c+d x)+\csc (c+d x)+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] + 2*Log[Sin[c + d*x]] - Sin[c + d*x])/(a^2*d))","A",1
549,1,38,47,0.0474331,"\int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{-\csc ^2(c+d x)+4 \csc (c+d x)+2 \log (\sin (c+d x))}{2 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,"(4*Csc[c + d*x] - Csc[c + d*x]^2 + 2*Log[Sin[c + d*x]])/(2*a^2*d)","A",1
550,1,20,31,0.0423804,"\int \frac{\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{(\csc (c+d x)-1)^3}{3 a^2 d}","-\frac{\csc ^3(c+d x) (a-a \sin (c+d x))^3}{3 a^5 d}",1,"-1/3*(-1 + Csc[c + d*x])^3/(a^2*d)","A",1
551,1,38,55,0.0649026,"\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]","\frac{\csc ^4(c+d x) (8 \sin (c+d x)+3 \cos (2 (c+d x))-6)}{12 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]^4*(-6 + 3*Cos[2*(c + d*x)] + 8*Sin[c + d*x]))/(12*a^2*d)","A",1
552,1,38,55,0.0741395,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^5*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{\csc ^5(c+d x) (15 \sin (c+d x)+5 \cos (2 (c+d x))-11)}{30 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]^5*(-11 + 5*Cos[2*(c + d*x)] + 15*Sin[c + d*x]))/(30*a^2*d)","A",1
553,1,38,55,0.0764234,"\int \frac{\cot ^5(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^5*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^4(c+d x) \left(10 \csc ^2(c+d x)-24 \csc (c+d x)+15\right)}{60 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,"-1/60*(Csc[c + d*x]^4*(15 - 24*Csc[c + d*x] + 10*Csc[c + d*x]^2))/(a^2*d)","A",1
554,1,71,102,0.9678586,"\int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{192 \sin ^5(c+d x)-720 \sin ^4(c+d x)+1280 \sin ^3(c+d x)-1920 \sin ^2(c+d x)+3840 \sin (c+d x)-3840 \log (\sin (c+d x)+1)+45}{960 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,"(45 - 3840*Log[1 + Sin[c + d*x]] + 3840*Sin[c + d*x] - 1920*Sin[c + d*x]^2 + 1280*Sin[c + d*x]^3 - 720*Sin[c + d*x]^4 + 192*Sin[c + d*x]^5)/(960*a^3*d)","A",1
555,1,59,82,0.9384283,"\int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{-152 \sin (c+d x)+8 \sin (3 (c+d x))-36 \cos (2 (c+d x))+\cos (4 (c+d x))+128 \log (\sin (c+d x)+1)+35}{32 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,"(35 - 36*Cos[2*(c + d*x)] + Cos[4*(c + d*x)] + 128*Log[1 + Sin[c + d*x]] - 152*Sin[c + d*x] + 8*Sin[3*(c + d*x)])/(32*a^3*d)","A",1
556,1,51,68,0.3462237,"\int \frac{\cos ^5(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{32 \sin ^3(c+d x)-144 \sin ^2(c+d x)+384 \sin (c+d x)-384 \log (\sin (c+d x)+1)+15}{96 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,"(15 - 384*Log[1 + Sin[c + d*x]] + 384*Sin[c + d*x] - 144*Sin[c + d*x]^2 + 32*Sin[c + d*x]^3)/(96*a^3*d)","A",1
557,1,32,45,0.0399321,"\int \frac{\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{\sin (c+d x)+\log (\sin (c+d x))-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]] - 4*Log[1 + Sin[c + d*x]] + Sin[c + d*x])/(a^3*d)","A",1
558,1,35,47,0.0475932,"\int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc (c+d x)+3 \log (\sin (c+d x))-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] + 3*Log[Sin[c + d*x]] - 4*Log[1 + Sin[c + d*x]])/(a^3*d))","A",1
559,1,49,65,0.0801897,"\int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{-\csc ^2(c+d x)+6 \csc (c+d x)+8 \log (\sin (c+d x))-8 \log (\sin (c+d x)+1)}{2 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,"(6*Csc[c + d*x] - Csc[c + d*x]^2 + 8*Log[Sin[c + d*x]] - 8*Log[1 + Sin[c + d*x]])/(2*a^3*d)","A",1
560,1,59,83,0.1164196,"\int \frac{\cos (c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","-\frac{2 \csc ^3(c+d x)-9 \csc ^2(c+d x)+24 \csc (c+d x)+24 \log (\sin (c+d x))-24 \log (\sin (c+d x)+1)}{6 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,"-1/6*(24*Csc[c + d*x] - 9*Csc[c + d*x]^2 + 2*Csc[c + d*x]^3 + 24*Log[Sin[c + d*x]] - 24*Log[1 + Sin[c + d*x]])/(a^3*d)","A",1
561,1,69,96,0.3121674,"\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]","\frac{-\csc ^4(c+d x)+4 \csc ^3(c+d x)-8 \csc ^2(c+d x)+16 \csc (c+d x)+16 \log (\sin (c+d x))-16 \log (\sin (c+d x)+1)}{4 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,"(16*Csc[c + d*x] - 8*Csc[c + d*x]^2 + 4*Csc[c + d*x]^3 - Csc[c + d*x]^4 + 16*Log[Sin[c + d*x]] - 16*Log[1 + Sin[c + d*x]])/(4*a^3*d)","A",1
562,1,79,117,0.1303576,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^5*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{12 \csc ^5(c+d x)-45 \csc ^4(c+d x)+80 \csc ^3(c+d x)-120 \csc ^2(c+d x)+240 \csc (c+d x)+240 \log (\sin (c+d x))-240 \log (\sin (c+d x)+1)}{60 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,"-1/60*(240*Csc[c + d*x] - 120*Csc[c + d*x]^2 + 80*Csc[c + d*x]^3 - 45*Csc[c + d*x]^4 + 12*Csc[c + d*x]^5 + 240*Log[Sin[c + d*x]] - 240*Log[1 + Sin[c + d*x]])/(a^3*d)","A",1
563,1,81,120,0.7144383,"\int \frac{\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^4,x]","\frac{\frac{48}{\sin (c+d x)+1}-3 \csc ^4(c+d x)+16 \csc ^3(c+d x)-48 \csc ^2(c+d x)+144 \csc (c+d x)+192 \log (\sin (c+d x))-192 \log (\sin (c+d x)+1)}{12 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,"(144*Csc[c + d*x] - 48*Csc[c + d*x]^2 + 16*Csc[c + d*x]^3 - 3*Csc[c + d*x]^4 + 192*Log[Sin[c + d*x]] - 192*Log[1 + Sin[c + d*x]] + 48/(1 + Sin[c + d*x]))/(12*a^4*d)","A",1
564,1,91,135,0.2832044,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Cot[c + d*x]^5*Csc[c + d*x])/(a + a*Sin[c + d*x])^4,x]","-\frac{\frac{60}{\sin (c+d x)+1}+3 \csc ^5(c+d x)-15 \csc ^4(c+d x)+40 \csc ^3(c+d x)-90 \csc ^2(c+d x)+240 \csc (c+d x)+300 \log (\sin (c+d x))-300 \log (\sin (c+d x)+1)}{15 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,"-1/15*(240*Csc[c + d*x] - 90*Csc[c + d*x]^2 + 40*Csc[c + d*x]^3 - 15*Csc[c + d*x]^4 + 3*Csc[c + d*x]^5 + 300*Log[Sin[c + d*x]] - 300*Log[1 + Sin[c + d*x]] + 60/(1 + Sin[c + d*x]))/(a^4*d)","A",1
565,1,123,181,0.5853719,"\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[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) \left(\frac{\sin ^7(c+d x)}{n+8}+\frac{3 \sin ^6(c+d x)}{n+7}+\frac{\sin ^5(c+d x)}{n+6}-\frac{5 \sin ^4(c+d x)}{n+5}-\frac{5 \sin ^3(c+d x)}{n+4}+\frac{\sin ^2(c+d x)}{n+3}+\frac{3 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right)}{d}","\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)*((1 + n)^(-1) + (3*Sin[c + d*x])/(2 + n) + Sin[c + d*x]^2/(3 + n) - (5*Sin[c + d*x]^3)/(4 + n) - (5*Sin[c + d*x]^4)/(5 + n) + Sin[c + d*x]^5/(6 + n) + (3*Sin[c + d*x]^6)/(7 + n) + Sin[c + d*x]^7/(8 + n)))/d","A",1
566,1,110,160,0.3913257,"\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[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) \left(\frac{\sin ^6(c+d x)}{n+7}+\frac{2 \sin ^5(c+d x)}{n+6}-\frac{\sin ^4(c+d x)}{n+5}-\frac{4 \sin ^3(c+d x)}{n+4}-\frac{\sin ^2(c+d x)}{n+3}+\frac{2 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right)}{d}","\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)*((1 + n)^(-1) + (2*Sin[c + d*x])/(2 + n) - Sin[c + d*x]^2/(3 + n) - (4*Sin[c + d*x]^3)/(4 + n) - Sin[c + d*x]^4/(5 + n) + (2*Sin[c + d*x]^5)/(6 + n) + Sin[c + d*x]^6/(7 + n)))/d","A",1
567,1,345,123,1.3835846,"\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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) \left(2 n^5 \sin (c+d x)+3 n^5 \sin (3 (c+d x))+n^5 \sin (5 (c+d x))+46 n^4 \sin (c+d x)+61 n^4 \sin (3 (c+d x))+15 n^4 \sin (5 (c+d x))+474 n^3 \sin (c+d x)+431 n^3 \sin (3 (c+d x))+85 n^3 \sin (5 (c+d x))+2258 n^2 \sin (c+d x)+1331 n^2 \sin (3 (c+d x))+225 n^2 \sin (5 (c+d x))+8 \left(n^5+20 n^4+147 n^3+484 n^2+692 n+336\right) \cos (2 (c+d x))+2 \left(n^5+16 n^4+95 n^3+260 n^2+324 n+144\right) \cos (4 (c+d x))+4468 n \sin (c+d x)+1798 n \sin (3 (c+d x))+274 n \sin (5 (c+d x))+2640 \sin (c+d x)+840 \sin (3 (c+d x))+120 \sin (5 (c+d x))+6 n^5+128 n^4+1114 n^3+4888 n^2+10520 n+8544\right)}{16 d (n+1) (n+2) (n+3) (n+4) (n+5) (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)*(8544 + 10520*n + 4888*n^2 + 1114*n^3 + 128*n^4 + 6*n^5 + 8*(336 + 692*n + 484*n^2 + 147*n^3 + 20*n^4 + n^5)*Cos[2*(c + d*x)] + 2*(144 + 324*n + 260*n^2 + 95*n^3 + 16*n^4 + n^5)*Cos[4*(c + d*x)] + 2640*Sin[c + d*x] + 4468*n*Sin[c + d*x] + 2258*n^2*Sin[c + d*x] + 474*n^3*Sin[c + d*x] + 46*n^4*Sin[c + d*x] + 2*n^5*Sin[c + d*x] + 840*Sin[3*(c + d*x)] + 1798*n*Sin[3*(c + d*x)] + 1331*n^2*Sin[3*(c + d*x)] + 431*n^3*Sin[3*(c + d*x)] + 61*n^4*Sin[3*(c + d*x)] + 3*n^5*Sin[3*(c + d*x)] + 120*Sin[5*(c + d*x)] + 274*n*Sin[5*(c + d*x)] + 225*n^2*Sin[5*(c + d*x)] + 85*n^3*Sin[5*(c + d*x)] + 15*n^4*Sin[5*(c + d*x)] + n^5*Sin[5*(c + d*x)]))/(16*d*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n))","B",1
568,1,74,91,0.7005385,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^n)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^{n+1}(c+d x) \left(-\frac{(n+4) \sin ^2(c+d x)}{n+3}-\frac{(n+4) \sin (c+d x)}{n+2}+\sin ^3(c+d x)+\frac{n+4}{n+1}\right)}{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)*((4 + n)/(1 + n) - ((4 + n)*Sin[c + d*x])/(2 + n) - ((4 + n)*Sin[c + d*x]^2)/(3 + n) + Sin[c + d*x]^3))/(a*d*(4 + n))","A",1
569,1,50,68,0.1187055,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(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) \left(\frac{\sin ^2(c+d x)}{n+3}-\frac{2 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right)}{a^2 d}","\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)*((1 + n)^(-1) - (2*Sin[c + d*x])/(2 + n) + Sin[c + d*x]^2/(3 + n)))/(a^2*d)","A",1
570,1,64,85,0.1052045,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^3,x]","\frac{\sin ^{n+1}(c+d x) (4 (n+2) \, _2F_1(1,n+1;n+2;-\sin (c+d x))+(n+1) \sin (c+d x)-3 (n+2))}{a^3 d (n+1) (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,"(Sin[c + d*x]^(1 + n)*(-3*(2 + n) + 4*(2 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]] + (1 + n)*Sin[c + d*x]))/(a^3*d*(1 + n)*(2 + n))","A",1
571,1,72,88,0.1048825,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^4,x]","\frac{\sin ^{n+1}(c+d x) (-4 (n+1) (\sin (c+d x)+1) \, _2F_1(1,n+1;n+2;-\sin (c+d x))+\sin (c+d x)+4 n+5)}{a^4 d (n+1) (\sin (c+d x)+1)}","-\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)*(5 + 4*n + Sin[c + d*x] - 4*(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*(1 + Sin[c + d*x])))/(a^4*d*(1 + n)*(1 + Sin[c + d*x]))","A",1
572,1,121,165,0.5673122,"\int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{a (13860 \sin (2 (c+d x))-27720 \sin (4 (c+d x))-6930 \sin (6 (c+d x))+3465 \sin (8 (c+d x))+1386 \sin (10 (c+d x))-69300 \cos (c+d x)-23100 \cos (3 (c+d x))+6930 \cos (5 (c+d x))+4950 \cos (7 (c+d x))-770 \cos (9 (c+d x))-630 \cos (11 (c+d x))+83160 d x)}{7096320 d}","-\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,"(a*(83160*d*x - 69300*Cos[c + d*x] - 23100*Cos[3*(c + d*x)] + 6930*Cos[5*(c + d*x)] + 4950*Cos[7*(c + d*x)] - 770*Cos[9*(c + d*x)] - 630*Cos[11*(c + d*x)] + 13860*Sin[2*(c + d*x)] - 27720*Sin[4*(c + d*x)] - 6930*Sin[6*(c + d*x)] + 3465*Sin[8*(c + d*x)] + 1386*Sin[10*(c + d*x)]))/(7096320*d)","A",1
573,1,101,149,0.3482024,"\int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a (1260 \sin (2 (c+d x))-2520 \sin (4 (c+d x))-630 \sin (6 (c+d x))+315 \sin (8 (c+d x))+126 \sin (10 (c+d x))-15120 \cos (c+d x)-6720 \cos (3 (c+d x))+1080 \cos (7 (c+d x))+280 \cos (9 (c+d x))+7560 d x)}{645120 d}","\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,"(a*(7560*d*x - 15120*Cos[c + d*x] - 6720*Cos[3*(c + d*x)] + 1080*Cos[7*(c + d*x)] + 280*Cos[9*(c + d*x)] + 1260*Sin[2*(c + d*x)] - 2520*Sin[4*(c + d*x)] - 630*Sin[6*(c + d*x)] + 315*Sin[8*(c + d*x)] + 126*Sin[10*(c + d*x)]))/(645120*d)","A",1
574,1,91,125,0.3207957,"\int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a (1008 \sin (2 (c+d x))-504 \sin (4 (c+d x))-336 \sin (6 (c+d x))-63 \sin (8 (c+d x))-1512 \cos (c+d x)-672 \cos (3 (c+d x))+108 \cos (7 (c+d x))+28 \cos (9 (c+d x))+2520 d x)}{64512 d}","\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,"(a*(2520*d*x - 1512*Cos[c + d*x] - 672*Cos[3*(c + d*x)] + 108*Cos[7*(c + d*x)] + 28*Cos[9*(c + d*x)] + 1008*Sin[2*(c + d*x)] - 504*Sin[4*(c + d*x)] - 336*Sin[6*(c + d*x)] - 63*Sin[8*(c + d*x)]))/(64512*d)","A",1
575,1,91,109,0.2626874,"\int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a (-336 \sin (2 (c+d x))+168 \sin (4 (c+d x))+112 \sin (6 (c+d x))+21 \sin (8 (c+d x))+1680 \cos (c+d x)+1008 \cos (3 (c+d x))+336 \cos (5 (c+d x))+48 \cos (7 (c+d x))-840 d x)}{21504 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,"-1/21504*(a*(-840*d*x + 1680*Cos[c + d*x] + 1008*Cos[3*(c + d*x)] + 336*Cos[5*(c + d*x)] + 48*Cos[7*(c + d*x)] - 336*Sin[2*(c + d*x)] + 168*Sin[4*(c + d*x)] + 112*Sin[6*(c + d*x)] + 21*Sin[8*(c + d*x)]))/d","A",1
576,1,100,127,0.1166731,"\int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Cot[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \left(225 \sin (2 (c+d x))+45 \sin (4 (c+d x))+5 \sin (6 (c+d x))+1320 \cos (c+d x)+140 \cos (3 (c+d x))+12 \cos (5 (c+d x))+960 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-960 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+300 c+300 d x\right)}{960 d}","\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,"(a*(300*c + 300*d*x + 1320*Cos[c + d*x] + 140*Cos[3*(c + d*x)] + 12*Cos[5*(c + d*x)] - 960*Log[Cos[(c + d*x)/2]] + 960*Log[Sin[(c + d*x)/2]] + 225*Sin[2*(c + d*x)] + 45*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)","A",1
577,1,98,121,0.2653102,"\int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \left(240 \sin (2 (c+d x))+15 \sin (4 (c+d x))-660 \cos (c+d x)-70 \cos (3 (c+d x))-6 \cos (5 (c+d x))+480 \cot (c+d x)-480 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+480 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+900 c+900 d x\right)}{480 d}","\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,"-1/480*(a*(900*c + 900*d*x - 660*Cos[c + d*x] - 70*Cos[3*(c + d*x)] - 6*Cos[5*(c + d*x)] + 480*Cot[c + d*x] + 480*Log[Cos[(c + d*x)/2]] - 480*Log[Sin[(c + d*x)/2]] + 240*Sin[2*(c + d*x)] + 15*Sin[4*(c + d*x)]))/d","A",1
578,1,117,134,2.7515183,"\int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \left(216 \cos (c+d x)+8 \cos (3 (c+d x))+3 \left(16 \sin (2 (c+d x))+\sin (4 (c+d x))+32 \cot (c+d x)+4 \csc ^2\left(\frac{1}{2} (c+d x)\right)-4 \sec ^2\left(\frac{1}{2} (c+d x)\right)+80 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-80 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+60 c+60 d x\right)\right)}{96 d}","-\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 ^4(c+d x) \cot (c+d x)}{4 d}-\frac{a \cos ^3(c+d x) \cot ^2(c+d x)}{2 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,"-1/96*(a*(216*Cos[c + d*x] + 8*Cos[3*(c + d*x)] + 3*(60*c + 60*d*x + 32*Cot[c + d*x] + 4*Csc[(c + d*x)/2]^2 - 80*Log[Cos[(c + d*x)/2]] + 80*Log[Sin[(c + d*x)/2]] - 4*Sec[(c + d*x)/2]^2 + 16*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])))/d","A",1
579,1,174,130,6.0961747,"\int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{5 a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{9 a \cos (c+d x)}{4 d}-\frac{a \cos (3 (c+d x))}{12 d}+\frac{7 a \cot (c+d x)}{3 d}-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{5 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{5 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}","-\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*(c + d*x))/(2*d) - (9*a*Cos[c + d*x])/(4*d) - (a*Cos[3*(c + d*x)])/(12*d) + (7*a*Cot[c + d*x])/(3*d) - (a*Csc[(c + d*x)/2]^2)/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) + (5*a*Log[Cos[(c + d*x)/2]])/(2*d) - (5*a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d) + (a*Sin[2*(c + d*x)])/(4*d)","A",1
580,1,138,134,1.4274802,"\int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a \left(192 \cos (c+d x)-64 \cot (c+d x) \left(\csc ^2(c+d x)-7\right)+3 \left(16 \sin (2 (c+d x))-\csc ^4\left(\frac{1}{2} (c+d x)\right)+18 \csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^4\left(\frac{1}{2} (c+d x)\right)-18 \sec ^2\left(\frac{1}{2} (c+d x)\right)+40 \left(3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 c+4 d x\right)\right)\right)}{192 d}","\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,"(a*(192*Cos[c + d*x] - 64*Cot[c + d*x]*(-7 + Csc[c + d*x]^2) + 3*(18*Csc[(c + d*x)/2]^2 - Csc[(c + d*x)/2]^4 + 40*(4*c + 4*d*x - 3*Log[Cos[(c + d*x)/2]] + 3*Log[Sin[(c + d*x)/2]]) - 18*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^4 + 16*Sin[2*(c + d*x)])))/(192*d)","A",1
581,1,164,122,0.0701761,"\int \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^5(c+d x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right)}{5 d}+\frac{a \cos (c+d x)}{d}-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{9 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{9 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{15 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{15 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}","\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*Cos[c + d*x])/d + (9*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d) - (15*a*Log[Cos[(c + d*x)/2]])/(8*d) + (15*a*Log[Sin[(c + d*x)/2]])/(8*d) - (9*a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)","C",1
582,1,193,128,0.0643646,"\int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \cot ^5(c+d x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right)}{5 d}-\frac{a \csc ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{11 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{11 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}+\frac{5 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{16 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{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,"(-11*a*Csc[(c + d*x)/2]^2)/(64*d) + (a*Csc[(c + d*x)/2]^4)/(32*d) - (a*Csc[(c + d*x)/2]^6)/(384*d) - (a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d) + (5*a*Log[Cos[(c + d*x)/2]])/(16*d) - (5*a*Log[Sin[(c + d*x)/2]])/(16*d) + (11*a*Sec[(c + d*x)/2]^2)/(64*d) - (a*Sec[(c + d*x)/2]^4)/(32*d) + (a*Sec[(c + d*x)/2]^6)/(384*d)","C",1
583,1,175,96,0.0475174,"\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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{a \csc ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{11 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{11 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}+\frac{5 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-1/7*(a*Cot[c + d*x]^7)/d - (11*a*Csc[(c + d*x)/2]^2)/(64*d) + (a*Csc[(c + d*x)/2]^4)/(32*d) - (a*Csc[(c + d*x)/2]^6)/(384*d) + (5*a*Log[Cos[(c + d*x)/2]])/(16*d) - (5*a*Log[Sin[(c + d*x)/2]])/(16*d) + (11*a*Sec[(c + d*x)/2]^2)/(64*d) - (a*Sec[(c + d*x)/2]^4)/(32*d) + (a*Sec[(c + d*x)/2]^6)/(384*d)","A",1
584,1,215,122,0.0617056,"\int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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{a \csc ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}+\frac{7 a \csc ^6\left(\frac{1}{2} (c+d x)\right)}{1536 d}-\frac{15 a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}+\frac{5 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}+\frac{a \sec ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}-\frac{7 a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{1536 d}+\frac{15 a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}-\frac{5 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}-\frac{5 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{128 d}+\frac{5 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-1/7*(a*Cot[c + d*x]^7)/d + (5*a*Csc[(c + d*x)/2]^2)/(512*d) - (15*a*Csc[(c + d*x)/2]^4)/(1024*d) + (7*a*Csc[(c + d*x)/2]^6)/(1536*d) - (a*Csc[(c + d*x)/2]^8)/(2048*d) + (5*a*Log[Cos[(c + d*x)/2]])/(128*d) - (5*a*Log[Sin[(c + d*x)/2]])/(128*d) - (5*a*Sec[(c + d*x)/2]^2)/(512*d) + (15*a*Sec[(c + d*x)/2]^4)/(1024*d) - (7*a*Sec[(c + d*x)/2]^6)/(1536*d) + (a*Sec[(c + d*x)/2]^8)/(2048*d)","A",1
585,1,301,138,0.0863933,"\int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{2 a \cot (c+d x)}{63 d}-\frac{a \csc ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}+\frac{7 a \csc ^6\left(\frac{1}{2} (c+d x)\right)}{1536 d}-\frac{15 a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}+\frac{5 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}+\frac{a \sec ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}-\frac{7 a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{1536 d}+\frac{15 a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}-\frac{5 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}-\frac{5 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{128 d}+\frac{5 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{128 d}-\frac{a \cot (c+d x) \csc ^8(c+d x)}{9 d}+\frac{19 a \cot (c+d x) \csc ^6(c+d x)}{63 d}-\frac{5 a \cot (c+d x) \csc ^4(c+d x)}{21 d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{63 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,"(2*a*Cot[c + d*x])/(63*d) + (5*a*Csc[(c + d*x)/2]^2)/(512*d) - (15*a*Csc[(c + d*x)/2]^4)/(1024*d) + (7*a*Csc[(c + d*x)/2]^6)/(1536*d) - (a*Csc[(c + d*x)/2]^8)/(2048*d) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(63*d) - (5*a*Cot[c + d*x]*Csc[c + d*x]^4)/(21*d) + (19*a*Cot[c + d*x]*Csc[c + d*x]^6)/(63*d) - (a*Cot[c + d*x]*Csc[c + d*x]^8)/(9*d) + (5*a*Log[Cos[(c + d*x)/2]])/(128*d) - (5*a*Log[Sin[(c + d*x)/2]])/(128*d) - (5*a*Sec[(c + d*x)/2]^2)/(512*d) + (15*a*Sec[(c + d*x)/2]^4)/(1024*d) - (7*a*Sec[(c + d*x)/2]^6)/(1536*d) + (a*Sec[(c + d*x)/2]^8)/(2048*d)","B",1
586,1,341,160,0.0855551,"\int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{2 a \cot (c+d x)}{63 d}-\frac{a \csc ^{10}\left(\frac{1}{2} (c+d x)\right)}{10240 d}+\frac{3 a \csc ^8\left(\frac{1}{2} (c+d x)\right)}{4096 d}-\frac{3 a \csc ^6\left(\frac{1}{2} (c+d x)\right)}{2048 d}-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}+\frac{3 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{1024 d}+\frac{a \sec ^{10}\left(\frac{1}{2} (c+d x)\right)}{10240 d}-\frac{3 a \sec ^8\left(\frac{1}{2} (c+d x)\right)}{4096 d}+\frac{3 a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{2048 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}-\frac{3 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{1024 d}-\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{256 d}+\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{256 d}-\frac{a \cot (c+d x) \csc ^8(c+d x)}{9 d}+\frac{19 a \cot (c+d x) \csc ^6(c+d x)}{63 d}-\frac{5 a \cot (c+d x) \csc ^4(c+d x)}{21 d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{63 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,"(2*a*Cot[c + d*x])/(63*d) + (3*a*Csc[(c + d*x)/2]^2)/(1024*d) - (a*Csc[(c + d*x)/2]^4)/(1024*d) - (3*a*Csc[(c + d*x)/2]^6)/(2048*d) + (3*a*Csc[(c + d*x)/2]^8)/(4096*d) - (a*Csc[(c + d*x)/2]^10)/(10240*d) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(63*d) - (5*a*Cot[c + d*x]*Csc[c + d*x]^4)/(21*d) + (19*a*Cot[c + d*x]*Csc[c + d*x]^6)/(63*d) - (a*Cot[c + d*x]*Csc[c + d*x]^8)/(9*d) + (3*a*Log[Cos[(c + d*x)/2]])/(256*d) - (3*a*Log[Sin[(c + d*x)/2]])/(256*d) - (3*a*Sec[(c + d*x)/2]^2)/(1024*d) + (a*Sec[(c + d*x)/2]^4)/(1024*d) + (3*a*Sec[(c + d*x)/2]^6)/(2048*d) - (3*a*Sec[(c + d*x)/2]^8)/(4096*d) + (a*Sec[(c + d*x)/2]^10)/(10240*d)","B",1
587,1,363,176,0.1009008,"\int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + a*Sin[c + d*x]),x]","\frac{8 a \cot (c+d x)}{693 d}-\frac{a \csc ^{10}\left(\frac{1}{2} (c+d x)\right)}{10240 d}+\frac{3 a \csc ^8\left(\frac{1}{2} (c+d x)\right)}{4096 d}-\frac{3 a \csc ^6\left(\frac{1}{2} (c+d x)\right)}{2048 d}-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}+\frac{3 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{1024 d}+\frac{a \sec ^{10}\left(\frac{1}{2} (c+d x)\right)}{10240 d}-\frac{3 a \sec ^8\left(\frac{1}{2} (c+d x)\right)}{4096 d}+\frac{3 a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{2048 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}-\frac{3 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{1024 d}-\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{256 d}+\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{256 d}-\frac{a \cot (c+d x) \csc ^{10}(c+d x)}{11 d}+\frac{23 a \cot (c+d x) \csc ^8(c+d x)}{99 d}-\frac{113 a \cot (c+d x) \csc ^6(c+d x)}{693 d}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{231 d}+\frac{4 a \cot (c+d x) \csc ^2(c+d x)}{693 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,"(8*a*Cot[c + d*x])/(693*d) + (3*a*Csc[(c + d*x)/2]^2)/(1024*d) - (a*Csc[(c + d*x)/2]^4)/(1024*d) - (3*a*Csc[(c + d*x)/2]^6)/(2048*d) + (3*a*Csc[(c + d*x)/2]^8)/(4096*d) - (a*Csc[(c + d*x)/2]^10)/(10240*d) + (4*a*Cot[c + d*x]*Csc[c + d*x]^2)/(693*d) + (a*Cot[c + d*x]*Csc[c + d*x]^4)/(231*d) - (113*a*Cot[c + d*x]*Csc[c + d*x]^6)/(693*d) + (23*a*Cot[c + d*x]*Csc[c + d*x]^8)/(99*d) - (a*Cot[c + d*x]*Csc[c + d*x]^10)/(11*d) + (3*a*Log[Cos[(c + d*x)/2]])/(256*d) - (3*a*Log[Sin[(c + d*x)/2]])/(256*d) - (3*a*Sec[(c + d*x)/2]^2)/(1024*d) + (a*Sec[(c + d*x)/2]^4)/(1024*d) + (3*a*Sec[(c + d*x)/2]^6)/(2048*d) - (3*a*Sec[(c + d*x)/2]^8)/(4096*d) + (a*Sec[(c + d*x)/2]^10)/(10240*d)","B",1
588,1,136,209,1.3872672,"\int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (55440 \sin (2 (c+d x))-162855 \sin (4 (c+d x))-27720 \sin (6 (c+d x))+24255 \sin (8 (c+d x))+5544 \sin (10 (c+d x))-1155 \sin (12 (c+d x))-554400 \cos (c+d x)-184800 \cos (3 (c+d x))+55440 \cos (5 (c+d x))+39600 \cos (7 (c+d x))-6160 \cos (9 (c+d x))-5040 \cos (11 (c+d x))+166320 c+471240 d x)}{28385280 d}","-\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,"(a^2*(166320*c + 471240*d*x - 554400*Cos[c + d*x] - 184800*Cos[3*(c + d*x)] + 55440*Cos[5*(c + d*x)] + 39600*Cos[7*(c + d*x)] - 6160*Cos[9*(c + d*x)] - 5040*Cos[11*(c + d*x)] + 55440*Sin[2*(c + d*x)] - 162855*Sin[4*(c + d*x)] - 27720*Sin[6*(c + d*x)] + 24255*Sin[8*(c + d*x)] + 5544*Sin[10*(c + d*x)] - 1155*Sin[12*(c + d*x)]))/(28385280*d)","A",1
589,1,126,183,0.9901142,"\int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (4620 \sin (2 (c+d x))-9240 \sin (4 (c+d x))-2310 \sin (6 (c+d x))+1155 \sin (8 (c+d x))+462 \sin (10 (c+d x))-39270 \cos (c+d x)-16170 \cos (3 (c+d x))+1155 \cos (5 (c+d x))+2805 \cos (7 (c+d x))+385 \cos (9 (c+d x))-105 \cos (11 (c+d x))+27720 c+27720 d x)}{1182720 d}","-\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,"(a^2*(27720*c + 27720*d*x - 39270*Cos[c + d*x] - 16170*Cos[3*(c + d*x)] + 1155*Cos[5*(c + d*x)] + 2805*Cos[7*(c + d*x)] + 385*Cos[9*(c + d*x)] - 105*Cos[11*(c + d*x)] + 4620*Sin[2*(c + d*x)] - 9240*Sin[4*(c + d*x)] - 2310*Sin[6*(c + d*x)] + 1155*Sin[8*(c + d*x)] + 462*Sin[10*(c + d*x)]))/(1182720*d)","A",1
590,1,106,165,0.6315011,"\int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (11340 \sin (2 (c+d x))-7560 \sin (4 (c+d x))-3990 \sin (6 (c+d x))-315 \sin (8 (c+d x))+126 \sin (10 (c+d x))-30240 \cos (c+d x)-13440 \cos (3 (c+d x))+2160 \cos (7 (c+d x))+560 \cos (9 (c+d x))+12600 c+32760 d x)}{645120 d}","\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,"(a^2*(12600*c + 32760*d*x - 30240*Cos[c + d*x] - 13440*Cos[3*(c + d*x)] + 2160*Cos[7*(c + d*x)] + 560*Cos[9*(c + d*x)] + 11340*Sin[2*(c + d*x)] - 7560*Sin[4*(c + d*x)] - 3990*Sin[6*(c + d*x)] - 315*Sin[8*(c + d*x)] + 126*Sin[10*(c + d*x)]))/(645120*d)","A",1
591,1,106,153,0.6286357,"\int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (1008 \sin (2 (c+d x))-504 \sin (4 (c+d x))-336 \sin (6 (c+d x))-63 \sin (8 (c+d x))-3276 \cos (c+d x)-1848 \cos (3 (c+d x))-504 \cos (5 (c+d x))-18 \cos (7 (c+d x))+14 \cos (9 (c+d x))+2520 c+2520 d x)}{32256 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,"(a^2*(2520*c + 2520*d*x - 3276*Cos[c + d*x] - 1848*Cos[3*(c + d*x)] - 504*Cos[5*(c + d*x)] - 18*Cos[7*(c + d*x)] + 14*Cos[9*(c + d*x)] + 1008*Sin[2*(c + d*x)] - 504*Sin[4*(c + d*x)] - 336*Sin[6*(c + d*x)] - 63*Sin[8*(c + d*x)]))/(32256*d)","A",1
592,1,112,161,0.4246056,"\int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*Cot[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(3150 \sin (2 (c+d x))+630 \sin (4 (c+d x))+70 \sin (6 (c+d x))+8715 \cos (c+d x)+665 \cos (3 (c+d x))-21 \cos (5 (c+d x))-15 \cos (7 (c+d x))+6720 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6720 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+4200 c+4200 d x\right)}{6720 d}","-\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,"(a^2*(4200*c + 4200*d*x + 8715*Cos[c + d*x] + 665*Cos[3*(c + d*x)] - 21*Cos[5*(c + d*x)] - 15*Cos[7*(c + d*x)] - 6720*Log[Cos[(c + d*x)/2]] + 6720*Log[Sin[(c + d*x)/2]] + 3150*Sin[2*(c + d*x)] + 630*Sin[4*(c + d*x)] + 70*Sin[6*(c + d*x)]))/(6720*d)","A",1
593,1,110,158,0.3123062,"\int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(-255 \sin (2 (c+d x))+15 \sin (4 (c+d x))+5 \sin (6 (c+d x))+2640 \cos (c+d x)+280 \cos (3 (c+d x))+24 \cos (5 (c+d x))-960 \cot (c+d x)+1920 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-1920 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-1500 c-1500 d x\right)}{960 d}","\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,"(a^2*(-1500*c - 1500*d*x + 2640*Cos[c + d*x] + 280*Cos[3*(c + d*x)] + 24*Cos[5*(c + d*x)] - 960*Cot[c + d*x] - 1920*Log[Cos[(c + d*x)/2]] + 1920*Log[Sin[(c + d*x)/2]] - 255*Sin[2*(c + d*x)] + 15*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)","A",1
594,1,174,140,5.5300967,"\int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{(a \sin (c+d x)+a)^2 \left(-300 (c+d x)-80 \sin (2 (c+d x))-5 \sin (4 (c+d x))-70 \cos (c+d x)+5 \cos (3 (c+d x))+\cos (5 (c+d x))+80 \tan \left(\frac{1}{2} (c+d x)\right)-80 \cot \left(\frac{1}{2} (c+d x)\right)-10 \csc ^2\left(\frac{1}{2} (c+d x)\right)+10 \sec ^2\left(\frac{1}{2} (c+d x)\right)-120 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+120 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{80 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^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,"((a + a*Sin[c + d*x])^2*(-300*(c + d*x) - 70*Cos[c + d*x] + 5*Cos[3*(c + d*x)] + Cos[5*(c + d*x)] - 80*Cot[(c + d*x)/2] - 10*Csc[(c + d*x)/2]^2 + 120*Log[Cos[(c + d*x)/2]] - 120*Log[Sin[(c + d*x)/2]] + 10*Sec[(c + d*x)/2]^2 - 80*Sin[2*(c + d*x)] - 5*Sin[4*(c + d*x)] + 80*Tan[(c + d*x)/2]))/(80*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
595,1,209,153,3.3683338,"\int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (\sin (c+d x)+1)^2 \left(60 (c+d x)-24 \sin (2 (c+d x))-3 \sin (4 (c+d x))-432 \cos (c+d x)-16 \cos (3 (c+d x))-64 \tan \left(\frac{1}{2} (c+d x)\right)+64 \cot \left(\frac{1}{2} (c+d x)\right)-24 \csc ^2\left(\frac{1}{2} (c+d x)\right)+24 \sec ^2\left(\frac{1}{2} (c+d x)\right)-480 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+480 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+32 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{96 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\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,"(a^2*(1 + Sin[c + d*x])^2*(60*(c + d*x) - 432*Cos[c + d*x] - 16*Cos[3*(c + d*x)] + 64*Cot[(c + d*x)/2] - 24*Csc[(c + d*x)/2]^2 + 480*Log[Cos[(c + d*x)/2]] - 480*Log[Sin[(c + d*x)/2]] + 24*Sec[(c + d*x)/2]^2 + 32*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 24*Sin[2*(c + d*x)] - 3*Sin[4*(c + d*x)] - 64*Tan[(c + d*x)/2]))/(96*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
596,1,227,153,1.294564,"\int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (\sin (c+d x)+1)^2 \left(960 (c+d x)+96 \sin (2 (c+d x))-240 \cos (c+d x)-16 \cos (3 (c+d x))-448 \tan \left(\frac{1}{2} (c+d x)\right)+448 \cot \left(\frac{1}{2} (c+d x)\right)-3 \csc ^4\left(\frac{1}{2} (c+d x)\right)+30 \csc ^2\left(\frac{1}{2} (c+d x)\right)+3 \sec ^4\left(\frac{1}{2} (c+d x)\right)-30 \sec ^2\left(\frac{1}{2} (c+d x)\right)-120 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+120 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-8 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+128 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{192 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\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,"(a^2*(1 + Sin[c + d*x])^2*(960*(c + d*x) - 240*Cos[c + d*x] - 16*Cos[3*(c + d*x)] + 448*Cot[(c + d*x)/2] + 30*Csc[(c + d*x)/2]^2 - 3*Csc[(c + d*x)/2]^4 + 120*Log[Cos[(c + d*x)/2]] - 120*Log[Sin[(c + d*x)/2]] - 30*Sec[(c + d*x)/2]^2 + 3*Sec[(c + d*x)/2]^4 + 128*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 8*Csc[(c + d*x)/2]^4*Sin[c + d*x] + 96*Sin[2*(c + d*x)] - 448*Tan[(c + d*x)/2]))/(192*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
597,1,264,139,1.410234,"\int \cot ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (\sin (c+d x)+1)^2 \left(240 (c+d x)+40 \sin (2 (c+d x))+320 \cos (c+d x)-64 \tan \left(\frac{1}{2} (c+d x)\right)+64 \cot \left(\frac{1}{2} (c+d x)\right)-5 \csc ^4\left(\frac{1}{2} (c+d x)\right)+90 \csc ^2\left(\frac{1}{2} (c+d x)\right)+5 \sec ^4\left(\frac{1}{2} (c+d x)\right)-90 \sec ^2\left(\frac{1}{2} (c+d x)\right)+600 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-600 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{1}{2} \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+\frac{7}{2} \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-56 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)\right)}{160 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","\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,"(a^2*(1 + Sin[c + d*x])^2*(240*(c + d*x) + 320*Cos[c + d*x] + 64*Cot[(c + d*x)/2] + 90*Csc[(c + d*x)/2]^2 - 5*Csc[(c + d*x)/2]^4 - 600*Log[Cos[(c + d*x)/2]] + 600*Log[Sin[(c + d*x)/2]] - 90*Sec[(c + d*x)/2]^2 + 5*Sec[(c + d*x)/2]^4 - 56*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + (7*Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 - (Csc[(c + d*x)/2]^6*Sin[c + d*x])/2 + 40*Sin[2*(c + d*x)] - 64*Tan[(c + d*x)/2] + Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(160*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
598,1,270,157,1.6107842,"\int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \sin (c+d x) (\sin (c+d x)+1)^2 \left(-1920 \cot (c+d x)+\csc ^6\left(\frac{1}{2} (c+d x)\right) (5 \csc (c+d x)+12)-2 \csc ^4\left(\frac{1}{2} (c+d x)\right) (15 \csc (c+d x)+82)+\csc ^2\left(\frac{1}{2} (c+d x)\right) (1472-210 \csc (c+d x))-2 (327 \cos (c+d x)+92 \cos (2 (c+d x))+241) \sec ^6\left(\frac{1}{2} (c+d x)\right)-320 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^7(c+d x)+480 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+840 \sin ^2\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+120 \csc (c+d x) \left(32 (c+d x)-25 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+25 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{1920 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","\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,"-1/1920*(a^2*(-1920*Cot[c + d*x] + Csc[(c + d*x)/2]^2*(1472 - 210*Csc[c + d*x]) + Csc[(c + d*x)/2]^6*(12 + 5*Csc[c + d*x]) - 2*Csc[(c + d*x)/2]^4*(82 + 15*Csc[c + d*x]) + 120*Csc[c + d*x]*(32*(c + d*x) + 25*Log[Cos[(c + d*x)/2]] - 25*Log[Sin[(c + d*x)/2]]) - 2*(241 + 327*Cos[c + d*x] + 92*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^6 + 840*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 + 480*Csc[c + d*x]^5*Sin[(c + d*x)/2]^4 - 320*Csc[c + d*x]^7*Sin[(c + d*x)/2]^6)*Sin[c + d*x]*(1 + Sin[c + d*x])^2)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
599,1,262,162,1.0861273,"\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(9344 \tan \left(\frac{1}{2} (c+d x)\right)-9344 \cot \left(\frac{1}{2} (c+d x)\right)-4620 \csc ^2\left(\frac{1}{2} (c+d x)\right)+70 \sec ^6\left(\frac{1}{2} (c+d x)\right)-840 \sec ^4\left(\frac{1}{2} (c+d x)\right)+4620 \sec ^2\left(\frac{1}{2} (c+d x)\right)-8400 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+8400 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{15}{2} \sin (c+d x) \csc ^8\left(\frac{1}{2} (c+d x)\right)+(33 \sin (c+d x)-70) \csc ^6\left(\frac{1}{2} (c+d x)\right)+(289 \sin (c+d x)+840) \csc ^4\left(\frac{1}{2} (c+d x)\right)-4624 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+15 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right)-66 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)-13440 c-13440 d x\right)}{13440 d}","-\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*(-13440*c - 13440*d*x - 9344*Cot[(c + d*x)/2] - 4620*Csc[(c + d*x)/2]^2 + 8400*Log[Cos[(c + d*x)/2]] - 8400*Log[Sin[(c + d*x)/2]] + 4620*Sec[(c + d*x)/2]^2 - 840*Sec[(c + d*x)/2]^4 + 70*Sec[(c + d*x)/2]^6 - 4624*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - (15*Csc[(c + d*x)/2]^8*Sin[c + d*x])/2 + Csc[(c + d*x)/2]^6*(-70 + 33*Sin[c + d*x]) + Csc[(c + d*x)/2]^4*(840 + 289*Sin[c + d*x]) + 9344*Tan[(c + d*x)/2] - 66*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] + 15*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2]))/(13440*d)","A",1
600,1,401,182,0.1099763,"\int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","a^2 \left(-\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{7 d}+\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{7 d}-\frac{\csc ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}+\frac{\csc ^6\left(\frac{1}{2} (c+d x)\right)}{512 d}+\frac{17 \csc ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}-\frac{83 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}+\frac{\sec ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}-\frac{\sec ^6\left(\frac{1}{2} (c+d x)\right)}{512 d}-\frac{17 \sec ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}+\frac{83 \sec ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}-\frac{45 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{128 d}+\frac{45 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{128 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^6\left(\frac{1}{2} (c+d x)\right)}{448 d}+\frac{5 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^4\left(\frac{1}{2} (c+d x)\right)}{224 d}-\frac{19 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{224 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right)}{448 d}-\frac{5 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)}{224 d}+\frac{19 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{224 d}\right)","-\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{a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 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,"a^2*(Cot[(c + d*x)/2]/(7*d) - (83*Csc[(c + d*x)/2]^2)/(512*d) - (19*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(224*d) + (17*Csc[(c + d*x)/2]^4)/(1024*d) + (5*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(224*d) + Csc[(c + d*x)/2]^6/(512*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^6)/(448*d) - Csc[(c + d*x)/2]^8/(2048*d) + (45*Log[Cos[(c + d*x)/2]])/(128*d) - (45*Log[Sin[(c + d*x)/2]])/(128*d) + (83*Sec[(c + d*x)/2]^2)/(512*d) - (17*Sec[(c + d*x)/2]^4)/(1024*d) - Sec[(c + d*x)/2]^6/(512*d) + Sec[(c + d*x)/2]^8/(2048*d) - Tan[(c + d*x)/2]/(7*d) + (19*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(224*d) - (5*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(224*d) + (Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2])/(448*d))","B",1
601,1,313,152,1.6277988,"\int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc ^9(c+d x) \left(36540 \sin (2 (c+d x))+20916 \sin (4 (c+d x))+16044 \sin (6 (c+d x))+630 \sin (8 (c+d x))+72576 \cos (c+d x)+37632 \cos (3 (c+d x))+6912 \cos (5 (c+d x))-1728 \cos (7 (c+d x))-704 \cos (9 (c+d x))+39690 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-26460 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+11340 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2835 \sin (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+315 \sin (9 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-39690 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+26460 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-11340 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2835 \sin (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-315 \sin (9 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1032192 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,"-1/1032192*(a^2*Csc[c + d*x]^9*(72576*Cos[c + d*x] + 37632*Cos[3*(c + d*x)] + 6912*Cos[5*(c + d*x)] - 1728*Cos[7*(c + d*x)] - 704*Cos[9*(c + d*x)] - 39690*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 39690*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 36540*Sin[2*(c + d*x)] + 26460*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 26460*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 20916*Sin[4*(c + d*x)] - 11340*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 11340*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 16044*Sin[6*(c + d*x)] + 2835*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] - 2835*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] + 630*Sin[8*(c + d*x)] - 315*Log[Cos[(c + d*x)/2]]*Sin[9*(c + d*x)] + 315*Log[Sin[(c + d*x)/2]]*Sin[9*(c + d*x)]))/d","B",1
602,1,353,228,1.2912783,"\int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \csc ^{10}(c+d x) \left(1075200 \sin (2 (c+d x))+1044480 \sin (4 (c+d x))+414720 \sin (6 (c+d x))+51200 \sin (8 (c+d x))-5120 \sin (10 (c+d x))+2732940 \cos (c+d x)+1151640 \cos (3 (c+d x))+388248 \cos (5 (c+d x))-135870 \cos (7 (c+d x))-8190 \cos (9 (c+d x))+515970 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+859950 \cos (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-491400 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+184275 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-40950 \cos (8 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+4095 \cos (10 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-515970 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-859950 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+491400 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-184275 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+40950 \cos (8 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4095 \cos (10 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{41287680 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,"-1/41287680*(a^2*Csc[c + d*x]^10*(2732940*Cos[c + d*x] + 1151640*Cos[3*(c + d*x)] + 388248*Cos[5*(c + d*x)] - 135870*Cos[7*(c + d*x)] - 8190*Cos[9*(c + d*x)] - 515970*Log[Cos[(c + d*x)/2]] + 859950*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 491400*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 184275*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 40950*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 4095*Cos[10*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 515970*Log[Sin[(c + d*x)/2]] - 859950*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 491400*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 184275*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 40950*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 4095*Cos[10*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 1075200*Sin[2*(c + d*x)] + 1044480*Sin[4*(c + d*x)] + 414720*Sin[6*(c + d*x)] + 51200*Sin[8*(c + d*x)] - 5120*Sin[10*(c + d*x)]))/d","A",1
603,1,187,194,3.0501812,"\int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (\sin (c+d x)+1)^2 \left(887040 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)-\cot (c+d x) \csc ^{10}(c+d x) (1073226 \sin (c+d x)+869484 \sin (3 (c+d x))+727188 \sin (5 (c+d x))+40425 \sin (7 (c+d x))-3465 \sin (9 (c+d x))+1798400 \cos (2 (c+d x))+440320 \cos (4 (c+d x))-81280 \cos (6 (c+d x))-38400 \cos (8 (c+d x))+3200 \cos (10 (c+d x))+1318400)\right)}{37847040 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\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,"(a^2*(1 + Sin[c + d*x])^2*(887040*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x]^10*(1318400 + 1798400*Cos[2*(c + d*x)] + 440320*Cos[4*(c + d*x)] - 81280*Cos[6*(c + d*x)] - 38400*Cos[8*(c + d*x)] + 3200*Cos[10*(c + d*x)] + 1073226*Sin[c + d*x] + 869484*Sin[3*(c + d*x)] + 727188*Sin[5*(c + d*x)] + 40425*Sin[7*(c + d*x)] - 3465*Sin[9*(c + d*x)])))/(37847040*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
604,1,197,270,4.7241024,"\int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (\sin (c+d x)+1)^2 \left(30159360 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)-\cot (c+d x) \csc ^{11}(c+d x) (29655040 \sin (c+d x)+51445760 \sin (3 (c+d x))+25600000 \sin (5 (c+d x))+3235840 \sin (7 (c+d x))-532480 \sin (9 (c+d x))+40960 \sin (11 (c+d x))+67499586 \cos (2 (c+d x))+25966248 \cos (4 (c+d x))-6944091 \cos (6 (c+d x))-746130 \cos (8 (c+d x))+58905 \cos (10 (c+d x))+65553642)\right)}{1816657920 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\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,"(a^2*(1 + Sin[c + d*x])^2*(30159360*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x]^11*(65553642 + 67499586*Cos[2*(c + d*x)] + 25966248*Cos[4*(c + d*x)] - 6944091*Cos[6*(c + d*x)] - 746130*Cos[8*(c + d*x)] + 58905*Cos[10*(c + d*x)] + 29655040*Sin[c + d*x] + 51445760*Sin[3*(c + d*x)] + 25600000*Sin[5*(c + d*x)] + 3235840*Sin[7*(c + d*x)] - 532480*Sin[9*(c + d*x)] + 40960*Sin[11*(c + d*x)])))/(1816657920*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
605,1,146,224,2.3042027,"\int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (80080 \sin (2 (c+d x))-385385 \sin (4 (c+d x))-40040 \sin (6 (c+d x))+65065 \sin (8 (c+d x))+8008 \sin (10 (c+d x))-5005 \sin (12 (c+d x))-1401400 \cos (c+d x)-450450 \cos (3 (c+d x))+150150 \cos (5 (c+d x))+94380 \cos (7 (c+d x))-20020 \cos (9 (c+d x))-11830 \cos (11 (c+d x))+770 \cos (13 (c+d x))+720720 c+1081080 d x)}{41000960 d}","\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,"(a^3*(720720*c + 1081080*d*x - 1401400*Cos[c + d*x] - 450450*Cos[3*(c + d*x)] + 150150*Cos[5*(c + d*x)] + 94380*Cos[7*(c + d*x)] - 20020*Cos[9*(c + d*x)] - 11830*Cos[11*(c + d*x)] + 770*Cos[13*(c + d*x)] + 80080*Sin[2*(c + d*x)] - 385385*Sin[4*(c + d*x)] - 40040*Sin[6*(c + d*x)] + 65065*Sin[8*(c + d*x)] + 8008*Sin[10*(c + d*x)] - 5005*Sin[12*(c + d*x)]))/(41000960*d)","A",1
606,1,136,209,1.6124754,"\int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (166320 \sin (2 (c+d x))-384615 \sin (4 (c+d x))-83160 \sin (6 (c+d x))+51975 \sin (8 (c+d x))+16632 \sin (10 (c+d x))-1155 \sin (12 (c+d x))-1496880 \cos (c+d x)-572880 \cos (3 (c+d x))+83160 \cos (5 (c+d x))+106920 \cos (7 (c+d x))+3080 \cos (9 (c+d x))-7560 \cos (11 (c+d x))+1247400 c+1136520 d x)}{28385280 d}","-\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,"(a^3*(1247400*c + 1136520*d*x - 1496880*Cos[c + d*x] - 572880*Cos[3*(c + d*x)] + 83160*Cos[5*(c + d*x)] + 106920*Cos[7*(c + d*x)] + 3080*Cos[9*(c + d*x)] - 7560*Cos[11*(c + d*x)] + 166320*Sin[2*(c + d*x)] - 384615*Sin[4*(c + d*x)] - 83160*Sin[6*(c + d*x)] + 51975*Sin[8*(c + d*x)] + 16632*Sin[10*(c + d*x)] - 1155*Sin[12*(c + d*x)]))/(28385280*d)","A",1
607,1,126,183,1.1393071,"\int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (152460 \sin (2 (c+d x))-138600 \sin (4 (c+d x))-57750 \sin (6 (c+d x))+3465 \sin (8 (c+d x))+4158 \sin (10 (c+d x))-568260 \cos (c+d x)-244860 \cos (3 (c+d x))+6930 \cos (5 (c+d x))+40590 \cos (7 (c+d x))+8470 \cos (9 (c+d x))-630 \cos (11 (c+d x))+415800 c+526680 d x)}{7096320 d}","-\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,"(a^3*(415800*c + 526680*d*x - 568260*Cos[c + d*x] - 244860*Cos[3*(c + d*x)] + 6930*Cos[5*(c + d*x)] + 40590*Cos[7*(c + d*x)] + 8470*Cos[9*(c + d*x)] - 630*Cos[11*(c + d*x)] + 152460*Sin[2*(c + d*x)] - 138600*Sin[4*(c + d*x)] - 57750*Sin[6*(c + d*x)] + 3465*Sin[8*(c + d*x)] + 4158*Sin[10*(c + d*x)]))/(7096320*d)","A",1
608,1,116,181,0.9173803,"\int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (10500 \sin (2 (c+d x))-5880 \sin (4 (c+d x))-3570 \sin (6 (c+d x))-525 \sin (8 (c+d x))+42 \sin (10 (c+d x))-31920 \cos (c+d x)-16800 \cos (3 (c+d x))-3360 \cos (5 (c+d x))+600 \cos (7 (c+d x))+280 \cos (9 (c+d x))+31500 c+27720 d x)}{215040 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,"(a^3*(31500*c + 27720*d*x - 31920*Cos[c + d*x] - 16800*Cos[3*(c + d*x)] - 3360*Cos[5*(c + d*x)] + 600*Cos[7*(c + d*x)] + 280*Cos[9*(c + d*x)] + 10500*Sin[2*(c + d*x)] - 5880*Sin[4*(c + d*x)] - 3570*Sin[6*(c + d*x)] - 525*Sin[8*(c + d*x)] + 42*Sin[10*(c + d*x)]))/(215040*d)","A",1
609,1,122,185,0.69925,"\int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^5*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(77280 \sin (2 (c+d x))+14280 \sin (4 (c+d x))+1120 \sin (6 (c+d x))-105 \sin (8 (c+d x))+122640 \cos (c+d x)+560 \cos (3 (c+d x))-3696 \cos (5 (c+d x))-720 \cos (7 (c+d x))+107520 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-107520 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+105000 c+105000 d x\right)}{107520 d}","-\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,"(a^3*(105000*c + 105000*d*x + 122640*Cos[c + d*x] + 560*Cos[3*(c + d*x)] - 3696*Cos[5*(c + d*x)] - 720*Cos[7*(c + d*x)] - 107520*Log[Cos[(c + d*x)/2]] + 107520*Log[Sin[(c + d*x)/2]] + 77280*Sin[2*(c + d*x)] + 14280*Sin[4*(c + d*x)] + 1120*Sin[6*(c + d*x)] - 105*Sin[8*(c + d*x)]))/(107520*d)","A",1
610,1,168,173,1.882029,"\int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{(a \sin (c+d x)+a)^3 \left(-2100 (c+d x)+455 \sin (2 (c+d x))+245 \sin (4 (c+d x))+35 \sin (6 (c+d x))+9065 \cos (c+d x)+875 \cos (3 (c+d x))+49 \cos (5 (c+d x))-5 \cos (7 (c+d x))+1120 \tan \left(\frac{1}{2} (c+d x)\right)-1120 \cot \left(\frac{1}{2} (c+d x)\right)+6720 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6720 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2240 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"((a + a*Sin[c + d*x])^3*(-2100*(c + d*x) + 9065*Cos[c + d*x] + 875*Cos[3*(c + d*x)] + 49*Cos[5*(c + d*x)] - 5*Cos[7*(c + d*x)] - 1120*Cot[(c + d*x)/2] - 6720*Log[Cos[(c + d*x)/2]] + 6720*Log[Sin[(c + d*x)/2]] + 455*Sin[2*(c + d*x)] + 245*Sin[4*(c + d*x)] + 35*Sin[6*(c + d*x)] + 1120*Tan[(c + d*x)/2]))/(2240*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
611,1,664,181,6.3735392,"\int \cos ^3(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{81 \sin (2 (c+d x)) (a \sin (c+d x)+a)^3}{64 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}-\frac{3 \sin (4 (c+d x)) (a \sin (c+d x)+a)^3}{64 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}+\frac{\sin (6 (c+d x)) (a \sin (c+d x)+a)^3}{192 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}-\frac{85 (c+d x) (a \sin (c+d x)+a)^3}{16 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}+\frac{15 \cos (c+d x) (a \sin (c+d x)+a)^3}{8 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}+\frac{17 \cos (3 (c+d x)) (a \sin (c+d x)+a)^3}{48 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}+\frac{3 \cos (5 (c+d x)) (a \sin (c+d x)+a)^3}{80 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}-\frac{(a \sin (c+d x)+a)^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}+\frac{(a \sin (c+d x)+a)^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}+\frac{3 \tan \left(\frac{1}{2} (c+d x)\right) (a \sin (c+d x)+a)^3}{2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}-\frac{3 \cot \left(\frac{1}{2} (c+d x)\right) (a \sin (c+d x)+a)^3}{2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right) (a \sin (c+d x)+a)^3}{8 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \sin (c+d x)+a)^3}{8 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\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*(c + d*x)*(a + a*Sin[c + d*x])^3)/(16*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (15*Cos[c + d*x]*(a + a*Sin[c + d*x])^3)/(8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (17*Cos[3*(c + d*x)]*(a + a*Sin[c + d*x])^3)/(48*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (3*Cos[5*(c + d*x)]*(a + a*Sin[c + d*x])^3)/(80*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (3*Cot[(c + d*x)/2]*(a + a*Sin[c + d*x])^3)/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (Csc[(c + d*x)/2]^2*(a + a*Sin[c + d*x])^3)/(8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (Log[Cos[(c + d*x)/2]]*(a + a*Sin[c + d*x])^3)/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (Log[Sin[(c + d*x)/2]]*(a + a*Sin[c + d*x])^3)/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (Sec[(c + d*x)/2]^2*(a + a*Sin[c + d*x])^3)/(8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (81*(a + a*Sin[c + d*x])^3*Sin[2*(c + d*x)])/(64*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (3*(a + a*Sin[c + d*x])^3*Sin[4*(c + d*x)])/(64*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + ((a + a*Sin[c + d*x])^3*Sin[6*(c + d*x)])/(192*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (3*(a + a*Sin[c + d*x])^3*Tan[(c + d*x)/2])/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","B",1
612,1,219,176,1.4308393,"\int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(-1500 (c+d x)-600 \sin (2 (c+d x))-45 \sin (4 (c+d x))-2580 \cos (c+d x)-50 \cos (3 (c+d x))+6 \cos (5 (c+d x))+160 \tan \left(\frac{1}{2} (c+d x)\right)-160 \cot \left(\frac{1}{2} (c+d x)\right)-180 \csc ^2\left(\frac{1}{2} (c+d x)\right)+180 \sec ^2\left(\frac{1}{2} (c+d x)\right)-3120 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+3120 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-10 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+160 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{480 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\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,"(a^3*(1 + Sin[c + d*x])^3*(-1500*(c + d*x) - 2580*Cos[c + d*x] - 50*Cos[3*(c + d*x)] + 6*Cos[5*(c + d*x)] - 160*Cot[(c + d*x)/2] - 180*Csc[(c + d*x)/2]^2 + 3120*Log[Cos[(c + d*x)/2]] - 3120*Log[Sin[(c + d*x)/2]] + 180*Sec[(c + d*x)/2]^2 + 160*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 10*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 600*Sin[2*(c + d*x)] - 45*Sin[4*(c + d*x)] + 160*Tan[(c + d*x)/2]))/(480*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
613,1,235,178,0.8556963,"\int \cos (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(360 (c+d x)+16 \sin (2 (c+d x))-2 \sin (4 (c+d x))-368 \cos (c+d x)-16 \cos (3 (c+d x))-192 \tan \left(\frac{1}{2} (c+d x)\right)+192 \cot \left(\frac{1}{2} (c+d x)\right)-\csc ^4\left(\frac{1}{2} (c+d x)\right)-6 \csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^4\left(\frac{1}{2} (c+d x)\right)+6 \sec ^2\left(\frac{1}{2} (c+d x)\right)-360 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+360 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+64 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{64 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"(a^3*(1 + Sin[c + d*x])^3*(360*(c + d*x) - 368*Cos[c + d*x] - 16*Cos[3*(c + d*x)] + 192*Cot[(c + d*x)/2] - 6*Csc[(c + d*x)/2]^2 - Csc[(c + d*x)/2]^4 + 360*Log[Cos[(c + d*x)/2]] - 360*Log[Sin[(c + d*x)/2]] + 6*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^4 + 64*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 4*Csc[(c + d*x)/2]^4*Sin[c + d*x] + 16*Sin[2*(c + d*x)] - 2*Sin[4*(c + d*x)] - 192*Tan[(c + d*x)/2]))/(64*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
614,1,271,175,1.7077993,"\int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(6240 (c+d x)+720 \sin (2 (c+d x))+720 \cos (c+d x)-80 \cos (3 (c+d x))-2624 \tan \left(\frac{1}{2} (c+d x)\right)+2624 \cot \left(\frac{1}{2} (c+d x)\right)-45 \csc ^4\left(\frac{1}{2} (c+d x)\right)+690 \csc ^2\left(\frac{1}{2} (c+d x)\right)+45 \sec ^4\left(\frac{1}{2} (c+d x)\right)-690 \sec ^2\left(\frac{1}{2} (c+d x)\right)+3000 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3000 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-3 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)-19 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+304 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+6 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)\right)}{960 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"(a^3*(1 + Sin[c + d*x])^3*(6240*(c + d*x) + 720*Cos[c + d*x] - 80*Cos[3*(c + d*x)] + 2624*Cot[(c + d*x)/2] + 690*Csc[(c + d*x)/2]^2 - 45*Csc[(c + d*x)/2]^4 - 3000*Log[Cos[(c + d*x)/2]] + 3000*Log[Sin[(c + d*x)/2]] - 690*Sec[(c + d*x)/2]^2 + 45*Sec[(c + d*x)/2]^4 + 304*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 19*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 3*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 720*Sin[2*(c + d*x)] - 2624*Tan[(c + d*x)/2] + 6*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(960*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
615,1,289,182,1.8619124,"\int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(-960 (c+d x)+480 \sin (2 (c+d x))+5760 \cos (c+d x)+2176 \tan \left(\frac{1}{2} (c+d x)\right)-2176 \cot \left(\frac{1}{2} (c+d x)\right)-5 \csc ^6\left(\frac{1}{2} (c+d x)\right)-30 \csc ^4\left(\frac{1}{2} (c+d x)\right)+1290 \csc ^2\left(\frac{1}{2} (c+d x)\right)+5 \sec ^6\left(\frac{1}{2} (c+d x)\right)+30 \sec ^4\left(\frac{1}{2} (c+d x)\right)-1290 \sec ^2\left(\frac{1}{2} (c+d x)\right)+10200 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-10200 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-18 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+206 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-3296 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+36 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)\right)}{1920 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\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*(1 + Sin[c + d*x])^3*(-960*(c + d*x) + 5760*Cos[c + d*x] - 2176*Cot[(c + d*x)/2] + 1290*Csc[(c + d*x)/2]^2 - 30*Csc[(c + d*x)/2]^4 - 5*Csc[(c + d*x)/2]^6 - 10200*Log[Cos[(c + d*x)/2]] + 10200*Log[Sin[(c + d*x)/2]] - 1290*Sec[(c + d*x)/2]^2 + 30*Sec[(c + d*x)/2]^4 + 5*Sec[(c + d*x)/2]^6 - 3296*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 206*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 18*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 480*Sin[2*(c + d*x)] + 2176*Tan[(c + d*x)/2] + 36*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(1920*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
616,1,292,172,1.4013473,"\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(4480 \cos (c+d x)+9984 \tan \left(\frac{1}{2} (c+d x)\right)-9984 \cot \left(\frac{1}{2} (c+d x)\right)-35 \csc ^6\left(\frac{1}{2} (c+d x)\right)+350 \csc ^4\left(\frac{1}{2} (c+d x)\right)-1050 \csc ^2\left(\frac{1}{2} (c+d x)\right)+35 \sec ^6\left(\frac{1}{2} (c+d x)\right)-350 \sec ^4\left(\frac{1}{2} (c+d x)\right)+1050 \sec ^2\left(\frac{1}{2} (c+d x)\right)+4200 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4200 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{5}{2} \sin (c+d x) \csc ^8\left(\frac{1}{2} (c+d x)\right)-17 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+479 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-7664 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+5 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right)+34 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)-13440 c-13440 d x\right)}{4480 d}","\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,"(a^3*(-13440*c - 13440*d*x + 4480*Cos[c + d*x] - 9984*Cot[(c + d*x)/2] - 1050*Csc[(c + d*x)/2]^2 + 350*Csc[(c + d*x)/2]^4 - 35*Csc[(c + d*x)/2]^6 - 4200*Log[Cos[(c + d*x)/2]] + 4200*Log[Sin[(c + d*x)/2]] + 1050*Sec[(c + d*x)/2]^2 - 350*Sec[(c + d*x)/2]^4 + 35*Sec[(c + d*x)/2]^6 - 7664*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 479*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 17*Csc[(c + d*x)/2]^6*Sin[c + d*x] - (5*Csc[(c + d*x)/2]^8*Sin[c + d*x])/2 + 9984*Tan[(c + d*x)/2] + 34*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] + 5*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2]))/(4480*d)","A",1
617,1,279,238,1.211851,"\int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(118784 \tan \left(\frac{1}{2} (c+d x)\right)-118784 \cot \left(\frac{1}{2} (c+d x)\right)-108780 \csc ^2\left(\frac{1}{2} (c+d x)\right)+105 \sec ^8\left(\frac{1}{2} (c+d x)\right)+700 \sec ^6\left(\frac{1}{2} (c+d x)\right)-17010 \sec ^4\left(\frac{1}{2} (c+d x)\right)+108780 \sec ^2\left(\frac{1}{2} (c+d x)\right)-210000 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+210000 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-15 (24 \sin (c+d x)+7) \csc ^8\left(\frac{1}{2} (c+d x)\right)+4 (732 \sin (c+d x)-175) \csc ^6\left(\frac{1}{2} (c+d x)\right)+(17010-4496 \sin (c+d x)) \csc ^4\left(\frac{1}{2} (c+d x)\right)+71936 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+720 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right)-5856 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)-215040 c-215040 d x\right)}{215040 d}","-\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{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 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*(-215040*c - 215040*d*x - 118784*Cot[(c + d*x)/2] - 108780*Csc[(c + d*x)/2]^2 + 210000*Log[Cos[(c + d*x)/2]] - 210000*Log[Sin[(c + d*x)/2]] + 108780*Sec[(c + d*x)/2]^2 - 17010*Sec[(c + d*x)/2]^4 + 700*Sec[(c + d*x)/2]^6 + 105*Sec[(c + d*x)/2]^8 + 71936*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + Csc[(c + d*x)/2]^4*(17010 - 4496*Sin[c + d*x]) - 15*Csc[(c + d*x)/2]^8*(7 + 24*Sin[c + d*x]) + 4*Csc[(c + d*x)/2]^6*(-175 + 732*Sin[c + d*x]) + 118784*Tan[(c + d*x)/2] - 5856*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] + 720*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2]))/(215040*d)","A",1
618,1,459,200,0.1378071,"\int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","a^3 \left(-\frac{29 \tan \left(\frac{1}{2} (c+d x)\right)}{126 d}+\frac{29 \cot \left(\frac{1}{2} (c+d x)\right)}{126 d}-\frac{3 \csc ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}+\frac{17 \csc ^6\left(\frac{1}{2} (c+d x)\right)}{1536 d}-\frac{13 \csc ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}-\frac{73 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}+\frac{3 \sec ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}-\frac{17 \sec ^6\left(\frac{1}{2} (c+d x)\right)}{1536 d}+\frac{13 \sec ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}+\frac{73 \sec ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}-\frac{55 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{128 d}+\frac{55 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{128 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^8\left(\frac{1}{2} (c+d x)\right)}{4608 d}-\frac{53 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^6\left(\frac{1}{2} (c+d x)\right)}{32256 d}+\frac{319 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^4\left(\frac{1}{2} (c+d x)\right)}{10752 d}-\frac{4163 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32256 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^8\left(\frac{1}{2} (c+d x)\right)}{4608 d}+\frac{53 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right)}{32256 d}-\frac{319 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)}{10752 d}+\frac{4163 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32256 d}\right)","-\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{a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 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,"a^3*((29*Cot[(c + d*x)/2])/(126*d) - (73*Csc[(c + d*x)/2]^2)/(512*d) - (4163*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(32256*d) - (13*Csc[(c + d*x)/2]^4)/(1024*d) + (319*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(10752*d) + (17*Csc[(c + d*x)/2]^6)/(1536*d) - (53*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^6)/(32256*d) - (3*Csc[(c + d*x)/2]^8)/(2048*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^8)/(4608*d) + (55*Log[Cos[(c + d*x)/2]])/(128*d) - (55*Log[Sin[(c + d*x)/2]])/(128*d) + (73*Sec[(c + d*x)/2]^2)/(512*d) + (13*Sec[(c + d*x)/2]^4)/(1024*d) - (17*Sec[(c + d*x)/2]^6)/(1536*d) + (3*Sec[(c + d*x)/2]^8)/(2048*d) - (29*Tan[(c + d*x)/2])/(126*d) + (4163*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(32256*d) - (319*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(10752*d) + (53*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2])/(32256*d) + (Sec[(c + d*x)/2]^8*Tan[(c + d*x)/2])/(4608*d))","B",1
619,1,365,228,2.0644782,"\int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(-51200 \tan \left(\frac{1}{2} (c+d x)\right)+51200 \cot \left(\frac{1}{2} (c+d x)\right)+13860 \csc ^2\left(\frac{1}{2} (c+d x)\right)+42 \sec ^{10}\left(\frac{1}{2} (c+d x)\right)+315 \sec ^8\left(\frac{1}{2} (c+d x)\right)-5250 \sec ^6\left(\frac{1}{2} (c+d x)\right)+19320 \sec ^4\left(\frac{1}{2} (c+d x)\right)-13860 \sec ^2\left(\frac{1}{2} (c+d x)\right)-55440 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+55440 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-14 (10 \sin (c+d x)+3) \csc ^{10}\left(\frac{1}{2} (c+d x)\right)+5 (172 \sin (c+d x)-63) \csc ^8\left(\frac{1}{2} (c+d x)\right)+(5250-60 \sin (c+d x)) \csc ^6\left(\frac{1}{2} (c+d x)\right)+3840 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)-20 (515 \sin (c+d x)+966) \csc ^4\left(\frac{1}{2} (c+d x)\right)+164800 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+280 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^8\left(\frac{1}{2} (c+d x)\right)-1720 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right)\right)}{430080 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"(a^3*(1 + Sin[c + d*x])^3*(51200*Cot[(c + d*x)/2] + 13860*Csc[(c + d*x)/2]^2 + 55440*Log[Cos[(c + d*x)/2]] - 55440*Log[Sin[(c + d*x)/2]] - 13860*Sec[(c + d*x)/2]^2 + 19320*Sec[(c + d*x)/2]^4 - 5250*Sec[(c + d*x)/2]^6 + 315*Sec[(c + d*x)/2]^8 + 42*Sec[(c + d*x)/2]^10 + 164800*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 3840*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 + Csc[(c + d*x)/2]^6*(5250 - 60*Sin[c + d*x]) - 14*Csc[(c + d*x)/2]^10*(3 + 10*Sin[c + d*x]) + 5*Csc[(c + d*x)/2]^8*(-63 + 172*Sin[c + d*x]) - 20*Csc[(c + d*x)/2]^4*(966 + 515*Sin[c + d*x]) - 51200*Tan[(c + d*x)/2] - 1720*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2] + 280*Sec[(c + d*x)/2]^8*Tan[(c + d*x)/2]))/(430080*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
620,1,187,246,3.5783954,"\int \cot ^6(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(16853760 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)-\cot (c+d x) \csc ^{10}(c+d x) (14477694 \sin (c+d x)+5875716 \sin (3 (c+d x))+7902972 \sin (5 (c+d x))-414645 \sin (7 (c+d x))-65835 \sin (9 (c+d x))+12423680 \cos (2 (c+d x))+839680 \cos (4 (c+d x))-2149120 \cos (6 (c+d x))-568320 \cos (8 (c+d x))+47360 \cos (10 (c+d x))+10050560)\right)}{227082240 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"(a^3*(1 + Sin[c + d*x])^3*(16853760*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x]^10*(10050560 + 12423680*Cos[2*(c + d*x)] + 839680*Cos[4*(c + d*x)] - 2149120*Cos[6*(c + d*x)] - 568320*Cos[8*(c + d*x)] + 47360*Cos[10*(c + d*x)] + 14477694*Sin[c + d*x] + 5875716*Sin[3*(c + d*x)] + 7902972*Sin[5*(c + d*x)] - 414645*Sin[7*(c + d*x)] - 65835*Sin[9*(c + d*x)])))/(227082240*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
621,1,197,270,4.7151606,"\int \cot ^6(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(72737280 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)-\cot (c+d x) \csc ^{11}(c+d x) (49776640 \sin (c+d x)+84039680 \sin (3 (c+d x))+38118400 \sin (5 (c+d x))+2206720 \sin (7 (c+d x))-1530880 \sin (9 (c+d x))+117760 \sin (11 (c+d x))+62609778 \cos (2 (c+d x))+22551144 \cos (4 (c+d x))-23426403 \cos (6 (c+d x))-1799490 \cos (8 (c+d x))+142065 \cos (10 (c+d x))+91311066)\right)}{1816657920 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"(a^3*(1 + Sin[c + d*x])^3*(72737280*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x]^11*(91311066 + 62609778*Cos[2*(c + d*x)] + 22551144*Cos[4*(c + d*x)] - 23426403*Cos[6*(c + d*x)] - 1799490*Cos[8*(c + d*x)] + 142065*Cos[10*(c + d*x)] + 49776640*Sin[c + d*x] + 84039680*Sin[3*(c + d*x)] + 38118400*Sin[5*(c + d*x)] + 2206720*Sin[7*(c + d*x)] - 1530880*Sin[9*(c + d*x)] + 117760*Sin[11*(c + d*x)])))/(1816657920*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
622,1,283,286,6.5041754,"\int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^8*(a + a*Sin[c + d*x])^3,x]","\frac{27 (a \sin (c+d x)+a)^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{1024 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}-\frac{27 (a \sin (c+d x)+a)^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{1024 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}+\frac{\cot (c+d x) \csc ^{12}(c+d x) (a \sin (c+d x)+a)^3 (-194159966 \sin (c+d x)-182107926 \sin (3 (c+d x))-123736613 \sin (5 (c+d x))+4571567 \sin (7 (c+d x))+1846845 \sin (9 (c+d x))-135135 \sin (11 (c+d x))-243712000 \cos (2 (c+d x))-11079680 \cos (4 (c+d x))+43294720 \cos (6 (c+d x))+9420800 \cos (8 (c+d x))-1433600 \cos (10 (c+d x))+102400 \cos (12 (c+d x))-200294400)}{5248122880 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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*Log[Cos[(c + d*x)/2]]*(a + a*Sin[c + d*x])^3)/(1024*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) - (27*Log[Sin[(c + d*x)/2]]*(a + a*Sin[c + d*x])^3)/(1024*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (Cot[c + d*x]*Csc[c + d*x]^12*(a + a*Sin[c + d*x])^3*(-200294400 - 243712000*Cos[2*(c + d*x)] - 11079680*Cos[4*(c + d*x)] + 43294720*Cos[6*(c + d*x)] + 9420800*Cos[8*(c + d*x)] - 1433600*Cos[10*(c + d*x)] + 102400*Cos[12*(c + d*x)] - 194159966*Sin[c + d*x] - 182107926*Sin[3*(c + d*x)] - 123736613*Sin[5*(c + d*x)] + 4571567*Sin[7*(c + d*x)] + 1846845*Sin[9*(c + d*x)] - 135135*Sin[11*(c + d*x)]))/(5248122880*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
623,1,229,178,1.660189,"\int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4,x]","\frac{a^4 (\sin (c+d x)+1)^4 \left(-8100 (c+d x)-2415 \sin (2 (c+d x))-135 \sin (4 (c+d x))+5 \sin (6 (c+d x))-3360 \cos (c+d x)+240 \cos (3 (c+d x))+48 \cos (5 (c+d x))+1760 \tan \left(\frac{1}{2} (c+d x)\right)-1760 \cot \left(\frac{1}{2} (c+d x)\right)-480 \csc ^2\left(\frac{1}{2} (c+d x)\right)+480 \sec ^2\left(\frac{1}{2} (c+d x)\right)-5760 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+5760 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-20 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+320 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)\right)}{960 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8}","\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,"(a^4*(1 + Sin[c + d*x])^4*(-8100*(c + d*x) - 3360*Cos[c + d*x] + 240*Cos[3*(c + d*x)] + 48*Cos[5*(c + d*x)] - 1760*Cot[(c + d*x)/2] - 480*Csc[(c + d*x)/2]^2 + 5760*Log[Cos[(c + d*x)/2]] - 5760*Log[Sin[(c + d*x)/2]] + 480*Sec[(c + d*x)/2]^2 + 320*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 20*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 2415*Sin[2*(c + d*x)] - 135*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)] + 1760*Tan[(c + d*x)/2]))/(960*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)","A",1
624,1,429,159,8.4058347,"\int \frac{\cos ^6(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{15120 d x \sin \left(\frac{c}{2}\right)-7560 \sin \left(\frac{c}{2}+d x\right)+7560 \sin \left(\frac{3 c}{2}+d x\right)-1680 \sin \left(\frac{5 c}{2}+3 d x\right)+1680 \sin \left(\frac{7 c}{2}+3 d x\right)-2520 \sin \left(\frac{7 c}{2}+4 d x\right)-2520 \sin \left(\frac{9 c}{2}+4 d x\right)+1008 \sin \left(\frac{9 c}{2}+5 d x\right)-1008 \sin \left(\frac{11 c}{2}+5 d x\right)+180 \sin \left(\frac{13 c}{2}+7 d x\right)-180 \sin \left(\frac{15 c}{2}+7 d x\right)+315 \sin \left(\frac{15 c}{2}+8 d x\right)+315 \sin \left(\frac{17 c}{2}+8 d x\right)-140 \sin \left(\frac{17 c}{2}+9 d x\right)+140 \sin \left(\frac{19 c}{2}+9 d x\right)+2520 \cos \left(\frac{c}{2}\right) (5 c+6 d x)+7560 \cos \left(\frac{c}{2}+d x\right)+7560 \cos \left(\frac{3 c}{2}+d x\right)+1680 \cos \left(\frac{5 c}{2}+3 d x\right)+1680 \cos \left(\frac{7 c}{2}+3 d x\right)-2520 \cos \left(\frac{7 c}{2}+4 d x\right)+2520 \cos \left(\frac{9 c}{2}+4 d x\right)-1008 \cos \left(\frac{9 c}{2}+5 d x\right)-1008 \cos \left(\frac{11 c}{2}+5 d x\right)-180 \cos \left(\frac{13 c}{2}+7 d x\right)-180 \cos \left(\frac{15 c}{2}+7 d x\right)+315 \cos \left(\frac{15 c}{2}+8 d x\right)-315 \cos \left(\frac{17 c}{2}+8 d x\right)+140 \cos \left(\frac{17 c}{2}+9 d x\right)+140 \cos \left(\frac{19 c}{2}+9 d x\right)+12600 c \sin \left(\frac{c}{2}\right)+12600 \sin \left(\frac{c}{2}\right)}{645120 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","\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,"(2520*(5*c + 6*d*x)*Cos[c/2] + 7560*Cos[c/2 + d*x] + 7560*Cos[(3*c)/2 + d*x] + 1680*Cos[(5*c)/2 + 3*d*x] + 1680*Cos[(7*c)/2 + 3*d*x] - 2520*Cos[(7*c)/2 + 4*d*x] + 2520*Cos[(9*c)/2 + 4*d*x] - 1008*Cos[(9*c)/2 + 5*d*x] - 1008*Cos[(11*c)/2 + 5*d*x] - 180*Cos[(13*c)/2 + 7*d*x] - 180*Cos[(15*c)/2 + 7*d*x] + 315*Cos[(15*c)/2 + 8*d*x] - 315*Cos[(17*c)/2 + 8*d*x] + 140*Cos[(17*c)/2 + 9*d*x] + 140*Cos[(19*c)/2 + 9*d*x] + 12600*Sin[c/2] + 12600*c*Sin[c/2] + 15120*d*x*Sin[c/2] - 7560*Sin[c/2 + d*x] + 7560*Sin[(3*c)/2 + d*x] - 1680*Sin[(5*c)/2 + 3*d*x] + 1680*Sin[(7*c)/2 + 3*d*x] - 2520*Sin[(7*c)/2 + 4*d*x] - 2520*Sin[(9*c)/2 + 4*d*x] + 1008*Sin[(9*c)/2 + 5*d*x] - 1008*Sin[(11*c)/2 + 5*d*x] + 180*Sin[(13*c)/2 + 7*d*x] - 180*Sin[(15*c)/2 + 7*d*x] + 315*Sin[(15*c)/2 + 8*d*x] + 315*Sin[(17*c)/2 + 8*d*x] - 140*Sin[(17*c)/2 + 9*d*x] + 140*Sin[(19*c)/2 + 9*d*x])/(645120*a*d*(Cos[c/2] + Sin[c/2]))","B",1
625,1,375,141,8.7284761,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{-1680 d x \sin \left(\frac{c}{2}\right)+1680 \sin \left(\frac{c}{2}+d x\right)-1680 \sin \left(\frac{3 c}{2}+d x\right)+560 \sin \left(\frac{5 c}{2}+3 d x\right)-560 \sin \left(\frac{7 c}{2}+3 d x\right)+280 \sin \left(\frac{7 c}{2}+4 d x\right)+280 \sin \left(\frac{9 c}{2}+4 d x\right)-112 \sin \left(\frac{9 c}{2}+5 d x\right)+112 \sin \left(\frac{11 c}{2}+5 d x\right)-80 \sin \left(\frac{13 c}{2}+7 d x\right)+80 \sin \left(\frac{15 c}{2}+7 d x\right)-35 \sin \left(\frac{15 c}{2}+8 d x\right)-35 \sin \left(\frac{17 c}{2}+8 d x\right)+1680 \cos \left(\frac{c}{2}\right) (c-d x)-1680 \cos \left(\frac{c}{2}+d x\right)-1680 \cos \left(\frac{3 c}{2}+d x\right)-560 \cos \left(\frac{5 c}{2}+3 d x\right)-560 \cos \left(\frac{7 c}{2}+3 d x\right)+280 \cos \left(\frac{7 c}{2}+4 d x\right)-280 \cos \left(\frac{9 c}{2}+4 d x\right)+112 \cos \left(\frac{9 c}{2}+5 d x\right)+112 \cos \left(\frac{11 c}{2}+5 d x\right)+80 \cos \left(\frac{13 c}{2}+7 d x\right)+80 \cos \left(\frac{15 c}{2}+7 d x\right)-35 \cos \left(\frac{15 c}{2}+8 d x\right)+35 \cos \left(\frac{17 c}{2}+8 d x\right)+1680 c \sin \left(\frac{c}{2}\right)-3360 \sin \left(\frac{c}{2}\right)}{71680 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","\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,"(1680*(c - d*x)*Cos[c/2] - 1680*Cos[c/2 + d*x] - 1680*Cos[(3*c)/2 + d*x] - 560*Cos[(5*c)/2 + 3*d*x] - 560*Cos[(7*c)/2 + 3*d*x] + 280*Cos[(7*c)/2 + 4*d*x] - 280*Cos[(9*c)/2 + 4*d*x] + 112*Cos[(9*c)/2 + 5*d*x] + 112*Cos[(11*c)/2 + 5*d*x] + 80*Cos[(13*c)/2 + 7*d*x] + 80*Cos[(15*c)/2 + 7*d*x] - 35*Cos[(15*c)/2 + 8*d*x] + 35*Cos[(17*c)/2 + 8*d*x] - 3360*Sin[c/2] + 1680*c*Sin[c/2] - 1680*d*x*Sin[c/2] + 1680*Sin[c/2 + d*x] - 1680*Sin[(3*c)/2 + d*x] + 560*Sin[(5*c)/2 + 3*d*x] - 560*Sin[(7*c)/2 + 3*d*x] + 280*Sin[(7*c)/2 + 4*d*x] + 280*Sin[(9*c)/2 + 4*d*x] - 112*Sin[(9*c)/2 + 5*d*x] + 112*Sin[(11*c)/2 + 5*d*x] - 80*Sin[(13*c)/2 + 7*d*x] + 80*Sin[(15*c)/2 + 7*d*x] - 35*Sin[(15*c)/2 + 8*d*x] - 35*Sin[(17*c)/2 + 8*d*x])/(71680*a*d*(Cos[c/2] + Sin[c/2]))","B",1
626,1,715,115,11.3890132,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{5 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d (a \sin (c+d x)+a)}-\frac{-\frac{50 \sin (c) \sin (d x)}{d}+\frac{10 \sin (3 c) \sin (3 d x)}{d}-\frac{2 \sin (5 c) \sin (5 d x)}{d}+\frac{50 \cos (c) \cos (d x)}{d}-\frac{10 \cos (3 c) \cos (3 d x)}{d}+\frac{2 \cos (5 c) \cos (5 d x)}{d}-\frac{20 \sin (2 c) \cos (2 d x)}{d}+\frac{5 \sin (4 c) \cos (4 d x)}{d}-\frac{20 \cos (2 c) \sin (2 d x)}{d}+\frac{5 \cos (4 c) \sin (4 d x)}{d}-\frac{10 \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+30 x}{160 a}+\frac{-\frac{\sin (c) \sin (d x)}{d}+\frac{\cos (c) \cos (d x)}{d}-\frac{\sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+x}{16 a}-\frac{\frac{9 \sin (c) \sin (d x)}{d}-\frac{\sin (3 c) \sin (3 d x)}{d}-\frac{9 \cos (c) \cos (d x)}{d}+\frac{\cos (3 c) \cos (3 d x)}{d}+\frac{3 \sin (2 c) \cos (2 d x)}{d}+\frac{3 \cos (2 c) \sin (2 d x)}{d}+\frac{3 \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-6 x}{48 a}-\frac{\frac{735 \sin (c) \sin (d x)}{d}-\frac{175 \sin (3 c) \sin (3 d x)}{d}+\frac{63 \sin (5 c) \sin (5 d x)}{d}-\frac{15 \sin (7 c) \sin (7 d x)}{d}-\frac{735 \cos (c) \cos (d x)}{d}+\frac{175 \cos (3 c) \cos (3 d x)}{d}-\frac{63 \cos (5 c) \cos (5 d x)}{d}+\frac{15 \cos (7 c) \cos (7 d x)}{d}+\frac{315 \sin (2 c) \cos (2 d x)}{d}-\frac{105 \sin (4 c) \cos (4 d x)}{d}+\frac{35 \sin (6 c) \cos (6 d x)}{d}+\frac{315 \cos (2 c) \sin (2 d x)}{d}-\frac{105 \cos (4 c) \sin (4 d x)}{d}+\frac{35 \cos (6 c) \sin (6 d x)}{d}+\frac{105 \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-420 x}{6720 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,"-1/160*(30*x + (50*Cos[c]*Cos[d*x])/d - (10*Cos[3*c]*Cos[3*d*x])/d + (2*Cos[5*c]*Cos[5*d*x])/d - (20*Cos[2*d*x]*Sin[2*c])/d + (5*Cos[4*d*x]*Sin[4*c])/d - (50*Sin[c]*Sin[d*x])/d - (20*Cos[2*c]*Sin[2*d*x])/d + (10*Sin[3*c]*Sin[3*d*x])/d + (5*Cos[4*c]*Sin[4*d*x])/d - (2*Sin[5*c]*Sin[5*d*x])/d - (10*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/a + (x + (Cos[c]*Cos[d*x])/d - (Sin[c]*Sin[d*x])/d - Sin[(d*x)/2]/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(16*a) - (-6*x - (9*Cos[c]*Cos[d*x])/d + (Cos[3*c]*Cos[3*d*x])/d + (3*Cos[2*d*x]*Sin[2*c])/d + (9*Sin[c]*Sin[d*x])/d + (3*Cos[2*c]*Sin[2*d*x])/d - (Sin[3*c]*Sin[3*d*x])/d + (3*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(48*a) - (-420*x - (735*Cos[c]*Cos[d*x])/d + (175*Cos[3*c]*Cos[3*d*x])/d - (63*Cos[5*c]*Cos[5*d*x])/d + (15*Cos[7*c]*Cos[7*d*x])/d + (315*Cos[2*d*x]*Sin[2*c])/d - (105*Cos[4*d*x]*Sin[4*c])/d + (35*Cos[6*d*x]*Sin[6*c])/d + (735*Sin[c]*Sin[d*x])/d + (315*Cos[2*c]*Sin[2*d*x])/d - (175*Sin[3*c]*Sin[3*d*x])/d - (105*Cos[4*c]*Sin[4*d*x])/d + (63*Sin[5*c]*Sin[5*d*x])/d + (35*Cos[6*c]*Sin[6*d*x])/d - (15*Sin[7*c]*Sin[7*d*x])/d + (105*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(6720*a) + (5*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(64*d*(a + a*Sin[c + d*x]))","B",1
627,1,377,97,5.0784215,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{120 d x \sin \left(\frac{c}{2}\right)-120 \sin \left(\frac{c}{2}+d x\right)+120 \sin \left(\frac{3 c}{2}+d x\right)+15 \sin \left(\frac{3 c}{2}+2 d x\right)+15 \sin \left(\frac{5 c}{2}+2 d x\right)-60 \sin \left(\frac{5 c}{2}+3 d x\right)+60 \sin \left(\frac{7 c}{2}+3 d x\right)-15 \sin \left(\frac{7 c}{2}+4 d x\right)-15 \sin \left(\frac{9 c}{2}+4 d x\right)-12 \sin \left(\frac{9 c}{2}+5 d x\right)+12 \sin \left(\frac{11 c}{2}+5 d x\right)-5 \sin \left(\frac{11 c}{2}+6 d x\right)-5 \sin \left(\frac{13 c}{2}+6 d x\right)-30 \cos \left(\frac{c}{2}\right) (5 c-4 d x)+120 \cos \left(\frac{c}{2}+d x\right)+120 \cos \left(\frac{3 c}{2}+d x\right)+15 \cos \left(\frac{3 c}{2}+2 d x\right)-15 \cos \left(\frac{5 c}{2}+2 d x\right)+60 \cos \left(\frac{5 c}{2}+3 d x\right)+60 \cos \left(\frac{7 c}{2}+3 d x\right)-15 \cos \left(\frac{7 c}{2}+4 d x\right)+15 \cos \left(\frac{9 c}{2}+4 d x\right)+12 \cos \left(\frac{9 c}{2}+5 d x\right)+12 \cos \left(\frac{11 c}{2}+5 d x\right)-5 \cos \left(\frac{11 c}{2}+6 d x\right)+5 \cos \left(\frac{13 c}{2}+6 d x\right)-150 c \sin \left(\frac{c}{2}\right)+300 \sin \left(\frac{c}{2}\right)}{1920 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"-1/1920*(-30*(5*c - 4*d*x)*Cos[c/2] + 120*Cos[c/2 + d*x] + 120*Cos[(3*c)/2 + d*x] + 15*Cos[(3*c)/2 + 2*d*x] - 15*Cos[(5*c)/2 + 2*d*x] + 60*Cos[(5*c)/2 + 3*d*x] + 60*Cos[(7*c)/2 + 3*d*x] - 15*Cos[(7*c)/2 + 4*d*x] + 15*Cos[(9*c)/2 + 4*d*x] + 12*Cos[(9*c)/2 + 5*d*x] + 12*Cos[(11*c)/2 + 5*d*x] - 5*Cos[(11*c)/2 + 6*d*x] + 5*Cos[(13*c)/2 + 6*d*x] + 300*Sin[c/2] - 150*c*Sin[c/2] + 120*d*x*Sin[c/2] - 120*Sin[c/2 + d*x] + 120*Sin[(3*c)/2 + d*x] + 15*Sin[(3*c)/2 + 2*d*x] + 15*Sin[(5*c)/2 + 2*d*x] - 60*Sin[(5*c)/2 + 3*d*x] + 60*Sin[(7*c)/2 + 3*d*x] - 15*Sin[(7*c)/2 + 4*d*x] - 15*Sin[(9*c)/2 + 4*d*x] - 12*Sin[(9*c)/2 + 5*d*x] + 12*Sin[(11*c)/2 + 5*d*x] - 5*Sin[(11*c)/2 + 6*d*x] - 5*Sin[(13*c)/2 + 6*d*x])/(a*d*(Cos[c/2] + Sin[c/2]))","B",1
628,1,86,101,0.3708491,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{120 \cos (c+d x)+8 \cos (3 (c+d x))-3 \left(8 \sin (2 (c+d x))+\sin (4 (c+d x))+4 \left(-8 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 c+3 d x\right)\right)}{96 a d}","\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,"(120*Cos[c + d*x] + 8*Cos[3*(c + d*x)] - 3*(4*(3*c + 3*d*x + 8*Log[Cos[(c + d*x)/2]] - 8*Log[Sin[(c + d*x)/2]]) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(96*a*d)","A",1
629,1,122,95,0.7892232,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right)^2 \left(27 \cos (c+d x)+(2 \sin (c+d x)-3) \cos (3 (c+d x))+6 \sin (c+d x) \left(5 \cos (c+d x)+4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+6 c+6 d x\right)\right)}{48 a d (\sin (c+d x)+1)}","-\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,"-1/48*((1 + Cot[(c + d*x)/2])^2*(27*Cos[c + d*x] + 6*(6*c + 6*d*x + 5*Cos[c + d*x] - 4*Log[Cos[(c + d*x)/2]] + 4*Log[Sin[(c + d*x)/2]])*Sin[c + d*x] + Cos[3*(c + d*x)]*(-3 + 2*Sin[c + d*x]))*Tan[(c + d*x)/2])/(a*d*(1 + Sin[c + d*x]))","A",1
630,1,152,106,0.4848032,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{\left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-10 \sin (2 (c+d x))+\sin (4 (c+d x))+12 \cos (c+d x)-4 \cos (3 (c+d x))+12 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-12 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 \cos (2 (c+d x)) \left(-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)-12 c-12 d x\right)}{64 a d (\sin (c+d x)+1)}","-\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,"-1/64*((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(-12*c - 12*d*x + 12*Cos[c + d*x] - 4*Cos[3*(c + d*x)] - 12*Log[Cos[(c + d*x)/2]] + 12*Cos[2*(c + d*x)]*(c + d*x + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + 12*Log[Sin[(c + d*x)/2]] - 10*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","A",1
631,1,138,94,0.9097626,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(9 \sin (2 (c+d x))-2 (3 \sin (c+d x)+4) \cos (3 (c+d x))+12 \sin ^3(c+d x) \left(3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 c+2 d x\right)\right)}{192 a d (\sin (c+d x)+1)}","\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,"(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(12*(2*c + 2*d*x - 3*Log[Cos[(c + d*x)/2]] + 3*Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^3 - 2*Cos[3*(c + d*x)]*(4 + 3*Sin[c + d*x]) + 9*Sin[2*(c + d*x)]))/(192*a*d*(1 + Sin[c + d*x]))","A",1
632,1,232,102,0.6722414,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^4(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(32 \sin (2 (c+d x))-32 \sin (4 (c+d x))+24 c \cos (4 (c+d x))+18 \cos (c+d x)+30 \cos (3 (c+d x))+24 d x \cos (4 (c+d x))-27 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+27 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-12 \cos (2 (c+d x)) \left(-3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 c+8 d x\right)-9 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+72 c+72 d x\right)}{192 a d (\sin (c+d x)+1)}","\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,"-1/192*(Csc[c + d*x]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(72*c + 72*d*x + 18*Cos[c + d*x] + 30*Cos[3*(c + d*x)] + 24*c*Cos[4*(c + d*x)] + 24*d*x*Cos[4*(c + d*x)] + 27*Log[Cos[(c + d*x)/2]] + 9*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 12*Cos[2*(c + d*x)]*(8*c + 8*d*x + 3*Log[Cos[(c + d*x)/2]] - 3*Log[Sin[(c + d*x)/2]]) - 27*Log[Sin[(c + d*x)/2]] - 9*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 32*Sin[2*(c + d*x)] - 32*Sin[4*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","B",1
633,1,189,82,0.7794119,"\int \frac{\cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^6/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^5(c+d x) \left(20 \sin (2 (c+d x))-50 \sin (4 (c+d x))+80 \cos (c+d x)+40 \cos (3 (c+d x))+8 \cos (5 (c+d x))+150 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-75 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-150 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+75 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-15 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{640 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,"-1/640*(Csc[c + d*x]^5*(80*Cos[c + d*x] + 40*Cos[3*(c + d*x)] + 8*Cos[5*(c + d*x)] - 150*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 150*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 20*Sin[2*(c + d*x)] + 75*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 75*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 50*Sin[4*(c + d*x)] - 15*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 15*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(a*d)","B",1
634,1,418,135,3.151124,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{1680 d x \sin \left(\frac{c}{2}\right)-1365 \sin \left(\frac{c}{2}+d x\right)+1365 \sin \left(\frac{3 c}{2}+d x\right)-210 \sin \left(\frac{3 c}{2}+2 d x\right)-210 \sin \left(\frac{5 c}{2}+2 d x\right)-175 \sin \left(\frac{5 c}{2}+3 d x\right)+175 \sin \left(\frac{7 c}{2}+3 d x\right)-210 \sin \left(\frac{7 c}{2}+4 d x\right)-210 \sin \left(\frac{9 c}{2}+4 d x\right)+147 \sin \left(\frac{9 c}{2}+5 d x\right)-147 \sin \left(\frac{11 c}{2}+5 d x\right)+70 \sin \left(\frac{11 c}{2}+6 d x\right)+70 \sin \left(\frac{13 c}{2}+6 d x\right)-15 \sin \left(\frac{13 c}{2}+7 d x\right)+15 \sin \left(\frac{15 c}{2}+7 d x\right)+210 \cos \left(\frac{c}{2}\right) (8 d x+1)+1365 \cos \left(\frac{c}{2}+d x\right)+1365 \cos \left(\frac{3 c}{2}+d x\right)-210 \cos \left(\frac{3 c}{2}+2 d x\right)+210 \cos \left(\frac{5 c}{2}+2 d x\right)+175 \cos \left(\frac{5 c}{2}+3 d x\right)+175 \cos \left(\frac{7 c}{2}+3 d x\right)-210 \cos \left(\frac{7 c}{2}+4 d x\right)+210 \cos \left(\frac{9 c}{2}+4 d x\right)-147 \cos \left(\frac{9 c}{2}+5 d x\right)-147 \cos \left(\frac{11 c}{2}+5 d x\right)+70 \cos \left(\frac{11 c}{2}+6 d x\right)-70 \cos \left(\frac{13 c}{2}+6 d x\right)+15 \cos \left(\frac{13 c}{2}+7 d x\right)+15 \cos \left(\frac{15 c}{2}+7 d x\right)-210 \sin \left(\frac{c}{2}\right)}{13440 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"-1/13440*(210*(1 + 8*d*x)*Cos[c/2] + 1365*Cos[c/2 + d*x] + 1365*Cos[(3*c)/2 + d*x] - 210*Cos[(3*c)/2 + 2*d*x] + 210*Cos[(5*c)/2 + 2*d*x] + 175*Cos[(5*c)/2 + 3*d*x] + 175*Cos[(7*c)/2 + 3*d*x] - 210*Cos[(7*c)/2 + 4*d*x] + 210*Cos[(9*c)/2 + 4*d*x] - 147*Cos[(9*c)/2 + 5*d*x] - 147*Cos[(11*c)/2 + 5*d*x] + 70*Cos[(11*c)/2 + 6*d*x] - 70*Cos[(13*c)/2 + 6*d*x] + 15*Cos[(13*c)/2 + 7*d*x] + 15*Cos[(15*c)/2 + 7*d*x] - 210*Sin[c/2] + 1680*d*x*Sin[c/2] - 1365*Sin[c/2 + d*x] + 1365*Sin[(3*c)/2 + d*x] - 210*Sin[(3*c)/2 + 2*d*x] - 210*Sin[(5*c)/2 + 2*d*x] - 175*Sin[(5*c)/2 + 3*d*x] + 175*Sin[(7*c)/2 + 3*d*x] - 210*Sin[(7*c)/2 + 4*d*x] - 210*Sin[(9*c)/2 + 4*d*x] + 147*Sin[(9*c)/2 + 5*d*x] - 147*Sin[(11*c)/2 + 5*d*x] + 70*Sin[(11*c)/2 + 6*d*x] + 70*Sin[(13*c)/2 + 6*d*x] - 15*Sin[(13*c)/2 + 7*d*x] + 15*Sin[(15*c)/2 + 7*d*x])/(a^2*d*(Cos[c/2] + Sin[c/2]))","B",1
635,1,362,104,2.0648026,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{360 d x \sin \left(\frac{c}{2}\right)-240 \sin \left(\frac{c}{2}+d x\right)+240 \sin \left(\frac{3 c}{2}+d x\right)-15 \sin \left(\frac{3 c}{2}+2 d x\right)-15 \sin \left(\frac{5 c}{2}+2 d x\right)-40 \sin \left(\frac{5 c}{2}+3 d x\right)+40 \sin \left(\frac{7 c}{2}+3 d x\right)-45 \sin \left(\frac{7 c}{2}+4 d x\right)-45 \sin \left(\frac{9 c}{2}+4 d x\right)+24 \sin \left(\frac{9 c}{2}+5 d x\right)-24 \sin \left(\frac{11 c}{2}+5 d x\right)+5 \sin \left(\frac{11 c}{2}+6 d x\right)+5 \sin \left(\frac{13 c}{2}+6 d x\right)+360 d x \cos \left(\frac{c}{2}\right)+240 \cos \left(\frac{c}{2}+d x\right)+240 \cos \left(\frac{3 c}{2}+d x\right)-15 \cos \left(\frac{3 c}{2}+2 d x\right)+15 \cos \left(\frac{5 c}{2}+2 d x\right)+40 \cos \left(\frac{5 c}{2}+3 d x\right)+40 \cos \left(\frac{7 c}{2}+3 d x\right)-45 \cos \left(\frac{7 c}{2}+4 d x\right)+45 \cos \left(\frac{9 c}{2}+4 d x\right)-24 \cos \left(\frac{9 c}{2}+5 d x\right)-24 \cos \left(\frac{11 c}{2}+5 d x\right)+5 \cos \left(\frac{11 c}{2}+6 d x\right)-5 \cos \left(\frac{13 c}{2}+6 d x\right)+50 \sin \left(\frac{c}{2}\right)}{1920 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","\frac{\cos ^3(c+d x) (a-a \sin (c+d x))^3}{6 a^5 d}+\frac{\cos ^5(c+d x)}{10 a^2 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,"(360*d*x*Cos[c/2] + 240*Cos[c/2 + d*x] + 240*Cos[(3*c)/2 + d*x] - 15*Cos[(3*c)/2 + 2*d*x] + 15*Cos[(5*c)/2 + 2*d*x] + 40*Cos[(5*c)/2 + 3*d*x] + 40*Cos[(7*c)/2 + 3*d*x] - 45*Cos[(7*c)/2 + 4*d*x] + 45*Cos[(9*c)/2 + 4*d*x] - 24*Cos[(9*c)/2 + 5*d*x] - 24*Cos[(11*c)/2 + 5*d*x] + 5*Cos[(11*c)/2 + 6*d*x] - 5*Cos[(13*c)/2 + 6*d*x] + 50*Sin[c/2] + 360*d*x*Sin[c/2] - 240*Sin[c/2 + d*x] + 240*Sin[(3*c)/2 + d*x] - 15*Sin[(3*c)/2 + 2*d*x] - 15*Sin[(5*c)/2 + 2*d*x] - 40*Sin[(5*c)/2 + 3*d*x] + 40*Sin[(7*c)/2 + 3*d*x] - 45*Sin[(7*c)/2 + 4*d*x] - 45*Sin[(9*c)/2 + 4*d*x] + 24*Sin[(9*c)/2 + 5*d*x] - 24*Sin[(11*c)/2 + 5*d*x] + 5*Sin[(11*c)/2 + 6*d*x] + 5*Sin[(13*c)/2 + 6*d*x])/(1920*a^2*d*(Cos[c/2] + Sin[c/2]))","B",1
636,1,262,100,1.1328144,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{-120 d x \sin \left(\frac{c}{2}\right)+90 \sin \left(\frac{c}{2}+d x\right)-90 \sin \left(\frac{3 c}{2}+d x\right)+25 \sin \left(\frac{5 c}{2}+3 d x\right)-25 \sin \left(\frac{7 c}{2}+3 d x\right)+15 \sin \left(\frac{7 c}{2}+4 d x\right)+15 \sin \left(\frac{9 c}{2}+4 d x\right)-3 \sin \left(\frac{9 c}{2}+5 d x\right)+3 \sin \left(\frac{11 c}{2}+5 d x\right)-5 \cos \left(\frac{c}{2}\right) (24 d x+5)-90 \cos \left(\frac{c}{2}+d x\right)-90 \cos \left(\frac{3 c}{2}+d x\right)-25 \cos \left(\frac{5 c}{2}+3 d x\right)-25 \cos \left(\frac{7 c}{2}+3 d x\right)+15 \cos \left(\frac{7 c}{2}+4 d x\right)-15 \cos \left(\frac{9 c}{2}+4 d x\right)+3 \cos \left(\frac{9 c}{2}+5 d x\right)+3 \cos \left(\frac{11 c}{2}+5 d x\right)+25 \sin \left(\frac{c}{2}\right)}{480 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"(-5*(5 + 24*d*x)*Cos[c/2] - 90*Cos[c/2 + d*x] - 90*Cos[(3*c)/2 + d*x] - 25*Cos[(5*c)/2 + 3*d*x] - 25*Cos[(7*c)/2 + 3*d*x] + 15*Cos[(7*c)/2 + 4*d*x] - 15*Cos[(9*c)/2 + 4*d*x] + 3*Cos[(9*c)/2 + 5*d*x] + 3*Cos[(11*c)/2 + 5*d*x] + 25*Sin[c/2] - 120*d*x*Sin[c/2] + 90*Sin[c/2 + d*x] - 90*Sin[(3*c)/2 + d*x] + 25*Sin[(5*c)/2 + 3*d*x] - 25*Sin[(7*c)/2 + 3*d*x] + 15*Sin[(7*c)/2 + 4*d*x] + 15*Sin[(9*c)/2 + 4*d*x] - 3*Sin[(9*c)/2 + 5*d*x] + 3*Sin[(11*c)/2 + 5*d*x])/(480*a^2*d*(Cos[c/2] + Sin[c/2]))","B",1
637,1,69,73,0.3181451,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{-9 \cos (c+d x)+\cos (3 (c+d x))+6 \left(\sin (2 (c+d x))+2 \left(-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)\right)}{12 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 (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{x}{a^2}",1,"-1/12*(-9*Cos[c + d*x] + Cos[3*(c + d*x)] + 6*(2*(c + d*x + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + Sin[2*(c + d*x)]))/(a^2*d)","A",1
638,1,116,74,0.5339557,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(-2 (c+d x)+\sin (2 (c+d x))-8 \cos (c+d x)+2 \tan \left(\frac{1}{2} (c+d x)\right)-2 \cot \left(\frac{1}{2} (c+d x)\right)-8 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d (a \sin (c+d x)+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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(-2*(c + d*x) - 8*Cos[c + d*x] - 2*Cot[(c + d*x)/2] + 8*Log[Cos[(c + d*x)/2]] - 8*Log[Sin[(c + d*x)/2]] + Sin[2*(c + d*x)] + 2*Tan[(c + d*x)/2]))/(4*d*(a + a*Sin[c + d*x])^2)","A",1
639,1,134,73,0.5776702,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(16 (c+d x)+8 \cos (c+d x)-8 \tan \left(\frac{1}{2} (c+d x)\right)+8 \cot \left(\frac{1}{2} (c+d x)\right)-\csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^2\left(\frac{1}{2} (c+d x)\right)+4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d (a \sin (c+d 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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(16*(c + d*x) + 8*Cos[c + d*x] + 8*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 - 4*Log[Cos[(c + d*x)/2]] + 4*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 - 8*Tan[(c + d*x)/2]))/(8*d*(a + a*Sin[c + d*x])^2)","A",1
640,1,124,73,1.2841725,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right)^4 \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-6 \sin (2 (c+d x))+6 \cos (c+d x)-2 \cos (3 (c+d x))+12 \sin ^3(c+d x) \left(-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)\right)}{96 a^2 d (\sin (c+d x)+1)^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,"-1/96*((1 + Cot[(c + d*x)/2])^4*Sec[(c + d*x)/2]^2*(6*Cos[c + d*x] - 2*Cos[3*(c + d*x)] + 12*(c + d*x + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^3 - 6*Sin[2*(c + d*x)])*Tan[(c + d*x)/2])/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
641,1,116,82,1.4175112,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]","\frac{\left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(24 \sin (2 (c+d x))-33 \cos (c+d x)+(16 \sin (c+d x)+9) \cos (3 (c+d x))+60 \sin ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{1536 a^2 d (\sin (c+d x)+1)^2}","\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,"((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^4*(-33*Cos[c + d*x] + 60*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^4 + Cos[3*(c + d*x)]*(9 + 16*Sin[c + d*x]) + 24*Sin[2*(c + d*x)]))/(1536*a^2*d*(1 + Sin[c + d*x])^2)","A",1
642,1,189,100,0.560481,"\int \frac{\cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^5(c+d x) \left(-180 \sin (2 (c+d x))-30 \sin (4 (c+d x))+200 \cos (c+d x)+20 \cos (3 (c+d x))-28 \cos (5 (c+d x))-150 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+75 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-15 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+150 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-75 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+15 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{960 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,"-1/960*(Csc[c + d*x]^5*(200*Cos[c + d*x] + 20*Cos[3*(c + d*x)] - 28*Cos[5*(c + d*x)] + 150*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 150*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 180*Sin[2*(c + d*x)] - 75*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 75*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 30*Sin[4*(c + d*x)] + 15*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 15*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(a^2*d)","A",1
643,1,229,124,0.6775888,"\int \frac{\cot ^6(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^6*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^6(c+d x) \left(-960 \sin (2 (c+d x))-384 \sin (4 (c+d x))+64 \sin (6 (c+d x))+1500 \cos (c+d x)-130 \cos (3 (c+d x))-90 \cos (5 (c+d x))+450 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+675 \cos (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-270 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+45 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-450 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-675 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+270 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-45 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{7680 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,"-1/7680*(Csc[c + d*x]^6*(1500*Cos[c + d*x] - 130*Cos[3*(c + d*x)] - 90*Cos[5*(c + d*x)] - 450*Log[Cos[(c + d*x)/2]] + 675*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 270*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 45*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 450*Log[Sin[(c + d*x)/2]] - 675*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 270*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 45*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 960*Sin[2*(c + d*x)] - 384*Sin[4*(c + d*x)] + 64*Sin[6*(c + d*x)]))/(a^2*d)","A",1
644,1,366,129,2.1871972,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{-2760 d x \sin \left(\frac{c}{2}\right)+2520 \sin \left(\frac{c}{2}+d x\right)-2520 \sin \left(\frac{3 c}{2}+d x\right)+945 \sin \left(\frac{3 c}{2}+2 d x\right)+945 \sin \left(\frac{5 c}{2}+2 d x\right)-380 \sin \left(\frac{5 c}{2}+3 d x\right)+380 \sin \left(\frac{7 c}{2}+3 d x\right)-135 \sin \left(\frac{7 c}{2}+4 d x\right)-135 \sin \left(\frac{9 c}{2}+4 d x\right)+36 \sin \left(\frac{9 c}{2}+5 d x\right)-36 \sin \left(\frac{11 c}{2}+5 d x\right)+5 \sin \left(\frac{11 c}{2}+6 d x\right)+5 \sin \left(\frac{13 c}{2}+6 d x\right)-3 \cos \left(\frac{c}{2}\right) (920 d x+3)-2520 \cos \left(\frac{c}{2}+d x\right)-2520 \cos \left(\frac{3 c}{2}+d x\right)+945 \cos \left(\frac{3 c}{2}+2 d x\right)-945 \cos \left(\frac{5 c}{2}+2 d x\right)+380 \cos \left(\frac{5 c}{2}+3 d x\right)+380 \cos \left(\frac{7 c}{2}+3 d x\right)-135 \cos \left(\frac{7 c}{2}+4 d x\right)+135 \cos \left(\frac{9 c}{2}+4 d x\right)-36 \cos \left(\frac{9 c}{2}+5 d x\right)-36 \cos \left(\frac{11 c}{2}+5 d x\right)+5 \cos \left(\frac{11 c}{2}+6 d x\right)-5 \cos \left(\frac{13 c}{2}+6 d x\right)+9 \sin \left(\frac{c}{2}\right)}{1920 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"(-3*(3 + 920*d*x)*Cos[c/2] - 2520*Cos[c/2 + d*x] - 2520*Cos[(3*c)/2 + d*x] + 945*Cos[(3*c)/2 + 2*d*x] - 945*Cos[(5*c)/2 + 2*d*x] + 380*Cos[(5*c)/2 + 3*d*x] + 380*Cos[(7*c)/2 + 3*d*x] - 135*Cos[(7*c)/2 + 4*d*x] + 135*Cos[(9*c)/2 + 4*d*x] - 36*Cos[(9*c)/2 + 5*d*x] - 36*Cos[(11*c)/2 + 5*d*x] + 5*Cos[(11*c)/2 + 6*d*x] - 5*Cos[(13*c)/2 + 6*d*x] + 9*Sin[c/2] - 2760*d*x*Sin[c/2] + 2520*Sin[c/2 + d*x] - 2520*Sin[(3*c)/2 + d*x] + 945*Sin[(3*c)/2 + 2*d*x] + 945*Sin[(5*c)/2 + 2*d*x] - 380*Sin[(5*c)/2 + 3*d*x] + 380*Sin[(7*c)/2 + 3*d*x] - 135*Sin[(7*c)/2 + 4*d*x] - 135*Sin[(9*c)/2 + 4*d*x] + 36*Sin[(9*c)/2 + 5*d*x] - 36*Sin[(11*c)/2 + 5*d*x] + 5*Sin[(11*c)/2 + 6*d*x] + 5*Sin[(13*c)/2 + 6*d*x])/(1920*a^3*d*(Cos[c/2] + Sin[c/2]))","B",1
645,1,310,105,1.7670954,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{1560 d x \sin \left(\frac{c}{2}\right)-1380 \sin \left(\frac{c}{2}+d x\right)+1380 \sin \left(\frac{3 c}{2}+d x\right)-480 \sin \left(\frac{3 c}{2}+2 d x\right)-480 \sin \left(\frac{5 c}{2}+2 d x\right)+170 \sin \left(\frac{5 c}{2}+3 d x\right)-170 \sin \left(\frac{7 c}{2}+3 d x\right)+45 \sin \left(\frac{7 c}{2}+4 d x\right)+45 \sin \left(\frac{9 c}{2}+4 d x\right)-6 \sin \left(\frac{9 c}{2}+5 d x\right)+6 \sin \left(\frac{11 c}{2}+5 d x\right)+1560 d x \cos \left(\frac{c}{2}\right)+1380 \cos \left(\frac{c}{2}+d x\right)+1380 \cos \left(\frac{3 c}{2}+d x\right)-480 \cos \left(\frac{3 c}{2}+2 d x\right)+480 \cos \left(\frac{5 c}{2}+2 d x\right)-170 \cos \left(\frac{5 c}{2}+3 d x\right)-170 \cos \left(\frac{7 c}{2}+3 d x\right)+45 \cos \left(\frac{7 c}{2}+4 d x\right)-45 \cos \left(\frac{9 c}{2}+4 d x\right)+6 \cos \left(\frac{9 c}{2}+5 d x\right)+6 \cos \left(\frac{11 c}{2}+5 d x\right)+10 \sin \left(\frac{c}{2}\right)}{960 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","\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,"(1560*d*x*Cos[c/2] + 1380*Cos[c/2 + d*x] + 1380*Cos[(3*c)/2 + d*x] - 480*Cos[(3*c)/2 + 2*d*x] + 480*Cos[(5*c)/2 + 2*d*x] - 170*Cos[(5*c)/2 + 3*d*x] - 170*Cos[(7*c)/2 + 3*d*x] + 45*Cos[(7*c)/2 + 4*d*x] - 45*Cos[(9*c)/2 + 4*d*x] + 6*Cos[(9*c)/2 + 5*d*x] + 6*Cos[(11*c)/2 + 5*d*x] + 10*Sin[c/2] + 1560*d*x*Sin[c/2] - 1380*Sin[c/2 + d*x] + 1380*Sin[(3*c)/2 + d*x] - 480*Sin[(3*c)/2 + 2*d*x] - 480*Sin[(5*c)/2 + 2*d*x] + 170*Sin[(5*c)/2 + 3*d*x] - 170*Sin[(7*c)/2 + 3*d*x] + 45*Sin[(7*c)/2 + 4*d*x] + 45*Sin[(9*c)/2 + 4*d*x] - 6*Sin[(9*c)/2 + 5*d*x] + 6*Sin[(11*c)/2 + 5*d*x])/(960*a^3*d*(Cos[c/2] + Sin[c/2]))","B",1
646,1,255,84,1.3746502,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{120 d x \sin \left(\frac{c}{2}\right)-104 \sin \left(\frac{c}{2}+d x\right)+104 \sin \left(\frac{3 c}{2}+d x\right)-32 \sin \left(\frac{3 c}{2}+2 d x\right)-32 \sin \left(\frac{5 c}{2}+2 d x\right)+8 \sin \left(\frac{5 c}{2}+3 d x\right)-8 \sin \left(\frac{7 c}{2}+3 d x\right)+\sin \left(\frac{7 c}{2}+4 d x\right)+\sin \left(\frac{9 c}{2}+4 d x\right)+\cos \left(\frac{c}{2}\right) (120 d x+1)+104 \cos \left(\frac{c}{2}+d x\right)+104 \cos \left(\frac{3 c}{2}+d x\right)-32 \cos \left(\frac{3 c}{2}+2 d x\right)+32 \cos \left(\frac{5 c}{2}+2 d x\right)-8 \cos \left(\frac{5 c}{2}+3 d x\right)-8 \cos \left(\frac{7 c}{2}+3 d x\right)+\cos \left(\frac{7 c}{2}+4 d x\right)-\cos \left(\frac{9 c}{2}+4 d x\right)-\sin \left(\frac{c}{2}\right)}{64 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","\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,"-1/64*((1 + 120*d*x)*Cos[c/2] + 104*Cos[c/2 + d*x] + 104*Cos[(3*c)/2 + d*x] - 32*Cos[(3*c)/2 + 2*d*x] + 32*Cos[(5*c)/2 + 2*d*x] - 8*Cos[(5*c)/2 + 3*d*x] - 8*Cos[(7*c)/2 + 3*d*x] + Cos[(7*c)/2 + 4*d*x] - Cos[(9*c)/2 + 4*d*x] - Sin[c/2] + 120*d*x*Sin[c/2] - 104*Sin[c/2 + d*x] + 104*Sin[(3*c)/2 + d*x] - 32*Sin[(3*c)/2 + 2*d*x] - 32*Sin[(5*c)/2 + 2*d*x] + 8*Sin[(5*c)/2 + 3*d*x] - 8*Sin[(7*c)/2 + 3*d*x] + Sin[(7*c)/2 + 4*d*x] + Sin[(9*c)/2 + 4*d*x])/(a^3*d*(Cos[c/2] + Sin[c/2]))","B",1
647,1,63,60,0.213668,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^5*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{\sin (2 (c+d x))-12 \cos (c+d x)-2 \left(-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+7 c+7 d x\right)}{4 a^3 d}","-\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,"(-12*Cos[c + d*x] - 2*(7*c + 7*d*x + 2*Log[Cos[(c + d*x)/2]] - 2*Log[Sin[(c + d*x)/2]]) + Sin[2*(c + d*x)])/(4*a^3*d)","A",1
648,1,106,49,0.4953225,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6 \left(6 (c+d x)+2 \cos (c+d x)+\tan \left(\frac{1}{2} (c+d x)\right)-\cot \left(\frac{1}{2} (c+d x)\right)-6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 d (a \sin (c+d 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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(6*(c + d*x) + 2*Cos[c + d*x] - Cot[(c + d*x)/2] + 6*Log[Cos[(c + d*x)/2]] - 6*Log[Sin[(c + d*x)/2]] + Tan[(c + d*x)/2]))/(2*d*(a + a*Sin[c + d*x])^3)","B",1
649,1,126,60,0.4675917,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6 \left(-8 (c+d x)-12 \tan \left(\frac{1}{2} (c+d x)\right)+12 \cot \left(\frac{1}{2} (c+d x)\right)-\csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^2\left(\frac{1}{2} (c+d x)\right)+28 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-28 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d (a \sin (c+d 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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(-8*(c + d*x) + 12*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 - 28*Log[Cos[(c + d*x)/2]] + 28*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 - 12*Tan[(c + d*x)/2]))/(8*d*(a + a*Sin[c + d*x])^3)","B",1
650,1,115,72,1.3012185,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^3(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6 \left(-18 \sin (2 (c+d x))+30 \cos (c+d x)-22 \cos (3 (c+d x))-60 \sin ^3(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{24 a^3 d (\sin (c+d x)+1)^3}","-\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,"-1/24*(Csc[c + d*x]^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(30*Cos[c + d*x] - 22*Cos[3*(c + d*x)] - 60*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^3 - 18*Sin[2*(c + d*x)]))/(a^3*d*(1 + Sin[c + d*x])^3)","A",1
651,1,125,93,2.1044829,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^4(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6 \left(-56 \sin (2 (c+d x))+46 \cos (c+d x)+6 (8 \sin (c+d x)-5) \cos (3 (c+d x))+120 \sin ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{64 a^3 d (\sin (c+d x)+1)^3}","\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,"-1/64*(Csc[c + d*x]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(46*Cos[c + d*x] + 120*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^4 + 6*Cos[3*(c + d*x)]*(-5 + 8*Sin[c + d*x]) - 56*Sin[2*(c + d*x)]))/(a^3*d*(1 + Sin[c + d*x])^3)","A",1
652,1,189,114,1.8005355,"\int \frac{\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^3,x]","\frac{\csc ^5(c+d x) \left(1500 \sin (2 (c+d x))-390 \sin (4 (c+d x))-1600 \cos (c+d x)+1520 \cos (3 (c+d x))-304 \cos (5 (c+d x))-1950 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+975 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-195 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+1950 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-975 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+195 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1920 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,"(Csc[c + d*x]^5*(-1600*Cos[c + d*x] + 1520*Cos[3*(c + d*x)] - 304*Cos[5*(c + d*x)] + 1950*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 1950*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 1500*Sin[2*(c + d*x)] - 975*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 975*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 390*Sin[4*(c + d*x)] + 195*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 195*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(1920*a^3*d)","A",1
653,1,188,267,0.6169027,"\int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{n+1}(c+d x) \left(\frac{\, _2F_1\left(-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{n+1}+\sin (c+d x) \left(\frac{3 \, _2F_1\left(-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{n+2}+\sin (c+d x) \left(\frac{3 \, _2F_1\left(-\frac{5}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right)}{n+3}+\frac{\sin (c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+4}{2};\frac{n+6}{2};\sin ^2(c+d x)\right)}{n+4}\right)\right)\right)}{d}","\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*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + n)*(Hypergeometric2F1[-5/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]/(1 + n) + Sin[c + d*x]*((3*Hypergeometric2F1[-5/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2])/(2 + n) + Sin[c + d*x]*((3*Hypergeometric2F1[-5/2, (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2])/(3 + n) + (Hypergeometric2F1[-5/2, (4 + n)/2, (6 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x])/(4 + n)))))/d","A",1
654,1,164,200,0.2884125,"\int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{n+1}(c+d x) \left(\left(n^2+5 n+6\right) \, _2F_1\left(-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)+(n+1) \sin (c+d x) \left(2 (n+3) \, _2F_1\left(-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)+(n+2) \sin (c+d x) \, _2F_1\left(-\frac{5}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right)\right)\right)}{d (n+1) (n+2) (n+3)}","\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*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + n)*((6 + 5*n + n^2)*Hypergeometric2F1[-5/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2] + (1 + n)*Sin[c + d*x]*(2*(3 + n)*Hypergeometric2F1[-5/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2] + (2 + n)*Hypergeometric2F1[-5/2, (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x])))/(d*(1 + n)*(2 + n)*(3 + n))","A",1
655,0,0,129,0.2984902,"\int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]","\int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","\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,"Integrate[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x]), x]","F",-1
656,1,117,129,1.0292579,"\int \cos ^7(c+d x) \sin ^6(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*Sin[c + d*x]^6*(a + a*Sin[c + d*x]),x]","-\frac{a (-1201200 \sin (c+d x)+300300 \sin (3 (c+d x))+180180 \sin (5 (c+d x))-51480 \sin (7 (c+d x))-40040 \sin (9 (c+d x))+5460 \sin (11 (c+d x))+4620 \sin (13 (c+d x))+525525 \cos (2 (c+d x))-105105 \cos (6 (c+d x))+21021 \cos (10 (c+d x))-2145 \cos (14 (c+d x)))}{246005760 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,"-1/246005760*(a*(525525*Cos[2*(c + d*x)] - 105105*Cos[6*(c + d*x)] + 21021*Cos[10*(c + d*x)] - 2145*Cos[14*(c + d*x)] - 1201200*Sin[c + d*x] + 300300*Sin[3*(c + d*x)] + 180180*Sin[5*(c + d*x)] - 51480*Sin[7*(c + d*x)] - 40040*Sin[9*(c + d*x)] + 5460*Sin[11*(c + d*x)] + 4620*Sin[13*(c + d*x)]))/d","A",1
657,1,137,113,0.7316697,"\int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*Sin[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a (-600600 \sin (c+d x)+150150 \sin (3 (c+d x))+90090 \sin (5 (c+d x))-25740 \sin (7 (c+d x))-20020 \sin (9 (c+d x))+2730 \sin (11 (c+d x))+2310 \sin (13 (c+d x))+600600 \cos (2 (c+d x))+75075 \cos (4 (c+d x))-100100 \cos (6 (c+d x))-30030 \cos (8 (c+d x))+12012 \cos (10 (c+d x))+5005 \cos (12 (c+d x)))}{123002880 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,"-1/123002880*(a*(600600*Cos[2*(c + d*x)] + 75075*Cos[4*(c + d*x)] - 100100*Cos[6*(c + d*x)] - 30030*Cos[8*(c + d*x)] + 12012*Cos[10*(c + d*x)] + 5005*Cos[12*(c + d*x)] - 600600*Sin[c + d*x] + 150150*Sin[3*(c + d*x)] + 90090*Sin[5*(c + d*x)] - 25740*Sin[7*(c + d*x)] - 20020*Sin[9*(c + d*x)] + 2730*Sin[11*(c + d*x)] + 2310*Sin[13*(c + d*x)]))/d","A",1
658,1,127,113,0.5551675,"\int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a (-129360 \sin (c+d x)+18480 \sin (3 (c+d x))+20328 \sin (5 (c+d x))+1320 \sin (7 (c+d x))-3080 \sin (9 (c+d x))-840 \sin (11 (c+d x))+46200 \cos (2 (c+d x))+5775 \cos (4 (c+d x))-7700 \cos (6 (c+d x))-2310 \cos (8 (c+d x))+924 \cos (10 (c+d x))+385 \cos (12 (c+d x)))}{9461760 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,"-1/9461760*(a*(46200*Cos[2*(c + d*x)] + 5775*Cos[4*(c + d*x)] - 7700*Cos[6*(c + d*x)] - 2310*Cos[8*(c + d*x)] + 924*Cos[10*(c + d*x)] + 385*Cos[12*(c + d*x)] - 129360*Sin[c + d*x] + 18480*Sin[3*(c + d*x)] + 20328*Sin[5*(c + d*x)] + 1320*Sin[7*(c + d*x)] - 3080*Sin[9*(c + d*x)] - 840*Sin[11*(c + d*x)]))/d","A",1
659,1,117,97,0.6057478,"\int \cos ^7(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a (16170 \sin (c+d x)-2310 \sin (3 (c+d x))-2541 \sin (5 (c+d x))-165 \sin (7 (c+d x))+385 \sin (9 (c+d x))+105 \sin (11 (c+d x))-16170 \cos (2 (c+d x))-4620 \cos (4 (c+d x))+1155 \cos (6 (c+d x))+1155 \cos (8 (c+d x))+231 \cos (10 (c+d x)))}{1182720 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*(-16170*Cos[2*(c + d*x)] - 4620*Cos[4*(c + d*x)] + 1155*Cos[6*(c + d*x)] + 1155*Cos[8*(c + d*x)] + 231*Cos[10*(c + d*x)] + 16170*Sin[c + d*x] - 2310*Sin[3*(c + d*x)] - 2541*Sin[5*(c + d*x)] - 165*Sin[7*(c + d*x)] + 385*Sin[9*(c + d*x)] + 105*Sin[11*(c + d*x)]))/(1182720*d)","A",1
660,1,97,97,0.4525296,"\int \cos ^7(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a (-17640 \sin (c+d x)+2016 \sin (5 (c+d x))+900 \sin (7 (c+d x))+140 \sin (9 (c+d x))+4410 \cos (2 (c+d x))+1260 \cos (4 (c+d x))-315 \cos (6 (c+d x))-315 \cos (8 (c+d x))-63 \cos (10 (c+d x)))}{322560 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,"-1/322560*(a*(4410*Cos[2*(c + d*x)] + 1260*Cos[4*(c + d*x)] - 315*Cos[6*(c + d*x)] - 315*Cos[8*(c + d*x)] - 63*Cos[10*(c + d*x)] - 17640*Sin[c + d*x] + 2016*Sin[5*(c + d*x)] + 900*Sin[7*(c + d*x)] + 140*Sin[9*(c + d*x)]))/d","A",1
661,1,60,81,0.3831342,"\int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \left(\sin ^3(c+d x) (1389 \cos (2 (c+d x))+330 \cos (4 (c+d x))+35 \cos (6 (c+d x))+1606)-1260 \cos ^8(c+d x)\right)}{10080 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*(-1260*Cos[c + d*x]^8 + (1606 + 1389*Cos[2*(c + d*x)] + 330*Cos[4*(c + d*x)] + 35*Cos[6*(c + d*x)])*Sin[c + d*x]^3))/(10080*d)","A",1
662,1,106,118,0.1338192,"\int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{a \left(-2 \sin ^6(c+d x)+9 \sin ^4(c+d x)-18 \sin ^2(c+d x)+12 \log (\sin (c+d x))\right)}{12 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*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d) + (a*(12*Log[Sin[c + d*x]] - 18*Sin[c + d*x]^2 + 9*Sin[c + d*x]^4 - 2*Sin[c + d*x]^6))/(12*d)","A",1
663,1,102,114,0.1311988,"\int \cos ^5(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Cot[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \sin ^3(c+d x)}{d}-\frac{3 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \left(-2 \sin ^6(c+d x)+9 \sin ^4(c+d x)-18 \sin ^2(c+d x)+12 \log (\sin (c+d x))\right)}{12 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) - (3*a*Sin[c + d*x])/d + (a*Sin[c + d*x]^3)/d - (a*Sin[c + d*x]^5)/(5*d) + (a*(12*Log[Sin[c + d*x]] - 18*Sin[c + d*x]^2 + 9*Sin[c + d*x]^4 - 2*Sin[c + d*x]^6))/(12*d)","A",1
664,1,100,115,0.1221786,"\int \cos ^4(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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 ^3(c+d x)}{d}-\frac{3 a \sin (c+d x)}{d}-\frac{a \csc (c+d x)}{d}-\frac{a \left(\sin ^4(c+d x)-6 \sin ^2(c+d x)+2 \csc ^2(c+d x)+12 \log (\sin (c+d x))\right)}{4 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) - (3*a*Sin[c + d*x])/d + (a*Sin[c + d*x]^3)/d - (a*Sin[c + d*x]^5)/(5*d) - (a*(2*Csc[c + d*x]^2 + 12*Log[Sin[c + d*x]] - 6*Sin[c + d*x]^2 + Sin[c + d*x]^4))/(4*d)","A",1
665,1,103,118,0.2141779,"\int \cos ^3(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{3 a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{3 a \csc (c+d x)}{d}-\frac{a \left(\sin ^4(c+d x)-6 \sin ^2(c+d x)+2 \csc ^2(c+d x)+12 \log (\sin (c+d x))\right)}{4 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]^3)/(3*d) + (3*a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) - (a*(2*Csc[c + d*x]^2 + 12*Log[Sin[c + d*x]] - 6*Sin[c + d*x]^2 + Sin[c + d*x]^4))/(4*d)","A",1
666,1,105,118,0.5432434,"\int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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{3 a \sin (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}+\frac{3 a \csc (c+d x)}{d}+\frac{a \left(-2 \sin ^2(c+d x)-\csc ^4(c+d x)+6 \csc ^2(c+d x)+12 \log (\sin (c+d x))\right)}{4 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 - (a*Csc[c + d*x]^3)/(3*d) + (3*a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) + (a*(6*Csc[c + d*x]^2 - Csc[c + d*x]^4 + 12*Log[Sin[c + d*x]] - 2*Sin[c + d*x]^2))/(4*d)","A",1
667,1,102,115,0.1948763,"\int \cos (c+d x) \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^6*(a + a*Sin[c + d*x]),x]","-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc (c+d x)}{d}+\frac{a \left(-2 \sin ^2(c+d x)-\csc ^4(c+d x)+6 \csc ^2(c+d x)+12 \log (\sin (c+d x))\right)}{4 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 + (a*Csc[c + d*x]^3)/d - (a*Csc[c + d*x]^5)/(5*d) - (a*Sin[c + d*x])/d + (a*(6*Csc[c + d*x]^2 - Csc[c + d*x]^4 + 12*Log[Sin[c + d*x]] - 2*Sin[c + d*x]^2))/(4*d)","A",1
668,1,111,115,0.3865632,"\int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^7*(a + a*Sin[c + d*x]),x]","-\frac{a \sin (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}+\frac{a \csc ^3(c+d x)}{d}-\frac{3 a \csc (c+d x)}{d}-\frac{a \left(2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right)}{12 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 + (a*Csc[c + d*x]^3)/d - (a*Csc[c + d*x]^5)/(5*d) - (a*(6*Cot[c + d*x]^2 - 3*Cot[c + d*x]^4 + 2*Cot[c + d*x]^6 + 12*Log[Cos[c + d*x]] + 12*Log[Tan[c + d*x]]))/(12*d) - (a*Sin[c + d*x])/d","A",1
669,1,115,119,0.3917902,"\int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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{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 \left(2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right)}{12 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 - (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*Cot[c + d*x]^2 - 3*Cot[c + d*x]^4 + 2*Cot[c + d*x]^6 + 12*Log[Cos[c + d*x]] + 12*Log[Tan[c + d*x]]))/(12*d)","A",1
670,1,74,74,0.032478,"\int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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,"-1/8*(a*Cot[c + d*x]^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",1
671,1,81,81,0.0714713,"\int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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,"-1/8*(a*Cot[c + d*x]^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",1
672,1,86,97,0.1846933,"\int \cot ^7(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^7*Csc[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^3(c+d x) \left(252 \csc ^7(c+d x)+280 \csc ^6(c+d x)-945 \csc ^5(c+d x)-1080 \csc ^4(c+d x)+1260 \csc ^3(c+d x)+1512 \csc ^2(c+d x)-630 \csc (c+d x)-840\right)}{2520 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,"-1/2520*(a*Csc[c + d*x]^3*(-840 - 630*Csc[c + d*x] + 1512*Csc[c + d*x]^2 + 1260*Csc[c + d*x]^3 - 1080*Csc[c + d*x]^4 - 945*Csc[c + d*x]^5 + 280*Csc[c + d*x]^6 + 252*Csc[c + d*x]^7))/d","A",1
673,1,86,97,0.1396394,"\int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^7*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^4(c+d x) \left(840 \csc ^7(c+d x)+924 \csc ^6(c+d x)-3080 \csc ^5(c+d x)-3465 \csc ^4(c+d x)+3960 \csc ^3(c+d x)+4620 \csc ^2(c+d x)-1848 \csc (c+d x)-2310\right)}{9240 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,"-1/9240*(a*Csc[c + d*x]^4*(-2310 - 1848*Csc[c + d*x] + 4620*Csc[c + d*x]^2 + 3960*Csc[c + d*x]^3 - 3465*Csc[c + d*x]^4 - 3080*Csc[c + d*x]^5 + 924*Csc[c + d*x]^6 + 840*Csc[c + d*x]^7))/d","A",1
674,1,86,113,0.2254742,"\int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^7*Csc[c + d*x]^6*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^{12}(c+d x) (-45 \sin (c+d x)+1111 \sin (3 (c+d x))+363 \sin (5 (c+d x))+231 \sin (7 (c+d x))+3003 \cos (2 (c+d x))+1155 \cos (4 (c+d x))+385 \cos (6 (c+d x))+1617)}{73920 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,"-1/73920*(a*Csc[c + d*x]^12*(1617 + 3003*Cos[2*(c + d*x)] + 1155*Cos[4*(c + d*x)] + 385*Cos[6*(c + d*x)] - 45*Sin[c + d*x] + 1111*Sin[3*(c + d*x)] + 363*Sin[5*(c + d*x)] + 231*Sin[7*(c + d*x)]))/d","A",1
675,1,86,113,0.2209866,"\int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^7*Csc[c + d*x]^7*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^{13}(c+d x) (3003 \sin (c+d x)+24024 \sin (3 (c+d x))+10010 \sin (5 (c+d x))+5005 \sin (7 (c+d x))+70460 \cos (2 (c+d x))+28600 \cos (4 (c+d x))+8580 \cos (6 (c+d x))+40200)}{1921920 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,"-1/1921920*(a*Csc[c + d*x]^13*(40200 + 70460*Cos[2*(c + d*x)] + 28600*Cos[4*(c + d*x)] + 8580*Cos[6*(c + d*x)] + 3003*Sin[c + d*x] + 24024*Sin[3*(c + d*x)] + 10010*Sin[5*(c + d*x)] + 5005*Sin[7*(c + d*x)]))/d","A",1
676,1,86,129,0.2361899,"\int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^7*Csc[c + d*x]^8*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^{14}(c+d x) (9940 \sin (c+d x)+41860 \sin (3 (c+d x))+20020 \sin (5 (c+d x))+8580 \sin (7 (c+d x))+129129 \cos (2 (c+d x))+54054 \cos (4 (c+d x))+15015 \cos (6 (c+d x))+76362)}{3843840 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,"-1/3843840*(a*Csc[c + d*x]^14*(76362 + 129129*Cos[2*(c + d*x)] + 54054*Cos[4*(c + d*x)] + 15015*Cos[6*(c + d*x)] + 9940*Sin[c + d*x] + 41860*Sin[3*(c + d*x)] + 20020*Sin[5*(c + d*x)] + 8580*Sin[7*(c + d*x)]))/d","A",1
677,1,68,109,0.597396,"\int \frac{\cos ^7(c+d x) \sin ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^7(c+d x) \left(-2310 \sin ^5(c+d x)+2520 \sin ^4(c+d x)+5544 \sin ^3(c+d x)-6160 \sin ^2(c+d x)-3465 \sin (c+d x)+3960\right)}{27720 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*(3960 - 3465*Sin[c + d*x] - 6160*Sin[c + d*x]^2 + 5544*Sin[c + d*x]^3 + 2520*Sin[c + d*x]^4 - 2310*Sin[c + d*x]^5))/(27720*a*d)","A",1
678,1,68,109,0.9170073,"\int \frac{\cos ^7(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^6(c+d x) \left(-1260 \sin ^5(c+d x)+1386 \sin ^4(c+d x)+3080 \sin ^3(c+d x)-3465 \sin ^2(c+d x)-1980 \sin (c+d x)+2310\right)}{13860 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*(2310 - 1980*Sin[c + d*x] - 3465*Sin[c + d*x]^2 + 3080*Sin[c + d*x]^3 + 1386*Sin[c + d*x]^4 - 1260*Sin[c + d*x]^5))/(13860*a*d)","A",1
679,1,68,109,0.4099725,"\int \frac{\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^5(c+d x) \left(-126 \sin ^5(c+d x)+140 \sin ^4(c+d x)+315 \sin ^3(c+d x)-360 \sin ^2(c+d x)-210 \sin (c+d x)+252\right)}{1260 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*(252 - 210*Sin[c + d*x] - 360*Sin[c + d*x]^2 + 315*Sin[c + d*x]^3 + 140*Sin[c + d*x]^4 - 126*Sin[c + d*x]^5))/(1260*a*d)","A",1
680,1,68,91,0.5643468,"\int \frac{\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^4(c+d x) \left(-280 \sin ^5(c+d x)+315 \sin ^4(c+d x)+720 \sin ^3(c+d x)-840 \sin ^2(c+d x)-504 \sin (c+d x)+630\right)}{2520 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,"(Sin[c + d*x]^4*(630 - 504*Sin[c + d*x] - 840*Sin[c + d*x]^2 + 720*Sin[c + d*x]^3 + 315*Sin[c + d*x]^4 - 280*Sin[c + d*x]^5))/(2520*a*d)","A",1
681,1,68,91,0.2895704,"\int \frac{\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^3(c+d x) \left(-105 \sin ^5(c+d x)+120 \sin ^4(c+d x)+280 \sin ^3(c+d x)-336 \sin ^2(c+d x)-210 \sin (c+d x)+280\right)}{840 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,"(Sin[c + d*x]^3*(280 - 210*Sin[c + d*x] - 336*Sin[c + d*x]^2 + 280*Sin[c + d*x]^3 + 120*Sin[c + d*x]^4 - 105*Sin[c + d*x]^5))/(840*a*d)","A",1
682,1,68,73,0.2637352,"\int \frac{\cos ^7(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^7*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\sin ^2(c+d x) \left(-30 \sin ^5(c+d x)+35 \sin ^4(c+d x)+84 \sin ^3(c+d x)-105 \sin ^2(c+d x)-70 \sin (c+d x)+105\right)}{210 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,"(Sin[c + d*x]^2*(105 - 70*Sin[c + d*x] - 105*Sin[c + d*x]^2 + 84*Sin[c + d*x]^3 + 35*Sin[c + d*x]^4 - 30*Sin[c + d*x]^5))/(210*a*d)","A",1
683,1,66,68,0.1872186,"\int \frac{\cos ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^7/(a + a*Sin[c + d*x]),x]","-\frac{\sin (c+d x) \left(5 \sin ^5(c+d x)-6 \sin ^4(c+d x)-15 \sin ^3(c+d x)+20 \sin ^2(c+d x)+15 \sin (c+d x)-30\right)}{30 a 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,"-1/30*(Sin[c + d*x]*(-30 + 15*Sin[c + d*x] + 20*Sin[c + d*x]^2 - 15*Sin[c + d*x]^3 - 6*Sin[c + d*x]^4 + 5*Sin[c + d*x]^5))/(a*d)","A",1
684,1,68,99,0.0629026,"\int \frac{\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^6*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{-12 \sin ^5(c+d x)+15 \sin ^4(c+d x)+40 \sin ^3(c+d x)-60 \sin ^2(c+d x)-60 \sin (c+d x)+60 \log (\sin (c+d x))}{60 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,"(60*Log[Sin[c + d*x]] - 60*Sin[c + d*x] - 60*Sin[c + d*x]^2 + 40*Sin[c + d*x]^3 + 15*Sin[c + d*x]^4 - 12*Sin[c + d*x]^5)/(60*a*d)","A",1
685,1,66,95,0.1252541,"\int \frac{\cos ^5(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{3 \sin ^4(c+d x)-4 \sin ^3(c+d x)-12 \sin ^2(c+d x)+24 \sin (c+d x)+12 \csc (c+d x)+12 \log (\sin (c+d x))}{12 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,"-1/12*(12*Csc[c + d*x] + 12*Log[Sin[c + d*x]] + 24*Sin[c + d*x] - 12*Sin[c + d*x]^2 - 4*Sin[c + d*x]^3 + 3*Sin[c + d*x]^4)/(a*d)","A",1
686,1,66,97,0.109857,"\int \frac{\cos ^4(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{-2 \sin ^3(c+d x)+3 \sin ^2(c+d x)+12 \sin (c+d x)-3 \csc ^2(c+d x)+6 \csc (c+d x)-12 \log (\sin (c+d x))}{6 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,"(6*Csc[c + d*x] - 3*Csc[c + d*x]^2 - 12*Log[Sin[c + d*x]] + 12*Sin[c + d*x] + 3*Sin[c + d*x]^2 - 2*Sin[c + d*x]^3)/(6*a*d)","A",1
687,1,66,97,0.1684974,"\int \frac{\cos ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{-3 \sin ^2(c+d x)+6 \sin (c+d x)-2 \csc ^3(c+d x)+3 \csc ^2(c+d x)+12 \csc (c+d x)+12 \log (\sin (c+d x))}{6 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,"(12*Csc[c + d*x] + 3*Csc[c + d*x]^2 - 2*Csc[c + d*x]^3 + 12*Log[Sin[c + d*x]] + 6*Sin[c + d*x] - 3*Sin[c + d*x]^2)/(6*a*d)","A",1
688,1,66,94,0.29515,"\int \frac{\cos ^2(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","-\frac{12 \sin (c+d x)+3 \csc ^4(c+d x)-4 \csc ^3(c+d x)-12 \csc ^2(c+d x)+24 \csc (c+d x)-12 \log (\sin (c+d x))}{12 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,"-1/12*(24*Csc[c + d*x] - 12*Csc[c + d*x]^2 - 4*Csc[c + d*x]^3 + 3*Csc[c + d*x]^4 - 12*Log[Sin[c + d*x]] + 12*Sin[c + d*x])/(a*d)","A",1
689,1,68,100,0.101422,"\int \frac{\cos (c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","-\frac{12 \csc ^5(c+d x)-15 \csc ^4(c+d x)-40 \csc ^3(c+d x)+60 \csc ^2(c+d x)+60 \csc (c+d x)+60 \log (\sin (c+d x))}{60 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,"-1/60*(60*Csc[c + d*x] + 60*Csc[c + d*x]^2 - 40*Csc[c + d*x]^3 - 15*Csc[c + d*x]^4 + 12*Csc[c + d*x]^5 + 60*Log[Sin[c + d*x]])/(a*d)","A",1
690,1,61,68,0.1344733,"\int \frac{\cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^7/(a + a*Sin[c + d*x]),x]","\frac{\csc ^6(c+d x) (78 \sin (c+d x)-5 (7 \sin (3 (c+d x))-3 \sin (5 (c+d x))+5)-15 \cos (4 (c+d x)))}{240 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,"(Csc[c + d*x]^6*(-15*Cos[4*(c + d*x)] + 78*Sin[c + d*x] - 5*(5 + 7*Sin[3*(c + d*x)] - 3*Sin[5*(c + d*x)])))/(240*a*d)","A",1
691,1,68,73,0.1569118,"\int \frac{\cot ^7(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^7*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\csc ^2(c+d x) \left(-30 \csc ^5(c+d x)+35 \csc ^4(c+d x)+84 \csc ^3(c+d x)-105 \csc ^2(c+d x)-70 \csc (c+d x)+105\right)}{210 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,"(Csc[c + d*x]^2*(105 - 70*Csc[c + d*x] - 105*Csc[c + d*x]^2 + 84*Csc[c + d*x]^3 + 35*Csc[c + d*x]^4 - 30*Csc[c + d*x]^5))/(210*a*d)","A",1
692,1,68,91,0.144861,"\int \frac{\cot ^7(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^7*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\csc ^3(c+d x) \left(-105 \csc ^5(c+d x)+120 \csc ^4(c+d x)+280 \csc ^3(c+d x)-336 \csc ^2(c+d x)-210 \csc (c+d x)+280\right)}{840 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,"(Csc[c + d*x]^3*(280 - 210*Csc[c + d*x] - 336*Csc[c + d*x]^2 + 280*Csc[c + d*x]^3 + 120*Csc[c + d*x]^4 - 105*Csc[c + d*x]^5))/(840*a*d)","A",1
693,1,68,91,0.1830532,"\int \frac{\cot ^7(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^7*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\csc ^4(c+d x) \left(-280 \csc ^5(c+d x)+315 \csc ^4(c+d x)+720 \csc ^3(c+d x)-840 \csc ^2(c+d x)-504 \csc (c+d x)+630\right)}{2520 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,"(Csc[c + d*x]^4*(630 - 504*Csc[c + d*x] - 840*Csc[c + d*x]^2 + 720*Csc[c + d*x]^3 + 315*Csc[c + d*x]^4 - 280*Csc[c + d*x]^5))/(2520*a*d)","A",1
694,1,68,109,0.1196578,"\int \frac{\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^7*Csc[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\csc ^5(c+d x) \left(-126 \csc ^5(c+d x)+140 \csc ^4(c+d x)+315 \csc ^3(c+d x)-360 \csc ^2(c+d x)-210 \csc (c+d x)+252\right)}{1260 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*(252 - 210*Csc[c + d*x] - 360*Csc[c + d*x]^2 + 315*Csc[c + d*x]^3 + 140*Csc[c + d*x]^4 - 126*Csc[c + d*x]^5))/(1260*a*d)","A",1
695,1,68,109,0.1278294,"\int \frac{\cot ^7(c+d x) \csc ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^7*Csc[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","\frac{\csc ^6(c+d x) \left(-1260 \csc ^5(c+d x)+1386 \csc ^4(c+d x)+3080 \csc ^3(c+d x)-3465 \csc ^2(c+d x)-1980 \csc (c+d x)+2310\right)}{13860 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*(2310 - 1980*Csc[c + d*x] - 3465*Csc[c + d*x]^2 + 3080*Csc[c + d*x]^3 + 1386*Csc[c + d*x]^4 - 1260*Csc[c + d*x]^5))/(13860*a*d)","A",1
696,1,68,109,0.1263587,"\int \frac{\cot ^7(c+d x) \csc ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^7*Csc[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","\frac{\csc ^7(c+d x) \left(-2310 \csc ^5(c+d x)+2520 \csc ^4(c+d x)+5544 \csc ^3(c+d x)-6160 \csc ^2(c+d x)-3465 \csc (c+d x)+3960\right)}{27720 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*(3960 - 3465*Csc[c + d*x] - 6160*Csc[c + d*x]^2 + 5544*Csc[c + d*x]^3 + 2520*Csc[c + d*x]^4 - 2310*Csc[c + d*x]^5))/(27720*a*d)","A",1
697,1,126,184,0.9552282,"\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[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) \left(-\frac{\sin ^9(c+d x)}{n+10}-\frac{3 \sin ^8(c+d x)}{n+9}+\frac{8 \sin ^6(c+d x)}{n+7}+\frac{6 \sin ^5(c+d x)}{n+6}-\frac{6 \sin ^4(c+d x)}{n+5}-\frac{8 \sin ^3(c+d x)}{n+4}+\frac{3 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right)}{d}","\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)*((1 + n)^(-1) + (3*Sin[c + d*x])/(2 + n) - (8*Sin[c + d*x]^3)/(4 + n) - (6*Sin[c + d*x]^4)/(5 + n) + (6*Sin[c + d*x]^5)/(6 + n) + (8*Sin[c + d*x]^6)/(7 + n) - (3*Sin[c + d*x]^8)/(9 + n) - Sin[c + d*x]^9/(10 + n)))/d","A",1
698,1,126,184,0.7799102,"\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[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) \left(-\frac{\sin ^8(c+d x)}{n+9}-\frac{2 \sin ^7(c+d x)}{n+8}+\frac{2 \sin ^6(c+d x)}{n+7}+\frac{6 \sin ^5(c+d x)}{n+6}-\frac{6 \sin ^3(c+d x)}{n+4}-\frac{2 \sin ^2(c+d x)}{n+3}+\frac{2 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right)}{d}","\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)*((1 + n)^(-1) + (2*Sin[c + d*x])/(2 + n) - (2*Sin[c + d*x]^2)/(3 + n) - (6*Sin[c + d*x]^3)/(4 + n) + (6*Sin[c + d*x]^5)/(6 + n) + (2*Sin[c + d*x]^6)/(7 + n) - (2*Sin[c + d*x]^7)/(8 + n) - Sin[c + d*x]^8/(9 + n)))/d","A",1
699,1,659,167,3.2392683,"\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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) \left(5 n^7 \sin (c+d x)+9 n^7 \sin (3 (c+d x))+5 n^7 \sin (5 (c+d x))+n^7 \sin (7 (c+d x))+2 n^7 \cos (6 (c+d x))+188 n^6 \sin (c+d x)+324 n^6 \sin (3 (c+d x))+164 n^6 \sin (5 (c+d x))+28 n^6 \sin (7 (c+d x))+58 n^6 \cos (6 (c+d x))+3050 n^5 \sin (c+d x)+4866 n^5 \sin (3 (c+d x))+2138 n^5 \sin (5 (c+d x))+322 n^5 \sin (7 (c+d x))+686 n^5 \cos (6 (c+d x))+28904 n^4 \sin (c+d x)+38232 n^4 \sin (3 (c+d x))+14360 n^4 \sin (5 (c+d x))+1960 n^4 \sin (7 (c+d x))+4270 n^4 \cos (6 (c+d x))+167669 n^3 \sin (c+d x)+165273 n^3 \sin (3 (c+d x))+53525 n^3 \sin (5 (c+d x))+6769 n^3 \sin (7 (c+d x))+15008 n^3 \cos (6 (c+d x))+552236 n^2 \sin (c+d x)+384948 n^2 \sin (3 (c+d x))+110036 n^2 \sin (5 (c+d x))+13132 n^2 \sin (7 (c+d x))+29512 n^2 \cos (6 (c+d x))+6 \left(5 n^7+177 n^6+2611 n^5+20499 n^4+90640 n^3+219828 n^2+262064 n+114816\right) \cos (2 (c+d x))+12 \left(n^7+33 n^6+439 n^5+3027 n^4+11584 n^3+24372 n^2+25776 n+10368\right) \cos (4 (c+d x))+879324 n \sin (c+d x)+439836 n \sin (3 (c+d x))+114252 n \sin (5 (c+d x))+13068 n \sin (7 (c+d x))+29664 n \cos (6 (c+d x))+468720 \sin (c+d x)+186480 \sin (3 (c+d x))+45360 \sin (5 (c+d x))+5040 \sin (7 (c+d x))+11520 \cos (6 (c+d x))+20 n^7+724 n^6+11084 n^5+94012 n^4+481280 n^3+1486096 n^2+2521536 n+1755648\right)}{64 d (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (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)*(1755648 + 2521536*n + 1486096*n^2 + 481280*n^3 + 94012*n^4 + 11084*n^5 + 724*n^6 + 20*n^7 + 6*(114816 + 262064*n + 219828*n^2 + 90640*n^3 + 20499*n^4 + 2611*n^5 + 177*n^6 + 5*n^7)*Cos[2*(c + d*x)] + 12*(10368 + 25776*n + 24372*n^2 + 11584*n^3 + 3027*n^4 + 439*n^5 + 33*n^6 + n^7)*Cos[4*(c + d*x)] + 11520*Cos[6*(c + d*x)] + 29664*n*Cos[6*(c + d*x)] + 29512*n^2*Cos[6*(c + d*x)] + 15008*n^3*Cos[6*(c + d*x)] + 4270*n^4*Cos[6*(c + d*x)] + 686*n^5*Cos[6*(c + d*x)] + 58*n^6*Cos[6*(c + d*x)] + 2*n^7*Cos[6*(c + d*x)] + 468720*Sin[c + d*x] + 879324*n*Sin[c + d*x] + 552236*n^2*Sin[c + d*x] + 167669*n^3*Sin[c + d*x] + 28904*n^4*Sin[c + d*x] + 3050*n^5*Sin[c + d*x] + 188*n^6*Sin[c + d*x] + 5*n^7*Sin[c + d*x] + 186480*Sin[3*(c + d*x)] + 439836*n*Sin[3*(c + d*x)] + 384948*n^2*Sin[3*(c + d*x)] + 165273*n^3*Sin[3*(c + d*x)] + 38232*n^4*Sin[3*(c + d*x)] + 4866*n^5*Sin[3*(c + d*x)] + 324*n^6*Sin[3*(c + d*x)] + 9*n^7*Sin[3*(c + d*x)] + 45360*Sin[5*(c + d*x)] + 114252*n*Sin[5*(c + d*x)] + 110036*n^2*Sin[5*(c + d*x)] + 53525*n^3*Sin[5*(c + d*x)] + 14360*n^4*Sin[5*(c + d*x)] + 2138*n^5*Sin[5*(c + d*x)] + 164*n^6*Sin[5*(c + d*x)] + 5*n^7*Sin[5*(c + d*x)] + 5040*Sin[7*(c + d*x)] + 13068*n*Sin[7*(c + d*x)] + 13132*n^2*Sin[7*(c + d*x)] + 6769*n^3*Sin[7*(c + d*x)] + 1960*n^4*Sin[7*(c + d*x)] + 322*n^5*Sin[7*(c + d*x)] + 28*n^6*Sin[7*(c + d*x)] + n^7*Sin[7*(c + d*x)]))/(64*d*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(8 + n))","B",1
700,1,95,137,0.3250784,"\int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x]),x]","\frac{\sin ^{n+1}(c+d x) \left(-\frac{\sin ^5(c+d x)}{n+6}+\frac{\sin ^4(c+d x)}{n+5}+\frac{2 \sin ^3(c+d x)}{n+4}-\frac{2 \sin ^2(c+d x)}{n+3}-\frac{\sin (c+d x)}{n+2}+\frac{1}{n+1}\right)}{a d}","\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)*((1 + n)^(-1) - Sin[c + d*x]/(2 + n) - (2*Sin[c + d*x]^2)/(3 + n) + (2*Sin[c + d*x]^3)/(4 + n) + Sin[c + d*x]^4/(5 + n) - Sin[c + d*x]^5/(6 + n)))/(a*d)","A",1
701,1,117,92,0.3609494,"\int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(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) \left(-\left(\left(n^3+7 n^2+14 n+8\right) \sin ^4(c+d x)\right)+2 \left(n^3+8 n^2+17 n+10\right) \sin ^3(c+d x)-2 \left(n^3+10 n^2+29 n+20\right) \sin (c+d x)+n^3+11 n^2+38 n+40\right)}{a^2 d (n+1) (n+2) (n+4) (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)*(40 + 38*n + 11*n^2 + n^3 - 2*(20 + 29*n + 10*n^2 + n^3)*Sin[c + d*x] + 2*(10 + 17*n + 8*n^2 + n^3)*Sin[c + d*x]^3 - (8 + 14*n + 7*n^2 + n^3)*Sin[c + d*x]^4))/(a^2*d*(1 + n)*(2 + n)*(4 + n)*(5 + n))","A",1
702,1,66,92,0.1969509,"\int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(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) \left(-\frac{\sin ^3(c+d x)}{n+4}+\frac{3 \sin ^2(c+d x)}{n+3}-\frac{3 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right)}{a^3 d}","\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)*((1 + n)^(-1) - (3*Sin[c + d*x])/(2 + n) + (3*Sin[c + d*x]^2)/(3 + n) - Sin[c + d*x]^3/(4 + n)))/(a^3*d)","A",1
703,1,104,109,0.2672342,"\int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^4,x]","\frac{\frac{8 a^3 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{n+1}-\frac{7 a^3 \sin ^{n+1}(c+d x)}{n+1}+\frac{4 a^3 \sin ^{n+2}(c+d x)}{n+2}-\frac{a^3 \sin ^{n+3}(c+d x)}{n+3}}{a^7 d}","\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*a^3*Sin[c + d*x]^(1 + n))/(1 + n) + (8*a^3*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(1 + n) + (4*a^3*Sin[c + d*x]^(2 + n))/(2 + n) - (a^3*Sin[c + d*x]^(3 + n))/(3 + n))/(a^7*d)","A",1
704,1,108,160,0.215686,"\int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx","Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^5,x]","\frac{\sin ^{n+1}(c+d x) \left(-4 \left(2 n^2+7 n+6\right) (\sin (c+d x)+1) \, _2F_1(1,n+1;n+2;-\sin (c+d x))-(n+1) \sin ^2(c+d x)+(4 n+9) \sin (c+d x)+8 n^2+29 n+26\right)}{a^5 d (n+1) (n+2) (\sin (c+d x)+1)}","-\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{\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{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)}",1,"(Sin[c + d*x]^(1 + n)*(26 + 29*n + 8*n^2 + (9 + 4*n)*Sin[c + d*x] - (1 + n)*Sin[c + d*x]^2 - 4*(6 + 7*n + 2*n^2)*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*(1 + Sin[c + d*x])))/(a^5*d*(1 + n)*(2 + n)*(1 + Sin[c + d*x]))","A",1
705,1,518,209,14.2786563,"\int \frac{\cos ^8(c+d x) \sin ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","-\frac{55440 d x \sin \left(\frac{c}{2}\right)-55440 \sin \left(\frac{c}{2}+d x\right)+55440 \sin \left(\frac{3 c}{2}+d x\right)-18480 \sin \left(\frac{5 c}{2}+3 d x\right)+18480 \sin \left(\frac{7 c}{2}+3 d x\right)-10395 \sin \left(\frac{7 c}{2}+4 d x\right)-10395 \sin \left(\frac{9 c}{2}+4 d x\right)+5544 \sin \left(\frac{9 c}{2}+5 d x\right)-5544 \sin \left(\frac{11 c}{2}+5 d x\right)+3960 \sin \left(\frac{13 c}{2}+7 d x\right)-3960 \sin \left(\frac{15 c}{2}+7 d x\right)+2079 \sin \left(\frac{15 c}{2}+8 d x\right)+2079 \sin \left(\frac{17 c}{2}+8 d x\right)-616 \sin \left(\frac{17 c}{2}+9 d x\right)+616 \sin \left(\frac{19 c}{2}+9 d x\right)-504 \sin \left(\frac{21 c}{2}+11 d x\right)+504 \sin \left(\frac{23 c}{2}+11 d x\right)-231 \sin \left(\frac{23 c}{2}+12 d x\right)-231 \sin \left(\frac{25 c}{2}+12 d x\right)+55440 d x \cos \left(\frac{c}{2}\right)+55440 \cos \left(\frac{c}{2}+d x\right)+55440 \cos \left(\frac{3 c}{2}+d x\right)+18480 \cos \left(\frac{5 c}{2}+3 d x\right)+18480 \cos \left(\frac{7 c}{2}+3 d x\right)-10395 \cos \left(\frac{7 c}{2}+4 d x\right)+10395 \cos \left(\frac{9 c}{2}+4 d x\right)-5544 \cos \left(\frac{9 c}{2}+5 d x\right)-5544 \cos \left(\frac{11 c}{2}+5 d x\right)-3960 \cos \left(\frac{13 c}{2}+7 d x\right)-3960 \cos \left(\frac{15 c}{2}+7 d x\right)+2079 \cos \left(\frac{15 c}{2}+8 d x\right)-2079 \cos \left(\frac{17 c}{2}+8 d x\right)+616 \cos \left(\frac{17 c}{2}+9 d x\right)+616 \cos \left(\frac{19 c}{2}+9 d x\right)+504 \cos \left(\frac{21 c}{2}+11 d x\right)+504 \cos \left(\frac{23 c}{2}+11 d x\right)-231 \cos \left(\frac{23 c}{2}+12 d x\right)+231 \cos \left(\frac{25 c}{2}+12 d x\right)+99792 \sin \left(\frac{c}{2}\right)}{11354112 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"-1/11354112*(55440*d*x*Cos[c/2] + 55440*Cos[c/2 + d*x] + 55440*Cos[(3*c)/2 + d*x] + 18480*Cos[(5*c)/2 + 3*d*x] + 18480*Cos[(7*c)/2 + 3*d*x] - 10395*Cos[(7*c)/2 + 4*d*x] + 10395*Cos[(9*c)/2 + 4*d*x] - 5544*Cos[(9*c)/2 + 5*d*x] - 5544*Cos[(11*c)/2 + 5*d*x] - 3960*Cos[(13*c)/2 + 7*d*x] - 3960*Cos[(15*c)/2 + 7*d*x] + 2079*Cos[(15*c)/2 + 8*d*x] - 2079*Cos[(17*c)/2 + 8*d*x] + 616*Cos[(17*c)/2 + 9*d*x] + 616*Cos[(19*c)/2 + 9*d*x] + 504*Cos[(21*c)/2 + 11*d*x] + 504*Cos[(23*c)/2 + 11*d*x] - 231*Cos[(23*c)/2 + 12*d*x] + 231*Cos[(25*c)/2 + 12*d*x] + 99792*Sin[c/2] + 55440*d*x*Sin[c/2] - 55440*Sin[c/2 + d*x] + 55440*Sin[(3*c)/2 + d*x] - 18480*Sin[(5*c)/2 + 3*d*x] + 18480*Sin[(7*c)/2 + 3*d*x] - 10395*Sin[(7*c)/2 + 4*d*x] - 10395*Sin[(9*c)/2 + 4*d*x] + 5544*Sin[(9*c)/2 + 5*d*x] - 5544*Sin[(11*c)/2 + 5*d*x] + 3960*Sin[(13*c)/2 + 7*d*x] - 3960*Sin[(15*c)/2 + 7*d*x] + 2079*Sin[(15*c)/2 + 8*d*x] + 2079*Sin[(17*c)/2 + 8*d*x] - 616*Sin[(17*c)/2 + 9*d*x] + 616*Sin[(19*c)/2 + 9*d*x] - 504*Sin[(21*c)/2 + 11*d*x] + 504*Sin[(23*c)/2 + 11*d*x] - 231*Sin[(23*c)/2 + 12*d*x] - 231*Sin[(25*c)/2 + 12*d*x])/(a*d*(Cos[c/2] + Sin[c/2]))","B",1
706,1,573,183,12.1607243,"\int \frac{\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\frac{97020 \sin ^2\left(\frac{1}{2} (c+d x)\right)}{d (a \sin (c+d x)+a)}+\frac{103950 \sin (c) \sin (d x)}{a d}-\frac{66990 \sin (3 c) \sin (3 d x)}{a d}+\frac{24948 \sin (5 c) \sin (5 d x)}{a d}-\frac{1980 \sin (7 c) \sin (7 d x)}{a d}-\frac{76230 \sin (2 (c+d x))}{a d}+\frac{27720 \sin (4 (c+d x))}{a d}-\frac{11550 \sin (6 (c+d x))}{a d}+\frac{3465 \sin (8 (c+d x))}{a d}+\frac{1386 \sin (10 (c+d x))}{a d}+\frac{48510 \sin (c+d x)}{a d (\sin (c+d x)+1)}-\frac{103950 \cos (c) \cos (d x)}{a d}+\frac{66990 \cos (3 c) \cos (3 d x)}{a d}-\frac{24948 \cos (5 c) \cos (5 d x)}{a d}+\frac{1980 \cos (7 c) \cos (7 d x)}{a d}+\frac{173250 \cos (c+d x)}{a d}-\frac{43890 \cos (3 (c+d x))}{a d}+\frac{18018 \cos (5 (c+d x))}{a d}-\frac{6930 \cos (7 (c+d x))}{a d}+\frac{770 \cos (9 (c+d x))}{a d}+\frac{630 \cos (11 (c+d x))}{a d}-\frac{20790 \sin \left(\frac{1}{2} (c+d x)\right)}{a d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{90090 \sin (2 c) \cos (2 d x)}{a d}-\frac{55440 \sin (4 c) \cos (4 d x)}{a d}+\frac{4620 \sin (6 c) \cos (6 d x)}{a d}+\frac{90090 \cos (2 c) \sin (2 d x)}{a d}-\frac{55440 \cos (4 c) \sin (4 d x)}{a d}+\frac{4620 \cos (6 c) \sin (6 d x)}{a d}-\frac{76230 \sin \left(\frac{d x}{2}\right)}{a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{97020 c}{a d}+\frac{83160 x}{a}}{7096320}","\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,"((97020*c)/(a*d) + (83160*x)/a - (103950*Cos[c]*Cos[d*x])/(a*d) + (66990*Cos[3*c]*Cos[3*d*x])/(a*d) - (24948*Cos[5*c]*Cos[5*d*x])/(a*d) + (1980*Cos[7*c]*Cos[7*d*x])/(a*d) + (173250*Cos[c + d*x])/(a*d) - (43890*Cos[3*(c + d*x)])/(a*d) + (18018*Cos[5*(c + d*x)])/(a*d) - (6930*Cos[7*(c + d*x)])/(a*d) + (770*Cos[9*(c + d*x)])/(a*d) + (630*Cos[11*(c + d*x)])/(a*d) + (90090*Cos[2*d*x]*Sin[2*c])/(a*d) - (55440*Cos[4*d*x]*Sin[4*c])/(a*d) + (4620*Cos[6*d*x]*Sin[6*c])/(a*d) + (103950*Sin[c]*Sin[d*x])/(a*d) + (90090*Cos[2*c]*Sin[2*d*x])/(a*d) - (66990*Sin[3*c]*Sin[3*d*x])/(a*d) - (55440*Cos[4*c]*Sin[4*d*x])/(a*d) + (24948*Sin[5*c]*Sin[5*d*x])/(a*d) + (4620*Cos[6*c]*Sin[6*d*x])/(a*d) - (1980*Sin[7*c]*Sin[7*d*x])/(a*d) - (76230*Sin[(d*x)/2])/(a*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) - (20790*Sin[(c + d*x)/2])/(a*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (48510*Sin[c + d*x])/(a*d*(1 + Sin[c + d*x])) + (97020*Sin[(c + d*x)/2]^2)/(d*(a + a*Sin[c + d*x])) - (76230*Sin[2*(c + d*x)])/(a*d) + (27720*Sin[4*(c + d*x)])/(a*d) - (11550*Sin[6*(c + d*x)])/(a*d) + (3465*Sin[8*(c + d*x)])/(a*d) + (1386*Sin[10*(c + d*x)])/(a*d))/7096320","B",1
707,1,533,165,14.1635934,"\int \frac{\cos ^8(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{15120 d x \sin \left(\frac{c}{2}\right)-15120 \sin \left(\frac{c}{2}+d x\right)+15120 \sin \left(\frac{3 c}{2}+d x\right)+1260 \sin \left(\frac{3 c}{2}+2 d x\right)+1260 \sin \left(\frac{5 c}{2}+2 d x\right)-6720 \sin \left(\frac{5 c}{2}+3 d x\right)+6720 \sin \left(\frac{7 c}{2}+3 d x\right)-2520 \sin \left(\frac{7 c}{2}+4 d x\right)-2520 \sin \left(\frac{9 c}{2}+4 d x\right)-630 \sin \left(\frac{11 c}{2}+6 d x\right)-630 \sin \left(\frac{13 c}{2}+6 d x\right)+1080 \sin \left(\frac{13 c}{2}+7 d x\right)-1080 \sin \left(\frac{15 c}{2}+7 d x\right)+315 \sin \left(\frac{15 c}{2}+8 d x\right)+315 \sin \left(\frac{17 c}{2}+8 d x\right)+280 \sin \left(\frac{17 c}{2}+9 d x\right)-280 \sin \left(\frac{19 c}{2}+9 d x\right)+126 \sin \left(\frac{19 c}{2}+10 d x\right)+126 \sin \left(\frac{21 c}{2}+10 d x\right)-1260 \cos \left(\frac{c}{2}\right) (25 c-12 d x)+15120 \cos \left(\frac{c}{2}+d x\right)+15120 \cos \left(\frac{3 c}{2}+d x\right)+1260 \cos \left(\frac{3 c}{2}+2 d x\right)-1260 \cos \left(\frac{5 c}{2}+2 d x\right)+6720 \cos \left(\frac{5 c}{2}+3 d x\right)+6720 \cos \left(\frac{7 c}{2}+3 d x\right)-2520 \cos \left(\frac{7 c}{2}+4 d x\right)+2520 \cos \left(\frac{9 c}{2}+4 d x\right)-630 \cos \left(\frac{11 c}{2}+6 d x\right)+630 \cos \left(\frac{13 c}{2}+6 d x\right)-1080 \cos \left(\frac{13 c}{2}+7 d x\right)-1080 \cos \left(\frac{15 c}{2}+7 d x\right)+315 \cos \left(\frac{15 c}{2}+8 d x\right)-315 \cos \left(\frac{17 c}{2}+8 d x\right)-280 \cos \left(\frac{17 c}{2}+9 d x\right)-280 \cos \left(\frac{19 c}{2}+9 d x\right)+126 \cos \left(\frac{19 c}{2}+10 d x\right)-126 \cos \left(\frac{21 c}{2}+10 d x\right)-31500 c \sin \left(\frac{c}{2}\right)+37800 \sin \left(\frac{c}{2}\right)}{1290240 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","\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,"-1/1290240*(-1260*(25*c - 12*d*x)*Cos[c/2] + 15120*Cos[c/2 + d*x] + 15120*Cos[(3*c)/2 + d*x] + 1260*Cos[(3*c)/2 + 2*d*x] - 1260*Cos[(5*c)/2 + 2*d*x] + 6720*Cos[(5*c)/2 + 3*d*x] + 6720*Cos[(7*c)/2 + 3*d*x] - 2520*Cos[(7*c)/2 + 4*d*x] + 2520*Cos[(9*c)/2 + 4*d*x] - 630*Cos[(11*c)/2 + 6*d*x] + 630*Cos[(13*c)/2 + 6*d*x] - 1080*Cos[(13*c)/2 + 7*d*x] - 1080*Cos[(15*c)/2 + 7*d*x] + 315*Cos[(15*c)/2 + 8*d*x] - 315*Cos[(17*c)/2 + 8*d*x] - 280*Cos[(17*c)/2 + 9*d*x] - 280*Cos[(19*c)/2 + 9*d*x] + 126*Cos[(19*c)/2 + 10*d*x] - 126*Cos[(21*c)/2 + 10*d*x] + 37800*Sin[c/2] - 31500*c*Sin[c/2] + 15120*d*x*Sin[c/2] - 15120*Sin[c/2 + d*x] + 15120*Sin[(3*c)/2 + d*x] + 1260*Sin[(3*c)/2 + 2*d*x] + 1260*Sin[(5*c)/2 + 2*d*x] - 6720*Sin[(5*c)/2 + 3*d*x] + 6720*Sin[(7*c)/2 + 3*d*x] - 2520*Sin[(7*c)/2 + 4*d*x] - 2520*Sin[(9*c)/2 + 4*d*x] - 630*Sin[(11*c)/2 + 6*d*x] - 630*Sin[(13*c)/2 + 6*d*x] + 1080*Sin[(13*c)/2 + 7*d*x] - 1080*Sin[(15*c)/2 + 7*d*x] + 315*Sin[(15*c)/2 + 8*d*x] + 315*Sin[(17*c)/2 + 8*d*x] + 280*Sin[(17*c)/2 + 9*d*x] - 280*Sin[(19*c)/2 + 9*d*x] + 126*Sin[(19*c)/2 + 10*d*x] + 126*Sin[(21*c)/2 + 10*d*x])/(a*d*(Cos[c/2] + Sin[c/2]))","B",1
708,1,479,139,9.6380011,"\int \frac{\cos ^8(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{-5040 d x \sin \left(\frac{c}{2}\right)+1512 \sin \left(\frac{c}{2}+d x\right)-1512 \sin \left(\frac{3 c}{2}+d x\right)-1008 \sin \left(\frac{3 c}{2}+2 d x\right)-1008 \sin \left(\frac{5 c}{2}+2 d x\right)+672 \sin \left(\frac{5 c}{2}+3 d x\right)-672 \sin \left(\frac{7 c}{2}+3 d x\right)+504 \sin \left(\frac{7 c}{2}+4 d x\right)+504 \sin \left(\frac{9 c}{2}+4 d x\right)+336 \sin \left(\frac{11 c}{2}+6 d x\right)+336 \sin \left(\frac{13 c}{2}+6 d x\right)-108 \sin \left(\frac{13 c}{2}+7 d x\right)+108 \sin \left(\frac{15 c}{2}+7 d x\right)+63 \sin \left(\frac{15 c}{2}+8 d x\right)+63 \sin \left(\frac{17 c}{2}+8 d x\right)-28 \sin \left(\frac{17 c}{2}+9 d x\right)+28 \sin \left(\frac{19 c}{2}+9 d x\right)+2520 \cos \left(\frac{c}{2}\right) (c-2 d x)-1512 \cos \left(\frac{c}{2}+d x\right)-1512 \cos \left(\frac{3 c}{2}+d x\right)-1008 \cos \left(\frac{3 c}{2}+2 d x\right)+1008 \cos \left(\frac{5 c}{2}+2 d x\right)-672 \cos \left(\frac{5 c}{2}+3 d x\right)-672 \cos \left(\frac{7 c}{2}+3 d x\right)+504 \cos \left(\frac{7 c}{2}+4 d x\right)-504 \cos \left(\frac{9 c}{2}+4 d x\right)+336 \cos \left(\frac{11 c}{2}+6 d x\right)-336 \cos \left(\frac{13 c}{2}+6 d x\right)+108 \cos \left(\frac{13 c}{2}+7 d x\right)+108 \cos \left(\frac{15 c}{2}+7 d x\right)+63 \cos \left(\frac{15 c}{2}+8 d x\right)-63 \cos \left(\frac{17 c}{2}+8 d x\right)+28 \cos \left(\frac{17 c}{2}+9 d x\right)+28 \cos \left(\frac{19 c}{2}+9 d x\right)+2520 c \sin \left(\frac{c}{2}\right)-7560 \sin \left(\frac{c}{2}\right)}{129024 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"-1/129024*(2520*(c - 2*d*x)*Cos[c/2] - 1512*Cos[c/2 + d*x] - 1512*Cos[(3*c)/2 + d*x] - 1008*Cos[(3*c)/2 + 2*d*x] + 1008*Cos[(5*c)/2 + 2*d*x] - 672*Cos[(5*c)/2 + 3*d*x] - 672*Cos[(7*c)/2 + 3*d*x] + 504*Cos[(7*c)/2 + 4*d*x] - 504*Cos[(9*c)/2 + 4*d*x] + 336*Cos[(11*c)/2 + 6*d*x] - 336*Cos[(13*c)/2 + 6*d*x] + 108*Cos[(13*c)/2 + 7*d*x] + 108*Cos[(15*c)/2 + 7*d*x] + 63*Cos[(15*c)/2 + 8*d*x] - 63*Cos[(17*c)/2 + 8*d*x] + 28*Cos[(17*c)/2 + 9*d*x] + 28*Cos[(19*c)/2 + 9*d*x] - 7560*Sin[c/2] + 2520*c*Sin[c/2] - 5040*d*x*Sin[c/2] + 1512*Sin[c/2 + d*x] - 1512*Sin[(3*c)/2 + d*x] - 1008*Sin[(3*c)/2 + 2*d*x] - 1008*Sin[(5*c)/2 + 2*d*x] + 672*Sin[(5*c)/2 + 3*d*x] - 672*Sin[(7*c)/2 + 3*d*x] + 504*Sin[(7*c)/2 + 4*d*x] + 504*Sin[(9*c)/2 + 4*d*x] + 336*Sin[(11*c)/2 + 6*d*x] + 336*Sin[(13*c)/2 + 6*d*x] - 108*Sin[(13*c)/2 + 7*d*x] + 108*Sin[(15*c)/2 + 7*d*x] + 63*Sin[(15*c)/2 + 8*d*x] + 63*Sin[(17*c)/2 + 8*d*x] - 28*Sin[(17*c)/2 + 9*d*x] + 28*Sin[(19*c)/2 + 9*d*x])/(a*d*(Cos[c/2] + Sin[c/2]))","B",1
709,1,481,121,10.354955,"\int \frac{\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{1680 d x \sin \left(\frac{c}{2}\right)-1680 \sin \left(\frac{c}{2}+d x\right)+1680 \sin \left(\frac{3 c}{2}+d x\right)+336 \sin \left(\frac{3 c}{2}+2 d x\right)+336 \sin \left(\frac{5 c}{2}+2 d x\right)-1008 \sin \left(\frac{5 c}{2}+3 d x\right)+1008 \sin \left(\frac{7 c}{2}+3 d x\right)-168 \sin \left(\frac{7 c}{2}+4 d x\right)-168 \sin \left(\frac{9 c}{2}+4 d x\right)-336 \sin \left(\frac{9 c}{2}+5 d x\right)+336 \sin \left(\frac{11 c}{2}+5 d x\right)-112 \sin \left(\frac{11 c}{2}+6 d x\right)-112 \sin \left(\frac{13 c}{2}+6 d x\right)-48 \sin \left(\frac{13 c}{2}+7 d x\right)+48 \sin \left(\frac{15 c}{2}+7 d x\right)-21 \sin \left(\frac{15 c}{2}+8 d x\right)-21 \sin \left(\frac{17 c}{2}+8 d x\right)-336 \cos \left(\frac{c}{2}\right) (7 c-5 d x)+1680 \cos \left(\frac{c}{2}+d x\right)+1680 \cos \left(\frac{3 c}{2}+d x\right)+336 \cos \left(\frac{3 c}{2}+2 d x\right)-336 \cos \left(\frac{5 c}{2}+2 d x\right)+1008 \cos \left(\frac{5 c}{2}+3 d x\right)+1008 \cos \left(\frac{7 c}{2}+3 d x\right)-168 \cos \left(\frac{7 c}{2}+4 d x\right)+168 \cos \left(\frac{9 c}{2}+4 d x\right)+336 \cos \left(\frac{9 c}{2}+5 d x\right)+336 \cos \left(\frac{11 c}{2}+5 d x\right)-112 \cos \left(\frac{11 c}{2}+6 d x\right)+112 \cos \left(\frac{13 c}{2}+6 d x\right)+48 \cos \left(\frac{13 c}{2}+7 d x\right)+48 \cos \left(\frac{15 c}{2}+7 d x\right)-21 \cos \left(\frac{15 c}{2}+8 d x\right)+21 \cos \left(\frac{17 c}{2}+8 d x\right)-2352 c \sin \left(\frac{c}{2}\right)+4704 \sin \left(\frac{c}{2}\right)}{43008 a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"-1/43008*(-336*(7*c - 5*d*x)*Cos[c/2] + 1680*Cos[c/2 + d*x] + 1680*Cos[(3*c)/2 + d*x] + 336*Cos[(3*c)/2 + 2*d*x] - 336*Cos[(5*c)/2 + 2*d*x] + 1008*Cos[(5*c)/2 + 3*d*x] + 1008*Cos[(7*c)/2 + 3*d*x] - 168*Cos[(7*c)/2 + 4*d*x] + 168*Cos[(9*c)/2 + 4*d*x] + 336*Cos[(9*c)/2 + 5*d*x] + 336*Cos[(11*c)/2 + 5*d*x] - 112*Cos[(11*c)/2 + 6*d*x] + 112*Cos[(13*c)/2 + 6*d*x] + 48*Cos[(13*c)/2 + 7*d*x] + 48*Cos[(15*c)/2 + 7*d*x] - 21*Cos[(15*c)/2 + 8*d*x] + 21*Cos[(17*c)/2 + 8*d*x] + 4704*Sin[c/2] - 2352*c*Sin[c/2] + 1680*d*x*Sin[c/2] - 1680*Sin[c/2 + d*x] + 1680*Sin[(3*c)/2 + d*x] + 336*Sin[(3*c)/2 + 2*d*x] + 336*Sin[(5*c)/2 + 2*d*x] - 1008*Sin[(5*c)/2 + 3*d*x] + 1008*Sin[(7*c)/2 + 3*d*x] - 168*Sin[(7*c)/2 + 4*d*x] - 168*Sin[(9*c)/2 + 4*d*x] - 336*Sin[(9*c)/2 + 5*d*x] + 336*Sin[(11*c)/2 + 5*d*x] - 112*Sin[(11*c)/2 + 6*d*x] - 112*Sin[(13*c)/2 + 6*d*x] - 48*Sin[(13*c)/2 + 7*d*x] + 48*Sin[(15*c)/2 + 7*d*x] - 21*Sin[(15*c)/2 + 8*d*x] - 21*Sin[(17*c)/2 + 8*d*x])/(a*d*(Cos[c/2] + Sin[c/2]))","B",1
710,1,102,143,0.265433,"\int \frac{\cos ^7(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^7*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{225 \sin (2 (c+d x))+45 \sin (4 (c+d x))+5 \sin (6 (c+d x))-1320 \cos (c+d x)-140 \cos (3 (c+d x))-12 \cos (5 (c+d x))-960 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+960 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+300 c+300 d x}{960 a d}","\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,"-1/960*(300*c + 300*d*x - 1320*Cos[c + d*x] - 140*Cos[3*(c + d*x)] - 12*Cos[5*(c + d*x)] + 960*Log[Cos[(c + d*x)/2]] - 960*Log[Sin[(c + d*x)/2]] + 225*Sin[2*(c + d*x)] + 45*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)])/(a*d)","A",1
711,1,146,137,0.6989403,"\int \frac{\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^6*Cot[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(1800 c \sin (c+d x)+1800 d x \sin (c+d x)+590 \sin (2 (c+d x))+64 \sin (4 (c+d x))+6 \sin (6 (c+d x))+1200 \cos (c+d x)-225 \cos (3 (c+d x))-15 \cos (5 (c+d x))+960 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-960 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1920 a d}","-\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,"-1/1920*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(1200*Cos[c + d*x] - 225*Cos[3*(c + d*x)] - 15*Cos[5*(c + d*x)] + 1800*c*Sin[c + d*x] + 1800*d*x*Sin[c + d*x] - 960*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 960*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 590*Sin[2*(c + d*x)] + 64*Sin[4*(c + d*x)] + 6*Sin[6*(c + d*x)]))/(a*d)","A",1
712,1,179,150,0.5320324,"\int \frac{\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{\left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-285 \sin (2 (c+d x))+42 \sin (4 (c+d x))+3 \sin (6 (c+d x))+400 \cos (c+d x)-200 \cos (3 (c+d x))-8 \cos (5 (c+d x))+480 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-480 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+120 \cos (2 (c+d x)) \left(-4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 c+3 d x\right)-360 c-360 d x\right)}{1536 a d (\sin (c+d x)+1)}","-\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 ^4(c+d x) \cot (c+d x)}{4 a d}-\frac{\cos ^3(c+d x) \cot ^2(c+d x)}{2 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,"-1/1536*((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(-360*c - 360*d*x + 400*Cos[c + d*x] - 200*Cos[3*(c + d*x)] - 8*Cos[5*(c + d*x)] - 480*Log[Cos[(c + d*x)/2]] + 120*Cos[2*(c + d*x)]*(3*c + 3*d*x + 4*Log[Cos[(c + d*x)/2]] - 4*Log[Sin[(c + d*x)/2]]) + 480*Log[Sin[(c + d*x)/2]] - 285*Sin[2*(c + d*x)] + 42*Sin[4*(c + d*x)] + 3*Sin[6*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","A",1
713,1,197,146,0.8088437,"\int \frac{\cos ^4(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^3(c+d x) \left(-180 c \sin (c+d x)-180 d x \sin (c+d x)-75 \sin (2 (c+d x))+60 c \sin (3 (c+d x))+60 d x \sin (3 (c+d x))+24 \sin (4 (c+d x))+\sin (6 (c+d x))-30 \cos (c+d x)+65 \cos (3 (c+d x))-3 \cos (5 (c+d x))-180 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+60 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+180 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-60 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{96 a d}","\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,"-1/96*(Csc[c + d*x]^3*(-30*Cos[c + d*x] + 65*Cos[3*(c + d*x)] - 3*Cos[5*(c + d*x)] - 180*c*Sin[c + d*x] - 180*d*x*Sin[c + d*x] + 180*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 180*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 75*Sin[2*(c + d*x)] + 60*c*Sin[3*(c + d*x)] + 60*d*x*Sin[3*(c + d*x)] - 60*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 60*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 24*Sin[4*(c + d*x)] + Sin[6*(c + d*x)]))/(a*d)","A",1
714,1,252,150,0.7489477,"\int \frac{\cos ^3(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^4(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(95 \sin (2 (c+d x))-68 \sin (4 (c+d x))+3 \sin (6 (c+d x))+60 c \cos (4 (c+d x))-30 \cos (c+d x)+90 \cos (3 (c+d x))+60 d x \cos (4 (c+d x))-12 \cos (5 (c+d x))-135 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+45 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+135 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-60 \cos (2 (c+d x)) \left(-3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 c+4 d x\right)-45 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+180 c+180 d x\right)}{192 a d (\sin (c+d x)+1)}","\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,"-1/192*(Csc[c + d*x]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(180*c + 180*d*x - 30*Cos[c + d*x] + 90*Cos[3*(c + d*x)] + 60*c*Cos[4*(c + d*x)] + 60*d*x*Cos[4*(c + d*x)] - 12*Cos[5*(c + d*x)] + 135*Log[Cos[(c + d*x)/2]] + 45*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 60*Cos[2*(c + d*x)]*(4*c + 4*d*x + 3*Log[Cos[(c + d*x)/2]] - 3*Log[Sin[(c + d*x)/2]]) - 135*Log[Sin[(c + d*x)/2]] - 45*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 95*Sin[2*(c + d*x)] - 68*Sin[4*(c + d*x)] + 3*Sin[6*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","A",1
715,1,264,138,0.9990427,"\int \frac{\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^5(c+d x) \left(1200 c \sin (c+d x)+1200 d x \sin (c+d x)+600 \sin (2 (c+d x))-600 c \sin (3 (c+d x))-600 d x \sin (3 (c+d x))-510 \sin (4 (c+d x))+120 c \sin (5 (c+d x))+120 d x \sin (5 (c+d x))+60 \sin (6 (c+d x))+400 \cos (c+d x)-200 \cos (3 (c+d x))+184 \cos (5 (c+d x))+2250 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-1125 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+225 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2250 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+1125 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-225 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1920 a d}","-\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,"-1/1920*(Csc[c + d*x]^5*(400*Cos[c + d*x] - 200*Cos[3*(c + d*x)] + 184*Cos[5*(c + d*x)] + 1200*c*Sin[c + d*x] + 1200*d*x*Sin[c + d*x] - 2250*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 2250*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 600*Sin[2*(c + d*x)] - 600*c*Sin[3*(c + d*x)] - 600*d*x*Sin[3*(c + d*x)] + 1125*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 1125*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 510*Sin[4*(c + d*x)] + 120*c*Sin[5*(c + d*x)] + 120*d*x*Sin[5*(c + d*x)] - 225*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 225*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 60*Sin[6*(c + d*x)]))/(a*d)","A",1
716,1,317,142,1.0219728,"\int \frac{\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^6(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-1200 \sin (2 (c+d x))+768 \sin (4 (c+d x))-368 \sin (6 (c+d x))-1440 c \cos (4 (c+d x))+240 c \cos (6 (c+d x))+900 \cos (c+d x)+50 \cos (3 (c+d x))-1440 d x \cos (4 (c+d x))+330 \cos (5 (c+d x))+240 d x \cos (6 (c+d x))+750 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-450 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+75 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-750 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+225 \cos (2 (c+d x)) \left(16 (c+d x)-5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+450 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-75 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2400 c-2400 d x\right)}{7680 a d (\sin (c+d x)+1)}","\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,"-1/7680*(Csc[c + d*x]^6*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(-2400*c - 2400*d*x + 900*Cos[c + d*x] + 50*Cos[3*(c + d*x)] - 1440*c*Cos[4*(c + d*x)] - 1440*d*x*Cos[4*(c + d*x)] + 330*Cos[5*(c + d*x)] + 240*c*Cos[6*(c + d*x)] + 240*d*x*Cos[6*(c + d*x)] - 750*Log[Cos[(c + d*x)/2]] - 450*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 75*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 225*Cos[2*(c + d*x)]*(16*(c + d*x) + 5*Log[Cos[(c + d*x)/2]] - 5*Log[Sin[(c + d*x)/2]]) + 750*Log[Sin[(c + d*x)/2]] + 450*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 75*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 1200*Sin[2*(c + d*x)] + 768*Sin[4*(c + d*x)] - 368*Sin[6*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","B",1
717,1,284,106,0.9670856,"\int \frac{\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^8/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^5(c+d x) \left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-1190 \sin (2 (c+d x))+392 \sin (4 (c+d x))-462 \sin (6 (c+d x))+1680 \cos (c+d x)+1008 \cos (3 (c+d x))+336 \cos (5 (c+d x))+48 \cos (7 (c+d x))-3675 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2205 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-735 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+105 \sin (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+3675 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2205 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+735 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-105 \sin (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{86016 a d (\sin (c+d x)+1)}","-\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,"-1/86016*(Csc[c + d*x]^5*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(1680*Cos[c + d*x] + 1008*Cos[3*(c + d*x)] + 336*Cos[5*(c + d*x)] + 48*Cos[7*(c + d*x)] + 3675*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 3675*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 1190*Sin[2*(c + d*x)] - 2205*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 2205*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 392*Sin[4*(c + d*x)] + 735*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 735*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 462*Sin[6*(c + d*x)] - 105*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] + 105*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","B",1
718,1,291,134,1.0040427,"\int \frac{\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^8*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\csc ^8(c+d x) \left(5376 \sin (2 (c+d x))+5376 \sin (4 (c+d x))+2304 \sin (6 (c+d x))+384 \sin (8 (c+d x))-24710 \cos (c+d x)-12530 \cos (3 (c+d x))-5558 \cos (5 (c+d x))-210 \cos (7 (c+d x))-3675 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-5880 \cos (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2940 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-840 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+105 \cos (8 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3675 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+5880 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2940 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+840 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-105 \cos (8 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{344064 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,"(Csc[c + d*x]^8*(-24710*Cos[c + d*x] - 12530*Cos[3*(c + d*x)] - 5558*Cos[5*(c + d*x)] - 210*Cos[7*(c + d*x)] + 3675*Log[Cos[(c + d*x)/2]] - 5880*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 2940*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 840*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 105*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 3675*Log[Sin[(c + d*x)/2]] + 5880*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 2940*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 840*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 105*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 5376*Sin[2*(c + d*x)] + 5376*Sin[4*(c + d*x)] + 2304*Sin[6*(c + d*x)] + 384*Sin[8*(c + d*x)]))/(344064*a*d)","B",1
719,1,313,152,1.3661362,"\int \frac{\cot ^8(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^9(c+d x) \left(-36540 \sin (2 (c+d x))-20916 \sin (4 (c+d x))-16044 \sin (6 (c+d x))-630 \sin (8 (c+d x))+129024 \cos (c+d x)+75264 \cos (3 (c+d x))+23040 \cos (5 (c+d x))+2304 \cos (7 (c+d x))-256 \cos (9 (c+d x))-39690 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+26460 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-11340 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2835 \sin (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-315 \sin (9 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+39690 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-26460 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+11340 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2835 \sin (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+315 \sin (9 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2064384 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,"-1/2064384*(Csc[c + d*x]^9*(129024*Cos[c + d*x] + 75264*Cos[3*(c + d*x)] + 23040*Cos[5*(c + d*x)] + 2304*Cos[7*(c + d*x)] - 256*Cos[9*(c + d*x)] + 39690*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 39690*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 36540*Sin[2*(c + d*x)] - 26460*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 26460*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 20916*Sin[4*(c + d*x)] + 11340*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 11340*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 16044*Sin[6*(c + d*x)] - 2835*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] + 2835*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 630*Sin[8*(c + d*x)] + 315*Log[Cos[(c + d*x)/2]]*Sin[9*(c + d*x)] - 315*Log[Sin[(c + d*x)/2]]*Sin[9*(c + d*x)]))/(a*d)","B",1
720,1,386,176,1.5457313,"\int \frac{\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^9(c+d x) \left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-537600 \sin (2 (c+d x))-522240 \sin (4 (c+d x))-207360 \sin (6 (c+d x))-25600 \sin (8 (c+d x))+2560 \sin (10 (c+d x))+2367540 \cos (c+d x)+1307880 \cos (3 (c+d x))+436968 \cos (5 (c+d x))+18270 \cos (7 (c+d x))-1890 \cos (9 (c+d x))+119070 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+198450 \cos (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-113400 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+42525 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-9450 \cos (8 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+945 \cos (10 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-119070 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-198450 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+113400 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-42525 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+9450 \cos (8 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-945 \cos (10 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{165150720 a d (\csc (c+d x)+1)}","\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,"-1/165150720*(Csc[c + d*x]^9*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(2367540*Cos[c + d*x] + 1307880*Cos[3*(c + d*x)] + 436968*Cos[5*(c + d*x)] + 18270*Cos[7*(c + d*x)] - 1890*Cos[9*(c + d*x)] - 119070*Log[Cos[(c + d*x)/2]] + 198450*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 113400*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 42525*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 9450*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 945*Cos[10*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 119070*Log[Sin[(c + d*x)/2]] - 198450*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 113400*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 42525*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 9450*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 945*Cos[10*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 537600*Sin[2*(c + d*x)] - 522240*Sin[4*(c + d*x)] - 207360*Sin[6*(c + d*x)] - 25600*Sin[8*(c + d*x)] + 2560*Sin[10*(c + d*x)]))/(a*d*(1 + Csc[c + d*x]))","B",1
721,1,187,194,2.9419906,"\int \frac{\cot ^8(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-2661120 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)-\cot (c+d x) \csc ^{10}(c+d x) (-3219678 \sin (c+d x)-2608452 \sin (3 (c+d x))-2181564 \sin (5 (c+d x))-121275 \sin (7 (c+d x))+10395 \sin (9 (c+d x))+9973760 \cos (2 (c+d x))+3543040 \cos (4 (c+d x))+343040 \cos (6 (c+d x))-61440 \cos (8 (c+d x))+5120 \cos (10 (c+d x))+6840320)\right)}{227082240 a d (\sin (c+d x)+1)}","-\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(-2661120*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - Cot[c + d*x]*Csc[c + d*x]^10*(6840320 + 9973760*Cos[2*(c + d*x)] + 3543040*Cos[4*(c + d*x)] + 343040*Cos[6*(c + d*x)] - 61440*Cos[8*(c + d*x)] + 5120*Cos[10*(c + d*x)] - 3219678*Sin[c + d*x] - 2608452*Sin[3*(c + d*x)] - 2181564*Sin[5*(c + d*x)] - 121275*Sin[7*(c + d*x)] + 10395*Sin[9*(c + d*x)])))/(227082240*a*d*(1 + Sin[c + d*x]))","A",1
722,1,1453,203,10.5916276,"\int \frac{\cos ^8(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]","-\frac{5 \cos (c+d x) (2 \sin (c+d x)+1)}{3072 a^2 d (\sin (c+d x)+1)^2}+\frac{27720 (c+d x)+41580 \cos (c+d x)-7056 \cos (3 (c+d x))+1764 \cos (5 (c+d x))-360 \cos (7 (c+d x))+28 \cos (9 (c+d x))-15120 \sin (2 (c+d x))+3528 \sin (4 (c+d x))-840 \sin (6 (c+d x))+126 \sin (8 (c+d x))-\frac{15204 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{21}{\left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{42 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}}{86016 a^2 d}+\frac{-360360 (c+d x)-566280 \cos (c+d x)+108900 \cos (3 (c+d x))-33264 \cos (5 (c+d x))+9900 \cos (7 (c+d x))-2200 \cos (9 (c+d x))+180 \cos (11 (c+d x))+217800 \sin (2 (c+d x))-59400 \sin (4 (c+d x))+18480 \sin (6 (c+d x))-4950 \sin (8 (c+d x))+792 \sin (10 (c+d x))+\frac{166980 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{165}{\left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{330 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}}{2027520 a^2 d}+\frac{25 \left(36 d x \cos \left(\frac{d x}{2}\right)-21 \cos \left(c+\frac{d x}{2}\right)+35 \cos \left(c+\frac{3 d x}{2}\right)-12 d x \cos \left(2 c+\frac{3 d x}{2}\right)-3 \cos \left(3 c+\frac{5 d x}{2}\right)-57 \sin \left(\frac{d x}{2}\right)+36 d x \sin \left(c+\frac{d x}{2}\right)+12 d x \sin \left(c+\frac{3 d x}{2}\right)+9 \sin \left(2 c+\frac{3 d x}{2}\right)+3 \sin \left(2 c+\frac{5 d x}{2}\right)\right)}{12288 a^2 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{5 \left(180 d x \cos \left(\frac{d x}{2}\right)-21 \cos \left(c+\frac{d x}{2}\right)+147 \cos \left(c+\frac{3 d x}{2}\right)-60 d x \cos \left(2 c+\frac{3 d x}{2}\right)-15 \cos \left(3 c+\frac{5 d x}{2}\right)+3 \cos \left(3 c+\frac{7 d x}{2}\right)+\cos \left(5 c+\frac{9 d x}{2}\right)-201 \sin \left(\frac{d x}{2}\right)+180 d x \sin \left(c+\frac{d x}{2}\right)+60 d x \sin \left(c+\frac{3 d x}{2}\right)+73 \sin \left(2 c+\frac{3 d x}{2}\right)+15 \sin \left(2 c+\frac{5 d x}{2}\right)+3 \sin \left(4 c+\frac{7 d x}{2}\right)-\sin \left(4 c+\frac{9 d x}{2}\right)\right)}{12288 a^2 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{7 \left(2520 d x \cos \left(\frac{d x}{2}\right)+165 \cos \left(c+\frac{d x}{2}\right)+1905 \cos \left(c+\frac{3 d x}{2}\right)-840 d x \cos \left(2 c+\frac{3 d x}{2}\right)-210 \cos \left(3 c+\frac{5 d x}{2}\right)+42 \cos \left(3 c+\frac{7 d x}{2}\right)+14 \cos \left(5 c+\frac{9 d x}{2}\right)-6 \cos \left(5 c+\frac{11 d x}{2}\right)-3 \cos \left(7 c+\frac{13 d x}{2}\right)-2355 \sin \left(\frac{d x}{2}\right)+2520 d x \sin \left(c+\frac{d x}{2}\right)+840 d x \sin \left(c+\frac{3 d x}{2}\right)+1175 \sin \left(2 c+\frac{3 d x}{2}\right)+210 \sin \left(2 c+\frac{5 d x}{2}\right)+42 \sin \left(4 c+\frac{7 d x}{2}\right)-14 \sin \left(4 c+\frac{9 d x}{2}\right)-6 \sin \left(6 c+\frac{11 d x}{2}\right)+3 \sin \left(6 c+\frac{13 d x}{2}\right)\right)}{30720 a^2 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{7560 d x \cos \left(\frac{d x}{2}\right)+1239 \cos \left(c+\frac{d x}{2}\right)+5467 \cos \left(c+\frac{3 d x}{2}\right)-2520 d x \cos \left(2 c+\frac{3 d x}{2}\right)-630 \cos \left(3 c+\frac{5 d x}{2}\right)+126 \cos \left(3 c+\frac{7 d x}{2}\right)+42 \cos \left(5 c+\frac{9 d x}{2}\right)-18 \cos \left(5 c+\frac{11 d x}{2}\right)-9 \cos \left(7 c+\frac{13 d x}{2}\right)+5 \cos \left(7 c+\frac{15 d x}{2}\right)+3 \cos \left(9 c+\frac{17 d x}{2}\right)-6321 \sin \left(\frac{d x}{2}\right)+7560 d x \sin \left(c+\frac{d x}{2}\right)+2520 d x \sin \left(c+\frac{3 d x}{2}\right)+3773 \sin \left(2 c+\frac{3 d x}{2}\right)+630 \sin \left(2 c+\frac{5 d x}{2}\right)+126 \sin \left(4 c+\frac{7 d x}{2}\right)-42 \sin \left(4 c+\frac{9 d x}{2}\right)-18 \sin \left(6 c+\frac{11 d x}{2}\right)+9 \sin \left(6 c+\frac{13 d x}{2}\right)+5 \sin \left(8 c+\frac{15 d x}{2}\right)-3 \sin \left(8 c+\frac{17 d x}{2}\right)}{43008 a^2 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","\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,"(-5*Cos[c + d*x]*(1 + 2*Sin[c + d*x]))/(3072*a^2*d*(1 + Sin[c + d*x])^2) + (27720*(c + d*x) + 41580*Cos[c + d*x] - 7056*Cos[3*(c + d*x)] + 1764*Cos[5*(c + d*x)] - 360*Cos[7*(c + d*x)] + 28*Cos[9*(c + d*x)] + (42*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 21/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (15204*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 15120*Sin[2*(c + d*x)] + 3528*Sin[4*(c + d*x)] - 840*Sin[6*(c + d*x)] + 126*Sin[8*(c + d*x)])/(86016*a^2*d) + (-360360*(c + d*x) - 566280*Cos[c + d*x] + 108900*Cos[3*(c + d*x)] - 33264*Cos[5*(c + d*x)] + 9900*Cos[7*(c + d*x)] - 2200*Cos[9*(c + d*x)] + 180*Cos[11*(c + d*x)] - (330*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 165/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (166980*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 217800*Sin[2*(c + d*x)] - 59400*Sin[4*(c + d*x)] + 18480*Sin[6*(c + d*x)] - 4950*Sin[8*(c + d*x)] + 792*Sin[10*(c + d*x)])/(2027520*a^2*d) + (25*(36*d*x*Cos[(d*x)/2] - 21*Cos[c + (d*x)/2] + 35*Cos[c + (3*d*x)/2] - 12*d*x*Cos[2*c + (3*d*x)/2] - 3*Cos[3*c + (5*d*x)/2] - 57*Sin[(d*x)/2] + 36*d*x*Sin[c + (d*x)/2] + 12*d*x*Sin[c + (3*d*x)/2] + 9*Sin[2*c + (3*d*x)/2] + 3*Sin[2*c + (5*d*x)/2]))/(12288*a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (5*(180*d*x*Cos[(d*x)/2] - 21*Cos[c + (d*x)/2] + 147*Cos[c + (3*d*x)/2] - 60*d*x*Cos[2*c + (3*d*x)/2] - 15*Cos[3*c + (5*d*x)/2] + 3*Cos[3*c + (7*d*x)/2] + Cos[5*c + (9*d*x)/2] - 201*Sin[(d*x)/2] + 180*d*x*Sin[c + (d*x)/2] + 60*d*x*Sin[c + (3*d*x)/2] + 73*Sin[2*c + (3*d*x)/2] + 15*Sin[2*c + (5*d*x)/2] + 3*Sin[4*c + (7*d*x)/2] - Sin[4*c + (9*d*x)/2]))/(12288*a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (7*(2520*d*x*Cos[(d*x)/2] + 165*Cos[c + (d*x)/2] + 1905*Cos[c + (3*d*x)/2] - 840*d*x*Cos[2*c + (3*d*x)/2] - 210*Cos[3*c + (5*d*x)/2] + 42*Cos[3*c + (7*d*x)/2] + 14*Cos[5*c + (9*d*x)/2] - 6*Cos[5*c + (11*d*x)/2] - 3*Cos[7*c + (13*d*x)/2] - 2355*Sin[(d*x)/2] + 2520*d*x*Sin[c + (d*x)/2] + 840*d*x*Sin[c + (3*d*x)/2] + 1175*Sin[2*c + (3*d*x)/2] + 210*Sin[2*c + (5*d*x)/2] + 42*Sin[4*c + (7*d*x)/2] - 14*Sin[4*c + (9*d*x)/2] - 6*Sin[6*c + (11*d*x)/2] + 3*Sin[6*c + (13*d*x)/2]))/(30720*a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (7560*d*x*Cos[(d*x)/2] + 1239*Cos[c + (d*x)/2] + 5467*Cos[c + (3*d*x)/2] - 2520*d*x*Cos[2*c + (3*d*x)/2] - 630*Cos[3*c + (5*d*x)/2] + 126*Cos[3*c + (7*d*x)/2] + 42*Cos[5*c + (9*d*x)/2] - 18*Cos[5*c + (11*d*x)/2] - 9*Cos[7*c + (13*d*x)/2] + 5*Cos[7*c + (15*d*x)/2] + 3*Cos[9*c + (17*d*x)/2] - 6321*Sin[(d*x)/2] + 7560*d*x*Sin[c + (d*x)/2] + 2520*d*x*Sin[c + (3*d*x)/2] + 3773*Sin[2*c + (3*d*x)/2] + 630*Sin[2*c + (5*d*x)/2] + 126*Sin[4*c + (7*d*x)/2] - 42*Sin[4*c + (9*d*x)/2] - 18*Sin[6*c + (11*d*x)/2] + 9*Sin[6*c + (13*d*x)/2] + 5*Sin[8*c + (15*d*x)/2] - 3*Sin[8*c + (17*d*x)/2])/(43008*a^2*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","B",1
723,1,585,185,8.3819408,"\int \frac{\cos ^8(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{45360 d x \sin \left(\frac{c}{2}\right)-30240 \sin \left(\frac{c}{2}+d x\right)+30240 \sin \left(\frac{3 c}{2}+d x\right)-1260 \sin \left(\frac{3 c}{2}+2 d x\right)-1260 \sin \left(\frac{5 c}{2}+2 d x\right)-6720 \sin \left(\frac{5 c}{2}+3 d x\right)+6720 \sin \left(\frac{7 c}{2}+3 d x\right)-7560 \sin \left(\frac{7 c}{2}+4 d x\right)-7560 \sin \left(\frac{9 c}{2}+4 d x\right)+4032 \sin \left(\frac{9 c}{2}+5 d x\right)-4032 \sin \left(\frac{11 c}{2}+5 d x\right)+630 \sin \left(\frac{11 c}{2}+6 d x\right)+630 \sin \left(\frac{13 c}{2}+6 d x\right)+720 \sin \left(\frac{13 c}{2}+7 d x\right)-720 \sin \left(\frac{15 c}{2}+7 d x\right)+945 \sin \left(\frac{15 c}{2}+8 d x\right)+945 \sin \left(\frac{17 c}{2}+8 d x\right)-560 \sin \left(\frac{17 c}{2}+9 d x\right)+560 \sin \left(\frac{19 c}{2}+9 d x\right)-126 \sin \left(\frac{19 c}{2}+10 d x\right)-126 \sin \left(\frac{21 c}{2}+10 d x\right)-2520 \cos \left(\frac{c}{2}\right) (187 c-18 d x)+30240 \cos \left(\frac{c}{2}+d x\right)+30240 \cos \left(\frac{3 c}{2}+d x\right)-1260 \cos \left(\frac{3 c}{2}+2 d x\right)+1260 \cos \left(\frac{5 c}{2}+2 d x\right)+6720 \cos \left(\frac{5 c}{2}+3 d x\right)+6720 \cos \left(\frac{7 c}{2}+3 d x\right)-7560 \cos \left(\frac{7 c}{2}+4 d x\right)+7560 \cos \left(\frac{9 c}{2}+4 d x\right)-4032 \cos \left(\frac{9 c}{2}+5 d x\right)-4032 \cos \left(\frac{11 c}{2}+5 d x\right)+630 \cos \left(\frac{11 c}{2}+6 d x\right)-630 \cos \left(\frac{13 c}{2}+6 d x\right)-720 \cos \left(\frac{13 c}{2}+7 d x\right)-720 \cos \left(\frac{15 c}{2}+7 d x\right)+945 \cos \left(\frac{15 c}{2}+8 d x\right)-945 \cos \left(\frac{17 c}{2}+8 d x\right)+560 \cos \left(\frac{17 c}{2}+9 d x\right)+560 \cos \left(\frac{19 c}{2}+9 d x\right)-126 \cos \left(\frac{19 c}{2}+10 d x\right)+126 \cos \left(\frac{21 c}{2}+10 d x\right)-471240 c \sin \left(\frac{c}{2}\right)+327180 \sin \left(\frac{c}{2}\right)}{1290240 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","\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,"(-2520*(187*c - 18*d*x)*Cos[c/2] + 30240*Cos[c/2 + d*x] + 30240*Cos[(3*c)/2 + d*x] - 1260*Cos[(3*c)/2 + 2*d*x] + 1260*Cos[(5*c)/2 + 2*d*x] + 6720*Cos[(5*c)/2 + 3*d*x] + 6720*Cos[(7*c)/2 + 3*d*x] - 7560*Cos[(7*c)/2 + 4*d*x] + 7560*Cos[(9*c)/2 + 4*d*x] - 4032*Cos[(9*c)/2 + 5*d*x] - 4032*Cos[(11*c)/2 + 5*d*x] + 630*Cos[(11*c)/2 + 6*d*x] - 630*Cos[(13*c)/2 + 6*d*x] - 720*Cos[(13*c)/2 + 7*d*x] - 720*Cos[(15*c)/2 + 7*d*x] + 945*Cos[(15*c)/2 + 8*d*x] - 945*Cos[(17*c)/2 + 8*d*x] + 560*Cos[(17*c)/2 + 9*d*x] + 560*Cos[(19*c)/2 + 9*d*x] - 126*Cos[(19*c)/2 + 10*d*x] + 126*Cos[(21*c)/2 + 10*d*x] + 327180*Sin[c/2] - 471240*c*Sin[c/2] + 45360*d*x*Sin[c/2] - 30240*Sin[c/2 + d*x] + 30240*Sin[(3*c)/2 + d*x] - 1260*Sin[(3*c)/2 + 2*d*x] - 1260*Sin[(5*c)/2 + 2*d*x] - 6720*Sin[(5*c)/2 + 3*d*x] + 6720*Sin[(7*c)/2 + 3*d*x] - 7560*Sin[(7*c)/2 + 4*d*x] - 7560*Sin[(9*c)/2 + 4*d*x] + 4032*Sin[(9*c)/2 + 5*d*x] - 4032*Sin[(11*c)/2 + 5*d*x] + 630*Sin[(11*c)/2 + 6*d*x] + 630*Sin[(13*c)/2 + 6*d*x] + 720*Sin[(13*c)/2 + 7*d*x] - 720*Sin[(15*c)/2 + 7*d*x] + 945*Sin[(15*c)/2 + 8*d*x] + 945*Sin[(17*c)/2 + 8*d*x] - 560*Sin[(17*c)/2 + 9*d*x] + 560*Sin[(19*c)/2 + 9*d*x] - 126*Sin[(19*c)/2 + 10*d*x] - 126*Sin[(21*c)/2 + 10*d*x])/(1290240*a^2*d*(Cos[c/2] + Sin[c/2]))","B",1
724,1,430,159,6.9106972,"\int \frac{\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","-\frac{15120 d x \sin \left(\frac{c}{2}\right)-11340 \sin \left(\frac{c}{2}+d x\right)+11340 \sin \left(\frac{3 c}{2}+d x\right)-3360 \sin \left(\frac{5 c}{2}+3 d x\right)+3360 \sin \left(\frac{7 c}{2}+3 d x\right)-2520 \sin \left(\frac{7 c}{2}+4 d x\right)-2520 \sin \left(\frac{9 c}{2}+4 d x\right)+1008 \sin \left(\frac{9 c}{2}+5 d x\right)-1008 \sin \left(\frac{11 c}{2}+5 d x\right)+450 \sin \left(\frac{13 c}{2}+7 d x\right)-450 \sin \left(\frac{15 c}{2}+7 d x\right)+315 \sin \left(\frac{15 c}{2}+8 d x\right)+315 \sin \left(\frac{17 c}{2}+8 d x\right)-70 \sin \left(\frac{17 c}{2}+9 d x\right)+70 \sin \left(\frac{19 c}{2}+9 d x\right)+420 \cos \left(\frac{c}{2}\right) (330 c+36 d x+7)+11340 \cos \left(\frac{c}{2}+d x\right)+11340 \cos \left(\frac{3 c}{2}+d x\right)+3360 \cos \left(\frac{5 c}{2}+3 d x\right)+3360 \cos \left(\frac{7 c}{2}+3 d x\right)-2520 \cos \left(\frac{7 c}{2}+4 d x\right)+2520 \cos \left(\frac{9 c}{2}+4 d x\right)-1008 \cos \left(\frac{9 c}{2}+5 d x\right)-1008 \cos \left(\frac{11 c}{2}+5 d x\right)-450 \cos \left(\frac{13 c}{2}+7 d x\right)-450 \cos \left(\frac{15 c}{2}+7 d x\right)+315 \cos \left(\frac{15 c}{2}+8 d x\right)-315 \cos \left(\frac{17 c}{2}+8 d x\right)+70 \cos \left(\frac{17 c}{2}+9 d x\right)+70 \cos \left(\frac{19 c}{2}+9 d x\right)+138600 c \sin \left(\frac{c}{2}\right)-78960 \sin \left(\frac{c}{2}\right)}{322560 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"-1/322560*(420*(7 + 330*c + 36*d*x)*Cos[c/2] + 11340*Cos[c/2 + d*x] + 11340*Cos[(3*c)/2 + d*x] + 3360*Cos[(5*c)/2 + 3*d*x] + 3360*Cos[(7*c)/2 + 3*d*x] - 2520*Cos[(7*c)/2 + 4*d*x] + 2520*Cos[(9*c)/2 + 4*d*x] - 1008*Cos[(9*c)/2 + 5*d*x] - 1008*Cos[(11*c)/2 + 5*d*x] - 450*Cos[(13*c)/2 + 7*d*x] - 450*Cos[(15*c)/2 + 7*d*x] + 315*Cos[(15*c)/2 + 8*d*x] - 315*Cos[(17*c)/2 + 8*d*x] + 70*Cos[(17*c)/2 + 9*d*x] + 70*Cos[(19*c)/2 + 9*d*x] - 78960*Sin[c/2] + 138600*c*Sin[c/2] + 15120*d*x*Sin[c/2] - 11340*Sin[c/2 + d*x] + 11340*Sin[(3*c)/2 + d*x] - 3360*Sin[(5*c)/2 + 3*d*x] + 3360*Sin[(7*c)/2 + 3*d*x] - 2520*Sin[(7*c)/2 + 4*d*x] - 2520*Sin[(9*c)/2 + 4*d*x] + 1008*Sin[(9*c)/2 + 5*d*x] - 1008*Sin[(11*c)/2 + 5*d*x] + 450*Sin[(13*c)/2 + 7*d*x] - 450*Sin[(15*c)/2 + 7*d*x] + 315*Sin[(15*c)/2 + 8*d*x] + 315*Sin[(17*c)/2 + 8*d*x] - 70*Sin[(17*c)/2 + 9*d*x] + 70*Sin[(19*c)/2 + 9*d*x])/(a^2*d*(Cos[c/2] + Sin[c/2]))","B",1
725,1,481,141,3.8849154,"\int \frac{\cos ^8(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{18480 d x \sin \left(\frac{c}{2}\right)-10080 \sin \left(\frac{c}{2}+d x\right)+10080 \sin \left(\frac{3 c}{2}+d x\right)+1680 \sin \left(\frac{3 c}{2}+2 d x\right)+1680 \sin \left(\frac{5 c}{2}+2 d x\right)-3360 \sin \left(\frac{5 c}{2}+3 d x\right)+3360 \sin \left(\frac{7 c}{2}+3 d x\right)-2520 \sin \left(\frac{7 c}{2}+4 d x\right)-2520 \sin \left(\frac{9 c}{2}+4 d x\right)+672 \sin \left(\frac{9 c}{2}+5 d x\right)-672 \sin \left(\frac{11 c}{2}+5 d x\right)-560 \sin \left(\frac{11 c}{2}+6 d x\right)-560 \sin \left(\frac{13 c}{2}+6 d x\right)+480 \sin \left(\frac{13 c}{2}+7 d x\right)-480 \sin \left(\frac{15 c}{2}+7 d x\right)+105 \sin \left(\frac{15 c}{2}+8 d x\right)+105 \sin \left(\frac{17 c}{2}+8 d x\right)+9240 \cos \left(\frac{c}{2}\right) (15 c+2 d x)+10080 \cos \left(\frac{c}{2}+d x\right)+10080 \cos \left(\frac{3 c}{2}+d x\right)+1680 \cos \left(\frac{3 c}{2}+2 d x\right)-1680 \cos \left(\frac{5 c}{2}+2 d x\right)+3360 \cos \left(\frac{5 c}{2}+3 d x\right)+3360 \cos \left(\frac{7 c}{2}+3 d x\right)-2520 \cos \left(\frac{7 c}{2}+4 d x\right)+2520 \cos \left(\frac{9 c}{2}+4 d x\right)-672 \cos \left(\frac{9 c}{2}+5 d x\right)-672 \cos \left(\frac{11 c}{2}+5 d x\right)-560 \cos \left(\frac{11 c}{2}+6 d x\right)+560 \cos \left(\frac{13 c}{2}+6 d x\right)-480 \cos \left(\frac{13 c}{2}+7 d x\right)-480 \cos \left(\frac{15 c}{2}+7 d x\right)+105 \cos \left(\frac{15 c}{2}+8 d x\right)-105 \cos \left(\frac{17 c}{2}+8 d x\right)+138600 c \sin \left(\frac{c}{2}\right)-79800 \sin \left(\frac{c}{2}\right)}{215040 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"(9240*(15*c + 2*d*x)*Cos[c/2] + 10080*Cos[c/2 + d*x] + 10080*Cos[(3*c)/2 + d*x] + 1680*Cos[(3*c)/2 + 2*d*x] - 1680*Cos[(5*c)/2 + 2*d*x] + 3360*Cos[(5*c)/2 + 3*d*x] + 3360*Cos[(7*c)/2 + 3*d*x] - 2520*Cos[(7*c)/2 + 4*d*x] + 2520*Cos[(9*c)/2 + 4*d*x] - 672*Cos[(9*c)/2 + 5*d*x] - 672*Cos[(11*c)/2 + 5*d*x] - 560*Cos[(11*c)/2 + 6*d*x] + 560*Cos[(13*c)/2 + 6*d*x] - 480*Cos[(13*c)/2 + 7*d*x] - 480*Cos[(15*c)/2 + 7*d*x] + 105*Cos[(15*c)/2 + 8*d*x] - 105*Cos[(17*c)/2 + 8*d*x] - 79800*Sin[c/2] + 138600*c*Sin[c/2] + 18480*d*x*Sin[c/2] - 10080*Sin[c/2 + d*x] + 10080*Sin[(3*c)/2 + d*x] + 1680*Sin[(3*c)/2 + 2*d*x] + 1680*Sin[(5*c)/2 + 2*d*x] - 3360*Sin[(5*c)/2 + 3*d*x] + 3360*Sin[(7*c)/2 + 3*d*x] - 2520*Sin[(7*c)/2 + 4*d*x] - 2520*Sin[(9*c)/2 + 4*d*x] + 672*Sin[(9*c)/2 + 5*d*x] - 672*Sin[(11*c)/2 + 5*d*x] - 560*Sin[(11*c)/2 + 6*d*x] - 560*Sin[(13*c)/2 + 6*d*x] + 480*Sin[(13*c)/2 + 7*d*x] - 480*Sin[(15*c)/2 + 7*d*x] + 105*Sin[(15*c)/2 + 8*d*x] + 105*Sin[(17*c)/2 + 8*d*x])/(215040*a^2*d*(Cos[c/2] + Sin[c/2]))","B",1
726,1,418,124,4.8578061,"\int \frac{\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{1680 d x \sin \left(\frac{c}{2}\right)-1155 \sin \left(\frac{c}{2}+d x\right)+1155 \sin \left(\frac{3 c}{2}+d x\right)+210 \sin \left(\frac{3 c}{2}+2 d x\right)+210 \sin \left(\frac{5 c}{2}+2 d x\right)-525 \sin \left(\frac{5 c}{2}+3 d x\right)+525 \sin \left(\frac{7 c}{2}+3 d x\right)-210 \sin \left(\frac{7 c}{2}+4 d x\right)-210 \sin \left(\frac{9 c}{2}+4 d x\right)-63 \sin \left(\frac{9 c}{2}+5 d x\right)+63 \sin \left(\frac{11 c}{2}+5 d x\right)-70 \sin \left(\frac{11 c}{2}+6 d x\right)-70 \sin \left(\frac{13 c}{2}+6 d x\right)+15 \sin \left(\frac{13 c}{2}+7 d x\right)-15 \sin \left(\frac{15 c}{2}+7 d x\right)+70 \cos \left(\frac{c}{2}\right) (24 d x+7)+1155 \cos \left(\frac{c}{2}+d x\right)+1155 \cos \left(\frac{3 c}{2}+d x\right)+210 \cos \left(\frac{3 c}{2}+2 d x\right)-210 \cos \left(\frac{5 c}{2}+2 d x\right)+525 \cos \left(\frac{5 c}{2}+3 d x\right)+525 \cos \left(\frac{7 c}{2}+3 d x\right)-210 \cos \left(\frac{7 c}{2}+4 d x\right)+210 \cos \left(\frac{9 c}{2}+4 d x\right)+63 \cos \left(\frac{9 c}{2}+5 d x\right)+63 \cos \left(\frac{11 c}{2}+5 d x\right)-70 \cos \left(\frac{11 c}{2}+6 d x\right)+70 \cos \left(\frac{13 c}{2}+6 d x\right)-15 \cos \left(\frac{13 c}{2}+7 d x\right)-15 \cos \left(\frac{15 c}{2}+7 d x\right)-490 \sin \left(\frac{c}{2}\right)}{13440 a^2 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"-1/13440*(70*(7 + 24*d*x)*Cos[c/2] + 1155*Cos[c/2 + d*x] + 1155*Cos[(3*c)/2 + d*x] + 210*Cos[(3*c)/2 + 2*d*x] - 210*Cos[(5*c)/2 + 2*d*x] + 525*Cos[(5*c)/2 + 3*d*x] + 525*Cos[(7*c)/2 + 3*d*x] - 210*Cos[(7*c)/2 + 4*d*x] + 210*Cos[(9*c)/2 + 4*d*x] + 63*Cos[(9*c)/2 + 5*d*x] + 63*Cos[(11*c)/2 + 5*d*x] - 70*Cos[(11*c)/2 + 6*d*x] + 70*Cos[(13*c)/2 + 6*d*x] - 15*Cos[(13*c)/2 + 7*d*x] - 15*Cos[(15*c)/2 + 7*d*x] - 490*Sin[c/2] + 1680*d*x*Sin[c/2] - 1155*Sin[c/2 + d*x] + 1155*Sin[(3*c)/2 + d*x] + 210*Sin[(3*c)/2 + 2*d*x] + 210*Sin[(5*c)/2 + 2*d*x] - 525*Sin[(5*c)/2 + 3*d*x] + 525*Sin[(7*c)/2 + 3*d*x] - 210*Sin[(7*c)/2 + 4*d*x] - 210*Sin[(9*c)/2 + 4*d*x] - 63*Sin[(9*c)/2 + 5*d*x] + 63*Sin[(11*c)/2 + 5*d*x] - 70*Sin[(11*c)/2 + 6*d*x] - 70*Sin[(13*c)/2 + 6*d*x] + 15*Sin[(13*c)/2 + 7*d*x] - 15*Sin[(15*c)/2 + 7*d*x])/(a^2*d*(Cos[c/2] + Sin[c/2]))","B",1
727,1,93,119,0.7107953,"\int \frac{\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^7*Cot[c + d*x])/(a + a*Sin[c + d*x])^2,x]","\frac{270 \cos (c+d x)+5 \cos (3 (c+d x))-3 \left(40 \sin (2 (c+d x))+5 \sin (4 (c+d x))+\cos (5 (c+d x))-80 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+80 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+60 c+60 d x\right)}{240 a^2 d}","-\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,"(270*Cos[c + d*x] + 5*Cos[3*(c + d*x)] - 3*(60*c + 60*d*x + Cos[5*(c + d*x)] + 80*Log[Cos[(c + d*x)/2]] - 80*Log[Sin[(c + d*x)/2]] + 40*Sin[2*(c + d*x)] + 5*Sin[4*(c + d*x)]))/(240*a^2*d)","A",1
728,1,128,116,1.5207267,"\int \frac{\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^6*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(-108 (c+d x)+3 \sin (4 (c+d x))-240 \cos (c+d x)-16 \cos (3 (c+d x))+48 \tan \left(\frac{1}{2} (c+d x)\right)-48 \cot \left(\frac{1}{2} (c+d x)\right)-192 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+192 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{96 d (a \sin (c+d x)+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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(-108*(c + d*x) - 240*Cos[c + d*x] - 16*Cos[3*(c + d*x)] - 48*Cot[(c + d*x)/2] + 192*Log[Cos[(c + d*x)/2]] - 192*Log[Sin[(c + d*x)/2]] + 3*Sin[4*(c + d*x)] + 48*Tan[(c + d*x)/2]))/(96*d*(a + a*Sin[c + d*x])^2)","A",1
729,1,158,97,1.9941457,"\int \frac{\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(6 \cos (c+d x)+2 \cos (3 (c+d x))+3 \left(4 \sin (2 (c+d x))-8 \tan \left(\frac{1}{2} (c+d x)\right)+8 \cot \left(\frac{1}{2} (c+d x)\right)-\csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^2\left(\frac{1}{2} (c+d x)\right)-4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 c+24 d x\right)\right)}{24 a^2 d (\sin (c+d x)+1)^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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(6*Cos[c + d*x] + 2*Cos[3*(c + d*x)] + 3*(24*c + 24*d*x + 8*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 + 4*Log[Cos[(c + d*x)/2]] - 4*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 + 4*Sin[2*(c + d*x)] - 8*Tan[(c + d*x)/2])))/(24*a^2*d*(1 + Sin[c + d*x])^2)","A",1
730,1,184,97,2.4510237,"\int \frac{\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right)^4 \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(30 \cos (c+d x)-\cos (3 (c+d x))+3 \left(\cos (5 (c+d x))+8 \sin (c+d x) \left(-6 \cos (c+d x)+2 \cos (3 (c+d x))-6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\cos (2 (c+d x)) \left(-6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)+c+d x\right)\right)\right)}{768 a^2 d (\sin (c+d x)+1)^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,"-1/768*((1 + Cot[(c + d*x)/2])^4*Sec[(c + d*x)/2]^2*(30*Cos[c + d*x] - Cos[3*(c + d*x)] + 3*(Cos[5*(c + d*x)] + 8*(c + d*x - 6*Cos[c + d*x] + 2*Cos[3*(c + d*x)] + 6*Log[Cos[(c + d*x)/2]] - Cos[2*(c + d*x)]*(c + d*x + 6*Log[Cos[(c + d*x)/2]] - 6*Log[Sin[(c + d*x)/2]]) - 6*Log[Sin[(c + d*x)/2]])*Sin[c + d*x]))*Tan[(c + d*x)/2])/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
731,1,219,116,1.8754441,"\int \frac{\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^2,x]","-\frac{\sin ^5(c+d x) \left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(192 \cot (c+d x)+(3 \csc (c+d x)-8) \csc ^4\left(\frac{1}{2} (c+d x)\right)+(128-6 \csc (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)+8 \left(-(8 \cos (c+d x)+7) \sec ^4\left(\frac{1}{2} (c+d x)\right)-6 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+3 \sin ^2\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+3 \csc (c+d x) \left(16 (c+d x)+9 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-9 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}{3072 a^2 d (\sin (c+d x)+1)^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,"-1/3072*((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^4*(192*Cot[c + d*x] + Csc[(c + d*x)/2]^2*(128 - 6*Csc[c + d*x]) + Csc[(c + d*x)/2]^4*(-8 + 3*Csc[c + d*x]) + 8*(3*Csc[c + d*x]*(16*(c + d*x) - 9*Log[Cos[(c + d*x)/2]] + 9*Log[Sin[(c + d*x)/2]]) - (7 + 8*Cos[c + d*x])*Sec[(c + d*x)/2]^4 + 3*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 - 6*Csc[c + d*x]^5*Sin[(c + d*x)/2]^4))*Sin[c + d*x]^5)/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
732,1,254,118,1.1212968,"\int \frac{\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^6)/(a + a*Sin[c + d*x])^2,x]","\frac{\csc ^5(c+d x) \left(600 c \sin (c+d x)+600 d x \sin (c+d x)-60 \sin (2 (c+d x))-300 c \sin (3 (c+d x))-300 d x \sin (3 (c+d x))+150 \sin (4 (c+d x))+60 c \sin (5 (c+d x))+60 d x \sin (5 (c+d x))-40 \cos (c+d x)-220 \cos (3 (c+d x))+68 \cos (5 (c+d x))-450 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+225 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-45 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+450 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-225 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+45 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{960 a^2 d}","-\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,"(Csc[c + d*x]^5*(-40*Cos[c + d*x] - 220*Cos[3*(c + d*x)] + 68*Cos[5*(c + d*x)] + 600*c*Sin[c + d*x] + 600*d*x*Sin[c + d*x] + 450*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 450*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 60*Sin[2*(c + d*x)] - 300*c*Sin[3*(c + d*x)] - 300*d*x*Sin[3*(c + d*x)] - 225*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 225*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 150*Sin[4*(c + d*x)] + 60*c*Sin[5*(c + d*x)] + 60*d*x*Sin[5*(c + d*x)] + 45*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 45*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(960*a^2*d)","B",1
733,1,145,132,1.8926746,"\int \frac{\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^2,x]","\frac{\csc ^6(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(60 (32 \sin (c+d x)-11) \cos (c+d x)+6 (32 \sin (c+d x)+45) \cos (5 (c+d x))+10 (96 \sin (c+d x)-89) \cos (3 (c+d x))+3360 \sin ^6(c+d x) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{7680 a^2 d (\sin (c+d x)+1)^2}","\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,"(Csc[c + d*x]^6*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(3360*(-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^6 + 60*Cos[c + d*x]*(-11 + 32*Sin[c + d*x]) + 6*Cos[5*(c + d*x)]*(45 + 32*Sin[c + d*x]) + 10*Cos[3*(c + d*x)]*(-89 + 96*Sin[c + d*x])))/(7680*a^2*d*(1 + Sin[c + d*x])^2)","A",1
734,1,251,124,1.1107466,"\int \frac{\cot ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^7(c+d x) \left(-2170 \sin (2 (c+d x))-3080 \sin (4 (c+d x))-210 \sin (6 (c+d x))+5880 \cos (c+d x)+2184 \cos (3 (c+d x))-168 \cos (5 (c+d x))-216 \cos (7 (c+d x))+3675 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2205 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+735 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-105 \sin (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3675 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2205 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-735 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+105 \sin (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{53760 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,"-1/53760*(Csc[c + d*x]^7*(5880*Cos[c + d*x] + 2184*Cos[3*(c + d*x)] - 168*Cos[5*(c + d*x)] - 216*Cos[7*(c + d*x)] - 3675*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 3675*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 2170*Sin[2*(c + d*x)] + 2205*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 2205*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 3080*Sin[4*(c + d*x)] - 735*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 735*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 210*Sin[6*(c + d*x)] + 105*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] - 105*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/(a^2*d)","B",1
735,1,291,176,0.9458729,"\int \frac{\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^8*Csc[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^8(c+d x) \left(-86016 \sin (2 (c+d x))-64512 \sin (4 (c+d x))-12288 \sin (6 (c+d x))+1536 \sin (8 (c+d x))+158270 \cos (c+d x)+77210 \cos (3 (c+d x))-18130 \cos (5 (c+d x))-2310 \cos (7 (c+d x))-40425 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-64680 \cos (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+32340 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-9240 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+1155 \cos (8 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+40425 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+64680 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-32340 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+9240 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-1155 \cos (8 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1720320 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,"-1/1720320*(Csc[c + d*x]^8*(158270*Cos[c + d*x] + 77210*Cos[3*(c + d*x)] - 18130*Cos[5*(c + d*x)] - 2310*Cos[7*(c + d*x)] + 40425*Log[Cos[(c + d*x)/2]] - 64680*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 32340*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 9240*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 1155*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 40425*Log[Sin[(c + d*x)/2]] + 64680*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 32340*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 9240*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 1155*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 86016*Sin[2*(c + d*x)] - 64512*Sin[4*(c + d*x)] - 12288*Sin[6*(c + d*x)] + 1536*Sin[8*(c + d*x)]))/(a^2*d)","A",1
736,1,313,168,1.8705491,"\int \frac{\cot ^8(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\csc ^9(c+d x) \left(212940 \sin (2 (c+d x))+195300 \sin (4 (c+d x))+16380 \sin (6 (c+d x))-1890 \sin (8 (c+d x))-451584 \cos (c+d x)-155904 \cos (3 (c+d x))+20736 \cos (5 (c+d x))+14976 \cos (7 (c+d x))-1664 \cos (9 (c+d x))-119070 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+79380 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-34020 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+8505 \sin (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-945 \sin (9 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+119070 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-79380 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+34020 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-8505 \sin (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+945 \sin (9 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{5160960 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,"(Csc[c + d*x]^9*(-451584*Cos[c + d*x] - 155904*Cos[3*(c + d*x)] + 20736*Cos[5*(c + d*x)] + 14976*Cos[7*(c + d*x)] - 1664*Cos[9*(c + d*x)] + 119070*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 119070*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 212940*Sin[2*(c + d*x)] - 79380*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 79380*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 195300*Sin[4*(c + d*x)] + 34020*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 34020*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 16380*Sin[6*(c + d*x)] - 8505*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] + 8505*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 1890*Sin[8*(c + d*x)] + 945*Log[Cos[(c + d*x)/2]]*Sin[9*(c + d*x)] - 945*Log[Sin[(c + d*x)/2]]*Sin[9*(c + d*x)]))/(5160960*a^2*d)","A",1
737,1,353,218,1.8143663,"\int \frac{\cot ^8(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{\csc ^{10}(c+d x) \left(1720320 \sin (2 (c+d x))+1228800 \sin (4 (c+d x))+184320 \sin (6 (c+d x))-40960 \sin (8 (c+d x))+4096 \sin (10 (c+d x))-3219300 \cos (c+d x)-1237320 \cos (3 (c+d x))+278712 \cos (5 (c+d x))+54810 \cos (7 (c+d x))-5670 \cos (9 (c+d x))+357210 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+595350 \cos (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-340200 \cos (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+127575 \cos (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-28350 \cos (8 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2835 \cos (10 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-357210 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-595350 \cos (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+340200 \cos (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-127575 \cos (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+28350 \cos (8 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2835 \cos (10 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{41287680 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,"(Csc[c + d*x]^10*(-3219300*Cos[c + d*x] - 1237320*Cos[3*(c + d*x)] + 278712*Cos[5*(c + d*x)] + 54810*Cos[7*(c + d*x)] - 5670*Cos[9*(c + d*x)] - 357210*Log[Cos[(c + d*x)/2]] + 595350*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 340200*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 127575*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 28350*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 2835*Cos[10*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 357210*Log[Sin[(c + d*x)/2]] - 595350*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 340200*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 127575*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 28350*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 2835*Cos[10*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 1720320*Sin[2*(c + d*x)] + 1228800*Sin[4*(c + d*x)] + 184320*Sin[6*(c + d*x)] - 40960*Sin[8*(c + d*x)] + 4096*Sin[10*(c + d*x)]))/(41287680*a^2*d)","A",1
738,1,186,210,4.2264514,"\int \frac{\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^8*Csc[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(2661120 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+\cot (c+d x) \csc ^{10}(c+d x) (2457378 \sin (c+d x)+5907132 \sin (3 (c+d x))+656964 \sin (5 (c+d x))-121275 \sin (7 (c+d x))+10395 \sin (9 (c+d x))-5752832 \cos (2 (c+d x))+346112 \cos (4 (c+d x))+583168 \cos (6 (c+d x))-104448 \cos (8 (c+d x))+8704 \cos (10 (c+d x))-5402624)\right)}{113541120 a^2 d (\sin (c+d x)+1)^2}","-\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(2661120*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + Cot[c + d*x]*Csc[c + d*x]^10*(-5402624 - 5752832*Cos[2*(c + d*x)] + 346112*Cos[4*(c + d*x)] + 583168*Cos[6*(c + d*x)] - 104448*Cos[8*(c + d*x)] + 8704*Cos[10*(c + d*x)] + 2457378*Sin[c + d*x] + 5907132*Sin[3*(c + d*x)] + 656964*Sin[5*(c + d*x)] - 121275*Sin[7*(c + d*x)] + 10395*Sin[9*(c + d*x)])))/(113541120*a^2*d*(1 + Sin[c + d*x])^2)","A",1
739,1,482,161,3.9935166,"\int \frac{\cos ^8(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{-48720 d x \sin \left(\frac{c}{2}\right)+38640 \sin \left(\frac{c}{2}+d x\right)-38640 \sin \left(\frac{3 c}{2}+d x\right)+6720 \sin \left(\frac{3 c}{2}+2 d x\right)+6720 \sin \left(\frac{5 c}{2}+2 d x\right)+3920 \sin \left(\frac{5 c}{2}+3 d x\right)-3920 \sin \left(\frac{7 c}{2}+3 d x\right)+5880 \sin \left(\frac{7 c}{2}+4 d x\right)+5880 \sin \left(\frac{9 c}{2}+4 d x\right)-4368 \sin \left(\frac{9 c}{2}+5 d x\right)+4368 \sin \left(\frac{11 c}{2}+5 d x\right)-2240 \sin \left(\frac{11 c}{2}+6 d x\right)-2240 \sin \left(\frac{13 c}{2}+6 d x\right)+720 \sin \left(\frac{13 c}{2}+7 d x\right)-720 \sin \left(\frac{15 c}{2}+7 d x\right)+105 \sin \left(\frac{15 c}{2}+8 d x\right)+105 \sin \left(\frac{17 c}{2}+8 d x\right)+84 \cos \left(\frac{c}{2}\right) (12870 c-580 d x-7)-38640 \cos \left(\frac{c}{2}+d x\right)-38640 \cos \left(\frac{3 c}{2}+d x\right)+6720 \cos \left(\frac{3 c}{2}+2 d x\right)-6720 \cos \left(\frac{5 c}{2}+2 d x\right)-3920 \cos \left(\frac{5 c}{2}+3 d x\right)-3920 \cos \left(\frac{7 c}{2}+3 d x\right)+5880 \cos \left(\frac{7 c}{2}+4 d x\right)-5880 \cos \left(\frac{9 c}{2}+4 d x\right)+4368 \cos \left(\frac{9 c}{2}+5 d x\right)+4368 \cos \left(\frac{11 c}{2}+5 d x\right)-2240 \cos \left(\frac{11 c}{2}+6 d x\right)+2240 \cos \left(\frac{13 c}{2}+6 d x\right)-720 \cos \left(\frac{13 c}{2}+7 d x\right)-720 \cos \left(\frac{15 c}{2}+7 d x\right)+105 \cos \left(\frac{15 c}{2}+8 d x\right)-105 \cos \left(\frac{17 c}{2}+8 d x\right)+1081080 c \sin \left(\frac{c}{2}\right)-998928 \sin \left(\frac{c}{2}\right)}{215040 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"(84*(-7 + 12870*c - 580*d*x)*Cos[c/2] - 38640*Cos[c/2 + d*x] - 38640*Cos[(3*c)/2 + d*x] + 6720*Cos[(3*c)/2 + 2*d*x] - 6720*Cos[(5*c)/2 + 2*d*x] - 3920*Cos[(5*c)/2 + 3*d*x] - 3920*Cos[(7*c)/2 + 3*d*x] + 5880*Cos[(7*c)/2 + 4*d*x] - 5880*Cos[(9*c)/2 + 4*d*x] + 4368*Cos[(9*c)/2 + 5*d*x] + 4368*Cos[(11*c)/2 + 5*d*x] - 2240*Cos[(11*c)/2 + 6*d*x] + 2240*Cos[(13*c)/2 + 6*d*x] - 720*Cos[(13*c)/2 + 7*d*x] - 720*Cos[(15*c)/2 + 7*d*x] + 105*Cos[(15*c)/2 + 8*d*x] - 105*Cos[(17*c)/2 + 8*d*x] - 998928*Sin[c/2] + 1081080*c*Sin[c/2] - 48720*d*x*Sin[c/2] + 38640*Sin[c/2 + d*x] - 38640*Sin[(3*c)/2 + d*x] + 6720*Sin[(3*c)/2 + 2*d*x] + 6720*Sin[(5*c)/2 + 2*d*x] + 3920*Sin[(5*c)/2 + 3*d*x] - 3920*Sin[(7*c)/2 + 3*d*x] + 5880*Sin[(7*c)/2 + 4*d*x] + 5880*Sin[(9*c)/2 + 4*d*x] - 4368*Sin[(9*c)/2 + 5*d*x] + 4368*Sin[(11*c)/2 + 5*d*x] - 2240*Sin[(11*c)/2 + 6*d*x] - 2240*Sin[(13*c)/2 + 6*d*x] + 720*Sin[(13*c)/2 + 7*d*x] - 720*Sin[(15*c)/2 + 7*d*x] + 105*Sin[(15*c)/2 + 8*d*x] + 105*Sin[(17*c)/2 + 8*d*x])/(215040*a^3*d*(Cos[c/2] + Sin[c/2]))","B",1
740,1,429,133,8.9219737,"\int \frac{\cos ^8(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{840 d x \sin \left(\frac{c}{2}\right)-609 \sin \left(\frac{c}{2}+d x\right)+609 \sin \left(\frac{3 c}{2}+d x\right)-63 \sin \left(\frac{3 c}{2}+2 d x\right)-63 \sin \left(\frac{5 c}{2}+2 d x\right)-91 \sin \left(\frac{5 c}{2}+3 d x\right)+91 \sin \left(\frac{7 c}{2}+3 d x\right)-105 \sin \left(\frac{7 c}{2}+4 d x\right)-105 \sin \left(\frac{9 c}{2}+4 d x\right)+63 \sin \left(\frac{9 c}{2}+5 d x\right)-63 \sin \left(\frac{11 c}{2}+5 d x\right)+21 \sin \left(\frac{11 c}{2}+6 d x\right)+21 \sin \left(\frac{13 c}{2}+6 d x\right)-3 \sin \left(\frac{13 c}{2}+7 d x\right)+3 \sin \left(\frac{15 c}{2}+7 d x\right)-168 \cos \left(\frac{c}{2}\right) (99 c-5 d x)+609 \cos \left(\frac{c}{2}+d x\right)+609 \cos \left(\frac{3 c}{2}+d x\right)-63 \cos \left(\frac{3 c}{2}+2 d x\right)+63 \cos \left(\frac{5 c}{2}+2 d x\right)+91 \cos \left(\frac{5 c}{2}+3 d x\right)+91 \cos \left(\frac{7 c}{2}+3 d x\right)-105 \cos \left(\frac{7 c}{2}+4 d x\right)+105 \cos \left(\frac{9 c}{2}+4 d x\right)-63 \cos \left(\frac{9 c}{2}+5 d x\right)-63 \cos \left(\frac{11 c}{2}+5 d x\right)+21 \cos \left(\frac{11 c}{2}+6 d x\right)-21 \cos \left(\frac{13 c}{2}+6 d x\right)+3 \cos \left(\frac{13 c}{2}+7 d x\right)+3 \cos \left(\frac{15 c}{2}+7 d x\right)-16632 c \sin \left(\frac{c}{2}\right)+16996 \sin \left(\frac{c}{2}\right)}{2688 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","\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,"(-168*(99*c - 5*d*x)*Cos[c/2] + 609*Cos[c/2 + d*x] + 609*Cos[(3*c)/2 + d*x] - 63*Cos[(3*c)/2 + 2*d*x] + 63*Cos[(5*c)/2 + 2*d*x] + 91*Cos[(5*c)/2 + 3*d*x] + 91*Cos[(7*c)/2 + 3*d*x] - 105*Cos[(7*c)/2 + 4*d*x] + 105*Cos[(9*c)/2 + 4*d*x] - 63*Cos[(9*c)/2 + 5*d*x] - 63*Cos[(11*c)/2 + 5*d*x] + 21*Cos[(11*c)/2 + 6*d*x] - 21*Cos[(13*c)/2 + 6*d*x] + 3*Cos[(13*c)/2 + 7*d*x] + 3*Cos[(15*c)/2 + 7*d*x] + 16996*Sin[c/2] - 16632*c*Sin[c/2] + 840*d*x*Sin[c/2] - 609*Sin[c/2 + d*x] + 609*Sin[(3*c)/2 + d*x] - 63*Sin[(3*c)/2 + 2*d*x] - 63*Sin[(5*c)/2 + 2*d*x] - 91*Sin[(5*c)/2 + 3*d*x] + 91*Sin[(7*c)/2 + 3*d*x] - 105*Sin[(7*c)/2 + 4*d*x] - 105*Sin[(9*c)/2 + 4*d*x] + 63*Sin[(9*c)/2 + 5*d*x] - 63*Sin[(11*c)/2 + 5*d*x] + 21*Sin[(11*c)/2 + 6*d*x] + 21*Sin[(13*c)/2 + 6*d*x] - 3*Sin[(13*c)/2 + 7*d*x] + 3*Sin[(15*c)/2 + 7*d*x])/(2688*a^3*d*(Cos[c/2] + Sin[c/2]))","B",1
741,1,366,131,1.9919516,"\int \frac{\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{-840 d x \sin \left(\frac{c}{2}\right)+600 \sin \left(\frac{c}{2}+d x\right)-600 \sin \left(\frac{3 c}{2}+d x\right)+15 \sin \left(\frac{3 c}{2}+2 d x\right)+15 \sin \left(\frac{5 c}{2}+2 d x\right)+140 \sin \left(\frac{5 c}{2}+3 d x\right)-140 \sin \left(\frac{7 c}{2}+3 d x\right)+105 \sin \left(\frac{7 c}{2}+4 d x\right)+105 \sin \left(\frac{9 c}{2}+4 d x\right)-36 \sin \left(\frac{9 c}{2}+5 d x\right)+36 \sin \left(\frac{11 c}{2}+5 d x\right)-5 \sin \left(\frac{11 c}{2}+6 d x\right)-5 \sin \left(\frac{13 c}{2}+6 d x\right)-21 \cos \left(\frac{c}{2}\right) (40 d x+1)-600 \cos \left(\frac{c}{2}+d x\right)-600 \cos \left(\frac{3 c}{2}+d x\right)+15 \cos \left(\frac{3 c}{2}+2 d x\right)-15 \cos \left(\frac{5 c}{2}+2 d x\right)-140 \cos \left(\frac{5 c}{2}+3 d x\right)-140 \cos \left(\frac{7 c}{2}+3 d x\right)+105 \cos \left(\frac{7 c}{2}+4 d x\right)-105 \cos \left(\frac{9 c}{2}+4 d x\right)+36 \cos \left(\frac{9 c}{2}+5 d x\right)+36 \cos \left(\frac{11 c}{2}+5 d x\right)-5 \cos \left(\frac{11 c}{2}+6 d x\right)+5 \cos \left(\frac{13 c}{2}+6 d x\right)+21 \sin \left(\frac{c}{2}\right)}{1920 a^3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\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,"(-21*(1 + 40*d*x)*Cos[c/2] - 600*Cos[c/2 + d*x] - 600*Cos[(3*c)/2 + d*x] + 15*Cos[(3*c)/2 + 2*d*x] - 15*Cos[(5*c)/2 + 2*d*x] - 140*Cos[(5*c)/2 + 3*d*x] - 140*Cos[(7*c)/2 + 3*d*x] + 105*Cos[(7*c)/2 + 4*d*x] - 105*Cos[(9*c)/2 + 4*d*x] + 36*Cos[(9*c)/2 + 5*d*x] + 36*Cos[(11*c)/2 + 5*d*x] - 5*Cos[(11*c)/2 + 6*d*x] + 5*Cos[(13*c)/2 + 6*d*x] + 21*Sin[c/2] - 840*d*x*Sin[c/2] + 600*Sin[c/2 + d*x] - 600*Sin[(3*c)/2 + d*x] + 15*Sin[(3*c)/2 + 2*d*x] + 15*Sin[(5*c)/2 + 2*d*x] + 140*Sin[(5*c)/2 + 3*d*x] - 140*Sin[(7*c)/2 + 3*d*x] + 105*Sin[(7*c)/2 + 4*d*x] + 105*Sin[(9*c)/2 + 4*d*x] - 36*Sin[(9*c)/2 + 5*d*x] + 36*Sin[(11*c)/2 + 5*d*x] - 5*Sin[(11*c)/2 + 6*d*x] - 5*Sin[(13*c)/2 + 6*d*x])/(1920*a^3*d*(Cos[c/2] + Sin[c/2]))","B",1
742,1,80,99,0.4092558,"\int \frac{\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^7*Cot[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{-24 \sin (2 (c+d x))+\sin (4 (c+d x))+8 \cos (c+d x)-8 \cos (3 (c+d x))+32 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-32 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-52 c-52 d x}{32 a^3 d}","-\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,"(-52*c - 52*d*x + 8*Cos[c + d*x] - 8*Cos[3*(c + d*x)] - 32*Log[Cos[(c + d*x)/2]] + 32*Log[Sin[(c + d*x)/2]] - 24*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])/(32*a^3*d)","A",1
743,1,126,92,0.9878576,"\int \frac{\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^6*Cot[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6 \left(6 (c+d x)+9 \sin (2 (c+d x))-33 \cos (c+d x)+\cos (3 (c+d x))+6 \tan \left(\frac{1}{2} (c+d x)\right)-6 \cot \left(\frac{1}{2} (c+d x)\right)-36 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+36 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{12 d (a \sin (c+d x)+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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(6*(c + d*x) - 33*Cos[c + d*x] + Cos[3*(c + d*x)] - 6*Cot[(c + d*x)/2] + 36*Log[Cos[(c + d*x)/2]] - 36*Log[Sin[(c + d*x)/2]] + 9*Sin[2*(c + d*x)] + 6*Tan[(c + d*x)/2]))/(12*d*(a + a*Sin[c + d*x])^3)","A",1
744,1,144,98,0.8941015,"\int \frac{\cos ^5(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^5*Cot[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6 \left(20 (c+d x)-2 \sin (2 (c+d x))+24 \cos (c+d x)-12 \tan \left(\frac{1}{2} (c+d x)\right)+12 \cot \left(\frac{1}{2} (c+d x)\right)-\csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^2\left(\frac{1}{2} (c+d x)\right)+20 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-20 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d (a \sin (c+d x)+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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(20*(c + d*x) + 24*Cos[c + d*x] + 12*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 - 20*Log[Cos[(c + d*x)/2]] + 20*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 - 2*Sin[2*(c + d*x)] - 12*Tan[(c + d*x)/2]))/(8*d*(a + a*Sin[c + d*x])^3)","A",1
745,1,132,92,2.6501596,"\int \frac{\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","\frac{\csc ^3(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6 \left(2 (3 \sin (c+d x)+8) \cos (3 (c+d x))+6 (5 \sin (c+d x)-4) \cos (c+d x)-12 \sin ^3(c+d x) \left(6 (c+d x)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{24 a^3 d (\sin (c+d x)+1)^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,"(Csc[c + d*x]^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(-12*(6*(c + d*x) + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^3 + 2*Cos[3*(c + d*x)]*(8 + 3*Sin[c + d*x]) + 6*Cos[c + d*x]*(-4 + 5*Sin[c + d*x])))/(24*a^3*d*(1 + Sin[c + d*x])^3)","A",1
746,1,165,97,2.453946,"\int \frac{\cos ^3(c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6 \left(-22 \csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^4\left(\frac{1}{2} (c+d x)\right)+22 \sec ^2\left(\frac{1}{2} (c+d x)\right)+(4 \sin (c+d x)-1) \csc ^4\left(\frac{1}{2} (c+d x)\right)+8 \left(-13 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+13 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-8 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+8 c+8 d x\right)\right)}{64 a^3 d (\sin (c+d x)+1)^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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(-22*Csc[(c + d*x)/2]^2 + 22*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^4 + 8*(8*c + 8*d*x + 13*Log[Cos[(c + d*x)/2]] - 13*Log[Sin[(c + d*x)/2]] - 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4) + Csc[(c + d*x)/2]^4*(-1 + 4*Sin[c + d*x])))/(64*a^3*d*(1 + Sin[c + d*x])^3)","A",1
747,1,189,100,1.7629295,"\int \frac{\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^6)/(a + a*Sin[c + d*x])^3,x]","-\frac{\csc ^5(c+d x) \left(-780 \sin (2 (c+d x))+30 \sin (4 (c+d x))+560 \cos (c+d x)-40 \cos (3 (c+d x))-136 \cos (5 (c+d x))-1050 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+525 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-105 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+1050 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-525 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+105 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1920 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,"-1/1920*(Csc[c + d*x]^5*(560*Cos[c + d*x] - 40*Cos[3*(c + d*x)] - 136*Cos[5*(c + d*x)] + 1050*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 1050*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 780*Sin[2*(c + d*x)] - 525*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 525*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 30*Sin[4*(c + d*x)] + 105*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 105*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(a^3*d)","A",1
748,1,242,124,1.0128561,"\int \frac{\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^3,x]","\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6 \left(704 \tan \left(\frac{1}{2} (c+d x)\right)-704 \cot \left(\frac{1}{2} (c+d x)\right)+210 \csc ^2\left(\frac{1}{2} (c+d x)\right)+5 \sec ^6\left(\frac{1}{2} (c+d x)\right)+90 \sec ^4\left(\frac{1}{2} (c+d x)\right)-210 \sec ^2\left(\frac{1}{2} (c+d x)\right)-840 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+840 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+(18 \sin (c+d x)-5) \csc ^6\left(\frac{1}{2} (c+d x)\right)+(34 \sin (c+d x)-90) \csc ^4\left(\frac{1}{2} (c+d x)\right)-544 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-36 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)\right)}{1920 a^3 d (\sin (c+d x)+1)^3}","\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,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(-704*Cot[(c + d*x)/2] + 210*Csc[(c + d*x)/2]^2 + 840*Log[Cos[(c + d*x)/2]] - 840*Log[Sin[(c + d*x)/2]] - 210*Sec[(c + d*x)/2]^2 + 90*Sec[(c + d*x)/2]^4 + 5*Sec[(c + d*x)/2]^6 - 544*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + Csc[(c + d*x)/2]^6*(-5 + 18*Sin[c + d*x]) + Csc[(c + d*x)/2]^4*(-90 + 34*Sin[c + d*x]) + 704*Tan[(c + d*x)/2] - 36*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(1920*a^3*d*(1 + Sin[c + d*x])^3)","A",1
749,1,251,140,0.9923377,"\int \frac{\cot ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]","\frac{\csc ^7(c+d x) \left(4998 \sin (2 (c+d x))+504 \sin (4 (c+d x))-210 \sin (6 (c+d x))-4704 \cos (c+d x)+672 \cos (3 (c+d x))+1120 \cos (5 (c+d x))-160 \cos (7 (c+d x))+3675 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2205 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+735 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-105 \sin (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3675 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2205 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-735 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+105 \sin (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{21504 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,"(Csc[c + d*x]^7*(-4704*Cos[c + d*x] + 672*Cos[3*(c + d*x)] + 1120*Cos[5*(c + d*x)] - 160*Cos[7*(c + d*x)] - 3675*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 3675*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 4998*Sin[2*(c + d*x)] + 2205*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 2205*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 504*Sin[4*(c + d*x)] - 735*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 735*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 210*Sin[6*(c + d*x)] + 105*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] - 105*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/(21504*a^3*d)","A",1
750,1,317,166,5.5071564,"\int \frac{\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^8*Csc[c + d*x])/(a + a*Sin[c + d*x])^3,x]","-\frac{\sin ^7(c+d x) \left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)^6 \left(15 (7 \csc (c+d x)-24) \csc ^8\left(\frac{1}{2} (c+d x)\right)+4 (455 \csc (c+d x)-276) \csc ^6\left(\frac{1}{2} (c+d x)\right)+(1328-210 \csc (c+d x)) \csc ^4\left(\frac{1}{2} (c+d x)\right)-4 (3045 \csc (c+d x)-4864) \csc ^2\left(\frac{1}{2} (c+d x)\right)-8 \left(\frac{1}{4} (4616 \cos (c+d x)+1907 \cos (2 (c+d x))+304 \cos (3 (c+d x))+2833) \sec ^8\left(\frac{1}{2} (c+d x)\right)+3360 \sin ^8\left(\frac{1}{2} (c+d x)\right) \csc ^9(c+d x)+14560 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^7(c+d x)-420 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)-6090 \sin ^2\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+6090 \csc (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}{13762560 a^3 d (\sin (c+d x)+1)^3}","\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,"-1/13762560*((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^6*(Csc[(c + d*x)/2]^4*(1328 - 210*Csc[c + d*x]) + 15*Csc[(c + d*x)/2]^8*(-24 + 7*Csc[c + d*x]) + 4*Csc[(c + d*x)/2]^6*(-276 + 455*Csc[c + d*x]) - 4*Csc[(c + d*x)/2]^2*(-4864 + 3045*Csc[c + d*x]) - 8*(6090*Csc[c + d*x]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + ((2833 + 4616*Cos[c + d*x] + 1907*Cos[2*(c + d*x)] + 304*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^8)/4 - 6090*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 - 420*Csc[c + d*x]^5*Sin[(c + d*x)/2]^4 + 14560*Csc[c + d*x]^7*Sin[(c + d*x)/2]^6 + 3360*Csc[c + d*x]^9*Sin[(c + d*x)/2]^8))*Sin[c + d*x]^7)/(a^3*d*(1 + Sin[c + d*x])^3)","A",1
751,1,82,82,0.3096994,"\int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Integrate[Sin[c + d*x]^2*(a + a*Sin[c + d*x])*Tan[c + d*x]^2,x]","-\frac{3 a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{7 a \cos (c+d x)}{4 d}-\frac{a \cos (3 (c+d x))}{12 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}","-\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*(c + d*x))/(2*d) + (7*a*Cos[c + d*x])/(4*d) - (a*Cos[3*(c + d*x)])/(12*d) + (a*Sec[c + d*x])/d + (a*Sin[2*(c + d*x)])/(4*d) + (a*Tan[c + d*x])/d","A",1
752,1,63,65,0.1062798,"\int \sin (c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Integrate[Sin[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^2,x]","-\frac{3 a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \cos (c+d x)}{d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}","\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*(c + d*x))/(2*d) + (a*Cos[c + d*x])/d + (a*Sec[c + d*x])/d + (a*Sin[2*(c + d*x)])/(4*d) + (a*Tan[c + d*x])/d","A",1
753,1,47,39,0.0354523,"\int (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{a \cos (c+d x)}{d}-\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}","\frac{a \cos (c+d x)}{d}+\frac{a \cos (c+d x)}{d (1-\sin (c+d x))}-a x",1,"-((a*ArcTan[Tan[c + d*x]])/d) + (a*Cos[c + d*x])/d + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",1
754,1,36,27,0.0198778,"\int \sec (c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x],x]","-\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}","\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-a x",1,"-((a*ArcTan[Tan[c + d*x]])/d) + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",1
755,1,56,36,0.0327836,"\int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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*Log[Cos[(c + d*x)/2]])/d) + (a*Log[Sin[(c + d*x)/2]])/d + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",1
756,1,68,48,0.0780919,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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*Cot[c + d*x])/d) - (a*Log[Cos[(c + d*x)/2]])/d + (a*Log[Sin[(c + d*x)/2]])/d + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",1
757,1,172,75,1.4991015,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{2 a \cot (2 (c+d x))}{d}-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{a \sin \left(\frac{1}{2} (c+d x)\right)}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{a \sin \left(\frac{1}{2} (c+d x)\right)}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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,"(-2*a*Cot[2*(c + d*x)])/d - (a*Csc[(c + d*x)/2]^2)/(8*d) - (3*a*Log[Cos[(c + d*x)/2]])/(2*d) + (3*a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d) + (a*Sin[(c + d*x)/2])/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (a*Sin[(c + d*x)/2])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
758,1,205,91,4.7215021,"\int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[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{5 a \cot (c+d x)}{3 d}-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{a \sin \left(\frac{1}{2} (c+d x)\right)}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{a \sin \left(\frac{1}{2} (c+d x)\right)}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 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,"(-5*a*Cot[c + d*x])/(3*d) - (a*Csc[(c + d*x)/2]^2)/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) - (3*a*Log[Cos[(c + d*x)/2]])/(2*d) + (3*a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d) + (a*Sin[(c + d*x)/2])/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (a*Sin[(c + d*x)/2])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (a*Tan[c + d*x])/d","B",1
759,1,161,89,0.5150324,"\int \sin (c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Integrate[Sin[c + d*x]*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","-\frac{a^2 (\sin (c+d x)+1)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right) (-6 \sin (2 (c+d x))-33 \cos (c+d x)+\cos (3 (c+d x))+36 c+36 d x)-\sin \left(\frac{1}{2} (c+d x)\right) (-6 \sin (2 (c+d x))-33 \cos (c+d x)+\cos (3 (c+d x))+36 c+36 d x+48)\right)}{12 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\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,"-1/12*(a^2*(1 + Sin[c + d*x])^2*(Cos[(c + d*x)/2]*(36*c + 36*d*x - 33*Cos[c + d*x] + Cos[3*(c + d*x)] - 6*Sin[2*(c + d*x)]) - Sin[(c + d*x)/2]*(48 + 36*c + 36*d*x - 33*Cos[c + d*x] + Cos[3*(c + d*x)] - 6*Sin[2*(c + d*x)])))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
760,1,145,71,0.3938656,"\int (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","-\frac{a^2 (\sin (c+d x)+1)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right) (10 (c+d x)-\sin (2 (c+d x))-8 \cos (c+d x))+\sin \left(\frac{1}{2} (c+d x)\right) (-2 (5 c+5 d x+8)+\sin (2 (c+d x))+8 \cos (c+d x))\right)}{4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","\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,"-1/4*(a^2*(1 + Sin[c + d*x])^2*(Cos[(c + d*x)/2]*(10*(c + d*x) - 8*Cos[c + d*x] - Sin[2*(c + d*x)]) + Sin[(c + d*x)/2]*(-2*(8 + 5*c + 5*d*x) + 8*Cos[c + d*x] + Sin[2*(c + d*x)])))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","B",1
761,1,90,43,0.3684573,"\int \sec (c+d x) (a+a \sin (c+d x))^2 \tan (c+d x) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^2*Tan[c + d*x],x]","\frac{(a \sin (c+d x)+a)^2 \left(-2 (c+d x)+\cos (c+d x)+\frac{4 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","\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*(c + d*x) + Cos[c + d*x] + (4*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))*(a + a*Sin[c + d*x])^2)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","B",1
762,1,69,44,0.1143465,"\int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{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*(-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]] + (4*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])))/d","A",1
763,1,96,58,0.392163,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(\tan \left(\frac{1}{2} (c+d x)\right)-\cot \left(\frac{1}{2} (c+d x)\right)+4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{8 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{2 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,"(a^2*(-Cot[(c + d*x)/2] - 4*Log[Cos[(c + d*x)/2]] + 4*Log[Sin[(c + d*x)/2]] + (8*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + Tan[(c + d*x)/2]))/(2*d)","A",1
764,1,124,86,1.0964867,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(8 \tan \left(\frac{1}{2} (c+d x)\right)-8 \cot \left(\frac{1}{2} (c+d x)\right)-\csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^2\left(\frac{1}{2} (c+d x)\right)+20 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-20 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{32 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{8 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,"(a^2*(-8*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 - 20*Log[Cos[(c + d*x)/2]] + 20*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 + (32*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + 8*Tan[(c + d*x)/2]))/(8*d)","A",1
765,1,125,111,0.7887472,"\int \sin (c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Integrate[Sin[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{(a \sin (c+d x)+a)^3 \left(-204 (c+d x)+40 \sin (2 (c+d x))-\sin (4 (c+d x))+200 \cos (c+d x)-8 \cos (3 (c+d x))+\frac{256 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{32 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"((a + a*Sin[c + d*x])^3*(-204*(c + d*x) + 200*Cos[c + d*x] - 8*Cos[3*(c + d*x)] + (256*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + 40*Sin[2*(c + d*x)] - Sin[4*(c + d*x)]))/(32*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
766,1,115,89,0.4830824,"\int (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{(a \sin (c+d x)+a)^3 \left(-66 (c+d x)+9 \sin (2 (c+d x))+57 \cos (c+d x)-\cos (3 (c+d x))+\frac{96 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{12 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\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,"((a + a*Sin[c + d*x])^3*(-66*(c + d*x) + 57*Cos[c + d*x] - Cos[3*(c + d*x)] + (96*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + 9*Sin[2*(c + d*x)]))/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
767,1,145,67,0.4971216,"\int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]","-\frac{a^3 (\sin (c+d x)+1)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right) (18 (c+d x)-\sin (2 (c+d x))-12 \cos (c+d x))+\sin \left(\frac{1}{2} (c+d x)\right) (-2 (9 c+9 d x+16)+\sin (2 (c+d x))+12 \cos (c+d x))\right)}{4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\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,"-1/4*(a^3*(1 + Sin[c + d*x])^3*(Cos[(c + d*x)/2]*(18*(c + d*x) - 12*Cos[c + d*x] - Sin[2*(c + d*x)]) + Sin[(c + d*x)/2]*(-2*(16 + 9*c + 9*d*x) + 12*Cos[c + d*x] + Sin[2*(c + d*x)])))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","B",1
768,1,74,48,0.1382758,"\int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \left(-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{8 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+c+d x\right)}{d}","\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*(c + d*x + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]] - (8*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])))/d)","A",1
769,1,96,56,0.3715619,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(\tan \left(\frac{1}{2} (c+d x)\right)-\cot \left(\frac{1}{2} (c+d x)\right)+6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{16 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{2 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,"(a^3*(-Cot[(c + d*x)/2] - 6*Log[Cos[(c + d*x)/2]] + 6*Log[Sin[(c + d*x)/2]] + (16*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + Tan[(c + d*x)/2]))/(2*d)","A",1
770,1,124,80,1.0927755,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(12 \tan \left(\frac{1}{2} (c+d x)\right)-12 \cot \left(\frac{1}{2} (c+d x)\right)-\csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^2\left(\frac{1}{2} (c+d x)\right)+36 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-36 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{64 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{8 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,"(a^3*(-12*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 - 36*Log[Cos[(c + d*x)/2]] + 36*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 + (64*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + 12*Tan[(c + d*x)/2]))/(8*d)","A",1
771,1,211,98,6.1410239,"\int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","a^3 \left(\frac{7 \tan \left(\frac{1}{2} (c+d x)\right)}{3 d}-\frac{7 \cot \left(\frac{1}{2} (c+d x)\right)}{3 d}-\frac{3 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{3 \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{11 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{11 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{8 \sin \left(\frac{1}{2} (c+d x)\right)}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}\right)","-\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,"a^3*((-7*Cot[(c + d*x)/2])/(3*d) - (3*Csc[(c + d*x)/2]^2)/(8*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*d) - (11*Log[Cos[(c + d*x)/2]])/(2*d) + (11*Log[Sin[(c + d*x)/2]])/(2*d) + (3*Sec[(c + d*x)/2]^2)/(8*d) + (8*Sin[(c + d*x)/2])/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (7*Tan[(c + d*x)/2])/(3*d) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*d))","B",1
772,1,148,83,0.3927473,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{11 \sin (c+d x)+6 c \sin (2 (c+d x))+6 d x \sin (2 (c+d x))-11 \sin (2 (c+d x))+3 \sin (3 (c+d x))+2 (6 c+6 d x-11) \cos (c+d x)+14 \cos (2 (c+d x))+18}{12 a d (\sin (c+d x)+1) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{\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,"(18 + 2*(-11 + 6*c + 6*d*x)*Cos[c + d*x] + 14*Cos[2*(c + d*x)] + 11*Sin[c + d*x] - 11*Sin[2*(c + d*x)] + 6*c*Sin[2*(c + d*x)] + 6*d*x*Sin[2*(c + d*x)] + 3*Sin[3*(c + d*x)])/(12*a*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(1 + Sin[c + d*x]))","A",1
773,1,111,70,0.3558195,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{-2 \sin (c+d x)+4 \cos (2 (c+d x))+(6 c+6 d x-5) (\sin (c+d x)+1) \cos (c+d x)}{6 a d (\sin (c+d x)+1) \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{\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,"(4*Cos[2*(c + d*x)] - 2*Sin[c + d*x] + (-5 + 6*c + 6*d*x)*Cos[c + d*x]*(1 + Sin[c + d*x]))/(6*a*d*(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(1 + Sin[c + d*x]))","A",1
774,1,106,50,0.1369094,"\int \frac{\tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^2/(a + a*Sin[c + d*x]),x]","\frac{8 \sin (c+d x)-5 \sin (2 (c+d x))-10 \cos (c+d x)+2 \cos (2 (c+d x))+6}{12 a d (\sin (c+d x)+1) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{\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,"(6 - 10*Cos[c + d*x] + 2*Cos[2*(c + d*x)] + 8*Sin[c + d*x] - 5*Sin[2*(c + d*x)])/(12*a*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(1 + Sin[c + d*x]))","B",1
775,1,104,37,0.1359996,"\int \frac{\sec (c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{-2 \sin (c+d x)+\frac{1}{2} \sin (2 (c+d x))+\cos (c+d x)+\cos (2 (c+d x))-3}{6 a d (\sin (c+d x)+1) \left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{\sec ^3(c+d x)}{3 a d}-\frac{\tan ^3(c+d x)}{3 a d}",1,"(-3 + Cos[c + d*x] + Cos[2*(c + d*x)] - 2*Sin[c + d*x] + Sin[2*(c + d*x)]/2)/(6*a*d*(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(1 + Sin[c + d*x]))","B",1
776,1,149,79,0.6098919,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\sin \left(\frac{1}{2} (c+d x)\right) \left(-\frac{11}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-\frac{2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{3}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)+\frac{1}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{6 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,"(-6*Log[Cos[(c + d*x)/2]] + 6*Log[Sin[(c + d*x)/2]] + (Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^(-2) + Sin[(c + d*x)/2]*(3/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - 2/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 11/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(6*a*d)","A",1
777,1,245,93,0.6284237,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\csc ^3(c+d x) \left(4 \sin (c+d x)-16 \sin (2 (c+d x))+8 \sin (3 (c+d x))+10 \cos (2 (c+d x))+8 \cos (3 (c+d x))+6 \sin (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-3 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\cos (c+d x) \left(3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-8\right)-6 \sin (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2\right)}{3 a d (\sin (c+d x)+1) \left(\csc \left(\frac{1}{2} (c+d x)\right)-\sec \left(\frac{1}{2} (c+d x)\right)\right) \left(\csc \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{1}{2} (c+d x)\right)\right)}","\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,"-1/3*(Csc[c + d*x]^3*(2 + 10*Cos[2*(c + d*x)] + 8*Cos[3*(c + d*x)] + 3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 3*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] + Cos[c + d*x]*(-8 - 3*Log[Cos[(c + d*x)/2]] + 3*Log[Sin[(c + d*x)/2]]) + 4*Sin[c + d*x] - 16*Sin[2*(c + d*x)] - 6*Log[Cos[(c + d*x)/2]]*Sin[2*(c + d*x)] + 6*Log[Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] + 8*Sin[3*(c + d*x)]))/(a*d*(Csc[(c + d*x)/2] - Sec[(c + d*x)/2])*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])*(1 + Sin[c + d*x]))","B",1
778,1,191,149,0.5863394,"\int \frac{\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sin[c + d*x]^4*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{250 \sin (c+d x)+720 c \sin (2 (c+d x))+720 d x \sin (2 (c+d x))-824 \sin (2 (c+d x))+351 \sin (3 (c+d x))+5 \sin (5 (c+d x))+10 (90 c+90 d x-103) \cos (c+d x)+544 \cos (2 (c+d x))-180 c \cos (3 (c+d x))-180 d x \cos (3 (c+d x))+206 \cos (3 (c+d x))-20 \cos (4 (c+d x))+500}{160 a^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\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,"-1/160*(500 + 10*(-103 + 90*c + 90*d*x)*Cos[c + d*x] + 544*Cos[2*(c + d*x)] + 206*Cos[3*(c + d*x)] - 180*c*Cos[3*(c + d*x)] - 180*d*x*Cos[3*(c + d*x)] - 20*Cos[4*(c + d*x)] + 250*Sin[c + d*x] - 824*Sin[2*(c + d*x)] + 720*c*Sin[2*(c + d*x)] + 720*d*x*Sin[2*(c + d*x)] + 351*Sin[3*(c + d*x)] + 5*Sin[5*(c + d*x)])/(a^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","A",1
779,1,148,120,0.5738304,"\int \frac{\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sin[c + d*x]^3*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\sec (c+d x) (400 \sin (c+d x)+480 c \sin (2 (c+d x))+480 d x \sin (2 (c+d x))-796 \sin (2 (c+d x))+304 \sin (3 (c+d x))+(600 c+600 d x-995) \cos (c+d x)+376 \cos (2 (c+d x))-120 c \cos (3 (c+d x))-120 d x \cos (3 (c+d x))+199 \cos (3 (c+d x))-30 \cos (4 (c+d x))+550)}{240 a^2 d (\sin (c+d x)+1)^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,"(Sec[c + d*x]*(550 + (-995 + 600*c + 600*d*x)*Cos[c + d*x] + 376*Cos[2*(c + d*x)] + 199*Cos[3*(c + d*x)] - 120*c*Cos[3*(c + d*x)] - 120*d*x*Cos[3*(c + d*x)] - 30*Cos[4*(c + d*x)] + 400*Sin[c + d*x] - 796*Sin[2*(c + d*x)] + 480*c*Sin[2*(c + d*x)] + 480*d*x*Sin[2*(c + d*x)] + 304*Sin[3*(c + d*x)]))/(240*a^2*d*(1 + Sin[c + d*x])^2)","A",1
780,1,143,106,0.5578316,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{\sec (c+d x) \left(-10 \sin (c+d x)+60 c \sin (2 (c+d x))+60 d x \sin (2 (c+d x))-89 \sin (2 (c+d x))+26 \sin (3 (c+d x))+\frac{5}{4} (60 c+60 d x-89) \cos (c+d x)+44 \cos (2 (c+d x))-15 c \cos (3 (c+d x))-15 d x \cos (3 (c+d x))+\frac{89}{4} \cos (3 (c+d x))+20\right)}{60 a^2 d (\sin (c+d x)+1)^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,"-1/60*(Sec[c + d*x]*(20 + (5*(-89 + 60*c + 60*d*x)*Cos[c + d*x])/4 + 44*Cos[2*(c + d*x)] + (89*Cos[3*(c + d*x)])/4 - 15*c*Cos[3*(c + d*x)] - 15*d*x*Cos[3*(c + d*x)] - 10*Sin[c + d*x] - 89*Sin[2*(c + d*x)] + 60*c*Sin[2*(c + d*x)] + 60*d*x*Sin[2*(c + d*x)] + 26*Sin[3*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
781,1,84,66,0.2786938,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\sec (c+d x) (40 \sin (c+d x)-52 \sin (2 (c+d x))+8 \sin (3 (c+d x))-65 \cos (c+d x)-8 \cos (2 (c+d x))+13 \cos (3 (c+d x))+40)}{80 a^2 d (\sin (c+d x)+1)^2}","-\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]*(40 - 65*Cos[c + d*x] - 8*Cos[2*(c + d*x)] + 13*Cos[3*(c + d*x)] + 40*Sin[c + d*x] - 52*Sin[2*(c + d*x)] + 8*Sin[3*(c + d*x)]))/(80*a^2*d*(1 + Sin[c + d*x])^2)","A",1
782,1,86,73,0.2575036,"\int \frac{\tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^2/(a + a*Sin[c + d*x])^2,x]","-\frac{\sec (c+d x) \left(-35 \sin (c+d x)+11 \sin (2 (c+d x))+\sin (3 (c+d x))+\frac{55}{4} \cos (c+d x)+4 \cos (2 (c+d x))-\frac{11}{4} \cos (3 (c+d x))-20\right)}{60 a^2 d (\sin (c+d x)+1)^2}","\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,"-1/60*(Sec[c + d*x]*(-20 + (55*Cos[c + d*x])/4 + 4*Cos[2*(c + d*x)] - (11*Cos[3*(c + d*x)])/4 - 35*Sin[c + d*x] + 11*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
783,1,82,71,0.2477706,"\int \frac{\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\sec (c+d x) (-80 \sin (c+d x)-4 \sin (2 (c+d x))+16 \sin (3 (c+d x))-5 \cos (c+d x)+64 \cos (2 (c+d x))+\cos (3 (c+d x))-80)}{240 a^2 d (\sin (c+d x)+1)^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,"-1/240*(Sec[c + d*x]*(-80 - 5*Cos[c + d*x] + 64*Cos[2*(c + d*x)] + Cos[3*(c + d*x)] - 80*Sin[c + d*x] - 4*Sin[2*(c + d*x)] + 16*Sin[3*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
784,1,196,115,0.5185314,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\sec (c+d x) \left(160 \sin (c+d x)-316 \sin (2 (c+d x))+64 \sin (3 (c+d x))+136 \cos (2 (c+d x))+79 \cos (3 (c+d x))+240 \sin (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+60 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-5 \cos (c+d x) \left(-60 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+60 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+79\right)-60 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-240 \sin (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+280\right)}{240 a^2 d (\sin (c+d x)+1)^2}","-\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,"(Sec[c + d*x]*(280 + 136*Cos[2*(c + d*x)] + 79*Cos[3*(c + d*x)] + 60*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 5*Cos[c + d*x]*(79 + 60*Log[Cos[(c + d*x)/2]] - 60*Log[Sin[(c + d*x)/2]]) - 60*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 160*Sin[c + d*x] - 316*Sin[2*(c + d*x)] - 240*Log[Cos[(c + d*x)/2]]*Sin[2*(c + d*x)] + 240*Log[Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] + 64*Sin[3*(c + d*x)]))/(240*a^2*d*(1 + Sin[c + d*x])^2)","A",1
785,1,289,130,0.8330538,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^3(c+d x) \left(58 \sin (c+d x)-168 \sin (2 (c+d x))+82 \sin (3 (c+d x))+28 \sin (4 (c+d x))+48 \cos (2 (c+d x))+112 \cos (3 (c+d x))-28 \cos (4 (c+d x))+90 \sin (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-15 \sin (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+60 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 \cos (c+d x) \left(-15 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+28\right)-60 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-90 \sin (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+15 \sin (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+40\right)}{15 a^2 d (\sin (c+d x)+1)^2 \left(\csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^2\left(\frac{1}{2} (c+d x)\right)\right)}","\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,"-1/15*(Csc[c + d*x]^3*(40 + 48*Cos[2*(c + d*x)] + 112*Cos[3*(c + d*x)] - 28*Cos[4*(c + d*x)] + 60*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 4*Cos[c + d*x]*(28 + 15*Log[Cos[(c + d*x)/2]] - 15*Log[Sin[(c + d*x)/2]]) - 60*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 58*Sin[c + d*x] - 168*Sin[2*(c + d*x)] - 90*Log[Cos[(c + d*x)/2]]*Sin[2*(c + d*x)] + 90*Log[Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] + 82*Sin[3*(c + d*x)] + 28*Sin[4*(c + d*x)] + 15*Log[Cos[(c + d*x)/2]]*Sin[4*(c + d*x)] - 15*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)]))/(a^2*d*(Csc[(c + d*x)/2]^2 - Sec[(c + d*x)/2]^2)*(1 + Sin[c + d*x])^2)","B",1
786,1,328,158,0.7303518,"\int \frac{\csc ^3(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","-\frac{\csc ^2(c+d x) \sec (c+d x) \left(-432 \sin (c+d x)+744 \sin (2 (c+d x))-176 \sin (3 (c+d x))-372 \sin (4 (c+d x))+128 \sin (5 (c+d x))+176 \cos (2 (c+d x))-651 \cos (3 (c+d x))+332 \cos (4 (c+d x))+93 \cos (5 (c+d x))-720 \sin (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+360 \sin (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-630 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+90 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+18 \cos (c+d x) \left(-30 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+30 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+31\right)+630 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-90 \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+720 \sin (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-360 \sin (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-348\right)}{320 a^2 d (\sin (c+d x)+1)^2}","-\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,"-1/320*(Csc[c + d*x]^2*Sec[c + d*x]*(-348 + 176*Cos[2*(c + d*x)] - 651*Cos[3*(c + d*x)] + 332*Cos[4*(c + d*x)] + 93*Cos[5*(c + d*x)] - 630*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 90*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 18*Cos[c + d*x]*(31 + 30*Log[Cos[(c + d*x)/2]] - 30*Log[Sin[(c + d*x)/2]]) + 630*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 90*Cos[5*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 432*Sin[c + d*x] + 744*Sin[2*(c + d*x)] + 720*Log[Cos[(c + d*x)/2]]*Sin[2*(c + d*x)] - 720*Log[Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] - 176*Sin[3*(c + d*x)] - 372*Sin[4*(c + d*x)] - 360*Log[Cos[(c + d*x)/2]]*Sin[4*(c + d*x)] + 360*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)] + 128*Sin[5*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)","B",1
787,1,224,151,0.6826694,"\int \frac{\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]^4*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{8008 \sin (c+d x)+11760 c \sin (2 (c+d x))+11760 d x \sin (2 (c+d x))-20762 \sin (2 (c+d x))+6588 \sin (3 (c+d x))-840 c \sin (4 (c+d x))-840 d x \sin (4 (c+d x))+1483 \sin (4 (c+d x))-140 \sin (5 (c+d x))+14 (840 c+840 d x-1483) \cos (c+d x)+5152 \cos (2 (c+d x))-5040 c \cos (3 (c+d x))-5040 d x \cos (3 (c+d x))+8898 \cos (3 (c+d x))-2288 \cos (4 (c+d x))+8400}{2240 a^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}","\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,"(8400 + 14*(-1483 + 840*c + 840*d*x)*Cos[c + d*x] + 5152*Cos[2*(c + d*x)] + 8898*Cos[3*(c + d*x)] - 5040*c*Cos[3*(c + d*x)] - 5040*d*x*Cos[3*(c + d*x)] - 2288*Cos[4*(c + d*x)] + 8008*Sin[c + d*x] - 20762*Sin[2*(c + d*x)] + 11760*c*Sin[2*(c + d*x)] + 11760*d*x*Sin[2*(c + d*x)] + 6588*Sin[3*(c + d*x)] + 1483*Sin[4*(c + d*x)] - 840*c*Sin[4*(c + d*x)] - 840*d*x*Sin[4*(c + d*x)] - 140*Sin[5*(c + d*x)])/(2240*a^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)","A",1
788,1,214,142,0.7999303,"\int \frac{\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]^3*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","-\frac{2688 \sin (c+d x)+11760 c \sin (2 (c+d x))+11760 d x \sin (2 (c+d x))-23282 \sin (2 (c+d x))+5568 \sin (3 (c+d x))-840 c \sin (4 (c+d x))-840 d x \sin (4 (c+d x))+1663 \sin (4 (c+d x))+14 (840 c+840 d x-1663) \cos (c+d x)+6272 \cos (2 (c+d x))-5040 c \cos (3 (c+d x))-5040 d x \cos (3 (c+d x))+9978 \cos (3 (c+d x))-1768 \cos (4 (c+d x))+4200}{6720 a^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}","-\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,"-1/6720*(4200 + 14*(-1663 + 840*c + 840*d*x)*Cos[c + d*x] + 6272*Cos[2*(c + d*x)] + 9978*Cos[3*(c + d*x)] - 5040*c*Cos[3*(c + d*x)] - 5040*d*x*Cos[3*(c + d*x)] - 1768*Cos[4*(c + d*x)] + 2688*Sin[c + d*x] - 23282*Sin[2*(c + d*x)] + 11760*c*Sin[2*(c + d*x)] + 11760*d*x*Sin[2*(c + d*x)] + 5568*Sin[3*(c + d*x)] + 1663*Sin[4*(c + d*x)] - 840*c*Sin[4*(c + d*x)] - 840*d*x*Sin[4*(c + d*x)])/(a^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)","A",1
789,1,104,102,0.4954436,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\sec (c+d x) (1344 \sin (c+d x)-1946 \sin (2 (c+d x))+64 \sin (3 (c+d x))+139 \sin (4 (c+d x))-1946 \cos (c+d x)-224 \cos (2 (c+d x))+834 \cos (3 (c+d x))-104 \cos (4 (c+d x))+840)}{2240 a^3 d (\sin (c+d x)+1)^3}","\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]*(840 - 1946*Cos[c + d*x] - 224*Cos[2*(c + d*x)] + 834*Cos[3*(c + d*x)] - 104*Cos[4*(c + d*x)] + 1344*Sin[c + d*x] - 1946*Sin[2*(c + d*x)] + 64*Sin[3*(c + d*x)] + 139*Sin[4*(c + d*x)]))/(2240*a^3*d*(1 + Sin[c + d*x])^3)","A",1
790,1,104,88,0.3821687,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\sec (c+d x) (1008 \sin (c+d x)-602 \sin (2 (c+d x))+48 \sin (3 (c+d x))+43 \sin (4 (c+d x))-602 \cos (c+d x)-448 \cos (2 (c+d x))+258 \cos (3 (c+d x))-8 \cos (4 (c+d x))+840)}{2240 a^3 d (\sin (c+d x)+1)^3}","-\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]*(840 - 602*Cos[c + d*x] - 448*Cos[2*(c + d*x)] + 258*Cos[3*(c + d*x)] - 8*Cos[4*(c + d*x)] + 1008*Sin[c + d*x] - 602*Sin[2*(c + d*x)] + 48*Sin[3*(c + d*x)] + 43*Sin[4*(c + d*x)]))/(2240*a^3*d*(1 + Sin[c + d*x])^3)","A",1
791,1,104,103,0.3472526,"\int \frac{\tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^2/(a + a*Sin[c + d*x])^3,x]","\frac{\sec (c+d x) (672 \sin (c+d x)-70 \sin (2 (c+d x))-96 \sin (3 (c+d x))+5 \sin (4 (c+d x))-70 \cos (c+d x)-224 \cos (2 (c+d x))+30 \cos (3 (c+d x))+16 \cos (4 (c+d x))+336)}{1344 a^3 d (\sin (c+d x)+1)^3}","\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]*(336 - 70*Cos[c + d*x] - 224*Cos[2*(c + d*x)] + 30*Cos[3*(c + d*x)] + 16*Cos[4*(c + d*x)] + 672*Sin[c + d*x] - 70*Sin[2*(c + d*x)] - 96*Sin[3*(c + d*x)] + 5*Sin[4*(c + d*x)]))/(1344*a^3*d*(1 + Sin[c + d*x])^3)","A",1
792,1,104,99,0.3791571,"\int \frac{\sec (c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{\sec (c+d x) (672 \sin (c+d x)+182 \sin (2 (c+d x))-288 \sin (3 (c+d x))-13 \sin (4 (c+d x))+182 \cos (c+d x)-672 \cos (2 (c+d x))-78 \cos (3 (c+d x))+48 \cos (4 (c+d x))+560)}{2240 a^3 d (\sin (c+d x)+1)^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]*(560 + 182*Cos[c + d*x] - 672*Cos[2*(c + d*x)] - 78*Cos[3*(c + d*x)] + 48*Cos[4*(c + d*x)] + 672*Sin[c + d*x] + 182*Sin[2*(c + d*x)] - 288*Sin[3*(c + d*x)] - 13*Sin[4*(c + d*x)]))/(2240*a^3*d*(1 + Sin[c + d*x])^3)","A",1
793,1,341,151,0.4294995,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\frac{105 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-2281 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5+353 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4-706 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3+162 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2-324 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{120 \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-840 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6+840 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6+60}{840 d (a \sin (c+d x)+a)^3}","-\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,"(60 - (120*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 324*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 162*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 706*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 353*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - 2281*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 - 840*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6 + 840*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6 + (105*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(840*d*(a + a*Sin[c + d*x])^3)","B",1
794,1,351,162,0.8849389,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{\csc ^3(c+d x) \left(-1316 \sin (c+d x)+3520 \sin (2 (c+d x))-1380 \sin (3 (c+d x))-1056 \sin (4 (c+d x))+176 \sin (5 (c+d x))-440 \cos (2 (c+d x))-2640 \cos (3 (c+d x))+846 \cos (4 (c+d x))+176 \cos (5 (c+d x))-2100 \sin (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+630 \sin (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-1575 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+105 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+14 \cos (c+d x) \left(-105 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+105 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+176\right)+1575 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-105 \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2100 \sin (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-630 \sin (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-966\right)}{140 a^3 d (\sin (c+d x)+1)^3 \left(\csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^2\left(\frac{1}{2} (c+d x)\right)\right)}","\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,"(Csc[c + d*x]^3*(-966 - 440*Cos[2*(c + d*x)] - 2640*Cos[3*(c + d*x)] + 846*Cos[4*(c + d*x)] + 176*Cos[5*(c + d*x)] - 1575*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 105*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 14*Cos[c + d*x]*(176 + 105*Log[Cos[(c + d*x)/2]] - 105*Log[Sin[(c + d*x)/2]]) + 1575*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 105*Cos[5*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 1316*Sin[c + d*x] + 3520*Sin[2*(c + d*x)] + 2100*Log[Cos[(c + d*x)/2]]*Sin[2*(c + d*x)] - 2100*Log[Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] - 1380*Sin[3*(c + d*x)] - 1056*Sin[4*(c + d*x)] - 630*Log[Cos[(c + d*x)/2]]*Sin[4*(c + d*x)] + 630*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)] + 176*Sin[5*(c + d*x)]))/(140*a^3*d*(Csc[(c + d*x)/2]^2 - Sec[(c + d*x)/2]^2)*(1 + Sin[c + d*x])^3)","B",1
795,1,84,117,0.408882,"\int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx","Integrate[Sin[c + d*x]^2*(a + a*Sin[c + d*x])*Tan[c + d*x]^4,x]","\frac{a \left(-3 \sin (2 (c+d x))-33 \cos (c+d x)+\cos (3 (c+d x))-28 \tan (c+d x)+4 \sec ^3(c+d x)-36 \sec (c+d x)+4 \tan (c+d x) \sec ^2(c+d x)+30 c+30 d x\right)}{12 d}","\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,"(a*(30*c + 30*d*x - 33*Cos[c + d*x] + Cos[3*(c + d*x)] - 36*Sec[c + d*x] + 4*Sec[c + d*x]^3 - 3*Sin[2*(c + d*x)] - 28*Tan[c + d*x] + 4*Sec[c + d*x]^2*Tan[c + d*x]))/(12*d)","A",1
796,1,76,101,0.2539192,"\int \sin (c+d x) (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx","Integrate[Sin[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^4,x]","\frac{a \left(-3 \sin (2 (c+d x))-12 \cos (c+d x)-28 \tan (c+d x)+4 \sec ^3(c+d x)-24 \sec (c+d x)+4 \tan (c+d x) \sec ^2(c+d x)+30 c+30 d x\right)}{12 d}","-\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,"(a*(30*c + 30*d*x - 12*Cos[c + d*x] - 24*Sec[c + d*x] + 4*Sec[c + d*x]^3 - 3*Sin[2*(c + d*x)] - 28*Tan[c + d*x] + 4*Sec[c + d*x]^2*Tan[c + d*x]))/(12*d)","A",1
797,1,81,72,0.0432604,"\int (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])*Tan[c + d*x]^4,x]","-\frac{a \cos (c+d x)}{d}+\frac{a \tan ^{-1}(\tan (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}","-\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*ArcTan[Tan[c + d*x]])/d - (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",1
798,1,69,60,0.0419778,"\int \sec (c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^3,x]","\frac{a \tan ^{-1}(\tan (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 \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*ArcTan[Tan[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",1
799,1,45,45,0.0367013,"\int \sec ^2(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Integrate[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",1
800,1,33,33,0.0221419,"\int \sec ^3(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx","Integrate[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",1
801,1,85,68,0.1214874,"\int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{a \sec ^3(c+d x)}{3 d}+\frac{a \sec (c+d x)}{d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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*Log[Cos[(c + d*x)/2]])/d) + (a*Log[Sin[(c + d*x)/2]])/d + (a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) + (a*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
802,1,109,81,0.0574372,"\int \csc ^2(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{5 a \tan (c+d x)}{3 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 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{a \tan (c+d x) \sec ^2(c+d x)}{3 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*Cot[c + d*x])/d) - (a*Log[Cos[(c + d*x)/2]])/d + (a*Log[Sin[(c + d*x)/2]])/d + (a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) + (5*a*Tan[c + d*x])/(3*d) + (a*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",1
803,1,359,110,6.103471,"\int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{5 a \tan (c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{5 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{5 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{13 a \sin \left(\frac{1}{2} (c+d x)\right)}{6 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{a \sin \left(\frac{1}{2} (c+d x)\right)}{6 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{13 a \sin \left(\frac{1}{2} (c+d x)\right)}{6 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{a}{12 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{a}{12 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a \sin \left(\frac{1}{2} (c+d x)\right)}{6 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{a \tan (c+d x) \sec ^2(c+d x)}{3 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,"-((a*Cot[c + d*x])/d) - (a*Csc[(c + d*x)/2]^2)/(8*d) - (5*a*Log[Cos[(c + d*x)/2]])/(2*d) + (5*a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d) + a/(12*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (a*Sin[(c + d*x)/2])/(6*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (13*a*Sin[(c + d*x)/2])/(6*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (a*Sin[(c + d*x)/2])/(6*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + a/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (13*a*Sin[(c + d*x)/2])/(6*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (5*a*Tan[c + d*x])/(3*d) + (a*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","B",1
804,1,159,101,1.2635414,"\int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^4,x]","-\frac{a^2 \left(-21 (12 c+12 d x+7) \cos \left(\frac{1}{2} (c+d x)\right)+(84 c+84 d x+239) \cos \left(\frac{3}{2} (c+d x)\right)+3 \left(-5 \cos \left(\frac{5}{2} (c+d x)\right)+\cos \left(\frac{7}{2} (c+d x)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((28 c+28 d x-27) \cos (c+d x)-6 \cos (2 (c+d x))-\cos (3 (c+d x))+56 c+56 d x+50)\right)\right)}{48 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\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,"-1/48*(a^2*(-21*(7 + 12*c + 12*d*x)*Cos[(c + d*x)/2] + (239 + 84*c + 84*d*x)*Cos[(3*(c + d*x))/2] + 3*(-5*Cos[(5*(c + d*x))/2] + Cos[(7*(c + d*x))/2] + 2*(50 + 56*c + 56*d*x + (-27 + 28*c + 28*d*x)*Cos[c + d*x] - 6*Cos[2*(c + d*x)] - Cos[3*(c + d*x)])*Sin[(c + d*x)/2])))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)","A",1
805,1,131,86,1.0206805,"\int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{a^2 (\sin (c+d x)+1)^2 \left(-3 \cos (c+d x)+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) (8 \sin (c+d x)-7)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{1}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+6 c+6 d x\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","\frac{a^4 \sin ^2(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac{4 a^2 \cos (c+d x)}{3 d}-\frac{2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}+2 a^2 x",1,"(a^2*(1 + Sin[c + d*x])^2*(6*c + 6*d*x - 3*Cos[c + d*x] + (Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^(-2) + (2*Sin[(c + d*x)/2]*(-7 + 8*Sin[c + d*x]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3))/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
806,1,79,63,0.0523833,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{a^2 \tan ^{-1}(\tan (c+d x))}{d}+\frac{2 a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \sec ^3(c+d x)}{3 d}-\frac{2 a^2 \sec (c+d x)}{d}","\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*ArcTan[Tan[c + d*x]])/d - (2*a^2*Sec[c + d*x])/d + (2*a^2*Sec[c + d*x]^3)/(3*d) - (a^2*Tan[c + d*x])/d + (2*a^2*Tan[c + d*x]^3)/(3*d)","A",1
807,1,72,60,0.2866731,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^2 \tan (c+d x) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2*Tan[c + d*x],x]","\frac{a^2 \left(-3 \sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)-2 \cos \left(\frac{3}{2} (c+d x)\right)\right)}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\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,"(a^2*(3*Cos[(c + d*x)/2] - 2*Cos[(3*(c + d*x))/2] - 3*Sin[(c + d*x)/2]))/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)","A",1
808,1,142,73,0.509095,"\int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (\sin (c+d x)+1)^2 \left(3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) (4 \sin (c+d x)-5)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{1}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","\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*(1 + Sin[c + d*x])^2*(-3*Log[Cos[(c + d*x)/2]] + 3*Log[Sin[(c + d*x)/2]] + (Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^(-2) + (2*Sin[(c + d*x)/2]*(-5 + 4*Sin[c + d*x]))/(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3))/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
809,1,135,87,0.9241334,"\int \csc ^2(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(3 \tan \left(\frac{1}{2} (c+d x)\right)-3 \cot \left(\frac{1}{2} (c+d x)\right)+12 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-12 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 \sin \left(\frac{1}{2} (c+d x)\right) (7 \sin (c+d x)-8)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{6 d}","\frac{a^4 \cot (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac{10 a^2 \cot (c+d x)}{3 d}-\frac{2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{2 a^2 \cot (c+d x)}{d (1-\sin (c+d x))}",1,"(a^2*(-3*Cot[(c + d*x)/2] - 12*Log[Cos[(c + d*x)/2]] + 12*Log[Sin[(c + d*x)/2]] + 2/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*Sin[(c + d*x)/2]*(-8 + 7*Sin[c + d*x]))/(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 3*Tan[(c + d*x)/2]))/(6*d)","A",1
810,1,190,125,1.9884067,"\int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(24 \tan \left(\frac{1}{2} (c+d x)\right)-24 \cot \left(\frac{1}{2} (c+d x)\right)-3 \csc ^2\left(\frac{1}{2} (c+d x)\right)+3 \sec ^2\left(\frac{1}{2} (c+d x)\right)+84 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-84 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{160 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{16 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{8}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{24 d}","\frac{a^4 \cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}-\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{8 a^2 \cot (c+d x) \csc (c+d x)}{3 d (1-\sin (c+d x))}",1,"(a^2*(-24*Cot[(c + d*x)/2] - 3*Csc[(c + d*x)/2]^2 - 84*Log[Cos[(c + d*x)/2]] + 84*Log[Sin[(c + d*x)/2]] + 3*Sec[(c + d*x)/2]^2 + 8/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (16*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (160*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + 24*Tan[(c + d*x)/2]))/(24*d)","A",1
811,1,177,119,2.1332325,"\int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^4,x]","\frac{(a \sin (c+d x)+a)^3 \left(102 (c+d x)-9 \sin (2 (c+d x))-69 \cos (c+d x)+\cos (3 (c+d x))-\frac{200 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{16 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{8}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{12 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\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,"((a + a*Sin[c + d*x])^3*(102*(c + d*x) - 69*Cos[c + d*x] + Cos[3*(c + d*x)] + 8/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (16*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 - (200*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - 9*Sin[2*(c + d*x)]))/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
812,1,159,101,1.4803612,"\int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^3,x]","-\frac{a^3 \left(-3 (132 c+132 d x+89) \cos \left(\frac{1}{2} (c+d x)\right)+(132 c+132 d x+403) \cos \left(\frac{3}{2} (c+d x)\right)+3 \left(-9 \cos \left(\frac{5}{2} (c+d x)\right)+\cos \left(\frac{7}{2} (c+d x)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((44 c+44 d x-43) \cos (c+d x)-10 \cos (2 (c+d x))-\cos (3 (c+d x))+88 c+88 d x+86)\right)\right)}{48 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\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,"-1/48*(a^3*(-3*(89 + 132*c + 132*d*x)*Cos[(c + d*x)/2] + (403 + 132*c + 132*d*x)*Cos[(3*(c + d*x))/2] + 3*(-9*Cos[(5*(c + d*x))/2] + Cos[(7*(c + d*x))/2] + 2*(86 + 88*c + 88*d*x + (-43 + 44*c + 44*d*x)*Cos[c + d*x] - 10*Cos[2*(c + d*x)] - Cos[3*(c + d*x)])*Sin[(c + d*x)/2])))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)","A",1
813,1,133,77,1.2431818,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(-3 \cos (c+d x)+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) (13 \sin (c+d x)-11)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+9 c+9 d x\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","-\frac{2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}-\frac{3 a^3 \cos (c+d x)}{d}+3 a^3 x+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}",1,"(a^3*(1 + Sin[c + d*x])^3*(9*c + 9*d*x - 3*Cos[c + d*x] + 2/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*Sin[(c + d*x)/2]*(-11 + 13*Sin[c + d*x]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3))/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
814,1,107,64,0.7238224,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]","-\frac{a^3 \left(-9 (c+d x+2) \cos \left(\frac{1}{2} (c+d x)\right)+(3 c+3 d x+14) \cos \left(\frac{3}{2} (c+d x)\right)+6 \sin \left(\frac{1}{2} (c+d x)\right) (2 (c+d x+2)+(c+d x) \cos (c+d x))\right)}{6 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","a^3 x-\frac{2 a^5 \cos (c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}",1,"-1/6*(a^3*(-9*(2 + c + d*x)*Cos[(c + d*x)/2] + (14 + 3*c + 3*d*x)*Cos[(3*(c + d*x))/2] + 6*(2*(2 + c + d*x) + (c + d*x)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)","A",1
815,1,144,72,0.6277938,"\int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \left(3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) (5 \sin (c+d x)-7)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\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*(1 + Sin[c + d*x])^3*(-3*Log[Cos[(c + d*x)/2]] + 3*Log[Sin[(c + d*x)/2]] + 2/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*Sin[(c + d*x)/2]*(-7 + 5*Sin[c + d*x]))/(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3))/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","A",1
816,1,135,86,0.9567862,"\int \csc ^2(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(3 \tan \left(\frac{1}{2} (c+d x)\right)-3 \cot \left(\frac{1}{2} (c+d x)\right)+18 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-18 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 \sin \left(\frac{1}{2} (c+d x)\right) (11 \sin (c+d x)-13)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)-\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{4}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{6 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,"(a^3*(-3*Cot[(c + d*x)/2] - 18*Log[Cos[(c + d*x)/2]] + 18*Log[Sin[(c + d*x)/2]] + 4/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*Sin[(c + d*x)/2]*(-13 + 11*Sin[c + d*x]))/(-Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 3*Tan[(c + d*x)/2]))/(6*d)","A",1
817,1,190,110,2.0155691,"\int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(36 \tan \left(\frac{1}{2} (c+d x)\right)-36 \cot \left(\frac{1}{2} (c+d x)\right)-3 \csc ^2\left(\frac{1}{2} (c+d x)\right)+3 \sec ^2\left(\frac{1}{2} (c+d x)\right)+132 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-132 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{272 \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{32 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{16}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{24 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,"(a^3*(-36*Cot[(c + d*x)/2] - 3*Csc[(c + d*x)/2]^2 - 132*Log[Cos[(c + d*x)/2]] + 132*Log[Sin[(c + d*x)/2]] + 3*Sec[(c + d*x)/2]^2 + 16/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (32*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (272*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + 36*Tan[(c + d*x)/2]))/(24*d)","A",1
818,1,287,128,6.177676,"\int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^4*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","a^3 \left(\frac{17 \tan \left(\frac{1}{2} (c+d x)\right)}{6 d}-\frac{17 \cot \left(\frac{1}{2} (c+d x)\right)}{6 d}-\frac{3 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{3 \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{17 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{17 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{46 \sin \left(\frac{1}{2} (c+d x)\right)}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 \sin \left(\frac{1}{2} (c+d x)\right)}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}\right)","-\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,"a^3*((-17*Cot[(c + d*x)/2])/(6*d) - (3*Csc[(c + d*x)/2]^2)/(8*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*d) - (17*Log[Cos[(c + d*x)/2]])/(2*d) + (17*Log[Sin[(c + d*x)/2]])/(2*d) + (3*Sec[(c + d*x)/2]^2)/(8*d) + 2/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*Sin[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (46*Sin[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (17*Tan[(c + d*x)/2])/(6*d) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*d))","B",0
819,1,252,143,1.5690222,"\int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^4,x]","\frac{a^4 \left(-11736 c \sin \left(\frac{1}{2} (c+d x)\right)-11736 d x \sin \left(\frac{1}{2} (c+d x)\right)-16488 \sin \left(\frac{1}{2} (c+d x)\right)-3912 c \sin \left(\frac{3}{2} (c+d x)\right)-3912 d x \sin \left(\frac{3}{2} (c+d x)\right)+3704 \sin \left(\frac{3}{2} (c+d x)\right)+885 \sin \left(\frac{5}{2} (c+d x)\right)+129 \sin \left(\frac{7}{2} (c+d x)\right)-23 \sin \left(\frac{9}{2} (c+d x)\right)-3 \sin \left(\frac{11}{2} (c+d x)\right)+24 (489 c+489 d x+209) \cos \left(\frac{1}{2} (c+d x)\right)-24 (163 c+163 d x+453) \cos \left(\frac{3}{2} (c+d x)\right)+885 \cos \left(\frac{5}{2} (c+d x)\right)-129 \cos \left(\frac{7}{2} (c+d x)\right)-23 \cos \left(\frac{9}{2} (c+d x)\right)+3 \cos \left(\frac{11}{2} (c+d x)\right)\right)}{384 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","\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,"(a^4*(24*(209 + 489*c + 489*d*x)*Cos[(c + d*x)/2] - 24*(453 + 163*c + 163*d*x)*Cos[(3*(c + d*x))/2] + 885*Cos[(5*(c + d*x))/2] - 129*Cos[(7*(c + d*x))/2] - 23*Cos[(9*(c + d*x))/2] + 3*Cos[(11*(c + d*x))/2] - 16488*Sin[(c + d*x)/2] - 11736*c*Sin[(c + d*x)/2] - 11736*d*x*Sin[(c + d*x)/2] + 3704*Sin[(3*(c + d*x))/2] - 3912*c*Sin[(3*(c + d*x))/2] - 3912*d*x*Sin[(3*(c + d*x))/2] + 885*Sin[(5*(c + d*x))/2] + 129*Sin[(7*(c + d*x))/2] - 23*Sin[(9*(c + d*x))/2] - 3*Sin[(11*(c + d*x))/2]))/(384*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)","A",1
820,1,158,101,1.7956896,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^4*Tan[c + d*x]^2,x]","-\frac{a^4 \left(-3 (204 c+204 d x+161) \cos \left(\frac{1}{2} (c+d x)\right)+(204 c+204 d x+647) \cos \left(\frac{3}{2} (c+d x)\right)-39 \cos \left(\frac{5}{2} (c+d x)\right)+3 \cos \left(\frac{7}{2} (c+d x)\right)+6 \sin \left(\frac{1}{2} (c+d x)\right) ((68 c+68 d x-59) \cos (c+d x)-14 \cos (2 (c+d x))-\cos (3 (c+d x))+136 c+136 d x+146)\right)}{48 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\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,"-1/48*(a^4*(-3*(161 + 204*c + 204*d*x)*Cos[(c + d*x)/2] + (647 + 204*c + 204*d*x)*Cos[(3*(c + d*x))/2] - 39*Cos[(5*(c + d*x))/2] + 3*Cos[(7*(c + d*x))/2] + 6*(146 + 136*c + 136*d*x + (-59 + 68*c + 68*d*x)*Cos[c + d*x] - 14*Cos[2*(c + d*x)] - Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)","A",1
821,1,224,117,0.6847449,"\int \frac{\sin ^2(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{216 \sin (c+d x)+240 c \sin (2 (c+d x))+240 d x \sin (2 (c+d x))-618 \sin (2 (c+d x))+532 \sin (3 (c+d x))+120 c \sin (4 (c+d x))+120 d x \sin (4 (c+d x))-309 \sin (4 (c+d x))+60 \sin (5 (c+d x))+18 (40 c+40 d x-103) \cos (c+d x)+1568 \cos (2 (c+d x))+240 c \cos (3 (c+d x))+240 d x \cos (3 (c+d x))-618 \cos (3 (c+d x))+304 \cos (4 (c+d x))+1200}{960 a d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\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,"-1/960*(1200 + 18*(-103 + 40*c + 40*d*x)*Cos[c + d*x] + 1568*Cos[2*(c + d*x)] - 618*Cos[3*(c + d*x)] + 240*c*Cos[3*(c + d*x)] + 240*d*x*Cos[3*(c + d*x)] + 304*Cos[4*(c + d*x)] + 216*Sin[c + d*x] - 618*Sin[2*(c + d*x)] + 240*c*Sin[2*(c + d*x)] + 240*d*x*Sin[2*(c + d*x)] + 532*Sin[3*(c + d*x)] - 309*Sin[4*(c + d*x)] + 120*c*Sin[4*(c + d*x)] + 120*d*x*Sin[4*(c + d*x)] + 60*Sin[5*(c + d*x)])/(a*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","A",1
822,1,191,105,0.6058619,"\int \frac{\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^3(c+d x) \left(8 \sin (c+d x)-30 c \sin (2 (c+d x))-30 d x \sin (2 (c+d x))+\frac{89}{4} \sin (2 (c+d x))+16 \sin (3 (c+d x))-15 c \sin (4 (c+d x))-15 d x \sin (4 (c+d x))+\frac{89}{8} \sin (4 (c+d x))+\left(-90 c-90 d x+\frac{267}{4}\right) \cos (c+d x)-16 \cos (2 (c+d x))-30 c \cos (3 (c+d x))-30 d x \cos (3 (c+d x))+\frac{89}{4} \cos (3 (c+d x))-23 \cos (4 (c+d x))-25\right)}{120 a d (\sin (c+d x)+1)}","-\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,"-1/120*(Sec[c + d*x]^3*(-25 + (267/4 - 90*c - 90*d*x)*Cos[c + d*x] - 16*Cos[2*(c + d*x)] + (89*Cos[3*(c + d*x)])/4 - 30*c*Cos[3*(c + d*x)] - 30*d*x*Cos[3*(c + d*x)] - 23*Cos[4*(c + d*x)] + 8*Sin[c + d*x] + (89*Sin[2*(c + d*x)])/4 - 30*c*Sin[2*(c + d*x)] - 30*d*x*Sin[2*(c + d*x)] + 16*Sin[3*(c + d*x)] + (89*Sin[4*(c + d*x)])/8 - 15*c*Sin[4*(c + d*x)] - 15*d*x*Sin[4*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","A",1
823,1,106,69,0.2679989,"\int \frac{\tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^4/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^3(c+d x) (-64 \sin (c+d x)-178 \sin (2 (c+d x))+192 \sin (3 (c+d x))-89 \sin (4 (c+d x))-534 \cos (c+d x)+288 \cos (2 (c+d x))-178 \cos (3 (c+d x))+24 \cos (4 (c+d x))+200)}{960 a d (\sin (c+d x)+1)}","\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,"-1/960*(Sec[c + d*x]^3*(200 - 534*Cos[c + d*x] + 288*Cos[2*(c + d*x)] - 178*Cos[3*(c + d*x)] + 24*Cos[4*(c + d*x)] - 64*Sin[c + d*x] - 178*Sin[2*(c + d*x)] + 192*Sin[3*(c + d*x)] - 89*Sin[4*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","A",1
824,1,106,55,0.2931348,"\int \frac{\sec (c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\sec ^3(c+d x) (16 \sin (c+d x)+22 \sin (2 (c+d x))-48 \sin (3 (c+d x))+11 \sin (4 (c+d x))+66 \cos (c+d x)-192 \cos (2 (c+d x))+22 \cos (3 (c+d x))+24 \cos (4 (c+d x))+40)}{960 a d (\sin (c+d x)+1)}","-\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*(40 + 66*Cos[c + d*x] - 192*Cos[2*(c + d*x)] + 22*Cos[3*(c + d*x)] + 24*Cos[4*(c + d*x)] + 16*Sin[c + d*x] + 22*Sin[2*(c + d*x)] - 48*Sin[3*(c + d*x)] + 11*Sin[4*(c + d*x)]))/(960*a*d*(1 + Sin[c + d*x]))","A",1
825,1,106,73,0.3365408,"\int \frac{\sec ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^3(c+d x) (-224 \sin (c+d x)+22 \sin (2 (c+d x))+32 \sin (3 (c+d x))+11 \sin (4 (c+d x))+66 \cos (c+d x)-32 \cos (2 (c+d x))+22 \cos (3 (c+d x))-16 \cos (4 (c+d x))-80)}{960 a d (\sin (c+d x)+1)}","\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,"-1/960*(Sec[c + d*x]^3*(-80 + 66*Cos[c + d*x] - 32*Cos[2*(c + d*x)] + 22*Cos[3*(c + d*x)] - 16*Cos[4*(c + d*x)] - 224*Sin[c + d*x] + 22*Sin[2*(c + d*x)] + 32*Sin[3*(c + d*x)] + 11*Sin[4*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","A",1
826,1,106,55,0.2637014,"\int \frac{\sec ^3(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*Tan[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^3(c+d x) (-96 \sin (c+d x)+18 \sin (2 (c+d x))-32 \sin (3 (c+d x))+9 \sin (4 (c+d x))+54 \cos (c+d x)+32 \cos (2 (c+d x))+18 \cos (3 (c+d x))+16 \cos (4 (c+d x))-240)}{960 a d (\sin (c+d x)+1)}","-\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,"-1/960*(Sec[c + d*x]^3*(-240 + 54*Cos[c + d*x] + 32*Cos[2*(c + d*x)] + 18*Cos[3*(c + d*x)] + 16*Cos[4*(c + d*x)] - 96*Sin[c + d*x] + 18*Sin[2*(c + d*x)] - 32*Sin[3*(c + d*x)] + 9*Sin[4*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","A",1
827,1,267,115,0.6584956,"\int \frac{\csc (c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^3(c+d x) \left(-22 \sin (c+d x)+\frac{149}{4} \sin (2 (c+d x))-14 \sin (3 (c+d x))+\frac{149}{8} \sin (4 (c+d x))-76 \cos (2 (c+d x))+\frac{149}{4} \cos (3 (c+d x))-8 \cos (4 (c+d x))-30 \sin (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-15 \sin (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+30 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\cos (c+d x) \left(-90 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+90 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{447}{4}\right)-30 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+30 \sin (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+15 \sin (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-100\right)}{120 a d (\sin (c+d x)+1)}","-\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,"-1/120*(Sec[c + d*x]^3*(-100 - 76*Cos[2*(c + d*x)] + (149*Cos[3*(c + d*x)])/4 - 8*Cos[4*(c + d*x)] + 30*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] + Cos[c + d*x]*(447/4 + 90*Log[Cos[(c + d*x)/2]] - 90*Log[Sin[(c + d*x)/2]]) - 30*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 22*Sin[c + d*x] + (149*Sin[2*(c + d*x)])/4 + 30*Log[Cos[(c + d*x)/2]]*Sin[2*(c + d*x)] - 30*Log[Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] - 14*Sin[3*(c + d*x)] + (149*Sin[4*(c + d*x)])/8 + 15*Log[Cos[(c + d*x)/2]]*Sin[4*(c + d*x)] - 15*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","B",1
828,1,341,126,0.603308,"\int \frac{\csc ^2(c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(352 \sin (c+d x)-596 \sin (2 (c+d x))+864 \sin (3 (c+d x))-298 \sin (4 (c+d x))+384 \sin (5 (c+d x))+1216 \cos (2 (c+d x))+149 \cos (3 (c+d x))+528 \cos (4 (c+d x))+149 \cos (5 (c+d x))+480 \sin (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+240 \sin (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+120 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+120 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-120 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-120 \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\cos (c+d x) \left(240 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-240 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-298\right)-480 \sin (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-240 \sin (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+176\right)}{3840 a d (\sin (c+d x)+1)}","\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,"-1/3840*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*Sec[c + d*x]^3*(176 + 1216*Cos[2*(c + d*x)] + 149*Cos[3*(c + d*x)] + 528*Cos[4*(c + d*x)] + 149*Cos[5*(c + d*x)] + 120*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 120*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 120*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 120*Cos[5*(c + d*x)]*Log[Sin[(c + d*x)/2]] + Cos[c + d*x]*(-298 - 240*Log[Cos[(c + d*x)/2]] + 240*Log[Sin[(c + d*x)/2]]) + 352*Sin[c + d*x] - 596*Sin[2*(c + d*x)] - 480*Log[Cos[(c + d*x)/2]]*Sin[2*(c + d*x)] + 480*Log[Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] + 864*Sin[3*(c + d*x)] - 298*Sin[4*(c + d*x)] - 240*Log[Cos[(c + d*x)/2]]*Sin[4*(c + d*x)] + 240*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)] + 384*Sin[5*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))","B",1
829,1,267,155,0.7879325,"\int \frac{\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sin[c + d*x]^3*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{5488 \sin (c+d x)+6720 c \sin (2 (c+d x))+6720 d x \sin (2 (c+d x))-13224 \sin (2 (c+d x))+8376 \sin (3 (c+d x))+3360 c \sin (4 (c+d x))+3360 d x \sin (4 (c+d x))-6612 \sin (4 (c+d x))+2248 \sin (5 (c+d x))+42 (280 c+280 d x-551) \cos (c+d x)+14834 \cos (2 (c+d x))+2520 c \cos (3 (c+d x))+2520 d x \cos (3 (c+d x))-4959 \cos (3 (c+d x))+1852 \cos (4 (c+d x))-840 c \cos (5 (c+d x))-840 d x \cos (5 (c+d x))+1653 \cos (5 (c+d x))-210 \cos (6 (c+d x))+11172}{6720 a^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}","-\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,"-1/6720*(11172 + 42*(-551 + 280*c + 280*d*x)*Cos[c + d*x] + 14834*Cos[2*(c + d*x)] - 4959*Cos[3*(c + d*x)] + 2520*c*Cos[3*(c + d*x)] + 2520*d*x*Cos[3*(c + d*x)] + 1852*Cos[4*(c + d*x)] + 1653*Cos[5*(c + d*x)] - 840*c*Cos[5*(c + d*x)] - 840*d*x*Cos[5*(c + d*x)] - 210*Cos[6*(c + d*x)] + 5488*Sin[c + d*x] - 13224*Sin[2*(c + d*x)] + 6720*c*Sin[2*(c + d*x)] + 6720*d*x*Sin[2*(c + d*x)] + 8376*Sin[3*(c + d*x)] - 6612*Sin[4*(c + d*x)] + 3360*c*Sin[4*(c + d*x)] + 3360*d*x*Sin[4*(c + d*x)] + 2248*Sin[5*(c + d*x)])/(a^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)","A",1
830,1,257,140,0.5712897,"\int \frac{\sin ^2(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{2128 \sin (c+d x)+6720 c \sin (2 (c+d x))+6720 d x \sin (2 (c+d x))-9144 \sin (2 (c+d x))+456 \sin (3 (c+d x))+3360 c \sin (4 (c+d x))+3360 d x \sin (4 (c+d x))-4572 \sin (4 (c+d x))+1528 \sin (5 (c+d x))+42 (280 c+280 d x-381) \cos (c+d x)+5504 \cos (2 (c+d x))+2520 c \cos (3 (c+d x))+2520 d x \cos (3 (c+d x))-3429 \cos (3 (c+d x))+2752 \cos (4 (c+d x))-840 c \cos (5 (c+d x))-840 d x \cos (5 (c+d x))+1143 \cos (5 (c+d x))+4032}{13440 a^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}","\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,"(4032 + 42*(-381 + 280*c + 280*d*x)*Cos[c + d*x] + 5504*Cos[2*(c + d*x)] - 3429*Cos[3*(c + d*x)] + 2520*c*Cos[3*(c + d*x)] + 2520*d*x*Cos[3*(c + d*x)] + 2752*Cos[4*(c + d*x)] + 1143*Cos[5*(c + d*x)] - 840*c*Cos[5*(c + d*x)] - 840*d*x*Cos[5*(c + d*x)] + 2128*Sin[c + d*x] - 9144*Sin[2*(c + d*x)] + 6720*c*Sin[2*(c + d*x)] + 6720*d*x*Sin[2*(c + d*x)] + 456*Sin[3*(c + d*x)] - 4572*Sin[4*(c + d*x)] + 3360*c*Sin[4*(c + d*x)] + 3360*d*x*Sin[4*(c + d*x)] + 1528*Sin[5*(c + d*x)])/(13440*a^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)","A",1
831,1,126,85,0.263303,"\int \frac{\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","-\frac{\sec ^3(c+d x) (28 \sin (c+d x)-104 \sin (2 (c+d x))+66 \sin (3 (c+d x))-52 \sin (4 (c+d x))+6 \sin (5 (c+d x))-182 \cos (c+d x)+104 \cos (2 (c+d x))-39 \cos (3 (c+d x))-18 \cos (4 (c+d x))+13 \cos (5 (c+d x))+42)}{336 a^2 d (\sin (c+d x)+1)^2}","-\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,"-1/336*(Sec[c + d*x]^3*(42 - 182*Cos[c + d*x] + 104*Cos[2*(c + d*x)] - 39*Cos[3*(c + d*x)] - 18*Cos[4*(c + d*x)] + 13*Cos[5*(c + d*x)] + 28*Sin[c + d*x] - 104*Sin[2*(c + d*x)] + 66*Sin[3*(c + d*x)] - 52*Sin[4*(c + d*x)] + 6*Sin[5*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
832,1,126,91,0.2589601,"\int \frac{\tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^4/(a + a*Sin[c + d*x])^2,x]","-\frac{\sec ^3(c+d x) (-1232 \sin (c+d x)-824 \sin (2 (c+d x))+1896 \sin (3 (c+d x))-412 \sin (4 (c+d x))-72 \sin (5 (c+d x))-1442 \cos (c+d x)+1664 \cos (2 (c+d x))-309 \cos (3 (c+d x))-288 \cos (4 (c+d x))+103 \cos (5 (c+d x))+672)}{13440 a^2 d (\sin (c+d x)+1)^2}","\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,"-1/13440*(Sec[c + d*x]^3*(672 - 1442*Cos[c + d*x] + 1664*Cos[2*(c + d*x)] - 309*Cos[3*(c + d*x)] - 288*Cos[4*(c + d*x)] + 103*Cos[5*(c + d*x)] - 1232*Sin[c + d*x] - 824*Sin[2*(c + d*x)] + 1896*Sin[3*(c + d*x)] - 412*Sin[4*(c + d*x)] - 72*Sin[5*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
833,1,126,91,0.3278616,"\int \frac{\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + a*Sin[c + d*x])^2,x]","\frac{\sec ^3(c+d x) (448 \sin (c+d x)-104 \sin (2 (c+d x))-144 \sin (3 (c+d x))-52 \sin (4 (c+d x))+48 \sin (5 (c+d x))-182 \cos (c+d x)-736 \cos (2 (c+d x))-39 \cos (3 (c+d x))+192 \cos (4 (c+d x))+13 \cos (5 (c+d x))+672)}{6720 a^2 d (\sin (c+d x)+1)^2}","-\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*(672 - 182*Cos[c + d*x] - 736*Cos[2*(c + d*x)] - 39*Cos[3*(c + d*x)] + 192*Cos[4*(c + d*x)] + 13*Cos[5*(c + d*x)] + 448*Sin[c + d*x] - 104*Sin[2*(c + d*x)] - 144*Sin[3*(c + d*x)] - 52*Sin[4*(c + d*x)] + 48*Sin[5*(c + d*x)]))/(6720*a^2*d*(1 + Sin[c + d*x])^2)","A",1
834,1,126,91,0.3741602,"\int \frac{\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]","\frac{\sec ^3(c+d x) (3136 \sin (c+d x)-408 \sin (2 (c+d x))-48 \sin (3 (c+d x))-204 \sin (4 (c+d x))+16 \sin (5 (c+d x))-714 \cos (c+d x)+128 \cos (2 (c+d x))-153 \cos (3 (c+d x))+64 \cos (4 (c+d x))+51 \cos (5 (c+d x))+1344)}{13440 a^2 d (\sin (c+d x)+1)^2}","\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,"(Sec[c + d*x]^3*(1344 - 714*Cos[c + d*x] + 128*Cos[2*(c + d*x)] - 153*Cos[3*(c + d*x)] + 64*Cos[4*(c + d*x)] + 51*Cos[5*(c + d*x)] + 3136*Sin[c + d*x] - 408*Sin[2*(c + d*x)] - 48*Sin[3*(c + d*x)] - 204*Sin[4*(c + d*x)] + 16*Sin[5*(c + d*x)]))/(13440*a^2*d*(1 + Sin[c + d*x])^2)","A",1
835,1,134,93,0.3021272,"\int \frac{\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^3*Tan[c + d*x])/(a + a*Sin[c + d*x])^2,x]","-\frac{\sec ^3(c+d x) \left(-56 \sin (c+d x)+3 \sin (2 (c+d x))-12 \sin (3 (c+d x))+\frac{3}{2} \sin (4 (c+d x))+4 \sin (5 (c+d x))+\frac{21}{4} \cos (c+d x)+32 \cos (2 (c+d x))+\frac{9}{8} \cos (3 (c+d x))+16 \cos (4 (c+d x))-\frac{3}{8} \cos (5 (c+d x))-84\right)}{420 a^2 d (\sin (c+d x)+1)^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,"-1/420*(Sec[c + d*x]^3*(-84 + (21*Cos[c + d*x])/4 + 32*Cos[2*(c + d*x)] + (9*Cos[3*(c + d*x)])/8 + 16*Cos[4*(c + d*x)] - (3*Cos[5*(c + d*x)])/8 - 56*Sin[c + d*x] + 3*Sin[2*(c + d*x)] - 12*Sin[3*(c + d*x)] + (3*Sin[4*(c + d*x)])/2 + 4*Sin[5*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
836,1,352,149,0.6076059,"\int \frac{\csc (c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{2464 \sin (c+d x)-4472 \sin (2 (c+d x))+2208 \sin (3 (c+d x))-2236 \sin (4 (c+d x))+384 \sin (5 (c+d x))+5312 \cos (2 (c+d x))-1677 \cos (3 (c+d x))+696 \cos (4 (c+d x))+559 \cos (5 (c+d x))+3360 \sin (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+1680 \sin (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-1260 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+420 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-14 \cos (c+d x) \left(-420 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+420 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+559\right)+1260 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-420 \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3360 \sin (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-1680 \sin (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+6216}{6720 a^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}","-\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,"(6216 + 5312*Cos[2*(c + d*x)] - 1677*Cos[3*(c + d*x)] + 696*Cos[4*(c + d*x)] + 559*Cos[5*(c + d*x)] - 1260*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 420*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 14*Cos[c + d*x]*(559 + 420*Log[Cos[(c + d*x)/2]] - 420*Log[Sin[(c + d*x)/2]]) + 1260*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 420*Cos[5*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 2464*Sin[c + d*x] - 4472*Sin[2*(c + d*x)] - 3360*Log[Cos[(c + d*x)/2]]*Sin[2*(c + d*x)] + 3360*Log[Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] + 2208*Sin[3*(c + d*x)] - 2236*Sin[4*(c + d*x)] - 1680*Log[Cos[(c + d*x)/2]]*Sin[4*(c + d*x)] + 1680*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)] + 384*Sin[5*(c + d*x)])/(6720*a^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)","B",1
837,1,442,164,6.0953057,"\int \frac{\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{16 \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{13 \sin \left(\frac{1}{2} (c+d x)\right)}{384 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{384 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{4777 \sin \left(\frac{1}{2} (c+d x)\right)}{13440 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{768 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{997}{26880 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{997 \sin \left(\frac{1}{2} (c+d x)\right)}{13440 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{3}{280 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{3 \sin \left(\frac{1}{2} (c+d x)\right)}{140 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}-\frac{1}{448 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{224 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}\right)}{a^2}","\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,"(16*(-1/32*Cot[(c + d*x)/2]/d + Log[Cos[(c + d*x)/2]]/(8*d) - Log[Sin[(c + d*x)/2]]/(8*d) + 1/(768*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + Sin[(c + d*x)/2]/(384*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (13*Sin[(c + d*x)/2])/(384*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + Sin[(c + d*x)/2]/(224*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7) - 1/(448*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6) + (3*Sin[(c + d*x)/2])/(140*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5) - 3/(280*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (997*Sin[(c + d*x)/2])/(13440*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - 997/(26880*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4777*Sin[(c + d*x)/2])/(13440*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + Tan[(c + d*x)/2]/(32*d)))/a^2","B",1
838,1,277,194,0.5735772,"\int \frac{\csc ^3(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^4)/(a + a*Sin[c + d*x])^2,x]","\frac{-36960 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7+36960 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7+\frac{\csc ^2(c+d x) (4488 \sin (c+d x)-7536 \sin (2 (c+d x))+3836 \sin (3 (c+d x))-780 \sin (5 (c+d x))+2512 \sin (6 (c+d x))-768 \sin (7 (c+d x))-6908 \cos (c+d x)-563 \cos (2 (c+d x))+4396 \cos (3 (c+d x))-5390 \cos (4 (c+d x))+3140 \cos (5 (c+d x))-1917 \cos (6 (c+d x))-628 \cos (7 (c+d x))+4510)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}}{6720 d (a \sin (c+d x)+a)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\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,"(-36960*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7 + 36960*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7 + (Csc[c + d*x]^2*(4510 - 6908*Cos[c + d*x] - 563*Cos[2*(c + d*x)] + 4396*Cos[3*(c + d*x)] - 5390*Cos[4*(c + d*x)] + 3140*Cos[5*(c + d*x)] - 1917*Cos[6*(c + d*x)] - 628*Cos[7*(c + d*x)] + 4488*Sin[c + d*x] - 7536*Sin[2*(c + d*x)] + 3836*Sin[3*(c + d*x)] - 780*Sin[5*(c + d*x)] + 2512*Sin[6*(c + d*x)] - 768*Sin[7*(c + d*x)]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)/(6720*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + a*Sin[c + d*x])^2)","A",1
839,1,273,178,0.5141109,"\int \frac{\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]^3*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","\frac{93312 \sin (c+d x)+272160 (c+d x) \sin (2 (c+d x))-506277 \sin (2 (c+d x))+125248 \sin (3 (c+d x))+120960 (c+d x) \sin (4 (c+d x))-225012 \sin (4 (c+d x))+67776 \sin (5 (c+d x))-10080 (c+d x) \sin (6 (c+d x))+18751 \sin (6 (c+d x))+362880 (c+d x) \cos (c+d x)-675036 \cos (c+d x)+173952 \cos (2 (c+d x))+20160 (c+d x) \cos (3 (c+d x))-37502 \cos (3 (c+d x))+54912 \cos (4 (c+d x))-60480 (c+d x) \cos (5 (c+d x))+112506 \cos (5 (c+d x))-21376 \cos (6 (c+d x))+169344}{322560 d (a \sin (c+d x)+a)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^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,"(169344 - 675036*Cos[c + d*x] + 362880*(c + d*x)*Cos[c + d*x] + 173952*Cos[2*(c + d*x)] - 37502*Cos[3*(c + d*x)] + 20160*(c + d*x)*Cos[3*(c + d*x)] + 54912*Cos[4*(c + d*x)] + 112506*Cos[5*(c + d*x)] - 60480*(c + d*x)*Cos[5*(c + d*x)] - 21376*Cos[6*(c + d*x)] + 93312*Sin[c + d*x] - 506277*Sin[2*(c + d*x)] + 272160*(c + d*x)*Sin[2*(c + d*x)] + 125248*Sin[3*(c + d*x)] - 225012*Sin[4*(c + d*x)] + 120960*(c + d*x)*Sin[4*(c + d*x)] + 67776*Sin[5*(c + d*x)] + 18751*Sin[6*(c + d*x)] - 10080*(c + d*x)*Sin[6*(c + d*x)])/(322560*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + a*Sin[c + d*x])^3)","A",1
840,1,185,121,0.3783295,"\int \frac{\sin ^2(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","\frac{-2304 \sin (c+d x)+27189 \sin (2 (c+d x))-16256 \sin (3 (c+d x))+12084 \sin (4 (c+d x))+384 \sin (5 (c+d x))-1007 \sin (6 (c+d x))+36252 \cos (c+d x)-12384 \cos (2 (c+d x))+2014 \cos (3 (c+d x))+4800 \cos (4 (c+d x))-6042 \cos (5 (c+d x))+608 \cos (6 (c+d x))-9408}{64512 d (a \sin (c+d x)+a)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","\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,"(-9408 + 36252*Cos[c + d*x] - 12384*Cos[2*(c + d*x)] + 2014*Cos[3*(c + d*x)] + 4800*Cos[4*(c + d*x)] - 6042*Cos[5*(c + d*x)] + 608*Cos[6*(c + d*x)] - 2304*Sin[c + d*x] + 27189*Sin[2*(c + d*x)] - 16256*Sin[3*(c + d*x)] + 12084*Sin[4*(c + d*x)] + 384*Sin[5*(c + d*x)] - 1007*Sin[6*(c + d*x)])/(64512*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + a*Sin[c + d*x])^3)","A",1
841,1,185,105,0.2053421,"\int \frac{\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","\frac{-1152 \sin (c+d x)+6507 \sin (2 (c+d x))-8128 \sin (3 (c+d x))+2892 \sin (4 (c+d x))+192 \sin (5 (c+d x))-241 \sin (6 (c+d x))+8676 \cos (c+d x)-11232 \cos (2 (c+d x))+482 \cos (3 (c+d x))+4416 \cos (4 (c+d x))-1446 \cos (5 (c+d x))-32 \cos (6 (c+d x))-1344}{64512 d (a \sin (c+d x)+a)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\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,"(-1344 + 8676*Cos[c + d*x] - 11232*Cos[2*(c + d*x)] + 482*Cos[3*(c + d*x)] + 4416*Cos[4*(c + d*x)] - 1446*Cos[5*(c + d*x)] - 32*Cos[6*(c + d*x)] - 1152*Sin[c + d*x] + 6507*Sin[2*(c + d*x)] - 8128*Sin[3*(c + d*x)] + 2892*Sin[4*(c + d*x)] + 192*Sin[5*(c + d*x)] - 241*Sin[6*(c + d*x)])/(64512*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + a*Sin[c + d*x])^3)","A",1
842,1,185,127,0.292668,"\int \frac{\tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]","\frac{39168 \sin (c+d x)+837 \sin (2 (c+d x))-28288 \sin (3 (c+d x))+372 \sin (4 (c+d x))+4224 \sin (5 (c+d x))-31 \sin (6 (c+d x))+1116 \cos (c+d x)-21312 \cos (2 (c+d x))+62 \cos (3 (c+d x))+8448 \cos (4 (c+d x))-186 \cos (5 (c+d x))-704 \cos (6 (c+d x))+5376}{322560 d (a \sin (c+d x)+a)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","\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,"(5376 + 1116*Cos[c + d*x] - 21312*Cos[2*(c + d*x)] + 62*Cos[3*(c + d*x)] + 8448*Cos[4*(c + d*x)] - 186*Cos[5*(c + d*x)] - 704*Cos[6*(c + d*x)] + 39168*Sin[c + d*x] + 837*Sin[2*(c + d*x)] - 28288*Sin[3*(c + d*x)] + 372*Sin[4*(c + d*x)] + 4224*Sin[5*(c + d*x)] - 31*Sin[6*(c + d*x)])/(322560*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + a*Sin[c + d*x])^3)","A",1
843,1,185,105,0.3182974,"\int \frac{\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + a*Sin[c + d*x])^3,x]","\frac{4608 \sin (c+d x)-1323 \sin (2 (c+d x))-128 \sin (3 (c+d x))-588 \sin (4 (c+d x))+384 \sin (5 (c+d x))+49 \sin (6 (c+d x))-1764 \cos (c+d x)-4032 \cos (2 (c+d x))-98 \cos (3 (c+d x))+768 \cos (4 (c+d x))+294 \cos (5 (c+d x))-64 \cos (6 (c+d x))+5376}{46080 d (a \sin (c+d x)+a)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\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,"(5376 - 1764*Cos[c + d*x] - 4032*Cos[2*(c + d*x)] - 98*Cos[3*(c + d*x)] + 768*Cos[4*(c + d*x)] + 294*Cos[5*(c + d*x)] - 64*Cos[6*(c + d*x)] + 4608*Sin[c + d*x] - 1323*Sin[2*(c + d*x)] - 128*Sin[3*(c + d*x)] - 588*Sin[4*(c + d*x)] + 384*Sin[5*(c + d*x)] + 49*Sin[6*(c + d*x)])/(46080*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + a*Sin[c + d*x])^3)","A",1
844,1,185,127,0.3262287,"\int \frac{\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]","\frac{73728 \sin (c+d x)-7263 \sin (2 (c+d x))+512 \sin (3 (c+d x))-3228 \sin (4 (c+d x))-1536 \sin (5 (c+d x))+269 \sin (6 (c+d x))-9684 \cos (c+d x)-6912 \cos (2 (c+d x))-538 \cos (3 (c+d x))-3072 \cos (4 (c+d x))+1614 \cos (5 (c+d x))+256 \cos (6 (c+d x))+32256}{322560 d (a \sin (c+d x)+a)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","\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,"(32256 - 9684*Cos[c + d*x] - 6912*Cos[2*(c + d*x)] - 538*Cos[3*(c + d*x)] - 3072*Cos[4*(c + d*x)] + 1614*Cos[5*(c + d*x)] + 256*Cos[6*(c + d*x)] + 73728*Sin[c + d*x] - 7263*Sin[2*(c + d*x)] + 512*Sin[3*(c + d*x)] - 3228*Sin[4*(c + d*x)] - 1536*Sin[5*(c + d*x)] + 269*Sin[6*(c + d*x)])/(322560*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + a*Sin[c + d*x])^3)","A",1
845,1,185,123,0.2819631,"\int \frac{\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^3*Tan[c + d*x])/(a + a*Sin[c + d*x])^3,x]","\frac{9216 \sin (c+d x)+675 \sin (2 (c+d x))+512 \sin (3 (c+d x))+300 \sin (4 (c+d x))-1536 \sin (5 (c+d x))-25 \sin (6 (c+d x))+900 \cos (c+d x)-6912 \cos (2 (c+d x))+50 \cos (3 (c+d x))-3072 \cos (4 (c+d x))-150 \cos (5 (c+d x))+256 \cos (6 (c+d x))+10752}{64512 d (a \sin (c+d x)+a)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^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,"(10752 + 900*Cos[c + d*x] - 6912*Cos[2*(c + d*x)] + 50*Cos[3*(c + d*x)] - 3072*Cos[4*(c + d*x)] - 150*Cos[5*(c + d*x)] + 256*Cos[6*(c + d*x)] + 9216*Sin[c + d*x] + 675*Sin[2*(c + d*x)] + 512*Sin[3*(c + d*x)] + 300*Sin[4*(c + d*x)] - 1536*Sin[5*(c + d*x)] - 25*Sin[6*(c + d*x)])/(64512*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + a*Sin[c + d*x])^3)","A",1
846,1,204,187,1.3769215,"\int \frac{\csc (c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","\frac{322560 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-322560 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{196992 \sin (c+d x)-383157 \sin (2 (c+d x))+211648 \sin (3 (c+d x))-170292 \sin (4 (c+d x))+50496 \sin (5 (c+d x))+14191 \sin (6 (c+d x))-510876 \cos (c+d x)+317952 \cos (2 (c+d x))-28382 \cos (3 (c+d x))+20352 \cos (4 (c+d x))+85146 \cos (5 (c+d x))-11776 \cos (6 (c+d x))+357504}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^9}}{322560 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,"(-322560*Log[Cos[(c + d*x)/2]] + 322560*Log[Sin[(c + d*x)/2]] + (357504 - 510876*Cos[c + d*x] + 317952*Cos[2*(c + d*x)] - 28382*Cos[3*(c + d*x)] + 20352*Cos[4*(c + d*x)] + 85146*Cos[5*(c + d*x)] - 11776*Cos[6*(c + d*x)] + 196992*Sin[c + d*x] - 383157*Sin[2*(c + d*x)] + 211648*Sin[3*(c + d*x)] - 170292*Sin[4*(c + d*x)] + 50496*Sin[5*(c + d*x)] + 14191*Sin[6*(c + d*x)])/((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^9))/(322560*a^3*d)","A",1
847,1,230,200,0.6390307,"\int \frac{\csc ^2(c+d x) \sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]","\frac{-1935360 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+1935360 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{\csc (c+d x) (-707328 \sin (c+d x)+1364182 \sin (2 (c+d x))-1161600 \sin (3 (c+d x))+320984 \sin (4 (c+d x))-329344 \sin (5 (c+d x))-240738 \sin (6 (c+d x))+53248 \sin (7 (c+d x))+1083321 \cos (c+d x)-653248 \cos (2 (c+d x))-601845 \cos (3 (c+d x))+340096 \cos (4 (c+d x))-521599 \cos (5 (c+d x))+259008 \cos (6 (c+d x))+40123 \cos (7 (c+d x))-590976)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^9}}{645120 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,"(1935360*Log[Cos[(c + d*x)/2]] - 1935360*Log[Sin[(c + d*x)/2]] + (Csc[c + d*x]*(-590976 + 1083321*Cos[c + d*x] - 653248*Cos[2*(c + d*x)] - 601845*Cos[3*(c + d*x)] + 340096*Cos[4*(c + d*x)] - 521599*Cos[5*(c + d*x)] + 259008*Cos[6*(c + d*x)] + 40123*Cos[7*(c + d*x)] - 707328*Sin[c + d*x] + 1364182*Sin[2*(c + d*x)] - 1161600*Sin[3*(c + d*x)] + 320984*Sin[4*(c + d*x)] - 329344*Sin[5*(c + d*x)] - 240738*Sin[6*(c + d*x)] + 53248*Sin[7*(c + d*x)]))/((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^9))/(645120*a^3*d)","A",1
848,1,166,145,0.452598,"\int \frac{\tan ^4(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^4/(a + a*Sin[c + d*x])^4,x]","\frac{\sec ^3(c+d x) (501600 \sin (c+d x)-70136 \sin (2 (c+d x))-200288 \sin (3 (c+d x))-25504 \sin (4 (c+d x))+48800 \sin (5 (c+d x))+6376 \sin (6 (c+d x))-1952 \sin (7 (c+d x))-78903 \cos (c+d x)-183040 \cos (2 (c+d x))+8767 \cos (3 (c+d x))+62464 \cos (4 (c+d x))+19925 \cos (5 (c+d x))-15616 \cos (6 (c+d x))-797 \cos (7 (c+d x))+168960)}{3548160 a^4 d (\sin (c+d x)+1)^4}","\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,"(Sec[c + d*x]^3*(168960 - 78903*Cos[c + d*x] - 183040*Cos[2*(c + d*x)] + 8767*Cos[3*(c + d*x)] + 62464*Cos[4*(c + d*x)] + 19925*Cos[5*(c + d*x)] - 15616*Cos[6*(c + d*x)] - 797*Cos[7*(c + d*x)] + 501600*Sin[c + d*x] - 70136*Sin[2*(c + d*x)] - 200288*Sin[3*(c + d*x)] - 25504*Sin[4*(c + d*x)] + 48800*Sin[5*(c + d*x)] + 6376*Sin[6*(c + d*x)] - 1952*Sin[7*(c + d*x)]))/(3548160*a^4*d*(1 + Sin[c + d*x])^4)","A",1
849,1,166,145,0.4561317,"\int \frac{\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + a*Sin[c + d*x])^4,x]","\frac{\sec ^3(c+d x) (844800 \sin (c+d x)-191752 \sin (2 (c+d x))+11264 \sin (3 (c+d x))-69728 \sin (4 (c+d x))+25600 \sin (5 (c+d x))+17432 \sin (6 (c+d x))-1024 \sin (7 (c+d x))-215721 \cos (c+d x)-619520 \cos (2 (c+d x))+23969 \cos (3 (c+d x))+32768 \cos (4 (c+d x))+54475 \cos (5 (c+d x))-8192 \cos (6 (c+d x))-2179 \cos (7 (c+d x))+844800)}{7096320 a^4 d (\sin (c+d x)+1)^4}","-\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]^3*(844800 - 215721*Cos[c + d*x] - 619520*Cos[2*(c + d*x)] + 23969*Cos[3*(c + d*x)] + 32768*Cos[4*(c + d*x)] + 54475*Cos[5*(c + d*x)] - 8192*Cos[6*(c + d*x)] - 2179*Cos[7*(c + d*x)] + 844800*Sin[c + d*x] - 191752*Sin[2*(c + d*x)] + 11264*Sin[3*(c + d*x)] - 69728*Sin[4*(c + d*x)] + 25600*Sin[5*(c + d*x)] + 17432*Sin[6*(c + d*x)] - 1024*Sin[7*(c + d*x)]))/(7096320*a^4*d*(1 + Sin[c + d*x])^4)","A",1
850,1,166,143,0.4913527,"\int \frac{\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^2*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^4,x]","\frac{\sec ^3(c+d x) (26048 \sin (c+d x)-1144 \sin (2 (c+d x))-704 \sin (3 (c+d x))-416 \sin (4 (c+d x))-1600 \sin (5 (c+d x))+104 \sin (6 (c+d x))+64 \sin (7 (c+d x))-1287 \cos (c+d x)-5632 \cos (2 (c+d x))+143 \cos (3 (c+d x))-2048 \cos (4 (c+d x))+325 \cos (5 (c+d x))+512 \cos (6 (c+d x))-13 \cos (7 (c+d x))+11264)}{118272 a^4 d (\sin (c+d x)+1)^4}","\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,"(Sec[c + d*x]^3*(11264 - 1287*Cos[c + d*x] - 5632*Cos[2*(c + d*x)] + 143*Cos[3*(c + d*x)] - 2048*Cos[4*(c + d*x)] + 325*Cos[5*(c + d*x)] + 512*Cos[6*(c + d*x)] - 13*Cos[7*(c + d*x)] + 26048*Sin[c + d*x] - 1144*Sin[2*(c + d*x)] - 704*Sin[3*(c + d*x)] - 416*Sin[4*(c + d*x)] - 1600*Sin[5*(c + d*x)] + 104*Sin[6*(c + d*x)] + 64*Sin[7*(c + d*x)]))/(118272*a^4*d*(1 + Sin[c + d*x])^4)","A",1
851,1,133,133,0.4512849,"\int \sin (c+d x) (a+a \sin (c+d x)) \tan ^5(c+d x) \, dx","Integrate[Sin[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^5,x]","-\frac{a \sin (c+d x) \tan ^4(c+d x)}{d}-\frac{a \left(2 \sin ^2(c+d x)-\sec ^4(c+d x)+6 \sec ^2(c+d x)+12 \log (\cos (c+d x))\right)}{4 d}-\frac{5 a \left(6 \tan (c+d x) \sec ^3(c+d x)-8 \tan ^3(c+d x) \sec (c+d x)-3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)}{8 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,"-1/4*(a*(12*Log[Cos[c + d*x]] + 6*Sec[c + d*x]^2 - Sec[c + d*x]^4 + 2*Sin[c + d*x]^2))/d - (a*Sin[c + d*x]*Tan[c + d*x]^4)/d - (5*a*(6*Sec[c + d*x]^3*Tan[c + d*x] - 8*Sec[c + d*x]*Tan[c + d*x]^3 - 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x])))/(8*d)","A",1
852,1,123,115,0.2708788,"\int (a+a \sin (c+d x)) \tan ^5(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])*Tan[c + d*x]^5,x]","-\frac{a \sin (c+d x) \tan ^4(c+d x)}{d}-\frac{a \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 d}-\frac{5 a \left(6 \tan (c+d x) \sec ^3(c+d x)-8 \tan ^3(c+d x) \sec (c+d x)-3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)}{8 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,"-((a*Sin[c + d*x]*Tan[c + d*x]^4)/d) - (a*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d) - (5*a*(6*Sec[c + d*x]^3*Tan[c + d*x] - 8*Sec[c + d*x]*Tan[c + d*x]^3 - 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x])))/(8*d)","A",1
853,1,106,105,0.2387117,"\int \sec (c+d x) (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^4,x]","\frac{a \tan ^3(c+d x) \sec (c+d x)}{d}-\frac{a \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 d}-\frac{a \left(6 \tan (c+d x) \sec ^3(c+d x)-3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)}{8 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,"(a*Sec[c + d*x]*Tan[c + d*x]^3)/d - (a*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d) - (a*(6*Sec[c + d*x]^3*Tan[c + d*x] - 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x])))/(8*d)","A",1
854,1,84,84,0.2076706,"\int \sec ^2(c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])*Tan[c + d*x]^3,x]","\frac{a \tan ^4(c+d x)}{4 d}+\frac{a \tan ^3(c+d x) \sec (c+d x)}{d}-\frac{a \left(6 \tan (c+d x) \sec ^3(c+d x)-3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)}{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,"(a*Sec[c + d*x]*Tan[c + d*x]^3)/d + (a*Tan[c + d*x]^4)/(4*d) - (a*(6*Sec[c + d*x]^3*Tan[c + d*x] - 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x])))/(8*d)","A",1
855,1,74,84,0.0295875,"\int \sec ^3(c+d x) (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])*Tan[c + d*x]^2,x]","\frac{a \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{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,"-1/8*(a*ArcTanh[Sin[c + d*x]])/d - (a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*Tan[c + d*x]^4)/(4*d)","A",1
856,1,74,61,0.0292888,"\int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])*Tan[c + d*x],x]","\frac{a \sec ^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{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,"-1/8*(a*ArcTanh[Sin[c + d*x]])/d + (a*Sec[c + d*x]^4)/(4*d) - (a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",1
857,1,99,117,0.2161367,"\int \csc (c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{a \left(-\sec ^4(c+d x)-2 \sec ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right)}{4 d}+\frac{3 a \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{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{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,"-1/4*(a*(4*Log[Cos[c + d*x]] - 4*Log[Sin[c + d*x]] - 2*Sec[c + d*x]^2 - Sec[c + d*x]^4))/d + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*a*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d)","A",1
858,1,76,129,0.1944729,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a \csc (c+d x) \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\sin ^2(c+d x)\right)}{d}-\frac{a \left(-\sec ^4(c+d x)-2 \sec ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right)}{4 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]*Hypergeometric2F1[-1/2, 3, 1/2, Sin[c + d*x]^2])/d) - (a*(4*Log[Cos[c + d*x]] - 4*Log[Sin[c + d*x]] - 2*Sec[c + d*x]^2 - Sec[c + d*x]^4))/(4*d)","C",1
859,1,86,143,0.7386833,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a \csc (c+d x) \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\sin ^2(c+d x)\right)}{d}-\frac{a \left(2 \csc ^2(c+d x)-\sec ^4(c+d x)-4 \sec ^2(c+d x)-12 \log (\sin (c+d x))+12 \log (\cos (c+d x))\right)}{4 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]*Hypergeometric2F1[-1/2, 3, 1/2, Sin[c + d*x]^2])/d) - (a*(2*Csc[c + d*x]^2 + 12*Log[Cos[c + d*x]] - 12*Log[Sin[c + d*x]] - 4*Sec[c + d*x]^2 - Sec[c + d*x]^4))/(4*d)","C",1
860,1,90,162,1.1630721,"\int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^4*Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]","-\frac{a \csc ^3(c+d x) \, _2F_1\left(-\frac{3}{2},3;-\frac{1}{2};\sin ^2(c+d x)\right)}{3 d}-\frac{a \left(2 \csc ^2(c+d x)-\sec ^4(c+d x)-4 \sec ^2(c+d x)-12 \log (\sin (c+d x))+12 \log (\cos (c+d x))\right)}{4 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,"-1/3*(a*Csc[c + d*x]^3*Hypergeometric2F1[-3/2, 3, -1/2, Sin[c + d*x]^2])/d - (a*(2*Csc[c + d*x]^2 + 12*Log[Cos[c + d*x]] - 12*Log[Sin[c + d*x]] - 4*Sec[c + d*x]^2 - Sec[c + d*x]^4))/(4*d)","C",1
861,1,75,119,0.240802,"\int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^5,x]","-\frac{a^2 \left(4 \sin ^2(c+d x)+16 \sin (c+d x)-\frac{18}{\sin (c+d x)-1}-\frac{2}{(\sin (c+d x)-1)^2}+31 \log (1-\sin (c+d x))+\log (\sin (c+d x)+1)\right)}{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 \sin ^2(c+d x)}{2 d}-\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,"-1/8*(a^2*(31*Log[1 - Sin[c + d*x]] + Log[1 + Sin[c + d*x]] - 2/(-1 + Sin[c + d*x])^2 - 18/(-1 + Sin[c + d*x]) + 16*Sin[c + d*x] + 4*Sin[c + d*x]^2))/d","A",1
862,1,67,101,0.1221114,"\int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^4,x]","-\frac{a^2 \left(8 \sin (c+d x)-\frac{14}{\sin (c+d x)-1}-\frac{2}{(\sin (c+d x)-1)^2}+17 \log (1-\sin (c+d x))-\log (\sin (c+d x)+1)\right)}{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,"-1/8*(a^2*(17*Log[1 - Sin[c + d*x]] - Log[1 + Sin[c + d*x]] - 2/(-1 + Sin[c + d*x])^2 - 14/(-1 + Sin[c + d*x]) + 8*Sin[c + d*x]))/d","A",1
863,1,91,87,0.3924234,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{a^2 \left(3 \tanh ^{-1}(\sin (c+d x))-6 \tan (c+d x) \sec ^3(c+d x)+\tan (c+d x) \left(8 \tan ^2(c+d x)+3\right) \sec (c+d x)-2 \left(-\tan ^4(c+d x)+\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)\right)}{4 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,"(a^2*(3*ArcTanh[Sin[c + d*x]] - 6*Sec[c + d*x]^3*Tan[c + d*x] + Sec[c + d*x]*Tan[c + d*x]*(3 + 8*Tan[c + d*x]^2) - 2*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2 - Tan[c + d*x]^4)))/(4*d)","A",1
864,1,39,64,0.10844,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{a^2 \left(\frac{3 \sin (c+d x)-2}{(\sin (c+d x)-1)^2}+\tanh ^{-1}(\sin (c+d x))\right)}{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]] + (-2 + 3*Sin[c + d*x])/(-1 + Sin[c + d*x])^2))/(4*d)","A",1
865,1,36,64,0.0770912,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^2 \tan (c+d x) \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2*Tan[c + d*x],x]","-\frac{a^2 \left(\tanh ^{-1}(\sin (c+d x))-\frac{\sin (c+d x)}{(\sin (c+d x)-1)^2}\right)}{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,"-1/4*(a^2*(ArcTanh[Sin[c + d*x]] - Sin[c + d*x]/(-1 + Sin[c + d*x])^2))/d","A",1
866,1,66,101,0.2909037,"\int \csc (c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \left(\frac{6}{\sin (c+d x)-1}-\frac{2}{(\sin (c+d x)-1)^2}+7 \log (1-\sin (c+d x))-8 \log (\sin (c+d x))+\log (\sin (c+d x)+1)\right)}{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,"-1/8*(a^2*(7*Log[1 - Sin[c + d*x]] - 8*Log[Sin[c + d*x]] + Log[1 + Sin[c + d*x]] - 2/(-1 + Sin[c + d*x])^2 + 6/(-1 + Sin[c + d*x])))/d","A",1
867,1,74,116,0.2673107,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \left(-\frac{10}{\sin (c+d x)-1}+\frac{2}{(\sin (c+d x)-1)^2}-8 \csc (c+d x)-17 \log (1-\sin (c+d x))+16 \log (\sin (c+d x))+\log (\sin (c+d x)+1)\right)}{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*(-8*Csc[c + d*x] - 17*Log[1 - Sin[c + d*x]] + 16*Log[Sin[c + d*x]] + Log[1 + Sin[c + d*x]] + 2/(-1 + Sin[c + d*x])^2 - 10/(-1 + Sin[c + d*x])))/(8*d)","A",1
868,1,84,134,1.1306937,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \left(\frac{14}{\sin (c+d x)-1}-\frac{2}{(\sin (c+d x)-1)^2}+4 \csc ^2(c+d x)+16 \csc (c+d x)+31 \log (1-\sin (c+d x))-32 \log (\sin (c+d x))+\log (\sin (c+d x)+1)\right)}{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,"-1/8*(a^2*(16*Csc[c + d*x] + 4*Csc[c + d*x]^2 + 31*Log[1 - Sin[c + d*x]] - 32*Log[Sin[c + d*x]] + Log[1 + Sin[c + d*x]] - 2/(-1 + Sin[c + d*x])^2 + 14/(-1 + Sin[c + d*x])))/d","A",1
869,1,133,150,6.0573831,"\int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^4*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^9 \left(-\frac{\csc ^3(c+d x)}{3 a^7}-\frac{\csc ^2(c+d x)}{a^7}-\frac{4 \csc (c+d x)}{a^7}-\frac{49 \log (1-\sin (c+d x))}{8 a^7}+\frac{6 \log (\sin (c+d x))}{a^7}+\frac{\log (\sin (c+d x)+1)}{8 a^7}+\frac{9}{4 a^6 (a-a \sin (c+d x))}+\frac{1}{4 a^5 (a-a \sin (c+d x))^2}\right)}{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,"(a^9*((-4*Csc[c + d*x])/a^7 - Csc[c + d*x]^2/a^7 - Csc[c + d*x]^3/(3*a^7) - (49*Log[1 - Sin[c + d*x]])/(8*a^7) + (6*Log[Sin[c + d*x]])/a^7 + Log[1 + Sin[c + d*x]]/(8*a^7) + 1/(4*a^5*(a - a*Sin[c + d*x])^2) + 9/(4*a^6*(a - a*Sin[c + d*x]))))/d","A",1
870,1,73,114,0.306958,"\int (a+a \sin (c+d x))^3 \tan ^5(c+d x) \, dx","Integrate[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^5,x]","-\frac{a^3 \left(2 \sin ^3(c+d x)+9 \sin ^2(c+d x)+36 \sin (c+d x)+\frac{27-30 \sin (c+d x)}{(\sin (c+d x)-1)^2}+60 \log (1-\sin (c+d x))\right)}{6 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{a^3 \sin ^3(c+d x)}{3 d}-\frac{3 a^3 \sin ^2(c+d x)}{2 d}-\frac{6 a^3 \sin (c+d x)}{d}-\frac{10 a^3 \log (1-\sin (c+d x))}{d}",1,"-1/6*(a^3*(60*Log[1 - Sin[c + d*x]] + (27 - 30*Sin[c + d*x])/(-1 + Sin[c + d*x])^2 + 36*Sin[c + d*x] + 9*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3))/d","A",1
871,1,61,96,0.2396878,"\int \sec (c+d x) (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^4,x]","-\frac{a^3 \left(\sin ^2(c+d x)+6 \sin (c+d x)+\frac{7-8 \sin (c+d x)}{(\sin (c+d x)-1)^2}+12 \log (1-\sin (c+d x))\right)}{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{a^3 \sin ^2(c+d x)}{2 d}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{6 a^3 \log (1-\sin (c+d x))}{d}",1,"-1/2*(a^3*(12*Log[1 - Sin[c + d*x]] + (7 - 8*Sin[c + d*x])/(-1 + Sin[c + d*x])^2 + 6*Sin[c + d*x] + Sin[c + d*x]^2))/d","A",1
872,1,53,78,0.2276209,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^3,x]","-\frac{a^3 \left(2 \sin (c+d x)+\frac{5-6 \sin (c+d x)}{(\sin (c+d x)-1)^2}+6 \log (1-\sin (c+d x))\right)}{2 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,"-1/2*(a^3*(6*Log[1 - Sin[c + d*x]] + (5 - 6*Sin[c + d*x])/(-1 + Sin[c + d*x])^2 + 2*Sin[c + d*x]))/d","A",1
873,1,45,64,0.1441081,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","-\frac{a^3 \left(\frac{3-4 \sin (c+d x)}{(\sin (c+d x)-1)^2}+2 \log (1-\sin (c+d x))\right)}{2 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,"-1/2*(a^3*(2*Log[1 - Sin[c + d*x]] + (3 - 4*Sin[c + d*x])/(-1 + Sin[c + d*x])^2))/d","A",1
874,1,30,31,0.0303356,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]","\frac{a^3 \sin ^2(c+d x)}{2 d (1-\sin (c+d x))^2}","\frac{a^5 \sin ^2(c+d x)}{2 d (a-a \sin (c+d x))^2}",1,"(a^3*Sin[c + d*x]^2)/(2*d*(1 - Sin[c + d*x])^2)","A",1
875,1,54,77,0.2313608,"\int \csc (c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(\frac{3-2 \sin (c+d x)}{(\sin (c+d x)-1)^2}-2 \log (1-\sin (c+d x))+2 \log (\sin (c+d x))\right)}{2 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*(-2*Log[1 - Sin[c + d*x]] + 2*Log[Sin[c + d*x]] + (3 - 2*Sin[c + d*x])/(-1 + Sin[c + d*x])^2))/(2*d)","A",1
876,1,63,93,0.2322435,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(-\frac{4}{\sin (c+d x)-1}+\frac{1}{(\sin (c+d x)-1)^2}-2 \csc (c+d x)-6 \log (1-\sin (c+d x))+6 \log (\sin (c+d x))\right)}{2 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*(-2*Csc[c + d*x] - 6*Log[1 - Sin[c + d*x]] + 6*Log[Sin[c + d*x]] + (-1 + Sin[c + d*x])^(-2) - 4/(-1 + Sin[c + d*x])))/(2*d)","A",1
877,1,73,111,0.7873573,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \left(\frac{6}{\sin (c+d x)-1}-\frac{1}{(\sin (c+d x)-1)^2}+\csc ^2(c+d x)+6 \csc (c+d x)+12 \log (1-\sin (c+d x))-12 \log (\sin (c+d x))\right)}{2 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,"-1/2*(a^3*(6*Csc[c + d*x] + Csc[c + d*x]^2 + 12*Log[1 - Sin[c + d*x]] - 12*Log[Sin[c + d*x]] - (-1 + Sin[c + d*x])^(-2) + 6/(-1 + Sin[c + d*x])))/d","A",1
878,1,153,236,6.1255183,"\int \frac{\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^4*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{256 \sin ^3(c+d x)-384 \sin ^2(c+d x)+3840 \sin (c+d x)+\frac{750}{1-\sin (c+d x)}-\frac{3840}{\sin (c+d x)+1}-\frac{102}{(1-\sin (c+d x))^2}+\frac{852}{(\sin (c+d x)+1)^2}+\frac{8}{(1-\sin (c+d x))^3}-\frac{144}{(\sin (c+d x)+1)^3}+\frac{12}{(\sin (c+d x)+1)^4}+1545 \log (1-\sin (c+d x))-5385 \log (\sin (c+d x)+1)}{768 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,"(1545*Log[1 - Sin[c + d*x]] - 5385*Log[1 + Sin[c + d*x]] + 8/(1 - Sin[c + d*x])^3 - 102/(1 - Sin[c + d*x])^2 + 750/(1 - Sin[c + d*x]) + 3840*Sin[c + d*x] - 384*Sin[c + d*x]^2 + 256*Sin[c + d*x]^3 + 12/(1 + Sin[c + d*x])^4 - 144/(1 + Sin[c + d*x])^3 + 852/(1 + Sin[c + d*x])^2 - 3840/(1 + Sin[c + d*x]))/(768*a*d)","A",1
879,1,143,220,6.147418,"\int \frac{\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^3*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{384 \sin ^2(c+d x)-768 \sin (c+d x)+\frac{570}{1-\sin (c+d x)}+\frac{2520}{\sin (c+d x)+1}-\frac{90}{(1-\sin (c+d x))^2}-\frac{660}{(\sin (c+d x)+1)^2}+\frac{8}{(1-\sin (c+d x))^3}+\frac{128}{(\sin (c+d x)+1)^3}-\frac{12}{(\sin (c+d x)+1)^4}+975 \log (1-\sin (c+d x))+2865 \log (\sin (c+d x)+1)}{768 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,"(975*Log[1 - Sin[c + d*x]] + 2865*Log[1 + Sin[c + d*x]] + 8/(1 - Sin[c + d*x])^3 - 90/(1 - Sin[c + d*x])^2 + 570/(1 - Sin[c + d*x]) - 768*Sin[c + d*x] + 384*Sin[c + d*x]^2 - 12/(1 + Sin[c + d*x])^4 + 128/(1 + Sin[c + d*x])^3 - 660/(1 + Sin[c + d*x])^2 + 2520/(1 + Sin[c + d*x]))/(768*a*d)","A",1
880,1,133,199,6.1364701,"\int \frac{\sin ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{768 \sin (c+d x)+\frac{414}{1-\sin (c+d x)}-\frac{1536}{\sin (c+d x)+1}-\frac{78}{(1-\sin (c+d x))^2}+\frac{492}{(\sin (c+d x)+1)^2}+\frac{8}{(1-\sin (c+d x))^3}-\frac{112}{(\sin (c+d x)+1)^3}+\frac{12}{(\sin (c+d x)+1)^4}+561 \log (1-\sin (c+d x))-1329 \log (\sin (c+d x)+1)}{768 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,"(561*Log[1 - Sin[c + d*x]] - 1329*Log[1 + Sin[c + d*x]] + 8/(1 - Sin[c + d*x])^3 - 78/(1 - Sin[c + d*x])^2 + 414/(1 - Sin[c + d*x]) + 768*Sin[c + d*x] + 12/(1 + Sin[c + d*x])^4 - 112/(1 + Sin[c + d*x])^3 + 492/(1 + Sin[c + d*x])^2 - 1536/(1 + Sin[c + d*x]))/(768*a*d)","A",1
881,1,117,188,3.7835502,"\int \frac{\sin (c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{\frac{2 \left(279 \sin ^6(c+d x)-489 \sin ^5(c+d x)-1000 \sin ^4(c+d x)+728 \sin ^3(c+d x)+1113 \sin ^2(c+d x)-295 \sin (c+d x)-400\right)}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+279 \log (1-\sin (c+d x))+489 \log (\sin (c+d x)+1)}{768 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,"(279*Log[1 - Sin[c + d*x]] + 489*Log[1 + Sin[c + d*x]] + (2*(-400 - 295*Sin[c + d*x] + 1113*Sin[c + d*x]^2 + 728*Sin[c + d*x]^3 - 1000*Sin[c + d*x]^4 - 489*Sin[c + d*x]^5 + 279*Sin[c + d*x]^6))/((-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^4))/(768*a*d)","A",1
882,1,101,130,0.9225941,"\int \frac{\tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^7/(a + a*Sin[c + d*x]),x]","-\frac{\frac{279 \sin ^6(c+d x)+87 \sin ^5(c+d x)-424 \sin ^4(c+d x)-136 \sin ^3(c+d x)+249 \sin ^2(c+d x)+57 \sin (c+d x)-48}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+105 \tanh ^{-1}(\sin (c+d x))}{384 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,"-1/384*(105*ArcTanh[Sin[c + d*x]] + (-48 + 57*Sin[c + d*x] + 249*Sin[c + d*x]^2 - 136*Sin[c + d*x]^3 - 424*Sin[c + d*x]^4 + 87*Sin[c + d*x]^5 + 279*Sin[c + d*x]^6)/((-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^4))/(a*d)","A",1
883,1,101,134,0.9016657,"\int \frac{\sec (c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","-\frac{\frac{-15 \sin ^6(c+d x)+177 \sin ^5(c+d x)+104 \sin ^4(c+d x)-184 \sin ^3(c+d x)-129 \sin ^2(c+d x)+63 \sin (c+d x)+48}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+15 \tanh ^{-1}(\sin (c+d x))}{384 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,"-1/384*(15*ArcTanh[Sin[c + d*x]] + (48 + 63*Sin[c + d*x] - 129*Sin[c + d*x]^2 - 184*Sin[c + d*x]^3 + 104*Sin[c + d*x]^4 + 177*Sin[c + d*x]^5 - 15*Sin[c + d*x]^6)/((-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^4))/(a*d)","A",1
884,1,92,152,1.11606,"\int \frac{\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*Tan[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","\frac{-\frac{15}{\sin (c+d x)-1}-\frac{15}{(\sin (c+d x)-1)^2}+\frac{30}{(\sin (c+d x)+1)^2}-\frac{4}{(\sin (c+d x)-1)^3}-\frac{24}{(\sin (c+d x)+1)^3}+\frac{6}{(\sin (c+d x)+1)^4}+15 \tanh ^{-1}(\sin (c+d x))}{384 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,"(15*ArcTanh[Sin[c + d*x]] - 4/(-1 + Sin[c + d*x])^3 - 15/(-1 + Sin[c + d*x])^2 - 15/(-1 + Sin[c + d*x]) + 6/(1 + Sin[c + d*x])^4 - 24/(1 + Sin[c + d*x])^3 + 30/(1 + Sin[c + d*x])^2)/(384*a*d)","A",1
885,1,101,150,0.671845,"\int \frac{\sec ^3(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{\frac{-9 \sin ^6(c+d x)-9 \sin ^5(c+d x)+24 \sin ^4(c+d x)-72 \sin ^3(c+d x)-39 \sin ^2(c+d x)+25 \sin (c+d x)+16}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+9 \tanh ^{-1}(\sin (c+d x))}{384 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,"(9*ArcTanh[Sin[c + d*x]] + (16 + 25*Sin[c + d*x] - 39*Sin[c + d*x]^2 - 72*Sin[c + d*x]^3 + 24*Sin[c + d*x]^4 - 9*Sin[c + d*x]^5 - 9*Sin[c + d*x]^6)/((-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^4))/(384*a*d)","A",1
886,1,92,150,0.8533716,"\int \frac{\sec ^4(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^4*Tan[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{-\frac{9}{\sin (c+d x)-1}+\frac{3}{(\sin (c+d x)-1)^2}+\frac{6}{(\sin (c+d x)+1)^2}+\frac{4}{(\sin (c+d x)-1)^3}+\frac{8}{(\sin (c+d x)+1)^3}-\frac{6}{(\sin (c+d x)+1)^4}+9 \tanh ^{-1}(\sin (c+d x))}{384 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,"-1/384*(9*ArcTanh[Sin[c + d*x]] + 4/(-1 + Sin[c + d*x])^3 + 3/(-1 + Sin[c + d*x])^2 - 9/(-1 + Sin[c + d*x]) - 6/(1 + Sin[c + d*x])^4 + 8/(1 + Sin[c + d*x])^3 + 6/(1 + Sin[c + d*x])^2)/(a*d)","A",1
887,1,92,148,0.5240798,"\int \frac{\sec ^5(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^5*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{-\frac{3}{\sin (c+d x)-1}-\frac{12}{\sin (c+d x)+1}-\frac{3}{(\sin (c+d x)-1)^2}-\frac{6}{(\sin (c+d x)+1)^2}+\frac{4}{(\sin (c+d x)-1)^3}+\frac{6}{(\sin (c+d x)+1)^4}+15 \tanh ^{-1}(\sin (c+d x))}{384 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,"-1/384*(15*ArcTanh[Sin[c + d*x]] + 4/(-1 + Sin[c + d*x])^3 - 3/(-1 + Sin[c + d*x])^2 - 3/(-1 + Sin[c + d*x]) + 6/(1 + Sin[c + d*x])^4 - 6/(1 + Sin[c + d*x])^2 - 12/(1 + Sin[c + d*x]))/(a*d)","A",1
888,1,92,130,0.9525901,"\int \frac{\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^6*Tan[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{-\frac{15}{\sin (c+d x)-1}+\frac{9}{(\sin (c+d x)-1)^2}+\frac{6}{(\sin (c+d x)+1)^2}-\frac{4}{(\sin (c+d x)-1)^3}+\frac{8}{(\sin (c+d x)+1)^3}+\frac{6}{(\sin (c+d x)+1)^4}+15 \tanh ^{-1}(\sin (c+d x))}{384 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,"(15*ArcTanh[Sin[c + d*x]] - 4/(-1 + Sin[c + d*x])^3 + 9/(-1 + Sin[c + d*x])^2 - 15/(-1 + Sin[c + d*x]) + 6/(1 + Sin[c + d*x])^4 + 8/(1 + Sin[c + d*x])^3 + 6/(1 + Sin[c + d*x])^2)/(384*a*d)","A",1
889,1,145,165,0.5135203,"\int \frac{\sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^7/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^6(c+d x) \left(-105 \sin ^6(c+d x)-105 \sin ^5(c+d x)+280 \sin ^4(c+d x)+280 \sin ^3(c+d x)-231 \sin ^2(c+d x)-231 \sin (c+d x)-105 \tanh ^{-1}(\sin (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^6 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8+48\right)}{384 a d (\sin (c+d x)+1)}","-\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,"-1/384*(Sec[c + d*x]^6*(48 - 105*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^6*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8 - 231*Sin[c + d*x] - 231*Sin[c + d*x]^2 + 280*Sin[c + d*x]^3 + 280*Sin[c + d*x]^4 - 105*Sin[c + d*x]^5 - 105*Sin[c + d*x]^6))/(a*d*(1 + Sin[c + d*x]))","A",1
890,1,189,202,6.1353541,"\int \frac{\csc (c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{a^8 \left(-\frac{93 \log (1-\sin (c+d x))}{256 a^9}+\frac{\log (\sin (c+d x))}{a^9}-\frac{163 \log (\sin (c+d x)+1)}{256 a^9}+\frac{29}{128 a^8 (a-a \sin (c+d x))}+\frac{1}{2 a^8 (a \sin (c+d x)+a)}+\frac{7}{128 a^7 (a-a \sin (c+d x))^2}+\frac{11}{64 a^7 (a \sin (c+d x)+a)^2}+\frac{1}{96 a^6 (a-a \sin (c+d x))^3}+\frac{1}{16 a^6 (a \sin (c+d x)+a)^3}+\frac{1}{64 a^5 (a \sin (c+d x)+a)^4}\right)}{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,"(a^8*((-93*Log[1 - Sin[c + d*x]])/(256*a^9) + Log[Sin[c + d*x]]/a^9 - (163*Log[1 + Sin[c + d*x]])/(256*a^9) + 1/(96*a^6*(a - a*Sin[c + d*x])^3) + 7/(128*a^7*(a - a*Sin[c + d*x])^2) + 29/(128*a^8*(a - a*Sin[c + d*x])) + 1/(64*a^5*(a + a*Sin[c + d*x])^4) + 1/(16*a^6*(a + a*Sin[c + d*x])^3) + 11/(64*a^7*(a + a*Sin[c + d*x])^2) + 1/(2*a^8*(a + a*Sin[c + d*x]))))/d","A",1
891,1,201,217,6.1411578,"\int \frac{\csc ^2(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{a^9 \left(-\frac{\csc (c+d x)}{a^{10}}-\frac{187 \log (1-\sin (c+d x))}{256 a^{10}}-\frac{\log (\sin (c+d x))}{a^{10}}+\frac{443 \log (\sin (c+d x)+1)}{256 a^{10}}+\frac{47}{128 a^9 (a-a \sin (c+d x))}-\frac{35}{32 a^9 (a \sin (c+d x)+a)}+\frac{9}{128 a^8 (a-a \sin (c+d x))^2}-\frac{19}{64 a^8 (a \sin (c+d x)+a)^2}+\frac{1}{96 a^7 (a-a \sin (c+d x))^3}-\frac{1}{12 a^7 (a \sin (c+d x)+a)^3}-\frac{1}{64 a^6 (a \sin (c+d x)+a)^4}\right)}{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,"(a^9*(-(Csc[c + d*x]/a^10) - (187*Log[1 - Sin[c + d*x]])/(256*a^10) - Log[Sin[c + d*x]]/a^10 + (443*Log[1 + Sin[c + d*x]])/(256*a^10) + 1/(96*a^7*(a - a*Sin[c + d*x])^3) + 9/(128*a^8*(a - a*Sin[c + d*x])^2) + 47/(128*a^9*(a - a*Sin[c + d*x])) - 1/(64*a^6*(a + a*Sin[c + d*x])^4) - 1/(12*a^7*(a + a*Sin[c + d*x])^3) - 19/(64*a^8*(a + a*Sin[c + d*x])^2) - 35/(32*a^9*(a + a*Sin[c + d*x]))))/d","A",1
892,1,213,232,6.1884577,"\int \frac{\csc ^3(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{a^{10} \left(-\frac{\csc ^2(c+d x)}{2 a^{11}}+\frac{\csc (c+d x)}{a^{11}}-\frac{325 \log (1-\sin (c+d x))}{256 a^{11}}+\frac{5 \log (\sin (c+d x))}{a^{11}}-\frac{955 \log (\sin (c+d x)+1)}{256 a^{11}}+\frac{69}{128 a^{10} (a-a \sin (c+d x))}+\frac{2}{a^{10} (a \sin (c+d x)+a)}+\frac{11}{128 a^9 (a-a \sin (c+d x))^2}+\frac{29}{64 a^9 (a \sin (c+d x)+a)^2}+\frac{1}{96 a^8 (a-a \sin (c+d x))^3}+\frac{5}{48 a^8 (a \sin (c+d x)+a)^3}+\frac{1}{64 a^7 (a \sin (c+d x)+a)^4}\right)}{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,"(a^10*(Csc[c + d*x]/a^11 - Csc[c + d*x]^2/(2*a^11) - (325*Log[1 - Sin[c + d*x]])/(256*a^11) + (5*Log[Sin[c + d*x]])/a^11 - (955*Log[1 + Sin[c + d*x]])/(256*a^11) + 1/(96*a^8*(a - a*Sin[c + d*x])^3) + 11/(128*a^9*(a - a*Sin[c + d*x])^2) + 69/(128*a^10*(a - a*Sin[c + d*x])) + 1/(64*a^7*(a + a*Sin[c + d*x])^4) + 5/(48*a^8*(a + a*Sin[c + d*x])^3) + 29/(64*a^9*(a + a*Sin[c + d*x])^2) + 2/(a^10*(a + a*Sin[c + d*x]))))/d","A",1
893,1,231,253,6.1359701,"\int \frac{\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^4*Sec[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","\frac{a^{11} \left(-\frac{\csc ^3(c+d x)}{3 a^{12}}+\frac{\csc ^2(c+d x)}{2 a^{12}}-\frac{5 \csc (c+d x)}{a^{12}}-\frac{515 \log (1-\sin (c+d x))}{256 a^{12}}-\frac{5 \log (\sin (c+d x))}{a^{12}}+\frac{1795 \log (\sin (c+d x)+1)}{256 a^{12}}+\frac{95}{128 a^{11} (a-a \sin (c+d x))}-\frac{105}{32 a^{11} (a \sin (c+d x)+a)}+\frac{13}{128 a^{10} (a-a \sin (c+d x))^2}-\frac{41}{64 a^{10} (a \sin (c+d x)+a)^2}+\frac{1}{96 a^9 (a-a \sin (c+d x))^3}-\frac{1}{8 a^9 (a \sin (c+d x)+a)^3}-\frac{1}{64 a^8 (a \sin (c+d x)+a)^4}\right)}{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,"(a^11*((-5*Csc[c + d*x])/a^12 + Csc[c + d*x]^2/(2*a^12) - Csc[c + d*x]^3/(3*a^12) - (515*Log[1 - Sin[c + d*x]])/(256*a^12) - (5*Log[Sin[c + d*x]])/a^12 + (1795*Log[1 + Sin[c + d*x]])/(256*a^12) + 1/(96*a^9*(a - a*Sin[c + d*x])^3) + 13/(128*a^10*(a - a*Sin[c + d*x])^2) + 95/(128*a^11*(a - a*Sin[c + d*x])) - 1/(64*a^8*(a + a*Sin[c + d*x])^4) - 1/(8*a^9*(a + a*Sin[c + d*x])^3) - 41/(64*a^10*(a + a*Sin[c + d*x])^2) - 105/(32*a^11*(a + a*Sin[c + d*x]))))/d","A",1
894,1,139,91,1.0173782,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","-\frac{a^2 \sec ^7(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 (448 \sin (c+d x)-104 \sin (2 (c+d x))-144 \sin (3 (c+d x))-52 \sin (4 (c+d x))+48 \sin (5 (c+d x))+182 \cos (c+d x)+736 \cos (2 (c+d x))+39 \cos (3 (c+d x))-192 \cos (4 (c+d x))-13 \cos (5 (c+d x))-672)}{6720 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,"-1/6720*(a^2*Sec[c + d*x]^7*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(-672 + 182*Cos[c + d*x] + 736*Cos[2*(c + d*x)] + 39*Cos[3*(c + d*x)] - 192*Cos[4*(c + d*x)] - 13*Cos[5*(c + d*x)] + 448*Sin[c + d*x] - 104*Sin[2*(c + d*x)] - 144*Sin[3*(c + d*x)] - 52*Sin[4*(c + d*x)] + 48*Sin[5*(c + d*x)]))/d","A",1
895,1,169,264,6.1680023,"\int \frac{\sin ^3(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^3*Tan[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","-\frac{1280 \sin ^2(c+d x)-2560 \sin (c+d x)+\frac{3120}{1-\sin (c+d x)}+\frac{11550}{\sin (c+d x)+1}-\frac{705}{(1-\sin (c+d x))^2}-\frac{3825}{(\sin (c+d x)+1)^2}+\frac{120}{(1-\sin (c+d x))^3}+\frac{1060}{(\sin (c+d x)+1)^3}-\frac{10}{(1-\sin (c+d x))^4}-\frac{190}{(\sin (c+d x)+1)^4}+\frac{16}{(\sin (c+d x)+1)^5}+4215 \log (1-\sin (c+d x))+11145 \log (\sin (c+d x)+1)}{2560 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,"-1/2560*(4215*Log[1 - Sin[c + d*x]] + 11145*Log[1 + Sin[c + d*x]] - 10/(1 - Sin[c + d*x])^4 + 120/(1 - Sin[c + d*x])^3 - 705/(1 - Sin[c + d*x])^2 + 3120/(1 - Sin[c + d*x]) - 2560*Sin[c + d*x] + 1280*Sin[c + d*x]^2 + 16/(1 + Sin[c + d*x])^5 - 190/(1 + Sin[c + d*x])^4 + 1060/(1 + Sin[c + d*x])^3 - 3825/(1 + Sin[c + d*x])^2 + 11550/(1 + Sin[c + d*x]))/(a*d)","A",1
896,1,159,247,6.1884146,"\int \frac{\sin ^2(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","-\frac{7680 \sin (c+d x)+\frac{6090}{1-\sin (c+d x)}-\frac{19200}{\sin (c+d x)+1}-\frac{1635}{(1-\sin (c+d x))^2}+\frac{7725}{(\sin (c+d x)+1)^2}+\frac{320}{(1-\sin (c+d x))^3}-\frac{2500}{(\sin (c+d x)+1)^3}-\frac{30}{(1-\sin (c+d x))^4}+\frac{510}{(\sin (c+d x)+1)^4}-\frac{48}{(\sin (c+d x)+1)^5}+6555 \log (1-\sin (c+d x))-14235 \log (\sin (c+d x)+1)}{7680 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,"-1/7680*(6555*Log[1 - Sin[c + d*x]] - 14235*Log[1 + Sin[c + d*x]] - 30/(1 - Sin[c + d*x])^4 + 320/(1 - Sin[c + d*x])^3 - 1635/(1 - Sin[c + d*x])^2 + 6090/(1 - Sin[c + d*x]) + 7680*Sin[c + d*x] - 48/(1 + Sin[c + d*x])^5 + 510/(1 + Sin[c + d*x])^4 - 2500/(1 + Sin[c + d*x])^3 + 7725/(1 + Sin[c + d*x])^2 - 19200/(1 + Sin[c + d*x]))/(a*d)","A",1
897,1,137,233,4.9680796,"\int \frac{\sin (c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","-\frac{\frac{2 \left(2895 \sin ^8(c+d x)-6705 \sin ^7(c+d x)-13815 \sin ^6(c+d x)+14985 \sin ^5(c+d x)+23049 \sin ^4(c+d x)-12151 \sin ^3(c+d x)-16561 \sin ^2(c+d x)+3439 \sin (c+d x)+4384\right)}{(\sin (c+d x)-1)^4 (\sin (c+d x)+1)^5}+2895 \log (1-\sin (c+d x))+4785 \log (\sin (c+d x)+1)}{7680 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,"-1/7680*(2895*Log[1 - Sin[c + d*x]] + 4785*Log[1 + Sin[c + d*x]] + (2*(4384 + 3439*Sin[c + d*x] - 16561*Sin[c + d*x]^2 - 12151*Sin[c + d*x]^3 + 23049*Sin[c + d*x]^4 + 14985*Sin[c + d*x]^5 - 13815*Sin[c + d*x]^6 - 6705*Sin[c + d*x]^7 + 2895*Sin[c + d*x]^8))/((-1 + Sin[c + d*x])^4*(1 + Sin[c + d*x])^5))/(a*d)","A",1
898,1,122,154,2.4189414,"\int \frac{\tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^9/(a + a*Sin[c + d*x]),x]","\frac{\frac{2 \left(965 \sin ^8(c+d x)+325 \sin ^7(c+d x)-2045 \sin ^6(c+d x)-765 \sin ^5(c+d x)+1923 \sin ^4(c+d x)+643 \sin ^3(c+d x)-827 \sin ^2(c+d x)-187 \sin (c+d x)+128\right)}{(\sin (c+d x)-1)^4 (\sin (c+d x)+1)^5}+630 \tanh ^{-1}(\sin (c+d x))}{2560 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,"(630*ArcTanh[Sin[c + d*x]] + (2*(128 - 187*Sin[c + d*x] - 827*Sin[c + d*x]^2 + 643*Sin[c + d*x]^3 + 1923*Sin[c + d*x]^4 - 765*Sin[c + d*x]^5 - 2045*Sin[c + d*x]^6 + 325*Sin[c + d*x]^7 + 965*Sin[c + d*x]^8))/((-1 + Sin[c + d*x])^4*(1 + Sin[c + d*x])^5))/(2560*a*d)","A",1
899,1,121,160,2.4222223,"\int \frac{\sec (c+d x) \tan ^8(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x]^8)/(a + a*Sin[c + d*x]),x]","\frac{\frac{-210 \sin ^8(c+d x)+3630 \sin ^7(c+d x)+2050 \sin ^6(c+d x)-5630 \sin ^5(c+d x)-3838 \sin ^4(c+d x)+3842 \sin ^3(c+d x)+2862 \sin ^2(c+d x)-978 \sin (c+d x)-768}{(\sin (c+d x)-1)^4 (\sin (c+d x)+1)^5}+210 \tanh ^{-1}(\sin (c+d x))}{7680 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,"(210*ArcTanh[Sin[c + d*x]] + (-768 - 978*Sin[c + d*x] + 2862*Sin[c + d*x]^2 + 3842*Sin[c + d*x]^3 - 3838*Sin[c + d*x]^4 - 5630*Sin[c + d*x]^5 + 2050*Sin[c + d*x]^6 + 3630*Sin[c + d*x]^7 - 210*Sin[c + d*x]^8)/((-1 + Sin[c + d*x])^4*(1 + Sin[c + d*x])^5))/(7680*a*d)","A",1
900,1,124,178,1.6768928,"\int \frac{\sec ^2(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*Tan[c + d*x]^7)/(a + a*Sin[c + d*x]),x]","-\frac{\frac{210}{1-\sin (c+d x)}-\frac{315}{(1-\sin (c+d x))^2}+\frac{525}{(\sin (c+d x)+1)^2}+\frac{160}{(1-\sin (c+d x))^3}-\frac{580}{(\sin (c+d x)+1)^3}-\frac{30}{(1-\sin (c+d x))^4}+\frac{270}{(\sin (c+d x)+1)^4}-\frac{48}{(\sin (c+d x)+1)^5}+210 \tanh ^{-1}(\sin (c+d x))}{7680 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,"-1/7680*(210*ArcTanh[Sin[c + d*x]] - 30/(1 - Sin[c + d*x])^4 + 160/(1 - Sin[c + d*x])^3 - 315/(1 - Sin[c + d*x])^2 + 210/(1 - Sin[c + d*x]) - 48/(1 + Sin[c + d*x])^5 + 270/(1 + Sin[c + d*x])^4 - 580/(1 + Sin[c + d*x])^3 + 525/(1 + Sin[c + d*x])^2)/(a*d)","A",1
901,1,122,176,2.7569624,"\int \frac{\sec ^3(c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*Tan[c + d*x]^6)/(a + a*Sin[c + d*x]),x]","-\frac{30 \tanh ^{-1}(\sin (c+d x))-\frac{2 \left(15 \sin ^8(c+d x)+15 \sin ^7(c+d x)-55 \sin ^6(c+d x)+265 \sin ^5(c+d x)+137 \sin ^4(c+d x)-183 \sin ^3(c+d x)-113 \sin ^2(c+d x)+47 \sin (c+d x)+32\right)}{(\sin (c+d x)-1)^4 (\sin (c+d x)+1)^5}}{2560 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,"-1/2560*(30*ArcTanh[Sin[c + d*x]] - (2*(32 + 47*Sin[c + d*x] - 113*Sin[c + d*x]^2 - 183*Sin[c + d*x]^3 + 137*Sin[c + d*x]^4 + 265*Sin[c + d*x]^5 - 55*Sin[c + d*x]^6 + 15*Sin[c + d*x]^7 + 15*Sin[c + d*x]^8))/((-1 + Sin[c + d*x])^4*(1 + Sin[c + d*x])^5))/(a*d)","A",1
902,1,116,194,5.4853375,"\int \frac{\sec ^4(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^4*Tan[c + d*x]^5)/(a + a*Sin[c + d*x]),x]","\frac{-\frac{90}{\sin (c+d x)-1}+\frac{15}{(\sin (c+d x)-1)^2}+\frac{75}{(\sin (c+d x)+1)^2}+\frac{80}{(\sin (c+d x)-1)^3}+\frac{100}{(\sin (c+d x)+1)^3}+\frac{30}{(\sin (c+d x)-1)^4}-\frac{150}{(\sin (c+d x)+1)^4}+\frac{48}{(\sin (c+d x)+1)^5}+90 \tanh ^{-1}(\sin (c+d x))}{7680 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,"(90*ArcTanh[Sin[c + d*x]] + 30/(-1 + Sin[c + d*x])^4 + 80/(-1 + Sin[c + d*x])^3 + 15/(-1 + Sin[c + d*x])^2 - 90/(-1 + Sin[c + d*x]) + 48/(1 + Sin[c + d*x])^5 - 150/(1 + Sin[c + d*x])^4 + 100/(1 + Sin[c + d*x])^3 + 75/(1 + Sin[c + d*x])^2)/(7680*a*d)","A",1
903,1,116,192,5.3967334,"\int \frac{\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^5*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{-\frac{90}{\sin (c+d x)+1}-\frac{45}{(\sin (c+d x)-1)^2}-\frac{45}{(\sin (c+d x)+1)^2}+\frac{40}{(\sin (c+d x)-1)^3}+\frac{20}{(\sin (c+d x)+1)^3}+\frac{30}{(\sin (c+d x)-1)^4}+\frac{90}{(\sin (c+d x)+1)^4}-\frac{48}{(\sin (c+d x)+1)^5}+90 \tanh ^{-1}(\sin (c+d x))}{7680 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,"(90*ArcTanh[Sin[c + d*x]] + 30/(-1 + Sin[c + d*x])^4 + 40/(-1 + Sin[c + d*x])^3 - 45/(-1 + Sin[c + d*x])^2 - 48/(1 + Sin[c + d*x])^5 + 90/(1 + Sin[c + d*x])^4 + 20/(1 + Sin[c + d*x])^3 - 45/(1 + Sin[c + d*x])^2 - 90/(1 + Sin[c + d*x]))/(7680*a*d)","A",1
904,1,104,174,2.8092047,"\int \frac{\sec ^6(c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^6*Tan[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{-\frac{30}{\sin (c+d x)-1}+\frac{15}{(\sin (c+d x)-1)^2}+\frac{15}{(\sin (c+d x)+1)^2}+\frac{20}{(\sin (c+d x)+1)^3}-\frac{10}{(\sin (c+d x)-1)^4}+\frac{10}{(\sin (c+d x)+1)^4}-\frac{16}{(\sin (c+d x)+1)^5}+30 \tanh ^{-1}(\sin (c+d x))}{2560 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,"-1/2560*(30*ArcTanh[Sin[c + d*x]] - 10/(-1 + Sin[c + d*x])^4 + 15/(-1 + Sin[c + d*x])^2 - 30/(-1 + Sin[c + d*x]) - 16/(1 + Sin[c + d*x])^5 + 10/(1 + Sin[c + d*x])^4 + 20/(1 + Sin[c + d*x])^3 + 15/(1 + Sin[c + d*x])^2)/(a*d)","A",1
905,1,122,172,2.6150014,"\int \frac{\sec ^7(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^7*Tan[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{210 \tanh ^{-1}(\sin (c+d x))-\frac{2 \left(105 \sin ^8(c+d x)+105 \sin ^7(c+d x)-385 \sin ^6(c+d x)-385 \sin ^5(c+d x)+511 \sin ^4(c+d x)+511 \sin ^3(c+d x)-279 \sin ^2(c+d x)+201 \sin (c+d x)+96\right)}{(\sin (c+d x)-1)^4 (\sin (c+d x)+1)^5}}{7680 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,"-1/7680*(210*ArcTanh[Sin[c + d*x]] - (2*(96 + 201*Sin[c + d*x] - 279*Sin[c + d*x]^2 + 511*Sin[c + d*x]^3 + 511*Sin[c + d*x]^4 - 385*Sin[c + d*x]^5 - 385*Sin[c + d*x]^6 + 105*Sin[c + d*x]^7 + 105*Sin[c + d*x]^8))/((-1 + Sin[c + d*x])^4*(1 + Sin[c + d*x])^5))/(a*d)","A",1
906,1,116,154,5.5259379,"\int \frac{\sec ^8(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^8*Tan[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{-\frac{210}{\sin (c+d x)-1}+\frac{135}{(\sin (c+d x)-1)^2}+\frac{75}{(\sin (c+d x)+1)^2}-\frac{80}{(\sin (c+d x)-1)^3}+\frac{100}{(\sin (c+d x)+1)^3}+\frac{30}{(\sin (c+d x)-1)^4}+\frac{90}{(\sin (c+d x)+1)^4}+\frac{48}{(\sin (c+d x)+1)^5}+210 \tanh ^{-1}(\sin (c+d x))}{7680 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,"(210*ArcTanh[Sin[c + d*x]] + 30/(-1 + Sin[c + d*x])^4 - 80/(-1 + Sin[c + d*x])^3 + 135/(-1 + Sin[c + d*x])^2 - 210/(-1 + Sin[c + d*x]) + 48/(1 + Sin[c + d*x])^5 + 90/(1 + Sin[c + d*x])^4 + 100/(1 + Sin[c + d*x])^3 + 75/(1 + Sin[c + d*x])^2)/(7680*a*d)","A",1
907,1,165,210,1.4012744,"\int \frac{\sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^9/(a + a*Sin[c + d*x]),x]","\frac{\sec ^8(c+d x) \left(-315 \sin ^8(c+d x)-315 \sin ^7(c+d x)+1155 \sin ^6(c+d x)+1155 \sin ^5(c+d x)-1533 \sin ^4(c+d x)-1533 \sin ^3(c+d x)+837 \sin ^2(c+d x)+837 \sin (c+d x)+315 \tanh ^{-1}(\sin (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^8 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^{10}-128\right)}{1280 a d (\sin (c+d x)+1)}","-\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,"(Sec[c + d*x]^8*(-128 + 315*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^8*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^10 + 837*Sin[c + d*x] + 837*Sin[c + d*x]^2 - 1533*Sin[c + d*x]^3 - 1533*Sin[c + d*x]^4 + 1155*Sin[c + d*x]^5 + 1155*Sin[c + d*x]^6 - 315*Sin[c + d*x]^7 - 315*Sin[c + d*x]^8))/(1280*a*d*(1 + Sin[c + d*x]))","A",1
908,1,228,247,6.2054704,"\int \frac{\csc (c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","\frac{a^{10} \left(-\frac{193 \log (1-\sin (c+d x))}{512 a^{11}}+\frac{\log (\sin (c+d x))}{a^{11}}-\frac{319 \log (\sin (c+d x)+1)}{512 a^{11}}+\frac{65}{256 a^{10} (a-a \sin (c+d x))}+\frac{1}{2 a^{10} (a \sin (c+d x)+a)}+\frac{37}{512 a^9 (a-a \sin (c+d x))^2}+\frac{93}{512 a^9 (a \sin (c+d x)+a)^2}+\frac{1}{48 a^8 (a-a \sin (c+d x))^3}+\frac{29}{384 a^8 (a \sin (c+d x)+a)^3}+\frac{1}{256 a^7 (a-a \sin (c+d x))^4}+\frac{7}{256 a^7 (a \sin (c+d x)+a)^4}+\frac{1}{160 a^6 (a \sin (c+d x)+a)^5}\right)}{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,"(a^10*((-193*Log[1 - Sin[c + d*x]])/(512*a^11) + Log[Sin[c + d*x]]/a^11 - (319*Log[1 + Sin[c + d*x]])/(512*a^11) + 1/(256*a^7*(a - a*Sin[c + d*x])^4) + 1/(48*a^8*(a - a*Sin[c + d*x])^3) + 37/(512*a^9*(a - a*Sin[c + d*x])^2) + 65/(256*a^10*(a - a*Sin[c + d*x])) + 1/(160*a^6*(a + a*Sin[c + d*x])^5) + 7/(256*a^7*(a + a*Sin[c + d*x])^4) + 29/(384*a^8*(a + a*Sin[c + d*x])^3) + 93/(512*a^9*(a + a*Sin[c + d*x])^2) + 1/(2*a^10*(a + a*Sin[c + d*x]))))/d","A",1
909,1,240,262,6.2062001,"\int \frac{\csc ^2(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","\frac{a^{11} \left(-\frac{\csc (c+d x)}{a^{12}}-\frac{437 \log (1-\sin (c+d x))}{512 a^{12}}-\frac{\log (\sin (c+d x))}{a^{12}}+\frac{949 \log (\sin (c+d x)+1)}{512 a^{12}}+\frac{61}{128 a^{11} (a-a \sin (c+d x))}-\frac{315}{256 a^{11} (a \sin (c+d x)+a)}+\frac{57}{512 a^{10} (a-a \sin (c+d x))^2}-\frac{187}{512 a^{10} (a \sin (c+d x)+a)^2}+\frac{5}{192 a^9 (a-a \sin (c+d x))^3}-\frac{47}{384 a^9 (a \sin (c+d x)+a)^3}+\frac{1}{256 a^8 (a-a \sin (c+d x))^4}-\frac{9}{256 a^8 (a \sin (c+d x)+a)^4}-\frac{1}{160 a^7 (a \sin (c+d x)+a)^5}\right)}{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,"(a^11*(-(Csc[c + d*x]/a^12) - (437*Log[1 - Sin[c + d*x]])/(512*a^12) - Log[Sin[c + d*x]]/a^12 + (949*Log[1 + Sin[c + d*x]])/(512*a^12) + 1/(256*a^8*(a - a*Sin[c + d*x])^4) + 5/(192*a^9*(a - a*Sin[c + d*x])^3) + 57/(512*a^10*(a - a*Sin[c + d*x])^2) + 61/(128*a^11*(a - a*Sin[c + d*x])) - 1/(160*a^7*(a + a*Sin[c + d*x])^5) - 9/(256*a^8*(a + a*Sin[c + d*x])^4) - 47/(384*a^9*(a + a*Sin[c + d*x])^3) - 187/(512*a^10*(a + a*Sin[c + d*x])^2) - 315/(256*a^11*(a + a*Sin[c + d*x]))))/d","A",1
910,1,254,279,6.2333414,"\int \frac{\csc ^3(c+d x) \sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^9)/(a + a*Sin[c + d*x]),x]","\frac{a^{12} \left(-\frac{\csc ^2(c+d x)}{2 a^{13}}+\frac{\csc (c+d x)}{a^{13}}-\frac{843 \log (1-\sin (c+d x))}{512 a^{13}}+\frac{6 \log (\sin (c+d x))}{a^{13}}-\frac{2229 \log (\sin (c+d x)+1)}{512 a^{13}}+\frac{203}{256 a^{12} (a-a \sin (c+d x))}+\frac{5}{2 a^{12} (a \sin (c+d x)+a)}+\frac{81}{512 a^{11} (a-a \sin (c+d x))^2}+\frac{325}{512 a^{11} (a \sin (c+d x)+a)^2}+\frac{1}{32 a^{10} (a-a \sin (c+d x))^3}+\frac{23}{128 a^{10} (a \sin (c+d x)+a)^3}+\frac{1}{256 a^9 (a-a \sin (c+d x))^4}+\frac{11}{256 a^9 (a \sin (c+d x)+a)^4}+\frac{1}{160 a^8 (a \sin (c+d x)+a)^5}\right)}{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,"(a^12*(Csc[c + d*x]/a^13 - Csc[c + d*x]^2/(2*a^13) - (843*Log[1 - Sin[c + d*x]])/(512*a^13) + (6*Log[Sin[c + d*x]])/a^13 - (2229*Log[1 + Sin[c + d*x]])/(512*a^13) + 1/(256*a^9*(a - a*Sin[c + d*x])^4) + 1/(32*a^10*(a - a*Sin[c + d*x])^3) + 81/(512*a^11*(a - a*Sin[c + d*x])^2) + 203/(256*a^12*(a - a*Sin[c + d*x])) + 1/(160*a^8*(a + a*Sin[c + d*x])^5) + 11/(256*a^9*(a + a*Sin[c + d*x])^4) + 23/(128*a^10*(a + a*Sin[c + d*x])^3) + 325/(512*a^11*(a + a*Sin[c + d*x])^2) + 5/(2*a^12*(a + a*Sin[c + d*x]))))/d","A",1
911,1,347,127,3.3211629,"\int (g \sec (e+f x))^p (d \sin (e+f x))^n (a+a \sin (e+f x))^m \, dx","Integrate[(g*Sec[e + f*x])^p*(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^m,x]","\frac{g (p-3) (a (\sin (e+f x)+1))^m (d \sin (e+f x))^n (g \sec (e+f x))^{p-1} F_1\left(\frac{1-p}{2};-n,m+n-p+1;\frac{3-p}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f (p-1) \left(2 \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(n F_1\left(\frac{3-p}{2};1-n,m+n-p+1;\frac{5-p}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)+(m+n-p+1) F_1\left(\frac{3-p}{2};-n,m+n-p+2;\frac{5-p}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)\right)+(p-3) F_1\left(\frac{1-p}{2};-n,m+n-p+1;\frac{3-p}{2};\cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)\right)}","\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,"(g*(-3 + p)*AppellF1[(1 - p)/2, -n, 1 + m + n - p, (3 - p)/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2]*(g*Sec[e + f*x])^(-1 + p)*(d*Sin[e + f*x])^n*(a*(1 + Sin[e + f*x]))^m)/(f*(-1 + p)*((-3 + p)*AppellF1[(1 - p)/2, -n, 1 + m + n - p, (3 - p)/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] + 2*(n*AppellF1[(3 - p)/2, 1 - n, 1 + m + n - p, (5 - p)/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2] + (1 + m + n - p)*AppellF1[(3 - p)/2, -n, 2 + m + n - p, (5 - p)/2, Cot[(2*e + Pi + 2*f*x)/4]^2, -Tan[(2*e - Pi + 2*f*x)/4]^2])*Tan[(2*e - Pi + 2*f*x)/4]^2))","B",0
912,1,88,88,0.1416799,"\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Integrate[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",1
913,1,130,175,0.6042386,"\int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^4*(c + d*Sin[e + f*x])^n,x]","\frac{a^4 (c+d \sin (e+f x))^{n+1} \left(-\frac{4 (c-d)^3 (c+d \sin (e+f x))}{n+2}+\frac{6 (c-d)^2 (c+d \sin (e+f x))^2}{n+3}-\frac{4 (c-d) (c+d \sin (e+f x))^3}{n+4}+\frac{(c+d \sin (e+f x))^4}{n+5}+\frac{(c-d)^4}{n+1}\right)}{d^5 f}","\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*Sin[e + f*x])^(1 + n)*((c - d)^4/(1 + n) - (4*(c - d)^3*(c + d*Sin[e + f*x]))/(2 + n) + (6*(c - d)^2*(c + d*Sin[e + f*x])^2)/(3 + n) - (4*(c - d)*(c + d*Sin[e + f*x])^3)/(4 + n) + (c + d*Sin[e + f*x])^4/(5 + n)))/(d^5*f)","A",1
914,1,105,139,0.338698,"\int \cos (e+f x) (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n,x]","\frac{a^3 (c+d \sin (e+f x))^{n+1} \left(\frac{3 (c-d)^2 (c+d \sin (e+f x))}{n+2}-\frac{3 (c-d) (c+d \sin (e+f x))^2}{n+3}+\frac{(c+d \sin (e+f x))^3}{n+4}-\frac{(c-d)^3}{n+1}\right)}{d^4 f}","-\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*Sin[e + f*x])^(1 + n)*(-((c - d)^3/(1 + n)) + (3*(c - d)^2*(c + d*Sin[e + f*x]))/(2 + n) - (3*(c - d)*(c + d*Sin[e + f*x])^2)/(3 + n) + (c + d*Sin[e + f*x])^3/(4 + n)))/(d^4*f)","A",1
915,1,78,101,0.3647291,"\int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","\frac{a^2 (c+d \sin (e+f x))^{n+1} \left(-\frac{2 (c-d) (c+d \sin (e+f x))}{n+2}+\frac{(c+d \sin (e+f x))^2}{n+3}+\frac{(c-d)^2}{n+1}\right)}{d^3 f}","\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*Sin[e + f*x])^(1 + n)*((c - d)^2/(1 + n) - (2*(c - d)*(c + d*Sin[e + f*x]))/(2 + n) + (c + d*Sin[e + f*x])^2/(3 + n)))/(d^3*f)","A",1
916,1,52,61,0.5010651,"\int \cos (e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[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+1} (-c+d (n+1) \sin (e+f x)+d (n+2))}{d^2 f (n+1) (n+2)}","\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*Sin[e + f*x])^(1 + n)*(-c + d*(2 + n) + d*(1 + n)*Sin[e + f*x]))/(d^2*f*(1 + n)*(2 + n))","A",1
917,1,60,60,0.0710957,"\int \frac{\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","Integrate[(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",1
918,1,61,60,0.0656956,"\int \frac{\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Integrate[(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)}{d-c}\right)}{a^2 f (n+1) (d-c)^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",1
919,1,63,63,0.0678259,"\int \frac{\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","Integrate[(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)}{d-c}\right)}{a^3 f (n+1) (d-c)^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",1
920,1,143,170,0.6909751,"\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^4 \, dx","Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^4,x]","\frac{(a (\sin (e+f x)+1))^{m+1} \left(\frac{4 a^4 d^3 (c-d) (\sin (e+f x)+1)^3}{m+4}+\frac{6 a^4 d^2 (c-d)^2 (\sin (e+f x)+1)^2}{m+3}+\frac{4 a^4 d (c-d)^3 (\sin (e+f x)+1)}{m+2}+\frac{a^4 (c-d)^4}{m+1}+\frac{d^4 (a \sin (e+f x)+a)^4}{m+5}\right)}{a^5 f}","\frac{d^4 (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac{4 d^3 (c-d) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac{6 d^2 (c-d)^2 (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{4 d (c-d)^3 (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{(c-d)^4 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}",1,"((a*(1 + Sin[e + f*x]))^(1 + m)*((a^4*(c - d)^4)/(1 + m) + (4*a^4*(c - d)^3*d*(1 + Sin[e + f*x]))/(2 + m) + (6*a^4*(c - d)^2*d^2*(1 + Sin[e + f*x])^2)/(3 + m) + (4*a^4*(c - d)*d^3*(1 + Sin[e + f*x])^3)/(4 + m) + (d^4*(a + a*Sin[e + f*x])^4)/(5 + m)))/(a^5*f)","A",1
921,1,113,133,0.3939314,"\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^3 \, dx","Integrate[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)+1))^{m+1} \left(\frac{3 a^3 d^2 (c-d) (\sin (e+f x)+1)^2}{m+3}+\frac{3 a^3 d (c-d)^2 (\sin (e+f x)+1)}{m+2}+\frac{a^3 (c-d)^3}{m+1}+\frac{d^3 (a \sin (e+f x)+a)^3}{m+4}\right)}{a^4 f}","\frac{d^3 (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\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{(c-d)^3 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}",1,"((a*(1 + Sin[e + f*x]))^(1 + m)*((a^3*(c - d)^3)/(1 + m) + (3*a^3*(c - d)^2*d*(1 + Sin[e + f*x]))/(2 + m) + (3*a^3*(c - d)*d^2*(1 + Sin[e + f*x])^2)/(3 + m) + (d^3*(a + a*Sin[e + f*x])^3)/(4 + m)))/(a^4*f)","A",1
922,1,83,96,0.4125477,"\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx","Integrate[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)+1))^{m+1} \left(\frac{2 a^2 d (c-d) (\sin (e+f x)+1)}{m+2}+\frac{a^2 (c-d)^2}{m+1}+\frac{d^2 (a \sin (e+f x)+a)^2}{m+3}\right)}{a^3 f}","\frac{d^2 (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{2 d (c-d) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac{(c-d)^2 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)}",1,"((a*(1 + Sin[e + f*x]))^(1 + m)*((a^2*(c - d)^2)/(1 + m) + (2*a^2*(c - d)*d*(1 + Sin[e + f*x]))/(2 + m) + (d^2*(a + a*Sin[e + f*x])^2)/(3 + m)))/(a^3*f)","A",1
923,1,51,59,0.1208787,"\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x)) \, dx","Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x]),x]","\frac{(a (\sin (e+f x)+1))^{m+1} (c (m+2)+d (m+1) \sin (e+f x)-d)}{a f (m+1) (m+2)}","\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,"((a*(1 + Sin[e + f*x]))^(1 + m)*(-d + c*(2 + m) + d*(1 + m)*Sin[e + f*x]))/(a*f*(1 + m)*(2 + m))","A",1
924,1,59,59,0.0966792,"\int \frac{\cos (e+f x) (a+a \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx","Integrate[(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",1
925,1,59,59,0.093729,"\int \frac{\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx","Integrate[(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",1
926,1,59,59,0.0753586,"\int \frac{\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx","Integrate[(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",1
927,1,61,54,0.074577,"\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[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",1
928,1,150,134,1.395877,"\int \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^m,x]","\frac{(a (\sin (c+d x)+1))^{m+1} \left(\frac{3 \left(-2 \left(m^2+3 m+2\right) \cos (2 (c+d x))-8 (m+1) \sin (c+d x)+m^2+m+6\right)}{(m+1) (m+2) (m+3)}+\frac{16 (\sin (c+d x)+1)^4}{m+5}-\frac{64 (\sin (c+d x)+1)^3}{m+4}+\frac{84 (\sin (c+d x)+1)^2}{m+3}-\frac{40 (\sin (c+d x)+1)}{m+2}+\frac{7}{m+1}\right)}{16 a d}","\frac{(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}-\frac{4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac{6 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac{4 (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*(1 + Sin[c + d*x]))^(1 + m)*(7/(1 + m) - (40*(1 + Sin[c + d*x]))/(2 + m) + (84*(1 + Sin[c + d*x])^2)/(3 + m) - (64*(1 + Sin[c + d*x])^3)/(4 + m) + (16*(1 + Sin[c + d*x])^4)/(5 + m) + (3*(6 + m + m^2 - 2*(2 + 3*m + m^2)*Cos[2*(c + d*x)] - 8*(1 + m)*Sin[c + d*x]))/((1 + m)*(2 + m)*(3 + m))))/(16*a*d)","A",1
929,1,94,108,0.5716156,"\int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^m,x]","\frac{\left(-3 \left(m^2+3 m+2\right) \sin ^2(c+d x)+\left(m^3+6 m^2+11 m+6\right) \sin ^3(c+d x)+6 (m+1) \sin (c+d x)-6\right) (a (\sin (c+d x)+1))^{m+1}}{a d (m+1) (m+2) (m+3) (m+4)}","\frac{(a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}-\frac{3 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}+\frac{3 (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*(1 + Sin[c + d*x]))^(1 + m)*(-6 + 6*(1 + m)*Sin[c + d*x] - 3*(2 + 3*m + m^2)*Sin[c + d*x]^2 + (6 + 11*m + 6*m^2 + m^3)*Sin[c + d*x]^3))/(a*d*(1 + m)*(2 + m)*(3 + m)*(4 + m))","A",1
930,1,77,80,0.1879091,"\int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^m,x]","-\frac{(a (\sin (c+d x)+1))^{m+1} \left(\left(m^2+3 m+2\right) \cos (2 (c+d x))+4 (m+1) \sin (c+d x)-m^2-3 m-6\right)}{2 a d (m+1) (m+2) (m+3)}","\frac{(a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac{2 (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,"-1/2*((a*(1 + Sin[c + d*x]))^(1 + m)*(-6 - 3*m - m^2 + (2 + 3*m + m^2)*Cos[2*(c + d*x)] + 4*(1 + m)*Sin[c + d*x]))/(a*d*(1 + m)*(2 + m)*(3 + m))","A",1
931,1,43,54,0.0259925,"\int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]*Sin[c + d*x]*(a + a*Sin[c + d*x])^m,x]","\frac{((m+1) \sin (c+d x)-1) (a (\sin (c+d x)+1))^{m+1}}{a d (m+1) (m+2)}","\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*(1 + Sin[c + d*x]))^(1 + m)*(-1 + (1 + m)*Sin[c + d*x]))/(a*d*(1 + m)*(2 + m))","A",1
932,1,43,43,0.0530094,"\int \cot (c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[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",1
933,1,42,42,0.0542708,"\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[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",1
934,1,43,43,0.0699653,"\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[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",1
935,1,64,79,0.5969983,"\int \cos ^2(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x)) \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]","-\frac{a (24 (c+d) \cos (e+f x)+8 (c+d) \cos (3 (e+f x))-12 f x (4 c+d)-24 c \sin (2 (e+f x))+3 d \sin (4 (e+f x)))}{96 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,"-1/96*(a*(-12*(4*c + d)*f*x + 24*(c + d)*Cos[e + f*x] + 8*(c + d)*Cos[3*(e + f*x)] - 24*c*Sin[2*(e + f*x)] + 3*d*Sin[4*(e + f*x)]))/f","A",1
936,1,220,123,2.7953341,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))} \, dx","Integrate[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])),x]","\frac{(-1)^{3/4} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(\sqrt[4]{-1} \sqrt{c+d} \left(\log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}-\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-\log \left(\sec ^2\left(\frac{1}{4} (e+f x)\right) \left(\sqrt{c+d}+\sqrt{d} \sin \left(\frac{1}{2} (e+f x)\right)-\sqrt{d} \cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)-(4+4 i) \sqrt{d} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (e+f x)\right)-1\right)\right)\right)}{\sqrt{d} f (d-c) (a (\sin (e+f x)+1))^{3/2}}","\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,"((-1)^(3/4)*((-4 - 4*I)*Sqrt[d]*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])] + (-1)^(1/4)*Sqrt[c + d]*(Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] + Sqrt[d]*Cos[(e + f*x)/2] - Sqrt[d]*Sin[(e + f*x)/2])] - Log[Sec[(e + f*x)/4]^2*(Sqrt[c + d] - Sqrt[d]*Cos[(e + f*x)/2] + Sqrt[d]*Sin[(e + f*x)/2])]))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/(Sqrt[d]*(-c + d)*f*(a*(1 + Sin[e + f*x]))^(3/2))","C",1
937,1,208404,141,33.3109032,"\int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx","Integrate[Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]),x]","\text{Result too large to show}","\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,"Result too large to show","C",0
938,1,158,135,0.7426819,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","-\frac{4 \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \cos (e+f x) \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{-m-\frac{1}{2}} (a (\sin (e+f x)+1))^m (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};-m-\frac{1}{2},-n;\frac{5}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{3 f}","\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,"(-4*AppellF1[3/2, -1/2 - m, -n, 5/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)]*Cos[e + f*x]*(a*(1 + Sin[e + f*x]))^m*(c + d*Sin[e + f*x])^n*Sin[(2*e - Pi + 2*f*x)/4]^2*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(-1/2 - m))/(3*f*((c + d*Sin[e + f*x])/(c + d))^n)","A",0
939,0,0,119,116.952141,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n,x]","\int \cos ^2(e+f x) (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx","-\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,"Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n, x]","F",-1
940,0,0,119,77.4272017,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","\int \cos ^2(e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","-\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,"Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x]","F",-1
941,0,0,117,16.1195757,"\int \cos ^2(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\int \cos ^2(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","-\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,"Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x]","F",-1
942,1,229,119,1.0089377,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]),x]","-\frac{\sec (e+f x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2 \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} (c+d \sin (e+f x))^{n+1} \left((n+1) (c+d \sin (e+f x)) F_1\left(n+2;\frac{1}{2},\frac{1}{2};n+3;\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right)-(n+2) (c+d) 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)\right)}{a d f (n+1) (n+2) (d-c) \sqrt{\frac{d (\sin (e+f x)+1)}{d-c}}}","-\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,"-((Sec[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*(c + d*Sin[e + f*x])^(1 + n)*(-((c + d)*(2 + n)*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)]) + (1 + n)*AppellF1[2 + n, 1/2, 1/2, 3 + n, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])))/(a*d*(-c + d)*f*(1 + n)*(2 + n)*Sqrt[(d*(1 + Sin[e + f*x]))/(-c + d)]))","A",0
943,0,0,119,8.8298072,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2,x]","\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","-\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,"Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2, x]","F",-1
944,0,0,119,20.6872417,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^3,x]","\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","-\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,"Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^3, x]","F",-1
945,1,160,135,1.3754225,"\int \cos ^4(e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]^4*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","-\frac{4 \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \cos ^3(e+f x) \sin ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)^{-m-\frac{3}{2}} (a (\sin (e+f x)+1))^m (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};-m-\frac{3}{2},-n;\frac{7}{2};\cos ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right),\frac{2 d \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)}{5 f}","\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*AppellF1[5/2, -3/2 - m, -n, 7/2, Cos[(2*e + Pi + 2*f*x)/4]^2, (2*d*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d)]*Cos[e + f*x]^3*(a*(1 + Sin[e + f*x]))^m*(c + d*Sin[e + f*x])^n*Sin[(2*e - Pi + 2*f*x)/4]^2*(Sin[(2*e + Pi + 2*f*x)/4]^2)^(-3/2 - m))/(5*f*((c + d*Sin[e + f*x])/(c + d))^n)","A",0
946,0,0,121,1.6344886,"\int \cos ^4(e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]^4*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","\int \cos ^4(e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","-\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,"Integrate[Cos[e + f*x]^4*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x]","F",-1
947,0,0,119,0.7458261,"\int \cos ^4(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]^4*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\int \cos ^4(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","-\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,"Integrate[Cos[e + f*x]^4*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x]","F",-1
948,0,0,121,30.7099251,"\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]),x]","\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","-\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,"Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]), x]","F",-1
949,0,0,121,17.6120663,"\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2,x]","\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","-\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,"Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2, x]","F",-1
950,0,0,121,19.8029974,"\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^3,x]","\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","-\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,"Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^3, x]","F",-1
951,0,0,121,25.131825,"\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^4} \, dx","Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^4,x]","\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^4} \, dx","-\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,"Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^4, x]","F",-1
952,0,0,121,2.4279338,"\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^5} \, dx","Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^5,x]","\int \frac{\cos ^4(e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^5} \, dx","-\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,"Integrate[(Cos[e + f*x]^4*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^5, x]","F",-1
953,1,194,134,0.8051101,"\int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a (\sin (c+d x)+1) (-17640 (A+B) \cos (2 (c+d x))-8820 (A+B) \cos (4 (c+d x))+176400 A \sin (c+d x)+35280 A \sin (3 (c+d x))+7056 A \sin (5 (c+d x))+720 A \sin (7 (c+d x))-2520 A \cos (6 (c+d x))-315 A \cos (8 (c+d x))+17640 B \sin (c+d x)-2016 B \sin (5 (c+d x))-900 B \sin (7 (c+d x))-140 B \sin (9 (c+d x))-2520 B \cos (6 (c+d x))-315 B \cos (8 (c+d x)))}{322560 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}","-\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}",1,"(a*(1 + Sin[c + d*x])*(-17640*(A + B)*Cos[2*(c + d*x)] - 8820*(A + B)*Cos[4*(c + d*x)] - 2520*A*Cos[6*(c + d*x)] - 2520*B*Cos[6*(c + d*x)] - 315*A*Cos[8*(c + d*x)] - 315*B*Cos[8*(c + d*x)] + 176400*A*Sin[c + d*x] + 17640*B*Sin[c + d*x] + 35280*A*Sin[3*(c + d*x)] + 7056*A*Sin[5*(c + d*x)] - 2016*B*Sin[5*(c + d*x)] + 720*A*Sin[7*(c + d*x)] - 900*B*Sin[7*(c + d*x)] - 140*B*Sin[9*(c + d*x)]))/(322560*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)","A",1
954,1,130,102,0.6621888,"\int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{a (525 (A+B) \cos (2 (c+d x))+210 (A+B) \cos (4 (c+d x))-4200 A \sin (c+d x)-700 A \sin (3 (c+d x))-84 A \sin (5 (c+d x))+35 A \cos (6 (c+d x))-525 B \sin (c+d x)+35 B \sin (3 (c+d x))+63 B \sin (5 (c+d x))+15 B \sin (7 (c+d x))+35 B \cos (6 (c+d x)))}{6720 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}",1,"-1/6720*(a*(525*(A + B)*Cos[2*(c + d*x)] + 210*(A + B)*Cos[4*(c + d*x)] + 35*A*Cos[6*(c + d*x)] + 35*B*Cos[6*(c + d*x)] - 4200*A*Sin[c + d*x] - 525*B*Sin[c + d*x] - 700*A*Sin[3*(c + d*x)] + 35*B*Sin[3*(c + d*x)] - 84*A*Sin[5*(c + d*x)] + 63*B*Sin[5*(c + d*x)] + 15*B*Sin[7*(c + d*x)]))/d","A",1
955,1,78,78,0.8106745,"\int \cos ^3(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{a (-4 (100 A+11 B) \sin (c+d x)+3 \cos (4 (c+d x)) (5 (A+B)+4 B \sin (c+d x))+\cos (2 (c+d x)) ((32 B-80 A) \sin (c+d x)+60 (A+B)))}{480 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}",1,"-1/480*(a*(-4*(100*A + 11*B)*Sin[c + d*x] + 3*Cos[4*(c + d*x)]*(5*(A + B) + 4*B*Sin[c + d*x]) + Cos[2*(c + d*x)]*(60*(A + B) + (-80*A + 32*B)*Sin[c + d*x])))/d","A",1
956,1,46,49,0.3907399,"\int \cos (c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{a (\cos (2 (c+d x)) (3 (A+B)+2 B \sin (c+d x))-2 (6 A+B) \sin (c+d x))}{12 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,"-1/12*(a*(-2*(6*A + B)*Sin[c + d*x] + Cos[2*(c + d*x)]*(3*(A + B) + 2*B*Sin[c + d*x])))/d","A",1
957,1,68,34,0.0348838,"\int \sec (c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a A \log (\cos (c+d x))}{d}-\frac{a B \sin (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a B \log (\cos (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*ArcTanh[Sin[c + d*x]])/d + (a*B*ArcTanh[Sin[c + d*x]])/d - (a*A*Log[Cos[c + d*x]])/d - (a*B*Log[Cos[c + d*x]])/d - (a*B*Sin[c + d*x])/d","A",1
958,1,260,47,0.6320765,"\int \sec ^3(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a \left(2 i (A-B) (\sin (c+d x)-1) \tan ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)+(A-B) \sin (c+d x) \left(2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2\right)-i d x\right)-2 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A \log \left(\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2\right)+i A d x+2 A+2 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-B \log \left(\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2\right)-i B d x+2 B\right)}{4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}","\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*(2*A + 2*B + I*A*d*x - I*B*d*x - 2*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*Log[(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2] - B*Log[(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2] + (2*I)*(A - B)*ArcTan[Tan[(c + d*x)/2]]*(-1 + Sin[c + d*x]) + (A - B)*((-I)*d*x + 2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2])*Sin[c + d*x]))/(4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2)","C",1
959,1,357,100,1.5633361,"\int \sec ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a (\sin (c+d x)+1) \left(i x (3 A-B) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+\frac{2 (A+B) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{2 (3 A-B) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{(3 A-B) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \log \left(\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2\right)}{d}-\frac{2 i (3 A-B) \tan ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{d}+\frac{2 (B-A)}{d}+\frac{4 A \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{16 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","\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*((2*(-A + B))/d + I*(3*A - B)*x*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - ((2*I)*(3*A - B)*ArcTan[Tan[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/d - (2*(3*A - B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/d + ((3*A - B)*Log[(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/d + (2*(A + B)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + (4*A*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2))*(1 + Sin[c + d*x]))/(16*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","C",1
960,1,451,157,1.7549237,"\int \sec ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^7*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a (\sin (c+d x)+1) \left(3 i x (5 A-B) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4+\frac{3 (3 A+B) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{4 (A+B) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^6}-\frac{6 (2 A-B) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{d}-\frac{6 (5 A-B) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{3 (5 A-B) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \log \left(\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2\right)}{d}-\frac{6 i (5 A-B) \tan ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{d}+\frac{3 (B-A)}{d}+\frac{18 A \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{96 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6}","\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*((3*(-A + B))/d - (6*(2*A - B)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/d + (3*I)*(5*A - B)*x*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - ((6*I)*(5*A - B)*ArcTan[Tan[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/d - (6*(5*A - B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/d + (3*(5*A - B)*Log[(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/d + (4*(A + B)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^6) + (3*(3*A + B)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + (18*A*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2))*(1 + Sin[c + d*x]))/(96*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)","C",1
961,1,164,138,0.8411872,"\int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{a (1680 (A+B) \cos (c+d x)+1008 (A+B) \cos (3 (c+d x))-5040 A \sin (2 (c+d x))-1008 A \sin (4 (c+d x))-112 A \sin (6 (c+d x))+336 A \cos (5 (c+d x))+48 A \cos (7 (c+d x))-6720 A d x-336 B \sin (2 (c+d x))+168 B \sin (4 (c+d x))+112 B \sin (6 (c+d x))+21 B \sin (8 (c+d x))+336 B \cos (5 (c+d x))+48 B \cos (7 (c+d x))-840 B d x)}{21504 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,"-1/21504*(a*(-6720*A*d*x - 840*B*d*x + 1680*(A + B)*Cos[c + d*x] + 1008*(A + B)*Cos[3*(c + d*x)] + 336*A*Cos[5*(c + d*x)] + 336*B*Cos[5*(c + d*x)] + 48*A*Cos[7*(c + d*x)] + 48*B*Cos[7*(c + d*x)] - 5040*A*Sin[2*(c + d*x)] - 336*B*Sin[2*(c + d*x)] - 1008*A*Sin[4*(c + d*x)] + 168*B*Sin[4*(c + d*x)] - 112*A*Sin[6*(c + d*x)] + 112*B*Sin[6*(c + d*x)] + 21*B*Sin[8*(c + d*x)]))/d","A",1
962,1,120,111,0.5912016,"\int \cos ^4(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{a (120 (A+B) \cos (c+d x)+60 (A+B) \cos (3 (c+d x))-240 A \sin (2 (c+d x))-30 A \sin (4 (c+d x))+12 A \cos (5 (c+d x))-360 A d x-15 B \sin (2 (c+d x))+15 B \sin (4 (c+d x))+5 B \sin (6 (c+d x))+12 B \cos (5 (c+d x))-60 B d x)}{960 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,"-1/960*(a*(-360*A*d*x - 60*B*d*x + 120*(A + B)*Cos[c + d*x] + 60*(A + B)*Cos[3*(c + d*x)] + 12*A*Cos[5*(c + d*x)] + 12*B*Cos[5*(c + d*x)] - 240*A*Sin[2*(c + d*x)] - 15*B*Sin[2*(c + d*x)] - 30*A*Sin[4*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 5*B*Sin[6*(c + d*x)]))/d","A",1
963,1,64,84,0.6705238,"\int \cos ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{a (24 (A+B) \cos (c+d x)+8 (A+B) \cos (3 (c+d x))-12 d x (4 A+B)-24 A \sin (2 (c+d x))+3 B \sin (4 (c+d x)))}{96 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,"-1/96*(a*(-12*(4*A + B)*d*x + 24*(A + B)*Cos[c + d*x] + 8*(A + B)*Cos[3*(c + d*x)] - 24*A*Sin[2*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/d","A",1
964,1,85,29,0.3539612,"\int \sec ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a \left(2 (A+B) \sin \left(\frac{d x}{2}\right)+B d x \sin \left(c+\frac{d x}{2}\right)-B d x \cos \left(\frac{d x}{2}\right)\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{(A+B) \sec (c+d x) (a \sin (c+d x)+a)}{d}-a B x",1,"(a*(-(B*d*x*Cos[(d*x)/2]) + 2*(A + B)*Sin[(d*x)/2] + B*d*x*Sin[c + (d*x)/2]))/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))","B",1
965,1,97,50,0.6259408,"\int \sec ^4(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a \sec (c) (\sin (c+d x)+1) \sec ^3(c+d x) (-2 (A+B) \cos (c+d x)+A \sin (2 (c+d x))+4 A \cos (c+2 d x)+8 A \sin (d x)+B \sin (2 (c+d x))-2 B \cos (c+2 d x)+6 B \cos (c)-4 B \sin (d x))}{12 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*Sec[c]*Sec[c + d*x]^3*(1 + Sin[c + d*x])*(6*B*Cos[c] - 2*(A + B)*Cos[c + d*x] + 4*A*Cos[c + 2*d*x] - 2*B*Cos[c + 2*d*x] + 8*A*Sin[d*x] - 4*B*Sin[d*x] + A*Sin[2*(c + d*x)] + B*Sin[2*(c + d*x)]))/(12*d)","A",1
966,1,223,73,1.2716776,"\int \sec ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^6*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a \sec (c) (-54 (A+B) \cos (c+d x)+18 A \sin (2 (c+d x))+9 A \sin (4 (c+d x))+128 A \sin (2 c+3 d x)-18 A \cos (3 (c+d x))+128 A \cos (c+2 d x)+64 A \cos (3 c+4 d x)+384 A \sin (d x)+18 B \sin (2 (c+d x))+9 B \sin (4 (c+d x))-32 B \sin (2 c+3 d x)-18 B \cos (3 (c+d x))-32 B \cos (c+2 d x)-16 B \cos (3 c+4 d x)+240 B \cos (c)-96 B \sin (d x))}{960 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","\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*Sec[c]*(240*B*Cos[c] - 54*(A + B)*Cos[c + d*x] - 18*A*Cos[3*(c + d*x)] - 18*B*Cos[3*(c + d*x)] + 128*A*Cos[c + 2*d*x] - 32*B*Cos[c + 2*d*x] + 64*A*Cos[3*c + 4*d*x] - 16*B*Cos[3*c + 4*d*x] + 384*A*Sin[d*x] - 96*B*Sin[d*x] + 18*A*Sin[2*(c + d*x)] + 18*B*Sin[2*(c + d*x)] + 9*A*Sin[4*(c + d*x)] + 9*B*Sin[4*(c + d*x)] + 128*A*Sin[2*c + 3*d*x] - 32*B*Sin[2*c + 3*d*x]))/(960*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","B",1
967,1,315,96,2.0334499,"\int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^8*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a \sec (c) (-1500 (A+B) \cos (c+d x)+375 A \sin (2 (c+d x))+300 A \sin (4 (c+d x))+75 A \sin (6 (c+d x))+7680 A \sin (2 c+3 d x)+1536 A \sin (4 c+5 d x)-750 A \cos (3 (c+d x))-150 A \cos (5 (c+d x))+3840 A \cos (c+2 d x)+3072 A \cos (3 c+4 d x)+768 A \cos (5 c+6 d x)+15360 A \sin (d x)+375 B \sin (2 (c+d x))+300 B \sin (4 (c+d x))+75 B \sin (6 (c+d x))-1280 B \sin (2 c+3 d x)-256 B \sin (4 c+5 d x)-750 B \cos (3 (c+d x))-150 B \cos (5 (c+d x))-640 B \cos (c+2 d x)-512 B \cos (3 c+4 d x)-128 B \cos (5 c+6 d x)+8960 B \cos (c)-2560 B \sin (d x))}{53760 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^7 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","\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*Sec[c]*(8960*B*Cos[c] - 1500*(A + B)*Cos[c + d*x] - 750*A*Cos[3*(c + d*x)] - 750*B*Cos[3*(c + d*x)] - 150*A*Cos[5*(c + d*x)] - 150*B*Cos[5*(c + d*x)] + 3840*A*Cos[c + 2*d*x] - 640*B*Cos[c + 2*d*x] + 3072*A*Cos[3*c + 4*d*x] - 512*B*Cos[3*c + 4*d*x] + 768*A*Cos[5*c + 6*d*x] - 128*B*Cos[5*c + 6*d*x] + 15360*A*Sin[d*x] - 2560*B*Sin[d*x] + 375*A*Sin[2*(c + d*x)] + 375*B*Sin[2*(c + d*x)] + 300*A*Sin[4*(c + d*x)] + 300*B*Sin[4*(c + d*x)] + 75*A*Sin[6*(c + d*x)] + 75*B*Sin[6*(c + d*x)] + 7680*A*Sin[2*c + 3*d*x] - 1280*B*Sin[2*c + 3*d*x] + 1536*A*Sin[4*c + 5*d*x] - 256*B*Sin[4*c + 5*d*x]))/(53760*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","B",1
968,1,407,119,4.3400647,"\int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^10*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a \sec (c) (-85750 (A+B) \cos (c+d x)+17150 A \sin (2 (c+d x))+17150 A \sin (4 (c+d x))+7350 A \sin (6 (c+d x))+1225 A \sin (8 (c+d x))+688128 A \sin (2 c+3 d x)+229376 A \sin (4 c+5 d x)+32768 A \sin (6 c+7 d x)-51450 A \cos (3 (c+d x))-17150 A \cos (5 (c+d x))-2450 A \cos (7 (c+d x))+229376 A \cos (c+2 d x)+229376 A \cos (3 c+4 d x)+98304 A \cos (5 c+6 d x)+16384 A \cos (7 c+8 d x)+1146880 A \sin (d x)+17150 B \sin (2 (c+d x))+17150 B \sin (4 (c+d x))+7350 B \sin (6 (c+d x))+1225 B \sin (8 (c+d x))-86016 B \sin (2 c+3 d x)-28672 B \sin (4 c+5 d x)-4096 B \sin (6 c+7 d x)-51450 B \cos (3 (c+d x))-17150 B \cos (5 (c+d x))-2450 B \cos (7 (c+d x))-28672 B \cos (c+2 d x)-28672 B \cos (3 c+4 d x)-12288 B \cos (5 c+6 d x)-2048 B \cos (7 c+8 d x)+645120 B \cos (c)-143360 B \sin (d x))}{5160960 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^9 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}","\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*Sec[c]*(645120*B*Cos[c] - 85750*(A + B)*Cos[c + d*x] - 51450*A*Cos[3*(c + d*x)] - 51450*B*Cos[3*(c + d*x)] - 17150*A*Cos[5*(c + d*x)] - 17150*B*Cos[5*(c + d*x)] - 2450*A*Cos[7*(c + d*x)] - 2450*B*Cos[7*(c + d*x)] + 229376*A*Cos[c + 2*d*x] - 28672*B*Cos[c + 2*d*x] + 229376*A*Cos[3*c + 4*d*x] - 28672*B*Cos[3*c + 4*d*x] + 98304*A*Cos[5*c + 6*d*x] - 12288*B*Cos[5*c + 6*d*x] + 16384*A*Cos[7*c + 8*d*x] - 2048*B*Cos[7*c + 8*d*x] + 1146880*A*Sin[d*x] - 143360*B*Sin[d*x] + 17150*A*Sin[2*(c + d*x)] + 17150*B*Sin[2*(c + d*x)] + 17150*A*Sin[4*(c + d*x)] + 17150*B*Sin[4*(c + d*x)] + 7350*A*Sin[6*(c + d*x)] + 7350*B*Sin[6*(c + d*x)] + 1225*A*Sin[8*(c + d*x)] + 1225*B*Sin[8*(c + d*x)] + 688128*A*Sin[2*c + 3*d*x] - 86016*B*Sin[2*c + 3*d*x] + 229376*A*Sin[4*c + 5*d*x] - 28672*B*Sin[4*c + 5*d*x] + 32768*A*Sin[6*c + 7*d*x] - 4096*B*Sin[6*c + 7*d*x]))/(5160960*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^9*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)","B",1
969,1,86,134,1.1855311,"\int \cos ^7(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{a^2 (\sin (c+d x)+1)^6 \left(28 (5 A-17 B) \sin ^3(c+d x)+(651 B-525 A) \sin ^2(c+d x)+6 (115 A-61 B) \sin (c+d x)-325 A+126 B \sin ^4(c+d x)+61 B\right)}{1260 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}",1,"-1/1260*(a^2*(1 + Sin[c + d*x])^6*(-325*A + 61*B + 6*(115*A - 61*B)*Sin[c + d*x] + (-525*A + 651*B)*Sin[c + d*x]^2 + 28*(5*A - 17*B)*Sin[c + d*x]^3 + 126*B*Sin[c + d*x]^4))/d","A",1
970,1,70,105,0.3576093,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 (\sin (c+d x)+1)^5 \left(15 (8 A-19 B) \sin ^2(c+d x)-5 (64 A-47 B) \sin (c+d x)+232 A+105 B \sin ^3(c+d x)-47 B\right)}{840 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}",1,"(a^2*(1 + Sin[c + d*x])^5*(232*A - 47*B - 5*(64*A - 47*B)*Sin[c + d*x] + 15*(8*A - 19*B)*Sin[c + d*x]^2 + 105*B*Sin[c + d*x]^3))/(840*d)","A",1
971,1,66,78,0.4318787,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8 (-4 (3 A-4 B) \sin (c+d x)+18 A+5 B \cos (2 (c+d x))-9 B)}{60 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}",1,"(a^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*(18*A - 9*B + 5*B*Cos[2*(c + d*x)] - 4*(3*A - 4*B)*Sin[c + d*x]))/(60*d)","A",1
972,1,49,51,0.0953846,"\int \cos (c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{\frac{1}{3} (A-B) (a \sin (c+d x)+a)^3+\frac{B (a \sin (c+d x)+a)^4}{4 a}}{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 + (B*(a + a*Sin[c + d*x])^4)/(4*a))/(a*d)","A",1
973,1,51,60,0.0819575,"\int \sec (c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a \left(-a (A+2 B) \sin (c+d x)-2 a (A+B) \log (1-\sin (c+d x))-\frac{1}{2} a B \sin ^2(c+d x)\right)}{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,"(a*(-2*a*(A + B)*Log[1 - Sin[c + d*x]] - a*(A + 2*B)*Sin[c + d*x] - (a*B*Sin[c + d*x]^2)/2))/d","A",1
974,1,41,43,0.0814044,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^3 \left(\frac{A+B}{a-a \sin (c+d x)}+\frac{B \log (1-\sin (c+d x))}{a}\right)}{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^3*((B*Log[1 - Sin[c + d*x]])/a + (A + B)/(a - a*Sin[c + d*x])))/d","A",1
975,1,75,77,0.1399283,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^5 \left(\frac{(A-B) \tanh ^{-1}(\sin (c+d x))}{4 a^3}+\frac{A-B}{4 a^2 (a-a \sin (c+d x))}+\frac{A+B}{4 a (a-a \sin (c+d x))^2}\right)}{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^5*(((A - B)*ArcTanh[Sin[c + d*x]])/(4*a^3) + (A + B)/(4*a*(a - a*Sin[c + d*x])^2) + (A - B)/(4*a^2*(a - a*Sin[c + d*x]))))/d","A",1
976,1,90,132,0.7063467,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 \left(\frac{3 B-9 A}{\sin (c+d x)-1}-\frac{3 (A-B)}{\sin (c+d x)+1}-\frac{4 (A+B)}{(\sin (c+d x)-1)^3}+6 (2 A-B) \tanh ^{-1}(\sin (c+d x))+\frac{6 A}{(\sin (c+d x)-1)^2}\right)}{48 d}","\frac{a^5 (A+B)}{12 d (a-a \sin (c+d x))^3}+\frac{a^4 A}{8 d (a-a \sin (c+d x))^2}+\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}",1,"(a^2*(6*(2*A - B)*ArcTanh[Sin[c + d*x]] - (4*(A + B))/(-1 + Sin[c + d*x])^3 + (6*A)/(-1 + Sin[c + d*x])^2 + (-9*A + 3*B)/(-1 + Sin[c + d*x]) - (3*(A - B))/(1 + Sin[c + d*x])))/(48*d)","A",1
977,1,216,196,5.098658,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{a^2 \cos (c+d x) \left(32 (135 A+86 B) \cos (2 (c+d x))+16 (108 A+59 B) \cos (4 (c+d x))+\frac{2520 (9 A+2 B) \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)}{\sqrt{\cos ^2(c+d x)}}-13671 A \sin (c+d x)-2457 A \sin (3 (c+d x))-63 A \sin (5 (c+d x))+63 A \sin (7 (c+d x))+288 A \cos (6 (c+d x))+2880 A-2478 B \sin (c+d x)+462 B \sin (3 (c+d x))+546 B \sin (5 (c+d x))+126 B \sin (7 (c+d x))+64 B \cos (6 (c+d x))-28 B \cos (8 (c+d x))+1900 B\right)}{32256 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,"-1/32256*(a^2*Cos[c + d*x]*(2880*A + 1900*B + (2520*(9*A + 2*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/Sqrt[Cos[c + d*x]^2] + 32*(135*A + 86*B)*Cos[2*(c + d*x)] + 16*(108*A + 59*B)*Cos[4*(c + d*x)] + 288*A*Cos[6*(c + d*x)] + 64*B*Cos[6*(c + d*x)] - 28*B*Cos[8*(c + d*x)] - 13671*A*Sin[c + d*x] - 2478*B*Sin[c + d*x] - 2457*A*Sin[3*(c + d*x)] + 462*B*Sin[3*(c + d*x)] - 63*A*Sin[5*(c + d*x)] + 546*B*Sin[5*(c + d*x)] + 63*A*Sin[7*(c + d*x)] + 126*B*Sin[7*(c + d*x)]))/d","A",1
978,1,171,165,1.730874,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{a^2 \cos (c+d x) \left((672 A+447 B) \cos (2 (c+d x))+6 (28 A+13 B) \cos (4 (c+d x))+\frac{420 (7 A+2 B) \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)}{\sqrt{\cos ^2(c+d x)}}-1645 A \sin (c+d x)-140 A \sin (3 (c+d x))+35 A \sin (5 (c+d x))+504 A-350 B \sin (c+d x)+140 B \sin (3 (c+d x))+70 B \sin (5 (c+d x))-15 B \cos (6 (c+d x))+354 B\right)}{3360 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,"-1/3360*(a^2*Cos[c + d*x]*(504*A + 354*B + (420*(7*A + 2*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/Sqrt[Cos[c + d*x]^2] + (672*A + 447*B)*Cos[2*(c + d*x)] + 6*(28*A + 13*B)*Cos[4*(c + d*x)] - 15*B*Cos[6*(c + d*x)] - 1645*A*Sin[c + d*x] - 350*B*Sin[c + d*x] - 140*A*Sin[3*(c + d*x)] + 140*B*Sin[3*(c + d*x)] + 35*A*Sin[5*(c + d*x)] + 70*B*Sin[5*(c + d*x)]))/d","A",1
979,1,133,134,0.9776671,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{a^2 \cos (c+d x) \left(8 (10 A+7 B) \cos (2 (c+d x))+\frac{60 (5 A+2 B) \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)}{\sqrt{\cos ^2(c+d x)}}-135 A \sin (c+d x)+15 A \sin (3 (c+d x))+80 A-30 B \sin (c+d x)+30 B \sin (3 (c+d x))-6 B \cos (4 (c+d x))+62 B\right)}{240 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,"-1/240*(a^2*Cos[c + d*x]*(80*A + 62*B + (60*(5*A + 2*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/Sqrt[Cos[c + d*x]^2] + 8*(10*A + 7*B)*Cos[2*(c + d*x)] - 6*B*Cos[4*(c + d*x)] - 135*A*Sin[c + d*x] - 30*B*Sin[c + d*x] + 15*A*Sin[3*(c + d*x)] + 30*B*Sin[3*(c + d*x)]))/d","A",1
980,1,91,55,0.2447278,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 \sec (c+d x) \left(4 (A+2 B) \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \sqrt{\cos ^2(c+d x)}+4 A \sin (c+d x)+4 A+4 B \sin (c+d x)+B \cos (2 (c+d x))+5 B\right)}{2 d}","\frac{a^2 (A+2 B) \cos (c+d x)}{d}-\left(a^2 x (A+2 B)\right)+\frac{(A+B) \sec (c+d x) (a \sin (c+d x)+a)^2}{d}",1,"(a^2*Sec[c + d*x]*(4*A + 5*B + 4*(A + 2*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[Cos[c + d*x]^2] + B*Cos[2*(c + d*x)] + 4*A*Sin[c + d*x] + 4*B*Sin[c + d*x]))/(2*d)","A",1
981,1,121,73,0.0188789,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{a^2 A \tan ^3(c+d x)}{3 d}+\frac{2 a^2 A \sec ^3(c+d x)}{3 d}+\frac{a^2 A \tan (c+d x) \sec ^2(c+d x)}{d}+\frac{2 a^2 B \tan ^3(c+d x)}{3 d}-\frac{a^2 B \sec ^3(c+d x)}{3 d}+\frac{a^2 B \tan ^2(c+d x) \sec (c+d x)}{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,"(2*a^2*A*Sec[c + d*x]^3)/(3*d) - (a^2*B*Sec[c + d*x]^3)/(3*d) + (a^2*A*Sec[c + d*x]^2*Tan[c + d*x])/d + (a^2*B*Sec[c + d*x]*Tan[c + d*x]^2)/d - (a^2*A*Tan[c + d*x]^3)/(3*d) + (2*a^2*B*Tan[c + d*x]^3)/(3*d)","A",1
982,1,178,104,0.0203265,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{2 a^2 A \tan ^5(c+d x)}{5 d}+\frac{2 a^2 A \sec ^5(c+d x)}{5 d}+\frac{a^2 A \tan (c+d x) \sec ^4(c+d x)}{d}-\frac{a^2 A \tan ^3(c+d x) \sec ^2(c+d x)}{d}-\frac{4 a^2 B \tan ^5(c+d x)}{15 d}+\frac{a^2 B \sec ^5(c+d x)}{15 d}+\frac{a^2 B \tan ^2(c+d x) \sec ^3(c+d x)}{3 d}+\frac{2 a^2 B \tan ^3(c+d x) \sec ^2(c+d x)}{3 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,"(2*a^2*A*Sec[c + d*x]^5)/(5*d) + (a^2*B*Sec[c + d*x]^5)/(15*d) + (a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/d + (a^2*B*Sec[c + d*x]^3*Tan[c + d*x]^2)/(3*d) - (a^2*A*Sec[c + d*x]^2*Tan[c + d*x]^3)/d + (2*a^2*B*Sec[c + d*x]^2*Tan[c + d*x]^3)/(3*d) + (2*a^2*A*Tan[c + d*x]^5)/(5*d) - (4*a^2*B*Tan[c + d*x]^5)/(15*d)","A",1
983,1,130,129,0.3403965,"\int \sec ^8(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^8*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 \left(8 (2 B-5 A) \tan ^7(c+d x)+(30 A+9 B) \sec ^7(c+d x)-35 (5 A-2 B) \tan ^3(c+d x) \sec ^4(c+d x)+28 (5 A-2 B) \tan ^5(c+d x) \sec ^2(c+d x)+105 A \tan (c+d x) \sec ^6(c+d x)+21 B \tan ^2(c+d x) \sec ^5(c+d x)\right)}{105 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*((30*A + 9*B)*Sec[c + d*x]^7 + 105*A*Sec[c + d*x]^6*Tan[c + d*x] + 21*B*Sec[c + d*x]^5*Tan[c + d*x]^2 - 35*(5*A - 2*B)*Sec[c + d*x]^4*Tan[c + d*x]^3 + 28*(5*A - 2*B)*Sec[c + d*x]^2*Tan[c + d*x]^5 + 8*(-5*A + 2*B)*Tan[c + d*x]^7))/(105*d)","A",1
984,1,156,154,0.4521657,"\int \sec ^{10}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^10*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 \left(16 (7 A-2 B) \tan ^9(c+d x)+5 (14 A+5 B) \sec ^9(c+d x)-105 (7 A-2 B) \tan ^3(c+d x) \sec ^6(c+d x)+126 (7 A-2 B) \tan ^5(c+d x) \sec ^4(c+d x)-72 (7 A-2 B) \tan ^7(c+d x) \sec ^2(c+d x)+315 A \tan (c+d x) \sec ^8(c+d x)+45 B \tan ^2(c+d x) \sec ^7(c+d x)\right)}{315 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*(5*(14*A + 5*B)*Sec[c + d*x]^9 + 315*A*Sec[c + d*x]^8*Tan[c + d*x] + 45*B*Sec[c + d*x]^7*Tan[c + d*x]^2 - 105*(7*A - 2*B)*Sec[c + d*x]^6*Tan[c + d*x]^3 + 126*(7*A - 2*B)*Sec[c + d*x]^4*Tan[c + d*x]^5 - 72*(7*A - 2*B)*Sec[c + d*x]^2*Tan[c + d*x]^7 + 16*(7*A - 2*B)*Tan[c + d*x]^9))/(315*d)","A",1
985,1,181,179,0.915172,"\int \sec ^{12}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^12*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{a^2 \left(128 (2 B-9 A) \tan ^{11}(c+d x)+35 (18 A+7 B) \sec ^{11}(c+d x)-1155 (9 A-2 B) \tan ^3(c+d x) \sec ^8(c+d x)+1848 (9 A-2 B) \tan ^5(c+d x) \sec ^6(c+d x)-1584 (9 A-2 B) \tan ^7(c+d x) \sec ^4(c+d x)+704 (9 A-2 B) \tan ^9(c+d x) \sec ^2(c+d x)+3465 A \tan (c+d x) \sec ^{10}(c+d x)+385 B \tan ^2(c+d x) \sec ^9(c+d x)\right)}{3465 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*(35*(18*A + 7*B)*Sec[c + d*x]^11 + 3465*A*Sec[c + d*x]^10*Tan[c + d*x] + 385*B*Sec[c + d*x]^9*Tan[c + d*x]^2 - 1155*(9*A - 2*B)*Sec[c + d*x]^8*Tan[c + d*x]^3 + 1848*(9*A - 2*B)*Sec[c + d*x]^6*Tan[c + d*x]^5 - 1584*(9*A - 2*B)*Sec[c + d*x]^4*Tan[c + d*x]^7 + 704*(9*A - 2*B)*Sec[c + d*x]^2*Tan[c + d*x]^9 + 128*(-9*A + 2*B)*Tan[c + d*x]^11))/(3465*d)","A",1
986,1,86,134,1.5269885,"\int \cos ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{a^3 (\sin (c+d x)+1)^7 \left(21 (11 A-37 B) \sin ^3(c+d x)+(1029 B-847 A) \sin ^2(c+d x)+14 (77 A-39 B) \sin (c+d x)-484 A+210 B \sin ^4(c+d x)+78 B\right)}{2310 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}",1,"-1/2310*(a^3*(1 + Sin[c + d*x])^7*(-484*A + 78*B + 14*(77*A - 39*B)*Sin[c + d*x] + (-847*A + 1029*B)*Sin[c + d*x]^2 + 21*(11*A - 37*B)*Sin[c + d*x]^3 + 210*B*Sin[c + d*x]^4))/d","A",1
987,1,70,105,0.4368677,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^3 (\sin (c+d x)+1)^6 \left(21 (3 A-7 B) \sin ^2(c+d x)-6 (27 A-19 B) \sin (c+d x)+111 A+56 B \sin ^3(c+d x)-19 B\right)}{504 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}",1,"(a^3*(1 + Sin[c + d*x])^6*(111*A - 19*B - 6*(27*A - 19*B)*Sin[c + d*x] + 21*(3*A - 7*B)*Sin[c + d*x]^2 + 56*B*Sin[c + d*x]^3))/(504*d)","A",1
988,1,53,78,0.2431183,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{a^3 (\sin (c+d x)+1)^5 \left(5 (7 A-9 B) \sin (c+d x)-49 A+30 B \sin ^2(c+d x)+9 B\right)}{210 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}",1,"-1/210*(a^3*(1 + Sin[c + d*x])^5*(-49*A + 9*B + 5*(7*A - 9*B)*Sin[c + d*x] + 30*B*Sin[c + d*x]^2))/d","A",1
989,1,36,51,0.087839,"\int \cos (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^3 (\sin (c+d x)+1)^4 (5 A+4 B \sin (c+d x)-B)}{20 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^3*(1 + Sin[c + d*x])^4*(5*A - B + 4*B*Sin[c + d*x]))/(20*d)","A",1
990,1,68,81,0.1300963,"\int \sec (c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{a^3 \left(3 (A+3 B) \sin ^2(c+d x)+6 (3 A+4 B) \sin (c+d x)+24 (A+B) \log (1-\sin (c+d x))+2 B \sin ^3(c+d x)\right)}{6 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,"-1/6*(a^3*(24*(A + B)*Log[1 - Sin[c + d*x]] + 6*(3*A + 4*B)*Sin[c + d*x] + 3*(A + 3*B)*Sin[c + d*x]^2 + 2*B*Sin[c + d*x]^3))/d","A",1
991,1,48,62,0.1193244,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^3 \left(-\frac{2 (A+B)}{\sin (c+d x)-1}+(A+3 B) \log (1-\sin (c+d x))+B \sin (c+d x)\right)}{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]] - (2*(A + B))/(-1 + Sin[c + d*x]) + B*Sin[c + d*x]))/d","A",1
992,1,37,43,0.047226,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^3 (A+B \sin (c+d x))^2}{2 d (A+B) (\sin (c+d x)-1)^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 + B*Sin[c + d*x])^2)/(2*(A + B)*d*(-1 + Sin[c + d*x])^2)","A",1
993,1,95,105,0.2560405,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^3 \left(-3 (A-B) \sin ^2(c+d x)+9 (A-B) \sin (c+d x)-3 (A-B) \tanh ^{-1}(\sin (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^6+2 (B-5 A)\right)}{24 d (\sin (c+d x)-1)^3}","\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*(2*(-5*A + B) - 3*(A - B)*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^6 + 9*(A - B)*Sin[c + d*x] - 3*(A - B)*Sin[c + d*x]^2))/(24*d*(-1 + Sin[c + d*x])^3)","A",1
994,1,151,162,0.6604193,"\int \sec ^9(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^9*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^9 \left(\frac{(5 A-3 B) \tanh ^{-1}(\sin (c+d x))}{32 a^6}+\frac{2 A-B}{16 a^5 (a-a \sin (c+d x))}-\frac{A-B}{32 a^5 (a \sin (c+d x)+a)}+\frac{3 A-B}{32 a^4 (a-a \sin (c+d x))^2}+\frac{A}{12 a^3 (a-a \sin (c+d x))^3}+\frac{A+B}{16 a^2 (a-a \sin (c+d x))^4}\right)}{d}","\frac{a^7 (A+B)}{16 d (a-a \sin (c+d x))^4}+\frac{a^6 A}{12 d (a-a \sin (c+d x))^3}+\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}",1,"(a^9*(((5*A - 3*B)*ArcTanh[Sin[c + d*x]])/(32*a^6) + (A + B)/(16*a^2*(a - a*Sin[c + d*x])^4) + A/(12*a^3*(a - a*Sin[c + d*x])^3) + (3*A - B)/(32*a^4*(a - a*Sin[c + d*x])^2) + (2*A - B)/(16*a^5*(a - a*Sin[c + d*x])) - (A - B)/(32*a^5*(a + a*Sin[c + d*x]))))/d","A",1
995,1,344,231,6.0529441,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{32 \sqrt{2} a^2 (10 a A+3 a B) \left(\frac{1}{2} (\sin (c+d x)-1)+1\right)^{13/2} \left(\frac{385 \left(\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \sqrt{1-\sin (c+d x)}}{\sqrt{\frac{1}{2} (\sin (c+d x)-1)+1}}-\frac{2}{15} (1-\sin (c+d x))^3-\frac{1}{3} (1-\sin (c+d x))^2+\sin (c+d x)-1\right)}{8192 \left(\frac{1}{2} (\sin (c+d x)-1)+1\right)^6 (1-\sin (c+d x))^4}+\frac{7}{18} \left(\frac{1}{\frac{1}{2} (\sin (c+d x)-1)+1}+\frac{11}{16 \left(\frac{1}{2} (\sin (c+d x)-1)+1\right)^2}+\frac{99}{224 \left(\frac{1}{2} (\sin (c+d x)-1)+1\right)^3}+\frac{33}{128 \left(\frac{1}{2} (\sin (c+d x)-1)+1\right)^4}+\frac{33}{256 \left(\frac{1}{2} (\sin (c+d x)-1)+1\right)^5}+\frac{99}{2048 \left(\frac{1}{2} (\sin (c+d x)-1)+1\right)^6}\right)\right) \cos ^7(c+d x)}{35 d (\sin (c+d x)+1)^{7/2}}-\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,"-1/10*(B*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3)/d - (32*Sqrt[2]*a^2*(10*a*A + 3*a*B)*Cos[c + d*x]^7*(1 + (-1 + Sin[c + d*x])/2)^(13/2)*((7*(99/(2048*(1 + (-1 + Sin[c + d*x])/2)^6) + 33/(256*(1 + (-1 + Sin[c + d*x])/2)^5) + 33/(128*(1 + (-1 + Sin[c + d*x])/2)^4) + 99/(224*(1 + (-1 + Sin[c + d*x])/2)^3) + 11/(16*(1 + (-1 + Sin[c + d*x])/2)^2) + (1 + (-1 + Sin[c + d*x])/2)^(-1)))/18 + (385*(-1 + (Sqrt[2]*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]])/Sqrt[1 + (-1 + Sin[c + d*x])/2] - (1 - Sin[c + d*x])^2/3 - (2*(1 - Sin[c + d*x])^3)/15 + Sin[c + d*x]))/(8192*(1 + (-1 + Sin[c + d*x])/2)^6*(1 - Sin[c + d*x])^4)))/(35*d*(1 + Sin[c + d*x])^(7/2))","A",1
996,1,183,200,2.2478258,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{a^3 \cos (c+d x) \left(16 (373 A+223 B) \cos (2 (c+d x))+32 (41 A+11 B) \cos (4 (c+d x))+\frac{2520 (8 A+3 B) \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)}{\sqrt{\cos ^2(c+d x)}}-10640 A \sin (c+d x)+560 A \sin (5 (c+d x))-80 A \cos (6 (c+d x))+4576 A-3045 B \sin (c+d x)+1365 B \sin (3 (c+d x))+595 B \sin (5 (c+d x))-35 B \sin (7 (c+d x))-240 B \cos (6 (c+d x))+2976 B\right)}{17920 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,"-1/17920*(a^3*Cos[c + d*x]*(4576*A + 2976*B + (2520*(8*A + 3*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/Sqrt[Cos[c + d*x]^2] + 16*(373*A + 223*B)*Cos[2*(c + d*x)] + 32*(41*A + 11*B)*Cos[4*(c + d*x)] - 80*A*Cos[6*(c + d*x)] - 240*B*Cos[6*(c + d*x)] - 10640*A*Sin[c + d*x] - 3045*B*Sin[c + d*x] + 1365*B*Sin[3*(c + d*x)] + 560*A*Sin[5*(c + d*x)] + 595*B*Sin[5*(c + d*x)] - 35*B*Sin[7*(c + d*x)]))/d","A",1
997,1,146,159,1.4192155,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{a^3 \cos (c+d x) \left(16 (17 A+11 B) \cos (2 (c+d x))-12 (A+3 B) \cos (4 (c+d x))+\frac{420 (2 A+B) \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)}{\sqrt{\cos ^2(c+d x)}}-330 A \sin (c+d x)+90 A \sin (3 (c+d x))+284 A-95 B \sin (c+d x)+110 B \sin (3 (c+d x))-5 B \sin (5 (c+d x))+212 B\right)}{480 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,"-1/480*(a^3*Cos[c + d*x]*(284*A + 212*B + (420*(2*A + B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/Sqrt[Cos[c + d*x]^2] + 16*(17*A + 11*B)*Cos[2*(c + d*x)] - 12*(A + 3*B)*Cos[4*(c + d*x)] - 330*A*Sin[c + d*x] - 95*B*Sin[c + d*x] + 90*A*Sin[3*(c + d*x)] + 110*B*Sin[3*(c + d*x)] - 5*B*Sin[5*(c + d*x)]))/d","A",1
998,1,82,91,0.2575199,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{\sec (c+d x) \left(4 \sqrt{2} a^3 (2 A+3 B) \sqrt{\sin (c+d x)+1} \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right)-B (a \sin (c+d x)+a)^3\right)}{2 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,"(Sec[c + d*x]*(4*Sqrt[2]*a^3*(2*A + 3*B)*Hypergeometric2F1[-3/2, -1/2, 1/2, (1 - Sin[c + d*x])/2]*Sqrt[1 + Sin[c + d*x]] - B*(a + a*Sin[c + d*x])^3))/(2*d)","C",1
999,1,121,69,1.1109325,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{a^3 \left(-3 \cos \left(\frac{1}{2} (c+d x)\right) (2 A+3 B (c+d x+2))+\cos \left(\frac{3}{2} (c+d x)\right) (2 A+B (3 c+3 d x+14))+6 B \sin \left(\frac{1}{2} (c+d x)\right) (2 (c+d x+2)+(c+d x) \cos (c+d x))\right)}{6 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","a^3 B x-\frac{2 a^5 B \cos (c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)}+\frac{(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}",1,"-1/6*(a^3*(-3*(2*A + 3*B*(2 + c + d*x))*Cos[(c + d*x)/2] + (2*A + B*(14 + 3*c + 3*d*x))*Cos[(3*(c + d*x))/2] + 6*B*(2*(2 + c + d*x) + (c + d*x)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)","A",1
1000,1,94,107,0.178464,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{a^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (6 (2 A-3 B) \sin (c+d x)+(2 A-3 B) \cos (2 (c+d x))-16 A+9 B)}{30 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}","\frac{a^5 (2 A-3 B) \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\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+B) \sec ^5(c+d x) (a \sin (c+d x)+a)^3}{5 d}",1,"-1/30*(a^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-16*A + 9*B + (2*A - 3*B)*Cos[2*(c + d*x)] + 6*(2*A - 3*B)*Sin[c + d*x]))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5)","A",1
1001,1,135,115,0.4948894,"\int \sec ^8(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^8*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","\frac{a^3 (14 (4 A-3 B) \cos (2 (c+d x))+(3 B-4 A) \cos (4 (c+d x))+56 A \sin (c+d x)-24 A \sin (3 (c+d x))-42 B \sin (c+d x)+18 B \sin (3 (c+d x))+35 B)}{140 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^7 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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^3*(35*B + 14*(4*A - 3*B)*Cos[2*(c + d*x)] + (-4*A + 3*B)*Cos[4*(c + d*x)] + 56*A*Sin[c + d*x] - 42*B*Sin[c + d*x] - 24*A*Sin[3*(c + d*x)] + 18*B*Sin[3*(c + d*x)]))/(140*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
1002,1,176,140,0.6068798,"\int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^10*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]","-\frac{a^3 (27 (B-2 A) \cos (2 (c+d x))+12 (B-2 A) \cos (4 (c+d x))-72 A \sin (c+d x)-4 A \sin (3 (c+d x))+12 A \sin (5 (c+d x))+2 A \cos (6 (c+d x))+36 B \sin (c+d x)+2 B \sin (3 (c+d x))-6 B \sin (5 (c+d x))+B (-\cos (6 (c+d x)))-42 B)}{252 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^9 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","\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,"-1/252*(a^3*(-42*B + 27*(-2*A + B)*Cos[2*(c + d*x)] + 12*(-2*A + B)*Cos[4*(c + d*x)] + 2*A*Cos[6*(c + d*x)] - B*Cos[6*(c + d*x)] - 72*A*Sin[c + d*x] + 36*B*Sin[c + d*x] - 4*A*Sin[3*(c + d*x)] + 2*B*Sin[3*(c + d*x)] + 12*A*Sin[5*(c + d*x)] - 6*B*Sin[5*(c + d*x)]))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^9*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","A",1
1003,1,69,105,0.2265103,"\int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","-\frac{(\sin (c+d x)-1)^4 \left(5 (7 A+17 B) \sin ^2(c+d x)+(98 A+76 B) \sin (c+d x)+77 A+30 B \sin ^3(c+d x)+19 B\right)}{210 a 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}",1,"-1/210*((-1 + Sin[c + d*x])^4*(77*A + 19*B + (98*A + 76*B)*Sin[c + d*x] + 5*(7*A + 17*B)*Sin[c + d*x]^2 + 30*B*Sin[c + d*x]^3))/(a*d)","A",1
1004,1,72,79,0.1504358,"\int \frac{\cos ^5(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","\frac{\sin (c+d x) \left(15 (A-B) \sin ^3(c+d x)-20 (A+B) \sin ^2(c+d x)-30 (A-B) \sin (c+d x)+60 A+12 B \sin ^4(c+d x)\right)}{60 a 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}",1,"(Sin[c + d*x]*(60*A - 30*(A - B)*Sin[c + d*x] - 20*(A + B)*Sin[c + d*x]^2 + 15*(A - B)*Sin[c + d*x]^3 + 12*B*Sin[c + d*x]^4))/(60*a*d)","A",1
1005,1,44,57,0.0999319,"\int \frac{\cos ^3(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","\frac{\sin (c+d x) \left(-3 (A-B) \sin (c+d x)+6 A-2 B \sin ^2(c+d x)\right)}{6 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,"(Sin[c + d*x]*(6*A - 3*(A - B)*Sin[c + d*x] - 2*B*Sin[c + d*x]^2))/(6*a*d)","A",1
1006,1,31,36,0.0359039,"\int \frac{\cos (c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Integrate[(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)+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]] + B*Sin[c + d*x])/(a*d)","A",1
1007,1,44,45,0.0644884,"\int \frac{\sec (c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","\frac{(A+B) (\sin (c+d x)+1) \tanh ^{-1}(\sin (c+d x))-A+B}{2 a d (\sin (c+d x)+1)}","\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 + (A + B)*ArcTanh[Sin[c + d*x]]*(1 + Sin[c + d*x]))/(2*a*d*(1 + Sin[c + d*x]))","A",1
1008,1,75,91,0.2872817,"\int \frac{\sec ^3(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","\frac{\frac{A+B}{a-a \sin (c+d x)}+\frac{B-A}{a (\sin (c+d x)+1)^2}+\frac{(3 A+B) \tanh ^{-1}(\sin (c+d x))}{a}-\frac{2 A}{a \sin (c+d x)+a}}{8 d}","\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]])/a + (-A + B)/(a*(1 + Sin[c + d*x])^2) + (A + B)/(a - a*Sin[c + d*x]) - (2*A)/(a + a*Sin[c + d*x]))/(8*d)","A",1
1009,1,105,146,0.5659569,"\int \frac{\sec ^5(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","\frac{-\frac{6 (2 A+B)}{\sin (c+d x)-1}+\frac{3 (A+B)}{(\sin (c+d x)-1)^2}+\frac{3 B-9 A}{(\sin (c+d x)+1)^2}-\frac{4 (A-B)}{(\sin (c+d x)+1)^3}+6 (5 A+B) \tanh ^{-1}(\sin (c+d x))-\frac{18 A}{\sin (c+d x)+1}}{96 a d}","-\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,"(6*(5*A + B)*ArcTanh[Sin[c + d*x]] + (3*(A + B))/(-1 + Sin[c + d*x])^2 - (6*(2*A + B))/(-1 + Sin[c + d*x]) - (4*(A - B))/(1 + Sin[c + d*x])^3 + (-9*A + 3*B)/(1 + Sin[c + d*x])^2 - (18*A)/(1 + Sin[c + d*x]))/(96*a*d)","A",1
1010,1,142,205,0.8835321,"\int \frac{\sec ^7(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]),x]","\frac{\frac{-15 (7 A+B) \sin ^6(c+d x)-15 (7 A+B) \sin ^5(c+d x)+40 (7 A+B) \sin ^4(c+d x)+40 (7 A+B) \sin ^3(c+d x)-33 (7 A+B) \sin ^2(c+d x)-33 (7 A+B) \sin (c+d x)+48 (A-B)}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+15 (7 A+B) \tanh ^{-1}(\sin (c+d x))}{384 a d}","-\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,"(15*(7*A + B)*ArcTanh[Sin[c + d*x]] + (48*(A - B) - 33*(7*A + B)*Sin[c + d*x] - 33*(7*A + B)*Sin[c + d*x]^2 + 40*(7*A + B)*Sin[c + d*x]^3 + 40*(7*A + B)*Sin[c + d*x]^4 - 15*(7*A + B)*Sin[c + d*x]^5 - 15*(7*A + B)*Sin[c + d*x]^6)/((-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^4))/(384*a*d)","A",1
1011,1,52,79,0.1658418,"\int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","-\frac{(\sin (c+d x)-1)^4 \left((6 A+8 B) \sin (c+d x)+9 A+5 B \sin ^2(c+d x)+2 B\right)}{30 a^2 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}",1,"-1/30*((-1 + Sin[c + d*x])^4*(9*A + 2*B + (6*A + 8*B)*Sin[c + d*x] + 5*B*Sin[c + d*x]^2))/(a^2*d)","A",1
1012,1,34,51,0.0565901,"\int \frac{\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","\frac{(\sin (c+d x)-1)^3 (4 A+3 B \sin (c+d x)+B)}{12 a^2 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,"((-1 + Sin[c + d*x])^3*(4*A + B + 3*B*Sin[c + d*x]))/(12*a^2*d)","A",1
1013,1,51,66,0.096682,"\int \frac{\cos ^3(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","-\frac{2 (A-2 B) \sin (c+d x)-4 (A-B) \log (\sin (c+d x)+1)+B \sin ^2(c+d x)+B}{2 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}",1,"-1/2*(B - 4*(A - B)*Log[1 + Sin[c + d*x]] + 2*(A - 2*B)*Sin[c + d*x] + B*Sin[c + d*x]^2)/(a^2*d)","A",1
1014,1,41,44,0.0730386,"\int \frac{\cos (c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","\frac{\frac{B \log (\sin (c+d x)+1)}{a}-\frac{A-B}{a \sin (c+d x)+a}}{a d}","\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 - (A - B)/(a + a*Sin[c + d*x]))/(a*d)","A",1
1015,1,69,71,0.1239158,"\int \frac{\sec (c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","\frac{a \left(\frac{(A+B) \tanh ^{-1}(\sin (c+d x))}{4 a^3}-\frac{A+B}{4 a^2 (a \sin (c+d x)+a)}-\frac{A-B}{4 a (a \sin (c+d x)+a)^2}\right)}{d}","-\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*(((A + B)*ArcTanh[Sin[c + d*x]])/(4*a^3) - (A - B)/(4*a*(a + a*Sin[c + d*x])^2) - (A + B)/(4*a^2*(a + a*Sin[c + d*x]))))/d","A",1
1016,1,87,123,0.7412934,"\int \frac{\sec ^3(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","-\frac{\frac{3 (A+B)}{\sin (c+d x)-1}+\frac{3 (3 A+B)}{\sin (c+d x)+1}+\frac{4 (A-B)}{(\sin (c+d x)+1)^3}-6 (2 A+B) \tanh ^{-1}(\sin (c+d x))+\frac{6 A}{(\sin (c+d x)+1)^2}}{48 a^2 d}","\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,"-1/48*(-6*(2*A + B)*ArcTanh[Sin[c + d*x]] + (3*(A + B))/(-1 + Sin[c + d*x]) + (4*(A - B))/(1 + Sin[c + d*x])^3 + (6*A)/(1 + Sin[c + d*x])^2 + (3*(3*A + B))/(1 + Sin[c + d*x]))/(a^2*d)","A",1
1017,1,123,179,0.6916113,"\int \frac{\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","\frac{-\frac{3 (5 A+3 B)}{\sin (c+d x)-1}-\frac{6 (5 A+B)}{\sin (c+d x)+1}+\frac{3 (A+B)}{(\sin (c+d x)-1)^2}+\frac{4 (B-3 A)}{(\sin (c+d x)+1)^3}-\frac{6 (A-B)}{(\sin (c+d x)+1)^4}+15 (3 A+B) \tanh ^{-1}(\sin (c+d x))-\frac{18 A}{(\sin (c+d x)+1)^2}}{192 a^2 d}","-\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,"(15*(3*A + B)*ArcTanh[Sin[c + d*x]] + (3*(A + B))/(-1 + Sin[c + d*x])^2 - (3*(5*A + 3*B))/(-1 + Sin[c + d*x]) - (6*(A - B))/(1 + Sin[c + d*x])^4 + (4*(-3*A + B))/(1 + Sin[c + d*x])^3 - (18*A)/(1 + Sin[c + d*x])^2 - (6*(5*A + B))/(1 + Sin[c + d*x]))/(192*a^2*d)","A",1
1018,1,160,236,1.4810106,"\int \frac{\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2,x]","\frac{210 (4 A+B) \tanh ^{-1}(\sin (c+d x))-\frac{2 \left(105 (4 A+B) \sin ^7(c+d x)+210 (4 A+B) \sin ^6(c+d x)-175 (4 A+B) \sin ^5(c+d x)-560 (4 A+B) \sin ^4(c+d x)-49 (4 A+B) \sin ^3(c+d x)+462 (4 A+B) \sin ^2(c+d x)+183 (4 A+B) \sin (c+d x)+48 (3 B-8 A)\right)}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^5}}{3840 a^2 d}","-\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,"(210*(4*A + B)*ArcTanh[Sin[c + d*x]] - (2*(48*(-8*A + 3*B) + 183*(4*A + B)*Sin[c + d*x] + 462*(4*A + B)*Sin[c + d*x]^2 - 49*(4*A + B)*Sin[c + d*x]^3 - 560*(4*A + B)*Sin[c + d*x]^4 - 175*(4*A + B)*Sin[c + d*x]^5 + 210*(4*A + B)*Sin[c + d*x]^6 + 105*(4*A + B)*Sin[c + d*x]^7))/((-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^5))/(3840*a^2*d)","A",1
1019,1,154,170,0.4649018,"\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{\cos (e+f x) (a (\sin (e+f x)+1))^m (g \cos (e+f x))^p (\sin (e+f x)+1)^{\frac{1}{2} (-2 m-p-1)} \left(2^{\frac{1}{2} (2 m+p+1)} (A (m+p+1)+B m) \, _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)+B (p+1) (\sin (e+f x)+1)^{\frac{1}{2} (2 m+p+1)}\right)}{f (p+1) (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,"-((Cos[e + f*x]*(g*Cos[e + f*x])^p*(1 + Sin[e + f*x])^((-1 - 2*m - p)/2)*(a*(1 + Sin[e + f*x]))^m*(2^((1 + 2*m + p)/2)*(B*m + A*(1 + m + p))*Hypergeometric2F1[(1 - 2*m - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[e + f*x])/2] + B*(1 + p)*(1 + Sin[e + f*x])^((1 + 2*m + p)/2)))/(f*(1 + p)*(1 + m + p)))","A",1
1020,1,132,159,0.7964769,"\int \cos ^7(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Cos[e + f*x]^7*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{(a (\sin (e+f x)+1))^{m+4} \left(-\frac{a^4 (A-7 B) (\sin (e+f x)+1)^3}{m+7}+\frac{6 a^4 (A-3 B) (\sin (e+f x)+1)^2}{m+6}-\frac{4 a^4 (3 A-5 B) (\sin (e+f x)+1)}{m+5}+\frac{8 a^4 (A-B)}{m+4}-\frac{B (a \sin (e+f x)+a)^4}{m+8}\right)}{a^8 f}","-\frac{B (a \sin (e+f x)+a)^{m+8}}{a^8 f (m+8)}-\frac{(A-7 B) (a \sin (e+f x)+a)^{m+7}}{a^7 f (m+7)}+\frac{6 (A-3 B) (a \sin (e+f x)+a)^{m+6}}{a^6 f (m+6)}-\frac{4 (3 A-5 B) (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}+\frac{8 (A-B) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}",1,"((a*(1 + Sin[e + f*x]))^(4 + m)*((8*a^4*(A - B))/(4 + m) - (4*a^4*(3*A - 5*B)*(1 + Sin[e + f*x]))/(5 + m) + (6*a^4*(A - 3*B)*(1 + Sin[e + f*x])^2)/(6 + m) - (a^4*(A - 7*B)*(1 + Sin[e + f*x])^3)/(7 + m) - (B*(a + a*Sin[e + f*x])^4)/(8 + m)))/(a^8*f)","A",1
1021,1,103,123,0.4009105,"\int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Cos[e + f*x]^5*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{(a (\sin (e+f x)+1))^{m+3} \left(\frac{a^3 (A-5 B) (\sin (e+f x)+1)^2}{m+5}-\frac{4 a^3 (A-2 B) (\sin (e+f x)+1)}{m+4}+\frac{4 a^3 (A-B)}{m+3}+\frac{B (a \sin (e+f x)+a)^3}{m+6}\right)}{a^6 f}","\frac{B (a \sin (e+f x)+a)^{m+6}}{a^6 f (m+6)}+\frac{(A-5 B) (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}-\frac{4 (A-2 B) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac{4 (A-B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}",1,"((a*(1 + Sin[e + f*x]))^(3 + m)*((4*a^3*(A - B))/(3 + m) - (4*a^3*(A - 2*B)*(1 + Sin[e + f*x]))/(4 + m) + (a^3*(A - 5*B)*(1 + Sin[e + f*x])^2)/(5 + m) + (B*(a + a*Sin[e + f*x])^3)/(6 + m)))/(a^6*f)","A",1
1022,1,93,93,0.3138621,"\int \cos ^3(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Cos[e + f*x]^3*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{B (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}-\frac{(A-3 B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{2 (A-B) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}","-\frac{B (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}-\frac{(A-3 B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac{2 (A-B) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}",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",1
1023,1,51,59,0.1250241,"\int \cos (e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{(a (\sin (e+f x)+1))^{m+1} (A (m+2)+B (m+1) \sin (e+f x)-B)}{a f (m+1) (m+2)}","\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*(1 + Sin[e + f*x]))^(1 + m)*(-B + A*(2 + m) + B*(1 + m)*Sin[e + f*x]))/(a*f*(1 + m)*(2 + m))","A",1
1024,1,71,80,0.1181258,"\int \sec (e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Sec[e + f*x]*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{(a (\sin (e+f x)+1))^m \left(m (A+B) (\sin (e+f x)+1) \, _2F_1\left(1,m+1;m+2;\frac{1}{2} (\sin (e+f x)+1)\right)+2 (m+1) (A-B)\right)}{4 f m (m+1)}","\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*(1 + Sin[e + f*x]))^m*(2*(A - B)*(1 + m) + (A + B)*m*Hypergeometric2F1[1, 1 + m, 2 + m, (1 + Sin[e + f*x])/2]*(1 + Sin[e + f*x])))/(4*f*m*(1 + m))","A",1
1025,1,82,100,0.1715936,"\int \sec ^3(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Sec[e + f*x]^3*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{a (a (\sin (e+f x)+1))^{m-1} \left((A (m-2)+B m) (\sin (e+f x)-1) \, _2F_1\left(1,m-1;m;\frac{1}{2} (\sin (e+f x)+1)\right)+2 (m-1) (A+B)\right)}{4 f (m-1) (\sin (e+f x)-1)}","\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,"-1/4*(a*(2*(A + B)*(-1 + m) + (A*(-2 + m) + B*m)*Hypergeometric2F1[1, -1 + m, m, (1 + Sin[e + f*x])/2]*(-1 + Sin[e + f*x]))*(a*(1 + Sin[e + f*x]))^(-1 + m))/(f*(-1 + m)*(-1 + Sin[e + f*x]))","A",1
1026,1,76,104,0.1801575,"\int \sec ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Sec[e + f*x]^5*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\frac{a^2 (a (\sin (e+f x)+1))^{m-2} \left(\frac{4 (A+B)}{(\sin (e+f x)-1)^2}-\frac{(A (m-4)+B m) \, _2F_1\left(2,m-2;m-1;\frac{1}{2} (\sin (e+f x)+1)\right)}{m-2}\right)}{16 f}","\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])/(-2 + m)) + (4*(A + B))/(-1 + Sin[e + f*x])^2)*(a*(1 + Sin[e + f*x]))^(-2 + m))/(16*f)","A",1
1027,1,111,129,0.8866569,"\int \cos ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Cos[e + f*x]^6*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{\cos ^7(e+f x) (\sin (e+f x)+1)^{-m-\frac{7}{2}} (a (\sin (e+f x)+1))^m \left(2^{m+\frac{7}{2}} (A (m+7)+B m) \, _2F_1\left(\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{1}{2} (1-\sin (e+f x))\right)+7 B (\sin (e+f x)+1)^{m+\frac{7}{2}}\right)}{7 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,"-1/7*(Cos[e + f*x]^7*(1 + Sin[e + f*x])^(-7/2 - m)*(a*(1 + Sin[e + f*x]))^m*(2^(7/2 + m)*(B*m + A*(7 + m))*Hypergeometric2F1[7/2, -5/2 - m, 9/2, (1 - Sin[e + f*x])/2] + 7*B*(1 + Sin[e + f*x])^(7/2 + m)))/(f*(7 + m))","A",1
1028,1,111,129,0.4865768,"\int \cos ^4(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Cos[e + f*x]^4*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{\cos ^5(e+f x) (\sin (e+f x)+1)^{-m-\frac{5}{2}} (a (\sin (e+f x)+1))^m \left(2^{m+\frac{5}{2}} (A (m+5)+B m) \, _2F_1\left(\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right)+5 B (\sin (e+f x)+1)^{m+\frac{5}{2}}\right)}{5 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,"-1/5*(Cos[e + f*x]^5*(1 + Sin[e + f*x])^(-5/2 - m)*(a*(1 + Sin[e + f*x]))^m*(2^(5/2 + m)*(B*m + A*(5 + m))*Hypergeometric2F1[5/2, -3/2 - m, 7/2, (1 - Sin[e + f*x])/2] + 5*B*(1 + Sin[e + f*x])^(5/2 + m)))/(f*(5 + m))","A",1
1029,1,111,127,0.3299288,"\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","-\frac{\cos ^3(e+f x) (\sin (e+f x)+1)^{-m-\frac{3}{2}} (a (\sin (e+f x)+1))^m \left(2^{m+\frac{3}{2}} (A (m+3)+B m) \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{1}{2} (1-\sin (e+f x))\right)+3 B (\sin (e+f x)+1)^{m+\frac{3}{2}}\right)}{3 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,"-1/3*(Cos[e + f*x]^3*(1 + Sin[e + f*x])^(-3/2 - m)*(a*(1 + Sin[e + f*x]))^m*(2^(3/2 + m)*(B*m + A*(3 + m))*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (1 - Sin[e + f*x])/2] + 3*B*(1 + Sin[e + f*x])^(3/2 + m)))/(f*(3 + m))","A",1
1030,1,3925,123,6.5457968,"\int \sec ^2(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Sec[e + f*x]^2*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\text{Result too large to show}","\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,"-1/4*((A + B)*(Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Cot[(-e + Pi/2 - f*x)/4]*(a + a*Sin[e + f*x])^m*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)))/(f*(Cos[Pi/4 + (e - Pi/2 + f*x)/2] - Sin[Pi/4 + (e - Pi/2 + f*x)/2])^2*(-1/2*(m*(Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2))) - ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Csc[(-e + Pi/2 - f*x)/4]^2*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)))/8 + ((Cos[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Cot[(-e + Pi/2 - f*x)/4]*(-(m*AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*(Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*Tan[(-e + Pi/2 - f*x)/4]) - (Sec[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(m*AppellF1[1/2, 1 - 2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4] + m*AppellF1[1/2, -2*m, 1 + 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(2*(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)) + (3*Tan[(-e + Pi/2 - f*x)/4]^2*(-1/3*(m*AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) - (m*AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/3)*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2) - (3*m*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]^3*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(-1 + 2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2) - (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Tan[(-e + Pi/2 - f*x)/4]^2*(1 - Tan[(-e + Pi/2 - f*x)/4]^2)^(2*m)*(-2*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4] + 3*(-1/3*(m*AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4]) - (m*AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/3) - 4*m*Tan[(-e + Pi/2 - f*x)/4]^2*((-6*m*AppellF1[5/2, 1 - 2*m, 1 + 2*m, 7/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/5 + (3*(1 - 2*m)*AppellF1[5/2, 2 - 2*m, 2*m, 7/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/10 - (3*(1 + 2*m)*AppellF1[5/2, -2*m, 2 + 2*m, 7/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2]*Sec[(-e + Pi/2 - f*x)/4]^2*Tan[(-e + Pi/2 - f*x)/4])/10)))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-e + Pi/2 - f*x)/4]^2, -Tan[(-e + Pi/2 - f*x)/4]^2])*Tan[(-e + Pi/2 - f*x)/4]^2)^2))/2)) + ((A - B)*Hypergeometric2F1[1/2, (-1 + 2*m)/2, (1 + 2*m)/2, Cos[(-e + Pi/2 - f*x)/2]^2]*(a + a*Sin[e + f*x])^m*Tan[(-e + Pi/2 - f*x)/2])/(2*f*(-1 + 2*m)*Sqrt[Sin[(-e + Pi/2 - f*x)/2]^2])","C",0
1031,0,0,135,1.7664332,"\int \sec ^4(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Sec[e + f*x]^4*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\int \sec ^4(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","\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,"Integrate[Sec[e + f*x]^4*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x]","F",-1
1032,0,0,135,3.9133871,"\int \sec ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[Sec[e + f*x]^6*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]","\int \sec ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","\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,"Integrate[Sec[e + f*x]^6*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x]","F",-1
1033,1,160,239,0.5729094,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-p} \, dx","Integrate[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-4 - p),x]","\frac{\cos (e+f x) (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^p \left(\left(p^2+8 p+13\right) (B (p+4)-3 A) \sin (e+f x)+(2 B (p+4)-6 A) \sin ^3(e+f x)-2 (p+4) (B (p+4)-3 A) \sin ^2(e+f x)+A \left(p^3+12 p^2+41 p+36\right)-B \left(p^2+8 p+13\right)\right)}{c^4 f (p+1) (p+3) (p+5) (p+7) (\sin (e+f x)-1)^4}","\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{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{(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,"(Cos[e + f*x]*(g*Cos[e + f*x])^p*(-(B*(13 + 8*p + p^2)) + A*(36 + 41*p + 12*p^2 + p^3) + (13 + 8*p + p^2)*(-3*A + B*(4 + p))*Sin[e + f*x] - 2*(4 + p)*(-3*A + B*(4 + p))*Sin[e + f*x]^2 + (-6*A + 2*B*(4 + p))*Sin[e + f*x]^3))/(c^4*f*(1 + p)*(3 + p)*(5 + p)*(7 + p)*(-1 + Sin[e + f*x])^4*(c - c*Sin[e + f*x])^p)","A",1
1034,1,119,168,0.296201,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-3-p} \, dx","Integrate[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-3 - p),x]","-\frac{\cos (e+f x) (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^p \left((2 A-B (p+3)) \sin ^2(e+f x)+(p+3) (B (p+3)-2 A) \sin (e+f x)+A \left(p^2+6 p+7\right)-B (p+3)\right)}{c^3 f (p+1) (p+3) (p+5) (\sin (e+f x)-1)^3}","\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,"-((Cos[e + f*x]*(g*Cos[e + f*x])^p*(-(B*(3 + p)) + A*(7 + 6*p + p^2) + (3 + p)*(-2*A + B*(3 + p))*Sin[e + f*x] + (2*A - B*(3 + p))*Sin[e + f*x]^2))/(c^3*f*(1 + p)*(3 + p)*(5 + p)*(-1 + Sin[e + f*x])^3*(c - c*Sin[e + f*x])^p))","A",1
1035,1,83,102,0.178693,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-2-p} \, dx","Integrate[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-2 - p),x]","\frac{\cos (e+f x) (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^p ((B (p+2)-A) \sin (e+f x)+A (p+2)-B)}{c^2 f (p+1) (p+3) (\sin (e+f x)-1)^2}","\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,"(Cos[e + f*x]*(g*Cos[e + f*x])^p*(-B + A*(2 + p) + (-A + B*(2 + p))*Sin[e + f*x]))/(c^2*f*(1 + p)*(3 + p)*(-1 + Sin[e + f*x])^2*(c - c*Sin[e + f*x])^p)","A",1
1036,1,300,151,3.7505124,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-1-p} \, dx","Integrate[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-1 - p),x]","-\frac{2^{-p} (c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^p \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^{-2 (-p-1)-2 p} \left(\frac{1-\tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\frac{1}{\cos (e+f x)+1}}}\right)^{2 p} \left(\frac{1-\tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)}}\right)^{-2 p} \left(p (A+B) \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)-i B (p+1) \left(\tan \left(\frac{1}{2} (e+f x)\right)-1\right) \, _2F_1\left(1,-p;1-p;-\frac{i \left(\tan \left(\frac{1}{2} (e+f x)\right)-1\right)}{\tan \left(\frac{1}{2} (e+f x)\right)+1}\right)+i B (p+1) \left(\tan \left(\frac{1}{2} (e+f x)\right)-1\right) \, _2F_1\left(1,-p;1-p;\frac{i \left(\tan \left(\frac{1}{2} (e+f x)\right)-1\right)}{\tan \left(\frac{1}{2} (e+f x)\right)+1}\right)\right)}{f p (p+1) \left(\tan \left(\frac{1}{2} (e+f x)\right)-1\right)}","\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,"-(((g*Cos[e + f*x])^p*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^(-2*(-1 - p) - 2*p)*(c - c*Sin[e + f*x])^(-1 - p)*((1 - Tan[(e + f*x)/2])/Sqrt[(1 + Cos[e + f*x])^(-1)])^(2*p)*((-I)*B*(1 + p)*Hypergeometric2F1[1, -p, 1 - p, ((-I)*(-1 + Tan[(e + f*x)/2]))/(1 + Tan[(e + f*x)/2])]*(-1 + Tan[(e + f*x)/2]) + I*B*(1 + p)*Hypergeometric2F1[1, -p, 1 - p, (I*(-1 + Tan[(e + f*x)/2]))/(1 + Tan[(e + f*x)/2])]*(-1 + Tan[(e + f*x)/2]) + (A + B)*p*(1 + Tan[(e + f*x)/2])))/(2^p*f*p*(1 + p)*((1 - Tan[(e + f*x)/2])/Sqrt[Sec[(e + f*x)/2]^2])^(2*p)*(-1 + Tan[(e + f*x)/2])))","C",1
1037,1,144,147,0.5594277,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-p} \, dx","Integrate[((g*Cos[e + f*x])^p*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^p,x]","-\frac{2^{\frac{1}{2} (-p-1)} \cos (e+f x) (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^p \left(2 (A+B p) (1-\sin (e+f x))^{\frac{p+1}{2}} \, _2F_1\left(\frac{p+1}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)+B 2^{\frac{p+1}{2}} (p+1) (\sin (e+f x)-1)\right)}{f (p+1) (\sin (e+f x)-1)}","\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 - p)/2)*Cos[e + f*x]*(g*Cos[e + f*x])^p*(2*(A + B*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) + 2^((1 + p)/2)*B*(1 + p)*(-1 + Sin[e + f*x])))/(f*(1 + p)*(-1 + Sin[e + f*x])*(c - c*Sin[e + f*x])^p))","A",1
1038,1,150,160,0.6589388,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{1-p} \, dx","Integrate[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1 - p),x]","\frac{c 2^{\frac{1}{2} (-p-3)} \cos (e+f x) (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^p \left(B 2^{\frac{p+1}{2}} (p+1) (\sin (e+f x)-1)^2-4 (2 A+B (p-1)) (1-\sin (e+f x))^{\frac{p+1}{2}} \, _2F_1\left(\frac{p-1}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)\right)}{f (p+1) (\sin (e+f x)-1)}","\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^((-3 - p)/2)*c*Cos[e + f*x]*(g*Cos[e + f*x])^p*(-4*(2*A + B*(-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) + 2^((1 + p)/2)*B*(1 + p)*(-1 + Sin[e + f*x])^2))/(f*(1 + p)*(-1 + Sin[e + f*x])*(c - c*Sin[e + f*x])^p)","A",1
1039,1,155,163,0.9070793,"\int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{2-p} \, dx","Integrate[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(2 - p),x]","-\frac{c^2 2^{\frac{1}{2} (-p-1)} \cos (e+f x) (c-c \sin (e+f x))^{-p} (g \cos (e+f x))^p \left(8 (3 A+B (p-2)) (1-\sin (e+f x))^{\frac{p+1}{2}} \, _2F_1\left(\frac{p-3}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (\sin (e+f x)+1)\right)+B 2^{\frac{p+1}{2}} (p+1) (\sin (e+f x)-1)^3\right)}{3 f (p+1) (\sin (e+f x)-1)}","\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,"-1/3*(2^((-1 - p)/2)*c^2*Cos[e + f*x]*(g*Cos[e + f*x])^p*(8*(3*A + B*(-2 + 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) + 2^((1 + p)/2)*B*(1 + p)*(-1 + Sin[e + f*x])^3))/(f*(1 + p)*(-1 + Sin[e + f*x])*(c - c*Sin[e + f*x])^p)","A",1
1040,1,33,32,0.168175,"\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","Integrate[(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 \cos (e+f x) (a (\sin (e+f x)+1))^m (g \cos (e+f x))^p}{f}","\frac{A (a \sin (e+f x)+a)^m (g \cos (e+f x))^{p+1}}{f g}",1,"(A*Cos[e + f*x]*(g*Cos[e + f*x])^p*(a*(1 + Sin[e + f*x]))^m)/f","A",1
1041,1,35,34,0.0652907,"\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","Integrate[(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 \cos (e+f x) (a-a \sin (e+f x))^m (g \cos (e+f x))^p}{f}","-\frac{A (a-a \sin (e+f x))^m (g \cos (e+f x))^{p+1}}{f g}",1,"-((A*Cos[e + f*x]*(g*Cos[e + f*x])^p*(a - a*Sin[e + f*x])^m)/f)","A",1
1042,1,798,168,10.3971615,"\int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","-\frac{2 F_1\left(\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right),-\frac{(c-d) \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right) (g \cos (e+f x))^p \cos \left(\frac{1}{4} (2 e+2 f x+\pi )\right) (a (\sin (e+f x)+1))^m (c+d \sin (e+f x))^n \sin \left(\frac{1}{4} (2 e+2 f x+\pi )\right)}{f \left(-\frac{d n F_1\left(\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right),-\frac{(c-d) \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right) \cos ^2(e+f x)}{c+d \sin (e+f x)}+\frac{2 (p+1) \left((c-d) n F_1\left(\frac{p+3}{2};m+n+p+1,1-n;\frac{p+5}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right),-\frac{(c-d) \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)-(c+d) (m+n+p+1) F_1\left(\frac{p+3}{2};m+n+p+2,-n;\frac{p+5}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right),-\frac{(c-d) \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)\right) \cot ^2\left(\frac{1}{4} (2 e+2 f x+\pi )\right)}{(c+d) (p+3)}+2 (n+p) F_1\left(\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right),-\frac{(c-d) \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right) \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)-\frac{2 (c-d) n F_1\left(\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right),-\frac{(c-d) \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right) \sin ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d \sin (e+f x)}+F_1\left(\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right),-\frac{(c-d) \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right)+p F_1\left(\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right),-\frac{(c-d) \tan ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{c+d}\right) \sin (e+f x)\right)}","\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*AppellF1[(1 + p)/2, 1 + m + n + p, -n, (3 + p)/2, -Tan[(2*e - Pi + 2*f*x)/4]^2, -(((c - d)*Tan[(2*e - Pi + 2*f*x)/4]^2)/(c + d))]*(g*Cos[e + f*x])^p*Cos[(2*e + Pi + 2*f*x)/4]*(a*(1 + Sin[e + f*x]))^m*(c + d*Sin[e + f*x])^n*Sin[(2*e + Pi + 2*f*x)/4])/(f*(AppellF1[(1 + p)/2, 1 + m + n + p, -n, (3 + p)/2, -Tan[(2*e - Pi + 2*f*x)/4]^2, -(((c - d)*Tan[(2*e - Pi + 2*f*x)/4]^2)/(c + d))] + (2*(1 + p)*((c - d)*n*AppellF1[(3 + p)/2, 1 + m + n + p, 1 - n, (5 + p)/2, -Tan[(2*e - Pi + 2*f*x)/4]^2, -(((c - d)*Tan[(2*e - Pi + 2*f*x)/4]^2)/(c + d))] - (c + d)*(1 + m + n + p)*AppellF1[(3 + p)/2, 2 + m + n + p, -n, (5 + p)/2, -Tan[(2*e - Pi + 2*f*x)/4]^2, -(((c - d)*Tan[(2*e - Pi + 2*f*x)/4]^2)/(c + d))])*Cot[(2*e + Pi + 2*f*x)/4]^2)/((c + d)*(3 + p)) + p*AppellF1[(1 + p)/2, 1 + m + n + p, -n, (3 + p)/2, -Tan[(2*e - Pi + 2*f*x)/4]^2, -(((c - d)*Tan[(2*e - Pi + 2*f*x)/4]^2)/(c + d))]*Sin[e + f*x] - (d*n*AppellF1[(1 + p)/2, 1 + m + n + p, -n, (3 + p)/2, -Tan[(2*e - Pi + 2*f*x)/4]^2, -(((c - d)*Tan[(2*e - Pi + 2*f*x)/4]^2)/(c + d))]*Cos[e + f*x]^2)/(c + d*Sin[e + f*x]) + 2*(n + p)*AppellF1[(1 + p)/2, 1 + m + n + p, -n, (3 + p)/2, -Tan[(2*e - Pi + 2*f*x)/4]^2, -(((c - d)*Tan[(2*e - Pi + 2*f*x)/4]^2)/(c + d))]*Sin[(2*e - Pi + 2*f*x)/4]^2 - (2*(c - d)*n*AppellF1[(1 + p)/2, 1 + m + n + p, -n, (3 + p)/2, -Tan[(2*e - Pi + 2*f*x)/4]^2, -(((c - d)*Tan[(2*e - Pi + 2*f*x)/4]^2)/(c + d))]*Sin[(2*e - Pi + 2*f*x)/4]^2)/(c + d*Sin[e + f*x])))","B",0
1043,0,0,149,22.607155,"\int (g \cos (e+f x))^p (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","\int (g \cos (e+f x))^p (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","-\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,"Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x]","F",-1
1044,0,0,145,4.2809274,"\int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","-\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,"Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x]","F",-1
1045,0,0,149,7.70133,"\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]),x]","\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx","-\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,"Integrate[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x]), x]","F",-1
1046,0,0,149,12.0559161,"\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","Integrate[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2,x]","\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx","-\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,"Integrate[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^2, x]","F",-1
1047,0,0,149,15.5694657,"\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","Integrate[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^3,x]","\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx","-\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,"Integrate[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^3, x]","F",-1
1048,0,0,149,20.9024514,"\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^4} \, dx","Integrate[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^4,x]","\int \frac{(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^4} \, dx","-\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,"Integrate[((g*Cos[e + f*x])^p*(c + d*Sin[e + f*x])^n)/(a + a*Sin[e + f*x])^4, x]","F",-1
1049,1,893,175,11.1299882,"\int (g \sec (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Integrate[(g*Sec[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n,x]","\frac{2 F_1\left(\frac{1-p}{2};m+n-p+1,-n;\frac{3-p}{2};-\tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{(d-c) \tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (g \sec (e+f x))^p \sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (\sin (e+f x) a+a)^m (c+d \sin (e+f x))^n}{f \left(\frac{d n F_1\left(\frac{1-p}{2};m+n-p+1,-n;\frac{3-p}{2};-\tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),-\frac{(c-d) \tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \cos ^2(e+f x)}{c+d \sin (e+f x)}-2 (n-p) F_1\left(\frac{1-p}{2};m+n-p+1,-n;\frac{3-p}{2};-\tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{(d-c) \tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)+\frac{2 (1-p) \left((m+n-p+1) F_1\left(\frac{3-p}{2};m+n-p+2,-n;\frac{5-p}{2};-\tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{(d-c) \tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)-\frac{(c-d) n F_1\left(\frac{3-p}{2};m+n-p+1,1-n;\frac{5-p}{2};-\tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{(d-c) \tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)}{c+d}\right) \tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{3-p}-F_1\left(\frac{1-p}{2};m+n-p+1,-n;\frac{3-p}{2};-\tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{(d-c) \tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right)+p F_1\left(\frac{1-p}{2};m+n-p+1,-n;\frac{3-p}{2};-\tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{(d-c) \tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \sin (e+f x)+\frac{2 (c-d) n F_1\left(\frac{1-p}{2};m+n-p+1,-n;\frac{3-p}{2};-\tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right),\frac{(d-c) \tan ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d}\right) \sin ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{c+d \sin (e+f x)}\right)}","\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*AppellF1[(1 - p)/2, 1 + m + n - p, -n, (3 - p)/2, -Tan[(-e + Pi/2 - f*x)/2]^2, ((-c + d)*Tan[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Cos[(-e + Pi/2 - f*x)/2]*(g*Sec[e + f*x])^p*Sin[(-e + Pi/2 - f*x)/2]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(f*(-AppellF1[(1 - p)/2, 1 + m + n - p, -n, (3 - p)/2, -Tan[(-e + Pi/2 - f*x)/2]^2, ((-c + d)*Tan[(-e + Pi/2 - f*x)/2]^2)/(c + d)] - 2*(n - p)*AppellF1[(1 - p)/2, 1 + m + n - p, -n, (3 - p)/2, -Tan[(-e + Pi/2 - f*x)/2]^2, ((-c + d)*Tan[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Sin[(-e + Pi/2 - f*x)/2]^2 + p*AppellF1[(1 - p)/2, 1 + m + n - p, -n, (3 - p)/2, -Tan[(-e + Pi/2 - f*x)/2]^2, ((-c + d)*Tan[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Sin[e + f*x] + (d*n*AppellF1[(1 - p)/2, 1 + m + n - p, -n, (3 - p)/2, -Tan[(-e + Pi/2 - f*x)/2]^2, -(((c - d)*Tan[(-e + Pi/2 - f*x)/2]^2)/(c + d))]*Cos[e + f*x]^2)/(c + d*Sin[e + f*x]) + (2*(c - d)*n*AppellF1[(1 - p)/2, 1 + m + n - p, -n, (3 - p)/2, -Tan[(-e + Pi/2 - f*x)/2]^2, ((-c + d)*Tan[(-e + Pi/2 - f*x)/2]^2)/(c + d)]*Sin[(-e + Pi/2 - f*x)/2]^2)/(c + d*Sin[e + f*x]) + (2*(1 - p)*(-(((c - d)*n*AppellF1[(3 - p)/2, 1 + m + n - p, 1 - n, (5 - p)/2, -Tan[(-e + Pi/2 - f*x)/2]^2, ((-c + d)*Tan[(-e + Pi/2 - f*x)/2]^2)/(c + d)])/(c + d)) + (1 + m + n - p)*AppellF1[(3 - p)/2, 2 + m + n - p, -n, (5 - p)/2, -Tan[(-e + Pi/2 - f*x)/2]^2, ((-c + d)*Tan[(-e + Pi/2 - f*x)/2]^2)/(c + d)])*Tan[(-e + Pi/2 - f*x)/2]^2)/(3 - p)))","B",0
1050,1,77,105,0.1768197,"\int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + b*Sin[c + d*x]),x]","\frac{-120 a \cos (c+d x)-20 a \cos (3 (c+d x))+12 a \cos (5 (c+d x))-15 b \sin (2 (c+d x))-15 b \sin (4 (c+d x))+5 b \sin (6 (c+d x))+60 b d x}{960 d}","\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,"(60*b*d*x - 120*a*Cos[c + d*x] - 20*a*Cos[3*(c + d*x)] + 12*a*Cos[5*(c + d*x)] - 15*b*Sin[2*(c + d*x)] - 15*b*Sin[4*(c + d*x)] + 5*b*Sin[6*(c + d*x)])/(960*d)","A",1
1051,1,59,81,0.1059026,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x]),x]","\frac{-15 a \sin (4 (c+d x))+60 a c+60 a d x-60 b \cos (c+d x)-10 b \cos (3 (c+d x))+6 b \cos (5 (c+d x))}{480 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,"(60*a*c + 60*a*d*x - 60*b*Cos[c + d*x] - 10*b*Cos[3*(c + d*x)] + 6*b*Cos[5*(c + d*x)] - 15*a*Sin[4*(c + d*x)])/(480*d)","A",1
1052,1,61,65,0.1160638,"\int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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{1}{8} b \left(-\frac{\sin (4 c) \cos (4 d x)}{4 d}-\frac{\cos (4 c) \sin (4 d x)}{4 d}\right)+\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*(-1/4*(Cos[4*d*x]*Sin[4*c])/d - (Cos[4*c]*Sin[4*d*x])/(4*d)))/8","A",1
1053,1,74,51,0.0561383,"\int \cos (c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{a \cos (c+d x)}{d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{b (c+d x)}{2 d}+\frac{b \sin (2 (c+d x))}{4 d}","\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*(c + d*x))/(2*d) + (a*Cos[c + d*x])/d - (a*Log[Cos[(c + d*x)/2]])/d + (a*Log[Sin[(c + d*x)/2]])/d + (b*Sin[2*(c + d*x)])/(4*d)","A",1
1054,1,75,41,0.0374026,"\int \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{a \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}+\frac{b \cos (c+d x)}{d}+\frac{b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"(b*Cos[c + d*x])/d - (a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d - (b*Log[Cos[(c + d*x)/2]])/d + (b*Log[Sin[(c + d*x)/2]])/d","C",1
1055,1,109,52,0.0493555,"\int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{b \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}","\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,"-1/8*(a*Csc[(c + d*x)/2]^2)/d - (b*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d + (a*Log[Cos[(c + d*x)/2]])/(2*d) - (a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)","C",1
1056,1,95,52,0.0355171,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-1/3*(a*Cot[c + d*x]^3)/d - (b*Csc[(c + d*x)/2]^2)/(8*d) + (b*Log[Cos[(c + d*x)/2]])/(2*d) - (b*Log[Sin[(c + d*x)/2]])/(2*d) + (b*Sec[(c + d*x)/2]^2)/(8*d)","A",1
1057,1,135,74,0.0403244,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^3*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-1/3*(b*Cot[c + d*x]^3)/d + (a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) + (a*Log[Cos[(c + d*x)/2]])/(8*d) - (a*Log[Sin[(c + d*x)/2]])/(8*d) - (a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)","A",1
1058,1,177,90,0.0690137,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + b*Sin[c + d*x]),x]","\frac{2 a \cot (c+d x)}{15 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{15 d}-\frac{b \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{b \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}-\frac{b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"(2*a*Cot[c + d*x])/(15*d) + (b*Csc[(c + d*x)/2]^2)/(32*d) - (b*Csc[(c + d*x)/2]^4)/(64*d) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d) + (b*Log[Cos[(c + d*x)/2]])/(8*d) - (b*Log[Sin[(c + d*x)/2]])/(8*d) - (b*Sec[(c + d*x)/2]^2)/(32*d) + (b*Sec[(c + d*x)/2]^4)/(64*d)","A",1
1059,1,132,190,0.5506973,"\int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{-105 \left(8 a^2+5 b^2\right) \cos (c+d x)-35 \left(4 a^2+b^2\right) \cos (3 (c+d x))+84 a^2 \cos (5 (c+d x))-210 a b \sin (2 (c+d x))-210 a b \sin (4 (c+d x))+70 a b \sin (6 (c+d x))+840 a b c+840 a b d x+63 b^2 \cos (5 (c+d x))-15 b^2 \cos (7 (c+d x))}{6720 d}","\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,"(840*a*b*c + 840*a*b*d*x - 105*(8*a^2 + 5*b^2)*Cos[c + d*x] - 35*(4*a^2 + b^2)*Cos[3*(c + d*x)] + 84*a^2*Cos[5*(c + d*x)] + 63*b^2*Cos[5*(c + d*x)] - 15*b^2*Cos[7*(c + d*x)] - 210*a*b*Sin[2*(c + d*x)] - 210*a*b*Sin[4*(c + d*x)] + 70*a*b*Sin[6*(c + d*x)])/(6720*d)","A",1
1060,1,120,163,0.2173077,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{-30 a^2 \sin (4 (c+d x))+120 a^2 c+120 a^2 d x-240 a b \cos (c+d x)-40 a b \cos (3 (c+d x))+24 a b \cos (5 (c+d x))-15 b^2 \sin (2 (c+d x))-15 b^2 \sin (4 (c+d x))+5 b^2 \sin (6 (c+d x))+60 b^2 c+60 b^2 d x}{960 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,"(120*a^2*c + 60*b^2*c + 120*a^2*d*x + 60*b^2*d*x - 240*a*b*Cos[c + d*x] - 40*a*b*Cos[3*(c + d*x)] + 24*a*b*Cos[5*(c + d*x)] - 15*b^2*Sin[2*(c + d*x)] - 30*a^2*Sin[4*(c + d*x)] - 15*b^2*Sin[4*(c + d*x)] + 5*b^2*Sin[6*(c + d*x)])/(960*d)","A",1
1061,1,77,106,0.3676315,"\int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{-30 \left(2 a^2+b^2\right) \cos (c+d x)-5 \left(4 a^2+b^2\right) \cos (3 (c+d x))+3 b (20 a (c+d x)-5 a \sin (4 (c+d x))+b \cos (5 (c+d x)))}{240 d}","-\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,"(-30*(2*a^2 + b^2)*Cos[c + d*x] - 5*(4*a^2 + b^2)*Cos[3*(c + d*x)] + 3*b*(20*a*(c + d*x) + b*Cos[5*(c + d*x)] - 5*a*Sin[4*(c + d*x)]))/(240*d)","A",1
1062,1,91,90,0.2158995,"\int \cos (c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{3 \left(4 a^2-b^2\right) \cos (c+d x)+6 a \left(2 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+b \sin (2 (c+d x))+2 b c+2 b d x\right)+b^2 (-\cos (3 (c+d x)))}{12 d}","\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,"(3*(4*a^2 - b^2)*Cos[c + d*x] - b^2*Cos[3*(c + d*x)] + 6*a*(2*b*c + 2*b*d*x - 2*a*Log[Cos[(c + d*x)/2]] + 2*a*Log[Sin[(c + d*x)/2]] + b*Sin[2*(c + d*x)]))/(12*d)","A",1
1063,1,116,78,0.4080591,"\int \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{2 a^2 \tan \left(\frac{1}{2} (c+d x)\right)-2 a^2 \cot \left(\frac{1}{2} (c+d x)\right)-4 a^2 c-4 a^2 d x+8 a b \cos (c+d x)+8 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+b^2 \sin (2 (c+d x))+2 b^2 c+2 b^2 d x}{4 d}","-\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,"(-4*a^2*c + 2*b^2*c - 4*a^2*d*x + 2*b^2*d*x + 8*a*b*Cos[c + d*x] - 2*a^2*Cot[(c + d*x)/2] - 8*a*b*Log[Cos[(c + d*x)/2]] + 8*a*b*Log[Sin[(c + d*x)/2]] + b^2*Sin[2*(c + d*x)] + 2*a^2*Tan[(c + d*x)/2])/(4*d)","A",1
1064,1,155,89,0.8896513,"\int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{a^2 \left(-\csc ^2\left(\frac{1}{2} (c+d x)\right)\right)+a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)-4 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 a b \tan \left(\frac{1}{2} (c+d x)\right)-8 a b \cot \left(\frac{1}{2} (c+d x)\right)-16 a b c-16 a b d x+8 b^2 \cos (c+d x)+8 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 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,"(-16*a*b*c - 16*a*b*d*x + 8*b^2*Cos[c + d*x] - 8*a*b*Cot[(c + d*x)/2] - a^2*Csc[(c + d*x)/2]^2 + 4*a^2*Log[Cos[(c + d*x)/2]] - 8*b^2*Log[Cos[(c + d*x)/2]] - 4*a^2*Log[Sin[(c + d*x)/2]] + 8*b^2*Log[Sin[(c + d*x)/2]] + a^2*Sec[(c + d*x)/2]^2 + 8*a*b*Tan[(c + d*x)/2])/(8*d)","A",1
1065,1,538,96,6.1873386,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\sin ^2(c+d x) \csc \left(\frac{1}{2} (c+d x)\right) \left(a^2 \cos \left(\frac{1}{2} (c+d x)\right)-3 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^2}{6 d (a+b \sin (c+d x))^2}+\frac{\sin ^2(c+d x) \sec \left(\frac{1}{2} (c+d x)\right) \left(3 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^2}{6 d (a+b \sin (c+d x))^2}-\frac{a^2 \sin ^2(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{24 d (a+b \sin (c+d x))^2}+\frac{a^2 \sin ^2(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{24 d (a+b \sin (c+d x))^2}-\frac{b^2 (c+d x) \sin ^2(c+d x) (a \csc (c+d x)+b)^2}{d (a+b \sin (c+d x))^2}-\frac{a b \sin ^2(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{4 d (a+b \sin (c+d x))^2}-\frac{a b \sin ^2(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^2}{d (a+b \sin (c+d x))^2}+\frac{a b \sin ^2(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{4 d (a+b \sin (c+d x))^2}+\frac{a b \sin ^2(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^2}{d (a+b \sin (c+d x))^2}","\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*(c + d*x)*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(d*(a + b*Sin[c + d*x])^2)) + ((a^2*Cos[(c + d*x)/2] - 3*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2]*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(6*d*(a + b*Sin[c + d*x])^2) - (a*b*Csc[(c + d*x)/2]^2*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(4*d*(a + b*Sin[c + d*x])^2) - (a^2*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(24*d*(a + b*Sin[c + d*x])^2) + (a*b*(b + a*Csc[c + d*x])^2*Log[Cos[(c + d*x)/2]]*Sin[c + d*x]^2)/(d*(a + b*Sin[c + d*x])^2) - (a*b*(b + a*Csc[c + d*x])^2*Log[Sin[(c + d*x)/2]]*Sin[c + d*x]^2)/(d*(a + b*Sin[c + d*x])^2) + (a*b*(b + a*Csc[c + d*x])^2*Sec[(c + d*x)/2]^2*Sin[c + d*x]^2)/(4*d*(a + b*Sin[c + d*x])^2) + ((b + a*Csc[c + d*x])^2*Sec[(c + d*x)/2]*(-(a^2*Sin[(c + d*x)/2]) + 3*b^2*Sin[(c + d*x)/2])*Sin[c + d*x]^2)/(6*d*(a + b*Sin[c + d*x])^2) + (a^2*(b + a*Csc[c + d*x])^2*Sec[(c + d*x)/2]^2*Sin[c + d*x]^2*Tan[(c + d*x)/2])/(24*d*(a + b*Sin[c + d*x])^2)","B",1
1066,1,579,123,6.1696962,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[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) \sin ^2(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{32 d (a+b \sin (c+d x))^2}+\frac{\left(-a^2-4 b^2\right) \sin ^2(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^2}{8 d (a+b \sin (c+d x))^2}+\frac{\left(4 b^2-a^2\right) \sin ^2(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{32 d (a+b \sin (c+d x))^2}+\frac{\left(a^2+4 b^2\right) \sin ^2(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^2}{8 d (a+b \sin (c+d x))^2}-\frac{a^2 \sin ^2(c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{64 d (a+b \sin (c+d x))^2}+\frac{a^2 \sin ^2(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{64 d (a+b \sin (c+d x))^2}-\frac{a b \sin ^2(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{12 d (a+b \sin (c+d x))^2}+\frac{a b \sin ^2(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{3 d (a+b \sin (c+d x))^2}-\frac{a b \sin ^2(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{3 d (a+b \sin (c+d x))^2}+\frac{a b \sin ^2(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^2}{12 d (a+b \sin (c+d x))^2}","\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*b*Cot[(c + d*x)/2]*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(3*d*(a + b*Sin[c + d*x])^2) + ((a^2 - 4*b^2)*Csc[(c + d*x)/2]^2*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(32*d*(a + b*Sin[c + d*x])^2) - (a*b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(12*d*(a + b*Sin[c + d*x])^2) - (a^2*Csc[(c + d*x)/2]^4*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(64*d*(a + b*Sin[c + d*x])^2) + ((a^2 + 4*b^2)*(b + a*Csc[c + d*x])^2*Log[Cos[(c + d*x)/2]]*Sin[c + d*x]^2)/(8*d*(a + b*Sin[c + d*x])^2) + ((-a^2 - 4*b^2)*(b + a*Csc[c + d*x])^2*Log[Sin[(c + d*x)/2]]*Sin[c + d*x]^2)/(8*d*(a + b*Sin[c + d*x])^2) + ((-a^2 + 4*b^2)*(b + a*Csc[c + d*x])^2*Sec[(c + d*x)/2]^2*Sin[c + d*x]^2)/(32*d*(a + b*Sin[c + d*x])^2) + (a^2*(b + a*Csc[c + d*x])^2*Sec[(c + d*x)/2]^4*Sin[c + d*x]^2)/(64*d*(a + b*Sin[c + d*x])^2) - (a*b*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2*Tan[(c + d*x)/2])/(3*d*(a + b*Sin[c + d*x])^2) + (a*b*(b + a*Csc[c + d*x])^2*Sec[(c + d*x)/2]^2*Sin[c + d*x]^2*Tan[(c + d*x)/2])/(12*d*(a + b*Sin[c + d*x])^2)","B",1
1067,1,236,148,0.8175855,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{\csc ^5(c+d x) \left(-40 \left(4 a^2+b^2\right) \cos (c+d x)+20 \left(b^2-2 a^2\right) \cos (3 (c+d x))+8 a^2 \cos (5 (c+d x))-180 a b \sin (2 (c+d x))-30 a b \sin (4 (c+d x))-150 a b \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+75 a b \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-15 a b \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+150 a b \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-75 a b \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+15 a b \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+20 b^2 \cos (5 (c+d x))\right)}{960 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,"(Csc[c + d*x]^5*(-40*(4*a^2 + b^2)*Cos[c + d*x] + 20*(-2*a^2 + b^2)*Cos[3*(c + d*x)] + 8*a^2*Cos[5*(c + d*x)] + 20*b^2*Cos[5*(c + d*x)] + 150*a*b*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 150*a*b*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 180*a*b*Sin[2*(c + d*x)] - 75*a*b*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 75*a*b*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 30*a*b*Sin[4*(c + d*x)] + 15*a*b*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 15*a*b*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(960*d)","A",1
1068,1,296,170,0.8005302,"\int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{30 \left(a^2+2 b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)+5 a^2 \sec ^6\left(\frac{1}{2} (c+d x)\right)-30 a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)-120 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+120 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\csc ^4\left(\frac{1}{2} (c+d x)\right) \left(4 a b \sin (c+d x)-30 b^2\right)-256 a b \tan \left(\frac{1}{2} (c+d x)\right)+256 a b \cot \left(\frac{1}{2} (c+d x)\right)-a \csc ^6\left(\frac{1}{2} (c+d x)\right) (5 a+12 b \sin (c+d x))+768 a b \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)-64 a b \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+30 b^2 \sec ^4\left(\frac{1}{2} (c+d x)\right)-60 b^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)-240 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+240 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{1920 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,"(256*a*b*Cot[(c + d*x)/2] + 30*(a^2 + 2*b^2)*Csc[(c + d*x)/2]^2 + 120*a^2*Log[Cos[(c + d*x)/2]] + 240*b^2*Log[Cos[(c + d*x)/2]] - 120*a^2*Log[Sin[(c + d*x)/2]] - 240*b^2*Log[Sin[(c + d*x)/2]] - 30*a^2*Sec[(c + d*x)/2]^2 - 60*b^2*Sec[(c + d*x)/2]^2 + 30*b^2*Sec[(c + d*x)/2]^4 + 5*a^2*Sec[(c + d*x)/2]^6 - 64*a*b*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 768*a*b*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 - a*Csc[(c + d*x)/2]^6*(5*a + 12*b*Sin[c + d*x]) + Csc[(c + d*x)/2]^4*(-30*b^2 + 4*a*b*Sin[c + d*x]) - 256*a*b*Tan[(c + d*x)/2])/(1920*d)","A",1
1069,1,157,232,0.8234641,"\int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{-35 \left(12 a^2 b+b^3\right) \cos (3 (c+d x))+63 \left(4 a^2 b+b^3\right) \cos (5 (c+d x))+105 a \left(-\left(2 a^2+3 b^2\right) \sin (4 (c+d x))+8 a^2 c+8 a^2 d x-3 b^2 \sin (2 (c+d x))+b^2 \sin (6 (c+d x))+12 b^2 c+12 b^2 d x\right)-105 b \left(24 a^2+5 b^2\right) \cos (c+d x)-15 b^3 \cos (7 (c+d x))}{6720 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,"(-105*b*(24*a^2 + 5*b^2)*Cos[c + d*x] - 35*(12*a^2*b + b^3)*Cos[3*(c + d*x)] + 63*(4*a^2*b + b^3)*Cos[5*(c + d*x)] - 15*b^3*Cos[7*(c + d*x)] + 105*a*(8*a^2*c + 12*b^2*c + 8*a^2*d*x + 12*b^2*d*x - 3*b^2*Sin[2*(c + d*x)] - (2*a^2 + 3*b^2)*Sin[4*(c + d*x)] + b^2*Sin[6*(c + d*x)]))/(6720*d)","A",1
1070,1,138,163,0.7609529,"\int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*Sin[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{-20 \left(4 a^3+3 a b^2\right) \cos (3 (c+d x))-120 a \left(2 a^2+3 b^2\right) \cos (c+d x)+b \left(5 \left(-3 \left(6 a^2+b^2\right) \sin (4 (c+d x))+72 a^2 c+72 a^2 d x-3 b^2 \sin (2 (c+d x))+b^2 \sin (6 (c+d x))+18 b^2 c+12 b^2 d x\right)+36 a b \cos (5 (c+d x))\right)}{960 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,"(-120*a*(2*a^2 + 3*b^2)*Cos[c + d*x] - 20*(4*a^3 + 3*a*b^2)*Cos[3*(c + d*x)] + b*(36*a*b*Cos[5*(c + d*x)] + 5*(72*a^2*c + 18*b^2*c + 72*a^2*d*x + 12*b^2*d*x - 3*b^2*Sin[2*(c + d*x)] - 3*(6*a^2 + b^2)*Sin[4*(c + d*x)] + b^2*Sin[6*(c + d*x)])))/(960*d)","A",1
1071,1,129,136,0.2942474,"\int \cos (c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{32 a^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-32 a^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 a \left(4 a^2-3 b^2\right) \cos (c+d x)+24 a^2 b \sin (2 (c+d x))+48 a^2 b c+48 a^2 b d x-8 a b^2 \cos (3 (c+d x))-b^3 \sin (4 (c+d x))+4 b^3 c+4 b^3 d x}{32 d}","-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{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 \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,"(48*a^2*b*c + 4*b^3*c + 48*a^2*b*d*x + 4*b^3*d*x + 8*a*(4*a^2 - 3*b^2)*Cos[c + d*x] - 8*a*b^2*Cos[3*(c + d*x)] - 32*a^3*Log[Cos[(c + d*x)/2]] + 32*a^3*Log[Sin[(c + d*x)/2]] + 24*a^2*b*Sin[2*(c + d*x)] - b^3*Sin[4*(c + d*x)])/(32*d)","A",1
1072,1,143,102,1.3189012,"\int \cot ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{-6 a^3 \cot \left(\frac{1}{2} (c+d x)\right)+\left(36 a^2 b-3 b^3\right) \cos (c+d x)+6 a \left(a^2 \tan \left(\frac{1}{2} (c+d x)\right)-2 a^2 c-2 a^2 d x+6 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 b^2 c+3 b^2 d x\right)+9 a b^2 \sin (2 (c+d x))-b^3 \cos (3 (c+d x))}{12 d}","-\frac{a^3 \cot (c+d x)}{d}+a^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{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,"((36*a^2*b - 3*b^3)*Cos[c + d*x] - b^3*Cos[3*(c + d*x)] - 6*a^3*Cot[(c + d*x)/2] + 9*a*b^2*Sin[2*(c + d*x)] + 6*a*(-2*a^2*c + 3*b^2*c - 2*a^2*d*x + 3*b^2*d*x - 6*a*b*Log[Cos[(c + d*x)/2]] + 6*a*b*Log[Sin[(c + d*x)/2]] + a^2*Tan[(c + d*x)/2]))/(12*d)","A",1
1073,1,192,138,1.3145723,"\int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{a^3 \left(-\csc ^2\left(\frac{1}{2} (c+d x)\right)\right)+a^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)-4 a^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 a^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 a^2 b \tan \left(\frac{1}{2} (c+d x)\right)-12 a^2 b \cot \left(\frac{1}{2} (c+d x)\right)-24 a^2 b c-24 a^2 b d x+24 a b^2 \cos (c+d x)+24 a b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-24 a b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 b^3 \sin (2 (c+d x))+4 b^3 c+4 b^3 d x}{8 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,"(-24*a^2*b*c + 4*b^3*c - 24*a^2*b*d*x + 4*b^3*d*x + 24*a*b^2*Cos[c + d*x] - 12*a^2*b*Cot[(c + d*x)/2] - a^3*Csc[(c + d*x)/2]^2 + 4*a^3*Log[Cos[(c + d*x)/2]] - 24*a*b^2*Log[Cos[(c + d*x)/2]] - 4*a^3*Log[Sin[(c + d*x)/2]] + 24*a*b^2*Log[Sin[(c + d*x)/2]] + a^3*Sec[(c + d*x)/2]^2 + 2*b^3*Sin[2*(c + d*x)] + 12*a^2*b*Tan[(c + d*x)/2])/(8*d)","A",1
1074,1,615,138,6.1885567,"\int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{\sin ^3(c+d x) \csc \left(\frac{1}{2} (c+d x)\right) \left(a^3 \cos \left(\frac{1}{2} (c+d x)\right)-9 a b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^3}{6 d (a+b \sin (c+d x))^3}+\frac{\sin ^3(c+d x) \sec \left(\frac{1}{2} (c+d x)\right) \left(9 a b^2 \sin \left(\frac{1}{2} (c+d x)\right)-a^3 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^3}{6 d (a+b \sin (c+d x))^3}-\frac{a^3 \sin ^3(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^3}{24 d (a+b \sin (c+d x))^3}+\frac{a^3 \sin ^3(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^3}{24 d (a+b \sin (c+d x))^3}+\frac{\left(2 b^3-3 a^2 b\right) \sin ^3(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^3}{2 d (a+b \sin (c+d x))^3}+\frac{\left(3 a^2 b-2 b^3\right) \sin ^3(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^3}{2 d (a+b \sin (c+d x))^3}-\frac{3 a^2 b \sin ^3(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}+\frac{3 a^2 b \sin ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}+\frac{b^3 \sin ^3(c+d x) \cos (c+d x) (a \csc (c+d x)+b)^3}{d (a+b \sin (c+d x))^3}-\frac{3 a b^2 (c+d x) \sin ^3(c+d x) (a \csc (c+d x)+b)^3}{d (a+b \sin (c+d x))^3}","\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*(c + d*x)*(b + a*Csc[c + d*x])^3*Sin[c + d*x]^3)/(d*(a + b*Sin[c + d*x])^3) + (b^3*Cos[c + d*x]*(b + a*Csc[c + d*x])^3*Sin[c + d*x]^3)/(d*(a + b*Sin[c + d*x])^3) + ((a^3*Cos[(c + d*x)/2] - 9*a*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2]*(b + a*Csc[c + d*x])^3*Sin[c + d*x]^3)/(6*d*(a + b*Sin[c + d*x])^3) - (3*a^2*b*Csc[(c + d*x)/2]^2*(b + a*Csc[c + d*x])^3*Sin[c + d*x]^3)/(8*d*(a + b*Sin[c + d*x])^3) - (a^3*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(b + a*Csc[c + d*x])^3*Sin[c + d*x]^3)/(24*d*(a + b*Sin[c + d*x])^3) + ((3*a^2*b - 2*b^3)*(b + a*Csc[c + d*x])^3*Log[Cos[(c + d*x)/2]]*Sin[c + d*x]^3)/(2*d*(a + b*Sin[c + d*x])^3) + ((-3*a^2*b + 2*b^3)*(b + a*Csc[c + d*x])^3*Log[Sin[(c + d*x)/2]]*Sin[c + d*x]^3)/(2*d*(a + b*Sin[c + d*x])^3) + (3*a^2*b*(b + a*Csc[c + d*x])^3*Sec[(c + d*x)/2]^2*Sin[c + d*x]^3)/(8*d*(a + b*Sin[c + d*x])^3) + ((b + a*Csc[c + d*x])^3*Sec[(c + d*x)/2]*(-(a^3*Sin[(c + d*x)/2]) + 9*a*b^2*Sin[(c + d*x)/2])*Sin[c + d*x]^3)/(6*d*(a + b*Sin[c + d*x])^3) + (a^3*(b + a*Csc[c + d*x])^3*Sec[(c + d*x)/2]^2*Sin[c + d*x]^3*Tan[(c + d*x)/2])/(24*d*(a + b*Sin[c + d*x])^3)","B",0
1075,1,690,152,6.2302298,"\int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","\frac{\left(a^3-12 a b^2\right) \sin ^3(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^3}{32 d (a+b \sin (c+d x))^3}+\frac{\left(-a^3-12 a b^2\right) \sin ^3(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}+\frac{\left(12 a b^2-a^3\right) \sin ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^3}{32 d (a+b \sin (c+d x))^3}+\frac{\left(a^3+12 a b^2\right) \sin ^3(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}-\frac{a^3 \sin ^3(c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^3}{64 d (a+b \sin (c+d x))^3}+\frac{a^3 \sin ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^3}{64 d (a+b \sin (c+d x))^3}+\frac{\sin ^3(c+d x) \csc \left(\frac{1}{2} (c+d x)\right) \left(a^2 b \cos \left(\frac{1}{2} (c+d x)\right)-b^3 \cos \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^3}{2 d (a+b \sin (c+d x))^3}+\frac{\sin ^3(c+d x) \sec \left(\frac{1}{2} (c+d x)\right) \left(b^3 \sin \left(\frac{1}{2} (c+d x)\right)-a^2 b \sin \left(\frac{1}{2} (c+d x)\right)\right) (a \csc (c+d x)+b)^3}{2 d (a+b \sin (c+d x))^3}-\frac{a^2 b \sin ^3(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}+\frac{a^2 b \sin ^3(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \csc (c+d x)+b)^3}{8 d (a+b \sin (c+d x))^3}-\frac{b^3 (c+d x) \sin ^3(c+d x) (a \csc (c+d x)+b)^3}{d (a+b \sin (c+d x))^3}","\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{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}-\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{4 d}+b^3 (-x)",1,"-((b^3*(c + d*x)*(b + a*Csc[c + d*x])^3*Sin[c + d*x]^3)/(d*(a + b*Sin[c + d*x])^3)) + ((a^2*b*Cos[(c + d*x)/2] - b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2]*(b + a*Csc[c + d*x])^3*Sin[c + d*x]^3)/(2*d*(a + b*Sin[c + d*x])^3) + ((a^3 - 12*a*b^2)*Csc[(c + d*x)/2]^2*(b + a*Csc[c + d*x])^3*Sin[c + d*x]^3)/(32*d*(a + b*Sin[c + d*x])^3) - (a^2*b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(b + a*Csc[c + d*x])^3*Sin[c + d*x]^3)/(8*d*(a + b*Sin[c + d*x])^3) - (a^3*Csc[(c + d*x)/2]^4*(b + a*Csc[c + d*x])^3*Sin[c + d*x]^3)/(64*d*(a + b*Sin[c + d*x])^3) + ((a^3 + 12*a*b^2)*(b + a*Csc[c + d*x])^3*Log[Cos[(c + d*x)/2]]*Sin[c + d*x]^3)/(8*d*(a + b*Sin[c + d*x])^3) + ((-a^3 - 12*a*b^2)*(b + a*Csc[c + d*x])^3*Log[Sin[(c + d*x)/2]]*Sin[c + d*x]^3)/(8*d*(a + b*Sin[c + d*x])^3) + ((-a^3 + 12*a*b^2)*(b + a*Csc[c + d*x])^3*Sec[(c + d*x)/2]^2*Sin[c + d*x]^3)/(32*d*(a + b*Sin[c + d*x])^3) + (a^3*(b + a*Csc[c + d*x])^3*Sec[(c + d*x)/2]^4*Sin[c + d*x]^3)/(64*d*(a + b*Sin[c + d*x])^3) + ((b + a*Csc[c + d*x])^3*Sec[(c + d*x)/2]*(-(a^2*b*Sin[(c + d*x)/2]) + b^3*Sin[(c + d*x)/2])*Sin[c + d*x]^3)/(2*d*(a + b*Sin[c + d*x])^3) + (a^2*b*(b + a*Csc[c + d*x])^3*Sec[(c + d*x)/2]^2*Sin[c + d*x]^3*Tan[(c + d*x)/2])/(8*d*(a + b*Sin[c + d*x])^3)","B",0
1076,1,344,183,1.2920706,"\int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","\frac{32 \left(2 a^3+15 a b^2\right) \cot \left(\frac{1}{2} (c+d x)\right)-64 a^3 \tan \left(\frac{1}{2} (c+d x)\right)-3 a^3 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)-16 a^3 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+6 a^3 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)+30 \left(3 a^2 b-4 b^3\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)+a \csc ^4\left(\frac{1}{2} (c+d x)\right) \left(\left(a^2-60 b^2\right) \sin (c+d x)-45 a b\right)+45 a^2 b \sec ^4\left(\frac{1}{2} (c+d x)\right)-90 a^2 b \sec ^2\left(\frac{1}{2} (c+d x)\right)-360 a^2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+360 a^2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-480 a b^2 \tan \left(\frac{1}{2} (c+d x)\right)+960 a b^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+120 b^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)-480 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+480 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{960 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,"(32*(2*a^3 + 15*a*b^2)*Cot[(c + d*x)/2] + 30*(3*a^2*b - 4*b^3)*Csc[(c + d*x)/2]^2 + 360*a^2*b*Log[Cos[(c + d*x)/2]] + 480*b^3*Log[Cos[(c + d*x)/2]] - 360*a^2*b*Log[Sin[(c + d*x)/2]] - 480*b^3*Log[Sin[(c + d*x)/2]] - 90*a^2*b*Sec[(c + d*x)/2]^2 + 120*b^3*Sec[(c + d*x)/2]^2 + 45*a^2*b*Sec[(c + d*x)/2]^4 - 16*a^3*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 960*a*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 3*a^3*Csc[(c + d*x)/2]^6*Sin[c + d*x] + a*Csc[(c + d*x)/2]^4*(-45*a*b + (a^2 - 60*b^2)*Sin[c + d*x]) - 64*a^3*Tan[(c + d*x)/2] - 480*a*b^2*Tan[(c + d*x)/2] + 6*a^3*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(960*d)","A",1
1077,1,369,212,2.0973393,"\int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","-\frac{-30 \left(a^3+6 a b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)-5 a^3 \sec ^6\left(\frac{1}{2} (c+d x)\right)+30 a^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)+120 a^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-120 a^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-64 \left(6 a^2 b+5 b^3\right) \cot \left(\frac{1}{2} (c+d x)\right)+2 b \csc ^4\left(\frac{1}{2} (c+d x)\right) \left(\left(20 b^2-3 a^2\right) \sin (c+d x)+45 a b\right)+384 a^2 b \tan \left(\frac{1}{2} (c+d x)\right)+a^2 \csc ^6\left(\frac{1}{2} (c+d x)\right) (5 a+18 b \sin (c+d x))+96 a^2 b \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-36 a^2 b \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)-90 a b^2 \sec ^4\left(\frac{1}{2} (c+d x)\right)+180 a b^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)+720 a b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-720 a b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+320 b^3 \tan \left(\frac{1}{2} (c+d x)\right)-640 b^3 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)}{1920 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,"-1/1920*(-64*(6*a^2*b + 5*b^3)*Cot[(c + d*x)/2] - 30*(a^3 + 6*a*b^2)*Csc[(c + d*x)/2]^2 - 120*a^3*Log[Cos[(c + d*x)/2]] - 720*a*b^2*Log[Cos[(c + d*x)/2]] + 120*a^3*Log[Sin[(c + d*x)/2]] + 720*a*b^2*Log[Sin[(c + d*x)/2]] + 30*a^3*Sec[(c + d*x)/2]^2 + 180*a*b^2*Sec[(c + d*x)/2]^2 - 90*a*b^2*Sec[(c + d*x)/2]^4 - 5*a^3*Sec[(c + d*x)/2]^6 + 96*a^2*b*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 640*b^3*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + a^2*Csc[(c + d*x)/2]^6*(5*a + 18*b*Sin[c + d*x]) + 2*b*Csc[(c + d*x)/2]^4*(45*a*b + (-3*a^2 + 20*b^2)*Sin[c + d*x]) + 384*a^2*b*Tan[(c + d*x)/2] + 320*b^3*Tan[(c + d*x)/2] - 36*a^2*b*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/d","A",1
1078,1,246,188,2.5250244,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{96 a^4 c+96 a^4 d x+96 a^3 b c \sin (c+d x)+96 a^3 b d x \sin (c+d x)+24 a^2 b^2 \sin (2 (c+d x))+12 a b \left(8 a^2-b^2\right) \cos (c+d x)-24 a^2 b^2 c-24 a^2 b^2 d x-24 a b^3 c \sin (c+d x)-24 a b^3 d x \sin (c+d x)+4 a b^3 \cos (3 (c+d x))-2 b^4 \sin (2 (c+d x))-b^4 \sin (4 (c+d x))}{a+b \sin (c+d x)}-\frac{48 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)}{\sqrt{a^2-b^2}}}{24 b^5 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{\left(12 a^2-b^2\right) \cos (c+d x)}{3 b^4 d}-\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,"((-48*a^2*(4*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (96*a^4*c - 24*a^2*b^2*c + 96*a^4*d*x - 24*a^2*b^2*d*x + 12*a*b*(8*a^2 - b^2)*Cos[c + d*x] + 4*a*b^3*Cos[3*(c + d*x)] + 96*a^3*b*c*Sin[c + d*x] - 24*a*b^3*c*Sin[c + d*x] + 96*a^3*b*d*x*Sin[c + d*x] - 24*a*b^3*d*x*Sin[c + d*x] + 24*a^2*b^2*Sin[2*(c + d*x)] - 2*b^4*Sin[2*(c + d*x)] - b^4*Sin[4*(c + d*x)])/(a + b*Sin[c + d*x]))/(24*b^5*d)","A",1
1079,1,129,153,0.4222461,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{2 \left(b^2-6 a^2\right) (c+d x)+\frac{8 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)}{\sqrt{a^2-b^2}}-\frac{4 a^2 b \cos (c+d x)}{a+b \sin (c+d x)}-8 a b \cos (c+d x)+b^2 \sin (2 (c+d x))}{4 b^4 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,"(2*(-6*a^2 + b^2)*(c + d*x) + (8*a*(3*a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 8*a*b*Cos[c + d*x] - (4*a^2*b*Cos[c + d*x])/(a + b*Sin[c + d*x]) + b^2*Sin[2*(c + d*x)])/(4*b^4*d)","A",1
1080,1,130,106,1.0681174,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{4 a^2 c+4 a^2 d x+4 a b (c+d x) \sin (c+d x)+4 a b \cos (c+d x)+b^2 \sin (2 (c+d x))}{a+b \sin (c+d x)}-\frac{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)}{\sqrt{a^2-b^2}}}{2 b^3 d}","-\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{2 a x}{b^3}+\frac{\cos (c+d x) (2 a+b \sin (c+d x))}{b^2 d (a+b \sin (c+d x))}",1,"((-4*(2*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (4*a^2*c + 4*a^2*d*x + 4*a*b*Cos[c + d*x] + 4*a*b*(c + d*x)*Sin[c + d*x] + b^2*Sin[2*(c + d*x)])/(a + b*Sin[c + d*x]))/(2*b^3*d)","A",1
1081,1,97,92,0.2230388,"\int \frac{\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{a \cos (c+d x)}{a+b \sin (c+d x)}+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^2 d}","-\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]])/Sqrt[a^2 - b^2] - Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]] + (a*Cos[c + d*x])/(a + b*Sin[c + d*x]))/(a^2*d)","A",1
1082,1,139,115,0.7597876,"\int \frac{\cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","-\frac{\frac{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)}{\sqrt{a^2-b^2}}+\frac{2 a b \cos (c+d x)}{a+b \sin (c+d x)}-a \tan \left(\frac{1}{2} (c+d x)\right)+a \cot \left(\frac{1}{2} (c+d x)\right)+4 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^3 d}","\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{2 \cot (c+d x)}{a^2 d}-\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{\cot (c+d x)}{a d (a+b \sin (c+d x))}",1,"-1/2*((4*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + a*Cot[(c + d*x)/2] - 4*b*Log[Cos[(c + d*x)/2]] + 4*b*Log[Sin[(c + d*x)/2]] + (2*a*b*Cos[c + d*x])/(a + b*Sin[c + d*x]) - a*Tan[(c + d*x)/2])/(a^3*d)","A",1
1083,1,196,157,3.0913861,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{-\frac{16 b \left(3 b^2-2 a^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-4 \left(a^2-6 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \left(a^2-6 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-a^2 \csc ^2\left(\frac{1}{2} (c+d x)\right)+a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)+\frac{8 a b^2 \cos (c+d x)}{a+b \sin (c+d x)}-8 a b \tan \left(\frac{1}{2} (c+d x)\right)+8 a b \cot \left(\frac{1}{2} (c+d x)\right)}{8 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{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{\cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}",1,"((-16*b*(-2*a^2 + 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 8*a*b*Cot[(c + d*x)/2] - a^2*Csc[(c + d*x)/2]^2 + 4*(a^2 - 6*b^2)*Log[Cos[(c + d*x)/2]] - 4*(a^2 - 6*b^2)*Log[Sin[(c + d*x)/2]] + a^2*Sec[(c + d*x)/2]^2 + (8*a*b^2*Cos[c + d*x])/(a + b*Sin[c + d*x]) - 8*a*b*Tan[(c + d*x)/2])/(8*a^4*d)","A",1
1084,1,385,193,6.341105,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","-\frac{b^3 \cos (c+d x)}{a^4 d (a+b \sin (c+d x))}+\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{4 a^3 d}-\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{4 a^3 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}+\frac{\left(a^2 b-4 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{\left(4 b^3-a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}-\frac{2 b^2 \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(a^2 \cos \left(\frac{1}{2} (c+d x)\right)-9 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(9 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 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{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{b \left(a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{\left(a^2-12 b^2\right) \cot (c+d x)}{3 a^4 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[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) + ((a^2*Cos[(c + d*x)/2] - 9*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + (b*Csc[(c + d*x)/2]^2)/(4*a^3*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^2*d) + ((-(a^2*b) + 4*b^3)*Log[Cos[(c + d*x)/2]])/(a^5*d) + ((a^2*b - 4*b^3)*Log[Sin[(c + d*x)/2]])/(a^5*d) - (b*Sec[(c + d*x)/2]^2)/(4*a^3*d) + (Sec[(c + d*x)/2]*(-(a^2*Sin[(c + d*x)/2]) + 9*b^2*Sin[(c + d*x)/2]))/(6*a^4*d) - (b^3*Cos[c + d*x])/(a^4*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^2*d)","A",0
1085,1,288,266,5.8784279,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{4 a \left(12 a^4-19 a^2 b^2+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)}{\sqrt{a^2-b^2}}+\frac{-2 b^2 \left(12 a^4-13 a^2 b^2+b^4\right) (c+d x) \sin ^2(c+d x)-\left(a^2 \left(\left(18 a^2 b^2-17 b^4\right) \sin (2 (c+d x))+2 \left(12 a^4-13 a^2 b^2+b^4\right) (c+d x)\right)\right)-4 a b \left(12 a^4-13 a^2 b^2+b^4\right) (c+d x) \sin (c+d x)+\cos (c+d x) \left(-24 a^5 b+22 a^3 b^3+2 b^4 \left(a^2-b^2\right) \sin ^3(c+d x)-8 a b^3 \left(a^2-b^2\right) \sin ^2(c+d x)\right)}{(a+b \sin (c+d x))^2}}{4 b^5 d (a-b) (a+b)}","-\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{x \left(12 a^2-b^2\right)}{2 b^5}-\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{\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{a \left(12 a^4-19 a^2 b^2+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{\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}",1,"((4*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]])/Sqrt[a^2 - b^2] + (-4*a*b*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x)*Sin[c + d*x] - 2*b^2*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x)*Sin[c + d*x]^2 + Cos[c + d*x]*(-24*a^5*b + 22*a^3*b^3 - 8*a*b^3*(a^2 - b^2)*Sin[c + d*x]^2 + 2*b^4*(a^2 - b^2)*Sin[c + d*x]^3) - a^2*(2*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x) + (18*a^2*b^2 - 17*b^4)*Sin[2*(c + d*x)]))/(a + b*Sin[c + d*x])^2)/(4*(a - b)*b^5*(a + b)*d)","A",1
1086,1,159,180,1.2213159,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{a b \left(5 a^2-4 b^2\right) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))}-\frac{a^2 b \cos (c+d x)}{(a+b \sin (c+d x))^2}-\frac{2 \left(6 a^4-9 a^2 b^2+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)}{\left(a^2-b^2\right)^{3/2}}+6 a (c+d x)+2 b \cos (c+d x)}{2 b^4 d}","\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{\left(6 a^4-9 a^2 b^2+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{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,"(6*a*(c + d*x) - (2*(6*a^4 - 9*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 2*b*Cos[c + d*x] - (a^2*b*Cos[c + d*x])/(a + b*Sin[c + d*x])^2 + (a*b*(5*a^2 - 4*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])))/(2*b^4*d)","A",1
1087,1,289,167,2.0578127,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{\frac{a b \left(4 a^2-3 b^2\right) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}+\frac{2 a \left(8 a^4-20 a^2 b^2+15 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{3 b \left(4 a^4-7 a^2 b^2+2 b^4\right) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-8 (c+d x)}{b^3}-\frac{\frac{6 a b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{\cos (c+d x) \left(b \left(a^2+2 b^2\right) \sin (c+d x)+a \left(2 a^2+b^2\right)\right)}{(a+b \sin (c+d x))^2}}{(a-b)^2 (a+b)^2}}{8 d}","-\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{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{x}{b^3}",1,"((-8*(c + d*x) + (2*a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a*b*(4*a^2 - 3*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) - (3*b*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/b^3 - ((6*a*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2)/((a - b)^2*(a + b)^2))/(8*d)","A",1
1088,1,154,154,1.1790198,"\int \frac{\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{2 b \left(2 b^2-3 a^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{a \cos (c+d x) \left(2 a^3+b \left(a^2-2 b^2\right) \sin (c+d x)-3 a b^2\right)}{(a-b) (a+b) (a+b \sin (c+d x))^2}+2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^3 d}","-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\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{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{\cos (c+d x)}{2 a d (a+b \sin (c+d x))^2}",1,"((2*b*(-3*a^2 + 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - 2*Log[Cos[(c + d*x)/2]] + 2*Log[Sin[(c + d*x)/2]] + (a*Cos[c + d*x]*(2*a^3 - 3*a*b^2 + b*(a^2 - 2*b^2)*Sin[c + d*x]))/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2))/(2*a^3*d)","A",1
1089,1,195,202,5.7816069,"\int \frac{\cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{a b \left(4 b^2-3 a^2\right) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))}-\frac{a^2 b \cos (c+d x)}{(a+b \sin (c+d x))^2}-\frac{2 \left(2 a^4-9 a^2 b^2+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)}{\left(a^2-b^2\right)^{3/2}}+a \tan \left(\frac{1}{2} (c+d x)\right)-a \cot \left(\frac{1}{2} (c+d x)\right)-6 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}","\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\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{\left(2 a^4-9 a^2 b^2+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{\cot (c+d x)}{2 a d (a+b \sin (c+d x))^2}",1,"((-2*(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^2 - b^2)^(3/2) - a*Cot[(c + d*x)/2] + 6*b*Log[Cos[(c + d*x)/2]] - 6*b*Log[Sin[(c + d*x)/2]] - (a^2*b*Cos[c + d*x])/(a + b*Sin[c + d*x])^2 + (a*b*(-3*a^2 + 4*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])) + a*Tan[(c + d*x)/2])/(2*a^4*d)","A",1
1090,1,330,269,6.334017,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x])/(a + b*Sin[c + d*x])^3,x]","-\frac{3 b \tan \left(\frac{1}{2} (c+d x)\right)}{2 a^4 d}+\frac{3 b \cot \left(\frac{1}{2} (c+d x)\right)}{2 a^4 d}+\frac{b^2 \cos (c+d x)}{2 a^3 d (a+b \sin (c+d x))^2}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^3 d}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^3 d}+\frac{\left(12 b^2-a^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^5 d}+\frac{\left(a^2-12 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^5 d}+\frac{5 a^2 b^2 \cos (c+d x)-6 b^4 \cos (c+d x)}{2 a^4 d (a-b) (a+b) (a+b \sin (c+d x))}+\frac{b \left(6 a^4-19 a^2 b^2+12 b^4\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^5 d \left(a^2-b^2\right)^{3/2}}","\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{\left(a^2-12 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}+\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(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{b \left(6 a^4-19 a^2 b^2+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{\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[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(3/2)*d) + (3*b*Cot[(c + d*x)/2])/(2*a^4*d) - Csc[(c + d*x)/2]^2/(8*a^3*d) + ((a^2 - 12*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^5*d) + ((-a^2 + 12*b^2)*Log[Sin[(c + d*x)/2]])/(2*a^5*d) + Sec[(c + d*x)/2]^2/(8*a^3*d) + (b^2*Cos[c + d*x])/(2*a^3*d*(a + b*Sin[c + d*x])^2) + (5*a^2*b^2*Cos[c + d*x] - 6*b^4*Cos[c + d*x])/(2*a^4*(a - b)*(a + b)*d*(a + b*Sin[c + d*x])) - (3*b*Tan[(c + d*x)/2])/(2*a^4*d)","A",1
1091,1,3348,347,23.0093804,"\int \frac{\cos ^2(e+f x)}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}} \, dx","Integrate[Cos[e + f*x]^2/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(5/2)),x]","\text{Result too large to show}","-\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{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{2 \cos (e+f x) \sqrt{d \sin (e+f x)}}{3 a d f (a+b \sin (e+f x))^{3/2}}",1,"(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((2*Cos[e + f*x])/(3*a*(a + b*Sin[e + f*x])^2) - (4*b^2*Cos[e + f*x])/(3*a^2*(a^2 - b^2)*(a + b*Sin[e + f*x]))))/(f*Sqrt[d*Sin[e + f*x]]) + (4*Sqrt[a + b*Sin[e + f*x]]*((2*Sqrt[a + b*Sin[e + f*x]])/(3*a*(a^2 - b^2)*Sqrt[Sin[e + f*x]]) - (4*b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(3*a^2*(a^2 - b^2)))*(-2*b*Sin[(e + f*x)/2]^2 - (2*a*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/(3*a^2*(a^2 - b^2)*f*Sqrt[d*Sin[e + f*x]]*((2*b*Cos[e + f*x]*(-2*b*Sin[(e + f*x)/2]^2 - (2*a*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/(3*a^2*(a^2 - b^2)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(-2*b*Sin[(e + f*x)/2]^2 - (2*a*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/(3*a^2*(a^2 - b^2)*Sin[e + f*x]^(3/2)) + (4*Sqrt[a + b*Sin[e + f*x]]*(-2*b*Cos[(e + f*x)/2]*Sin[(e + f*x)/2] + (a^2*Sec[(e + f*x)/2]^2*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(2*Sqrt[-a^2 + b^2]*(-b + Sqrt[-a^2 + b^2])*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*((a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2]))^(3/2)) + (a*((a*b*Cos[e + f*x]*Sec[(e + f*x)/2]^2)/(a^2 - b^2) + (a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x])*Tan[(e + f*x)/2])/(a^2 - b^2))*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(Sqrt[-a^2 + b^2]*((a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2))^(3/2)*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]) - (2*a*(-1/2*(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Sec[(e + f*x)/2]^2) - (a^2*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sec[(e + f*x)/2]^2*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])/(4*(b + Sqrt[-a^2 + b^2])*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]) + (a^2*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sec[(e + f*x)/2]^2*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))])/(4*(-b + Sqrt[-a^2 + b^2])*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]) + (a*b*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]*Sqrt[1 - (-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/(2*Sqrt[-a^2 + b^2])]) + (a^2*Sec[(e + f*x)/2]^2*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))])/(4*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(2*Sqrt[-a^2 + b^2])]*Sqrt[1 - (b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2])])))/(Sqrt[-a^2 + b^2]*Sqrt[(a*Sec[(e + f*x)/2]^2*(a + b*Sin[e + f*x]))/(a^2 - b^2)]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])])))/(3*a^2*(a^2 - b^2)*Sqrt[Sin[e + f*x]])))","B",0
1092,1,92,143,0.249475,"\int \cos ^4(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + b*Sin[c + d*x]),x]","\frac{-2520 a \sin (4 (c+d x))+315 a \sin (8 (c+d x))+7560 a c+7560 a d x-7560 b \cos (c+d x)-1680 b \cos (3 (c+d x))+1008 b \cos (5 (c+d x))+180 b \cos (7 (c+d x))-140 b \cos (9 (c+d x))}{322560 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,"(7560*a*c + 7560*a*d*x - 7560*b*Cos[c + d*x] - 1680*b*Cos[3*(c + d*x)] + 1008*b*Cos[5*(c + d*x)] + 180*b*Cos[7*(c + d*x)] - 140*b*Cos[9*(c + d*x)] - 2520*a*Sin[4*(c + d*x)] + 315*a*Sin[8*(c + d*x)])/(322560*d)","A",1
1093,1,77,127,0.1901576,"\int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + b*Sin[c + d*x]),x]","\frac{-1680 a \cos (c+d x)-560 a \cos (3 (c+d x))+112 a \cos (5 (c+d x))+80 a \cos (7 (c+d x))-280 b \sin (4 (c+d x))+35 b \sin (8 (c+d x))+840 b d x}{35840 d}","\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,"(840*b*d*x - 1680*a*Cos[c + d*x] - 560*a*Cos[3*(c + d*x)] + 112*a*Cos[5*(c + d*x)] + 80*a*Cos[7*(c + d*x)] - 280*b*Sin[4*(c + d*x)] + 35*b*Sin[8*(c + d*x)])/(35840*d)","A",1
1094,1,88,103,0.1873423,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x]),x]","\frac{105 a \sin (2 (c+d x))-105 a \sin (4 (c+d x))-35 a \sin (6 (c+d x))+420 a d x-315 b \cos (c+d x)-105 b \cos (3 (c+d x))+21 b \cos (5 (c+d x))+15 b \cos (7 (c+d x))}{6720 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,"(420*a*d*x - 315*b*Cos[c + d*x] - 105*b*Cos[3*(c + d*x)] + 21*b*Cos[5*(c + d*x)] + 15*b*Cos[7*(c + d*x)] + 105*a*Sin[2*(c + d*x)] - 105*a*Sin[4*(c + d*x)] - 35*a*Sin[6*(c + d*x)])/(6720*d)","A",1
1095,1,77,87,0.1671181,"\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x]),x]","-\frac{120 a \cos (c+d x)+60 a \cos (3 (c+d x))+12 a \cos (5 (c+d x))-15 b \sin (2 (c+d x))+15 b \sin (4 (c+d x))+5 b \sin (6 (c+d x))-60 b d x}{960 d}","-\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,"-1/960*(-60*b*d*x + 120*a*Cos[c + d*x] + 60*a*Cos[3*(c + d*x)] + 12*a*Cos[5*(c + d*x)] - 15*b*Sin[2*(c + d*x)] + 15*b*Sin[4*(c + d*x)] + 5*b*Sin[6*(c + d*x)])/d","A",1
1096,1,109,89,0.1240919,"\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{5 a \cos (c+d x)}{4 d}+\frac{a \cos (3 (c+d x))}{12 d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{3 b (c+d x)}{8 d}+\frac{b \sin (2 (c+d x))}{4 d}+\frac{b \sin (4 (c+d x))}{32 d}","\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*(c + d*x))/(8*d) + (5*a*Cos[c + d*x])/(4*d) + (a*Cos[3*(c + d*x)])/(12*d) - (a*Log[Cos[(c + d*x)/2]])/d + (a*Log[Sin[(c + d*x)/2]])/d + (b*Sin[2*(c + d*x)])/(4*d) + (b*Sin[4*(c + d*x)])/(32*d)","A",1
1097,1,105,83,0.3300585,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{3 a (c+d x)}{2 d}-\frac{a \sin (2 (c+d x))}{4 d}-\frac{a \cot (c+d x)}{d}+\frac{5 b \cos (c+d x)}{4 d}+\frac{b \cos (3 (c+d x))}{12 d}+\frac{b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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*(c + d*x))/(2*d) + (5*b*Cos[c + d*x])/(4*d) + (b*Cos[3*(c + d*x)])/(12*d) - (a*Cot[c + d*x])/d - (b*Log[Cos[(c + d*x)/2]])/d + (b*Log[Sin[(c + d*x)/2]])/d - (a*Sin[2*(c + d*x)])/(4*d)","A",1
1098,1,132,94,1.4759441,"\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x]),x]","-\frac{a \cos (c+d x)}{d}-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{3 b (c+d x)}{2 d}-\frac{b \sin (2 (c+d x))}{4 d}-\frac{b \cot (c+d x)}{d}","-\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*(c + d*x))/(2*d) - (a*Cos[c + d*x])/d - (b*Cot[c + d*x])/d - (a*Csc[(c + d*x)/2]^2)/(8*d) + (3*a*Log[Cos[(c + d*x)/2]])/(2*d) - (3*a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d) - (b*Sin[2*(c + d*x)])/(4*d)","A",1
1099,1,125,82,0.0424715,"\int \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x]),x]","-\frac{a \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 d}-\frac{b \cos (c+d x)}{d}-\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{3 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{3 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-((b*Cos[c + d*x])/d) - (b*Csc[(c + d*x)/2]^2)/(8*d) - (a*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + (3*b*Log[Cos[(c + d*x)/2]])/(2*d) - (3*b*Log[Sin[(c + d*x)/2]])/(2*d) + (b*Sec[(c + d*x)/2]^2)/(8*d)","C",1
1100,1,153,88,0.0498134,"\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{5 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{b \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 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{b \cot ^3(c+d x)}{3 d}+\frac{b \cot (c+d x)}{d}+b x",1,"(5*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (b*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) - (3*a*Log[Cos[(c + d*x)/2]])/(8*d) + (3*a*Log[Sin[(c + d*x)/2]])/(8*d) - (5*a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)","C",1
1101,1,135,74,0.0356041,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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{b \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{5 b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{b \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{3 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-1/5*(a*Cot[c + d*x]^5)/d + (5*b*Csc[(c + d*x)/2]^2)/(32*d) - (b*Csc[(c + d*x)/2]^4)/(64*d) - (3*b*Log[Cos[(c + d*x)/2]])/(8*d) + (3*b*Log[Sin[(c + d*x)/2]])/(8*d) - (5*b*Sec[(c + d*x)/2]^2)/(32*d) + (b*Sec[(c + d*x)/2]^4)/(64*d)","A",1
1102,1,175,98,0.0436136,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-1/5*(b*Cot[c + d*x]^5)/d - (a*Csc[(c + d*x)/2]^2)/(64*d) + (a*Csc[(c + d*x)/2]^4)/(64*d) - (a*Csc[(c + d*x)/2]^6)/(384*d) - (a*Log[Cos[(c + d*x)/2]])/(16*d) + (a*Log[Sin[(c + d*x)/2]])/(16*d) + (a*Sec[(c + d*x)/2]^2)/(64*d) - (a*Sec[(c + d*x)/2]^4)/(64*d) + (a*Sec[(c + d*x)/2]^6)/(384*d)","A",1
1103,1,239,114,0.0768411,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + b*Sin[c + d*x]),x]","-\frac{2 a \cot (c+d x)}{35 d}-\frac{a \cot (c+d x) \csc ^6(c+d x)}{7 d}+\frac{8 a \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{35 d}-\frac{b \csc ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{b \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{b \sec ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{b \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"(-2*a*Cot[c + d*x])/(35*d) - (b*Csc[(c + d*x)/2]^2)/(64*d) + (b*Csc[(c + d*x)/2]^4)/(64*d) - (b*Csc[(c + d*x)/2]^6)/(384*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(35*d) + (8*a*Cot[c + d*x]*Csc[c + d*x]^4)/(35*d) - (a*Cot[c + d*x]*Csc[c + d*x]^6)/(7*d) - (b*Log[Cos[(c + d*x)/2]])/(16*d) + (b*Log[Sin[(c + d*x)/2]])/(16*d) + (b*Sec[(c + d*x)/2]^2)/(64*d) - (b*Sec[(c + d*x)/2]^4)/(64*d) + (b*Sec[(c + d*x)/2]^6)/(384*d)","B",1
1104,1,279,136,0.0726518,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}+\frac{a \csc ^6\left(\frac{1}{2} (c+d x)\right)}{512 d}+\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}-\frac{3 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}+\frac{a \sec ^8\left(\frac{1}{2} (c+d x)\right)}{2048 d}-\frac{a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{512 d}-\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{1024 d}+\frac{3 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{512 d}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{128 d}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{128 d}-\frac{2 b \cot (c+d x)}{35 d}-\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 d}+\frac{8 b \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac{b \cot (c+d x) \csc ^2(c+d x)}{35 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,"(-2*b*Cot[c + d*x])/(35*d) - (3*a*Csc[(c + d*x)/2]^2)/(512*d) + (a*Csc[(c + d*x)/2]^4)/(1024*d) + (a*Csc[(c + d*x)/2]^6)/(512*d) - (a*Csc[(c + d*x)/2]^8)/(2048*d) - (b*Cot[c + d*x]*Csc[c + d*x]^2)/(35*d) + (8*b*Cot[c + d*x]*Csc[c + d*x]^4)/(35*d) - (b*Cot[c + d*x]*Csc[c + d*x]^6)/(7*d) - (3*a*Log[Cos[(c + d*x)/2]])/(128*d) + (3*a*Log[Sin[(c + d*x)/2]])/(128*d) + (3*a*Sec[(c + d*x)/2]^2)/(512*d) - (a*Sec[(c + d*x)/2]^4)/(1024*d) - (a*Sec[(c + d*x)/2]^6)/(512*d) + (a*Sec[(c + d*x)/2]^8)/(2048*d)","B",1
1105,1,144,301,0.9605387,"\int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{-3780 \left(2 a^2+b^2\right) \cos (c+d x)-840 \left(3 a^2+b^2\right) \cos (3 (c+d x))+504 a^2 \cos (5 (c+d x))+360 a^2 \cos (7 (c+d x))-2520 a b \sin (4 (c+d x))+315 a b \sin (8 (c+d x))+7560 a b c+7560 a b d x+504 b^2 \cos (5 (c+d x))+90 b^2 \cos (7 (c+d x))-70 b^2 \cos (9 (c+d x))}{161280 d}","\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(15 a^4-44 a^2 b^2+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,"(7560*a*b*c + 7560*a*b*d*x - 3780*(2*a^2 + b^2)*Cos[c + d*x] - 840*(3*a^2 + b^2)*Cos[3*(c + d*x)] + 504*a^2*Cos[5*(c + d*x)] + 504*b^2*Cos[5*(c + d*x)] + 360*a^2*Cos[7*(c + d*x)] + 90*b^2*Cos[7*(c + d*x)] - 70*b^2*Cos[9*(c + d*x)] - 2520*a*b*Sin[4*(c + d*x)] + 315*a*b*Sin[8*(c + d*x)])/(161280*d)","A",1
1106,1,141,278,0.6177545,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{840 a^2 \sin (2 (c+d x))-840 a^2 \sin (4 (c+d x))-280 a^2 \sin (6 (c+d x))+3360 a^2 d x-5040 a b \cos (c+d x)-1680 a b \cos (3 (c+d x))+336 a b \cos (5 (c+d x))+240 a b \cos (7 (c+d x))-420 b^2 \sin (4 (c+d x))+\frac{105}{2} b^2 \sin (8 (c+d x))+1680 b^2 c+1260 b^2 d x}{53760 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(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{\left(40 a^4-140 a^2 b^2+21 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{1344 b^2 d}+\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,"(1680*b^2*c + 3360*a^2*d*x + 1260*b^2*d*x - 5040*a*b*Cos[c + d*x] - 1680*a*b*Cos[3*(c + d*x)] + 336*a*b*Cos[5*(c + d*x)] + 240*a*b*Cos[7*(c + d*x)] + 840*a^2*Sin[2*(c + d*x)] - 840*a^2*Sin[4*(c + d*x)] - 420*b^2*Sin[4*(c + d*x)] - 280*a^2*Sin[6*(c + d*x)] + (105*b^2*Sin[8*(c + d*x)])/2)/(53760*d)","A",1
1107,1,132,129,0.4263546,"\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{-105 \left(8 a^2+3 b^2\right) \cos (c+d x)-105 \left(4 a^2+b^2\right) \cos (3 (c+d x))-84 a^2 \cos (5 (c+d x))+210 a b \sin (2 (c+d x))-210 a b \sin (4 (c+d x))-70 a b \sin (6 (c+d x))+840 a b c+840 a b d x+21 b^2 \cos (5 (c+d x))+15 b^2 \cos (7 (c+d x))}{6720 d}","-\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,"(840*a*b*c + 840*a*b*d*x - 105*(8*a^2 + 3*b^2)*Cos[c + d*x] - 105*(4*a^2 + b^2)*Cos[3*(c + d*x)] - 84*a^2*Cos[5*(c + d*x)] + 21*b^2*Cos[5*(c + d*x)] + 15*b^2*Cos[7*(c + d*x)] + 210*a*b*Sin[2*(c + d*x)] - 210*a*b*Sin[4*(c + d*x)] - 70*a*b*Sin[6*(c + d*x)])/(6720*d)","A",1
1108,1,125,116,0.5030017,"\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{30 \left(10 a^2-b^2\right) \cos (c+d x)+5 \left(4 a^2-3 b^2\right) \cos (3 (c+d x))+15 a \left(4 \left(4 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 b (c+d x)\right)+8 b \sin (2 (c+d x))+b \sin (4 (c+d x))\right)-3 b^2 \cos (5 (c+d x))}{240 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 ^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,"(30*(10*a^2 - b^2)*Cos[c + d*x] + 5*(4*a^2 - 3*b^2)*Cos[3*(c + d*x)] - 3*b^2*Cos[5*(c + d*x)] + 15*a*(4*(3*b*(c + d*x) - 4*a*Log[Cos[(c + d*x)/2]] + 4*a*Log[Sin[(c + d*x)/2]]) + 8*b*Sin[2*(c + d*x)] + b*Sin[4*(c + d*x)]))/(240*d)","A",1
1109,1,167,181,0.6985356,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","-\frac{3 a^2 (c+d x)}{2 d}-\frac{a^2 \sin (2 (c+d x))}{4 d}-\frac{a^2 \cot (c+d x)}{d}+\frac{5 a b \cos (c+d x)}{2 d}+\frac{a b \cos (3 (c+d x))}{6 d}+\frac{2 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{2 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{3 b^2 (c+d x)}{8 d}+\frac{b^2 \sin (2 (c+d x))}{4 d}+\frac{b^2 \sin (4 (c+d x))}{32 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*a^2*(c + d*x))/(2*d) + (3*b^2*(c + d*x))/(8*d) + (5*a*b*Cos[c + d*x])/(2*d) + (a*b*Cos[3*(c + d*x)])/(6*d) - (a^2*Cot[c + d*x])/d - (2*a*b*Log[Cos[(c + d*x)/2]])/d + (2*a*b*Log[Sin[(c + d*x)/2]])/d - (a^2*Sin[2*(c + d*x)])/(4*d) + (b^2*Sin[2*(c + d*x)])/(4*d) + (b^2*Sin[4*(c + d*x)])/(32*d)","A",1
1110,1,191,189,3.3567067,"\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{-6 \left(4 a^2-5 b^2\right) \cos (c+d x)+3 \left(a^2 \left(-\csc ^2\left(\frac{1}{2} (c+d x)\right)\right)+a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)-12 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 a b \sin (2 (c+d x))+8 a b \tan \left(\frac{1}{2} (c+d x)\right)-8 a b \cot \left(\frac{1}{2} (c+d x)\right)-24 a b c-24 a b d x+8 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 b^2 \cos (3 (c+d x))}{24 d}","-\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,"(-6*(4*a^2 - 5*b^2)*Cos[c + d*x] + 2*b^2*Cos[3*(c + d*x)] + 3*(-24*a*b*c - 24*a*b*d*x - 8*a*b*Cot[(c + d*x)/2] - a^2*Csc[(c + d*x)/2]^2 + 12*a^2*Log[Cos[(c + d*x)/2]] - 8*b^2*Log[Cos[(c + d*x)/2]] - 12*a^2*Log[Sin[(c + d*x)/2]] + 8*b^2*Log[Sin[(c + d*x)/2]] + a^2*Sec[(c + d*x)/2]^2 - 4*a*b*Sin[2*(c + d*x)] + 8*a*b*Tan[(c + d*x)/2]))/(24*d)","A",1
1111,1,293,133,6.1858705,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-3 b^2\right) (c+d x)}{2 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(4 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-3 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(3 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-4 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}-\frac{a^2 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}+\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}-\frac{2 a b \cos (c+d x)}{d}-\frac{a b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{4 d}+\frac{a b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{4 d}-\frac{3 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{3 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{b^2 \sin (2 (c+d x))}{4 d}","-\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,"((2*a^2 - 3*b^2)*(c + d*x))/(2*d) - (2*a*b*Cos[c + d*x])/d + ((4*a^2*Cos[(c + d*x)/2] - 3*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*d) - (a*b*Csc[(c + d*x)/2]^2)/(4*d) - (a^2*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*d) + (3*a*b*Log[Cos[(c + d*x)/2]])/d - (3*a*b*Log[Sin[(c + d*x)/2]])/d + (a*b*Sec[(c + d*x)/2]^2)/(4*d) + (Sec[(c + d*x)/2]*(-4*a^2*Sin[(c + d*x)/2] + 3*b^2*Sin[(c + d*x)/2]))/(6*d) - (b^2*Sin[2*(c + d*x)])/(4*d) + (a^2*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*d)","B",0
1112,1,270,178,2.7372524,"\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{-3 a^2 \csc ^4\left(\frac{1}{2} (c+d x)\right)+30 a^2 \csc ^2\left(\frac{1}{2} (c+d x)\right)+3 a^2 \sec ^4\left(\frac{1}{2} (c+d x)\right)-30 a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)+72 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-72 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-256 a b \tan \left(\frac{1}{2} (c+d x)\right)+256 a b \cot \left(\frac{1}{2} (c+d x)\right)-8 a b \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+128 a b \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+384 a b c+384 a b d x-192 b^2 \cos (c+d x)-24 b^2 \csc ^2\left(\frac{1}{2} (c+d x)\right)+24 b^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)-288 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+288 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{192 d}","-\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,"(384*a*b*c + 384*a*b*d*x - 192*b^2*Cos[c + d*x] + 256*a*b*Cot[(c + d*x)/2] + 30*a^2*Csc[(c + d*x)/2]^2 - 24*b^2*Csc[(c + d*x)/2]^2 - 3*a^2*Csc[(c + d*x)/2]^4 - 72*a^2*Log[Cos[(c + d*x)/2]] + 288*b^2*Log[Cos[(c + d*x)/2]] + 72*a^2*Log[Sin[(c + d*x)/2]] - 288*b^2*Log[Sin[(c + d*x)/2]] - 30*a^2*Sec[(c + d*x)/2]^2 + 24*b^2*Sec[(c + d*x)/2]^2 + 3*a^2*Sec[(c + d*x)/2]^4 + 128*a*b*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 8*a*b*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 256*a*b*Tan[(c + d*x)/2])/(192*d)","A",1
1113,1,285,209,1.5371795,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\left(640 b^2-96 a^2\right) \cot \left(\frac{1}{2} (c+d x)\right)+\csc ^4\left(\frac{1}{2} (c+d x)\right) \left(\left(21 a^2-20 b^2\right) \sin (c+d x)-30 a b\right)+96 a^2 \tan \left(\frac{1}{2} (c+d x)\right)-3 a^2 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+192 a^2 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)-336 a^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+300 a b \csc ^2\left(\frac{1}{2} (c+d x)\right)+30 a b \sec ^4\left(\frac{1}{2} (c+d x)\right)-300 a b \sec ^2\left(\frac{1}{2} (c+d x)\right)+720 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-720 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-640 b^2 \tan \left(\frac{1}{2} (c+d x)\right)+320 b^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+960 b^2 c+960 b^2 d x}{960 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{\left(3 a^4-14 a^2 b^2+b^4\right) \cot (c+d x)}{15 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,"(960*b^2*c + 960*b^2*d*x + (-96*a^2 + 640*b^2)*Cot[(c + d*x)/2] + 300*a*b*Csc[(c + d*x)/2]^2 - 720*a*b*Log[Cos[(c + d*x)/2]] + 720*a*b*Log[Sin[(c + d*x)/2]] - 300*a*b*Sec[(c + d*x)/2]^2 + 30*a*b*Sec[(c + d*x)/2]^4 - 336*a^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 320*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 192*a^2*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 - 3*a^2*Csc[(c + d*x)/2]^6*Sin[c + d*x] + Csc[(c + d*x)/2]^4*(-30*a*b + (21*a^2 - 20*b^2)*Sin[c + d*x]) + 96*a^2*Tan[(c + d*x)/2] - 640*b^2*Tan[(c + d*x)/2])/(960*d)","A",1
1114,1,319,236,0.8872957,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{-30 \left(a^2-10 b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)+6 \csc ^4\left(\frac{1}{2} (c+d x)\right) \left(5 a^2+14 a b \sin (c+d x)-5 b^2\right)+5 a^2 \sec ^6\left(\frac{1}{2} (c+d x)\right)-30 a^2 \sec ^4\left(\frac{1}{2} (c+d x)\right)+30 a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)+120 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-120 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+384 a b \tan \left(\frac{1}{2} (c+d x)\right)-384 a b \cot \left(\frac{1}{2} (c+d x)\right)-a \csc ^6\left(\frac{1}{2} (c+d x)\right) (5 a+12 b \sin (c+d x))+768 a b \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)-1344 a b \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+30 b^2 \sec ^4\left(\frac{1}{2} (c+d x)\right)-300 b^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)+720 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-720 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{1920 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(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{\left(15 a^4-80 a^2 b^2+12 b^4\right) \cot (c+d x) \csc (c+d x)}{240 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,"(-384*a*b*Cot[(c + d*x)/2] - 30*(a^2 - 10*b^2)*Csc[(c + d*x)/2]^2 - 120*a^2*Log[Cos[(c + d*x)/2]] - 720*b^2*Log[Cos[(c + d*x)/2]] + 120*a^2*Log[Sin[(c + d*x)/2]] + 720*b^2*Log[Sin[(c + d*x)/2]] + 30*a^2*Sec[(c + d*x)/2]^2 - 300*b^2*Sec[(c + d*x)/2]^2 - 30*a^2*Sec[(c + d*x)/2]^4 + 30*b^2*Sec[(c + d*x)/2]^4 + 5*a^2*Sec[(c + d*x)/2]^6 - 1344*a*b*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 768*a*b*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 - a*Csc[(c + d*x)/2]^6*(5*a + 12*b*Sin[c + d*x]) + 6*Csc[(c + d*x)/2]^4*(5*a^2 - 5*b^2 + 14*a*b*Sin[c + d*x]) + 384*a*b*Tan[(c + d*x)/2])/(1920*d)","A",1
1115,1,322,261,1.3259947,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","-\frac{\csc ^7(c+d x) \left(840 \left(6 a^2+b^2\right) \cos (c+d x)+168 \left(14 a^2-b^2\right) \cos (3 (c+d x))+336 a^2 \cos (5 (c+d x))-48 a^2 \cos (7 (c+d x))+2170 a b \sin (2 (c+d x))+3080 a b \sin (4 (c+d x))+210 a b \sin (6 (c+d x))-3675 a b \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2205 a b \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-735 a b \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+105 a b \sin (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+3675 a b \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2205 a b \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+735 a b \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-105 a b \sin (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-504 b^2 \cos (5 (c+d x))-168 b^2 \cos (7 (c+d x))\right)}{53760 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{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{\left(3 a^4-18 a^2 b^2+4 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{105 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,"-1/53760*(Csc[c + d*x]^7*(840*(6*a^2 + b^2)*Cos[c + d*x] + 168*(14*a^2 - b^2)*Cos[3*(c + d*x)] + 336*a^2*Cos[5*(c + d*x)] - 504*b^2*Cos[5*(c + d*x)] - 48*a^2*Cos[7*(c + d*x)] - 168*b^2*Cos[7*(c + d*x)] + 3675*a*b*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 3675*a*b*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 2170*a*b*Sin[2*(c + d*x)] - 2205*a*b*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 2205*a*b*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 3080*a*b*Sin[4*(c + d*x)] + 735*a*b*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 735*a*b*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 210*a*b*Sin[6*(c + d*x)] - 105*a*b*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] + 105*a*b*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/d","A",1
1116,1,204,354,1.2816846,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{2520 a^3 \sin (2 (c+d x))-2520 a^3 \sin (4 (c+d x))-840 a^3 \sin (6 (c+d x))+10080 a^3 d x-840 \left(9 a^2 b+b^3\right) \cos (3 (c+d x))-3780 b \left(6 a^2+b^2\right) \cos (c+d x)+1512 a^2 b \cos (5 (c+d x))+1080 a^2 b \cos (7 (c+d x))-3780 a b^2 \sin (4 (c+d x))+\frac{945}{2} a b^2 \sin (8 (c+d x))+15120 a b^2 c+11340 a b^2 d x+504 b^3 \cos (5 (c+d x))+90 b^3 \cos (7 (c+d x))-70 b^3 \cos (9 (c+d x))}{161280 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{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(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{\left(20 a^4-93 a^2 b^2+24 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{2520 b d}-\frac{a \left(40 a^4-188 a^2 b^2+189 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{4032 b^2 d}+\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,"(15120*a*b^2*c + 10080*a^3*d*x + 11340*a*b^2*d*x - 3780*b*(6*a^2 + b^2)*Cos[c + d*x] - 840*(9*a^2*b + b^3)*Cos[3*(c + d*x)] + 1512*a^2*b*Cos[5*(c + d*x)] + 504*b^3*Cos[5*(c + d*x)] + 1080*a^2*b*Cos[7*(c + d*x)] + 90*b^3*Cos[7*(c + d*x)] - 70*b^3*Cos[9*(c + d*x)] + 2520*a^3*Sin[2*(c + d*x)] - 2520*a^3*Sin[4*(c + d*x)] - 3780*a*b^2*Sin[4*(c + d*x)] - 840*a^3*Sin[6*(c + d*x)] + (945*a*b^2*Sin[8*(c + d*x)])/2)/(161280*d)","A",1
1117,1,189,194,0.8706977,"\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{-280 \left(4 a^3+3 a b^2\right) \cos (3 (c+d x))-224 a^3 \cos (5 (c+d x))-280 a \left(8 a^2+9 b^2\right) \cos (c+d x)+840 a^2 b \sin (2 (c+d x))-840 a^2 b \sin (4 (c+d x))-280 a^2 b \sin (6 (c+d x))+3360 a^2 b c+3360 a^2 b d x+168 a b^2 \cos (5 (c+d x))+120 a b^2 \cos (7 (c+d x))-140 b^3 \sin (4 (c+d x))+\frac{35}{2} b^3 \sin (8 (c+d x))+840 b^3 c+420 b^3 d x}{17920 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,"(3360*a^2*b*c + 840*b^3*c + 3360*a^2*b*d*x + 420*b^3*d*x - 280*a*(8*a^2 + 9*b^2)*Cos[c + d*x] - 280*(4*a^3 + 3*a*b^2)*Cos[3*(c + d*x)] - 224*a^3*Cos[5*(c + d*x)] + 168*a*b^2*Cos[5*(c + d*x)] + 120*a*b^2*Cos[7*(c + d*x)] + 840*a^2*b*Sin[2*(c + d*x)] - 840*a^2*b*Sin[4*(c + d*x)] - 140*b^3*Sin[4*(c + d*x)] - 280*a^2*b*Sin[6*(c + d*x)] + (35*b^3*Sin[8*(c + d*x)])/2)/(17920*d)","A",1
1118,1,191,250,0.525905,"\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{20 \left(4 a^3-9 a b^2\right) \cos (3 (c+d x))+960 a^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-960 a^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+120 a \left(10 a^2-3 b^2\right) \cos (c+d x)+720 a^2 b \sin (2 (c+d x))+90 a^2 b \sin (4 (c+d x))+1080 a^2 b c+1080 a^2 b d x-36 a b^2 \cos (5 (c+d x))+15 b^3 \sin (2 (c+d x))-15 b^3 \sin (4 (c+d x))-5 b^3 \sin (6 (c+d x))+60 b^3 c+60 b^3 d x}{960 d}","-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{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{1}{16} b x \left(18 a^2+b^2\right)-\frac{a \left(2 a^4-43 a^2 b^2+36 b^4\right) \cos (c+d x)}{60 b^2 d}-\frac{\left(4 a^4-84 a^2 b^2+15 b^4\right) \sin (c+d x) \cos (c+d x)}{240 b 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,"(1080*a^2*b*c + 60*b^3*c + 1080*a^2*b*d*x + 60*b^3*d*x + 120*a*(10*a^2 - 3*b^2)*Cos[c + d*x] + 20*(4*a^3 - 9*a*b^2)*Cos[3*(c + d*x)] - 36*a*b^2*Cos[5*(c + d*x)] - 960*a^3*Log[Cos[(c + d*x)/2]] + 960*a^3*Log[Sin[(c + d*x)/2]] + 720*a^2*b*Sin[2*(c + d*x)] + 15*b^3*Sin[2*(c + d*x)] + 90*a^2*b*Sin[4*(c + d*x)] - 15*b^3*Sin[4*(c + d*x)] - 5*b^3*Sin[6*(c + d*x)])/(960*d)","A",1
1119,1,194,229,3.0368079,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{-40 a^3 \sin (2 (c+d x))+80 a^3 \tan \left(\frac{1}{2} (c+d x)\right)-80 a^3 \cot \left(\frac{1}{2} (c+d x)\right)-240 a^3 c-240 a^3 d x+10 \left(4 a^2 b-b^3\right) \cos (3 (c+d x))-20 b \left(b^2-30 a^2\right) \cos (c+d x)+480 a^2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-480 a^2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+120 a b^2 \sin (2 (c+d x))+15 a b^2 \sin (4 (c+d x))+180 a b^2 c+180 a b^2 d x-2 b^3 \cos (5 (c+d x))}{160 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{\left(a^4+56 a^2 b^2-2 b^4\right) \cos (c+d x)}{10 b 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,"(-240*a^3*c + 180*a*b^2*c - 240*a^3*d*x + 180*a*b^2*d*x - 20*b*(-30*a^2 + b^2)*Cos[c + d*x] + 10*(4*a^2*b - b^3)*Cos[3*(c + d*x)] - 2*b^3*Cos[5*(c + d*x)] - 80*a^3*Cot[(c + d*x)/2] - 480*a^2*b*Log[Cos[(c + d*x)/2]] + 480*a^2*b*Log[Sin[(c + d*x)/2]] - 40*a^3*Sin[2*(c + d*x)] + 120*a*b^2*Sin[2*(c + d*x)] + 15*a*b^2*Sin[4*(c + d*x)] + 80*a^3*Tan[(c + d*x)/2])/(160*d)","A",1
1120,1,252,231,6.160631,"\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","-\frac{3 \left(a^3-2 a b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{3 \left(a^3-2 a b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{a^3 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{3 b \left(b^2-12 a^2\right) (c+d x)}{8 d}+\frac{b \left(b^2-3 a^2\right) \sin (2 (c+d x))}{4 d}-\frac{a \left(4 a^2-15 b^2\right) \cos (c+d x)}{4 d}+\frac{3 a^2 b \tan \left(\frac{1}{2} (c+d x)\right)}{2 d}-\frac{3 a^2 b \cot \left(\frac{1}{2} (c+d x)\right)}{2 d}+\frac{a b^2 \cos (3 (c+d x))}{4 d}+\frac{b^3 \sin (4 (c+d x))}{32 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)*(c + d*x))/(8*d) - (a*(4*a^2 - 15*b^2)*Cos[c + d*x])/(4*d) + (a*b^2*Cos[3*(c + d*x)])/(4*d) - (3*a^2*b*Cot[(c + d*x)/2])/(2*d) - (a^3*Csc[(c + d*x)/2]^2)/(8*d) + (3*(a^3 - 2*a*b^2)*Log[Cos[(c + d*x)/2]])/(2*d) - (3*(a^3 - 2*a*b^2)*Log[Sin[(c + d*x)/2]])/(2*d) + (a^3*Sec[(c + d*x)/2]^2)/(8*d) + (b*(-3*a^2 + b^2)*Sin[2*(c + d*x)])/(4*d) + (b^3*Sin[4*(c + d*x)])/(32*d) + (3*a^2*b*Tan[(c + d*x)/2])/(2*d)","A",1
1121,1,355,194,6.24959,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(4 a^3 \cos \left(\frac{1}{2} (c+d x)\right)-9 a b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(9 a b^2 \sin \left(\frac{1}{2} (c+d x)\right)-4 a^3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}-\frac{a^3 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}+\frac{a^3 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}+\frac{\left(2 b^3-9 a^2 b\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{\left(9 a^2 b-2 b^3\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{a \left(2 a^2-9 b^2\right) (c+d x)}{2 d}+\frac{b \left(5 b^2-12 a^2\right) \cos (c+d x)}{4 d}-\frac{3 a^2 b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{3 a^2 b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{3 a b^2 \sin (2 (c+d x))}{4 d}+\frac{b^3 \cos (3 (c+d x))}{12 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^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{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*(2*a^2 - 9*b^2)*(c + d*x))/(2*d) + (b*(-12*a^2 + 5*b^2)*Cos[c + d*x])/(4*d) + (b^3*Cos[3*(c + d*x)])/(12*d) + ((4*a^3*Cos[(c + d*x)/2] - 9*a*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*d) - (3*a^2*b*Csc[(c + d*x)/2]^2)/(8*d) - (a^3*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*d) + ((9*a^2*b - 2*b^3)*Log[Cos[(c + d*x)/2]])/(2*d) + ((-9*a^2*b + 2*b^3)*Log[Sin[(c + d*x)/2]])/(2*d) + (3*a^2*b*Sec[(c + d*x)/2]^2)/(8*d) + (Sec[(c + d*x)/2]*(-4*a^3*Sin[(c + d*x)/2] + 9*a*b^2*Sin[(c + d*x)/2]))/(6*d) - (3*a*b^2*Sin[2*(c + d*x)])/(4*d) + (a^3*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*d)","A",0
1122,1,381,187,6.2923223,"\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{\left(5 a^3-12 a b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{\left(12 a b^2-5 a^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 \left(a^3-12 a b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{3 \left(a^3-12 a b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{a^3 \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a^3 \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(4 a^2 b \cos \left(\frac{1}{2} (c+d x)\right)-b^3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(b^3 \sin \left(\frac{1}{2} (c+d x)\right)-4 a^2 b \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{3 b \left(b^2-2 a^2\right) (c+d x)}{2 d}-\frac{a^2 b \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a^2 b \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{3 a b^2 \cos (c+d x)}{d}-\frac{b^3 \sin (2 (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)*(c + d*x))/(2*d) - (3*a*b^2*Cos[c + d*x])/d + ((4*a^2*b*Cos[(c + d*x)/2] - b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(2*d) + ((5*a^3 - 12*a*b^2)*Csc[(c + d*x)/2]^2)/(32*d) - (a^2*b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(8*d) - (a^3*Csc[(c + d*x)/2]^4)/(64*d) - (3*(a^3 - 12*a*b^2)*Log[Cos[(c + d*x)/2]])/(8*d) + (3*(a^3 - 12*a*b^2)*Log[Sin[(c + d*x)/2]])/(8*d) + ((-5*a^3 + 12*a*b^2)*Sec[(c + d*x)/2]^2)/(32*d) + (a^3*Sec[(c + d*x)/2]^4)/(64*d) + (Sec[(c + d*x)/2]*(-4*a^2*b*Sin[(c + d*x)/2] + b^3*Sin[(c + d*x)/2]))/(2*d) - (b^3*Sin[2*(c + d*x)])/(4*d) + (a^2*b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(8*d)","B",0
1123,1,405,227,1.2972064,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{-32 \left(a^3-20 a b^2\right) \cot \left(\frac{1}{2} (c+d x)\right)+32 a^3 \tan \left(\frac{1}{2} (c+d x)\right)-a^3 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+64 a^3 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+7 a^3 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-112 a^3 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-15 a^2 b \csc ^4\left(\frac{1}{2} (c+d x)\right)+150 a^2 b \csc ^2\left(\frac{1}{2} (c+d x)\right)+15 a^2 b \sec ^4\left(\frac{1}{2} (c+d x)\right)-150 a^2 b \sec ^2\left(\frac{1}{2} (c+d x)\right)+360 a^2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-360 a^2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-640 a b^2 \tan \left(\frac{1}{2} (c+d x)\right)-20 a b^2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+320 a b^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+960 a b^2 c+960 a b^2 d x-320 b^3 \cos (c+d x)-40 b^3 \csc ^2\left(\frac{1}{2} (c+d x)\right)+40 b^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)-480 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+480 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{320 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^3 \left(83 a^2+2 b^2\right) \cos (c+d x)}{40 a^2 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,"(960*a*b^2*c + 960*a*b^2*d*x - 320*b^3*Cos[c + d*x] - 32*(a^3 - 20*a*b^2)*Cot[(c + d*x)/2] + 150*a^2*b*Csc[(c + d*x)/2]^2 - 40*b^3*Csc[(c + d*x)/2]^2 - 15*a^2*b*Csc[(c + d*x)/2]^4 - 360*a^2*b*Log[Cos[(c + d*x)/2]] + 480*b^3*Log[Cos[(c + d*x)/2]] + 360*a^2*b*Log[Sin[(c + d*x)/2]] - 480*b^3*Log[Sin[(c + d*x)/2]] - 150*a^2*b*Sec[(c + d*x)/2]^2 + 40*b^3*Sec[(c + d*x)/2]^2 + 15*a^2*b*Sec[(c + d*x)/2]^4 - 112*a^3*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 320*a*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 64*a^3*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 + 7*a^3*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 20*a*b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - a^3*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 32*a^3*Tan[(c + d*x)/2] - 640*a*b^2*Tan[(c + d*x)/2])/(320*d)","A",1
1124,1,408,275,1.8654909,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","\frac{-30 \left(a^3-30 a b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)+5 a^3 \sec ^6\left(\frac{1}{2} (c+d x)\right)-30 a^3 \sec ^4\left(\frac{1}{2} (c+d x)\right)+30 a^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)+120 a^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-120 a^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-64 \left(9 a^2 b-20 b^3\right) \cot \left(\frac{1}{2} (c+d x)\right)+576 a^2 b \tan \left(\frac{1}{2} (c+d x)\right)-a^2 \csc ^6\left(\frac{1}{2} (c+d x)\right) (5 a+18 b \sin (c+d x))-2016 a^2 b \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+36 a^2 b \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)+2 \csc ^4\left(\frac{1}{2} (c+d x)\right) \left(15 \left(a^3-3 a b^2\right)+b \left(63 a^2-20 b^2\right) \sin (c+d x)\right)+90 a b^2 \sec ^4\left(\frac{1}{2} (c+d x)\right)-900 a b^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)+2160 a b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2160 a b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-1280 b^3 \tan \left(\frac{1}{2} (c+d x)\right)+640 b^3 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+1920 b^3 c+1920 b^3 d x}{1920 d}","-\frac{a \left(a^2+18 b^2\right) \tanh ^{-1}(\cos (c+d x))}{16 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{b \left(36 a^4-43 a^2 b^2+2 b^4\right) \cot (c+d x)}{60 a^2 d}-\frac{\left(15 a^4-84 a^2 b^2+4 b^4\right) \cot (c+d x) \csc (c+d x)}{240 a 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,"(1920*b^3*c + 1920*b^3*d*x - 64*(9*a^2*b - 20*b^3)*Cot[(c + d*x)/2] - 30*(a^3 - 30*a*b^2)*Csc[(c + d*x)/2]^2 - 120*a^3*Log[Cos[(c + d*x)/2]] - 2160*a*b^2*Log[Cos[(c + d*x)/2]] + 120*a^3*Log[Sin[(c + d*x)/2]] + 2160*a*b^2*Log[Sin[(c + d*x)/2]] + 30*a^3*Sec[(c + d*x)/2]^2 - 900*a*b^2*Sec[(c + d*x)/2]^2 - 30*a^3*Sec[(c + d*x)/2]^4 + 90*a*b^2*Sec[(c + d*x)/2]^4 + 5*a^3*Sec[(c + d*x)/2]^6 - 2016*a^2*b*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 640*b^3*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - a^2*Csc[(c + d*x)/2]^6*(5*a + 18*b*Sin[c + d*x]) + 2*Csc[(c + d*x)/2]^4*(15*(a^3 - 3*a*b^2) + b*(63*a^2 - 20*b^2)*Sin[c + d*x]) + 576*a^2*b*Tan[(c + d*x)/2] - 1280*b^3*Tan[(c + d*x)/2] + 36*a^2*b*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(1920*d)","A",1
1125,1,324,303,0.9705075,"\int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","-\frac{112 a^3 \cos (5 (c+d x)) \csc ^7(c+d x)-16 a^3 \cos (7 (c+d x)) \csc ^7(c+d x)+56 a \left(14 a^2-3 b^2\right) \cos (3 (c+d x)) \csc ^7(c+d x)+70 \cot (c+d x) \csc ^6(c+d x) \left(b \left(31 a^2-18 b^2\right) \sin (c+d x)+12 a \left(2 a^2+b^2\right)\right)-3360 a^2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+3360 a^2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+1540 a^2 b \sin (4 (c+d x)) \csc ^7(c+d x)+105 a^2 b \sin (6 (c+d x)) \csc ^7(c+d x)-504 a b^2 \cos (5 (c+d x)) \csc ^7(c+d x)-168 a b^2 \cos (7 (c+d x)) \csc ^7(c+d x)-6720 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6720 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+840 b^3 \sin (4 (c+d x)) \csc ^7(c+d x)-350 b^3 \sin (6 (c+d x)) \csc ^7(c+d x)}{17920 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(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{\left(4 a^4-19 a^2 b^2+2 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{140 a d}-\frac{b \left(105 a^4-116 a^2 b^2+12 b^4\right) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}",1,"-1/17920*(56*a*(14*a^2 - 3*b^2)*Cos[3*(c + d*x)]*Csc[c + d*x]^7 + 112*a^3*Cos[5*(c + d*x)]*Csc[c + d*x]^7 - 504*a*b^2*Cos[5*(c + d*x)]*Csc[c + d*x]^7 - 16*a^3*Cos[7*(c + d*x)]*Csc[c + d*x]^7 - 168*a*b^2*Cos[7*(c + d*x)]*Csc[c + d*x]^7 + 3360*a^2*b*Log[Cos[(c + d*x)/2]] + 6720*b^3*Log[Cos[(c + d*x)/2]] - 3360*a^2*b*Log[Sin[(c + d*x)/2]] - 6720*b^3*Log[Sin[(c + d*x)/2]] + 70*Cot[c + d*x]*Csc[c + d*x]^6*(12*a*(2*a^2 + b^2) + b*(31*a^2 - 18*b^2)*Sin[c + d*x]) + 1540*a^2*b*Csc[c + d*x]^7*Sin[4*(c + d*x)] + 840*b^3*Csc[c + d*x]^7*Sin[4*(c + d*x)] + 105*a^2*b*Csc[c + d*x]^7*Sin[6*(c + d*x)] - 350*b^3*Csc[c + d*x]^7*Sin[6*(c + d*x)])/d","A",1
1126,1,268,334,1.6015603,"\int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","-\frac{-6720 a \left(a^2+8 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6720 a \left(a^2+8 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\csc ^8(c+d x) \left(35 \left(333 a^3+104 a b^2\right) \cos (3 (c+d x))+805 a^3 \cos (5 (c+d x))-105 a^3 \cos (7 (c+d x))+35 a \left(671 a^2+248 b^2\right) \cos (c+d x)+21504 a^2 b \sin (2 (c+d x))+16128 a^2 b \sin (4 (c+d x))+3072 a^2 b \sin (6 (c+d x))-384 a^2 b \sin (8 (c+d x))-11480 a b^2 \cos (5 (c+d x))-840 a b^2 \cos (7 (c+d x))+2688 b^3 \sin (2 (c+d x))+896 b^3 \sin (4 (c+d x))-896 b^3 \sin (6 (c+d x))-448 b^3 \sin (8 (c+d x))\right)}{286720 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{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{\left(35 a^4-148 a^2 b^2+24 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}-\frac{b \left(24 a^4-25 a^2 b^2+4 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}",1,"-1/286720*(6720*a*(a^2 + 8*b^2)*Log[Cos[(c + d*x)/2]] - 6720*a*(a^2 + 8*b^2)*Log[Sin[(c + d*x)/2]] + Csc[c + d*x]^8*(35*a*(671*a^2 + 248*b^2)*Cos[c + d*x] + 35*(333*a^3 + 104*a*b^2)*Cos[3*(c + d*x)] + 805*a^3*Cos[5*(c + d*x)] - 11480*a*b^2*Cos[5*(c + d*x)] - 105*a^3*Cos[7*(c + d*x)] - 840*a*b^2*Cos[7*(c + d*x)] + 21504*a^2*b*Sin[2*(c + d*x)] + 2688*b^3*Sin[2*(c + d*x)] + 16128*a^2*b*Sin[4*(c + d*x)] + 896*b^3*Sin[4*(c + d*x)] + 3072*a^2*b*Sin[6*(c + d*x)] - 896*b^3*Sin[6*(c + d*x)] - 384*a^2*b*Sin[8*(c + d*x)] - 448*b^3*Sin[8*(c + d*x)]))/d","A",1
1127,1,378,307,4.5734893,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{960 a^2 \left(2 a^4-3 a^2 b^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)}{\sqrt{a^2-b^2}}-\frac{960 a^6 c+960 a^6 d x+960 a^5 b c \sin (c+d x)+960 a^5 b d x \sin (c+d x)+240 a^4 b^2 \sin (2 (c+d x))-960 a^4 b^2 c-960 a^4 b^2 d x-960 a^3 b^3 c \sin (c+d x)-960 a^3 b^3 d x \sin (c+d x)+5 \left(8 a^3 b^3-5 a b^5\right) \cos (3 (c+d x))-200 a^2 b^4 \sin (2 (c+d x))-10 a^2 b^4 \sin (4 (c+d x))+120 a^2 b^4 c+120 a^2 b^4 d x+60 a b \left(16 a^4-14 a^2 b^2+b^4\right) \cos (c+d x)+120 a b^5 c \sin (c+d x)+120 a b^5 d x \sin (c+d x)-3 a b^5 \cos (5 (c+d x))+5 b^6 \sin (2 (c+d x))+4 b^6 \sin (4 (c+d x))+b^6 \sin (6 (c+d x))}{a+b \sin (c+d x)}}{160 b^7 d}","-\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{3 a \left(4 a^2-3 b^2\right) \sin (c+d x) \cos (c+d x)}{4 b^5 d}-\frac{\left(10 a^2-7 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{5 b^4 d}+\frac{\left(3 a^2-2 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{2 a b^3 d}+\frac{6 a^2 \left(2 a^4-3 a^2 b^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{3 a x \left(8 a^4-8 a^2 b^2+b^4\right)}{4 b^7}-\frac{\left(30 a^4-25 a^2 b^2+b^4\right) \cos (c+d x)}{5 b^6 d}-\frac{\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}",1,"((960*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]])/Sqrt[a^2 - b^2] - (960*a^6*c - 960*a^4*b^2*c + 120*a^2*b^4*c + 960*a^6*d*x - 960*a^4*b^2*d*x + 120*a^2*b^4*d*x + 60*a*b*(16*a^4 - 14*a^2*b^2 + b^4)*Cos[c + d*x] + 5*(8*a^3*b^3 - 5*a*b^5)*Cos[3*(c + d*x)] - 3*a*b^5*Cos[5*(c + d*x)] + 960*a^5*b*c*Sin[c + d*x] - 960*a^3*b^3*c*Sin[c + d*x] + 120*a*b^5*c*Sin[c + d*x] + 960*a^5*b*d*x*Sin[c + d*x] - 960*a^3*b^3*d*x*Sin[c + d*x] + 120*a*b^5*d*x*Sin[c + d*x] + 240*a^4*b^2*Sin[2*(c + d*x)] - 200*a^2*b^4*Sin[2*(c + d*x)] + 5*b^6*Sin[2*(c + d*x)] - 10*a^2*b^4*Sin[4*(c + d*x)] + 4*b^6*Sin[4*(c + d*x)] + b^6*Sin[6*(c + d*x)])/(a + b*Sin[c + d*x]))/(160*b^7*d)","A",1
1128,1,325,267,3.6726463,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{960 a^5 c+960 a^5 d x+960 a^4 b c \sin (c+d x)+960 a^4 b d x \sin (c+d x)+240 a^3 b^2 \sin (2 (c+d x))-864 a^3 b^2 c-864 a^3 b^2 d x-864 a^2 b^3 c \sin (c+d x)-864 a^2 b^3 d x \sin (c+d x)+\left(40 a^2 b^3-21 b^5\right) \cos (3 (c+d x))+24 b \left(40 a^4-31 a^2 b^2+b^4\right) \cos (c+d x)-176 a b^4 \sin (2 (c+d x))-10 a b^4 \sin (4 (c+d x))+72 a b^4 c+72 a b^4 d x+72 b^5 c \sin (c+d x)+72 b^5 d x \sin (c+d x)-3 b^5 \cos (5 (c+d x))}{a+b \sin (c+d x)}-\frac{384 a \left(5 a^4-7 a^2 b^2+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)}{\sqrt{a^2-b^2}}}{192 b^6 d}","-\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{a \left(15 a^2-11 b^2\right) \cos (c+d x)}{3 b^5 d}-\frac{\left(20 a^2-13 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 d}+\frac{\left(5 a^2-3 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{3 a b^3 d}-\frac{2 a \left(5 a^4-7 a^2 b^2+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{x \left(40 a^4-36 a^2 b^2+3 b^4\right)}{8 b^6}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b^2 d}",1,"((-384*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]])/Sqrt[a^2 - b^2] + (960*a^5*c - 864*a^3*b^2*c + 72*a*b^4*c + 960*a^5*d*x - 864*a^3*b^2*d*x + 72*a*b^4*d*x + 24*b*(40*a^4 - 31*a^2*b^2 + b^4)*Cos[c + d*x] + (40*a^2*b^3 - 21*b^5)*Cos[3*(c + d*x)] - 3*b^5*Cos[5*(c + d*x)] + 960*a^4*b*c*Sin[c + d*x] - 864*a^2*b^3*c*Sin[c + d*x] + 72*b^5*c*Sin[c + d*x] + 960*a^4*b*d*x*Sin[c + d*x] - 864*a^2*b^3*d*x*Sin[c + d*x] + 72*b^5*d*x*Sin[c + d*x] + 240*a^3*b^2*Sin[2*(c + d*x)] - 176*a*b^4*Sin[2*(c + d*x)] - 10*a*b^4*Sin[4*(c + d*x)])/(a + b*Sin[c + d*x]))/(192*b^6*d)","A",1
1129,1,247,163,2.5135189,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{48 \left(4 a^4-5 a^2 b^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)}{\sqrt{a^2-b^2}}+\frac{-96 a^4 c-96 a^4 d x+\left(60 a b^3-96 a^3 b\right) \cos (c+d x)-96 a^3 b c \sin (c+d x)-96 a^3 b d x \sin (c+d x)-24 a^2 b^2 \sin (2 (c+d x))+72 a^2 b^2 c+72 a^2 b^2 d x+72 a b^3 c \sin (c+d x)+72 a b^3 d x \sin (c+d x)-4 a b^3 \cos (3 (c+d x))+14 b^4 \sin (2 (c+d x))+b^4 \sin (4 (c+d x))}{a+b \sin (c+d x)}}{24 b^5 d}","-\frac{a x \left(4 a^2-3 b^2\right)}{b^5}-\frac{\cos (c+d x) \left(4 a^2-2 a b \sin (c+d x)-b^2\right)}{b^4 d}+\frac{2 \left(4 a^4-5 a^2 b^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^5 d \sqrt{a^2-b^2}}+\frac{\cos ^3(c+d x) (4 a+b \sin (c+d x))}{3 b^2 d (a+b \sin (c+d x))}",1,"((48*(4*a^4 - 5*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (-96*a^4*c + 72*a^2*b^2*c - 96*a^4*d*x + 72*a^2*b^2*d*x + (-96*a^3*b + 60*a*b^3)*Cos[c + d*x] - 4*a*b^3*Cos[3*(c + d*x)] - 96*a^3*b*c*Sin[c + d*x] + 72*a*b^3*c*Sin[c + d*x] - 96*a^3*b*d*x*Sin[c + d*x] + 72*a*b^3*d*x*Sin[c + d*x] - 24*a^2*b^2*Sin[2*(c + d*x)] + 14*b^4*Sin[2*(c + d*x)] + b^4*Sin[4*(c + d*x)])/(a + b*Sin[c + d*x]))/(24*b^5*d)","A",1
1130,1,161,137,0.7122174,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{\left(b^2-a^2\right) \cos (c+d x)}{a b^2 (a+b \sin (c+d x))}+\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^2}-\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^2}+\frac{2 \left(2 a^4-a^2 b^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^2 b^3 \sqrt{a^2-b^2}}-\frac{2 a (c+d x)}{b^3}-\frac{\cos (c+d x)}{b^2}}{d}","-\frac{\left(a^2-b^2\right) \cos (c+d x)}{a b^2 d (a+b \sin (c+d 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{\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*(c + d*x))/b^3 + (2*(2*a^4 - a^2*b^2 - b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^3*Sqrt[a^2 - b^2]) - Cos[c + d*x]/b^2 - Log[Cos[(c + d*x)/2]]/a^2 + Log[Sin[(c + d*x)/2]]/a^2 + ((-a^2 + b^2)*Cos[c + d*x])/(a*b^2*(a + b*Sin[c + d*x])))/d","A",1
1131,1,182,154,1.7864783,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{-\frac{4 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}+\frac{4 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}+\frac{2 \left(a^2-b^2\right) \cos (c+d x)}{a^2 b (a+b \sin (c+d x))}+\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{a^2}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{a^2}-\frac{4 \left(a^4+a^2 b^2-2 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 \sqrt{a^2-b^2}}+\frac{2 (c+d x)}{b^2}}{2 d}","\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{\left(a^2-2 b^2\right) \cos (c+d x)}{a^2 b d (a+b \sin (c+d x))}-\frac{2 \left(a^4+a^2 b^2-2 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{\cot (c+d x)}{a d (a+b \sin (c+d x))}+\frac{x}{b^2}",1,"((2*(c + d*x))/b^2 - (4*(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]) - Cot[(c + d*x)/2]/a^2 + (4*b*Log[Cos[(c + d*x)/2]])/a^3 - (4*b*Log[Sin[(c + d*x)/2]])/a^3 + (2*(a^2 - b^2)*Cos[c + d*x])/(a^2*b*(a + b*Sin[c + d*x])) + Tan[(c + d*x)/2]/a^2)/(2*d)","A",1
1132,1,191,158,4.3006187,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","\frac{48 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)-12 \left(a^2-2 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 \left(a^2-2 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{8 a \left(b^2-a^2\right) \cos (c+d x)}{a+b \sin (c+d x)}-a^2 \csc ^2\left(\frac{1}{2} (c+d x)\right)+a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)-8 a b \tan \left(\frac{1}{2} (c+d x)\right)+8 a b \cot \left(\frac{1}{2} (c+d x)\right)}{8 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)}+\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{3 \left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}-\frac{\left(a^2-b^2\right) \cos (c+d x)}{a^3 d (a+b \sin (c+d x))}",1,"(48*b*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 8*a*b*Cot[(c + d*x)/2] - a^2*Csc[(c + d*x)/2]^2 + 12*(a^2 - 2*b^2)*Log[Cos[(c + d*x)/2]] - 12*(a^2 - 2*b^2)*Log[Sin[(c + d*x)/2]] + a^2*Sec[(c + d*x)/2]^2 + (8*a*(-a^2 + b^2)*Cos[c + d*x])/(a + b*Sin[c + d*x]) - 8*a*b*Tan[(c + d*x)/2])/(8*a^4*d)","A",1
1133,1,403,238,6.2148702,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^2,x]","\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{4 a^3 d}-\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{4 a^3 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}+\frac{\left(3 a^2 b-4 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{\left(4 b^3-3 a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{a^4 d (a+b \sin (c+d x))}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(4 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-9 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(9 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-4 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{2 \left(a^4-5 a^2 b^2+4 b^4\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}","\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{b \left(3 a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{\left(7 a^2-12 b^2\right) \cot (c+d x)}{3 a^4 d}-\frac{\left(a^2-2 b^2\right) \cot (c+d x) \csc (c+d x)}{a^3 b d}+\frac{2 \left(a^4-5 a^2 b^2+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{\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[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) + ((4*a^2*Cos[(c + d*x)/2] - 9*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + (b*Csc[(c + d*x)/2]^2)/(4*a^3*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^2*d) + ((-3*a^2*b + 4*b^3)*Log[Cos[(c + d*x)/2]])/(a^5*d) + ((3*a^2*b - 4*b^3)*Log[Sin[(c + d*x)/2]])/(a^5*d) - (b*Sec[(c + d*x)/2]^2)/(4*a^3*d) + (Sec[(c + d*x)/2]*(-4*a^2*Sin[(c + d*x)/2] + 9*b^2*Sin[(c + d*x)/2]))/(6*a^4*d) + (a^2*b*Cos[c + d*x] - b^3*Cos[c + d*x])/(a^4*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^2*d)","A",0
1134,1,496,292,6.2941877,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{b \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{12 a^3 d}-\frac{b \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{12 a^3 d}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 a^2 d}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 a^2 d}-\frac{2 \csc \left(\frac{1}{2} (c+d x)\right) \left(2 a^2 b \cos \left(\frac{1}{2} (c+d x)\right)-3 b^3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{3 a^5 d}+\frac{2 \sec \left(\frac{1}{2} (c+d x)\right) \left(2 a^2 b \sin \left(\frac{1}{2} (c+d x)\right)-3 b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 a^5 d}+\frac{b^4 \cos (c+d x)-a^2 b^2 \cos (c+d x)}{a^5 d (a+b \sin (c+d x))}+\frac{\left(5 a^2-12 b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 a^4 d}+\frac{\left(12 b^2-5 a^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 a^4 d}+\frac{\left(3 a^4-36 a^2 b^2+40 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^6 d}+\frac{\left(-3 a^4+36 a^2 b^2-40 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^6 d}-\frac{2 b \left(2 a^4-7 a^2 b^2+5 b^4\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2}}","\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{b \left(11 a^2-15 b^2\right) \cot (c+d x)}{3 a^5 d}+\frac{\left(13 a^2-20 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 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{2 b \left(2 a^4-7 a^2 b^2+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{\left(3 a^4-36 a^2 b^2+40 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\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[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) - (2*(2*a^2*b*Cos[(c + d*x)/2] - 3*b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(3*a^5*d) + ((5*a^2 - 12*b^2)*Csc[(c + d*x)/2]^2)/(32*a^4*d) + (b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(12*a^3*d) - Csc[(c + d*x)/2]^4/(64*a^2*d) + ((-3*a^4 + 36*a^2*b^2 - 40*b^4)*Log[Cos[(c + d*x)/2]])/(8*a^6*d) + ((3*a^4 - 36*a^2*b^2 + 40*b^4)*Log[Sin[(c + d*x)/2]])/(8*a^6*d) + ((-5*a^2 + 12*b^2)*Sec[(c + d*x)/2]^2)/(32*a^4*d) + Sec[(c + d*x)/2]^4/(64*a^2*d) + (2*Sec[(c + d*x)/2]*(2*a^2*b*Sin[(c + d*x)/2] - 3*b^3*Sin[(c + d*x)/2]))/(3*a^5*d) + (-(a^2*b^2*Cos[c + d*x]) + b^4*Cos[c + d*x])/(a^5*d*(a + b*Sin[c + d*x])) - (b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(12*a^3*d)","A",0
1135,1,1250,331,10.5617255,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]","-\frac{-\frac{6 \left(-8 (c+d x)+\frac{2 a \left(8 a^4-20 b^2 a^2+15 b^4\right) \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{3 b \left(4 a^4-7 b^2 a^2+2 b^4\right) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{a b \left(4 a^2-3 b^2\right) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}\right)}{b^3}+\frac{6 \left(\frac{6 a b \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{\cos (c+d x) \left(a \left(2 a^2+b^2\right)+b \left(a^2+2 b^2\right) \sin (c+d x)\right)}{(a+b \sin (c+d x))^2}\right)}{(a-b)^2 (a+b)^2}+\frac{2 \left(-8 \sin (2 (c+d x)) b^2+96 a \cos (c+d x) b+\frac{\left(112 a^6-220 b^2 a^4+115 b^4 a^2-10 b^6\right) \cos (c+d x) b}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{a \left(-16 a^4+20 b^2 a^2-5 b^4\right) \cos (c+d x) b}{(a-b) (a+b) (a+b \sin (c+d x))^2}-24 \left(b^2-8 a^2\right) (c+d x)-\frac{6 a \left(64 a^6-168 b^2 a^4+140 b^4 a^2-35 b^6\right) \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}\right)}{b^5}+\frac{\frac{12 a \left(640 a^8-1920 b^2 a^6+2016 b^4 a^4-840 b^6 a^2+105 b^8\right) \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{-3840 (c+d x) a^{10}-3840 b \cos (c+d x) a^9-7680 b (c+d x) \sin (c+d x) a^9+7680 b^2 (c+d x) a^8+1920 b^2 (c+d x) \cos (2 (c+d x)) a^8-2880 b^2 \sin (2 (c+d x)) a^8+8640 b^3 \cos (c+d x) a^7+320 b^3 \cos (3 (c+d x)) a^7+19200 b^3 (c+d x) \sin (c+d x) a^7-2976 b^4 (c+d x) a^6-4800 b^4 (c+d x) \cos (2 (c+d x)) a^6+6880 b^4 \sin (2 (c+d x)) a^6-40 b^4 \sin (4 (c+d x)) a^6-5696 b^5 \cos (c+d x) a^5-760 b^5 \cos (3 (c+d x)) a^5-8 b^5 \cos (5 (c+d x)) a^5-15552 b^5 (c+d x) \sin (c+d x) a^5-1776 b^6 (c+d x) a^4+3888 b^6 (c+d x) \cos (2 (c+d x)) a^4-5182 b^6 \sin (2 (c+d x)) a^4+88 b^6 \sin (4 (c+d x)) a^4+2 b^6 \sin (6 (c+d x)) a^4+788 b^7 \cos (c+d x) a^3+560 b^7 \cos (3 (c+d x)) a^3+16 b^7 \cos (5 (c+d x)) a^3+4224 b^7 (c+d x) \sin (c+d x) a^3+960 b^8 (c+d x) a^2-1056 b^8 (c+d x) \cos (2 (c+d x)) a^2+1221 b^8 \sin (2 (c+d x)) a^2-56 b^8 \sin (4 (c+d x)) a^2-4 b^8 \sin (6 (c+d x)) a^2+114 b^9 \cos (c+d x) a-120 b^9 \cos (3 (c+d x)) a-8 b^9 \cos (5 (c+d x)) a-192 b^9 (c+d x) \sin (c+d x) a-48 b^{10} (c+d x)+48 b^{10} (c+d x) \cos (2 (c+d x))-36 b^{10} \sin (2 (c+d x))+8 b^{10} \sin (4 (c+d x))+2 b^{10} \sin (6 (c+d x))}{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}}{b^7}}{256 d}","\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{a \left(30 a^2-13 b^2\right) \cos (c+d x)}{2 b^6 d}-\frac{3 \left(20 a^2-7 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^5 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{\left(15 a^2-4 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{4 a^2 b^3 d}-\frac{3 a \left(10 a^4-11 a^2 b^2+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{3 x \left(40 a^4-24 a^2 b^2+b^4\right)}{8 b^7}",1,"-1/256*((-6*(-8*(c + d*x) + (2*a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a*b*(4*a^2 - 3*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) - (3*b*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x]))))/b^3 + (6*((6*a*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2))/((a - b)^2*(a + b)^2) + (2*(-24*(-8*a^2 + b^2)*(c + d*x) - (6*a*(64*a^6 - 168*a^4*b^2 + 140*a^2*b^4 - 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 96*a*b*Cos[c + d*x] + (a*b*(-16*a^4 + 20*a^2*b^2 - 5*b^4)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) + (b*(112*a^6 - 220*a^4*b^2 + 115*a^2*b^4 - 10*b^6)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])) - 8*b^2*Sin[2*(c + d*x)]))/b^5 + ((12*a*(640*a^8 - 1920*a^6*b^2 + 2016*a^4*b^4 - 840*a^2*b^6 + 105*b^8)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (-3840*a^10*(c + d*x) + 7680*a^8*b^2*(c + d*x) - 2976*a^6*b^4*(c + d*x) - 1776*a^4*b^6*(c + d*x) + 960*a^2*b^8*(c + d*x) - 48*b^10*(c + d*x) - 3840*a^9*b*Cos[c + d*x] + 8640*a^7*b^3*Cos[c + d*x] - 5696*a^5*b^5*Cos[c + d*x] + 788*a^3*b^7*Cos[c + d*x] + 114*a*b^9*Cos[c + d*x] + 1920*a^8*b^2*(c + d*x)*Cos[2*(c + d*x)] - 4800*a^6*b^4*(c + d*x)*Cos[2*(c + d*x)] + 3888*a^4*b^6*(c + d*x)*Cos[2*(c + d*x)] - 1056*a^2*b^8*(c + d*x)*Cos[2*(c + d*x)] + 48*b^10*(c + d*x)*Cos[2*(c + d*x)] + 320*a^7*b^3*Cos[3*(c + d*x)] - 760*a^5*b^5*Cos[3*(c + d*x)] + 560*a^3*b^7*Cos[3*(c + d*x)] - 120*a*b^9*Cos[3*(c + d*x)] - 8*a^5*b^5*Cos[5*(c + d*x)] + 16*a^3*b^7*Cos[5*(c + d*x)] - 8*a*b^9*Cos[5*(c + d*x)] - 7680*a^9*b*(c + d*x)*Sin[c + d*x] + 19200*a^7*b^3*(c + d*x)*Sin[c + d*x] - 15552*a^5*b^5*(c + d*x)*Sin[c + d*x] + 4224*a^3*b^7*(c + d*x)*Sin[c + d*x] - 192*a*b^9*(c + d*x)*Sin[c + d*x] - 2880*a^8*b^2*Sin[2*(c + d*x)] + 6880*a^6*b^4*Sin[2*(c + d*x)] - 5182*a^4*b^6*Sin[2*(c + d*x)] + 1221*a^2*b^8*Sin[2*(c + d*x)] - 36*b^10*Sin[2*(c + d*x)] - 40*a^6*b^4*Sin[4*(c + d*x)] + 88*a^4*b^6*Sin[4*(c + d*x)] - 56*a^2*b^8*Sin[4*(c + d*x)] + 8*b^10*Sin[4*(c + d*x)] + 2*a^4*b^6*Sin[6*(c + d*x)] - 4*a^2*b^8*Sin[6*(c + d*x)] + 2*b^10*Sin[6*(c + d*x)])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2))/b^7)/d","B",1
1136,1,1030,284,6.4442461,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{-\frac{12 \left(-48 a (c+d x)+\frac{6 \left(16 a^6-40 b^2 a^4+30 b^4 a^2-5 b^6\right) \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-16 b \cos (c+d x)+\frac{a b \left(-40 a^4+72 b^2 a^2-29 b^4\right) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{b \left(8 a^4-8 b^2 a^2+b^4\right) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}\right)}{b^4}+12 \left(\frac{2 \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{b \cos (c+d x) \left(4 a^2+3 b \sin (c+d x) a-b^2\right)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))^2}\right)+\frac{6 \left(\frac{\cos (c+d x) \left(a \left(2 a^2-5 b^2\right) \sin (c+d x)-b \left(2 a^2+b^2\right)\right)}{(a+b \sin (c+d x))^2}-\frac{6 b^2 \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}\right)}{(a-b)^2 (a+b)^2}-\frac{\frac{3840 (c+d x) a^9+3840 b \cos (c+d x) a^8+7680 b (c+d x) \sin (c+d x) a^8-6912 b^2 (c+d x) a^7-1920 b^2 (c+d x) \cos (2 (c+d x)) a^7+2880 b^2 \sin (2 (c+d x)) a^7-7872 b^3 \cos (c+d x) a^6-320 b^3 \cos (3 (c+d x)) a^6-17664 b^3 (c+d x) \sin (c+d x) a^6+1728 b^4 (c+d x) a^5+4416 b^4 (c+d x) \cos (2 (c+d x)) a^5-6304 b^4 \sin (2 (c+d x)) a^5+40 b^4 \sin (4 (c+d x)) a^5+4256 b^5 \cos (c+d x) a^4+696 b^5 \cos (3 (c+d x)) a^4+8 b^5 \cos (5 (c+d x)) a^4+12288 b^5 (c+d x) \sin (c+d x) a^4+1920 b^6 (c+d x) a^3-3072 b^6 (c+d x) \cos (2 (c+d x)) a^3+4022 b^6 \sin (2 (c+d x)) a^3-80 b^6 \sin (4 (c+d x)) a^3-172 b^7 \cos (c+d x) a^2-432 b^7 \cos (3 (c+d x)) a^2-16 b^7 \cos (5 (c+d x)) a^2-2304 b^7 (c+d x) \sin (c+d x) a^2-576 b^8 (c+d x) a+576 b^8 (c+d x) \cos (2 (c+d x)) a-607 b^8 \sin (2 (c+d x)) a+40 b^8 \sin (4 (c+d x)) a-70 b^9 \cos (c+d x)+56 b^9 \cos (3 (c+d x))+8 b^9 \cos (5 (c+d x))}{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{12 \left(640 a^8-1792 b^2 a^6+1680 b^4 a^4-560 b^6 a^2+35 b^8\right) \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}}{b^6}}{384 d}","\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(60 a^2-17 b^2\right) \cos (c+d x)}{6 b^5 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(20 a^2-3 b^2\right) \sin ^2(c+d x) \cos (c+d x)}{6 a^2 b^3 d}+\frac{\left(20 a^4-19 a^2 b^2+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}}",1,"((-12*(-48*a*(c + d*x) + (6*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 16*b*Cos[c + d*x] + (b*(8*a^4 - 8*a^2*b^2 + b^4)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) + (a*b*(-40*a^4 + 72*a^2*b^2 - 29*b^4)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x]))))/b^4 + 12*((2*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (b*Cos[c + d*x]*(4*a^2 - b^2 + 3*a*b*Sin[c + d*x]))/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^2)) + (6*((-6*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(-(b*(2*a^2 + b^2)) + a*(2*a^2 - 5*b^2)*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2))/((a - b)^2*(a + b)^2) - ((-12*(640*a^8 - 1792*a^6*b^2 + 1680*a^4*b^4 - 560*a^2*b^6 + 35*b^8)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (3840*a^9*(c + d*x) - 6912*a^7*b^2*(c + d*x) + 1728*a^5*b^4*(c + d*x) + 1920*a^3*b^6*(c + d*x) - 576*a*b^8*(c + d*x) + 3840*a^8*b*Cos[c + d*x] - 7872*a^6*b^3*Cos[c + d*x] + 4256*a^4*b^5*Cos[c + d*x] - 172*a^2*b^7*Cos[c + d*x] - 70*b^9*Cos[c + d*x] - 1920*a^7*b^2*(c + d*x)*Cos[2*(c + d*x)] + 4416*a^5*b^4*(c + d*x)*Cos[2*(c + d*x)] - 3072*a^3*b^6*(c + d*x)*Cos[2*(c + d*x)] + 576*a*b^8*(c + d*x)*Cos[2*(c + d*x)] - 320*a^6*b^3*Cos[3*(c + d*x)] + 696*a^4*b^5*Cos[3*(c + d*x)] - 432*a^2*b^7*Cos[3*(c + d*x)] + 56*b^9*Cos[3*(c + d*x)] + 8*a^4*b^5*Cos[5*(c + d*x)] - 16*a^2*b^7*Cos[5*(c + d*x)] + 8*b^9*Cos[5*(c + d*x)] + 7680*a^8*b*(c + d*x)*Sin[c + d*x] - 17664*a^6*b^3*(c + d*x)*Sin[c + d*x] + 12288*a^4*b^5*(c + d*x)*Sin[c + d*x] - 2304*a^2*b^7*(c + d*x)*Sin[c + d*x] + 2880*a^7*b^2*Sin[2*(c + d*x)] - 6304*a^5*b^4*Sin[2*(c + d*x)] + 4022*a^3*b^6*Sin[2*(c + d*x)] - 607*a*b^8*Sin[2*(c + d*x)] + 40*a^5*b^4*Sin[4*(c + d*x)] - 80*a^3*b^6*Sin[4*(c + d*x)] + 40*a*b^8*Sin[4*(c + d*x)])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2))/b^6)/(384*d)","B",1
1137,1,274,173,3.4643268,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{96 a^4 c+96 a^4 d x+192 a^3 b c \sin (c+d x)+192 a^3 b d x \sin (c+d x)+96 a^3 b \cos (c+d x)+72 a^2 b^2 \sin (2 (c+d x))+12 b^2 \left(b^2-4 a^2\right) (c+d x) \cos (2 (c+d x))+24 a^2 b^2 c+24 a^2 b^2 d x-48 a b^3 c \sin (c+d x)-48 a b^3 d x \sin (c+d x)-8 a b^3 \cos (3 (c+d x))-10 b^4 \sin (2 (c+d x))+b^4 \sin (4 (c+d x))-12 b^4 c-12 b^4 d x}{(a+b \sin (c+d x))^2}-\frac{48 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)}{\sqrt{a^2-b^2}}}{16 b^5 d}","-\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 x \left(4 a^2-b^2\right)}{2 b^5}+\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{\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}",1,"((-48*a*(4*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (96*a^4*c + 24*a^2*b^2*c - 12*b^4*c + 96*a^4*d*x + 24*a^2*b^2*d*x - 12*b^4*d*x + 96*a^3*b*Cos[c + d*x] + 12*b^2*(-4*a^2 + b^2)*(c + d*x)*Cos[2*(c + d*x)] - 8*a*b^3*Cos[3*(c + d*x)] + 192*a^3*b*c*Sin[c + d*x] - 48*a*b^3*c*Sin[c + d*x] + 192*a^3*b*d*x*Sin[c + d*x] - 48*a*b^3*d*x*Sin[c + d*x] + 72*a^2*b^2*Sin[2*(c + d*x)] - 10*b^4*Sin[2*(c + d*x)] + b^4*Sin[4*(c + d*x)])/(a + b*Sin[c + d*x])^2)/(16*b^5*d)","A",1
1138,1,176,175,1.7775639,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{2 \left(\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}-\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}+\frac{c+d x}{b^3}\right)+\frac{\cos (c+d x) \left(2 a^3+b \left(3 a^2+2 b^2\right) \sin (c+d x)+3 a b^2\right)}{a^2 b^2 (a+b \sin (c+d x))^2}-\frac{2 \left(2 a^4-a^2 b^2+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 \sqrt{a^2-b^2}}}{2 d}","-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\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{\left(2 a^4-a^2 b^2+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{x}{b^3}",1,"((-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^3*Sqrt[a^2 - b^2]) + 2*((c + d*x)/b^3 - Log[Cos[(c + d*x)/2]]/a^3 + Log[Sin[(c + d*x)/2]]/a^3) + (Cos[c + d*x]*(2*a^3 + 3*a*b^2 + b*(3*a^2 + 2*b^2)*Sin[c + d*x]))/(a^2*b^2*(a + b*Sin[c + d*x])^2))/(2*d)","A",1
1139,1,184,182,2.5505263,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{-\frac{6 \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)}{\sqrt{a^2-b^2}}+\frac{a^2 \left(a^2-b^2\right) \cos (c+d x)}{b (a+b \sin (c+d x))^2}-\frac{a \left(a^2+4 b^2\right) \cos (c+d x)}{b (a+b \sin (c+d x))}+a \tan \left(\frac{1}{2} (c+d x)\right)-a \cot \left(\frac{1}{2} (c+d x)\right)-6 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}","\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\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 \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{\cot (c+d x)}{a d (a+b \sin (c+d x))^2}",1,"((-6*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - a*Cot[(c + d*x)/2] + 6*b*Log[Cos[(c + d*x)/2]] - 6*b*Log[Sin[(c + d*x)/2]] + (a^2*(a^2 - b^2)*Cos[c + d*x])/(b*(a + b*Sin[c + d*x])^2) - (a*(a^2 + 4*b^2)*Cos[c + d*x])/(b*(a + b*Sin[c + d*x])) + a*Tan[(c + d*x)/2])/(2*a^4*d)","A",1
1140,1,319,218,6.1844811,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]","-\frac{3 b \tan \left(\frac{1}{2} (c+d x)\right)}{2 a^4 d}+\frac{3 b \cot \left(\frac{1}{2} (c+d x)\right)}{2 a^4 d}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^3 d}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^3 d}-\frac{3 \left(a^2-4 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^5 d}+\frac{3 \left(a^2-4 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^5 d}+\frac{3 b \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^5 d \sqrt{a^2-b^2}}+\frac{6 b^2 \cos (c+d x)-a^2 \cos (c+d x)}{2 a^4 d (a+b \sin (c+d x))}+\frac{b^2 \cos (c+d x)-a^2 \cos (c+d x)}{2 a^3 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{\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 \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{3 \left(a^2-4 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}-\frac{\left(a^2-12 b^2\right) \cot (c+d x)}{2 a^4 b d}-\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[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) + (3*b*Cot[(c + d*x)/2])/(2*a^4*d) - Csc[(c + d*x)/2]^2/(8*a^3*d) + (3*(a^2 - 4*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^5*d) - (3*(a^2 - 4*b^2)*Log[Sin[(c + d*x)/2]])/(2*a^5*d) + Sec[(c + d*x)/2]^2/(8*a^3*d) + (-(a^2*Cos[c + d*x]) + b^2*Cos[c + d*x])/(2*a^3*d*(a + b*Sin[c + d*x])^2) + (-(a^2*Cos[c + d*x]) + 6*b^2*Cos[c + d*x])/(2*a^4*d*(a + b*Sin[c + d*x])) - (3*b*Tan[(c + d*x)/2])/(2*a^4*d)","A",1
1141,1,459,289,6.2038611,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]","\frac{3 b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^4 d}-\frac{3 b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^4 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^3 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^3 d}+\frac{\left(9 a^2 b-20 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^6 d}+\frac{\left(20 b^3-9 a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^6 d}+\frac{3 a^2 b \cos (c+d x)-8 b^3 \cos (c+d x)}{2 a^5 d (a+b \sin (c+d x))}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(2 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-9 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{3 a^5 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(9 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-2 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 a^5 d}+\frac{a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{2 a^4 d (a+b \sin (c+d x))^2}+\frac{\left(2 a^4-19 a^2 b^2+20 b^4\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2}}","\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{b \left(9 a^2-20 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac{\left(17 a^2-60 b^2\right) \cot (c+d x)}{6 a^5 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(2 a^4-19 a^2 b^2+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{\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[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) + ((2*a^2*Cos[(c + d*x)/2] - 9*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(3*a^5*d) + (3*b*Csc[(c + d*x)/2]^2)/(8*a^4*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^3*d) + ((-9*a^2*b + 20*b^3)*Log[Cos[(c + d*x)/2]])/(2*a^6*d) + ((9*a^2*b - 20*b^3)*Log[Sin[(c + d*x)/2]])/(2*a^6*d) - (3*b*Sec[(c + d*x)/2]^2)/(8*a^4*d) + (Sec[(c + d*x)/2]*(-2*a^2*Sin[(c + d*x)/2] + 9*b^2*Sin[(c + d*x)/2]))/(3*a^5*d) + (a^2*b*Cos[c + d*x] - b^3*Cos[c + d*x])/(2*a^4*d*(a + b*Sin[c + d*x])^2) + (3*a^2*b*Cos[c + d*x] - 8*b^3*Cos[c + d*x])/(2*a^5*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^3*d)","A",0
1142,1,347,340,4.7649783,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x])^3,x]","-\frac{\frac{384 b \left(2 a^4-11 a^2 b^2+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)}{\sqrt{a^2-b^2}}-48 \left(a^4-24 a^2 b^2+40 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+48 \left(a^4-24 a^2 b^2+40 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 a \cot (c+d x) \csc ^5(c+d x) \left(-4 a^5+100 a^4 b \sin (c+d x)-44 a^4 b \sin (3 (c+d x))+289 a^3 b^2+\left(83 a^3 b^2-180 a b^4\right) \cos (4 (c+d x))+20 a^2 b^3 \sin (c+d x)-50 a^2 b^3 \sin (3 (c+d x))+26 a^2 b^3 \sin (5 (c+d x))+4 \left(5 a^5-93 a^3 b^2+180 a b^4\right) \cos (2 (c+d x))-540 a b^4-600 b^5 \sin (c+d x)+300 b^5 \sin (3 (c+d x))-60 b^5 \sin (5 (c+d x))\right)}{(a \csc (c+d x)+b)^2}}{128 a^7 d}","\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{b \left(13 a^2-30 b^2\right) \cot (c+d x)}{2 a^6 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(3 a^2-10 b^2\right) \cot (c+d x) \csc ^2(c+d x)}{2 a^4 b 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{3 b \left(2 a^4-11 a^2 b^2+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{3 \left(a^4-24 a^2 b^2+40 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d (a+b \sin (c+d x))^2}",1,"-1/128*((384*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]])/Sqrt[a^2 - b^2] + 48*(a^4 - 24*a^2*b^2 + 40*b^4)*Log[Cos[(c + d*x)/2]] - 48*(a^4 - 24*a^2*b^2 + 40*b^4)*Log[Sin[(c + d*x)/2]] + (2*a*Cot[c + d*x]*Csc[c + d*x]^5*(-4*a^5 + 289*a^3*b^2 - 540*a*b^4 + 4*(5*a^5 - 93*a^3*b^2 + 180*a*b^4)*Cos[2*(c + d*x)] + (83*a^3*b^2 - 180*a*b^4)*Cos[4*(c + d*x)] + 100*a^4*b*Sin[c + d*x] + 20*a^2*b^3*Sin[c + d*x] - 600*b^5*Sin[c + d*x] - 44*a^4*b*Sin[3*(c + d*x)] - 50*a^2*b^3*Sin[3*(c + d*x)] + 300*b^5*Sin[3*(c + d*x)] + 26*a^2*b^3*Sin[5*(c + d*x)] - 60*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x])^2)/(a^7*d)","A",1
1143,1,327,463,5.0979989,"\int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]","\frac{\sqrt{a+b \sin (c+d x)} \left(128 \left(320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-256 a \left(160 a^5-160 a^4 b-279 a^3 b^2+279 a^2 b^3+27 a b^4-27 b^5\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-2 b \cos (c+d x) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left(10240 a^5-7680 a^4 b \sin (c+d x)-21056 a^3 b^2-1600 \left(2 a^3 b^2-3 a b^4\right) \cos (2 (c+d x))+13592 a^2 b^3 \sin (c+d x)+1400 a^2 b^3 \sin (3 (c+d x))+630 a b^4 \cos (4 (c+d x))+5898 a b^4-19866 b^5 \sin (c+d x)+5775 b^5 \sin (3 (c+d x))+3465 b^5 \sin (5 (c+d x))\right)\right)}{720720 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","\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{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 \left(480 a^4-937 a^2 b^2+231 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{16 a \left(160 a^4-279 a^2 b^2+27 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{16 a \left(160 a^6-439 a^4 b^2+306 a^2 b^4-27 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(320 a^6-798 a^4 b^2+435 a^2 b^4-693 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,"(Sqrt[a + b*Sin[c + d*x]]*(128*(320*a^6 - 798*a^4*b^2 + 435*a^2*b^4 - 693*b^6)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - 256*a*(160*a^5 - 160*a^4*b - 279*a^3*b^2 + 279*a^2*b^3 + 27*a*b^4 - 27*b^5)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - 2*b*Cos[c + d*x]*Sqrt[(a + b*Sin[c + d*x])/(a + b)]*(10240*a^5 - 21056*a^3*b^2 + 5898*a*b^4 - 1600*(2*a^3*b^2 - 3*a*b^4)*Cos[2*(c + d*x)] + 630*a*b^4*Cos[4*(c + d*x)] - 7680*a^4*b*Sin[c + d*x] + 13592*a^2*b^3*Sin[c + d*x] - 19866*b^5*Sin[c + d*x] + 1400*a^2*b^3*Sin[3*(c + d*x)] + 5775*b^5*Sin[3*(c + d*x)] + 3465*b^5*Sin[5*(c + d*x)])))/(720720*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])","A",1
1144,1,326,332,4.1163053,"\int \cos ^4(c+d x) \sin (c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]","\frac{64 \left(32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+b \cos (c+d x) \left(1024 a^5+256 a^4 b \sin (c+d x)-2912 a^3 b^2+16 \left(4 a^3 b^2-183 a b^4\right) \cos (2 (c+d x))-692 a^2 b^3 \sin (c+d x)-20 a^2 b^3 \sin (3 (c+d x))-700 a b^4 \cos (4 (c+d x))+748 a b^4+990 b^5 \sin (c+d x)-765 b^5 \sin (3 (c+d x))-315 b^5 \sin (5 (c+d x))\right)-64 a \left(32 a^5+32 a^4 b-93 a^3 b^2-93 a^2 b^3+93 a b^4+93 b^5\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{27720 b^5 d \sqrt{a+b \sin (c+d 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(32 a^4-24 a b \left(a^2-2 b^2\right) \sin (c+d x)-69 a^2 b^2+45 b^4\right)}{3465 b^4 d}+\frac{8 a \left(32 a^4-93 a^2 b^2+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{8 \left(32 a^6-101 a^4 b^2+114 a^2 b^4-45 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{2 \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}",1,"(-64*a*(32*a^5 + 32*a^4*b - 93*a^3*b^2 - 93*a^2*b^3 + 93*a*b^4 + 93*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 64*(32*a^6 - 101*a^4*b^2 + 114*a^2*b^4 - 45*b^6)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(1024*a^5 - 2912*a^3*b^2 + 748*a*b^4 + 16*(4*a^3*b^2 - 183*a*b^4)*Cos[2*(c + d*x)] - 700*a*b^4*Cos[4*(c + d*x)] + 256*a^4*b*Sin[c + d*x] - 692*a^2*b^3*Sin[c + d*x] + 990*b^5*Sin[c + d*x] - 20*a^2*b^3*Sin[3*(c + d*x)] - 765*b^5*Sin[3*(c + d*x)] - 315*b^5*Sin[5*(c + d*x)]))/(27720*b^5*d*Sqrt[a + b*Sin[c + d*x]])","A",1
1145,1,435,338,3.5063477,"\int \cos ^3(c+d x) \cot (c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]","\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(8 a^2-6 a b \sin (c+d x)+15 b^2 \cos (2 (c+d x))+75 b^2\right)-\frac{8 b \left(a^2+30 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 a \left(8 a^2+159 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 i \left(51 b^2-8 a^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{b^2 \sqrt{-\frac{1}{a+b}}}}{210 b^2 d}","-\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 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{2 \left(8 a^4-53 a^2 b^2-60 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{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*I)*(-8*a^2 + 51*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(b^2*Sqrt[-(a + b)^(-1)]) - (8*b*(a^2 + 30*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*a*(8*a^2 + 159*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + 2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(8*a^2 + 75*b^2 + 15*b^2*Cos[2*(c + d*x)] - 6*a*b*Sin[c + d*x]))/(210*b^2*d)","C",1
1146,1,422,323,3.6500845,"\int \cos ^2(c+d x) \cot ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]","\frac{\frac{2 \left(4 a^2+27 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{b \sqrt{a+b \sin (c+d x)}}+\frac{2 i \left(4 a^2+57 b^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^3 \sqrt{-\frac{1}{a+b}}}+\frac{184 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{4 \sqrt{a+b \sin (c+d x)} (2 a \cos (c+d x)+3 b (\sin (2 (c+d x))+5 \cot (c+d x)))}{b}}{60 d}","\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,"(((2*I)*(4*a^2 + 57*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b^3*Sqrt[-(a + b)^(-1)]) + (184*a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*(4*a^2 + 27*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b*Sqrt[a + b*Sin[c + d*x]]) - (4*Sqrt[a + b*Sin[c + d*x]]*(2*a*Cos[c + d*x] + 3*b*(5*Cot[c + d*x] + Sin[2*(c + d*x)])))/b)/(60*d)","C",1
1147,1,450,345,3.575583,"\int \cos (c+d x) \cot ^3(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]],x]","\frac{\frac{2 \left(64 a^2+9 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{a \sqrt{a+b \sin (c+d x)}}+\frac{2 i \left(8 a^2-3 b^2\right) \cos (2 (c+d x)) \csc ^2(c+d x) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a^2 b^2 \sqrt{-\frac{1}{a+b}} \left(\csc ^2(c+d x)-2\right)}+\frac{136 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{4 \sqrt{a+b \sin (c+d x)} (3 \cot (c+d x) (2 a \csc (c+d x)+b)+8 a \cos (c+d x))}{a}}{48 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,"(((2*I)*(8*a^2 - 3*b^2)*Cos[2*(c + d*x)]*Csc[c + d*x]^2*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a^2*b^2*Sqrt[-(a + b)^(-1)]*(-2 + Csc[c + d*x]^2)) - (4*(8*a*Cos[c + d*x] + 3*Cot[c + d*x]*(b + 2*a*Csc[c + d*x]))*Sqrt[a + b*Sin[c + d*x]])/a + (136*b*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*(64*a^2 + 9*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a*Sqrt[a + b*Sin[c + d*x]]))/(48*d)","C",1
1148,1,473,351,5.705791,"\int \cot ^4(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]],x]","\frac{-\frac{4 \cot (c+d x) \sqrt{a+b \sin (c+d x)} \left(8 a^2 \csc ^2(c+d x)-32 a^2+2 a b \csc (c+d x)-3 b^2\right)}{a^2}+\frac{-\frac{8 a \left(24 a^2+b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 b \left(8 a^2+9 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 i \left(80 a^2+3 b^2\right) \cos (2 (c+d x)) \csc ^2(c+d x) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a b \sqrt{-\frac{1}{a+b}} \left(\csc ^2(c+d x)-2\right)}}{a^2}}{96 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,"((-4*Cot[c + d*x]*(-32*a^2 - 3*b^2 + 2*a*b*Csc[c + d*x] + 8*a^2*Csc[c + d*x]^2)*Sqrt[a + b*Sin[c + d*x]])/a^2 + (((2*I)*(80*a^2 + 3*b^2)*Cos[2*(c + d*x)]*Csc[c + d*x]^2*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]*(-2 + Csc[c + d*x]^2)) - (8*a*(24*a^2 + b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*b*(8*a^2 + 9*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]])/a^2)/(96*d)","C",1
1149,1,643,412,6.6221379,"\int \cot ^4(c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{5 \csc ^2(c+d x) \left(12 a^2 \cos (c+d x)+b^2 \cos (c+d x)\right)}{96 a^2}+\frac{\csc (c+d x) \left(68 a^2 b \cos (c+d x)-15 b^3 \cos (c+d x)\right)}{192 a^3}-\frac{b \cot (c+d x) \csc ^2(c+d x)}{24 a}-\frac{1}{4} \cot (c+d x) \csc ^3(c+d x)\right)}{d}+\frac{-\frac{2 \left(528 a^3 b-20 a b^3\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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(15 b^4-68 a^2 b^2\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}-\frac{2 \left(288 a^4+212 a^2 b^2-45 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)}{\sqrt{a+b \sin (c+d x)}}}{768 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{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{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(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(48 a^4+24 a^2 b^2-5 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{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d}",1,"((((68*a^2*b*Cos[c + d*x] - 15*b^3*Cos[c + d*x])*Csc[c + d*x])/(192*a^3) + (5*(12*a^2*Cos[c + d*x] + b^2*Cos[c + d*x])*Csc[c + d*x]^2)/(96*a^2) - (b*Cot[c + d*x]*Csc[c + d*x]^2)/(24*a) - (Cot[c + d*x]*Csc[c + d*x]^3)/4)*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(528*a^3*b - 20*a*b^3)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(288*a^4 + 212*a^2*b^2 - 45*b^4)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(-68*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(768*a^3*d)","C",1
1150,1,702,484,6.696644,"\int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{\csc ^3(c+d x) \left(96 a^2 \cos (c+d x)+7 b^2 \cos (c+d x)\right)}{240 a^2}+\frac{\csc (c+d x) \left(-384 a^4 \cos (c+d x)-332 a^2 b^2 \cos (c+d x)+105 b^4 \cos (c+d x)\right)}{1920 a^4}+\frac{\csc ^2(c+d x) \left(108 a^2 b \cos (c+d x)-35 b^3 \cos (c+d x)\right)}{960 a^3}-\frac{b \cot (c+d x) \csc ^3(c+d x)}{40 a}-\frac{1}{5} \cot (c+d x) \csc ^4(c+d x)\right)}{d}+\frac{b \left(-\frac{2 \left(140 a b^3-432 a^3 b\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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left(1056 a^4-1052 a^2 b^2+315 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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(384 a^4+332 a^2 b^2-105 b^4\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}\right)}{7680 a^4 d}","\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{7 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{40 a^2 d}-\frac{\left(384 a^4+332 a^2 b^2-105 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1920 a^4 d}-\frac{\left(384 a^4+332 a^2 b^2-105 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(48 a^4-24 a^2 b^2+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{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{\left(384 a^4+116 a^2 b^2-35 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{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{5 a d}",1,"((((-384*a^4*Cos[c + d*x] - 332*a^2*b^2*Cos[c + d*x] + 105*b^4*Cos[c + d*x])*Csc[c + d*x])/(1920*a^4) + ((108*a^2*b*Cos[c + d*x] - 35*b^3*Cos[c + d*x])*Csc[c + d*x]^2)/(960*a^3) + ((96*a^2*Cos[c + d*x] + 7*b^2*Cos[c + d*x])*Csc[c + d*x]^3)/(240*a^2) - (b*Cot[c + d*x]*Csc[c + d*x]^3)/(40*a) - (Cot[c + d*x]*Csc[c + d*x]^4)/5)*Sqrt[a + b*Sin[c + d*x]])/d + (b*((-2*(-432*a^3*b + 140*a*b^3)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(1056*a^4 - 1052*a^2*b^2 + 315*b^4)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(384*a^4 + 332*a^2*b^2 - 105*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)])))/(7680*a^4*d)","C",1
1151,1,382,528,15.6617465,"\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(512 \left(32 a^7-111 a^5 b^2+102 a^3 b^4-471 a b^6\right) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-2 b \cos (c+d x) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left(4096 a^6-3072 a^5 b \sin (c+d x)-12416 a^4 b^2+8432 a^3 b^3 \sin (c+d x)+560 a^3 b^3 \sin (3 (c+d x))+8100 a^2 b^4+42 \left(6 a^2 b^4-13 b^6\right) \cos (4 (c+d x))+\left(-1280 a^4 b^2+3168 a^2 b^4+21723 b^6\right) \cos (2 (c+d x))-41424 a b^5 \sin (c+d x)+13776 a b^5 \sin (3 (c+d x))+7392 a b^5 \sin (5 (c+d x))-3003 b^6 \cos (6 (c+d x))+6786 b^6\right)-256 \left(64 a^7-64 a^6 b-174 a^5 b^2+174 a^4 b^3+81 a^3 b^4-81 a^2 b^5-195 a b^6+195 b^7\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)}{1441440 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","\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{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 \left(160 a^4-375 a^2 b^2+117 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac{16 a \left(32 a^4-47 a^2 b^2-27 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac{16 a \left(32 a^6-111 a^4 b^2+102 a^2 b^4-471 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{8 \left(64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{8 \left(64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+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{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,"(Sqrt[a + b*Sin[c + d*x]]*(512*(32*a^7 - 111*a^5*b^2 + 102*a^3*b^4 - 471*a*b^6)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - 256*(64*a^7 - 64*a^6*b - 174*a^5*b^2 + 174*a^4*b^3 + 81*a^3*b^4 - 81*a^2*b^5 - 195*a*b^6 + 195*b^7)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - 2*b*Cos[c + d*x]*Sqrt[(a + b*Sin[c + d*x])/(a + b)]*(4096*a^6 - 12416*a^4*b^2 + 8100*a^2*b^4 + 6786*b^6 + (-1280*a^4*b^2 + 3168*a^2*b^4 + 21723*b^6)*Cos[2*(c + d*x)] + 42*(6*a^2*b^4 - 13*b^6)*Cos[4*(c + d*x)] - 3003*b^6*Cos[6*(c + d*x)] - 3072*a^5*b*Sin[c + d*x] + 8432*a^3*b^3*Sin[c + d*x] - 41424*a*b^5*Sin[c + d*x] + 560*a^3*b^3*Sin[3*(c + d*x)] + 13776*a*b^5*Sin[3*(c + d*x)] + 7392*a*b^5*Sin[5*(c + d*x)])))/(1441440*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])","A",1
1152,1,382,394,12.0719746,"\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^(3/2),x]","\frac{384 a \left(32 a^6-145 a^4 b^2+290 a^2 b^4-177 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-3 b \cos (c+d x) \left(-2048 a^6-512 a^5 b \sin (c+d x)+8640 a^4 b^2+2088 a^3 b^3 \sin (c+d x)+40 a^3 b^3 \sin (3 (c+d x))+1980 a^2 b^4+70 \left(86 a^2 b^4-11 b^6\right) \cos (4 (c+d x))+\left(-128 a^4 b^2+24512 a^2 b^4+8547 b^6\right) \cos (2 (c+d x))-19492 a b^5 \sin (c+d x)+11870 a b^5 \sin (3 (c+d x))+5250 a b^5 \sin (5 (c+d x))-1155 b^6 \cos (6 (c+d x))-6622 b^6\right)-384 \left(32 a^7+32 a^6 b-137 a^5 b^2-137 a^4 b^3+258 a^3 b^4+258 a^2 b^5+231 a b^6+231 b^7\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{720720 b^5 d \sqrt{a+b \sin (c+d 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(32 a^4-113 a^2 b^2+177 b^4\right)-3 b \left(8 a^4-27 a^2 b^2-77 b^4\right) \sin (c+d x)\right)}{15015 b^4 d}-\frac{8 a \left(32 a^6-145 a^4 b^2+290 a^2 b^4-177 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(32 a^6-137 a^4 b^2+258 a^2 b^4+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,"(-384*(32*a^7 + 32*a^6*b - 137*a^5*b^2 - 137*a^4*b^3 + 258*a^3*b^4 + 258*a^2*b^5 + 231*a*b^6 + 231*b^7)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 384*a*(32*a^6 - 145*a^4*b^2 + 290*a^2*b^4 - 177*b^6)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 3*b*Cos[c + d*x]*(-2048*a^6 + 8640*a^4*b^2 + 1980*a^2*b^4 - 6622*b^6 + (-128*a^4*b^2 + 24512*a^2*b^4 + 8547*b^6)*Cos[2*(c + d*x)] + 70*(86*a^2*b^4 - 11*b^6)*Cos[4*(c + d*x)] - 1155*b^6*Cos[6*(c + d*x)] - 512*a^5*b*Sin[c + d*x] + 2088*a^3*b^3*Sin[c + d*x] - 19492*a*b^5*Sin[c + d*x] + 40*a^3*b^3*Sin[3*(c + d*x)] + 11870*a*b^5*Sin[3*(c + d*x)] + 5250*a*b^5*Sin[5*(c + d*x)]))/(720720*b^5*d*Sqrt[a + b*Sin[c + d*x]])","A",1
1153,1,477,390,3.8220414,"\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2),x]","\frac{-\frac{8 a b \left(a^2+156 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left(8 a^4+537 a^2 b^2+84 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(8 a^4-93 a^2 b^2+84 b^4\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^2 \sqrt{-\frac{1}{a+b}}}+\cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(16 a^3+\left(203 b^3-12 a^2 b\right) \sin (c+d x)+100 a b^2 \cos (2 (c+d x))+556 a b^2+35 b^3 \sin (3 (c+d x))\right)}{630 b^2 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^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^4-95 a^2 b^2-228 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(8 a^4-93 a^2 b^2+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{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*I)*(8*a^4 - 93*a^2*b^2 + 84*b^4)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(a*b^2*Sqrt[-(a + b)^(-1)]) - (8*a*b*(a^2 + 156*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(8*a^4 + 537*a^2*b^2 + 84*b^4)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(16*a^3 + 556*a*b^2 + 100*a*b^2*Cos[2*(c + d*x)] + (-12*a^2*b + 203*b^3)*Sin[c + d*x] + 35*b^3*Sin[3*(c + d*x)]))/(630*b^2*d)","C",1
1154,1,452,374,4.2349511,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]","\frac{\frac{8 \left(53 a^2-20 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 a \left(4 a^2-43 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{b \sqrt{a+b \sin (c+d x)}}-\frac{2 \sqrt{a+b \sin (c+d x)} \left(\left(4 a^2-55 b^2\right) \cos (c+d x)+b (16 a \sin (2 (c+d x))+70 a \cot (c+d x)-5 b \cos (3 (c+d x)))\right)}{b}+\frac{2 i \left(4 a^2+167 b^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{b^3 \sqrt{-\frac{1}{a+b}}}}{140 d}","\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{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{\left(4 a^4+61 a^2 b^2+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{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,"(((2*I)*(4*a^2 + 167*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(b^3*Sqrt[-(a + b)^(-1)]) + (8*(53*a^2 - 20*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*a*(4*a^2 - 43*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b*Sqrt[a + b*Sin[c + d*x]]) - (2*Sqrt[a + b*Sin[c + d*x]]*((4*a^2 - 55*b^2)*Cos[c + d*x] + b*(-5*b*Cos[3*(c + d*x)] + 70*a*Cot[c + d*x] + 16*a*Sin[2*(c + d*x)])))/b)/(140*d)","C",1
1155,1,434,383,3.4090706,"\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2),x]","\frac{\frac{2 \left(112 a^2+51 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 i \left(81 b^2-8 a^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^2 \sqrt{-\frac{1}{a+b}}}+\frac{472 a b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+4 \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)} (8 a \cos (2 (c+d x))-18 a-31 b \sin (c+d x)+2 b \sin (3 (c+d x)))}{80 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,"(((2*I)*(-8*a^2 + 81*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b^2*Sqrt[-(a + b)^(-1)]) + (472*a*b*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*(112*a^2 + 51*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + 4*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(-18*a + 8*a*Cos[2*(c + d*x)] - 31*b*Sin[c + d*x] + 2*b*Sin[3*(c + d*x)]))/(80*d)","C",1
1156,1,600,386,6.6196389,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{\csc (c+d x) \left(32 a^2 \cos (c+d x)-3 b^2 \cos (c+d x)\right)}{24 a}-\frac{1}{3} a \cot (c+d x) \csc ^2(c+d x)-\frac{2}{3} b \cos (c+d x)-\frac{7}{12} b \cot (c+d x) \csc (c+d x)\right)}{d}+\frac{-\frac{2 \left(32 a^3-44 a 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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left(-40 a^2 b-3 b^3\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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(b^3-32 a^2 b\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}}{32 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,"(((-2*b*Cos[c + d*x])/3 + ((32*a^2*Cos[c + d*x] - 3*b^2*Cos[c + d*x])*Csc[c + d*x])/(24*a) - (7*b*Cot[c + d*x]*Csc[c + d*x])/12 - (a*Cot[c + d*x]*Csc[c + d*x]^2)/3)*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(32*a^3 - 44*a*b^2)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(-40*a^2*b - 3*b^3)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(-32*a^2*b + b^3)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(32*a*d)","C",1
1157,1,641,408,6.7028756,"\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{3 \csc (c+d x) \left(36 a^2 b \cos (c+d x)+b^3 \cos (c+d x)\right)}{64 a^2}+\frac{\csc ^2(c+d x) \left(20 a^2 \cos (c+d x)-b^2 \cos (c+d x)\right)}{32 a}-\frac{1}{4} a \cot (c+d x) \csc ^3(c+d x)-\frac{3}{8} b \cot (c+d x) \csc ^2(c+d x)\right)}{d}+\frac{-\frac{2 \left(432 a^3 b+4 a b^3\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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(-236 a^2 b^2-3 b^4\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}-\frac{2 \left(96 a^4+92 a^2 b^2+9 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)}{\sqrt{a+b \sin (c+d x)}}}{256 a^2 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{\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{3 \left(16 a^4-24 a^2 b^2+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{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}",1,"(((3*(36*a^2*b*Cos[c + d*x] + b^3*Cos[c + d*x])*Csc[c + d*x])/(64*a^2) + ((20*a^2*Cos[c + d*x] - b^2*Cos[c + d*x])*Csc[c + d*x]^2)/(32*a) - (3*b*Cot[c + d*x]*Csc[c + d*x]^2)/8 - (a*Cot[c + d*x]*Csc[c + d*x]^3)/4)*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(432*a^3*b + 4*a*b^3)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(96*a^4 + 92*a^2*b^2 + 9*b^4)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(-236*a^2*b^2 - 3*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(256*a^2*d)","C",0
1158,1,700,484,6.7564983,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{\csc ^2(c+d x) \left(236 a^2 b \cos (c+d x)+5 b^3 \cos (c+d x)\right)}{320 a^2}+\frac{\csc ^3(c+d x) \left(32 a^2 \cos (c+d x)-b^2 \cos (c+d x)\right)}{80 a}+\frac{\csc (c+d x) \left(-128 a^4 \cos (c+d x)+116 a^2 b^2 \cos (c+d x)-15 b^4 \cos (c+d x)\right)}{640 a^3}-\frac{1}{5} a \cot (c+d x) \csc ^4(c+d x)-\frac{11}{40} b \cot (c+d x) \csc ^3(c+d x)\right)}{d}+\frac{b \left(-\frac{2 \left(1616 a^3 b-20 a b^3\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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left(1312 a^4+356 a^2 b^2-45 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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(128 a^4-116 a^2 b^2+15 b^4\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}\right)}{2560 a^3 d}","\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{\left(128 a^4+692 a^2 b^2+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(128 a^4-116 a^2 b^2+15 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^3 d}-\frac{\left(128 a^4-116 a^2 b^2+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(48 a^4+8 a^2 b^2-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{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{5 a d}",1,"((((-128*a^4*Cos[c + d*x] + 116*a^2*b^2*Cos[c + d*x] - 15*b^4*Cos[c + d*x])*Csc[c + d*x])/(640*a^3) + ((236*a^2*b*Cos[c + d*x] + 5*b^3*Cos[c + d*x])*Csc[c + d*x]^2)/(320*a^2) + ((32*a^2*Cos[c + d*x] - b^2*Cos[c + d*x])*Csc[c + d*x]^3)/(80*a) - (11*b*Cot[c + d*x]*Csc[c + d*x]^3)/40 - (a*Cot[c + d*x]*Csc[c + d*x]^4)/5)*Sqrt[a + b*Sin[c + d*x]])/d + (b*((-2*(1616*a^3*b - 20*a*b^3)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(1312*a^4 + 356*a^2*b^2 - 45*b^4)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(128*a^4 - 116*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)])))/(2560*a^3*d)","C",1
1159,1,771,551,6.7825309,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{\csc ^3(c+d x) \left(436 a^2 b \cos (c+d x)+7 b^3 \cos (c+d x)\right)}{960 a^2}+\frac{\csc ^4(c+d x) \left(140 a^2 \cos (c+d x)-3 b^2 \cos (c+d x)\right)}{480 a}+\frac{\csc (c+d x) \left(-2064 a^4 b \cos (c+d x)-512 a^2 b^3 \cos (c+d x)+105 b^5 \cos (c+d x)\right)}{7680 a^4}+\frac{\csc ^2(c+d x) \left(-240 a^4 \cos (c+d x)+168 a^2 b^2 \cos (c+d x)-35 b^4 \cos (c+d x)\right)}{3840 a^3}-\frac{1}{6} a \cot (c+d x) \csc ^5(c+d x)-\frac{13}{60} b \cot (c+d x) \csc ^4(c+d x)\right)}{d}+\frac{-\frac{2 \left(960 a^5 b-672 a^3 b^3+140 a b^5\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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(2064 a^4 b^2+512 a^2 b^4-105 b^6\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}-\frac{2 \left(1920 a^6+2256 a^4 b^2-1592 a^2 b^4+315 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)}{\sqrt{a+b \sin (c+d x)}}}{30720 a^4 d}","\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{7 b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{60 a^2 d}-\frac{b \left(2064 a^4+512 a^2 b^2-105 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{7680 a^4 d}-\frac{b \left(2064 a^4+512 a^2 b^2-105 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(64 a^6+144 a^4 b^2-36 a^2 b^4+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{b \left(2544 a^4+176 a^2 b^2-35 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{\left(240 a^4-168 a^2 b^2+35 b^4\right) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{3840 a^3 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2}}{6 a d}",1,"((((-2064*a^4*b*Cos[c + d*x] - 512*a^2*b^3*Cos[c + d*x] + 105*b^5*Cos[c + d*x])*Csc[c + d*x])/(7680*a^4) + ((-240*a^4*Cos[c + d*x] + 168*a^2*b^2*Cos[c + d*x] - 35*b^4*Cos[c + d*x])*Csc[c + d*x]^2)/(3840*a^3) + ((436*a^2*b*Cos[c + d*x] + 7*b^3*Cos[c + d*x])*Csc[c + d*x]^3)/(960*a^2) + ((140*a^2*Cos[c + d*x] - 3*b^2*Cos[c + d*x])*Csc[c + d*x]^4)/(480*a) - (13*b*Cot[c + d*x]*Csc[c + d*x]^4)/60 - (a*Cot[c + d*x]*Csc[c + d*x]^5)/6)*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(960*a^5*b - 672*a^3*b^3 + 140*a*b^5)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(1920*a^6 + 2256*a^4*b^2 - 1592*a^2*b^4 + 315*b^6)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(2064*a^4*b^2 + 512*a^2*b^4 - 105*b^6)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(30720*a^4*d)","C",1
1160,1,450,451,21.6450743,"\int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]","\frac{256 \left(32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 b^8\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+b \cos (c+d x) \left(4096 a^7+1024 a^6 b \sin (c+d x)-23936 a^5 b^2-5840 a^4 b^3 \sin (c+d x)-80 a^4 b^3 \sin (3 (c+d x))-36512 a^3 b^4-224 \left(161 a^3 b^4-54 a b^6\right) \cos (4 (c+d x))+186768 a^2 b^5 \sin (c+d x)-101688 a^2 b^5 \sin (3 (c+d x))-46536 a^2 b^5 \sin (5 (c+d x))+8 \left(32 a^5 b^2-18192 a^3 b^4-18741 a b^6\right) \cos (2 (c+d x))+20328 a b^6 \cos (6 (c+d x))+67584 a b^6+8151 b^7 \sin (c+d x)-22269 b^7 \sin (3 (c+d x))-2457 b^7 \sin (5 (c+d x))+3003 b^7 \sin (7 (c+d x))\right)-256 a \left(32 a^7+32 a^6 b-189 a^5 b^2-189 a^4 b^3+570 a^3 b^4+570 a^2 b^5+1635 a b^6+1635 b^7\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{1441440 b^5 d \sqrt{a+b \sin (c+d 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(8 a^4-7 a b \left(a^2+63 b^2\right) \sin (c+d x)-33 a^2 b^2-39 b^4\right)}{9009 b^2 d}+\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(32 a^6-165 a^4 b^2+450 a^2 b^4-24 a b \left(a^4-5 a^2 b^2-60 b^4\right) \sin (c+d x)+195 b^6\right)}{45045 b^4 d}+\frac{8 a \left(32 a^6-189 a^4 b^2+570 a^2 b^4+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{8 \left(32 a^8-197 a^6 b^2+615 a^4 b^4-255 a^2 b^6-195 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{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,"(-256*a*(32*a^7 + 32*a^6*b - 189*a^5*b^2 - 189*a^4*b^3 + 570*a^3*b^4 + 570*a^2*b^5 + 1635*a*b^6 + 1635*b^7)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 256*(32*a^8 - 197*a^6*b^2 + 615*a^4*b^4 - 255*a^2*b^6 - 195*b^8)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(4096*a^7 - 23936*a^5*b^2 - 36512*a^3*b^4 + 67584*a*b^6 + 8*(32*a^5*b^2 - 18192*a^3*b^4 - 18741*a*b^6)*Cos[2*(c + d*x)] - 224*(161*a^3*b^4 - 54*a*b^6)*Cos[4*(c + d*x)] + 20328*a*b^6*Cos[6*(c + d*x)] + 1024*a^6*b*Sin[c + d*x] - 5840*a^4*b^3*Sin[c + d*x] + 186768*a^2*b^5*Sin[c + d*x] + 8151*b^7*Sin[c + d*x] - 80*a^4*b^3*Sin[3*(c + d*x)] - 101688*a^2*b^5*Sin[3*(c + d*x)] - 22269*b^7*Sin[3*(c + d*x)] - 46536*a^2*b^5*Sin[5*(c + d*x)] - 2457*b^7*Sin[5*(c + d*x)] + 3003*b^7*Sin[7*(c + d*x)]))/(1441440*b^5*d*Sqrt[a + b*Sin[c + d*x]])","A",1
1161,1,521,447,4.0220629,"\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]","\frac{\cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(32 a^4-24 a^3 b \sin (c+d x)+2660 a^2 b^2+4 \left(113 a^2 b^2-54 b^4\right) \cos (2 (c+d x))+1954 a b^3 \sin (c+d x)+322 a b^3 \sin (3 (c+d x))-63 b^4 \cos (4 (c+d x))-9 b^4\right)-2 \left(\frac{8 b \left(a^4+480 a^2 b^2+18 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 a \left(8 a^4+1239 a^2 b^2+444 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 i \left(8 a^4-147 a^2 b^2+444 b^4\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{b^2 \sqrt{-\frac{1}{a+b}}}\right)}{2772 b^2 d}","\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{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(8 a^4-141 a^2 b^2+36 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^2 d}+\frac{2 a \left(8 a^4-147 a^2 b^2+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 \left(8 a^6-149 a^4 b^2-516 a^2 b^4-36 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{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*(((2*I)*(8*a^4 - 147*a^2*b^2 + 444*b^4)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(b^2*Sqrt[-(a + b)^(-1)]) + (8*b*(a^4 + 480*a^2*b^2 + 18*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*a*(8*a^4 + 1239*a^2*b^2 + 444*b^4)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]]) + Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^4 + 2660*a^2*b^2 - 9*b^4 + 4*(113*a^2*b^2 - 54*b^4)*Cos[2*(c + d*x)] - 63*b^4*Cos[4*(c + d*x)] - 24*a^3*b*Sin[c + d*x] + 1954*a*b^3*Sin[c + d*x] + 322*a*b^3*Sin[3*(c + d*x)]))/(2772*b^2*d)","C",1
1162,1,496,426,4.7855339,"\int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2),x]","\frac{\frac{8 a b \left(475 a^2-492 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 \left(20 a^4-1461 a^2 b^2-168 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(-20 a^4-1689 a^2 b^2+168 b^4\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^2 \sqrt{-\frac{1}{a+b}}}-\sqrt{a+b \sin (c+d x)} \left(\left(40 a^3-2202 a b^2\right) \cos (c+d x)+2 b \left(\sin (2 (c+d x)) \left(150 a^2-35 b^2 \cos (2 (c+d x))-119 b^2\right)+630 a^2 \cot (c+d x)-95 a b \cos (3 (c+d x))\right)\right)}{1260 b 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{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{a \left(20 a^4+739 a^2 b^2+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(20 a^4+1689 a^2 b^2-168 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{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,"(((-2*I)*(-20*a^4 - 1689*a^2*b^2 + 168*b^4)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(a*b^2*Sqrt[-(a + b)^(-1)]) + (8*a*b*(475*a^2 - 492*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*(20*a^4 - 1461*a^2*b^2 - 168*b^4)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - Sqrt[a + b*Sin[c + d*x]]*((40*a^3 - 2202*a*b^2)*Cos[c + d*x] + 2*b*(-95*a*b*Cos[3*(c + d*x)] + 630*a^2*Cot[c + d*x] + (150*a^2 - 119*b^2 - 35*b^2*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])))/(1260*b*d)","C",1
1163,1,460,430,5.5384428,"\int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]","\frac{\frac{8 b \left(125 a^2-16 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 a \left(160 a^2+37 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+4 \sqrt{a+b \sin (c+d x)} \left(\left(22 b^2-24 a^2\right) \cos (c+d x)-12 a b \sin (2 (c+d x))-7 a \cot (c+d x) (2 a \csc (c+d x)+9 b)+2 b^2 \cos (3 (c+d x))\right)+\frac{2 i \left(247 b^2-8 a^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{b^2 \sqrt{-\frac{1}{a+b}}}}{112 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{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{\left(8 a^4+3 a^2 b^2-32 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{\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}",1,"(((2*I)*(-8*a^2 + 247*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(b^2*Sqrt[-(a + b)^(-1)]) + (8*b*(125*a^2 - 16*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*a*(160*a^2 + 37*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + 4*Sqrt[a + b*Sin[c + d*x]]*((-24*a^2 + 22*b^2)*Cos[c + d*x] + 2*b^2*Cos[3*(c + d*x)] - 7*a*Cot[c + d*x]*(9*b + 2*a*Csc[c + d*x]) - 12*a*b*Sin[2*(c + d*x)]))/(112*d)","C",1
1164,1,466,429,3.6530165,"\int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2),x]","\frac{-\frac{8 a \left(40 a^2-173 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 b \left(424 a^2+117 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{4}{3} \sqrt{a+b \sin (c+d x)} \left(5 \cot (c+d x) \left(8 a^2 \csc ^2(c+d x)-32 a^2+26 a b \csc (c+d x)+33 b^2\right)+176 a b \cos (c+d x)+24 b^2 \sin (2 (c+d x))\right)+\frac{2 i \left(167 b^2-176 a^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}}{160 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,"(((2*I)*(-176*a^2 + 167*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]) - (8*a*(40*a^2 - 173*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*b*(424*a^2 + 117*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (4*Sqrt[a + b*Sin[c + d*x]]*(176*a*b*Cos[c + d*x] + 5*Cot[c + d*x]*(-32*a^2 + 33*b^2 + 26*a*b*Csc[c + d*x] + 8*a^2*Csc[c + d*x]^2) + 24*b^2*Sin[2*(c + d*x)]))/3)/(160*d)","C",1
1165,1,655,449,6.697251,"\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{5 \csc (c+d x) \left(116 a^2 b \cos (c+d x)-3 b^3 \cos (c+d x)\right)}{192 a}+\frac{1}{96} \csc ^2(c+d x) \left(60 a^2 \cos (c+d x)-59 b^2 \cos (c+d x)\right)-\frac{1}{4} a^2 \cot (c+d x) \csc ^3(c+d x)-\frac{17}{24} a b \cot (c+d x) \csc ^2(c+d x)-\frac{2}{3} b^2 \cos (c+d x)\right)}{d}+\frac{-\frac{2 \left(688 a^3 b-348 a b^3\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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(5 b^4-492 a^2 b^2\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}-\frac{2 \left(96 a^4-228 a^2 b^2-15 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)}{\sqrt{a+b \sin (c+d x)}}}{256 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(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{\left(48 a^4-360 a^2 b^2-5 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{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}",1,"(((-2*b^2*Cos[c + d*x])/3 + (5*(116*a^2*b*Cos[c + d*x] - 3*b^3*Cos[c + d*x])*Csc[c + d*x])/(192*a) + ((60*a^2*Cos[c + d*x] - 59*b^2*Cos[c + d*x])*Csc[c + d*x]^2)/96 - (17*a*b*Cot[c + d*x]*Csc[c + d*x]^2)/24 - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/4)*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(688*a^3*b - 348*a*b^3)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(96*a^4 - 228*a^2*b^2 - 15*b^4)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(-492*a^2*b^2 + 5*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(256*a*d)","C",1
1166,1,700,482,6.8403019,"\int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{\csc ^2(c+d x) \left(436 a^2 b \cos (c+d x)-5 b^3 \cos (c+d x)\right)}{320 a}+\frac{1}{80} \csc ^3(c+d x) \left(32 a^2 \cos (c+d x)-31 b^2 \cos (c+d x)\right)-\frac{1}{5} a^2 \cot (c+d x) \csc ^4(c+d x)+\frac{\csc (c+d x) \left(-128 a^4 \cos (c+d x)+1196 a^2 b^2 \cos (c+d x)+15 b^4 \cos (c+d x)\right)}{640 a^2}-\frac{21}{40} a b \cot (c+d x) \csc ^3(c+d x)\right)}{d}+\frac{b \left(-\frac{2 \left(5936 a^3 b+20 a b^3\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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left(2272 a^4+1276 a^2 b^2+45 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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(128 a^4-2476 a^2 b^2-15 b^4\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}\right)}{2560 a^2 d}","\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{\left(128 a^4-580 a^2 b^2+15 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{640 a^2 d}+\frac{\left(128 a^4+492 a^2 b^2-5 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(128 a^4-2476 a^2 b^2-15 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(80 a^4-40 a^2 b^2+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{\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{5 a d}",1,"((((-128*a^4*Cos[c + d*x] + 1196*a^2*b^2*Cos[c + d*x] + 15*b^4*Cos[c + d*x])*Csc[c + d*x])/(640*a^2) + ((436*a^2*b*Cos[c + d*x] - 5*b^3*Cos[c + d*x])*Csc[c + d*x]^2)/(320*a) + ((32*a^2*Cos[c + d*x] - 31*b^2*Cos[c + d*x])*Csc[c + d*x]^3)/80 - (21*a*b*Cot[c + d*x]*Csc[c + d*x]^3)/40 - (a^2*Cot[c + d*x]*Csc[c + d*x]^4)/5)*Sqrt[a + b*Sin[c + d*x]])/d + (b*((-2*(5936*a^3*b + 20*a*b^3)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(2272*a^4 + 1276*a^2*b^2 + 45*b^4)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(128*a^4 - 2476*a^2*b^2 - 15*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)])))/(2560*a^2*d)","C",1
1167,1,771,551,6.8128136,"\int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{\csc ^3(c+d x) \left(164 a^2 b \cos (c+d x)-b^3 \cos (c+d x)\right)}{192 a}+\frac{1}{96} \csc ^4(c+d x) \left(28 a^2 \cos (c+d x)-27 b^2 \cos (c+d x)\right)-\frac{1}{6} a^2 \cot (c+d x) \csc ^5(c+d x)+\frac{\csc ^2(c+d x) \left(-48 a^4 \cos (c+d x)+600 a^2 b^2 \cos (c+d x)+5 b^4 \cos (c+d x)\right)}{768 a^2}+\frac{\csc (c+d x) \left(-720 a^4 b \cos (c+d x)+176 a^2 b^3 \cos (c+d x)-15 b^5 \cos (c+d x)\right)}{1536 a^3}-\frac{5}{12} a b \cot (c+d x) \csc ^4(c+d x)\right)}{d}+\frac{-\frac{2 \left(192 a^5 b+3744 a^3 b^3-20 a b^5\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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(720 a^4 b^2-176 a^2 b^4+15 b^6\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}-\frac{2 \left(384 a^6+3600 a^4 b^2+536 a^2 b^4-45 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)}{\sqrt{a+b \sin (c+d x)}}}{6144 a^3 d}","\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{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}+\frac{b \left(816 a^4+1696 a^2 b^2+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{\left(16 a^4-56 a^2 b^2+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 \left(720 a^4-176 a^2 b^2+15 b^4\right) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1536 a^3 d}-\frac{b \left(720 a^4-176 a^2 b^2+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(64 a^6+720 a^4 b^2+60 a^2 b^4-5 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{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}",1,"((((-720*a^4*b*Cos[c + d*x] + 176*a^2*b^3*Cos[c + d*x] - 15*b^5*Cos[c + d*x])*Csc[c + d*x])/(1536*a^3) + ((-48*a^4*Cos[c + d*x] + 600*a^2*b^2*Cos[c + d*x] + 5*b^4*Cos[c + d*x])*Csc[c + d*x]^2)/(768*a^2) + ((164*a^2*b*Cos[c + d*x] - b^3*Cos[c + d*x])*Csc[c + d*x]^3)/(192*a) + ((28*a^2*Cos[c + d*x] - 27*b^2*Cos[c + d*x])*Csc[c + d*x]^4)/96 - (5*a*b*Cot[c + d*x]*Csc[c + d*x]^4)/12 - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/6)*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(192*a^5*b + 3744*a^3*b^3 - 20*a*b^5)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(384*a^6 + 3600*a^4*b^2 + 536*a^2*b^4 - 45*b^6)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(720*a^4*b^2 - 176*a^2*b^4 + 15*b^6)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(6144*a^3*d)","C",1
1168,1,382,471,5.5933597,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/Sqrt[a + b*Sin[c + d*x]],x]","\frac{384 a \left(1280 a^6-2368 a^4 b^2+875 a^2 b^4+213 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+3 b \cos (c+d x) \left(81920 a^6+20480 a^5 b \sin (c+d x)-125952 a^4 b^2-28608 a^3 b^3 \sin (c+d x)-1600 a^3 b^3 \sin (3 (c+d x))+23760 a^2 b^4-70 \left(8 a^2 b^4-11 b^6\right) \cos (4 (c+d x))+\left(5120 a^4 b^2-5792 a^2 b^4-8547 b^6\right) \cos (2 (c+d x))+2332 a b^5 \sin (c+d x)+1390 a b^5 \sin (3 (c+d x))+210 a b^5 \sin (5 (c+d x))+1155 b^6 \cos (6 (c+d x))+6622 b^6\right)-384 \left(1280 a^7+1280 a^6 b-2048 a^5 b^2-2048 a^4 b^3+453 a^3 b^4+453 a^2 b^5+231 a b^6+231 b^7\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{720720 b^7 d \sqrt{a+b \sin (c+d x)}}","\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{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{64 a \left(80 a^4-118 a^2 b^2+17 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^6 d}-\frac{8 \left(480 a^4-683 a^2 b^2+77 b^4\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15015 b^5 d}-\frac{8 a \left(1280 a^6-2368 a^4 b^2+875 a^2 b^4+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(1280 a^6-2048 a^4 b^2+453 a^2 b^4+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,"(-384*(1280*a^7 + 1280*a^6*b - 2048*a^5*b^2 - 2048*a^4*b^3 + 453*a^3*b^4 + 453*a^2*b^5 + 231*a*b^6 + 231*b^7)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 384*a*(1280*a^6 - 2368*a^4*b^2 + 875*a^2*b^4 + 213*b^6)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 3*b*Cos[c + d*x]*(81920*a^6 - 125952*a^4*b^2 + 23760*a^2*b^4 + 6622*b^6 + (5120*a^4*b^2 - 5792*a^2*b^4 - 8547*b^6)*Cos[2*(c + d*x)] - 70*(8*a^2*b^4 - 11*b^6)*Cos[4*(c + d*x)] + 1155*b^6*Cos[6*(c + d*x)] + 20480*a^5*b*Sin[c + d*x] - 28608*a^3*b^3*Sin[c + d*x] + 2332*a*b^5*Sin[c + d*x] - 1600*a^3*b^3*Sin[3*(c + d*x)] + 1390*a*b^5*Sin[3*(c + d*x)] + 210*a*b^5*Sin[5*(c + d*x)]))/(720720*b^7*d*Sqrt[a + b*Sin[c + d*x]])","A",1
1169,1,326,405,4.3573608,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/Sqrt[a + b*Sin[c + d*x]],x]","\frac{-64 \left(320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+b \cos (c+d x) \left(-10240 a^5-2560 a^4 b \sin (c+d x)+16448 a^3 b^2-128 \left(5 a^3 b^2-6 a b^4\right) \cos (2 (c+d x))+3752 a^2 b^3 \sin (c+d x)+200 a^2 b^3 \sin (3 (c+d x))+70 a b^4 \cos (4 (c+d x))-3718 a b^4+990 b^5 \sin (c+d x)-765 b^5 \sin (3 (c+d x))-315 b^5 \sin (5 (c+d x))\right)+128 a \left(160 a^5+160 a^4 b-267 a^3 b^2-267 a^2 b^3+69 a b^4+69 b^5\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{27720 b^6 d \sqrt{a+b \sin (c+d x)}}","\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{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{16 a \left(160 a^4-267 a^2 b^2+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{8 \left(160 a^4-247 a^2 b^2+45 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3465 b^5 d}+\frac{8 \left(320 a^6-614 a^4 b^2+249 a^2 b^4+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{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,"(128*a*(160*a^5 + 160*a^4*b - 267*a^3*b^2 - 267*a^2*b^3 + 69*a*b^4 + 69*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 64*(320*a^6 - 614*a^4*b^2 + 249*a^2*b^4 + 45*b^6)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(-10240*a^5 + 16448*a^3*b^2 - 3718*a*b^4 - 128*(5*a^3*b^2 - 6*a*b^4)*Cos[2*(c + d*x)] + 70*a*b^4*Cos[4*(c + d*x)] - 2560*a^4*b*Sin[c + d*x] + 3752*a^2*b^3*Sin[c + d*x] + 990*b^5*Sin[c + d*x] + 200*a^2*b^3*Sin[3*(c + d*x)] - 765*b^5*Sin[3*(c + d*x)] - 315*b^5*Sin[5*(c + d*x)]))/(27720*b^6*d*Sqrt[a + b*Sin[c + d*x]])","A",1
1170,1,275,283,3.1366306,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/Sqrt[a + b*Sin[c + d*x]],x]","\frac{32 a \left(32 a^4-65 a^2 b^2+33 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-b \cos (c+d x) \left(-512 a^4-128 a^3 b \sin (c+d x)+880 a^2 b^2-8 \left(4 a^2 b^2-21 b^4\right) \cos (2 (c+d x))+202 a b^3 \sin (c+d x)+10 a b^3 \sin (3 (c+d x))+35 b^4 \cos (4 (c+d x))-203 b^4\right)-32 \left(32 a^5+32 a^4 b-57 a^3 b^2-57 a^2 b^3+21 a b^4+21 b^5\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{1260 b^5 d \sqrt{a+b \sin (c+d 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(32 a^4-65 a^2 b^2+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(32 a^4-57 a^2 b^2+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,"(-32*(32*a^5 + 32*a^4*b - 57*a^3*b^2 - 57*a^2*b^3 + 21*a*b^4 + 21*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 32*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*Cos[c + d*x]*(-512*a^4 + 880*a^2*b^2 - 203*b^4 - 8*(4*a^2*b^2 - 21*b^4)*Cos[2*(c + d*x)] + 35*b^4*Cos[4*(c + d*x)] - 128*a^3*b*Sin[c + d*x] + 202*a*b^3*Sin[c + d*x] + 10*a*b^3*Sin[3*(c + d*x)]))/(1260*b^5*d*Sqrt[a + b*Sin[c + d*x]])","A",1
1171,1,408,288,3.7515935,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/Sqrt[a + b*Sin[c + d*x]],x]","\frac{-\frac{2 \left(8 a^2+9 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 i \left(21 b^2-8 a^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^2 \sqrt{-\frac{1}{a+b}}}+4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}-\frac{8 a b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}}{30 b^2 d}","-\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,"(((2*I)*(-8*a^2 + 21*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(a*b^2*Sqrt[-(a + b)^(-1)]) + 4*Cos[c + d*x]*(4*a - 3*b*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]] - (8*a*b*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(8*a^2 + 9*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]])/(30*b^2*d)","C",1
1172,1,416,285,3.639405,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/Sqrt[a + b*Sin[c + d*x]],x]","\frac{\frac{2 \left(4 a^2+9 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{a b \sqrt{a+b \sin (c+d x)}}+\frac{2 i \left(4 a^2+3 b^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a^2 b^3 \sqrt{-\frac{1}{a+b}}}-\frac{4 \cot (c+d x) (2 a \sin (c+d x)+3 b) \sqrt{a+b \sin (c+d x)}}{a b}+\frac{40 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}}{12 d}","\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*I)*(4*a^2 + 3*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a^2*b^3*Sqrt[-(a + b)^(-1)]) - (4*Cot[c + d*x]*(3*b + 2*a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(a*b) + (40*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*(4*a^2 + 9*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a*b*Sqrt[a + b*Sin[c + d*x]]))/(12*d)","C",1
1173,1,443,307,3.3477268,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/Sqrt[a + b*Sin[c + d*x]],x]","\frac{\frac{2 \left(16 a^2-9 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{a^2 \sqrt{a+b \sin (c+d x)}}-\frac{4 \cot (c+d x) \sqrt{a+b \sin (c+d x)} (2 a \csc (c+d x)-3 b)}{a^2}+\frac{2 i \left(8 a^2+3 b^2\right) \cos (2 (c+d x)) \csc ^2(c+d x) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a^3 b^2 \sqrt{-\frac{1}{a+b}} \left(\csc ^2(c+d x)-2\right)}-\frac{8 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{a \sqrt{a+b \sin (c+d x)}}}{16 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,"(((2*I)*(8*a^2 + 3*b^2)*Cos[2*(c + d*x)]*Csc[c + d*x]^2*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a^3*b^2*Sqrt[-(a + b)^(-1)]*(-2 + Csc[c + d*x]^2)) - (4*Cot[c + d*x]*(-3*b + 2*a*Csc[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/a^2 - (8*b*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a*Sqrt[a + b*Sin[c + d*x]]) + (2*(16*a^2 - 9*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a^2*Sqrt[a + b*Sin[c + d*x]]))/(16*d)","C",1
1174,1,475,353,5.7201465,"\int \frac{\cot ^4(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[Cot[c + d*x]^4/Sqrt[a + b*Sin[c + d*x]],x]","\frac{-\frac{4 \cot (c+d x) \sqrt{a+b \sin (c+d x)} \left(8 a^2 \csc ^2(c+d x)-32 a^2-10 a b \csc (c+d x)+15 b^2\right)}{a^3}+\frac{-\frac{8 a \left(24 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 b \left(45 b^2-104 a^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 i \left(32 a^2-15 b^2\right) \cos (2 (c+d x)) \csc ^2(c+d x) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a b \sqrt{-\frac{1}{a+b}} \left(\csc ^2(c+d x)-2\right)}}{a^3}}{96 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{5 b \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^2 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(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{\cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3 a d}",1,"((-4*Cot[c + d*x]*(-32*a^2 + 15*b^2 - 10*a*b*Csc[c + d*x] + 8*a^2*Csc[c + d*x]^2)*Sqrt[a + b*Sin[c + d*x]])/a^3 + (((2*I)*(32*a^2 - 15*b^2)*Cos[2*(c + d*x)]*Csc[c + d*x]^2*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]*(-2 + Csc[c + d*x]^2)) - (8*a*(24*a^2 - 5*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*b*(-104*a^2 + 45*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]])/a^3)/(96*d)","C",1
1175,1,647,412,6.7007115,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/Sqrt[a + b*Sin[c + d*x]],x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{7 b \cot (c+d x) \csc ^2(c+d x)}{24 a^2}+\frac{\csc (c+d x) \left(105 b^3 \cos (c+d x)-188 a^2 b \cos (c+d x)\right)}{192 a^4}+\frac{5 \csc ^2(c+d x) \left(12 a^2 \cos (c+d x)-7 b^2 \cos (c+d x)\right)}{96 a^3}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a}\right)}{d}+\frac{-\frac{2 \left(140 a b^3-240 a^3 b\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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(188 a^2 b^2-105 b^4\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}-\frac{2 \left(288 a^4-620 a^2 b^2+315 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)}{\sqrt{a+b \sin (c+d x)}}}{768 a^4 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{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(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(48 a^4-72 a^2 b^2+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{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{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{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d}",1,"((((-188*a^2*b*Cos[c + d*x] + 105*b^3*Cos[c + d*x])*Csc[c + d*x])/(192*a^4) + (5*(12*a^2*Cos[c + d*x] - 7*b^2*Cos[c + d*x])*Csc[c + d*x]^2)/(96*a^3) + (7*b*Cot[c + d*x]*Csc[c + d*x]^2)/(24*a^2) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a))*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(-240*a^3*b + 140*a*b^3)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(288*a^4 - 620*a^2*b^2 + 315*b^4)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(188*a^2*b^2 - 105*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(768*a^4*d)","C",1
1176,1,326,466,6.8842921,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^(3/2),x]","\frac{-64 \left(1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+b \cos (c+d x) \left(-40960 a^5-10240 a^4 b \sin (c+d x)+40448 a^3 b^2-16 \left(160 a^3 b^2-93 a b^4\right) \cos (2 (c+d x))+8672 a^2 b^3 \sin (c+d x)+800 a^2 b^3 \sin (3 (c+d x))+280 a b^4 \cos (4 (c+d x))-2728 a b^4+330 b^5 \sin (c+d x)-255 b^5 \sin (3 (c+d x))-105 b^5 \sin (5 (c+d x))\right)+64 a \left(1280 a^5+1280 a^4 b-1344 a^3 b^2-1344 a^2 b^3+123 a b^4+123 b^5\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{9240 b^7 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)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\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{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{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{8 a \left(1280 a^4-1344 a^2 b^2+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{8 \left(640 a^4-592 a^2 b^2+15 b^4\right) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{1155 b^6 d}+\frac{8 \left(1280 a^6-1664 a^4 b^2+369 a^2 b^4+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{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{11 b^2 d}",1,"(64*a*(1280*a^5 + 1280*a^4*b - 1344*a^3*b^2 - 1344*a^2*b^3 + 123*a*b^4 + 123*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 64*(1280*a^6 - 1664*a^4*b^2 + 369*a^2*b^4 + 15*b^6)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(-40960*a^5 + 40448*a^3*b^2 - 2728*a*b^4 - 16*(160*a^3*b^2 - 93*a*b^4)*Cos[2*(c + d*x)] + 280*a*b^4*Cos[4*(c + d*x)] - 10240*a^4*b*Sin[c + d*x] + 8672*a^2*b^3*Sin[c + d*x] + 330*b^5*Sin[c + d*x] + 800*a^2*b^3*Sin[3*(c + d*x)] - 255*b^5*Sin[3*(c + d*x)] - 105*b^5*Sin[5*(c + d*x)]))/(9240*b^7*d*Sqrt[a + b*Sin[c + d*x]])","A",1
1177,1,275,401,5.3024966,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^(3/2),x]","\frac{64 a \left(160 a^4-199 a^2 b^2+39 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-b \cos (c+d x) \left(-5120 a^4-1280 a^3 b \sin (c+d x)+4768 a^2 b^2-8 \left(40 a^2 b^2-21 b^4\right) \cos (2 (c+d x))+1012 a b^3 \sin (c+d x)+100 a b^3 \sin (3 (c+d x))+35 b^4 \cos (4 (c+d x))-203 b^4\right)-32 \left(320 a^5+320 a^4 b-318 a^3 b^2-318 a^2 b^3+21 a b^4+21 b^5\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{1260 b^6 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)}{a b^2 d \sqrt{a+b \sin (c+d x)}}+\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 \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{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 a \left(160 a^4-199 a^2 b^2+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(320 a^4-318 a^2 b^2+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,"(-32*(320*a^5 + 320*a^4*b - 318*a^3*b^2 - 318*a^2*b^3 + 21*a*b^4 + 21*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 64*a*(160*a^4 - 199*a^2*b^2 + 39*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*Cos[c + d*x]*(-5120*a^4 + 4768*a^2*b^2 - 203*b^4 - 8*(40*a^2*b^2 - 21*b^4)*Cos[2*(c + d*x)] + 35*b^4*Cos[4*(c + d*x)] - 1280*a^3*b*Sin[c + d*x] + 1012*a*b^3*Sin[c + d*x] + 100*a*b^3*Sin[3*(c + d*x)]))/(1260*b^6*d*Sqrt[a + b*Sin[c + d*x]])","A",1
1178,1,222,261,4.1567824,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^(3/2),x]","\frac{-16 \left(32 a^4-37 a^2 b^2+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+b \cos (c+d x) \left(-256 a^3+\left(45 b^3-64 a^2 b\right) \sin (c+d x)-16 a b^2 \cos (2 (c+d x))+216 a b^2+5 b^3 \sin (3 (c+d x))\right)+16 a \left(32 a^3+32 a^2 b-29 a b^2-29 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{70 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{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(32 a^4-37 a^2 b^2+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{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,"(16*a*(32*a^3 + 32*a^2*b - 29*a*b^2 - 29*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 16*(32*a^4 - 37*a^2*b^2 + 5*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(-256*a^3 + 216*a*b^2 - 16*a*b^2*Cos[2*(c + d*x)] + (-64*a^2*b + 45*b^3)*Sin[c + d*x] + 5*b^3*Sin[3*(c + d*x)]))/(70*b^5*d*Sqrt[a + b*Sin[c + d*x]])","A",1
1179,1,419,296,3.8660694,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x])^(3/2),x]","\frac{-\frac{4 \cos (c+d x) \left(4 a^2+a b \sin (c+d x)-3 b^2\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 \left(8 a^2-9 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(3 b^2-8 a^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^2 \sqrt{-\frac{1}{a+b}}}+\frac{8 a b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}}{6 a b^2 d}","-\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*I)*(-8*a^2 + 3*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b^2*Sqrt[-(a + b)^(-1)]) + (8*a*b*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*(8*a^2 - 9*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (4*Cos[c + d*x]*(4*a^2 - 3*b^2 + a*b*Sin[c + d*x]))/Sqrt[a + b*Sin[c + d*x]])/(6*a*b^2*d)","C",1
1180,1,433,294,3.6017594,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^(3/2),x]","\frac{\frac{4 a \left(a^2-b^2\right) \cos (c+d x)}{b \sqrt{a+b \sin (c+d x)}}-\frac{a \left(4 a^2-9 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{b \sqrt{a+b \sin (c+d x)}}+\frac{i \left(3 b^2-4 a^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{b^3 \sqrt{-\frac{1}{a+b}}}+\frac{4 a^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-2 a \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{2 a^3 d}","\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,"((I*(-4*a^2 + 3*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(b^3*Sqrt[-(a + b)^(-1)]) + (4*a*(a^2 - b^2)*Cos[c + d*x])/(b*Sqrt[a + b*Sin[c + d*x]]) - 2*a*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]] + (4*a^2*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (a*(4*a^2 - 9*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b*Sqrt[a + b*Sin[c + d*x]]))/(2*a^3*d)","C",1
1181,1,435,366,5.0338882,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^(3/2),x]","\frac{\frac{\left(60 b^2-32 a^2\right) \cos (c+d x)+4 a \cot (c+d x) (5 b-2 a \csc (c+d x))}{a^3 \sqrt{a+b \sin (c+d x)}}+\frac{\frac{2 \left(32 a^2-45 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(15 b^2-8 a^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^2 \sqrt{-\frac{1}{a+b}}}-\frac{40 a b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}}{a^3}}{16 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) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^3 b d}-\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,"(((-32*a^2 + 60*b^2)*Cos[c + d*x] + 4*a*Cot[c + d*x]*(5*b - 2*a*Csc[c + d*x]))/(a^3*Sqrt[a + b*Sin[c + d*x]]) + (((-2*I)*(-8*a^2 + 15*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b^2*Sqrt[-(a + b)^(-1)]) - (40*a*b*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*(32*a^2 - 45*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]])/a^3)/(16*d)","C",1
1182,1,468,416,5.8463533,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^(3/2),x]","\frac{-\frac{4 \left(\left(105 b^3-80 a^2 b\right) \cos (c+d x)+a \cot (c+d x) \left(8 a^2 \csc ^2(c+d x)-32 a^2-14 a b \csc (c+d x)+35 b^2\right)\right)}{a^4 \sqrt{a+b \sin (c+d x)}}+\frac{-\frac{8 a \left(24 a^2-35 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{2 b \left(315 b^2-296 a^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{10 i \left(21 b^2-16 a^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}}{a^4}}{96 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{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{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(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{\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{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}}",1,"((-4*((-80*a^2*b + 105*b^3)*Cos[c + d*x] + a*Cot[c + d*x]*(-32*a^2 + 35*b^2 - 14*a*b*Csc[c + d*x] + 8*a^2*Csc[c + d*x]^2)))/(a^4*Sqrt[a + b*Sin[c + d*x]]) + (((10*I)*(-16*a^2 + 21*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]) - (8*a*(24*a^2 - 35*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*b*(-296*a^2 + 315*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]])/a^4)/(96*d)","C",1
1183,1,1044,469,9.8310036,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^(5/2),x]","\frac{315 \left(\frac{\left(\left(a^2+3 b^2\right) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+a (b-a) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right) \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2}}{(a-b)^2 b}-\frac{\cos (c+d x) \left(2 a \left(a^2+b^2\right)+b \left(a^2+3 b^2\right) \sin (c+d x)\right)}{\left(a^2-b^2\right)^2}\right)+\frac{315 \left(\frac{\left(\left(32 a^4-57 b^2 a^2+21 b^4\right) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+a \left(-32 a^3+32 b a^2+33 b^2 a-33 b^3\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right) \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2}}{(a-b)^2}-\frac{b \left(4 a \left(8 a^4-13 b^2 a^2+3 b^4\right) \cos (c+d x)+b \left(20 a^4-33 b^2 a^2+9 b^4\right) \sin (2 (c+d x))\right)}{2 \left(a^2-b^2\right)^2}\right)}{b^3}-\frac{21 \left(\frac{\left(\left(-2048 a^6+4192 b^2 a^4-2355 b^4 a^2+231 b^6\right) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+a \left(2048 a^5-2048 b a^4-2656 b^2 a^3+2656 b^3 a^2+603 b^4 a-603 b^5\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right) \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2}}{(a-b)^2}+\frac{b \cos (c+d x) \left(-64 a b^2 \cos (2 (c+d x)) \left(a^2-b^2\right)^2+b \left(1280 a^6-2536 b^2 a^4+1347 b^4 a^2-111 b^6\right) \sin (c+d x)+2 \left(512 a^7-952 b^2 a^5+423 b^4 a^3+7 b^6 a+6 b^3 \left(a^2-b^2\right)^2 \sin (3 (c+d x))\right)\right)}{\left(a^2-b^2\right)^2}\right)}{b^5}-\frac{5 (a+b \sin (c+d x)) \left(\frac{\sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left(\left(65536 a^8-161792 b^2 a^6+129664 b^4 a^4-35109 b^6 a^2+1617 b^8\right) \left((a+b) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)-4 b \left(-4096 b a^7+8960 b^3 a^5-5884 b^5 a^3+1041 b^7 a\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)}{(a-b)^2 (a+b)^2}+b (a+b \sin (c+d x)) \left(-56 \sin (4 (c+d x)) b^3+416 a \cos (3 (c+d x)) b^2-8 \left(35 b^2-276 a^2\right) \sin (2 (c+d x)) b-128 a \left(88 a^2-27 b^2\right) \cos (c+d x)-\frac{21 \left(1088 a^8-2576 b^2 a^6+1960 b^4 a^4-497 b^6 a^2+21 b^8\right) \cos (c+d x)}{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{21 a \left(64 a^6-112 b^2 a^4+56 b^4 a^2-7 b^6\right) \cos (c+d x)}{\left(a^2-b^2\right) (a+b \sin (c+d x))^2}\right)\right)}{b^7}}{10080 d (a+b \sin (c+d x))^{3/2}}","\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{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 \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{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{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{8 a \left(1280 a^4-1088 a^2 b^2+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(1280 a^4-768 a^2 b^2+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,"(315*((((a^2 + 3*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + a*(-a + b)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a + b))^(3/2))/((a - b)^2*b) - (Cos[c + d*x]*(2*a*(a^2 + b^2) + b*(a^2 + 3*b^2)*Sin[c + d*x]))/(a^2 - b^2)^2) + (315*((((32*a^4 - 57*a^2*b^2 + 21*b^4)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + a*(-32*a^3 + 32*a^2*b + 33*a*b^2 - 33*b^3)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a + b))^(3/2))/(a - b)^2 - (b*(4*a*(8*a^4 - 13*a^2*b^2 + 3*b^4)*Cos[c + d*x] + b*(20*a^4 - 33*a^2*b^2 + 9*b^4)*Sin[2*(c + d*x)]))/(2*(a^2 - b^2)^2)))/b^3 - (21*((((-2048*a^6 + 4192*a^4*b^2 - 2355*a^2*b^4 + 231*b^6)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + a*(2048*a^5 - 2048*a^4*b - 2656*a^3*b^2 + 2656*a^2*b^3 + 603*a*b^4 - 603*b^5)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a + b))^(3/2))/(a - b)^2 + (b*Cos[c + d*x]*(-64*a*b^2*(a^2 - b^2)^2*Cos[2*(c + d*x)] + b*(1280*a^6 - 2536*a^4*b^2 + 1347*a^2*b^4 - 111*b^6)*Sin[c + d*x] + 2*(512*a^7 - 952*a^5*b^2 + 423*a^3*b^4 + 7*a*b^6 + 6*b^3*(a^2 - b^2)^2*Sin[3*(c + d*x)])))/(a^2 - b^2)^2))/b^5 - (5*(a + b*Sin[c + d*x])*(((-4*b*(-4096*a^7*b + 8960*a^5*b^3 - 5884*a^3*b^5 + 1041*a*b^7)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (65536*a^8 - 161792*a^6*b^2 + 129664*a^4*b^4 - 35109*a^2*b^6 + 1617*b^8)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]))*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/((a - b)^2*(a + b)^2) + b*(a + b*Sin[c + d*x])*(-128*a*(88*a^2 - 27*b^2)*Cos[c + d*x] + 416*a*b^2*Cos[3*(c + d*x)] + (21*a*(64*a^6 - 112*a^4*b^2 + 56*a^2*b^4 - 7*b^6)*Cos[c + d*x])/((a^2 - b^2)*(a + b*Sin[c + d*x])^2) - (21*(1088*a^8 - 2576*a^6*b^2 + 1960*a^4*b^4 - 497*a^2*b^6 + 21*b^8)*Cos[c + d*x])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])) - 8*b*(-276*a^2 + 35*b^2)*Sin[2*(c + d*x)] - 56*b^3*Sin[4*(c + d*x)])))/b^7)/(10080*d*(a + b*Sin[c + d*x])^(3/2))","B",1
1184,1,257,411,7.9643629,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^(5/2),x]","\frac{32 a \left(32 a^2-15 b^2\right) (a+b)^2 \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-16 \left(64 a^4-46 a^2 b^2+3 b^4\right) (a+b) \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-\frac{1}{2} b \cos (c+d x) \left(1024 a^4+1280 a^3 b \sin (c+d x)-288 a^2 b^2-8 \left(8 a^2 b^2-3 b^4\right) \cos (2 (c+d x))-516 a b^3 \sin (c+d x)+12 a b^3 \sin (3 (c+d x))+3 b^4 \cos (4 (c+d x))-27 b^4\right)}{42 b^6 d (a+b \sin (c+d x))^{3/2}}","\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{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{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(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{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(64 a^4-46 a^2 b^2+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)}}",1,"(32*a*(a + b)^2*(32*a^2 - 15*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[c + d*x])/(a + b))^(3/2) - 16*(a + b)*(64*a^4 - 46*a^2*b^2 + 3*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[c + d*x])/(a + b))^(3/2) - (b*Cos[c + d*x]*(1024*a^4 - 288*a^2*b^2 - 27*b^4 - 8*(8*a^2*b^2 - 3*b^4)*Cos[2*(c + d*x)] + 3*b^4*Cos[4*(c + d*x)] + 1280*a^3*b*Sin[c + d*x] - 516*a*b^3*Sin[c + d*x] + 12*a*b^3*Sin[3*(c + d*x)]))/2)/(42*b^6*d*(a + b*Sin[c + d*x])^(3/2))","A",1
1185,1,211,254,6.2782953,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^(5/2),x]","\frac{32 a \left(32 a^2-17 b^2\right) (a+b) \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-32 \left(32 a^2-9 b^2\right) (a+b)^2 \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+2 b \cos (c+d x) \left(256 a^3+b \left(320 a^2-69 b^2\right) \sin (c+d x)-16 a b^2 \cos (2 (c+d x))-24 a b^2+3 b^3 \sin (3 (c+d x))\right)}{60 b^5 d (a+b \sin (c+d x))^{3/2}}","-\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{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{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,"(-32*(a + b)^2*(32*a^2 - 9*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[c + d*x])/(a + b))^(3/2) + 32*a*(a + b)*(32*a^2 - 17*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[c + d*x])/(a + b))^(3/2) + 2*b*Cos[c + d*x]*(256*a^3 - 24*a*b^2 - 16*a*b^2*Cos[2*(c + d*x)] + b*(320*a^2 - 69*b^2)*Sin[c + d*x] + 3*b^3*Sin[3*(c + d*x)]))/(60*b^5*d*(a + b*Sin[c + d*x])^(3/2))","A",1
1186,1,443,313,5.1811474,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{\frac{2 a^2 \left(a^2-b^2\right) \cos (c+d x)}{(a+b \sin (c+d x))^{3/2}}-\frac{2 a \left(5 a^2+3 b^2\right) \cos (c+d x)}{\sqrt{a+b \sin (c+d x)}}+\frac{a \left(8 a^2+9 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}+\frac{i \left(8 a^2+3 b^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{b^2 \sqrt{-\frac{1}{a+b}}}+\frac{4 a^2 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}}{3 a^3 b^2 d}","\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,"-1/3*((I*(8*a^2 + 3*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(b^2*Sqrt[-(a + b)^(-1)]) + (2*a^2*(a^2 - b^2)*Cos[c + d*x])/(a + b*Sin[c + d*x])^(3/2) - (2*a*(5*a^2 + 3*b^2)*Cos[c + d*x])/Sqrt[a + b*Sin[c + d*x]] + (4*a^2*b*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (a*(8*a^2 + 9*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]])/(a^3*b^2*d)","C",1
1187,1,445,346,5.4638431,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^(5/2),x]","\frac{\frac{2 \left(4 a^2+45 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left(2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left(b \left(4 a^2+15 b^2\right) \sin (2 (c+d x))+4 a \left(a^2+10 b^2\right) \cos (c+d x)+6 a^2 b \cot (c+d x)\right)}{(a+b \sin (c+d x))^{3/2}}+\frac{2 i \left(4 a^2+15 b^2\right) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^2 \sqrt{-\frac{1}{a+b}}}+\frac{40 a b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}}{12 a^3 b d}","-\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{\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) \cos (c+d x)}{3 a^3 b 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{\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}",1,"(((2*I)*(4*a^2 + 15*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b^2*Sqrt[-(a + b)^(-1)]) + (40*a*b*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*(4*a^2 + 45*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(4*a*(a^2 + 10*b^2)*Cos[c + d*x] + 6*a^2*b*Cot[c + d*x] + b*(4*a^2 + 15*b^2)*Sin[2*(c + d*x)]))/(a + b*Sin[c + d*x])^(3/2))/(12*a^3*b*d)","C",1
1188,1,622,407,6.7077444,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^(5/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{11 b \cot (c+d x)}{4 a^4}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^3}-\frac{2 \left(a^2 \cos (c+d x)-9 b^2 \cos (c+d x)\right)}{3 a^4 (a+b \sin (c+d x))}-\frac{2 \left(a^2 \cos (c+d x)-b^2 \cos (c+d x)\right)}{3 a^3 (a+b \sin (c+d x))^2}\right)}{d}+\frac{-\frac{2 \left(315 b^2-80 a^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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(8 a^2-105 b^2\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}-\frac{280 a b \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)}{\sqrt{a+b \sin (c+d x)}}}{48 a^4 d}","\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^2-105 b^2\right) \cos (c+d x)}{12 a^4 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{\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(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{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^{3/2}}",1,"(Sqrt[a + b*Sin[c + d*x]]*((11*b*Cot[c + d*x])/(4*a^4) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3) - (2*(a^2*Cos[c + d*x] - b^2*Cos[c + d*x]))/(3*a^3*(a + b*Sin[c + d*x])^2) - (2*(a^2*Cos[c + d*x] - 9*b^2*Cos[c + d*x]))/(3*a^4*(a + b*Sin[c + d*x]))))/d + ((-280*a*b*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(-80*a^2 + 315*b^2)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(8*a^2 - 105*b^2)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(48*a^4*d)","C",0
1189,1,680,458,6.970067,"\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{17 b \cot (c+d x) \csc (c+d x)}{12 a^4}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^3}+\frac{8 \left(a^2 b \cos (c+d x)-3 b^3 \cos (c+d x)\right)}{3 a^5 (a+b \sin (c+d x))}+\frac{\csc (c+d x) \left(32 a^2 \cos (c+d x)-123 b^2 \cos (c+d x)\right)}{24 a^5}+\frac{2 \left(a^2 b \cos (c+d x)-b^3 \cos (c+d x)\right)}{3 a^4 (a+b \sin (c+d x))^2}\right)}{d}+\frac{-\frac{2 \left(32 a^3-140 a 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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left(152 a^2 b-315 b^3\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)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left(105 b^3-32 a^2 b\right) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left(-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}}{32 a^5 d}","\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{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(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(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(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{\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^{3/2}}",1,"(Sqrt[a + b*Sin[c + d*x]]*(((32*a^2*Cos[c + d*x] - 123*b^2*Cos[c + d*x])*Csc[c + d*x])/(24*a^5) + (17*b*Cot[c + d*x]*Csc[c + d*x])/(12*a^4) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^3) + (2*(a^2*b*Cos[c + d*x] - b^3*Cos[c + d*x]))/(3*a^4*(a + b*Sin[c + d*x])^2) + (8*(a^2*b*Cos[c + d*x] - 3*b^3*Cos[c + d*x]))/(3*a^5*(a + b*Sin[c + d*x]))))/d + ((-2*(32*a^3 - 140*a*b^2)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(152*a^2*b - 315*b^3)*EllipticPi[2, (-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(-32*a^2*b + 105*b^3)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(32*a^5*d)","C",0
1190,1,1670,510,6.5544136,"\int \frac{\cos ^4(e+f x)}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx","Integrate[Cos[e + f*x]^4/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(9/2)),x]","\frac{\sin (e+f x) \sqrt{a+b \sin (e+f x)} \left(-\frac{2 \left(a^2 \cos (e+f x)-b^2 \cos (e+f x)\right)}{7 a b^2 (a+b \sin (e+f x))^4}-\frac{32 \left(2 a^2 b^2 \cos (e+f x)-b^4 \cos (e+f x)\right)}{35 a^4 \left(a^2-b^2\right)^2 (a+b \sin (e+f x))}-\frac{2 \left(5 \cos (e+f x) a^4-9 b^2 \cos (e+f x) a^2+8 b^4 \cos (e+f x)\right)}{35 a^3 b^2 \left(a^2-b^2\right) (a+b \sin (e+f x))^2}+\frac{4 \left(5 \cos (e+f x) a^2+3 b^2 \cos (e+f x)\right)}{35 a^2 b^2 (a+b \sin (e+f x))^3}\right)}{f \sqrt{d \sin (e+f x)}}+\frac{4 \sqrt{\sin (e+f x)} \left(\frac{4 a \left(5 a^4-9 b^2 a^2+4 b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}} \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}+4 a \left(4 a b^3-8 a^3 b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{b \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}\right)+2 \left(8 a^2 b^2-4 b^4\right) \left(\frac{\sqrt{a+b \sin (e+f x)} \cos (e+f x)}{b \sqrt{\sin (e+f x)}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{b \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}\right)}{b}+\frac{i \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc (e+f x) E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{\sqrt{\sin (e+f x)}}\right)|-\frac{2 a}{-a-b}\right) \sqrt{a+b \sin (e+f x)}}{b \sqrt{\cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc (e+f x)} \sqrt{\frac{\csc (e+f x) (a+b \sin (e+f x))}{a+b}}}\right)\right)}{35 a^4 (a-b)^2 (a+b)^2 f \sqrt{d \sin (e+f x)}}","\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{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{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) \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{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,"(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((-2*(a^2*Cos[e + f*x] - b^2*Cos[e + f*x]))/(7*a*b^2*(a + b*Sin[e + f*x])^4) + (4*(5*a^2*Cos[e + f*x] + 3*b^2*Cos[e + f*x]))/(35*a^2*b^2*(a + b*Sin[e + f*x])^3) - (2*(5*a^4*Cos[e + f*x] - 9*a^2*b^2*Cos[e + f*x] + 8*b^4*Cos[e + f*x]))/(35*a^3*b^2*(a^2 - b^2)*(a + b*Sin[e + f*x])^2) - (32*(2*a^2*b^2*Cos[e + f*x] - b^4*Cos[e + f*x]))/(35*a^4*(a^2 - b^2)^2*(a + b*Sin[e + f*x]))))/(f*Sqrt[d*Sin[e + f*x]]) + (4*Sqrt[Sin[e + f*x]]*((4*a*(5*a^4 - 9*a^2*b^2 + 4*b^4)*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + 4*a*(-8*a^3*b + 4*a*b^3)*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])) + 2*(8*a^2*b^2 - 4*b^4)*((Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Sin[e + f*x]]) + (I*Cos[(-e + Pi/2 - f*x)/2]*Csc[e + f*x]*EllipticE[I*ArcSinh[Sin[(-e + Pi/2 - f*x)/2]/Sqrt[Sin[e + f*x]]], (-2*a)/(-a - b)]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Cos[(-e + Pi/2 - f*x)/2]^2*Csc[e + f*x]]*Sqrt[(Csc[e + f*x]*(a + b*Sin[e + f*x]))/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])))/b)))/(35*a^4*(a - b)^2*(a + b)^2*f*Sqrt[d*Sin[e + f*x]])","C",0
1191,0,0,36,24.5201352,"\int \frac{\cos ^4(c+d x) \sqrt[3]{\sin (c+d x)}}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[(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,"Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]], x]","A",-1
1192,0,0,32,6.4959968,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx","Integrate[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,"Integrate[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p, x]","A",-1
1193,0,0,36,4.9945199,"\int \cos ^4(c+d x) \sin ^{-3-p}(c+d x) (a+b \sin (c+d x))^p \, dx","Integrate[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,"Integrate[Cos[c + d*x]^4*Sin[c + d*x]^(-3 - p)*(a + b*Sin[c + d*x])^p, x]","A",-1
1194,0,0,36,5.6501301,"\int \cos ^4(c+d x) \sin ^{-4-p}(c+d x) (a+b \sin (c+d x))^p \, dx","Integrate[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,"Integrate[Cos[c + d*x]^4*Sin[c + d*x]^(-4 - p)*(a + b*Sin[c + d*x])^p, x]","A",-1
1195,1,195,623,0.8485259,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^3,x]","\frac{\sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{n+1}(c+d x) \left(\frac{a^3 \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)}{n+1}+b \sin (c+d x) \left(\frac{3 a^2 \, _2F_1\left(-\frac{3}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)}{n+2}+b \sin (c+d x) \left(\frac{3 a \, _2F_1\left(-\frac{3}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right)}{n+3}+\frac{b \sin (c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+4}{2};\frac{n+6}{2};\sin ^2(c+d x)\right)}{n+4}\right)\right)\right)}{d}","\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{\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{3 a \left(2 a^4 \left(n^2+5 n+6\right)-2 a^2 b^2 \left(n^2+16 n+58\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^4 \left(n^2+5 n+6\right)-2 a^2 b^2 \left(n^2+16 n+57\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{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,"(Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + n)*((a^3*Hypergeometric2F1[-3/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2])/(1 + n) + b*Sin[c + d*x]*((3*a^2*Hypergeometric2F1[-3/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2])/(2 + n) + b*Sin[c + d*x]*((3*a*Hypergeometric2F1[-3/2, (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2])/(3 + n) + (b*Hypergeometric2F1[-3/2, (4 + n)/2, (6 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x])/(4 + n)))))/d","A",1
1196,1,167,487,0.3411577,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^2,x]","\frac{\sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{n+1}(c+d x) \left(a^2 \left(n^2+5 n+6\right) \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)+b (n+1) \sin (c+d x) \left(2 a (n+3) \, _2F_1\left(-\frac{3}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)+b (n+2) \sin (c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right)\right)\right)}{d (n+1) (n+2) (n+3)}","\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{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{\left(2 a^4 \left(n^2+5 n+6\right)-2 a^2 b^2 \left(n^2+13 n+40\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{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,"(Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + n)*(a^2*(6 + 5*n + n^2)*Hypergeometric2F1[-3/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2] + b*(1 + n)*Sin[c + d*x]*(2*a*(3 + n)*Hypergeometric2F1[-3/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2] + b*(2 + n)*Hypergeometric2F1[-3/2, (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x])))/(d*(1 + n)*(2 + n)*(3 + n))","A",1
1197,1,111,129,0.1624754,"\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x]),x]","\frac{\sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{n+1}(c+d x) \left(a (n+2) \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)+b (n+1) \sin (c+d x) \, _2F_1\left(-\frac{3}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)\right)}{d (n+1) (n+2)}","\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,"(Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + n)*(a*(2 + n)*Hypergeometric2F1[-3/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2] + b*(1 + n)*Hypergeometric2F1[-3/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]))/(d*(1 + n)*(2 + n))","A",1
1198,1,105,97,0.3956309,"\int \cos ^5(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^5*(a + b*Sin[c + d*x]),x]","\frac{-34650 a \cos (2 (c+d x))+5775 a \cos (6 (c+d x))-693 a \cos (10 (c+d x))+34650 b \sin (c+d x)-11550 b \sin (3 (c+d x))-3465 b \sin (5 (c+d x))+2475 b \sin (7 (c+d x))+385 b \sin (9 (c+d x))-315 b \sin (11 (c+d x))}{3548160 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,"(-34650*a*Cos[2*(c + d*x)] + 5775*a*Cos[6*(c + d*x)] - 693*a*Cos[10*(c + d*x)] + 34650*b*Sin[c + d*x] - 11550*b*Sin[3*(c + d*x)] - 3465*b*Sin[5*(c + d*x)] + 2475*b*Sin[7*(c + d*x)] + 385*b*Sin[9*(c + d*x)] - 315*b*Sin[11*(c + d*x)])/(3548160*d)","A",1
1199,1,94,97,0.3060926,"\int \cos ^5(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^4*(a + b*Sin[c + d*x]),x]","\frac{7560 a \sin (c+d x)-1680 a \sin (3 (c+d x))-1008 a \sin (5 (c+d x))+180 a \sin (7 (c+d x))+140 a \sin (9 (c+d x))-3150 b \cos (2 (c+d x))+525 b \cos (6 (c+d x))-63 b \cos (10 (c+d x))}{322560 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,"(-3150*b*Cos[2*(c + d*x)] + 525*b*Cos[6*(c + d*x)] - 63*b*Cos[10*(c + d*x)] + 7560*a*Sin[c + d*x] - 1680*a*Sin[3*(c + d*x)] - 1008*a*Sin[5*(c + d*x)] + 180*a*Sin[7*(c + d*x)] + 140*a*Sin[9*(c + d*x)])/(322560*d)","A",1
1200,1,105,81,0.2885084,"\int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + b*Sin[c + d*x]),x]","\frac{-7560 a \cos (2 (c+d x))-1260 a \cos (4 (c+d x))+840 a \cos (6 (c+d x))+315 a \cos (8 (c+d x))+7560 b \sin (c+d x)-1680 b \sin (3 (c+d x))-1008 b \sin (5 (c+d x))+180 b \sin (7 (c+d x))+140 b \sin (9 (c+d x))}{322560 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,"(-7560*a*Cos[2*(c + d*x)] - 1260*a*Cos[4*(c + d*x)] + 840*a*Cos[6*(c + d*x)] + 315*a*Cos[8*(c + d*x)] + 7560*b*Sin[c + d*x] - 1680*b*Sin[3*(c + d*x)] - 1008*b*Sin[5*(c + d*x)] + 180*b*Sin[7*(c + d*x)] + 140*b*Sin[9*(c + d*x)])/(322560*d)","A",1
1201,1,94,81,0.3035235,"\int \cos ^5(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{-8400 a \sin (c+d x)+560 a \sin (3 (c+d x))+1008 a \sin (5 (c+d x))+240 a \sin (7 (c+d x))+2520 b \cos (2 (c+d x))+420 b \cos (4 (c+d x))-280 b \cos (6 (c+d x))-105 b \cos (8 (c+d x))}{107520 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,"-1/107520*(2520*b*Cos[2*(c + d*x)] + 420*b*Cos[4*(c + d*x)] - 280*b*Cos[6*(c + d*x)] - 105*b*Cos[8*(c + d*x)] - 8400*a*Sin[c + d*x] + 560*a*Sin[3*(c + d*x)] + 1008*a*Sin[5*(c + d*x)] + 240*a*Sin[7*(c + d*x)])/d","A",1
1202,1,86,65,0.2213868,"\int \cos ^5(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]*(a + b*Sin[c + d*x]),x]","-\frac{525 a \cos (2 (c+d x))+210 a \cos (4 (c+d x))+35 a \cos (6 (c+d x))+350 a-525 b \sin (c+d x)+35 b \sin (3 (c+d x))+63 b \sin (5 (c+d x))+15 b \sin (7 (c+d x))}{6720 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,"-1/6720*(350*a + 525*a*Cos[2*(c + d*x)] + 210*a*Cos[4*(c + d*x)] + 35*a*Cos[6*(c + d*x)] - 525*b*Sin[c + d*x] + 35*b*Sin[3*(c + d*x)] + 63*b*Sin[5*(c + d*x)] + 15*b*Sin[7*(c + d*x)])/d","A",1
1203,1,86,86,0.0334709,"\int \cos ^4(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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",1
1204,1,83,83,0.0317392,"\int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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",1
1205,1,77,86,0.2271003,"\int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^3*(a + b*Sin[c + d*x]),x]","-\frac{a \left(-\sin ^2(c+d x)+\csc ^2(c+d x)+4 \log (\sin (c+d x))\right)}{2 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) - (2*b*Sin[c + d*x])/d + (b*Sin[c + d*x]^3)/(3*d) - (a*(Csc[c + d*x]^2 + 4*Log[Sin[c + d*x]] - Sin[c + d*x]^2))/(2*d)","A",1
1206,1,76,85,0.1514163,"\int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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 \left(-\sin ^2(c+d x)+\csc ^2(c+d x)+4 \log (\sin (c+d x))\right)}{2 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 - (a*Csc[c + d*x]^3)/(3*d) + (a*Sin[c + d*x])/d - (b*(Csc[c + d*x]^2 + 4*Log[Sin[c + d*x]] - Sin[c + d*x]^2))/(2*d)","A",1
1207,1,87,81,0.2649213,"\int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Sin[c + d*x]),x]","\frac{a \left(-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right)}{4 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 - (b*Csc[c + d*x]^3)/(3*d) + (a*(2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/(4*d) + (b*Sin[c + d*x])/d","A",1
1208,1,92,86,0.1845831,"\int \cot ^5(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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 \left(-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right)}{4 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) + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d) + (b*(2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/(4*d)","A",1
1209,1,61,61,0.0276392,"\int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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,"-1/6*(a*Cot[c + d*x]^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",1
1210,1,65,65,0.0276859,"\int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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,"-1/6*(b*Cot[c + d*x]^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",1
1211,1,88,81,0.1084518,"\int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + b*Sin[c + d*x]),x]","-\frac{a \left(3 \csc ^8(c+d x)-8 \csc ^6(c+d x)+6 \csc ^4(c+d x)\right)}{24 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,"-1/3*(b*Csc[c + d*x]^3)/d + (2*b*Csc[c + d*x]^5)/(5*d) - (b*Csc[c + d*x]^7)/(7*d) - (a*(6*Csc[c + d*x]^4 - 8*Csc[c + d*x]^6 + 3*Csc[c + d*x]^8))/(24*d)","A",1
1212,1,88,81,0.1173558,"\int \cot ^5(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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 \left(3 \csc ^8(c+d x)-8 \csc ^6(c+d x)+6 \csc ^4(c+d x)\right)}{24 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,"-1/5*(a*Csc[c + d*x]^5)/d + (2*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d) - (b*(6*Csc[c + d*x]^4 - 8*Csc[c + d*x]^6 + 3*Csc[c + d*x]^8))/(24*d)","A",1
1213,1,88,97,0.1058924,"\int \cot ^5(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]^6*(a + b*Sin[c + d*x]),x]","-\frac{a \left(6 \csc ^{10}(c+d x)-15 \csc ^8(c+d x)+10 \csc ^6(c+d x)\right)}{60 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,"-1/5*(b*Csc[c + d*x]^5)/d + (2*b*Csc[c + d*x]^7)/(7*d) - (b*Csc[c + d*x]^9)/(9*d) - (a*(10*Csc[c + d*x]^6 - 15*Csc[c + d*x]^8 + 6*Csc[c + d*x]^10))/(60*d)","A",1
1214,1,88,97,0.1067726,"\int \cot ^5(c+d x) \csc ^7(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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 \left(6 \csc ^{10}(c+d x)-15 \csc ^8(c+d x)+10 \csc ^6(c+d x)\right)}{60 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,"-1/7*(a*Csc[c + d*x]^7)/d + (2*a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^11)/(11*d) - (b*(10*Csc[c + d*x]^6 - 15*Csc[c + d*x]^8 + 6*Csc[c + d*x]^10))/(60*d)","A",1
1215,1,169,138,0.7764145,"\int \cos ^5(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{12600 a^2 \sin (c+d x)-840 a^2 \sin (3 (c+d x))-1512 a^2 \sin (5 (c+d x))-360 a^2 \sin (7 (c+d x))-7560 a b \cos (2 (c+d x))-1260 a b \cos (4 (c+d x))+840 a b \cos (6 (c+d x))+315 a b \cos (8 (c+d x))+3780 b^2 \sin (c+d x)-840 b^2 \sin (3 (c+d x))-504 b^2 \sin (5 (c+d x))+90 b^2 \sin (7 (c+d x))+70 b^2 \sin (9 (c+d x))}{161280 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,"(-7560*a*b*Cos[2*(c + d*x)] - 1260*a*b*Cos[4*(c + d*x)] + 840*a*b*Cos[6*(c + d*x)] + 315*a*b*Cos[8*(c + d*x)] + 12600*a^2*Sin[c + d*x] + 3780*b^2*Sin[c + d*x] - 840*a^2*Sin[3*(c + d*x)] - 840*b^2*Sin[3*(c + d*x)] - 1512*a^2*Sin[5*(c + d*x)] - 504*b^2*Sin[5*(c + d*x)] - 360*a^2*Sin[7*(c + d*x)] + 90*b^2*Sin[7*(c + d*x)] + 70*b^2*Sin[9*(c + d*x)])/(161280*d)","A",1
1216,1,138,138,0.6835774,"\int \cos ^5(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]*(a + b*Sin[c + d*x])^2,x]","-\frac{840 \left(10 a^2+3 b^2\right) \cos (2 (c+d x))+420 \left(8 a^2+b^2\right) \cos (4 (c+d x))+560 a^2 \cos (6 (c+d x))-16800 a b \sin (c+d x)+1120 a b \sin (3 (c+d x))+2016 a b \sin (5 (c+d x))+480 a b \sin (7 (c+d x))-280 b^2 \cos (6 (c+d x))-105 b^2 \cos (8 (c+d x))-2590 b^2}{107520 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,"-1/107520*(-2590*b^2 + 840*(10*a^2 + 3*b^2)*Cos[2*(c + d*x)] + 420*(8*a^2 + b^2)*Cos[4*(c + d*x)] + 560*a^2*Cos[6*(c + d*x)] - 280*b^2*Cos[6*(c + d*x)] - 105*b^2*Cos[8*(c + d*x)] - 16800*a*b*Sin[c + d*x] + 1120*a*b*Sin[3*(c + d*x)] + 2016*a*b*Sin[5*(c + d*x)] + 480*a*b*Sin[7*(c + d*x)])/d","A",1
1217,1,105,130,0.1505676,"\int \cos ^4(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Cot[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{15 \left(a^2-2 b^2\right) \sin ^4(c+d x)+30 \left(b^2-2 a^2\right) \sin ^2(c+d x)+60 a^2 \log (\sin (c+d x))+24 a b \sin ^5(c+d x)-80 a b \sin ^3(c+d x)+120 a b \sin (c+d x)+10 b^2 \sin ^6(c+d x)}{60 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,"(60*a^2*Log[Sin[c + d*x]] + 120*a*b*Sin[c + d*x] + 30*(-2*a^2 + b^2)*Sin[c + d*x]^2 - 80*a*b*Sin[c + d*x]^3 + 15*(a^2 - 2*b^2)*Sin[c + d*x]^4 + 24*a*b*Sin[c + d*x]^5 + 10*b^2*Sin[c + d*x]^6)/(60*d)","A",1
1218,1,142,125,0.0521768,"\int \cos ^3(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \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{2 b^2 \sin ^3(c+d x)}{3 d}+\frac{b^2 \sin (c+d x)}{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*Sin[c + d*x])/d + (b^2*Sin[c + d*x])/d - (2*a*b*Sin[c + d*x]^2)/d + (a^2*Sin[c + d*x]^3)/(3*d) - (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",1
1219,1,103,127,0.2952298,"\int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{6 \left(a^2-2 b^2\right) \sin ^2(c+d x)+12 \left(b^2-2 a^2\right) \log (\sin (c+d x))-6 a^2 \csc ^2(c+d x)+8 a b \sin ^3(c+d x)-48 a b \sin (c+d x)-24 a b \csc (c+d x)+3 b^2 \sin ^4(c+d x)}{12 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,"(-24*a*b*Csc[c + d*x] - 6*a^2*Csc[c + d*x]^2 + 12*(-2*a^2 + b^2)*Log[Sin[c + d*x]] - 48*a*b*Sin[c + d*x] + 6*(a^2 - 2*b^2)*Sin[c + d*x]^2 + 8*a*b*Sin[c + d*x]^3 + 3*b^2*Sin[c + d*x]^4)/(12*d)","A",1
1220,1,103,120,0.2774745,"\int \cos (c+d x) \cot ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{3 \left(a^2-2 b^2\right) \sin (c+d x)+\left(6 a^2-3 b^2\right) \csc (c+d x)-a^2 \csc ^3(c+d x)+3 a b \sin ^2(c+d x)-3 a b \csc ^2(c+d x)-12 a b \log (\sin (c+d x))+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,"((6*a^2 - 3*b^2)*Csc[c + d*x] - 3*a*b*Csc[c + d*x]^2 - a^2*Csc[c + d*x]^3 - 12*a*b*Log[Sin[c + d*x]] + 3*(a^2 - 2*b^2)*Sin[c + d*x] + 3*a*b*Sin[c + d*x]^2 + b^2*Sin[c + d*x]^3)/(3*d)","A",1
1221,1,107,126,0.7381147,"\int \cot ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{6 \left(2 a^2-b^2\right) \csc ^2(c+d x)+6 \left(2 \left(a^2-2 b^2\right) \log (\sin (c+d x))+4 a b \sin (c+d x)+b^2 \sin ^2(c+d x)\right)-3 a^2 \csc ^4(c+d x)-8 a b \csc ^3(c+d x)+48 a b \csc (c+d x)}{12 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,"(48*a*b*Csc[c + d*x] + 6*(2*a^2 - b^2)*Csc[c + d*x]^2 - 8*a*b*Csc[c + d*x]^3 - 3*a^2*Csc[c + d*x]^4 + 6*(2*(a^2 - 2*b^2)*Log[Sin[c + d*x]] + 4*a*b*Sin[c + d*x] + b^2*Sin[c + d*x]^2))/(12*d)","A",1
1222,1,105,124,0.1873855,"\int \cot ^5(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{10 \left(2 a^2-b^2\right) \csc ^3(c+d x)-30 \left(a^2-2 b^2\right) \csc (c+d x)-6 a^2 \csc ^5(c+d x)-15 a b \csc ^4(c+d x)+60 a b \csc ^2(c+d x)+30 b (2 a \log (\sin (c+d x))+b \sin (c+d x))}{30 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,"(-30*(a^2 - 2*b^2)*Csc[c + d*x] + 60*a*b*Csc[c + d*x]^2 + 10*(2*a^2 - b^2)*Csc[c + d*x]^3 - 15*a*b*Csc[c + d*x]^4 - 6*a^2*Csc[c + d*x]^5 + 30*b*(2*a*Log[Sin[c + d*x]] + b*Sin[c + d*x]))/(30*d)","A",1
1223,1,107,130,0.1729387,"\int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{15 \left(2 a^2-b^2\right) \csc ^4(c+d x)-30 \left(a^2-2 b^2\right) \csc ^2(c+d x)-10 a^2 \csc ^6(c+d x)-24 a b \csc ^5(c+d x)+80 a b \csc ^3(c+d x)-120 a b \csc (c+d x)+60 b^2 \log (\sin (c+d x))}{60 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,"(-120*a*b*Csc[c + d*x] - 30*(a^2 - 2*b^2)*Csc[c + d*x]^2 + 80*a*b*Csc[c + d*x]^3 + 15*(2*a^2 - b^2)*Csc[c + d*x]^4 - 24*a*b*Csc[c + d*x]^5 - 10*a^2*Csc[c + d*x]^6 + 60*b^2*Log[Sin[c + d*x]])/(60*d)","A",1
1224,1,104,129,0.1996475,"\int \cot ^5(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","-\frac{\csc (c+d x) \left(21 \left(b^2-2 a^2\right) \csc ^4(c+d x)+35 \left(a^2-2 b^2\right) \csc ^2(c+d x)+15 a^2 \csc ^6(c+d x)+35 a b \csc ^5(c+d x)-105 a b \csc ^3(c+d x)+105 a b \csc (c+d x)+105 b^2\right)}{105 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,"-1/105*(Csc[c + d*x]*(105*b^2 + 105*a*b*Csc[c + d*x] + 35*(a^2 - 2*b^2)*Csc[c + d*x]^2 - 105*a*b*Csc[c + d*x]^3 + 21*(-2*a^2 + b^2)*Csc[c + d*x]^4 + 35*a*b*Csc[c + d*x]^5 + 15*a^2*Csc[c + d*x]^6))/d","A",1
1225,1,108,138,0.2488672,"\int \cot ^5(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","-\frac{\csc ^2(c+d x) \left(-140 \left(2 a^2-b^2\right) \csc ^4(c+d x)+210 \left(a^2-2 b^2\right) \csc ^2(c+d x)+105 a^2 \csc ^6(c+d x)+240 a b \csc ^5(c+d x)-672 a b \csc ^3(c+d x)+560 a b \csc (c+d x)+420 b^2\right)}{840 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,"-1/840*(Csc[c + d*x]^2*(420*b^2 + 560*a*b*Csc[c + d*x] + 210*(a^2 - 2*b^2)*Csc[c + d*x]^2 - 672*a*b*Csc[c + d*x]^3 - 140*(2*a^2 - b^2)*Csc[c + d*x]^4 + 240*a*b*Csc[c + d*x]^5 + 105*a^2*Csc[c + d*x]^6))/d","A",1
1226,1,264,235,2.1923806,"\int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","\frac{\left(50 a b^6-35 a^3 b^4\right) \sin ^4(c+d x)+60 a^2 b \left(a^2-b^2\right) \sin (c+d x) \left(\left(7 a^2-3 b^2\right) \log (a+b \sin (c+d x))-6 a^2+2 b^2\right)+3 b^5 \left(7 a^2-10 b^2\right) \sin ^5(c+d x)-30 a b^2 \left(7 a^4-10 a^2 b^2+3 b^4\right) \sin ^2(c+d x)+10 b^3 \left(7 a^4-10 a^2 b^2+3 b^4\right) \sin ^3(c+d x)+60 a^3 \left(a^2-b^2\right) \left(\left(7 a^2-3 b^2\right) \log (a+b \sin (c+d x))+a^2-b^2\right)-14 a b^6 \sin ^6(c+d x)+10 b^7 \sin ^7(c+d x)}{60 b^8 d (a+b \sin (c+d x))}","-\frac{4 a \left(a^2-b^2\right) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left(3 a^2-2 b^2\right) \sin ^4(c+d x)}{4 b^4 d}+\frac{a^2 \left(7 a^4-10 a^2 b^2+3 b^4\right) \log (a+b \sin (c+d x))}{b^8 d}-\frac{2 a \left(3 a^4-4 a^2 b^2+b^4\right) \sin (c+d x)}{b^7 d}+\frac{\left(5 a^4-6 a^2 b^2+b^4\right) \sin ^2(c+d x)}{2 b^6 d}+\frac{a^3 \left(a^2-b^2\right)^2}{b^8 d (a+b \sin (c+d x))}-\frac{2 a \sin ^5(c+d x)}{5 b^3 d}+\frac{\sin ^6(c+d x)}{6 b^2 d}",1,"(60*a^3*(a^2 - b^2)*(a^2 - b^2 + (7*a^2 - 3*b^2)*Log[a + b*Sin[c + d*x]]) + 60*a^2*b*(a^2 - b^2)*(-6*a^2 + 2*b^2 + (7*a^2 - 3*b^2)*Log[a + b*Sin[c + d*x]])*Sin[c + d*x] - 30*a*b^2*(7*a^4 - 10*a^2*b^2 + 3*b^4)*Sin[c + d*x]^2 + 10*b^3*(7*a^4 - 10*a^2*b^2 + 3*b^4)*Sin[c + d*x]^3 + (-35*a^3*b^4 + 50*a*b^6)*Sin[c + d*x]^4 + 3*b^5*(7*a^2 - 10*b^2)*Sin[c + d*x]^5 - 14*a*b^6*Sin[c + d*x]^6 + 10*b^7*Sin[c + d*x]^7)/(60*b^8*d*(a + b*Sin[c + d*x]))","A",1
1227,1,225,193,1.4928317,"\int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{\left(40 a b^5-30 a^3 b^3\right) \sin ^3(c+d x)-30 a b \left(a^2-b^2\right) \sin (c+d x) \left(\left(6 a^2-2 b^2\right) \log (a+b \sin (c+d x))-5 a^2+b^2\right)-30 a^2 \left(a^2-b^2\right) \left(\left(6 a^2-2 b^2\right) \log (a+b \sin (c+d x))+a^2-b^2\right)+5 b^4 \left(3 a^2-4 b^2\right) \sin ^4(c+d x)+30 b^2 \left(3 a^4-4 a^2 b^2+b^4\right) \sin ^2(c+d x)-9 a b^5 \sin ^5(c+d x)+6 b^6 \sin ^6(c+d x)}{30 b^7 d (a+b \sin (c+d x))}","-\frac{\left(2-\frac{3 a^2}{b^2}\right) \sin ^3(c+d x)}{3 b^2 d}-\frac{a^2 \left(a^2-b^2\right)^2}{b^7 d (a+b \sin (c+d x))}-\frac{2 a \left(a^2-b^2\right) \sin ^2(c+d x)}{b^5 d}-\frac{2 a \left(3 a^4-4 a^2 b^2+b^4\right) \log (a+b \sin (c+d x))}{b^7 d}+\frac{\left(5 a^4-6 a^2 b^2+b^4\right) \sin (c+d x)}{b^6 d}-\frac{a \sin ^4(c+d x)}{2 b^3 d}+\frac{\sin ^5(c+d x)}{5 b^2 d}",1,"(-30*a^2*(a^2 - b^2)*(a^2 - b^2 + (6*a^2 - 2*b^2)*Log[a + b*Sin[c + d*x]]) - 30*a*b*(a^2 - b^2)*(-5*a^2 + b^2 + (6*a^2 - 2*b^2)*Log[a + b*Sin[c + d*x]])*Sin[c + d*x] + 30*b^2*(3*a^4 - 4*a^2*b^2 + b^4)*Sin[c + d*x]^2 + (-30*a^3*b^3 + 40*a*b^5)*Sin[c + d*x]^3 + 5*b^4*(3*a^2 - 4*b^2)*Sin[c + d*x]^4 - 9*a*b^5*Sin[c + d*x]^5 + 6*b^6*Sin[c + d*x]^6)/(30*b^7*d*(a + b*Sin[c + d*x]))","A",1
1228,1,188,157,1.0363111,"\int \frac{\cos ^5(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{-6 a b^2 \left(5 a^2-6 b^2\right) \sin ^2(c+d x)+12 b \left(b^2-a^2\right) \sin (c+d x) \left(\left(b^2-5 a^2\right) \log (a+b \sin (c+d x))+4 a^2\right)+12 a \left(a^2-b^2\right) \left(\left(5 a^2-b^2\right) \log (a+b \sin (c+d x))+a^2-b^2\right)+2 b^3 \left(5 a^2-6 b^2\right) \sin ^3(c+d x)-5 a b^4 \sin ^4(c+d x)+3 b^5 \sin ^5(c+d x)}{12 b^6 d (a+b \sin (c+d x))}","\frac{a \left(a^2-b^2\right)^2}{b^6 d (a+b \sin (c+d x))}-\frac{4 a \left(a^2-b^2\right) \sin (c+d x)}{b^5 d}+\frac{\left(3 a^2-2 b^2\right) \sin ^2(c+d x)}{2 b^4 d}+\frac{\left(5 a^4-6 a^2 b^2+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,"(12*a*(a^2 - b^2)*(a^2 - b^2 + (5*a^2 - b^2)*Log[a + b*Sin[c + d*x]]) + 12*b*(-a^2 + b^2)*(4*a^2 + (-5*a^2 + b^2)*Log[a + b*Sin[c + d*x]])*Sin[c + d*x] - 6*a*b^2*(5*a^2 - 6*b^2)*Sin[c + d*x]^2 + 2*b^3*(5*a^2 - 6*b^2)*Sin[c + d*x]^3 - 5*a*b^4*Sin[c + d*x]^4 + 3*b^5*Sin[c + d*x]^5)/(12*b^6*d*(a + b*Sin[c + d*x]))","A",1
1229,1,111,120,0.5154794,"\int \frac{\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{2 \left(a^2-b^2\right)^2}{a b^4 (a+b \sin (c+d x))}+\frac{2 (a-b) (a+b) \left(3 a^2+b^2\right) \log (a+b \sin (c+d x))}{a^2 b^4}+\frac{2 \log (\sin (c+d x))}{a^2}-\frac{4 a \sin (c+d x)}{b^3}+\frac{\sin ^2(c+d x)}{b^2}}{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,"((2*Log[Sin[c + d*x]])/a^2 + (2*(a - b)*(a + b)*(3*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/(a^2*b^4) - (4*a*Sin[c + d*x])/b^3 + Sin[c + d*x]^2/b^2 + (2*(a^2 - b^2)^2)/(a*b^4*(a + b*Sin[c + d*x])))/(2*d)","A",1
1230,1,95,109,0.6207641,"\int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","-\frac{2 \left(\frac{a}{b^3}-\frac{b}{a^3}\right) \log (a+b \sin (c+d x))+\frac{2 b \log (\sin (c+d x))}{a^3}+\frac{\left(a^2-b^2\right)^2}{a^2 b^3 (a+b \sin (c+d x))}+\frac{\csc (c+d x)}{a^2}-\frac{\sin (c+d x)}{b^2}}{d}","-\frac{2 b \log (\sin (c+d x))}{a^3 d}-\frac{\left(a^2-b^2\right)^2}{a^2 b^3 d (a+b \sin (c+d x))}-\frac{\csc (c+d x)}{a^2 d}-\frac{2 \left(a^4-b^4\right) \log (a+b \sin (c+d x))}{a^3 b^3 d}+\frac{\sin (c+d x)}{b^2 d}",1,"-((Csc[c + d*x]/a^2 + (2*b*Log[Sin[c + d*x]])/a^3 + 2*(a/b^3 - b/a^3)*Log[a + b*Sin[c + d*x]] - Sin[c + d*x]/b^2 + (a^2 - b^2)^2/(a^2*b^3*(a + b*Sin[c + d*x])))/d)","A",1
1231,1,116,131,0.7558618,"\int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{2 a \left(a^2-b^2\right)^2}{b^2 (a+b \sin (c+d x))}-2 \left(2 a^2-3 b^2\right) \log (\sin (c+d x))-a^2 \csc ^2(c+d x)+\frac{2 \left(a^4+2 a^2 b^2-3 b^4\right) \log (a+b \sin (c+d x))}{b^2}+4 a b \csc (c+d x)}{2 a^4 d}","\frac{2 b \csc (c+d x)}{a^3 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}-\frac{\left(2 a^2-3 b^2\right) \log (\sin (c+d x))}{a^4 d}+\frac{\left(a^4+2 a^2 b^2-3 b^4\right) \log (a+b \sin (c+d x))}{a^4 b^2 d}+\frac{\left(a^2-b^2\right)^2}{a^3 b^2 d (a+b \sin (c+d x))}",1,"(4*a*b*Csc[c + d*x] - a^2*Csc[c + d*x]^2 - 2*(2*a^2 - 3*b^2)*Log[Sin[c + d*x]] + (2*(a^4 + 2*a^2*b^2 - 3*b^4)*Log[a + b*Sin[c + d*x]])/b^2 + (2*a*(a^2 - b^2)^2)/(b^2*(a + b*Sin[c + d*x])))/(2*a^4*d)","A",1
1232,1,127,147,1.8985126,"\int \frac{\cos (c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^2,x]","\frac{-a^3 \csc ^3(c+d x)-\frac{3 a \left(a^2-b^2\right)^2}{b (a+b \sin (c+d x))}+3 a \left(2 a^2-3 b^2\right) \csc (c+d x)+3 a^2 b \csc ^2(c+d x)+12 b (a-b) (a+b) \log (\sin (c+d x))-12 b (a-b) (a+b) \log (a+b \sin (c+d x))}{3 a^5 d}","\frac{b \csc ^2(c+d x)}{a^3 d}-\frac{\csc ^3(c+d x)}{3 a^2 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{\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}",1,"(3*a*(2*a^2 - 3*b^2)*Csc[c + d*x] + 3*a^2*b*Csc[c + d*x]^2 - a^3*Csc[c + d*x]^3 + 12*(a - b)*b*(a + b)*Log[Sin[c + d*x]] - 12*(a - b)*b*(a + b)*Log[a + b*Sin[c + d*x]] - (3*a*(a^2 - b^2)^2)/(b*(a + b*Sin[c + d*x])))/(3*a^5*d)","A",1
1233,1,187,188,6.1387316,"\int \frac{\cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^5/(a + b*Sin[c + d*x])^2,x]","-\frac{4 b (a-b) (a+b) \csc (c+d x)}{a^5 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{\left(a^4-6 a^2 b^2+5 b^4\right) \log (\sin (c+d x))}{a^6 d}-\frac{\left(a^4-6 a^2 b^2+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{4 b \left(a^2-b^2\right) \csc (c+d x)}{a^5 d}+\frac{\left(2 a^2-3 b^2\right) \csc ^2(c+d x)}{2 a^4 d}+\frac{\left(a^4-6 a^2 b^2+5 b^4\right) \log (\sin (c+d x))}{a^6 d}-\frac{\left(a^4-6 a^2 b^2+5 b^4\right) \log (a+b \sin (c+d x))}{a^6 d}",1,"(-4*(a - b)*b*(a + b)*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",1
1234,1,220,226,3.1702843,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^5*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{-6 a^6 \csc ^6(c+d x)+9 a^5 b \csc ^5(c+d x)+\left(30 a^3 b^3-40 a^5 b\right) \csc ^3(c+d x)+5 a^4 \left(4 a^2-3 b^2\right) \csc ^4(c+d x)-30 a^2 \left(a^4-4 a^2 b^2+3 b^4\right) \csc ^2(c+d x)-60 b^2 \left(a^4-4 a^2 b^2+3 b^4\right) (\log (\sin (c+d x))-\log (a+b \sin (c+d x)))-60 a b \left(a^4-4 a^2 b^2+3 b^4\right) \csc (c+d x) (-\log (a+b \sin (c+d x))+\log (\sin (c+d x))+1)}{30 a^7 d (a \csc (c+d x)+b)}","\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{2 b \left(a^2-b^2\right) \csc ^2(c+d x)}{a^5 d}+\frac{\left(2 a^2-3 b^2\right) \csc ^3(c+d x)}{3 a^4 d}-\frac{2 b \left(a^4-4 a^2 b^2+3 b^4\right) \log (\sin (c+d x))}{a^7 d}+\frac{2 b \left(a^4-4 a^2 b^2+3 b^4\right) \log (a+b \sin (c+d x))}{a^7 d}-\frac{\left(a^4-6 a^2 b^2+5 b^4\right) \csc (c+d x)}{a^6 d}",1,"(-30*a^2*(a^4 - 4*a^2*b^2 + 3*b^4)*Csc[c + d*x]^2 + (-40*a^5*b + 30*a^3*b^3)*Csc[c + d*x]^3 + 5*a^4*(4*a^2 - 3*b^2)*Csc[c + d*x]^4 + 9*a^5*b*Csc[c + d*x]^5 - 6*a^6*Csc[c + d*x]^6 - 60*b^2*(a^4 - 4*a^2*b^2 + 3*b^4)*(Log[Sin[c + d*x]] - Log[a + b*Sin[c + d*x]]) - 60*a*b*(a^4 - 4*a^2*b^2 + 3*b^4)*Csc[c + d*x]*(1 + Log[Sin[c + d*x]] - Log[a + b*Sin[c + d*x]]))/(30*a^7*d*(b + a*Csc[c + d*x]))","A",1
1235,1,139,170,0.7631721,"\int \cos ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^2,x]","\frac{\sin ^{n+1}(c+d x) \left(\frac{\left(a^2-2 b^2\right) \sin ^4(c+d x)}{n+5}-\frac{\left(2 a^2-b^2\right) \sin ^2(c+d x)}{n+3}+\frac{a^2}{n+1}+\frac{2 a b \sin ^5(c+d x)}{n+6}-\frac{4 a b \sin ^3(c+d x)}{n+4}+\frac{2 a b \sin (c+d x)}{n+2}+\frac{b^2 \sin ^6(c+d x)}{n+7}\right)}{d}","-\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,"(Sin[c + d*x]^(1 + n)*(a^2/(1 + n) + (2*a*b*Sin[c + d*x])/(2 + n) - ((2*a^2 - b^2)*Sin[c + d*x]^2)/(3 + n) - (4*a*b*Sin[c + d*x]^3)/(4 + n) + ((a^2 - 2*b^2)*Sin[c + d*x]^4)/(5 + n) + (2*a*b*Sin[c + d*x]^5)/(6 + n) + (b^2*Sin[c + d*x]^6)/(7 + n)))/d","A",1
1236,1,97,123,0.2090578,"\int \cos ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x]),x]","\frac{\sin ^{n+1}(c+d x) \left(\frac{a \sin ^4(c+d x)}{n+5}-\frac{2 a \sin ^2(c+d x)}{n+3}+\frac{a}{n+1}+\frac{b \sin ^5(c+d x)}{n+6}-\frac{2 b \sin ^3(c+d x)}{n+4}+\frac{b \sin (c+d x)}{n+2}\right)}{d}","\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,"(Sin[c + d*x]^(1 + n)*(a/(1 + n) + (b*Sin[c + d*x])/(2 + n) - (2*a*Sin[c + d*x]^2)/(3 + n) - (2*b*Sin[c + d*x]^3)/(4 + n) + (a*Sin[c + d*x]^4)/(5 + n) + (b*Sin[c + d*x]^5)/(6 + n)))/d","A",1
1237,1,133,167,0.5817303,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^n)/(a + b*Sin[c + d*x]),x]","\frac{\sin ^{n+1}(c+d x) \left(-\frac{a^3-2 a b^2}{n+1}+\frac{\left(a^2-b^2\right)^2 \, _2F_1\left(1,n+1;n+2;-\frac{b \sin (c+d x)}{a}\right)}{a (n+1)}+\frac{b \left(a^2-2 b^2\right) \sin (c+d x)}{n+2}-\frac{a b^2 \sin ^2(c+d x)}{n+3}+\frac{b^3 \sin ^3(c+d x)}{n+4}\right)}{b^4 d}","\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,"(Sin[c + d*x]^(1 + n)*(-((a^3 - 2*a*b^2)/(1 + n)) + ((a^2 - b^2)^2*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Sin[c + d*x])/a)])/(a*(1 + n)) + (b*(a^2 - 2*b^2)*Sin[c + d*x])/(2 + n) - (a*b^2*Sin[c + d*x]^2)/(3 + n) + (b^3*Sin[c + d*x]^3)/(4 + n)))/(b^4*d)","A",1
1238,1,143,191,0.4406035,"\int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^n)/(a + b*Sin[c + d*x])^2,x]","\frac{\sin ^{n+1}(c+d x) \left(\frac{\left(a^2-b^2\right)^2 \, _2F_1\left(2,n+1;n+2;-\frac{b \sin (c+d x)}{a}\right)}{a^2 (n+1)}-\frac{4 \left(a^2-b^2\right) \, _2F_1\left(1,n+1;n+2;-\frac{b \sin (c+d x)}{a}\right)}{n+1}+\frac{3 a^2-2 b^2}{n+1}-\frac{2 a b \sin (c+d x)}{n+2}+\frac{b^2 \sin ^2(c+d x)}{n+3}\right)}{b^4 d}","\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,"(Sin[c + d*x]^(1 + n)*((3*a^2 - 2*b^2)/(1 + n) - (4*(a^2 - b^2)*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Sin[c + d*x])/a)])/(1 + n) + ((a^2 - b^2)^2*Hypergeometric2F1[2, 1 + n, 2 + n, -((b*Sin[c + d*x])/a)])/(a^2*(1 + n)) - (2*a*b*Sin[c + d*x])/(2 + n) + (b^2*Sin[c + d*x]^2)/(3 + n)))/(b^4*d)","A",1
1239,1,210,238,2.2319493,"\int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{-180180 \left(2 a^2+b^2\right) \cos (c+d x)-15015 \left(8 a^2+3 b^2\right) \cos (3 (c+d x))+36036 a^2 \cos (5 (c+d x))+25740 a^2 \cos (7 (c+d x))-4004 a^2 \cos (9 (c+d x))-3276 a^2 \cos (11 (c+d x))-135135 a b \sin (4 (c+d x))+27027 a b \sin (8 (c+d x))-3003 a b \sin (12 (c+d x))+360360 a b c+360360 a b d x+27027 b^2 \cos (5 (c+d x))+7722 b^2 \cos (7 (c+d x))-6006 b^2 \cos (9 (c+d x))-819 b^2 \cos (11 (c+d x))+693 b^2 \cos (13 (c+d x))}{36900864 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,"(360360*a*b*c + 360360*a*b*d*x - 180180*(2*a^2 + b^2)*Cos[c + d*x] - 15015*(8*a^2 + 3*b^2)*Cos[3*(c + d*x)] + 36036*a^2*Cos[5*(c + d*x)] + 27027*b^2*Cos[5*(c + d*x)] + 25740*a^2*Cos[7*(c + d*x)] + 7722*b^2*Cos[7*(c + d*x)] - 4004*a^2*Cos[9*(c + d*x)] - 6006*b^2*Cos[9*(c + d*x)] - 3276*a^2*Cos[11*(c + d*x)] - 819*b^2*Cos[11*(c + d*x)] + 693*b^2*Cos[13*(c + d*x)] - 135135*a*b*Sin[4*(c + d*x)] + 27027*a*b*Sin[8*(c + d*x)] - 3003*a*b*Sin[12*(c + d*x)])/(36900864*d)","A",1
1240,1,202,250,1.4694278,"\int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{55440 a^2 \sin (2 (c+d x))-110880 a^2 \sin (4 (c+d x))-27720 a^2 \sin (6 (c+d x))+13860 a^2 \sin (8 (c+d x))+5544 a^2 \sin (10 (c+d x))+332640 a^2 d x-554400 a b \cos (c+d x)-184800 a b \cos (3 (c+d x))+55440 a b \cos (5 (c+d x))+39600 a b \cos (7 (c+d x))-6160 a b \cos (9 (c+d x))-5040 a b \cos (11 (c+d x))-51975 b^2 \sin (4 (c+d x))+10395 b^2 \sin (8 (c+d x))-1155 b^2 \sin (12 (c+d x))+166320 b^2 c+138600 b^2 d x}{28385280 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,"(166320*b^2*c + 332640*a^2*d*x + 138600*b^2*d*x - 554400*a*b*Cos[c + d*x] - 184800*a*b*Cos[3*(c + d*x)] + 55440*a*b*Cos[5*(c + d*x)] + 39600*a*b*Cos[7*(c + d*x)] - 6160*a*b*Cos[9*(c + d*x)] - 5040*a*b*Cos[11*(c + d*x)] + 55440*a^2*Sin[2*(c + d*x)] - 110880*a^2*Sin[4*(c + d*x)] - 51975*b^2*Sin[4*(c + d*x)] - 27720*a^2*Sin[6*(c + d*x)] + 13860*a^2*Sin[8*(c + d*x)] + 10395*b^2*Sin[8*(c + d*x)] + 5544*a^2*Sin[10*(c + d*x)] - 1155*b^2*Sin[12*(c + d*x)])/(28385280*d)","A",1
1241,1,197,187,1.132649,"\int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{-6930 \left(12 a^2+5 b^2\right) \cos (c+d x)-2310 \left(16 a^2+5 b^2\right) \cos (3 (c+d x))+5940 a^2 \cos (7 (c+d x))+1540 a^2 \cos (9 (c+d x))+13860 a b \sin (2 (c+d x))-27720 a b \sin (4 (c+d x))-6930 a b \sin (6 (c+d x))+3465 a b \sin (8 (c+d x))+1386 a b \sin (10 (c+d x))+83160 a b c+83160 a b d x+3465 b^2 \cos (5 (c+d x))+2475 b^2 \cos (7 (c+d x))-385 b^2 \cos (9 (c+d x))-315 b^2 \cos (11 (c+d x))}{3548160 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,"(83160*a*b*c + 83160*a*b*d*x - 6930*(12*a^2 + 5*b^2)*Cos[c + d*x] - 2310*(16*a^2 + 5*b^2)*Cos[3*(c + d*x)] + 3465*b^2*Cos[5*(c + d*x)] + 5940*a^2*Cos[7*(c + d*x)] + 2475*b^2*Cos[7*(c + d*x)] + 1540*a^2*Cos[9*(c + d*x)] - 385*b^2*Cos[9*(c + d*x)] - 315*b^2*Cos[11*(c + d*x)] + 13860*a*b*Sin[2*(c + d*x)] - 27720*a*b*Sin[4*(c + d*x)] - 6930*a*b*Sin[6*(c + d*x)] + 3465*a*b*Sin[8*(c + d*x)] + 1386*a*b*Sin[10*(c + d*x)])/(3548160*d)","A",1
1242,1,193,201,0.9021862,"\int \cos ^6(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{5040 a^2 \sin (2 (c+d x))-2520 a^2 \sin (4 (c+d x))-1680 a^2 \sin (6 (c+d x))-315 a^2 \sin (8 (c+d x))+12600 a^2 d x-15120 a b \cos (c+d x)-6720 a b \cos (3 (c+d x))+1080 a b \cos (7 (c+d x))+280 a b \cos (9 (c+d x))+630 b^2 \sin (2 (c+d x))-1260 b^2 \sin (4 (c+d x))-315 b^2 \sin (6 (c+d x))+\frac{315}{2} b^2 \sin (8 (c+d x))+63 b^2 \sin (10 (c+d x))+6300 b^2 c+3780 b^2 d x}{322560 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,"(6300*b^2*c + 12600*a^2*d*x + 3780*b^2*d*x - 15120*a*b*Cos[c + d*x] - 6720*a*b*Cos[3*(c + d*x)] + 1080*a*b*Cos[7*(c + d*x)] + 280*a*b*Cos[9*(c + d*x)] + 5040*a^2*Sin[2*(c + d*x)] + 630*b^2*Sin[2*(c + d*x)] - 2520*a^2*Sin[4*(c + d*x)] - 1260*b^2*Sin[4*(c + d*x)] - 1680*a^2*Sin[6*(c + d*x)] - 315*b^2*Sin[6*(c + d*x)] - 315*a^2*Sin[8*(c + d*x)] + (315*b^2*Sin[8*(c + d*x)])/2 + 63*b^2*Sin[10*(c + d*x)])/(322560*d)","A",1
1243,1,161,152,1.0203542,"\int \cos ^6(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*Sin[c + d*x]*(a + b*Sin[c + d*x])^2,x]","-\frac{126 \left(10 a^2+3 b^2\right) \cos (c+d x)+84 \left(9 a^2+2 b^2\right) \cos (3 (c+d x))+252 a^2 \cos (5 (c+d x))+36 a^2 \cos (7 (c+d x))-504 a b \sin (2 (c+d x))+252 a b \sin (4 (c+d x))+168 a b \sin (6 (c+d x))+\frac{63}{2} a b \sin (8 (c+d x))-1260 a b c-1260 a b d x-27 b^2 \cos (7 (c+d x))-7 b^2 \cos (9 (c+d x))}{16128 d}","-\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,"-1/16128*(-1260*a*b*c - 1260*a*b*d*x + 126*(10*a^2 + 3*b^2)*Cos[c + d*x] + 84*(9*a^2 + 2*b^2)*Cos[3*(c + d*x)] + 252*a^2*Cos[5*(c + d*x)] + 36*a^2*Cos[7*(c + d*x)] - 27*b^2*Cos[7*(c + d*x)] - 7*b^2*Cos[9*(c + d*x)] - 504*a*b*Sin[2*(c + d*x)] + 252*a*b*Sin[4*(c + d*x)] + 168*a*b*Sin[6*(c + d*x)] + (63*a*b*Sin[8*(c + d*x)])/2)/d","A",1
1244,1,166,157,0.2848005,"\int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*Cot[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{105 \left(88 a^2-5 b^2\right) \cos (c+d x)+35 \left(28 a^2-9 b^2\right) \cos (3 (c+d x))+84 a^2 \cos (5 (c+d x))+6720 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6720 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3150 a b \sin (2 (c+d x))+630 a b \sin (4 (c+d x))+70 a b \sin (6 (c+d x))+4200 a b c+4200 a b d x-105 b^2 \cos (5 (c+d x))-15 b^2 \cos (7 (c+d x))}{6720 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,"(4200*a*b*c + 4200*a*b*d*x + 105*(88*a^2 - 5*b^2)*Cos[c + d*x] + 35*(28*a^2 - 9*b^2)*Cos[3*(c + d*x)] + 84*a^2*Cos[5*(c + d*x)] - 105*b^2*Cos[5*(c + d*x)] - 15*b^2*Cos[7*(c + d*x)] - 6720*a^2*Log[Cos[(c + d*x)/2]] + 6720*a^2*Log[Sin[(c + d*x)/2]] + 3150*a*b*Sin[2*(c + d*x)] + 630*a*b*Sin[4*(c + d*x)] + 70*a*b*Sin[6*(c + d*x)])/(6720*d)","A",1
1245,1,220,178,0.4014942,"\int \cos ^4(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","-\frac{15 a^2 (c+d x)}{8 d}-\frac{a^2 \sin (2 (c+d x))}{2 d}-\frac{a^2 \sin (4 (c+d x))}{32 d}-\frac{a^2 \cot (c+d x)}{d}+\frac{11 a b \cos (c+d x)}{4 d}+\frac{7 a b \cos (3 (c+d x))}{24 d}+\frac{a b \cos (5 (c+d x))}{40 d}+\frac{2 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{2 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{5 b^2 (c+d x)}{16 d}+\frac{15 b^2 \sin (2 (c+d x))}{64 d}+\frac{3 b^2 \sin (4 (c+d x))}{64 d}+\frac{b^2 \sin (6 (c+d x))}{192 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,"(-15*a^2*(c + d*x))/(8*d) + (5*b^2*(c + d*x))/(16*d) + (11*a*b*Cos[c + d*x])/(4*d) + (7*a*b*Cos[3*(c + d*x)])/(24*d) + (a*b*Cos[5*(c + d*x)])/(40*d) - (a^2*Cot[c + d*x])/d - (2*a*b*Log[Cos[(c + d*x)/2]])/d + (2*a*b*Log[Sin[(c + d*x)/2]])/d - (a^2*Sin[2*(c + d*x)])/(2*d) + (15*b^2*Sin[2*(c + d*x)])/(64*d) - (a^2*Sin[4*(c + d*x)])/(32*d) + (3*b^2*Sin[4*(c + d*x)])/(64*d) + (b^2*Sin[6*(c + d*x)])/(192*d)","A",1
1246,1,250,180,6.188421,"\int \cos ^3(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(18 a^2-11 b^2\right) \cos (c+d x)}{8 d}-\frac{\left(4 a^2-7 b^2\right) \cos (3 (c+d x))}{48 d}+\frac{\left(2 b^2-5 a^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{\left(5 a^2-2 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{a^2 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{15 a b (c+d x)}{4 d}-\frac{a b \sin (2 (c+d x))}{d}-\frac{a b \sin (4 (c+d x))}{16 d}+\frac{a b \tan \left(\frac{1}{2} (c+d x)\right)}{d}-\frac{a b \cot \left(\frac{1}{2} (c+d x)\right)}{d}+\frac{b^2 \cos (5 (c+d x))}{80 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*(c + d*x))/(4*d) - ((18*a^2 - 11*b^2)*Cos[c + d*x])/(8*d) - ((4*a^2 - 7*b^2)*Cos[3*(c + d*x)])/(48*d) + (b^2*Cos[5*(c + d*x)])/(80*d) - (a*b*Cot[(c + d*x)/2])/d - (a^2*Csc[(c + d*x)/2]^2)/(8*d) + ((5*a^2 - 2*b^2)*Log[Cos[(c + d*x)/2]])/(2*d) + ((-5*a^2 + 2*b^2)*Log[Sin[(c + d*x)/2]])/(2*d) + (a^2*Sec[(c + d*x)/2]^2)/(8*d) - (a*b*Sin[2*(c + d*x)])/d - (a*b*Sin[4*(c + d*x)])/(16*d) + (a*b*Tan[(c + d*x)/2])/d","A",1
1247,1,336,177,6.2834139,"\int \cos ^2(c+d x) \cot ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*Cot[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{5 \left(4 a^2-3 b^2\right) (c+d x)}{8 d}+\frac{\left(a^2-2 b^2\right) \sin (2 (c+d x))}{4 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(7 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-3 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(3 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-7 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}-\frac{a^2 \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}+\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 d}-\frac{9 a b \cos (c+d x)}{2 d}-\frac{a b \cos (3 (c+d x))}{6 d}-\frac{a b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{4 d}+\frac{a b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{4 d}-\frac{5 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{5 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{b^2 \sin (4 (c+d x))}{32 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)*(c + d*x))/(8*d) - (9*a*b*Cos[c + d*x])/(2*d) - (a*b*Cos[3*(c + d*x)])/(6*d) + ((7*a^2*Cos[(c + d*x)/2] - 3*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*d) - (a*b*Csc[(c + d*x)/2]^2)/(4*d) - (a^2*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*d) + (5*a*b*Log[Cos[(c + d*x)/2]])/d - (5*a*b*Log[Sin[(c + d*x)/2]])/d + (a*b*Sec[(c + d*x)/2]^2)/(4*d) + (Sec[(c + d*x)/2]*(-7*a^2*Sin[(c + d*x)/2] + 3*b^2*Sin[(c + d*x)/2]))/(6*d) + ((a^2 - 2*b^2)*Sin[2*(c + d*x)])/(4*d) - (b^2*Sin[4*(c + d*x)])/(32*d) + (a^2*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*d)","A",0
1248,1,337,174,6.1796844,"\int \cos (c+d x) \cot ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*Cot[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{\left(9 a^2-4 b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{\left(4 b^2-9 a^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{5 \left(3 a^2-4 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{5 \left(3 a^2-4 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{a^2 \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a^2 \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{5 a b (c+d x)}{d}+\frac{a b \sin (2 (c+d x))}{2 d}+\frac{(2 a-3 b) (2 a+3 b) \cos (c+d x)}{4 d}-\frac{7 a b \tan \left(\frac{1}{2} (c+d x)\right)}{3 d}+\frac{7 a b \cot \left(\frac{1}{2} (c+d x)\right)}{3 d}-\frac{a b \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{12 d}+\frac{a b \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{12 d}-\frac{b^2 \cos (3 (c+d x))}{12 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*(c + d*x))/d + ((2*a - 3*b)*(2*a + 3*b)*Cos[c + d*x])/(4*d) - (b^2*Cos[3*(c + d*x)])/(12*d) + (7*a*b*Cot[(c + d*x)/2])/(3*d) + ((9*a^2 - 4*b^2)*Csc[(c + d*x)/2]^2)/(32*d) - (a*b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(12*d) - (a^2*Csc[(c + d*x)/2]^4)/(64*d) - (5*(3*a^2 - 4*b^2)*Log[Cos[(c + d*x)/2]])/(8*d) + (5*(3*a^2 - 4*b^2)*Log[Sin[(c + d*x)/2]])/(8*d) + ((-9*a^2 + 4*b^2)*Sec[(c + d*x)/2]^2)/(32*d) + (a^2*Sec[(c + d*x)/2]^4)/(64*d) + (a*b*Sin[2*(c + d*x)])/(2*d) - (7*a*b*Tan[(c + d*x)/2])/(3*d) + (a*b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(12*d)","A",0
1249,1,351,202,1.1213144,"\int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]","\frac{\left(560 b^2-368 a^2\right) \cot \left(\frac{1}{2} (c+d x)\right)+368 a^2 \tan \left(\frac{1}{2} (c+d x)\right)-\frac{3}{2} a^2 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+96 a^2 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+\frac{41}{2} a^2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-328 a^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-480 a^2 c-480 a^2 d x+960 a b \cos (c+d x)-15 a b \csc ^4\left(\frac{1}{2} (c+d x)\right)+270 a b \csc ^2\left(\frac{1}{2} (c+d x)\right)+15 a b \sec ^4\left(\frac{1}{2} (c+d x)\right)-270 a b \sec ^2\left(\frac{1}{2} (c+d x)\right)+1800 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-1800 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+120 b^2 \sin (2 (c+d x))-560 b^2 \tan \left(\frac{1}{2} (c+d x)\right)-10 b^2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+160 b^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+1200 b^2 c+1200 b^2 d x}{480 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}-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,"(-480*a^2*c + 1200*b^2*c - 480*a^2*d*x + 1200*b^2*d*x + 960*a*b*Cos[c + d*x] + (-368*a^2 + 560*b^2)*Cot[(c + d*x)/2] + 270*a*b*Csc[(c + d*x)/2]^2 - 15*a*b*Csc[(c + d*x)/2]^4 - 1800*a*b*Log[Cos[(c + d*x)/2]] + 1800*a*b*Log[Sin[(c + d*x)/2]] - 270*a*b*Sec[(c + d*x)/2]^2 + 15*a*b*Sec[(c + d*x)/2]^4 - 328*a^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 160*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 96*a^2*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 + (41*a^2*Csc[(c + d*x)/2]^4*Sin[c + d*x])/2 - 10*b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - (3*a^2*Csc[(c + d*x)/2]^6*Sin[c + d*x])/2 + 120*b^2*Sin[2*(c + d*x)] + 368*a^2*Tan[(c + d*x)/2] - 560*b^2*Tan[(c + d*x)/2])/(480*d)","A",1
1250,1,384,175,1.0494736,"\int \cot ^6(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{-5 a^2 \csc ^6\left(\frac{1}{2} (c+d x)\right)+60 a^2 \csc ^4\left(\frac{1}{2} (c+d x)\right)-330 a^2 \csc ^2\left(\frac{1}{2} (c+d x)\right)+5 a^2 \sec ^6\left(\frac{1}{2} (c+d x)\right)-60 a^2 \sec ^4\left(\frac{1}{2} (c+d x)\right)+330 a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)-600 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+600 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2944 a b \tan \left(\frac{1}{2} (c+d x)\right)-2944 a b \cot \left(\frac{1}{2} (c+d x)\right)-12 a b \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+768 a b \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+164 a b \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-2624 a b \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-3840 a b c-3840 a b d x+1920 b^2 \cos (c+d x)-30 b^2 \csc ^4\left(\frac{1}{2} (c+d x)\right)+540 b^2 \csc ^2\left(\frac{1}{2} (c+d x)\right)+30 b^2 \sec ^4\left(\frac{1}{2} (c+d x)\right)-540 b^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)+3600 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3600 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{1920 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,"(-3840*a*b*c - 3840*a*b*d*x + 1920*b^2*Cos[c + d*x] - 2944*a*b*Cot[(c + d*x)/2] - 330*a^2*Csc[(c + d*x)/2]^2 + 540*b^2*Csc[(c + d*x)/2]^2 + 60*a^2*Csc[(c + d*x)/2]^4 - 30*b^2*Csc[(c + d*x)/2]^4 - 5*a^2*Csc[(c + d*x)/2]^6 + 600*a^2*Log[Cos[(c + d*x)/2]] - 3600*b^2*Log[Cos[(c + d*x)/2]] - 600*a^2*Log[Sin[(c + d*x)/2]] + 3600*b^2*Log[Sin[(c + d*x)/2]] + 330*a^2*Sec[(c + d*x)/2]^2 - 540*b^2*Sec[(c + d*x)/2]^2 - 60*a^2*Sec[(c + d*x)/2]^4 + 30*b^2*Sec[(c + d*x)/2]^4 + 5*a^2*Sec[(c + d*x)/2]^6 - 2624*a*b*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 768*a*b*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 + 164*a*b*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 12*a*b*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 2944*a*b*Tan[(c + d*x)/2])/(1920*d)","B",1
1251,1,280,158,1.3969608,"\int \cot ^6(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\csc ^7(c+d x) \left(-84 \left(15 a^2-41 b^2\right) \cos (3 (c+d x))-28 \left(15 a^2+71 b^2\right) \cos (5 (c+d x))-60 a^2 \cos (7 (c+d x))+980 a b \sin (4 (c+d x))-1155 a b \sin (6 (c+d x))+8820 b^2 c \sin (3 (c+d x))+8820 b^2 d x \sin (3 (c+d x))-2940 b^2 c \sin (5 (c+d x))-2940 b^2 d x \sin (5 (c+d x))+420 b^2 c \sin (7 (c+d x))+420 b^2 d x \sin (7 (c+d x))+644 b^2 \cos (7 (c+d x))\right)-350 \cot (c+d x) \csc ^6(c+d x) \left(6 \left(a^2+b^2\right)+17 a b \sin (c+d x)\right)+16800 a b \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)-14700 b^2 (c+d x) \csc ^6(c+d x)}{26880 d}","-\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,"(-14700*b^2*(c + d*x)*Csc[c + d*x]^6 + 16800*a*b*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - 350*Cot[c + d*x]*Csc[c + d*x]^6*(6*(a^2 + b^2) + 17*a*b*Sin[c + d*x]) + Csc[c + d*x]^7*(-84*(15*a^2 - 41*b^2)*Cos[3*(c + d*x)] - 28*(15*a^2 + 71*b^2)*Cos[5*(c + d*x)] - 60*a^2*Cos[7*(c + d*x)] + 644*b^2*Cos[7*(c + d*x)] + 8820*b^2*c*Sin[3*(c + d*x)] + 8820*b^2*d*x*Sin[3*(c + d*x)] + 980*a*b*Sin[4*(c + d*x)] - 2940*b^2*c*Sin[5*(c + d*x)] - 2940*b^2*d*x*Sin[5*(c + d*x)] - 1155*a*b*Sin[6*(c + d*x)] + 420*b^2*c*Sin[7*(c + d*x)] + 420*b^2*d*x*Sin[7*(c + d*x)]))/(26880*d)","A",1
1252,1,282,159,0.8141171,"\int \cot ^6(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","-\frac{7 \left(895 a^2-904 b^2\right) \cos (3 (c+d x)) \csc ^8(c+d x)+7 \cot (c+d x) \csc ^7(c+d x) \left(1765 a^2+1536 a b \sin (c+d x)+680 b^2\right)+6720 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6720 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2779 a^2 \cos (5 (c+d x)) \csc ^8(c+d x)+105 a^2 \cos (7 (c+d x)) \csc ^8(c+d x)+5376 a b \sin (4 (c+d x)) \csc ^8(c+d x)+2304 a b \sin (6 (c+d x)) \csc ^8(c+d x)+384 a b \sin (8 (c+d x)) \csc ^8(c+d x)+53760 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-53760 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3416 b^2 \cos (5 (c+d x)) \csc ^8(c+d x)-1848 b^2 \cos (7 (c+d x)) \csc ^8(c+d x)}{172032 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,"-1/172032*(7*(895*a^2 - 904*b^2)*Cos[3*(c + d*x)]*Csc[c + d*x]^8 + 2779*a^2*Cos[5*(c + d*x)]*Csc[c + d*x]^8 + 3416*b^2*Cos[5*(c + d*x)]*Csc[c + d*x]^8 + 105*a^2*Cos[7*(c + d*x)]*Csc[c + d*x]^8 - 1848*b^2*Cos[7*(c + d*x)]*Csc[c + d*x]^8 - 6720*a^2*Log[Cos[(c + d*x)/2]] - 53760*b^2*Log[Cos[(c + d*x)/2]] + 6720*a^2*Log[Sin[(c + d*x)/2]] + 53760*b^2*Log[Sin[(c + d*x)/2]] + 7*Cot[c + d*x]*Csc[c + d*x]^7*(1765*a^2 + 680*b^2 + 1536*a*b*Sin[c + d*x]) + 5376*a*b*Csc[c + d*x]^8*Sin[4*(c + d*x)] + 2304*a*b*Csc[c + d*x]^8*Sin[6*(c + d*x)] + 384*a*b*Csc[c + d*x]^8*Sin[8*(c + d*x)])/d","A",1
1253,1,204,151,1.1613404,"\int \cot ^6(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","-\frac{\csc ^9(c+d x) \left(4032 \left(8 a^2+b^2\right) \cos (c+d x)+18816 a^2 \cos (3 (c+d x))+5760 a^2 \cos (5 (c+d x))+576 a^2 \cos (7 (c+d x))-64 a^2 \cos (9 (c+d x))+18270 a b \sin (2 (c+d x))+10458 a b \sin (4 (c+d x))+8022 a b \sin (6 (c+d x))+315 a b \sin (8 (c+d x))-2304 b^2 \cos (5 (c+d x))-1440 b^2 \cos (7 (c+d x))-288 b^2 \cos (9 (c+d x))\right)+40320 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-40320 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{516096 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,"-1/516096*(-40320*a*b*Log[Cos[(c + d*x)/2]] + 40320*a*b*Log[Sin[(c + d*x)/2]] + Csc[c + d*x]^9*(4032*(8*a^2 + b^2)*Cos[c + d*x] + 18816*a^2*Cos[3*(c + d*x)] + 5760*a^2*Cos[5*(c + d*x)] - 2304*b^2*Cos[5*(c + d*x)] + 576*a^2*Cos[7*(c + d*x)] - 1440*b^2*Cos[7*(c + d*x)] - 64*a^2*Cos[9*(c + d*x)] - 288*b^2*Cos[9*(c + d*x)] + 18270*a*b*Sin[2*(c + d*x)] + 10458*a*b*Sin[4*(c + d*x)] + 8022*a*b*Sin[6*(c + d*x)] + 315*a*b*Sin[8*(c + d*x)]))/d","A",1
1254,1,244,210,1.4708019,"\int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","-\frac{80640 \left(3 a^2+10 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-80640 \left(3 a^2+10 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\csc ^{10}(c+d x) \left(630 \left(1879 a^2+290 b^2\right) \cos (c+d x)+1260 \left(519 a^2-62 b^2\right) \cos (3 (c+d x))+218484 a^2 \cos (5 (c+d x))+9135 a^2 \cos (7 (c+d x))-945 a^2 \cos (9 (c+d x))+537600 a b \sin (2 (c+d x))+522240 a b \sin (4 (c+d x))+207360 a b \sin (6 (c+d x))+25600 a b \sin (8 (c+d x))-2560 a b \sin (10 (c+d x))-24360 b^2 \cos (5 (c+d x))-77070 b^2 \cos (7 (c+d x))-3150 b^2 \cos (9 (c+d x))\right)}{20643840 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,"-1/20643840*(-80640*(3*a^2 + 10*b^2)*Log[Cos[(c + d*x)/2]] + 80640*(3*a^2 + 10*b^2)*Log[Sin[(c + d*x)/2]] + Csc[c + d*x]^10*(630*(1879*a^2 + 290*b^2)*Cos[c + d*x] + 1260*(519*a^2 - 62*b^2)*Cos[3*(c + d*x)] + 218484*a^2*Cos[5*(c + d*x)] - 24360*b^2*Cos[5*(c + d*x)] + 9135*a^2*Cos[7*(c + d*x)] - 77070*b^2*Cos[7*(c + d*x)] - 945*a^2*Cos[9*(c + d*x)] - 3150*b^2*Cos[9*(c + d*x)] + 537600*a*b*Sin[2*(c + d*x)] + 522240*a*b*Sin[4*(c + d*x)] + 207360*a*b*Sin[6*(c + d*x)] + 25600*a*b*Sin[8*(c + d*x)] - 2560*a*b*Sin[10*(c + d*x)]))/d","A",1
1255,1,250,198,1.672177,"\int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]","-\frac{\csc ^{11}(c+d x) \left(1478400 \left(8 a^2+b^2\right) \cos (c+d x)+42240 \left(160 a^2-b^2\right) \cos (3 (c+d x))+1943040 a^2 \cos (5 (c+d x))+140800 a^2 \cos (7 (c+d x))-28160 a^2 \cos (9 (c+d x))+2560 a^2 \cos (11 (c+d x))+5828130 a b \sin (2 (c+d x))+4790016 a b \sin (4 (c+d x))+2302839 a b \sin (6 (c+d x))+110880 a b \sin (8 (c+d x))-10395 a b \sin (10 (c+d x))-865920 b^2 \cos (5 (c+d x))-499840 b^2 \cos (7 (c+d x))-77440 b^2 \cos (9 (c+d x))+7040 b^2 \cos (11 (c+d x))\right)+5322240 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-5322240 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{227082240 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,"-1/227082240*(-5322240*a*b*Log[Cos[(c + d*x)/2]] + 5322240*a*b*Log[Sin[(c + d*x)/2]] + Csc[c + d*x]^11*(1478400*(8*a^2 + b^2)*Cos[c + d*x] + 42240*(160*a^2 - b^2)*Cos[3*(c + d*x)] + 1943040*a^2*Cos[5*(c + d*x)] - 865920*b^2*Cos[5*(c + d*x)] + 140800*a^2*Cos[7*(c + d*x)] - 499840*b^2*Cos[7*(c + d*x)] - 28160*a^2*Cos[9*(c + d*x)] - 77440*b^2*Cos[9*(c + d*x)] + 2560*a^2*Cos[11*(c + d*x)] + 7040*b^2*Cos[11*(c + d*x)] + 5828130*a*b*Sin[2*(c + d*x)] + 4790016*a*b*Sin[4*(c + d*x)] + 2302839*a*b*Sin[6*(c + d*x)] + 110880*a*b*Sin[8*(c + d*x)] - 10395*a*b*Sin[10*(c + d*x)]))/d","A",1
1256,1,531,525,9.1786116,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{107520 a^8 c+107520 a^8 d x+107520 a^7 b c \sin (c+d x)+107520 a^7 b d x \sin (c+d x)+26880 a^6 b^2 \sin (2 (c+d x))-201600 a^6 b^2 c-201600 a^6 b^2 d x-201600 a^5 b^3 c \sin (c+d x)-201600 a^5 b^3 d x \sin (c+d x)-45920 a^4 b^4 \sin (2 (c+d x))-1120 a^4 b^4 \sin (4 (c+d x))+100800 a^4 b^4 c+100800 a^4 b^4 d x+100800 a^3 b^5 c \sin (c+d x)+100800 a^3 b^5 d x \sin (c+d x)-336 a^3 b^5 \cos (5 (c+d x))+18480 a^2 b^6 \sin (2 (c+d x))+1428 a^2 b^6 \sin (4 (c+d x))+112 a^2 b^6 \sin (6 (c+d x))-8400 a^2 b^6 c-8400 a^2 b^6 d x+70 \left(64 a^5 b^3-96 a^3 b^5+27 a b^7\right) \cos (3 (c+d x))+840 a b \left(128 a^6-224 a^4 b^2+98 a^2 b^4-5 b^6\right) \cos (c+d x)-8400 a b^7 c \sin (c+d x)-8400 a b^7 d x \sin (c+d x)+350 a b^7 \cos (5 (c+d x))+40 a b^7 \cos (7 (c+d x))-210 b^8 \sin (2 (c+d x))-210 b^8 \sin (4 (c+d x))-90 b^8 \sin (6 (c+d x))-15 b^8 \sin (8 (c+d x))}{a+b \sin (c+d x)}-26880 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)}{13440 b^9 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{3 b \sin ^5(c+d x) \cos (c+d x)}{20 a^2 d (a+b \sin (c+d x))}+\frac{\left(224 a^4-340 a^2 b^2+105 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^4 d}-\frac{a \left(32 a^4-52 a^2 b^2+19 b^4\right) \sin (c+d x) \cos (c+d x)}{8 b^7 d}+\frac{\left(280 a^4-441 a^2 b^2+150 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}-\frac{\left(24 a^4-37 a^2 b^2+12 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{12 a b^5 d}-\frac{\left(20 a^4-30 a^2 b^2+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{a x \left(64 a^6-120 a^4 b^2+60 a^2 b^4-5 b^6\right)}{8 b^9}+\frac{\left(840 a^6-1435 a^4 b^2+588 a^2 b^4-15 b^6\right) \cos (c+d x)}{105 b^8 d}-\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,"(-26880*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]] + (107520*a^8*c - 201600*a^6*b^2*c + 100800*a^4*b^4*c - 8400*a^2*b^6*c + 107520*a^8*d*x - 201600*a^6*b^2*d*x + 100800*a^4*b^4*d*x - 8400*a^2*b^6*d*x + 840*a*b*(128*a^6 - 224*a^4*b^2 + 98*a^2*b^4 - 5*b^6)*Cos[c + d*x] + 70*(64*a^5*b^3 - 96*a^3*b^5 + 27*a*b^7)*Cos[3*(c + d*x)] - 336*a^3*b^5*Cos[5*(c + d*x)] + 350*a*b^7*Cos[5*(c + d*x)] + 40*a*b^7*Cos[7*(c + d*x)] + 107520*a^7*b*c*Sin[c + d*x] - 201600*a^5*b^3*c*Sin[c + d*x] + 100800*a^3*b^5*c*Sin[c + d*x] - 8400*a*b^7*c*Sin[c + d*x] + 107520*a^7*b*d*x*Sin[c + d*x] - 201600*a^5*b^3*d*x*Sin[c + d*x] + 100800*a^3*b^5*d*x*Sin[c + d*x] - 8400*a*b^7*d*x*Sin[c + d*x] + 26880*a^6*b^2*Sin[2*(c + d*x)] - 45920*a^4*b^4*Sin[2*(c + d*x)] + 18480*a^2*b^6*Sin[2*(c + d*x)] - 210*b^8*Sin[2*(c + d*x)] - 1120*a^4*b^4*Sin[4*(c + d*x)] + 1428*a^2*b^6*Sin[4*(c + d*x)] - 210*b^8*Sin[4*(c + d*x)] + 112*a^2*b^6*Sin[6*(c + d*x)] - 90*b^8*Sin[6*(c + d*x)] - 15*b^8*Sin[8*(c + d*x)])/(a + b*Sin[c + d*x]))/(13440*b^9*d)","A",1
1257,1,462,471,8.1661818,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{3840 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)-\frac{13440 a^7 c+13440 a^7 d x+13440 a^6 b c \sin (c+d x)+13440 a^6 b d x \sin (c+d x)+3360 a^5 b^2 \sin (2 (c+d x))-24000 a^5 b^2 c-24000 a^5 b^2 d x-24000 a^4 b^3 c \sin (c+d x)-24000 a^4 b^3 d x \sin (c+d x)-5440 a^3 b^4 \sin (2 (c+d x))-140 a^3 b^4 \sin (4 (c+d x))+10800 a^3 b^4 c+10800 a^3 b^4 d x+10800 a^2 b^5 c \sin (c+d x)+10800 a^2 b^5 d x \sin (c+d x)-42 a^2 b^5 \cos (5 (c+d x))+10 \left(56 a^4 b^3-79 a^2 b^5+18 b^7\right) \cos (3 (c+d x))+15 b \left(896 a^6-1488 a^4 b^2+576 a^2 b^4-15 b^6\right) \cos (c+d x)+1910 a b^6 \sin (2 (c+d x))+166 a b^6 \sin (4 (c+d x))+14 a b^6 \sin (6 (c+d x))-600 a b^6 c-600 a b^6 d x-600 b^7 c \sin (c+d x)-600 b^7 d x \sin (c+d x)+40 b^7 \cos (5 (c+d x))+5 b^7 \cos (7 (c+d x))}{a+b \sin (c+d x)}}{1920 b^8 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{b \sin ^4(c+d x) \cos (c+d x)}{6 a^2 d (a+b \sin (c+d x))}+\frac{\left(42 a^4-61 a^2 b^2+16 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^4 d}-\frac{a \left(105 a^4-170 a^2 b^2+61 b^4\right) \cos (c+d x)}{15 b^7 d}+\frac{\left(56 a^4-86 a^2 b^2+27 b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^6 d}-\frac{\left(35 a^4-52 a^2 b^2+15 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{15 a b^5 d}-\frac{\left(14 a^4-20 a^2 b^2+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{x \left(112 a^6-200 a^4 b^2+90 a^2 b^4-5 b^6\right)}{16 b^8}-\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,"(3840*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]] - (13440*a^7*c - 24000*a^5*b^2*c + 10800*a^3*b^4*c - 600*a*b^6*c + 13440*a^7*d*x - 24000*a^5*b^2*d*x + 10800*a^3*b^4*d*x - 600*a*b^6*d*x + 15*b*(896*a^6 - 1488*a^4*b^2 + 576*a^2*b^4 - 15*b^6)*Cos[c + d*x] + 10*(56*a^4*b^3 - 79*a^2*b^5 + 18*b^7)*Cos[3*(c + d*x)] - 42*a^2*b^5*Cos[5*(c + d*x)] + 40*b^7*Cos[5*(c + d*x)] + 5*b^7*Cos[7*(c + d*x)] + 13440*a^6*b*c*Sin[c + d*x] - 24000*a^4*b^3*c*Sin[c + d*x] + 10800*a^2*b^5*c*Sin[c + d*x] - 600*b^7*c*Sin[c + d*x] + 13440*a^6*b*d*x*Sin[c + d*x] - 24000*a^4*b^3*d*x*Sin[c + d*x] + 10800*a^2*b^5*d*x*Sin[c + d*x] - 600*b^7*d*x*Sin[c + d*x] + 3360*a^5*b^2*Sin[2*(c + d*x)] - 5440*a^3*b^4*Sin[2*(c + d*x)] + 1910*a*b^6*Sin[2*(c + d*x)] - 140*a^3*b^4*Sin[4*(c + d*x)] + 166*a*b^6*Sin[4*(c + d*x)] + 14*a*b^6*Sin[6*(c + d*x)])/(a + b*Sin[c + d*x]))/(1920*b^8*d)","A",1
1258,1,371,231,4.3548938,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{2880 a^6 c+2880 a^6 d x+2880 a^5 b c \sin (c+d x)+2880 a^5 b d x \sin (c+d x)+720 a^4 b^2 \sin (2 (c+d x))-4800 a^4 b^2 c-4800 a^4 b^2 d x-4800 a^3 b^3 c \sin (c+d x)-4800 a^3 b^3 d x \sin (c+d x)+5 \left(24 a^3 b^3-31 a b^5\right) \cos (3 (c+d x))-1080 a^2 b^4 \sin (2 (c+d x))-30 a^2 b^4 \sin (4 (c+d x))+1800 a^2 b^4 c+1800 a^2 b^4 d x+60 a b \left(48 a^4-74 a^2 b^2+23 b^4\right) \cos (c+d x)+1800 a b^5 c \sin (c+d x)+1800 a b^5 d x \sin (c+d x)-9 a b^5 \cos (5 (c+d x))+295 b^6 \sin (2 (c+d x))+32 b^6 \sin (4 (c+d x))+3 b^6 \sin (6 (c+d x))}{a+b \sin (c+d x)}-960 \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)}{480 b^7 d}","-\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{a x \left(24 a^4-40 a^2 b^2+15 b^4\right)}{4 b^7}+\frac{\cos (c+d x) \left(4 \left(6 a^4-7 a^2 b^2+b^4\right)-a b \left(12 a^2-11 b^2\right) \sin (c+d x)\right)}{4 b^6 d}+\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}",1,"(-960*(a^2 - b^2)^(3/2)*(6*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + (2880*a^6*c - 4800*a^4*b^2*c + 1800*a^2*b^4*c + 2880*a^6*d*x - 4800*a^4*b^2*d*x + 1800*a^2*b^4*d*x + 60*a*b*(48*a^4 - 74*a^2*b^2 + 23*b^4)*Cos[c + d*x] + 5*(24*a^3*b^3 - 31*a*b^5)*Cos[3*(c + d*x)] - 9*a*b^5*Cos[5*(c + d*x)] + 2880*a^5*b*c*Sin[c + d*x] - 4800*a^3*b^3*c*Sin[c + d*x] + 1800*a*b^5*c*Sin[c + d*x] + 2880*a^5*b*d*x*Sin[c + d*x] - 4800*a^3*b^3*d*x*Sin[c + d*x] + 1800*a*b^5*d*x*Sin[c + d*x] + 720*a^4*b^2*Sin[2*(c + d*x)] - 1080*a^2*b^4*Sin[2*(c + d*x)] + 295*b^6*Sin[2*(c + d*x)] - 30*a^2*b^4*Sin[4*(c + d*x)] + 32*b^6*Sin[4*(c + d*x)] + 3*b^6*Sin[6*(c + d*x)])/(a + b*Sin[c + d*x]))/(480*b^7*d)","A",1
1259,1,207,266,1.740835,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{12 a \left(4 a^2-5 b^2\right) (c+d x)}{b^5}-\frac{24 \left(4 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^2 b^5}+\frac{9 \left(4 a^2-3 b^2\right) \cos (c+d x)}{b^4}+\frac{12 \left(a^2-b^2\right)^2 \cos (c+d x)}{a b^4 (a+b \sin (c+d x))}+\frac{12 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^2}-\frac{12 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^2}-\frac{6 a \sin (2 (c+d x))}{b^3}-\frac{\cos (3 (c+d x))}{b^2}}{12 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{2 a x \left(2 a^2-3 b^2\right)}{b^5}+\frac{3 \left(a^2-b^2\right) \cos (c+d x)}{b^4 d}+\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a b^4 d (a+b \sin (c+d x))}-\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,"((12*a*(4*a^2 - 5*b^2)*(c + d*x))/b^5 - (24*(a^2 - b^2)^(3/2)*(4*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^5) + (9*(4*a^2 - 3*b^2)*Cos[c + d*x])/b^4 - Cos[3*(c + d*x)]/b^2 - (12*Log[Cos[(c + d*x)/2]])/a^2 + (12*Log[Sin[(c + d*x)/2]])/a^2 + (12*(a^2 - b^2)^2*Cos[c + d*x])/(a*b^4*(a + b*Sin[c + d*x])) - (6*a*Sin[2*(c + d*x)])/b^3)/(12*d)","A",1
1260,1,215,254,2.7582465,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{-\frac{8 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}+\frac{8 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}+\frac{2 \left(5 b^2-6 a^2\right) (c+d x)}{b^4}-\frac{4 \left(a^2-b^2\right)^2 \cos (c+d x)}{a^2 b^3 (a+b \sin (c+d x))}+\frac{2 \tan \left(\frac{1}{2} (c+d x)\right)}{a^2}-\frac{2 \cot \left(\frac{1}{2} (c+d x)\right)}{a^2}+\frac{8 \left(3 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)}{a^3 b^4}-\frac{8 a \cos (c+d x)}{b^3}+\frac{\sin (2 (c+d x))}{b^2}}{4 d}","\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^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 b^4 d}-\frac{3 x \left(a^2-b^2\right)}{b^4}-\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a^2 b^3 d (a+b \sin (c+d x))}-\frac{\cot (c+d x)}{a^2 d}+\frac{4 \left(2 a^6-3 a^4 b^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^3 b^4 d \sqrt{a^2-b^2}}-\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,"((2*(-6*a^2 + 5*b^2)*(c + d*x))/b^4 + (8*(a^2 - b^2)^(3/2)*(3*a^2 + 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b^4) - (8*a*Cos[c + d*x])/b^3 - (2*Cot[(c + d*x)/2])/a^2 + (8*b*Log[Cos[(c + d*x)/2]])/a^3 - (8*b*Log[Sin[(c + d*x)/2]])/a^3 - (4*(a^2 - b^2)^2*Cos[c + d*x])/(a^2*b^3*(a + b*Sin[c + d*x])) + Sin[2*(c + d*x)]/b^2 + (2*Tan[(c + d*x)/2])/a^2)/(4*d)","A",1
1261,1,315,251,6.1941923,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]","-\frac{b \tan \left(\frac{1}{2} (c+d x)\right)}{a^3 d}+\frac{b \cot \left(\frac{1}{2} (c+d x)\right)}{a^3 d}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}+\frac{\left(6 b^2-5 a^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}+\frac{\left(5 a^2-6 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}-\frac{2 \left(a^2-b^2\right)^{3/2} \left(2 a^2+3 b^2\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^4 b^3 d}+\frac{a^4 \cos (c+d x)-2 a^2 b^2 \cos (c+d x)+b^4 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac{2 a (c+d x)}{b^3 d}+\frac{\cos (c+d x)}{b^2 d}","\frac{2 b \cot (c+d x)}{a^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{\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 \left(a^2-b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^4 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{\left(a^2-b^2\right)^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac{2 a x}{b^3}+\frac{\cos (c+d x)}{b^2 d}",1,"(2*a*(c + d*x))/(b^3*d) - (2*(a^2 - b^2)^(3/2)*(2*a^2 + 3*b^2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^4*b^3*d) + Cos[c + d*x]/(b^2*d) + (b*Cot[(c + d*x)/2])/(a^3*d) - Csc[(c + d*x)/2]^2/(8*a^2*d) + ((5*a^2 - 6*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^4*d) + ((-5*a^2 + 6*b^2)*Log[Sin[(c + d*x)/2]])/(2*a^4*d) + Sec[(c + d*x)/2]^2/(8*a^2*d) + (a^4*Cos[c + d*x] - 2*a^2*b^2*Cos[c + d*x] + b^4*Cos[c + d*x])/(a^3*b^2*d*(a + b*Sin[c + d*x])) - (b*Tan[(c + d*x)/2])/(a^3*d)","A",1
1262,1,428,287,6.296038,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^2,x]","\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{4 a^3 d}-\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{4 a^3 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}+\frac{\left(5 a^2 b-4 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{\left(4 b^3-5 a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{2 \left(a^2-b^2\right)^{3/2} \left(a^2+4 b^2\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^5 b^2 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(7 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-9 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(9 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-7 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{a^4 (-\cos (c+d x))+2 a^2 b^2 \cos (c+d x)-b^4 \cos (c+d x)}{a^4 b d (a+b \sin (c+d x))}-\frac{c+d x}{b^2 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{2 b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\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 \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(a^6-3 a^2 b^4+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{x}{b^2}",1,"-((c + d*x)/(b^2*d)) + (2*(a^2 - b^2)^(3/2)*(a^2 + 4*b^2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*b^2*d) + ((7*a^2*Cos[(c + d*x)/2] - 9*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + (b*Csc[(c + d*x)/2]^2)/(4*a^3*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^2*d) + ((-5*a^2*b + 4*b^3)*Log[Cos[(c + d*x)/2]])/(a^5*d) + ((5*a^2*b - 4*b^3)*Log[Sin[(c + d*x)/2]])/(a^5*d) - (b*Sec[(c + d*x)/2]^2)/(4*a^3*d) + (Sec[(c + d*x)/2]*(-7*a^2*Sin[(c + d*x)/2] + 9*b^2*Sin[(c + d*x)/2]))/(6*a^4*d) + (-(a^4*Cos[c + d*x]) + 2*a^2*b^2*Cos[c + d*x] - b^4*Cos[c + d*x])/(a^4*b*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^2*d)","A",0
1263,1,487,303,6.2272071,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x])^2,x]","\frac{b \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{12 a^3 d}-\frac{b \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{12 a^3 d}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 a^2 d}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 a^2 d}-\frac{10 b \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^6 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(6 b^3 \cos \left(\frac{1}{2} (c+d x)\right)-7 a^2 b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{3 a^5 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(7 a^2 b \sin \left(\frac{1}{2} (c+d x)\right)-6 b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 a^5 d}+\frac{3 \left(3 a^2-4 b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 a^4 d}-\frac{3 \left(3 a^2-4 b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 a^4 d}+\frac{5 \left(3 a^4-12 a^2 b^2+8 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^6 d}-\frac{5 \left(3 a^4-12 a^2 b^2+8 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^6 d}+\frac{a^4 \cos (c+d x)-2 a^2 b^2 \cos (c+d x)+b^4 \cos (c+d x)}{a^5 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{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{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 \left(3 a^4-12 a^2 b^2+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}+\frac{\left(3 a^4-20 a^2 b^2+15 b^4\right) \cot (c+d x)}{3 a^5 b d}-\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[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^6*d) + ((-7*a^2*b*Cos[(c + d*x)/2] + 6*b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(3*a^5*d) + (3*(3*a^2 - 4*b^2)*Csc[(c + d*x)/2]^2)/(32*a^4*d) + (b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(12*a^3*d) - Csc[(c + d*x)/2]^4/(64*a^2*d) - (5*(3*a^4 - 12*a^2*b^2 + 8*b^4)*Log[Cos[(c + d*x)/2]])/(8*a^6*d) + (5*(3*a^4 - 12*a^2*b^2 + 8*b^4)*Log[Sin[(c + d*x)/2]])/(8*a^6*d) - (3*(3*a^2 - 4*b^2)*Sec[(c + d*x)/2]^2)/(32*a^4*d) + Sec[(c + d*x)/2]^4/(64*a^2*d) + (Sec[(c + d*x)/2]*(7*a^2*b*Sin[(c + d*x)/2] - 6*b^3*Sin[(c + d*x)/2]))/(3*a^5*d) + (a^4*Cos[c + d*x] - 2*a^2*b^2*Cos[c + d*x] + b^4*Cos[c + d*x])/(a^5*d*(a + b*Sin[c + d*x])) - (b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(12*a^3*d)","A",0
1264,1,361,424,1.6055233,"\int \frac{\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^6/(a + b*Sin[c + d*x])^2,x]","-\frac{1920 \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)+240 b \left(15 a^4-40 a^2 b^2+24 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-240 b \left(15 a^4-40 a^2 b^2+24 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 a \cot (c+d x) \csc ^5(c+d x) \left(196 a^5+1162 a^4 b \sin (c+d x)-562 a^4 b \sin (3 (c+d x))+76 a^4 b \sin (5 (c+d x))-735 a^3 b^2-3060 a^2 b^3 \sin (c+d x)+1470 a^2 b^3 \sin (3 (c+d x))-270 a^2 b^3 \sin (5 (c+d x))-12 \left(16 a^5-85 a^3 b^2+60 a b^4\right) \cos (2 (c+d x))+\left(92 a^5-285 a^3 b^2+180 a b^4\right) \cos (4 (c+d x))+540 a b^4+1800 b^5 \sin (c+d x)-900 b^5 \sin (3 (c+d x))+180 b^5 \sin (5 (c+d x))\right)}{a \csc (c+d x)+b}}{960 a^7 d}","\frac{3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 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(15 a^4-82 a^2 b^2+60 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}+\frac{b \left(15 a^4-40 a^2 b^2+24 b^4\right) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac{\left(38 a^4-135 a^2 b^2+90 b^4\right) \cot (c+d x)}{15 a^6 d}+\frac{\left(4 a^4-17 a^2 b^2+12 b^4\right) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}+\frac{\left(2 a^4-12 a^2 b^2+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{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,"-1/960*(1920*(a^2 - 6*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 240*b*(15*a^4 - 40*a^2*b^2 + 24*b^4)*Log[Cos[(c + d*x)/2]] + 240*b*(15*a^4 - 40*a^2*b^2 + 24*b^4)*Log[Sin[(c + d*x)/2]] + (2*a*Cot[c + d*x]*Csc[c + d*x]^5*(196*a^5 - 735*a^3*b^2 + 540*a*b^4 - 12*(16*a^5 - 85*a^3*b^2 + 60*a*b^4)*Cos[2*(c + d*x)] + (92*a^5 - 285*a^3*b^2 + 180*a*b^4)*Cos[4*(c + d*x)] + 1162*a^4*b*Sin[c + d*x] - 3060*a^2*b^3*Sin[c + d*x] + 1800*b^5*Sin[c + d*x] - 562*a^4*b*Sin[3*(c + d*x)] + 1470*a^2*b^3*Sin[3*(c + d*x)] - 900*b^5*Sin[3*(c + d*x)] + 76*a^4*b*Sin[5*(c + d*x)] - 270*a^2*b^3*Sin[5*(c + d*x)] + 180*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x]))/(a^7*d)","A",1
1265,1,447,480,1.7236223,"\int \frac{\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^6*Csc[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{15360 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)+480 \left(-5 a^6+90 a^4 b^2-200 a^2 b^4+112 b^6\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+480 \left(5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{2 a \cot (c+d x) \csc ^6(c+d x) \left(590 a^6-3942 a^5 b \sin (c+d x)+1967 a^5 b \sin (3 (c+d x))-571 a^5 b \sin (5 (c+d x))+488 a^4 b^2 \cos (6 (c+d x))-6956 a^4 b^2+12620 a^3 b^3 \sin (c+d x)-6590 a^3 b^3 \sin (3 (c+d x))+1430 a^3 b^3 \sin (5 (c+d x))-1360 a^2 b^4 \cos (6 (c+d x))+15280 a^2 b^4-8 \left(35 a^6-1289 a^4 b^2+2830 a^2 b^4-1575 b^6\right) \cos (2 (c+d x))+\left(330 a^6-3844 a^4 b^2+8720 a^2 b^4-5040 b^6\right) \cos (4 (c+d x))-8400 a b^5 \sin (c+d x)+4200 a b^5 \sin (3 (c+d x))-840 a b^5 \sin (5 (c+d x))+840 b^6 \cos (6 (c+d x))-8400 b^6\right)}{a \csc (c+d x)+b}}{7680 a^8 d}","\frac{7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 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{\left(16 a^4-61 a^2 b^2+42 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}+\frac{b \left(61 a^4-170 a^2 b^2+105 b^4\right) \cot (c+d x)}{15 a^7 d}-\frac{\left(27 a^4-86 a^2 b^2+56 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac{\left(15 a^4-52 a^2 b^2+35 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}+\frac{\left(5 a^4-20 a^2 b^2+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{\left(5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^8 d}+\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,"(15360*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]] + 480*(5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*Log[Cos[(c + d*x)/2]] + 480*(-5*a^6 + 90*a^4*b^2 - 200*a^2*b^4 + 112*b^6)*Log[Sin[(c + d*x)/2]] - (2*a*Cot[c + d*x]*Csc[c + d*x]^6*(590*a^6 - 6956*a^4*b^2 + 15280*a^2*b^4 - 8400*b^6 - 8*(35*a^6 - 1289*a^4*b^2 + 2830*a^2*b^4 - 1575*b^6)*Cos[2*(c + d*x)] + (330*a^6 - 3844*a^4*b^2 + 8720*a^2*b^4 - 5040*b^6)*Cos[4*(c + d*x)] + 488*a^4*b^2*Cos[6*(c + d*x)] - 1360*a^2*b^4*Cos[6*(c + d*x)] + 840*b^6*Cos[6*(c + d*x)] - 3942*a^5*b*Sin[c + d*x] + 12620*a^3*b^3*Sin[c + d*x] - 8400*a*b^5*Sin[c + d*x] + 1967*a^5*b*Sin[3*(c + d*x)] - 6590*a^3*b^3*Sin[3*(c + d*x)] + 4200*a*b^5*Sin[3*(c + d*x)] - 571*a^5*b*Sin[5*(c + d*x)] + 1430*a^3*b^3*Sin[5*(c + d*x)] - 840*a*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x]))/(7680*a^8*d)","A",1
1266,1,2015,536,15.3065502,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]","\text{Result too large to show}","-\frac{b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac{\left(112 a^4-110 a^2 b^2+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{a \sqrt{a^2-b^2} \left(56 a^4-47 a^2 b^2+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{a \left(840 a^4-985 a^2 b^2+213 b^4\right) \cos (c+d x)}{30 b^8 d}+\frac{\left(224 a^4-244 a^2 b^2+43 b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^7 d}-\frac{\left(280 a^4-291 a^2 b^2+45 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{30 a b^6 d}+\frac{\left(168 a^4-169 a^2 b^2+24 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{24 a^2 b^5 d}-\frac{\left(56 a^4-60 a^2 b^2+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{x \left(448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right)}{16 b^9}-\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,"(-8*(c + d*x) + (2*a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a*b*(4*a^2 - 3*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) - (3*b*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/(64*b^3*d) - (3*((6*a*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2))/(256*(a - b)^2*(a + b)^2*d) - (3*((12*a*(640*a^8 - 1920*a^6*b^2 + 2016*a^4*b^4 - 840*a^2*b^6 + 105*b^8)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (-3840*a^10*(c + d*x) + 7680*a^8*b^2*(c + d*x) - 2976*a^6*b^4*(c + d*x) - 1776*a^4*b^6*(c + d*x) + 960*a^2*b^8*(c + d*x) - 48*b^10*(c + d*x) - 3840*a^9*b*Cos[c + d*x] + 8640*a^7*b^3*Cos[c + d*x] - 5696*a^5*b^5*Cos[c + d*x] + 788*a^3*b^7*Cos[c + d*x] + 114*a*b^9*Cos[c + d*x] + 1920*a^8*b^2*(c + d*x)*Cos[2*(c + d*x)] - 4800*a^6*b^4*(c + d*x)*Cos[2*(c + d*x)] + 3888*a^4*b^6*(c + d*x)*Cos[2*(c + d*x)] - 1056*a^2*b^8*(c + d*x)*Cos[2*(c + d*x)] + 48*b^10*(c + d*x)*Cos[2*(c + d*x)] + 320*a^7*b^3*Cos[3*(c + d*x)] - 760*a^5*b^5*Cos[3*(c + d*x)] + 560*a^3*b^7*Cos[3*(c + d*x)] - 120*a*b^9*Cos[3*(c + d*x)] - 8*a^5*b^5*Cos[5*(c + d*x)] + 16*a^3*b^7*Cos[5*(c + d*x)] - 8*a*b^9*Cos[5*(c + d*x)] - 7680*a^9*b*(c + d*x)*Sin[c + d*x] + 19200*a^7*b^3*(c + d*x)*Sin[c + d*x] - 15552*a^5*b^5*(c + d*x)*Sin[c + d*x] + 4224*a^3*b^7*(c + d*x)*Sin[c + d*x] - 192*a*b^9*(c + d*x)*Sin[c + d*x] - 2880*a^8*b^2*Sin[2*(c + d*x)] + 6880*a^6*b^4*Sin[2*(c + d*x)] - 5182*a^4*b^6*Sin[2*(c + d*x)] + 1221*a^2*b^8*Sin[2*(c + d*x)] - 36*b^10*Sin[2*(c + d*x)] - 40*a^6*b^4*Sin[4*(c + d*x)] + 88*a^4*b^6*Sin[4*(c + d*x)] - 56*a^2*b^8*Sin[4*(c + d*x)] + 8*b^10*Sin[4*(c + d*x)] + 2*a^4*b^6*Sin[6*(c + d*x)] - 4*a^2*b^8*Sin[6*(c + d*x)] + 2*b^10*Sin[6*(c + d*x)])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2)))/(1024*b^7*d) - ((-60*a*(14336*a^10 - 49280*a^8*b^2 + 63360*a^6*b^4 - 36960*a^4*b^6 + 9240*a^2*b^8 - 693*b^10)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (430080*a^12*(c + d*x) - 1048320*a^10*b^2*(c + d*x) + 691200*a^8*b^4*(c + d*x) + 83040*a^6*b^6*(c + d*x) - 198000*a^4*b^8*(c + d*x) + 43200*a^2*b^10*(c + d*x) - 1200*b^12*(c + d*x) + 430080*a^11*b*Cos[c + d*x] - 1155840*a^9*b^3*Cos[c + d*x] + 1042880*a^7*b^5*Cos[c + d*x] - 332800*a^5*b^7*Cos[c + d*x] + 11060*a^3*b^9*Cos[c + d*x] + 4530*a*b^11*Cos[c + d*x] - 215040*a^10*b^2*(c + d*x)*Cos[2*(c + d*x)] + 631680*a^8*b^4*(c + d*x)*Cos[2*(c + d*x)] - 661440*a^6*b^6*(c + d*x)*Cos[2*(c + d*x)] + 289200*a^4*b^8*(c + d*x)*Cos[2*(c + d*x)] - 45600*a^2*b^10*(c + d*x)*Cos[2*(c + d*x)] + 1200*b^12*(c + d*x)*Cos[2*(c + d*x)] - 35840*a^9*b^3*Cos[3*(c + d*x)] + 100800*a^7*b^5*Cos[3*(c + d*x)] - 98424*a^5*b^7*Cos[3*(c + d*x)] + 37808*a^3*b^9*Cos[3*(c + d*x)] - 4344*a*b^11*Cos[3*(c + d*x)] + 896*a^7*b^5*Cos[5*(c + d*x)] - 2184*a^5*b^7*Cos[5*(c + d*x)] + 1680*a^3*b^9*Cos[5*(c + d*x)] - 392*a*b^11*Cos[5*(c + d*x)] - 64*a^5*b^7*Cos[7*(c + d*x)] + 128*a^3*b^9*Cos[7*(c + d*x)] - 64*a*b^11*Cos[7*(c + d*x)] + 860160*a^11*b*(c + d*x)*Sin[c + d*x] - 2526720*a^9*b^3*(c + d*x)*Sin[c + d*x] + 2645760*a^7*b^5*(c + d*x)*Sin[c + d*x] - 1156800*a^5*b^7*(c + d*x)*Sin[c + d*x] + 182400*a^3*b^9*(c + d*x)*Sin[c + d*x] - 4800*a*b^11*(c + d*x)*Sin[c + d*x] + 322560*a^10*b^2*Sin[2*(c + d*x)] - 911680*a^8*b^4*Sin[2*(c + d*x)] + 903680*a^6*b^6*Sin[2*(c + d*x)] - 362830*a^4*b^8*Sin[2*(c + d*x)] + 49125*a^2*b^10*Sin[2*(c + d*x)] - 900*b^12*Sin[2*(c + d*x)] + 4480*a^8*b^4*Sin[4*(c + d*x)] - 11816*a^6*b^6*Sin[4*(c + d*x)] + 10392*a^4*b^8*Sin[4*(c + d*x)] - 3256*a^2*b^10*Sin[4*(c + d*x)] + 200*b^12*Sin[4*(c + d*x)] - 224*a^6*b^6*Sin[6*(c + d*x)] + 498*a^4*b^8*Sin[6*(c + d*x)] - 324*a^2*b^10*Sin[6*(c + d*x)] + 50*b^12*Sin[6*(c + d*x)] + 20*a^4*b^8*Sin[8*(c + d*x)] - 40*a^2*b^10*Sin[8*(c + d*x)] + 20*b^12*Sin[8*(c + d*x)])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2))/(15360*b^9*d)","B",1
1267,1,517,485,12.866748,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{\left(a^2-b^2\right)^2 \left(40320 a^7 c+40320 a^7 d x+80640 a^6 b c \sin (c+d x)+80640 a^6 b d x \sin (c+d x)+30240 a^5 b^2 \sin (2 (c+d x))-27840 a^5 b^2 c-27840 a^5 b^2 d x-96000 a^4 b^3 c \sin (c+d x)-96000 a^4 b^3 d x \sin (c+d x)-3360 a^4 b^3 \cos (3 (c+d x))-32640 a^3 b^4 \sin (2 (c+d x))+420 a^3 b^4 \sin (4 (c+d x))-13200 a^3 b^4 c-13200 a^3 b^4 d x+21600 a^2 b^5 c \sin (c+d x)+21600 a^2 b^5 d x \sin (c+d x)+3580 a^2 b^5 \cos (3 (c+d x))+84 a^2 b^5 \cos (5 (c+d x))-120 a b^2 \left(168 a^4-200 a^2 b^2+45 b^4\right) (c+d x) \cos (2 (c+d x))+10 b \left(4032 a^6-3792 a^4 b^2+216 a^2 b^4+59 b^6\right) \cos (c+d x)+5675 a b^6 \sin (2 (c+d x))-374 a b^6 \sin (4 (c+d x))-21 a b^6 \sin (6 (c+d x))+5400 a b^6 c+5400 a b^6 d x-526 b^7 \cos (3 (c+d x))-58 b^7 \cos (5 (c+d x))-6 b^7 \cos (7 (c+d x))\right)}{(a+b \sin (c+d x))^2}-1920 \left(a^2-b^2\right)^{5/2} \left(42 a^4-29 a^2 b^2+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)}{1920 b^8 d (a-b)^2 (a+b)^2}","-\frac{b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{\left(63 a^4-54 a^2 b^2+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{\sqrt{a^2-b^2} \left(42 a^4-29 a^2 b^2+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{a x \left(168 a^4-200 a^2 b^2+45 b^4\right)}{8 b^8}+\frac{\left(630 a^4-645 a^2 b^2+91 b^4\right) \cos (c+d x)}{30 b^7 d}-\frac{\left(84 a^4-79 a^2 b^2+8 b^4\right) \sin (c+d x) \cos (c+d x)}{8 a b^6 d}+\frac{\left(210 a^4-187 a^2 b^2+15 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{30 a^2 b^5 d}-\frac{\left(63 a^4-60 a^2 b^2+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{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,"(-1920*(a^2 - b^2)^(5/2)*(42*a^4 - 29*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + ((a^2 - b^2)^2*(40320*a^7*c - 27840*a^5*b^2*c - 13200*a^3*b^4*c + 5400*a*b^6*c + 40320*a^7*d*x - 27840*a^5*b^2*d*x - 13200*a^3*b^4*d*x + 5400*a*b^6*d*x + 10*b*(4032*a^6 - 3792*a^4*b^2 + 216*a^2*b^4 + 59*b^6)*Cos[c + d*x] - 120*a*b^2*(168*a^4 - 200*a^2*b^2 + 45*b^4)*(c + d*x)*Cos[2*(c + d*x)] - 3360*a^4*b^3*Cos[3*(c + d*x)] + 3580*a^2*b^5*Cos[3*(c + d*x)] - 526*b^7*Cos[3*(c + d*x)] + 84*a^2*b^5*Cos[5*(c + d*x)] - 58*b^7*Cos[5*(c + d*x)] - 6*b^7*Cos[7*(c + d*x)] + 80640*a^6*b*c*Sin[c + d*x] - 96000*a^4*b^3*c*Sin[c + d*x] + 21600*a^2*b^5*c*Sin[c + d*x] + 80640*a^6*b*d*x*Sin[c + d*x] - 96000*a^4*b^3*d*x*Sin[c + d*x] + 21600*a^2*b^5*d*x*Sin[c + d*x] + 30240*a^5*b^2*Sin[2*(c + d*x)] - 32640*a^3*b^4*Sin[2*(c + d*x)] + 5675*a*b^6*Sin[2*(c + d*x)] + 420*a^3*b^4*Sin[4*(c + d*x)] - 374*a*b^6*Sin[4*(c + d*x)] - 21*a*b^6*Sin[6*(c + d*x)]))/(a + b*Sin[c + d*x])^2)/(1920*(a - b)^2*b^8*(a + b)^2*d)","A",1
1268,1,1250,237,8.0541327,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{18 \left(-8 (c+d x)+\frac{2 a \left(8 a^4-20 b^2 a^2+15 b^4\right) \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{3 b \left(4 a^4-7 b^2 a^2+2 b^4\right) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{a b \left(4 a^2-3 b^2\right) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}\right)}{b^3}-\frac{10 \left(\frac{6 a b \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{\cos (c+d x) \left(a \left(2 a^2+b^2\right)+b \left(a^2+2 b^2\right) \sin (c+d x)\right)}{(a+b \sin (c+d x))^2}\right)}{(a-b)^2 (a+b)^2}+\frac{10 \left(-8 \sin (2 (c+d x)) b^2+96 a \cos (c+d x) b+\frac{\left(112 a^6-220 b^2 a^4+115 b^4 a^2-10 b^6\right) \cos (c+d x) b}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{a \left(-16 a^4+20 b^2 a^2-5 b^4\right) \cos (c+d x) b}{(a-b) (a+b) (a+b \sin (c+d x))^2}-24 \left(b^2-8 a^2\right) (c+d x)-\frac{6 a \left(64 a^6-168 b^2 a^4+140 b^4 a^2-35 b^6\right) \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}\right)}{b^5}+\frac{\frac{12 a \left(640 a^8-1920 b^2 a^6+2016 b^4 a^4-840 b^6 a^2+105 b^8\right) \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{-3840 (c+d x) a^{10}-3840 b \cos (c+d x) a^9-7680 b (c+d x) \sin (c+d x) a^9+7680 b^2 (c+d x) a^8+1920 b^2 (c+d x) \cos (2 (c+d x)) a^8-2880 b^2 \sin (2 (c+d x)) a^8+8640 b^3 \cos (c+d x) a^7+320 b^3 \cos (3 (c+d x)) a^7+19200 b^3 (c+d x) \sin (c+d x) a^7-2976 b^4 (c+d x) a^6-4800 b^4 (c+d x) \cos (2 (c+d x)) a^6+6880 b^4 \sin (2 (c+d x)) a^6-40 b^4 \sin (4 (c+d x)) a^6-5696 b^5 \cos (c+d x) a^5-760 b^5 \cos (3 (c+d x)) a^5-8 b^5 \cos (5 (c+d x)) a^5-15552 b^5 (c+d x) \sin (c+d x) a^5-1776 b^6 (c+d x) a^4+3888 b^6 (c+d x) \cos (2 (c+d x)) a^4-5182 b^6 \sin (2 (c+d x)) a^4+88 b^6 \sin (4 (c+d x)) a^4+2 b^6 \sin (6 (c+d x)) a^4+788 b^7 \cos (c+d x) a^3+560 b^7 \cos (3 (c+d x)) a^3+16 b^7 \cos (5 (c+d x)) a^3+4224 b^7 (c+d x) \sin (c+d x) a^3+960 b^8 (c+d x) a^2-1056 b^8 (c+d x) \cos (2 (c+d x)) a^2+1221 b^8 \sin (2 (c+d x)) a^2-56 b^8 \sin (4 (c+d x)) a^2-4 b^8 \sin (6 (c+d x)) a^2+114 b^9 \cos (c+d x) a-120 b^9 \cos (3 (c+d x)) a-8 b^9 \cos (5 (c+d x)) a-192 b^9 (c+d x) \sin (c+d x) a-48 b^{10} (c+d x)+48 b^{10} (c+d x) \cos (2 (c+d x))-36 b^{10} \sin (2 (c+d x))+8 b^{10} \sin (4 (c+d x))+2 b^{10} \sin (6 (c+d x))}{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}}{b^7}}{256 d}","-\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{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 a \left(2 a^4-3 a^2 b^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{15 x \left(8 a^4-8 a^2 b^2+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,"((18*(-8*(c + d*x) + (2*a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a*b*(4*a^2 - 3*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) - (3*b*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x]))))/b^3 - (10*((6*a*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2))/((a - b)^2*(a + b)^2) + (10*(-24*(-8*a^2 + b^2)*(c + d*x) - (6*a*(64*a^6 - 168*a^4*b^2 + 140*a^2*b^4 - 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 96*a*b*Cos[c + d*x] + (a*b*(-16*a^4 + 20*a^2*b^2 - 5*b^4)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) + (b*(112*a^6 - 220*a^4*b^2 + 115*a^2*b^4 - 10*b^6)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])) - 8*b^2*Sin[2*(c + d*x)]))/b^5 + ((12*a*(640*a^8 - 1920*a^6*b^2 + 2016*a^4*b^4 - 840*a^2*b^6 + 105*b^8)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (-3840*a^10*(c + d*x) + 7680*a^8*b^2*(c + d*x) - 2976*a^6*b^4*(c + d*x) - 1776*a^4*b^6*(c + d*x) + 960*a^2*b^8*(c + d*x) - 48*b^10*(c + d*x) - 3840*a^9*b*Cos[c + d*x] + 8640*a^7*b^3*Cos[c + d*x] - 5696*a^5*b^5*Cos[c + d*x] + 788*a^3*b^7*Cos[c + d*x] + 114*a*b^9*Cos[c + d*x] + 1920*a^8*b^2*(c + d*x)*Cos[2*(c + d*x)] - 4800*a^6*b^4*(c + d*x)*Cos[2*(c + d*x)] + 3888*a^4*b^6*(c + d*x)*Cos[2*(c + d*x)] - 1056*a^2*b^8*(c + d*x)*Cos[2*(c + d*x)] + 48*b^10*(c + d*x)*Cos[2*(c + d*x)] + 320*a^7*b^3*Cos[3*(c + d*x)] - 760*a^5*b^5*Cos[3*(c + d*x)] + 560*a^3*b^7*Cos[3*(c + d*x)] - 120*a*b^9*Cos[3*(c + d*x)] - 8*a^5*b^5*Cos[5*(c + d*x)] + 16*a^3*b^7*Cos[5*(c + d*x)] - 8*a*b^9*Cos[5*(c + d*x)] - 7680*a^9*b*(c + d*x)*Sin[c + d*x] + 19200*a^7*b^3*(c + d*x)*Sin[c + d*x] - 15552*a^5*b^5*(c + d*x)*Sin[c + d*x] + 4224*a^3*b^7*(c + d*x)*Sin[c + d*x] - 192*a*b^9*(c + d*x)*Sin[c + d*x] - 2880*a^8*b^2*Sin[2*(c + d*x)] + 6880*a^6*b^4*Sin[2*(c + d*x)] - 5182*a^4*b^6*Sin[2*(c + d*x)] + 1221*a^2*b^8*Sin[2*(c + d*x)] - 36*b^10*Sin[2*(c + d*x)] - 40*a^6*b^4*Sin[4*(c + d*x)] + 88*a^4*b^6*Sin[4*(c + d*x)] - 56*a^2*b^8*Sin[4*(c + d*x)] + 8*b^10*Sin[4*(c + d*x)] + 2*a^4*b^6*Sin[6*(c + d*x)] - 4*a^2*b^8*Sin[6*(c + d*x)] + 2*b^10*Sin[6*(c + d*x)])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2))/b^7)/(256*d)","B",1
1269,1,243,399,1.9497446,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^5*Cot[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}-\frac{4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}+\frac{2 \left(5 b^2-12 a^2\right) (c+d x)}{b^5}+\frac{2 \left(a^2-b^2\right)^2 \cos (c+d x)}{a b^4 (a+b \sin (c+d x))^2}+\frac{2 \left(-7 a^4+5 a^2 b^2+2 b^4\right) \cos (c+d x)}{a^2 b^4 (a+b \sin (c+d x))}+\frac{4 \left(12 a^6-11 a^4 b^2+a^2 b^4-2 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^3 b^5 \sqrt{a^2-b^2}}-\frac{12 a \cos (c+d x)}{b^4}+\frac{\sin (2 (c+d x))}{b^3}}{4 d}","-\frac{\tanh ^{-1}(\cos (c+d x))}{a^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 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{3 x \left(2 a^2-b^2\right)}{b^5}+\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{2 \left(10 a^6-9 a^4 b^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^3 b^5 d \sqrt{a^2-b^2}}-\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,"((2*(-12*a^2 + 5*b^2)*(c + d*x))/b^5 + (4*(12*a^6 - 11*a^4*b^2 + a^2*b^4 - 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]) - (12*a*Cos[c + d*x])/b^4 - (4*Log[Cos[(c + d*x)/2]])/a^3 + (4*Log[Sin[(c + d*x)/2]])/a^3 + (2*(a^2 - b^2)^2*Cos[c + d*x])/(a*b^4*(a + b*Sin[c + d*x])^2) + (2*(-7*a^4 + 5*a^2*b^2 + 2*b^4)*Cos[c + d*x])/(a^2*b^4*(a + b*Sin[c + d*x])) + Sin[2*(c + d*x)]/b^3)/(4*d)","A",1
1270,1,332,314,6.2089998,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","-\frac{3 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^4 d}+\frac{3 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^4 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{2 a^3 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{2 a^3 d}+\frac{a^4 (-\cos (c+d x))+2 a^2 b^2 \cos (c+d x)-b^4 \cos (c+d x)}{2 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{3 \left(2 a^6-a^4 b^2+a^2 b^4-2 b^6\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^4 b^4 d \sqrt{a^2-b^2}}+\frac{5 a^4 \cos (c+d x)-a^2 b^2 \cos (c+d x)-4 b^4 \cos (c+d x)}{2 a^3 b^3 d (a+b \sin (c+d x))}+\frac{3 a (c+d x)}{b^4 d}+\frac{\cos (c+d x)}{b^3 d}","\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a^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{\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{3 \left(a^2-b^2\right) \cos (c+d x)}{2 a b^3 d (a+b \sin (c+d x))}+\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{6 \left(2 a^6-a^4 b^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^4 b^4 d \sqrt{a^2-b^2}}+\frac{3 a x}{b^4}+\frac{\cos (c+d x)}{b^3 d}",1,"(3*a*(c + d*x))/(b^4*d) - (3*(2*a^6 - a^4*b^2 + a^2*b^4 - 2*b^6)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^4*b^4*Sqrt[a^2 - b^2]*d) + Cos[c + d*x]/(b^3*d) - Cot[(c + d*x)/2]/(2*a^3*d) + (3*b*Log[Cos[(c + d*x)/2]])/(a^4*d) - (3*b*Log[Sin[(c + d*x)/2]])/(a^4*d) + (-(a^4*Cos[c + d*x]) + 2*a^2*b^2*Cos[c + d*x] - b^4*Cos[c + d*x])/(2*a^2*b^3*d*(a + b*Sin[c + d*x])^2) + (5*a^4*Cos[c + d*x] - a^2*b^2*Cos[c + d*x] - 4*b^4*Cos[c + d*x])/(2*a^3*b^3*d*(a + b*Sin[c + d*x])) + Tan[(c + d*x)/2]/(2*a^3*d)","A",1
1271,1,384,395,6.2602903,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^3,x]","-\frac{3 b \tan \left(\frac{1}{2} (c+d x)\right)}{2 a^4 d}+\frac{3 b \cot \left(\frac{1}{2} (c+d x)\right)}{2 a^4 d}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^3 d}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^3 d}+\frac{\left(12 b^2-5 a^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^5 d}+\frac{\left(5 a^2-12 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^5 d}-\frac{3 \left(a^4 \cos (c+d x)+a^2 b^2 \cos (c+d x)-2 b^4 \cos (c+d x)\right)}{2 a^4 b^2 d (a+b \sin (c+d x))}+\frac{a^4 \cos (c+d x)-2 a^2 b^2 \cos (c+d x)+b^4 \cos (c+d x)}{2 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{\left(2 a^6-a^4 b^2+11 a^2 b^4-12 b^6\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^5 b^3 d \sqrt{a^2-b^2}}-\frac{c+d x}{b^3 d}","-\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 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{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^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\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{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^6+a^2 b^4-2 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{x}{b^3}",1,"-((c + d*x)/(b^3*d)) + ((2*a^6 - a^4*b^2 + 11*a^2*b^4 - 12*b^6)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*b^3*Sqrt[a^2 - b^2]*d) + (3*b*Cot[(c + d*x)/2])/(2*a^4*d) - Csc[(c + d*x)/2]^2/(8*a^3*d) + ((5*a^2 - 12*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^5*d) + ((-5*a^2 + 12*b^2)*Log[Sin[(c + d*x)/2]])/(2*a^5*d) + Sec[(c + d*x)/2]^2/(8*a^3*d) + (a^4*Cos[c + d*x] - 2*a^2*b^2*Cos[c + d*x] + b^4*Cos[c + d*x])/(2*a^3*b^2*d*(a + b*Sin[c + d*x])^2) - (3*(a^4*Cos[c + d*x] + a^2*b^2*Cos[c + d*x] - 2*b^4*Cos[c + d*x]))/(2*a^4*b^2*d*(a + b*Sin[c + d*x])) - (3*b*Tan[(c + d*x)/2])/(2*a^4*d)","A",1
1272,1,490,329,6.2670329,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^3,x]","\frac{3 b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^4 d}-\frac{3 b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^4 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^3 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^3 d}+\frac{5 \left(3 a^2 b-4 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^6 d}-\frac{5 \left(3 a^2 b-4 b^3\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^6 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(7 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-18 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^5 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(18 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-7 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^5 d}+\frac{a^4 (-\cos (c+d x))+2 a^2 b^2 \cos (c+d x)-b^4 \cos (c+d x)}{2 a^4 b d (a+b \sin (c+d x))^2}+\frac{5 \left(a^4-5 a^2 b^2+4 b^4\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2}}+\frac{a^4 \cos (c+d x)+7 a^2 b^2 \cos (c+d x)-8 b^4 \cos (c+d x)}{2 a^5 b d (a+b \sin (c+d x))}","\frac{5 b \cot (c+d x) \csc (c+d x)}{6 a^2 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{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{\left(3 a^4+35 a^2 b^2-60 b^4\right) \cot (c+d x)}{6 a^5 b^2 d}-\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^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) + ((7*a^2*Cos[(c + d*x)/2] - 18*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^5*d) + (3*b*Csc[(c + d*x)/2]^2)/(8*a^4*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^3*d) - (5*(3*a^2*b - 4*b^3)*Log[Cos[(c + d*x)/2]])/(2*a^6*d) + (5*(3*a^2*b - 4*b^3)*Log[Sin[(c + d*x)/2]])/(2*a^6*d) - (3*b*Sec[(c + d*x)/2]^2)/(8*a^4*d) + (Sec[(c + d*x)/2]*(-7*a^2*Sin[(c + d*x)/2] + 18*b^2*Sin[(c + d*x)/2]))/(6*a^5*d) + (-(a^4*Cos[c + d*x]) + 2*a^2*b^2*Cos[c + d*x] - b^4*Cos[c + d*x])/(2*a^4*b*d*(a + b*Sin[c + d*x])^2) + (a^4*Cos[c + d*x] + 7*a^2*b^2*Cos[c + d*x] - 8*b^4*Cos[c + d*x])/(2*a^5*b*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^3*d)","A",0
1273,1,363,355,1.3853049,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x])^3,x]","\frac{-\frac{1920 b \left(a^4-3 a^2 b^2+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)}{\sqrt{a^2-b^2}}+240 \left(a^4-8 a^2 b^2+8 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-240 \left(a^4-8 a^2 b^2+8 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 a \cot (c+d x) \csc ^5(c+d x) \left(44 a^5-176 a^4 b \sin (c+d x)+66 a^4 b \sin (3 (c+d x))+2 a^4 b \sin (5 (c+d x))-505 a^3 b^2-260 a^2 b^3 \sin (c+d x)+170 a^2 b^3 \sin (3 (c+d x))-50 a^2 b^3 \sin (5 (c+d x))+\left(-68 a^5+660 a^3 b^2-720 a b^4\right) \cos (2 (c+d x))+\left(8 a^5-155 a^3 b^2+180 a b^4\right) \cos (4 (c+d x))+540 a b^4+600 b^5 \sin (c+d x)-300 b^5 \sin (3 (c+d x))+60 b^5 \sin (5 (c+d x))\right)}{(a \csc (c+d x)+b)^2}}{128 a^7 d}","\frac{b \cot (c+d x) \csc ^2(c+d x)}{2 a^2 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{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{15 \left(a^4-8 a^2 b^2+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^7 d}+\frac{\left(a^4-25 a^2 b^2+30 b^4\right) \cot (c+d x)}{2 a^6 b d}-\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,"((-1920*b*(a^4 - 3*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 240*(a^4 - 8*a^2*b^2 + 8*b^4)*Log[Cos[(c + d*x)/2]] + 240*(a^4 - 8*a^2*b^2 + 8*b^4)*Log[Sin[(c + d*x)/2]] + (2*a*Cot[c + d*x]*Csc[c + d*x]^5*(44*a^5 - 505*a^3*b^2 + 540*a*b^4 + (-68*a^5 + 660*a^3*b^2 - 720*a*b^4)*Cos[2*(c + d*x)] + (8*a^5 - 155*a^3*b^2 + 180*a*b^4)*Cos[4*(c + d*x)] - 176*a^4*b*Sin[c + d*x] - 260*a^2*b^3*Sin[c + d*x] + 600*b^5*Sin[c + d*x] + 66*a^4*b*Sin[3*(c + d*x)] + 170*a^2*b^3*Sin[3*(c + d*x)] - 300*b^5*Sin[3*(c + d*x)] + 2*a^4*b*Sin[5*(c + d*x)] - 50*a^2*b^3*Sin[5*(c + d*x)] + 60*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x])^2)/(128*a^7*d)","A",1
1274,1,448,492,1.7866343,"\int \frac{\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^6/(a + b*Sin[c + d*x])^3,x]","\frac{-480 b \left(45 a^4-200 a^2 b^2+168 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+480 b \left(45 a^4-200 a^2 b^2+168 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{3840 \left(2 a^6-31 a^4 b^2+71 a^2 b^4-42 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{2 a \cot (c+d x) \csc ^6(c+d x) \left(-784 a^6-8156 a^5 b \sin (c+d x)+3956 a^5 b \sin (3 (c+d x))-608 a^5 b \sin (5 (c+d x))+182 a^4 b^2 \cos (6 (c+d x))+3256 a^4 b^2+42270 a^3 b^3 \sin (c+d x)-20715 a^3 b^3 \sin (3 (c+d x))+3975 a^3 b^3 \sin (5 (c+d x))-1290 a^2 b^4 \cos (6 (c+d x))+7860 a^2 b^4+2 \left(384 a^6-2131 a^4 b^2-6315 a^2 b^4+9450 b^6\right) \cos (2 (c+d x))+\left(-368 a^6+824 a^4 b^2+6060 a^2 b^4-7560 b^6\right) \cos (4 (c+d x))-37800 a b^5 \sin (c+d x)+18900 a b^5 \sin (3 (c+d x))-3780 a b^5 \sin (5 (c+d x))+1260 b^6 \cos (6 (c+d x))-12600 b^6\right)}{(a \csc (c+d x)+b)^2}}{3840 a^8 d}","\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac{\left(4 a^4-54 a^2 b^2+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{\sqrt{a^2-b^2} \left(2 a^4-29 a^2 b^2+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{b \left(45 a^4-200 a^2 b^2+168 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac{\left(91 a^4-645 a^2 b^2+630 b^4\right) \cot (c+d x)}{30 a^7 d}+\frac{\left(8 a^4-79 a^2 b^2+84 b^4\right) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac{\left(15 a^4-187 a^2 b^2+210 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}+\frac{\left(5 a^4-60 a^2 b^2+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{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,"((-3840*(2*a^6 - 31*a^4*b^2 + 71*a^2*b^4 - 42*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 480*b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*Log[Cos[(c + d*x)/2]] - 480*b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*Log[Sin[(c + d*x)/2]] + (2*a*Cot[c + d*x]*Csc[c + d*x]^6*(-784*a^6 + 3256*a^4*b^2 + 7860*a^2*b^4 - 12600*b^6 + 2*(384*a^6 - 2131*a^4*b^2 - 6315*a^2*b^4 + 9450*b^6)*Cos[2*(c + d*x)] + (-368*a^6 + 824*a^4*b^2 + 6060*a^2*b^4 - 7560*b^6)*Cos[4*(c + d*x)] + 182*a^4*b^2*Cos[6*(c + d*x)] - 1290*a^2*b^4*Cos[6*(c + d*x)] + 1260*b^6*Cos[6*(c + d*x)] - 8156*a^5*b*Sin[c + d*x] + 42270*a^3*b^3*Sin[c + d*x] - 37800*a*b^5*Sin[c + d*x] + 3956*a^5*b*Sin[3*(c + d*x)] - 20715*a^3*b^3*Sin[3*(c + d*x)] + 18900*a*b^5*Sin[3*(c + d*x)] - 608*a^5*b*Sin[5*(c + d*x)] + 3975*a^3*b^3*Sin[5*(c + d*x)] - 3780*a*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x])^2)/(3840*a^8*d)","A",1
1275,1,728,600,3.1455036,"\int \frac{\cot ^6(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^6*Csc[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{215040 b^2 \left(-4 a^6+27 a^4 b^2-47 a^2 b^4+24 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+13440 b \left(5 a^6-100 a^4 b^2+280 a^2 b^4-192 b^6\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+13440 b \left(-5 a^6+100 a^4 b^2-280 a^2 b^4+192 b^6\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{a \csc ^9(c+d x) \left(1120 a^8 \cos (5 (c+d x))+160 a^8 \cos (7 (c+d x))-9660 a^7 b \sin (2 (c+d x))+6160 a^7 b \sin (4 (c+d x))-3660 a^7 b \sin (6 (c+d x))+160 a^7 b \sin (8 (c+d x))+22948 a^6 b^2 \cos (5 (c+d x))-5884 a^6 b^2 \cos (7 (c+d x))-40 a^6 b^2 \cos (9 (c+d x))+194334 a^5 b^3 \sin (2 (c+d x))-190582 a^5 b^3 \sin (4 (c+d x))+77462 a^5 b^3 \sin (6 (c+d x))-11389 a^5 b^3 \sin (8 (c+d x))-18144 a^4 b^4 \cos (5 (c+d x))-5964 a^4 b^4 \cos (7 (c+d x))+3556 a^4 b^4 \cos (9 (c+d x))-592200 a^3 b^5 \sin (2 (c+d x))+585480 a^3 b^5 \sin (4 (c+d x))-246120 a^3 b^5 \sin (6 (c+d x))+39900 a^3 b^5 \sin (8 (c+d x))-193200 a^2 b^6 \cos (5 (c+d x))+77700 a^2 b^6 \cos (7 (c+d x))-13020 a^2 b^6 \cos (9 (c+d x))+28 \left(200 a^8+795 a^6 b^2-1218 a^4 b^4-4110 a^2 b^6+5040 b^8\right) \cos (c+d x)+28 \left(120 a^8-1403 a^6 b^2+1952 a^4 b^4+8700 a^2 b^6-10080 b^8\right) \cos (3 (c+d x))+423360 a b^7 \sin (2 (c+d x))-423360 a b^7 \sin (4 (c+d x))+181440 a b^7 \sin (6 (c+d x))-30240 a b^7 \sin (8 (c+d x))+201600 b^8 \cos (5 (c+d x))-70560 b^8 \cos (7 (c+d x))+10080 b^8 \cos (9 (c+d x))\right)}{(a \csc (c+d x)+b)^2}}{71680 a^{10} d}","\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}+\frac{\left(12 a^4-65 a^2 b^2+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{3 b^2 \sqrt{a^2-b^2} \left(4 a^4-23 a^2 b^2+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{3 b \left(27 a^4-116 a^2 b^2+96 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^8 d}-\frac{\left(205 a^4-973 a^2 b^2+840 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{\left(16 a^4-81 a^2 b^2+72 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{3 \left(35 a^4-185 a^2 b^2+168 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}+\frac{\left(7 a^4-35 a^2 b^2+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 \left(5 a^6-100 a^4 b^2+280 a^2 b^4-192 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^{10} d}+\frac{\left(10 a^6-889 a^4 b^2+3255 a^2 b^4-2520 b^6\right) \cot (c+d x)}{70 a^9 d}+\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,"((215040*b^2*(-4*a^6 + 27*a^4*b^2 - 47*a^2*b^4 + 24*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 13440*b*(-5*a^6 + 100*a^4*b^2 - 280*a^2*b^4 + 192*b^6)*Log[Cos[(c + d*x)/2]] + 13440*b*(5*a^6 - 100*a^4*b^2 + 280*a^2*b^4 - 192*b^6)*Log[Sin[(c + d*x)/2]] - (a*Csc[c + d*x]^9*(28*(200*a^8 + 795*a^6*b^2 - 1218*a^4*b^4 - 4110*a^2*b^6 + 5040*b^8)*Cos[c + d*x] + 28*(120*a^8 - 1403*a^6*b^2 + 1952*a^4*b^4 + 8700*a^2*b^6 - 10080*b^8)*Cos[3*(c + d*x)] + 1120*a^8*Cos[5*(c + d*x)] + 22948*a^6*b^2*Cos[5*(c + d*x)] - 18144*a^4*b^4*Cos[5*(c + d*x)] - 193200*a^2*b^6*Cos[5*(c + d*x)] + 201600*b^8*Cos[5*(c + d*x)] + 160*a^8*Cos[7*(c + d*x)] - 5884*a^6*b^2*Cos[7*(c + d*x)] - 5964*a^4*b^4*Cos[7*(c + d*x)] + 77700*a^2*b^6*Cos[7*(c + d*x)] - 70560*b^8*Cos[7*(c + d*x)] - 40*a^6*b^2*Cos[9*(c + d*x)] + 3556*a^4*b^4*Cos[9*(c + d*x)] - 13020*a^2*b^6*Cos[9*(c + d*x)] + 10080*b^8*Cos[9*(c + d*x)] - 9660*a^7*b*Sin[2*(c + d*x)] + 194334*a^5*b^3*Sin[2*(c + d*x)] - 592200*a^3*b^5*Sin[2*(c + d*x)] + 423360*a*b^7*Sin[2*(c + d*x)] + 6160*a^7*b*Sin[4*(c + d*x)] - 190582*a^5*b^3*Sin[4*(c + d*x)] + 585480*a^3*b^5*Sin[4*(c + d*x)] - 423360*a*b^7*Sin[4*(c + d*x)] - 3660*a^7*b*Sin[6*(c + d*x)] + 77462*a^5*b^3*Sin[6*(c + d*x)] - 246120*a^3*b^5*Sin[6*(c + d*x)] + 181440*a*b^7*Sin[6*(c + d*x)] + 160*a^7*b*Sin[8*(c + d*x)] - 11389*a^5*b^3*Sin[8*(c + d*x)] + 39900*a^3*b^5*Sin[8*(c + d*x)] - 30240*a*b^7*Sin[8*(c + d*x)]))/(b + a*Csc[c + d*x])^2)/(71680*a^10*d)","A",1
1276,1,1906,712,6.9754533,"\int \frac{\cos ^6(e+f x)}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx","Integrate[Cos[e + f*x]^6/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(13/2)),x]","\frac{\sin (e+f x) \sqrt{a+b \sin (e+f x)} \left(-\frac{4 \left(18 \cos (e+f x) a^4-13 b^2 \cos (e+f x) a^2-5 b^4 \cos (e+f x)\right)}{99 a^2 b^4 (a+b \sin (e+f x))^5}-\frac{16 \left(32 \cos (e+f x) b^6-93 a^2 \cos (e+f x) b^4+93 a^4 \cos (e+f x) b^2\right)}{693 a^6 \left(a^2-b^2\right)^3 (a+b \sin (e+f x))}+\frac{2 \left(63 \cos (e+f x) a^8-146 b^2 \cos (e+f x) a^6+151 b^4 \cos (e+f x) a^4-260 b^6 \cos (e+f x) a^2+128 b^8 \cos (e+f x)\right)}{693 a^5 b^4 \left(a^2-b^2\right)^2 (a+b \sin (e+f x))^2}-\frac{4 \left(42 \cos (e+f x) a^6-37 b^2 \cos (e+f x) a^4-17 b^4 \cos (e+f x) a^2+16 b^6 \cos (e+f x)\right)}{231 a^4 b^4 \left(a^2-b^2\right) (a+b \sin (e+f x))^3}+\frac{4 \left(189 \cos (e+f x) a^4-3 b^2 \cos (e+f x) a^2+40 b^4 \cos (e+f x)\right)}{693 a^3 b^4 (a+b \sin (e+f x))^4}+\frac{2 \left(\cos (e+f x) a^4-2 b^2 \cos (e+f x) a^2+b^4 \cos (e+f x)\right)}{11 a b^4 (a+b \sin (e+f x))^6}\right)}{f \sqrt{d \sin (e+f x)}}+\frac{8 \sqrt{\sin (e+f x)} \left(\frac{4 a \left(45 a^6-114 b^2 a^4+101 b^4 a^2-32 b^6\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}} \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}+4 a \left(-93 b a^5+93 b^3 a^3-32 b^5 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{b \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}\right)+2 \left(32 b^6-93 a^2 b^4+93 a^4 b^2\right) \left(\frac{\sqrt{a+b \sin (e+f x)} \cos (e+f x)}{b \sqrt{\sin (e+f x)}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{b \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}\right)}{b}+\frac{i \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc (e+f x) E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{\sqrt{\sin (e+f x)}}\right)|-\frac{2 a}{-a-b}\right) \sqrt{a+b \sin (e+f x)}}{b \sqrt{\cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc (e+f x)} \sqrt{\frac{\csc (e+f x) (a+b \sin (e+f x))}{a+b}}}\right)\right)}{693 a^6 (a-b)^3 (a+b)^3 f \sqrt{d \sin (e+f x)}}","-\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{4 \left(5 a^4-17 a^2 b^2+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{16 b \left(93 a^4-93 a^2 b^2+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{16 b \left(93 a^4-93 a^2 b^2+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(5 a^6-22 a^4 b^2+65 a^2 b^4-32 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{16 \left(45 a^4-48 a^3 b-69 a^2 b^2+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{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,"(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((2*(a^4*Cos[e + f*x] - 2*a^2*b^2*Cos[e + f*x] + b^4*Cos[e + f*x]))/(11*a*b^4*(a + b*Sin[e + f*x])^6) - (4*(18*a^4*Cos[e + f*x] - 13*a^2*b^2*Cos[e + f*x] - 5*b^4*Cos[e + f*x]))/(99*a^2*b^4*(a + b*Sin[e + f*x])^5) + (4*(189*a^4*Cos[e + f*x] - 3*a^2*b^2*Cos[e + f*x] + 40*b^4*Cos[e + f*x]))/(693*a^3*b^4*(a + b*Sin[e + f*x])^4) - (4*(42*a^6*Cos[e + f*x] - 37*a^4*b^2*Cos[e + f*x] - 17*a^2*b^4*Cos[e + f*x] + 16*b^6*Cos[e + f*x]))/(231*a^4*b^4*(a^2 - b^2)*(a + b*Sin[e + f*x])^3) + (2*(63*a^8*Cos[e + f*x] - 146*a^6*b^2*Cos[e + f*x] + 151*a^4*b^4*Cos[e + f*x] - 260*a^2*b^6*Cos[e + f*x] + 128*b^8*Cos[e + f*x]))/(693*a^5*b^4*(a^2 - b^2)^2*(a + b*Sin[e + f*x])^2) - (16*(93*a^4*b^2*Cos[e + f*x] - 93*a^2*b^4*Cos[e + f*x] + 32*b^6*Cos[e + f*x]))/(693*a^6*(a^2 - b^2)^3*(a + b*Sin[e + f*x]))))/(f*Sqrt[d*Sin[e + f*x]]) + (8*Sqrt[Sin[e + f*x]]*((4*a*(45*a^6 - 114*a^4*b^2 + 101*a^2*b^4 - 32*b^6)*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + 4*a*(-93*a^5*b + 93*a^3*b^3 - 32*a*b^5)*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])) + 2*(93*a^4*b^2 - 93*a^2*b^4 + 32*b^6)*((Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Sin[e + f*x]]) + (I*Cos[(-e + Pi/2 - f*x)/2]*Csc[e + f*x]*EllipticE[I*ArcSinh[Sin[(-e + Pi/2 - f*x)/2]/Sqrt[Sin[e + f*x]]], (-2*a)/(-a - b)]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Cos[(-e + Pi/2 - f*x)/2]^2*Csc[e + f*x]]*Sqrt[(Csc[e + f*x]*(a + b*Sin[e + f*x]))/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])))/b)))/(693*a^6*(a - b)^3*(a + b)^3*f*Sqrt[d*Sin[e + f*x]])","C",0
1277,1,127,159,0.450009,"\int \frac{(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt{d \sin (e+f x)}} \, dx","Integrate[(a + b*Sin[e + f*x])^2/((g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]]),x]","\frac{2 \tan (e+f x) \left(15 a^2 \cos ^2(e+f x)^{3/4} \, _2F_1\left(\frac{1}{4},\frac{7}{4};\frac{5}{4};\sin ^2(e+f x)\right)+b \sin (e+f x) \left(10 a+3 b \sin (e+f x) \cos ^2(e+f x)^{3/4} \, _2F_1\left(\frac{5}{4},\frac{7}{4};\frac{9}{4};\sin ^2(e+f x)\right)\right)\right)}{15 f g^2 \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}","\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*(15*a^2*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[1/4, 7/4, 5/4, Sin[e + f*x]^2] + b*Sin[e + f*x]*(10*a + 3*b*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[5/4, 7/4, 9/4, Sin[e + f*x]^2]*Sin[e + f*x]))*Tan[e + f*x])/(15*f*g^2*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","C",1
1278,1,105,193,0.6577914,"\int \frac{(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt{d \sin (e+f x)}} \, dx","Integrate[(a + b*Sin[e + f*x])^2/((g*Cos[e + f*x])^(7/2)*Sqrt[d*Sin[e + f*x]]),x]","\frac{2 \tan (e+f x) \left(3 \left(b^2-4 a^2\right) \sin ^2(e+f x)+15 a^2+10 a b \sin (e+f x) \cos ^2(e+f x)^{5/4} \, _2F_1\left(\frac{3}{4},\frac{9}{4};\frac{7}{4};\sin ^2(e+f x)\right)\right)}{15 f g^2 \sqrt{d \sin (e+f x)} (g \cos (e+f x))^{3/2}}","\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,"(2*(15*a^2 + 10*a*b*(Cos[e + f*x]^2)^(5/4)*Hypergeometric2F1[3/4, 9/4, 7/4, Sin[e + f*x]^2]*Sin[e + f*x] + 3*(-4*a^2 + b^2)*Sin[e + f*x]^2)*Tan[e + f*x])/(15*f*g^2*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])","C",1
1279,1,66,76,0.1991246,"\int \frac{\cos (c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{-6 a^3 \log (a+b \sin (c+d x))+6 a^2 b \sin (c+d x)-3 a b^2 \sin ^2(c+d x)+2 b^3 \sin ^3(c+d x)}{6 b^4 d}","-\frac{a^3 \log (a+b \sin (c+d x))}{b^4 d}+\frac{a^2 \sin (c+d x)}{b^3 d}-\frac{a \sin ^2(c+d x)}{2 b^2 d}+\frac{\sin ^3(c+d x)}{3 b d}",1,"(-6*a^3*Log[a + b*Sin[c + d*x]] + 6*a^2*b*Sin[c + d*x] - 3*a*b^2*Sin[c + d*x]^2 + 2*b^3*Sin[c + d*x]^3)/(6*b^4*d)","A",1
1280,1,49,55,0.1079892,"\int \frac{\cos (c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{2 a^2 \log (a+b \sin (c+d x))-2 a b \sin (c+d x)+b^2 \sin ^2(c+d x)}{2 b^3 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,"(2*a^2*Log[a + b*Sin[c + d*x]] - 2*a*b*Sin[c + d*x] + b^2*Sin[c + d*x]^2)/(2*b^3*d)","A",1
1281,1,33,34,0.0224059,"\int \frac{\cos (c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{\frac{a \log (a+b \sin (c+d x))}{b^2}-\frac{\sin (c+d x)}{b}}{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 - Sin[c + d*x]/b)/d)","A",1
1282,1,34,34,0.0165418,"\int \frac{\cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[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",1
1283,1,50,50,0.0390273,"\int \frac{\cot (c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(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",1
1284,1,72,72,0.0490209,"\int \frac{\cot (c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(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",1
1285,1,177,235,1.766295,"\int \frac{\cos ^2(c+d x) \sin ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","\frac{-10 \left(4 a^2 b^3+b^5\right) \cos (3 (c+d x))-960 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)+15 a \left(-8 a^2 b^2 \sin (2 (c+d x))+4 \left(8 a^4-4 a^2 b^2-b^4\right) (c+d x)+b^4 \sin (4 (c+d x))\right)-60 b \left(-8 a^4+2 a^2 b^2+b^4\right) \cos (c+d x)+6 b^5 \cos (5 (c+d x))}{480 b^6 d}","-\frac{a \left(4 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 d}+\frac{\left(5 a^2-b^2\right) \sin ^2(c+d x) \cos (c+d x)}{15 b^3 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{a x \left(8 a^4-4 a^2 b^2-b^4\right)}{8 b^6}+\frac{\left(15 a^4-5 a^2 b^2-2 b^4\right) \cos (c+d x)}{15 b^5 d}-\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,"(-960*a^4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 60*b*(-8*a^4 + 2*a^2*b^2 + b^4)*Cos[c + d*x] - 10*(4*a^2*b^3 + b^5)*Cos[3*(c + d*x)] + 6*b^5*Cos[5*(c + d*x)] + 15*a*(4*(8*a^4 - 4*a^2*b^2 - b^4)*(c + d*x) - 8*a^2*b^2*Sin[2*(c + d*x)] + b^4*Sin[4*(c + d*x)]))/(480*b^6*d)","A",1
1286,1,146,191,1.1129559,"\int \frac{\cos ^2(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{24 a^2 b^2 \sin (2 (c+d x))+24 a b \left(b^2-4 a^2\right) \cos (c+d x)-12 \left(8 a^4-4 a^2 b^2-b^4\right) (c+d x)+192 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)+8 a b^3 \cos (3 (c+d x))-3 b^4 \sin (4 (c+d x))}{96 b^5 d}","-\frac{a \left(3 a^2-b^2\right) \cos (c+d x)}{3 b^4 d}+\frac{\left(4 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}-\frac{x \left(8 a^4-4 a^2 b^2-b^4\right)}{8 b^5}+\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{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,"(-12*(8*a^4 - 4*a^2*b^2 - b^4)*(c + d*x) + 192*a^3*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 24*a*b*(-4*a^2 + b^2)*Cos[c + d*x] + 8*a*b^3*Cos[3*(c + d*x)] + 24*a^2*b^2*Sin[2*(c + d*x)] - 3*b^4*Sin[4*(c + d*x)])/(96*b^5*d)","A",1
1287,1,130,148,0.2630963,"\int \frac{\cos ^2(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{-12 a^3 c-12 a^3 d x+3 b \left(b^2-4 a^2\right) \cos (c+d x)+24 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)+3 a b^2 \sin (2 (c+d x))+6 a b^2 c+6 a b^2 d x+b^3 \cos (3 (c+d x))}{12 b^4 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{\left(3 a^2-b^2\right) \cos (c+d x)}{3 b^3 d}-\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,"-1/12*(-12*a^3*c + 6*a*b^2*c - 12*a^3*d*x + 6*a*b^2*d*x + 24*a^2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 3*b*(-4*a^2 + b^2)*Cos[c + d*x] + b^3*Cos[3*(c + d*x)] + 3*a*b^2*Sin[2*(c + d*x)])/(b^4*d)","A",1
1288,1,104,100,0.2327236,"\int \frac{\cos ^2(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{8 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)-4 a^2 c-4 a^2 d x-4 a b \cos (c+d x)+b^2 \sin (2 (c+d x))+2 b^2 c+2 b^2 d x}{4 b^3 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,"(-4*a^2*c + 2*b^2*c - 4*a^2*d*x + 2*b^2*d*x + 8*a*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 4*a*b*Cos[c + d*x] + b^2*Sin[2*(c + d*x)])/(4*b^3*d)","A",1
1289,1,90,75,0.0975923,"\int \frac{\cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(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 c+a d x-b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a b 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 b d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{b}",1,"-((a*c + a*d*x - 2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + b*Log[Cos[(c + d*x)/2]] - b*Log[Sin[(c + d*x)/2]])/(a*b*d))","A",1
1290,1,108,80,0.237955,"\int \frac{\cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Sin[c + d*x]),x]","\frac{-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 \tan \left(\frac{1}{2} (c+d x)\right)-a \cot \left(\frac{1}{2} (c+d x)\right)-2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^2 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,"(-4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - a*Cot[(c + d*x)/2] + 2*b*Log[Cos[(c + d*x)/2]] - 2*b*Log[Sin[(c + d*x)/2]] + a*Tan[(c + d*x)/2])/(2*a^2*d)","A",1
1291,1,181,114,0.8481429,"\int \frac{\cot ^2(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{16 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^2 \left(-\csc ^2\left(\frac{1}{2} (c+d x)\right)\right)+a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)-4 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 a b \tan \left(\frac{1}{2} (c+d x)\right)+4 a b \cot \left(\frac{1}{2} (c+d x)\right)+8 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^3 d}","\frac{b \cot (c+d x)}{a^2 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{\cot (c+d x) \csc (c+d x)}{2 a d}",1,"(16*b*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 4*a*b*Cot[(c + d*x)/2] - a^2*Csc[(c + d*x)/2]^2 + 4*a^2*Log[Cos[(c + d*x)/2]] - 8*b^2*Log[Cos[(c + d*x)/2]] - 4*a^2*Log[Sin[(c + d*x)/2]] + 8*b^2*Log[Sin[(c + d*x)/2]] + a^2*Sec[(c + d*x)/2]^2 - 4*a*b*Tan[(c + d*x)/2])/(8*a^3*d)","A",1
1292,1,351,153,6.2323327,"\int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}-\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}+\frac{\left(a^2 b-2 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}+\frac{\left(2 b^3-a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}-\frac{2 b^2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^4 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(a^2 \cos \left(\frac{1}{2} (c+d x)\right)-3 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^3 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(3 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^3 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a d}","\frac{b \cot (c+d x) \csc (c+d x)}{2 a^2 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{b \left(a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{\left(a^2-3 b^2\right) \cot (c+d x)}{3 a^3 d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d}",1,"(-2*b^2*Sqrt[a^2 - b^2]*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^4*d) + ((a^2*Cos[(c + d*x)/2] - 3*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^3*d) + (b*Csc[(c + d*x)/2]^2)/(8*a^2*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a*d) + ((-(a^2*b) + 2*b^3)*Log[Cos[(c + d*x)/2]])/(2*a^4*d) + ((a^2*b - 2*b^3)*Log[Sin[(c + d*x)/2]])/(2*a^4*d) - (b*Sec[(c + d*x)/2]^2)/(8*a^2*d) + (Sec[(c + d*x)/2]*(-(a^2*Sin[(c + d*x)/2]) + 3*b^2*Sin[(c + d*x)/2]))/(6*a^3*d) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a*d)","B",0
1293,1,430,194,6.2650536,"\int \frac{\cot ^2(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{b \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}-\frac{b \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}+\frac{2 b^3 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^5 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(3 b^3 \cos \left(\frac{1}{2} (c+d x)\right)-a^2 b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a^2 b \sin \left(\frac{1}{2} (c+d x)\right)-3 b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{\left(a^2-4 b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 a^3 d}+\frac{\left(4 b^2-a^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 a^3 d}+\frac{\left(-a^4-4 a^2 b^2+8 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^5 d}+\frac{\left(a^4+4 a^2 b^2-8 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^5 d}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 a d}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 a d}","\frac{b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 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(a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac{\left(a^4+4 a^2 b^2-8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}",1,"(2*b^3*Sqrt[a^2 - b^2]*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*d) + ((-(a^2*b*Cos[(c + d*x)/2]) + 3*b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + ((a^2 - 4*b^2)*Csc[(c + d*x)/2]^2)/(32*a^3*d) + (b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^2*d) - Csc[(c + d*x)/2]^4/(64*a*d) + ((a^4 + 4*a^2*b^2 - 8*b^4)*Log[Cos[(c + d*x)/2]])/(8*a^5*d) + ((-a^4 - 4*a^2*b^2 + 8*b^4)*Log[Sin[(c + d*x)/2]])/(8*a^5*d) + ((-a^2 + 4*b^2)*Sec[(c + d*x)/2]^2)/(32*a^3*d) + Sec[(c + d*x)/2]^4/(64*a*d) + (Sec[(c + d*x)/2]*(a^2*b*Sin[(c + d*x)/2] - 3*b^3*Sin[(c + d*x)/2]))/(6*a^4*d) - (b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^2*d)","B",0
1294,1,506,238,1.8513634,"\int \frac{\cot ^2(c+d x) \csc ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","\frac{-64 a^5 \tan \left(\frac{1}{2} (c+d x)\right)-3 a^5 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+a^5 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-16 a^5 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+6 a^5 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)+15 a^4 b \csc ^4\left(\frac{1}{2} (c+d x)\right)-30 a^4 b \csc ^2\left(\frac{1}{2} (c+d x)\right)-15 a^4 b \sec ^4\left(\frac{1}{2} (c+d x)\right)+30 a^4 b \sec ^2\left(\frac{1}{2} (c+d x)\right)+120 a^4 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-120 a^4 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-160 a^3 b^2 \tan \left(\frac{1}{2} (c+d x)\right)-20 a^3 b^2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+320 a^3 b^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+120 a^2 b^3 \csc ^2\left(\frac{1}{2} (c+d x)\right)-120 a^2 b^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)+480 a^2 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-480 a^2 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-1920 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)+32 \left(2 a^5+5 a^3 b^2-15 a b^4\right) \cot \left(\frac{1}{2} (c+d x)\right)+480 a b^4 \tan \left(\frac{1}{2} (c+d x)\right)-960 b^5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+960 b^5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{960 a^6 d}","\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 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{b \left(a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 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^4+4 a^2 b^2-8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}+\frac{\left(2 a^4+5 a^2 b^2-15 b^4\right) \cot (c+d x)}{15 a^5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}",1,"(-1920*b^4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 32*(2*a^5 + 5*a^3*b^2 - 15*a*b^4)*Cot[(c + d*x)/2] - 30*a^4*b*Csc[(c + d*x)/2]^2 + 120*a^2*b^3*Csc[(c + d*x)/2]^2 + 15*a^4*b*Csc[(c + d*x)/2]^4 - 120*a^4*b*Log[Cos[(c + d*x)/2]] - 480*a^2*b^3*Log[Cos[(c + d*x)/2]] + 960*b^5*Log[Cos[(c + d*x)/2]] + 120*a^4*b*Log[Sin[(c + d*x)/2]] + 480*a^2*b^3*Log[Sin[(c + d*x)/2]] - 960*b^5*Log[Sin[(c + d*x)/2]] + 30*a^4*b*Sec[(c + d*x)/2]^2 - 120*a^2*b^3*Sec[(c + d*x)/2]^2 - 15*a^4*b*Sec[(c + d*x)/2]^4 - 16*a^5*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 320*a^3*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + a^5*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 20*a^3*b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 3*a^5*Csc[(c + d*x)/2]^6*Sin[c + d*x] - 64*a^5*Tan[(c + d*x)/2] - 160*a^3*b^2*Tan[(c + d*x)/2] + 480*a*b^4*Tan[(c + d*x)/2] + 6*a^5*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(960*a^6*d)","B",1
1295,1,127,149,0.2826417,"\int \frac{\cos ^3(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{\frac{60 a^3 (a-b) (a+b) \log (a+b \sin (c+d x))}{b^6}-\frac{60 a^2 (a-b) (a+b) \sin (c+d x)}{b^5}+\frac{30 a (a-b) (a+b) \sin ^2(c+d x)}{b^4}-\frac{20 (a-b) (a+b) \sin ^3(c+d x)}{b^3}+\frac{15 a \sin ^4(c+d x)}{b^2}-\frac{12 \sin ^5(c+d x)}{b}}{60 d}","-\frac{a^2 \left(a^2-b^2\right) \sin (c+d x)}{b^5 d}+\frac{a \left(a^2-b^2\right) \sin ^2(c+d x)}{2 b^4 d}-\frac{\left(a^2-b^2\right) \sin ^3(c+d x)}{3 b^3 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,"((60*a^3*(a - b)*(a + b)*Log[a + b*Sin[c + d*x]])/b^6 - (60*a^2*(a - b)*(a + b)*Sin[c + d*x])/b^5 + (30*a*(a - b)*(a + b)*Sin[c + d*x]^2)/b^4 - (20*(a - b)*(a + b)*Sin[c + d*x]^3)/b^3 + (15*a*Sin[c + d*x]^4)/b^2 - (12*Sin[c + d*x]^5)/b)/(60*d)","A",1
1296,1,104,119,0.3562913,"\int \frac{\cos ^3(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{6 b^2 \left(b^2-a^2\right) \sin ^2(c+d x)+12 a b \left(a^2-b^2\right) \sin (c+d x)+12 a^2 \left(b^2-a^2\right) \log (a+b \sin (c+d x))+4 a b^3 \sin ^3(c+d x)-3 b^4 \sin ^4(c+d x)}{12 b^5 d}","-\frac{a^2 \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^5 d}+\frac{a \left(a^2-b^2\right) \sin (c+d x)}{b^4 d}-\frac{\left(a^2-b^2\right) \sin ^2(c+d x)}{2 b^3 d}+\frac{a \sin ^3(c+d x)}{3 b^2 d}-\frac{\sin ^4(c+d x)}{4 b d}",1,"(12*a^2*(-a^2 + b^2)*Log[a + b*Sin[c + d*x]] + 12*a*b*(a^2 - b^2)*Sin[c + d*x] + 6*b^2*(-a^2 + b^2)*Sin[c + d*x]^2 + 4*a*b^3*Sin[c + d*x]^3 - 3*b^4*Sin[c + d*x]^4)/(12*b^5*d)","A",1
1297,1,79,89,0.2014986,"\int \frac{\cos ^3(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{6 b \left(b^2-a^2\right) \sin (c+d x)+6 a \left(a^2-b^2\right) \log (a+b \sin (c+d x))+3 a b^2 \sin ^2(c+d x)-2 b^3 \sin ^3(c+d x)}{6 b^4 d}","\frac{a \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^4 d}-\frac{\left(a^2-b^2\right) \sin (c+d x)}{b^3 d}+\frac{a \sin ^2(c+d x)}{2 b^2 d}-\frac{\sin ^3(c+d x)}{3 b d}",1,"(6*a*(a^2 - b^2)*Log[a + b*Sin[c + d*x]] + 6*b*(-a^2 + b^2)*Sin[c + d*x] + 3*a*b^2*Sin[c + d*x]^2 - 2*b^3*Sin[c + d*x]^3)/(6*b^4*d)","A",1
1298,1,53,59,0.0669831,"\int \frac{\cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(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 \sin (c+d x)+b^2 \log (\sin (c+d x))}{a b^2 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,"(b^2*Log[Sin[c + d*x]] + (a^2 - b^2)*Log[a + b*Sin[c + d*x]] - a*b*Sin[c + d*x])/(a*b^2*d)","A",1
1299,1,54,60,0.0797357,"\int \frac{\cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\left(b^2-a^2\right) \log (a+b \sin (c+d x))-a b \csc (c+d x)+b^2 (-\log (\sin (c+d x)))}{a^2 b 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,"(-(a*b*Csc[c + d*x]) - b^2*Log[Sin[c + d*x]] + (-a^2 + b^2)*Log[a + b*Sin[c + d*x]])/(a^2*b*d)","A",1
1300,1,65,84,0.1564382,"\int \frac{\cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^3/(a + b*Sin[c + d*x]),x]","-\frac{2 \left(a^2-b^2\right) (\log (\sin (c+d x))-\log (a+b \sin (c+d x)))+a^2 \csc ^2(c+d x)-2 a b \csc (c+d x)}{2 a^3 d}","\frac{b \csc (c+d x)}{a^2 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{\csc ^2(c+d x)}{2 a d}",1,"-1/2*(-2*a*b*Csc[c + d*x] + a^2*Csc[c + d*x]^2 + 2*(a^2 - b^2)*(Log[Sin[c + d*x]] - Log[a + b*Sin[c + d*x]]))/(a^3*d)","A",1
1301,1,274,282,2.2069284,"\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{960 a^6 c+960 a^6 d x-240 a^4 b^2 \sin (2 (c+d x))-1440 a^4 b^2 c-1440 a^4 b^2 d x+\left(60 a b^5-80 a^3 b^3\right) \cos (3 (c+d x))+240 a^2 b^4 \sin (2 (c+d x))+30 a^2 b^4 \sin (4 (c+d x))+360 a^2 b^4 c+360 a^2 b^4 d x+120 a b \left(8 a^4-10 a^2 b^2+b^4\right) \cos (c+d x)-1920 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)+12 a b^5 \cos (5 (c+d x))+15 b^6 \sin (2 (c+d x))-15 b^6 \sin (4 (c+d x))-5 b^6 \sin (6 (c+d x))+60 b^6 c+60 b^6 d x}{960 b^7 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(6 a^2-7 b^2\right) \sin ^3(c+d x) \cos (c+d x)}{24 b^3 d}+\frac{a \left(15 a^4-20 a^2 b^2+3 b^4\right) \cos (c+d x)}{15 b^6 d}-\frac{\left(8 a^4-10 a^2 b^2+b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^5 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{x \left(16 a^6-24 a^4 b^2+6 a^2 b^4+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,"(960*a^6*c - 1440*a^4*b^2*c + 360*a^2*b^4*c + 60*b^6*c + 960*a^6*d*x - 1440*a^4*b^2*d*x + 360*a^2*b^4*d*x + 60*b^6*d*x - 1920*a^3*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 120*a*b*(8*a^4 - 10*a^2*b^2 + b^4)*Cos[c + d*x] + (-80*a^3*b^3 + 60*a*b^5)*Cos[3*(c + d*x)] + 12*a*b^5*Cos[5*(c + d*x)] - 240*a^4*b^2*Sin[2*(c + d*x)] + 240*a^2*b^4*Sin[2*(c + d*x)] + 15*b^6*Sin[2*(c + d*x)] + 30*a^2*b^4*Sin[4*(c + d*x)] - 15*b^6*Sin[4*(c + d*x)] - 5*b^6*Sin[6*(c + d*x)])/(960*b^7*d)","A",1
1302,1,186,235,2.0440686,"\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{960 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)+10 \left(4 a^2 b^3-3 b^5\right) \cos (3 (c+d x))-15 a \left(\left(8 b^4-8 a^2 b^2\right) \sin (2 (c+d x))+4 \left(8 a^4-12 a^2 b^2+3 b^4\right) (c+d x)+b^4 \sin (4 (c+d x))\right)-60 b \left(8 a^4-10 a^2 b^2+b^4\right) \cos (c+d x)-6 b^5 \cos (5 (c+d x))}{480 b^6 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{a \left(4 a^2-5 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 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 x \left(8 a^4-12 a^2 b^2+3 b^4\right)}{8 b^6}-\frac{\left(15 a^4-20 a^2 b^2+3 b^4\right) \cos (c+d x)}{15 b^5 d}+\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,"(960*a^2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 60*b*(8*a^4 - 10*a^2*b^2 + b^4)*Cos[c + d*x] + 10*(4*a^2*b^3 - 3*b^5)*Cos[3*(c + d*x)] - 6*b^5*Cos[5*(c + d*x)] - 15*a*(4*(8*a^4 - 12*a^2*b^2 + 3*b^4)*(c + d*x) + (-8*a^2*b^2 + 8*b^4)*Sin[2*(c + d*x)] + b^4*Sin[4*(c + d*x)]))/(480*b^6*d)","A",1
1303,1,155,159,1.0786406,"\int \frac{\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{24 a b \left(4 a^2-5 b^2\right) \cos (c+d x)-192 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)+3 \left(\left(8 b^4-8 a^2 b^2\right) \sin (2 (c+d x))+4 \left(8 a^4-12 a^2 b^2+3 b^4\right) (c+d x)+b^4 \sin (4 (c+d x))\right)-8 a b^3 \cos (3 (c+d x))}{96 b^5 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(8 a^4-12 a^2 b^2+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,"(-192*a*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 24*a*b*(4*a^2 - 5*b^2)*Cos[c + d*x] - 8*a*b^3*Cos[3*(c + d*x)] + 3*(4*(8*a^4 - 12*a^2*b^2 + 3*b^4)*(c + d*x) + (-8*a^2*b^2 + 8*b^4)*Sin[2*(c + d*x)] + b^4*Sin[4*(c + d*x)]))/(96*b^5*d)","A",1
1304,1,143,124,0.2631236,"\int \frac{\cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{-4 a^3 c-4 a^3 d x+8 \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)-4 a^2 b \cos (c+d x)+a b^2 \sin (2 (c+d x))+6 a b^2 c+6 a b^2 d x-4 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 a 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 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,"-1/4*(-4*a^3*c + 6*a*b^2*c - 4*a^3*d*x + 6*a*b^2*d*x + 8*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 4*a^2*b*Cos[c + d*x] + 4*b^3*Log[Cos[(c + d*x)/2]] - 4*b^3*Log[Sin[(c + d*x)/2]] + a*b^2*Sin[2*(c + d*x)])/(a*b^3*d)","A",1
1305,1,146,104,0.7777008,"\int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{2 a^3 c+2 a^3 d x-4 \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)+2 a^2 b \cos (c+d x)-a b^2 \tan \left(\frac{1}{2} (c+d x)\right)+a b^2 \cot \left(\frac{1}{2} (c+d x)\right)+2 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^2 b^2 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,"-1/2*(2*a^3*c + 2*a^3*d*x - 4*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 2*a^2*b*Cos[c + d*x] + a*b^2*Cot[(c + d*x)/2] - 2*b^3*Log[Cos[(c + d*x)/2]] + 2*b^3*Log[Sin[(c + d*x)/2]] - a*b^2*Tan[(c + d*x)/2])/(a^2*b^2*d)","A",1
1306,1,204,123,1.6857036,"\int \frac{\cos (c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{8 a^3 c+8 a^3 d x-16 \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 \csc ^2\left(\frac{1}{2} (c+d x)\right)+a^2 b \sec ^2\left(\frac{1}{2} (c+d x)\right)-12 a^2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 a^2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 a b^2 \tan \left(\frac{1}{2} (c+d x)\right)+4 a b^2 \cot \left(\frac{1}{2} (c+d x)\right)+8 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^3 b d}","\frac{b \cot (c+d x)}{a^2 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^3 b d}+\frac{\left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}+\frac{x}{b}",1,"(8*a^3*c + 8*a^3*d*x - 16*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 4*a*b^2*Cot[(c + d*x)/2] - a^2*b*Csc[(c + d*x)/2]^2 + 12*a^2*b*Log[Cos[(c + d*x)/2]] - 8*b^3*Log[Cos[(c + d*x)/2]] - 12*a^2*b*Log[Sin[(c + d*x)/2]] + 8*b^3*Log[Sin[(c + d*x)/2]] + a^2*b*Sec[(c + d*x)/2]^2 - 4*a*b^2*Tan[(c + d*x)/2])/(8*a^3*b*d)","A",1
1307,1,350,154,6.1254562,"\int \frac{\cot ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}-\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}+\frac{\left(3 a^2 b-2 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}+\frac{\left(2 b^3-3 a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}+\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^4 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(4 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-3 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^3 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(3 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-4 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^3 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a d}","\frac{b \cot (c+d x) \csc (c+d x)}{2 a^2 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{b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{\left(4 a^2-3 b^2\right) \cot (c+d x)}{3 a^3 d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d}",1,"(2*(a^2 - b^2)^(3/2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^4*d) + ((4*a^2*Cos[(c + d*x)/2] - 3*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^3*d) + (b*Csc[(c + d*x)/2]^2)/(8*a^2*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a*d) + ((-3*a^2*b + 2*b^3)*Log[Cos[(c + d*x)/2]])/(2*a^4*d) + ((3*a^2*b - 2*b^3)*Log[Sin[(c + d*x)/2]])/(2*a^4*d) - (b*Sec[(c + d*x)/2]^2)/(8*a^2*d) + (Sec[(c + d*x)/2]*(-4*a^2*Sin[(c + d*x)/2] + 3*b^2*Sin[(c + d*x)/2]))/(6*a^3*d) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a*d)","B",0
1308,1,433,198,6.2107825,"\int \frac{\cot ^4(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{b \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}-\frac{b \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}-\frac{2 b \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^5 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(3 b^3 \cos \left(\frac{1}{2} (c+d x)\right)-4 a^2 b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(4 a^2 b \sin \left(\frac{1}{2} (c+d x)\right)-3 b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{\left(5 a^2-4 b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 a^3 d}+\frac{\left(4 b^2-5 a^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 a^3 d}+\frac{\left(3 a^4-12 a^2 b^2+8 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^5 d}+\frac{\left(-3 a^4+12 a^2 b^2-8 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^5 d}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 a d}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 a d}","\frac{b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 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(5 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac{\left(3 a^4-12 a^2 b^2+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}",1,"(-2*b*(a^2 - b^2)^(3/2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*d) + ((-4*a^2*b*Cos[(c + d*x)/2] + 3*b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + ((5*a^2 - 4*b^2)*Csc[(c + d*x)/2]^2)/(32*a^3*d) + (b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^2*d) - Csc[(c + d*x)/2]^4/(64*a*d) + ((-3*a^4 + 12*a^2*b^2 - 8*b^4)*Log[Cos[(c + d*x)/2]])/(8*a^5*d) + ((3*a^4 - 12*a^2*b^2 + 8*b^4)*Log[Sin[(c + d*x)/2]])/(8*a^5*d) + ((-5*a^2 + 4*b^2)*Sec[(c + d*x)/2]^2)/(32*a^3*d) + Sec[(c + d*x)/2]^4/(64*a*d) + (Sec[(c + d*x)/2]*(4*a^2*b*Sin[(c + d*x)/2] - 3*b^3*Sin[(c + d*x)/2]))/(6*a^4*d) - (b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^2*d)","B",0
1309,1,507,244,1.8239943,"\int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^4*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{96 a^5 \tan \left(\frac{1}{2} (c+d x)\right)-3 a^5 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+21 a^5 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-336 a^5 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+6 a^5 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)+15 a^4 b \csc ^4\left(\frac{1}{2} (c+d x)\right)-150 a^4 b \csc ^2\left(\frac{1}{2} (c+d x)\right)-15 a^4 b \sec ^4\left(\frac{1}{2} (c+d x)\right)+150 a^4 b \sec ^2\left(\frac{1}{2} (c+d x)\right)-360 a^4 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+360 a^4 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-640 a^3 b^2 \tan \left(\frac{1}{2} (c+d x)\right)-20 a^3 b^2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+320 a^3 b^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+120 a^2 b^3 \csc ^2\left(\frac{1}{2} (c+d x)\right)-120 a^2 b^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)+1440 a^2 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-1440 a^2 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+1920 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)-32 \left(3 a^5-20 a^3 b^2+15 a b^4\right) \cot \left(\frac{1}{2} (c+d x)\right)+480 a b^4 \tan \left(\frac{1}{2} (c+d x)\right)-960 b^5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+960 b^5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{960 a^6 d}","\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 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{b \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 ^2(c+d x)}{15 a^3 d}+\frac{b \left(3 a^4-12 a^2 b^2+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left(3 a^4-20 a^2 b^2+15 b^4\right) \cot (c+d x)}{15 a^5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}",1,"(1920*b^2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 32*(3*a^5 - 20*a^3*b^2 + 15*a*b^4)*Cot[(c + d*x)/2] - 150*a^4*b*Csc[(c + d*x)/2]^2 + 120*a^2*b^3*Csc[(c + d*x)/2]^2 + 15*a^4*b*Csc[(c + d*x)/2]^4 + 360*a^4*b*Log[Cos[(c + d*x)/2]] - 1440*a^2*b^3*Log[Cos[(c + d*x)/2]] + 960*b^5*Log[Cos[(c + d*x)/2]] - 360*a^4*b*Log[Sin[(c + d*x)/2]] + 1440*a^2*b^3*Log[Sin[(c + d*x)/2]] - 960*b^5*Log[Sin[(c + d*x)/2]] + 150*a^4*b*Sec[(c + d*x)/2]^2 - 120*a^2*b^3*Sec[(c + d*x)/2]^2 - 15*a^4*b*Sec[(c + d*x)/2]^4 - 336*a^5*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 320*a^3*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 21*a^5*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 20*a^3*b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 3*a^5*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 96*a^5*Tan[(c + d*x)/2] - 640*a^3*b^2*Tan[(c + d*x)/2] + 480*a*b^4*Tan[(c + d*x)/2] + 6*a^5*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(960*a^6*d)","B",1
1310,1,180,212,1.2673167,"\int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{420 b \left(a^3-a b^2\right)^2 \sin (c+d x)-210 a b^2 \left(a^2-b^2\right)^2 \sin ^2(c+d x)+84 b^5 \left(a^2-2 b^2\right) \sin ^5(c+d x)-105 a b^4 \left(a^2-2 b^2\right) \sin ^4(c+d x)+140 b^3 \left(a^2-b^2\right)^2 \sin ^3(c+d x)-420 a^3 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))-70 a b^6 \sin ^6(c+d x)+60 b^7 \sin ^7(c+d x)}{420 b^8 d}","\frac{a^2 \left(a^2-b^2\right)^2 \sin (c+d x)}{b^7 d}-\frac{a \left(a^2-b^2\right)^2 \sin ^2(c+d x)}{2 b^6 d}+\frac{\left(a^2-b^2\right)^2 \sin ^3(c+d x)}{3 b^5 d}-\frac{a \left(a^2-2 b^2\right) \sin ^4(c+d x)}{4 b^4 d}+\frac{\left(a^2-2 b^2\right) \sin ^5(c+d x)}{5 b^3 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,"(-420*a^3*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]] + 420*b*(a^3 - a*b^2)^2*Sin[c + d*x] - 210*a*b^2*(a^2 - b^2)^2*Sin[c + d*x]^2 + 140*b^3*(a^2 - b^2)^2*Sin[c + d*x]^3 - 105*a*b^4*(a^2 - 2*b^2)*Sin[c + d*x]^4 + 84*b^5*(a^2 - 2*b^2)*Sin[c + d*x]^5 - 70*a*b^6*Sin[c + d*x]^6 + 60*b^7*Sin[c + d*x]^7)/(420*b^8*d)","A",1
1311,1,153,180,0.7181651,"\int \frac{\cos ^5(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{60 \left(a^3-a b^2\right)^2 \log (a+b \sin (c+d x))+30 b^2 \left(a^2-b^2\right)^2 \sin ^2(c+d x)-60 a b \left(a^2-b^2\right)^2 \sin (c+d x)+15 b^4 \left(a^2-2 b^2\right) \sin ^4(c+d x)-20 a b^3 \left(a^2-2 b^2\right) \sin ^3(c+d x)-12 a b^5 \sin ^5(c+d x)+10 b^6 \sin ^6(c+d x)}{60 b^7 d}","\frac{a^2 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^7 d}-\frac{a \left(a^2-b^2\right)^2 \sin (c+d x)}{b^6 d}+\frac{\left(a^2-b^2\right)^2 \sin ^2(c+d x)}{2 b^5 d}-\frac{a \left(a^2-2 b^2\right) \sin ^3(c+d x)}{3 b^4 d}+\frac{\left(a^2-2 b^2\right) \sin ^4(c+d x)}{4 b^3 d}-\frac{a \sin ^5(c+d x)}{5 b^2 d}+\frac{\sin ^6(c+d x)}{6 b d}",1,"(60*(a^3 - a*b^2)^2*Log[a + b*Sin[c + d*x]] - 60*a*b*(a^2 - b^2)^2*Sin[c + d*x] + 30*b^2*(a^2 - b^2)^2*Sin[c + d*x]^2 - 20*a*b^3*(a^2 - 2*b^2)*Sin[c + d*x]^3 + 15*b^4*(a^2 - 2*b^2)*Sin[c + d*x]^4 - 12*a*b^5*Sin[c + d*x]^5 + 10*b^6*Sin[c + d*x]^6)/(60*b^7*d)","A",1
1312,1,128,148,0.6384795,"\int \frac{\cos ^5(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{-30 a b^2 \left(a^2-2 b^2\right) \sin ^2(c+d x)+60 b \left(a^2-b^2\right)^2 \sin (c+d x)-60 a \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))+20 b^3 \left(a^2-2 b^2\right) \sin ^3(c+d x)-15 a b^4 \sin ^4(c+d x)+12 b^5 \sin ^5(c+d x)}{60 b^6 d}","-\frac{a \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^6 d}+\frac{\left(a^2-b^2\right)^2 \sin (c+d x)}{b^5 d}-\frac{a \left(a^2-2 b^2\right) \sin ^2(c+d x)}{2 b^4 d}+\frac{\left(a^2-2 b^2\right) \sin ^3(c+d x)}{3 b^3 d}-\frac{a \sin ^4(c+d x)}{4 b^2 d}+\frac{\sin ^5(c+d x)}{5 b d}",1,"(-60*a*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]] + 60*b*(a^2 - b^2)^2*Sin[c + d*x] - 30*a*b^2*(a^2 - 2*b^2)*Sin[c + d*x]^2 + 20*b^3*(a^2 - 2*b^2)*Sin[c + d*x]^3 - 15*a*b^4*Sin[c + d*x]^4 + 12*b^5*Sin[c + d*x]^5)/(60*b^6*d)","A",1
1313,1,101,107,0.1428689,"\int \frac{\cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{-3 a^2 b^2 \sin ^2(c+d x)+6 a b \left(a^2-2 b^2\right) \sin (c+d x)+6 \left(b^4 \log (\sin (c+d x))-\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))\right)+2 a b^3 \sin ^3(c+d x)}{6 a b^4 d}","-\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a b^4 d}+\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{b^3 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,"(6*(b^4*Log[Sin[c + d*x]] - (a^2 - b^2)^2*Log[a + b*Sin[c + d*x]]) + 6*a*b*(a^2 - 2*b^2)*Sin[c + d*x] - 3*a^2*b^2*Sin[c + d*x]^2 + 2*a*b^3*Sin[c + d*x]^3)/(6*a*b^4*d)","A",1
1314,1,86,96,0.1756967,"\int \frac{\cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\frac{2 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^2 b^3}-\frac{2 b \log (\sin (c+d x))}{a^2}-\frac{2 a \sin (c+d x)}{b^2}-\frac{2 \csc (c+d x)}{a}+\frac{\sin ^2(c+d x)}{b}}{2 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,"((-2*Csc[c + d*x])/a - (2*b*Log[Sin[c + d*x]])/a^2 + (2*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^2*b^3) - (2*a*Sin[c + d*x])/b^2 + Sin[c + d*x]^2/b)/(2*d)","A",1
1315,1,97,105,0.3901398,"\int \frac{\cos ^2(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{\frac{2 b \csc (c+d x)}{a^2}+\frac{\frac{2 b^2 \left(b^2-2 a^2\right) \log (\sin (c+d x))-2 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{a^3}+2 b \sin (c+d x)}{b^2}-\frac{\csc ^2(c+d x)}{a}}{2 d}","\frac{b \csc (c+d x)}{a^2 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{\csc ^2(c+d x)}{2 a d}+\frac{\sin (c+d x)}{b d}",1,"((2*b*Csc[c + d*x])/a^2 - Csc[c + d*x]^2/a + ((2*b^2*(-2*a^2 + b^2)*Log[Sin[c + d*x]] - 2*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/a^3 + 2*b*Sin[c + d*x])/b^2)/(2*d)","A",1
1316,1,110,120,0.2846428,"\int \frac{\cos (c+d x) \cot ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","\frac{-2 a^3 b \csc ^3(c+d x)+3 a^2 b^2 \csc ^2(c+d x)+6 a b \left(2 a^2-b^2\right) \csc (c+d x)-6 b^2 \left(b^2-2 a^2\right) \log (\sin (c+d x))+6 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{6 a^4 b d}","\frac{b \csc ^2(c+d x)}{2 a^2 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{\left(2 a^2-b^2\right) \csc (c+d x)}{a^3 d}-\frac{\csc ^3(c+d x)}{3 a d}",1,"(6*a*b*(2*a^2 - b^2)*Csc[c + d*x] + 3*a^2*b^2*Csc[c + d*x]^2 - 2*a^3*b*Csc[c + d*x]^3 - 6*b^2*(-2*a^2 + b^2)*Log[Sin[c + d*x]] + 6*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(6*a^4*b*d)","A",1
1317,1,115,148,1.0430809,"\int \frac{\cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^5/(a + b*Sin[c + d*x]),x]","\frac{-3 a^4 \csc ^4(c+d x)+4 a^3 b \csc ^3(c+d x)+6 a^2 \left(2 a^2-b^2\right) \csc ^2(c+d x)+12 a b \left(b^2-2 a^2\right) \csc (c+d x)+12 \left(a^2-b^2\right)^2 (\log (\sin (c+d x))-\log (a+b \sin (c+d x)))}{12 a^5 d}","\frac{b \csc ^3(c+d x)}{3 a^2 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 \left(2 a^2-b^2\right) \csc (c+d x)}{a^4 d}+\frac{\left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 a^3 d}-\frac{\csc ^4(c+d x)}{4 a d}",1,"(12*a*b*(-2*a^2 + b^2)*Csc[c + d*x] + 6*a^2*(2*a^2 - b^2)*Csc[c + d*x]^2 + 4*a^3*b*Csc[c + d*x]^3 - 3*a^4*Csc[c + d*x]^4 + 12*(a^2 - b^2)^2*(Log[Sin[c + d*x]] - Log[a + b*Sin[c + d*x]]))/(12*a^5*d)","A",1
1318,1,179,179,6.1211419,"\int \frac{\cot ^5(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^5*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{b \csc ^4(c+d x)}{4 a^2 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{\left(a^2-b^2\right)^2 \csc (c+d x)}{a^5 d}-\frac{b \left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 a^4 d}+\frac{\left(2 a^2-b^2\right) \csc ^3(c+d x)}{3 a^3 d}-\frac{\csc ^5(c+d x)}{5 a d}","\frac{b \csc ^4(c+d x)}{4 a^2 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{\left(a^2-b^2\right)^2 \csc (c+d x)}{a^5 d}-\frac{b \left(2 a^2-b^2\right) \csc ^2(c+d x)}{2 a^4 d}+\frac{\left(2 a^2-b^2\right) \csc ^3(c+d x)}{3 a^3 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",1
1319,1,165,212,2.8600139,"\int \frac{\cot ^5(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^5*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{-10 a^6 \csc ^6(c+d x)+12 a^5 b \csc ^5(c+d x)+60 \left(b^3-a^2 b\right)^2 (\log (\sin (c+d x))-\log (a+b \sin (c+d x)))-30 a^2 \left(a^2-b^2\right)^2 \csc ^2(c+d x)+60 a b \left(a^2-b^2\right)^2 \csc (c+d x)+15 a^4 \left(2 a^2-b^2\right) \csc ^4(c+d x)+20 a^3 b \left(b^2-2 a^2\right) \csc ^3(c+d x)}{60 a^7 d}","\frac{b \csc ^5(c+d x)}{5 a^2 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 \left(a^2-b^2\right)^2 \csc (c+d x)}{a^6 d}-\frac{\left(a^2-b^2\right)^2 \csc ^2(c+d x)}{2 a^5 d}-\frac{b \left(2 a^2-b^2\right) \csc ^3(c+d x)}{3 a^4 d}+\frac{\left(2 a^2-b^2\right) \csc ^4(c+d x)}{4 a^3 d}-\frac{\csc ^6(c+d x)}{6 a d}",1,"(60*a*b*(a^2 - b^2)^2*Csc[c + d*x] - 30*a^2*(a^2 - b^2)^2*Csc[c + d*x]^2 + 20*a^3*b*(-2*a^2 + b^2)*Csc[c + d*x]^3 + 15*a^4*(2*a^2 - b^2)*Csc[c + d*x]^4 + 12*a^5*b*Csc[c + d*x]^5 - 10*a^6*Csc[c + d*x]^6 + 60*(-(a^2*b) + b^3)^2*(Log[Sin[c + d*x]] - Log[a + b*Sin[c + d*x]]))/(60*a^7*d)","A",1
1320,1,403,467,3.3177687,"\int \frac{\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{-107520 a^8 c-107520 a^8 d x+26880 a^6 b^2 \sin (2 (c+d x))+268800 a^6 b^2 c+268800 a^6 b^2 d x-53760 a^4 b^4 \sin (2 (c+d x))-3360 a^4 b^4 \sin (4 (c+d x))-201600 a^4 b^4 c-201600 a^4 b^4 d x-1344 a^3 b^5 \cos (5 (c+d x))+25200 a^2 b^6 \sin (2 (c+d x))+5040 a^2 b^6 \sin (4 (c+d x))+560 a^2 b^6 \sin (6 (c+d x))+33600 a^2 b^6 c+33600 a^2 b^6 d x+560 \left(16 a^5 b^3-28 a^3 b^5+9 a b^7\right) \cos (3 (c+d x))+215040 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)-1680 a b \left(64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right) \cos (c+d x)+1680 a b^7 \cos (5 (c+d x))+240 a b^7 \cos (7 (c+d x))+1680 b^8 \sin (2 (c+d x))-840 b^8 \sin (4 (c+d x))-560 b^8 \sin (6 (c+d x))-105 b^8 \sin (8 (c+d x))+4200 b^8 c+4200 b^8 d x}{107520 b^9 d}","-\frac{b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac{\left(28 a^4-60 a^2 b^2+35 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{140 a b^4 d}-\frac{a \left(35 a^4-77 a^2 b^2+45 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{105 b^6 d}+\frac{\left(48 a^4-104 a^2 b^2+59 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{192 b^5 d}+\frac{\left(40 a^4-85 a^2 b^2+48 b^4\right) \sin ^5(c+d x) \cos (c+d x)}{240 a^2 b^3 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{a \left(105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right) \cos (c+d x)}{105 b^8 d}+\frac{\left(64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right) \sin (c+d x) \cos (c+d x)}{128 b^7 d}-\frac{x \left(128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right)}{128 b^9}-\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,"(-107520*a^8*c + 268800*a^6*b^2*c - 201600*a^4*b^4*c + 33600*a^2*b^6*c + 4200*b^8*c - 107520*a^8*d*x + 268800*a^6*b^2*d*x - 201600*a^4*b^4*d*x + 33600*a^2*b^6*d*x + 4200*b^8*d*x + 215040*a^3*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 1680*a*b*(64*a^6 - 144*a^4*b^2 + 88*a^2*b^4 - 5*b^6)*Cos[c + d*x] + 560*(16*a^5*b^3 - 28*a^3*b^5 + 9*a*b^7)*Cos[3*(c + d*x)] - 1344*a^3*b^5*Cos[5*(c + d*x)] + 1680*a*b^7*Cos[5*(c + d*x)] + 240*a*b^7*Cos[7*(c + d*x)] + 26880*a^6*b^2*Sin[2*(c + d*x)] - 53760*a^4*b^4*Sin[2*(c + d*x)] + 25200*a^2*b^6*Sin[2*(c + d*x)] + 1680*b^8*Sin[2*(c + d*x)] - 3360*a^4*b^4*Sin[4*(c + d*x)] + 5040*a^2*b^6*Sin[4*(c + d*x)] - 840*b^8*Sin[4*(c + d*x)] + 560*a^2*b^6*Sin[6*(c + d*x)] - 560*b^8*Sin[6*(c + d*x)] - 105*b^8*Sin[8*(c + d*x)])/(107520*b^9*d)","A",1
1321,1,324,408,3.0689121,"\int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{-6720 a^7 c-6720 a^7 d x+1680 a^5 b^2 \sin (2 (c+d x))+16800 a^5 b^2 c+16800 a^5 b^2 d x-3360 a^3 b^4 \sin (2 (c+d x))-210 a^3 b^4 \sin (4 (c+d x))-12600 a^3 b^4 c-12600 a^3 b^4 d x-84 a^2 b^5 \cos (5 (c+d x))+13440 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)+35 \left(16 a^4 b^3-28 a^2 b^5+9 b^7\right) \cos (3 (c+d x))+105 b \left(-64 a^6+144 a^4 b^2-88 a^2 b^4+5 b^6\right) \cos (c+d x)+1575 a b^6 \sin (2 (c+d x))+315 a b^6 \sin (4 (c+d x))+35 a b^6 \sin (6 (c+d x))+2100 a b^6 c+2100 a b^6 d x+105 b^7 \cos (5 (c+d x))+15 b^7 \cos (7 (c+d x))}{6720 b^8 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{b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac{\left(6 a^4-13 a^2 b^2+8 b^4\right) \sin ^3(c+d x) \cos (c+d x)}{24 a b^4 d}-\frac{a \left(8 a^4-18 a^2 b^2+11 b^4\right) \sin (c+d x) \cos (c+d x)}{16 b^6 d}+\frac{\left(35 a^4-77 a^2 b^2+45 b^4\right) \sin ^2(c+d x) \cos (c+d x)}{105 b^5 d}+\frac{\left(28 a^4-60 a^2 b^2+35 b^4\right) \sin ^4(c+d x) \cos (c+d x)}{140 a^2 b^3 d}+\frac{a x \left(16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right)}{16 b^8}+\frac{\left(105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right) \cos (c+d x)}{105 b^7 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,"-1/6720*(-6720*a^7*c + 16800*a^5*b^2*c - 12600*a^3*b^4*c + 2100*a*b^6*c - 6720*a^7*d*x + 16800*a^5*b^2*d*x - 12600*a^3*b^4*d*x + 2100*a*b^6*d*x + 13440*a^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 105*b*(-64*a^6 + 144*a^4*b^2 - 88*a^2*b^4 + 5*b^6)*Cos[c + d*x] + 35*(16*a^4*b^3 - 28*a^2*b^5 + 9*b^7)*Cos[3*(c + d*x)] - 84*a^2*b^5*Cos[5*(c + d*x)] + 105*b^7*Cos[5*(c + d*x)] + 15*b^7*Cos[7*(c + d*x)] + 1680*a^5*b^2*Sin[2*(c + d*x)] - 3360*a^3*b^4*Sin[2*(c + d*x)] + 1575*a*b^6*Sin[2*(c + d*x)] - 210*a^3*b^4*Sin[4*(c + d*x)] + 315*a*b^6*Sin[4*(c + d*x)] + 35*a*b^6*Sin[6*(c + d*x)])/(b^8*d)","A",1
1322,1,275,228,2.2571713,"\int \frac{\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{-960 a^6 c-960 a^6 d x+240 a^4 b^2 \sin (2 (c+d x))+2400 a^4 b^2 c+2400 a^4 b^2 d x+20 \left(4 a^3 b^3-7 a b^5\right) \cos (3 (c+d x))-480 a^2 b^4 \sin (2 (c+d x))-30 a^2 b^4 \sin (4 (c+d x))-1800 a^2 b^4 c-1800 a^2 b^4 d x+1920 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)-120 a b \left(8 a^4-18 a^2 b^2+11 b^4\right) \cos (c+d x)-12 a b^5 \cos (5 (c+d x))+225 b^6 \sin (2 (c+d x))+45 b^6 \sin (4 (c+d x))+5 b^6 \sin (6 (c+d x))+300 b^6 c+300 b^6 d x}{960 b^7 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(8 a^4-14 a^2 b^2+5 b^4\right) \sin (c+d x)\right)}{16 b^6 d}-\frac{x \left(16 a^6-40 a^4 b^2+30 a^2 b^4-5 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,"(-960*a^6*c + 2400*a^4*b^2*c - 1800*a^2*b^4*c + 300*b^6*c - 960*a^6*d*x + 2400*a^4*b^2*d*x - 1800*a^2*b^4*d*x + 300*b^6*d*x + 1920*a*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 120*a*b*(8*a^4 - 18*a^2*b^2 + 11*b^4)*Cos[c + d*x] + 20*(4*a^3*b^3 - 7*a*b^5)*Cos[3*(c + d*x)] - 12*a*b^5*Cos[5*(c + d*x)] + 240*a^4*b^2*Sin[2*(c + d*x)] - 480*a^2*b^4*Sin[2*(c + d*x)] + 225*b^6*Sin[2*(c + d*x)] - 30*a^2*b^4*Sin[4*(c + d*x)] + 45*b^6*Sin[4*(c + d*x)] + 5*b^6*Sin[6*(c + d*x)])/(960*b^7*d)","A",1
1323,1,220,252,0.513146,"\int \frac{\cos ^5(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{96 a^5 c+96 a^5 d x-24 a^3 b^2 \sin (2 (c+d x))-240 a^3 b^2 c-240 a^3 b^2 d x-8 a^2 b^3 \cos (3 (c+d x))+24 a^2 b \left(4 a^2-9 b^2\right) \cos (c+d x)-192 \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)+48 a b^4 \sin (2 (c+d x))+3 a b^4 \sin (4 (c+d x))+180 a b^4 c+180 a b^4 d x-96 b^5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+96 b^5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{96 a b^5 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{a \left(a^2-3 b^2\right) \cos (c+d x)}{b^4 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(a^4-3 a^2 b^2+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,"-1/96*(96*a^5*c - 240*a^3*b^2*c + 180*a*b^4*c + 96*a^5*d*x - 240*a^3*b^2*d*x + 180*a*b^4*d*x - 192*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 24*a^2*b*(4*a^2 - 9*b^2)*Cos[c + d*x] - 8*a^2*b^3*Cos[3*(c + d*x)] + 96*b^5*Log[Cos[(c + d*x)/2]] - 96*b^5*Log[Sin[(c + d*x)/2]] - 24*a^3*b^2*Sin[2*(c + d*x)] + 48*a*b^4*Sin[2*(c + d*x)] + 3*a*b^4*Sin[4*(c + d*x)])/(a*b^5*d)","A",1
1324,1,208,183,1.4054062,"\int \frac{\cos ^4(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{-12 a^5 c-12 a^5 d x+3 a^3 b^2 \sin (2 (c+d x))+30 a^3 b^2 c+30 a^3 b^2 d x+a^2 b^3 \cos (3 (c+d x))-3 a^2 b \left(4 a^2-9 b^2\right) \cos (c+d x)+24 \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)-6 a b^4 \tan \left(\frac{1}{2} (c+d x)\right)+6 a b^4 \cot \left(\frac{1}{2} (c+d x)\right)+12 b^5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-12 b^5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{12 a^2 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^2 b^4 d}+\frac{a x \left(a^2-3 b^2\right)}{b^4}+\frac{\left(a^2-3 b^2\right) \cos (c+d x)}{b^3 d}+\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,"-1/12*(-12*a^5*c + 30*a^3*b^2*c - 12*a^5*d*x + 30*a^3*b^2*d*x + 24*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 3*a^2*b*(4*a^2 - 9*b^2)*Cos[c + d*x] + a^2*b^3*Cos[3*(c + d*x)] + 6*a*b^4*Cot[(c + d*x)/2] - 12*b^5*Log[Cos[(c + d*x)/2]] + 12*b^5*Log[Sin[(c + d*x)/2]] + 3*a^3*b^2*Sin[2*(c + d*x)] - 6*a*b^4*Tan[(c + d*x)/2])/(a^2*b^4*d)","A",1
1325,1,259,174,5.2635428,"\int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*Cot[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{-8 a^5 c-8 a^5 d x-8 a^4 b \cos (c+d x)+2 a^3 b^2 \sin (2 (c+d x))+20 a^3 b^2 c+20 a^3 b^2 d x-a^2 b^3 \csc ^2\left(\frac{1}{2} (c+d x)\right)+a^2 b^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)-20 a^2 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+20 a^2 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+16 \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)-4 a b^4 \tan \left(\frac{1}{2} (c+d x)\right)+4 a b^4 \cot \left(\frac{1}{2} (c+d x)\right)+8 b^5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 b^5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^3 b^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{\left(5 a^2-2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^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^3 b^3 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,"(-8*a^5*c + 20*a^3*b^2*c - 8*a^5*d*x + 20*a^3*b^2*d*x + 16*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 8*a^4*b*Cos[c + d*x] + 4*a*b^4*Cot[(c + d*x)/2] - a^2*b^3*Csc[(c + d*x)/2]^2 + 20*a^2*b^3*Log[Cos[(c + d*x)/2]] - 8*b^5*Log[Cos[(c + d*x)/2]] - 20*a^2*b^3*Log[Sin[(c + d*x)/2]] + 8*b^5*Log[Sin[(c + d*x)/2]] + a^2*b^3*Sec[(c + d*x)/2]^2 + 2*a^3*b^2*Sin[2*(c + d*x)] - 4*a*b^4*Tan[(c + d*x)/2])/(8*a^3*b^3*d)","A",1
1326,1,379,197,6.1977169,"\int \frac{\cos ^2(c+d x) \cot ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","\frac{b \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}-\frac{b \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}+\frac{\left(5 a^2 b-2 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}+\frac{\left(2 b^3-5 a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^4 b^2 d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(7 a^2 \cos \left(\frac{1}{2} (c+d x)\right)-3 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^3 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(3 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-7 a^2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^3 d}+\frac{a (c+d x)}{b^2 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a d}+\frac{\cos (c+d x)}{b 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{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{b \left(3 a^2-b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{\left(3 a^2-b^2\right) \cot (c+d x)}{a^3 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*(c + d*x))/(b^2*d) - (2*(a^2 - b^2)^(5/2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^4*b^2*d) + Cos[c + d*x]/(b*d) + ((7*a^2*Cos[(c + d*x)/2] - 3*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^3*d) + (b*Csc[(c + d*x)/2]^2)/(8*a^2*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a*d) + ((-5*a^2*b + 2*b^3)*Log[Cos[(c + d*x)/2]])/(2*a^4*d) + ((5*a^2*b - 2*b^3)*Log[Sin[(c + d*x)/2]])/(2*a^4*d) - (b*Sec[(c + d*x)/2]^2)/(8*a^2*d) + (Sec[(c + d*x)/2]*(-7*a^2*Sin[(c + d*x)/2] + 3*b^2*Sin[(c + d*x)/2]))/(6*a^3*d) + (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a*d)","A",0
1327,1,448,195,6.191174,"\int \frac{\cos (c+d x) \cot ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","\frac{b \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}-\frac{b \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{24 a^2 d}+\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^5 b d}+\frac{\csc \left(\frac{1}{2} (c+d x)\right) \left(3 b^3 \cos \left(\frac{1}{2} (c+d x)\right)-7 a^2 b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(7 a^2 b \sin \left(\frac{1}{2} (c+d x)\right)-3 b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}+\frac{\left(9 a^2-4 b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 a^3 d}+\frac{\left(4 b^2-9 a^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 a^3 d}+\frac{\left(15 a^4-20 a^2 b^2+8 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^5 d}+\frac{\left(-15 a^4+20 a^2 b^2-8 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^5 d}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 a d}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 a d}-\frac{c+d x}{b d}","\frac{b \cot ^3(c+d x)}{3 a^2 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^5 b d}+\frac{b \left(b^2-2 a^2\right) \cot (c+d x)}{a^4 d}+\frac{\left(7 a^2-4 b^2\right) \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac{\left(15 a^4-20 a^2 b^2+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac{\cot ^3(c+d x) \csc (c+d x)}{4 a d}-\frac{x}{b}",1,"-((c + d*x)/(b*d)) + (2*(a^2 - b^2)^(5/2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*b*d) + ((-7*a^2*b*Cos[(c + d*x)/2] + 3*b^3*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + ((9*a^2 - 4*b^2)*Csc[(c + d*x)/2]^2)/(32*a^3*d) + (b*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^2*d) - Csc[(c + d*x)/2]^4/(64*a*d) + ((-15*a^4 + 20*a^2*b^2 - 8*b^4)*Log[Cos[(c + d*x)/2]])/(8*a^5*d) + ((15*a^4 - 20*a^2*b^2 + 8*b^4)*Log[Sin[(c + d*x)/2]])/(8*a^5*d) + ((-9*a^2 + 4*b^2)*Sec[(c + d*x)/2]^2)/(32*a^3*d) + Sec[(c + d*x)/2]^4/(64*a*d) + (Sec[(c + d*x)/2]*(7*a^2*b*Sin[(c + d*x)/2] - 3*b^3*Sin[(c + d*x)/2]))/(6*a^4*d) - (b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(24*a^2*d)","B",0
1328,1,504,241,1.3944464,"\int \frac{\cot ^6(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cot[c + d*x]^6/(a + b*Sin[c + d*x]),x]","\frac{736 a^5 \tan \left(\frac{1}{2} (c+d x)\right)-3 a^5 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)+41 a^5 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-656 a^5 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+6 a^5 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right)+15 a^4 b \csc ^4\left(\frac{1}{2} (c+d x)\right)-270 a^4 b \csc ^2\left(\frac{1}{2} (c+d x)\right)-15 a^4 b \sec ^4\left(\frac{1}{2} (c+d x)\right)+270 a^4 b \sec ^2\left(\frac{1}{2} (c+d x)\right)-1800 a^4 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+1800 a^4 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-1120 a^3 b^2 \tan \left(\frac{1}{2} (c+d x)\right)-20 a^3 b^2 \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)+320 a^3 b^2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+120 a^2 b^3 \csc ^2\left(\frac{1}{2} (c+d x)\right)-120 a^2 b^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)+2400 a^2 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2400 a^2 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-1920 \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)-32 \left(23 a^5-35 a^3 b^2+15 a b^4\right) \cot \left(\frac{1}{2} (c+d x)\right)+480 a b^4 \tan \left(\frac{1}{2} (c+d x)\right)-960 b^5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+960 b^5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{960 a^6 d}","\frac{b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 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{b \left(4 b^2-9 a^2\right) \cot (c+d x) \csc (c+d x)}{8 a^4 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(15 a^4-20 a^2 b^2+8 b^4\right) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac{\left(23 a^4-35 a^2 b^2+15 b^4\right) \cot (c+d x)}{15 a^5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d}",1,"(-1920*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 32*(23*a^5 - 35*a^3*b^2 + 15*a*b^4)*Cot[(c + d*x)/2] - 270*a^4*b*Csc[(c + d*x)/2]^2 + 120*a^2*b^3*Csc[(c + d*x)/2]^2 + 15*a^4*b*Csc[(c + d*x)/2]^4 + 1800*a^4*b*Log[Cos[(c + d*x)/2]] - 2400*a^2*b^3*Log[Cos[(c + d*x)/2]] + 960*b^5*Log[Cos[(c + d*x)/2]] - 1800*a^4*b*Log[Sin[(c + d*x)/2]] + 2400*a^2*b^3*Log[Sin[(c + d*x)/2]] - 960*b^5*Log[Sin[(c + d*x)/2]] + 270*a^4*b*Sec[(c + d*x)/2]^2 - 120*a^2*b^3*Sec[(c + d*x)/2]^2 - 15*a^4*b*Sec[(c + d*x)/2]^4 - 656*a^5*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 320*a^3*b^2*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 41*a^5*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 20*a^3*b^2*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 3*a^5*Csc[(c + d*x)/2]^6*Sin[c + d*x] + 736*a^5*Tan[(c + d*x)/2] - 1120*a^3*b^2*Tan[(c + d*x)/2] + 480*a*b^4*Tan[(c + d*x)/2] + 6*a^5*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(960*a^6*d)","B",1
1329,1,356,363,1.5177927,"\int \frac{\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^6*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{7680 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)+240 \left(-5 a^6+30 a^4 b^2-40 a^2 b^4+16 b^6\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+240 \left(5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 a \cot (c+d x) \csc ^5(c+d x) \left(-295 a^5+1168 a^4 b \sin (c+d x)-568 a^4 b \sin (3 (c+d x))+184 a^4 b \sin (5 (c+d x))+570 a^3 b^2-2320 a^2 b^3 \sin (c+d x)+1240 a^2 b^3 \sin (3 (c+d x))-280 a^2 b^3 \sin (5 (c+d x))+20 \left(7 a^5-42 a^3 b^2+24 a b^4\right) \cos (2 (c+d x))-15 \left(11 a^5-18 a^3 b^2+8 a b^4\right) \cos (4 (c+d x))-360 a b^4+1200 b^5 \sin (c+d x)-600 b^5 \sin (3 (c+d x))+120 b^5 \sin (5 (c+d x))\right)}{3840 a^7 d}","\frac{b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 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{\left(15 a^4-22 a^2 b^2+10 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac{b \left(23 a^4-35 a^2 b^2+15 b^4\right) \cot (c+d x)}{15 a^6 d}-\frac{\left(11 a^4-18 a^2 b^2+8 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac{\left(8 a^4-13 a^2 b^2+6 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac{\left(5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^7 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,"(7680*b*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 240*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*Log[Cos[(c + d*x)/2]] + 240*(-5*a^6 + 30*a^4*b^2 - 40*a^2*b^4 + 16*b^6)*Log[Sin[(c + d*x)/2]] + 2*a*Cot[c + d*x]*Csc[c + d*x]^5*(-295*a^5 + 570*a^3*b^2 - 360*a*b^4 + 20*(7*a^5 - 42*a^3*b^2 + 24*a*b^4)*Cos[2*(c + d*x)] - 15*(11*a^5 - 18*a^3*b^2 + 8*a*b^4)*Cos[4*(c + d*x)] + 1168*a^4*b*Sin[c + d*x] - 2320*a^2*b^3*Sin[c + d*x] + 1200*b^5*Sin[c + d*x] - 568*a^4*b*Sin[3*(c + d*x)] + 1240*a^2*b^3*Sin[3*(c + d*x)] - 600*b^5*Sin[3*(c + d*x)] + 184*a^4*b*Sin[5*(c + d*x)] - 280*a^2*b^3*Sin[5*(c + d*x)] + 120*b^5*Sin[5*(c + d*x)]))/(3840*a^7*d)","A",1
1330,1,442,417,1.9899646,"\int \frac{\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^6*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{-107520 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)+3360 \left(-5 a^6 b+30 a^4 b^3-40 a^2 b^5+16 b^7\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3360 b \left(5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 a \cot (c+d x) \csc ^6(c+d x) \left(120 a^6 \cos (6 (c+d x))+1200 a^6-5110 a^5 b \sin (c+d x)+2135 a^5 b \sin (3 (c+d x))-1155 a^5 b \sin (5 (c+d x))-1288 a^4 b^2 \cos (6 (c+d x))+8176 a^4 b^2+13860 a^3 b^3 \sin (c+d x)-7770 a^3 b^3 \sin (3 (c+d x))+1890 a^3 b^3 \sin (5 (c+d x))+1960 a^2 b^4 \cos (6 (c+d x))-16240 a^2 b^4+8 \left(225 a^6-1519 a^4 b^2+3115 a^2 b^4-1575 b^6\right) \cos (2 (c+d x))+16 \left(45 a^6+329 a^4 b^2-665 a^2 b^4+315 b^6\right) \cos (4 (c+d x))-8400 a b^5 \sin (c+d x)+4200 a b^5 \sin (3 (c+d x))-840 a b^5 \sin (5 (c+d x))-840 b^6 \cos (6 (c+d x))+8400 b^6\right)}{53760 a^8 d}","\frac{b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 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(8 a^4-13 a^2 b^2+6 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac{b \left(11 a^4-18 a^2 b^2+8 b^4\right) \cot (c+d x) \csc (c+d x)}{16 a^6 d}-\frac{\left(45 a^4-77 a^2 b^2+35 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}-\frac{\left(35 a^4-60 a^2 b^2+28 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}-\frac{b \left(5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right) \tanh ^{-1}(\cos (c+d x))}{16 a^8 d}+\frac{\left(15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right) \cot (c+d x)}{105 a^7 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,"(-107520*b^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 3360*(-5*a^6*b + 30*a^4*b^3 - 40*a^2*b^5 + 16*b^7)*Log[Cos[(c + d*x)/2]] + 3360*b*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*Log[Sin[(c + d*x)/2]] - 2*a*Cot[c + d*x]*Csc[c + d*x]^6*(1200*a^6 + 8176*a^4*b^2 - 16240*a^2*b^4 + 8400*b^6 + 8*(225*a^6 - 1519*a^4*b^2 + 3115*a^2*b^4 - 1575*b^6)*Cos[2*(c + d*x)] + 16*(45*a^6 + 329*a^4*b^2 - 665*a^2*b^4 + 315*b^6)*Cos[4*(c + d*x)] + 120*a^6*Cos[6*(c + d*x)] - 1288*a^4*b^2*Cos[6*(c + d*x)] + 1960*a^2*b^4*Cos[6*(c + d*x)] - 840*b^6*Cos[6*(c + d*x)] - 5110*a^5*b*Sin[c + d*x] + 13860*a^3*b^3*Sin[c + d*x] - 8400*a*b^5*Sin[c + d*x] + 2135*a^5*b*Sin[3*(c + d*x)] - 7770*a^3*b^3*Sin[3*(c + d*x)] + 4200*a*b^5*Sin[3*(c + d*x)] - 1155*a^5*b*Sin[5*(c + d*x)] + 1890*a^3*b^3*Sin[5*(c + d*x)] - 840*a*b^5*Sin[5*(c + d*x)]))/(53760*a^8*d)","A",1
1331,1,593,476,3.5867205,"\int \frac{\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Cot[c + d*x]^6*Csc[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{1720320 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)-6720 \left(5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6720 \left(5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+a \csc ^8(c+d x) \left(-13895 a^7 \cos (5 (c+d x))-525 a^7 \cos (7 (c+d x))+13440 a^6 b \sin (2 (c+d x))+13440 a^6 b \sin (4 (c+d x))+5760 a^6 b \sin (6 (c+d x))+960 a^6 b \sin (8 (c+d x))-17080 a^5 b^2 \cos (5 (c+d x))+9240 a^5 b^2 \cos (7 (c+d x))+88704 a^4 b^3 \sin (2 (c+d x))-86912 a^4 b^3 \sin (4 (c+d x))+42112 a^4 b^3 \sin (6 (c+d x))-10304 a^4 b^3 \sin (8 (c+d x))+62160 a^3 b^4 \cos (5 (c+d x))-15120 a^3 b^4 \cos (7 (c+d x))-174720 a^2 b^5 \sin (2 (c+d x))+183680 a^2 b^5 \sin (4 (c+d x))-85120 a^2 b^5 \sin (6 (c+d x))+15680 a^2 b^5 \sin (8 (c+d x))-35 \left(895 a^7-904 a^5 b^2+2736 a^3 b^4-1728 a b^6\right) \cos (3 (c+d x))-35 a \left(1765 a^6+680 a^4 b^2-1392 a^2 b^4+960 b^6\right) \cos (c+d x)-33600 a b^6 \cos (5 (c+d x))+6720 a b^6 \cos (7 (c+d x))+94080 b^7 \sin (2 (c+d x))-94080 b^7 \sin (4 (c+d x))+40320 b^7 \sin (6 (c+d x))-6720 b^7 \sin (8 (c+d x))\right)}{860160 a^9 d}","\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 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{\left(35 a^4-60 a^2 b^2+28 b^4\right) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac{b \left(45 a^4-77 a^2 b^2+35 b^4\right) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac{\left(59 a^4-104 a^2 b^2+48 b^4\right) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac{\left(48 a^4-85 a^2 b^2+40 b^4\right) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}-\frac{b \left(15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right) \cot (c+d x)}{105 a^8 d}+\frac{\left(5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac{\left(5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right) \tanh ^{-1}(\cos (c+d x))}{128 a^9 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,"(1720320*b^3*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 6720*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*Log[Cos[(c + d*x)/2]] - 6720*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*Log[Sin[(c + d*x)/2]] + a*Csc[c + d*x]^8*(-35*a*(1765*a^6 + 680*a^4*b^2 - 1392*a^2*b^4 + 960*b^6)*Cos[c + d*x] - 35*(895*a^7 - 904*a^5*b^2 + 2736*a^3*b^4 - 1728*a*b^6)*Cos[3*(c + d*x)] - 13895*a^7*Cos[5*(c + d*x)] - 17080*a^5*b^2*Cos[5*(c + d*x)] + 62160*a^3*b^4*Cos[5*(c + d*x)] - 33600*a*b^6*Cos[5*(c + d*x)] - 525*a^7*Cos[7*(c + d*x)] + 9240*a^5*b^2*Cos[7*(c + d*x)] - 15120*a^3*b^4*Cos[7*(c + d*x)] + 6720*a*b^6*Cos[7*(c + d*x)] + 13440*a^6*b*Sin[2*(c + d*x)] + 88704*a^4*b^3*Sin[2*(c + d*x)] - 174720*a^2*b^5*Sin[2*(c + d*x)] + 94080*b^7*Sin[2*(c + d*x)] + 13440*a^6*b*Sin[4*(c + d*x)] - 86912*a^4*b^3*Sin[4*(c + d*x)] + 183680*a^2*b^5*Sin[4*(c + d*x)] - 94080*b^7*Sin[4*(c + d*x)] + 5760*a^6*b*Sin[6*(c + d*x)] + 42112*a^4*b^3*Sin[6*(c + d*x)] - 85120*a^2*b^5*Sin[6*(c + d*x)] + 40320*b^7*Sin[6*(c + d*x)] + 960*a^6*b*Sin[8*(c + d*x)] - 10304*a^4*b^3*Sin[8*(c + d*x)] + 15680*a^2*b^5*Sin[8*(c + d*x)] - 6720*b^7*Sin[8*(c + d*x)]))/(860160*a^9*d)","A",1
1332,1,83,93,0.2081487,"\int \frac{\sin ^2(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{-\frac{2 a^3 \log (a+b \sin (c+d x))}{b^2 \left(a^2-b^2\right)}+\frac{\log (1-\sin (c+d x))}{a+b}+\frac{\log (\sin (c+d x)+1)}{a-b}+\frac{2 \sin (c+d x)}{b}}{2 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,"-1/2*(Log[1 - Sin[c + d*x]]/(a + b) + Log[1 + Sin[c + d*x]]/(a - b) - (2*a^3*Log[a + b*Sin[c + d*x]])/(b^2*(a^2 - b^2)) + (2*Sin[c + d*x])/b)/d","A",1
1333,1,72,80,0.0659948,"\int \frac{\sin (c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{-2 a^2 \log (a+b \sin (c+d x))-b (a-b) \log (1-\sin (c+d x))+b (a+b) \log (\sin (c+d x)+1)}{2 b d (a-b) (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,"(-((a - b)*b*Log[1 - Sin[c + d*x]]) + b*(a + b)*Log[1 + Sin[c + d*x]] - 2*a^2*Log[a + b*Sin[c + d*x]])/(2*(a - b)*b*(a + b)*d)","A",1
1334,1,87,74,0.0810212,"\int \frac{\tan (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{a \log (a+b \sin (c+d x))+(b-a) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-(a+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d (a-b) (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,"((-a + b)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - (a + b)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a*Log[a + b*Sin[c + d*x]])/((a - b)*(a + b)*d)","A",1
1335,1,84,93,0.1099359,"\int \frac{\csc (c+d x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{-\frac{2 b^2 \log (a+b \sin (c+d x))}{a \left(a^2-b^2\right)}+\frac{\log (1-\sin (c+d x))}{a+b}+\frac{\log (\sin (c+d x)+1)}{a-b}-\frac{2 \log (\sin (c+d x))}{a}}{2 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,"-1/2*(Log[1 - Sin[c + d*x]]/(a + b) - (2*Log[Sin[c + d*x]])/a + Log[1 + Sin[c + d*x]]/(a - b) - (2*b^2*Log[a + b*Sin[c + d*x]])/(a*(a^2 - b^2)))/d","A",1
1336,1,97,110,0.2434743,"\int \frac{\csc ^2(c+d x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\frac{2 b^3 \log (a+b \sin (c+d x))}{a^2 \left(b^2-a^2\right)}-\frac{2 b \log (\sin (c+d x))}{a^2}-\frac{\log (1-\sin (c+d x))}{a+b}+\frac{\log (\sin (c+d x)+1)}{a-b}-\frac{2 \csc (c+d x)}{a}}{2 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,"((-2*Csc[c + d*x])/a - Log[1 - Sin[c + d*x]]/(a + b) - (2*b*Log[Sin[c + d*x]])/a^2 + Log[1 + Sin[c + d*x]]/(a - b) + (2*b^3*Log[a + b*Sin[c + d*x]])/(a^2*(-a^2 + b^2)))/(2*d)","A",1
1337,1,132,132,0.5323603,"\int \frac{\csc ^3(c+d x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^3*Sec[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{b^4 \left(\frac{\csc (c+d x)}{a^2 b^3}+\frac{\log (a+b \sin (c+d x))}{a^3 \left(a^2-b^2\right)}+\frac{\left(a^2+b^2\right) \log (\sin (c+d x))}{a^3 b^4}-\frac{\csc ^2(c+d x)}{2 a b^4}-\frac{\log (1-\sin (c+d x))}{2 b^4 (a+b)}-\frac{\log (\sin (c+d x)+1)}{2 b^4 (a-b)}\right)}{d}","\frac{b \csc (c+d x)}{a^2 d}+\frac{\left(a^2+b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{b^4 \log (a+b \sin (c+d x))}{a^3 d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}-\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}-\frac{\csc ^2(c+d x)}{2 a d}",1,"(b^4*(Csc[c + d*x]/(a^2*b^3) - Csc[c + d*x]^2/(2*a*b^4) - Log[1 - Sin[c + d*x]]/(2*b^4*(a + b)) + ((a^2 + b^2)*Log[Sin[c + d*x]])/(a^3*b^4) - Log[1 + Sin[c + d*x]]/(2*(a - b)*b^4) + Log[a + b*Sin[c + d*x]]/(a^3*(a^2 - b^2))))/d","A",1
1338,1,221,268,1.5319425,"\int \frac{\sin ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^3*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\frac{8 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 \left(a^2-b^2\right)^{3/2}}+\frac{4 a^4 (c+d x)+2 a^2 b^2 (c+d x)-4 a b^3-6 b^4 (c+d x)}{b^5-a^2 b^3}-\frac{4 a \cos (c+d x)}{b^2}+\frac{4 \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{4 \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\sin (2 (c+d x))}{b}}{4 d}","\frac{a \cos (c+d x)}{d \left(a^2-b^2\right)}-\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{3 b x}{2 \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{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{a^3 \cos (c+d x)}{b^2 d \left(a^2-b^2\right)}",1,"((-4*a*b^3 + 4*a^4*(c + d*x) + 2*a^2*b^2*(c + d*x) - 6*b^4*(c + d*x))/(-(a^2*b^3) + b^5) + (8*a^5*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)) - (4*a*Cos[c + d*x])/b^2 + (4*Sin[(c + d*x)/2])/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (4*Sin[(c + d*x)/2])/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + Sin[2*(c + d*x)]/b)/(4*d)","A",1
1339,1,186,183,1.1353926,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{-\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 \left(a^2-b^2\right)^{3/2}}+\frac{-\left(a^3 (c+d x)\right)+a b^2 (c+d x)+b^3}{b^4-a^2 b^2}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\cos (c+d x)}{b}}{d}","\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{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 x}{a^2-b^2}-\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^3 x}{b^2 \left(a^2-b^2\right)}",1,"((b^3 - a^3*(c + d*x) + a*b^2*(c + d*x))/(-(a^2*b^2) + b^4) - (2*a^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)) + Cos[c + d*x]/b + Sin[(c + d*x)/2]/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + Sin[(c + d*x)/2]/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/d","A",1
1340,1,152,133,0.8356278,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\frac{b (a-b \sin (c+d x))-\left(a^2-b^2\right) (c+d x) \cos (c+d x)}{(a-b) (a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 a^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}}{b d}","-\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}}",1,"((2*a^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (-((a^2 - b^2)*(c + d*x)*Cos[c + d*x]) + b*(a - b*Sin[c + d*x]))/((a - b)*(a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(b*d)","A",1
1341,1,152,96,0.1880969,"\int \frac{\tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Sin[c + d*x]),x]","\frac{\sqrt{a^2-b^2} (a \sin (c+d x)+b \cos (c+d x)-b)-2 a^2 \cos (c+d x) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d (a-b) (a+b) \sqrt{a^2-b^2} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 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]]*Cos[c + d*x] + Sqrt[a^2 - b^2]*(-b + b*Cos[c + d*x] + a*Sin[c + d*x]))/((a - b)*(a + b)*Sqrt[a^2 - b^2]*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
1342,1,151,82,0.181366,"\int \frac{\sec (c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\sqrt{a^2-b^2} (a (-\cos (c+d x))+a-b \sin (c+d x))+2 a b \cos (c+d x) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d (a-b) (a+b) \sqrt{a^2-b^2} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{2 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]]*Cos[c + d*x] + Sqrt[a^2 - b^2]*(a - a*Cos[c + d*x] - b*Sin[c + d*x]))/((a - b)*(a + b)*Sqrt[a^2 - b^2]*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
1343,1,191,118,0.3770562,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\sqrt{a^2-b^2} \left(a (a-b \sin (c+d x))-\left(a^2-b^2\right) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)+2 b^3 \cos (c+d x) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d (a-b) (a+b) \sqrt{a^2-b^2} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{b \sec (c+d x) (b-a \sin (c+d x))}{a d \left(a^2-b^2\right)}+\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{\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]]*Cos[c + d*x] + Sqrt[a^2 - b^2]*(-((a^2 - b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])) + a*(a - b*Sin[c + d*x])))/(a*(a - b)*(a + b)*Sqrt[a^2 - b^2]*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
1344,1,205,128,1.0280126,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{-\frac{4 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 \left(a^2-b^2\right)^{3/2}}-\frac{2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^2}+\frac{2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^2}+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{a}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{a}}{2 d}","\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 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 \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}",1,"((-4*b^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2)) - Cot[(c + d*x)/2]/a + (2*b*Log[Cos[(c + d*x)/2]])/a^2 - (2*b*Log[Sin[(c + d*x)/2]])/a^2 + (2*Sin[(c + d*x)/2])/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*Sin[(c + d*x)/2])/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + Tan[(c + d*x)/2]/a)/(2*d)","A",1
1345,1,261,181,3.0959695,"\int \frac{\csc ^3(c+d x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{-\frac{4 b \tan \left(\frac{1}{2} (c+d x)\right)}{a^2}+\frac{4 b \cot \left(\frac{1}{2} (c+d x)\right)}{a^2}+\frac{4 \left(3 a^2+2 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}-\frac{4 \left(3 a^2+2 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}+\frac{16 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 \left(a^2-b^2\right)^{3/2}}+\frac{8 \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{8 \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{a}}{8 d}","-\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{b \cot (c+d x)}{a^2 d}-\frac{\left(3 a^2+2 b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 a^3 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{\csc ^2(c+d x) \sec (c+d x)}{2 a d}",1,"((16*b^5*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(3/2)) + (4*b*Cot[(c + d*x)/2])/a^2 - Csc[(c + d*x)/2]^2/a - (4*(3*a^2 + 2*b^2)*Log[Cos[(c + d*x)/2]])/a^3 + (4*(3*a^2 + 2*b^2)*Log[Sin[(c + d*x)/2]])/a^3 + Sec[(c + d*x)/2]^2/a + (8*Sin[(c + d*x)/2])/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (8*Sin[(c + d*x)/2])/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) - (4*b*Tan[(c + d*x)/2])/a^2)/(8*d)","A",1
1346,1,117,126,0.5520396,"\int \frac{\tan ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Sin[c + d*x]),x]","\frac{-\frac{4 a^3 \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}-\frac{1}{(a+b) (\sin (c+d x)-1)}+\frac{1}{(a-b) (\sin (c+d x)+1)}+\frac{(2 a+b) \log (1-\sin (c+d x))}{(a+b)^2}+\frac{(2 a-b) \log (\sin (c+d x)+1)}{(a-b)^2}}{4 d}","\frac{\sec ^2(c+d x) (a-b \sin (c+d x))}{2 d \left(a^2-b^2\right)}-\frac{a^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}+\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]])/(a + b)^2 + ((2*a - b)*Log[1 + Sin[c + d*x]])/(a - b)^2 - (4*a^3*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) - 1/((a + b)*(-1 + Sin[c + d*x])) + 1/((a - b)*(1 + Sin[c + d*x])))/(4*d)","A",1
1347,1,108,116,0.4721758,"\int \frac{\sec (c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","-\frac{-\frac{4 a^2 b \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}+\frac{1}{(a+b) (\sin (c+d x)-1)}+\frac{1}{(a-b) (\sin (c+d x)+1)}-\frac{a \log (1-\sin (c+d x))}{(a+b)^2}+\frac{a \log (\sin (c+d x)+1)}{(a-b)^2}}{4 d}","\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,"-1/4*(-((a*Log[1 - Sin[c + d*x]])/(a + b)^2) + (a*Log[1 + Sin[c + d*x]])/(a - b)^2 - (4*a^2*b*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) + 1/((a + b)*(-1 + Sin[c + d*x])) + 1/((a - b)*(1 + Sin[c + d*x])))/d","A",1
1348,1,162,117,0.4054782,"\int \frac{\sec ^2(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{-\frac{4 a b^2 \log (a+b \sin (c+d x))}{\left(a^2-b^2\right)^2}+\frac{1}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{1}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{(a+b)^2}+\frac{2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{(a-b)^2}}{4 d}","-\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,"((-2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a + b)^2 + (2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a - b)^2 - (4*a*b^2*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^2 + 1/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + 1/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2))/(4*d)","A",1
1349,1,151,156,0.6925963,"\int \frac{\csc (c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{b^4 \left(-\frac{1}{b^4 (a+b) (\sin (c+d x)-1)}+\frac{1}{b^4 (a-b) (\sin (c+d x)+1)}-\frac{(2 a+3 b) \log (1-\sin (c+d x))}{b^4 (a+b)^2}+\frac{4 \log (\sin (c+d x))}{a b^4}-\frac{(2 a-3 b) \log (\sin (c+d x)+1)}{b^4 (a-b)^2}-\frac{4 \log (a+b \sin (c+d x))}{a (a-b)^2 (a+b)^2}\right)}{4 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,"(b^4*(-(((2*a + 3*b)*Log[1 - Sin[c + d*x]])/(b^4*(a + b)^2)) + (4*Log[Sin[c + d*x]])/(a*b^4) - ((2*a - 3*b)*Log[1 + Sin[c + d*x]])/((a - b)^2*b^4) - (4*Log[a + b*Sin[c + d*x]])/(a*(a - b)^2*(a + b)^2) - 1/(b^4*(a + b)*(-1 + Sin[c + d*x])) + 1/((a - b)*b^4*(1 + Sin[c + d*x]))))/(4*d)","A",1
1350,1,174,171,0.7777289,"\int \frac{\csc ^2(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","-\frac{\csc (c+d x) (a+b \sin (c+d x)) \left(-\frac{4 b^5 \log (a+b \sin (c+d x))}{a^2 (a-b)^2 (a+b)^2}+\frac{4 b \log (\sin (c+d x))}{a^2}+\frac{1}{(a+b) (\sin (c+d x)-1)}+\frac{1}{(a-b) (\sin (c+d x)+1)}+\frac{(3 a+4 b) \log (1-\sin (c+d x))}{(a+b)^2}-\frac{(3 a-4 b) \log (\sin (c+d x)+1)}{(a-b)^2}+\frac{4 \csc (c+d x)}{a}\right)}{4 d (a \csc (c+d x)+b)}","\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,"-1/4*(Csc[c + d*x]*(a + b*Sin[c + d*x])*((4*Csc[c + d*x])/a + ((3*a + 4*b)*Log[1 - Sin[c + d*x]])/(a + b)^2 + (4*b*Log[Sin[c + d*x]])/a^2 - ((3*a - 4*b)*Log[1 + Sin[c + d*x]])/(a - b)^2 - (4*b^5*Log[a + b*Sin[c + d*x]])/(a^2*(a - b)^2*(a + b)^2) + 1/((a + b)*(-1 + Sin[c + d*x])) + 1/((a - b)*(1 + Sin[c + d*x]))))/(d*(b + a*Csc[c + d*x]))","A",1
1351,1,168,197,1.4418311,"\int \frac{\csc ^3(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","-\frac{\frac{4 b^6 \log (a+b \sin (c+d x))}{a^3 (a-b)^2 (a+b)^2}-\frac{4 b \csc (c+d x)}{a^2}-\frac{4 \left(2 a^2+b^2\right) \log (\sin (c+d x))}{a^3}+\frac{1}{(a+b) (\sin (c+d x)-1)}-\frac{1}{(a-b) (\sin (c+d x)+1)}+\frac{(4 a+5 b) \log (1-\sin (c+d x))}{(a+b)^2}+\frac{(4 a-5 b) \log (\sin (c+d x)+1)}{(a-b)^2}+\frac{2 \csc ^2(c+d x)}{a}}{4 d}","\frac{b \csc (c+d x)}{a^2 d}+\frac{\left(2 a^2+b^2\right) \log (\sin (c+d x))}{a^3 d}-\frac{b^6 \log (a+b \sin (c+d x))}{a^3 d \left(a^2-b^2\right)^2}+\frac{1}{4 d (a+b) (1-\sin (c+d x))}+\frac{1}{4 d (a-b) (\sin (c+d x)+1)}-\frac{(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,"-1/4*((-4*b*Csc[c + d*x])/a^2 + (2*Csc[c + d*x]^2)/a + ((4*a + 5*b)*Log[1 - Sin[c + d*x]])/(a + b)^2 - (4*(2*a^2 + b^2)*Log[Sin[c + d*x]])/a^3 + ((4*a - 5*b)*Log[1 + Sin[c + d*x]])/(a - b)^2 + (4*b^6*Log[a + b*Sin[c + d*x]])/(a^3*(a - b)^2*(a + b)^2) + 1/((a + b)*(-1 + Sin[c + d*x])) - 1/((a - b)*(1 + Sin[c + d*x])))/d","A",1
1352,1,195,177,1.4173564,"\int \frac{\tan ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{\frac{48 a^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{\sec ^3(c+d x) \left(8 a^3 \sin (3 (c+d x))+3 b \left(11 a^2-5 b^2\right) \cos (c+d x)+12 b \left(b^2-2 a^2\right) \cos (2 (c+d x))+11 a^2 b \cos (3 (c+d x))-16 a^2 b+6 a b^2 \sin (c+d x)-2 a b^2 \sin (3 (c+d x))-5 b^3 \cos (3 (c+d x))+4 b^3\right)}{(a-b)^2 (a+b)^2}}{24 d}","\frac{a \tan ^3(c+d x)}{3 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 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^3 \tan (c+d x)}{d \left(a^2-b^2\right)^2}",1,"((48*a^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - (Sec[c + d*x]^3*(-16*a^2*b + 4*b^3 + 3*b*(11*a^2 - 5*b^2)*Cos[c + d*x] + 12*b*(-2*a^2 + b^2)*Cos[2*(c + d*x)] + 11*a^2*b*Cos[3*(c + d*x)] - 5*b^3*Cos[3*(c + d*x)] + 6*a*b^2*Sin[c + d*x] + 8*a^3*Sin[3*(c + d*x)] - 2*a*b^2*Sin[3*(c + d*x)]))/((a - b)^2*(a + b)^2))/(24*d)","A",1
1353,1,184,142,1.4456324,"\int \frac{\sec (c+d x) \tan ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{\frac{\sec ^3(c+d x) \left(-12 a^3 \cos (2 (c+d x))+5 a^3 \cos (3 (c+d x))-4 a^3+3 a \left(5 a^2+b^2\right) \cos (c+d x)+8 a^2 b \sin (3 (c+d x))+a b^2 \cos (3 (c+d x))-8 a b^2+6 b^3 \sin (c+d x)-2 b^3 \sin (3 (c+d x))\right)}{(a-b)^2 (a+b)^2}-\frac{48 a^3 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}}{24 d}","-\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}}",1,"((-48*a^3*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (Sec[c + d*x]^3*(-4*a^3 - 8*a*b^2 + 3*a*(5*a^2 + b^2)*Cos[c + d*x] - 12*a^3*Cos[2*(c + d*x)] + 5*a^3*Cos[3*(c + d*x)] + a*b^2*Cos[3*(c + d*x)] + 6*b^3*Sin[c + d*x] + 8*a^2*b*Sin[3*(c + d*x)] - 2*b^3*Sin[3*(c + d*x)]))/((a - b)^2*(a + b)^2))/(24*d)","A",1
1354,1,200,165,1.2830759,"\int \frac{\sec ^2(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\frac{48 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)}{\left(a^2-b^2\right)^{5/2}}-\frac{\sec ^3(c+d x) \left(-6 a^3 \sin (c+d x)+2 a^3 \sin (3 (c+d x))+3 b \left(5 a^2+b^2\right) \cos (c+d x)-12 a^2 b \cos (2 (c+d x))+5 a^2 b \cos (3 (c+d x))-4 a^2 b+12 a b^2 \sin (c+d x)+4 a b^2 \sin (3 (c+d x))+b^3 \cos (3 (c+d x))-8 b^3\right)}{(a-b)^2 (a+b)^2}}{24 d}","\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 ^3(c+d x)}{3 d \left(a^2-b^2\right)}+\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,"((48*a^2*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - (Sec[c + d*x]^3*(-4*a^2*b - 8*b^3 + 3*b*(5*a^2 + b^2)*Cos[c + d*x] - 12*a^2*b*Cos[2*(c + d*x)] + 5*a^2*b*Cos[3*(c + d*x)] + b^3*Cos[3*(c + d*x)] - 6*a^3*Sin[c + d*x] + 12*a*b^2*Sin[c + d*x] + 2*a^3*Sin[3*(c + d*x)] + 4*a*b^2*Sin[3*(c + d*x)]))/((a - b)^2*(a + b)^2))/(24*d)","A",1
1355,1,203,138,1.3048747,"\int \frac{\sec ^3(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\frac{\sec ^3(c+d x) \left(-\frac{1}{2} a^3 \cos (3 (c+d x))+4 a^3-\frac{3}{2} a \left(a^2-7 b^2\right) \cos (c+d x)-3 a^2 b \sin (c+d x)+a^2 b \sin (3 (c+d x))-6 a b^2 \cos (2 (c+d x))+\frac{7}{2} a b^2 \cos (3 (c+d x))-10 a b^2+6 b^3 \sin (c+d x)+2 b^3 \sin (3 (c+d x))\right)}{(a-b)^2 (a+b)^2}-\frac{24 a b^3 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}}{12 d}","\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}}",1,"((-24*a*b^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (Sec[c + d*x]^3*(4*a^3 - 10*a*b^2 - (3*a*(a^2 - 7*b^2)*Cos[c + d*x])/2 - 6*a*b^2*Cos[2*(c + d*x)] - (a^3*Cos[3*(c + d*x)])/2 + (7*a*b^2*Cos[3*(c + d*x)])/2 - 3*a^2*b*Sin[c + d*x] + 6*b^3*Sin[c + d*x] + a^2*b*Sin[3*(c + d*x)] + 2*b^3*Sin[3*(c + d*x)]))/((a - b)^2*(a + b)^2))/(12*d)","A",1
1356,1,334,194,4.8767316,"\int \frac{\csc (c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","\frac{-\frac{24 b^5 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a \left(a^2-b^2\right)^{5/2}}+\frac{2 (7 a+10 b) \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 (10 b-7 a) \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{1}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{12 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a}-\frac{12 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a}}{12 d}","\frac{b \sec ^3(c+d x) (b-a \sin (c+d x))}{3 a d \left(a^2-b^2\right)}-\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 (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,"((-24*b^5*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(5/2)) - (12*Log[Cos[(c + d*x)/2]])/a + (12*Log[Sin[(c + d*x)/2]])/a + 1/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (2*Sin[(c + d*x)/2])/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (2*(7*a + 10*b)*Sin[(c + d*x)/2])/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (2*Sin[(c + d*x)/2])/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + 1/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (2*(-7*a + 10*b)*Sin[(c + d*x)/2])/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(12*d)","A",1
1357,1,450,220,6.436483,"\int \frac{\csc ^2(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(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{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{5/2}}-\frac{b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^2 d}+\frac{b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^2 d}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{6 d (a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{10 a \sin \left(\frac{1}{2} (c+d x)\right)-13 b \sin \left(\frac{1}{2} (c+d x)\right)}{6 d (a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{10 a \sin \left(\frac{1}{2} (c+d x)\right)+13 b \sin \left(\frac{1}{2} (c+d x)\right)}{6 d (a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{12 d (a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{12 d (a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{6 d (a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{2 a d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{2 a d}","\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{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 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\left(6 a^4-10 a^2 b^2+b^4\right) \tan (c+d x)}{3 a d \left(a^2-b^2\right)^2}+\frac{\tan ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}",1,"(2*b^6*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)*d) - Cot[(c + d*x)/2]/(2*a*d) + (b*Log[Cos[(c + d*x)/2]])/(a^2*d) - (b*Log[Sin[(c + d*x)/2]])/(a^2*d) + 1/(12*(a + b)*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + Sin[(c + d*x)/2]/(6*(a + b)*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + Sin[(c + d*x)/2]/(6*(a - b)*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - 1/(12*(a - b)*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (10*a*Sin[(c + d*x)/2] - 13*b*Sin[(c + d*x)/2])/(6*(a - b)^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (10*a*Sin[(c + d*x)/2] + 13*b*Sin[(c + d*x)/2])/(6*(a + b)^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + Tan[(c + d*x)/2]/(2*a*d)","B",1
1358,1,947,332,6.2664281,"\int \frac{\csc ^3(c+d x) \sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","16 \left(-\frac{\tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(b \cos \left(\frac{1}{2} (c+d x)\right)+a \sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right) \csc (c+d x) (a+b \sin (c+d x)) b^7}{8 a^3 \left(a^2-b^2\right)^{5/2} d (b+a \csc (c+d x))}+\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc (c+d x) (a+b \sin (c+d x)) b}{32 a^2 d (b+a \csc (c+d x))}-\frac{\csc (c+d x) (a+b \sin (c+d x)) \tan \left(\frac{1}{2} (c+d x)\right) b}{32 a^2 d (b+a \csc (c+d x))}+\frac{\csc (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \sin (c+d x))}{128 a d (b+a \csc (c+d x))}+\frac{\left(-5 a^2-2 b^2\right) \csc (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sin (c+d x))}{32 a^3 d (b+a \csc (c+d x))}+\frac{\left(5 a^2+2 b^2\right) \csc (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sin (c+d x))}{32 a^3 d (b+a \csc (c+d x))}+\frac{\csc (c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \sin (c+d x))}{96 (a+b) d (b+a \csc (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\csc (c+d x) \left(16 b \sin \left(\frac{1}{2} (c+d x)\right)-13 a \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sin (c+d x))}{96 (a-b)^2 d (b+a \csc (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\csc (c+d x) \left(13 a \sin \left(\frac{1}{2} (c+d x)\right)+16 b \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sin (c+d x))}{96 (a+b)^2 d (b+a \csc (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right) \csc (c+d x) (a+b \sin (c+d x))}{128 a d (b+a \csc (c+d x))}+\frac{a \left(13 a^2-19 b^2\right) \csc (c+d x) (a+b \sin (c+d x))}{96 \left(a^2-b^2\right)^2 d (b+a \csc (c+d x))}+\frac{\csc (c+d x) (a+b \sin (c+d x))}{192 (a+b) d (b+a \csc (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\csc (c+d x) (a+b \sin (c+d x))}{192 (a-b) d (b+a \csc (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{\csc (c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \sin (c+d x))}{96 (a-b) d (b+a \csc (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)","\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 \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{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^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{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,"16*((a*(13*a^2 - 19*b^2)*Csc[c + d*x]*(a + b*Sin[c + d*x]))/(96*(a^2 - b^2)^2*d*(b + a*Csc[c + d*x])) - (b^7*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]]*Csc[c + d*x]*(a + b*Sin[c + d*x]))/(8*a^3*(a^2 - b^2)^(5/2)*d*(b + a*Csc[c + d*x])) + (b*Cot[(c + d*x)/2]*Csc[c + d*x]*(a + b*Sin[c + d*x]))/(32*a^2*d*(b + a*Csc[c + d*x])) - (Csc[(c + d*x)/2]^2*Csc[c + d*x]*(a + b*Sin[c + d*x]))/(128*a*d*(b + a*Csc[c + d*x])) + ((-5*a^2 - 2*b^2)*Csc[c + d*x]*Log[Cos[(c + d*x)/2]]*(a + b*Sin[c + d*x]))/(32*a^3*d*(b + a*Csc[c + d*x])) + ((5*a^2 + 2*b^2)*Csc[c + d*x]*Log[Sin[(c + d*x)/2]]*(a + b*Sin[c + d*x]))/(32*a^3*d*(b + a*Csc[c + d*x])) + (Csc[c + d*x]*Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x]))/(128*a*d*(b + a*Csc[c + d*x])) + (Csc[c + d*x]*(a + b*Sin[c + d*x]))/(192*(a + b)*d*(b + a*Csc[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (Csc[c + d*x]*Sin[(c + d*x)/2]*(a + b*Sin[c + d*x]))/(96*(a + b)*d*(b + a*Csc[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) - (Csc[c + d*x]*Sin[(c + d*x)/2]*(a + b*Sin[c + d*x]))/(96*(a - b)*d*(b + a*Csc[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (Csc[c + d*x]*(a + b*Sin[c + d*x]))/(192*(a - b)*d*(b + a*Csc[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (Csc[c + d*x]*(-13*a*Sin[(c + d*x)/2] + 16*b*Sin[(c + d*x)/2])*(a + b*Sin[c + d*x]))/(96*(a - b)^2*d*(b + a*Csc[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (Csc[c + d*x]*(13*a*Sin[(c + d*x)/2] + 16*b*Sin[(c + d*x)/2])*(a + b*Sin[c + d*x]))/(96*(a + b)^2*d*(b + a*Csc[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (b*Csc[c + d*x]*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2])/(32*a^2*d*(b + a*Csc[c + d*x])))","B",1
1359,1,212,240,3.0081706,"\int \frac{\sin ^3(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^3*Tan[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","\frac{-\frac{16 a^8 \log (a+b \sin (c+d x))}{b^3 (a-b)^3 (a+b)^3}-\frac{\left(35 a^2+57 a b+24 b^2\right) \log (1-\sin (c+d x))}{(a+b)^3}+\frac{\left(35 a^2-57 a b+24 b^2\right) \log (\sin (c+d x)+1)}{(a-b)^3}+\frac{16 a \sin (c+d x)}{b^2}+\frac{13 a+11 b}{(a+b)^2 (\sin (c+d x)-1)}+\frac{13 a-11 b}{(a-b)^2 (\sin (c+d x)+1)}+\frac{1}{(a+b) (\sin (c+d x)-1)^2}-\frac{1}{(a-b) (\sin (c+d x)+1)^2}-\frac{8 \sin ^2(c+d x)}{b}}{16 d}","-\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^8 \log (a+b \sin (c+d x))}{b^3 d \left(a^2-b^2\right)^3}+\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]])/(a + b)^3) + ((35*a^2 - 57*a*b + 24*b^2)*Log[1 + Sin[c + d*x]])/(a - b)^3 - (16*a^8*Log[a + b*Sin[c + d*x]])/((a - b)^3*b^3*(a + b)^3) + 1/((a + b)*(-1 + Sin[c + d*x])^2) + (13*a + 11*b)/((a + b)^2*(-1 + Sin[c + d*x])) + (16*a*Sin[c + d*x])/b^2 - (8*Sin[c + d*x]^2)/b - 1/((a - b)*(1 + Sin[c + d*x])^2) + (13*a - 11*b)/((a - b)^2*(1 + Sin[c + d*x])))/(16*d)","A",1
1360,1,198,221,2.495688,"\int \frac{\sin ^2(c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","\frac{\frac{16 a^7 \log (a+b \sin (c+d x))}{b^2 (a-b)^3 (a+b)^3}-\frac{\left(24 a^2+37 a b+15 b^2\right) \log (1-\sin (c+d x))}{(a+b)^3}-\frac{\left(24 a^2-37 a b+15 b^2\right) \log (\sin (c+d x)+1)}{(a-b)^3}+\frac{11 a+9 b}{(a+b)^2 (\sin (c+d x)-1)}+\frac{9 b-11 a}{(a-b)^2 (\sin (c+d x)+1)}+\frac{1}{(a+b) (\sin (c+d x)-1)^2}+\frac{1}{(a-b) (\sin (c+d x)+1)^2}-\frac{16 \sin (c+d x)}{b}}{16 d}","-\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{a^7 \log (a+b \sin (c+d x))}{b^2 d \left(a^2-b^2\right)^3}-\frac{\sin (c+d x)}{b d}",1,"(-(((24*a^2 + 37*a*b + 15*b^2)*Log[1 - Sin[c + d*x]])/(a + b)^3) - ((24*a^2 - 37*a*b + 15*b^2)*Log[1 + Sin[c + d*x]])/(a - b)^3 + (16*a^7*Log[a + b*Sin[c + d*x]])/((a - b)^3*b^2*(a + b)^3) + 1/((a + b)*(-1 + Sin[c + d*x])^2) + (11*a + 9*b)/((a + b)^2*(-1 + Sin[c + d*x])) - (16*Sin[c + d*x])/b + 1/((a - b)*(1 + Sin[c + d*x])^2) + (-11*a + 9*b)/((a - b)^2*(1 + Sin[c + d*x])))/(16*d)","A",1
1361,1,187,208,1.7524296,"\int \frac{\sin (c+d x) \tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","\frac{-\frac{16 a^6 \log (a+b \sin (c+d x))}{b (a-b)^3 (a+b)^3}-\frac{\left(15 a^2+21 a b+8 b^2\right) \log (1-\sin (c+d x))}{(a+b)^3}+\frac{\left(15 a^2-21 a b+8 b^2\right) \log (\sin (c+d x)+1)}{(a-b)^3}+\frac{9 a+7 b}{(a+b)^2 (\sin (c+d x)-1)}+\frac{9 a-7 b}{(a-b)^2 (\sin (c+d x)+1)}+\frac{1}{(a+b) (\sin (c+d x)-1)^2}-\frac{1}{(a-b) (\sin (c+d x)+1)^2}}{16 d}","-\frac{\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}",1,"(-(((15*a^2 + 21*a*b + 8*b^2)*Log[1 - Sin[c + d*x]])/(a + b)^3) + ((15*a^2 - 21*a*b + 8*b^2)*Log[1 + Sin[c + d*x]])/(a - b)^3 - (16*a^6*Log[a + b*Sin[c + d*x]])/((a - b)^3*b*(a + b)^3) + 1/((a + b)*(-1 + Sin[c + d*x])^2) + (9*a + 7*b)/((a + b)^2*(-1 + Sin[c + d*x])) - 1/((a - b)*(1 + Sin[c + d*x])^2) + (9*a - 7*b)/((a - b)^2*(1 + Sin[c + d*x])))/(16*d)","A",1
1362,1,184,204,1.4046271,"\int \frac{\tan ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Sin[c + d*x]),x]","\frac{\frac{16 a^5 \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}-\frac{\left(8 a^2+9 a b+3 b^2\right) \log (1-\sin (c+d x))}{(a+b)^3}-\frac{\left(8 a^2-9 a b+3 b^2\right) \log (\sin (c+d x)+1)}{(a-b)^3}+\frac{7 a+5 b}{(a+b)^2 (\sin (c+d x)-1)}+\frac{5 b-7 a}{(a-b)^2 (\sin (c+d x)+1)}+\frac{1}{(a+b) (\sin (c+d x)-1)^2}+\frac{1}{(a-b) (\sin (c+d x)+1)^2}}{16 d}","-\frac{\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}",1,"(-(((8*a^2 + 9*a*b + 3*b^2)*Log[1 - Sin[c + d*x]])/(a + b)^3) - ((8*a^2 - 9*a*b + 3*b^2)*Log[1 + Sin[c + d*x]])/(a - b)^3 + (16*a^5*Log[a + b*Sin[c + d*x]])/((a - b)^3*(a + b)^3) + 1/((a + b)*(-1 + Sin[c + d*x])^2) + (7*a + 5*b)/((a + b)^2*(-1 + Sin[c + d*x])) + 1/((a - b)*(1 + Sin[c + d*x])^2) + (-7*a + 5*b)/((a - b)^2*(1 + Sin[c + d*x])))/(16*d)","A",1
1363,1,169,190,1.5842634,"\int \frac{\sec (c+d x) \tan ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x]^4)/(a + b*Sin[c + d*x]),x]","\frac{-\frac{16 a^4 b \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac{5 a+3 b}{(a+b)^2 (\sin (c+d x)-1)}+\frac{5 a-3 b}{(a-b)^2 (\sin (c+d x)+1)}+\frac{1}{(a+b) (\sin (c+d x)-1)^2}-\frac{1}{(a-b) (\sin (c+d x)+1)^2}-\frac{a (3 a+b) \log (1-\sin (c+d x))}{(a+b)^3}+\frac{a (3 a-b) \log (\sin (c+d x)+1)}{(a-b)^3}}{16 d}","-\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^4 b \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\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]])/(a + b)^3) + (a*(3*a - b)*Log[1 + Sin[c + d*x]])/(a - b)^3 - (16*a^4*b*Log[a + b*Sin[c + d*x]])/((a - b)^3*(a + b)^3) + 1/((a + b)*(-1 + Sin[c + d*x])^2) + (5*a + 3*b)/((a + b)^2*(-1 + Sin[c + d*x])) - 1/((a - b)*(1 + Sin[c + d*x])^2) + (5*a - 3*b)/((a - b)^2*(1 + Sin[c + d*x])))/(16*d)","A",1
1364,1,166,182,1.2457768,"\int \frac{\sec ^2(c+d x) \tan ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*Tan[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{\frac{16 a^3 b^2 \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac{3 a+b}{(a+b)^2 (\sin (c+d x)-1)}+\frac{b-3 a}{(a-b)^2 (\sin (c+d x)+1)}+\frac{1}{(a+b) (\sin (c+d x)-1)^2}+\frac{1}{(a-b) (\sin (c+d x)+1)^2}+\frac{b (3 a+b) \log (1-\sin (c+d x))}{(a+b)^3}-\frac{b (3 a-b) \log (\sin (c+d x)+1)}{(a-b)^3}}{16 d}","\frac{\sec ^4(c+d x) (a-b \sin (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{a^3 b^2 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\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]])/(a + b)^3 - ((3*a - b)*b*Log[1 + Sin[c + d*x]])/(a - b)^3 + (16*a^3*b^2*Log[a + b*Sin[c + d*x]])/((a - b)^3*(a + b)^3) + 1/((a + b)*(-1 + Sin[c + d*x])^2) + (3*a + b)/((a + b)^2*(-1 + Sin[c + d*x])) + 1/((a - b)*(1 + Sin[c + d*x])^2) + (-3*a + b)/((a - b)^2*(1 + Sin[c + d*x])))/(16*d)","A",1
1365,1,163,178,1.2392716,"\int \frac{\sec ^3(c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*Tan[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{-\frac{16 a^2 b^3 \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac{a-b}{(a+b)^2 (\sin (c+d x)-1)}+\frac{a+b}{(a-b)^2 (\sin (c+d x)+1)}+\frac{1}{(a+b) (\sin (c+d x)-1)^2}-\frac{1}{(a-b) (\sin (c+d x)+1)^2}+\frac{a (a+3 b) \log (1-\sin (c+d x))}{(a+b)^3}-\frac{a (a-3 b) \log (\sin (c+d x)+1)}{(a-b)^3}}{16 d}","-\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^2 b^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}+\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]])/(a + b)^3 - (a*(a - 3*b)*Log[1 + Sin[c + d*x]])/(a - b)^3 - (16*a^2*b^3*Log[a + b*Sin[c + d*x]])/((a - b)^3*(a + b)^3) + 1/((a + b)*(-1 + Sin[c + d*x])^2) + (a - b)/((a + b)^2*(-1 + Sin[c + d*x])) - 1/((a - b)*(1 + Sin[c + d*x])^2) + (a + b)/((a - b)^2*(1 + Sin[c + d*x])))/(16*d)","A",1
1366,1,244,177,0.938674,"\int \frac{\sec ^4(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^4*Tan[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\frac{16 a b^4 \log (a+b \sin (c+d x))}{\left(a^2-b^2\right)^3}+\frac{a+3 b}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{a-3 b}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{1}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{1}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{2 b (a+3 b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{(a+b)^3}+\frac{2 b (a-3 b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{(a-b)^3}}{16 d}","\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{a b^4 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\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,"((-2*b*(a + 3*b)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a + b)^3 + (2*(a - 3*b)*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a - b)^3 + (16*a*b^4*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^3 + 1/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + (a + 3*b)/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + 1/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (a - 3*b)/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2))/(16*d)","A",1
1367,1,220,233,2.8895097,"\int \frac{\csc (c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","\frac{b^6 \left(-\frac{\left(8 a^2+21 a b+15 b^2\right) \log (1-\sin (c+d x))}{b^6 (a+b)^3}-\frac{\left(8 a^2-21 a b+15 b^2\right) \log (\sin (c+d x)+1)}{b^6 (a-b)^3}+\frac{-5 a-7 b}{b^6 (a+b)^2 (\sin (c+d x)-1)}+\frac{5 a-7 b}{b^6 (a-b)^2 (\sin (c+d x)+1)}+\frac{1}{b^6 (a+b) (\sin (c+d x)-1)^2}+\frac{1}{b^6 (a-b) (\sin (c+d x)+1)^2}+\frac{16 \log (\sin (c+d x))}{a b^6}+\frac{16 \log (a+b \sin (c+d x))}{a (a-b)^3 (a+b)^3}\right)}{16 d}","-\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{b^6 \log (a+b \sin (c+d x))}{a d \left(a^2-b^2\right)^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,"(b^6*(-(((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sin[c + d*x]])/(b^6*(a + b)^3)) + (16*Log[Sin[c + d*x]])/(a*b^6) - ((8*a^2 - 21*a*b + 15*b^2)*Log[1 + Sin[c + d*x]])/((a - b)^3*b^6) + (16*Log[a + b*Sin[c + d*x]])/(a*(a - b)^3*(a + b)^3) + 1/(b^6*(a + b)*(-1 + Sin[c + d*x])^2) + (-5*a - 7*b)/(b^6*(a + b)^2*(-1 + Sin[c + d*x])) + 1/((a - b)*b^6*(1 + Sin[c + d*x])^2) + (5*a - 7*b)/((a - b)^2*b^6*(1 + Sin[c + d*x]))))/(16*d)","A",1
1368,1,257,250,6.2117888,"\int \frac{\csc ^2(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","\frac{b^7 \left(-\frac{\log (\sin (c+d x))}{a^2 b^6}-\frac{\left(15 a^2+37 a b+24 b^2\right) \log (1-\sin (c+d x))}{16 b^7 (a+b)^3}+\frac{\left(15 a^2-37 a b+24 b^2\right) \log (\sin (c+d x)+1)}{16 b^7 (a-b)^3}-\frac{\log (a+b \sin (c+d x))}{a^2 (a-b)^3 (a+b)^3}-\frac{\csc (c+d x)}{a b^7}-\frac{7 a-9 b}{16 b^6 (a-b)^2 (b \sin (c+d x)+b)}+\frac{7 a+9 b}{16 b^6 (a+b)^2 (b-b \sin (c+d x))}+\frac{1}{16 b^5 (a+b) (b-b \sin (c+d x))^2}-\frac{1}{16 b^5 (a-b) (b \sin (c+d x)+b)^2}\right)}{d}","-\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^7 \log (a+b \sin (c+d x))}{a^2 d \left(a^2-b^2\right)^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,"(b^7*(-(Csc[c + d*x]/(a*b^7)) - ((15*a^2 + 37*a*b + 24*b^2)*Log[1 - Sin[c + d*x]])/(16*b^7*(a + b)^3) - Log[Sin[c + d*x]]/(a^2*b^6) + ((15*a^2 - 37*a*b + 24*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*b^7) - Log[a + b*Sin[c + d*x]]/(a^2*(a - b)^3*(a + b)^3) + 1/(16*b^5*(a + b)*(b - b*Sin[c + d*x])^2) + (7*a + 9*b)/(16*b^6*(a + b)^2*(b - b*Sin[c + d*x])) - 1/(16*(a - b)*b^5*(b + b*Sin[c + d*x])^2) - (7*a - 9*b)/(16*(a - b)^2*b^6*(b + b*Sin[c + d*x]))))/d","A",1
1369,1,281,274,6.2560346,"\int \frac{\csc ^3(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^5)/(a + b*Sin[c + d*x]),x]","\frac{b^8 \left(\frac{\log (a+b \sin (c+d x))}{a^3 (a-b)^3 (a+b)^3}+\frac{\csc (c+d x)}{a^2 b^7}-\frac{\left(24 a^2+57 a b+35 b^2\right) \log (1-\sin (c+d x))}{16 b^8 (a+b)^3}-\frac{\left(24 a^2-57 a b+35 b^2\right) \log (\sin (c+d x)+1)}{16 b^8 (a-b)^3}+\frac{\left(3 a^2+b^2\right) \log (\sin (c+d x))}{a^3 b^8}-\frac{\csc ^2(c+d x)}{2 a b^8}+\frac{9 a+11 b}{16 b^7 (a+b)^2 (b-b \sin (c+d x))}+\frac{9 a-11 b}{16 b^7 (a-b)^2 (b \sin (c+d x)+b)}+\frac{1}{16 b^6 (a+b) (b-b \sin (c+d x))^2}+\frac{1}{16 b^6 (a-b) (b \sin (c+d x)+b)^2}\right)}{d}","-\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(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{\left(3 a^2+b^2\right) \log (\sin (c+d x))}{a^3 d}+\frac{b^8 \log (a+b \sin (c+d x))}{a^3 d \left(a^2-b^2\right)^3}+\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^8*(Csc[c + d*x]/(a^2*b^7) - Csc[c + d*x]^2/(2*a*b^8) - ((24*a^2 + 57*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*b^8*(a + b)^3) + ((3*a^2 + b^2)*Log[Sin[c + d*x]])/(a^3*b^8) - ((24*a^2 - 57*a*b + 35*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*b^8) + Log[a + b*Sin[c + d*x]]/(a^3*(a - b)^3*(a + b)^3) + 1/(16*b^6*(a + b)*(b - b*Sin[c + d*x])^2) + (9*a + 11*b)/(16*b^7*(a + b)^2*(b - b*Sin[c + d*x])) + 1/(16*(a - b)*b^6*(b + b*Sin[c + d*x])^2) + (9*a - 11*b)/(16*(a - b)^2*b^7*(b + b*Sin[c + d*x]))))/d","A",1
1370,1,816,500,26.8539662,"\int \frac{\sqrt{g \cos (e+f x)} \sin ^4(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^4)/(a + b*Sin[e + f*x]),x]","\frac{\sqrt{g \cos (e+f x)} \left(-\frac{\left(28 a^2+19 b^2\right) \cos (e+f x)}{42 b^3}+\frac{\cos (3 (e+f x))}{14 b}+\frac{a \sin (2 (e+f x))}{5 b^2}\right)}{f}-\frac{a \sqrt{g \cos (e+f x)} \left(-\frac{\left(5 a^2+2 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(e+f x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{4 a b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right)}{5 b^3 f \sqrt{\cos (e+f x)}}","-\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{2 a^2 (g \cos (e+f x))^{3/2}}{3 b^3 f g}+\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{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 \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,"-1/5*(a*Sqrt[g*Cos[e + f*x]]*((-4*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) - ((5*a^2 + 2*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*Sin[e + f*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(b^3*f*Sqrt[Cos[e + f*x]]) + (Sqrt[g*Cos[e + f*x]]*(-1/42*((28*a^2 + 19*b^2)*Cos[e + f*x])/b^3 + Cos[3*(e + f*x)]/(14*b) + (a*Sin[2*(e + f*x)])/(5*b^2)))/f","C",0
1371,1,789,448,26.6877287,"\int \frac{\sqrt{g \cos (e+f x)} \sin ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","\frac{\sqrt{g \cos (e+f x)} \left(\frac{2 a \cos (e+f x)}{3 b^2}-\frac{\sin (2 (e+f x))}{5 b}\right)}{f}+\frac{\sqrt{g \cos (e+f x)} \left(-\frac{\left(5 a^2+2 b^2\right) \sin ^2(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{4 a b \sin (e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right)}{5 b^2 f \sqrt{\cos (e+f x)}}","\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{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 (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,"(Sqrt[g*Cos[e + f*x]]*((-4*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) - ((5*a^2 + 2*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*Sin[e + f*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(5*b^2*f*Sqrt[Cos[e + f*x]]) + (Sqrt[g*Cos[e + f*x]]*((2*a*Cos[e + f*x])/(3*b^2) - Sin[2*(e + f*x)]/(5*b)))/f","C",0
1372,1,372,369,16.8773054,"\int \frac{\sqrt{g \cos (e+f x)} \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","\frac{\sqrt{g \cos (e+f x)} \left(-\frac{a \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{\left(a^2-b^2\right) (a+b \sin (e+f x))}-8 b^{3/2} \cos ^{\frac{3}{2}}(e+f x)\right)}{12 b^{5/2} f \sqrt{\cos (e+f 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}",1,"(Sqrt[g*Cos[e + f*x]]*(-8*b^(3/2)*Cos[e + f*x]^(3/2) - (a*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*(a + b*Sqrt[Sin[e + f*x]^2]))/((a^2 - b^2)*(a + b*Sin[e + f*x]))))/(12*b^(5/2)*f*Sqrt[Cos[e + f*x]])","C",0
1373,1,351,341,19.7963714,"\int \frac{\sqrt{g \cos (e+f x)} \sin (e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(a + b*Sin[e + f*x]),x]","-\frac{\sqrt{g \cos (e+f x)} \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 b^{3/2} f \left(b^2-a^2\right) \sqrt{\cos (e+f x)} (a+b \sin (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(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{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{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,"-1/12*(Sqrt[g*Cos[e + f*x]]*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*(a + b*Sqrt[Sin[e + f*x]^2]))/(b^(3/2)*(-a^2 + b^2)*f*Sqrt[Cos[e + f*x]]*(a + b*Sin[e + f*x]))","C",0
1374,1,534,355,14.3628386,"\int \frac{\sqrt{g \cos (e+f x)} \csc (e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Cos[e + f*x]]*Csc[e + f*x])/(a + b*Sin[e + f*x]),x]","\frac{\csc (e+f x) \sqrt{g \cos (e+f x)} \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(8 a b \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+3 \left(-\sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+\sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+2 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+2 a^2 \log \left(1-\sqrt{\cos (e+f x)}\right)-2 a^2 \log \left(\sqrt{\cos (e+f x)}+1\right)+4 a^2 \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)-2 b^2 \log \left(1-\sqrt{\cos (e+f x)}\right)+2 b^2 \log \left(\sqrt{\cos (e+f x)}+1\right)-4 b^2 \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)\right)\right)}{12 a f \left(a^2-b^2\right) \sqrt{\cos (e+f x)} (a \csc (e+f x)+b)}","-\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*Cos[e + f*x]]*Csc[e + f*x]*(8*a*b*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*(2*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 4*a^2*ArcTan[Sqrt[Cos[e + f*x]]] - 4*b^2*ArcTan[Sqrt[Cos[e + f*x]]] + 2*a^2*Log[1 - Sqrt[Cos[e + f*x]]] - 2*b^2*Log[1 - Sqrt[Cos[e + f*x]]] - 2*a^2*Log[1 + Sqrt[Cos[e + f*x]]] + 2*b^2*Log[1 + Sqrt[Cos[e + f*x]]] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*(a + b*Sqrt[Sin[e + f*x]^2]))/(12*a*(a^2 - b^2)*f*Sqrt[Cos[e + f*x]]*(b + a*Csc[e + f*x]))","C",0
1375,1,1550,433,27.1039246,"\int \frac{\sqrt{g \cos (e+f x)} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","\frac{\sqrt{g \cos (e+f x)} \left(\frac{4 a \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(e+f x)} (b+a \csc (e+f x))}+\frac{5 b \left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \csc (e+f x) \left(6 \log \left(1-\sqrt{\cos (e+f x)}\right) a^2-6 \log \left(\sqrt{\cos (e+f x)}+1\right) a^2+8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x) a+6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+12 \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)-6 b^2 \log \left(1-\sqrt{\cos (e+f x)}\right)+6 b^2 \log \left(\sqrt{\cos (e+f x)}+1\right)-3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{12 \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) (b+a \csc (e+f x))}-\frac{\left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \csc (e+f x) \left(48 a b^{5/2} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{7}{2}}(e+f x)-56 a b^{5/2} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)-42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+84 b^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(a^2-b^2\right) \log \left(1-\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(b^2-a^2\right) \log \left(\sqrt{\cos (e+f x)}+1\right)+21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)-21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{84 \sqrt{b} \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) \left(2 \cos ^2(e+f x)-1\right) (b+a \csc (e+f x))}\right)}{4 a f \sqrt{\cos (e+f x)}}-\frac{\sqrt{g \cos (e+f x)} \cot (e+f x)}{a f}","\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^{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 \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,"-((Sqrt[g*Cos[e + f*x]]*Cot[e + f*x])/(a*f)) + (Sqrt[g*Cos[e + f*x]]*((4*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4))))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) + (5*b*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*(6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 12*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 6*a^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*b^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*a^2*Log[1 + Sqrt[Cos[e + f*x]]] + 6*b^2*Log[1 + Sqrt[Cos[e + f*x]]] - 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(12*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - ((-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*(-42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 84*b^(3/2)*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] - 56*a*b^(5/2)*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 48*a*b^(5/2)*AppellF1[7/4, 1/2, 1, 11/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(7/2) + 42*b^(3/2)*(a^2 - b^2)*Log[1 - Sqrt[Cos[e + f*x]]] + 42*b^(3/2)*(-a^2 + b^2)*Log[1 + Sqrt[Cos[e + f*x]]] + 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] - 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(84*Sqrt[b]*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/(4*a*f*Sqrt[Cos[e + f*x]])","C",0
1376,1,1582,544,29.1853427,"\int \frac{\sqrt{g \cos (e+f x)} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","\frac{\sqrt{g \cos (e+f x)} \left(\frac{b \cot (e+f x)}{a^2}-\frac{\cot (e+f x) \csc (e+f x)}{2 a}\right)}{f}-\frac{\sqrt{g \cos (e+f x)} \left(\frac{6 a b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(e+f x)} (b+a \csc (e+f x))}-\frac{\left(-a^2-5 b^2\right) \left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \csc (e+f x) \left(6 \log \left(1-\sqrt{\cos (e+f x)}\right) a^2-6 \log \left(\sqrt{\cos (e+f x)}+1\right) a^2+8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x) a+6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+12 \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)-6 b^2 \log \left(1-\sqrt{\cos (e+f x)}\right)+6 b^2 \log \left(\sqrt{\cos (e+f x)}+1\right)-3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{12 \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) (b+a \csc (e+f x))}-\frac{\sqrt{b} \left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \csc (e+f x) \left(48 a b^{5/2} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{7}{2}}(e+f x)-56 a b^{5/2} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)-42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+84 b^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(a^2-b^2\right) \log \left(1-\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(b^2-a^2\right) \log \left(\sqrt{\cos (e+f x)}+1\right)+21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)-21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{84 \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) \left(2 \cos ^2(e+f x)-1\right) (b+a \csc (e+f x))}\right)}{4 a^2 f \sqrt{\cos (e+f x)}}","\frac{b^2 \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f}-\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{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^{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{\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*Cos[e + f*x]]*((b*Cot[e + f*x])/a^2 - (Cot[e + f*x]*Csc[e + f*x])/(2*a)))/f - (Sqrt[g*Cos[e + f*x]]*((6*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4))))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) - ((-a^2 - 5*b^2)*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*(6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 12*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 6*a^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*b^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*a^2*Log[1 + Sqrt[Cos[e + f*x]]] + 6*b^2*Log[1 + Sqrt[Cos[e + f*x]]] - 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(12*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - (Sqrt[b]*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*(-42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 84*b^(3/2)*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] - 56*a*b^(5/2)*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 48*a*b^(5/2)*AppellF1[7/4, 1/2, 1, 11/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(7/2) + 42*b^(3/2)*(a^2 - b^2)*Log[1 - Sqrt[Cos[e + f*x]]] + 42*b^(3/2)*(-a^2 + b^2)*Log[1 + Sqrt[Cos[e + f*x]]] + 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] - 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(84*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/(4*a^2*f*Sqrt[Cos[e + f*x]])","C",0
1377,1,1991,621,27.8390403,"\int \frac{(g \cos (e+f x))^{3/2} \sin ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","\frac{(g \cos (e+f x))^{3/2} \sec (e+f x) \left(\frac{a \cos (2 (e+f x))}{5 b^2}+\frac{\left(28 a^2+5 b^2\right) \sin (e+f x)}{42 b^3}-\frac{\sin (3 (e+f x))}{14 b}\right)}{f}-\frac{(g \cos (e+f x))^{3/2} \left(-\frac{2 \left(-40 b^3-98 a^2 b\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \sqrt{1-\cos ^2(e+f x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(e+f x)}{\left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{2 \left(70 a^3-19 a b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}+\frac{\left(210 a^3-21 a b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{5}{2}}(e+f x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\cos (e+f x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} \left(2 \cos ^2(e+f x)-1\right) (a+b \sin (e+f x))}\right)}{420 b^3 f \cos ^{\frac{3}{2}}(e+f x)}","-\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^3 g \sqrt{g \cos (e+f x)}}{b^4 f}+\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{2 a^2 g \sin (e+f x) \sqrt{g \cos (e+f x)}}{3 b^3 f}+\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{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 (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,"-1/420*((g*Cos[e + f*x])^(3/2)*((-2*(70*a^3 - 19*a*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) + ((210*a^3 - 21*a*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[e + f*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(-1 + 2*Cos[e + f*x]^2)*(a + b*Sin[e + f*x])) - (2*(-98*a^2*b - 40*b^3)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]]*Sqrt[1 - Cos[e + f*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[e + f*x]^2)/((1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(b^3*f*Cos[e + f*x]^(3/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*((a*Cos[2*(e + f*x)])/(5*b^2) + ((28*a^2 + 5*b^2)*Sin[e + f*x])/(42*b^3) - Sin[3*(e + f*x)]/(14*b)))/f","C",0
1378,1,1953,514,27.4286375,"\int \frac{(g \cos (e+f x))^{3/2} \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","\frac{\sec (e+f x) \left(-\frac{\cos (2 (e+f x))}{5 b}-\frac{2 a \sin (e+f x)}{3 b^2}\right) (g \cos (e+f x))^{3/2}}{f}+\frac{\left(\frac{28 a b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \sqrt{1-\cos ^2(e+f x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(e+f x)}{\left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{2 \left(10 a^2+3 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}+\frac{\left(30 a^2-3 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{5}{2}}(e+f x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\cos (e+f x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} \left(2 \cos ^2(e+f x)-1\right) (a+b \sin (e+f x))}\right) (g \cos (e+f x))^{3/2}}{60 b^2 f \cos ^{\frac{3}{2}}(e+f x)}","\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{2 a^2 g \sqrt{g \cos (e+f x)}}{b^3 f}-\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{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 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,"((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(-1/5*Cos[2*(e + f*x)]/b - (2*a*Sin[e + f*x])/(3*b^2)))/f + ((g*Cos[e + f*x])^(3/2)*((-2*(10*a^2 + 3*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) + ((30*a^2 - 3*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[e + f*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(-1 + 2*Cos[e + f*x]^2)*(a + b*Sin[e + f*x])) + (28*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]]*Sqrt[1 - Cos[e + f*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[e + f*x]^2)/((1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(60*b^2*f*Cos[e + f*x]^(3/2))","C",0
1379,1,1909,426,26.9546158,"\int \frac{(g \cos (e+f x))^{3/2} \sin (e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(a + b*Sin[e + f*x]),x]","\frac{2 (g \cos (e+f x))^{3/2} \tan (e+f x)}{3 b f}-\frac{(g \cos (e+f x))^{3/2} \left(\frac{4 b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \sqrt{1-\cos ^2(e+f x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(e+f x)}{\left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{2 a \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}+\frac{3 a \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{5}{2}}(e+f x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\cos (e+f x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} \left(2 \cos ^2(e+f x)-1\right) (a+b \sin (e+f x))}\right)}{6 b f \cos ^{\frac{3}{2}}(e+f 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}",1,"-1/6*((g*Cos[e + f*x])^(3/2)*((-2*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) + (3*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[e + f*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(-1 + 2*Cos[e + f*x]^2)*(a + b*Sin[e + f*x])) + (4*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]]*Sqrt[1 - Cos[e + f*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[e + f*x]^2)/((1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(b*f*Cos[e + f*x]^(3/2)) + (2*(g*Cos[e + f*x])^(3/2)*Tan[e + f*x])/(3*b*f)","C",0
1380,1,484,439,6.3316238,"\int \frac{(g \cos (e+f x))^{3/2} \csc (e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*Csc[e + f*x])/(a + b*Sin[e + f*x]),x]","\frac{\csc (e+f x) (g \cos (e+f x))^{3/2} \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(8 a b^{3/2} \cos ^{\frac{5}{2}}(e+f x) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-5 \left(a^2-b^2\right) \left(\sqrt{2} \sqrt[4]{a^2-b^2} \log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)-\sqrt{2} \sqrt[4]{a^2-b^2} \log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+2 \sqrt{2} \sqrt[4]{a^2-b^2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \sqrt{2} \sqrt[4]{a^2-b^2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-2 \sqrt{b} \log \left(1-\sqrt{\cos (e+f x)}\right)+2 \sqrt{b} \log \left(\sqrt{\cos (e+f x)}+1\right)+4 \sqrt{b} \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)\right)\right)}{20 a \sqrt{b} f \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(e+f x) (a \csc (e+f x)+b)}","\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*Cos[e + f*x])^(3/2)*Csc[e + f*x]*(8*a*b^(3/2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2) - 5*(a^2 - b^2)*(2*Sqrt[2]*(a^2 - b^2)^(1/4)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*Sqrt[2]*(a^2 - b^2)^(1/4)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 4*Sqrt[b]*ArcTan[Sqrt[Cos[e + f*x]]] - 2*Sqrt[b]*Log[1 - Sqrt[Cos[e + f*x]]] + 2*Sqrt[b]*Log[1 + Sqrt[Cos[e + f*x]]] + Sqrt[2]*(a^2 - b^2)^(1/4)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] - Sqrt[2]*(a^2 - b^2)^(1/4)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*(a + b*Sqrt[Sin[e + f*x]^2]))/(20*a*Sqrt[b]*(a^2 - b^2)*f*Cos[e + f*x]^(3/2)*(b + a*Csc[e + f*x]))","C",0
1381,1,2099,469,27.1096586,"\int \frac{(g \cos (e+f x))^{3/2} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*Csc[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","\text{Result too large to show}","-\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,"-1/4*((g*Cos[e + f*x])^(3/2)*((-4*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4)))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) - (b*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*((-10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) - (20*ArcTan[Sqrt[Cos[e + f*x]]])/a - (16*b*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(-a^2 + b^2) - (200*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (10*Log[1 - Sqrt[Cos[e + f*x]]])/a - (10*Log[1 + Sqrt[Cos[e + f*x]]])/a - (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4))))/(20*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - (6*b*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*((5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - (-2*Sqrt[2]*b^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*Sqrt[2]*b^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 4*(a^2 - b^2)^(3/4)*ArcTan[Sqrt[Cos[e + f*x]]] - 2*(a^2 - b^2)^(3/4)*Log[1 - Sqrt[Cos[e + f*x]]] + 2*(a^2 - b^2)^(3/4)*Log[1 + Sqrt[Cos[e + f*x]]] - Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(8*a*(a^2 - b^2)^(3/4))))/((1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/(a*f*Cos[e + f*x]^(3/2)) - ((g*Cos[e + f*x])^(3/2)*Csc[e + f*x]*Sec[e + f*x])/(a*f)","C",0
1382,1,2129,574,29.4758818,"\int \frac{(g \cos (e+f x))^{3/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*Csc[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","\text{Result too large to show}","-\frac{b^2 g^{3/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\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 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{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^{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{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*Cos[e + f*x])^(3/2)*((-2*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4)))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) - (b^2*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*((-10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) - (20*ArcTan[Sqrt[Cos[e + f*x]]])/a - (16*b*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(-a^2 + b^2) - (200*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (10*Log[1 - Sqrt[Cos[e + f*x]]])/a - (10*Log[1 + Sqrt[Cos[e + f*x]]])/a - (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4))))/(20*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - (2*(-a^2 + 3*b^2)*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*((5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - (-2*Sqrt[2]*b^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*Sqrt[2]*b^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 4*(a^2 - b^2)^(3/4)*ArcTan[Sqrt[Cos[e + f*x]]] - 2*(a^2 - b^2)^(3/4)*Log[1 - Sqrt[Cos[e + f*x]]] + 2*(a^2 - b^2)^(3/4)*Log[1 + Sqrt[Cos[e + f*x]]] - Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(8*a*(a^2 - b^2)^(3/4))))/((1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/(4*a^2*f*Cos[e + f*x]^(3/2)) + ((g*Cos[e + f*x])^(3/2)*((b*Csc[e + f*x])/a^2 - Csc[e + f*x]^2/(2*a))*Sec[e + f*x])/f","C",0
1383,1,867,610,27.2014741,"\int \frac{(g \cos (e+f x))^{5/2} \sin ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","\frac{(g \cos (e+f x))^{5/2} \sec ^2(e+f x) \left(-\frac{a \left(28 a^2-9 b^2\right) \cos (e+f x)}{42 b^4}+\frac{a \cos (3 (e+f x))}{14 b^2}-\frac{\left(b^2-18 a^2\right) \sin (2 (e+f x))}{90 b^3}-\frac{\sin (4 (e+f x))}{36 b}\right)}{f}-\frac{(g \cos (e+f x))^{5/2} \left(-\frac{\left(15 a^4-9 b^2 a^2-2 b^4\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(e+f x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{2 \left(6 a^3 b-2 a b^3\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right)}{15 b^4 f \cos ^{\frac{5}{2}}(e+f x)}","-\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{2 a^3 g (g \cos (e+f x))^{3/2}}{3 b^4 f}+\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{2 a^2 g \sin (e+f x) (g \cos (e+f x))^{3/2}}{5 b^3 f}+\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{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 (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,"-1/15*((g*Cos[e + f*x])^(5/2)*((-2*(6*a^3*b - 2*a*b^3)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) - ((15*a^4 - 9*a^2*b^2 - 2*b^4)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*Sin[e + f*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(b^4*f*Cos[e + f*x]^(5/2)) + ((g*Cos[e + f*x])^(5/2)*Sec[e + f*x]^2*(-1/42*(a*(28*a^2 - 9*b^2)*Cos[e + f*x])/b^4 + (a*Cos[3*(e + f*x)])/(14*b^2) - ((-18*a^2 + b^2)*Sin[2*(e + f*x)])/(90*b^3) - Sin[4*(e + f*x)]/(36*b)))/f","C",0
1384,1,824,501,27.1157089,"\int \frac{(g \cos (e+f x))^{5/2} \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","\frac{a \left(-\frac{\left(5 a^2-3 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(e+f x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{4 a b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right) (g \cos (e+f x))^{5/2}}{5 b^3 f \cos ^{\frac{5}{2}}(e+f x)}+\frac{\sec ^2(e+f x) \left(-\frac{\left(9 b^2-28 a^2\right) \cos (e+f x)}{42 b^3}-\frac{\cos (3 (e+f x))}{14 b}-\frac{a \sin (2 (e+f x))}{5 b^2}\right) (g \cos (e+f x))^{5/2}}{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{2 a^2 g (g \cos (e+f x))^{3/2}}{3 b^3 f}+\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{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{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*(g*Cos[e + f*x])^(5/2)*((-4*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) - ((5*a^2 - 3*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*Sin[e + f*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(5*b^3*f*Cos[e + f*x]^(5/2)) + ((g*Cos[e + f*x])^(5/2)*Sec[e + f*x]^2*(-1/42*((-28*a^2 + 9*b^2)*Cos[e + f*x])/b^3 - Cos[3*(e + f*x)]/(14*b) - (a*Sin[2*(e + f*x)])/(5*b^2)))/f","C",0
1385,1,737,413,24.8089789,"\int \frac{(g \cos (e+f x))^{5/2} \sin (e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(5/2)*Sin[e + f*x])/(a + b*Sin[e + f*x]),x]","\frac{(g \cos (e+f x))^{5/2} \left(\frac{4 a \sin (e+f x) \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(\frac{a \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{b \sqrt{\sin ^2(e+f x)} (a+b \sin (e+f x))}+\frac{\left(5 a^2-3 b^2\right) \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 b^{7/2} \left(b^2-a^2\right) (a+b \sin (e+f x))}+\frac{2 \cos ^{\frac{3}{2}}(e+f x) (3 b \sin (e+f x)-5 a)}{3 b^2}\right)}{5 f \cos ^{\frac{5}{2}}(e+f 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{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^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{2 g (g \cos (e+f x))^{3/2} (5 a-3 b \sin (e+f x))}{15 b^2 f}",1,"((g*Cos[e + f*x])^(5/2)*((2*Cos[e + f*x]^(3/2)*(-5*a + 3*b*Sin[e + f*x]))/(3*b^2) + ((5*a^2 - 3*b^2)*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*(a + b*Sqrt[Sin[e + f*x]^2]))/(12*b^(7/2)*(-a^2 + b^2)*(a + b*Sin[e + f*x])) + (4*a*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[e + f*x]*(a + b*Sqrt[Sin[e + f*x]^2]))/(b*Sqrt[Sin[e + f*x]^2]*(a + b*Sin[e + f*x]))))/(5*f*Cos[e + f*x]^(5/2))","C",0
1386,1,484,425,24.4600255,"\int \frac{(g \cos (e+f x))^{5/2} \csc (e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(5/2)*Csc[e + f*x])/(a + b*Sin[e + f*x]),x]","\frac{\csc (e+f x) (g \cos (e+f x))^{5/2} \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(8 a b^{5/2} \cos ^{\frac{7}{2}}(e+f x) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+7 \left(a^2-b^2\right) \left(\sqrt{2} \left(a^2-b^2\right)^{3/4} \log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)-\sqrt{2} \left(a^2-b^2\right)^{3/4} \log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)-2 \sqrt{2} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \sqrt{2} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+2 b^{3/2} \log \left(1-\sqrt{\cos (e+f x)}\right)-2 b^{3/2} \log \left(\sqrt{\cos (e+f x)}+1\right)+4 b^{3/2} \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)\right)\right)}{28 a b^{3/2} f \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(e+f x) (a \csc (e+f x)+b)}","\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} \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^{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*Cos[e + f*x])^(5/2)*Csc[e + f*x]*(8*a*b^(5/2)*AppellF1[7/4, 1/2, 1, 11/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(7/2) + 7*(a^2 - b^2)*(-2*Sqrt[2]*(a^2 - b^2)^(3/4)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*Sqrt[2]*(a^2 - b^2)^(3/4)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 4*b^(3/2)*ArcTan[Sqrt[Cos[e + f*x]]] + 2*b^(3/2)*Log[1 - Sqrt[Cos[e + f*x]]] - 2*b^(3/2)*Log[1 + Sqrt[Cos[e + f*x]]] + Sqrt[2]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] - Sqrt[2]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*(a + b*Sqrt[Sin[e + f*x]^2]))/(28*a*b^(3/2)*(a^2 - b^2)*f*Cos[e + f*x]^(5/2)*(b + a*Csc[e + f*x]))","C",0
1387,1,1556,462,27.2783157,"\int \frac{(g \cos (e+f x))^{5/2} \csc ^2(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(5/2)*Csc[e + f*x]^2)/(a + b*Sin[e + f*x]),x]","\frac{(g \cos (e+f x))^{5/2} \left(\frac{12 a \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(e+f x)} (b+a \csc (e+f x))}+\frac{5 b \left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \csc (e+f x) \left(6 \log \left(1-\sqrt{\cos (e+f x)}\right) a^2-6 \log \left(\sqrt{\cos (e+f x)}+1\right) a^2+8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x) a+6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+12 \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)-6 b^2 \log \left(1-\sqrt{\cos (e+f x)}\right)+6 b^2 \log \left(\sqrt{\cos (e+f x)}+1\right)-3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{12 \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) (b+a \csc (e+f x))}-\frac{\left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \csc (e+f x) \left(48 a b^{5/2} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{7}{2}}(e+f x)-56 a b^{5/2} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)-42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+84 b^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(a^2-b^2\right) \log \left(1-\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(b^2-a^2\right) \log \left(\sqrt{\cos (e+f x)}+1\right)+21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)-21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{84 \sqrt{b} \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) \left(2 \cos ^2(e+f x)-1\right) (b+a \csc (e+f x))}\right)}{4 a f \cos ^{\frac{5}{2}}(e+f x)}-\frac{(g \cos (e+f x))^{5/2} \csc (e+f x) \sec (e+f x)}{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,"((g*Cos[e + f*x])^(5/2)*((12*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4))))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) + (5*b*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*(6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 12*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 6*a^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*b^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*a^2*Log[1 + Sqrt[Cos[e + f*x]]] + 6*b^2*Log[1 + Sqrt[Cos[e + f*x]]] - 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(12*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - ((-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*(-42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 84*b^(3/2)*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] - 56*a*b^(5/2)*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 48*a*b^(5/2)*AppellF1[7/4, 1/2, 1, 11/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(7/2) + 42*b^(3/2)*(a^2 - b^2)*Log[1 - Sqrt[Cos[e + f*x]]] + 42*b^(3/2)*(-a^2 + b^2)*Log[1 + Sqrt[Cos[e + f*x]]] + 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] - 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(84*Sqrt[b]*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/(4*a*f*Cos[e + f*x]^(5/2)) - ((g*Cos[e + f*x])^(5/2)*Csc[e + f*x]*Sec[e + f*x])/(a*f)","C",0
1388,1,1590,557,29.4240743,"\int \frac{(g \cos (e+f x))^{5/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(5/2)*Csc[e + f*x]^3)/(a + b*Sin[e + f*x]),x]","\frac{(g \cos (e+f x))^{5/2} \left(\frac{b \cot (e+f x)}{a^2}-\frac{\cot (e+f x) \csc (e+f x)}{2 a}\right) \sec ^2(e+f x)}{f}-\frac{(g \cos (e+f x))^{5/2} \left(\frac{6 a b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(e+f x)} (b+a \csc (e+f x))}-\frac{\left(3 a^2-5 b^2\right) \left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \csc (e+f x) \left(6 \log \left(1-\sqrt{\cos (e+f x)}\right) a^2-6 \log \left(\sqrt{\cos (e+f x)}+1\right) a^2+8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x) a+6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+12 \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)-6 b^2 \log \left(1-\sqrt{\cos (e+f x)}\right)+6 b^2 \log \left(\sqrt{\cos (e+f x)}+1\right)-3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{12 \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) (b+a \csc (e+f x))}-\frac{\sqrt{b} \left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \csc (e+f x) \left(48 a b^{5/2} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{7}{2}}(e+f x)-56 a b^{5/2} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)-42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+84 b^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(a^2-b^2\right) \log \left(1-\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(b^2-a^2\right) \log \left(\sqrt{\cos (e+f x)}+1\right)+21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)-21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{84 \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) \left(2 \cos ^2(e+f x)-1\right) (b+a \csc (e+f x))}\right)}{4 a^2 f \cos ^{\frac{5}{2}}(e+f x)}","\frac{b^2 g^{5/2} \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\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{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{\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{\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{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,"-1/4*((g*Cos[e + f*x])^(5/2)*((6*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4))))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) - ((3*a^2 - 5*b^2)*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*(6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 12*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 6*a^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*b^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*a^2*Log[1 + Sqrt[Cos[e + f*x]]] + 6*b^2*Log[1 + Sqrt[Cos[e + f*x]]] - 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(12*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - (Sqrt[b]*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*(-42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 84*b^(3/2)*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] - 56*a*b^(5/2)*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 48*a*b^(5/2)*AppellF1[7/4, 1/2, 1, 11/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(7/2) + 42*b^(3/2)*(a^2 - b^2)*Log[1 - Sqrt[Cos[e + f*x]]] + 42*b^(3/2)*(-a^2 + b^2)*Log[1 + Sqrt[Cos[e + f*x]]] + 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] - 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(84*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/(a^2*f*Cos[e + f*x]^(5/2)) + ((g*Cos[e + f*x])^(5/2)*((b*Cot[e + f*x])/a^2 - (Cot[e + f*x]*Csc[e + f*x])/(2*a))*Sec[e + f*x]^2)/f","C",0
1389,1,1953,509,26.9175451,"\int \frac{\sin ^4(e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]^4/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{\cos (e+f x) \left(\frac{\cos (2 (e+f x))}{5 b}+\frac{2 a \sin (e+f x)}{3 b^2}\right)}{f \sqrt{g \cos (e+f x)}}-\frac{\sqrt{\cos (e+f x)} \left(\frac{28 a b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \sqrt{1-\cos ^2(e+f x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(e+f x)}{\left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{2 \left(10 a^2-27 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}+\frac{\left(30 a^2+27 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{5}{2}}(e+f x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\cos (e+f x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} \left(2 \cos ^2(e+f x)-1\right) (a+b \sin (e+f x))}\right)}{60 b^2 f \sqrt{g \cos (e+f x)}}","-\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{2 a^2 \sqrt{g \cos (e+f x)}}{b^3 f g}+\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{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 \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,"(Cos[e + f*x]*(Cos[2*(e + f*x)]/(5*b) + (2*a*Sin[e + f*x])/(3*b^2)))/(f*Sqrt[g*Cos[e + f*x]]) - (Sqrt[Cos[e + f*x]]*((-2*(10*a^2 - 27*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) + ((30*a^2 + 27*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[e + f*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(-1 + 2*Cos[e + f*x]^2)*(a + b*Sin[e + f*x])) + (28*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]]*Sqrt[1 - Cos[e + f*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[e + f*x]^2)/((1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(60*b^2*f*Sqrt[g*Cos[e + f*x]])","C",0
1390,1,1915,457,26.5241612,"\int \frac{\sin ^3(e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]^3/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{\sqrt{\cos (e+f x)} \left(-\frac{8 b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \sqrt{1-\cos ^2(e+f x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(e+f x)}{\left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{2 a \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}+\frac{3 a \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{5}{2}}(e+f x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\cos (e+f x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} \left(2 \cos ^2(e+f x)-1\right) (a+b \sin (e+f x))}\right)}{6 b f \sqrt{g \cos (e+f x)}}-\frac{2 \cos (e+f x) \sin (e+f x)}{3 b f \sqrt{g \cos (e+f x)}}","\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{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 \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,"(-2*Cos[e + f*x]*Sin[e + f*x])/(3*b*f*Sqrt[g*Cos[e + f*x]]) + (Sqrt[Cos[e + f*x]]*((-2*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) + (3*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[e + f*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(-1 + 2*Cos[e + f*x]^2)*(a + b*Sin[e + f*x])) - (8*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]]*Sqrt[1 - Cos[e + f*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[e + f*x]^2)/((1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(6*b*f*Sqrt[g*Cos[e + f*x]])","C",0
1391,1,1326,380,25.4139481,"\int \frac{\sin ^2(e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]^2/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{\sqrt{\cos (e+f x)} \left(-\frac{2 \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}-\frac{\left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{5}{2}}(e+f x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\cos (e+f x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} \left(2 \cos ^2(e+f x)-1\right) (a+b \sin (e+f x))}\right)}{2 f \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)}{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,"(Sqrt[Cos[e + f*x]]*((-2*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) - ((a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[e + f*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(-1 + 2*Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(2*f*Sqrt[g*Cos[e + f*x]])","C",0
1392,1,546,352,6.2043491,"\int \frac{\sin (e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","-\frac{2 \sqrt{\cos (e+f x)} \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\sin ^2(e+f x)} \sqrt{\cos (e+f x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(a^2+b^2 \cos ^2(e+f x)-b^2\right) \left(2 \cos ^2(e+f x) \left(2 b^2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)}+\frac{a \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right)}{f \sqrt{g \cos (e+f x)} (a+b \sin (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,"(-2*Sqrt[Cos[e + f*x]]*(a + b*Sqrt[Sin[e + f*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]^2])/((a^2 - b^2 + b^2*Cos[e + f*x]^2)*(-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2))))/(f*Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x]))","C",0
1393,1,698,369,20.6194235,"\int \frac{\csc (e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[Csc[e + f*x]/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","-\frac{2 \sqrt{\cos (e+f x)} \left(\cos ^2(e+f x)-1\right) \csc (e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (e+f x)} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\sqrt{1-\cos ^2(e+f x)} \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right) \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \cos ^2(e+f x) \left(2 b^2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)\right)}-\frac{-2 \left(a^2-b^2\right)^{3/4} \log \left(1-\sqrt{\cos (e+f x)}\right)+2 \left(a^2-b^2\right)^{3/4} \log \left(\sqrt{\cos (e+f x)}+1\right)+4 \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)-\sqrt{2} b^{3/2} \log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+\sqrt{2} b^{3/2} \log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)-2 \sqrt{2} b^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \sqrt{2} b^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)}{8 a \left(a^2-b^2\right)^{3/4}}\right)}{f \left(1-\cos ^2(e+f x)\right) \sqrt{g \cos (e+f x)} (a \csc (e+f x)+b)}","-\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{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{\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,"(-2*Sqrt[Cos[e + f*x]]*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*((5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - (-2*Sqrt[2]*b^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*Sqrt[2]*b^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 4*(a^2 - b^2)^(3/4)*ArcTan[Sqrt[Cos[e + f*x]]] - 2*(a^2 - b^2)^(3/4)*Log[1 - Sqrt[Cos[e + f*x]]] + 2*(a^2 - b^2)^(3/4)*Log[1 + Sqrt[Cos[e + f*x]]] - Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(8*a*(a^2 - b^2)^(3/4))))/(f*Sqrt[g*Cos[e + f*x]]*(1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))","C",0
1394,1,2093,448,30.0021644,"\int \frac{\csc ^2(e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[Csc[e + f*x]^2/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\text{Result too large to show}","\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^{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 \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,"-(Cot[e + f*x]/(a*f*Sqrt[g*Cos[e + f*x]])) - (Sqrt[Cos[e + f*x]]*((4*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4)))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) - (b*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*((-10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) - (20*ArcTan[Sqrt[Cos[e + f*x]]])/a - (16*b*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(-a^2 + b^2) - (200*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (10*Log[1 - Sqrt[Cos[e + f*x]]])/a - (10*Log[1 + Sqrt[Cos[e + f*x]]])/a - (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4))))/(20*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - (6*b*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*((5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - (-2*Sqrt[2]*b^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*Sqrt[2]*b^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 4*(a^2 - b^2)^(3/4)*ArcTan[Sqrt[Cos[e + f*x]]] - 2*(a^2 - b^2)^(3/4)*Log[1 - Sqrt[Cos[e + f*x]]] + 2*(a^2 - b^2)^(3/4)*Log[1 + Sqrt[Cos[e + f*x]]] - Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(8*a*(a^2 - b^2)^(3/4))))/((1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/(4*a*f*Sqrt[g*Cos[e + f*x]])","C",0
1395,1,2129,557,30.8458159,"\int \frac{\csc ^3(e+f x)}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[Csc[e + f*x]^3/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\text{Result too large to show}","-\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right)}{a^3 f \sqrt{g}}-\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{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^{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{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,"(Cos[e + f*x]*((b*Csc[e + f*x])/a^2 - Csc[e + f*x]^2/(2*a)))/(f*Sqrt[g*Cos[e + f*x]]) + (Sqrt[Cos[e + f*x]]*((-2*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4)))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) - (b^2*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*((-10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) - (20*ArcTan[Sqrt[Cos[e + f*x]]])/a - (16*b*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(-a^2 + b^2) - (200*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (10*Log[1 - Sqrt[Cos[e + f*x]]])/a - (10*Log[1 + Sqrt[Cos[e + f*x]]])/a - (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4))))/(20*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - (2*(3*a^2 + 3*b^2)*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*((5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - (-2*Sqrt[2]*b^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*Sqrt[2]*b^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 4*(a^2 - b^2)^(3/4)*ArcTan[Sqrt[Cos[e + f*x]]] - 2*(a^2 - b^2)^(3/4)*Log[1 - Sqrt[Cos[e + f*x]]] + 2*(a^2 - b^2)^(3/4)*Log[1 + Sqrt[Cos[e + f*x]]] - Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(8*a*(a^2 - b^2)^(3/4))))/((1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/(4*a^2*f*Sqrt[g*Cos[e + f*x]])","C",0
1396,1,820,584,26.8189849,"\int \frac{\sin ^4(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]^4/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{\left(\frac{2 \cos (e+f x)}{3 b}+\frac{2 \sec (e+f x) (a \sin (e+f x)-b)}{a^2-b^2}\right) \cos ^2(e+f x)}{f (g \cos (e+f x))^{3/2}}+\frac{a \left(\frac{4 a b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}-\frac{\left(a^2-2 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(e+f x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}\right) \cos ^{\frac{3}{2}}(e+f x)}{(a-b) b (a+b) f (g \cos (e+f x))^{3/2}}","\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{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{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)}}",1,"(Cos[e + f*x]^2*((2*Cos[e + f*x])/(3*b) + (2*Sec[e + f*x]*(-b + a*Sin[e + f*x]))/(a^2 - b^2)))/(f*(g*Cos[e + f*x])^(3/2)) + (a*Cos[e + f*x]^(3/2)*((4*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) - ((a^2 - 2*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*Sin[e + f*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/((a - b)*b*(a + b)*f*(g*Cos[e + f*x])^(3/2))","C",0
1397,1,793,509,26.8600959,"\int \frac{\sin ^3(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]^3/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{2 \cos (e+f x) (a-b \sin (e+f x))}{f \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}-\frac{\cos ^{\frac{3}{2}}(e+f x) \left(-\frac{\left(a^2-2 b^2\right) \sin ^2(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}+\frac{4 a b \sin (e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right)}{f (a-b) (a+b) (g \cos (e+f x))^{3/2}}","-\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}}",1,"(2*Cos[e + f*x]*(a - b*Sin[e + f*x]))/((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/2)) - (Cos[e + f*x]^(3/2)*((4*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) - ((a^2 - 2*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*Sin[e + f*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/((a - b)*(a + b)*f*(g*Cos[e + f*x])^(3/2))","C",0
1398,1,785,453,17.0063044,"\int \frac{\sin ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]^2/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{2 \cos (e+f x) (a \sin (e+f x)-b)}{f \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}-\frac{a \cos ^{\frac{3}{2}}(e+f x) \left(-\frac{\sin ^2(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 \sqrt{b} \left(b^2-a^2\right) \left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{4 a \sin (e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right)}{f (a-b) (a+b) (g \cos (e+f x))^{3/2}}","\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,"(2*Cos[e + f*x]*(-b + a*Sin[e + f*x]))/((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/2)) - (a*Cos[e + f*x]^(3/2)*((-4*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) - ((a + b*Sqrt[1 - Cos[e + f*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*Sin[e + f*x]^2)/(12*Sqrt[b]*(-a^2 + b^2)*(1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/((a - b)*(a + b)*f*(g*Cos[e + f*x])^(3/2))","C",0
1399,1,783,413,16.53307,"\int \frac{\sin (e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{2 \cos (e+f x) (a-b \sin (e+f x))}{f \left(a^2-b^2\right) (g \cos (e+f x))^{3/2}}+\frac{b \cos ^{\frac{3}{2}}(e+f x) \left(-\frac{\sin ^2(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}+b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 \sqrt{b} \left(b^2-a^2\right) \left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{4 a \sin (e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a \cos ^{\frac{3}{2}}(e+f x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}+i b \cos (e+f x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right)}{f (a-b) (a+b) (g \cos (e+f x))^{3/2}}","-\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,"(2*Cos[e + f*x]*(a - b*Sin[e + f*x]))/((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/2)) + (b*Cos[e + f*x]^(3/2)*((-4*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) - ((a + b*Sqrt[1 - Cos[e + f*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))*Sin[e + f*x]^2)/(12*Sqrt[b]*(-a^2 + b^2)*(1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/((a - b)*(a + b)*f*(g*Cos[e + f*x])^(3/2))","C",0
1400,1,1587,507,27.9801838,"\int \frac{\csc (e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[Csc[e + f*x]/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{2 \cos (e+f x) (a-b \sin (e+f x))}{\left(a^2-b^2\right) f (g \cos (e+f x))^{3/2}}-\frac{\cos ^{\frac{3}{2}}(e+f x) \left(\frac{8 a b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(e+f x)} (b+a \csc (e+f x))}-\frac{\left(b^2-2 a^2\right) \left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \csc (e+f x) \left(6 \log \left(1-\sqrt{\cos (e+f x)}\right) a^2-6 \log \left(\sqrt{\cos (e+f x)}+1\right) a^2+8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x) a+6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+12 \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)-6 b^2 \log \left(1-\sqrt{\cos (e+f x)}\right)+6 b^2 \log \left(\sqrt{\cos (e+f x)}+1\right)-3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{12 \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) (b+a \csc (e+f x))}-\frac{\sqrt{b} \left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \csc (e+f x) \left(48 a b^{5/2} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{7}{2}}(e+f x)-56 a b^{5/2} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)-42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+84 b^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(a^2-b^2\right) \log \left(1-\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(b^2-a^2\right) \log \left(\sqrt{\cos (e+f x)}+1\right)+21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)-21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{84 \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) \left(2 \cos ^2(e+f x)-1\right) (b+a \csc (e+f x))}\right)}{2 (a-b) (a+b) f (g \cos (e+f x))^{3/2}}","\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{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{\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,"-1/2*(Cos[e + f*x]^(3/2)*((8*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4))))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) - ((-2*a^2 + b^2)*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*(6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 12*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 6*a^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*b^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*a^2*Log[1 + Sqrt[Cos[e + f*x]]] + 6*b^2*Log[1 + Sqrt[Cos[e + f*x]]] - 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(12*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - (Sqrt[b]*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*(-42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 84*b^(3/2)*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] - 56*a*b^(5/2)*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 48*a*b^(5/2)*AppellF1[7/4, 1/2, 1, 11/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(7/2) + 42*b^(3/2)*(a^2 - b^2)*Log[1 - Sqrt[Cos[e + f*x]]] + 42*b^(3/2)*(-a^2 + b^2)*Log[1 + Sqrt[Cos[e + f*x]]] + 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] - 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(84*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/((a - b)*(a + b)*f*(g*Cos[e + f*x])^(3/2)) + (2*Cos[e + f*x]*(a - b*Sin[e + f*x]))/((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/2))","C",0
1401,1,1635,627,28.4362943,"\int \frac{\csc ^2(e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[Csc[e + f*x]^2/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{\cos ^2(e+f x) \left(\frac{2 \sec (e+f x) (a \sin (e+f x)-b)}{a^2-b^2}-\frac{\cot (e+f x)}{a}\right)}{f (g \cos (e+f x))^{3/2}}-\frac{\cos ^{\frac{3}{2}}(e+f x) \left(-\frac{2 \left(6 a^3+2 b^2 a\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(e+f x)} (b+a \csc (e+f x))}-\frac{\left(7 a^2 b-5 b^3\right) \left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \csc (e+f x) \left(6 \log \left(1-\sqrt{\cos (e+f x)}\right) a^2-6 \log \left(\sqrt{\cos (e+f x)}+1\right) a^2+8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x) a+6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)-6 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+12 \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)-6 b^2 \log \left(1-\sqrt{\cos (e+f x)}\right)+6 b^2 \log \left(\sqrt{\cos (e+f x)}+1\right)-3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+3 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4} \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{12 \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) (b+a \csc (e+f x))}-\frac{\left(b^3-3 a^2 b\right) \left(\cos ^2(e+f x)-1\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \csc (e+f x) \left(48 a b^{5/2} F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{7}{2}}(e+f x)-56 a b^{5/2} F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(e+f x)-42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+42 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)+84 b^{3/2} \left(a^2-b^2\right) \tan ^{-1}\left(\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(a^2-b^2\right) \log \left(1-\sqrt{\cos (e+f x)}\right)+42 b^{3/2} \left(b^2-a^2\right) \log \left(\sqrt{\cos (e+f x)}+1\right)+21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)-21 \sqrt{2} \left(a^2-b^2\right)^{3/4} \left(2 a^2-b^2\right) \log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{84 b^{3/2} \left(a^3-a b^2\right) \left(1-\cos ^2(e+f x)\right) \left(2 \cos ^2(e+f x)-1\right) (b+a \csc (e+f x))}\right)}{4 a (a-b) (a+b) f (g \cos (e+f x))^{3/2}}","-\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^{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{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,"-1/4*(Cos[e + f*x]^(3/2)*((-2*(6*a^3 + 2*a*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4))))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) - ((7*a^2*b - 5*b^3)*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*(6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 6*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 12*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] + 8*a*b*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 6*a^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*b^2*Log[1 - Sqrt[Cos[e + f*x]]] - 6*a^2*Log[1 + Sqrt[Cos[e + f*x]]] + 6*b^2*Log[1 + Sqrt[Cos[e + f*x]]] - 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + 3*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(12*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - ((-3*a^2*b + b^3)*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*(-42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 42*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 84*b^(3/2)*(a^2 - b^2)*ArcTan[Sqrt[Cos[e + f*x]]] - 56*a*b^(5/2)*AppellF1[3/4, 1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 48*a*b^(5/2)*AppellF1[7/4, 1/2, 1, 11/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(7/2) + 42*b^(3/2)*(a^2 - b^2)*Log[1 - Sqrt[Cos[e + f*x]]] + 42*b^(3/2)*(-a^2 + b^2)*Log[1 + Sqrt[Cos[e + f*x]]] + 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] - 21*Sqrt[2]*(a^2 - b^2)^(3/4)*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(84*b^(3/2)*(a^3 - a*b^2)*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/(a*(a - b)*(a + b)*f*(g*Cos[e + f*x])^(3/2)) + (Cos[e + f*x]^2*(-(Cot[e + f*x]/a) + (2*Sec[e + f*x]*(-b + a*Sin[e + f*x]))/(a^2 - b^2)))/(f*(g*Cos[e + f*x])^(3/2))","C",0
1402,1,1958,601,26.8368661,"\int \frac{\sin ^4(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]^4/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{\left(-\frac{4 a b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \sqrt{1-\cos ^2(e+f x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(e+f x)}{\left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{2 \left(3 b^2-7 a^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}+\frac{\left(3 a^2-3 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \cos (2 (e+f x)) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \cos ^{\frac{5}{2}}(e+f x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\cos (e+f x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} \left(2 \cos ^2(e+f x)-1\right) (a+b \sin (e+f x))}\right) \cos ^{\frac{5}{2}}(e+f x)}{6 (a-b) (a+b) f (g \cos (e+f x))^{5/2}}+\frac{2 (a \sin (e+f x)-b) \cos (e+f x)}{3 \left(a^2-b^2\right) f (g \cos (e+f x))^{5/2}}","\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{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{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^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{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)}}",1,"(2*Cos[e + f*x]*(-b + a*Sin[e + f*x]))/(3*(a^2 - b^2)*f*(g*Cos[e + f*x])^(5/2)) + (Cos[e + f*x]^(5/2)*((-2*(-7*a^2 + 3*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) + ((3*a^2 - 3*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[e + f*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(-1 + 2*Cos[e + f*x]^2)*(a + b*Sin[e + f*x])) - (4*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]]*Sqrt[1 - Cos[e + f*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[e + f*x]^2)/((1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(6*(a - b)*(a + b)*f*(g*Cos[e + f*x])^(5/2))","C",0
1403,1,1193,528,23.8646767,"\int \frac{\sin ^3(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]^3/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{2 \cos (e+f x) (a-b \sin (e+f x))}{3 \left(a^2-b^2\right) f (g \cos (e+f x))^{5/2}}-\frac{\cos ^{\frac{5}{2}}(e+f x) \left(\frac{4 a b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}-\frac{2 \left(3 a^2-2 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \sqrt{1-\cos ^2(e+f x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(e+f x)}{\left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}\right)}{3 (a-b) (a+b) f (g \cos (e+f x))^{5/2}}","-\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{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^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{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}}",1,"(2*Cos[e + f*x]*(a - b*Sin[e + f*x]))/(3*(a^2 - b^2)*f*(g*Cos[e + f*x])^(5/2)) - (Cos[e + f*x]^(5/2)*((4*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) - (2*(3*a^2 - 2*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]]*Sqrt[1 - Cos[e + f*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[e + f*x]^2)/((1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(3*(a - b)*(a + b)*f*(g*Cos[e + f*x])^(5/2))","C",0
1404,1,1184,468,23.3448987,"\int \frac{\sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]^2/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{2 \cos (e+f x) (a \sin (e+f x)-b)}{3 \left(a^2-b^2\right) f (g \cos (e+f x))^{5/2}}-\frac{a \cos ^{\frac{5}{2}}(e+f x) \left(\frac{2 b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \sqrt{1-\cos ^2(e+f x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(e+f x)}{\left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{4 a \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right)}{3 (a-b) (a+b) f (g \cos (e+f x))^{5/2}}","-\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{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^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)}}",1,"(2*Cos[e + f*x]*(-b + a*Sin[e + f*x]))/(3*(a^2 - b^2)*f*(g*Cos[e + f*x])^(5/2)) - (a*Cos[e + f*x]^(5/2)*((-4*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) + (2*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]]*Sqrt[1 - Cos[e + f*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[e + f*x]^2)/((1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(3*(a - b)*(a + b)*f*(g*Cos[e + f*x])^(5/2))","C",0
1405,1,1183,432,23.249172,"\int \frac{\sin (e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Integrate[Sin[e + f*x]/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{b \left(\frac{2 b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \sqrt{1-\cos ^2(e+f x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (e+f x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (e+f x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (e+f x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(e+f x)}{\left(1-\cos ^2(e+f x)\right) (a+b \sin (e+f x))}-\frac{4 a \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\sqrt{1-\cos ^2(e+f x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (e+f x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (e+f x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (e+f x)}{\sqrt{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right) \cos ^{\frac{5}{2}}(e+f x)}{3 (a-b) (a+b) f (g \cos (e+f x))^{5/2}}+\frac{2 (a-b \sin (e+f x)) \cos (e+f x)}{3 \left(a^2-b^2\right) f (g \cos (e+f x))^{5/2}}","-\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}}",1,"(2*Cos[e + f*x]*(a - b*Sin[e + f*x]))/(3*(a^2 - b^2)*f*(g*Cos[e + f*x])^(5/2)) + (b*Cos[e + f*x]^(5/2)*((-4*a*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4))*Sin[e + f*x])/(Sqrt[1 - Cos[e + f*x]^2]*(a + b*Sin[e + f*x])) + (2*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]]*Sqrt[1 - Cos[e + f*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[e + f*x]^2)/((1 - Cos[e + f*x]^2)*(a + b*Sin[e + f*x]))))/(3*(a - b)*(a + b)*f*(g*Cos[e + f*x])^(5/2))","C",0
1406,1,2136,527,30.572846,"\int \frac{\csc (e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Integrate[Csc[e + f*x]/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\text{Result too large to show}","-\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{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{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{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{\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,"(Cos[e + f*x]^(5/2)*((-8*a*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4)))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) - (b^2*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*((-10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) - (20*ArcTan[Sqrt[Cos[e + f*x]]])/a - (16*b*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(-a^2 + b^2) - (200*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (10*Log[1 - Sqrt[Cos[e + f*x]]])/a - (10*Log[1 + Sqrt[Cos[e + f*x]]])/a - (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4))))/(20*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - (2*(6*a^2 - 7*b^2)*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*((5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - (-2*Sqrt[2]*b^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*Sqrt[2]*b^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 4*(a^2 - b^2)^(3/4)*ArcTan[Sqrt[Cos[e + f*x]]] - 2*(a^2 - b^2)^(3/4)*Log[1 - Sqrt[Cos[e + f*x]]] + 2*(a^2 - b^2)^(3/4)*Log[1 + Sqrt[Cos[e + f*x]]] - Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(8*a*(a^2 - b^2)^(3/4))))/((1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/(6*(a - b)*(a + b)*f*(g*Cos[e + f*x])^(5/2)) + (2*Cos[e + f*x]*(a - b*Sin[e + f*x]))/(3*(a^2 - b^2)*f*(g*Cos[e + f*x])^(5/2))","C",0
1407,1,2183,651,29.9612938,"\int \frac{\csc ^2(e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Integrate[Csc[e + f*x]^2/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\text{Result too large to show}","\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{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^{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{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{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,"(Cos[e + f*x]^(5/2)*((-2*(10*a^3 - 18*a*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]))/(-a^2 + b^2)^(3/4)))/(Sqrt[1 - Cos[e + f*x]^2]*(b + a*Csc[e + f*x])) - ((-5*a^2*b + 3*b^3)*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Cos[2*(e + f*x)]*Csc[e + f*x]*((-10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (10*Sqrt[2]*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) - (20*ArcTan[Sqrt[Cos[e + f*x]]])/a - (16*b*AppellF1[5/4, 1/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(5/2))/(-a^2 + b^2) - (200*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) + (10*Log[1 - Sqrt[Cos[e + f*x]]])/a - (10*Log[1 + Sqrt[Cos[e + f*x]]])/a - (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*Sqrt[2]*(2*a^2 - b^2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(a*Sqrt[b]*(a^2 - b^2)^(3/4))))/(20*(1 - Cos[e + f*x]^2)*(-1 + 2*Cos[e + f*x]^2)*(b + a*Csc[e + f*x])) - (2*(-7*a^2*b + 9*b^3)*(-1 + Cos[e + f*x]^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Csc[e + f*x]*((5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/(Sqrt[1 - Cos[e + f*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - (-2*Sqrt[2]*b^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*Sqrt[2]*b^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] + 4*(a^2 - b^2)^(3/4)*ArcTan[Sqrt[Cos[e + f*x]]] - 2*(a^2 - b^2)^(3/4)*Log[1 - Sqrt[Cos[e + f*x]]] + 2*(a^2 - b^2)^(3/4)*Log[1 + Sqrt[Cos[e + f*x]]] - Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]] + Sqrt[2]*b^(3/2)*Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e + f*x]])/(8*a*(a^2 - b^2)^(3/4))))/((1 - Cos[e + f*x]^2)*(b + a*Csc[e + f*x]))))/(12*a*(a - b)*(a + b)*f*(g*Cos[e + f*x])^(5/2)) + (Cos[e + f*x]^3*(-(Csc[e + f*x]/a) + (2*Sec[e + f*x]^2*(-b + a*Sin[e + f*x]))/(3*(a^2 - b^2))))/(f*(g*Cos[e + f*x])^(5/2))","C",0
1408,1,1623,926,27.5866346,"\int \frac{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2))/(a + b*Sin[e + f*x]),x]","\frac{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{5/2} \left(-\frac{2 b \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}-\frac{\sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{42 a b^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{4 b f \sqrt{\cos (e+f x)} \sin ^{\frac{5}{2}}(e+f x)}-\frac{\sqrt{g \cos (e+f x)} \cot (e+f x) \csc (e+f x) (d \sin (e+f x))^{5/2}}{2 b f}","-\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,"-1/2*(Sqrt[g*Cos[e + f*x]]*Cot[e + f*x]*Csc[e + f*x]*(d*Sin[e + f*x])^(5/2))/(b*f) + (Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2)*((-2*b*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) - (Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(42*a*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(4*b*f*Sqrt[Cos[e + f*x]]*Sin[e + f*x]^(5/2))","C",0
1409,1,176,578,18.0966502,"\int \frac{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(3/2))/(a + b*Sin[e + f*x]),x]","\frac{2 d \sqrt{d \sin (e+f x)} (g \cos (e+f x))^{3/2} \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(b F_1\left(\frac{3}{4};-\frac{3}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-a F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)}{3 f g \left(a^2-b^2\right) \sqrt[4]{\sin ^2(e+f x)} (a+b \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{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,"(2*d*(b*AppellF1[3/4, -3/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - a*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]]*(a + b*Sqrt[Sin[e + f*x]^2]))/(3*(a^2 - b^2)*f*g*(Sin[e + f*x]^2)^(1/4)*(a + b*Sin[e + f*x]))","C",0
1410,1,178,509,11.0322134,"\int \frac{\sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Integrate[(Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(a + b*Sin[e + f*x]),x]","\frac{2 (d \sin (e+f x))^{3/2} (g \cos (e+f x))^{3/2} \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)}{3 d f g \left(a^2-b^2\right) \sin ^2(e+f x)^{3/4} (a+b \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{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,"(2*(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*(g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2)*(a + b*Sqrt[Sin[e + f*x]^2]))/(3*(a^2 - b^2)*d*f*g*(Sin[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x]))","C",0
1411,1,182,208,7.8261289,"\int \frac{\sqrt{g \cos (e+f x)}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[Sqrt[g*Cos[e + f*x]]/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]","-\frac{4 \sqrt{2} g \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)-1}} \tan ^{\frac{3}{2}}\left(\frac{1}{2} (e+f x)\right) \left(-\Pi \left(\frac{a}{\sqrt{b^2-a^2}-b};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}}\right)\right|-1\right)-\Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}}\right)\right|-1\right)+F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}}\right)\right|-1\right)\right)}{a f \sqrt{d \sin (e+f x)} \sqrt{g \cos (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,"(-4*Sqrt[2]*g*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(-1 + Cos[e + f*x])]*(EllipticF[ArcSin[1/Sqrt[Tan[(e + f*x)/2]]], -1] - EllipticPi[a/(-b + Sqrt[-a^2 + b^2]), ArcSin[1/Sqrt[Tan[(e + f*x)/2]]], -1] - EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[1/Sqrt[Tan[(e + f*x)/2]]], -1])*Tan[(e + f*x)/2]^(3/2))/(a*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",1
1412,1,1619,320,24.7244457,"\int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{\sqrt{g \cos (e+f x)} \sin ^{\frac{3}{2}}(e+f x) \left(\frac{4 a \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}-\frac{b \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{6 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{a f \sqrt{\cos (e+f x)} (d \sin (e+f x))^{3/2}}-\frac{2 \cos (e+f x) \sqrt{g \cos (e+f x)} \sin (e+f x)}{a f (d \sin (e+f x))^{3/2}}","-\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*Cos[e + f*x]*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(a*f*(d*Sin[e + f*x])^(3/2)) + (Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^(3/2)*((4*a*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) - (b*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(6*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(a*f*Sqrt[Cos[e + f*x]]*(d*Sin[e + f*x])^(3/2))","C",0
1413,1,1645,366,22.0778545,"\int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Integrate[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{\sqrt{g \cos (e+f x)} \left(\frac{2 b \cot (e+f x)}{a^2}-\frac{2 \cot (e+f x) \csc (e+f x)}{3 a}\right) \sin ^3(e+f x)}{f (d \sin (e+f x))^{5/2}}-\frac{b \sqrt{g \cos (e+f x)} \sin ^{\frac{5}{2}}(e+f x) \left(\frac{4 a \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}-\frac{b \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{6 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{a^2 f \sqrt{\cos (e+f x)} (d \sin (e+f x))^{5/2}}","\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 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 b (g \cos (e+f x))^{3/2}}{a^2 d^2 f g \sqrt{d \sin (e+f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{3 a d f g (d \sin (e+f x))^{3/2}}",1,"(Sqrt[g*Cos[e + f*x]]*((2*b*Cot[e + f*x])/a^2 - (2*Cot[e + f*x]*Csc[e + f*x])/(3*a))*Sin[e + f*x]^3)/(f*(d*Sin[e + f*x])^(5/2)) - (b*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^(5/2)*((4*a*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) - (b*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(6*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(a^2*f*Sqrt[Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2))","C",0
1414,1,1726,513,23.6130768,"\int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx","Integrate[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(7/2)*(a + b*Sin[e + f*x])),x]","\frac{\sqrt{g \cos (e+f x)} \left(-\frac{2 \cot (e+f x) \csc ^2(e+f x)}{5 a}-\frac{2 \left(2 \cos (e+f x) a^2+5 b^2 \cos (e+f x)\right) \csc (e+f x)}{5 a^3}+\frac{2 b \cot (e+f x) \csc (e+f x)}{3 a^2}\right) \sin ^4(e+f x)}{f (d \sin (e+f x))^{7/2}}-\frac{\sqrt{g \cos (e+f x)} \sin ^{\frac{7}{2}}(e+f x) \left(-\frac{2 \left(4 a^3+10 b^2 a\right) \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}+\frac{\left(10 b^3+2 a^2 b\right) \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\left(-5 b^3-2 a^2 b\right) \cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{5 a^3 f \sqrt{\cos (e+f x)} (d \sin (e+f x))^{7/2}}","-\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^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 b^2 (g \cos (e+f x))^{3/2}}{a^3 d^3 f g \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 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 \cos (e+f x))^{3/2}}{5 a d^3 f g \sqrt{d \sin (e+f x)}}-\frac{2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}}",1,"(Sqrt[g*Cos[e + f*x]]*((-2*(2*a^2*Cos[e + f*x] + 5*b^2*Cos[e + f*x])*Csc[e + f*x])/(5*a^3) + (2*b*Cot[e + f*x]*Csc[e + f*x])/(3*a^2) - (2*Cot[e + f*x]*Csc[e + f*x]^2)/(5*a))*Sin[e + f*x]^4)/(f*(d*Sin[e + f*x])^(7/2)) - (Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^(7/2)*((-2*(4*a^3 + 10*a*b^2)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((2*a^2*b + 10*b^3)*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + ((-2*a^2*b - 5*b^3)*Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(5*a^3*f*Sqrt[Cos[e + f*x]]*(d*Sin[e + f*x])^(7/2))","C",0
1415,1,1768,598,22.5513425,"\int \frac{\sqrt{g \cos (e+f x)}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx","Integrate[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(9/2)*(a + b*Sin[e + f*x])),x]","\frac{\sqrt{g \cos (e+f x)} \left(-\frac{2 \cot (e+f x) \csc ^3(e+f x)}{7 a}-\frac{2 \left(4 \cos (e+f x) a^2+7 b^2 \cos (e+f x)\right) \csc ^2(e+f x)}{21 a^3}+\frac{2 b \cot (e+f x) \csc ^2(e+f x)}{5 a^2}+\frac{2 \left(5 \cos (e+f x) b^3+2 a^2 \cos (e+f x) b\right) \csc (e+f x)}{5 a^4}\right) \sin ^5(e+f x)}{f (d \sin (e+f x))^{9/2}}+\frac{b \sqrt{g \cos (e+f x)} \left(-\frac{2 \left(4 a^3+10 b^2 a\right) \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}+\frac{\left(10 b^3+2 a^2 b\right) \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\left(-5 b^3-2 a^2 b\right) \cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right) \sin ^{\frac{9}{2}}(e+f x)}{5 a^4 f \sqrt{\cos (e+f x)} (d \sin (e+f x))^{9/2}}","\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{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 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{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{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{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,"(Sqrt[g*Cos[e + f*x]]*((2*(2*a^2*b*Cos[e + f*x] + 5*b^3*Cos[e + f*x])*Csc[e + f*x])/(5*a^4) - (2*(4*a^2*Cos[e + f*x] + 7*b^2*Cos[e + f*x])*Csc[e + f*x]^2)/(21*a^3) + (2*b*Cot[e + f*x]*Csc[e + f*x]^2)/(5*a^2) - (2*Cot[e + f*x]*Csc[e + f*x]^3)/(7*a))*Sin[e + f*x]^5)/(f*(d*Sin[e + f*x])^(9/2)) + (b*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^(9/2)*((-2*(4*a^3 + 10*a*b^2)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((2*a^2*b + 10*b^3)*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + ((-2*a^2*b - 5*b^3)*Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(5*a^4*f*Sqrt[Cos[e + f*x]]*(d*Sin[e + f*x])^(9/2))","C",0
1416,1,1898,982,28.8089347,"\int \frac{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2))/(a + b*Sin[e + f*x]),x]","\frac{(g \cos (e+f x))^{3/2} \sec (e+f x) (d \sin (e+f x))^{3/2}}{2 b f}-\frac{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} \left(\frac{10 b \left(a^2-b^2\right) \sqrt{\cos (e+f x)} \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{b \sqrt{1-\cos ^2(e+f x)} F_1\left(\frac{1}{4};-\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(4 F_1\left(\frac{5}{4};-\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) b^2+3 \left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}+\frac{a F_1\left(\frac{1}{4};-\frac{1}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-4 b^2 F_1\left(\frac{5}{4};-\frac{1}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}\right) \sin ^{\frac{5}{2}}(e+f x)}{\left(1-\cos ^2(e+f x)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right) (a+b \sin (e+f x))}+\frac{2 a \left(\frac{\sqrt{a} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)+\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)-\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{4 \sqrt{2} \left(a^2-b^2\right)^{3/4}}-\frac{b F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{5}{2}}(e+f x)}{5 a^2}\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \sqrt{\sin (e+f x)}}{\cos ^{\frac{5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt{\tan (e+f x)} \left(\tan ^2(e+f x)+1\right)^{3/2}}-\frac{a \cos (2 (e+f x)) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(\frac{200 b F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sqrt{\tan (e+f x)} a^4}{\sqrt{\tan ^2(e+f x)+1} \left(2 \left(F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) a^2+2 \left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-5 a^2 F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right) \left(a^2 \left(\tan ^2(e+f x)+1\right)-b^2 \tan ^2(e+f x)\right)}-20 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a+20 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a+10 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a-10 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a+\frac{10 \sqrt{2} \left(2 a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) \sqrt{a}}{\left(a^2-b^2\right)^{3/4}}-\frac{10 \sqrt{2} \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) \sqrt{a}}{\left(a^2-b^2\right)^{3/4}}-\frac{5 \sqrt{2} \left(2 a^2-b^2\right) \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) \sqrt{a}}{\left(a^2-b^2\right)^{3/4}}+\frac{5 \sqrt{2} \left(2 a^2-b^2\right) \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) \sqrt{a}}{\left(a^2-b^2\right)^{3/4}}+8 b F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{5}{2}}(e+f x)+\frac{40 b \sqrt{\tan (e+f x)}}{\sqrt{\tan ^2(e+f x)+1}}\right) \sqrt{\sin (e+f x)}}{10 b^2 \cos ^{\frac{5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt{\tan (e+f x)} \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1}}\right)}{4 b f \cos ^{\frac{3}{2}}(e+f x) \sin ^{\frac{3}{2}}(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{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,"((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(d*Sin[e + f*x])^(3/2))/(2*b*f) - ((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2)*((10*b*(a^2 - b^2)*Sqrt[Cos[e + f*x]]*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((b*AppellF1[1/4, -3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[1 - Cos[e + f*x]^2])/(-5*(a^2 - b^2)*AppellF1[1/4, -3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (4*b^2*AppellF1[5/4, -3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3*(a^2 - b^2)*AppellF1[5/4, 1/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2) + (a*AppellF1[1/4, -1/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])/(5*(a^2 - b^2)*AppellF1[1/4, -1/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-4*b^2*AppellF1[5/4, -1/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2))*Sin[e + f*x]^(5/2))/((1 - Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))*(a + b*Sin[e + f*x])) + (2*a*Sqrt[Sin[e + f*x]]*((Sqrt[a]*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] - Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(4*Sqrt[2]*(a^2 - b^2)^(3/4)) - (b*AppellF1[5/4, 1/2, 1, 9/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(5/2))/(5*a^2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(Cos[e + f*x]^(5/2)*(a + b*Sin[e + f*x])*Sqrt[Tan[e + f*x]]*(1 + Tan[e + f*x]^2)^(3/2)) - (a*Cos[2*(e + f*x)]*Sqrt[Sin[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(-20*Sqrt[2]*a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] + 20*Sqrt[2]*a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + (10*Sqrt[2]*Sqrt[a]*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(3/4) - (10*Sqrt[2]*Sqrt[a]*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(3/4) + 10*Sqrt[2]*a*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 10*Sqrt[2]*a*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (5*Sqrt[2]*Sqrt[a]*(2*a^2 - b^2)*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(3/4) + (5*Sqrt[2]*Sqrt[a]*(2*a^2 - b^2)*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(3/4) + 8*b*AppellF1[5/4, 1/2, 1, 9/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(5/2) + (40*b*Sqrt[Tan[e + f*x]])/Sqrt[1 + Tan[e + f*x]^2] + (200*a^4*b*AppellF1[1/4, 1/2, 1, 5/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sqrt[Tan[e + f*x]])/(Sqrt[1 + Tan[e + f*x]^2]*(-5*a^2*AppellF1[1/4, 1/2, 1, 5/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + 2*(2*(a^2 - b^2)*AppellF1[5/4, 1/2, 2, 9/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + a^2*AppellF1[5/4, 3/2, 1, 9/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Tan[e + f*x]^2)*(-(b^2*Tan[e + f*x]^2) + a^2*(1 + Tan[e + f*x]^2)))))/(10*b^2*Cos[e + f*x]^(5/2)*(a + b*Sin[e + f*x])*Sqrt[Tan[e + f*x]]*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(4*b*f*Cos[e + f*x]^(3/2)*Sin[e + f*x]^(3/2))","C",0
1417,1,604,611,21.4053835,"\int \frac{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])/(a + b*Sin[e + f*x]),x]","-\frac{(d \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2} \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(\frac{2 a F_1\left(\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{a^2-b^2}+\frac{5 \left(\sin ^2(e+f x) \left(a^2-b^2 \sin ^2(e+f x)\right) \left(3 \left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-4 b^2 F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)-5 \left(a^2-b^2\right) \left(a^2+b^2 \cos ^2(e+f x)-2 b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)}{b \sin ^2(e+f x)^{3/4} \left(a^2-b^2 \sin ^2(e+f x)\right) \left(\cos ^2(e+f x) \left(4 b^2 F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+3 \left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)}+\frac{\left(2 a^2-b^2\right) F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{b^3-a^2 b}\right)}{5 d f g \sin ^2(e+f x)^{3/4} (a+b \sin (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{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,"-1/5*((g*Cos[e + f*x])^(5/2)*(d*Sin[e + f*x])^(3/2)*(a + b*Sqrt[Sin[e + f*x]^2])*((2*a*AppellF1[5/4, 1/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])/(a^2 - b^2) + ((2*a^2 - b^2)*AppellF1[5/4, 3/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])/(-(a^2*b) + b^3) + (5*(-5*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*(a^2 - 2*b^2 + b^2*Cos[e + f*x]^2) + (-4*b^2*AppellF1[5/4, 3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3*(a^2 - b^2)*AppellF1[5/4, 7/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Sin[e + f*x]^2*(a^2 - b^2*Sin[e + f*x]^2)))/(b*(-5*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (4*b^2*AppellF1[5/4, 3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3*(-a^2 + b^2)*AppellF1[5/4, 7/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(Sin[e + f*x]^2)^(3/4)*(a^2 - b^2*Sin[e + f*x]^2))))/(d*f*g*(Sin[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x]))","C",0
1418,1,178,577,11.7474917,"\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{2 \sqrt{d \sin (e+f x)} (g \cos (e+f x))^{5/2} \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(b F_1\left(\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-a F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)}{5 d f g \left(a^2-b^2\right) \sqrt[4]{\sin ^2(e+f x)} (a+b \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 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,"(2*(b*AppellF1[5/4, 1/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - a*AppellF1[5/4, 3/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*(g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]]*(a + b*Sqrt[Sin[e + f*x]^2]))/(5*(a^2 - b^2)*d*f*g*(Sin[e + f*x]^2)^(1/4)*(a + b*Sin[e + f*x]))","C",0
1419,1,1095,321,20.4983177,"\int \frac{(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)/((d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","-\frac{2 \tan (e+f x) (g \cos (e+f x))^{3/2}}{a f (d \sin (e+f x))^{3/2}}-\frac{\sin ^{\frac{3}{2}}(e+f x) \left(\frac{2 a \sqrt{\sin (e+f x)} \left(\frac{\sqrt{a} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)+\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)-\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{4 \sqrt{2} \left(a^2-b^2\right)^{3/4}}-\frac{b F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{5}{2}}(e+f x)}{5 a^2}\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{\cos ^{\frac{5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt{\tan (e+f x)} \left(\tan ^2(e+f x)+1\right)^{3/2}}-\frac{2 b \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\left(1-\cos ^2(e+f x)\right)^{3/4} \left(\left(3 \left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-4 b^2 F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) b \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}+1\right)+\log \left(\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}-\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right)-\log \left(\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}+\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{a} \left(b^2-a^2\right)^{3/4}}\right) \sqrt{\sin (e+f x)}}{\sqrt[4]{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right) (g \cos (e+f x))^{3/2}}{a f \cos ^{\frac{3}{2}}(e+f x) (d \sin (e+f x))^{3/2}}","-\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*(g*Cos[e + f*x])^(3/2)*Tan[e + f*x])/(a*f*(d*Sin[e + f*x])^(3/2)) - ((g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^(3/2)*((-2*b*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/((1 - Cos[e + f*x]^2)^(3/4)*(5*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-4*b^2*AppellF1[5/4, 3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3*(a^2 - b^2)*AppellF1[5/4, 7/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*b*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] + Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] - ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] + ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)]))/(Sqrt[a]*(-a^2 + b^2)^(3/4)))*Sqrt[Sin[e + f*x]])/((1 - Cos[e + f*x]^2)^(1/4)*(a + b*Sin[e + f*x])) + (2*a*Sqrt[Sin[e + f*x]]*((Sqrt[a]*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] - Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(4*Sqrt[2]*(a^2 - b^2)^(3/4)) - (b*AppellF1[5/4, 1/2, 1, 9/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(5/2))/(5*a^2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(Cos[e + f*x]^(5/2)*(a + b*Sin[e + f*x])*Sqrt[Tan[e + f*x]]*(1 + Tan[e + f*x]^2)^(3/2))))/(a*f*Cos[e + f*x]^(3/2)*(d*Sin[e + f*x])^(3/2))","C",0
1420,1,1138,435,21.5064624,"\int \frac{(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)/((d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{(g \cos (e+f x))^{3/2} \left(\frac{2 b \csc (e+f x)}{a^2}-\frac{2 \csc ^2(e+f x)}{3 a}\right) \sin ^2(e+f x) \tan (e+f x)}{f (d \sin (e+f x))^{5/2}}-\frac{(g \cos (e+f x))^{3/2} \sin ^{\frac{5}{2}}(e+f x) \left(-\frac{2 \left(a^2-3 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sqrt{\sin (e+f x)} \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\left(1-\cos ^2(e+f x)\right)^{3/4} \left(\left(3 \left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-4 b^2 F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) b \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}+1\right)+\log \left(\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}-\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right)-\log \left(\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}+\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{a} \left(b^2-a^2\right)^{3/4}}\right)}{\sqrt[4]{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}-\frac{4 a b \sqrt{\sin (e+f x)} \left(\frac{\sqrt{a} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)+\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)-\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{4 \sqrt{2} \left(a^2-b^2\right)^{3/4}}-\frac{b F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{5}{2}}(e+f x)}{5 a^2}\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{\cos ^{\frac{5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt{\tan (e+f x)} \left(\tan ^2(e+f x)+1\right)^{3/2}}\right)}{3 a^2 f \cos ^{\frac{3}{2}}(e+f x) (d \sin (e+f x))^{5/2}}","\frac{2 b g \sqrt{g \cos (e+f x)}}{a^2 d^2 f \sqrt{d \sin (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{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 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,"((g*Cos[e + f*x])^(3/2)*((2*b*Csc[e + f*x])/a^2 - (2*Csc[e + f*x]^2)/(3*a))*Sin[e + f*x]^2*Tan[e + f*x])/(f*(d*Sin[e + f*x])^(5/2)) - ((g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^(5/2)*((-2*(a^2 - 3*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/((1 - Cos[e + f*x]^2)^(3/4)*(5*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-4*b^2*AppellF1[5/4, 3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3*(a^2 - b^2)*AppellF1[5/4, 7/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*b*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] + Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] - ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] + ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)]))/(Sqrt[a]*(-a^2 + b^2)^(3/4)))*Sqrt[Sin[e + f*x]])/((1 - Cos[e + f*x]^2)^(1/4)*(a + b*Sin[e + f*x])) - (4*a*b*Sqrt[Sin[e + f*x]]*((Sqrt[a]*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] - Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(4*Sqrt[2]*(a^2 - b^2)^(3/4)) - (b*AppellF1[5/4, 1/2, 1, 9/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(5/2))/(5*a^2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(Cos[e + f*x]^(5/2)*(a + b*Sin[e + f*x])*Sqrt[Tan[e + f*x]]*(1 + Tan[e + f*x]^2)^(3/2))))/(3*a^2*f*Cos[e + f*x]^(3/2)*(d*Sin[e + f*x])^(5/2))","C",0
1421,1,1165,525,21.5574298,"\int \frac{(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)/((d*Sin[e + f*x])^(7/2)*(a + b*Sin[e + f*x])),x]","\frac{b (g \cos (e+f x))^{3/2} \left(-\frac{2 \left(a^2-3 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sqrt{\sin (e+f x)} \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\left(1-\cos ^2(e+f x)\right)^{3/4} \left(\left(3 \left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-4 b^2 F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) b \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}+1\right)+\log \left(\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}-\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right)-\log \left(\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}+\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{a} \left(b^2-a^2\right)^{3/4}}\right)}{\sqrt[4]{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}-\frac{4 a b \sqrt{\sin (e+f x)} \left(\frac{\sqrt{a} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)+\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)-\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{4 \sqrt{2} \left(a^2-b^2\right)^{3/4}}-\frac{b F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{5}{2}}(e+f x)}{5 a^2}\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{\cos ^{\frac{5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt{\tan (e+f x)} \left(\tan ^2(e+f x)+1\right)^{3/2}}\right) \sin ^{\frac{7}{2}}(e+f x)}{3 a^3 f \cos ^{\frac{3}{2}}(e+f x) (d \sin (e+f x))^{7/2}}+\frac{(g \cos (e+f x))^{3/2} \left(-\frac{2 \csc ^3(e+f x)}{5 a}+\frac{2 b \csc ^2(e+f x)}{3 a^2}+\frac{2 \left(a^2-5 b^2\right) \csc (e+f x)}{5 a^3}\right) \tan (e+f x) \sin ^3(e+f x)}{f (d \sin (e+f x))^{7/2}}","-\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{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{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 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{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,"((g*Cos[e + f*x])^(3/2)*((2*(a^2 - 5*b^2)*Csc[e + f*x])/(5*a^3) + (2*b*Csc[e + f*x]^2)/(3*a^2) - (2*Csc[e + f*x]^3)/(5*a))*Sin[e + f*x]^3*Tan[e + f*x])/(f*(d*Sin[e + f*x])^(7/2)) + (b*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^(7/2)*((-2*(a^2 - 3*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/((1 - Cos[e + f*x]^2)^(3/4)*(5*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-4*b^2*AppellF1[5/4, 3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3*(a^2 - b^2)*AppellF1[5/4, 7/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*b*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] + Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] - ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] + ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)]))/(Sqrt[a]*(-a^2 + b^2)^(3/4)))*Sqrt[Sin[e + f*x]])/((1 - Cos[e + f*x]^2)^(1/4)*(a + b*Sin[e + f*x])) - (4*a*b*Sqrt[Sin[e + f*x]]*((Sqrt[a]*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] - Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(4*Sqrt[2]*(a^2 - b^2)^(3/4)) - (b*AppellF1[5/4, 1/2, 1, 9/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(5/2))/(5*a^2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(Cos[e + f*x]^(5/2)*(a + b*Sin[e + f*x])*Sqrt[Tan[e + f*x]]*(1 + Tan[e + f*x]^2)^(3/2))))/(3*a^3*f*Cos[e + f*x]^(3/2)*(d*Sin[e + f*x])^(7/2))","C",0
1422,1,1210,688,21.2201273,"\int \frac{(g \cos (e+f x))^{3/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)/((d*Sin[e + f*x])^(9/2)*(a + b*Sin[e + f*x])),x]","\frac{(g \cos (e+f x))^{3/2} \left(-\frac{2 \csc ^4(e+f x)}{7 a}+\frac{2 b \csc ^3(e+f x)}{5 a^2}+\frac{2 \left(a^2-7 b^2\right) \csc ^2(e+f x)}{21 a^3}-\frac{2 b \left(a^2-5 b^2\right) \csc (e+f x)}{5 a^4}\right) \sin ^4(e+f x) \tan (e+f x)}{f (d \sin (e+f x))^{9/2}}-\frac{(g \cos (e+f x))^{3/2} \sin ^{\frac{9}{2}}(e+f x) \left(\frac{2 \left(2 a^3 b-14 a b^3\right) \sqrt{\sin (e+f x)} \left(\frac{\sqrt{a} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)+\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)-\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{4 \sqrt{2} \left(a^2-b^2\right)^{3/4}}-\frac{b F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{5}{2}}(e+f x)}{5 a^2}\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{\cos ^{\frac{5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt{\tan (e+f x)} \left(\tan ^2(e+f x)+1\right)^{3/2}}-\frac{2 \left(2 a^4+7 b^2 a^2-21 b^4\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\left(1-\cos ^2(e+f x)\right)^{3/4} \left(\left(3 \left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-4 b^2 F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) b \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}+1\right)+\log \left(\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}-\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right)-\log \left(\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}+\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{a} \left(b^2-a^2\right)^{3/4}}\right) \sqrt{\sin (e+f x)}}{\sqrt[4]{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right)}{21 a^4 f \cos ^{\frac{3}{2}}(e+f x) (d \sin (e+f x))^{9/2}}","\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{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 \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^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 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{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,"((g*Cos[e + f*x])^(3/2)*((-2*b*(a^2 - 5*b^2)*Csc[e + f*x])/(5*a^4) + (2*(a^2 - 7*b^2)*Csc[e + f*x]^2)/(21*a^3) + (2*b*Csc[e + f*x]^3)/(5*a^2) - (2*Csc[e + f*x]^4)/(7*a))*Sin[e + f*x]^4*Tan[e + f*x])/(f*(d*Sin[e + f*x])^(9/2)) - ((g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^(9/2)*((-2*(2*a^4 + 7*a^2*b^2 - 21*b^4)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/((1 - Cos[e + f*x]^2)^(3/4)*(5*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-4*b^2*AppellF1[5/4, 3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3*(a^2 - b^2)*AppellF1[5/4, 7/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*b*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] + Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] - ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] + ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)]))/(Sqrt[a]*(-a^2 + b^2)^(3/4)))*Sqrt[Sin[e + f*x]])/((1 - Cos[e + f*x]^2)^(1/4)*(a + b*Sin[e + f*x])) + (2*(2*a^3*b - 14*a*b^3)*Sqrt[Sin[e + f*x]]*((Sqrt[a]*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] - Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(4*Sqrt[2]*(a^2 - b^2)^(3/4)) - (b*AppellF1[5/4, 1/2, 1, 9/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(5/2))/(5*a^2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(Cos[e + f*x]^(5/2)*(a + b*Sin[e + f*x])*Sqrt[Tan[e + f*x]]*(1 + Tan[e + f*x]^2)^(3/2))))/(21*a^4*f*Cos[e + f*x]^(3/2)*(d*Sin[e + f*x])^(9/2))","C",0
1423,1,1615,936,27.0346318,"\int \frac{(g \cos (e+f x))^{5/2} \sqrt{d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx","Integrate[((g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]])/(a + b*Sin[e + f*x]),x]","\frac{(g \cos (e+f x))^{5/2} \sec (e+f x) \sqrt{d \sin (e+f x)}}{2 b f}-\frac{(g \cos (e+f x))^{5/2} \sqrt{d \sin (e+f x)} \left(\frac{2 b \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{\left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}-\frac{\sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{42 a b^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{4 b f \cos ^{\frac{5}{2}}(e+f x) \sqrt{\sin (e+f 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}",1,"((g*Cos[e + f*x])^(5/2)*Sec[e + f*x]*Sqrt[d*Sin[e + f*x]])/(2*b*f) - ((g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]]*((2*b*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/((a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) - (Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(42*a*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(4*b*f*Cos[e + f*x]^(5/2)*Sqrt[Sin[e + f*x]])","C",0
1424,1,1399,572,25.8060564,"\int \frac{(g \cos (e+f x))^{5/2}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^(5/2)/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{(g \cos (e+f x))^{5/2} \sqrt{\sin (e+f x)} \left(\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{84 a^2 b^2 \cos ^{\frac{3}{2}}(e+f x) \sqrt{\sin (e+f x)} (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1}}+\frac{\sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a^2 \cos ^{\frac{3}{2}}(e+f x) \sqrt{\sin (e+f x)} (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2}}\right)}{2 f \cos ^{\frac{5}{2}}(e+f x) \sqrt{d \sin (e+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,"((g*Cos[e + f*x])^(5/2)*Sqrt[Sin[e + f*x]]*((Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(2*f*Cos[e + f*x]^(5/2)*Sqrt[d*Sin[e + f*x]])","C",0
1425,1,1611,616,26.6492382,"\int \frac{(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{(g \cos (e+f x))^{5/2} \sin ^{\frac{3}{2}}(e+f x) \left(\frac{2 a \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{\left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}-\frac{b \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{6 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{a f \cos ^{\frac{5}{2}}(e+f x) (d \sin (e+f x))^{3/2}}-\frac{2 (g \cos (e+f x))^{5/2} \tan (e+f x)}{a f (d \sin (e+f x))^{3/2}}","-\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,"(-2*(g*Cos[e + f*x])^(5/2)*Tan[e + f*x])/(a*f*(d*Sin[e + f*x])^(3/2)) + ((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^(3/2)*((2*a*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/((a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) - (b*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(6*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(a*f*Cos[e + f*x]^(5/2)*(d*Sin[e + f*x])^(3/2))","C",0
1426,1,1656,359,26.3535047,"\int \frac{(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{(g \cos (e+f x))^{5/2} \left(\frac{2 b \cot (e+f x)}{a^2}-\frac{2 \cot (e+f x) \csc (e+f x)}{3 a}\right) \sin (e+f x) \tan ^2(e+f x)}{f (d \sin (e+f x))^{5/2}}-\frac{(g \cos (e+f x))^{5/2} \sin ^{\frac{5}{2}}(e+f x) \left(\frac{4 a b \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}+\frac{\left(a^2-2 b^2\right) \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{a^2 f \cos ^{\frac{5}{2}}(e+f x) (d \sin (e+f x))^{5/2}}","\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,"((g*Cos[e + f*x])^(5/2)*((2*b*Cot[e + f*x])/a^2 - (2*Cot[e + f*x]*Csc[e + f*x])/(3*a))*Sin[e + f*x]*Tan[e + f*x]^2)/(f*(d*Sin[e + f*x])^(5/2)) - ((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^(5/2)*((4*a*b*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((a^2 - 2*b^2)*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(a^2*f*Cos[e + f*x]^(5/2)*(d*Sin[e + f*x])^(5/2))","C",0
1427,1,1734,519,24.6135271,"\int \frac{(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(7/2)*(a + b*Sin[e + f*x])),x]","\frac{(g \cos (e+f x))^{5/2} \left(-\frac{2 \left(6 a^3-10 a b^2\right) \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}+\frac{\left(8 a^2 b-10 b^3\right) \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\left(5 b^3-3 a^2 b\right) \cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right) \sin ^{\frac{7}{2}}(e+f x)}{5 a^3 f \cos ^{\frac{5}{2}}(e+f x) (d \sin (e+f x))^{7/2}}+\frac{(g \cos (e+f x))^{5/2} \left(-\frac{2 \cot (e+f x) \csc ^2(e+f x)}{5 a}+\frac{2 \left(3 a^2 \cos (e+f x)-5 b^2 \cos (e+f x)\right) \csc (e+f x)}{5 a^3}+\frac{2 b \cot (e+f x) \csc (e+f x)}{3 a^2}\right) \tan ^2(e+f x) \sin ^2(e+f x)}{f (d \sin (e+f x))^{7/2}}","-\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{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{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,"((g*Cos[e + f*x])^(5/2)*((2*(3*a^2*Cos[e + f*x] - 5*b^2*Cos[e + f*x])*Csc[e + f*x])/(5*a^3) + (2*b*Cot[e + f*x]*Csc[e + f*x])/(3*a^2) - (2*Cot[e + f*x]*Csc[e + f*x]^2)/(5*a))*Sin[e + f*x]^2*Tan[e + f*x]^2)/(f*(d*Sin[e + f*x])^(7/2)) + ((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^(7/2)*((-2*(6*a^3 - 10*a*b^2)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((8*a^2*b - 10*b^3)*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + ((-3*a^2*b + 5*b^3)*Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(5*a^3*f*Cos[e + f*x]^(5/2)*(d*Sin[e + f*x])^(7/2))","C",0
1428,1,1776,612,23.7458924,"\int \frac{(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{9/2} (a+b \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(9/2)*(a + b*Sin[e + f*x])),x]","\frac{(g \cos (e+f x))^{5/2} \left(-\frac{2 \cot (e+f x) \csc ^3(e+f x)}{7 a}+\frac{2 \left(3 a^2 \cos (e+f x)-7 b^2 \cos (e+f x)\right) \csc ^2(e+f x)}{21 a^3}+\frac{2 b \cot (e+f x) \csc ^2(e+f x)}{5 a^2}-\frac{2 \left(3 a^2 b \cos (e+f x)-5 b^3 \cos (e+f x)\right) \csc (e+f x)}{5 a^4}\right) \sin ^3(e+f x) \tan ^2(e+f x)}{f (d \sin (e+f x))^{9/2}}-\frac{b (g \cos (e+f x))^{5/2} \sin ^{\frac{9}{2}}(e+f x) \left(-\frac{2 \left(6 a^3-10 a b^2\right) \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}+\frac{\left(8 a^2 b-10 b^3\right) \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\left(5 b^3-3 a^2 b\right) \cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{5 a^4 f \cos ^{\frac{5}{2}}(e+f x) (d \sin (e+f x))^{9/2}}","\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{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{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 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{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,"((g*Cos[e + f*x])^(5/2)*((-2*(3*a^2*b*Cos[e + f*x] - 5*b^3*Cos[e + f*x])*Csc[e + f*x])/(5*a^4) + (2*(3*a^2*Cos[e + f*x] - 7*b^2*Cos[e + f*x])*Csc[e + f*x]^2)/(21*a^3) + (2*b*Cot[e + f*x]*Csc[e + f*x]^2)/(5*a^2) - (2*Cot[e + f*x]*Csc[e + f*x]^3)/(7*a))*Sin[e + f*x]^3*Tan[e + f*x]^2)/(f*(d*Sin[e + f*x])^(9/2)) - (b*(g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^(9/2)*((-2*(6*a^3 - 10*a*b^2)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((8*a^2*b - 10*b^3)*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + ((-3*a^2*b + 5*b^3)*Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(5*a^4*f*Cos[e + f*x]^(5/2)*(d*Sin[e + f*x])^(9/2))","C",0
1429,1,1850,822,25.0176377,"\int \frac{(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(11/2)*(a + b*Sin[e + f*x])),x]","\frac{(g \cos (e+f x))^{5/2} \left(-\frac{2 \left(4 a^5+18 b^2 a^3-30 b^4 a\right) \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}+\frac{\left(-30 b^5+24 a^2 b^3+2 a^4 b\right) \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\left(15 b^5-9 a^2 b^3-2 a^4 b\right) \cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right) \sin ^{\frac{11}{2}}(e+f x)}{15 a^5 f \cos ^{\frac{5}{2}}(e+f x) (d \sin (e+f x))^{11/2}}+\frac{(g \cos (e+f x))^{5/2} \left(-\frac{2 \cot (e+f x) \csc ^4(e+f x)}{9 a}+\frac{2 \left(a^2 \cos (e+f x)-3 b^2 \cos (e+f x)\right) \csc ^3(e+f x)}{15 a^3}+\frac{2 b \cot (e+f x) \csc ^3(e+f x)}{7 a^2}-\frac{2 \left(3 a^2 b \cos (e+f x)-7 b^3 \cos (e+f x)\right) \csc ^2(e+f x)}{21 a^4}+\frac{2 \left(2 \cos (e+f x) a^4+9 b^2 \cos (e+f x) a^2-15 b^4 \cos (e+f x)\right) \csc (e+f x)}{15 a^5}\right) \tan ^2(e+f x) \sin ^4(e+f x)}{f (d \sin (e+f x))^{11/2}}","-\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,"((g*Cos[e + f*x])^(5/2)*((2*(2*a^4*Cos[e + f*x] + 9*a^2*b^2*Cos[e + f*x] - 15*b^4*Cos[e + f*x])*Csc[e + f*x])/(15*a^5) - (2*(3*a^2*b*Cos[e + f*x] - 7*b^3*Cos[e + f*x])*Csc[e + f*x]^2)/(21*a^4) + (2*(a^2*Cos[e + f*x] - 3*b^2*Cos[e + f*x])*Csc[e + f*x]^3)/(15*a^3) + (2*b*Cot[e + f*x]*Csc[e + f*x]^3)/(7*a^2) - (2*Cot[e + f*x]*Csc[e + f*x]^4)/(9*a))*Sin[e + f*x]^4*Tan[e + f*x]^2)/(f*(d*Sin[e + f*x])^(11/2)) + ((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^(11/2)*((-2*(4*a^5 + 18*a^3*b^2 - 30*a*b^4)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((2*a^4*b + 24*a^2*b^3 - 30*b^5)*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + ((-2*a^4*b - 9*a^2*b^3 + 15*b^5)*Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(15*a^5*f*Cos[e + f*x]^(5/2)*(d*Sin[e + f*x])^(11/2))","C",0
1430,1,1318,616,27.0750013,"\int \frac{(d \sin (e+f x))^{5/2}}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[(d*Sin[e + f*x])^(5/2)/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{\sqrt{\cos (e+f x)} (d \sin (e+f x))^{5/2} \left(\frac{2 \sqrt{\sin (e+f x)} \left(\frac{\sqrt{a} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)+\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)-\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{4 \sqrt{2} \left(a^2-b^2\right)^{3/4}}-\frac{b F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{5}{2}}(e+f x)}{5 a^2}\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{\cos ^{\frac{5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt{\tan (e+f x)} \left(\tan ^2(e+f x)+1\right)^{3/2}}+\frac{\cos (2 (e+f x)) \sqrt{\sin (e+f x)} \left(\frac{200 b F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \sqrt{\tan (e+f x)} a^4}{\sqrt{\tan ^2(e+f x)+1} \left(2 \left(F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) a^2+2 \left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right) \tan ^2(e+f x)-5 a^2 F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right)\right) \left(a^2 \left(\tan ^2(e+f x)+1\right)-b^2 \tan ^2(e+f x)\right)}-20 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a+20 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a+10 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a-10 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a+\frac{10 \sqrt{2} \left(2 a^2-b^2\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) \sqrt{a}}{\left(a^2-b^2\right)^{3/4}}-\frac{10 \sqrt{2} \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) \sqrt{a}}{\left(a^2-b^2\right)^{3/4}}-\frac{5 \sqrt{2} \left(2 a^2-b^2\right) \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) \sqrt{a}}{\left(a^2-b^2\right)^{3/4}}+\frac{5 \sqrt{2} \left(2 a^2-b^2\right) \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) \sqrt{a}}{\left(a^2-b^2\right)^{3/4}}+8 b F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{5}{2}}(e+f x)+\frac{40 b \sqrt{\tan (e+f x)}}{\sqrt{\tan ^2(e+f x)+1}}\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{20 b^2 \cos ^{\frac{5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt{\tan (e+f x)} \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1}}\right)}{2 f \sqrt{g \cos (e+f x)} \sin ^{\frac{5}{2}}(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^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{d^2 \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{b f g}",1,"(Sqrt[Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2)*((2*Sqrt[Sin[e + f*x]]*((Sqrt[a]*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] - Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(4*Sqrt[2]*(a^2 - b^2)^(3/4)) - (b*AppellF1[5/4, 1/2, 1, 9/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(5/2))/(5*a^2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(Cos[e + f*x]^(5/2)*(a + b*Sin[e + f*x])*Sqrt[Tan[e + f*x]]*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Sin[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(-20*Sqrt[2]*a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] + 20*Sqrt[2]*a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + (10*Sqrt[2]*Sqrt[a]*(2*a^2 - b^2)*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(3/4) - (10*Sqrt[2]*Sqrt[a]*(2*a^2 - b^2)*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(3/4) + 10*Sqrt[2]*a*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 10*Sqrt[2]*a*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (5*Sqrt[2]*Sqrt[a]*(2*a^2 - b^2)*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(3/4) + (5*Sqrt[2]*Sqrt[a]*(2*a^2 - b^2)*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(3/4) + 8*b*AppellF1[5/4, 1/2, 1, 9/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(5/2) + (40*b*Sqrt[Tan[e + f*x]])/Sqrt[1 + Tan[e + f*x]^2] + (200*a^4*b*AppellF1[1/4, 1/2, 1, 5/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sqrt[Tan[e + f*x]])/(Sqrt[1 + Tan[e + f*x]^2]*(-5*a^2*AppellF1[1/4, 1/2, 1, 5/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + 2*(2*(a^2 - b^2)*AppellF1[5/4, 1/2, 2, 9/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + a^2*AppellF1[5/4, 3/2, 1, 9/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])*Tan[e + f*x]^2)*(-(b^2*Tan[e + f*x]^2) + a^2*(1 + Tan[e + f*x]^2)))))/(20*b^2*Cos[e + f*x]^(5/2)*(a + b*Sin[e + f*x])*Sqrt[Tan[e + f*x]]*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(2*f*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^(5/2))","C",0
1431,1,518,508,16.4517205,"\int \frac{(d \sin (e+f x))^{3/2}}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[(d*Sin[e + f*x])^(3/2)/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{10 \left(a^2-b^2\right) \cot (e+f x) (d \sin (e+f x))^{3/2} \left(a+b \sqrt{\sin ^2(e+f x)}\right) \left(\frac{a F_1\left(\frac{1}{4};-\frac{1}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\cos ^2(e+f x) \left(\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-4 b^2 F_1\left(\frac{5}{4};-\frac{1}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}+\frac{b \sqrt{\sin ^2(e+f x)} F_1\left(\frac{1}{4};-\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\cos ^2(e+f x) \left(4 b^2 F_1\left(\frac{5}{4};-\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)+3 \left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}\right)}{f \sqrt{g \cos (e+f x)} (b \sin (e+f x)-a) (a+b \sin (e+f x))^2}","\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,"(10*(a^2 - b^2)*Cot[e + f*x]*(d*Sin[e + f*x])^(3/2)*(a + b*Sqrt[Sin[e + f*x]^2])*((a*AppellF1[1/4, -1/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])/(5*(a^2 - b^2)*AppellF1[1/4, -1/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-4*b^2*AppellF1[5/4, -1/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2) + (b*AppellF1[1/4, -3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[e + f*x]^2])/(-5*(a^2 - b^2)*AppellF1[1/4, -3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (4*b^2*AppellF1[5/4, -3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3*(a^2 - b^2)*AppellF1[5/4, 1/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)))/(f*Sqrt[g*Cos[e + f*x]]*(-a + b*Sin[e + f*x])*(a + b*Sin[e + f*x])^2)","C",0
1432,1,166,209,4.4421186,"\int \frac{\sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[Sqrt[d*Sin[e + f*x]]/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]","-\frac{2 \sqrt{2} \sqrt{\tan \left(\frac{1}{2} (e+f x)\right)} \cot (e+f x) \sqrt{d \sin (e+f x)} \left(\Pi \left(\frac{a}{\sqrt{b^2-a^2}-b};\left.\sin ^{-1}\left(\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}\right)\right|-1\right)\right)}{f \sqrt{b^2-a^2} \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \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]*Cot[e + f*x]*(EllipticPi[a/(-b + Sqrt[-a^2 + b^2]), ArcSin[Sqrt[Tan[(e + f*x)/2]]], -1] - EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[Tan[(e + f*x)/2]]], -1])*Sqrt[d*Sin[e + f*x]]*Sqrt[Tan[(e + f*x)/2]])/(Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])])","A",1
1433,1,209,273,10.098243,"\int \frac{1}{\sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[1/(Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]","-\frac{4 \sqrt{2} \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)-1}} \tan ^{\frac{3}{2}}\left(\frac{1}{2} (e+f x)\right) \left(\sqrt{b^2-a^2} F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}}\right)\right|-1\right)+b \left(\Pi \left(\frac{a}{\sqrt{b^2-a^2}-b};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}}\right)\right|-1\right)-\Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}}\right)\right|-1\right)\right)\right)}{a f \sqrt{b^2-a^2} \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,"(-4*Sqrt[2]*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(-1 + Cos[e + f*x])]*(Sqrt[-a^2 + b^2]*EllipticF[ArcSin[1/Sqrt[Tan[(e + f*x)/2]]], -1] + b*(EllipticPi[a/(-b + Sqrt[-a^2 + b^2]), ArcSin[1/Sqrt[Tan[(e + f*x)/2]]], -1] - EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[1/Sqrt[Tan[(e + f*x)/2]]], -1]))*Tan[(e + f*x)/2]^(3/2))/(a*Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",1
1434,1,225,320,10.8014441,"\int \frac{1}{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[1/(Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{2 \left(\frac{2 \sqrt{2} b \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)-1}} \tan ^{\frac{3}{2}}\left(\frac{1}{2} (e+f x)\right) \left(\sqrt{b^2-a^2} F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}}\right)\right|-1\right)+b \left(\Pi \left(\frac{a}{\sqrt{b^2-a^2}-b};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}}\right)\right|-1\right)-\Pi \left(-\frac{a}{b+\sqrt{b^2-a^2}};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{2} (e+f x)\right)}}\right)\right|-1\right)\right)\right)}{\sqrt{b^2-a^2}}-a \cos (e+f x)\right)}{a^2 d f \sqrt{d \sin (e+f x)} \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{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*(-(a*Cos[e + f*x]) + (2*Sqrt[2]*b*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(-1 + Cos[e + f*x])]*(Sqrt[-a^2 + b^2]*EllipticF[ArcSin[1/Sqrt[Tan[(e + f*x)/2]]], -1] + b*(EllipticPi[a/(-b + Sqrt[-a^2 + b^2]), ArcSin[1/Sqrt[Tan[(e + f*x)/2]]], -1] - EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[1/Sqrt[Tan[(e + f*x)/2]]], -1]))*Tan[(e + f*x)/2]^(3/2))/Sqrt[-a^2 + b^2]))/(a^2*d*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])","A",1
1435,1,1140,424,20.2252104,"\int \frac{1}{\sqrt{g \cos (e+f x)} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Integrate[1/(Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{\cos (e+f x) \left(\frac{2 b \csc (e+f x)}{a^2}-\frac{2 \csc ^2(e+f x)}{3 a}\right) \sin ^3(e+f x)}{f \sqrt{g \cos (e+f x)} (d \sin (e+f x))^{5/2}}+\frac{\sqrt{\cos (e+f x)} \left(\frac{4 a b \sqrt{\sin (e+f x)} \left(\frac{\sqrt{a} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)+\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)-\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{4 \sqrt{2} \left(a^2-b^2\right)^{3/4}}-\frac{b F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{5}{2}}(e+f x)}{5 a^2}\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{\cos ^{\frac{5}{2}}(e+f x) (a+b \sin (e+f x)) \sqrt{\tan (e+f x)} \left(\tan ^2(e+f x)+1\right)^{3/2}}-\frac{2 \left(2 a^2+3 b^2\right) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right) \sqrt{\cos (e+f x)}}{\left(1-\cos ^2(e+f x)\right)^{3/4} \left(\left(3 \left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-4 b^2 F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^2(e+f x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) b \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}+1\right)+\log \left(\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}-\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right)-\log \left(\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}+\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{a} \left(b^2-a^2\right)^{3/4}}\right) \sqrt{\sin (e+f x)}}{\sqrt[4]{1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right) \sin ^{\frac{5}{2}}(e+f x)}{3 a^2 f \sqrt{g \cos (e+f x)} (d \sin (e+f x))^{5/2}}","\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 b \sqrt{g \cos (e+f x)}}{a^2 d^2 f g \sqrt{d \sin (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 \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,"(Cos[e + f*x]*((2*b*Csc[e + f*x])/a^2 - (2*Csc[e + f*x]^2)/(3*a))*Sin[e + f*x]^3)/(f*Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2)) + (Sqrt[Cos[e + f*x]]*Sin[e + f*x]^(5/2)*((-2*(2*a^2 + 3*b^2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/((1 - Cos[e + f*x]^2)^(3/4)*(5*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-4*b^2*AppellF1[5/4, 3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3*(a^2 - b^2)*AppellF1[5/4, 7/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*b*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] + Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] - ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] + ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)]))/(Sqrt[a]*(-a^2 + b^2)^(3/4)))*Sqrt[Sin[e + f*x]])/((1 - Cos[e + f*x]^2)^(1/4)*(a + b*Sin[e + f*x])) + (4*a*b*Sqrt[Sin[e + f*x]]*((Sqrt[a]*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] - Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(4*Sqrt[2]*(a^2 - b^2)^(3/4)) - (b*AppellF1[5/4, 1/2, 1, 9/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(5/2))/(5*a^2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(Cos[e + f*x]^(5/2)*(a + b*Sin[e + f*x])*Sqrt[Tan[e + f*x]]*(1 + Tan[e + f*x]^2)^(3/2))))/(3*a^2*f*Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(5/2))","C",0
1436,1,1290,1064,80.6576066,"\int \frac{(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[(d*Sin[e + f*x])^(5/2)/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{2 \cot (e+f x) \csc (e+f x) (d \sin (e+f x))^{5/2} (a \sin (e+f x)-b)}{\left(a^2-b^2\right) f (g \cos (e+f x))^{3/2}}-\frac{\cos ^{\frac{3}{2}}(e+f x) (d \sin (e+f x))^{5/2} \left(-\frac{2 \left(3 a^2-b^2\right) \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}-\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a b \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{(a-b) (a+b) f (g \cos (e+f x))^{3/2} \sin ^{\frac{5}{2}}(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,"(2*Cot[e + f*x]*Csc[e + f*x]*(d*Sin[e + f*x])^(5/2)*(-b + a*Sin[e + f*x]))/((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/2)) - (Cos[e + f*x]^(3/2)*(d*Sin[e + f*x])^(5/2)*((-2*(3*a^2 - b^2)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) - (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a*b*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/((a - b)*(a + b)*f*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^(5/2))","C",0
1437,1,1648,379,23.8749134,"\int \frac{(d \sin (e+f x))^{3/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[(d*Sin[e + f*x])^(3/2)/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{2 \cot (e+f x) (d \sin (e+f x))^{3/2} (a-b \sin (e+f x))}{\left(a^2-b^2\right) f (g \cos (e+f x))^{3/2}}-\frac{\cos ^{\frac{3}{2}}(e+f x) (d \sin (e+f x))^{3/2} \left(\frac{4 a b \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}+\frac{\left(a^2-b^2\right) \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{(a-b) (a+b) f (g \cos (e+f x))^{3/2} \sin ^{\frac{3}{2}}(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*Cot[e + f*x]*(d*Sin[e + f*x])^(3/2)*(a - b*Sin[e + f*x]))/((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/2)) - (Cos[e + f*x]^(3/2)*(d*Sin[e + f*x])^(3/2)*((4*a*b*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((a^2 - b^2)*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/((a - b)*(a + b)*f*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^(3/2))","C",0
1438,1,1274,374,23.1195739,"\int \frac{\sqrt{d \sin (e+f x)}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[Sqrt[d*Sin[e + f*x]]/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{a \sqrt{d \sin (e+f x)} \left(\frac{4 a \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}+\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right) \cos ^{\frac{3}{2}}(e+f x)}{(a-b) (a+b) f (g \cos (e+f x))^{3/2} \sqrt{\sin (e+f x)}}+\frac{2 \sqrt{d \sin (e+f x)} (a \sin (e+f x)-b) \cos (e+f x)}{\left(a^2-b^2\right) f (g \cos (e+f x))^{3/2}}","-\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*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]]*(-b + a*Sin[e + f*x]))/((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/2)) + (a*Cos[e + f*x]^(3/2)*Sqrt[d*Sin[e + f*x]]*((4*a*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/((a - b)*(a + b)*f*(g*Cos[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]])","C",0
1439,1,1279,380,21.8060206,"\int \frac{1}{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx","Integrate[1/((g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]","\frac{b \sqrt{\sin (e+f x)} \left(\frac{4 a \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}+\frac{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right) \cos ^{\frac{3}{2}}(e+f x)}{(b-a) (a+b) f (g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}}+\frac{2 \sin (e+f x) (a-b \sin (e+f x)) \cos (e+f x)}{\left(a^2-b^2\right) f (g \cos (e+f x))^{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*Cos[e + f*x]*Sin[e + f*x]*(a - b*Sin[e + f*x]))/((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]]) + (b*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*((4*a*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/((-a + b)*(a + b)*f*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])","C",0
1440,1,1707,568,25.0069239,"\int \frac{1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx","Integrate[1/((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]","\frac{\cos ^2(e+f x) \sin ^2(e+f x) \left(\frac{2 \sec (e+f x) (a \sin (e+f x)-b)}{a^2-b^2}-\frac{2 \cot (e+f x)}{a}\right)}{f (g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}-\frac{\cos ^{\frac{3}{2}}(e+f x) \sin ^{\frac{3}{2}}(e+f x) \left(-\frac{2 \left(4 a^3-2 a b^2\right) \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}+\frac{\left(2 a^2 b-2 b^3\right) \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\left(b^3-2 a^2 b\right) \cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{a (a-b) (a+b) f (g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}","\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 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{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,"(Cos[e + f*x]^2*Sin[e + f*x]^2*((-2*Cot[e + f*x])/a + (2*Sec[e + f*x]*(-b + a*Sin[e + f*x]))/(a^2 - b^2)))/(f*(g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2)) - (Cos[e + f*x]^(3/2)*Sin[e + f*x]^(3/2)*((-2*(4*a^3 - 2*a*b^2)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((2*a^2*b - 2*b^3)*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + ((-2*a^2*b + b^3)*Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(a*(a - b)*(a + b)*f*(g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2))","C",0
1441,1,1727,673,24.3685761,"\int \frac{1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx","Integrate[1/((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]","\frac{\cos ^2(e+f x) \sin ^3(e+f x) \left(\frac{2 b \cot (e+f x)}{a^2}-\frac{2 \csc (e+f x) \cot (e+f x)}{3 a}+\frac{2 \sec (e+f x) (a-b \sin (e+f x))}{a^2-b^2}\right)}{f (g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2}}-\frac{b \cos ^{\frac{3}{2}}(e+f x) \sin ^{\frac{5}{2}}(e+f x) \left(-\frac{2 \left(4 a^3-2 a b^2\right) \left(a F_1\left(\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-b F_1\left(\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right) \cos ^{\frac{3}{2}}(e+f x) \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \sin ^{\frac{3}{2}}(e+f x)}{3 \left(a^2-b^2\right) \left(1-\cos ^2(e+f x)\right)^{3/4} (a+b \sin (e+f x))}+\frac{\left(2 a^2 b-2 b^3\right) \sqrt{\tan (e+f x)} \left(\frac{3 \sqrt{2} a^{3/2} \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)-\log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)+\log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)\right)}{\sqrt[4]{a^2-b^2}}-8 b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left(b^2-a^2\right) \tan ^2(e+f x)}{a^2}\right) \tan ^{\frac{3}{2}}(e+f x)\right) \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right)}{12 a^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)^{3/2} \sqrt{\sin (e+f x)}}+\frac{\left(b^3-2 a^2 b\right) \cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left(\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right) \left(24 b \left(b^2-a^2\right) F_1\left(\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{7}{2}}(e+f x)+56 b \left(b^2-3 a^2\right) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\left(\frac{b^2}{a^2}-1\right) \tan ^2(e+f x)\right) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left(-\frac{4 \sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right) a^2}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right) a^2}{\sqrt[4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+2 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right)}{\sqrt[4]{a^2-b^2}}-\frac{2 \sqrt{2} b^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right)}{\sqrt[4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left(-a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}-\frac{\sqrt{2} b^2 \log \left(a+\sqrt{2} \sqrt[4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right)}{\sqrt[4]{a^2-b^2}}\right)\right)}{84 a^2 b^2 \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left(\tan ^2(e+f x)-1\right) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right)}{a^2 (b-a) (a+b) f (g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2}}","\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{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{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}{d^2 f g \left(a^2-b^2\right) \sqrt{d \sin (e+f x)} \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 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{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)}}",1,"(Cos[e + f*x]^2*Sin[e + f*x]^3*((2*b*Cot[e + f*x])/a^2 - (2*Cot[e + f*x]*Csc[e + f*x])/(3*a) + (2*Sec[e + f*x]*(a - b*Sin[e + f*x]))/(a^2 - b^2)))/(f*(g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(5/2)) - (b*Cos[e + f*x]^(3/2)*Sin[e + f*x]^(5/2)*((-2*(4*a^3 - 2*a*b^2)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((2*a^2*b - 2*b^3)*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + ((-2*a^2*b + b^3)*Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(a^2*(-a + b)*(a + b)*f*(g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(5/2))","C",0
1442,1,717,331,16.0878575,"\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^2} \, dx","Integrate[(g*Cos[e + f*x])^(3/2)/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^2),x]","\frac{\tan (e+f x) (g \cos (e+f x))^{3/2}}{a f \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))}-\frac{\sin (e+f x) (g \cos (e+f x))^{3/2} \left(a+b \sqrt{1-\cos ^2(e+f x)}\right) \left(\frac{5 a \left(a^2-b^2\right) \sqrt{\cos (e+f x)} F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)}{\left(1-\cos ^2(e+f x)\right)^{3/4} \left(a^2+b^2 \left(\cos ^2(e+f x)-1\right)\right) \left(\cos ^2(e+f x) \left(3 \left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{7}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)-4 b^2 F_1\left(\frac{5}{4};\frac{3}{4},2;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{3}{4},1;\frac{5}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) b \left(\log \left(-\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}+\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}\right)-\log \left(\frac{(1+i) \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\cos (e+f x)}}{\sqrt[4]{\cos ^2(e+f x)-1}}+\sqrt{b^2-a^2}+\frac{i a \cos (e+f x)}{\sqrt{\cos ^2(e+f x)-1}}\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{a} \sqrt{\cos (e+f x)}}{\sqrt[4]{b^2-a^2} \sqrt[4]{\cos ^2(e+f x)-1}}\right)\right)}{\sqrt{a} \left(b^2-a^2\right)^{3/4}}\right)}{a f \cos ^{\frac{3}{2}}(e+f x) \sqrt[4]{1-\cos ^2(e+f x)} \sqrt{d \sin (e+f x)} (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,"-(((g*Cos[e + f*x])^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[e + f*x]])/((1 - Cos[e + f*x]^2)^(3/4)*(5*(a^2 - b^2)*AppellF1[1/4, 3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-4*b^2*AppellF1[5/4, 3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + 3*(a^2 - b^2)*AppellF1[5/4, 7/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)*(a^2 + b^2*(-1 + Cos[e + f*x]^2))) - ((1/8 - I/8)*b*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*(-1 + Cos[e + f*x]^2)^(1/4))] + Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] - ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] + (I*a*Cos[e + f*x])/Sqrt[-1 + Cos[e + f*x]^2] + ((1 + I)*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]])/(-1 + Cos[e + f*x]^2)^(1/4)]))/(Sqrt[a]*(-a^2 + b^2)^(3/4)))*Sin[e + f*x])/(a*f*Cos[e + f*x]^(3/2)*(1 - Cos[e + f*x]^2)^(1/4)*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x]))) + ((g*Cos[e + f*x])^(3/2)*Tan[e + f*x])/(a*f*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x]))","C",0
1443,1,82,82,0.4158168,"\int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx","Integrate[Sin[c + d*x]^2*(a + b*Sin[c + d*x])*Tan[c + d*x]^2,x]","-\frac{3 a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \tan (c+d x)}{d}+\frac{7 b \cos (c+d x)}{4 d}-\frac{b \cos (3 (c+d x))}{12 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*(c + d*x))/(2*d) + (7*b*Cos[c + d*x])/(4*d) - (b*Cos[3*(c + d*x)])/(12*d) + (b*Sec[c + d*x])/d + (a*Sin[2*(c + d*x)])/(4*d) + (a*Tan[c + d*x])/d","A",1
1444,1,63,65,0.1867091,"\int \sin (c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx","Integrate[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 (c+d x)}{2 d}+\frac{b \sin (2 (c+d x))}{4 d}+\frac{b \tan (c+d x)}{d}","\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*(c + d*x))/(2*d) + (a*Cos[c + d*x])/d + (a*Sec[c + d*x])/d + (b*Sin[2*(c + d*x)])/(4*d) + (b*Tan[c + d*x])/d","A",1
1445,1,47,38,0.0319198,"\int (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])*Tan[c + d*x]^2,x]","-\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\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*ArcTan[Tan[c + d*x]])/d) + (b*Cos[c + d*x])/d + (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",1
1446,1,36,27,0.0155537,"\int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx","Integrate[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 ^{-1}(\tan (c+d x))}{d}+\frac{b \tan (c+d x)}{d}","\frac{a \sec (c+d x)}{d}+\frac{b \tan (c+d x)}{d}-b x",1,"-((b*ArcTan[Tan[c + d*x]])/d) + (a*Sec[c + d*x])/d + (b*Tan[c + d*x])/d","A",1
1447,1,56,36,0.0319708,"\int \csc (c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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*Log[Cos[(c + d*x)/2]])/d) + (a*Log[Sin[(c + d*x)/2]])/d + (a*Sec[c + d*x])/d + (b*Tan[c + d*x])/d","A",1
1448,1,68,48,0.0767896,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[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 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-((a*Cot[c + d*x])/d) - (b*Log[Cos[(c + d*x)/2]])/d + (b*Log[Sin[(c + d*x)/2]])/d + (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",1
1449,1,172,75,0.3428226,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{a \sin \left(\frac{1}{2} (c+d x)\right)}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{a \sin \left(\frac{1}{2} (c+d x)\right)}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{2 b \cot (2 (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,"(-2*b*Cot[2*(c + d*x)])/d - (a*Csc[(c + d*x)/2]^2)/(8*d) - (3*a*Log[Cos[(c + d*x)/2]])/(2*d) + (3*a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d) + (a*Sin[(c + d*x)/2])/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (a*Sin[(c + d*x)/2])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
1450,1,104,94,0.4266353,"\int \sin (c+d x) (a+b \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Integrate[Sin[c + d*x]*(a + b*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{\sec (c+d x) \left(-24 \cos (c+d x) \left(a^2+3 a b (c+d x)+b^2\right)+4 \left(3 a^2+5 b^2\right) \cos (2 (c+d x))+36 a^2+54 a b \sin (c+d x)+6 a b \sin (3 (c+d x))-b^2 \cos (4 (c+d x))+45 b^2\right)}{24 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,"(Sec[c + d*x]*(36*a^2 + 45*b^2 - 24*(a^2 + b^2 + 3*a*b*(c + d*x))*Cos[c + d*x] + 4*(3*a^2 + 5*b^2)*Cos[2*(c + d*x)] - b^2*Cos[4*(c + d*x)] + 54*a*b*Sin[c + d*x] + 6*a*b*Sin[3*(c + d*x)]))/(24*d)","A",1
1451,1,77,94,0.4264325,"\int (a+b \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^2,x]","\frac{-4 \left(2 a^2+3 b^2\right) (c+d x)+\left(8 a^2+9 b^2\right) \tan (c+d x)+b \sec (c+d x) (8 a \cos (2 (c+d x))+24 a+b \sin (3 (c+d x)))}{8 d}","\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,"(-4*(2*a^2 + 3*b^2)*(c + d*x) + b*Sec[c + d*x]*(24*a + 8*a*Cos[2*(c + d*x)] + b*Sin[3*(c + d*x)]) + (8*a^2 + 9*b^2)*Tan[c + d*x])/(8*d)","A",1
1452,1,66,42,0.3102852,"\int \sec (c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^2*Tan[c + d*x],x]","\frac{\sec (c+d x) \left(2 a^2+b^2 \cos (2 (c+d x))+3 b^2\right)-2 \left(a^2+2 a b (c+d x)-2 a b \tan (c+d x)+b^2\right)}{2 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^2 + 3*b^2 + b^2*Cos[2*(c + d*x)])*Sec[c + d*x] - 2*(a^2 + b^2 + 2*a*b*(c + d*x) - 2*a*b*Tan[c + d*x]))/(2*d)","A",1
1453,1,58,46,0.1974991,"\int \csc (c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[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)+a \left(a \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 b \tan (c+d x)\right)}{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] + a*(a*(-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]) + 2*b*Tan[c + d*x]))/d","A",1
1454,1,102,59,0.3470855,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\left(2 a^2+b^2\right) \cos (2 (c+d x))-b \left(4 a \sin (c+d x)-2 a \sin (2 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+b\right)\right)}{4 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,"-1/4*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*Sec[c + d*x]*((2*a^2 + b^2)*Cos[2*(c + d*x)] - b*(b + 4*a*Sin[c + d*x] - 2*a*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[2*(c + d*x)])))/d","A",1
1455,1,238,100,0.4846922,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\csc ^4(c+d x) \left(-2 \left(3 a^2+2 b^2\right) \cos (2 (c+d x))-\left(3 a^2+2 b^2\right) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 a^2 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-3 a^2 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 a^2+8 a b \sin (c+d x)-8 a b \sin (3 (c+d x))+2 b^2 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 b^2 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 b^2\right)}{2 d \left(\csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^2\left(\frac{1}{2} (c+d x)\right)\right)}","\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,"(Csc[c + d*x]^4*(2*a^2 + 4*b^2 - 2*(3*a^2 + 2*b^2)*Cos[2*(c + d*x)] + 3*a^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 2*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] - (3*a^2 + 2*b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - 3*a^2*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 2*b^2*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 8*a*b*Sin[c + d*x] - 8*a*b*Sin[3*(c + d*x)]))/(2*d*(Csc[(c + d*x)/2]^2 - Sec[(c + d*x)/2]^2))","B",1
1456,1,196,104,0.9945661,"\int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\csc ^5\left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(-4 \left(4 a^2+3 b^2\right) \cos (2 (c+d x))+\left(8 a^2+6 b^2\right) \cos (4 (c+d x))+3 b \left(10 a \sin (c+d x)-6 a \sin (3 (c+d x))-3 a \sin (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 a \sin (2 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 a \sin (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 b\right)\right)}{192 d \left(\cot ^2\left(\frac{1}{2} (c+d x)\right)-1\right)}","\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,"(Csc[(c + d*x)/2]^5*Sec[(c + d*x)/2]^3*(-4*(4*a^2 + 3*b^2)*Cos[2*(c + d*x)] + (8*a^2 + 6*b^2)*Cos[4*(c + d*x)] + 3*b*(2*b + 10*a*Sin[c + d*x] - 6*a*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[2*(c + d*x)] - 6*a*Sin[3*(c + d*x)] + 3*a*Log[Cos[(c + d*x)/2]]*Sin[4*(c + d*x)] - 3*a*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)])))/(192*d*(-1 + Cot[(c + d*x)/2]^2))","A",1
1457,1,147,197,0.7570492,"\int \sin (c+d x) (a+b \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Integrate[Sin[c + d*x]*(a + b*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{\sec (c+d x) \left(32 \left(a^3+5 a b^2\right) \cos (2 (c+d x))+96 a^3-24 b \left(12 a^2+5 b^2\right) (c+d x) \cos (c+d x)+216 a^2 b \sin (c+d x)+24 a^2 b \sin (3 (c+d x))-8 a b^2 \cos (4 (c+d x))+360 a b^2+80 b^3 \sin (c+d x)+15 b^3 \sin (3 (c+d x))-b^3 \sin (5 (c+d x))\right)}{64 d}","\frac{a^3 \cos (c+d x)}{d}+\frac{a^3 \sec (c+d x)}{d}+\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 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,"(Sec[c + d*x]*(96*a^3 + 360*a*b^2 - 24*b*(12*a^2 + 5*b^2)*(c + d*x)*Cos[c + d*x] + 32*(a^3 + 5*a*b^2)*Cos[2*(c + d*x)] - 8*a*b^2*Cos[4*(c + d*x)] + 216*a^2*b*Sin[c + d*x] + 80*b^3*Sin[c + d*x] + 24*a^2*b*Sin[3*(c + d*x)] + 15*b^3*Sin[3*(c + d*x)] - b^3*Sin[5*(c + d*x)]))/(64*d)","A",1
1458,1,113,146,0.7415731,"\int (a+b \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^3*Tan[c + d*x]^2,x]","\frac{3 a \left(\left(8 a^2+27 b^2\right) \tan (c+d x)-4 \left(2 a^2+9 b^2\right) (c+d x)\right)+b \sec (c+d x) \left(4 \left(9 a^2+5 b^2\right) \cos (2 (c+d x))+108 a^2+9 a b \sin (3 (c+d x))-b^2 \cos (4 (c+d x))+45 b^2\right)}{24 d}","\frac{a^3 \tan (c+d x)}{d}+a^3 (-x)+\frac{3 a^2 b \cos (c+d x)}{d}+\frac{3 a^2 b \sec (c+d x)}{d}+\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,"(b*Sec[c + d*x]*(108*a^2 + 45*b^2 + 4*(9*a^2 + 5*b^2)*Cos[2*(c + d*x)] - b^2*Cos[4*(c + d*x)] + 9*a*b*Sin[3*(c + d*x)]) + 3*a*(-4*(2*a^2 + 9*b^2)*(c + d*x) + (8*a^2 + 27*b^2)*Tan[c + d*x]))/(24*d)","A",1
1459,1,91,75,0.5499332,"\int \sec (c+d x) (a+b \sin (c+d x))^3 \tan (c+d x) \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^3*Tan[c + d*x],x]","\frac{\sec (c+d x) \left(8 a^3+12 a b^2 \cos (2 (c+d x))+36 a b^2+b^3 \sin (3 (c+d x))\right)+3 b \left(\left(8 a^2+3 b^2\right) \tan (c+d x)-4 \left(2 a^2+b^2\right) (c+d x)\right)}{8 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,"(Sec[c + d*x]*(8*a^3 + 36*a*b^2 + 12*a*b^2*Cos[2*(c + d*x)] + b^3*Sin[3*(c + d*x)]) + 3*b*(-4*(2*a^2 + b^2)*(c + d*x) + (8*a^2 + 3*b^2)*Tan[c + d*x]))/(8*d)","A",1
1460,1,83,78,0.2997236,"\int \csc (c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{a^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+a^3 \left(-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b \left(3 a^2+b^2\right) \tan (c+d x)+a \left(a^2+3 b^2\right) \sec (c+d x)-b^3 c-b^3 d x}{d}","\frac{a^3 \sec (c+d x)}{d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^2 b \tan (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*c) - b^3*d*x - a^3*Log[Cos[(c + d*x)/2]] + a^3*Log[Sin[(c + d*x)/2]] + a*(a^2 + 3*b^2)*Sec[c + d*x] + b*(3*a^2 + b^2)*Tan[c + d*x])/d","A",1
1461,1,114,87,0.3964355,"\int \csc ^2(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\left(2 a^3+3 a b^2\right) \cos (2 (c+d x))-2 b \left(3 a^2+b^2\right) \sin (c+d x)-3 a b \left(a \sin (2 (c+d x)) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b\right)\right)}{4 d}","\frac{a^3 \tan (c+d x)}{d}-\frac{a^3 \cot (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{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \sec (c+d x)}{d}",1,"-1/4*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*Sec[c + d*x]*((2*a^3 + 3*a*b^2)*Cos[2*(c + d*x)] - 2*b*(3*a^2 + b^2)*Sin[c + d*x] - 3*a*b*(b + a*(-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]])*Sin[2*(c + d*x)])))/d","A",1
1462,1,267,132,0.5718442,"\int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{\csc ^4(c+d x) \left(-6 \left(a^3+2 a b^2\right) \cos (2 (c+d x))+3 a^3 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-3 a^3 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 a^3-3 a \left(a^2+2 b^2\right) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+12 a^2 b \sin (c+d x)-12 a^2 b \sin (3 (c+d x))+6 a b^2 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-6 a b^2 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 a b^2+6 b^3 \sin (c+d x)-2 b^3 \sin (3 (c+d x))\right)}{2 d \left(\csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^2\left(\frac{1}{2} (c+d x)\right)\right)}","\frac{3 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^2 b \tan (c+d x)}{d}-\frac{3 a^2 b \cot (c+d x)}{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,"(Csc[c + d*x]^4*(2*a^3 + 12*a*b^2 - 6*(a^3 + 2*a*b^2)*Cos[2*(c + d*x)] + 3*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 6*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 3*a*(a^2 + 2*b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) - 3*a^3*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 6*a*b^2*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 12*a^2*b*Sin[c + d*x] + 6*b^3*Sin[c + d*x] - 12*a^2*b*Sin[3*(c + d*x)] - 2*b^3*Sin[3*(c + d*x)]))/(2*d*(Csc[(c + d*x)/2]^2 - Sec[(c + d*x)/2]^2))","B",1
1463,1,287,164,1.3592875,"\int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{\csc ^5\left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(-8 \left(4 a^3+9 a b^2\right) \cos (2 (c+d x))+4 \left(4 a^3+9 a b^2\right) \cos (4 (c+d x))+3 b \left(6 \left(5 a^2+2 b^2\right) \sin (c+d x)-2 \left(9 a^2+2 b^2\right) \sin (2 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)-18 a^2 \sin (3 (c+d x))-9 a^2 \sin (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 a^2 \sin (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 a b-4 b^2 \sin (3 (c+d x))-2 b^2 \sin (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b^2 \sin (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{384 d \left(\cot ^2\left(\frac{1}{2} (c+d x)\right)-1\right)}","\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{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{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,"(Csc[(c + d*x)/2]^5*Sec[(c + d*x)/2]^3*(-8*(4*a^3 + 9*a*b^2)*Cos[2*(c + d*x)] + 4*(4*a^3 + 9*a*b^2)*Cos[4*(c + d*x)] + 3*b*(12*a*b + 6*(5*a^2 + 2*b^2)*Sin[c + d*x] - 2*(9*a^2 + 2*b^2)*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[2*(c + d*x)] - 18*a^2*Sin[3*(c + d*x)] - 4*b^2*Sin[3*(c + d*x)] + 9*a^2*Log[Cos[(c + d*x)/2]]*Sin[4*(c + d*x)] + 2*b^2*Log[Cos[(c + d*x)/2]]*Sin[4*(c + d*x)] - 9*a^2*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)] - 2*b^2*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)])))/(384*d*(-1 + Cot[(c + d*x)/2]^2))","A",1
1464,1,236,222,1.9770097,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{-\frac{a^4 \cos (c+d x)}{b (a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-\frac{a^4 (c+d x)-2 a^2 b^2 (c+d x)+2 a b^3+b^4 (c+d x)}{\left(b^3-a^2 b\right)^2}+\frac{2 a^3 \left(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)}{b^2 \left(a^2-b^2\right)^{5/2}}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}}{d}","-\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{a^4 \cos (c+d x)}{b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\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{\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,"(-((2*a*b^3 + a^4*(c + d*x) - 2*a^2*b^2*(c + d*x) + b^4*(c + d*x))/(-(a^2*b) + b^3)^2) + (2*a^3*(a^2 - 4*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(5/2)) + Sin[(c + d*x)/2]/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + Sin[(c + d*x)/2]/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) - (a^4*Cos[c + d*x])/((a - b)^2*b*(a + b)^2*(a + b*Sin[c + d*x])))/d","A",1
1465,1,162,212,0.9751262,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{a^3 \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{6 a^2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{1}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{d}","-\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{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{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,"((6*a^2*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + Sin[(c + d*x)/2]*(1/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 1/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (a^3*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/d","A",1
1466,1,169,200,0.910987,"\int \frac{\tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","\frac{-\frac{2 a \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)}{\left(a^2-b^2\right)^{5/2}}-\frac{a^2 b \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{d}","-\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{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{\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*(a^2 + 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + Sin[(c + d*x)/2]*(1/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + 1/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) - (a^2*b*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/d","A",1
1467,1,169,133,0.8569,"\int \frac{\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{\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)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^2 \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{1}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{d}","\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) + Sin[(c + d*x)/2]*(1/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 1/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (a*b^2*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/d","A",1
1468,1,203,229,2.2209095,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{\frac{a b^4 \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^2}-\frac{2 \left(b^5-4 a^2 b^3\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right)^{5/2}}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{1}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{d}","\frac{b^4 \cos (c+d x)}{a d \left(a^2-b^2\right)^2 (a+b \sin (c+d 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{\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*(-4*a^2*b^3 + b^5)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)) + Sin[(c + d*x)/2]*(1/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 1/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]] + (a*b^4*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/a^2)/d","A",1
1469,1,254,248,3.2473218,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{-\frac{4 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}+\frac{4 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^3}-\frac{2 b^5 \cos (c+d x)}{a^2 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{a^2}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{a^2}+\frac{4 b^4 \left(2 b^2-5 a^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 \left(a^2-b^2\right)^{5/2}}+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}}{2 d}","\frac{2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\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^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{\cot (c+d x)}{a^2 d}-\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{\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,"((4*b^4*(-5*a^2 + 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(5/2)) - Cot[(c + d*x)/2]/a^2 + (4*b*Log[Cos[(c + d*x)/2]])/a^3 - (4*b*Log[Sin[(c + d*x)/2]])/a^3 + (2*Sin[(c + d*x)/2])/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*Sin[(c + d*x)/2])/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) - (2*b^5*Cos[c + d*x])/(a^2*(a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])) + Tan[(c + d*x)/2]/a^2)/(2*d)","A",1
1470,1,356,295,6.4760311,"\int \frac{\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]","\frac{b^6 \cos (c+d x)}{a^3 d (a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-\frac{b \tan \left(\frac{1}{2} (c+d x)\right)}{a^3 d}+\frac{b \cot \left(\frac{1}{2} (c+d x)\right)}{a^3 d}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^2 d}+\frac{3 \left(a^2+2 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}-\frac{3 \left(a^2+2 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^4 d}+\frac{6 b^5 \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^4 d \left(a^2-b^2\right)^{5/2}}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{d (a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{d (a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{2 b \cot (c+d x)}{a^3 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^2 d \left(a^2-b^2\right)^{5/2}}-\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{\left(a^2+3 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\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{b^6 \cos (c+d x)}{a^3 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,"(6*b^5*(2*a^2 - b^2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(5/2)*d) + (b*Cot[(c + d*x)/2])/(a^3*d) - Csc[(c + d*x)/2]^2/(8*a^2*d) - (3*(a^2 + 2*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^4*d) + (3*(a^2 + 2*b^2)*Log[Sin[(c + d*x)/2]])/(2*a^4*d) + Sec[(c + d*x)/2]^2/(8*a^2*d) + Sin[(c + d*x)/2]/((a + b)^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - Sin[(c + d*x)/2]/((a - b)^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b^6*Cos[c + d*x])/(a^3*(a - b)^2*(a + b)^2*d*(a + b*Sin[c + d*x])) - (b*Tan[(c + d*x)/2])/(a^3*d)","A",1
1471,1,195,388,3.1883639,"\int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]^2*Tan[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{-\frac{6 a^2 \left(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)}{\left(a^2-b^2\right)^{7/2}}+\frac{a^3 \cos (c+d x) \left(\left(a^2-8 b^2\right) \sin (c+d x)-7 a b\right)}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^2}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{2}{(a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2}{(a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{2 d}","-\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 \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{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^2 \left(a^4-3 a^2 b^2+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{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,"((-6*a^2*(a^2 + 4*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + Sin[(c + d*x)/2]*(2/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + 2/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (a^3*Cos[c + d*x]*(-7*a*b + (a^2 - 8*b^2)*Sin[c + d*x]))/((a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^2))/(2*d)","A",1
1472,1,204,366,3.5113769,"\int \frac{\sin (c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Sin[c + d*x]*Tan[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{6 a 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)}{\left(a^2-b^2\right)^{7/2}}+\frac{a^2 \cos (c+d x) \left(2 a^3+b \left(a^2+6 b^2\right) \sin (c+d x)+5 a b^2\right)}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^2}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{2}{(a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{2}{(a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{2 d}","\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{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{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 \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{a^3 \cos (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\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,"((6*a*b*(3*a^2 + 2*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + Sin[(c + d*x)/2]*(2/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 2/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (a^2*Cos[c + d*x]*(2*a^3 + 5*a*b^2 + b*(a^2 + 6*b^2)*Sin[c + d*x]))/((a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^2))/(2*d)","A",1
1473,1,212,350,3.1648234,"\int \frac{\tan ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","\frac{-\frac{2 \left(2 a^4+11 a^2 b^2+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)}{\left(a^2-b^2\right)^{7/2}}-\frac{a b \cos (c+d x) \left(4 a^3+b \left(3 a^2+4 b^2\right) \sin (c+d x)+3 a b^2\right)}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^2}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{2}{(a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2}{(a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{2 d}","-\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{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{3 a^3 b \cos (c+d x)}{2 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*(2*a^4 + 11*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + Sin[(c + d*x)/2]*(2/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + 2/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) - (a*b*Cos[c + d*x]*(4*a^3 + 3*a*b^2 + b*(3*a^2 + 4*b^2)*Sin[c + d*x]))/((a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^2))/(2*d)","A",1
1474,1,206,204,3.0305032,"\int \frac{\sec (c+d x) \tan (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]*Tan[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{6 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)}{\left(a^2-b^2\right)^{7/2}}+\frac{b^2 \cos (c+d x) \left(b \left(5 a^2+2 b^2\right) \sin (c+d x)+a \left(6 a^2+b^2\right)\right)}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^2}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{2}{(a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{2}{(a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{2 d}","\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,"((6*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) + Sin[(c + d*x)/2]*(2/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 2/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (b^2*Cos[c + d*x]*(a*(6*a^2 + b^2) + b*(5*a^2 + 2*b^2)*Sin[c + d*x]))/((a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^2))/(2*d)","A",1
1475,1,322,402,6.5915798,"\int \frac{\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Csc[c + d*x]*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^3 d}-\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^3 d}+\frac{9 a^2 b^4 \cos (c+d x)-2 b^6 \cos (c+d x)}{2 a^2 d (a-b)^3 (a+b)^3 (a+b \sin (c+d x))}+\frac{b^3 \left(20 a^4-7 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{7/2}}+\frac{b^4 \cos (c+d x)}{2 a d (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^2}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{d (a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{d (a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\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{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(6 a^4-3 a^2 b^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 d \left(a^2-b^2\right)^{7/2}}+\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,"(b^3*(20*a^4 - 7*a^2*b^2 + 2*b^4)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) - Log[Cos[(c + d*x)/2]]/(a^3*d) + Log[Sin[(c + d*x)/2]]/(a^3*d) + Sin[(c + d*x)/2]/((a + b)^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - Sin[(c + d*x)/2]/((a - b)^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b^4*Cos[c + d*x])/(2*a*(a - b)^2*(a + b)^2*d*(a + b*Sin[c + d*x])^2) + (9*a^2*b^4*Cos[c + d*x] - 2*b^6*Cos[c + d*x])/(2*a^2*(a - b)^3*(a + b)^3*d*(a + b*Sin[c + d*x]))","A",1
1476,1,379,424,6.3522627,"\int \frac{\csc ^2(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Csc[c + d*x]^2*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","4 \left(-\frac{3 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 a^4 d}+\frac{3 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 a^4 d}+\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{8 a^3 d}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{8 a^3 d}-\frac{b^5 \cos (c+d x)}{8 a^2 d (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^2}-\frac{3 b^4 \left(10 a^4-7 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{4 a^4 d \left(a^2-b^2\right)^{7/2}}+\frac{4 b^7 \cos (c+d x)-11 a^2 b^5 \cos (c+d x)}{8 a^3 d (a-b)^3 (a+b)^3 (a+b \sin (c+d x))}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{4 d (a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{4 d (a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)","\frac{3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a^3 d}-\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{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(10 a^4-9 a^2 b^2+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{\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*((-3*b^4*(10*a^4 - 7*a^2*b^2 + 2*b^4)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(4*a^4*(a^2 - b^2)^(7/2)*d) - Cot[(c + d*x)/2]/(8*a^3*d) + (3*b*Log[Cos[(c + d*x)/2]])/(4*a^4*d) - (3*b*Log[Sin[(c + d*x)/2]])/(4*a^4*d) + Sin[(c + d*x)/2]/(4*(a + b)^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + Sin[(c + d*x)/2]/(4*(a - b)^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) - (b^5*Cos[c + d*x])/(8*a^2*(a - b)^2*(a + b)^2*d*(a + b*Sin[c + d*x])^2) + (-11*a^2*b^5*Cos[c + d*x] + 4*b^7*Cos[c + d*x])/(8*a^3*(a - b)^3*(a + b)^3*d*(a + b*Sin[c + d*x])) + Tan[(c + d*x)/2]/(8*a^3*d))","A",1
1477,1,432,470,6.7789105,"\int \frac{\csc ^3(c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[(Csc[c + d*x]^3*Sec[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]","-\frac{3 b \tan \left(\frac{1}{2} (c+d x)\right)}{2 a^4 d}+\frac{3 b \cot \left(\frac{1}{2} (c+d x)\right)}{2 a^4 d}+\frac{b^6 \cos (c+d x)}{2 a^3 d (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^2}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^3 d}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 a^3 d}+\frac{3 \left(a^2+4 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^5 d}-\frac{3 \left(a^2+4 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^5 d}+\frac{13 a^2 b^6 \cos (c+d x)-6 b^8 \cos (c+d x)}{2 a^4 d (a-b)^3 (a+b)^3 (a+b \sin (c+d x))}+\frac{3 b^5 \left(14 a^4-13 a^2 b^2+4 b^4\right) \tan ^{-1}\left(\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(a \sin \left(\frac{1}{2} (c+d x)\right)+b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^5 d \left(a^2-b^2\right)^{7/2}}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{d (a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{d (a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\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{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{\left(a^2+6 b^2\right) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\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{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{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(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{2 b^5 \left(15 a^4-17 a^2 b^2+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{\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,"(3*b^5*(14*a^4 - 13*a^2*b^2 + 4*b^4)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(7/2)*d) + (3*b*Cot[(c + d*x)/2])/(2*a^4*d) - Csc[(c + d*x)/2]^2/(8*a^3*d) - (3*(a^2 + 4*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^5*d) + (3*(a^2 + 4*b^2)*Log[Sin[(c + d*x)/2]])/(2*a^5*d) + Sec[(c + d*x)/2]^2/(8*a^3*d) + Sin[(c + d*x)/2]/((a + b)^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - Sin[(c + d*x)/2]/((a - b)^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b^6*Cos[c + d*x])/(2*a^3*(a - b)^2*(a + b)^2*d*(a + b*Sin[c + d*x])^2) + (13*a^2*b^6*Cos[c + d*x] - 6*b^8*Cos[c + d*x])/(2*a^4*(a - b)^3*(a + b)^3*d*(a + b*Sin[c + d*x])) - (3*b*Tan[(c + d*x)/2])/(2*a^4*d)","A",1
1478,1,198,158,6.2260154,"\int \frac{\sec ^2(e+f x) \sqrt{a+b \sin (e+f x)}}{\sqrt{d \sin (e+f x)}} \, dx","Integrate[(Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]])/Sqrt[d*Sin[e + f*x]],x]","\frac{4 a^2 \sin ^4\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \sec (e+f x) \sqrt{-\frac{(a+b) \sin (e+f x) (a+b \sin (e+f x))}{a^2 (\sin (e+f x)-1)^2}} \sqrt{-\frac{(a+b) \cot ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{a-b}} F\left(\sin ^{-1}\left(\sqrt{-\frac{a+b \sin (e+f x)}{a (\sin (e+f x)-1)}}\right)|\frac{2 a}{a-b}\right)+(a+b) \tan (e+f x) (a+b \sin (e+f x))}{f (a+b) \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f 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}",1,"(4*a^2*Sqrt[-(((a + b)*Cot[(2*e - Pi + 2*f*x)/4]^2)/(a - b))]*EllipticF[ArcSin[Sqrt[-((a + b*Sin[e + f*x])/(a*(-1 + Sin[e + f*x])))]], (2*a)/(a - b)]*Sec[e + f*x]*Sqrt[-(((a + b)*Sin[e + f*x]*(a + b*Sin[e + f*x]))/(a^2*(-1 + Sin[e + f*x])^2))]*Sin[(2*e - Pi + 2*f*x)/4]^4 + (a + b)*(a + b*Sin[e + f*x])*Tan[e + f*x])/((a + b)*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])","A",1
1479,1,602,312,22.9053398,"\int \frac{\sec ^2(e+f x) (a+b \sin (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)}} \, dx","Integrate[(Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2))/Sqrt[d*Sin[e + f*x]],x]","\frac{\sqrt{\sin (e+f x)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{\tan \left(\frac{1}{2} (e+f x)\right)}{2 \tan ^2\left(\frac{1}{2} (e+f x)\right)+2}} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+2 b \tan \left(\frac{1}{2} (e+f x)\right)}{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}} \left(\frac{2 \sqrt{b^2-a^2} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{a \left(a \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+2 b \tan \left(\frac{1}{2} (e+f x)\right)\right)}{a^2-b^2}} \left(a \sqrt{\frac{a \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}-b}} \sqrt{-\frac{a \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}+b}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+a \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{b^2-a^2}}{\sqrt{b^2-a^2}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-a^2}}{b+\sqrt{b^2-a^2}}\right)-b \tan \left(\frac{1}{2} (e+f x)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{-b-a \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{b^2-a^2}}{\sqrt{b^2-a^2}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-a^2}}{\sqrt{b^2-a^2}-b}\right)\right)}{\sqrt{\frac{a \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}-b}} \left(a \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+2 b \tan \left(\frac{1}{2} (e+f x)\right)\right)}-2 b \tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f \left(\tan ^3\left(\frac{1}{2} (e+f x)\right)+\tan \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{d \sin (e+f x)}}+\frac{\tan (e+f x) (a+b \sin (e+f x))^{3/2}}{f \sqrt{d \sin (e+f x)}}","\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,"(Sqrt[Sin[e + f*x]]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[Tan[(e + f*x)/2]/(2 + 2*Tan[(e + f*x)/2]^2)]*Sqrt[(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)]*(-2*b*Tan[(e + f*x)/2]^2 + (2*Sqrt[-a^2 + b^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))))/(f*Sqrt[d*Sin[e + f*x]]*(Tan[(e + f*x)/2] + Tan[(e + f*x)/2]^3)) + ((a + b*Sin[e + f*x])^(3/2)*Tan[e + f*x])/(f*Sqrt[d*Sin[e + f*x]])","A",1
1480,1,667,366,22.9460428,"\int \frac{\sec ^4(e+f x) (a+b \sin (e+f x))^{5/2}}{\sqrt{d \sin (e+f x)}} \, dx","Integrate[(Sec[e + f*x]^4*(a + b*Sin[e + f*x])^(5/2))/Sqrt[d*Sin[e + f*x]],x]","\frac{\sin (e+f x) \sqrt{a+b \sin (e+f x)} \left(\frac{1}{3} \sec ^3(e+f x) \left(a^2+2 a b \sin (e+f x)+b^2\right)+\frac{1}{6} \sec (e+f x) \left(5 a^2+5 a b \sin (e+f x)-2 b^2\right)\right)}{f \sqrt{d \sin (e+f x)}}+\frac{5 a \sqrt{\sin (e+f x)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{\tan \left(\frac{1}{2} (e+f x)\right)}{2 \tan ^2\left(\frac{1}{2} (e+f x)\right)+2}} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+2 b \tan \left(\frac{1}{2} (e+f x)\right)}{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}} \left(\frac{2 \sqrt{b^2-a^2} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{a \left(a \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+2 b \tan \left(\frac{1}{2} (e+f x)\right)\right)}{a^2-b^2}} \left(a \sqrt{\frac{a \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}-b}} \sqrt{-\frac{a \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}+b}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{b+a \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{b^2-a^2}}{\sqrt{b^2-a^2}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-a^2}}{b+\sqrt{b^2-a^2}}\right)-b \tan \left(\frac{1}{2} (e+f x)\right) E\left(\sin ^{-1}\left(\frac{\sqrt{\frac{-b-a \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{b^2-a^2}}{\sqrt{b^2-a^2}}}}{\sqrt{2}}\right)|\frac{2 \sqrt{b^2-a^2}}{\sqrt{b^2-a^2}-b}\right)\right)}{\sqrt{\frac{a \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}-b}} \left(a \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+2 b \tan \left(\frac{1}{2} (e+f x)\right)\right)}-2 b \tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{6 f \left(\tan ^3\left(\frac{1}{2} (e+f x)\right)+\tan \left(\frac{1}{2} (e+f x)\right)\right) \sqrt{d \sin (e+f x)}}","\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,"(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((Sec[e + f*x]^3*(a^2 + b^2 + 2*a*b*Sin[e + f*x]))/3 + (Sec[e + f*x]*(5*a^2 - 2*b^2 + 5*a*b*Sin[e + f*x]))/6))/(f*Sqrt[d*Sin[e + f*x]]) + (5*a*Sqrt[Sin[e + f*x]]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[Tan[(e + f*x)/2]/(2 + 2*Tan[(e + f*x)/2]^2)]*Sqrt[(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)]*(-2*b*Tan[(e + f*x)/2]^2 + (2*Sqrt[-a^2 + b^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))/(a^2 - b^2)]*(-(b*EllipticE[ArcSin[Sqrt[(-b + Sqrt[-a^2 + b^2] - a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(-b + Sqrt[-a^2 + b^2])]*Tan[(e + f*x)/2]) + a*EllipticF[ArcSin[Sqrt[(b + Sqrt[-a^2 + b^2] + a*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]/Sqrt[2]], (2*Sqrt[-a^2 + b^2])/(b + Sqrt[-a^2 + b^2])]*Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*Sqrt[-((a*Tan[(e + f*x)/2])/(b + Sqrt[-a^2 + b^2]))]))/(Sqrt[(a*Tan[(e + f*x)/2])/(-b + Sqrt[-a^2 + b^2])]*(a + 2*b*Tan[(e + f*x)/2] + a*Tan[(e + f*x)/2]^2))))/(6*f*Sqrt[d*Sin[e + f*x]]*(Tan[(e + f*x)/2] + Tan[(e + f*x)/2]^3))","A",0
1481,1,156,155,0.5950102,"\int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx","Integrate[Sin[c + d*x]^2*(a + b*Sin[c + d*x])*Tan[c + d*x]^5,x]","-\frac{a \left(2 \sin ^2(c+d x)-\sec ^4(c+d x)+6 \sec ^2(c+d x)+12 \log (\cos (c+d x))\right)}{4 d}-\frac{b \sin ^3(c+d x) \tan ^4(c+d x)}{3 d}-\frac{7 b \left(8 \sin (c+d x) \tan ^4(c+d x)+5 \left(6 \tan (c+d x) \sec ^3(c+d x)-8 \tan ^3(c+d x) \sec (c+d x)-3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)\right)}{24 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,"-1/4*(a*(12*Log[Cos[c + d*x]] + 6*Sec[c + d*x]^2 - Sec[c + d*x]^4 + 2*Sin[c + d*x]^2))/d - (b*Sin[c + d*x]^3*Tan[c + d*x]^4)/(3*d) - (7*b*(8*Sin[c + d*x]*Tan[c + d*x]^4 + 5*(6*Sec[c + d*x]^3*Tan[c + d*x] - 8*Sec[c + d*x]*Tan[c + d*x]^3 - 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))))/(24*d)","A",1
1482,1,133,135,0.3613329,"\int \sin (c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx","Integrate[Sin[c + d*x]*(a + b*Sin[c + d*x])*Tan[c + d*x]^5,x]","-\frac{a \sin (c+d x) \tan ^4(c+d x)}{d}-\frac{5 a \left(6 \tan (c+d x) \sec ^3(c+d x)-8 \tan ^3(c+d x) \sec (c+d x)-3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)}{8 d}-\frac{b \left(2 \sin ^2(c+d x)-\sec ^4(c+d x)+6 \sec ^2(c+d x)+12 \log (\cos (c+d x))\right)}{4 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,"-1/4*(b*(12*Log[Cos[c + d*x]] + 6*Sec[c + d*x]^2 - Sec[c + d*x]^4 + 2*Sin[c + d*x]^2))/d - (a*Sin[c + d*x]*Tan[c + d*x]^4)/d - (5*a*(6*Sec[c + d*x]^3*Tan[c + d*x] - 8*Sec[c + d*x]*Tan[c + d*x]^3 - 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x])))/(8*d)","A",1
1483,1,123,116,0.3153943,"\int (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])*Tan[c + d*x]^5,x]","-\frac{a \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 d}-\frac{b \sin (c+d x) \tan ^4(c+d x)}{d}-\frac{5 b \left(6 \tan (c+d x) \sec ^3(c+d x)-8 \tan ^3(c+d x) \sec (c+d x)-3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)}{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,"-((b*Sin[c + d*x]*Tan[c + d*x]^4)/d) - (a*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d) - (5*b*(6*Sec[c + d*x]^3*Tan[c + d*x] - 8*Sec[c + d*x]*Tan[c + d*x]^3 - 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x])))/(8*d)","A",1
1484,1,106,103,0.3135222,"\int \sec (c+d x) (a+b \sin (c+d x)) \tan ^4(c+d x) \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])*Tan[c + d*x]^4,x]","\frac{a \tan ^3(c+d x) \sec (c+d x)}{d}-\frac{a \left(6 \tan (c+d x) \sec ^3(c+d x)-3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)}{8 d}-\frac{b \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 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,"(a*Sec[c + d*x]*Tan[c + d*x]^3)/d - (b*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d) - (a*(6*Sec[c + d*x]^3*Tan[c + d*x] - 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x])))/(8*d)","A",1
1485,1,84,74,0.2691051,"\int \sec ^2(c+d x) (a+b \sin (c+d x)) \tan ^3(c+d x) \, dx","Integrate[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{b \tan ^3(c+d x) \sec (c+d x)}{d}-\frac{b \left(6 \tan (c+d x) \sec ^3(c+d x)-3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)}{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,"(b*Sec[c + d*x]*Tan[c + d*x]^3)/d + (a*Tan[c + d*x]^4)/(4*d) - (b*(6*Sec[c + d*x]^3*Tan[c + d*x] - 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x])))/(8*d)","A",1
1486,1,74,74,0.0241225,"\int \sec ^3(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx","Integrate[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,"-1/8*(a*ArcTanh[Sin[c + d*x]])/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",1
1487,1,74,74,0.0241885,"\int \sec ^4(c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx","Integrate[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,"-1/8*(b*ArcTanh[Sin[c + d*x]])/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",1
1488,1,99,99,1.2626746,"\int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^5*(a + b*Sin[c + d*x]),x]","-\frac{a \left(-\sec ^4(c+d x)-2 \sec ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right)}{4 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{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,"-1/4*(a*(4*Log[Cos[c + d*x]] - 4*Log[Sin[c + d*x]] - 2*Sec[c + d*x]^2 - Sec[c + d*x]^4))/d + (b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*b*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d)","A",1
1489,1,76,115,0.3857814,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^5*(a + b*Sin[c + d*x]),x]","-\frac{a \csc (c+d x) \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\sin ^2(c+d x)\right)}{d}-\frac{b \left(-\sec ^4(c+d x)-2 \sec ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right)}{4 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,"-((a*Csc[c + d*x]*Hypergeometric2F1[-1/2, 3, 1/2, Sin[c + d*x]^2])/d) - (b*(4*Log[Cos[c + d*x]] - 4*Log[Sin[c + d*x]] - 2*Sec[c + d*x]^2 - Sec[c + d*x]^4))/(4*d)","C",1
1490,1,86,135,0.6077257,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + b*Sin[c + d*x]),x]","-\frac{a \left(2 \csc ^2(c+d x)-\sec ^4(c+d x)-4 \sec ^2(c+d x)-12 \log (\sin (c+d x))+12 \log (\cos (c+d x))\right)}{4 d}-\frac{b \csc (c+d x) \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\sin ^2(c+d x)\right)}{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,"-((b*Csc[c + d*x]*Hypergeometric2F1[-1/2, 3, 1/2, Sin[c + d*x]^2])/d) - (a*(2*Csc[c + d*x]^2 + 12*Log[Cos[c + d*x]] - 12*Log[Sin[c + d*x]] - 4*Sec[c + d*x]^2 - Sec[c + d*x]^4))/(4*d)","C",1
1491,1,90,155,0.8171087,"\int \csc ^4(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Csc[c + d*x]^4*Sec[c + d*x]^5*(a + b*Sin[c + d*x]),x]","-\frac{a \csc ^3(c+d x) \, _2F_1\left(-\frac{3}{2},3;-\frac{1}{2};\sin ^2(c+d x)\right)}{3 d}-\frac{b \left(2 \csc ^2(c+d x)-\sec ^4(c+d x)-4 \sec ^2(c+d x)-12 \log (\sin (c+d x))+12 \log (\cos (c+d x))\right)}{4 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,"-1/3*(a*Csc[c + d*x]^3*Hypergeometric2F1[-3/2, 3, -1/2, Sin[c + d*x]^2])/d - (b*(2*Csc[c + d*x]^2 + 12*Log[Cos[c + d*x]] - 12*Log[Sin[c + d*x]] - 4*Sec[c + d*x]^2 - Sec[c + d*x]^4))/(4*d)","C",1
1492,1,186,189,1.5578205,"\int \sin (c+d x) (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx","Integrate[Sin[c + d*x]*(a + b*Sin[c + d*x])^2*Tan[c + d*x]^5,x]","\frac{-48 \left(a^2+3 b^2\right) \sin (c+d x)-3 \left(15 a^2+48 a b+35 b^2\right) \log (1-\sin (c+d x))+3 \left(15 a^2-48 a b+35 b^2\right) \log (\sin (c+d x)+1)-48 a b \sin ^2(c+d x)+\frac{3 (a+b) (9 a+13 b)}{\sin (c+d x)-1}+\frac{3 (9 a-13 b) (a-b)}{\sin (c+d x)+1}+\frac{3 (a+b)^2}{(\sin (c+d x)-1)^2}-\frac{3 (a-b)^2}{(\sin (c+d x)+1)^2}-16 b^2 \sin ^3(c+d x)}{48 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,"(-3*(15*a^2 + 48*a*b + 35*b^2)*Log[1 - Sin[c + d*x]] + 3*(15*a^2 - 48*a*b + 35*b^2)*Log[1 + Sin[c + d*x]] + (3*(a + b)^2)/(-1 + Sin[c + d*x])^2 + (3*(a + b)*(9*a + 13*b))/(-1 + Sin[c + d*x]) - 48*(a^2 + 3*b^2)*Sin[c + d*x] - 48*a*b*Sin[c + d*x]^2 - 16*b^2*Sin[c + d*x]^3 - (3*(a - b)^2)/(1 + Sin[c + d*x])^2 + (3*(9*a - 13*b)*(a - b))/(1 + Sin[c + d*x]))/(48*d)","A",1
1493,1,164,162,2.1521034,"\int (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^2*Tan[c + d*x]^5,x]","\frac{-2 \left(4 a^2+15 a b+12 b^2\right) \log (1-\sin (c+d x))-2 \left(4 a^2-15 a b+12 b^2\right) \log (\sin (c+d x)+1)+\frac{(a-b)^2}{(\sin (c+d x)+1)^2}-\frac{(7 a-11 b) (a-b)}{\sin (c+d x)+1}-32 a b \sin (c+d x)+\frac{(a+b) (7 a+11 b)}{\sin (c+d x)-1}+\frac{(a+b)^2}{(\sin (c+d x)-1)^2}-8 b^2 \sin ^2(c+d x)}{16 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,"(-2*(4*a^2 + 15*a*b + 12*b^2)*Log[1 - Sin[c + d*x]] - 2*(4*a^2 - 15*a*b + 12*b^2)*Log[1 + Sin[c + d*x]] + (a + b)^2/(-1 + Sin[c + d*x])^2 + ((a + b)*(7*a + 11*b))/(-1 + Sin[c + d*x]) - 32*a*b*Sin[c + d*x] - 8*b^2*Sin[c + d*x]^2 + (a - b)^2/(1 + Sin[c + d*x])^2 - ((7*a - 11*b)*(a - b))/(1 + Sin[c + d*x]))/(16*d)","A",1
1494,1,151,150,1.0211254,"\int \sec (c+d x) (a+b \sin (c+d x))^2 \tan ^4(c+d x) \, dx","Integrate[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))+\left(3 a^2-16 a b+15 b^2\right) \log (\sin (c+d x)+1)-\frac{(a-b)^2}{(\sin (c+d x)+1)^2}+\frac{(5 a-9 b) (a-b)}{\sin (c+d x)+1}+\frac{(a+b) (5 a+9 b)}{\sin (c+d x)-1}+\frac{(a+b)^2}{(\sin (c+d x)-1)^2}-16 b^2 \sin (c+d x)}{16 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]]) + (3*a^2 - 16*a*b + 15*b^2)*Log[1 + Sin[c + d*x]] + (a + b)^2/(-1 + Sin[c + d*x])^2 + ((a + b)*(5*a + 9*b))/(-1 + Sin[c + d*x]) - 16*b^2*Sin[c + d*x] - (a - b)^2/(1 + Sin[c + d*x])^2 + ((5*a - 9*b)*(a - b))/(1 + Sin[c + d*x]))/(16*d)","A",1
1495,1,129,116,0.378046,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^2 \tan ^3(c+d x) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2*Tan[c + d*x]^3,x]","\frac{a^2 \tan ^4(c+d x)}{4 d}+\frac{2 a b \tan ^3(c+d x) \sec (c+d x)}{d}-\frac{a b \left(6 \tan (c+d x) \sec ^3(c+d x)-3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)}{4 d}-\frac{b^2 \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{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,"(2*a*b*Sec[c + d*x]*Tan[c + d*x]^3)/d + (a^2*Tan[c + d*x]^4)/(4*d) - (b^2*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d) - (a*b*(6*Sec[c + d*x]^3*Tan[c + d*x] - 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x])))/(4*d)","A",1
1496,1,85,93,0.7685092,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^2 \tan ^2(c+d x) \, dx","Integrate[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))+\frac{1}{4} \sec ^4(c+d x) \left(2 \sin (c+d x) \left(\left(a^2+5 b^2\right) \cos (2 (c+d x))-3 a^2+b^2\right)+16 a b \cos (2 (c+d x))\right)}{8 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,"-1/8*((a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]] + (Sec[c + d*x]^4*(16*a*b*Cos[2*(c + d*x)] + 2*(-3*a^2 + b^2 + (a^2 + 5*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]))/4)/d","A",1
1497,1,215,72,2.9374595,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^2*Tan[c + d*x],x]","\frac{2 a^4 b^2 \sec ^2(c+d x)+a b \left(a^2-b^2\right)^2 (\log (1-\sin (c+d x))-\log (\sin (c+d x)+1))-2 a b \left(a^2-b^2\right) \tan (c+d x) \sec (c+d x) \left(a^2+2 b^2 \tan ^2(c+d x)+b^2\right)+2 b^4 \left(b^2-a^2\right) \tan ^4(c+d x)+b \left(4 a^2 b^3-6 a^4 b\right) \tan ^2(c+d x)+2 a^4 \left(a^2-b^2\right) \sec ^4(c+d x)+4 a^3 b \left(a^2-b^2\right) \tan (c+d x) \sec ^3(c+d x)}{8 d \left(a^2-b^2\right)^2}","-\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*(a^2 - b^2)^2*(Log[1 - Sin[c + d*x]] - Log[1 + Sin[c + d*x]]) + 2*a^4*b^2*Sec[c + d*x]^2 + 2*a^4*(a^2 - b^2)*Sec[c + d*x]^4 + 4*a^3*b*(a^2 - b^2)*Sec[c + d*x]^3*Tan[c + d*x] + b*(-6*a^4*b + 4*a^2*b^3)*Tan[c + d*x]^2 + 2*b^4*(-a^2 + b^2)*Tan[c + d*x]^4 - 2*a*b*(a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x]*(a^2 + b^2 + 2*b^2*Tan[c + d*x]^2))/(8*(a^2 - b^2)^2*d)","B",1
1498,1,137,126,0.9248074,"\int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{16 a^2 \log (\sin (c+d x))-\frac{(a+b) (5 a+b)}{\sin (c+d x)-1}+\frac{(a-b) (5 a-b)}{\sin (c+d x)+1}+\frac{(a+b)^2}{(\sin (c+d x)-1)^2}+\frac{(a-b)^2}{(\sin (c+d x)+1)^2}-2 a (4 a+3 b) \log (1-\sin (c+d x))-2 a (4 a-3 b) \log (\sin (c+d x)+1)}{16 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,"(-2*a*(4*a + 3*b)*Log[1 - Sin[c + d*x]] + 16*a^2*Log[Sin[c + d*x]] - 2*a*(4*a - 3*b)*Log[1 + Sin[c + d*x]] + (a + b)^2/(-1 + Sin[c + d*x])^2 - ((a + b)*(5*a + b))/(-1 + Sin[c + d*x]) + (a - b)^2/(1 + Sin[c + d*x])^2 + ((a - b)*(5*a - b))/(1 + Sin[c + d*x]))/(16*d)","A",1
1499,1,162,168,2.8254123,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[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))-\left(15 a^2-16 a b+3 b^2\right) \log (\sin (c+d x)+1)+16 a^2 \csc (c+d x)+\frac{(a+b) (7 a+3 b)}{\sin (c+d x)-1}+\frac{(7 a-3 b) (a-b)}{\sin (c+d x)+1}-\frac{(a+b)^2}{(\sin (c+d x)-1)^2}+\frac{(a-b)^2}{(\sin (c+d x)+1)^2}-32 a b \log (\sin (c+d x))}{16 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,"-1/16*(16*a^2*Csc[c + d*x] + (15*a^2 + 16*a*b + 3*b^2)*Log[1 - Sin[c + d*x]] - 32*a*b*Log[Sin[c + d*x]] - (15*a^2 - 16*a*b + 3*b^2)*Log[1 + Sin[c + d*x]] - (a + b)^2/(-1 + Sin[c + d*x])^2 + ((a + b)*(7*a + 3*b))/(-1 + Sin[c + d*x]) + (a - b)^2/(1 + Sin[c + d*x])^2 + ((7*a - 3*b)*(a - b))/(1 + Sin[c + d*x]))/d","A",1
1500,1,182,185,3.6629547,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{-2 \left(12 a^2+15 a b+4 b^2\right) \log (1-\sin (c+d x))+16 \left(3 a^2+b^2\right) \log (\sin (c+d x))-2 \left(12 a^2-15 a b+4 b^2\right) \log (\sin (c+d x)+1)-8 a^2 \csc ^2(c+d x)+\frac{(a-b)^2}{(\sin (c+d x)+1)^2}+\frac{(9 a-5 b) (a-b)}{\sin (c+d x)+1}-\frac{(a+b) (9 a+5 b)}{\sin (c+d x)-1}+\frac{(a+b)^2}{(\sin (c+d x)-1)^2}-32 a b \csc (c+d x)}{16 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,"(-32*a*b*Csc[c + d*x] - 8*a^2*Csc[c + d*x]^2 - 2*(12*a^2 + 15*a*b + 4*b^2)*Log[1 - Sin[c + d*x]] + 16*(3*a^2 + b^2)*Log[Sin[c + d*x]] - 2*(12*a^2 - 15*a*b + 4*b^2)*Log[1 + Sin[c + d*x]] + (a + b)^2/(-1 + Sin[c + d*x])^2 - ((a + b)*(9*a + 5*b))/(-1 + Sin[c + d*x]) + (a - b)^2/(1 + Sin[c + d*x])^2 + ((9*a - 5*b)*(a - b))/(1 + Sin[c + d*x]))/(16*d)","A",1
1501,1,199,202,1.0190462,"\int (a+b \sin (c+d x))^3 \tan ^5(c+d x) \, dx","Integrate[(a + b*Sin[c + d*x])^3*Tan[c + d*x]^5,x]","-\frac{144 b \left(a^2+b^2\right) \sin (c+d x)+3 \left(8 a^2-37 a b+35 b^2\right) (a-b) \log (\sin (c+d x)+1)+3 (a+b) \left(8 a^2+37 a b+35 b^2\right) \log (1-\sin (c+d x))+72 a b^2 \sin ^2(c+d x)-\frac{3 (a-b)^3}{(\sin (c+d x)+1)^2}+\frac{3 (7 a-13 b) (a-b)^2}{\sin (c+d x)+1}-\frac{3 (a+b)^2 (7 a+13 b)}{\sin (c+d x)-1}-\frac{3 (a+b)^3}{(\sin (c+d x)-1)^2}+16 b^3 \sin ^3(c+d x)}{48 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,"-1/48*(3*(a + b)*(8*a^2 + 37*a*b + 35*b^2)*Log[1 - Sin[c + d*x]] + 3*(a - b)*(8*a^2 - 37*a*b + 35*b^2)*Log[1 + Sin[c + d*x]] - (3*(a + b)^3)/(-1 + Sin[c + d*x])^2 - (3*(a + b)^2*(7*a + 13*b))/(-1 + Sin[c + d*x]) + 144*b*(a^2 + b^2)*Sin[c + d*x] + 72*a*b^2*Sin[c + d*x]^2 + 16*b^3*Sin[c + d*x]^3 - (3*(a - b)^3)/(1 + Sin[c + d*x])^2 + (3*(7*a - 13*b)*(a - b)^2)/(1 + Sin[c + d*x]))/d","A",1
1502,1,174,177,0.5666703,"\int \sec (c+d x) (a+b \sin (c+d x))^3 \tan ^4(c+d x) \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^3*Tan[c + d*x]^4,x]","\frac{3 \left(a^2-7 a b+8 b^2\right) (a-b) \log (\sin (c+d x)+1)-3 (a+b) \left(a^2+7 a b+8 b^2\right) \log (1-\sin (c+d x))-48 a b^2 \sin (c+d x)-\frac{(a-b)^3}{(\sin (c+d x)+1)^2}+\frac{(5 a-11 b) (a-b)^2}{\sin (c+d x)+1}+\frac{(a+b)^2 (5 a+11 b)}{\sin (c+d x)-1}+\frac{(a+b)^3}{(\sin (c+d x)-1)^2}-8 b^3 \sin ^2(c+d x)}{16 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]] + 3*(a - b)*(a^2 - 7*a*b + 8*b^2)*Log[1 + Sin[c + d*x]] + (a + b)^3/(-1 + Sin[c + d*x])^2 + ((a + b)^2*(5*a + 11*b))/(-1 + Sin[c + d*x]) - 48*a*b^2*Sin[c + d*x] - 8*b^3*Sin[c + d*x]^2 - (a - b)^3/(1 + Sin[c + d*x])^2 + ((5*a - 11*b)*(a - b)^2)/(1 + Sin[c + d*x]))/(16*d)","A",1
1503,1,147,142,0.4444305,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^3 \tan ^3(c+d x) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3*Tan[c + d*x]^3,x]","\frac{\frac{(a-b)^3}{(\sin (c+d x)+1)^2}-\frac{3 (a-3 b) (a-b)^2}{\sin (c+d x)+1}+\frac{3 (a+b)^2 (a+3 b)}{\sin (c+d x)-1}+\frac{(a+b)^3}{(\sin (c+d x)-1)^2}+3 b (3 a-5 b) (a-b) \log (\sin (c+d x)+1)-3 b (a+b) (3 a+5 b) \log (1-\sin (c+d x))-16 b^3 \sin (c+d x)}{16 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]] + 3*(3*a - 5*b)*(a - b)*b*Log[1 + Sin[c + d*x]] + (a + b)^3/(-1 + Sin[c + d*x])^2 + (3*(a + b)^2*(a + 3*b))/(-1 + Sin[c + d*x]) - 16*b^3*Sin[c + d*x] + (a - b)^3/(1 + Sin[c + d*x])^2 - (3*(a - 3*b)*(a - b)^2)/(1 + Sin[c + d*x]))/(16*d)","A",1
1504,1,140,144,0.3886324,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^3 \tan ^2(c+d x) \, dx","Integrate[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))-\left(a^3-9 a b^2+8 b^3\right) \log (\sin (c+d x)+1)-\frac{(a-b)^3}{(\sin (c+d x)+1)^2}+\frac{(a-7 b) (a-b)^2}{\sin (c+d x)+1}+\frac{(a+b)^2 (a+7 b)}{\sin (c+d x)-1}+\frac{(a+b)^3}{(\sin (c+d x)-1)^2}}{16 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]] - (a^3 - 9*a*b^2 + 8*b^3)*Log[1 + Sin[c + d*x]] + (a + b)^3/(-1 + Sin[c + d*x])^2 + ((a + b)^2*(a + 7*b))/(-1 + Sin[c + d*x]) - (a - b)^3/(1 + Sin[c + d*x])^2 + ((a - 7*b)*(a - b)^2)/(1 + Sin[c + d*x]))/(16*d)","A",1
1505,1,370,90,1.4701613,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^3 \tan (c+d x) \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3*Tan[c + d*x],x]","\frac{2 b \left(a^2-b^2\right) \sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^5+2 a \left(a^2-b^2\right)^2 \sec ^4(c+d x) (a+b \sin (c+d x))^4+b \sec ^2(c+d x) \left(-3 a \left(a^2+b^2\right) \sin (c+d x)+5 a^2 b+b^3\right) (a+b \sin (c+d x))^5-a b \left(a^2+b^2\right) \left(60 a b^2 \left(2 a^2+b^2\right) \sin (c+d x)+6 b^3 \left(10 a^2+b^2\right) \sin ^2(c+d x)+20 a b^4 \sin ^3(c+d x)-6 (a-b)^5 \log (\sin (c+d x)+1)+6 (a+b)^5 \log (1-\sin (c+d x))+3 b^5 \sin ^4(c+d x)\right)+\frac{1}{2} \left(5 a^4 b+10 a^2 b^3+b^5\right) \left(6 b^2 \left(6 a^2+b^2\right) \sin (c+d x)+12 a b^3 \sin ^2(c+d x)+3 \left((a+b)^4 \log (1-\sin (c+d x))-(a-b)^4 \log (\sin (c+d x)+1)\right)+2 b^4 \sin ^3(c+d x)\right)}{8 d \left(a^2-b^2\right)^3}","-\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,"(2*a*(a^2 - b^2)^2*Sec[c + d*x]^4*(a + b*Sin[c + d*x])^4 + 2*b*(a^2 - b^2)*Sec[c + d*x]^4*(b - a*Sin[c + d*x])*(a + b*Sin[c + d*x])^5 + b*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^5*(5*a^2*b + b^3 - 3*a*(a^2 + b^2)*Sin[c + d*x]) + ((5*a^4*b + 10*a^2*b^3 + b^5)*(3*((a + b)^4*Log[1 - Sin[c + d*x]] - (a - b)^4*Log[1 + Sin[c + d*x]]) + 6*b^2*(6*a^2 + b^2)*Sin[c + d*x] + 12*a*b^3*Sin[c + d*x]^2 + 2*b^4*Sin[c + d*x]^3))/2 - a*b*(a^2 + b^2)*(6*(a + b)^5*Log[1 - Sin[c + d*x]] - 6*(a - b)^5*Log[1 + Sin[c + d*x]] + 60*a*b^2*(2*a^2 + b^2)*Sin[c + d*x] + 6*b^3*(10*a^2 + b^2)*Sin[c + d*x]^2 + 20*a*b^4*Sin[c + d*x]^3 + 3*b^5*Sin[c + d*x]^4))/(8*(a^2 - b^2)^3*d)","B",1
1506,1,157,165,0.5684591,"\int \csc (c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","\frac{16 a^3 \log (\sin (c+d x))-\left(8 a^3+9 a^2 b-b^3\right) \log (1-\sin (c+d x))-\left(8 a^3-9 a^2 b+b^3\right) \log (\sin (c+d x)+1)-\frac{(5 a-b) (a+b)^2}{\sin (c+d x)-1}+\frac{(a-b)^2 (5 a+b)}{\sin (c+d x)+1}+\frac{(a+b)^3}{(\sin (c+d x)-1)^2}+\frac{(a-b)^3}{(\sin (c+d x)+1)^2}}{16 d}","\frac{a^3 \log (\sin (c+d x))}{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{\left(8 a^3+9 a^2 b-b^3\right) \log (1-\sin (c+d x))}{16 d}-\frac{\left(8 a^3-9 a^2 b+b^3\right) \log (\sin (c+d x)+1)}{16 d}+\frac{\sec ^2(c+d x) \left(4 a^3+b \left(9 a^2-b^2\right) \sin (c+d x)\right)}{8 d}",1,"(-((8*a^3 + 9*a^2*b - b^3)*Log[1 - Sin[c + d*x]]) + 16*a^3*Log[Sin[c + d*x]] - (8*a^3 - 9*a^2*b + b^3)*Log[1 + Sin[c + d*x]] + (a + b)^3/(-1 + Sin[c + d*x])^2 - ((5*a - b)*(a + b)^2)/(-1 + Sin[c + d*x]) + (a - b)^3/(1 + Sin[c + d*x])^2 + ((a - b)^2*(5*a + b))/(1 + Sin[c + d*x]))/(16*d)","A",1
1507,1,161,171,1.4490087,"\int \csc ^2(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^2*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","-\frac{16 a^3 \csc (c+d x)-48 a^2 b \log (\sin (c+d x))+\frac{(a+b)^2 (7 a+b)}{\sin (c+d x)-1}+\frac{(a-b)^2 (7 a-b)}{\sin (c+d x)+1}-\frac{(a+b)^3}{(\sin (c+d x)-1)^2}+\frac{(a-b)^3}{(\sin (c+d x)+1)^2}+3 a (a+b) (5 a+3 b) \log (1-\sin (c+d x))-3 a (5 a-3 b) (a-b) \log (\sin (c+d x)+1)}{16 d}","-\frac{a^3 \csc (c+d x)}{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{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,"-1/16*(16*a^3*Csc[c + d*x] + 3*a*(a + b)*(5*a + 3*b)*Log[1 - Sin[c + d*x]] - 48*a^2*b*Log[Sin[c + d*x]] - 3*a*(5*a - 3*b)*(a - b)*Log[1 + Sin[c + d*x]] - (a + b)^3/(-1 + Sin[c + d*x])^2 + ((a + b)^2*(7*a + b))/(-1 + Sin[c + d*x]) + (a - b)^3/(1 + Sin[c + d*x])^2 + ((a - b)^2*(7*a - b))/(1 + Sin[c + d*x]))/d","A",1
1508,1,190,221,3.0966656,"\int \csc ^3(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^3*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","\frac{-8 a^3 \csc ^2(c+d x)+48 a \left(a^2+b^2\right) \log (\sin (c+d x))-3 (a+b) \left(8 a^2+7 a b+b^2\right) \log (1-\sin (c+d x))-3 (a-b) \left(8 a^2-7 a b+b^2\right) \log (\sin (c+d x)+1)-48 a^2 b \csc (c+d x)-\frac{3 (a+b)^2 (3 a+b)}{\sin (c+d x)-1}+\frac{3 (a-b)^2 (3 a-b)}{\sin (c+d x)+1}+\frac{(a+b)^3}{(\sin (c+d x)-1)^2}+\frac{(a-b)^3}{(\sin (c+d x)+1)^2}}{16 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}",1,"(-48*a^2*b*Csc[c + d*x] - 8*a^3*Csc[c + d*x]^2 - 3*(a + b)*(8*a^2 + 7*a*b + b^2)*Log[1 - Sin[c + d*x]] + 48*a*(a^2 + b^2)*Log[Sin[c + d*x]] - 3*(a - b)*(8*a^2 - 7*a*b + b^2)*Log[1 + Sin[c + d*x]] + (a + b)^3/(-1 + Sin[c + d*x])^2 - (3*(a + b)^2*(3*a + b))/(-1 + Sin[c + d*x]) + (a - b)^3/(1 + Sin[c + d*x])^2 + (3*(a - b)^2*(3*a - b))/(1 + Sin[c + d*x]))/(16*d)","A",1
1509,1,164,295,0.1994525,"\int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^4,x]","\frac{\sin ^{n+1}(c+d x) \left(6 \left(a^2-b^2\right)^2 \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)+2 (a-b)^4 \, _2F_1(3,n+1;n+2;-\sin (c+d x))+(3 a+5 b) (a-b)^3 \, _2F_1(2,n+1;n+2;-\sin (c+d x))+(3 a-5 b) (a+b)^3 \, _2F_1(2,n+1;n+2;\sin (c+d x))+2 (a+b)^4 \, _2F_1(3,n+1;n+2;\sin (c+d x))\right)}{16 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{\left(-\left(a^4 \left(n^2-4 n+3\right)\right)+6 a^2 b^2 \left(1-n^2\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{\sec ^4(c+d x) \sin ^{n+1}(c+d x) \left(a^4+4 a b \left(a^2+b^2\right) \sin (c+d x)+6 a^2 b^2+b^4\right)}{4 d}+\frac{\sec ^2(c+d x) \sin ^{n+1}(c+d x) \left(a^4 (3-n)+4 a b \left(a^2 (2-n)-b^2 (n+2)\right) \sin (c+d x)-6 a^2 b^2 (n+1)-b^4 (n+5)\right)}{8 d}",1,"((6*(a^2 - b^2)^2*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2] + (a - b)^3*(3*a + 5*b)*Hypergeometric2F1[2, 1 + n, 2 + n, -Sin[c + d*x]] + (3*a - 5*b)*(a + b)^3*Hypergeometric2F1[2, 1 + n, 2 + n, Sin[c + d*x]] + 2*(a - b)^4*Hypergeometric2F1[3, 1 + n, 2 + n, -Sin[c + d*x]] + 2*(a + b)^4*Hypergeometric2F1[3, 1 + n, 2 + n, Sin[c + d*x]])*Sin[c + d*x]^(1 + n))/(16*d*(1 + n))","A",1
1510,1,158,186,0.161917,"\int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^3,x]","\frac{\sin ^{n+1}(c+d x) \left(6 a (a+b) (a-b) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)+2 (a-b)^3 \, _2F_1(3,n+1;n+2;-\sin (c+d x))+3 (a+b) (a-b)^2 \, _2F_1(2,n+1;n+2;-\sin (c+d x))+3 (a+b)^2 (a-b) \, _2F_1(2,n+1;n+2;\sin (c+d x))+2 (a+b)^3 \, _2F_1(3,n+1;n+2;\sin (c+d x))\right)}{16 d (n+1)}","\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,"((6*a*(a - b)*(a + b)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2] + 3*(a - b)^2*(a + b)*Hypergeometric2F1[2, 1 + n, 2 + n, -Sin[c + d*x]] + 3*(a - b)*(a + b)^2*Hypergeometric2F1[2, 1 + n, 2 + n, Sin[c + d*x]] + 2*(a - b)^3*Hypergeometric2F1[3, 1 + n, 2 + n, -Sin[c + d*x]] + 2*(a + b)^3*Hypergeometric2F1[3, 1 + n, 2 + n, Sin[c + d*x]])*Sin[c + d*x]^(1 + n))/(16*d*(1 + n))","A",1
1511,1,158,160,0.222627,"\int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^2,x]","\frac{\sin ^{n+1}(c+d x) \left(2 \left(3 a^2-b^2\right) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)+2 (a-b)^2 \, _2F_1(3,n+1;n+2;-\sin (c+d x))+(3 a+b) (a-b) \, _2F_1(2,n+1;n+2;-\sin (c+d x))+(3 a-b) (a+b) \, _2F_1(2,n+1;n+2;\sin (c+d x))+2 (a+b)^2 \, _2F_1(3,n+1;n+2;\sin (c+d x))\right)}{16 d (n+1)}","\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,"((2*(3*a^2 - b^2)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2] + (a - b)*(3*a + b)*Hypergeometric2F1[2, 1 + n, 2 + n, -Sin[c + d*x]] + (3*a - b)*(a + b)*Hypergeometric2F1[2, 1 + n, 2 + n, Sin[c + d*x]] + 2*(a - b)^2*Hypergeometric2F1[3, 1 + n, 2 + n, -Sin[c + d*x]] + 2*(a + b)^2*Hypergeometric2F1[3, 1 + n, 2 + n, Sin[c + d*x]])*Sin[c + d*x]^(1 + n))/(16*d*(1 + n))","A",1
1512,1,89,89,0.1155067,"\int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x]),x]","\frac{\sin ^{n+1}(c+d x) \left(a (n+2) \, _2F_1\left(3,\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right)+b (n+1) \sin (c+d x) \, _2F_1\left(3,\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right)\right)}{d (n+1) (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,"(Sin[c + d*x]^(1 + n)*(a*(2 + n)*Hypergeometric2F1[3, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2] + b*(1 + n)*Hypergeometric2F1[3, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]))/(d*(1 + n)*(2 + n))","A",1
1513,1,241,360,0.4612479,"\int \frac{\sec ^5(c+d x) \sin ^n(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^5*Sin[c + d*x]^n)/(a + b*Sin[c + d*x]),x]","\frac{\sin ^{n+1}(c+d x) \left(\frac{\left(3 a^2-9 a b+8 b^2\right) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{(a-b)^3}+\frac{\left(3 a^2+9 a b+8 b^2\right) \, _2F_1(1,n+1;n+2;\sin (c+d x))}{(a+b)^3}-\frac{16 b^6 \, _2F_1\left(1,n+1;n+2;-\frac{b \sin (c+d x)}{a}\right)}{a (a-b)^3 (a+b)^3}+\frac{(3 a-5 b) \, _2F_1(2,n+1;n+2;-\sin (c+d x))}{(a-b)^2}+\frac{(3 a+5 b) \, _2F_1(2,n+1;n+2;\sin (c+d x))}{(a+b)^2}+\frac{2 \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{a-b}+\frac{2 \, _2F_1(3,n+1;n+2;\sin (c+d x))}{a+b}\right)}{16 d (n+1)}","\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]])/(a - b)^3 + ((3*a^2 + 9*a*b + 8*b^2)*Hypergeometric2F1[1, 1 + n, 2 + n, Sin[c + d*x]])/(a + b)^3 - (16*b^6*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Sin[c + d*x])/a)])/(a*(a - b)^3*(a + b)^3) + ((3*a - 5*b)*Hypergeometric2F1[2, 1 + n, 2 + n, -Sin[c + d*x]])/(a - b)^2 + ((3*a + 5*b)*Hypergeometric2F1[2, 1 + n, 2 + n, Sin[c + d*x]])/(a + b)^2 + (2*Hypergeometric2F1[3, 1 + n, 2 + n, -Sin[c + d*x]])/(a - b) + (2*Hypergeometric2F1[3, 1 + n, 2 + n, Sin[c + d*x]])/(a + b))*Sin[c + d*x]^(1 + n))/(16*d*(1 + n))","A",1
1514,0,0,487,15.0137185,"\int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx","Integrate[Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p,x]","\int \sec ^5(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx","\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,"Integrate[Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p, x]","F",-1
1515,1,1600,502,9.9874093,"\int \frac{\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt{d \sin (e+f x)}} \, dx","Integrate[(Sec[e + f*x]^6*(a + b*Sin[e + f*x])^(9/2))/Sqrt[d*Sin[e + f*x]],x]","\frac{\sin (e+f x) \sqrt{a+b \sin (e+f x)} \left(\frac{1}{5} \left(a^4+4 b \sin (e+f x) a^3+6 b^2 a^2+4 b^3 \sin (e+f x) a+b^4\right) \sec ^5(e+f x)+\frac{1}{10} \left(3 a^4+9 b \sin (e+f x) a^3-3 b^2 a^2-5 b^3 \sin (e+f x) a-4 b^4\right) \sec ^3(e+f x)+\frac{1}{20} \left(15 a^4+24 b \sin (e+f x) a^3-15 b^2 a^2-12 b^3 \sin (e+f x) a+4 b^4\right) \sec (e+f x)\right)}{f \sqrt{d \sin (e+f x)}}+\frac{3 a \sqrt{\sin (e+f x)} \left(\frac{4 a \left(5 a^4-9 b^2 a^2+4 b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}} \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}+4 a \left(4 a b^3-8 a^3 b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{b \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}\right)+2 \left(8 a^2 b^2-4 b^4\right) \left(\frac{\sqrt{a+b \sin (e+f x)} \cos (e+f x)}{b \sqrt{\sin (e+f x)}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{b-a}} \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sec (e+f x) \sin ^4\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sqrt{-\frac{(a+b) \csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \sin (e+f x)}{a}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) (a+b \sin (e+f x))}{a}}}{b \sqrt{\sin (e+f x)} \sqrt{a+b \sin (e+f x)}}\right)}{b}+\frac{i \cos \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc (e+f x) E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right)}{\sqrt{\sin (e+f x)}}\right)|-\frac{2 a}{-a-b}\right) \sqrt{a+b \sin (e+f x)}}{b \sqrt{\cos ^2\left(\frac{1}{2} \left(-e-f x+\frac{\pi }{2}\right)\right) \csc (e+f x)} \sqrt{\frac{\csc (e+f x) (a+b \sin (e+f x))}{a+b}}}\right)\right)}{40 f \sqrt{d \sin (e+f x)}}","-\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 (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{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(2 a^4-3 a^2 b^2+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{\sec ^5(e+f x) \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f}",1,"(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((Sec[e + f*x]*(15*a^4 - 15*a^2*b^2 + 4*b^4 + 24*a^3*b*Sin[e + f*x] - 12*a*b^3*Sin[e + f*x]))/20 + (Sec[e + f*x]^3*(3*a^4 - 3*a^2*b^2 - 4*b^4 + 9*a^3*b*Sin[e + f*x] - 5*a*b^3*Sin[e + f*x]))/10 + (Sec[e + f*x]^5*(a^4 + 6*a^2*b^2 + b^4 + 4*a^3*b*Sin[e + f*x] + 4*a*b^3*Sin[e + f*x]))/5))/(f*Sqrt[d*Sin[e + f*x]]) + (3*a*Sqrt[Sin[e + f*x]]*((4*a*(5*a^4 - 9*a^2*b^2 + 4*b^4)*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + 4*a*(-8*a^3*b + 4*a*b^3)*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])) + 2*(8*a^2*b^2 - 4*b^4)*((Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Sin[e + f*x]]) + (I*Cos[(-e + Pi/2 - f*x)/2]*Csc[e + f*x]*EllipticE[I*ArcSinh[Sin[(-e + Pi/2 - f*x)/2]/Sqrt[Sin[e + f*x]]], (-2*a)/(-a - b)]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Cos[(-e + Pi/2 - f*x)/2]^2*Csc[e + f*x]]*Sqrt[(Csc[e + f*x]*(a + b*Sin[e + f*x]))/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])))/b)))/(40*f*Sqrt[d*Sin[e + f*x]])","C",1
1516,1,3522,458,7.0846757,"\int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{4/3} \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(4/3),x]","\text{Result too large to show}","-\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-\left(b^2 \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-\left(b^2 \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,"(513*a*b*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/(455*f) + (81*b^2*c^4*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/(7280*d^3*f) - (18*a*b*c^3*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/(455*d^2*f) + (54*a^2*c^2*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/(35*d*f) + (5211*b^2*c^2*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/(14560*d*f) + (9*a^2*d*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/(40*f) + (63*b^2*d*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/(640*f) + (9*a*b*c^2*((-3*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/d^2 + (3*AppellF1[4/3, 1/2, 1/2, 7/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(4/3))/(4*d^2)))/(65*f) + (81*b^2*c^5*((-3*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/d^2 + (3*AppellF1[4/3, 1/2, 1/2, 7/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(4/3))/(4*d^2)))/(7280*d^3*f) - (18*a*b*c^4*((-3*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/d^2 + (3*AppellF1[4/3, 1/2, 1/2, 7/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(4/3))/(4*d^2)))/(455*d^2*f) + (3*a^2*c^3*((-3*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/d^2 + (3*AppellF1[4/3, 1/2, 1/2, 7/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(4/3))/(4*d^2)))/(70*d*f) - (21*b^2*c^3*((-3*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/d^2 + (3*AppellF1[4/3, 1/2, 1/2, 7/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(4/3))/(4*d^2)))/(1040*d*f) + (153*a^2*c*d*((-3*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/d^2 + (3*AppellF1[4/3, 1/2, 1/2, 7/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(4/3))/(4*d^2)))/(280*f) + (9603*b^2*c*d*((-3*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/d^2 + (3*AppellF1[4/3, 1/2, 1/2, 7/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(4/3))/(4*d^2)))/(58240*f) + (24*a*b*d^2*((-3*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(1/3))/d^2 + (3*AppellF1[4/3, 1/2, 1/2, 7/3, -((c + d*Sin[e + f*x])/((1 - c/d)*d)), -((c + d*Sin[e + f*x])/((-1 - c/d)*d))]*Sec[e + f*x]*Sqrt[(-d - d*Sin[e + f*x])/(c - d)]*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*(c + d*Sin[e + f*x])^(4/3))/(4*d^2)))/(91*f) + ((c + d*Sin[e + f*x])^(1/3)*((-3*(-216*b^2*c^4 + 768*a*b*c^3*d - 832*a^2*c^2*d^2 + 332*b^2*c^2*d^2 + 7232*a*b*c*d^3 + 2912*a^2*d^4 + 1729*b^2*d^4)*Cos[e + f*x])/(58240*d^3) - (3*(8*b^2*c^2 + 896*a*b*c*d + 416*a^2*d^2 + 117*b^2*d^2)*Cos[3*(e + f*x)])/(16640*d) + (3*b^2*d*Cos[5*(e + f*x)])/256 + (3*(-18*b^2*c^3 + 64*a*b*c^2*d + 1144*a^2*c*d^2 + 23*b^2*c*d^2 + 80*a*b*d^3)*Sin[2*(e + f*x)])/(14560*d^2) - (3*b*(17*b*c + 32*a*d)*Sin[4*(e + f*x)])/1664))/f","B",0
1517,1,398,341,5.132375,"\int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{4/3} \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(4/3),x]","\frac{3 \sec (e+f x) \sqrt[3]{c+d \sin (e+f x)} \left(12 \left(d^2-c^2\right) \left(52 a c^2 d+91 a d^3-24 b c^3+68 b c d^2\right) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{-\frac{d (\sin (e+f x)+1)}{c-d}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right)+3 \left(52 a c^3 d+663 a c d^3-24 b c^4+84 b c^2 d^2+160 b d^4\right) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{-\frac{d (\sin (e+f x)+1)}{c-d}} (c+d \sin (e+f x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right)-4 d^2 \cos ^2(e+f x) \left(-2 d \left(286 a c d+8 b c^2+45 b d^2\right) \sin (e+f x)+14 d^2 (13 a d+14 b c) \cos (2 (e+f x))-52 a c^2 d+91 a d^3+24 b c^3+128 b c d^2+70 b d^3 \sin (3 (e+f x))\right)\right)}{14560 d^4 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*Sec[e + f*x]*(c + d*Sin[e + f*x])^(1/3)*(12*(-c^2 + d^2)*(-24*b*c^3 + 52*a*c^2*d + 68*b*c*d^2 + 91*a*d^3)*AppellF1[1/3, 1/2, 1/2, 4/3, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[-((d*(1 + Sin[e + f*x]))/(c - d))] + 3*(-24*b*c^4 + 52*a*c^3*d + 84*b*c^2*d^2 + 663*a*c*d^3 + 160*b*d^4)*AppellF1[4/3, 1/2, 1/2, 7/3, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[-((d*(1 + Sin[e + f*x]))/(c - d))]*(c + d*Sin[e + f*x]) - 4*d^2*Cos[e + f*x]^2*(24*b*c^3 - 52*a*c^2*d + 128*b*c*d^2 + 91*a*d^3 + 14*d^2*(14*b*c + 13*a*d)*Cos[2*(e + f*x)] - 2*d*(8*b*c^2 + 286*a*c*d + 45*b*d^2)*Sin[e + f*x] + 70*b*d^3*Sin[3*(e + f*x)])))/(14560*d^4*f)","A",0
1518,1,301,125,2.1312974,"\int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx","Integrate[Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3),x]","-\frac{3 \sec (e+f x) \sqrt[3]{c+d \sin (e+f x)} \left(-3 c \left(4 c^2+51 d^2\right) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{-\frac{d (\sin (e+f x)+1)}{c-d}} (c+d \sin (e+f x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right)+12 \left(4 c^4+3 c^2 d^2-7 d^4\right) \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} \sqrt{-\frac{d (\sin (e+f x)+1)}{c-d}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right)+4 d^2 \cos ^2(e+f x) \left(-4 c^2-44 c d \sin (e+f x)+14 d^2 \cos (2 (e+f x))+7 d^2\right)\right)}{1120 d^3 f}","\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*Sec[e + f*x]*(c + d*Sin[e + f*x])^(1/3)*(12*(4*c^4 + 3*c^2*d^2 - 7*d^4)*AppellF1[1/3, 1/2, 1/2, 4/3, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[-((d*(1 + Sin[e + f*x]))/(c - d))] - 3*c*(4*c^2 + 51*d^2)*AppellF1[4/3, 1/2, 1/2, 7/3, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Sqrt[-((d*(-1 + Sin[e + f*x]))/(c + d))]*Sqrt[-((d*(1 + Sin[e + f*x]))/(c - d))]*(c + d*Sin[e + f*x]) + 4*d^2*Cos[e + f*x]^2*(-4*c^2 + 7*d^2 + 14*d^2*Cos[2*(e + f*x)] - 44*c*d*Sin[e + f*x])))/(1120*d^3*f)","B",0
1519,0,0,38,62.9879515,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{a+b \sin (e+f x)} \, dx","Integrate[(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,"Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3))/(a + b*Sin[e + f*x]), x]","A",-1
1520,0,0,38,46.8637651,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx","Integrate[(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,"Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3))/(a + b*Sin[e + f*x])^2, x]","A",-1
1521,0,0,36,7.4661485,"\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx","Integrate[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,"Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x]","A",-1
1522,0,0,38,36.2867231,"\int \cos ^2(e+f x) (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{4/3} \, dx","Integrate[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,"Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(4/3), x]","A",-1
1523,0,0,552,8.8490587,"\int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]","\int \cos ^2(e+f x) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx","-\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)-\left(b^2 \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,"Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x]","F",-1
1524,0,0,375,3.5672166,"\int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]","\int \cos ^2(e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^n \, dx","-\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,"Integrate[Cos[e + f*x]^2*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x]","F",-1
1525,0,0,127,0.3251776,"\int \cos ^2(e+f x) (c+d \sin (e+f x))^n \, dx","Integrate[Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n,x]","\int \cos ^2(e+f x) (c+d \sin (e+f x))^n \, dx","\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,"Integrate[Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n, x]","F",-1
1526,0,0,36,3.3340838,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx","Integrate[(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,"Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x]), x]","A",-1
1527,0,0,36,35.5950641,"\int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx","Integrate[(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,"Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x])^2, x]","A",-1
1528,1,151,188,0.7959393,"\int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{\sin (c+d x) \left(-315 (a B+A b) \sin ^7(c+d x)-360 (a A-3 b B) \sin ^6(c+d x)+1260 (a B+A b) \sin ^5(c+d x)+1512 (a A-b B) \sin ^4(c+d x)-1890 (a B+A b) \sin ^3(c+d x)-840 (3 a A-b B) \sin ^2(c+d x)+1260 (a B+A b) \sin (c+d x)+2520 a A-280 b B \sin ^8(c+d x)\right)}{2520 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,"(Sin[c + d*x]*(2520*a*A + 1260*(A*b + a*B)*Sin[c + d*x] - 840*(3*a*A - b*B)*Sin[c + d*x]^2 - 1890*(A*b + a*B)*Sin[c + d*x]^3 + 1512*(a*A - b*B)*Sin[c + d*x]^4 + 1260*(A*b + a*B)*Sin[c + d*x]^5 - 360*(a*A - 3*b*B)*Sin[c + d*x]^6 - 315*(A*b + a*B)*Sin[c + d*x]^7 - 280*b*B*Sin[c + d*x]^8))/(2520*d)","A",1
1529,1,116,143,0.2947921,"\int \cos ^5(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{\sin (c+d x) \left(35 (a B+A b) \sin ^5(c+d x)+42 (a A-2 b B) \sin ^4(c+d x)-105 (a B+A b) \sin ^3(c+d x)-70 (2 a A-b B) \sin ^2(c+d x)+105 (a B+A b) \sin (c+d x)+210 a A+30 b B \sin ^6(c+d x)\right)}{210 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,"(Sin[c + d*x]*(210*a*A + 105*(A*b + a*B)*Sin[c + d*x] - 70*(2*a*A - b*B)*Sin[c + d*x]^2 - 105*(A*b + a*B)*Sin[c + d*x]^3 + 42*(a*A - 2*b*B)*Sin[c + d*x]^4 + 35*(A*b + a*B)*Sin[c + d*x]^5 + 30*b*B*Sin[c + d*x]^6))/(210*d)","A",1
1530,1,80,97,0.258093,"\int \cos ^3(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{\sin (c+d x) \left(-15 (a B+A b) \sin ^3(c+d x)-20 (a A-b B) \sin ^2(c+d x)+30 (a B+A b) \sin (c+d x)+60 a A-12 b B \sin ^4(c+d x)\right)}{60 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,"(Sin[c + d*x]*(60*a*A + 30*(A*b + a*B)*Sin[c + d*x] - 20*(a*A - b*B)*Sin[c + d*x]^2 - 15*(A*b + a*B)*Sin[c + d*x]^3 - 12*b*B*Sin[c + d*x]^4))/(60*d)","A",1
1531,1,45,52,0.0815337,"\int \cos (c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{\sin (c+d x) \left(3 (a B+A b) \sin (c+d x)+6 a A+2 b B \sin ^2(c+d x)\right)}{6 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,"(Sin[c + d*x]*(6*a*A + 3*(A*b + a*B)*Sin[c + d*x] + 2*b*B*Sin[c + d*x]^2))/(6*d)","A",1
1532,1,68,64,0.0291067,"\int \sec (c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a B \log (\cos (c+d x))}{d}-\frac{A b \log (\cos (c+d x))}{d}-\frac{b B \sin (c+d x)}{d}+\frac{b B \tanh ^{-1}(\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*A*ArcTanh[Sin[c + d*x]])/d + (b*B*ArcTanh[Sin[c + d*x]])/d - (A*b*Log[Cos[c + d*x]])/d - (a*B*Log[Cos[c + d*x]])/d - (b*B*Sin[c + d*x])/d","A",1
1533,1,54,59,0.2241231,"\int \sec ^3(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[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))+\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]] + Sec[c + d*x]^2*(A*b + a*B + (a*A + b*B)*Sin[c + d*x]))/(2*d)","A",1
1534,1,82,88,0.5851448,"\int \sec ^5(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^5*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","\frac{\sec ^4(c+d x) \left((b B-3 a A) \sin ^3(c+d x)+(5 a A+b B) \sin (c+d x)+(3 a A-b B) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))+2 (a B+A b)\right)}{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,"(Sec[c + d*x]^4*(2*(A*b + a*B) + (3*a*A - b*B)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + (5*a*A + b*B)*Sin[c + d*x] + (-3*a*A + b*B)*Sin[c + d*x]^3))/(8*d)","A",1
1535,1,104,118,0.8823038,"\int \sec ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^7*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]","-\frac{\sec ^6(c+d x) \left((3 b B-15 a A) \sin ^5(c+d x)+8 (5 a A-b B) \sin ^3(c+d x)-3 (11 a A+b B) \sin (c+d x)-3 (5 a A-b B) \cos ^6(c+d x) \tanh ^{-1}(\sin (c+d x))-8 (a B+A b)\right)}{48 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,"-1/48*(Sec[c + d*x]^6*(-8*(A*b + a*B) - 3*(5*a*A - b*B)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^6 - 3*(11*a*A + b*B)*Sin[c + d*x] + 8*(5*a*A - b*B)*Sin[c + d*x]^3 + (-15*a*A + 3*b*B)*Sin[c + d*x]^5))/d","A",1
1536,1,295,349,1.5011896,"\int \cos ^7(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{2520 a^2 A b^8 \sin (c+d x)-315 b^8 \left(a^2 B+2 a A b-3 b^2 B\right) \sin ^8(c+d x)+360 b^8 \left(a^2 (-A)+6 a b B+3 A b^2\right) \sin ^7(c+d x)+1260 b^8 \left(a^2 B+2 a A b-b^2 B\right) \sin ^6(c+d x)-1512 b^8 \left(a^2 (-A)+2 a b B+A b^2\right) \sin ^5(c+d x)+630 b^8 \left(-3 a^2 B-6 a A b+b^2 B\right) \sin ^4(c+d x)+840 b^8 \left(-3 a^2 A+2 a b B+A b^2\right) \sin ^3(c+d x)-3 a^4 B \left(a^6-9 a^4 b^2+42 a^2 b^4-210 b^6\right)-280 b^9 (2 a B+A b) \sin ^9(c+d x)+1260 a b^8 (a B+2 A b) \sin ^2(c+d x)-252 b^{10} B \sin ^{10}(c+d x)}{2520 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(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{\left(-35 a^3 B+15 a^2 A b+15 a b^2 B-3 A b^3\right) (a+b \sin (c+d x))^7}{7 b^8 d}-\frac{3 \left(a^2-b^2\right) \left(-7 a^3 B+5 a^2 A b+3 a b^2 B-A b^3\right) (a+b \sin (c+d x))^5}{5 b^8 d}+\frac{\left(-35 a^4 B+20 a^3 A b+30 a^2 b^2 B-12 a A b^3-3 b^4 B\right) (a+b \sin (c+d x))^6}{6 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,"(-3*a^4*(a^6 - 9*a^4*b^2 + 42*a^2*b^4 - 210*b^6)*B + 2520*a^2*A*b^8*Sin[c + d*x] + 1260*a*b^8*(2*A*b + a*B)*Sin[c + d*x]^2 + 840*b^8*(-3*a^2*A + A*b^2 + 2*a*b*B)*Sin[c + d*x]^3 + 630*b^8*(-6*a*A*b - 3*a^2*B + b^2*B)*Sin[c + d*x]^4 - 1512*b^8*(-(a^2*A) + A*b^2 + 2*a*b*B)*Sin[c + d*x]^5 + 1260*b^8*(2*a*A*b + a^2*B - b^2*B)*Sin[c + d*x]^6 + 360*b^8*(-(a^2*A) + 3*A*b^2 + 6*a*b*B)*Sin[c + d*x]^7 - 315*b^8*(2*a*A*b + a^2*B - 3*b^2*B)*Sin[c + d*x]^8 - 280*b^9*(A*b + 2*a*B)*Sin[c + d*x]^9 - 252*b^10*B*Sin[c + d*x]^10)/(2520*b^8*d)","A",1
1537,1,227,231,0.5014057,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{840 a^2 A b^6 \sin (c+d x)+140 b^6 \left(a^2 B+2 a A b-2 b^2 B\right) \sin ^6(c+d x)+168 b^6 \left(a^2 A-4 a b B-2 A b^2\right) \sin ^5(c+d x)+210 b^6 \left(-2 a^2 B-4 a A b+b^2 B\right) \sin ^4(c+d x)+280 b^6 \left(-2 a^2 A+2 a b B+A b^2\right) \sin ^3(c+d x)+a^4 B \left(3 a^4-28 a^2 b^2+210 b^4\right)+120 b^7 (2 a B+A b) \sin ^7(c+d x)+420 a b^6 (a B+2 A b) \sin ^2(c+d x)+105 b^8 B \sin ^8(c+d x)}{840 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{\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{2 \left(-5 a^3 B+3 a^2 A b+3 a b^2 B-A b^3\right) (a+b \sin (c+d x))^5}{5 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^4*(3*a^4 - 28*a^2*b^2 + 210*b^4)*B + 840*a^2*A*b^6*Sin[c + d*x] + 420*a*b^6*(2*A*b + a*B)*Sin[c + d*x]^2 + 280*b^6*(-2*a^2*A + A*b^2 + 2*a*b*B)*Sin[c + d*x]^3 + 210*b^6*(-4*a*A*b - 2*a^2*B + b^2*B)*Sin[c + d*x]^4 + 168*b^6*(a^2*A - 2*A*b^2 - 4*a*b*B)*Sin[c + d*x]^5 + 140*b^6*(2*a*A*b + a^2*B - 2*b^2*B)*Sin[c + d*x]^6 + 120*b^7*(A*b + 2*a*B)*Sin[c + d*x]^7 + 105*b^8*B*Sin[c + d*x]^8)/(840*b^6*d)","A",1
1538,1,111,132,0.2483764,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{(a+b \sin (c+d x))^3 \left(a^3 B+3 b \left(a^2 (-B)+2 a A b+5 b^2 B\right) \sin (c+d x)-2 a^2 A b-6 b^2 (2 A b-a B) \sin ^2(c+d x)-5 a b^2 B+20 A b^3-10 b^3 B \sin ^3(c+d x)\right)}{60 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 + b*Sin[c + d*x])^3*(-2*a^2*A*b + 20*A*b^3 + a^3*B - 5*a*b^2*B + 3*b*(2*a*A*b - a^2*B + 5*b^2*B)*Sin[c + d*x] - 6*b^2*(2*A*b - a*B)*Sin[c + d*x]^2 - 10*b^3*B*Sin[c + d*x]^3))/(60*b^4*d)","A",1
1539,1,41,54,0.0685874,"\int \cos (c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{(a+b \sin (c+d x))^3 (-a B+4 A b+3 b B \sin (c+d x))}{12 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*Sin[c + d*x])^3*(4*A*b - a*B + 3*b*B*Sin[c + d*x]))/(12*b^2*d)","A",1
1540,1,81,94,0.2074554,"\int \sec (c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","-\frac{2 b (2 a B+A b) \sin (c+d x)-\left((a-b)^2 (A-B) \log (\sin (c+d x)+1)\right)+(a+b)^2 (A+B) \log (1-\sin (c+d x))+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,"-1/2*((a + b)^2*(A + B)*Log[1 - Sin[c + d*x]] - (a - b)^2*(A - B)*Log[1 + Sin[c + d*x]] + 2*b*(A*b + 2*a*B)*Sin[c + d*x] + b^2*B*Sin[c + d*x]^2)/d","A",1
1541,1,174,112,1.5349345,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{\left(-6 a^3 A b+4 a A b^3+2 b^4 B\right) \tan ^2(c+d x)-2 a^3 (a B-A b) \sec ^2(c+d x)+\left(a^2-b^2\right) ((a+b) (a A-b (A+2 B)) \log (1-\sin (c+d x))-(a-b) (a A+b (A-2 B)) \log (\sin (c+d x)+1))-2 \left(a^2-b^2\right) \left(a^2 A+2 a b B+A b^2\right) \tan (c+d x) \sec (c+d x)}{4 d \left(b^2-a^2\right)}","-\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^2 - b^2)*((a + b)*(a*A - b*(A + 2*B))*Log[1 - Sin[c + d*x]] - (a - b)*(a*A + b*(A - 2*B))*Log[1 + Sin[c + d*x]]) - 2*a^3*(-(A*b) + a*B)*Sec[c + d*x]^2 - 2*(a^2 - b^2)*(a^2*A + A*b^2 + 2*a*b*B)*Sec[c + d*x]*Tan[c + d*x] + (-6*a^3*A*b + 4*a*A*b^3 + 2*b^4*B)*Tan[c + d*x]^2)/(4*(-a^2 + b^2)*d)","A",1
1542,1,186,122,1.7835718,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{4 \left(b^2-a^2\right) \sec ^4(c+d x) (a+b \sin (c+d x))^3 ((b B-a A) \sin (c+d x)-a B+A b)+\left(-3 a^2 A+2 a b B+A b^2\right) \left(-2 \left(a^4-b^4\right) \tan (c+d x) \sec (c+d x)+\left(4 a b^3-6 a^3 b\right) \tan ^2(c+d x)+2 a^3 b \sec ^2(c+d x)+\left(a^2-b^2\right)^2 (\log (1-\sin (c+d x))-\log (\sin (c+d x)+1))\right)}{16 d \left(a^2-b^2\right)^2}","\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,"(4*(-a^2 + b^2)*Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3*(A*b - a*B + (-(a*A) + b*B)*Sin[c + d*x]) + (-3*a^2*A + A*b^2 + 2*a*b*B)*((a^2 - b^2)^2*(Log[1 - Sin[c + d*x]] - Log[1 + Sin[c + d*x]]) + 2*a^3*b*Sec[c + d*x]^2 - 2*(a^4 - b^4)*Sec[c + d*x]*Tan[c + d*x] + (-6*a^3*b + 4*a*b^3)*Tan[c + d*x]^2))/(16*(a^2 - b^2)^2*d)","A",1
1543,1,242,160,1.5380127,"\int \sec ^7(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^7*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]","\frac{-\frac{3 b \left(-5 a^2 A+2 a b B+A b^2\right) \left(-2 \left(a^4-b^4\right) \tan (c+d x) \sec (c+d x)+\left(4 a b^3-6 a^3 b\right) \tan ^2(c+d x)+2 a^3 b \sec ^2(c+d x)+\left(a^2-b^2\right)^2 (\log (1-\sin (c+d x))-\log (\sin (c+d x)+1))\right)}{16 (a-b) (a+b)}+b \sec ^6(c+d x) (a+b \sin (c+d x))^3 ((b B-a A) \sin (c+d x)-a B+A b)+\frac{1}{4} b \sec ^4(c+d x) (a+b \sin (c+d x))^3 ((2 b B-5 a A) \sin (c+d x)+3 A b)}{6 b d \left(b^2-a^2\right)}","\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,"(b*Sec[c + d*x]^6*(a + b*Sin[c + d*x])^3*(A*b - a*B + (-(a*A) + b*B)*Sin[c + d*x]) + (b*Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3*(3*A*b + (-5*a*A + 2*b*B)*Sin[c + d*x]))/4 - (3*b*(-5*a^2*A + A*b^2 + 2*a*b*B)*((a^2 - b^2)^2*(Log[1 - Sin[c + d*x]] - Log[1 + Sin[c + d*x]]) + 2*a^3*b*Sec[c + d*x]^2 - 2*(a^4 - b^4)*Sec[c + d*x]*Tan[c + d*x] + (-6*a^3*b + 4*a*b^3)*Tan[c + d*x]^2))/(16*(a - b)*(a + b)))/(6*b*(-a^2 + b^2)*d)","A",1
1544,1,218,315,0.8657317,"\int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{\frac{(A b-a B) \left(-30 b^2 \left(a^2-b^2\right)^2 \sin ^2(c+d x)-60 \left(a^2-b^2\right)^3 \log (a+b \sin (c+d x))+15 b^4 \left(b^2-a^2\right) \cos ^4(c+d x)+20 a b^3 \left(a^2-3 b^2\right) \sin ^3(c+d x)+60 a b \left(a^4-3 a^2 b^2+3 b^4\right) \sin (c+d x)+12 a b^5 \sin ^5(c+d x)+10 b^6 \cos ^6(c+d x)\right)}{60 b}+\frac{b^6 B (1225 \sin (c+d x)+245 \sin (3 (c+d x))+49 \sin (5 (c+d x))+5 \sin (7 (c+d x)))}{2240}}{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{\left(a^2-3 b^2\right) (A b-a B) \sin ^4(c+d x)}{4 b^4 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^4-3 a^2 b^2+3 b^4\right) (A b-a B) \sin ^2(c+d x)}{2 b^6 d}+\frac{\left(a^4 (-B)+a^3 A b+3 a^2 b^2 B-3 a A b^3-3 b^4 B\right) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left(a^6 (-B)+a^5 A b+3 a^4 b^2 B-3 a^3 A b^3-3 a^2 b^4 B+3 a A b^5+b^6 B\right) \sin (c+d x)}{b^7 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*b - a*B)*(15*b^4*(-a^2 + b^2)*Cos[c + d*x]^4 + 10*b^6*Cos[c + d*x]^6 - 60*(a^2 - b^2)^3*Log[a + b*Sin[c + d*x]] + 60*a*b*(a^4 - 3*a^2*b^2 + 3*b^4)*Sin[c + d*x] - 30*b^2*(a^2 - b^2)^2*Sin[c + d*x]^2 + 20*a*b^3*(a^2 - 3*b^2)*Sin[c + d*x]^3 + 12*a*b^5*Sin[c + d*x]^5))/(60*b) + (b^6*B*(1225*Sin[c + d*x] + 245*Sin[3*(c + d*x)] + 49*Sin[5*(c + d*x)] + 5*Sin[7*(c + d*x)]))/2240)/(b^7*d)","A",1
1545,1,148,202,0.4312907,"\int \frac{\cos ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{20 (A b-a B) \left(6 b^2 \left(a^2-b^2\right) \sin ^2(c+d x)-12 a b \left(a^2-2 b^2\right) \sin (c+d x)+12 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))-4 a b^3 \sin ^3(c+d x)+3 b^4 \cos ^4(c+d x)\right)+b^5 B (150 \sin (c+d x)+25 \sin (3 (c+d x))+3 \sin (5 (c+d x)))}{240 b^6 d}","\frac{\left(a^2-b^2\right)^2 (A b-a B) \log (a+b \sin (c+d x))}{b^6 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^2 (-B)+a A b+2 b^2 B\right) \sin ^3(c+d x)}{3 b^3 d}-\frac{\left(a^4 (-B)+a^3 A b+2 a^2 b^2 B-2 a A b^3-b^4 B\right) \sin (c+d x)}{b^5 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,"(20*(A*b - a*B)*(3*b^4*Cos[c + d*x]^4 + 12*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]] - 12*a*b*(a^2 - 2*b^2)*Sin[c + d*x] + 6*b^2*(a^2 - b^2)*Sin[c + d*x]^2 - 4*a*b^3*Sin[c + d*x]^3) + b^5*B*(150*Sin[c + d*x] + 25*Sin[3*(c + d*x)] + 3*Sin[5*(c + d*x)]))/(240*b^6*d)","A",1
1546,1,89,111,0.3757951,"\int \frac{\cos ^3(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{\left(A-\frac{a B}{b}\right) \left(\left(b^2-a^2\right) \log (a+b \sin (c+d x))+a b \sin (c+d x)-\frac{1}{2} b^2 \sin ^2(c+d x)\right)+\frac{1}{12} b^2 B (9 \sin (c+d x)+\sin (3 (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{\left(a^2 (-B)+a A b+b^2 B\right) \sin (c+d x)}{b^3 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 - (a*B)/b)*((-a^2 + b^2)*Log[a + b*Sin[c + d*x]] + a*b*Sin[c + d*x] - (b^2*Sin[c + d*x]^2)/2) + (b^2*B*(9*Sin[c + d*x] + Sin[3*(c + d*x)]))/12)/(b^3*d)","A",1
1547,1,39,41,0.0456652,"\int \frac{\cos (c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{\frac{(A b-a B) \log (a+b \sin (c+d x))}{b}+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 + B*Sin[c + d*x])/(b*d)","A",1
1548,1,99,90,0.1850012,"\int \frac{\sec (c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{\frac{(a B-A b) \log (a+b \sin (c+d x))+(a+b) (A-B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a-b}-(A+B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{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[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]) + ((a + b)*(A - B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (-(A*b) + a*B)*Log[a + b*Sin[c + d*x]])/(a - b))/((a + b)*d)","A",1
1549,1,197,159,0.7652064,"\int \frac{\sec ^3(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{\frac{4 b^2 (A b-a B) \log (a+b \sin (c+d x))}{\left(a^2-b^2\right)^2}+\frac{A+B}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{B-A}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{2 (a A+b (2 A+B)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{(a+b)^2}+\frac{2 (a A+b (B-2 A)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{(a-b)^2}}{4 d}","\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,"((-2*(a*A + b*(2*A + B))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a + b)^2 + (2*(a*A + b*(-2*A + B))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a - b)^2 + (4*b^2*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^2 + (A + B)/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (-A + B)/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2))/(4*d)","A",1
1550,1,321,263,1.3123872,"\int \frac{\sec ^5(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{-\frac{2 \left(3 a^2 A+a b (9 A+B)+b^2 (8 A+3 B)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{(a+b)^3}+\frac{2 \left(3 a^2 A+a b (B-9 A)+b^2 (8 A-3 B)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{(a-b)^3}+\frac{16 b^4 (A b-a B) \log (a+b \sin (c+d x))}{\left(b^2-a^2\right)^3}+\frac{3 a A+a B+5 A b+3 b B}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{-3 a A+a B+5 A b-3 b B}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{A+B}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{B-A}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}}{16 d}","-\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{b^4 (A b-a B) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}+\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,"((-2*(3*a^2*A + a*b*(9*A + B) + b^2*(8*A + 3*B))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a + b)^3 + (2*(3*a^2*A + b^2*(8*A - 3*B) + a*b*(-9*A + B))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a - b)^3 + (16*b^4*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(-a^2 + b^2)^3 + (A + B)/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + (3*a*A + 5*A*b + a*B + 3*b*B)/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (-A + B)/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (-3*a*A + 5*A*b + a*B - 3*b*B)/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2))/(16*d)","A",0
1551,1,565,383,2.4892049,"\int \frac{\sec ^7(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx","Integrate[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]),x]","\frac{\frac{768 b^6 (A b-a B) \log (a+b \sin (c+d x))}{\left(a^2-b^2\right)^4}-\frac{48 \left(5 a^3 A+a^2 b (20 A+B)+a b^2 (29 A+4 B)+b^3 (16 A+5 B)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{(a+b)^4}+\frac{48 \left(5 a^3 A+a^2 b (B-20 A)+a b^2 (29 A-4 B)+b^3 (5 B-16 A)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{(a-b)^4}+\frac{\sec ^6(c+d x) \left(198 a^5 A \sin (c+d x)+85 a^5 A \sin (3 (c+d x))+15 a^5 A \sin (5 (c+d x))+128 a^5 B-128 a^4 A b-114 a^4 b B \sin (c+d x)+17 a^4 b B \sin (3 (c+d x))+3 a^4 b B \sin (5 (c+d x))-480 a^3 A b^2 \sin (c+d x)-272 a^3 A b^2 \sin (3 (c+d x))-48 a^3 A b^2 \sin (5 (c+d x))-352 a^3 b^2 B+352 a^2 A b^3-96 b^2 \left(a^2-3 b^2\right) (a B-A b) \cos (2 (c+d x))+264 a^2 b^3 B \sin (c+d x)-4 a^2 b^3 B \sin (3 (c+d x))-12 a^2 b^3 B \sin (5 (c+d x))-48 b^4 (A b-a B) \cos (4 (c+d x))+330 a A b^4 \sin (c+d x)+259 a A b^4 \sin (3 (c+d x))+57 a A b^4 \sin (5 (c+d x))+368 a b^4 B-368 A b^5-198 b^5 B \sin (c+d x)-85 b^5 B \sin (3 (c+d x))-15 b^5 B \sin (5 (c+d x))\right)}{\left(a^2-b^2\right)^3}}{768 d}","-\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{b^6 (A b-a B) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}-\frac{\left(5 a^3 A+a^2 b (20 A+B)+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(5 a^3 A-a^2 b (20 A-B)+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 ^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(5 a^5 A+a^4 b B-16 a^3 A b^2-4 a^2 b^3 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,"((-48*(5*a^3*A + a^2*b*(20*A + B) + a*b^2*(29*A + 4*B) + b^3*(16*A + 5*B))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a + b)^4 + (48*(5*a^3*A + a*b^2*(29*A - 4*B) + a^2*b*(-20*A + B) + b^3*(-16*A + 5*B))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a - b)^4 + (768*b^6*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^4 + (Sec[c + d*x]^6*(-128*a^4*A*b + 352*a^2*A*b^3 - 368*A*b^5 + 128*a^5*B - 352*a^3*b^2*B + 368*a*b^4*B - 96*b^2*(a^2 - 3*b^2)*(-(A*b) + a*B)*Cos[2*(c + d*x)] - 48*b^4*(A*b - a*B)*Cos[4*(c + d*x)] + 198*a^5*A*Sin[c + d*x] - 480*a^3*A*b^2*Sin[c + d*x] + 330*a*A*b^4*Sin[c + d*x] - 114*a^4*b*B*Sin[c + d*x] + 264*a^2*b^3*B*Sin[c + d*x] - 198*b^5*B*Sin[c + d*x] + 85*a^5*A*Sin[3*(c + d*x)] - 272*a^3*A*b^2*Sin[3*(c + d*x)] + 259*a*A*b^4*Sin[3*(c + d*x)] + 17*a^4*b*B*Sin[3*(c + d*x)] - 4*a^2*b^3*B*Sin[3*(c + d*x)] - 85*b^5*B*Sin[3*(c + d*x)] + 15*a^5*A*Sin[5*(c + d*x)] - 48*a^3*A*b^2*Sin[5*(c + d*x)] + 57*a*A*b^4*Sin[5*(c + d*x)] + 3*a^4*b*B*Sin[5*(c + d*x)] - 12*a^2*b^3*B*Sin[5*(c + d*x)] - 15*b^5*B*Sin[5*(c + d*x)]))/(a^2 - b^2)^3)/(768*d)","A",1
1552,1,396,324,1.6251591,"\int \frac{\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{6 (A b-a B) \left(-4 a^2 b^4 \sin ^4(c+d x)+4 \left(a^2-b^2\right)^2 \left(15 a^2 \log (a+b \sin (c+d x))+4 a^2-4 b^2\right)+b^4 \cos ^4(c+d x) \left(-a^2+3 a b \sin (c+d x)+4 b^2\right)+2 a b^3 \left(5 a^2-7 b^2\right) \sin ^3(c+d x)-2 b^2 \left(15 a^4-29 a^2 b^2+8 b^4\right) \sin ^2(c+d x)+4 a b \sin (c+d x) \left(-11 a^4+15 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))+18 a^2 b^2-4 b^4\right)+2 b^6 \cos ^6(c+d x)\right)}{a+b \sin (c+d x)}+B \left(-30 b^2 \left(a^2-b^2\right)^2 \sin ^2(c+d x)-60 \left(a^2-b^2\right)^3 \log (a+b \sin (c+d x))+15 b^4 \left(b^2-a^2\right) \cos ^4(c+d x)+20 a b^3 \left(a^2-3 b^2\right) \sin ^3(c+d x)+60 a b \left(a^4-3 a^2 b^2+3 b^4\right) \sin (c+d x)+12 a b^5 \sin ^5(c+d x)+10 b^6 \cos ^6(c+d x)\right)}{60 b^8 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{\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(-4 a^3 B+3 a^2 A b+6 a b^2 B-3 A b^3\right) \sin ^3(c+d x)}{3 b^5 d}+\frac{\left(-5 a^4 B+4 a^3 A b+9 a^2 b^2 B-6 a A b^3-3 b^4 B\right) \sin ^2(c+d x)}{2 b^6 d}-\frac{\left(-6 a^5 B+5 a^4 A b+12 a^3 b^2 B-9 a^2 A b^3-6 a b^4 B+3 A b^5\right) \sin (c+d x)}{b^7 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,"(B*(15*b^4*(-a^2 + b^2)*Cos[c + d*x]^4 + 10*b^6*Cos[c + d*x]^6 - 60*(a^2 - b^2)^3*Log[a + b*Sin[c + d*x]] + 60*a*b*(a^4 - 3*a^2*b^2 + 3*b^4)*Sin[c + d*x] - 30*b^2*(a^2 - b^2)^2*Sin[c + d*x]^2 + 20*a*b^3*(a^2 - 3*b^2)*Sin[c + d*x]^3 + 12*a*b^5*Sin[c + d*x]^5) + (6*(A*b - a*B)*(2*b^6*Cos[c + d*x]^6 + 4*(a^2 - b^2)^2*(4*a^2 - 4*b^2 + 15*a^2*Log[a + b*Sin[c + d*x]]) + 4*a*b*(-11*a^4 + 18*a^2*b^2 - 4*b^4 + 15*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])*Sin[c + d*x] - 2*b^2*(15*a^4 - 29*a^2*b^2 + 8*b^4)*Sin[c + d*x]^2 + 2*a*b^3*(5*a^2 - 7*b^2)*Sin[c + d*x]^3 - 4*a^2*b^4*Sin[c + d*x]^4 + b^4*Cos[c + d*x]^4*(-a^2 + 4*b^2 + 3*a*b*Sin[c + d*x])))/(a + b*Sin[c + d*x]))/(60*b^8*d)","A",1
1553,1,234,206,2.1622189,"\int \frac{\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","\frac{4 \left(A-\frac{a B}{b}\right) \left(\left(8 a^2 b-4 b^3\right) \sin (c+d x)+\frac{b^4 \cos ^4(c+d x)-4 \left(a^2-b^2\right) \left(3 a^2 \log (a+b \sin (c+d x))+a^2+3 a b \sin (c+d x) \log (a+b \sin (c+d x))-b^2\right)}{a+b \sin (c+d x)}-2 a b^2 \sin ^2(c+d x)\right)+B \left(6 b \left(a^2-b^2\right) \sin ^2(c+d x)-12 a \left(a^2-2 b^2\right) \sin (c+d x)+\frac{12 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b}-4 a b^2 \sin ^3(c+d x)+3 b^3 \cos ^4(c+d x)\right)}{12 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{\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(-4 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right) \sin (c+d x)}{b^5 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,"(B*(3*b^3*Cos[c + d*x]^4 + (12*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/b - 12*a*(a^2 - 2*b^2)*Sin[c + d*x] + 6*b*(a^2 - b^2)*Sin[c + d*x]^2 - 4*a*b^2*Sin[c + d*x]^3) + 4*(A - (a*B)/b)*((8*a^2*b - 4*b^3)*Sin[c + d*x] - 2*a*b^2*Sin[c + d*x]^2 + (b^4*Cos[c + d*x]^4 - 4*(a^2 - b^2)*(a^2 - b^2 + 3*a^2*Log[a + b*Sin[c + d*x]] + 3*a*b*Log[a + b*Sin[c + d*x]]*Sin[c + d*x]))/(a + b*Sin[c + d*x])))/(12*b^5*d)","A",1
1554,1,111,113,0.536748,"\int \frac{\cos ^3(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{B \left(b^2-a^2\right) \log (a+b \sin (c+d x))}{b}+\left(A-\frac{a B}{b}\right) \left(\frac{(a-b) (a+b)}{a+b \sin (c+d x)}+2 a \log (a+b \sin (c+d x))-b \sin (c+d x)\right)+a B \sin (c+d x)-\frac{1}{2} b B \sin ^2(c+d x)}{b^3 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,"(((-a^2 + b^2)*B*Log[a + b*Sin[c + d*x]])/b + a*B*Sin[c + d*x] - (b*B*Sin[c + d*x]^2)/2 + (A - (a*B)/b)*(2*a*Log[a + b*Sin[c + d*x]] - b*Sin[c + d*x] + ((a - b)*(a + b))/(a + b*Sin[c + d*x])))/(b^3*d)","A",1
1555,1,42,48,0.101455,"\int \frac{\cos (c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{a B-A b}{a+b \sin (c+d x)}+B \log (a+b \sin (c+d x))}{b^2 d}","\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]] + (-(A*b) + a*B)/(a + b*Sin[c + d*x]))/(b^2*d)","A",1
1556,1,178,135,1.3665528,"\int \frac{\sec (c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","\frac{b \left(A-\frac{a B}{b}\right) \left(\frac{1}{\left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right)-\frac{B ((b-a) \log (1-\sin (c+d x))+(a+b) \log (\sin (c+d x)+1)-2 b \log (a+b \sin (c+d x)))}{2 b (b-a) (a+b)}}{d}","\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,"(-1/2*(B*((-a + b)*Log[1 - Sin[c + d*x]] + (a + b)*Log[1 + Sin[c + d*x]] - 2*b*Log[a + b*Sin[c + d*x]]))/(b*(-a + b)*(a + b)) + b*(A - (a*B)/b)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*b) - (2*a*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x]))))/d","A",1
1557,1,246,228,1.7166463,"\int \frac{\sec ^3(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Integrate[(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) \left(\frac{1}{\left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right)+\frac{(a A-b B) ((a-b) \log (1-\sin (c+d x))-(a+b) \log (\sin (c+d x)+1)+2 b \log (a+b \sin (c+d x)))}{(a-b) (a+b)}+\frac{\sec ^2(c+d x) ((b B-a A) \sin (c+d x)-a B+A b)}{a+b \sin (c+d x)}}{2 d \left(b^2-a^2\right)}","-\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 - b*B)*((a - b)*Log[1 - Sin[c + d*x]] - (a + b)*Log[1 + Sin[c + d*x]] + 2*b*Log[a + b*Sin[c + d*x]]))/((a - b)*(a + b)) + (Sec[c + d*x]^2*(A*b - a*B + (-(a*A) + b*B)*Sin[c + d*x]))/(a + b*Sin[c + d*x]) + b*(a^2*A + 3*A*b^2 - 4*a*b*B)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*b) - (2*a*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x]))))/(2*(-a^2 + b^2)*d)","A",1
1558,1,370,372,4.2785969,"\int \frac{\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{2 \left(b^2-a^2\right) \sec ^4(c+d x) ((b B-a A) \sin (c+d x)-a B+A b)}{a+b \sin (c+d x)}-\frac{\left(3 a^3 A+2 a^2 b B-9 a A b^2+4 b^3 B\right) ((a-b) \log (1-\sin (c+d x))-(a+b) \log (\sin (c+d x)+1)+2 b \log (a+b \sin (c+d x)))}{(a-b) (a+b)}+\frac{\sec ^2(c+d x) \left(b \left(a^2 A-6 a b B+5 A b^2\right)+\left(3 a^3 A+2 a^2 b B-9 a A b^2+4 b^3 B\right) \sin (c+d x)\right)}{a+b \sin (c+d x)}+b \left(-3 a^4 A-2 a^3 b B+12 a^2 A b^2-22 a b^3 B+15 A b^4\right) \left(\frac{1}{\left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right)}{8 d \left(a^2-b^2\right)^2}","-\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{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{\sec ^2(c+d x) \left(b \left(a^2 A-6 a b B+5 A b^2\right)+\left(3 a^3 A+2 a^2 b B-9 a A b^2+4 b^3 B\right) \sin (c+d x)\right)}{8 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{b \left(3 a^4 A+2 a^3 b B-12 a^2 A b^2+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))}",1,"(-(((3*a^3*A - 9*a*A*b^2 + 2*a^2*b*B + 4*b^3*B)*((a - b)*Log[1 - Sin[c + d*x]] - (a + b)*Log[1 + Sin[c + d*x]] + 2*b*Log[a + b*Sin[c + d*x]]))/((a - b)*(a + b))) + (2*(-a^2 + b^2)*Sec[c + d*x]^4*(A*b - a*B + (-(a*A) + b*B)*Sin[c + d*x]))/(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]))/(a + b*Sin[c + d*x]) + 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)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*b) - (2*a*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x]))))/(8*(a^2 - b^2)^2*d)","A",1
1559,1,766,550,6.2063499,"\int \frac{\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]","\frac{b^7 \left(\frac{\frac{\frac{\left(6 a \left(5 a^5 A+2 a^4 b B-18 a^3 A b^2-10 a^2 b^3 B+29 a A b^4-8 b^5 B\right)-3 \left(5 a^6 A+2 a^5 b B-13 a^4 A b^2-8 a^3 b^3 B+11 a^2 A b^4+38 a b^5 B-35 A b^6\right)\right) \left(\frac{1}{\left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right)-6 \left(5 a^5 A+2 a^4 b B-18 a^3 A b^2-10 a^2 b^3 B+29 a A b^4-8 b^5 B\right) \left(-\frac{\log (a+b \sin (c+d x))}{a^2-b^2}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)}\right)}{2 b^2 \left(b^2-a^2\right)}-\frac{\sec ^2(c+d x) \left(4 a b^2 \left(5 a^3 A+2 a^2 b B-13 a A b^2+6 b^3 B\right)-b \left(4 b^2 \left(5 a^3 A+2 a^2 b B-13 a A b^2+6 b^3 B\right)-a \left(15 a^4 A+6 a^3 b B-34 a^2 A b^2-22 a b^3 B+35 A b^4\right)\right) \sin (c+d x)-b^2 \left(15 a^4 A+6 a^3 b B-34 a^2 A b^2-22 a b^3 B+35 A b^4\right)\right)}{2 b^4 \left(b^2-a^2\right) (a+b \sin (c+d x))}}{4 b^2 \left(b^2-a^2\right)}-\frac{\sec ^4(c+d x) \left(-b \left(-a \left(-5 a^2 A-2 a b B+7 A b^2\right)-6 b^2 (a A-b B)\right) \sin (c+d x)-b^2 \left(-5 a^2 A-2 a b B+7 A b^2\right)-6 a b^2 (a A-b B)\right)}{4 b^6 \left(b^2-a^2\right) (a+b \sin (c+d x))}}{6 b^2 \left(b^2-a^2\right)}-\frac{\sec ^6(c+d x) \left(-b (b B-a A) \sin (c+d x)+a b B-A b^2\right)}{6 b^8 \left(b^2-a^2\right) (a+b \sin (c+d x))}\right)}{d}","-\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{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(5 a^3 A+a^2 b (25 A+2 B)+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(5 a^3 A-a^2 (25 A b-2 b B)+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 ^4(c+d x) \left(b \left(a^2 A-8 a b B+7 A b^2\right)+\left(5 a^3 A+2 a^2 b B-13 a A b^2+6 b^3 B\right) \sin (c+d x)\right)}{24 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left(b \left(5 a^4 A+2 a^3 b B-18 a^2 A b^2+46 a b^3 B-35 A b^4\right)+3 \left(5 a^5 A+2 a^4 b B-18 a^3 A b^2-10 a^2 b^3 B+29 a A b^4-8 b^5 B\right) \sin (c+d x)\right)}{48 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{b \left(5 a^6 A+2 a^5 b B-23 a^4 A b^2-12 a^3 b^3 B+47 a^2 A b^4-54 a b^5 B+35 A b^6\right)}{16 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}",1,"(b^7*(-1/6*(Sec[c + d*x]^6*(-(A*b^2) + a*b*B - b*(-(a*A) + b*B)*Sin[c + d*x]))/(b^8*(-a^2 + b^2)*(a + b*Sin[c + d*x])) + (-1/4*(Sec[c + d*x]^4*(-6*a*b^2*(a*A - b*B) - b^2*(-5*a^2*A + 7*A*b^2 - 2*a*b*B) - b*(-6*b^2*(a*A - b*B) - a*(-5*a^2*A + 7*A*b^2 - 2*a*b*B))*Sin[c + d*x]))/(b^6*(-a^2 + b^2)*(a + b*Sin[c + d*x])) + (-1/2*(Sec[c + d*x]^2*(4*a*b^2*(5*a^3*A - 13*a*A*b^2 + 2*a^2*b*B + 6*b^3*B) - b^2*(15*a^4*A - 34*a^2*A*b^2 + 35*A*b^4 + 6*a^3*b*B - 22*a*b^3*B) - b*(4*b^2*(5*a^3*A - 13*a*A*b^2 + 2*a^2*b*B + 6*b^3*B) - a*(15*a^4*A - 34*a^2*A*b^2 + 35*A*b^4 + 6*a^3*b*B - 22*a*b^3*B))*Sin[c + d*x]))/(b^4*(-a^2 + b^2)*(a + b*Sin[c + d*x])) + (-6*(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)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)) + Log[1 + Sin[c + d*x]]/(2*(a - b)*b) - Log[a + b*Sin[c + d*x]]/(a^2 - b^2)) + (6*a*(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) - 3*(5*a^6*A - 13*a^4*A*b^2 + 11*a^2*A*b^4 - 35*A*b^6 + 2*a^5*b*B - 8*a^3*b^3*B + 38*a*b^5*B))*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*b) - (2*a*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x]))))/(2*b^2*(-a^2 + b^2)))/(4*b^2*(-a^2 + b^2)))/(6*b^2*(-a^2 + b^2))))/d","A",1
1560,0,0,40,5.1677086,"\int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx","Integrate[(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,"Integrate[(g*Cos[e + f*x])^(-1 - m)*(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x]","A",-1
1561,1,5085,330,34.7241604,"\int \frac{(g \cos (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx","Integrate[(g*Cos[e + f*x])^p/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])),x]","\text{Result too large to show}","\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,"Result too large to show","B",0
1562,1,12568,508,55.2529201,"\int \frac{(g \cos (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx","Integrate[(g*Cos[e + f*x])^p/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2),x]","\text{Result too large to show}","-\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,"Result too large to show","B",0
1563,1,5113,308,30.0480909,"\int \frac{(g \sec (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx","Integrate[(g*Sec[e + f*x])^p/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])),x]","\text{Result too large to show}","\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,"Result too large to show","B",0