1,1,77,92,0.3861223,"\int (c+d x)^4 \sin (a+b x) \, dx","Integrate[(c + d*x)^4*Sin[a + b*x],x]","\frac{4 b d (c+d x) \sin (a+b x) \left(b^2 (c+d x)^2-6 d^2\right)-\cos (a+b x) \left(b^4 (c+d x)^4-12 b^2 d^2 (c+d x)^2+24 d^4\right)}{b^5}","-\frac{24 d^4 \cos (a+b x)}{b^5}-\frac{24 d^3 (c+d x) \sin (a+b x)}{b^4}+\frac{12 d^2 (c+d x)^2 \cos (a+b x)}{b^3}+\frac{4 d (c+d x)^3 \sin (a+b x)}{b^2}-\frac{(c+d x)^4 \cos (a+b x)}{b}",1,"(-((24*d^4 - 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Cos[a + b*x]) + 4*b*d*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Sin[a + b*x])/b^5","A",1
2,1,62,71,0.2244577,"\int (c+d x)^3 \sin (a+b x) \, dx","Integrate[(c + d*x)^3*Sin[a + b*x],x]","\frac{3 d \sin (a+b x) \left(b^2 (c+d x)^2-2 d^2\right)-b (c+d x) \cos (a+b x) \left(b^2 (c+d x)^2-6 d^2\right)}{b^4}","-\frac{6 d^3 \sin (a+b x)}{b^4}+\frac{6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac{3 d (c+d x)^2 \sin (a+b x)}{b^2}-\frac{(c+d x)^3 \cos (a+b x)}{b}",1,"(-(b*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x]) + 3*d*(-2*d^2 + b^2*(c + d*x)^2)*Sin[a + b*x])/b^4","A",1
3,1,45,50,0.1949768,"\int (c+d x)^2 \sin (a+b x) \, dx","Integrate[(c + d*x)^2*Sin[a + b*x],x]","\frac{2 b d (c+d x) \sin (a+b x)-\cos (a+b x) \left(b^2 (c+d x)^2-2 d^2\right)}{b^3}","\frac{2 d^2 \cos (a+b x)}{b^3}+\frac{2 d (c+d x) \sin (a+b x)}{b^2}-\frac{(c+d x)^2 \cos (a+b x)}{b}",1,"(-((-2*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x]) + 2*b*d*(c + d*x)*Sin[a + b*x])/b^3","A",1
4,1,27,28,0.0773474,"\int (c+d x) \sin (a+b x) \, dx","Integrate[(c + d*x)*Sin[a + b*x],x]","\frac{d \sin (a+b x)-b (c+d x) \cos (a+b x)}{b^2}","\frac{d \sin (a+b x)}{b^2}-\frac{(c+d x) \cos (a+b x)}{b}",1,"(-(b*(c + d*x)*Cos[a + b*x]) + d*Sin[a + b*x])/b^2","A",1
5,1,49,51,0.1000503,"\int \frac{\sin (a+b x)}{c+d x} \, dx","Integrate[Sin[a + b*x]/(c + d*x),x]","\frac{\sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)+\cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d}","\frac{\sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d}+\frac{\cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d}",1,"(CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d] + Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d","A",1
6,1,66,72,0.2287296,"\int \frac{\sin (a+b x)}{(c+d x)^2} \, dx","Integrate[Sin[a + b*x]/(c + d*x)^2,x]","\frac{b \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)-b \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)-\frac{d \sin (a+b x)}{c+d x}}{d^2}","\frac{b \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d^2}-\frac{b \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d^2}-\frac{\sin (a+b x)}{d (c+d x)}",1,"(b*Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] - (d*Sin[a + b*x])/(c + d*x) - b*Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)])/d^2","A",1
7,1,87,104,0.727557,"\int \frac{\sin (a+b x)}{(c+d x)^3} \, dx","Integrate[Sin[a + b*x]/(c + d*x)^3,x]","-\frac{b^2 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+b^2 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+\frac{d (b (c+d x) \cos (a+b x)+d \sin (a+b x))}{(c+d x)^2}}{2 d^3}","-\frac{b^2 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{2 d^3}-\frac{b^2 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{2 d^3}-\frac{b \cos (a+b x)}{2 d^2 (c+d x)}-\frac{\sin (a+b x)}{2 d (c+d x)^2}",1,"-1/2*(b^2*CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + (d*(b*(c + d*x)*Cos[a + b*x] + d*Sin[a + b*x]))/(c + d*x)^2 + b^2*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)])/d^3","A",1
8,1,132,161,0.6777274,"\int (c+d x)^4 \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^4*Sin[a + b*x]^2,x]","\frac{-20 b d (c+d x) \cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-3 d^2\right)-10 \sin (2 (a+b x)) \left(2 b^4 (c+d x)^4-6 b^2 d^2 (c+d x)^2+3 d^4\right)+8 b^5 x \left(5 c^4+10 c^3 d x+10 c^2 d^2 x^2+5 c d^3 x^3+d^4 x^4\right)}{80 b^5}","-\frac{3 d^4 \sin (a+b x) \cos (a+b x)}{4 b^5}-\frac{3 d^3 (c+d x) \sin ^2(a+b x)}{2 b^4}+\frac{3 d^2 (c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b^3}+\frac{d (c+d x)^3 \sin ^2(a+b x)}{b^2}-\frac{(c+d x)^4 \sin (a+b x) \cos (a+b x)}{2 b}+\frac{3 d^4 x}{4 b^4}-\frac{d (c+d x)^3}{2 b^2}+\frac{(c+d x)^5}{10 d}",1,"(8*b^5*x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4) - 20*b*d*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] - 10*(3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Sin[2*(a + b*x)])/(80*b^5)","A",1
9,1,106,134,0.4429193,"\int (c+d x)^3 \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Sin[a + b*x]^2,x]","\frac{-2 b (c+d x) \sin (2 (a+b x)) \left(2 b^2 (c+d x)^2-3 d^2\right)-3 d \cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-d^2\right)+2 b^4 x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)}{16 b^4}","-\frac{3 d^3 \sin ^2(a+b x)}{8 b^4}+\frac{3 d^2 (c+d x) \sin (a+b x) \cos (a+b x)}{4 b^3}+\frac{3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac{(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}-\frac{3 c d^2 x}{4 b^2}-\frac{3 d^3 x^2}{8 b^2}+\frac{(c+d x)^4}{8 d}",1,"(2*b^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) - 3*d*(-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] - 2*b*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Sin[2*(a + b*x)])/(16*b^4)","A",1
10,1,77,95,0.323064,"\int (c+d x)^2 \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Sin[a + b*x]^2,x]","\frac{-3 \sin (2 (a+b x)) \left(2 b^2 (c+d x)^2-d^2\right)-6 b d (c+d x) \cos (2 (a+b x))+4 b^3 x \left(3 c^2+3 c d x+d^2 x^2\right)}{24 b^3}","\frac{d^2 \sin (a+b x) \cos (a+b x)}{4 b^3}+\frac{d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac{(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}-\frac{d^2 x}{4 b^2}+\frac{(c+d x)^3}{6 d}",1,"(4*b^3*x*(3*c^2 + 3*c*d*x + d^2*x^2) - 6*b*d*(c + d*x)*Cos[2*(a + b*x)] - 3*(-d^2 + 2*b^2*(c + d*x)^2)*Sin[2*(a + b*x)])/(24*b^3)","A",1
11,1,52,55,0.1551578,"\int (c+d x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)*Sin[a + b*x]^2,x]","\frac{2 b (-(c+d x) \sin (2 (a+b x))+2 a c+b x (2 c+d x))-d \cos (2 (a+b x))}{8 b^2}","\frac{d \sin ^2(a+b x)}{4 b^2}-\frac{(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac{c x}{2}+\frac{d x^2}{4}",1,"(-(d*Cos[2*(a + b*x)]) + 2*b*(2*a*c + b*x*(2*c + d*x) - (c + d*x)*Sin[2*(a + b*x)]))/(8*b^2)","A",1
12,1,65,78,0.1096708,"\int \frac{\sin ^2(a+b x)}{c+d x} \, dx","Integrate[Sin[a + b*x]^2/(c + d*x),x]","\frac{-\cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)+\sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\log (c+d x)}{2 d}","-\frac{\cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{2 d}+\frac{\sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{2 d}+\frac{\log (c+d x)}{2 d}",1,"(-(Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d]) + Log[c + d*x] + Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/(2*d)","A",1
13,1,75,81,0.4188327,"\int \frac{\sin ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[Sin[a + b*x]^2/(c + d*x)^2,x]","\frac{b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)+b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)-\frac{d \sin ^2(a+b x)}{c+d x}}{d^2}","\frac{b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}+\frac{b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}-\frac{\sin ^2(a+b x)}{d (c+d x)}",1,"(b*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] - (d*Sin[a + b*x]^2)/(c + d*x) + b*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/d^2","A",1
14,1,101,113,1.1995258,"\int \frac{\sin ^2(a+b x)}{(c+d x)^3} \, dx","Integrate[Sin[a + b*x]^2/(c + d*x)^3,x]","-\frac{-2 b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)+2 b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\frac{d \left(b (c+d x) \sin (2 (a+b x))+d \sin ^2(a+b x)\right)}{(c+d x)^2}}{2 d^3}","\frac{b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d^3}-\frac{b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d^3}-\frac{b \sin (a+b x) \cos (a+b x)}{d^2 (c+d x)}-\frac{\sin ^2(a+b x)}{2 d (c+d x)^2}",1,"-1/2*(-2*b^2*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] + (d*(d*Sin[a + b*x]^2 + b*(c + d*x)*Sin[2*(a + b*x)]))/(c + d*x)^2 + 2*b^2*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/d^3","A",1
15,1,122,162,1.2445227,"\int \frac{\sin ^2(a+b x)}{(c+d x)^4} \, dx","Integrate[Sin[a + b*x]^2/(c + d*x)^4,x]","-\frac{4 b^3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)+4 b^3 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\frac{d \left(\cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-d^2\right)+d (b (c+d x) \sin (2 (a+b x))+d)\right)}{(c+d x)^3}}{6 d^4}","-\frac{2 b^3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{3 d^4}-\frac{2 b^3 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{3 d^4}+\frac{2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}-\frac{b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac{\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac{b^2}{3 d^3 (c+d x)}",1,"-1/6*(4*b^3*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] + (d*((-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + d*(d + b*(c + d*x)*Sin[2*(a + b*x)])))/(c + d*x)^3 + 4*b^3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/d^4","A",1
16,1,150,225,1.0588059,"\int (c+d x)^4 \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^4*Sin[a + b*x]^3,x]","\frac{-24 b d (c+d x) \sin (a+b x) \left(\cos (2 (a+b x)) \left(3 b^2 (c+d x)^2-2 d^2\right)-39 b^2 (c+d x)^2+242 d^2\right)-243 \cos (a+b x) \left(b^4 (c+d x)^4-12 b^2 d^2 (c+d x)^2+24 d^4\right)+\cos (3 (a+b x)) \left(27 b^4 (c+d x)^4-36 b^2 d^2 (c+d x)^2+8 d^4\right)}{324 b^5}","\frac{8 d^4 \cos ^3(a+b x)}{81 b^5}-\frac{488 d^4 \cos (a+b x)}{27 b^5}-\frac{8 d^3 (c+d x) \sin ^3(a+b x)}{27 b^4}-\frac{160 d^3 (c+d x) \sin (a+b x)}{9 b^4}+\frac{80 d^2 (c+d x)^2 \cos (a+b x)}{9 b^3}+\frac{4 d^2 (c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{9 b^3}+\frac{4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac{8 d (c+d x)^3 \sin (a+b x)}{3 b^2}-\frac{2 (c+d x)^4 \cos (a+b x)}{3 b}-\frac{(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b}",1,"(-243*(24*d^4 - 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Cos[a + b*x] + (8*d^4 - 36*b^2*d^2*(c + d*x)^2 + 27*b^4*(c + d*x)^4)*Cos[3*(a + b*x)] - 24*b*d*(c + d*x)*(242*d^2 - 39*b^2*(c + d*x)^2 + (-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[a + b*x])/(324*b^5)","A",1
17,1,127,175,0.9928965,"\int (c+d x)^3 \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Sin[a + b*x]^3,x]","\frac{-162 b (c+d x) \cos (a+b x) \left(b^2 (c+d x)^2-6 d^2\right)+6 b (c+d x) \cos (3 (a+b x)) \left(3 b^2 (c+d x)^2-2 d^2\right)-4 d \sin (a+b x) \left(\cos (2 (a+b x)) \left(9 b^2 (c+d x)^2-2 d^2\right)-117 b^2 (c+d x)^2+242 d^2\right)}{216 b^4}","-\frac{2 d^3 \sin ^3(a+b x)}{27 b^4}-\frac{40 d^3 \sin (a+b x)}{9 b^4}+\frac{40 d^2 (c+d x) \cos (a+b x)}{9 b^3}+\frac{2 d^2 (c+d x) \sin ^2(a+b x) \cos (a+b x)}{9 b^3}+\frac{d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac{2 d (c+d x)^2 \sin (a+b x)}{b^2}-\frac{2 (c+d x)^3 \cos (a+b x)}{3 b}-\frac{(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}",1,"(-162*b*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + 6*b*(c + d*x)*(-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] - 4*d*(242*d^2 - 117*b^2*(c + d*x)^2 + (-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[a + b*x])/(216*b^4)","A",1
18,1,86,123,0.4582491,"\int (c+d x)^2 \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Sin[a + b*x]^3,x]","\frac{-81 \cos (a+b x) \left(b^2 (c+d x)^2-2 d^2\right)+\cos (3 (a+b x)) \left(9 b^2 (c+d x)^2-2 d^2\right)-6 b d (c+d x) (\sin (3 (a+b x))-27 \sin (a+b x))}{108 b^3}","-\frac{2 d^2 \cos ^3(a+b x)}{27 b^3}+\frac{14 d^2 \cos (a+b x)}{9 b^3}+\frac{2 d (c+d x) \sin ^3(a+b x)}{9 b^2}+\frac{4 d (c+d x) \sin (a+b x)}{3 b^2}-\frac{2 (c+d x)^2 \cos (a+b x)}{3 b}-\frac{(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}",1,"(-81*(-2*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + (-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] - 6*b*d*(c + d*x)*(-27*Sin[a + b*x] + Sin[3*(a + b*x)]))/(108*b^3)","A",1
19,1,59,75,0.1865979,"\int (c+d x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)*Sin[a + b*x]^3,x]","\frac{-27 b (c+d x) \cos (a+b x)+3 b (c+d x) \cos (3 (a+b x))+d (27 \sin (a+b x)-\sin (3 (a+b x)))}{36 b^2}","\frac{d \sin ^3(a+b x)}{9 b^2}+\frac{2 d \sin (a+b x)}{3 b^2}-\frac{2 (c+d x) \cos (a+b x)}{3 b}-\frac{(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}",1,"(-27*b*(c + d*x)*Cos[a + b*x] + 3*b*(c + d*x)*Cos[3*(a + b*x)] + d*(27*Sin[a + b*x] - Sin[3*(a + b*x)]))/(36*b^2)","A",1
20,1,102,121,0.2504499,"\int \frac{\sin ^3(a+b x)}{c+d x} \, dx","Integrate[Sin[a + b*x]^3/(c + d*x),x]","-\frac{\sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)-3 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)-3 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+\cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)}{4 d}","-\frac{\sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{4 d}+\frac{3 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{4 d}+\frac{3 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{4 d}-\frac{\cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{4 d}",1,"-1/4*(CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] - 3*CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] - 3*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/d","A",1
21,1,175,145,1.0699011,"\int \frac{\sin ^3(a+b x)}{(c+d x)^2} \, dx","Integrate[Sin[a + b*x]^3/(c + d*x)^2,x]","\frac{3 b (c+d x) \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)-3 b (c+d x) \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)-3 b (c+d x) \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+3 b (c+d x) \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)-3 d \sin (a) \cos (b x)+d \sin (3 a) \cos (3 b x)-3 d \cos (a) \sin (b x)+d \cos (3 a) \sin (3 b x)}{4 d^2 (c+d x)}","\frac{3 b \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{4 d^2}-\frac{3 b \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{4 d^2}-\frac{3 b \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{4 d^2}+\frac{3 b \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{4 d^2}-\frac{\sin ^3(a+b x)}{d (c+d x)}",1,"(3*b*(c + d*x)*Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] - 3*b*(c + d*x)*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] - 3*d*Cos[b*x]*Sin[a] + d*Cos[3*b*x]*Sin[3*a] - 3*d*Cos[a]*Sin[b*x] + d*Cos[3*a]*Sin[3*b*x] - 3*b*(c + d*x)*Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + 3*b*(c + d*x)*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/(4*d^2*(c + d*x))","A",1
22,1,221,184,0.8213985,"\int \frac{\sin ^3(a+b x)}{(c+d x)^3} \, dx","Integrate[Sin[a + b*x]^3/(c + d*x)^3,x]","\frac{6 b^2 (c+d x)^2 \left(3 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)-\sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)-\cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+3 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)\right)-6 d \cos (b x) (b \cos (a) (c+d x)+d \sin (a))+2 d \cos (3 b x) (3 b \cos (3 a) (c+d x)+d \sin (3 a))+6 d \sin (b x) (b \sin (a) (c+d x)-d \cos (a))+2 d \sin (3 b x) (d \cos (3 a)-3 b \sin (3 a) (c+d x))}{16 d^3 (c+d x)^2}","\frac{9 b^2 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{8 d^3}-\frac{3 b^2 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{8 d^3}-\frac{3 b^2 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{8 d^3}+\frac{9 b^2 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{8 d^3}-\frac{3 b \sin ^2(a+b x) \cos (a+b x)}{2 d^2 (c+d x)}-\frac{\sin ^3(a+b x)}{2 d (c+d x)^2}",1,"(-6*d*Cos[b*x]*(b*(c + d*x)*Cos[a] + d*Sin[a]) + 2*d*Cos[3*b*x]*(3*b*(c + d*x)*Cos[3*a] + d*Sin[3*a]) + 6*d*(-(d*Cos[a]) + b*(c + d*x)*Sin[a])*Sin[b*x] + 2*d*(d*Cos[3*a] - 3*b*(c + d*x)*Sin[3*a])*Sin[3*b*x] + 6*b^2*(c + d*x)^2*(3*CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] - CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] - Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + 3*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d]))/(16*d^3*(c + d*x)^2)","A",1
23,1,221,185,0.5268521,"\int (c+d x)^3 \csc (a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x],x]","\frac{-2 b^3 (c+d x)^3 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x))+3 i d \left(b^2 (c+d x)^2 \text{Li}_2(-\cos (a+b x)-i \sin (a+b x))+2 i b d (c+d x) \text{Li}_3(-\cos (a+b x)-i \sin (a+b x))-2 d^2 \text{Li}_4(-\cos (a+b x)-i \sin (a+b x))\right)-3 i d \left(b^2 (c+d x)^2 \text{Li}_2(\cos (a+b x)+i \sin (a+b x))+2 i b d (c+d x) \text{Li}_3(\cos (a+b x)+i \sin (a+b x))-2 d^2 \text{Li}_4(\cos (a+b x)+i \sin (a+b x))\right)}{b^4}","-\frac{6 i d^3 \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^4}+\frac{6 i d^3 \text{Li}_4\left(e^{i (a+b x)}\right)}{b^4}-\frac{6 d^2 (c+d x) \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{6 d^2 (c+d x) \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{2 (c+d x)^3 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(-2*b^3*(c + d*x)^3*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]] + (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, -Cos[a + b*x] - I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]] - 2*d^2*PolyLog[4, -Cos[a + b*x] - I*Sin[a + b*x]]) - (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]] - 2*d^2*PolyLog[4, Cos[a + b*x] + I*Sin[a + b*x]]))/b^4","A",0
24,1,148,123,0.3230165,"\int (c+d x)^2 \csc (a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x],x]","\frac{\frac{2 i d \left(b (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)+i d \text{Li}_3\left(-e^{i (a+b x)}\right)\right)}{b^2}+\frac{2 d \left(d \text{Li}_3\left(e^{i (a+b x)}\right)-i b (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)\right)}{b^2}+(c+d x)^2 \log \left(1-e^{i (a+b x)}\right)-(c+d x)^2 \log \left(1+e^{i (a+b x)}\right)}{b}","-\frac{2 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{2 d^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}+\frac{2 i d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{2 i d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{2 (c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"((c + d*x)^2*Log[1 - E^(I*(a + b*x))] - (c + d*x)^2*Log[1 + E^(I*(a + b*x))] + ((2*I)*d*(b*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] + I*d*PolyLog[3, -E^(I*(a + b*x))]))/b^2 + (2*d*((-I)*b*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] + d*PolyLog[3, E^(I*(a + b*x))]))/b^2)/b","A",1
25,1,134,67,0.0743947,"\int (c+d x) \csc (a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x],x]","\frac{d \left(i \left(\text{Li}_2\left(-e^{i (a+b x)}\right)-\text{Li}_2\left(e^{i (a+b x)}\right)\right)+(a+b x) \left(\log \left(1-e^{i (a+b x)}\right)-\log \left(1+e^{i (a+b x)}\right)\right)-a \log \left(\tan \left(\frac{1}{2} (a+b x)\right)\right)\right)}{b^2}+\frac{c \log \left(\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}{b}-\frac{c \log \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}{b}","\frac{i d \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{i d \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{2 (c+d x) \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"-((c*Log[Cos[a/2 + (b*x)/2]])/b) + (c*Log[Sin[a/2 + (b*x)/2]])/b + (d*((a + b*x)*(Log[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(a + b*x))]) - a*Log[Tan[(a + b*x)/2]] + I*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^(I*(a + b*x))])))/b^2","A",1
26,0,0,17,6.5220892,"\int \frac{\csc (a+b x)}{c+d x} \, dx","Integrate[Csc[a + b*x]/(c + d*x),x]","\int \frac{\csc (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\csc (a+b x)}{c+d x},x\right)",0,"Integrate[Csc[a + b*x]/(c + d*x), x]","A",-1
27,0,0,17,7.3527446,"\int \frac{\csc (a+b x)}{(c+d x)^2} \, dx","Integrate[Csc[a + b*x]/(c + d*x)^2,x]","\int \frac{\csc (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\csc (a+b x)}{(c+d x)^2},x\right)",0,"Integrate[Csc[a + b*x]/(c + d*x)^2, x]","A",-1
28,1,478,113,6.9585162,"\int (c+d x)^3 \csc ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]^2,x]","\frac{3 c^2 d \csc (a) (\sin (a) \log (\sin (a) \cos (b x)+\cos (a) \sin (b x))-b x \cos (a))}{b^2 \left(\sin ^2(a)+\cos ^2(a)\right)}-\frac{3 c d^2 \csc (a) \sec (a) \left(b^2 x^2 e^{i \tan ^{-1}(\tan (a))}+\frac{\tan (a) \left(i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\tan ^2(a)+1}}\right)}{b^3 \sqrt{\sec ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}-\frac{e^{i a} d^3 \csc (a) \left(2 e^{-2 i a} b^3 x^3+3 i \left(1-e^{-2 i a}\right) b^2 x^2 \log \left(1-e^{-i (a+b x)}\right)+3 i \left(1-e^{-2 i a}\right) b^2 x^2 \log \left(1+e^{-i (a+b x)}\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right)}{2 b^4}+\frac{\csc (a) \csc (a+b x) \left(c^3 \sin (b x)+3 c^2 d x \sin (b x)+3 c d^2 x^2 \sin (b x)+d^3 x^3 \sin (b x)\right)}{b}","\frac{3 d^3 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 i d^2 (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^3}+\frac{3 d (c+d x)^2 \log \left(1-e^{2 i (a+b x)}\right)}{b^2}-\frac{(c+d x)^3 \cot (a+b x)}{b}-\frac{i (c+d x)^3}{b}",1,"-1/2*(d^3*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^4 + (3*c^2*d*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^2*(Cos[a]^2 + Sin[a]^2)) + (Csc[a]*Csc[a + b*x]*(c^3*Sin[b*x] + 3*c^2*d*x*Sin[b*x] + 3*c*d^2*x^2*Sin[b*x] + d^3*x^3*Sin[b*x]))/b - (3*c*d^2*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^3*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
29,1,181,83,4.1934925,"\int (c+d x)^2 \csc ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]^2,x]","\frac{\csc (a) \left(b^2 \sin (b x) (c+d x)^2 \csc (a+b x)+d^2 \left(-b^2 x^2 \cos (a) e^{i \tan ^{-1}(\tan (a))} \sqrt{\sec ^2(a)}-\sin (a) \left(i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)-i b x \left(\pi -2 \tan ^{-1}(\tan (a))\right)-2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)\right)-2 b c d (b x \cos (a)-\sin (a) \log (\sin (a+b x)))\right)}{b^3}","-\frac{i d^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^3}+\frac{2 d (c+d x) \log \left(1-e^{2 i (a+b x)}\right)}{b^2}-\frac{(c+d x)^2 \cot (a+b x)}{b}-\frac{i (c+d x)^2}{b}",1,"(Csc[a]*(-2*b*c*d*(b*x*Cos[a] - Log[Sin[a + b*x]]*Sin[a]) + d^2*(-(b^2*E^(I*ArcTan[Tan[a]])*x^2*Cos[a]*Sqrt[Sec[a]^2]) - ((-I)*b*x*(Pi - 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Sin[a]) + b^2*(c + d*x)^2*Csc[a + b*x]*Sin[b*x]))/b^3","B",0
30,1,52,29,0.0875248,"\int (c+d x) \csc ^2(a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]^2,x]","\frac{d \log (\sin (a+b x))}{b^2}-\frac{c \cot (a+b x)}{b}-\frac{d x \cot (a)}{b}+\frac{d x \csc (a) \sin (b x) \csc (a+b x)}{b}","\frac{d \log (\sin (a+b x))}{b^2}-\frac{(c+d x) \cot (a+b x)}{b}",1,"-((d*x*Cot[a])/b) - (c*Cot[a + b*x])/b + (d*Log[Sin[a + b*x]])/b^2 + (d*x*Csc[a]*Csc[a + b*x]*Sin[b*x])/b","A",1
31,0,0,19,6.6575097,"\int \frac{\csc ^2(a+b x)}{c+d x} \, dx","Integrate[Csc[a + b*x]^2/(c + d*x),x]","\int \frac{\csc ^2(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\csc ^2(a+b x)}{c+d x},x\right)",0,"Integrate[Csc[a + b*x]^2/(c + d*x), x]","A",-1
32,0,0,19,6.7234959,"\int \frac{\csc ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[Csc[a + b*x]^2/(c + d*x)^2,x]","\int \frac{\csc ^2(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\csc ^2(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[Csc[a + b*x]^2/(c + d*x)^2, x]","A",-1
33,1,528,309,5.5401411,"\int (c+d x)^3 \csc ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]^3,x]","-\frac{b^3 \left(-c^3\right) \log \left(1-e^{i (a+b x)}\right)+b^3 c^3 \log \left(1+e^{i (a+b x)}\right)-3 b^3 c^2 d x \log \left(1-e^{i (a+b x)}\right)+3 b^3 c^2 d x \log \left(1+e^{i (a+b x)}\right)-3 b^3 c d^2 x^2 \log \left(1-e^{i (a+b x)}\right)+3 b^3 c d^2 x^2 \log \left(1+e^{i (a+b x)}\right)-b^3 d^3 x^3 \log \left(1-e^{i (a+b x)}\right)+b^3 d^3 x^3 \log \left(1+e^{i (a+b x)}\right)-3 i d \text{Li}_2\left(-e^{i (a+b x)}\right) \left(b^2 (c+d x)^2+2 d^2\right)+3 i d \text{Li}_2\left(e^{i (a+b x)}\right) \left(b^2 (c+d x)^2+2 d^2\right)+b^2 (c+d x)^2 \csc (a+b x) (b (c+d x) \cot (a+b x)+3 d)+6 b c d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)-6 b c d^2 \text{Li}_3\left(e^{i (a+b x)}\right)-6 b c d^2 \log \left(1-e^{i (a+b x)}\right)+6 b c d^2 \log \left(1+e^{i (a+b x)}\right)+6 b d^3 x \text{Li}_3\left(-e^{i (a+b x)}\right)-6 b d^3 x \text{Li}_3\left(e^{i (a+b x)}\right)+6 i d^3 \text{Li}_4\left(-e^{i (a+b x)}\right)-6 i d^3 \text{Li}_4\left(e^{i (a+b x)}\right)-6 b d^3 x \log \left(1-e^{i (a+b x)}\right)+6 b d^3 x \log \left(1+e^{i (a+b x)}\right)}{2 b^4}","\frac{3 i d^3 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^4}-\frac{3 i d^3 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^4}-\frac{3 i d^3 \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^4}+\frac{3 i d^3 \text{Li}_4\left(e^{i (a+b x)}\right)}{b^4}-\frac{3 d^2 (c+d x) \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{3 d^2 (c+d x) \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{6 d^2 (c+d x) \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{2 b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{2 b^2}-\frac{3 d (c+d x)^2 \csc (a+b x)}{2 b^2}-\frac{(c+d x)^3 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b}",1,"-1/2*(b^2*(c + d*x)^2*(3*d + b*(c + d*x)*Cot[a + b*x])*Csc[a + b*x] - b^3*c^3*Log[1 - E^(I*(a + b*x))] - 6*b*c*d^2*Log[1 - E^(I*(a + b*x))] - 3*b^3*c^2*d*x*Log[1 - E^(I*(a + b*x))] - 6*b*d^3*x*Log[1 - E^(I*(a + b*x))] - 3*b^3*c*d^2*x^2*Log[1 - E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 - E^(I*(a + b*x))] + b^3*c^3*Log[1 + E^(I*(a + b*x))] + 6*b*c*d^2*Log[1 + E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 + E^(I*(a + b*x))] + 6*b*d^3*x*Log[1 + E^(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 + E^(I*(a + b*x))] + b^3*d^3*x^3*Log[1 + E^(I*(a + b*x))] - (3*I)*d*(2*d^2 + b^2*(c + d*x)^2)*PolyLog[2, -E^(I*(a + b*x))] + (3*I)*d*(2*d^2 + b^2*(c + d*x)^2)*PolyLog[2, E^(I*(a + b*x))] + 6*b*c*d^2*PolyLog[3, -E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, -E^(I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, -E^(I*(a + b*x))] - (6*I)*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4","A",1
34,1,471,180,7.6443951,"\int (c+d x)^2 \csc ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]^3,x]","\frac{\csc \left(\frac{a}{2}\right) \csc \left(\frac{a}{2}+\frac{b x}{2}\right) \left(c d \sin \left(\frac{b x}{2}\right)+d^2 x \sin \left(\frac{b x}{2}\right)\right)}{2 b^2}+\frac{\sec \left(\frac{a}{2}\right) \sec \left(\frac{a}{2}+\frac{b x}{2}\right) \left(d^2 (-x) \sin \left(\frac{b x}{2}\right)-c d \sin \left(\frac{b x}{2}\right)\right)}{2 b^2}-\frac{d \csc (a) (c+d x)}{b^2}+\frac{b^2 c^2 \log \left(1-e^{i (a+b x)}\right)-b^2 c^2 \log \left(1+e^{i (a+b x)}\right)+2 b^2 c d x \log \left(1-e^{i (a+b x)}\right)-2 b^2 c d x \log \left(1+e^{i (a+b x)}\right)+b^2 d^2 x^2 \log \left(1-e^{i (a+b x)}\right)-b^2 d^2 x^2 \log \left(1+e^{i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)-2 i b d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)-2 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)+2 d^2 \text{Li}_3\left(e^{i (a+b x)}\right)+2 d^2 \log \left(1-e^{i (a+b x)}\right)-2 d^2 \log \left(1+e^{i (a+b x)}\right)}{2 b^3}+\frac{\left(-c^2-2 c d x-d^2 x^2\right) \csc ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}+\frac{\left(c^2+2 c d x+d^2 x^2\right) \sec ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}","-\frac{d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{d^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{d^2 \tanh ^{-1}(\cos (a+b x))}{b^3}+\frac{i d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{i d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{d (c+d x) \csc (a+b x)}{b^2}-\frac{(c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{(c+d x)^2 \cot (a+b x) \csc (a+b x)}{2 b}",1,"-((d*(c + d*x)*Csc[a])/b^2) + ((-c^2 - 2*c*d*x - d^2*x^2)*Csc[a/2 + (b*x)/2]^2)/(8*b) + (b^2*c^2*Log[1 - E^(I*(a + b*x))] + 2*d^2*Log[1 - E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 - E^(I*(a + b*x))] + b^2*d^2*x^2*Log[1 - E^(I*(a + b*x))] - b^2*c^2*Log[1 + E^(I*(a + b*x))] - 2*d^2*Log[1 + E^(I*(a + b*x))] - 2*b^2*c*d*x*Log[1 + E^(I*(a + b*x))] - b^2*d^2*x^2*Log[1 + E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] - 2*d^2*PolyLog[3, -E^(I*(a + b*x))] + 2*d^2*PolyLog[3, E^(I*(a + b*x))])/(2*b^3) + ((c^2 + 2*c*d*x + d^2*x^2)*Sec[a/2 + (b*x)/2]^2)/(8*b) + (Sec[a/2]*Sec[a/2 + (b*x)/2]*(-(c*d*Sin[(b*x)/2]) - d^2*x*Sin[(b*x)/2]))/(2*b^2) + (Csc[a/2]*Csc[a/2 + (b*x)/2]*(c*d*Sin[(b*x)/2] + d^2*x*Sin[(b*x)/2]))/(2*b^2)","B",1
35,1,292,109,2.0328586,"\int (c+d x) \csc ^3(a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]^3,x]","\frac{d \left(i \left(\text{Li}_2\left(-e^{i (a+b x)}\right)-\text{Li}_2\left(e^{i (a+b x)}\right)\right)+(a+b x) \left(\log \left(1-e^{i (a+b x)}\right)-\log \left(1+e^{i (a+b x)}\right)\right)-a \log \left(\tan \left(\frac{1}{2} (a+b x)\right)\right)\right)}{2 b^2}+\frac{d \csc \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) \csc \left(\frac{a}{2}+\frac{b x}{2}\right)}{4 b^2}-\frac{d \sec \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) \sec \left(\frac{a}{2}+\frac{b x}{2}\right)}{4 b^2}-\frac{c \csc ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{c \sec ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{c \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}-\frac{c \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}-\frac{d x \csc ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}+\frac{d x \sec ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}","\frac{i d \text{Li}_2\left(-e^{i (a+b x)}\right)}{2 b^2}-\frac{i d \text{Li}_2\left(e^{i (a+b x)}\right)}{2 b^2}-\frac{d \csc (a+b x)}{2 b^2}-\frac{(c+d x) \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}",1,"-1/8*(d*x*Csc[a/2 + (b*x)/2]^2)/b - (c*Csc[(a + b*x)/2]^2)/(8*b) - (c*Log[Cos[(a + b*x)/2]])/(2*b) + (c*Log[Sin[(a + b*x)/2]])/(2*b) + (d*((a + b*x)*(Log[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(a + b*x))]) - a*Log[Tan[(a + b*x)/2]] + I*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^(I*(a + b*x))])))/(2*b^2) + (d*x*Sec[a/2 + (b*x)/2]^2)/(8*b) + (c*Sec[(a + b*x)/2]^2)/(8*b) + (d*Csc[a/2]*Csc[a/2 + (b*x)/2]*Sin[(b*x)/2])/(4*b^2) - (d*Sec[a/2]*Sec[a/2 + (b*x)/2]*Sin[(b*x)/2])/(4*b^2)","B",1
36,0,0,19,32.3442441,"\int \frac{\csc ^3(a+b x)}{c+d x} \, dx","Integrate[Csc[a + b*x]^3/(c + d*x),x]","\int \frac{\csc ^3(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\csc ^3(a+b x)}{c+d x},x\right)",0,"Integrate[Csc[a + b*x]^3/(c + d*x), x]","A",-1
37,0,0,19,35.6435212,"\int \frac{\csc ^3(a+b x)}{(c+d x)^2} \, dx","Integrate[Csc[a + b*x]^3/(c + d*x)^2,x]","\int \frac{\csc ^3(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\csc ^3(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[Csc[a + b*x]^3/(c + d*x)^2, x]","A",-1
38,1,124,195,0.1226019,"\int (c+d x)^{5/2} \sin (a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Sin[a + b*x],x]","\frac{d^2 \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(\frac{e^{2 i a} \Gamma \left(\frac{7}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}+\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{7}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{2 b^3}","-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (a+b x)}{4 b^3}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{2 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x)}{b}",1,"(d^2*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[7/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] + (E^(((2*I)*b*c)/d)*Gamma[7/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(2*b^3*E^((I*(b*c + a*d))/d))","C",1
39,1,125,170,0.1060216,"\int (c+d x)^{3/2} \sin (a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Sin[a + b*x],x]","-\frac{i d \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(\frac{e^{2 i a} \Gamma \left(\frac{5}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{5}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{2 b^2}","-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{2 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{2 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (a+b x)}{2 b^2}-\frac{(c+d x)^{3/2} \cos (a+b x)}{b}",1,"((-1/2*I)*d*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[5/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[5/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b^2*E^((I*(b*c + a*d))/d))","C",1
40,1,123,142,0.1044378,"\int \sqrt{c+d x} \sin (a+b x) \, dx","Integrate[Sqrt[c + d*x]*Sin[a + b*x],x]","\frac{\sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(-\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{2 b}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{b^{3/2}}-\frac{\sqrt{c+d x} \cos (a+b x)}{b}",1,"(Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(2*b*E^((I*(b*c + a*d))/d))","C",1
41,1,121,117,0.0641654,"\int \frac{\sin (a+b x)}{\sqrt{c+d x}} \, dx","Integrate[Sin[a + b*x]/Sqrt[c + d*x],x]","-\frac{e^{-\frac{i (a d+b c)}{d}} \left(e^{2 i a} \sqrt{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{1}{2},-\frac{i b (c+d x)}{d}\right)+e^{\frac{2 i b c}{d}} \sqrt{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{1}{2},\frac{i b (c+d x)}{d}\right)\right)}{2 b \sqrt{c+d x}}","\frac{\sqrt{2 \pi } \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{\sqrt{b} \sqrt{d}}+\frac{\sqrt{2 \pi } \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{\sqrt{b} \sqrt{d}}",1,"-1/2*(E^((2*I)*a)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[1/2, ((-I)*b*(c + d*x))/d] + E^(((2*I)*b*c)/d)*Sqrt[(I*b*(c + d*x))/d]*Gamma[1/2, (I*b*(c + d*x))/d])/(b*E^((I*(b*c + a*d))/d)*Sqrt[c + d*x])","C",1
42,1,148,139,0.336528,"\int \frac{\sin (a+b x)}{(c+d x)^{3/2}} \, dx","Integrate[Sin[a + b*x]/(c + d*x)^(3/2),x]","\frac{i e^{-\frac{i (a d+b c)}{d}} \left(2 i e^{\frac{i (a d+b c)}{d}} \sin (a+b x)-e^{2 i a} \sqrt{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{1}{2},-\frac{i b (c+d x)}{d}\right)+e^{\frac{2 i b c}{d}} \sqrt{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{1}{2},\frac{i b (c+d x)}{d}\right)\right)}{d \sqrt{c+d x}}","\frac{2 \sqrt{2 \pi } \sqrt{b} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{3/2}}-\frac{2 \sqrt{2 \pi } \sqrt{b} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{3/2}}-\frac{2 \sin (a+b x)}{d \sqrt{c+d x}}",1,"(I*(-(E^((2*I)*a)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[1/2, ((-I)*b*(c + d*x))/d]) + E^(((2*I)*b*c)/d)*Sqrt[(I*b*(c + d*x))/d]*Gamma[1/2, (I*b*(c + d*x))/d] + (2*I)*E^((I*(b*c + a*d))/d)*Sin[a + b*x]))/(d*E^((I*(b*c + a*d))/d)*Sqrt[c + d*x])","C",1
43,1,162,168,0.6384264,"\int \frac{\sin (a+b x)}{(c+d x)^{5/2}} \, dx","Integrate[Sin[a + b*x]/(c + d*x)^(5/2),x]","\frac{2 \left(-d \sin (a+b x)-b (c+d x) \left(e^{-i (a+b x)} \left(e^{2 i (a+b x)}-e^{\frac{i b (c+d x)}{d}} \sqrt{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{1}{2},\frac{i b (c+d x)}{d}\right)+1\right)-e^{i \left(a-\frac{b c}{d}\right)} \sqrt{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{1}{2},-\frac{i b (c+d x)}{d}\right)\right)\right)}{3 d^2 (c+d x)^{3/2}}","-\frac{4 \sqrt{2 \pi } b^{3/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{3 d^{5/2}}-\frac{4 \sqrt{2 \pi } b^{3/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{3 d^{5/2}}-\frac{4 b \cos (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \sin (a+b x)}{3 d (c+d x)^{3/2}}",1,"(2*(-(b*(c + d*x)*(-(E^(I*(a - (b*c)/d))*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[1/2, ((-I)*b*(c + d*x))/d]) + (1 + E^((2*I)*(a + b*x)) - E^((I*b*(c + d*x))/d)*Sqrt[(I*b*(c + d*x))/d]*Gamma[1/2, (I*b*(c + d*x))/d])/E^(I*(a + b*x)))) - d*Sin[a + b*x]))/(3*d^2*(c + d*x)^(3/2))","C",1
44,1,208,193,0.4990424,"\int \frac{\sin (a+b x)}{(c+d x)^{7/2}} \, dx","Integrate[Sin[a + b*x]/(c + d*x)^(7/2),x]","-\frac{i \left(b (c+d x) \left(2 e^{i \left(a-\frac{b c}{d}\right)} \left(e^{\frac{i b (c+d x)}{d}} (2 b (c+d x)-i d)-2 i d \left(-\frac{i b (c+d x)}{d}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{i b (c+d x)}{d}\right)\right)-i e^{-i (a+b x)} \left(-4 i b (c+d x)+4 d e^{\frac{i b (c+d x)}{d}} \left(\frac{i b (c+d x)}{d}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{i b (c+d x)}{d}\right)+2 d\right)\right)-6 i d^2 \sin (a+b x)\right)}{15 d^3 (c+d x)^{5/2}}","-\frac{8 \sqrt{2 \pi } b^{5/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{15 d^{7/2}}+\frac{8 \sqrt{2 \pi } b^{5/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{15 d^{7/2}}+\frac{8 b^2 \sin (a+b x)}{15 d^3 \sqrt{c+d x}}-\frac{4 b \cos (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac{2 \sin (a+b x)}{5 d (c+d x)^{5/2}}",1,"((-1/15*I)*(b*(c + d*x)*(2*E^(I*(a - (b*c)/d))*(E^((I*b*(c + d*x))/d)*((-I)*d + 2*b*(c + d*x)) - (2*I)*d*(((-I)*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((-I)*b*(c + d*x))/d]) - (I*(2*d - (4*I)*b*(c + d*x) + 4*d*E^((I*b*(c + d*x))/d)*((I*b*(c + d*x))/d)^(3/2)*Gamma[1/2, (I*b*(c + d*x))/d]))/E^(I*(a + b*x))) - (6*I)*d^2*Sin[a + b*x]))/(d^3*(c + d*x)^(5/2))","C",1
45,1,194,231,2.319082,"\int (c+d x)^{5/2} \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Sin[a + b*x]^2,x]","\frac{\sqrt{\frac{b}{d}} \left(2 \sqrt{\frac{b}{d}} \sqrt{c+d x} \left(-7 d \sin (2 (a+b x)) \left(16 b^2 (c+d x)^2-15 d^2\right)-140 b d^2 (c+d x) \cos (2 (a+b x))+64 b^3 (c+d x)^3\right)-105 \sqrt{\pi } d^3 \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-105 \sqrt{\pi } d^3 \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)\right)}{896 b^4}","-\frac{15 \sqrt{\pi } d^{5/2} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{128 b^{7/2}}-\frac{15 \sqrt{\pi } d^{5/2} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{128 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \sin (2 a+2 b x)}{64 b^3}+\frac{5 d (c+d x)^{3/2} \sin ^2(a+b x)}{8 b^2}-\frac{(c+d x)^{5/2} \sin (a+b x) \cos (a+b x)}{2 b}-\frac{5 d (c+d x)^{3/2}}{16 b^2}+\frac{(c+d x)^{7/2}}{7 d}",1,"(Sqrt[b/d]*(-105*d^3*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 105*d^3*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 2*Sqrt[b/d]*Sqrt[c + d*x]*(64*b^3*(c + d*x)^3 - 140*b*d^2*(c + d*x)*Cos[2*(a + b*x)] - 7*d*(-15*d^2 + 16*b^2*(c + d*x)^2)*Sin[2*(a + b*x)])))/(896*b^4)","A",1
46,1,175,203,1.7855901,"\int (c+d x)^{3/2} \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Sin[a + b*x]^2,x]","\frac{\sqrt{\frac{b}{d}} \left(15 \sqrt{\pi } d^2 \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-15 \sqrt{\pi } d^2 \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+2 \sqrt{\frac{b}{d}} \sqrt{c+d x} \left(4 b (c+d x) (4 b (c+d x)-5 d \sin (2 (a+b x)))-15 d^2 \cos (2 (a+b x))\right)\right)}{160 b^3}","\frac{3 \sqrt{\pi } d^{3/2} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{32 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{32 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin ^2(a+b x)}{8 b^2}-\frac{(c+d x)^{3/2} \sin (a+b x) \cos (a+b x)}{2 b}-\frac{3 d \sqrt{c+d x}}{16 b^2}+\frac{(c+d x)^{5/2}}{5 d}",1,"(Sqrt[b/d]*(15*d^2*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 15*d^2*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 2*Sqrt[b/d]*Sqrt[c + d*x]*(-15*d^2*Cos[2*(a + b*x)] + 4*b*(c + d*x)*(4*b*(c + d*x) - 5*d*Sin[2*(a + b*x)]))))/(160*b^3)","A",1
47,1,149,158,0.5564055,"\int \sqrt{c+d x} \sin ^2(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Sin[a + b*x]^2,x]","\frac{3 \sqrt{\pi } d \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+3 \sqrt{\pi } d \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+2 \sqrt{\frac{b}{d}} \sqrt{c+d x} (4 b (c+d x)-3 d \sin (2 (a+b x)))}{24 d^2 \left(\frac{b}{d}\right)^{3/2}}","\frac{\sqrt{\pi } \sqrt{d} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{8 b^{3/2}}+\frac{\sqrt{\pi } \sqrt{d} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{8 b^{3/2}}-\frac{\sqrt{c+d x} \sin (2 a+2 b x)}{4 b}+\frac{(c+d x)^{3/2}}{3 d}",1,"(3*d*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + 3*d*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 2*Sqrt[b/d]*Sqrt[c + d*x]*(4*b*(c + d*x) - 3*d*Sin[2*(a + b*x)]))/(24*(b/d)^(3/2)*d^2)","A",1
48,1,126,130,0.2410774,"\int \frac{\sin ^2(a+b x)}{\sqrt{c+d x}} \, dx","Integrate[Sin[a + b*x]^2/Sqrt[c + d*x],x]","\frac{\sqrt{\frac{b}{d}} \left(-\sqrt{\pi } \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+\sqrt{\pi } \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+2 \sqrt{\frac{b}{d}} \sqrt{c+d x}\right)}{2 b}","-\frac{\sqrt{\pi } \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{2 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\pi } \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{2 \sqrt{b} \sqrt{d}}+\frac{\sqrt{c+d x}}{d}",1,"(Sqrt[b/d]*(2*Sqrt[b/d]*Sqrt[c + d*x] - Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d]))/(2*b)","A",1
49,1,149,135,0.4266043,"\int \frac{\sin ^2(a+b x)}{(c+d x)^{3/2}} \, dx","Integrate[Sin[a + b*x]^2/(c + d*x)^(3/2),x]","\frac{2 \sqrt{\pi } \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+2 \sqrt{\pi } \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+\cos (2 (a+b x))-1}{d \sqrt{c+d x}}","\frac{2 \sqrt{\pi } \sqrt{b} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{d^{3/2}}+\frac{2 \sqrt{\pi } \sqrt{b} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{d^{3/2}}-\frac{2 \sin ^2(a+b x)}{d \sqrt{c+d x}}",1,"(-1 + Cos[2*(a + b*x)] + 2*Sqrt[b/d]*Sqrt[Pi]*Sqrt[c + d*x]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + 2*Sqrt[b/d]*Sqrt[Pi]*Sqrt[c + d*x]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d])/(d*Sqrt[c + d*x])","A",1
50,1,158,170,1.486415,"\int \frac{\sin ^2(a+b x)}{(c+d x)^{5/2}} \, dx","Integrate[Sin[a + b*x]^2/(c + d*x)^(5/2),x]","\frac{2 \left(4 \sqrt{\pi } b \sqrt{\frac{b}{d}} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-4 \sqrt{\pi } b \sqrt{\frac{b}{d}} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-\frac{\sin (a+b x) (4 b (c+d x) \cos (a+b x)+d \sin (a+b x))}{(c+d x)^{3/2}}\right)}{3 d^2}","\frac{8 \sqrt{\pi } b^{3/2} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{3 d^{5/2}}-\frac{8 \sqrt{\pi } b^{3/2} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{3 d^{5/2}}-\frac{8 b \sin (a+b x) \cos (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \sin ^2(a+b x)}{3 d (c+d x)^{3/2}}",1,"(2*(4*b*Sqrt[b/d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 4*b*Sqrt[b/d]*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] - (Sin[a + b*x]*(4*b*(c + d*x)*Cos[a + b*x] + d*Sin[a + b*x]))/(c + d*x)^(3/2)))/(3*d^2)","A",1
51,1,244,216,2.1310342,"\int \frac{\sin ^2(a+b x)}{(c+d x)^{7/2}} \, dx","Integrate[Sin[a + b*x]^2/(c + d*x)^(7/2),x]","-\frac{16 b^2 c^2 \cos (2 (a+b x))+32 b^2 c d x \cos (2 (a+b x))+16 b^2 d^2 x^2 \cos (2 (a+b x))+32 \sqrt{\pi } b d \left(\frac{b}{d}\right)^{3/2} (c+d x)^{5/2} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+32 \sqrt{\pi } b d \left(\frac{b}{d}\right)^{3/2} (c+d x)^{5/2} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+4 b c d \sin (2 (a+b x))+4 b d^2 x \sin (2 (a+b x))-3 d^2 \cos (2 (a+b x))+3 d^2}{15 d^3 (c+d x)^{5/2}}","-\frac{32 \sqrt{\pi } b^{5/2} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{15 d^{7/2}}-\frac{32 \sqrt{\pi } b^{5/2} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{15 d^{7/2}}+\frac{32 b^2 \sin ^2(a+b x)}{15 d^3 \sqrt{c+d x}}-\frac{8 b \sin (a+b x) \cos (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac{2 \sin ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac{16 b^2}{15 d^3 \sqrt{c+d x}}",1,"-1/15*(3*d^2 + 16*b^2*c^2*Cos[2*(a + b*x)] - 3*d^2*Cos[2*(a + b*x)] + 32*b^2*c*d*x*Cos[2*(a + b*x)] + 16*b^2*d^2*x^2*Cos[2*(a + b*x)] + 32*b*(b/d)^(3/2)*d*Sqrt[Pi]*(c + d*x)^(5/2)*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + 32*b*(b/d)^(3/2)*d*Sqrt[Pi]*(c + d*x)^(5/2)*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 4*b*c*d*Sin[2*(a + b*x)] + 4*b*d^2*x*Sin[2*(a + b*x)])/(d^3*(c + d*x)^(5/2))","A",1
52,1,661,247,4.8354787,"\int \frac{\sin ^2(a+b x)}{(c+d x)^{9/2}} \, dx","Integrate[Sin[a + b*x]^2/(c + d*x)^(9/2),x]","\frac{\cos (2 a) \left(2 \cos \left(\frac{2 b c}{d}\right) \left(15 d^3 \cos \left(\frac{2 b (c+d x)}{d}\right)-4 b (c+d x) \left(3 d^2 \sin \left(\frac{2 b (c+d x)}{d}\right)+4 b (c+d x) \left(8 \sqrt{\pi } b \sqrt{\frac{b}{d}} (c+d x)^{3/2} C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-4 b (c+d x) \sin \left(\frac{2 b (c+d x)}{d}\right)+d \cos \left(\frac{2 b (c+d x)}{d}\right)\right)\right)\right)+4 \sin \left(\frac{b c}{d}\right) \cos \left(\frac{b c}{d}\right) \left(15 d^3 \sin \left(\frac{2 b (c+d x)}{d}\right)+4 b (c+d x) \left(3 d^2 \cos \left(\frac{2 b (c+d x)}{d}\right)-4 b (c+d x) \left(8 \sqrt{\pi } b \sqrt{\frac{b}{d}} (c+d x)^{3/2} S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+d \sin \left(\frac{2 b (c+d x)}{d}\right)+4 b (c+d x) \cos \left(\frac{2 b (c+d x)}{d}\right)\right)\right)\right)\right)-2 \sin (a) \cos (a) \left(2 \left(\cos \left(\frac{b c}{d}\right)-\sin \left(\frac{b c}{d}\right)\right) \left(\sin \left(\frac{b c}{d}\right)+\cos \left(\frac{b c}{d}\right)\right) \left(15 d^3 \sin \left(\frac{2 b (c+d x)}{d}\right)+4 b (c+d x) \left(3 d^2 \cos \left(\frac{2 b (c+d x)}{d}\right)-4 b (c+d x) \left(8 \sqrt{\pi } b \sqrt{\frac{b}{d}} (c+d x)^{3/2} S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+d \sin \left(\frac{2 b (c+d x)}{d}\right)+4 b (c+d x) \cos \left(\frac{2 b (c+d x)}{d}\right)\right)\right)\right)-2 \sin \left(\frac{2 b c}{d}\right) \left(15 d^3 \cos \left(\frac{2 b (c+d x)}{d}\right)-4 b (c+d x) \left(3 d^2 \sin \left(\frac{2 b (c+d x)}{d}\right)+4 b (c+d x) \left(8 \sqrt{\pi } b \sqrt{\frac{b}{d}} (c+d x)^{3/2} C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-4 b (c+d x) \sin \left(\frac{2 b (c+d x)}{d}\right)+d \cos \left(\frac{2 b (c+d x)}{d}\right)\right)\right)\right)\right)-30 d^3}{210 d^4 (c+d x)^{7/2}}","-\frac{128 \sqrt{\pi } b^{7/2} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{105 d^{9/2}}+\frac{128 \sqrt{\pi } b^{7/2} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{105 d^{9/2}}+\frac{128 b^3 \sin (a+b x) \cos (a+b x)}{105 d^4 \sqrt{c+d x}}+\frac{32 b^2 \sin ^2(a+b x)}{105 d^3 (c+d x)^{3/2}}-\frac{8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac{2 \sin ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac{16 b^2}{105 d^3 (c+d x)^{3/2}}",1,"(-30*d^3 + Cos[2*a]*(4*Cos[(b*c)/d]*Sin[(b*c)/d]*(15*d^3*Sin[(2*b*(c + d*x))/d] + 4*b*(c + d*x)*(3*d^2*Cos[(2*b*(c + d*x))/d] - 4*b*(c + d*x)*(4*b*(c + d*x)*Cos[(2*b*(c + d*x))/d] + 8*b*Sqrt[b/d]*Sqrt[Pi]*(c + d*x)^(3/2)*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + d*Sin[(2*b*(c + d*x))/d]))) + 2*Cos[(2*b*c)/d]*(15*d^3*Cos[(2*b*(c + d*x))/d] - 4*b*(c + d*x)*(3*d^2*Sin[(2*b*(c + d*x))/d] + 4*b*(c + d*x)*(d*Cos[(2*b*(c + d*x))/d] + 8*b*Sqrt[b/d]*Sqrt[Pi]*(c + d*x)^(3/2)*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 4*b*(c + d*x)*Sin[(2*b*(c + d*x))/d])))) - 2*Cos[a]*Sin[a]*(2*(Cos[(b*c)/d] - Sin[(b*c)/d])*(Cos[(b*c)/d] + Sin[(b*c)/d])*(15*d^3*Sin[(2*b*(c + d*x))/d] + 4*b*(c + d*x)*(3*d^2*Cos[(2*b*(c + d*x))/d] - 4*b*(c + d*x)*(4*b*(c + d*x)*Cos[(2*b*(c + d*x))/d] + 8*b*Sqrt[b/d]*Sqrt[Pi]*(c + d*x)^(3/2)*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + d*Sin[(2*b*(c + d*x))/d]))) - 2*Sin[(2*b*c)/d]*(15*d^3*Cos[(2*b*(c + d*x))/d] - 4*b*(c + d*x)*(3*d^2*Sin[(2*b*(c + d*x))/d] + 4*b*(c + d*x)*(d*Cos[(2*b*(c + d*x))/d] + 8*b*Sqrt[b/d]*Sqrt[Pi]*(c + d*x)^(3/2)*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 4*b*(c + d*x)*Sin[(2*b*(c + d*x))/d])))))/(210*d^4*(c + d*x)^(7/2))","B",1
53,1,542,410,3.3118989,"\int (c+d x)^{5/2} \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Sin[a + b*x]^3,x]","\frac{-648 b^3 c^2 \sqrt{c+d x} \cos (a+b x)+72 b^3 c^2 \sqrt{c+d x} \cos (3 (a+b x))-648 b^3 d^2 x^2 \sqrt{c+d x} \cos (a+b x)+72 b^3 d^2 x^2 \sqrt{c+d x} \cos (3 (a+b x))-1296 b^3 c d x \sqrt{c+d x} \cos (a+b x)+144 b^3 c d x \sqrt{c+d x} \cos (3 (a+b x))+1620 b^2 d^2 x \sqrt{c+d x} \sin (a+b x)-60 b^2 d^2 x \sqrt{c+d x} \sin (3 (a+b x))+1620 b^2 c d \sqrt{c+d x} \sin (a+b x)-60 b^2 c d \sqrt{c+d x} \sin (3 (a+b x))-1215 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left(a-\frac{b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+5 \sqrt{6 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-5 \sqrt{6 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+1215 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left(a-\frac{b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+2430 b d^2 \sqrt{c+d x} \cos (a+b x)-30 b d^2 \sqrt{c+d x} \cos (3 (a+b x))}{864 b^4}","-\frac{45 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}+\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{144 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{144 b^{7/2}}+\frac{45 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}+\frac{45 d^2 \sqrt{c+d x} \cos (a+b x)}{16 b^3}-\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac{2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac{(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}",1,"(-648*b^3*c^2*Sqrt[c + d*x]*Cos[a + b*x] + 2430*b*d^2*Sqrt[c + d*x]*Cos[a + b*x] - 1296*b^3*c*d*x*Sqrt[c + d*x]*Cos[a + b*x] - 648*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[a + b*x] + 72*b^3*c^2*Sqrt[c + d*x]*Cos[3*(a + b*x)] - 30*b*d^2*Sqrt[c + d*x]*Cos[3*(a + b*x)] + 144*b^3*c*d*x*Sqrt[c + d*x]*Cos[3*(a + b*x)] + 72*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[3*(a + b*x)] - 1215*Sqrt[b/d]*d^3*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] + 5*Sqrt[b/d]*d^3*Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] - 5*Sqrt[b/d]*d^3*Sqrt[6*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 1215*Sqrt[b/d]*d^3*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[a - (b*c)/d] + 1620*b^2*c*d*Sqrt[c + d*x]*Sin[a + b*x] + 1620*b^2*d^2*x*Sqrt[c + d*x]*Sin[a + b*x] - 60*b^2*c*d*Sqrt[c + d*x]*Sin[3*(a + b*x)] - 60*b^2*d^2*x*Sqrt[c + d*x]*Sin[3*(a + b*x)])/(864*b^4)","A",1
54,1,389,354,1.6902533,"\int (c+d x)^{3/2} \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Sin[a + b*x]^3,x]","\frac{\sqrt{6 \pi } d \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-81 \sqrt{2 \pi } d \sin \left(a-\frac{b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-81 \sqrt{2 \pi } d \cos \left(a-\frac{b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+\sqrt{6 \pi } d \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+162 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (a+b x)-6 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))-108 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (a+b x)-108 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (a+b x)+12 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))+12 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))}{144 b^2 \sqrt{\frac{b}{d}}}","\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{24 b^{5/2}}-\frac{9 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}-\frac{9 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}+\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{24 b^{5/2}}+\frac{d \sqrt{c+d x} \sin ^3(a+b x)}{6 b^2}+\frac{d \sqrt{c+d x} \sin (a+b x)}{b^2}-\frac{2 (c+d x)^{3/2} \cos (a+b x)}{3 b}-\frac{(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}",1,"(-108*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[a + b*x] - 108*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[a + b*x] + 12*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] + 12*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[3*(a + b*x)] - 81*d*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] + d*Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + d*Sqrt[6*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] - 81*d*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[a - (b*c)/d] + 162*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[a + b*x] - 6*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[3*(a + b*x)])/(144*b^2*Sqrt[b/d])","A",1
55,1,266,304,0.8133273,"\int \sqrt{c+d x} \sin ^3(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Sin[a + b*x]^3,x]","\frac{27 \sqrt{2 \pi } \cos \left(a-\frac{b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-\sqrt{6 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{6 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-27 \sqrt{2 \pi } \sin \left(a-\frac{b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-54 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (a+b x)+6 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))}{72 b \sqrt{\frac{b}{d}}}","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{12 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{12 b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{3/2}}-\frac{3 \sqrt{c+d x} \cos (a+b x)}{4 b}+\frac{\sqrt{c+d x} \cos (3 a+3 b x)}{12 b}",1,"(-54*Sqrt[b/d]*Sqrt[c + d*x]*Cos[a + b*x] + 6*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] + 27*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[6*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] - 27*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[a - (b*c)/d])/(72*b*Sqrt[b/d])","A",1
56,1,202,257,0.5996589,"\int \frac{\sin ^3(a+b x)}{\sqrt{c+d x}} \, dx","Integrate[Sin[a + b*x]^3/Sqrt[c + d*x],x]","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{\frac{b}{d}} \left(\sqrt{3} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-9 \sin \left(a-\frac{b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-9 \cos \left(a-\frac{b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+\sqrt{3} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)\right)}{6 b}","-\frac{\sqrt{\frac{\pi }{6}} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{2 \sqrt{b} \sqrt{d}}+\frac{3 \sqrt{\frac{\pi }{2}} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{2 \sqrt{b} \sqrt{d}}+\frac{3 \sqrt{\frac{\pi }{2}} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{2 \sqrt{b} \sqrt{d}}-\frac{\sqrt{\frac{\pi }{6}} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{2 \sqrt{b} \sqrt{d}}",1,"-1/6*(Sqrt[b/d]*Sqrt[Pi/2]*(-9*Cos[a - (b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] + Sqrt[3]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[3]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] - 9*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[a - (b*c)/d]))/b","A",1
57,1,300,270,1.01686,"\int \frac{\sin ^3(a+b x)}{(c+d x)^{3/2}} \, dx","Integrate[Sin[a + b*x]^3/(c + d*x)^(3/2),x]","\frac{3 \sqrt{2 \pi } \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos \left(a-\frac{b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-\sqrt{6 \pi } \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{6 \pi } \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-3 \sqrt{2 \pi } \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin \left(a-\frac{b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-3 \sin (a+b x)+\sin (3 (a+b x))}{2 d \sqrt{c+d x}}","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{b} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{3/2}}-\frac{\sqrt{\frac{3 \pi }{2}} \sqrt{b} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{3/2}}+\frac{\sqrt{\frac{3 \pi }{2}} \sqrt{b} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{b} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{3/2}}-\frac{2 \sin ^3(a+b x)}{d \sqrt{c+d x}}",1,"(3*Sqrt[b/d]*Sqrt[2*Pi]*Sqrt[c + d*x]*Cos[a - (b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - Sqrt[b/d]*Sqrt[6*Pi]*Sqrt[c + d*x]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[b/d]*Sqrt[6*Pi]*Sqrt[c + d*x]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] - 3*Sqrt[b/d]*Sqrt[2*Pi]*Sqrt[c + d*x]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[a - (b*c)/d] - 3*Sin[a + b*x] + Sin[3*(a + b*x)])/(2*d*Sqrt[c + d*x])","A",1
58,1,496,292,2.4165967,"\int \frac{\sin ^3(a+b x)}{(c+d x)^{5/2}} \, dx","Integrate[Sin[a + b*x]^3/(c + d*x)^(5/2),x]","\frac{6 \sqrt{6 \pi } b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+6 \sqrt{6 \pi } b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-6 \sqrt{2 \pi } b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin \left(a-\frac{b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-6 \sqrt{2 \pi } b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin \left(a-\frac{b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-6 \sqrt{2 \pi } b \sqrt{\frac{b}{d}} (c+d x)^{3/2} \cos \left(a-\frac{b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+6 \sqrt{6 \pi } b \sqrt{\frac{b}{d}} (c+d x)^{3/2} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-6 b c \cos (a+b x)+6 b c \cos (3 (a+b x))-3 d \sin (a+b x)+d \sin (3 (a+b x))-6 b d x \cos (a+b x)+6 b d x \cos (3 (a+b x))}{6 d^2 (c+d x)^{3/2}}","\frac{\sqrt{6 \pi } b^{3/2} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{5/2}}-\frac{\sqrt{2 \pi } b^{3/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{5/2}}-\frac{\sqrt{2 \pi } b^{3/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{5/2}}+\frac{\sqrt{6 \pi } b^{3/2} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{5/2}}-\frac{4 b \sin ^2(a+b x) \cos (a+b x)}{d^2 \sqrt{c+d x}}-\frac{2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}",1,"(-6*b*c*Cos[a + b*x] - 6*b*d*x*Cos[a + b*x] + 6*b*c*Cos[3*(a + b*x)] + 6*b*d*x*Cos[3*(a + b*x)] - 6*b*Sqrt[b/d]*Sqrt[2*Pi]*(c + d*x)^(3/2)*Cos[a - (b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] + 6*b*Sqrt[b/d]*Sqrt[6*Pi]*(c + d*x)^(3/2)*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + 6*b*c*Sqrt[b/d]*Sqrt[6*Pi]*Sqrt[c + d*x]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 6*b*Sqrt[b/d]*d*Sqrt[6*Pi]*x*Sqrt[c + d*x]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] - 6*b*c*Sqrt[b/d]*Sqrt[2*Pi]*Sqrt[c + d*x]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[a - (b*c)/d] - 6*b*Sqrt[b/d]*d*Sqrt[2*Pi]*x*Sqrt[c + d*x]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[a - (b*c)/d] - 3*d*Sin[a + b*x] + d*Sin[3*(a + b*x)])/(6*d^2*(c + d*x)^(3/2))","A",1
59,1,1429,356,6.4042727,"\int \frac{\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx","Integrate[Sin[a + b*x]^3/(c + d*x)^(7/2),x]","\frac{3}{4} \left(\cos (a) \left(\frac{2 \left(\frac{b}{d}\right)^{5/2} \sin \left(\frac{b c}{d}\right) \left(\frac{\cos \left(\frac{b (c+d x)}{d}\right)}{\left(\frac{b}{d}\right)^{5/2} (c+d x)^{5/2}}-\frac{2}{3} \left(2 \left(\frac{\cos \left(\frac{b (c+d x)}{d}\right)}{\sqrt{\frac{b}{d}} \sqrt{c+d x}}+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)\right)+\frac{\sin \left(\frac{b (c+d x)}{d}\right)}{\left(\frac{b}{d}\right)^{3/2} (c+d x)^{3/2}}\right)\right)}{5 d}-\frac{2 \left(\frac{b}{d}\right)^{5/2} \cos \left(\frac{b c}{d}\right) \left(\frac{\sin \left(\frac{b (c+d x)}{d}\right)}{\left(\frac{b}{d}\right)^{5/2} (c+d x)^{5/2}}+\frac{2}{3} \left(\frac{\cos \left(\frac{b (c+d x)}{d}\right)}{\left(\frac{b}{d}\right)^{3/2} (c+d x)^{3/2}}-2 \left(\frac{\sin \left(\frac{b (c+d x)}{d}\right)}{\sqrt{\frac{b}{d}} \sqrt{c+d x}}-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)\right)\right)\right)}{5 d}\right)+\sin (a) \left(-\frac{2 \cos \left(\frac{b c}{d}\right) \left(\frac{\cos \left(\frac{b (c+d x)}{d}\right)}{\left(\frac{b}{d}\right)^{5/2} (c+d x)^{5/2}}-\frac{2}{3} \left(2 \left(\frac{\cos \left(\frac{b (c+d x)}{d}\right)}{\sqrt{\frac{b}{d}} \sqrt{c+d x}}+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)\right)+\frac{\sin \left(\frac{b (c+d x)}{d}\right)}{\left(\frac{b}{d}\right)^{3/2} (c+d x)^{3/2}}\right)\right) \left(\frac{b}{d}\right)^{5/2}}{5 d}-\frac{2 \sin \left(\frac{b c}{d}\right) \left(\frac{\sin \left(\frac{b (c+d x)}{d}\right)}{\left(\frac{b}{d}\right)^{5/2} (c+d x)^{5/2}}+\frac{2}{3} \left(\frac{\cos \left(\frac{b (c+d x)}{d}\right)}{\left(\frac{b}{d}\right)^{3/2} (c+d x)^{3/2}}-2 \left(\frac{\sin \left(\frac{b (c+d x)}{d}\right)}{\sqrt{\frac{b}{d}} \sqrt{c+d x}}-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)\right)\right)\right) \left(\frac{b}{d}\right)^{5/2}}{5 d}\right)\right)+\frac{1}{4} \left(-\cos (3 a) \left(\frac{18 \sqrt{3} \left(\frac{b}{d}\right)^{5/2} \sin \left(\frac{3 b c}{d}\right) \left(\frac{\cos \left(\frac{3 b (c+d x)}{d}\right)}{9 \sqrt{3} \left(\frac{b}{d}\right)^{5/2} (c+d x)^{5/2}}-\frac{2}{3} \left(2 \left(\frac{\cos \left(\frac{3 b (c+d x)}{d}\right)}{\sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x}}+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)\right)+\frac{\sin \left(\frac{3 b (c+d x)}{d}\right)}{3 \sqrt{3} \left(\frac{b}{d}\right)^{3/2} (c+d x)^{3/2}}\right)\right)}{5 d}-\frac{18 \sqrt{3} \left(\frac{b}{d}\right)^{5/2} \cos \left(\frac{3 b c}{d}\right) \left(\frac{\sin \left(\frac{3 b (c+d x)}{d}\right)}{9 \sqrt{3} \left(\frac{b}{d}\right)^{5/2} (c+d x)^{5/2}}+\frac{2}{3} \left(\frac{\cos \left(\frac{3 b (c+d x)}{d}\right)}{3 \sqrt{3} \left(\frac{b}{d}\right)^{3/2} (c+d x)^{3/2}}-2 \left(\frac{\sin \left(\frac{3 b (c+d x)}{d}\right)}{\sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x}}-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)\right)\right)\right)}{5 d}\right)-\sin (3 a) \left(-\frac{18 \sqrt{3} \cos \left(\frac{3 b c}{d}\right) \left(\frac{\cos \left(\frac{3 b (c+d x)}{d}\right)}{9 \sqrt{3} \left(\frac{b}{d}\right)^{5/2} (c+d x)^{5/2}}-\frac{2}{3} \left(2 \left(\frac{\cos \left(\frac{3 b (c+d x)}{d}\right)}{\sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x}}+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)\right)+\frac{\sin \left(\frac{3 b (c+d x)}{d}\right)}{3 \sqrt{3} \left(\frac{b}{d}\right)^{3/2} (c+d x)^{3/2}}\right)\right) \left(\frac{b}{d}\right)^{5/2}}{5 d}-\frac{18 \sqrt{3} \sin \left(\frac{3 b c}{d}\right) \left(\frac{\sin \left(\frac{3 b (c+d x)}{d}\right)}{9 \sqrt{3} \left(\frac{b}{d}\right)^{5/2} (c+d x)^{5/2}}+\frac{2}{3} \left(\frac{\cos \left(\frac{3 b (c+d x)}{d}\right)}{3 \sqrt{3} \left(\frac{b}{d}\right)^{3/2} (c+d x)^{3/2}}-2 \left(\frac{\sin \left(\frac{3 b (c+d x)}{d}\right)}{\sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x}}-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)\right)\right)\right) \left(\frac{b}{d}\right)^{5/2}}{5 d}\right)\right)","-\frac{2 \sqrt{2 \pi } b^{5/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{5 d^{7/2}}+\frac{6 \sqrt{6 \pi } b^{5/2} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{5 d^{7/2}}-\frac{6 \sqrt{6 \pi } b^{5/2} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{5 d^{7/2}}+\frac{2 \sqrt{2 \pi } b^{5/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{5 d^{7/2}}+\frac{24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt{c+d x}}-\frac{16 b^2 \sin (a+b x)}{5 d^3 \sqrt{c+d x}}-\frac{4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac{2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}",1,"(3*(Cos[a]*((2*(b/d)^(5/2)*Sin[(b*c)/d]*(Cos[(b*(c + d*x))/d]/((b/d)^(5/2)*(c + d*x)^(5/2)) - (2*(2*(Cos[(b*(c + d*x))/d]/(Sqrt[b/d]*Sqrt[c + d*x]) + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]) + Sin[(b*(c + d*x))/d]/((b/d)^(3/2)*(c + d*x)^(3/2))))/3))/(5*d) - (2*(b/d)^(5/2)*Cos[(b*c)/d]*(Sin[(b*(c + d*x))/d]/((b/d)^(5/2)*(c + d*x)^(5/2)) + (2*(Cos[(b*(c + d*x))/d]/((b/d)^(3/2)*(c + d*x)^(3/2)) - 2*(-(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]) + Sin[(b*(c + d*x))/d]/(Sqrt[b/d]*Sqrt[c + d*x]))))/3))/(5*d)) + Sin[a]*((-2*(b/d)^(5/2)*Cos[(b*c)/d]*(Cos[(b*(c + d*x))/d]/((b/d)^(5/2)*(c + d*x)^(5/2)) - (2*(2*(Cos[(b*(c + d*x))/d]/(Sqrt[b/d]*Sqrt[c + d*x]) + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]) + Sin[(b*(c + d*x))/d]/((b/d)^(3/2)*(c + d*x)^(3/2))))/3))/(5*d) - (2*(b/d)^(5/2)*Sin[(b*c)/d]*(Sin[(b*(c + d*x))/d]/((b/d)^(5/2)*(c + d*x)^(5/2)) + (2*(Cos[(b*(c + d*x))/d]/((b/d)^(3/2)*(c + d*x)^(3/2)) - 2*(-(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]) + Sin[(b*(c + d*x))/d]/(Sqrt[b/d]*Sqrt[c + d*x]))))/3))/(5*d))))/4 + (-(Cos[3*a]*((18*Sqrt[3]*(b/d)^(5/2)*Sin[(3*b*c)/d]*(Cos[(3*b*(c + d*x))/d]/(9*Sqrt[3]*(b/d)^(5/2)*(c + d*x)^(5/2)) - (2*(2*(Cos[(3*b*(c + d*x))/d]/(Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]) + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) + Sin[(3*b*(c + d*x))/d]/(3*Sqrt[3]*(b/d)^(3/2)*(c + d*x)^(3/2))))/3))/(5*d) - (18*Sqrt[3]*(b/d)^(5/2)*Cos[(3*b*c)/d]*(Sin[(3*b*(c + d*x))/d]/(9*Sqrt[3]*(b/d)^(5/2)*(c + d*x)^(5/2)) + (2*(Cos[(3*b*(c + d*x))/d]/(3*Sqrt[3]*(b/d)^(3/2)*(c + d*x)^(3/2)) - 2*(-(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) + Sin[(3*b*(c + d*x))/d]/(Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]))))/3))/(5*d))) - Sin[3*a]*((-18*Sqrt[3]*(b/d)^(5/2)*Cos[(3*b*c)/d]*(Cos[(3*b*(c + d*x))/d]/(9*Sqrt[3]*(b/d)^(5/2)*(c + d*x)^(5/2)) - (2*(2*(Cos[(3*b*(c + d*x))/d]/(Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]) + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) + Sin[(3*b*(c + d*x))/d]/(3*Sqrt[3]*(b/d)^(3/2)*(c + d*x)^(3/2))))/3))/(5*d) - (18*Sqrt[3]*(b/d)^(5/2)*Sin[(3*b*c)/d]*(Sin[(3*b*(c + d*x))/d]/(9*Sqrt[3]*(b/d)^(5/2)*(c + d*x)^(5/2)) + (2*(Cos[(3*b*(c + d*x))/d]/(3*Sqrt[3]*(b/d)^(3/2)*(c + d*x)^(3/2)) - 2*(-(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) + Sin[(3*b*(c + d*x))/d]/(Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]))))/3))/(5*d)))/4","B",1
60,1,60,87,0.0139012,"\int (d x)^{3/2} \sin (f x) \, dx","Integrate[(d*x)^(3/2)*Sin[f*x],x]","\frac{d^2 \left(\sqrt{-i f x} \Gamma \left(\frac{5}{2},-i f x\right)+\sqrt{i f x} \Gamma \left(\frac{5}{2},i f x\right)\right)}{2 f^3 \sqrt{d x}}","-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} S\left(\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right)}{2 f^{5/2}}+\frac{3 d \sqrt{d x} \sin (f x)}{2 f^2}-\frac{(d x)^{3/2} \cos (f x)}{f}",1,"(d^2*(Sqrt[(-I)*f*x]*Gamma[5/2, (-I)*f*x] + Sqrt[I*f*x]*Gamma[5/2, I*f*x]))/(2*f^3*Sqrt[d*x])","C",1
61,1,69,65,0.0118303,"\int \sqrt{d x} \sin (f x) \, dx","Integrate[Sqrt[d*x]*Sin[f*x],x]","-\frac{\sqrt{d x} \Gamma \left(\frac{3}{2},-i f x\right)}{2 f \sqrt{-i f x}}-\frac{\sqrt{d x} \Gamma \left(\frac{3}{2},i f x\right)}{2 f \sqrt{i f x}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} C\left(\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right)}{f^{3/2}}-\frac{\sqrt{d x} \cos (f x)}{f}",1,"-1/2*(Sqrt[d*x]*Gamma[3/2, (-I)*f*x])/(f*Sqrt[(-I)*f*x]) - (Sqrt[d*x]*Gamma[3/2, I*f*x])/(2*f*Sqrt[I*f*x])","C",1
62,1,59,46,0.0084078,"\int \frac{\sin (f x)}{\sqrt{d x}} \, dx","Integrate[Sin[f*x]/Sqrt[d*x],x]","\frac{-\sqrt{-i f x} \Gamma \left(\frac{1}{2},-i f x\right)-\sqrt{i f x} \Gamma \left(\frac{1}{2},i f x\right)}{2 f \sqrt{d x}}","\frac{\sqrt{2 \pi } S\left(\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right)}{\sqrt{d} \sqrt{f}}",1,"(-(Sqrt[(-I)*f*x]*Gamma[1/2, (-I)*f*x]) - Sqrt[I*f*x]*Gamma[1/2, I*f*x])/(2*f*Sqrt[d*x])","C",1
63,1,64,64,0.0239082,"\int \frac{\sin (f x)}{(d x)^{3/2}} \, dx","Integrate[Sin[f*x]/(d*x)^(3/2),x]","\frac{x \left(-2 \sin (f x)-i \sqrt{-i f x} \Gamma \left(\frac{1}{2},-i f x\right)+i \sqrt{i f x} \Gamma \left(\frac{1}{2},i f x\right)\right)}{(d x)^{3/2}}","\frac{2 \sqrt{2 \pi } \sqrt{f} C\left(\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right)}{d^{3/2}}-\frac{2 \sin (f x)}{d \sqrt{d x}}",1,"(x*((-I)*Sqrt[(-I)*f*x]*Gamma[1/2, (-I)*f*x] + I*Sqrt[I*f*x]*Gamma[1/2, I*f*x] - 2*Sin[f*x]))/(d*x)^(3/2)","C",1
64,1,111,87,0.0878944,"\int \frac{\sin (f x)}{(d x)^{5/2}} \, dx","Integrate[Sin[f*x]/(d*x)^(5/2),x]","-\frac{2 x \sin (f x)}{3 (d x)^{5/2}}+\frac{2 f x^{5/2} \left(\frac{\sqrt{i f x} \Gamma \left(\frac{1}{2},i f x\right)-e^{-i f x}}{\sqrt{x}}-\frac{e^{i f x}-\sqrt{-i f x} \Gamma \left(\frac{1}{2},-i f x\right)}{\sqrt{x}}\right)}{3 (d x)^{5/2}}","-\frac{4 \sqrt{2 \pi } f^{3/2} S\left(\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right)}{3 d^{5/2}}-\frac{4 f \cos (f x)}{3 d^2 \sqrt{d x}}-\frac{2 \sin (f x)}{3 d (d x)^{3/2}}",1,"(2*f*x^(5/2)*(-((E^(I*f*x) - Sqrt[(-I)*f*x]*Gamma[1/2, (-I)*f*x])/Sqrt[x]) + (-E^((-I)*f*x) + Sqrt[I*f*x]*Gamma[1/2, I*f*x])/Sqrt[x]))/(3*(d*x)^(5/2)) - (2*x*Sin[f*x])/(3*(d*x)^(5/2))","C",1
65,0,0,19,15.9072383,"\int \sqrt{c+d x} \csc (a+b x) \, dx","Integrate[Sqrt[c + d*x]*Csc[a + b*x],x]","\int \sqrt{c+d x} \csc (a+b x) \, dx","\text{Int}\left(\sqrt{c+d x} \csc (a+b x),x\right)",0,"Integrate[Sqrt[c + d*x]*Csc[a + b*x], x]","A",-1
66,0,0,19,15.2702524,"\int \frac{\csc (a+b x)}{\sqrt{c+d x}} \, dx","Integrate[Csc[a + b*x]/Sqrt[c + d*x],x]","\int \frac{\csc (a+b x)}{\sqrt{c+d x}} \, dx","\text{Int}\left(\frac{\csc (a+b x)}{\sqrt{c+d x}},x\right)",0,"Integrate[Csc[a + b*x]/Sqrt[c + d*x], x]","A",-1
67,1,33,38,0.4703915,"\int \left(\frac{x}{\sin ^{\frac{3}{2}}(e+f x)}+x \sqrt{\sin (e+f x)}\right) \, dx","Integrate[x/Sin[e + f*x]^(3/2) + x*Sqrt[Sin[e + f*x]],x]","\frac{4 \sin (e+f x)-2 f x \cos (e+f x)}{f^2 \sqrt{\sin (e+f x)}}","\frac{4 \sqrt{\sin (e+f x)}}{f^2}-\frac{2 x \cos (e+f x)}{f \sqrt{\sin (e+f x)}}",1,"(-2*f*x*Cos[e + f*x] + 4*Sin[e + f*x])/(f^2*Sqrt[Sin[e + f*x]])","A",1
68,1,185,62,4.687263,"\int \left(\frac{x^2}{\sin ^{\frac{3}{2}}(e+f x)}+x^2 \sqrt{\sin (e+f x)}\right) \, dx","Integrate[x^2/Sin[e + f*x]^(3/2) + x^2*Sqrt[Sin[e + f*x]],x]","-\frac{\sec (e) \left(\left(f^2 x^2-8\right) \cos (2 e+f x)-8 f x \cos (e) \sin (e+f x)+\left(f^2 x^2+8\right) \cos (f x)\right)}{f^3 \sqrt{\sin (e+f x)}}+\frac{8 \sec (e) e^{-i f x} \sqrt{2-2 e^{2 i (e+f x)}} \left(3 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};e^{2 i (e+f x)}\right)+e^{2 i f x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{2 i (e+f x)}\right)\right)}{3 f^3 \sqrt{-i e^{-i (e+f x)} \left(-1+e^{2 i (e+f x)}\right)}}","-\frac{16 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{f^3}+\frac{8 x \sqrt{\sin (e+f x)}}{f^2}-\frac{2 x^2 \cos (e+f x)}{f \sqrt{\sin (e+f x)}}",1,"(8*Sqrt[2 - 2*E^((2*I)*(e + f*x))]*(3*Hypergeometric2F1[-1/4, 1/2, 3/4, E^((2*I)*(e + f*x))] + E^((2*I)*f*x)*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(e + f*x))])*Sec[e])/(3*E^(I*f*x)*Sqrt[((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x))]*f^3) - (Sec[e]*((8 + f^2*x^2)*Cos[f*x] + (-8 + f^2*x^2)*Cos[2*e + f*x] - 8*f*x*Cos[e]*Sin[e + f*x]))/(f^3*Sqrt[Sin[e + f*x]])","C",1
69,1,35,42,0.4649231,"\int \left(\frac{x}{\sin ^{\frac{5}{2}}(e+f x)}-\frac{x}{3 \sqrt{\sin (e+f x)}}\right) \, dx","Integrate[x/Sin[e + f*x]^(5/2) - x/(3*Sqrt[Sin[e + f*x]]),x]","-\frac{2 (2 \sin (e+f x)+f x \cos (e+f x))}{3 f^2 \sin ^{\frac{3}{2}}(e+f x)}","-\frac{4}{3 f^2 \sqrt{\sin (e+f x)}}-\frac{2 x \cos (e+f x)}{3 f \sin ^{\frac{3}{2}}(e+f x)}",1,"(-2*(f*x*Cos[e + f*x] + 2*Sin[e + f*x]))/(3*f^2*Sin[e + f*x]^(3/2))","A",1
70,1,58,83,0.6805672,"\int \left(\frac{x}{\sin ^{\frac{7}{2}}(e+f x)}+\frac{3}{5} x \sqrt{\sin (e+f x)}\right) \, dx","Integrate[x/Sin[e + f*x]^(7/2) + (3*x*Sqrt[Sin[e + f*x]])/5,x]","\frac{46 \sin (e+f x)-18 \sin (3 (e+f x))-21 f x \cos (e+f x)+9 f x \cos (3 (e+f x))}{30 f^2 \sin ^{\frac{5}{2}}(e+f x)}","-\frac{4}{15 f^2 \sin ^{\frac{3}{2}}(e+f x)}+\frac{12 \sqrt{\sin (e+f x)}}{5 f^2}-\frac{2 x \cos (e+f x)}{5 f \sin ^{\frac{5}{2}}(e+f x)}-\frac{6 x \cos (e+f x)}{5 f \sqrt{\sin (e+f x)}}",1,"(-21*f*x*Cos[e + f*x] + 9*f*x*Cos[3*(e + f*x)] + 46*Sin[e + f*x] - 18*Sin[3*(e + f*x)])/(30*f^2*Sin[e + f*x]^(5/2))","A",1
71,0,0,21,0.8730455,"\int (c+d x)^m (b \sin (e+f x))^n \, dx","Integrate[(c + d*x)^m*(b*Sin[e + f*x])^n,x]","\int (c+d x)^m (b \sin (e+f x))^n \, dx","\text{Int}\left((c+d x)^m (b \sin (e+f x))^n,x\right)",0,"Integrate[(c + d*x)^m*(b*Sin[e + f*x])^n, x]","A",-1
72,1,251,267,10.5773756,"\int (c+d x)^m \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^m*Sin[a + b*x]^3,x]","\frac{3^{-m-1} e^{-\frac{3 i (a d+b c)}{d}} (c+d x)^m \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(-3^{m+2} e^{2 i a+\frac{4 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)-3^{m+2} e^{2 i \left(2 a+\frac{b c}{d}\right)} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)+e^{6 i a} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{3 i b (c+d x)}{d}\right)+e^{\frac{6 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{3 i b (c+d x)}{d}\right)\right)}{8 b}","-\frac{3 e^{i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)}{8 b}+\frac{3^{-m-1} e^{3 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{3 i b (c+d x)}{d}\right)}{8 b}-\frac{3 e^{-i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)}{8 b}+\frac{3^{-m-1} e^{-3 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{3 i b (c+d x)}{d}\right)}{8 b}",1,"(3^(-1 - m)*(c + d*x)^m*(-(3^(2 + m)*E^((2*I)*(2*a + (b*c)/d))*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-I)*b*(c + d*x))/d]) - 3^(2 + m)*E^((2*I)*a + ((4*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, (I*b*(c + d*x))/d] + E^((6*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d] + E^(((6*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((3*I)*b*(c + d*x))/d]))/(8*b*E^(((3*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^m)","A",1
73,1,211,162,0.7096454,"\int (c+d x)^m \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Sin[a + b*x]^2,x]","\frac{2^{-m-3} (c+d x)^m \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(-i d (m+1) \left(-\frac{i b (c+d x)}{d}\right)^m \left(\cos \left(2 a-\frac{2 b c}{d}\right)-i \sin \left(2 a-\frac{2 b c}{d}\right)\right) \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)+i d (m+1) \left(\frac{i b (c+d x)}{d}\right)^m \left(\cos \left(2 a-\frac{2 b c}{d}\right)+i \sin \left(2 a-\frac{2 b c}{d}\right)\right) \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)+b 2^{m+2} (c+d x) \left(\frac{b^2 (c+d x)^2}{d^2}\right)^m\right)}{b d (m+1)}","\frac{i 2^{-m-3} e^{2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)}{b}-\frac{i 2^{-m-3} e^{-2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)}{b}+\frac{(c+d x)^{m+1}}{2 d (m+1)}",1,"(2^(-3 - m)*(c + d*x)^m*(2^(2 + m)*b*(c + d*x)*((b^2*(c + d*x)^2)/d^2)^m - I*d*(1 + m)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((2*I)*b*(c + d*x))/d]*(Cos[2*a - (2*b*c)/d] - I*Sin[2*a - (2*b*c)/d]) + I*d*(1 + m)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-2*I)*b*(c + d*x))/d]*(Cos[2*a - (2*b*c)/d] + I*Sin[2*a - (2*b*c)/d])))/(b*d*(1 + m)*((b^2*(c + d*x)^2)/d^2)^m)","A",1
74,1,121,127,0.0475577,"\int (c+d x)^m \sin (a+b x) \, dx","Integrate[(c + d*x)^m*Sin[a + b*x],x]","\frac{e^{-\frac{i (a d+b c)}{d}} (c+d x)^m \left(-e^{2 i a} \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)-e^{\frac{2 i b c}{d}} \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)\right)}{2 b}","-\frac{e^{i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)}{2 b}-\frac{e^{-i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)}{2 b}",1,"((c + d*x)^m*(-((E^((2*I)*a)*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^m) - (E^(((2*I)*b*c)/d)*Gamma[1 + m, (I*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^m))/(2*b*E^((I*(b*c + a*d))/d))","A",1
75,0,0,17,5.7077816,"\int (c+d x)^m \csc (a+b x) \, dx","Integrate[(c + d*x)^m*Csc[a + b*x],x]","\int (c+d x)^m \csc (a+b x) \, dx","\text{Int}\left(\csc (a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Csc[a + b*x], x]","A",-1
76,0,0,19,1.2335387,"\int (c+d x)^m \csc ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Csc[a + b*x]^2,x]","\int (c+d x)^m \csc ^2(a+b x) \, dx","\text{Int}\left(\csc ^2(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Csc[a + b*x]^2, x]","A",-1
77,1,79,79,0.0199342,"\int x^{3+m} \sin (a+b x) \, dx","Integrate[x^(3 + m)*Sin[a + b*x],x]","\frac{i e^{i a} x^m (-i b x)^{-m} \Gamma (m+4,-i b x)}{2 b^4}-\frac{i e^{-i a} x^m (i b x)^{-m} \Gamma (m+4,i b x)}{2 b^4}","\frac{i e^{i a} x^m (-i b x)^{-m} \Gamma (m+4,-i b x)}{2 b^4}-\frac{i e^{-i a} x^m (i b x)^{-m} \Gamma (m+4,i b x)}{2 b^4}",1,"((I/2)*E^(I*a)*x^m*Gamma[4 + m, (-I)*b*x])/(b^4*((-I)*b*x)^m) - ((I/2)*x^m*Gamma[4 + m, I*b*x])/(b^4*E^(I*a)*(I*b*x)^m)","A",1
78,1,75,75,0.0178373,"\int x^{2+m} \sin (a+b x) \, dx","Integrate[x^(2 + m)*Sin[a + b*x],x]","\frac{e^{i a} x^m (-i b x)^{-m} \Gamma (m+3,-i b x)}{2 b^3}+\frac{e^{-i a} x^m (i b x)^{-m} \Gamma (m+3,i b x)}{2 b^3}","\frac{e^{i a} x^m (-i b x)^{-m} \Gamma (m+3,-i b x)}{2 b^3}+\frac{e^{-i a} x^m (i b x)^{-m} \Gamma (m+3,i b x)}{2 b^3}",1,"(E^(I*a)*x^m*Gamma[3 + m, (-I)*b*x])/(2*b^3*((-I)*b*x)^m) + (x^m*Gamma[3 + m, I*b*x])/(2*b^3*E^(I*a)*(I*b*x)^m)","A",1
79,1,79,79,0.017652,"\int x^{1+m} \sin (a+b x) \, dx","Integrate[x^(1 + m)*Sin[a + b*x],x]","\frac{i e^{-i a} x^m (i b x)^{-m} \Gamma (m+2,i b x)}{2 b^2}-\frac{i e^{i a} x^m (-i b x)^{-m} \Gamma (m+2,-i b x)}{2 b^2}","\frac{i e^{-i a} x^m (i b x)^{-m} \Gamma (m+2,i b x)}{2 b^2}-\frac{i e^{i a} x^m (-i b x)^{-m} \Gamma (m+2,-i b x)}{2 b^2}",1,"((-1/2*I)*E^(I*a)*x^m*Gamma[2 + m, (-I)*b*x])/(b^2*((-I)*b*x)^m) + ((I/2)*x^m*Gamma[2 + m, I*b*x])/(b^2*E^(I*a)*(I*b*x)^m)","A",1
80,1,75,75,0.0147627,"\int x^m \sin (a+b x) \, dx","Integrate[x^m*Sin[a + b*x],x]","-\frac{e^{i a} x^m (-i b x)^{-m} \Gamma (m+1,-i b x)}{2 b}-\frac{e^{-i a} x^m (i b x)^{-m} \Gamma (m+1,i b x)}{2 b}","-\frac{e^{i a} x^m (-i b x)^{-m} \Gamma (m+1,-i b x)}{2 b}-\frac{e^{-i a} x^m (i b x)^{-m} \Gamma (m+1,i b x)}{2 b}",1,"-1/2*(E^(I*a)*x^m*Gamma[1 + m, (-I)*b*x])/(b*((-I)*b*x)^m) - (x^m*Gamma[1 + m, I*b*x])/(2*b*E^(I*a)*(I*b*x)^m)","A",1
81,1,63,69,0.0296474,"\int x^{-1+m} \sin (a+b x) \, dx","Integrate[x^(-1 + m)*Sin[a + b*x],x]","\frac{1}{2} i e^{-i a} x^m \left(e^{2 i a} (-i b x)^{-m} \Gamma (m,-i b x)-(i b x)^{-m} \Gamma (m,i b x)\right)","\frac{1}{2} i e^{i a} x^m (-i b x)^{-m} \Gamma (m,-i b x)-\frac{1}{2} i e^{-i a} x^m (i b x)^{-m} \Gamma (m,i b x)",1,"((I/2)*x^m*((E^((2*I)*a)*Gamma[m, (-I)*b*x])/((-I)*b*x)^m - Gamma[m, I*b*x]/(I*b*x)^m))/E^(I*a)","A",1
82,1,65,71,0.0187073,"\int x^{-2+m} \sin (a+b x) \, dx","Integrate[x^(-2 + m)*Sin[a + b*x],x]","\frac{1}{2} e^{-i a} b x^m \left(e^{2 i a} (-i b x)^{-m} \Gamma (m-1,-i b x)+(i b x)^{-m} \Gamma (m-1,i b x)\right)","\frac{1}{2} e^{i a} b x^m (-i b x)^{-m} \Gamma (m-1,-i b x)+\frac{1}{2} e^{-i a} b x^m (i b x)^{-m} \Gamma (m-1,i b x)",1,"(b*x^m*((E^((2*I)*a)*Gamma[-1 + m, (-I)*b*x])/((-I)*b*x)^m + Gamma[-1 + m, I*b*x]/(I*b*x)^m))/(2*E^(I*a))","A",1
83,1,79,79,0.016447,"\int x^{-3+m} \sin (a+b x) \, dx","Integrate[x^(-3 + m)*Sin[a + b*x],x]","\frac{1}{2} i e^{-i a} b^2 x^m (i b x)^{-m} \Gamma (m-2,i b x)-\frac{1}{2} i e^{i a} b^2 x^m (-i b x)^{-m} \Gamma (m-2,-i b x)","\frac{1}{2} i e^{-i a} b^2 x^m (i b x)^{-m} \Gamma (m-2,i b x)-\frac{1}{2} i e^{i a} b^2 x^m (-i b x)^{-m} \Gamma (m-2,-i b x)",1,"((-1/2*I)*b^2*E^(I*a)*x^m*Gamma[-2 + m, (-I)*b*x])/((-I)*b*x)^m + ((I/2)*b^2*x^m*Gamma[-2 + m, I*b*x])/(E^(I*a)*(I*b*x)^m)","A",1
84,1,118,97,0.3520837,"\int x^{3+m} \sin ^2(a+b x) \, dx","Integrate[x^(3 + m)*Sin[a + b*x]^2,x]","\frac{2^{-m-6} x^m \left(b^2 x^2\right)^{-m} \left((m+4) (\cos (a)-i \sin (a))^2 (-i b x)^m \Gamma (m+4,2 i b x)+(m+4) (\cos (a)+i \sin (a))^2 (i b x)^m \Gamma (m+4,-2 i b x)+b^4 2^{m+5} x^4 \left(b^2 x^2\right)^m\right)}{b^4 (m+4)}","\frac{e^{2 i a} 2^{-m-6} x^m (-i b x)^{-m} \Gamma (m+4,-2 i b x)}{b^4}+\frac{e^{-2 i a} 2^{-m-6} x^m (i b x)^{-m} \Gamma (m+4,2 i b x)}{b^4}+\frac{x^{m+4}}{2 (m+4)}",1,"(2^(-6 - m)*x^m*(2^(5 + m)*b^4*x^4*(b^2*x^2)^m + (4 + m)*((-I)*b*x)^m*Gamma[4 + m, (2*I)*b*x]*(Cos[a] - I*Sin[a])^2 + (4 + m)*(I*b*x)^m*Gamma[4 + m, (-2*I)*b*x]*(Cos[a] + I*Sin[a])^2))/(b^4*(4 + m)*(b^2*x^2)^m)","A",1
85,1,120,103,0.357242,"\int x^{2+m} \sin ^2(a+b x) \, dx","Integrate[x^(2 + m)*Sin[a + b*x]^2,x]","\frac{2^{-m-5} x^m \left(b^2 x^2\right)^{-m} \left((m+3) (\sin (2 a)+i \cos (2 a)) (-i b x)^m \Gamma (m+3,2 i b x)+(m+3) (\sin (2 a)-i \cos (2 a)) (i b x)^m \Gamma (m+3,-2 i b x)+b 2^{m+4} x \left(b^2 x^2\right)^{m+1}\right)}{b^3 (m+3)}","-\frac{i e^{2 i a} 2^{-m-5} x^m (-i b x)^{-m} \Gamma (m+3,-2 i b x)}{b^3}+\frac{i e^{-2 i a} 2^{-m-5} x^m (i b x)^{-m} \Gamma (m+3,2 i b x)}{b^3}+\frac{x^{m+3}}{2 (m+3)}",1,"(2^(-5 - m)*x^m*(2^(4 + m)*b*x*(b^2*x^2)^(1 + m) + (3 + m)*(I*b*x)^m*Gamma[3 + m, (-2*I)*b*x]*((-I)*Cos[2*a] + Sin[2*a]) + (3 + m)*((-I)*b*x)^m*Gamma[3 + m, (2*I)*b*x]*(I*Cos[2*a] + Sin[2*a])))/(b^3*(3 + m)*(b^2*x^2)^m)","A",1
86,1,116,99,0.325209,"\int x^{1+m} \sin ^2(a+b x) \, dx","Integrate[x^(1 + m)*Sin[a + b*x]^2,x]","\frac{2^{-m-4} x^m \left(b^2 x^2\right)^{-m} \left(-\left((m+2) (\cos (a)-i \sin (a))^2 (-i b x)^m \Gamma (m+2,2 i b x)\right)-(m+2) (\cos (a)+i \sin (a))^2 (i b x)^m \Gamma (m+2,-2 i b x)+2^{m+3} \left(b^2 x^2\right)^{m+1}\right)}{b^2 (m+2)}","-\frac{e^{2 i a} 2^{-m-4} x^m (-i b x)^{-m} \Gamma (m+2,-2 i b x)}{b^2}-\frac{e^{-2 i a} 2^{-m-4} x^m (i b x)^{-m} \Gamma (m+2,2 i b x)}{b^2}+\frac{x^{m+2}}{2 (m+2)}",1,"(2^(-4 - m)*x^m*(2^(3 + m)*(b^2*x^2)^(1 + m) - (2 + m)*((-I)*b*x)^m*Gamma[2 + m, (2*I)*b*x]*(Cos[a] - I*Sin[a])^2 - (2 + m)*(I*b*x)^m*Gamma[2 + m, (-2*I)*b*x]*(Cos[a] + I*Sin[a])^2))/(b^2*(2 + m)*(b^2*x^2)^m)","A",1
87,1,120,103,0.2976492,"\int x^m \sin ^2(a+b x) \, dx","Integrate[x^m*Sin[a + b*x]^2,x]","\frac{2^{-m-3} x^m \left(b^2 x^2\right)^{-m} \left(-i (m+1) (\cos (a)-i \sin (a))^2 (-i b x)^m \Gamma (m+1,2 i b x)+i (m+1) (\cos (a)+i \sin (a))^2 (i b x)^m \Gamma (m+1,-2 i b x)+b 2^{m+2} x \left(b^2 x^2\right)^m\right)}{b (m+1)}","\frac{i e^{2 i a} 2^{-m-3} x^m (-i b x)^{-m} \Gamma (m+1,-2 i b x)}{b}-\frac{i e^{-2 i a} 2^{-m-3} x^m (i b x)^{-m} \Gamma (m+1,2 i b x)}{b}+\frac{x^{m+1}}{2 (m+1)}",1,"(2^(-3 - m)*x^m*(2^(2 + m)*b*x*(b^2*x^2)^m - I*(1 + m)*((-I)*b*x)^m*Gamma[1 + m, (2*I)*b*x]*(Cos[a] - I*Sin[a])^2 + I*(1 + m)*(I*b*x)^m*Gamma[1 + m, (-2*I)*b*x]*(Cos[a] + I*Sin[a])^2))/(b*(1 + m)*(b^2*x^2)^m)","A",1
88,1,99,83,0.2492451,"\int x^{-1+m} \sin ^2(a+b x) \, dx","Integrate[x^(-1 + m)*Sin[a + b*x]^2,x]","\frac{2^{-m-2} x^m \left(b^2 x^2\right)^{-m} \left(m (\cos (a)-i \sin (a))^2 (-i b x)^m \Gamma (m,2 i b x)+m (\cos (a)+i \sin (a))^2 (i b x)^m \Gamma (m,-2 i b x)+2^{m+1} \left(b^2 x^2\right)^m\right)}{m}","e^{2 i a} 2^{-m-2} x^m (-i b x)^{-m} \Gamma (m,-2 i b x)+e^{-2 i a} 2^{-m-2} x^m (i b x)^{-m} \Gamma (m,2 i b x)+\frac{x^m}{2 m}",1,"(2^(-2 - m)*x^m*(2^(1 + m)*(b^2*x^2)^m + m*((-I)*b*x)^m*Gamma[m, (2*I)*b*x]*(Cos[a] - I*Sin[a])^2 + m*(I*b*x)^m*Gamma[m, (-2*I)*b*x]*(Cos[a] + I*Sin[a])^2))/(m*(b^2*x^2)^m)","A",1
89,1,117,101,0.3302367,"\int x^{-2+m} \sin ^2(a+b x) \, dx","Integrate[x^(-2 + m)*Sin[a + b*x]^2,x]","\frac{2^{-m-1} x^{m-1} \left(b^2 x^2\right)^{-m} \left(b (m-1) x (\sin (2 a)+i \cos (2 a)) (-i b x)^m \Gamma (m-1,2 i b x)+b (m-1) x (\sin (2 a)-i \cos (2 a)) (i b x)^m \Gamma (m-1,-2 i b x)+2^m \left(b^2 x^2\right)^m\right)}{m-1}","-i e^{2 i a} b 2^{-m-1} x^m (-i b x)^{-m} \Gamma (m-1,-2 i b x)+i e^{-2 i a} b 2^{-m-1} x^m (i b x)^{-m} \Gamma (m-1,2 i b x)-\frac{x^{m-1}}{2 (1-m)}",1,"(2^(-1 - m)*x^(-1 + m)*(2^m*(b^2*x^2)^m + b*(-1 + m)*x*(I*b*x)^m*Gamma[-1 + m, (-2*I)*b*x]*((-I)*Cos[2*a] + Sin[2*a]) + b*(-1 + m)*x*((-I)*b*x)^m*Gamma[-1 + m, (2*I)*b*x]*(I*Cos[2*a] + Sin[2*a])))/((-1 + m)*(b^2*x^2)^m)","A",1
90,1,121,97,0.4355509,"\int x^{-3+m} \sin ^2(a+b x) \, dx","Integrate[x^(-3 + m)*Sin[a + b*x]^2,x]","\frac{2^{-m-1} x^{m-2} \left(b^2 x^2\right)^{-m} \left(-2 b^2 (m-2) x^2 (\cos (a)-i \sin (a))^2 (-i b x)^m \Gamma (m-2,2 i b x)+2 (m-2) (\cos (2 a)+i \sin (2 a)) (i b x)^{m+2} \Gamma (m-2,-2 i b x)+2^m \left(b^2 x^2\right)^m\right)}{m-2}","-e^{2 i a} b^2 2^{-m} x^m (-i b x)^{-m} \Gamma (m-2,-2 i b x)-e^{-2 i a} b^2 2^{-m} x^m (i b x)^{-m} \Gamma (m-2,2 i b x)-\frac{x^{m-2}}{2 (2-m)}",1,"(2^(-1 - m)*x^(-2 + m)*(2^m*(b^2*x^2)^m - 2*b^2*(-2 + m)*x^2*((-I)*b*x)^m*Gamma[-2 + m, (2*I)*b*x]*(Cos[a] - I*Sin[a])^2 + 2*(-2 + m)*(I*b*x)^(2 + m)*Gamma[-2 + m, (-2*I)*b*x]*(Cos[2*a] + I*Sin[2*a])))/((-2 + m)*(b^2*x^2)^m)","A",1
91,1,29,42,0.5476932,"\int \left(\frac{x}{\csc ^{\frac{3}{2}}(e+f x)}-\frac{1}{3} x \sqrt{\csc (e+f x)}\right) \, dx","Integrate[x/Csc[e + f*x]^(3/2) - (x*Sqrt[Csc[e + f*x]])/3,x]","-\frac{2 (3 f x \cot (e+f x)-2)}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}","\frac{4}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{3 f \sqrt{\csc (e+f x)}}",1,"(-2*(-2 + 3*f*x*Cot[e + f*x]))/(9*f^2*Csc[e + f*x]^(3/2))","A",1
92,1,87,111,0.5894515,"\int \left(\frac{x^2}{\csc ^{\frac{3}{2}}(e+f x)}-\frac{1}{3} x^2 \sqrt{\csc (e+f x)}\right) \, dx","Integrate[x^2/Csc[e + f*x]^(3/2) - (x^2*Sqrt[Csc[e + f*x]])/3,x]","-\frac{\sqrt{\csc (e+f x)} \left(9 f^2 x^2 \sin (2 (e+f x))-8 \sin (2 (e+f x))+12 f x \cos (2 (e+f x))-16 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)-12 f x\right)}{27 f^3}","\frac{16 \cos (e+f x)}{27 f^3 \sqrt{\csc (e+f x)}}-\frac{16 \sqrt{\sin (e+f x)} \sqrt{\csc (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{27 f^3}+\frac{8 x}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}-\frac{2 x^2 \cos (e+f x)}{3 f \sqrt{\csc (e+f x)}}",1,"-1/27*(Sqrt[Csc[e + f*x]]*(-12*f*x + 12*f*x*Cos[2*(e + f*x)] - 16*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] - 8*Sin[2*(e + f*x)] + 9*f^2*x^2*Sin[2*(e + f*x)]))/f^3","A",1
93,1,29,42,0.502854,"\int \left(\frac{x}{\csc ^{\frac{5}{2}}(e+f x)}-\frac{3 x}{5 \sqrt{\csc (e+f x)}}\right) \, dx","Integrate[x/Csc[e + f*x]^(5/2) - (3*x)/(5*Sqrt[Csc[e + f*x]]),x]","-\frac{2 (5 f x \cot (e+f x)-2)}{25 f^2 \csc ^{\frac{5}{2}}(e+f x)}","\frac{4}{25 f^2 \csc ^{\frac{5}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{5 f \csc ^{\frac{3}{2}}(e+f x)}",1,"(-2*(-2 + 5*f*x*Cot[e + f*x]))/(25*f^2*Csc[e + f*x]^(5/2))","A",1
94,1,57,83,2.3987009,"\int \left(\frac{x}{\csc ^{\frac{7}{2}}(e+f x)}-\frac{5}{21} x \sqrt{\csc (e+f x)}\right) \, dx","Integrate[x/Csc[e + f*x]^(7/2) - (5*x*Sqrt[Csc[e + f*x]])/21,x]","\frac{-36 \cos (2 (e+f x))-483 f x \cot (e+f x)+63 f x \cos (3 (e+f x)) \csc (e+f x)+316}{882 f^2 \csc ^{\frac{3}{2}}(e+f x)}","\frac{20}{63 f^2 \csc ^{\frac{3}{2}}(e+f x)}+\frac{4}{49 f^2 \csc ^{\frac{7}{2}}(e+f x)}-\frac{2 x \cos (e+f x)}{7 f \csc ^{\frac{5}{2}}(e+f x)}-\frac{10 x \cos (e+f x)}{21 f \sqrt{\csc (e+f x)}}",1,"(316 - 36*Cos[2*(e + f*x)] - 483*f*x*Cot[e + f*x] + 63*f*x*Cos[3*(e + f*x)]*Csc[e + f*x])/(882*f^2*Csc[e + f*x]^(3/2))","A",1
95,1,123,90,0.9279182,"\int (c+d x)^3 (a+a \sin (e+f x)) \, dx","Integrate[(c + d*x)^3*(a + a*Sin[e + f*x]),x]","a \left(\frac{3 d \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-2\right)\right) \sin (e+f x)}{f^4}-\frac{(c+d x) \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-6\right)\right) \cos (e+f x)}{f^3}+\frac{1}{4} x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)\right)","\frac{6 a d^2 (c+d x) \cos (e+f x)}{f^3}+\frac{3 a d (c+d x)^2 \sin (e+f x)}{f^2}-\frac{a (c+d x)^3 \cos (e+f x)}{f}+\frac{a (c+d x)^4}{4 d}-\frac{6 a d^3 \sin (e+f x)}{f^4}",1,"a*((x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3))/4 - ((c + d*x)*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-6 + f^2*x^2))*Cos[e + f*x])/f^3 + (3*d*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Sin[e + f*x])/f^4)","A",1
96,1,81,68,0.5241058,"\int (c+d x)^2 (a+a \sin (e+f x)) \, dx","Integrate[(c + d*x)^2*(a + a*Sin[e + f*x]),x]","a \left(-\frac{\left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-2\right)\right) \cos (e+f x)}{f^3}+c^2 x+\frac{2 d (c+d x) \sin (e+f x)}{f^2}+c d x^2+\frac{d^2 x^3}{3}\right)","\frac{2 a d (c+d x) \sin (e+f x)}{f^2}-\frac{a (c+d x)^2 \cos (e+f x)}{f}+\frac{a (c+d x)^3}{3 d}+\frac{2 a d^2 \cos (e+f x)}{f^3}",1,"a*(c^2*x + c*d*x^2 + (d^2*x^3)/3 - ((c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Cos[e + f*x])/f^3 + (2*d*(c + d*x)*Sin[e + f*x])/f^2)","A",1
97,1,51,45,0.3886976,"\int (c+d x) (a+a \sin (e+f x)) \, dx","Integrate[(c + d*x)*(a + a*Sin[e + f*x]),x]","-\frac{a ((e+f x) (-2 c f+d e-d f x)+2 f (c+d x) \cos (e+f x)-2 d \sin (e+f x))}{2 f^2}","-\frac{a (c+d x) \cos (e+f x)}{f}+\frac{a (c+d x)^2}{2 d}+\frac{a d \sin (e+f x)}{f^2}",1,"-1/2*(a*((e + f*x)*(d*e - 2*c*f - d*f*x) + 2*f*(c + d*x)*Cos[e + f*x] - 2*d*Sin[e + f*x]))/f^2","A",1
98,1,54,64,0.3289423,"\int \frac{a+a \sin (e+f x)}{c+d x} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*x),x]","\frac{a \left(\text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \sin \left(e-\frac{c f}{d}\right)+\cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)+\log (c+d x)\right)}{d}","\frac{a \text{Ci}\left(x f+\frac{c f}{d}\right) \sin \left(e-\frac{c f}{d}\right)}{d}+\frac{a \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{d}+\frac{a \log (c+d x)}{d}",1,"(a*(Log[c + d*x] + CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)]))/d","A",1
99,1,110,88,0.5185481,"\int \frac{a+a \sin (e+f x)}{(c+d x)^2} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*x)^2,x]","\frac{a (\sin (e+f x)+1) \left(f (c+d x) \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \cos \left(e-\frac{c f}{d}\right)-f (c+d x) \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)-d (\sin (e+f x)+1)\right)}{d^2 (c+d x) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","\frac{a f \text{Ci}\left(x f+\frac{c f}{d}\right) \cos \left(e-\frac{c f}{d}\right)}{d^2}-\frac{a f \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{d^2}-\frac{a \sin (e+f x)}{d (c+d x)}-\frac{a}{d (c+d x)}",1,"(a*(1 + Sin[e + f*x])*(f*(c + d*x)*Cos[e - (c*f)/d]*CosIntegral[f*(c/d + x)] - d*(1 + Sin[e + f*x]) - f*(c + d*x)*Sin[e - (c*f)/d]*SinIntegral[f*(c/d + x)]))/(d^2*(c + d*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","A",1
100,1,104,123,0.7286024,"\int \frac{a+a \sin (e+f x)}{(c+d x)^3} \, dx","Integrate[(a + a*Sin[e + f*x])/(c + d*x)^3,x]","-\frac{a \left(f^2 (c+d x)^2 \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \sin \left(e-\frac{c f}{d}\right)+f^2 (c+d x)^2 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)+d (f (c+d x) \cos (e+f x)+d (\sin (e+f x)+1))\right)}{2 d^3 (c+d x)^2}","-\frac{a f^2 \text{Ci}\left(x f+\frac{c f}{d}\right) \sin \left(e-\frac{c f}{d}\right)}{2 d^3}-\frac{a f^2 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{2 d^3}-\frac{a f \cos (e+f x)}{2 d^2 (c+d x)}-\frac{a \sin (e+f x)}{2 d (c+d x)^2}-\frac{a}{2 d (c+d x)^2}",1,"-1/2*(a*(f^2*(c + d*x)^2*CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + d*(f*(c + d*x)*Cos[e + f*x] + d*(1 + Sin[e + f*x])) + f^2*(c + d*x)^2*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)]))/(d^3*(c + d*x)^2)","A",1
101,1,216,237,1.4479538,"\int (c+d x)^3 (a+a \sin (e+f x))^2 \, dx","Integrate[(c + d*x)^3*(a + a*Sin[e + f*x])^2,x]","\frac{a^2 \left(-2 f (c+d x) \left(2 c^2 f^2+4 c d f^2 x+d^2 \left(2 f^2 x^2-3\right)\right) \sin (2 (e+f x))+96 d \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-2\right)\right) \sin (e+f x)-32 f (c+d x) \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-6\right)\right) \cos (e+f x)-3 d \left(2 c^2 f^2+4 c d f^2 x+d^2 \left(2 f^2 x^2-1\right)\right) \cos (2 (e+f x))+6 f^4 x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)\right)}{16 f^4}","\frac{12 a^2 d^2 (c+d x) \cos (e+f x)}{f^3}+\frac{3 a^2 d^2 (c+d x) \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac{3 a^2 c d^2 x}{4 f^2}+\frac{3 a^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}+\frac{6 a^2 d (c+d x)^2 \sin (e+f x)}{f^2}-\frac{2 a^2 (c+d x)^3 \cos (e+f x)}{f}-\frac{a^2 (c+d x)^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{3 a^2 (c+d x)^4}{8 d}-\frac{3 a^2 d^3 \sin ^2(e+f x)}{8 f^4}-\frac{12 a^2 d^3 \sin (e+f x)}{f^4}-\frac{3 a^2 d^3 x^2}{8 f^2}",1,"(a^2*(6*f^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) - 32*f*(c + d*x)*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-6 + f^2*x^2))*Cos[e + f*x] - 3*d*(2*c^2*f^2 + 4*c*d*f^2*x + d^2*(-1 + 2*f^2*x^2))*Cos[2*(e + f*x)] + 96*d*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Sin[e + f*x] - 2*f*(c + d*x)*(2*c^2*f^2 + 4*c*d*f^2*x + d^2*(-3 + 2*f^2*x^2))*Sin[2*(e + f*x)]))/(16*f^4)","A",1
102,1,182,168,0.6721763,"\int (c+d x)^2 (a+a \sin (e+f x))^2 \, dx","Integrate[(c + d*x)^2*(a + a*Sin[e + f*x])^2,x]","\frac{a^2 \left(-16 \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-2\right)\right) \cos (e+f x)-2 c^2 f^2 \sin (2 (e+f x))+12 c^2 f^3 x-4 c d f^2 x \sin (2 (e+f x))+32 c d f \sin (e+f x)-2 d f (c+d x) \cos (2 (e+f x))+12 c d f^3 x^2-2 d^2 f^2 x^2 \sin (2 (e+f x))+32 d^2 f x \sin (e+f x)+d^2 \sin (2 (e+f x))+4 d^2 f^3 x^3\right)}{8 f^3}","\frac{a^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}+\frac{4 a^2 d (c+d x) \sin (e+f x)}{f^2}-\frac{2 a^2 (c+d x)^2 \cos (e+f x)}{f}-\frac{a^2 (c+d x)^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a^2 (c+d x)^3}{2 d}+\frac{4 a^2 d^2 \cos (e+f x)}{f^3}+\frac{a^2 d^2 \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac{a^2 d^2 x}{4 f^2}",1,"(a^2*(12*c^2*f^3*x + 12*c*d*f^3*x^2 + 4*d^2*f^3*x^3 - 16*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Cos[e + f*x] - 2*d*f*(c + d*x)*Cos[2*(e + f*x)] + 32*c*d*f*Sin[e + f*x] + 32*d^2*f*x*Sin[e + f*x] + d^2*Sin[2*(e + f*x)] - 2*c^2*f^2*Sin[2*(e + f*x)] - 4*c*d*f^2*x*Sin[2*(e + f*x)] - 2*d^2*f^2*x^2*Sin[2*(e + f*x)]))/(8*f^3)","A",1
103,1,80,118,1.1219025,"\int (c+d x) (a+a \sin (e+f x))^2 \, dx","Integrate[(c + d*x)*(a + a*Sin[e + f*x])^2,x]","-\frac{a^2 (6 (e+f x) (d (e-f x)-2 c f)+2 f (c+d x) \sin (2 (e+f x))+16 f (c+d x) \cos (e+f x)-16 d \sin (e+f x)+d \cos (2 (e+f x)))}{8 f^2}","-\frac{2 a^2 (c+d x) \cos (e+f x)}{f}-\frac{a^2 (c+d x) \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a^2 (c+d x)^2}{2 d}+\frac{1}{2} a^2 c x+\frac{a^2 d \sin ^2(e+f x)}{4 f^2}+\frac{2 a^2 d \sin (e+f x)}{f^2}+\frac{1}{4} a^2 d x^2",1,"-1/8*(a^2*(6*(e + f*x)*(-2*c*f + d*(e - f*x)) + 16*f*(c + d*x)*Cos[e + f*x] + d*Cos[2*(e + f*x)] - 16*d*Sin[e + f*x] + 2*f*(c + d*x)*Sin[2*(e + f*x)]))/f^2","A",1
104,1,114,145,0.314951,"\int \frac{(a+a \sin (e+f x))^2}{c+d x} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c + d*x),x]","\frac{a^2 \left(4 \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \sin \left(e-\frac{c f}{d}\right)+\text{Ci}\left(\frac{2 f (c+d x)}{d}\right) \left(-\cos \left(2 e-\frac{2 c f}{d}\right)\right)+\sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)+4 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)+3 \log (c+d x)\right)}{2 d}","\frac{2 a^2 \text{Ci}\left(x f+\frac{c f}{d}\right) \sin \left(e-\frac{c f}{d}\right)}{d}-\frac{a^2 \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)}{2 d}+\frac{a^2 \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{2 d}+\frac{2 a^2 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{d}+\frac{3 a^2 \log (c+d x)}{2 d}",1,"(a^2*(-(Cos[2*e - (2*c*f)/d]*CosIntegral[(2*f*(c + d*x))/d]) + 3*Log[c + d*x] + 4*CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + 4*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + Sin[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d]))/(2*d)","A",1
105,1,206,162,0.6304931,"\int \frac{(a+a \sin (e+f x))^2}{(c+d x)^2} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c + d*x)^2,x]","\frac{a^2 \left(2 f (c+d x) \text{Ci}\left(\frac{2 f (c+d x)}{d}\right) \sin \left(2 e-\frac{2 c f}{d}\right)+4 f (c+d x) \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \cos \left(e-\frac{c f}{d}\right)-4 d f x \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)-4 c f \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)+2 d f x \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)+2 c f \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)-4 d \sin (e+f x)+d \cos (2 (e+f x))-3 d\right)}{2 d^2 (c+d x)}","\frac{a^2 f \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \sin \left(2 e-\frac{2 c f}{d}\right)}{d^2}+\frac{2 a^2 f \text{Ci}\left(x f+\frac{c f}{d}\right) \cos \left(e-\frac{c f}{d}\right)}{d^2}-\frac{2 a^2 f \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{d^2}+\frac{a^2 f \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{d^2}-\frac{4 a^2 \sin ^4\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{d (c+d x)}",1,"(a^2*(-3*d + d*Cos[2*(e + f*x)] + 4*f*(c + d*x)*Cos[e - (c*f)/d]*CosIntegral[f*(c/d + x)] + 2*f*(c + d*x)*CosIntegral[(2*f*(c + d*x))/d]*Sin[2*e - (2*c*f)/d] - 4*d*Sin[e + f*x] - 4*c*f*Sin[e - (c*f)/d]*SinIntegral[f*(c/d + x)] - 4*d*f*x*Sin[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + 2*c*f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d] + 2*d*f*x*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d]))/(2*d^2*(c + d*x))","A",1
106,1,353,225,1.0200376,"\int \frac{(a+a \sin (e+f x))^2}{(c+d x)^3} \, dx","Integrate[(a + a*Sin[e + f*x])^2/(c + d*x)^3,x]","-\frac{a^2 \left(4 c^2 f^2 \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)+4 c^2 f^2 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)+4 f^2 (c+d x)^2 \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \sin \left(e-\frac{c f}{d}\right)-4 f^2 (c+d x)^2 \text{Ci}\left(\frac{2 f (c+d x)}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)+4 d^2 f^2 x^2 \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)+4 d^2 f^2 x^2 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)+8 c d f^2 x \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)+8 c d f^2 x \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)+2 c d f \sin (2 (e+f x))+4 c d f \cos (e+f x)+4 d^2 \sin (e+f x)+2 d^2 f x \sin (2 (e+f x))+4 d^2 f x \cos (e+f x)-d^2 \cos (2 (e+f x))+3 d^2\right)}{4 d^3 (c+d x)^2}","-\frac{a^2 f^2 \text{Ci}\left(x f+\frac{c f}{d}\right) \sin \left(e-\frac{c f}{d}\right)}{d^3}+\frac{a^2 f^2 \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)}{d^3}-\frac{a^2 f^2 \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{d^3}-\frac{a^2 f^2 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{d^3}-\frac{4 a^2 f \sin ^3\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \cos \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{d^2 (c+d x)}-\frac{2 a^2 \sin ^4\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{d (c+d x)^2}",1,"-1/4*(a^2*(3*d^2 + 4*c*d*f*Cos[e + f*x] + 4*d^2*f*x*Cos[e + f*x] - d^2*Cos[2*(e + f*x)] - 4*f^2*(c + d*x)^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*f*(c + d*x))/d] + 4*f^2*(c + d*x)^2*CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + 4*d^2*Sin[e + f*x] + 2*c*d*f*Sin[2*(e + f*x)] + 2*d^2*f*x*Sin[2*(e + f*x)] + 4*c^2*f^2*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + 8*c*d*f^2*x*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + 4*d^2*f^2*x^2*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + 4*c^2*f^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d] + 8*c*d*f^2*x*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d] + 4*d^2*f^2*x^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d]))/(d^3*(c + d*x)^2)","A",1
107,1,126,148,1.1873539,"\int \frac{(c+d x)^3}{a+a \sin (e+f x)} \, dx","Integrate[(c + d*x)^3/(a + a*Sin[e + f*x]),x]","\frac{-12 i d^2 f (c+d x) \text{Li}_2\left(i e^{i (e+f x)}\right)+f^2 (c+d x)^2 \left(f (c+d x) \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right)-i f (c+d x)+6 d \log \left(1-i e^{i (e+f x)}\right)\right)+12 d^3 \text{Li}_3\left(i e^{i (e+f x)}\right)}{a f^4}","-\frac{12 i d^2 (c+d x) \text{Li}_2\left(i e^{i (e+f x)}\right)}{a f^3}+\frac{6 d (c+d x)^2 \log \left(1-i e^{i (e+f x)}\right)}{a f^2}-\frac{(c+d x)^3 \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f}-\frac{i (c+d x)^3}{a f}+\frac{12 d^3 \text{Li}_3\left(i e^{i (e+f x)}\right)}{a f^4}",1,"((-12*I)*d^2*f*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))] + 12*d^3*PolyLog[3, I*E^(I*(e + f*x))] + f^2*(c + d*x)^2*((-I)*f*(c + d*x) + 6*d*Log[1 - I*E^(I*(e + f*x))] + f*(c + d*x)*Tan[(2*e - Pi + 2*f*x)/4]))/(a*f^4)","A",1
108,1,94,113,0.7380071,"\int \frac{(c+d x)^2}{a+a \sin (e+f x)} \, dx","Integrate[(c + d*x)^2/(a + a*Sin[e + f*x]),x]","\frac{f (c+d x) \left(f (c+d x) \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right)-i f (c+d x)+4 d \log \left(1-i e^{i (e+f x)}\right)\right)-4 i d^2 \text{Li}_2\left(i e^{i (e+f x)}\right)}{a f^3}","\frac{4 d (c+d x) \log \left(1-i e^{i (e+f x)}\right)}{a f^2}-\frac{(c+d x)^2 \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f}-\frac{i (c+d x)^2}{a f}-\frac{4 i d^2 \text{Li}_2\left(i e^{i (e+f x)}\right)}{a f^3}",1,"((-4*I)*d^2*PolyLog[2, I*E^(I*(e + f*x))] + f*(c + d*x)*((-I)*f*(c + d*x) + 4*d*Log[1 - I*E^(I*(e + f*x))] + f*(c + d*x)*Tan[(2*e - Pi + 2*f*x)/4]))/(a*f^3)","A",1
109,1,51,60,0.1632965,"\int \frac{c+d x}{a+a \sin (e+f x)} \, dx","Integrate[(c + d*x)/(a + a*Sin[e + f*x]),x]","\frac{f (c+d x) \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right)+2 d \log \left(\cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{a f^2}","\frac{2 d \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)\right)}{a f^2}-\frac{(c+d x) \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f}",1,"(2*d*Log[Cos[(2*e - Pi + 2*f*x)/4]] + f*(c + d*x)*Tan[(2*e - Pi + 2*f*x)/4])/(a*f^2)","A",1
110,0,0,23,5.3550078,"\int \frac{1}{(c+d x) (a+a \sin (e+f x))} \, dx","Integrate[1/((c + d*x)*(a + a*Sin[e + f*x])),x]","\int \frac{1}{(c+d x) (a+a \sin (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a \sin (e+f x)+a)},x\right)",0,"Integrate[1/((c + d*x)*(a + a*Sin[e + f*x])), x]","A",-1
111,0,0,23,5.0609662,"\int \frac{1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx","Integrate[1/((c + d*x)^2*(a + a*Sin[e + f*x])),x]","\int \frac{1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a \sin (e+f x)+a)},x\right)",0,"Integrate[1/((c + d*x)^2*(a + a*Sin[e + f*x])), x]","A",-1
112,1,257,309,2.131543,"\int \frac{(c+d x)^3}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c + d*x)^3/(a + a*Sin[e + f*x])^2,x]","\frac{\frac{24 d^2 \left(d \text{Li}_3\left(i e^{i (e+f x)}\right)-i f (c+d x) \text{Li}_2\left(i e^{i (e+f x)}\right)\right)}{f^2}+\frac{12 d^2 (c+d x) \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right)}{f}+12 d (c+d x)^2 \log \left(1-i e^{i (e+f x)}\right)+2 f (c+d x)^3 \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right)-3 d (c+d x)^2 \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)+f (c+d x)^3 \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right)-2 i f (c+d x)^3+\frac{24 d^3 \log \left(\cos \left(\frac{1}{4} (2 e+2 f x-\pi )\right)\right)}{f^2}}{6 a^2 f^2}","-\frac{4 i d^2 (c+d x) \text{Li}_2\left(i e^{i (e+f x)}\right)}{a^2 f^3}-\frac{2 d^2 (c+d x) \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a^2 f^3}+\frac{2 d (c+d x)^2 \log \left(1-i e^{i (e+f x)}\right)}{a^2 f^2}-\frac{d (c+d x)^2 \csc ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{2 a^2 f^2}-\frac{(c+d x)^3 \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{3 a^2 f}-\frac{(c+d x)^3 \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \csc ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{6 a^2 f}-\frac{i (c+d x)^3}{3 a^2 f}+\frac{4 d^3 \text{Li}_3\left(i e^{i (e+f x)}\right)}{a^2 f^4}+\frac{4 d^3 \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)\right)}{a^2 f^4}",1,"((-2*I)*f*(c + d*x)^3 + 12*d*(c + d*x)^2*Log[1 - I*E^(I*(e + f*x))] + (24*d^3*Log[Cos[(2*e - Pi + 2*f*x)/4]])/f^2 + (24*d^2*((-I)*f*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))] + d*PolyLog[3, I*E^(I*(e + f*x))]))/f^2 - 3*d*(c + d*x)^2*Sec[(2*e - Pi + 2*f*x)/4]^2 + (12*d^2*(c + d*x)*Tan[(2*e - Pi + 2*f*x)/4])/f + 2*f*(c + d*x)^3*Tan[(2*e - Pi + 2*f*x)/4] + f*(c + d*x)^3*Sec[(2*e - Pi + 2*f*x)/4]^2*Tan[(2*e - Pi + 2*f*x)/4])/(6*a^2*f^2)","A",1
113,1,175,243,2.45151,"\int \frac{(c+d x)^2}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c + d*x)^2/(a + a*Sin[e + f*x])^2,x]","\frac{2 \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2+2\right)\right) \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right)-2 i f (c+d x) \left(f (c+d x)+4 i d \log \left(1-i e^{i (e+f x)}\right)\right)+f (c+d x) \sec ^2\left(\frac{1}{4} (2 e+2 f x-\pi )\right) \left(f (c+d x) \tan \left(\frac{1}{4} (2 e+2 f x-\pi )\right)-2 d\right)-8 i d^2 \text{Li}_2\left(i e^{i (e+f x)}\right)}{6 a^2 f^3}","\frac{4 d (c+d x) \log \left(1-i e^{i (e+f x)}\right)}{3 a^2 f^2}-\frac{d (c+d x) \csc ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{3 a^2 f^2}-\frac{(c+d x)^2 \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{3 a^2 f}-\frac{(c+d x)^2 \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \csc ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{6 a^2 f}-\frac{i (c+d x)^2}{3 a^2 f}-\frac{4 i d^2 \text{Li}_2\left(i e^{i (e+f x)}\right)}{3 a^2 f^3}-\frac{2 d^2 \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{3 a^2 f^3}",1,"((-2*I)*f*(c + d*x)*(f*(c + d*x) + (4*I)*d*Log[1 - I*E^(I*(e + f*x))]) - (8*I)*d^2*PolyLog[2, I*E^(I*(e + f*x))] + 2*(c^2*f^2 + 2*c*d*f^2*x + d^2*(2 + f^2*x^2))*Tan[(2*e - Pi + 2*f*x)/4] + f*(c + d*x)*Sec[(2*e - Pi + 2*f*x)/4]^2*(-2*d + f*(c + d*x)*Tan[(2*e - Pi + 2*f*x)/4]))/(6*a^2*f^3)","A",1
114,1,225,148,1.1859924,"\int \frac{c+d x}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c + d*x)/(a + a*Sin[e + f*x])^2,x]","-\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(\cos \left(\frac{3}{2} (e+f x)\right) \left(2 c f+2 d \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-d e+d f x\right)+2 \sin \left(\frac{1}{2} (e+f x)\right) \left(-3 c f+d \cos (e+f x) \left(-2 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+e+f x\right)-4 d \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+2 d e-d f x+d\right)+d \cos \left(\frac{1}{2} (e+f x)\right) \left(-6 \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+3 e+3 f x+2\right)\right)}{6 a^2 f^2 (\sin (e+f x)+1)^2}","-\frac{(c+d x) \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{3 a^2 f}-\frac{(c+d x) \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \csc ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{6 a^2 f}-\frac{d \csc ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{6 a^2 f^2}+\frac{2 d \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)\right)}{3 a^2 f^2}",1,"-1/6*((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(d*Cos[(e + f*x)/2]*(2 + 3*e + 3*f*x - 6*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + Cos[(3*(e + f*x))/2]*(-(d*e) + 2*c*f + d*f*x + 2*d*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + 2*(d + 2*d*e - 3*c*f - d*f*x + d*Cos[e + f*x]*(e + f*x - 2*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 4*d*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])*Sin[(e + f*x)/2]))/(a^2*f^2*(1 + Sin[e + f*x])^2)","A",1
115,0,0,23,15.1525003,"\int \frac{1}{(c+d x) (a+a \sin (e+f x))^2} \, dx","Integrate[1/((c + d*x)*(a + a*Sin[e + f*x])^2),x]","\int \frac{1}{(c+d x) (a+a \sin (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a \sin (e+f x)+a)^2},x\right)",0,"Integrate[1/((c + d*x)*(a + a*Sin[e + f*x])^2), x]","A",-1
116,0,0,23,16.1626023,"\int \frac{1}{(c+d x)^2 (a+a \sin (e+f x))^2} \, dx","Integrate[1/((c + d*x)^2*(a + a*Sin[e + f*x])^2),x]","\int \frac{1}{(c+d x)^2 (a+a \sin (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a \sin (e+f x)+a)^2},x\right)",0,"Integrate[1/((c + d*x)^2*(a + a*Sin[e + f*x])^2), x]","A",-1
117,1,124,147,1.2612466,"\int \frac{(c+d x)^3}{a-a \sin (e+f x)} \, dx","Integrate[(c + d*x)^3/(a - a*Sin[e + f*x]),x]","\frac{-12 i d^2 f (c+d x) \text{Li}_2\left(-i e^{i (e+f x)}\right)+f^2 (c+d x)^2 \left(f (c+d x) \tan \left(\frac{1}{4} (2 e+2 f x+\pi )\right)-i f (c+d x)+6 d \log \left(1+i e^{i (e+f x)}\right)\right)+12 d^3 \text{Li}_3\left(-i e^{i (e+f x)}\right)}{a f^4}","-\frac{12 i d^2 (c+d x) \text{Li}_2\left(-i e^{i (e+f x)}\right)}{a f^3}+\frac{6 d (c+d x)^2 \log \left(1+i e^{i (e+f x)}\right)}{a f^2}+\frac{(c+d x)^3 \tan \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f}-\frac{i (c+d x)^3}{a f}+\frac{12 d^3 \text{Li}_3\left(-i e^{i (e+f x)}\right)}{a f^4}",1,"((-12*I)*d^2*f*(c + d*x)*PolyLog[2, (-I)*E^(I*(e + f*x))] + 12*d^3*PolyLog[3, (-I)*E^(I*(e + f*x))] + f^2*(c + d*x)^2*((-I)*f*(c + d*x) + 6*d*Log[1 + I*E^(I*(e + f*x))] + f*(c + d*x)*Tan[(2*e + Pi + 2*f*x)/4]))/(a*f^4)","A",1
118,1,92,112,0.7992274,"\int \frac{(c+d x)^2}{a-a \sin (e+f x)} \, dx","Integrate[(c + d*x)^2/(a - a*Sin[e + f*x]),x]","\frac{f (c+d x) \left(f (c+d x) \tan \left(\frac{1}{4} (2 e+2 f x+\pi )\right)-i f (c+d x)+4 d \log \left(1+i e^{i (e+f x)}\right)\right)-4 i d^2 \text{Li}_2\left(-i e^{i (e+f x)}\right)}{a f^3}","\frac{4 d (c+d x) \log \left(1+i e^{i (e+f x)}\right)}{a f^2}+\frac{(c+d x)^2 \tan \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f}-\frac{i (c+d x)^2}{a f}-\frac{4 i d^2 \text{Li}_2\left(-i e^{i (e+f x)}\right)}{a f^3}",1,"((-4*I)*d^2*PolyLog[2, (-I)*E^(I*(e + f*x))] + f*(c + d*x)*((-I)*f*(c + d*x) + 4*d*Log[1 + I*E^(I*(e + f*x))] + f*(c + d*x)*Tan[(2*e + Pi + 2*f*x)/4]))/(a*f^3)","A",1
119,1,47,59,0.1685714,"\int \frac{c+d x}{a-a \sin (e+f x)} \, dx","Integrate[(c + d*x)/(a - a*Sin[e + f*x]),x]","\frac{f (c+d x) \tan \left(\frac{1}{4} (2 e+2 f x+\pi )\right)+2 d \log \left(\cos \left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)}{a f^2}","\frac{(c+d x) \tan \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f}+\frac{2 d \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)\right)}{a f^2}",1,"(2*d*Log[Cos[(2*e + Pi + 2*f*x)/4]] + f*(c + d*x)*Tan[(2*e + Pi + 2*f*x)/4])/(a*f^2)","A",1
120,0,0,24,5.3754549,"\int \frac{1}{(c+d x) (a-a \sin (e+f x))} \, dx","Integrate[1/((c + d*x)*(a - a*Sin[e + f*x])),x]","\int \frac{1}{(c+d x) (a-a \sin (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a-a \sin (e+f x))},x\right)",0,"Integrate[1/((c + d*x)*(a - a*Sin[e + f*x])), x]","A",-1
121,0,0,24,5.0784003,"\int \frac{1}{(c+d x)^2 (a-a \sin (e+f x))} \, dx","Integrate[1/((c + d*x)^2*(a - a*Sin[e + f*x])),x]","\int \frac{1}{(c+d x)^2 (a-a \sin (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a-a \sin (e+f x))},x\right)",0,"Integrate[1/((c + d*x)^2*(a - a*Sin[e + f*x])), x]","A",-1
122,1,108,120,0.3392077,"\int x^3 \sqrt{a+a \sin (c+d x)} \, dx","Integrate[x^3*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \sqrt{a (\sin (c+d x)+1)} \left(\left(-d^3 x^3-6 d^2 x^2+24 d x+48\right) \sin \left(\frac{1}{2} (c+d x)\right)+\left(d^3 x^3-6 d^2 x^2-24 d x+48\right) \cos \left(\frac{1}{2} (c+d x)\right)\right)}{d^4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{96 \sqrt{a \sin (c+d x)+a}}{d^4}+\frac{48 x \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}}{d^3}+\frac{12 x^2 \sqrt{a \sin (c+d x)+a}}{d^2}-\frac{2 x^3 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}}{d}",1,"(-2*((48 - 24*d*x - 6*d^2*x^2 + d^3*x^3)*Cos[(c + d*x)/2] + (48 + 24*d*x - 6*d^2*x^2 - d^3*x^3)*Sin[(c + d*x)/2])*Sqrt[a*(1 + Sin[c + d*x])])/(d^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
123,1,92,98,0.2313671,"\int x^2 \sqrt{a+a \sin (c+d x)} \, dx","Integrate[x^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \sqrt{a (\sin (c+d x)+1)} \left(\left(d^2 x^2-4 d x-8\right) \cos \left(\frac{1}{2} (c+d x)\right)-\left(d^2 x^2+4 d x-8\right) \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{16 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}}{d^3}+\frac{8 x \sqrt{a \sin (c+d x)+a}}{d^2}-\frac{2 x^2 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}}{d}",1,"(-2*((-8 - 4*d*x + d^2*x^2)*Cos[(c + d*x)/2] - (-8 + 4*d*x + d^2*x^2)*Sin[(c + d*x)/2])*Sqrt[a*(1 + Sin[c + d*x])])/(d^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
124,1,76,58,0.1979841,"\int x \sqrt{a+a \sin (c+d x)} \, dx","Integrate[x*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \sqrt{a (\sin (c+d x)+1)} \left((d x-2) \cos \left(\frac{1}{2} (c+d x)\right)-(d x+2) \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{4 \sqrt{a \sin (c+d x)+a}}{d^2}-\frac{2 x \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}}{d}",1,"(-2*((-2 + d*x)*Cos[(c + d*x)/2] - (2 + d*x)*Sin[(c + d*x)/2])*Sqrt[a*(1 + Sin[c + d*x])])/(d^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
125,1,83,101,0.1795604,"\int \frac{\sqrt{a+a \sin (c+d x)}}{x} \, dx","Integrate[Sqrt[a + a*Sin[c + d*x]]/x,x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \text{Ci}\left(\frac{d x}{2}\right)+\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \text{Si}\left(\frac{d x}{2}\right)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}","\sin \left(\frac{1}{4} (2 c+\pi )\right) \text{Ci}\left(\frac{d x}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}+\cos \left(\frac{1}{4} (2 c+\pi )\right) \text{Si}\left(\frac{d x}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}",1,"(Sqrt[a*(1 + Sin[c + d*x])]*(CosIntegral[(d*x)/2]*(Cos[c/2] + Sin[c/2]) + (Cos[c/2] - Sin[c/2])*SinIntegral[(d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])","A",1
126,1,117,130,0.3509132,"\int \frac{\sqrt{a+a \sin (c+d x)}}{x^2} \, dx","Integrate[Sqrt[a + a*Sin[c + d*x]]/x^2,x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(d x \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \text{Ci}\left(\frac{d x}{2}\right)-d x \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \text{Si}\left(\frac{d x}{2}\right)-2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 x \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{1}{2} d \sin \left(\frac{1}{4} (2 c-\pi )\right) \text{Ci}\left(\frac{d x}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}-\frac{1}{2} d \sin \left(\frac{1}{4} (2 c+\pi )\right) \text{Si}\left(\frac{d x}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}-\frac{\sqrt{a \sin (c+d x)+a}}{x}",1,"(Sqrt[a*(1 + Sin[c + d*x])]*(d*x*CosIntegral[(d*x)/2]*(Cos[c/2] - Sin[c/2]) - 2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - d*x*(Cos[c/2] + Sin[c/2])*SinIntegral[(d*x)/2]))/(2*x*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
127,1,153,174,0.3619931,"\int \frac{\sqrt{a+a \sin (c+d x)}}{x^3} \, dx","Integrate[Sqrt[a + a*Sin[c + d*x]]/x^3,x]","-\frac{\sqrt{a (\sin (c+d x)+1)} \left(d^2 x^2 \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \text{Ci}\left(\frac{d x}{2}\right)+d^2 x^2 \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \text{Si}\left(\frac{d x}{2}\right)-2 d x \sin \left(\frac{1}{2} (c+d x)\right)+4 \sin \left(\frac{1}{2} (c+d x)\right)+2 d x \cos \left(\frac{1}{2} (c+d x)\right)+4 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 x^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{1}{8} d^2 \sin \left(\frac{1}{4} (2 c+\pi )\right) \text{Ci}\left(\frac{d x}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}-\frac{1}{8} d^2 \cos \left(\frac{1}{4} (2 c+\pi )\right) \text{Si}\left(\frac{d x}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}-\frac{\sqrt{a \sin (c+d x)+a}}{2 x^2}-\frac{d \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (c+d x)+a}}{4 x}",1,"-1/8*(Sqrt[a*(1 + Sin[c + d*x])]*(4*Cos[(c + d*x)/2] + 2*d*x*Cos[(c + d*x)/2] + d^2*x^2*CosIntegral[(d*x)/2]*(Cos[c/2] + Sin[c/2]) + 4*Sin[(c + d*x)/2] - 2*d*x*Sin[(c + d*x)/2] + d^2*x^2*(Cos[c/2] - Sin[c/2])*SinIntegral[(d*x)/2]))/(x^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
128,1,231,337,1.3331144,"\int x^3 (a+a \sin (e+f x))^{3/2} \, dx","Integrate[x^3*(a + a*Sin[e + f*x])^(3/2),x]","\frac{2 a \sqrt{a (\sin (e+f x)+1)} \left(-\cos (f x) \left(2 \sin (e) \left(8-9 f^2 x^2\right)+3 f x \cos (e) \left(3 f^2 x^2-8\right)\right)+\sin (f x) \left(3 f x \sin (e) \left(3 f^2 x^2-8\right)+2 \cos (e) \left(9 f^2 x^2-8\right)\right)+\frac{24 f x \left(3 f^2 x^2-80\right) \sin \left(\frac{f x}{2}\right)}{\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)}-\frac{2 \left(\sin \left(\frac{e}{2}\right) \left(-18 f^3 x^3-117 f^2 x^2+480 f x+968\right)+\cos \left(\frac{e}{2}\right) \left(18 f^3 x^3-117 f^2 x^2-480 f x+968\right)\right)}{\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)}\right)}{27 f^4}","-\frac{64 a \sin ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{27 f^4}-\frac{1280 a \sqrt{a \sin (e+f x)+a}}{9 f^4}+\frac{32 a x \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \cos \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{9 f^3}+\frac{640 a x \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{9 f^3}+\frac{8 a x^2 \sin ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{3 f^2}+\frac{16 a x^2 \sqrt{a \sin (e+f x)+a}}{f^2}-\frac{4 a x^3 \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \cos \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{8 a x^3 \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{3 f}",1,"(2*a*((-2*((968 - 480*f*x - 117*f^2*x^2 + 18*f^3*x^3)*Cos[e/2] + (968 + 480*f*x - 117*f^2*x^2 - 18*f^3*x^3)*Sin[e/2]))/(Cos[e/2] + Sin[e/2]) - Cos[f*x]*(3*f*x*(-8 + 3*f^2*x^2)*Cos[e] + 2*(8 - 9*f^2*x^2)*Sin[e]) + (2*(-8 + 9*f^2*x^2)*Cos[e] + 3*f*x*(-8 + 3*f^2*x^2)*Sin[e])*Sin[f*x] + (24*f*x*(-80 + 3*f^2*x^2)*Sin[(f*x)/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))*Sqrt[a*(1 + Sin[e + f*x])])/(27*f^4)","A",1
129,1,191,271,0.9813207,"\int x^2 (a+a \sin (e+f x))^{3/2} \, dx","Integrate[x^2*(a + a*Sin[e + f*x])^(3/2),x]","\frac{2 a \sqrt{a (\sin (e+f x)+1)} \left(-\frac{4 \left(\sin \left(\frac{e}{2}\right) \left(-9 f^2 x^2-39 f x+80\right)+\cos \left(\frac{e}{2}\right) \left(9 f^2 x^2-39 f x-80\right)\right)}{\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)}-\cos (f x) \left(\cos (e) \left(9 f^2 x^2-8\right)-12 f x \sin (e)\right)+\sin (f x) \left(\sin (e) \left(9 f^2 x^2-8\right)+12 f x \cos (e)\right)+\frac{8 \left(9 f^2 x^2-80\right) \sin \left(\frac{f x}{2}\right)}{\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)}\right)}{27 f^3}","\frac{224 a \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{9 f^3}-\frac{32 a \cos ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{27 f^3}+\frac{16 a x \sin ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{9 f^2}+\frac{32 a x \sqrt{a \sin (e+f x)+a}}{3 f^2}-\frac{4 a x^2 \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \cos \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{8 a x^2 \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{3 f}",1,"(2*a*((-4*((-80 - 39*f*x + 9*f^2*x^2)*Cos[e/2] + (80 - 39*f*x - 9*f^2*x^2)*Sin[e/2]))/(Cos[e/2] + Sin[e/2]) - Cos[f*x]*((-8 + 9*f^2*x^2)*Cos[e] - 12*f*x*Sin[e]) + (12*f*x*Cos[e] + (-8 + 9*f^2*x^2)*Sin[e])*Sin[f*x] + (8*(-80 + 9*f^2*x^2)*Sin[(f*x)/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))*Sqrt[a*(1 + Sin[e + f*x])])/(27*f^3)","A",1
130,1,113,165,0.7075946,"\int x (a+a \sin (e+f x))^{3/2} \, dx","Integrate[x*(a + a*Sin[e + f*x])^(3/2),x]","-\frac{(a (\sin (e+f x)+1))^{3/2} \left(27 (f x-2) \cos \left(\frac{1}{2} (e+f x)\right)+(3 f x+2) \cos \left(\frac{3}{2} (e+f x)\right)+2 \sin \left(\frac{1}{2} (e+f x)\right) ((3 f x-2) \cos (e+f x)-4 (3 f x+7))\right)}{9 f^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","\frac{8 a \sin ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{9 f^2}+\frac{16 a \sqrt{a \sin (e+f x)+a}}{3 f^2}-\frac{4 a x \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \cos \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{8 a x \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{3 f}",1,"-1/9*((27*(-2 + f*x)*Cos[(e + f*x)/2] + (2 + 3*f*x)*Cos[(3*(e + f*x))/2] + 2*(-4*(7 + 3*f*x) + (-2 + 3*f*x)*Cos[e + f*x])*Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(3/2))/(f^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
131,1,127,221,0.7305323,"\int \frac{(a+a \sin (e+f x))^{3/2}}{x} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/x,x]","\frac{(a (\sin (e+f x)+1))^{3/2} \left(3 \text{Ci}\left(\frac{f x}{2}\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right)+\text{Ci}\left(\frac{3 f x}{2}\right) \left(\sin \left(\frac{3 e}{2}\right)-\cos \left(\frac{3 e}{2}\right)\right)+\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left((2 \sin (e)+1) \text{Si}\left(\frac{3 f x}{2}\right)+3 \text{Si}\left(\frac{f x}{2}\right)\right)\right)}{2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","\frac{3}{2} a \sin \left(\frac{1}{4} (2 e+\pi )\right) \text{Ci}\left(\frac{f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}+\frac{1}{2} a \cos \left(\frac{3}{4} (2 e-\pi )\right) \text{Ci}\left(\frac{3 f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}-\frac{1}{2} a \sin \left(\frac{3}{4} (2 e-\pi )\right) \text{Si}\left(\frac{3 f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}+\frac{3}{2} a \cos \left(\frac{1}{4} (2 e+\pi )\right) \text{Si}\left(\frac{f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}",1,"((a*(1 + Sin[e + f*x]))^(3/2)*(3*CosIntegral[(f*x)/2]*(Cos[e/2] + Sin[e/2]) + CosIntegral[(3*f*x)/2]*(-Cos[(3*e)/2] + Sin[(3*e)/2]) + (Cos[e/2] - Sin[e/2])*(3*SinIntegral[(f*x)/2] + (1 + 2*Sin[e])*SinIntegral[(3*f*x)/2])))/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
132,1,226,263,0.9888256,"\int \frac{(a+a \sin (e+f x))^{3/2}}{x^2} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/x^2,x]","\frac{i \left(-i a e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2\right)^{3/2} \left(3 f x e^{i e+\frac{3 i f x}{2}} \text{Ei}\left(-\frac{1}{2} i f x\right)+3 i f x e^{2 i e+\frac{3 i f x}{2}} \text{Ei}\left(\frac{i f x}{2}\right)+3 f x e^{\frac{3}{2} i (2 e+f x)} \text{Ei}\left(\frac{3 i f x}{2}\right)-6 i e^{i (e+f x)}-6 e^{2 i (e+f x)}+2 i e^{3 i (e+f x)}+3 i f x e^{\frac{3 i f x}{2}} \text{Ei}\left(-\frac{3}{2} i f x\right)+2\right)}{4 \sqrt{2} x \left(e^{i (e+f x)}+i\right)^3}","-\frac{3}{4} a f \sin \left(\frac{1}{4} (2 e-\pi )\right) \text{Ci}\left(\frac{f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}+\frac{3}{4} a f \sin \left(\frac{1}{4} (6 e+\pi )\right) \text{Ci}\left(\frac{3 f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}-\frac{3}{4} a f \sin \left(\frac{1}{4} (2 e+\pi )\right) \text{Si}\left(\frac{f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}+\frac{3}{4} a f \cos \left(\frac{1}{4} (6 e+\pi )\right) \text{Si}\left(\frac{3 f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}-\frac{2 a \sin ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{x}",1,"((I/4)*(((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x)))^(3/2)*(2 - (6*I)*E^(I*(e + f*x)) - 6*E^((2*I)*(e + f*x)) + (2*I)*E^((3*I)*(e + f*x)) + 3*E^(I*e + ((3*I)/2)*f*x)*f*x*ExpIntegralEi[(-1/2*I)*f*x] + (3*I)*E^((2*I)*e + ((3*I)/2)*f*x)*f*x*ExpIntegralEi[(I/2)*f*x] + (3*I)*E^(((3*I)/2)*f*x)*f*x*ExpIntegralEi[((-3*I)/2)*f*x] + 3*E^(((3*I)/2)*(2*e + f*x))*f*x*ExpIntegralEi[((3*I)/2)*f*x]))/(Sqrt[2]*(I + E^(I*(e + f*x)))^3*x)","C",1
133,1,295,332,0.9113382,"\int \frac{(a+a \sin (e+f x))^{3/2}}{x^3} \, dx","Integrate[(a + a*Sin[e + f*x])^(3/2)/x^3,x]","-\frac{i \left(-i a e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2\right)^{3/2} \left(3 i f^2 x^2 e^{i e+\frac{3 i f x}{2}} \text{Ei}\left(-\frac{1}{2} i f x\right)+3 f^2 x^2 e^{2 i e+\frac{3 i f x}{2}} \text{Ei}\left(\frac{i f x}{2}\right)-9 i f^2 x^2 e^{\frac{3}{2} i (2 e+f x)} \text{Ei}\left(\frac{3 i f x}{2}\right)+6 f x e^{i (e+f x)}+6 i f x e^{2 i (e+f x)}+6 f x e^{3 i (e+f x)}+12 i e^{i (e+f x)}+12 e^{2 i (e+f x)}-4 i e^{3 i (e+f x)}-9 f^2 x^2 e^{\frac{3 i f x}{2}} \text{Ei}\left(-\frac{3}{2} i f x\right)+6 i f x-4\right)}{16 \sqrt{2} x^2 \left(e^{i (e+f x)}+i\right)^3}","-\frac{3}{16} a f^2 \sin \left(\frac{1}{4} (2 e+\pi )\right) \text{Ci}\left(\frac{f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}-\frac{9}{16} a f^2 \cos \left(\frac{3}{4} (2 e-\pi )\right) \text{Ci}\left(\frac{3 f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}+\frac{9}{16} a f^2 \sin \left(\frac{3}{4} (2 e-\pi )\right) \text{Si}\left(\frac{3 f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}-\frac{3}{16} a f^2 \cos \left(\frac{1}{4} (2 e+\pi )\right) \text{Si}\left(\frac{f x}{2}\right) \csc \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}-\frac{a \sin ^2\left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{x^2}-\frac{3 a f \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \cos \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \sqrt{a \sin (e+f x)+a}}{2 x}",1,"((-1/16*I)*(((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x)))^(3/2)*(-4 + (12*I)*E^(I*(e + f*x)) + 12*E^((2*I)*(e + f*x)) - (4*I)*E^((3*I)*(e + f*x)) + (6*I)*f*x + 6*E^(I*(e + f*x))*f*x + (6*I)*E^((2*I)*(e + f*x))*f*x + 6*E^((3*I)*(e + f*x))*f*x + (3*I)*E^(I*e + ((3*I)/2)*f*x)*f^2*x^2*ExpIntegralEi[(-1/2*I)*f*x] + 3*E^((2*I)*e + ((3*I)/2)*f*x)*f^2*x^2*ExpIntegralEi[(I/2)*f*x] - 9*E^(((3*I)/2)*f*x)*f^2*x^2*ExpIntegralEi[((-3*I)/2)*f*x] - (9*I)*E^(((3*I)/2)*(2*e + f*x))*f^2*x^2*ExpIntegralEi[((3*I)/2)*f*x]))/(Sqrt[2]*(I + E^(I*(e + f*x)))^3*x^2)","C",1
134,1,306,417,0.8801224,"\int \frac{x^3}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[x^3/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt[4]{-1} \sqrt{2} e^{-\frac{1}{2} i (c+d x)} \left(e^{i (c+d x)}+i\right) \left(-i d^3 x^3 \log \left(1-\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)+i d^3 x^3 \log \left(1+\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)+6 d^2 x^2 \text{Li}_2\left(-\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)-6 d^2 x^2 \text{Li}_2\left(\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)+24 i d x \text{Li}_3\left(-\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)-24 i d x \text{Li}_3\left(\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)-48 \text{Li}_4\left(-\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)+48 \text{Li}_4\left(\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)\right)}{d^4 \sqrt{-i a e^{-i (c+d x)} \left(e^{i (c+d x)}+i\right)^2}}","-\frac{96 i \text{Li}_4\left(-e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^4 \sqrt{a \sin (c+d x)+a}}+\frac{96 i \text{Li}_4\left(e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^4 \sqrt{a \sin (c+d x)+a}}-\frac{48 x \text{Li}_3\left(-e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^3 \sqrt{a \sin (c+d x)+a}}+\frac{48 x \text{Li}_3\left(e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^3 \sqrt{a \sin (c+d x)+a}}+\frac{12 i x^2 \text{Li}_2\left(-e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^2 \sqrt{a \sin (c+d x)+a}}-\frac{12 i x^2 \text{Li}_2\left(e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^2 \sqrt{a \sin (c+d x)+a}}-\frac{4 x^3 \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \tanh ^{-1}\left(e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right)}{d \sqrt{a \sin (c+d x)+a}}",1,"((-1)^(1/4)*Sqrt[2]*(I + E^(I*(c + d*x)))*((-I)*d^3*x^3*Log[1 - (-1)^(1/4)*E^((I/2)*(c + d*x))] + I*d^3*x^3*Log[1 + (-1)^(1/4)*E^((I/2)*(c + d*x))] + 6*d^2*x^2*PolyLog[2, -((-1)^(1/4)*E^((I/2)*(c + d*x)))] - 6*d^2*x^2*PolyLog[2, (-1)^(1/4)*E^((I/2)*(c + d*x))] + (24*I)*d*x*PolyLog[3, -((-1)^(1/4)*E^((I/2)*(c + d*x)))] - (24*I)*d*x*PolyLog[3, (-1)^(1/4)*E^((I/2)*(c + d*x))] - 48*PolyLog[4, -((-1)^(1/4)*E^((I/2)*(c + d*x)))] + 48*PolyLog[4, (-1)^(1/4)*E^((I/2)*(c + d*x))]))/(d^4*E^((I/2)*(c + d*x))*Sqrt[((-I)*a*(I + E^(I*(c + d*x)))^2)/E^(I*(c + d*x))])","A",1
135,1,245,293,0.6205636,"\int \frac{x^2}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[x^2/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt[4]{-1} \sqrt{2} e^{-\frac{1}{2} i (c+d x)} \left(e^{i (c+d x)}+i\right) \left(4 d x \text{Li}_2\left(-\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)-i \left(d^2 x^2 \log \left(1-\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)-d^2 x^2 \log \left(1+\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)-4 i d x \text{Li}_2\left(\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)-8 \text{Li}_3\left(-\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)+8 \text{Li}_3\left(\sqrt[4]{-1} e^{\frac{1}{2} i (c+d x)}\right)\right)\right)}{d^3 \sqrt{-i a e^{-i (c+d x)} \left(e^{i (c+d x)}+i\right)^2}}","-\frac{16 \text{Li}_3\left(-e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^3 \sqrt{a \sin (c+d x)+a}}+\frac{16 \text{Li}_3\left(e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^3 \sqrt{a \sin (c+d x)+a}}+\frac{8 i x \text{Li}_2\left(-e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^2 \sqrt{a \sin (c+d x)+a}}-\frac{8 i x \text{Li}_2\left(e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^2 \sqrt{a \sin (c+d x)+a}}-\frac{4 x^2 \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \tanh ^{-1}\left(e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right)}{d \sqrt{a \sin (c+d x)+a}}",1,"((-1)^(1/4)*Sqrt[2]*(I + E^(I*(c + d*x)))*(4*d*x*PolyLog[2, -((-1)^(1/4)*E^((I/2)*(c + d*x)))] - I*(d^2*x^2*Log[1 - (-1)^(1/4)*E^((I/2)*(c + d*x))] - d^2*x^2*Log[1 + (-1)^(1/4)*E^((I/2)*(c + d*x))] - (4*I)*d*x*PolyLog[2, (-1)^(1/4)*E^((I/2)*(c + d*x))] - 8*PolyLog[3, -((-1)^(1/4)*E^((I/2)*(c + d*x)))] + 8*PolyLog[3, (-1)^(1/4)*E^((I/2)*(c + d*x))])))/(d^3*E^((I/2)*(c + d*x))*Sqrt[((-I)*a*(I + E^(I*(c + d*x)))^2)/E^(I*(c + d*x))])","A",1
136,1,231,175,1.62936,"\int \frac{x}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[x/Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 \left(\frac{c \sin \left(\frac{1}{4} (2 c+2 d x-\pi )\right) \sin ^{-1}\left(\csc \left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)}{\sqrt{\frac{\sin (c+d x)-1}{\sin (c+d x)+1}}}+\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \left(2 i \left(\text{Li}_2\left(-e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right)-\text{Li}_2\left(e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right)\right)+\frac{1}{2} (2 c+2 d x+\pi ) \left(\log \left(1-e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right)-\log \left(1+e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right)\right)-\pi  \tanh ^{-1}\left(\frac{\tan \left(\frac{1}{4} (c+d x)\right)-1}{\sqrt{2}}\right)\right)}{\sqrt{2}}\right)}{d^2 \sqrt{a (\sin (c+d x)+1)}}","\frac{4 i \text{Li}_2\left(-e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^2 \sqrt{a \sin (c+d x)+a}}-\frac{4 i \text{Li}_2\left(e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{d^2 \sqrt{a \sin (c+d x)+a}}-\frac{4 x \sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right) \tanh ^{-1}\left(e^{\frac{1}{4} i (2 c+2 d x+\pi )}\right)}{d \sqrt{a \sin (c+d x)+a}}",1,"(2*(((-(Pi*ArcTanh[(-1 + Tan[(c + d*x)/4])/Sqrt[2]]) + ((2*c + Pi + 2*d*x)*(Log[1 - E^((I/4)*(2*c + Pi + 2*d*x))] - Log[1 + E^((I/4)*(2*c + Pi + 2*d*x))]))/2 + (2*I)*(PolyLog[2, -E^((I/4)*(2*c + Pi + 2*d*x))] - PolyLog[2, E^((I/4)*(2*c + Pi + 2*d*x))]))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/Sqrt[2] + (c*ArcSin[Csc[(2*c + Pi + 2*d*x)/4]]*Sin[(2*c - Pi + 2*d*x)/4])/Sqrt[(-1 + Sin[c + d*x])/(1 + Sin[c + d*x])]))/(d^2*Sqrt[a*(1 + Sin[c + d*x])])","A",1
137,0,0,21,3.2502814,"\int \frac{1}{x \sqrt{a+a \sin (c+d x)}} \, dx","Integrate[1/(x*Sqrt[a + a*Sin[c + d*x]]),x]","\int \frac{1}{x \sqrt{a+a \sin (c+d x)}} \, dx","\text{Int}\left(\frac{1}{x \sqrt{a \sin (c+d x)+a}},x\right)",0,"Integrate[1/(x*Sqrt[a + a*Sin[c + d*x]]), x]","A",-1
138,0,0,21,0.8018921,"\int \frac{1}{x^2 \sqrt{a+a \sin (c+d x)}} \, dx","Integrate[1/(x^2*Sqrt[a + a*Sin[c + d*x]]),x]","\int \frac{1}{x^2 \sqrt{a+a \sin (c+d x)}} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{a \sin (c+d x)+a}},x\right)",0,"Integrate[1/(x^2*Sqrt[a + a*Sin[c + d*x]]), x]","A",-1
139,1,455,691,2.9342027,"\int \frac{x^3}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[x^3/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{x^2 \sqrt{a (\sin (e+f x)+1)} \left((6-f x) \sin \left(\frac{1}{2} (e+f x)\right)+(f x+6) \cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 a^2 f^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}-\frac{(-1)^{3/4} e^{-\frac{3}{2} i (e+f x)} \left(e^{i (e+f x)}+i\right)^3 \left(-i \left(f^3 x^3 \log \left(1-\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)-f^3 x^3 \log \left(1+\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)-24 f x \text{Li}_3\left(-\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)+24 f x \text{Li}_3\left(\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)-48 i \text{Li}_4\left(-\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)+48 i \text{Li}_4\left(\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)+24 f x \log \left(1-\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)-24 f x \log \left(1+\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)\right)+6 \left(f^2 x^2+8\right) \text{Li}_2\left(-\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)-6 \left(f^2 x^2+8\right) \text{Li}_2\left(\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)\right)}{2 \sqrt{2} f^4 \left(-i a e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2\right)^{3/2}}","\frac{24 i \text{Li}_2\left(-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^4 \sqrt{a \sin (e+f x)+a}}-\frac{24 i \text{Li}_2\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^4 \sqrt{a \sin (e+f x)+a}}-\frac{24 i \text{Li}_4\left(-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^4 \sqrt{a \sin (e+f x)+a}}+\frac{24 i \text{Li}_4\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^4 \sqrt{a \sin (e+f x)+a}}-\frac{12 x \text{Li}_3\left(-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^3 \sqrt{a \sin (e+f x)+a}}+\frac{12 x \text{Li}_3\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^3 \sqrt{a \sin (e+f x)+a}}-\frac{24 x \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \tanh ^{-1}\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right)}{a f^3 \sqrt{a \sin (e+f x)+a}}+\frac{3 i x^2 \text{Li}_2\left(-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{3 i x^2 \text{Li}_2\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{3 x^2}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{x^3 \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \tanh ^{-1}\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right)}{a f \sqrt{a \sin (e+f x)+a}}-\frac{x^3 \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"-1/2*((-1)^(3/4)*(I + E^(I*(e + f*x)))^3*(6*(8 + f^2*x^2)*PolyLog[2, -((-1)^(1/4)*E^((I/2)*(e + f*x)))] - 6*(8 + f^2*x^2)*PolyLog[2, (-1)^(1/4)*E^((I/2)*(e + f*x))] - I*(24*f*x*Log[1 - (-1)^(1/4)*E^((I/2)*(e + f*x))] + f^3*x^3*Log[1 - (-1)^(1/4)*E^((I/2)*(e + f*x))] - 24*f*x*Log[1 + (-1)^(1/4)*E^((I/2)*(e + f*x))] - f^3*x^3*Log[1 + (-1)^(1/4)*E^((I/2)*(e + f*x))] - 24*f*x*PolyLog[3, -((-1)^(1/4)*E^((I/2)*(e + f*x)))] + 24*f*x*PolyLog[3, (-1)^(1/4)*E^((I/2)*(e + f*x))] - (48*I)*PolyLog[4, -((-1)^(1/4)*E^((I/2)*(e + f*x)))] + (48*I)*PolyLog[4, (-1)^(1/4)*E^((I/2)*(e + f*x))])))/(Sqrt[2]*E^(((3*I)/2)*(e + f*x))*(((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x)))^(3/2)*f^4) - (x^2*((6 + f*x)*Cos[(e + f*x)/2] + (6 - f*x)*Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])])/(2*a^2*f^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
140,1,352,435,2.1244218,"\int \frac{x^2}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[x^2/(a + a*Sin[e + f*x])^(3/2),x]","-\frac{x \sqrt{a (\sin (e+f x)+1)} \left((4-f x) \sin \left(\frac{1}{2} (e+f x)\right)+(f x+4) \cos \left(\frac{1}{2} (e+f x)\right)\right)}{2 a^2 f^2 \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{\sqrt[4]{-1} e^{-\frac{3}{2} i (e+f x)} \left(e^{i (e+f x)}+i\right)^3 \left(-f^2 x^2 \log \left(1-\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)+f^2 x^2 \log \left(1+\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)-4 i f x \text{Li}_2\left(-\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)+4 i f x \text{Li}_2\left(\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)+8 \text{Li}_3\left(-\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)-8 \text{Li}_3\left(\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)+16 \tanh ^{-1}\left(\sqrt[4]{-1} e^{\frac{1}{2} i (e+f x)}\right)\right)}{2 \sqrt{2} f^3 \left(-i a e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2\right)^{3/2}}","-\frac{4 \text{Li}_3\left(-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^3 \sqrt{a \sin (e+f x)+a}}+\frac{4 \text{Li}_3\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^3 \sqrt{a \sin (e+f x)+a}}-\frac{4 \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \tanh ^{-1}\left(\cos \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)\right)}{a f^3 \sqrt{a \sin (e+f x)+a}}+\frac{2 i x \text{Li}_2\left(-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{2 i x \text{Li}_2\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{2 x}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{x^2 \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \tanh ^{-1}\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right)}{a f \sqrt{a \sin (e+f x)+a}}-\frac{x^2 \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"((-1)^(1/4)*(I + E^(I*(e + f*x)))^3*(16*ArcTanh[(-1)^(1/4)*E^((I/2)*(e + f*x))] - f^2*x^2*Log[1 - (-1)^(1/4)*E^((I/2)*(e + f*x))] + f^2*x^2*Log[1 + (-1)^(1/4)*E^((I/2)*(e + f*x))] - (4*I)*f*x*PolyLog[2, -((-1)^(1/4)*E^((I/2)*(e + f*x)))] + (4*I)*f*x*PolyLog[2, (-1)^(1/4)*E^((I/2)*(e + f*x))] + 8*PolyLog[3, -((-1)^(1/4)*E^((I/2)*(e + f*x)))] - 8*PolyLog[3, (-1)^(1/4)*E^((I/2)*(e + f*x))]))/(2*Sqrt[2]*E^(((3*I)/2)*(e + f*x))*(((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x)))^(3/2)*f^3) - (x*((4 + f*x)*Cos[(e + f*x)/2] + (4 - f*x)*Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])])/(2*a^2*f^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)","A",1
141,1,308,249,2.7197972,"\int \frac{x}{(a+a \sin (e+f x))^{3/2}} \, dx","Integrate[x/(a + a*Sin[e + f*x])^(3/2),x]","\frac{\frac{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3 \left(2 i \left(\text{Li}_2\left(-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right)-\text{Li}_2\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right)\right)+\frac{1}{2} (2 e+2 f x+\pi ) \left(\log \left(1-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right)-\log \left(1+e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right)\right)-\pi  \tanh ^{-1}\left(\frac{\tan \left(\frac{1}{4} (e+f x)\right)-1}{\sqrt{2}}\right)\right)}{\sqrt{2}}-(f x+2) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2+2 f x \sin \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)+\frac{e (\sin (e+f x)+1) \sin \left(\frac{1}{4} (2 e+2 f x-\pi )\right) \sin ^{-1}\left(\csc \left(\frac{1}{4} (2 e+2 f x+\pi )\right)\right)}{\sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}}}}{2 f^2 (a (\sin (e+f x)+1))^{3/2}}","\frac{i \text{Li}_2\left(-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{i \text{Li}_2\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right) \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{1}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{x \sin \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right) \tanh ^{-1}\left(e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right)}{a f \sqrt{a \sin (e+f x)+a}}-\frac{x \cot \left(\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right)}{2 a f \sqrt{a \sin (e+f x)+a}}",1,"(2*f*x*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - (2 + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + ((-(Pi*ArcTanh[(-1 + Tan[(e + f*x)/4])/Sqrt[2]]) + ((2*e + Pi + 2*f*x)*(Log[1 - E^((I/4)*(2*e + Pi + 2*f*x))] - Log[1 + E^((I/4)*(2*e + Pi + 2*f*x))]))/2 + (2*I)*(PolyLog[2, -E^((I/4)*(2*e + Pi + 2*f*x))] - PolyLog[2, E^((I/4)*(2*e + Pi + 2*f*x))]))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)/Sqrt[2] + (e*ArcSin[Csc[(2*e + Pi + 2*f*x)/4]]*(1 + Sin[e + f*x])*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[(-1 + Sin[e + f*x])/(1 + Sin[e + f*x])])/(2*f^2*(a*(1 + Sin[e + f*x]))^(3/2))","A",1
142,0,0,21,35.5655552,"\int \frac{1}{x (a+a \sin (e+f x))^{3/2}} \, dx","Integrate[1/(x*(a + a*Sin[e + f*x])^(3/2)),x]","\int \frac{1}{x (a+a \sin (e+f x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{x (a \sin (e+f x)+a)^{3/2}},x\right)",0,"Integrate[1/(x*(a + a*Sin[e + f*x])^(3/2)), x]","A",-1
143,0,0,21,18.856407,"\int \frac{1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx","Integrate[1/(x^2*(a + a*Sin[e + f*x])^(3/2)),x]","\int \frac{1}{x^2 (a+a \sin (e+f x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{x^2 (a \sin (e+f x)+a)^{3/2}},x\right)",0,"Integrate[1/(x^2*(a + a*Sin[e + f*x])^(3/2)), x]","A",-1
144,0,0,21,3.2173945,"\int \frac{\sqrt[3]{a+a \sin (c+d x)}}{x} \, dx","Integrate[(a + a*Sin[c + d*x])^(1/3)/x,x]","\int \frac{\sqrt[3]{a+a \sin (c+d x)}}{x} \, dx","\text{Int}\left(\frac{\sqrt[3]{a \sin (c+d x)+a}}{x},x\right)",0,"Integrate[(a + a*Sin[c + d*x])^(1/3)/x, x]","A",-1
145,0,0,23,1.2798586,"\int (c+d x)^m (a+a \sin (e+f x))^n \, dx","Integrate[(c + d*x)^m*(a + a*Sin[e + f*x])^n,x]","\int (c+d x)^m (a+a \sin (e+f x))^n \, dx","\text{Int}\left((c+d x)^m (a \sin (e+f x)+a)^n,x\right)",0,"Integrate[(c + d*x)^m*(a + a*Sin[e + f*x])^n, x]","A",-1
146,1,376,449,0.9024199,"\int (c+d x)^m (a+a \sin (e+f x))^3 \, dx","Integrate[(c + d*x)^m*(a + a*Sin[e + f*x])^3,x]","\frac{1}{24} a^3 (c+d x)^m \left(-\frac{45 e^{i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)}{f}+\frac{9 i 2^{-m} e^{2 i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)}{f}+\frac{3^{-m} e^{3 i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{3 i f (c+d x)}{d}\right)}{f}-\frac{45 e^{-i \left(e-\frac{c f}{d}\right)} \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)}{f}-\frac{9 i 2^{-m} e^{-2 i \left(e-\frac{c f}{d}\right)} \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i f (c+d x)}{d}\right)}{f}+\frac{3^{-m} e^{-3 i \left(e-\frac{c f}{d}\right)} \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{3 i f (c+d x)}{d}\right)}{f}+\frac{60 (c+d x)}{d (m+1)}\right)","-\frac{15 a^3 e^{i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)}{8 f}+\frac{3 i a^3 2^{-m-3} e^{2 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)}{f}+\frac{a^3 3^{-m-1} e^{3 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{3 i f (c+d x)}{d}\right)}{8 f}-\frac{15 a^3 e^{-i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)}{8 f}-\frac{3 i a^3 2^{-m-3} e^{-2 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i f (c+d x)}{d}\right)}{f}+\frac{a^3 3^{-m-1} e^{-3 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{3 i f (c+d x)}{d}\right)}{8 f}+\frac{5 a^3 (c+d x)^{m+1}}{2 d (m+1)}",1,"(a^3*(c + d*x)^m*((60*(c + d*x))/(d*(1 + m)) - (45*E^(I*(e - (c*f)/d))*Gamma[1 + m, ((-I)*f*(c + d*x))/d])/(f*(((-I)*f*(c + d*x))/d)^m) - (45*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m) + ((9*I)*E^((2*I)*(e - (c*f)/d))*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d])/(2^m*f*(((-I)*f*(c + d*x))/d)^m) - ((9*I)*Gamma[1 + m, ((2*I)*f*(c + d*x))/d])/(2^m*E^((2*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m) + (E^((3*I)*(e - (c*f)/d))*Gamma[1 + m, ((-3*I)*f*(c + d*x))/d])/(3^m*f*(((-I)*f*(c + d*x))/d)^m) + Gamma[1 + m, ((3*I)*f*(c + d*x))/d]/(3^m*E^((3*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m)))/24","A",1
147,1,260,299,0.3050489,"\int (c+d x)^m (a+a \sin (e+f x))^2 \, dx","Integrate[(c + d*x)^m*(a + a*Sin[e + f*x])^2,x]","\frac{1}{8} a^2 (c+d x)^m \left(-\frac{8 e^{i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)}{f}+\frac{i 2^{-m} e^{2 i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)}{f}-\frac{8 e^{-i \left(e-\frac{c f}{d}\right)} \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)}{f}-\frac{i 2^{-m} e^{-2 i \left(e-\frac{c f}{d}\right)} \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i f (c+d x)}{d}\right)}{f}+\frac{12 (c+d x)}{d (m+1)}\right)","-\frac{a^2 e^{i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)}{f}+\frac{i a^2 2^{-m-3} e^{2 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)}{f}-\frac{a^2 e^{-i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)}{f}-\frac{i a^2 2^{-m-3} e^{-2 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i f (c+d x)}{d}\right)}{f}+\frac{3 a^2 (c+d x)^{m+1}}{2 d (m+1)}",1,"(a^2*(c + d*x)^m*((12*(c + d*x))/(d*(1 + m)) - (8*E^(I*(e - (c*f)/d))*Gamma[1 + m, ((-I)*f*(c + d*x))/d])/(f*(((-I)*f*(c + d*x))/d)^m) - (8*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m) + (I*E^((2*I)*(e - (c*f)/d))*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d])/(2^m*f*(((-I)*f*(c + d*x))/d)^m) - (I*Gamma[1 + m, ((2*I)*f*(c + d*x))/d])/(2^m*E^((2*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m)))/8","A",1
148,1,199,148,2.9322363,"\int (c+d x)^m (a+a \sin (e+f x)) \, dx","Integrate[(c + d*x)^m*(a + a*Sin[e + f*x]),x]","-\frac{a (c+d x)^m (\sin (e+f x)+1) \left(d (m+1) \left(-\frac{i f (c+d x)}{d}\right)^{-m} \left(\cos \left(e-\frac{c f}{d}\right)+i \sin \left(e-\frac{c f}{d}\right)\right) \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)+d (m+1) \left(\frac{i f (c+d x)}{d}\right)^{-m} \left(\cos \left(e-\frac{c f}{d}\right)-i \sin \left(e-\frac{c f}{d}\right)\right) \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)-2 c f-2 d (e+f x)+2 d e\right)}{2 d f (m+1) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}","-\frac{a e^{i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)}{2 f}-\frac{a e^{-i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)}{2 f}+\frac{a (c+d x)^{m+1}}{d (m+1)}",1,"-1/2*(a*(c + d*x)^m*(2*d*e - 2*c*f - 2*d*(e + f*x) + (d*(1 + m)*Gamma[1 + m, (I*f*(c + d*x))/d]*(Cos[e - (c*f)/d] - I*Sin[e - (c*f)/d]))/((I*f*(c + d*x))/d)^m + (d*(1 + m)*Gamma[1 + m, ((-I)*f*(c + d*x))/d]*(Cos[e - (c*f)/d] + I*Sin[e - (c*f)/d]))/(((-I)*f*(c + d*x))/d)^m)*(1 + Sin[e + f*x]))/(d*f*(1 + m)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2)","A",1
149,0,0,23,0.9161546,"\int \frac{(c+d x)^m}{a+a \sin (e+f x)} \, dx","Integrate[(c + d*x)^m/(a + a*Sin[e + f*x]),x]","\int \frac{(c+d x)^m}{a+a \sin (e+f x)} \, dx","\text{Int}\left(\frac{(c+d x)^m}{a \sin (e+f x)+a},x\right)",0,"Integrate[(c + d*x)^m/(a + a*Sin[e + f*x]), x]","A",-1
150,0,0,23,11.2221804,"\int \frac{(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx","Integrate[(c + d*x)^m/(a + a*Sin[e + f*x])^2,x]","\int \frac{(c+d x)^m}{(a+a \sin (e+f x))^2} \, dx","\text{Int}\left(\frac{(c+d x)^m}{(a \sin (e+f x)+a)^2},x\right)",0,"Integrate[(c + d*x)^m/(a + a*Sin[e + f*x])^2, x]","A",-1
151,1,124,90,0.5090296,"\int (c+d x)^3 (a+b \sin (e+f x)) \, dx","Integrate[(c + d*x)^3*(a + b*Sin[e + f*x]),x]","\frac{1}{4} a x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)+\frac{3 b d \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-2\right)\right) \sin (e+f x)}{f^4}-\frac{b (c+d x) \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-6\right)\right) \cos (e+f x)}{f^3}","\frac{a (c+d x)^4}{4 d}+\frac{6 b d^2 (c+d x) \cos (e+f x)}{f^3}+\frac{3 b d (c+d x)^2 \sin (e+f x)}{f^2}-\frac{b (c+d x)^3 \cos (e+f x)}{f}-\frac{6 b d^3 \sin (e+f x)}{f^4}",1,"(a*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3))/4 - (b*(c + d*x)*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-6 + f^2*x^2))*Cos[e + f*x])/f^3 + (3*b*d*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Sin[e + f*x])/f^4","A",1
152,1,84,68,0.3563303,"\int (c+d x)^2 (a+b \sin (e+f x)) \, dx","Integrate[(c + d*x)^2*(a + b*Sin[e + f*x]),x]","\frac{1}{3} a x \left(3 c^2+3 c d x+d^2 x^2\right)-\frac{b \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-2\right)\right) \cos (e+f x)}{f^3}+\frac{2 b d (c+d x) \sin (e+f x)}{f^2}","\frac{a (c+d x)^3}{3 d}+\frac{2 b d (c+d x) \sin (e+f x)}{f^2}-\frac{b (c+d x)^2 \cos (e+f x)}{f}+\frac{2 b d^2 \cos (e+f x)}{f^3}",1,"(a*x*(3*c^2 + 3*c*d*x + d^2*x^2))/3 - (b*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Cos[e + f*x])/f^3 + (2*b*d*(c + d*x)*Sin[e + f*x])/f^2","A",1
153,1,43,45,0.1277421,"\int (c+d x) (a+b \sin (e+f x)) \, dx","Integrate[(c + d*x)*(a + b*Sin[e + f*x]),x]","\frac{1}{2} a x (2 c+d x)-\frac{b (c+d x) \cos (e+f x)}{f}+\frac{b d \sin (e+f x)}{f^2}","\frac{a (c+d x)^2}{2 d}-\frac{b (c+d x) \cos (e+f x)}{f}+\frac{b d \sin (e+f x)}{f^2}",1,"(a*x*(2*c + d*x))/2 - (b*(c + d*x)*Cos[e + f*x])/f + (b*d*Sin[e + f*x])/f^2","A",1
154,1,57,64,0.1611099,"\int \frac{a+b \sin (e+f x)}{c+d x} \, dx","Integrate[(a + b*Sin[e + f*x])/(c + d*x),x]","\frac{a \log (c+d x)+b \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \sin \left(e-\frac{c f}{d}\right)+b \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)}{d}","\frac{a \log (c+d x)}{d}+\frac{b \text{Ci}\left(x f+\frac{c f}{d}\right) \sin \left(e-\frac{c f}{d}\right)}{d}+\frac{b \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{d}",1,"(a*Log[c + d*x] + b*CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + b*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)])/d","A",1
155,1,72,88,0.3788099,"\int \frac{a+b \sin (e+f x)}{(c+d x)^2} \, dx","Integrate[(a + b*Sin[e + f*x])/(c + d*x)^2,x]","\frac{-\frac{d (a+b \sin (e+f x))}{c+d x}+b f \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \cos \left(e-\frac{c f}{d}\right)-b f \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)}{d^2}","-\frac{a}{d (c+d x)}+\frac{b f \text{Ci}\left(x f+\frac{c f}{d}\right) \cos \left(e-\frac{c f}{d}\right)}{d^2}-\frac{b f \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{d^2}-\frac{b \sin (e+f x)}{d (c+d x)}",1,"(b*f*Cos[e - (c*f)/d]*CosIntegral[f*(c/d + x)] - (d*(a + b*Sin[e + f*x]))/(c + d*x) - b*f*Sin[e - (c*f)/d]*SinIntegral[f*(c/d + x)])/d^2","A",1
156,1,94,123,0.8816427,"\int \frac{a+b \sin (e+f x)}{(c+d x)^3} \, dx","Integrate[(a + b*Sin[e + f*x])/(c + d*x)^3,x]","-\frac{\frac{d (d (a+b \sin (e+f x))+b f (c+d x) \cos (e+f x))}{(c+d x)^2}+b f^2 \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \sin \left(e-\frac{c f}{d}\right)+b f^2 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)}{2 d^3}","-\frac{a}{2 d (c+d x)^2}-\frac{b f^2 \text{Ci}\left(x f+\frac{c f}{d}\right) \sin \left(e-\frac{c f}{d}\right)}{2 d^3}-\frac{b f^2 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{2 d^3}-\frac{b f \cos (e+f x)}{2 d^2 (c+d x)}-\frac{b \sin (e+f x)}{2 d (c+d x)^2}",1,"-1/2*(b*f^2*CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + (d*(b*f*(c + d*x)*Cos[e + f*x] + d*(a + b*Sin[e + f*x])))/(c + d*x)^2 + b*f^2*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)])/d^3","A",1
157,1,232,250,1.412619,"\int (c+d x)^3 (a+b \sin (e+f x))^2 \, dx","Integrate[(c + d*x)^3*(a + b*Sin[e + f*x])^2,x]","\frac{2 f^4 x \left(2 a^2+b^2\right) \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)+96 a b d \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-2\right)\right) \sin (e+f x)-32 a b f (c+d x) \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-6\right)\right) \cos (e+f x)-2 b^2 f (c+d x) \left(2 c^2 f^2+4 c d f^2 x+d^2 \left(2 f^2 x^2-3\right)\right) \sin (2 (e+f x))-3 b^2 d \left(2 c^2 f^2+4 c d f^2 x+d^2 \left(2 f^2 x^2-1\right)\right) \cos (2 (e+f x))}{16 f^4}","\frac{a^2 (c+d x)^4}{4 d}+\frac{12 a b d^2 (c+d x) \cos (e+f x)}{f^3}+\frac{6 a b d (c+d x)^2 \sin (e+f x)}{f^2}-\frac{2 a b (c+d x)^3 \cos (e+f x)}{f}-\frac{12 a b d^3 \sin (e+f x)}{f^4}+\frac{3 b^2 d^2 (c+d x) \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac{3 b^2 c d^2 x}{4 f^2}+\frac{3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}-\frac{b^2 (c+d x)^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{b^2 (c+d x)^4}{8 d}-\frac{3 b^2 d^3 \sin ^2(e+f x)}{8 f^4}-\frac{3 b^2 d^3 x^2}{8 f^2}",1,"(2*(2*a^2 + b^2)*f^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) - 32*a*b*f*(c + d*x)*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-6 + f^2*x^2))*Cos[e + f*x] - 3*b^2*d*(2*c^2*f^2 + 4*c*d*f^2*x + d^2*(-1 + 2*f^2*x^2))*Cos[2*(e + f*x)] + 96*a*b*d*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Sin[e + f*x] - 2*b^2*f*(c + d*x)*(2*c^2*f^2 + 4*c*d*f^2*x + d^2*(-3 + 2*f^2*x^2))*Sin[2*(e + f*x)])/(16*f^4)","A",1
158,1,249,182,0.9027667,"\int (c+d x)^2 (a+b \sin (e+f x))^2 \, dx","Integrate[(c + d*x)^2*(a + b*Sin[e + f*x])^2,x]","\frac{24 a^2 c^2 f^3 x+24 a^2 c d f^3 x^2+8 a^2 d^2 f^3 x^3-48 a b \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-2\right)\right) \cos (e+f x)+96 a b c d f \sin (e+f x)+96 a b d^2 f x \sin (e+f x)-6 b^2 c^2 f^2 \sin (2 (e+f x))+12 b^2 c^2 f^3 x-12 b^2 c d f^2 x \sin (2 (e+f x))-6 b^2 d f (c+d x) \cos (2 (e+f x))+12 b^2 c d f^3 x^2-6 b^2 d^2 f^2 x^2 \sin (2 (e+f x))+3 b^2 d^2 \sin (2 (e+f x))+4 b^2 d^2 f^3 x^3}{24 f^3}","\frac{a^2 (c+d x)^3}{3 d}+\frac{4 a b d (c+d x) \sin (e+f x)}{f^2}-\frac{2 a b (c+d x)^2 \cos (e+f x)}{f}+\frac{4 a b d^2 \cos (e+f x)}{f^3}+\frac{b^2 d (c+d x) \sin ^2(e+f x)}{2 f^2}-\frac{b^2 (c+d x)^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac{b^2 (c+d x)^3}{6 d}+\frac{b^2 d^2 \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac{b^2 d^2 x}{4 f^2}",1,"(24*a^2*c^2*f^3*x + 12*b^2*c^2*f^3*x + 24*a^2*c*d*f^3*x^2 + 12*b^2*c*d*f^3*x^2 + 8*a^2*d^2*f^3*x^3 + 4*b^2*d^2*f^3*x^3 - 48*a*b*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Cos[e + f*x] - 6*b^2*d*f*(c + d*x)*Cos[2*(e + f*x)] + 96*a*b*c*d*f*Sin[e + f*x] + 96*a*b*d^2*f*x*Sin[e + f*x] + 3*b^2*d^2*Sin[2*(e + f*x)] - 6*b^2*c^2*f^2*Sin[2*(e + f*x)] - 12*b^2*c*d*f^2*x*Sin[2*(e + f*x)] - 6*b^2*d^2*f^2*x^2*Sin[2*(e + f*x)])/(24*f^3)","A",1
159,1,96,116,0.7280685,"\int (c+d x) (a+b \sin (e+f x))^2 \, dx","Integrate[(c + d*x)*(a + b*Sin[e + f*x])^2,x]","-\frac{2 \left(2 a^2+b^2\right) (e+f x) (d (e-f x)-2 c f)+16 a b f (c+d x) \cos (e+f x)-16 a b d \sin (e+f x)+2 b^2 f (c+d x) \sin (2 (e+f x))+b^2 d \cos (2 (e+f x))}{8 f^2}","\frac{a^2 (c+d x)^2}{2 d}-\frac{2 a b (c+d x) \cos (e+f x)}{f}+\frac{2 a b d \sin (e+f x)}{f^2}-\frac{b^2 (c+d x) \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} b^2 c x+\frac{b^2 d \sin ^2(e+f x)}{4 f^2}+\frac{1}{4} b^2 d x^2",1,"-1/8*(2*(2*a^2 + b^2)*(e + f*x)*(-2*c*f + d*(e - f*x)) + 16*a*b*f*(c + d*x)*Cos[e + f*x] + b^2*d*Cos[2*(e + f*x)] - 16*a*b*d*Sin[e + f*x] + 2*b^2*f*(c + d*x)*Sin[2*(e + f*x)])/f^2","A",1
160,1,134,156,0.3443736,"\int \frac{(a+b \sin (e+f x))^2}{c+d x} \, dx","Integrate[(a + b*Sin[e + f*x])^2/(c + d*x),x]","\frac{2 a^2 \log (c+d x)+4 a b \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \sin \left(e-\frac{c f}{d}\right)+4 a b \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)-b^2 \text{Ci}\left(\frac{2 f (c+d x)}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)+b^2 \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)+b^2 \log (c+d x)}{2 d}","\frac{a^2 \log (c+d x)}{d}+\frac{2 a b \text{Ci}\left(x f+\frac{c f}{d}\right) \sin \left(e-\frac{c f}{d}\right)}{d}+\frac{2 a b \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{d}-\frac{b^2 \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)}{2 d}+\frac{b^2 \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{2 d}+\frac{b^2 \log (c+d x)}{2 d}",1,"(-(b^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*f*(c + d*x))/d]) + 2*a^2*Log[c + d*x] + b^2*Log[c + d*x] + 4*a*b*CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + 4*a*b*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + b^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d])/(2*d)","A",1
161,1,232,183,0.6462229,"\int \frac{(a+b \sin (e+f x))^2}{(c+d x)^2} \, dx","Integrate[(a + b*Sin[e + f*x])^2/(c + d*x)^2,x]","\frac{-2 a^2 d+4 a b f (c+d x) \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \cos \left(e-\frac{c f}{d}\right)-4 a b c f \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)-4 a b d f x \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)-4 a b d \sin (e+f x)+2 b^2 f (c+d x) \text{Ci}\left(\frac{2 f (c+d x)}{d}\right) \sin \left(2 e-\frac{2 c f}{d}\right)+2 b^2 c f \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)+2 b^2 d f x \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)+b^2 d \cos (2 (e+f x))-b^2 d}{2 d^2 (c+d x)}","-\frac{a^2}{d (c+d x)}+\frac{2 a b f \text{Ci}\left(x f+\frac{c f}{d}\right) \cos \left(e-\frac{c f}{d}\right)}{d^2}-\frac{2 a b f \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{d^2}-\frac{2 a b \sin (e+f x)}{d (c+d x)}+\frac{b^2 f \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \sin \left(2 e-\frac{2 c f}{d}\right)}{d^2}+\frac{b^2 f \cos \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{d^2}-\frac{b^2 \sin ^2(e+f x)}{d (c+d x)}",1,"(-2*a^2*d - b^2*d + b^2*d*Cos[2*(e + f*x)] + 4*a*b*f*(c + d*x)*Cos[e - (c*f)/d]*CosIntegral[f*(c/d + x)] + 2*b^2*f*(c + d*x)*CosIntegral[(2*f*(c + d*x))/d]*Sin[2*e - (2*c*f)/d] - 4*a*b*d*Sin[e + f*x] - 4*a*b*c*f*Sin[e - (c*f)/d]*SinIntegral[f*(c/d + x)] - 4*a*b*d*f*x*Sin[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + 2*b^2*c*f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d] + 2*b^2*d*f*x*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d])/(2*d^2*(c + d*x))","A",1
162,1,395,245,1.3029194,"\int \frac{(a+b \sin (e+f x))^2}{(c+d x)^3} \, dx","Integrate[(a + b*Sin[e + f*x])^2/(c + d*x)^3,x]","-\frac{2 a^2 d^2+4 a b c^2 f^2 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)+4 a b f^2 (c+d x)^2 \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \sin \left(e-\frac{c f}{d}\right)+4 a b d^2 f^2 x^2 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)+8 a b c d f^2 x \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)+4 a b c d f \cos (e+f x)+4 a b d^2 \sin (e+f x)+4 a b d^2 f x \cos (e+f x)+4 b^2 c^2 f^2 \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)-4 b^2 f^2 (c+d x)^2 \text{Ci}\left(\frac{2 f (c+d x)}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)+4 b^2 d^2 f^2 x^2 \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)+8 b^2 c d f^2 x \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)+2 b^2 c d f \sin (2 (e+f x))+2 b^2 d^2 f x \sin (2 (e+f x))-b^2 d^2 \cos (2 (e+f x))+b^2 d^2}{4 d^3 (c+d x)^2}","-\frac{a^2}{2 d (c+d x)^2}-\frac{a b f^2 \text{Ci}\left(x f+\frac{c f}{d}\right) \sin \left(e-\frac{c f}{d}\right)}{d^3}-\frac{a b f^2 \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{d^3}-\frac{a b f \cos (e+f x)}{d^2 (c+d x)}-\frac{a b \sin (e+f x)}{d (c+d x)^2}+\frac{b^2 f^2 \text{Ci}\left(2 x f+\frac{2 c f}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)}{d^3}-\frac{b^2 f^2 \sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(2 x f+\frac{2 c f}{d}\right)}{d^3}-\frac{b^2 f \sin (e+f x) \cos (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sin ^2(e+f x)}{2 d (c+d x)^2}",1,"-1/4*(2*a^2*d^2 + b^2*d^2 + 4*a*b*c*d*f*Cos[e + f*x] + 4*a*b*d^2*f*x*Cos[e + f*x] - b^2*d^2*Cos[2*(e + f*x)] - 4*b^2*f^2*(c + d*x)^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*f*(c + d*x))/d] + 4*a*b*f^2*(c + d*x)^2*CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + 4*a*b*d^2*Sin[e + f*x] + 2*b^2*c*d*f*Sin[2*(e + f*x)] + 2*b^2*d^2*f*x*Sin[2*(e + f*x)] + 4*a*b*c^2*f^2*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + 8*a*b*c*d*f^2*x*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + 4*a*b*d^2*f^2*x^2*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + 4*b^2*c^2*f^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d] + 8*b^2*c*d*f^2*x*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d] + 4*b^2*d^2*f^2*x^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d])/(d^3*(c + d*x)^2)","A",1
163,1,401,495,0.2554225,"\int \frac{(c+d x)^3}{a+b \sin (e+f x)} \, dx","Integrate[(c + d*x)^3/(a + b*Sin[e + f*x]),x]","-\frac{i \left(\frac{3 i d \left(f^2 (c+d x)^2 \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)+2 i d f (c+d x) \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)-2 d^2 \text{Li}_4\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)\right)}{f^3}+\frac{3 d \left(2 d \left(f (c+d x) \text{Li}_3\left(-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)+i d \text{Li}_4\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)\right)-i f^2 (c+d x)^2 \text{Li}_2\left(-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)\right)}{f^3}+(c+d x)^3 \log \left(1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)-(c+d x)^3 \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)\right)}{f \sqrt{a^2-b^2}}","-\frac{6 i d^2 (c+d x) \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f^3 \sqrt{a^2-b^2}}+\frac{6 i d^2 (c+d x) \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{f^3 \sqrt{a^2-b^2}}-\frac{3 d (c+d x)^2 \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f^2 \sqrt{a^2-b^2}}+\frac{3 d (c+d x)^2 \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{f^2 \sqrt{a^2-b^2}}-\frac{i (c+d x)^3 \log \left(1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2}}+\frac{i (c+d x)^3 \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)}{f \sqrt{a^2-b^2}}+\frac{6 d^3 \text{Li}_4\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f^4 \sqrt{a^2-b^2}}-\frac{6 d^3 \text{Li}_4\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{f^4 \sqrt{a^2-b^2}}",1,"((-I)*((c + d*x)^3*Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - (c + d*x)^3*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + (3*d*((-I)*f^2*(c + d*x)^2*PolyLog[2, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] + 2*d*(f*(c + d*x)*PolyLog[3, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] + I*d*PolyLog[4, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])))/f^3 + ((3*I)*d*(f^2*(c + d*x)^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + (2*I)*d*f*(c + d*x)*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] - 2*d^2*PolyLog[4, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])]))/f^3))/(Sqrt[a^2 - b^2]*f)","A",1
164,1,296,367,0.2068935,"\int \frac{(c+d x)^2}{a+b \sin (e+f x)} \, dx","Integrate[(c + d*x)^2/(a + b*Sin[e + f*x]),x]","-\frac{i \left(\frac{2 d \left(d \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)-i f (c+d x) \text{Li}_2\left(-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)\right)}{f^2}+\frac{2 i d \left(f (c+d x) \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)+i d \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)\right)}{f^2}+(c+d x)^2 \log \left(1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)-(c+d x)^2 \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)\right)}{f \sqrt{a^2-b^2}}","-\frac{2 d (c+d x) \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f^2 \sqrt{a^2-b^2}}+\frac{2 d (c+d x) \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{f^2 \sqrt{a^2-b^2}}-\frac{i (c+d x)^2 \log \left(1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2}}+\frac{i (c+d x)^2 \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)}{f \sqrt{a^2-b^2}}-\frac{2 i d^2 \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f^3 \sqrt{a^2-b^2}}+\frac{2 i d^2 \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{f^3 \sqrt{a^2-b^2}}",1,"((-I)*((c + d*x)^2*Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - (c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + (2*d*((-I)*f*(c + d*x)*PolyLog[2, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] + d*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])]))/f^2 + ((2*I)*d*(f*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + I*d*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])]))/f^2))/(Sqrt[a^2 - b^2]*f)","A",1
165,1,182,234,0.0434058,"\int \frac{c+d x}{a+b \sin (e+f x)} \, dx","Integrate[(c + d*x)/(a + b*Sin[e + f*x]),x]","\frac{-i f (c+d x) \left(\log \left(1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)-\log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)\right)-d \text{Li}_2\left(-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)+d \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{f^2 \sqrt{a^2-b^2}}","-\frac{i (c+d x) \log \left(1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f \sqrt{a^2-b^2}}+\frac{i (c+d x) \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)}{f \sqrt{a^2-b^2}}-\frac{d \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f^2 \sqrt{a^2-b^2}}+\frac{d \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{f^2 \sqrt{a^2-b^2}}",1,"((-I)*f*(c + d*x)*(Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])]) - d*PolyLog[2, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] + d*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^2)","A",1
166,0,0,23,0.4289345,"\int \frac{1}{(c+d x) (a+b \sin (e+f x))} \, dx","Integrate[1/((c + d*x)*(a + b*Sin[e + f*x])),x]","\int \frac{1}{(c+d x) (a+b \sin (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a+b \sin (e+f x))},x\right)",0,"Integrate[1/((c + d*x)*(a + b*Sin[e + f*x])), x]","A",-1
167,0,0,23,0.3684183,"\int \frac{1}{(c+d x)^2 (a+b \sin (e+f x))} \, dx","Integrate[1/((c + d*x)^2*(a + b*Sin[e + f*x])),x]","\int \frac{1}{(c+d x)^2 (a+b \sin (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a+b \sin (e+f x))},x\right)",0,"Integrate[1/((c + d*x)^2*(a + b*Sin[e + f*x])), x]","A",-1
168,1,742,925,3.5112303,"\int \frac{(c+d x)^3}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c + d*x)^3/(a + b*Sin[e + f*x])^2,x]","\frac{-\frac{i a \left(-3 i d \left(f^2 (c+d x)^2 \text{Li}_2\left(-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)+2 i d f (c+d x) \text{Li}_3\left(-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)-2 d^2 \text{Li}_4\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)\right)+3 i d \left(f^2 (c+d x)^2 \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)+2 i d f (c+d x) \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)-2 d^2 \text{Li}_4\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)\right)+f^3 (c+d x)^3 \log \left(1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)-f^3 (c+d x)^3 \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)\right)}{\sqrt{a^2-b^2}}+6 i d^2 \left(f (c+d x) \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)+i d \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)\right)+6 i d^2 \left(f (c+d x) \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)+i d \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)\right)-3 d f^2 (c+d x)^2 \log \left(1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)-3 d f^2 (c+d x)^2 \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)+\frac{b f^3 (c+d x)^3 \cos (e+f x)}{a+b \sin (e+f x)}+i f^3 (c+d x)^3}{f^4 \left(a^2-b^2\right)}","-\frac{6 \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right) d^3}{\left(a^2-b^2\right) f^4}-\frac{6 \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right) d^3}{\left(a^2-b^2\right) f^4}+\frac{6 a \text{Li}_4\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right) d^3}{\left(a^2-b^2\right)^{3/2} f^4}-\frac{6 a \text{Li}_4\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right) d^3}{\left(a^2-b^2\right)^{3/2} f^4}+\frac{6 i (c+d x) \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right) d^2}{\left(a^2-b^2\right) f^3}+\frac{6 i (c+d x) \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right) d^2}{\left(a^2-b^2\right) f^3}-\frac{6 i a (c+d x) \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right) d^2}{\left(a^2-b^2\right)^{3/2} f^3}+\frac{6 i a (c+d x) \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right) d^2}{\left(a^2-b^2\right)^{3/2} f^3}-\frac{3 (c+d x)^2 \log \left(1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right) d}{\left(a^2-b^2\right) f^2}-\frac{3 (c+d x)^2 \log \left(1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right) d}{\left(a^2-b^2\right) f^2}-\frac{3 a (c+d x)^2 \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right) d}{\left(a^2-b^2\right)^{3/2} f^2}+\frac{3 a (c+d x)^2 \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right) d}{\left(a^2-b^2\right)^{3/2} f^2}+\frac{i (c+d x)^3}{\left(a^2-b^2\right) f}-\frac{i a (c+d x)^3 \log \left(1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2} f}+\frac{i a (c+d x)^3 \log \left(1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2} f}+\frac{b (c+d x)^3 \cos (e+f x)}{\left(a^2-b^2\right) f (a+b \sin (e+f x))}",1,"(I*f^3*(c + d*x)^3 - 3*d*f^2*(c + d*x)^2*Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - 3*d*f^2*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + (6*I)*d^2*(f*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])] + I*d*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])]) + (6*I)*d^2*(f*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + I*d*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])]) - (I*a*(f^3*(c + d*x)^3*Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - f^3*(c + d*x)^3*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] - (3*I)*d*(f^2*(c + d*x)^2*PolyLog[2, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] + (2*I)*d*f*(c + d*x)*PolyLog[3, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - 2*d^2*PolyLog[4, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])]) + (3*I)*d*(f^2*(c + d*x)^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + (2*I)*d*f*(c + d*x)*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] - 2*d^2*PolyLog[4, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])))/Sqrt[a^2 - b^2] + (b*f^3*(c + d*x)^3*Cos[e + f*x])/(a + b*Sin[e + f*x]))/((a^2 - b^2)*f^4)","A",1
169,1,530,671,1.7712676,"\int \frac{(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c + d*x)^2/(a + b*Sin[e + f*x])^2,x]","\frac{-\frac{i a \left(f^2 (c+d x)^2 \log \left(1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)-f^2 (c+d x)^2 \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)-2 i d f (c+d x) \text{Li}_2\left(-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)+2 i d f (c+d x) \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)+2 d^2 \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)-2 d^2 \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)\right)}{\sqrt{a^2-b^2}}-2 d f (c+d x) \log \left(1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)-2 d f (c+d x) \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)+2 i d^2 \text{Li}_2\left(-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)+2 i d^2 \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)+\frac{b f^2 (c+d x)^2 \cos (e+f x)}{a+b \sin (e+f x)}+i f^2 (c+d x)^2}{f^3 \left(a^2-b^2\right)}","-\frac{2 a d (c+d x) \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f^2 \left(a^2-b^2\right)^{3/2}}+\frac{2 a d (c+d x) \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{f^2 \left(a^2-b^2\right)^{3/2}}-\frac{2 d (c+d x) \log \left(1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f^2 \left(a^2-b^2\right)}-\frac{2 d (c+d x) \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)}{f^2 \left(a^2-b^2\right)}-\frac{i a (c+d x)^2 \log \left(1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2}}+\frac{i a (c+d x)^2 \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)}{f \left(a^2-b^2\right)^{3/2}}+\frac{b (c+d x)^2 \cos (e+f x)}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}+\frac{i (c+d x)^2}{f \left(a^2-b^2\right)}+\frac{2 i d^2 \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f^3 \left(a^2-b^2\right)}+\frac{2 i d^2 \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{f^3 \left(a^2-b^2\right)}-\frac{2 i a d^2 \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f^3 \left(a^2-b^2\right)^{3/2}}+\frac{2 i a d^2 \text{Li}_3\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{f^3 \left(a^2-b^2\right)^{3/2}}",1,"(I*f^2*(c + d*x)^2 - 2*d*f*(c + d*x)*Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - 2*d*f*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + (2*I)*d^2*PolyLog[2, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] + (2*I)*d^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] - (I*a*(f^2*(c + d*x)^2*Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - f^2*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] - (2*I)*d*f*(c + d*x)*PolyLog[2, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] + (2*I)*d*f*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + 2*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])] - 2*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])]))/Sqrt[a^2 - b^2] + (b*f^2*(c + d*x)^2*Cos[e + f*x])/(a + b*Sin[e + f*x]))/((a^2 - b^2)*f^3)","A",1
170,1,236,305,1.0830764,"\int \frac{c+d x}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c + d*x)/(a + b*Sin[e + f*x])^2,x]","\frac{\frac{a \left(-i f (c+d x) \left(\log \left(1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)-\log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)\right)-d \text{Li}_2\left(-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right)+d \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)\right)}{\sqrt{a^2-b^2}}+\frac{b f (c+d x) \cos (e+f x)}{a+b \sin (e+f x)}-d \log (a+b \sin (e+f x))}{f^2 \left(a^2-b^2\right)}","-\frac{i a (c+d x) \log \left(1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f \left(a^2-b^2\right)^{3/2}}+\frac{i a (c+d x) \log \left(1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right)}{f \left(a^2-b^2\right)^{3/2}}+\frac{b (c+d x) \cos (e+f x)}{f \left(a^2-b^2\right) (a+b \sin (e+f x))}-\frac{a d \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right)}{f^2 \left(a^2-b^2\right)^{3/2}}+\frac{a d \text{Li}_2\left(\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right)}{f^2 \left(a^2-b^2\right)^{3/2}}-\frac{d \log (a+b \sin (e+f x))}{f^2 \left(a^2-b^2\right)}",1,"(-(d*Log[a + b*Sin[e + f*x]]) + (a*((-I)*f*(c + d*x)*(Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])]) - d*PolyLog[2, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] + d*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])]))/Sqrt[a^2 - b^2] + (b*f*(c + d*x)*Cos[e + f*x])/(a + b*Sin[e + f*x]))/((a^2 - b^2)*f^2)","A",1
171,0,0,23,37.8197908,"\int \frac{1}{(c+d x) (a+b \sin (e+f x))^2} \, dx","Integrate[1/((c + d*x)*(a + b*Sin[e + f*x])^2),x]","\int \frac{1}{(c+d x) (a+b \sin (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a+b \sin (e+f x))^2},x\right)",0,"Integrate[1/((c + d*x)*(a + b*Sin[e + f*x])^2), x]","A",-1
172,0,0,23,109.5066527,"\int \frac{1}{(c+d x)^2 (a+b \sin (e+f x))^2} \, dx","Integrate[1/((c + d*x)^2*(a + b*Sin[e + f*x])^2),x]","\int \frac{1}{(c+d x)^2 (a+b \sin (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a+b \sin (e+f x))^2},x\right)",0,"Integrate[1/((c + d*x)^2*(a + b*Sin[e + f*x])^2), x]","A",-1
173,0,0,23,1.0813572,"\int (c+d x)^m (a+b \sin (e+f x))^n \, dx","Integrate[(c + d*x)^m*(a + b*Sin[e + f*x])^n,x]","\int (c+d x)^m (a+b \sin (e+f x))^n \, dx","\text{Int}\left((c+d x)^m (a+b \sin (e+f x))^n,x\right)",0,"Integrate[(c + d*x)^m*(a + b*Sin[e + f*x])^n, x]","A",-1
174,1,415,607,6.1818984,"\int (c+d x)^m (a+b \sin (e+f x))^3 \, dx","Integrate[(c + d*x)^m*(a + b*Sin[e + f*x])^3,x]","\frac{i (c+d x)^m \left(9 i b \left(4 a^2+b^2\right) e^{i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)+9 i b \left(4 a^2+b^2\right) e^{-i \left(e-\frac{c f}{d}\right)} \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)-\frac{12 i a f \left(2 a^2+3 b^2\right) (c+d x)}{d (m+1)}+9 a b^2 2^{-m} e^{2 i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)-9 a b^2 2^{-m} e^{-2 i \left(e-\frac{c f}{d}\right)} \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i f (c+d x)}{d}\right)-i b^3 3^{-m} e^{3 i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{3 i f (c+d x)}{d}\right)-i b^3 3^{-m} e^{-3 i \left(e-\frac{c f}{d}\right)} \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{3 i f (c+d x)}{d}\right)\right)}{24 f}","\frac{a^3 (c+d x)^{m+1}}{d (m+1)}-\frac{3 a^2 b e^{i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)}{2 f}-\frac{3 a^2 b e^{-i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)}{2 f}+\frac{3 i a b^2 2^{-m-3} e^{2 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)}{f}-\frac{3 i a b^2 2^{-m-3} e^{-2 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i f (c+d x)}{d}\right)}{f}+\frac{3 a b^2 (c+d x)^{m+1}}{2 d (m+1)}-\frac{3 b^3 e^{i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)}{8 f}+\frac{b^3 3^{-m-1} e^{3 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{3 i f (c+d x)}{d}\right)}{8 f}-\frac{3 b^3 e^{-i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)}{8 f}+\frac{b^3 3^{-m-1} e^{-3 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{3 i f (c+d x)}{d}\right)}{8 f}",1,"((I/24)*(c + d*x)^m*(((-12*I)*a*(2*a^2 + 3*b^2)*f*(c + d*x))/(d*(1 + m)) + ((9*I)*b*(4*a^2 + b^2)*E^(I*(e - (c*f)/d))*Gamma[1 + m, ((-I)*f*(c + d*x))/d])/(((-I)*f*(c + d*x))/d)^m + ((9*I)*b*(4*a^2 + b^2)*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m) + (9*a*b^2*E^((2*I)*(e - (c*f)/d))*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d])/(2^m*(((-I)*f*(c + d*x))/d)^m) - (9*a*b^2*Gamma[1 + m, ((2*I)*f*(c + d*x))/d])/(2^m*E^((2*I)*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m) - (I*b^3*E^((3*I)*(e - (c*f)/d))*Gamma[1 + m, ((-3*I)*f*(c + d*x))/d])/(3^m*(((-I)*f*(c + d*x))/d)^m) - (I*b^3*Gamma[1 + m, ((3*I)*f*(c + d*x))/d])/(3^m*E^((3*I)*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m)))/f","A",1
175,1,268,318,4.1508301,"\int (c+d x)^m (a+b \sin (e+f x))^2 \, dx","Integrate[(c + d*x)^m*(a + b*Sin[e + f*x])^2,x]","-\frac{(c+d x)^m \left(-\frac{4 f \left(2 a^2+b^2\right) (c+d x)}{d (m+1)}+8 a b e^{i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)+8 a b e^{-i \left(e-\frac{c f}{d}\right)} \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)-i b^2 2^{-m} e^{2 i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)+i b^2 2^{-m} e^{-2 i \left(e-\frac{c f}{d}\right)} \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i f (c+d x)}{d}\right)\right)}{8 f}","\frac{a^2 (c+d x)^{m+1}}{d (m+1)}-\frac{a b e^{i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)}{f}-\frac{a b e^{-i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)}{f}+\frac{i b^2 2^{-m-3} e^{2 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i f (c+d x)}{d}\right)}{f}-\frac{i b^2 2^{-m-3} e^{-2 i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i f (c+d x)}{d}\right)}{f}+\frac{b^2 (c+d x)^{m+1}}{2 d (m+1)}",1,"-1/8*((c + d*x)^m*((-4*(2*a^2 + b^2)*f*(c + d*x))/(d*(1 + m)) + (8*a*b*E^(I*(e - (c*f)/d))*Gamma[1 + m, ((-I)*f*(c + d*x))/d])/(((-I)*f*(c + d*x))/d)^m + (8*a*b*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m) - (I*b^2*E^((2*I)*(e - (c*f)/d))*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d])/(2^m*(((-I)*f*(c + d*x))/d)^m) + (I*b^2*Gamma[1 + m, ((2*I)*f*(c + d*x))/d])/(2^m*E^((2*I)*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m)))/f","A",1
176,1,138,148,0.2061251,"\int (c+d x)^m (a+b \sin (e+f x)) \, dx","Integrate[(c + d*x)^m*(a + b*Sin[e + f*x]),x]","\frac{1}{2} (c+d x)^m \left(\frac{2 a (c+d x)}{d (m+1)}-\frac{b e^{i \left(e-\frac{c f}{d}\right)} \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)}{f}-\frac{b e^{-i \left(e-\frac{c f}{d}\right)} \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)}{f}\right)","\frac{a (c+d x)^{m+1}}{d (m+1)}-\frac{b e^{i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(-\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i f (c+d x)}{d}\right)}{2 f}-\frac{b e^{-i \left(e-\frac{c f}{d}\right)} (c+d x)^m \left(\frac{i f (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i f (c+d x)}{d}\right)}{2 f}",1,"((c + d*x)^m*((2*a*(c + d*x))/(d*(1 + m)) - (b*E^(I*(e - (c*f)/d))*Gamma[1 + m, ((-I)*f*(c + d*x))/d])/(f*(((-I)*f*(c + d*x))/d)^m) - (b*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m)))/2","A",1
177,0,0,23,0.4072514,"\int \frac{(c+d x)^m}{a+b \sin (e+f x)} \, dx","Integrate[(c + d*x)^m/(a + b*Sin[e + f*x]),x]","\int \frac{(c+d x)^m}{a+b \sin (e+f x)} \, dx","\text{Int}\left(\frac{(c+d x)^m}{a+b \sin (e+f x)},x\right)",0,"Integrate[(c + d*x)^m/(a + b*Sin[e + f*x]), x]","A",-1
178,0,0,23,4.1302601,"\int \frac{(c+d x)^m}{(a+b \sin (e+f x))^2} \, dx","Integrate[(c + d*x)^m/(a + b*Sin[e + f*x])^2,x]","\int \frac{(c+d x)^m}{(a+b \sin (e+f x))^2} \, dx","\text{Int}\left(\frac{(c+d x)^m}{(a+b \sin (e+f x))^2},x\right)",0,"Integrate[(c + d*x)^m/(a + b*Sin[e + f*x])^2, x]","A",-1
179,1,261,164,2.0353878,"\int \frac{(e+f x)^3 \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\frac{24 f (\cos (c)+i \sin (c)) \left(\frac{2 f (\cos (c)-i (\sin (c)+1)) (d (e+f x) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-i f \text{Li}_3(-i \cos (c+d x)-\sin (c+d x)))}{d^3}-\frac{(\sin (c)+i \cos (c)+1) (e+f x)^2 \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^3}{3 f}\right)}{d (\cos (c)+i (\sin (c)+1))}-\frac{8 \sin \left(\frac{d x}{2}\right) (e+f x)^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)}+x \left(4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right)}{4 a}","-\frac{12 f^3 \text{Li}_3\left(i e^{i (c+d x)}\right)}{a d^4}+\frac{12 i f^2 (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}-\frac{6 f (e+f x)^2 \log \left(1-i e^{i (c+d x)}\right)}{a d^2}+\frac{(e+f x)^3 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}+\frac{i (e+f x)^3}{a d}+\frac{(e+f x)^4}{4 a f}",1,"(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) + (24*f*(Cos[c] + I*Sin[c])*(((e + f*x)^3*(Cos[c] - I*Sin[c]))/(3*f) - ((e + f*x)^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (2*f*(d*(e + f*x)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*f*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*(1 + Sin[c])))/d^3))/(d*(Cos[c] + I*(1 + Sin[c]))) - (8*(e + f*x)^3*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(4*a)","A",1
180,1,213,129,1.3913027,"\int \frac{(e+f x)^2 \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\frac{12 f (\cos (c)+i \sin (c)) \left(\frac{f (\cos (c)-i (\sin (c)+1)) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))}{d^2}-\frac{(\sin (c)+i \cos (c)+1) (e+f x) \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^2}{2 f}\right)}{d (\cos (c)+i (\sin (c)+1))}-\frac{6 \sin \left(\frac{d x}{2}\right) (e+f x)^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)}+x \left(3 e^2+3 e f x+f^2 x^2\right)}{3 a}","\frac{4 i f^2 \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}-\frac{4 f (e+f x) \log \left(1-i e^{i (c+d x)}\right)}{a d^2}+\frac{(e+f x)^2 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}+\frac{i (e+f x)^2}{a d}+\frac{(e+f x)^3}{3 a f}",1,"(x*(3*e^2 + 3*e*f*x + f^2*x^2) + (12*f*(Cos[c] + I*Sin[c])*(((e + f*x)^2*(Cos[c] - I*Sin[c]))/(2*f) - ((e + f*x)*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2))/(d*(Cos[c] + I*(1 + Sin[c]))) - (6*(e + f*x)^2*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(3*a)","A",1
181,1,199,76,0.5662825,"\int \frac{(e+f x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\cos \left(\frac{d x}{2}\right) \left(d^2 x (2 e+f x)-4 f \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 d^2 e x \sin \left(c+\frac{d x}{2}\right)+d^2 f x^2 \sin \left(c+\frac{d x}{2}\right)+2 d f x \cos \left(c+\frac{d x}{2}\right)-4 f \sin \left(c+\frac{d x}{2}\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 d e \sin \left(\frac{d x}{2}\right)-2 d f x \sin \left(\frac{d x}{2}\right)}{2 a d^2 \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 f \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)\right)}{a d^2}+\frac{(e+f x) \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}+\frac{e x}{a}+\frac{f x^2}{2 a}",1,"(2*d*f*x*Cos[c + (d*x)/2] + Cos[(d*x)/2]*(d^2*x*(2*e + f*x) - 4*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 4*d*e*Sin[(d*x)/2] - 2*d*f*x*Sin[(d*x)/2] + 2*d^2*e*x*Sin[c + (d*x)/2] + d^2*f*x^2*Sin[c + (d*x)/2] - 4*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + (d*x)/2])/(2*a*d^2*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
182,1,72,28,0.1153277,"\int \frac{\sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sin[c + d*x]/(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((c+d x-2) \sin \left(\frac{1}{2} (c+d x)\right)+(c+d x) \cos \left(\frac{1}{2} (c+d x)\right)\right)}{a d (\sin (c+d x)+1)}","\frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}+\frac{x}{a}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*((c + d*x)*Cos[(c + d*x)/2] + (-2 + c + d*x)*Sin[(c + d*x)/2]))/(a*d*(1 + Sin[c + d*x]))","B",1
183,0,0,29,9.2775197,"\int \frac{\sin (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Sin[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])),x]","\int \frac{\sin (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sin (c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sin[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]","A",-1
184,0,0,29,8.9472858,"\int \frac{\sin (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Sin[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\int \frac{\sin (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sin (c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sin[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x]","A",-1
185,1,1314,247,3.245614,"\int \frac{(e+f x)^3 \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{-f^3 x^4 \cos \left(\frac{1}{2} (c+d x)\right) d^4-4 e f^2 x^3 \cos \left(\frac{1}{2} (c+d x)\right) d^4-6 e^2 f x^2 \cos \left(\frac{1}{2} (c+d x)\right) d^4-4 e^3 x \cos \left(\frac{1}{2} (c+d x)\right) d^4-f^3 x^4 \sin \left(\frac{1}{2} (c+d x)\right) d^4-4 e f^2 x^3 \sin \left(\frac{1}{2} (c+d x)\right) d^4-6 e^2 f x^2 \sin \left(\frac{1}{2} (c+d x)\right) d^4-4 e^3 x \sin \left(\frac{1}{2} (c+d x)\right) d^4-(6-4 i) e^3 \cos \left(\frac{1}{2} (c+d x)\right) d^3-(6-4 i) f^3 x^3 \cos \left(\frac{1}{2} (c+d x)\right) d^3-(18-12 i) e f^2 x^2 \cos \left(\frac{1}{2} (c+d x)\right) d^3-(18-12 i) e^2 f x \cos \left(\frac{1}{2} (c+d x)\right) d^3-2 e^3 \cos \left(\frac{3}{2} (c+d x)\right) d^3-2 f^3 x^3 \cos \left(\frac{3}{2} (c+d x)\right) d^3-6 e f^2 x^2 \cos \left(\frac{3}{2} (c+d x)\right) d^3-6 e^2 f x \cos \left(\frac{3}{2} (c+d x)\right) d^3+(6+4 i) e^3 \sin \left(\frac{1}{2} (c+d x)\right) d^3+(6+4 i) f^3 x^3 \sin \left(\frac{1}{2} (c+d x)\right) d^3+(18+12 i) e f^2 x^2 \sin \left(\frac{1}{2} (c+d x)\right) d^3+(18+12 i) e^2 f x \sin \left(\frac{1}{2} (c+d x)\right) d^3-2 e^3 \sin \left(\frac{3}{2} (c+d x)\right) d^3-2 f^3 x^3 \sin \left(\frac{3}{2} (c+d x)\right) d^3-6 e f^2 x^2 \sin \left(\frac{3}{2} (c+d x)\right) d^3-6 e^2 f x \sin \left(\frac{3}{2} (c+d x)\right) d^3+6 f^3 x^2 \cos \left(\frac{1}{2} (c+d x)\right) d^2+6 e^2 f \cos \left(\frac{1}{2} (c+d x)\right) d^2+12 e f^2 x \cos \left(\frac{1}{2} (c+d x)\right) d^2-6 f^3 x^2 \cos \left(\frac{3}{2} (c+d x)\right) d^2-6 e^2 f \cos \left(\frac{3}{2} (c+d x)\right) d^2-12 e f^2 x \cos \left(\frac{3}{2} (c+d x)\right) d^2+24 f^3 x^2 \cos \left(\frac{1}{2} (c+d x)\right) \log (i \cos (c+d x)+\sin (c+d x)+1) d^2+24 e^2 f \cos \left(\frac{1}{2} (c+d x)\right) \log (i \cos (c+d x)+\sin (c+d x)+1) d^2+48 e f^2 x \cos \left(\frac{1}{2} (c+d x)\right) \log (i \cos (c+d x)+\sin (c+d x)+1) d^2+6 f^3 x^2 \sin \left(\frac{1}{2} (c+d x)\right) d^2+6 e^2 f \sin \left(\frac{1}{2} (c+d x)\right) d^2+12 e f^2 x \sin \left(\frac{1}{2} (c+d x)\right) d^2+24 f^3 x^2 \log (i \cos (c+d x)+\sin (c+d x)+1) \sin \left(\frac{1}{2} (c+d x)\right) d^2+24 e^2 f \log (i \cos (c+d x)+\sin (c+d x)+1) \sin \left(\frac{1}{2} (c+d x)\right) d^2+48 e f^2 x \log (i \cos (c+d x)+\sin (c+d x)+1) \sin \left(\frac{1}{2} (c+d x)\right) d^2+6 f^3 x^2 \sin \left(\frac{3}{2} (c+d x)\right) d^2+6 e^2 f \sin \left(\frac{3}{2} (c+d x)\right) d^2+12 e f^2 x \sin \left(\frac{3}{2} (c+d x)\right) d^2+12 e f^2 \cos \left(\frac{1}{2} (c+d x)\right) d+12 f^3 x \cos \left(\frac{1}{2} (c+d x)\right) d+12 e f^2 \cos \left(\frac{3}{2} (c+d x)\right) d+12 f^3 x \cos \left(\frac{3}{2} (c+d x)\right) d-12 e f^2 \sin \left(\frac{1}{2} (c+d x)\right) d-12 f^3 x \sin \left(\frac{1}{2} (c+d x)\right) d+48 i f^2 (e+f x) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) d+12 e f^2 \sin \left(\frac{3}{2} (c+d x)\right) d+12 f^3 x \sin \left(\frac{3}{2} (c+d x)\right) d-12 f^3 \cos \left(\frac{1}{2} (c+d x)\right)+12 f^3 \cos \left(\frac{3}{2} (c+d x)\right)-12 f^3 \sin \left(\frac{1}{2} (c+d x)\right)+48 f^3 \text{Li}_3(-i \cos (c+d x)-\sin (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)-12 f^3 \sin \left(\frac{3}{2} (c+d x)\right)}{4 a d^4 \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{12 f^3 \text{Li}_3\left(i e^{i (c+d x)}\right)}{a d^4}-\frac{6 f^3 \sin (c+d x)}{a d^4}-\frac{12 i f^2 (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}+\frac{6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac{6 f (e+f x)^2 \log \left(1-i e^{i (c+d x)}\right)}{a d^2}+\frac{3 f (e+f x)^2 \sin (c+d x)}{a d^2}-\frac{(e+f x)^3 \cos (c+d x)}{a d}-\frac{(e+f x)^3 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{i (e+f x)^3}{a d}-\frac{(e+f x)^4}{4 a f}",1,"((-6 + 4*I)*d^3*e^3*Cos[(c + d*x)/2] + 6*d^2*e^2*f*Cos[(c + d*x)/2] + 12*d*e*f^2*Cos[(c + d*x)/2] - 12*f^3*Cos[(c + d*x)/2] - 4*d^4*e^3*x*Cos[(c + d*x)/2] - (18 - 12*I)*d^3*e^2*f*x*Cos[(c + d*x)/2] + 12*d^2*e*f^2*x*Cos[(c + d*x)/2] + 12*d*f^3*x*Cos[(c + d*x)/2] - 6*d^4*e^2*f*x^2*Cos[(c + d*x)/2] - (18 - 12*I)*d^3*e*f^2*x^2*Cos[(c + d*x)/2] + 6*d^2*f^3*x^2*Cos[(c + d*x)/2] - 4*d^4*e*f^2*x^3*Cos[(c + d*x)/2] - (6 - 4*I)*d^3*f^3*x^3*Cos[(c + d*x)/2] - d^4*f^3*x^4*Cos[(c + d*x)/2] - 2*d^3*e^3*Cos[(3*(c + d*x))/2] - 6*d^2*e^2*f*Cos[(3*(c + d*x))/2] + 12*d*e*f^2*Cos[(3*(c + d*x))/2] + 12*f^3*Cos[(3*(c + d*x))/2] - 6*d^3*e^2*f*x*Cos[(3*(c + d*x))/2] - 12*d^2*e*f^2*x*Cos[(3*(c + d*x))/2] + 12*d*f^3*x*Cos[(3*(c + d*x))/2] - 6*d^3*e*f^2*x^2*Cos[(3*(c + d*x))/2] - 6*d^2*f^3*x^2*Cos[(3*(c + d*x))/2] - 2*d^3*f^3*x^3*Cos[(3*(c + d*x))/2] + 24*d^2*e^2*f*Cos[(c + d*x)/2]*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]] + 48*d^2*e*f^2*x*Cos[(c + d*x)/2]*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]] + 24*d^2*f^3*x^2*Cos[(c + d*x)/2]*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]] + (6 + 4*I)*d^3*e^3*Sin[(c + d*x)/2] + 6*d^2*e^2*f*Sin[(c + d*x)/2] - 12*d*e*f^2*Sin[(c + d*x)/2] - 12*f^3*Sin[(c + d*x)/2] - 4*d^4*e^3*x*Sin[(c + d*x)/2] + (18 + 12*I)*d^3*e^2*f*x*Sin[(c + d*x)/2] + 12*d^2*e*f^2*x*Sin[(c + d*x)/2] - 12*d*f^3*x*Sin[(c + d*x)/2] - 6*d^4*e^2*f*x^2*Sin[(c + d*x)/2] + (18 + 12*I)*d^3*e*f^2*x^2*Sin[(c + d*x)/2] + 6*d^2*f^3*x^2*Sin[(c + d*x)/2] - 4*d^4*e*f^2*x^3*Sin[(c + d*x)/2] + (6 + 4*I)*d^3*f^3*x^3*Sin[(c + d*x)/2] - d^4*f^3*x^4*Sin[(c + d*x)/2] + 24*d^2*e^2*f*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*Sin[(c + d*x)/2] + 48*d^2*e*f^2*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*Sin[(c + d*x)/2] + 24*d^2*f^3*x^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*Sin[(c + d*x)/2] + (48*I)*d*f^2*(e + f*x)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 48*f^3*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 2*d^3*e^3*Sin[(3*(c + d*x))/2] + 6*d^2*e^2*f*Sin[(3*(c + d*x))/2] + 12*d*e*f^2*Sin[(3*(c + d*x))/2] - 12*f^3*Sin[(3*(c + d*x))/2] - 6*d^3*e^2*f*x*Sin[(3*(c + d*x))/2] + 12*d^2*e*f^2*x*Sin[(3*(c + d*x))/2] + 12*d*f^3*x*Sin[(3*(c + d*x))/2] - 6*d^3*e*f^2*x^2*Sin[(3*(c + d*x))/2] + 6*d^2*f^3*x^2*Sin[(3*(c + d*x))/2] - 2*d^3*f^3*x^3*Sin[(3*(c + d*x))/2])/(4*a*d^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
186,1,295,188,2.7639399,"\int \frac{(e+f x)^2 \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{\frac{12 f (\cos (c)+i \sin (c)) \left(\frac{f (\cos (c)-i (\sin (c)+1)) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))}{d^2}-\frac{(\sin (c)+i \cos (c)+1) (e+f x) \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^2}{2 f}\right)}{d (\cos (c)+i (\sin (c)+1))}+\frac{3 \cos (d x) \left(\cos (c) \left(d^2 (e+f x)^2-2 f^2\right)-2 d f \sin (c) (e+f x)\right)}{d^3}-\frac{3 \sin (d x) \left(\sin (c) \left(d^2 (e+f x)^2-2 f^2\right)+2 d f \cos (c) (e+f x)\right)}{d^3}-\frac{6 \sin \left(\frac{d x}{2}\right) (e+f x)^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)}+x \left(3 e^2+3 e f x+f^2 x^2\right)}{3 a}","-\frac{4 i f^2 \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}+\frac{2 f^2 \cos (c+d x)}{a d^3}+\frac{4 f (e+f x) \log \left(1-i e^{i (c+d x)}\right)}{a d^2}+\frac{2 f (e+f x) \sin (c+d x)}{a d^2}-\frac{(e+f x)^2 \cos (c+d x)}{a d}-\frac{(e+f x)^2 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{i (e+f x)^2}{a d}-\frac{(e+f x)^3}{3 a f}",1,"-1/3*(x*(3*e^2 + 3*e*f*x + f^2*x^2) + (3*Cos[d*x]*((-2*f^2 + d^2*(e + f*x)^2)*Cos[c] - 2*d*f*(e + f*x)*Sin[c]))/d^3 + (12*f*(Cos[c] + I*Sin[c])*(((e + f*x)^2*(Cos[c] - I*Sin[c]))/(2*f) - ((e + f*x)*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2))/(d*(Cos[c] + I*(1 + Sin[c]))) - (3*(2*d*f*(e + f*x)*Cos[c] + (-2*f^2 + d^2*(e + f*x)^2)*Sin[c])*Sin[d*x])/d^3 - (6*(e + f*x)^2*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/a","A",1
187,1,236,111,0.8279683,"\int \frac{(e+f x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Sin[c + d*x]^2)/(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(\sin \left(\frac{1}{2} (c+d x)\right) \left(c^2 (-f)+2 d (e+f x) \cos (c+d x)+2 c d e-2 f \sin (c+d x)-4 f \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 c f+2 d^2 e x+d^2 f x^2-4 d e-2 d f x\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left(c^2 (-f)+2 d (e+f x) \cos (c+d x)+2 c d e-2 f \sin (c+d x)-4 f \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 c f+2 d^2 e x+d^2 f x^2+2 d f x\right)\right)}{2 a d^2 (\sin (c+d x)+1)}","\frac{f \sin (c+d x)}{a d^2}+\frac{2 f \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)\right)}{a d^2}-\frac{(e+f x) \cos (c+d x)}{a d}-\frac{(e+f x) \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{e x}{a}-\frac{f x^2}{2 a}",1,"-1/2*((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Sin[(c + d*x)/2]*(-4*d*e + 2*c*d*e + 2*c*f - c^2*f + 2*d^2*e*x - 2*d*f*x + d^2*f*x^2 + 2*d*(e + f*x)*Cos[c + d*x] - 4*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2*f*Sin[c + d*x]) + Cos[(c + d*x)/2]*(2*c*d*e + 2*c*f - c^2*f + 2*d^2*e*x + 2*d*f*x + d^2*f*x^2 + 2*d*(e + f*x)*Cos[c + d*x] - 4*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2*f*Sin[c + d*x])))/(a*d^2*(1 + Sin[c + d*x]))","B",1
188,1,85,45,0.1597092,"\int \frac{\sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sin[c + d*x]^2/(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(\cos \left(\frac{1}{2} (c+d x)\right) (\cos (c+d x)+c+d x)+\sin \left(\frac{1}{2} (c+d x)\right) (\cos (c+d x)+c+d x-2)\right)}{a d (\sin (c+d x)+1)}","-\frac{\cos (c+d x)}{a d}-\frac{\cos (c+d x)}{a d (\sin (c+d x)+1)}-\frac{x}{a}",1,"-(((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Cos[(c + d*x)/2]*(c + d*x + Cos[c + d*x]) + (-2 + c + d*x + Cos[c + d*x])*Sin[(c + d*x)/2]))/(a*d*(1 + Sin[c + d*x])))","A",1
189,0,0,31,9.6227998,"\int \frac{\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Sin[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])),x]","\int \frac{\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sin ^2(c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sin[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])), x]","A",-1
190,0,0,31,10.4403273,"\int \frac{\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Sin[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\int \frac{\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sin ^2(c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sin[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])), x]","A",-1
191,1,538,382,2.9521742,"\int \frac{(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\frac{192 f (\cos (c)+i \sin (c)) \left(\frac{2 f (\cos (c)-i (\sin (c)+1)) (d (e+f x) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-i f \text{Li}_3(-i \cos (c+d x)-\sin (c+d x)))}{d^3}-\frac{(\sin (c)+i \cos (c)+1) (e+f x)^2 \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^3}{3 f}\right)}{d (\cos (c)+i (\sin (c)+1))}+\frac{16 \left(d^3 (e+f x)^3-3 i d^2 f (e+f x)^2-6 d f^2 (e+f x)+6 i f^3\right) (\cos (c+d x)-i \sin (c+d x))}{d^4}+\frac{16 \left(d^3 (e+f x)^3+3 i d^2 f (e+f x)^2-6 d f^2 (e+f x)-6 i f^3\right) (\cos (c+d x)+i \sin (c+d x))}{d^4}+\frac{\left(-4 i d^3 (e+f x)^3-6 d^2 f (e+f x)^2+6 i d f^2 (e+f x)+3 f^3\right) (\cos (2 (c+d x))-i \sin (2 (c+d x)))}{d^4}+\frac{\left(4 i d^3 (e+f x)^3-6 d^2 f (e+f x)^2-6 i d f^2 (e+f x)+3 f^3\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))}{d^4}-\frac{64 \sin \left(\frac{d x}{2}\right) (e+f x)^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)}+48 e^3 x+72 e^2 f x^2+48 e f^2 x^3+12 f^3 x^4}{32 a}","-\frac{12 f^3 \text{Li}_3\left(i e^{i (c+d x)}\right)}{a d^4}-\frac{3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac{6 f^3 \sin (c+d x)}{a d^4}+\frac{12 i f^2 (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}-\frac{6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac{3 f^2 (e+f x) \sin (c+d x) \cos (c+d x)}{4 a d^3}-\frac{6 f (e+f x)^2 \log \left(1-i e^{i (c+d x)}\right)}{a d^2}+\frac{3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2}-\frac{3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac{(e+f x)^3 \cos (c+d x)}{a d}+\frac{(e+f x)^3 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{3 e f^2 x}{4 a d^2}-\frac{3 f^3 x^2}{8 a d^2}+\frac{i (e+f x)^3}{a d}+\frac{3 (e+f x)^4}{8 a f}",1,"(48*e^3*x + 72*e^2*f*x^2 + 48*e*f^2*x^3 + 12*f^3*x^4 + (192*f*(Cos[c] + I*Sin[c])*(((e + f*x)^3*(Cos[c] - I*Sin[c]))/(3*f) - ((e + f*x)^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (2*f*(d*(e + f*x)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*f*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*(1 + Sin[c])))/d^3))/(d*(Cos[c] + I*(1 + Sin[c]))) - (64*(e + f*x)^3*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (16*((6*I)*f^3 - 6*d*f^2*(e + f*x) - (3*I)*d^2*f*(e + f*x)^2 + d^3*(e + f*x)^3)*(Cos[c + d*x] - I*Sin[c + d*x]))/d^4 + (16*((-6*I)*f^3 - 6*d*f^2*(e + f*x) + (3*I)*d^2*f*(e + f*x)^2 + d^3*(e + f*x)^3)*(Cos[c + d*x] + I*Sin[c + d*x]))/d^4 + ((3*f^3 + (6*I)*d*f^2*(e + f*x) - 6*d^2*f*(e + f*x)^2 - (4*I)*d^3*(e + f*x)^3)*(Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)]))/d^4 + ((3*f^3 - (6*I)*d*f^2*(e + f*x) - 6*d^2*f*(e + f*x)^2 + (4*I)*d^3*(e + f*x)^3)*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]))/d^4)/(32*a)","A",1
192,1,830,278,3.1071938,"\int \frac{(e+f x)^2 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","-\frac{-8 f^2 x^3 \sin \left(\frac{1}{2} (c+d x)\right) d^3-24 e f x^2 \sin \left(\frac{1}{2} (c+d x)\right) d^3-24 e^2 x \sin \left(\frac{1}{2} (c+d x)\right) d^3-6 e^2 \cos \left(\frac{3}{2} (c+d x)\right) d^2-6 f^2 x^2 \cos \left(\frac{3}{2} (c+d x)\right) d^2-12 e f x \cos \left(\frac{3}{2} (c+d x)\right) d^2-2 e^2 \cos \left(\frac{5}{2} (c+d x)\right) d^2-2 f^2 x^2 \cos \left(\frac{5}{2} (c+d x)\right) d^2-4 e f x \cos \left(\frac{5}{2} (c+d x)\right) d^2+(24+16 i) e^2 \sin \left(\frac{1}{2} (c+d x)\right) d^2+(24+16 i) f^2 x^2 \sin \left(\frac{1}{2} (c+d x)\right) d^2+(48+32 i) e f x \sin \left(\frac{1}{2} (c+d x)\right) d^2-6 e^2 \sin \left(\frac{3}{2} (c+d x)\right) d^2-6 f^2 x^2 \sin \left(\frac{3}{2} (c+d x)\right) d^2-12 e f x \sin \left(\frac{3}{2} (c+d x)\right) d^2+2 e^2 \sin \left(\frac{5}{2} (c+d x)\right) d^2+2 f^2 x^2 \sin \left(\frac{5}{2} (c+d x)\right) d^2+4 e f x \sin \left(\frac{5}{2} (c+d x)\right) d^2-14 e f \cos \left(\frac{3}{2} (c+d x)\right) d-14 f^2 x \cos \left(\frac{3}{2} (c+d x)\right) d+2 e f \cos \left(\frac{5}{2} (c+d x)\right) d+2 f^2 x \cos \left(\frac{5}{2} (c+d x)\right) d+16 e f \sin \left(\frac{1}{2} (c+d x)\right) d+16 f^2 x \sin \left(\frac{1}{2} (c+d x)\right) d+64 e f \log (i \cos (c+d x)+\sin (c+d x)+1) \sin \left(\frac{1}{2} (c+d x)\right) d+64 f^2 x \log (i \cos (c+d x)+\sin (c+d x)+1) \sin \left(\frac{1}{2} (c+d x)\right) d+14 e f \sin \left(\frac{3}{2} (c+d x)\right) d+14 f^2 x \sin \left(\frac{3}{2} (c+d x)\right) d+2 e f \sin \left(\frac{5}{2} (c+d x)\right) d+2 f^2 x \sin \left(\frac{5}{2} (c+d x)\right) d+15 f^2 \cos \left(\frac{3}{2} (c+d x)\right)+f^2 \cos \left(\frac{5}{2} (c+d x)\right)-8 \cos \left(\frac{1}{2} (c+d x)\right) \left(x \left(3 e^2+3 f x e+f^2 x^2\right) d^3+(3-2 i) (e+f x)^2 d^2-2 f (e+f x) d-8 f (e+f x) \log (i \cos (c+d x)+\sin (c+d x)+1) d-2 f^2\right)-16 f^2 \sin \left(\frac{1}{2} (c+d x)\right)+64 i f^2 \text{Li}_2(-i \cos (c+d x)-\sin (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 f^2 \sin \left(\frac{3}{2} (c+d x)\right)-f^2 \sin \left(\frac{5}{2} (c+d x)\right)}{16 a d^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{4 i f^2 \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}-\frac{2 f^2 \cos (c+d x)}{a d^3}+\frac{f^2 \sin (c+d x) \cos (c+d x)}{4 a d^3}-\frac{4 f (e+f x) \log \left(1-i e^{i (c+d x)}\right)}{a d^2}+\frac{f (e+f x) \sin ^2(c+d x)}{2 a d^2}-\frac{2 f (e+f x) \sin (c+d x)}{a d^2}+\frac{(e+f x)^2 \cos (c+d x)}{a d}+\frac{(e+f x)^2 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{f^2 x}{4 a d^2}+\frac{i (e+f x)^2}{a d}+\frac{(e+f x)^3}{2 a f}",1,"-1/16*(-6*d^2*e^2*Cos[(3*(c + d*x))/2] - 14*d*e*f*Cos[(3*(c + d*x))/2] + 15*f^2*Cos[(3*(c + d*x))/2] - 12*d^2*e*f*x*Cos[(3*(c + d*x))/2] - 14*d*f^2*x*Cos[(3*(c + d*x))/2] - 6*d^2*f^2*x^2*Cos[(3*(c + d*x))/2] - 2*d^2*e^2*Cos[(5*(c + d*x))/2] + 2*d*e*f*Cos[(5*(c + d*x))/2] + f^2*Cos[(5*(c + d*x))/2] - 4*d^2*e*f*x*Cos[(5*(c + d*x))/2] + 2*d*f^2*x*Cos[(5*(c + d*x))/2] - 2*d^2*f^2*x^2*Cos[(5*(c + d*x))/2] - 8*Cos[(c + d*x)/2]*(-2*f^2 - 2*d*f*(e + f*x) + (3 - 2*I)*d^2*(e + f*x)^2 + d^3*x*(3*e^2 + 3*e*f*x + f^2*x^2) - 8*d*f*(e + f*x)*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]) + (24 + 16*I)*d^2*e^2*Sin[(c + d*x)/2] + 16*d*e*f*Sin[(c + d*x)/2] - 16*f^2*Sin[(c + d*x)/2] - 24*d^3*e^2*x*Sin[(c + d*x)/2] + (48 + 32*I)*d^2*e*f*x*Sin[(c + d*x)/2] + 16*d*f^2*x*Sin[(c + d*x)/2] - 24*d^3*e*f*x^2*Sin[(c + d*x)/2] + (24 + 16*I)*d^2*f^2*x^2*Sin[(c + d*x)/2] - 8*d^3*f^2*x^3*Sin[(c + d*x)/2] + 64*d*e*f*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*Sin[(c + d*x)/2] + 64*d*f^2*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*Sin[(c + d*x)/2] + (64*I)*f^2*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 6*d^2*e^2*Sin[(3*(c + d*x))/2] + 14*d*e*f*Sin[(3*(c + d*x))/2] + 15*f^2*Sin[(3*(c + d*x))/2] - 12*d^2*e*f*x*Sin[(3*(c + d*x))/2] + 14*d*f^2*x*Sin[(3*(c + d*x))/2] - 6*d^2*f^2*x^2*Sin[(3*(c + d*x))/2] + 2*d^2*e^2*Sin[(5*(c + d*x))/2] + 2*d*e*f*Sin[(5*(c + d*x))/2] - f^2*Sin[(5*(c + d*x))/2] + 4*d^2*e*f*x*Sin[(5*(c + d*x))/2] + 2*d*f^2*x*Sin[(5*(c + d*x))/2] + 2*d^2*f^2*x^2*Sin[(5*(c + d*x))/2])/(a*d^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
193,1,298,158,1.543493,"\int \frac{(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Sin[c + d*x]^3)/(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(\sin \left(\frac{1}{2} (c+d x)\right) \left(2 \left(-3 c^2 f-d (e+f x) \sin (2 (c+d x))+6 c d e-4 f \sin (c+d x)-8 f \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 c f+6 d^2 e x+3 d^2 f x^2-8 d e-4 d f x\right)+8 d (e+f x) \cos (c+d x)-f \cos (2 (c+d x))\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left(2 \left(-3 c^2 f-d (e+f x) \sin (2 (c+d x))+6 c d e-4 f \sin (c+d x)-8 f \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 c f+6 d^2 e x+3 d^2 f x^2+4 d f x\right)+8 d (e+f x) \cos (c+d x)-f \cos (2 (c+d x))\right)\right)}{8 a d^2 (\sin (c+d x)+1)}","\frac{f \sin ^2(c+d x)}{4 a d^2}-\frac{f \sin (c+d x)}{a d^2}-\frac{2 f \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)\right)}{a d^2}+\frac{(e+f x) \cos (c+d x)}{a d}+\frac{(e+f x) \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{(e+f x) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 e x}{2 a}+\frac{3 f x^2}{4 a}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Sin[(c + d*x)/2]*(8*d*(e + f*x)*Cos[c + d*x] - f*Cos[2*(c + d*x)] + 2*(-8*d*e + 6*c*d*e + 4*c*f - 3*c^2*f + 6*d^2*e*x - 4*d*f*x + 3*d^2*f*x^2 - 8*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 4*f*Sin[c + d*x] - d*(e + f*x)*Sin[2*(c + d*x)])) + Cos[(c + d*x)/2]*(8*d*(e + f*x)*Cos[c + d*x] - f*Cos[2*(c + d*x)] + 2*(6*c*d*e + 4*c*f - 3*c^2*f + 6*d^2*e*x + 4*d*f*x + 3*d^2*f*x^2 - 8*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 4*f*Sin[c + d*x] - d*(e + f*x)*Sin[2*(c + d*x)]))))/(8*a*d^2*(1 + Sin[c + d*x]))","A",1
194,1,117,75,0.2026368,"\int \frac{\sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sin[c + d*x]^3/(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(\sin \left(\frac{1}{2} (c+d x)\right) (-\sin (2 (c+d x))+4 \cos (c+d x)+6 c+6 d x-8)+\cos \left(\frac{1}{2} (c+d x)\right) (-\sin (2 (c+d x))+4 \cos (c+d x)+6 c+6 d x)\right)}{4 a d (\sin (c+d x)+1)}","\frac{2 \cos (c+d x)}{a d}+\frac{\sin ^2(c+d x) \cos (c+d x)}{d (a \sin (c+d x)+a)}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x}{2 a}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Sin[(c + d*x)/2]*(-8 + 6*c + 6*d*x + 4*Cos[c + d*x] - Sin[2*(c + d*x)]) + Cos[(c + d*x)/2]*(6*c + 6*d*x + 4*Cos[c + d*x] - Sin[2*(c + d*x)])))/(4*a*d*(1 + Sin[c + d*x]))","A",1
195,0,0,31,6.7535367,"\int \frac{\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Sin[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])),x]","\int \frac{\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sin ^3(c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sin[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x]","A",-1
196,0,0,31,6.1924235,"\int \frac{\sin ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Sin[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\int \frac{\sin ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sin ^3(c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sin[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x]","A",-1
197,1,443,352,2.8247407,"\int \frac{(e+f x)^3 \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\frac{6 f (\cos (c)+i \sin (c)) \left(\frac{2 f (\cos (c)-i (\sin (c)+1)) (d (e+f x) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-i f \text{Li}_3(-i \cos (c+d x)-\sin (c+d x)))}{d^3}-\frac{(\sin (c)+i \cos (c)+1) (e+f x)^2 \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^3}{3 f}\right)}{\cos (c)+i (\sin (c)+1)}+\frac{3 i f \left(d^2 (e+f x)^2 \text{Li}_2(-\cos (c+d x)-i \sin (c+d x))+2 i d f (e+f x) \text{Li}_3(-\cos (c+d x)-i \sin (c+d x))-2 f^2 \text{Li}_4(-\cos (c+d x)-i \sin (c+d x))\right)}{d^3}-\frac{3 i f \left(d^2 (e+f x)^2 \text{Li}_2(\cos (c+d x)+i \sin (c+d x))+2 i d f (e+f x) \text{Li}_3(\cos (c+d x)+i \sin (c+d x))-2 f^2 \text{Li}_4(\cos (c+d x)+i \sin (c+d x))\right)}{d^3}-\frac{2 \sin \left(\frac{d x}{2}\right) (e+f x)^3}{\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)}-2 (e+f x)^3 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x))}{a d}","-\frac{12 f^3 \text{Li}_3\left(i e^{i (c+d x)}\right)}{a d^4}-\frac{6 i f^3 \text{Li}_4\left(-e^{i (c+d x)}\right)}{a d^4}+\frac{6 i f^3 \text{Li}_4\left(e^{i (c+d x)}\right)}{a d^4}+\frac{12 i f^2 (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}-\frac{6 f^2 (e+f x) \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}+\frac{3 i f (e+f x)^2 \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}-\frac{6 f (e+f x)^2 \log \left(1-i e^{i (c+d x)}\right)}{a d^2}+\frac{(e+f x)^3 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}+\frac{i (e+f x)^3}{a d}",1,"(-2*(e + f*x)^3*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] + ((3*I)*f*(d^2*(e + f*x)^2*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)*PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] - 2*f^2*PolyLog[4, -Cos[c + d*x] - I*Sin[c + d*x]]))/d^3 - ((3*I)*f*(d^2*(e + f*x)^2*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)*PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]] - 2*f^2*PolyLog[4, Cos[c + d*x] + I*Sin[c + d*x]]))/d^3 + (6*f*(Cos[c] + I*Sin[c])*(((e + f*x)^3*(Cos[c] - I*Sin[c]))/(3*f) - ((e + f*x)^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (2*f*(d*(e + f*x)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*f*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*(1 + Sin[c])))/d^3))/(Cos[c] + I*(1 + Sin[c])) - (2*(e + f*x)^3*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(a*d)","A",0
198,1,330,249,2.2032769,"\int \frac{(e+f x)^2 \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{\frac{2 i f \left(d (e+f x) \text{Li}_2\left(-e^{i (c+d x)}\right)+i f \text{Li}_3\left(-e^{i (c+d x)}\right)\right)}{d^2}+\frac{2 f \left(f \text{Li}_3\left(e^{i (c+d x)}\right)-i d (e+f x) \text{Li}_2\left(e^{i (c+d x)}\right)\right)}{d^2}+\frac{4 f (\cos (c)+i \sin (c)) \left(\frac{f (\cos (c)-i (\sin (c)+1)) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))}{d^2}-\frac{(\sin (c)+i \cos (c)+1) (e+f x) \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^2}{2 f}\right)}{\cos (c)+i (\sin (c)+1)}+(e+f x)^2 \log \left(1-e^{i (c+d x)}\right)-(e+f x)^2 \log \left(1+e^{i (c+d x)}\right)-\frac{2 \sin \left(\frac{d x}{2}\right) (e+f x)^2}{\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)}}{a d}","\frac{4 i f^2 \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}-\frac{2 f^2 \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^3}+\frac{2 f^2 \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}+\frac{2 i f (e+f x) \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}-\frac{2 i f (e+f x) \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}-\frac{4 f (e+f x) \log \left(1-i e^{i (c+d x)}\right)}{a d^2}+\frac{(e+f x)^2 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{2 (e+f x)^2 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}+\frac{i (e+f x)^2}{a d}",1,"((e + f*x)^2*Log[1 - E^(I*(c + d*x))] - (e + f*x)^2*Log[1 + E^(I*(c + d*x))] + ((2*I)*f*(d*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))] + I*f*PolyLog[3, -E^(I*(c + d*x))]))/d^2 + (2*f*((-I)*d*(e + f*x)*PolyLog[2, E^(I*(c + d*x))] + f*PolyLog[3, E^(I*(c + d*x))]))/d^2 + (4*f*(Cos[c] + I*Sin[c])*(((e + f*x)^2*(Cos[c] - I*Sin[c]))/(2*f) - ((e + f*x)*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2))/(Cos[c] + I*(1 + Sin[c])) - (2*(e + f*x)^2*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(a*d)","A",1
199,1,300,134,1.1004796,"\int \frac{(e+f x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Csc[c + d*x])/(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 d (e+f x) \sin \left(\frac{1}{2} (c+d x)\right)+d e \log \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)+f \left(i \left(\text{Li}_2\left(-e^{i (c+d x)}\right)-\text{Li}_2\left(e^{i (c+d x)}\right)\right)+(c+d x) \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+f (c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 f \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-c f \log \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)\right)}{a d^2 (\sin (c+d x)+1)}","\frac{i f \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}-\frac{i f \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}-\frac{2 f \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)\right)}{a d^2}+\frac{(e+f x) \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{2 (e+f x) \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-2*d*(e + f*x)*Sin[(c + d*x)/2] + f*(c + d*x)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 2*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + d*e*Log[Tan[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - c*f*Log[Tan[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))]))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(a*d^2*(1 + Sin[c + d*x]))","B",1
200,1,48,38,0.0697525,"\int \frac{\csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]/(a + a*Sin[c + d*x]),x]","-\frac{\sec (c+d x) \left(\sin (c+d x)+\sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left(\sqrt{\cos ^2(c+d x)}\right)-1\right)}{a d}","\frac{\cos (c+d x)}{d (a \sin (c+d x)+a)}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"-((Sec[c + d*x]*(-1 + ArcTanh[Sqrt[Cos[c + d*x]^2]]*Sqrt[Cos[c + d*x]^2] + Sin[c + d*x]))/(a*d))","A",1
201,0,0,29,11.9667455,"\int \frac{\csc (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Csc[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])),x]","\int \frac{\csc (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\csc (c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right)",0,"Integrate[Csc[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]","A",-1
202,0,0,29,13.5405912,"\int \frac{\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Csc[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\int \frac{\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\csc (c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right)",0,"Integrate[Csc[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x]","A",-1
203,1,1013,463,11.2220832,"\int \frac{(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{-d^3 x^3 \log \left(1-e^{-i (c+d x)}\right) f^3+d^3 x^3 \log \left(1+e^{-i (c+d x)}\right) f^3+3 \left(i d^2 \text{Li}_2\left(-e^{-i (c+d x)}\right) x^2+2 d \text{Li}_3\left(-e^{-i (c+d x)}\right) x-2 i \text{Li}_4\left(-e^{-i (c+d x)}\right)\right) f^3-3 i \left(d^2 \text{Li}_2\left(e^{-i (c+d x)}\right) x^2-2 i d \text{Li}_3\left(e^{-i (c+d x)}\right) x-2 \text{Li}_4\left(e^{-i (c+d x)}\right)\right) f^3-3 d^2 (d e-f) x^2 \log \left(1-e^{-i (c+d x)}\right) f^2+3 d^2 (d e+f) x^2 \log \left(1+e^{-i (c+d x)}\right) f^2+6 (d e+f) \left(i d x \text{Li}_2\left(-e^{-i (c+d x)}\right)+\text{Li}_3\left(-e^{-i (c+d x)}\right)\right) f^2-6 i (d e-f) \left(d x \text{Li}_2\left(e^{-i (c+d x)}\right)-i \text{Li}_3\left(e^{-i (c+d x)}\right)\right) f^2-3 d^2 e (d e-2 f) x \log \left(1-e^{-i (c+d x)}\right) f+3 d^2 e (d e+2 f) x \log \left(1+e^{-i (c+d x)}\right) f+3 i d e (d e+2 f) \text{Li}_2\left(-e^{-i (c+d x)}\right) f-3 i d e (d e-2 f) \text{Li}_2\left(e^{-i (c+d x)}\right) f-\frac{2 i d^3 (e+f x)^3}{-1+e^{2 i c}}+i d^2 e^2 (d e-3 f) \left(d x+i \log \left(1-e^{i (c+d x)}\right)\right)+d^2 e^2 (d e+3 f) \left(\log \left(1+e^{i (c+d x)}\right)-i d x\right)}{a d^4}-\frac{6 f (\cos (c)+i \sin (c)) \left(\frac{(\cos (c)-i \sin (c)) (e+f x)^3}{3 f}-\frac{\log (i \cos (c+d x)+\sin (c+d x)+1) (i \cos (c)+\sin (c)+1) (e+f x)^2}{d}+\frac{2 f (d (e+f x) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-i f \text{Li}_3(-i \cos (c+d x)-\sin (c+d x))) (\cos (c)-i (\sin (c)+1))}{d^3}\right)}{a d (\cos (c)+i (\sin (c)+1))}+\frac{\csc \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\sin \left(\frac{d x}{2}\right) e^3+3 f x \sin \left(\frac{d x}{2}\right) e^2+3 f^2 x^2 \sin \left(\frac{d x}{2}\right) e+f^3 x^3 \sin \left(\frac{d x}{2}\right)\right)}{2 a d}+\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\sin \left(\frac{d x}{2}\right) e^3+3 f x \sin \left(\frac{d x}{2}\right) e^2+3 f^2 x^2 \sin \left(\frac{d x}{2}\right) e+f^3 x^3 \sin \left(\frac{d x}{2}\right)\right)}{2 a d}+\frac{2 \left(\sin \left(\frac{d x}{2}\right) e^3+3 f x \sin \left(\frac{d x}{2}\right) e^2+3 f^2 x^2 \sin \left(\frac{d x}{2}\right) e+f^3 x^3 \sin \left(\frac{d x}{2}\right)\right)}{a d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}","\frac{12 f^3 \text{Li}_3\left(i e^{i (c+d x)}\right)}{a d^4}+\frac{3 f^3 \text{Li}_3\left(e^{2 i (c+d x)}\right)}{2 a d^4}+\frac{6 i f^3 \text{Li}_4\left(-e^{i (c+d x)}\right)}{a d^4}-\frac{6 i f^3 \text{Li}_4\left(e^{i (c+d x)}\right)}{a d^4}-\frac{12 i f^2 (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}-\frac{3 i f^2 (e+f x) \text{Li}_2\left(e^{2 i (c+d x)}\right)}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^3}-\frac{6 f^2 (e+f x) \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}-\frac{3 i f (e+f x)^2 \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}+\frac{3 i f (e+f x)^2 \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}+\frac{6 f (e+f x)^2 \log \left(1-i e^{i (c+d x)}\right)}{a d^2}+\frac{3 f (e+f x)^2 \log \left(1-e^{2 i (c+d x)}\right)}{a d^2}-\frac{(e+f x)^3 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{(e+f x)^3 \cot (c+d x)}{a d}+\frac{2 (e+f x)^3 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{2 i (e+f x)^3}{a d}",1,"(((-2*I)*d^3*(e + f*x)^3)/(-1 + E^((2*I)*c)) - 3*d^2*e*(d*e - 2*f)*f*x*Log[1 - E^((-I)*(c + d*x))] - 3*d^2*(d*e - f)*f^2*x^2*Log[1 - E^((-I)*(c + d*x))] - d^3*f^3*x^3*Log[1 - E^((-I)*(c + d*x))] + 3*d^2*e*f*(d*e + 2*f)*x*Log[1 + E^((-I)*(c + d*x))] + 3*d^2*f^2*(d*e + f)*x^2*Log[1 + E^((-I)*(c + d*x))] + d^3*f^3*x^3*Log[1 + E^((-I)*(c + d*x))] + I*d^2*e^2*(d*e - 3*f)*(d*x + I*Log[1 - E^(I*(c + d*x))]) + d^2*e^2*(d*e + 3*f)*((-I)*d*x + Log[1 + E^(I*(c + d*x))]) + (3*I)*d*e*f*(d*e + 2*f)*PolyLog[2, -E^((-I)*(c + d*x))] - (3*I)*d*e*(d*e - 2*f)*f*PolyLog[2, E^((-I)*(c + d*x))] + 6*f^2*(d*e + f)*(I*d*x*PolyLog[2, -E^((-I)*(c + d*x))] + PolyLog[3, -E^((-I)*(c + d*x))]) - (6*I)*(d*e - f)*f^2*(d*x*PolyLog[2, E^((-I)*(c + d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))]) + 3*f^3*(I*d^2*x^2*PolyLog[2, -E^((-I)*(c + d*x))] + 2*d*x*PolyLog[3, -E^((-I)*(c + d*x))] - (2*I)*PolyLog[4, -E^((-I)*(c + d*x))]) - (3*I)*f^3*(d^2*x^2*PolyLog[2, E^((-I)*(c + d*x))] - (2*I)*d*x*PolyLog[3, E^((-I)*(c + d*x))] - 2*PolyLog[4, E^((-I)*(c + d*x))]))/(a*d^4) - (6*f*(Cos[c] + I*Sin[c])*(((e + f*x)^3*(Cos[c] - I*Sin[c]))/(3*f) - ((e + f*x)^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (2*f*(d*(e + f*x)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*f*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*(1 + Sin[c])))/d^3))/(a*d*(Cos[c] + I*(1 + Sin[c]))) + (Csc[c/2]*Csc[c/2 + (d*x)/2]*(e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2]))/(2*a*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2]))/(2*a*d) + (2*(e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2]))/(a*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
204,1,693,327,8.5354651,"\int \frac{(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{4 f (\cos (c)+i \sin (c)) \left(\frac{f (\cos (c)-i (\sin (c)+1)) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))}{d^2}-\frac{(\sin (c)+i \cos (c)+1) (e+f x) \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^2}{2 f}\right)}{a d (\cos (c)+i (\sin (c)+1))}+\frac{-\frac{2 i d^2 (e+f x)^2}{-1+e^{2 i c}}-d^2 f^2 x^2 \log \left(1-e^{-i (c+d x)}\right)+d^2 f^2 x^2 \log \left(1+e^{-i (c+d x)}\right)+2 i f (d e+f) \text{Li}_2\left(-e^{-i (c+d x)}\right)-2 i f (d e-f) \text{Li}_2\left(e^{-i (c+d x)}\right)-2 d f x (d e-f) \log \left(1-e^{-i (c+d x)}\right)+2 d f x (d e+f) \log \left(1+e^{-i (c+d x)}\right)+i d e (d e-2 f) \left(d x+i \log \left(1-e^{i (c+d x)}\right)\right)+d e (d e+2 f) \left(\log \left(1+e^{i (c+d x)}\right)-i d x\right)+2 f^2 \left(i d x \text{Li}_2\left(-e^{-i (c+d x)}\right)+\text{Li}_3\left(-e^{-i (c+d x)}\right)\right)-2 i f^2 \left(d x \text{Li}_2\left(e^{-i (c+d x)}\right)-i \text{Li}_3\left(e^{-i (c+d x)}\right)\right)}{a d^3}+\frac{2 \left(e^2 \sin \left(\frac{d x}{2}\right)+2 e f x \sin \left(\frac{d x}{2}\right)+f^2 x^2 \sin \left(\frac{d x}{2}\right)\right)}{a d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\csc \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right) \left(e^2 \sin \left(\frac{d x}{2}\right)+2 e f x \sin \left(\frac{d x}{2}\right)+f^2 x^2 \sin \left(\frac{d x}{2}\right)\right)}{2 a d}+\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(e^2 \sin \left(\frac{d x}{2}\right)+2 e f x \sin \left(\frac{d x}{2}\right)+f^2 x^2 \sin \left(\frac{d x}{2}\right)\right)}{2 a d}","-\frac{4 i f^2 \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}-\frac{i f^2 \text{Li}_2\left(e^{2 i (c+d x)}\right)}{a d^3}+\frac{2 f^2 \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^3}-\frac{2 f^2 \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}-\frac{2 i f (e+f x) \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}+\frac{2 i f (e+f x) \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}+\frac{4 f (e+f x) \log \left(1-i e^{i (c+d x)}\right)}{a d^2}+\frac{2 f (e+f x) \log \left(1-e^{2 i (c+d x)}\right)}{a d^2}-\frac{(e+f x)^2 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{(e+f x)^2 \cot (c+d x)}{a d}+\frac{2 (e+f x)^2 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{2 i (e+f x)^2}{a d}",1,"(((-2*I)*d^2*(e + f*x)^2)/(-1 + E^((2*I)*c)) - 2*d*(d*e - f)*f*x*Log[1 - E^((-I)*(c + d*x))] - d^2*f^2*x^2*Log[1 - E^((-I)*(c + d*x))] + 2*d*f*(d*e + f)*x*Log[1 + E^((-I)*(c + d*x))] + d^2*f^2*x^2*Log[1 + E^((-I)*(c + d*x))] + I*d*e*(d*e - 2*f)*(d*x + I*Log[1 - E^(I*(c + d*x))]) + d*e*(d*e + 2*f)*((-I)*d*x + Log[1 + E^(I*(c + d*x))]) + (2*I)*f*(d*e + f)*PolyLog[2, -E^((-I)*(c + d*x))] - (2*I)*(d*e - f)*f*PolyLog[2, E^((-I)*(c + d*x))] + 2*f^2*(I*d*x*PolyLog[2, -E^((-I)*(c + d*x))] + PolyLog[3, -E^((-I)*(c + d*x))]) - (2*I)*f^2*(d*x*PolyLog[2, E^((-I)*(c + d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))]))/(a*d^3) - (4*f*(Cos[c] + I*Sin[c])*(((e + f*x)^2*(Cos[c] - I*Sin[c]))/(2*f) - ((e + f*x)*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2))/(a*d*(Cos[c] + I*(1 + Sin[c]))) + (Csc[c/2]*Csc[c/2 + (d*x)/2]*(e^2*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2]))/(2*a*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(e^2*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2]))/(2*a*d) + (2*(e^2*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2]))/(a*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
205,1,396,169,1.8035327,"\int \frac{(e+f x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Csc[c + d*x]^2)/(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 d (e+f x) \sin \left(\frac{1}{2} (c+d x)\right)+d (e+f x) \sin \left(\frac{1}{2} (c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right)+1\right)-d (e+f x) \cos \left(\frac{1}{2} (c+d x)\right) \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right)-2 d e \log \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 f \left(i \left(\text{Li}_2\left(-e^{i (c+d x)}\right)-\text{Li}_2\left(e^{i (c+d x)}\right)\right)+(c+d x) \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 f (c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 f \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 f \log (\sin (c+d x)) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 c f \log \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)\right)}{2 a d^2 (\sin (c+d x)+1)}","-\frac{i f \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}+\frac{i f \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}+\frac{2 f \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)\right)}{a d^2}+\frac{f \log (\sin (c+d x))}{a d^2}-\frac{(e+f x) \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}-\frac{(e+f x) \cot (c+d x)}{a d}+\frac{2 (e+f x) \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-(d*(e + f*x)*Cos[(c + d*x)/2]*(1 + Cot[(c + d*x)/2])) + 4*d*(e + f*x)*Sin[(c + d*x)/2] - 2*f*(c + d*x)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 2*f*Log[Sin[c + d*x]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 2*d*e*Log[Tan[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 2*c*f*Log[Tan[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 2*f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))]))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + d*(e + f*x)*Sin[(c + d*x)/2]*(1 + Tan[(c + d*x)/2])))/(2*a*d^2*(1 + Sin[c + d*x]))","B",1
206,1,57,51,0.199784,"\int \frac{\csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]^2/(a + a*Sin[c + d*x]),x]","\frac{\sec (c+d x) \left(2 \sin (c+d x)-\csc (c+d x)+\sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left(\sqrt{\cos ^2(c+d x)}\right)-1\right)}{a d}","-\frac{2 \cot (c+d x)}{a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}+\frac{\cot (c+d x)}{d (a \sin (c+d x)+a)}",1,"(Sec[c + d*x]*(-1 + ArcTanh[Sqrt[Cos[c + d*x]^2]]*Sqrt[Cos[c + d*x]^2] - Csc[c + d*x] + 2*Sin[c + d*x]))/(a*d)","A",1
207,0,0,31,25.2733077,"\int \frac{\csc ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Csc[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])),x]","\int \frac{\csc ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\csc ^2(c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right)",0,"Integrate[Csc[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])), x]","A",-1
208,0,0,31,49.3561271,"\int \frac{\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Csc[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\int \frac{\csc ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\csc ^2(c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right)",0,"Integrate[Csc[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])), x]","A",-1
209,1,1485,600,33.9628535,"\int \frac{(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{3 \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right) e^3}{2 a d}+\frac{9 f \left((c+d x) \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)\right)-c \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)+i \left(\text{Li}_2\left(-e^{i (c+d x)}\right)-\text{Li}_2\left(e^{i (c+d x)}\right)\right)\right) e^2}{2 a d^2}-\frac{3 f \csc (c) (\log (\cos (d x) \sin (c)+\cos (c) \sin (d x)) \sin (c)-d x \cos (c)) e^2}{a d^2 \left(\cos ^2(c)+\sin ^2(c)\right)}+\frac{3 f^2 \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right) e}{a d^3}-\frac{9 f^2 \left(d^2 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) x^2-i d \text{Li}_2(-\cos (c+d x)-i \sin (c+d x)) x+i d \text{Li}_2(\cos (c+d x)+i \sin (c+d x)) x+\text{Li}_3(-\cos (c+d x)-i \sin (c+d x))-\text{Li}_3(\cos (c+d x)+i \sin (c+d x))\right) e}{a d^3}+\frac{3 f^2 \csc (c) \sec (c) \left(d^2 e^{i \tan ^{-1}(\tan (c))} x^2+\frac{\left(i d x \left(2 \tan ^{-1}(\tan (c))-\pi \right)-\pi  \log \left(1+e^{-2 i d x}\right)-2 \left(d x+\tan ^{-1}(\tan (c))\right) \log \left(1-e^{2 i \left(d x+\tan ^{-1}(\tan (c))\right)}\right)+\pi  \log (\cos (d x))+2 \tan ^{-1}(\tan (c)) \log \left(\sin \left(d x+\tan ^{-1}(\tan (c))\right)\right)+i \text{Li}_2\left(e^{2 i \left(d x+\tan ^{-1}(\tan (c))\right)}\right)\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}\right) e}{a d^3 \sqrt{\sec ^2(c) \left(\cos ^2(c)+\sin ^2(c)\right)}}+\frac{3 f^3 \left((c+d x) \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)\right)-c \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)+i \left(\text{Li}_2\left(-e^{i (c+d x)}\right)-\text{Li}_2\left(e^{i (c+d x)}\right)\right)\right)}{a d^4}+\frac{e^{i c} f^3 \csc (c) \left(2 d^3 e^{-2 i c} x^3+3 i d^2 \left(1-e^{-2 i c}\right) \log \left(1-e^{-i (c+d x)}\right) x^2+3 i d^2 \left(1-e^{-2 i c}\right) \log \left(1+e^{-i (c+d x)}\right) x^2-6 e^{-2 i c} \left(-1+e^{2 i c}\right) \left(d x \text{Li}_2\left(-e^{-i (c+d x)}\right)-i \text{Li}_3\left(-e^{-i (c+d x)}\right)\right)-6 e^{-2 i c} \left(-1+e^{2 i c}\right) \left(d x \text{Li}_2\left(e^{-i (c+d x)}\right)-i \text{Li}_3\left(e^{-i (c+d x)}\right)\right)\right)}{2 a d^4}+\frac{3 f^3 \left(-2 d^3 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) x^3+3 i d^2 \text{Li}_2(-\cos (c+d x)-i \sin (c+d x)) x^2-3 i d^2 \text{Li}_2(\cos (c+d x)+i \sin (c+d x)) x^2-6 d \text{Li}_3(-\cos (c+d x)-i \sin (c+d x)) x+6 d \text{Li}_3(\cos (c+d x)+i \sin (c+d x)) x-6 i \text{Li}_4(-\cos (c+d x)-i \sin (c+d x))+6 i \text{Li}_4(\cos (c+d x)+i \sin (c+d x))\right)}{2 a d^4}+\frac{6 f (\cos (c)+i \sin (c)) \left(\frac{(\cos (c)-i \sin (c)) (e+f x)^3}{3 f}-\frac{\log (i \cos (c+d x)+\sin (c+d x)+1) (i \cos (c)+\sin (c)+1) (e+f x)^2}{d}+\frac{2 f (d (e+f x) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-i f \text{Li}_3(-i \cos (c+d x)-\sin (c+d x))) (\cos (c)-i (\sin (c)+1))}{d^3}\right)}{a d (\cos (c)+i (\sin (c)+1))}+\frac{\csc (c) \csc ^2(c+d x) \left(\sin (d x) e^3+3 f x \sin (d x) e^2+3 f^2 x^2 \sin (d x) e+f^3 x^3 \sin (d x)\right)}{2 a d}+\frac{\csc (c) \csc (c+d x) \left(-d \cos (c) e^3-2 d \sin (d x) e^3-3 d f x \cos (c) e^2-3 f \sin (c) e^2-6 d f x \sin (d x) e^2-3 d f^2 x^2 \cos (c) e-6 f^2 x \sin (c) e-6 d f^2 x^2 \sin (d x) e-d f^3 x^3 \cos (c)-3 f^3 x^2 \sin (c)-2 d f^3 x^3 \sin (d x)\right)}{2 a d^2}-\frac{2 \left(\sin \left(\frac{d x}{2}\right) e^3+3 f x \sin \left(\frac{d x}{2}\right) e^2+3 f^2 x^2 \sin \left(\frac{d x}{2}\right) e+f^3 x^3 \sin \left(\frac{d x}{2}\right)\right)}{a d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}","\frac{3 i f^3 \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^4}-\frac{3 i f^3 \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^4}-\frac{12 f^3 \text{Li}_3\left(i e^{i (c+d x)}\right)}{a d^4}-\frac{3 f^3 \text{Li}_3\left(e^{2 i (c+d x)}\right)}{2 a d^4}-\frac{9 i f^3 \text{Li}_4\left(-e^{i (c+d x)}\right)}{a d^4}+\frac{9 i f^3 \text{Li}_4\left(e^{i (c+d x)}\right)}{a d^4}+\frac{12 i f^2 (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}+\frac{3 i f^2 (e+f x) \text{Li}_2\left(e^{2 i (c+d x)}\right)}{a d^3}-\frac{9 f^2 (e+f x) \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^3}+\frac{9 f^2 (e+f x) \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d^3}+\frac{9 i f (e+f x)^2 \text{Li}_2\left(-e^{i (c+d x)}\right)}{2 a d^2}-\frac{9 i f (e+f x)^2 \text{Li}_2\left(e^{i (c+d x)}\right)}{2 a d^2}-\frac{6 f (e+f x)^2 \log \left(1-i e^{i (c+d x)}\right)}{a d^2}-\frac{3 f (e+f x)^2 \log \left(1-e^{2 i (c+d x)}\right)}{a d^2}-\frac{3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}+\frac{(e+f x)^3 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}+\frac{(e+f x)^3 \cot (c+d x)}{a d}-\frac{3 (e+f x)^3 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{2 i (e+f x)^3}{a d}",1,"(3*e^3*Log[Tan[(c + d*x)/2]])/(2*a*d) + (3*e*f^2*Log[Tan[(c + d*x)/2]])/(a*d^3) + (9*e^2*f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) - c*Log[Tan[(c + d*x)/2]] + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/(2*a*d^2) + (3*f^3*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) - c*Log[Tan[(c + d*x)/2]] + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/(a*d^4) + (E^(I*c)*f^3*Csc[c]*((2*d^3*x^3)/E^((2*I)*c) + (3*I)*d^2*(1 - E^((-2*I)*c))*x^2*Log[1 - E^((-I)*(c + d*x))] + (3*I)*d^2*(1 - E^((-2*I)*c))*x^2*Log[1 + E^((-I)*(c + d*x))] - (6*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, -E^((-I)*(c + d*x))] - I*PolyLog[3, -E^((-I)*(c + d*x))]))/E^((2*I)*c) - (6*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, E^((-I)*(c + d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))]))/E^((2*I)*c)))/(2*a*d^4) - (9*e*f^2*(d^2*x^2*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - I*d*x*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] + I*d*x*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] + PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] - PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]]))/(a*d^3) + (3*f^3*(-2*d^3*x^3*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] + (3*I)*d^2*x^2*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] - (3*I)*d^2*x^2*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] - 6*d*x*PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] + 6*d*x*PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]] - (6*I)*PolyLog[4, -Cos[c + d*x] - I*Sin[c + d*x]] + (6*I)*PolyLog[4, Cos[c + d*x] + I*Sin[c + d*x]]))/(2*a*d^4) - (3*e^2*f*Csc[c]*(-(d*x*Cos[c]) + Log[Cos[d*x]*Sin[c] + Cos[c]*Sin[d*x]]*Sin[c]))/(a*d^2*(Cos[c]^2 + Sin[c]^2)) + (6*f*(Cos[c] + I*Sin[c])*(((e + f*x)^3*(Cos[c] - I*Sin[c]))/(3*f) - ((e + f*x)^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (2*f*(d*(e + f*x)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*f*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*(1 + Sin[c])))/d^3))/(a*d*(Cos[c] + I*(1 + Sin[c]))) + (Csc[c]*Csc[c + d*x]^2*(e^3*Sin[d*x] + 3*e^2*f*x*Sin[d*x] + 3*e*f^2*x^2*Sin[d*x] + f^3*x^3*Sin[d*x]))/(2*a*d) + (Csc[c]*Csc[c + d*x]*(-(d*e^3*Cos[c]) - 3*d*e^2*f*x*Cos[c] - 3*d*e*f^2*x^2*Cos[c] - d*f^3*x^3*Cos[c] - 3*e^2*f*Sin[c] - 6*e*f^2*x*Sin[c] - 3*f^3*x^2*Sin[c] - 2*d*e^3*Sin[d*x] - 6*d*e^2*f*x*Sin[d*x] - 6*d*e*f^2*x^2*Sin[d*x] - 2*d*f^3*x^3*Sin[d*x]))/(2*a*d^2) - (2*(e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2]))/(a*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])) + (3*e*f^2*Csc[c]*Sec[c]*(d^2*E^(I*ArcTan[Tan[c]])*x^2 + ((I*d*x*(-Pi + 2*ArcTan[Tan[c]]) - Pi*Log[1 + E^((-2*I)*d*x)] - 2*(d*x + ArcTan[Tan[c]])*Log[1 - E^((2*I)*(d*x + ArcTan[Tan[c]]))] + Pi*Log[Cos[d*x]] + 2*ArcTan[Tan[c]]*Log[Sin[d*x + ArcTan[Tan[c]]]] + I*PolyLog[2, E^((2*I)*(d*x + ArcTan[Tan[c]]))])*Tan[c])/Sqrt[1 + Tan[c]^2]))/(a*d^3*Sqrt[Sec[c]^2*(Cos[c]^2 + Sin[c]^2)])","B",0
210,1,951,392,19.5069164,"\int \frac{(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\frac{32 f (\cos (c)+i \sin (c)) \left(\frac{(\cos (c)-i \sin (c)) (e+f x)^2}{2 f}-\frac{\log (i \cos (c+d x)+\sin (c+d x)+1) (i \cos (c)+\sin (c)+1) (e+f x)}{d}+\frac{f \text{Li}_2(-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (\sin (c)+1))}{d^2}\right) d^2}{\cos (c)+i (\sin (c)+1)}-\frac{(e+f x) \csc (c) \csc ^2(c+d x) \left(2 f \cos \left(\frac{d x}{2}\right)+2 f \cos \left(\frac{3 d x}{2}\right)+5 d e \cos \left(c-\frac{d x}{2}\right)+5 d f x \cos \left(c-\frac{d x}{2}\right)-d e \cos \left(c+\frac{d x}{2}\right)-d f x \cos \left(c+\frac{d x}{2}\right)-2 f \cos \left(2 c+\frac{d x}{2}\right)+d e \cos \left(c+\frac{3 d x}{2}\right)+d f x \cos \left(c+\frac{3 d x}{2}\right)-2 f \cos \left(2 c+\frac{3 d x}{2}\right)-3 d e \cos \left(3 c+\frac{3 d x}{2}\right)-3 d f x \cos \left(3 c+\frac{3 d x}{2}\right)-4 d e \cos \left(c+\frac{5 d x}{2}\right)-4 d f x \cos \left(c+\frac{5 d x}{2}\right)+2 d e \cos \left(3 c+\frac{5 d x}{2}\right)+2 d f x \cos \left(3 c+\frac{5 d x}{2}\right)+d e \sin \left(\frac{d x}{2}\right)+d f x \sin \left(\frac{d x}{2}\right)+d e \sin \left(\frac{3 d x}{2}\right)+d f x \sin \left(\frac{3 d x}{2}\right)+2 f \sin \left(c-\frac{d x}{2}\right)+2 f \sin \left(c+\frac{d x}{2}\right)+3 d e \sin \left(2 c+\frac{d x}{2}\right)+3 d f x \sin \left(2 c+\frac{d x}{2}\right)+2 f \sin \left(c+\frac{3 d x}{2}\right)+d e \sin \left(2 c+\frac{3 d x}{2}\right)+d f x \sin \left(2 c+\frac{3 d x}{2}\right)-2 f \sin \left(3 c+\frac{3 d x}{2}\right)-2 d e \sin \left(2 c+\frac{5 d x}{2}\right)-2 d f x \sin \left(2 c+\frac{5 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)}+8 \left(i f^2 x^2 d^2+2 i e f x d^2-3 e^2 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) d^2-3 f^2 x^2 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) d^2-6 e f x \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) d^2+f^2 x^2 \cot (c) d^2+2 e f x \cot (c) d^2-2 e f \log (-\cos (2 (c+d x))-i \sin (2 (c+d x))+1) d-2 f^2 x \log (-\cos (2 (c+d x))-i \sin (2 (c+d x))+1) d+3 i f (e+f x) \text{Li}_2(-\cos (c+d x)-i \sin (c+d x)) d-3 i f (e+f x) \text{Li}_2(\cos (c+d x)+i \sin (c+d x)) d-2 f^2 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x))+i f^2 \text{Li}_2(\cos (2 (c+d x))+i \sin (2 (c+d x)))-3 f^2 \text{Li}_3(-\cos (c+d x)-i \sin (c+d x))+3 f^2 \text{Li}_3(\cos (c+d x)+i \sin (c+d x))\right)}{8 a d^3}","\frac{4 i f^2 \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}+\frac{i f^2 \text{Li}_2\left(e^{2 i (c+d x)}\right)}{a d^3}-\frac{3 f^2 \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^3}+\frac{3 f^2 \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}-\frac{f^2 \tanh ^{-1}(\cos (c+d x))}{a d^3}+\frac{3 i f (e+f x) \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}-\frac{3 i f (e+f x) \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}-\frac{4 f (e+f x) \log \left(1-i e^{i (c+d x)}\right)}{a d^2}-\frac{2 f (e+f x) \log \left(1-e^{2 i (c+d x)}\right)}{a d^2}-\frac{f (e+f x) \csc (c+d x)}{a d^2}+\frac{(e+f x)^2 \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}+\frac{(e+f x)^2 \cot (c+d x)}{a d}-\frac{3 (e+f x)^2 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{2 i (e+f x)^2}{a d}",1,"(8*((2*I)*d^2*e*f*x + I*d^2*f^2*x^2 - 3*d^2*e^2*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - 2*f^2*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - 6*d^2*e*f*x*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - 3*d^2*f^2*x^2*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] + 2*d^2*e*f*x*Cot[c] + d^2*f^2*x^2*Cot[c] - 2*d*e*f*Log[1 - Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)]] - 2*d*f^2*x*Log[1 - Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)]] + (3*I)*d*f*(e + f*x)*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] - (3*I)*d*f*(e + f*x)*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] + I*f^2*PolyLog[2, Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]] - 3*f^2*PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] + 3*f^2*PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]]) + (32*d^2*f*(Cos[c] + I*Sin[c])*(((e + f*x)^2*(Cos[c] - I*Sin[c]))/(2*f) - ((e + f*x)*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2))/(Cos[c] + I*(1 + Sin[c])) - (d*(e + f*x)*Csc[c]*Csc[c + d*x]^2*(2*f*Cos[(d*x)/2] + 2*f*Cos[(3*d*x)/2] + 5*d*e*Cos[c - (d*x)/2] + 5*d*f*x*Cos[c - (d*x)/2] - d*e*Cos[c + (d*x)/2] - d*f*x*Cos[c + (d*x)/2] - 2*f*Cos[2*c + (d*x)/2] + d*e*Cos[c + (3*d*x)/2] + d*f*x*Cos[c + (3*d*x)/2] - 2*f*Cos[2*c + (3*d*x)/2] - 3*d*e*Cos[3*c + (3*d*x)/2] - 3*d*f*x*Cos[3*c + (3*d*x)/2] - 4*d*e*Cos[c + (5*d*x)/2] - 4*d*f*x*Cos[c + (5*d*x)/2] + 2*d*e*Cos[3*c + (5*d*x)/2] + 2*d*f*x*Cos[3*c + (5*d*x)/2] + d*e*Sin[(d*x)/2] + d*f*x*Sin[(d*x)/2] + d*e*Sin[(3*d*x)/2] + d*f*x*Sin[(3*d*x)/2] + 2*f*Sin[c - (d*x)/2] + 2*f*Sin[c + (d*x)/2] + 3*d*e*Sin[2*c + (d*x)/2] + 3*d*f*x*Sin[2*c + (d*x)/2] + 2*f*Sin[c + (3*d*x)/2] + d*e*Sin[2*c + (3*d*x)/2] + d*f*x*Sin[2*c + (3*d*x)/2] - 2*f*Sin[3*c + (3*d*x)/2] - 2*d*e*Sin[2*c + (5*d*x)/2] - 2*d*f*x*Sin[2*c + (5*d*x)/2]))/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(8*a*d^3)","B",0
211,1,484,216,3.8129285,"\int \frac{(e+f x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Csc[c + d*x]^3)/(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(-16 d (e+f x) \sin \left(\frac{1}{2} (c+d x)\right)-d (e+f x) \left(\cot \left(\frac{1}{2} (c+d x)\right)+1\right) \csc \left(\frac{1}{2} (c+d x)\right)+d (e+f x) \left(\tan \left(\frac{1}{2} (c+d x)\right)+1\right) \sec \left(\frac{1}{2} (c+d x)\right)-2 \tan \left(\frac{1}{2} (c+d x)\right) (2 d (e+f x)+f) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 \cot \left(\frac{1}{2} (c+d x)\right) (2 d (e+f x)-f) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 d e \log \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)+12 f \left(i \left(\text{Li}_2\left(-e^{i (c+d x)}\right)-\text{Li}_2\left(e^{i (c+d x)}\right)\right)+(c+d x) \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 f (c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-16 f \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-8 f \log (\sin (c+d x)) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-12 c f \log \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)\right)}{8 a d^2 (\sin (c+d x)+1)}","\frac{3 i f \text{Li}_2\left(-e^{i (c+d x)}\right)}{2 a d^2}-\frac{3 i f \text{Li}_2\left(e^{i (c+d x)}\right)}{2 a d^2}-\frac{f \csc (c+d x)}{2 a d^2}-\frac{2 f \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)\right)}{a d^2}-\frac{f \log (\sin (c+d x))}{a d^2}+\frac{(e+f x) \cot \left(\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right)}{a d}+\frac{(e+f x) \cot (c+d x)}{a d}-\frac{3 (e+f x) \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{(e+f x) \cot (c+d x) \csc (c+d x)}{2 a d}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-(d*(e + f*x)*(1 + Cot[(c + d*x)/2])*Csc[(c + d*x)/2]) - 16*d*(e + f*x)*Sin[(c + d*x)/2] + 8*f*(c + d*x)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 2*(-f + 2*d*(e + f*x))*Cot[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 16*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 8*f*Log[Sin[c + d*x]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*d*e*Log[Tan[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 12*c*f*Log[Tan[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))]))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 2*(f + 2*d*(e + f*x))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*Tan[(c + d*x)/2] + d*(e + f*x)*Sec[(c + d*x)/2]*(1 + Tan[(c + d*x)/2])))/(8*a*d^2*(1 + Sin[c + d*x]))","B",1
212,1,85,82,0.5494445,"\int \frac{\csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]^3/(a + a*Sin[c + d*x]),x]","-\frac{4 \tan (c+d x)-4 \csc (2 (c+d x))-3 \sec (c+d x)+\csc ^2(c+d x) \sec (c+d x)+3 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \tanh ^{-1}\left(\sqrt{\cos ^2(c+d x)}\right)}{2 a d}","\frac{2 \cot (c+d x)}{a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{\cot (c+d x) \csc (c+d x)}{d (a \sin (c+d x)+a)}",1,"-1/2*(-4*Csc[2*(c + d*x)] - 3*Sec[c + d*x] + 3*ArcTanh[Sqrt[Cos[c + d*x]^2]]*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x] + Csc[c + d*x]^2*Sec[c + d*x] + 4*Tan[c + d*x])/(a*d)","A",1
213,0,0,31,83.9376028,"\int \frac{\csc ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Csc[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])),x]","\int \frac{\csc ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\csc ^3(c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right)",0,"Integrate[Csc[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x]","A",-1
214,0,0,31,176.7806632,"\int \frac{\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Csc[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\int \frac{\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\csc ^3(c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right)",0,"Integrate[Csc[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x]","A",-1
215,0,0,31,8.7543216,"\int \frac{(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx","\text{Int}\left(\frac{\sin ^2(c+d x) (e+f x)^m}{a \sin (c+d x)+a},x\right)",0,"Integrate[((e + f*x)^m*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]), x]","A",-1
216,0,0,29,1.8586557,"\int \frac{(e+f x)^m \sin (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \sin (c+d x)}{a+a \sin (c+d x)} \, dx","\text{Int}\left(\frac{\sin (c+d x) (e+f x)^m}{a \sin (c+d x)+a},x\right)",0,"Integrate[((e + f*x)^m*Sin[c + d*x])/(a + a*Sin[c + d*x]), x]","A",-1
217,0,0,23,0.7898001,"\int \frac{(e+f x)^m}{a+a \sin (c+d x)} \, dx","Integrate[(e + f*x)^m/(a + a*Sin[c + d*x]),x]","\int \frac{(e+f x)^m}{a+a \sin (c+d x)} \, dx","\text{Int}\left(\frac{(e+f x)^m}{a \sin (c+d x)+a},x\right)",0,"Integrate[(e + f*x)^m/(a + a*Sin[c + d*x]), x]","A",-1
218,0,0,29,37.1909846,"\int \frac{(e+f x)^m \csc (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \csc (c+d x)}{a+a \sin (c+d x)} \, dx","\text{Int}\left(\frac{\csc (c+d x) (e+f x)^m}{a \sin (c+d x)+a},x\right)",0,"Integrate[((e + f*x)^m*Csc[c + d*x])/(a + a*Sin[c + d*x]), x]","A",-1
219,0,0,31,34.7653597,"\int \frac{(e+f x)^m \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx","\text{Int}\left(\frac{\csc ^2(c+d x) (e+f x)^m}{a \sin (c+d x)+a},x\right)",0,"Integrate[((e + f*x)^m*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]), x]","A",-1
220,1,956,544,3.6180019,"\int \frac{(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{x \left(4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right)}{4 b}-\frac{a \left(2 \sqrt{b^2-a^2} e^3 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) d^3+\sqrt{a^2-b^2} f^3 x^3 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e^2 f x \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3-\sqrt{a^2-b^2} f^3 x^3 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e^2 f x \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^2+3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d^2+6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d+6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d-6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d-6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d+6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{b \sqrt{-\left(a^2-b^2\right)^2} d^4}","-\frac{6 a f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^4 \sqrt{a^2-b^2}}+\frac{6 a f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^4 \sqrt{a^2-b^2}}+\frac{6 i a f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^3 \sqrt{a^2-b^2}}-\frac{6 i a f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^3 \sqrt{a^2-b^2}}+\frac{3 a f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^2 \sqrt{a^2-b^2}}-\frac{3 a f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^2 \sqrt{a^2-b^2}}+\frac{i a (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d \sqrt{a^2-b^2}}-\frac{i a (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d \sqrt{a^2-b^2}}+\frac{(e+f x)^4}{4 b f}",1,"(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))/(4*b) - (a*(2*Sqrt[-a^2 + b^2]*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))]))/(b*Sqrt[-(a^2 - b^2)^2]*d^4)","A",0
221,1,445,408,2.2855683,"\int \frac{(e+f x)^2 \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{x \left(3 e^2+3 e f x+f^2 x^2\right)}{3 b}-\frac{i a \left(-i \left(d^2 \left(2 e^2 \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right)+f x \sqrt{a^2-b^2} (2 e+f x) \left(\log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-\log \left(1+\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}+i a}\right)\right)\right)+2 f^2 \sqrt{a^2-b^2} \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-2 f^2 \sqrt{a^2-b^2} \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)-2 d f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)+2 d f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{b d^3 \sqrt{-\left(a^2-b^2\right)^2}}","\frac{2 i a f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^3 \sqrt{a^2-b^2}}-\frac{2 i a f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^3 \sqrt{a^2-b^2}}+\frac{2 a f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^2 \sqrt{a^2-b^2}}-\frac{2 a f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^2 \sqrt{a^2-b^2}}+\frac{i a (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d \sqrt{a^2-b^2}}-\frac{i a (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d \sqrt{a^2-b^2}}+\frac{(e+f x)^3}{3 b f}",1,"(x*(3*e^2 + 3*e*f*x + f^2*x^2))/(3*b) - (I*a*(-2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - I*(d^2*(2*Sqrt[-a^2 + b^2]*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(2*e + f*x)*(Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])])) + 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))])))/(b*Sqrt[-(a^2 - b^2)^2]*d^3)","A",0
222,1,299,267,1.7168639,"\int \frac{(e+f x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{x (2 e+f x)}{2 b}-\frac{i a \left(-i d \left(2 e \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right)+f x \sqrt{a^2-b^2} \left(\log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-\log \left(1+\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}+i a}\right)\right)\right)-f \sqrt{a^2-b^2} \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)+f \sqrt{a^2-b^2} \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{b d^2 \sqrt{-\left(a^2-b^2\right)^2}}","\frac{a f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^2 \sqrt{a^2-b^2}}-\frac{a f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^2 \sqrt{a^2-b^2}}+\frac{i a (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d \sqrt{a^2-b^2}}-\frac{i a (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d \sqrt{a^2-b^2}}+\frac{e x}{b}+\frac{f x^2}{2 b}",1,"(x*(2*e + f*x))/(2*b) - (I*a*((-I)*d*(2*Sqrt[-a^2 + b^2]*e*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])])) - Sqrt[a^2 - b^2]*f*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*f*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))]))/(b*Sqrt[-(a^2 - b^2)^2]*d^2)","A",0
223,1,59,57,0.1219454,"\int \frac{\sin (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Sin[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{-\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \sqrt{a^2-b^2}}+\frac{c}{d}+x}{b}","\frac{x}{b}-\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d \sqrt{a^2-b^2}}",1,"(c/d + x - (2*a*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d))/b","A",1
224,1,1020,643,7.6187354,"\int \frac{(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{-a x \left(4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right) d^4-4 b (e+f x) \left(d^2 (e+f x)^2-6 f^2\right) \cos (c+d x) d+\frac{4 a^2 \left(2 \sqrt{b^2-a^2} e^3 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) d^3+\sqrt{a^2-b^2} f^3 x^3 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e^2 f x \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3-\sqrt{a^2-b^2} f^3 x^3 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e^2 f x \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^2+3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d^2+6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d+6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d-6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d-6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d+6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{\sqrt{-\left(a^2-b^2\right)^2}}+12 b f \left(d^2 (e+f x)^2-2 f^2\right) \sin (c+d x)}{4 b^2 d^4}","\frac{6 a^2 f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^4 \sqrt{a^2-b^2}}-\frac{6 a^2 f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^4 \sqrt{a^2-b^2}}-\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^3 \sqrt{a^2-b^2}}+\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^3 \sqrt{a^2-b^2}}-\frac{3 a^2 f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^2 \sqrt{a^2-b^2}}+\frac{3 a^2 f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^2 \sqrt{a^2-b^2}}-\frac{i a^2 (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d \sqrt{a^2-b^2}}+\frac{i a^2 (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b^2 d \sqrt{a^2-b^2}}-\frac{a (e+f x)^4}{4 b^2 f}-\frac{6 f^3 \sin (c+d x)}{b d^4}+\frac{6 f^2 (e+f x) \cos (c+d x)}{b d^3}+\frac{3 f (e+f x)^2 \sin (c+d x)}{b d^2}-\frac{(e+f x)^3 \cos (c+d x)}{b d}",1,"(-(a*d^4*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)) - 4*b*d*(e + f*x)*(-6*f^2 + d^2*(e + f*x)^2)*Cos[c + d*x] + (4*a^2*(2*Sqrt[-a^2 + b^2]*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))]))/Sqrt[-(a^2 - b^2)^2] + 12*b*f*(-2*f^2 + d^2*(e + f*x)^2)*Sin[c + d*x])/(4*b^2*d^4)","A",0
225,1,531,479,3.3510569,"\int \frac{(e+f x)^2 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\frac{3 i a^2 \left(-i \left(d^2 \left(2 e^2 \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right)+f x \sqrt{a^2-b^2} (2 e+f x) \left(\log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-\log \left(1+\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}+i a}\right)\right)\right)+2 f^2 \sqrt{a^2-b^2} \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-2 f^2 \sqrt{a^2-b^2} \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)-2 d f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)+2 d f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{d^3 \sqrt{-\left(a^2-b^2\right)^2}}-a x \left(3 e^2+3 e f x+f^2 x^2\right)-\frac{3 b \cos (d x) \left(\cos (c) \left(d^2 (e+f x)^2-2 f^2\right)-2 d f \sin (c) (e+f x)\right)}{d^3}+\frac{3 b \sin (d x) \left(\sin (c) \left(d^2 (e+f x)^2-2 f^2\right)+2 d f \cos (c) (e+f x)\right)}{d^3}}{3 b^2}","-\frac{2 i a^2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^3 \sqrt{a^2-b^2}}+\frac{2 i a^2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^3 \sqrt{a^2-b^2}}-\frac{2 a^2 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^2 \sqrt{a^2-b^2}}+\frac{2 a^2 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^2 \sqrt{a^2-b^2}}-\frac{i a^2 (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d \sqrt{a^2-b^2}}+\frac{i a^2 (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b^2 d \sqrt{a^2-b^2}}-\frac{a (e+f x)^3}{3 b^2 f}+\frac{2 f^2 \cos (c+d x)}{b d^3}+\frac{2 f (e+f x) \sin (c+d x)}{b d^2}-\frac{(e+f x)^2 \cos (c+d x)}{b d}",1,"(-(a*x*(3*e^2 + 3*e*f*x + f^2*x^2)) + ((3*I)*a^2*(-2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - I*(d^2*(2*Sqrt[-a^2 + b^2]*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(2*e + f*x)*(Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])])) + 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))])))/(Sqrt[-(a^2 - b^2)^2]*d^3) - (3*b*Cos[d*x]*((-2*f^2 + d^2*(e + f*x)^2)*Cos[c] - 2*d*f*(e + f*x)*Sin[c]))/d^3 + (3*b*(2*d*f*(e + f*x)*Cos[c] + (-2*f^2 + d^2*(e + f*x)^2)*Sin[c])*Sin[d*x])/d^3)/(3*b^2)","A",1
226,1,709,311,7.3895027,"\int \frac{(e+f x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\frac{2 a^2 d (e+f x) \left(\frac{2 (d e-c f) \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{i f \left(\text{Li}_2\left(\frac{a \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a+i \left(b+\sqrt{b^2-a^2}\right)}\right)+\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{b^2-a^2}-i a+b}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\text{Li}_2\left(\frac{a \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)}{a-i \left(b+\sqrt{b^2-a^2}\right)}\right)+\log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{b^2-a^2}+i a+b}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\text{Li}_2\left(\frac{a \left(\tan \left(\frac{1}{2} (c+d x)\right)+i\right)}{i a-b+\sqrt{b^2-a^2}}\right)+\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}-a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{b^2-a^2}+i a-b}\right)\right)}{\sqrt{b^2-a^2}}-\frac{i f \left(\text{Li}_2\left(\frac{i \tan \left(\frac{1}{2} (c+d x)\right) a+a}{a+i \left(\sqrt{b^2-a^2}-b\right)}\right)+\log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{-\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{-\sqrt{b^2-a^2}+i a+b}\right)\right)}{\sqrt{b^2-a^2}}\right)}{i f \log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)-i f \log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right)-c f+d e}+a (c+d x) (c f-d (2 e+f x))-2 b d (e+f x) \cos (c+d x)+2 b f \sin (c+d x)}{2 b^2 d^2}","-\frac{a^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^2 \sqrt{a^2-b^2}}+\frac{a^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^2 \sqrt{a^2-b^2}}-\frac{i a^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d \sqrt{a^2-b^2}}+\frac{i a^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b^2 d \sqrt{a^2-b^2}}-\frac{a e x}{b^2}-\frac{a f x^2}{2 b^2}+\frac{f \sin (c+d x)}{b d^2}-\frac{(e+f x) \cos (c+d x)}{b d}",1,"(a*(c + d*x)*(c*f - d*(2*e + f*x)) - 2*b*d*(e + f*x)*Cos[c + d*x] + (2*a^2*d*(e + f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(-b + Sqrt[-a^2 + b^2] - a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] - (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2]))/(d*e - c*f + I*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1 + I*Tan[(c + d*x)/2]]) + 2*b*f*Sin[c + d*x])/(2*b^2*d^2)","B",0
227,1,71,75,0.1841496,"\int \frac{\sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Sin[c + d*x]),x]","-\frac{-\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)}{\sqrt{a^2-b^2}}+a (c+d x)+b \cos (c+d x)}{b^2 d}","\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)}{b^2 d \sqrt{a^2-b^2}}-\frac{a x}{b^2}-\frac{\cos (c+d x)}{b d}",1,"-((a*(c + d*x) - (2*a^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + b*Cos[c + d*x])/(b^2*d))","A",1
228,1,1923,802,5.644552,"\int \frac{(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{4 a^2 \sqrt{-\left(a^2-b^2\right)^2} f^3 x^4 d^4+2 b^2 \sqrt{-\left(b^2-a^2\right)^2} f^3 x^4 d^4+16 a^2 \sqrt{-\left(a^2-b^2\right)^2} e f^2 x^3 d^4+8 b^2 \sqrt{-\left(b^2-a^2\right)^2} e f^2 x^3 d^4+24 a^2 \sqrt{-\left(a^2-b^2\right)^2} e^2 f x^2 d^4+12 b^2 \sqrt{-\left(b^2-a^2\right)^2} e^2 f x^2 d^4+16 a^2 \sqrt{-\left(a^2-b^2\right)^2} e^3 x d^4+8 b^2 \sqrt{-\left(b^2-a^2\right)^2} e^3 x d^4-32 a^3 \sqrt{b^2-a^2} e^3 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) d^3+16 a b \sqrt{-\left(a^2-b^2\right)^2} e^3 \cos (c+d x) d^3+16 a b \sqrt{-\left(a^2-b^2\right)^2} f^3 x^3 \cos (c+d x) d^3+48 a b \sqrt{-\left(a^2-b^2\right)^2} e f^2 x^2 \cos (c+d x) d^3+48 a b \sqrt{-\left(a^2-b^2\right)^2} e^2 f x \cos (c+d x) d^3-16 a^3 \sqrt{a^2-b^2} f^3 x^3 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3-48 a^3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3-48 a^3 \sqrt{a^2-b^2} e^2 f x \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+16 a^3 \sqrt{a^2-b^2} f^3 x^3 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3+48 a^3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3+48 a^3 \sqrt{a^2-b^2} e^2 f x \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-4 b^2 \sqrt{-\left(a^2-b^2\right)^2} e^3 \sin (2 (c+d x)) d^3-4 b^2 \sqrt{-\left(a^2-b^2\right)^2} f^3 x^3 \sin (2 (c+d x)) d^3-12 b^2 \sqrt{-\left(a^2-b^2\right)^2} e f^2 x^2 \sin (2 (c+d x)) d^3-12 b^2 \sqrt{-\left(a^2-b^2\right)^2} e^2 f x \sin (2 (c+d x)) d^3-6 b^2 \sqrt{-\left(a^2-b^2\right)^2} f^3 x^2 \cos (2 (c+d x)) d^2-6 b^2 \sqrt{-\left(a^2-b^2\right)^2} e^2 f \cos (2 (c+d x)) d^2-12 b^2 \sqrt{-\left(a^2-b^2\right)^2} e f^2 x \cos (2 (c+d x)) d^2+48 i a^3 \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^2-48 i a^3 \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d^2-48 a b \sqrt{-\left(a^2-b^2\right)^2} f^3 x^2 \sin (c+d x) d^2-48 a b \sqrt{-\left(a^2-b^2\right)^2} e^2 f \sin (c+d x) d^2-96 a b \sqrt{-\left(a^2-b^2\right)^2} e f^2 x \sin (c+d x) d^2-96 a b \sqrt{-\left(a^2-b^2\right)^2} e f^2 \cos (c+d x) d-96 a b \sqrt{-\left(a^2-b^2\right)^2} f^3 x \cos (c+d x) d-96 a^3 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d-96 a^3 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d+96 a^3 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d+96 a^3 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d+6 b^2 \sqrt{-\left(a^2-b^2\right)^2} e f^2 \sin (2 (c+d x)) d+6 b^2 \sqrt{-\left(a^2-b^2\right)^2} f^3 x \sin (2 (c+d x)) d+3 b^2 \sqrt{-\left(a^2-b^2\right)^2} f^3 \cos (2 (c+d x))-96 i a^3 \sqrt{a^2-b^2} f^3 \text{Li}_4\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)+96 i a^3 \sqrt{a^2-b^2} f^3 \text{Li}_4\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)+96 a b \sqrt{-\left(a^2-b^2\right)^2} f^3 \sin (c+d x)}{16 b^3 \sqrt{-\left(a^2-b^2\right)^2} d^4}","\frac{(e+f x)^4}{8 b f}+\frac{a^2 (e+f x)^4}{4 b^3 f}+\frac{a \cos (c+d x) (e+f x)^3}{b^2 d}+\frac{i a^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^3}{b^3 \sqrt{a^2-b^2} d}-\frac{i a^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^3}{b^3 \sqrt{a^2-b^2} d}-\frac{\cos (c+d x) \sin (c+d x) (e+f x)^3}{2 b d}+\frac{3 f \sin ^2(c+d x) (e+f x)^2}{4 b d^2}+\frac{3 a^3 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^2}{b^3 \sqrt{a^2-b^2} d^2}-\frac{3 a^3 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^2}{b^3 \sqrt{a^2-b^2} d^2}-\frac{3 a f \sin (c+d x) (e+f x)^2}{b^2 d^2}-\frac{6 a f^2 \cos (c+d x) (e+f x)}{b^2 d^3}+\frac{6 i a^3 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)}{b^3 \sqrt{a^2-b^2} d^3}-\frac{6 i a^3 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)}{b^3 \sqrt{a^2-b^2} d^3}+\frac{3 f^2 \cos (c+d x) \sin (c+d x) (e+f x)}{4 b d^3}-\frac{3 f^3 x^2}{8 b d^2}-\frac{3 f^3 \sin ^2(c+d x)}{8 b d^4}-\frac{3 e f^2 x}{4 b d^2}-\frac{6 a^3 f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 \sqrt{a^2-b^2} d^4}+\frac{6 a^3 f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^3 \sqrt{a^2-b^2} d^4}+\frac{6 a f^3 \sin (c+d x)}{b^2 d^4}",1,"(16*a^2*Sqrt[-(a^2 - b^2)^2]*d^4*e^3*x + 8*b^2*Sqrt[-(-a^2 + b^2)^2]*d^4*e^3*x + 24*a^2*Sqrt[-(a^2 - b^2)^2]*d^4*e^2*f*x^2 + 12*b^2*Sqrt[-(-a^2 + b^2)^2]*d^4*e^2*f*x^2 + 16*a^2*Sqrt[-(a^2 - b^2)^2]*d^4*e*f^2*x^3 + 8*b^2*Sqrt[-(-a^2 + b^2)^2]*d^4*e*f^2*x^3 + 4*a^2*Sqrt[-(a^2 - b^2)^2]*d^4*f^3*x^4 + 2*b^2*Sqrt[-(-a^2 + b^2)^2]*d^4*f^3*x^4 - 32*a^3*Sqrt[-a^2 + b^2]*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 16*a*b*Sqrt[-(a^2 - b^2)^2]*d^3*e^3*Cos[c + d*x] - 96*a*b*Sqrt[-(a^2 - b^2)^2]*d*e*f^2*Cos[c + d*x] + 48*a*b*Sqrt[-(a^2 - b^2)^2]*d^3*e^2*f*x*Cos[c + d*x] - 96*a*b*Sqrt[-(a^2 - b^2)^2]*d*f^3*x*Cos[c + d*x] + 48*a*b*Sqrt[-(a^2 - b^2)^2]*d^3*e*f^2*x^2*Cos[c + d*x] + 16*a*b*Sqrt[-(a^2 - b^2)^2]*d^3*f^3*x^3*Cos[c + d*x] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*e^2*f*Cos[2*(c + d*x)] + 3*b^2*Sqrt[-(a^2 - b^2)^2]*f^3*Cos[2*(c + d*x)] - 12*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*e*f^2*x*Cos[2*(c + d*x)] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*f^3*x^2*Cos[2*(c + d*x)] - 48*a^3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 48*a^3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 16*a^3*Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 48*a^3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] + 48*a^3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] + 16*a^3*Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] + (48*I)*a^3*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - (48*I)*a^3*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 96*a^3*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 96*a^3*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 96*a^3*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + 96*a^3*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - (96*I)*a^3*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + (96*I)*a^3*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 48*a*b*Sqrt[-(a^2 - b^2)^2]*d^2*e^2*f*Sin[c + d*x] + 96*a*b*Sqrt[-(a^2 - b^2)^2]*f^3*Sin[c + d*x] - 96*a*b*Sqrt[-(a^2 - b^2)^2]*d^2*e*f^2*x*Sin[c + d*x] - 48*a*b*Sqrt[-(a^2 - b^2)^2]*d^2*f^3*x^2*Sin[c + d*x] - 4*b^2*Sqrt[-(a^2 - b^2)^2]*d^3*e^3*Sin[2*(c + d*x)] + 6*b^2*Sqrt[-(a^2 - b^2)^2]*d*e*f^2*Sin[2*(c + d*x)] - 12*b^2*Sqrt[-(a^2 - b^2)^2]*d^3*e^2*f*x*Sin[2*(c + d*x)] + 6*b^2*Sqrt[-(a^2 - b^2)^2]*d*f^3*x*Sin[2*(c + d*x)] - 12*b^2*Sqrt[-(a^2 - b^2)^2]*d^3*e*f^2*x^2*Sin[2*(c + d*x)] - 4*b^2*Sqrt[-(a^2 - b^2)^2]*d^3*f^3*x^3*Sin[2*(c + d*x)])/(16*b^3*Sqrt[-(a^2 - b^2)^2]*d^4)","B",0
229,1,1166,592,4.0467331,"\int \frac{(e+f x)^2 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\frac{-48 \sqrt{b^2-a^2} d^2 e^2 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) a^3-24 \sqrt{a^2-b^2} d^2 f^2 x^2 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) a^3-48 \sqrt{a^2-b^2} d^2 e f x \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) a^3+24 \sqrt{a^2-b^2} d^2 f^2 x^2 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) a^3+48 \sqrt{a^2-b^2} d^2 e f x \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) a^3+48 i \sqrt{a^2-b^2} d f (e+f x) \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) a^3-48 i \sqrt{a^2-b^2} d f (e+f x) \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) a^3-48 \sqrt{a^2-b^2} f^2 \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) a^3+48 \sqrt{a^2-b^2} f^2 \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) a^3+8 \sqrt{-\left(a^2-b^2\right)^2} d^3 f^2 x^3 a^2+24 \sqrt{-\left(a^2-b^2\right)^2} d^3 e f x^2 a^2+24 \sqrt{-\left(a^2-b^2\right)^2} d^3 e^2 x a^2+24 b \sqrt{-\left(a^2-b^2\right)^2} d^2 e^2 \cos (c+d x) a-48 b \sqrt{-\left(a^2-b^2\right)^2} f^2 \cos (c+d x) a+24 b \sqrt{-\left(a^2-b^2\right)^2} d^2 f^2 x^2 \cos (c+d x) a+48 b \sqrt{-\left(a^2-b^2\right)^2} d^2 e f x \cos (c+d x) a-48 b \sqrt{-\left(a^2-b^2\right)^2} d e f \sin (c+d x) a-48 b \sqrt{-\left(a^2-b^2\right)^2} d f^2 x \sin (c+d x) a+4 b^2 \sqrt{-\left(b^2-a^2\right)^2} d^3 f^2 x^3+12 b^2 \sqrt{-\left(b^2-a^2\right)^2} d^3 e f x^2+12 b^2 \sqrt{-\left(b^2-a^2\right)^2} d^3 e^2 x-6 b^2 \sqrt{-\left(a^2-b^2\right)^2} d e f \cos (2 (c+d x))-6 b^2 \sqrt{-\left(a^2-b^2\right)^2} d f^2 x \cos (2 (c+d x))-6 b^2 \sqrt{-\left(a^2-b^2\right)^2} d^2 e^2 \sin (2 (c+d x))+3 b^2 \sqrt{-\left(a^2-b^2\right)^2} f^2 \sin (2 (c+d x))-6 b^2 \sqrt{-\left(a^2-b^2\right)^2} d^2 f^2 x^2 \sin (2 (c+d x))-12 b^2 \sqrt{-\left(a^2-b^2\right)^2} d^2 e f x \sin (2 (c+d x))}{24 b^3 \sqrt{-\left(a^2-b^2\right)^2} d^3}","\frac{a^2 (e+f x)^3}{3 b^3 f}+\frac{2 i a^3 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d^3 \sqrt{a^2-b^2}}-\frac{2 i a^3 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^3 d^3 \sqrt{a^2-b^2}}+\frac{2 a^3 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d^2 \sqrt{a^2-b^2}}-\frac{2 a^3 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^3 d^2 \sqrt{a^2-b^2}}+\frac{i a^3 (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d \sqrt{a^2-b^2}}-\frac{i a^3 (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b^3 d \sqrt{a^2-b^2}}-\frac{2 a f^2 \cos (c+d x)}{b^2 d^3}-\frac{2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac{a (e+f x)^2 \cos (c+d x)}{b^2 d}+\frac{f^2 \sin (c+d x) \cos (c+d x)}{4 b d^3}+\frac{f (e+f x) \sin ^2(c+d x)}{2 b d^2}-\frac{(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 b d}-\frac{f^2 x}{4 b d^2}+\frac{(e+f x)^3}{6 b f}",1,"(24*a^2*Sqrt[-(a^2 - b^2)^2]*d^3*e^2*x + 12*b^2*Sqrt[-(-a^2 + b^2)^2]*d^3*e^2*x + 24*a^2*Sqrt[-(a^2 - b^2)^2]*d^3*e*f*x^2 + 12*b^2*Sqrt[-(-a^2 + b^2)^2]*d^3*e*f*x^2 + 8*a^2*Sqrt[-(a^2 - b^2)^2]*d^3*f^2*x^3 + 4*b^2*Sqrt[-(-a^2 + b^2)^2]*d^3*f^2*x^3 - 48*a^3*Sqrt[-a^2 + b^2]*d^2*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 24*a*b*Sqrt[-(a^2 - b^2)^2]*d^2*e^2*Cos[c + d*x] - 48*a*b*Sqrt[-(a^2 - b^2)^2]*f^2*Cos[c + d*x] + 48*a*b*Sqrt[-(a^2 - b^2)^2]*d^2*e*f*x*Cos[c + d*x] + 24*a*b*Sqrt[-(a^2 - b^2)^2]*d^2*f^2*x^2*Cos[c + d*x] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d*e*f*Cos[2*(c + d*x)] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d*f^2*x*Cos[2*(c + d*x)] - 48*a^3*Sqrt[a^2 - b^2]*d^2*e*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 24*a^3*Sqrt[a^2 - b^2]*d^2*f^2*x^2*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 48*a^3*Sqrt[a^2 - b^2]*d^2*e*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] + 24*a^3*Sqrt[a^2 - b^2]*d^2*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] + (48*I)*a^3*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - (48*I)*a^3*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 48*a^3*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 48*a^3*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 48*a*b*Sqrt[-(a^2 - b^2)^2]*d*e*f*Sin[c + d*x] - 48*a*b*Sqrt[-(a^2 - b^2)^2]*d*f^2*x*Sin[c + d*x] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*e^2*Sin[2*(c + d*x)] + 3*b^2*Sqrt[-(a^2 - b^2)^2]*f^2*Sin[2*(c + d*x)] - 12*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*e*f*x*Sin[2*(c + d*x)] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*f^2*x^2*Sin[2*(c + d*x)])/(24*b^3*Sqrt[-(a^2 - b^2)^2]*d^3)","A",0
230,1,752,382,8.7407454,"\int \frac{(e+f x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","-\frac{2 \left(2 a^2+b^2\right) (c+d x) (c f-d (2 e+f x))+\frac{8 a^3 d (e+f x) \left(\frac{2 (d e-c f) \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{i f \left(\text{Li}_2\left(\frac{a \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a+i \left(b+\sqrt{b^2-a^2}\right)}\right)+\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{b^2-a^2}-i a+b}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\text{Li}_2\left(\frac{a \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)}{a-i \left(b+\sqrt{b^2-a^2}\right)}\right)+\log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{b^2-a^2}+i a+b}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\text{Li}_2\left(\frac{a \left(\tan \left(\frac{1}{2} (c+d x)\right)+i\right)}{i a-b+\sqrt{b^2-a^2}}\right)+\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}-a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{b^2-a^2}+i a-b}\right)\right)}{\sqrt{b^2-a^2}}-\frac{i f \left(\text{Li}_2\left(\frac{i \tan \left(\frac{1}{2} (c+d x)\right) a+a}{a+i \left(\sqrt{b^2-a^2}-b\right)}\right)+\log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{-\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{-\sqrt{b^2-a^2}+i a+b}\right)\right)}{\sqrt{b^2-a^2}}\right)}{i f \log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)-i f \log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right)-c f+d e}-8 a b d (e+f x) \cos (c+d x)+8 a b f \sin (c+d x)+2 b^2 d (e+f x) \sin (2 (c+d x))+b^2 f \cos (2 (c+d x))}{8 b^3 d^2}","\frac{a^2 e x}{b^3}+\frac{a^2 f x^2}{2 b^3}+\frac{a^3 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d^2 \sqrt{a^2-b^2}}-\frac{a^3 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^3 d^2 \sqrt{a^2-b^2}}+\frac{i a^3 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d \sqrt{a^2-b^2}}-\frac{i a^3 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b^3 d \sqrt{a^2-b^2}}-\frac{a f \sin (c+d x)}{b^2 d^2}+\frac{a (e+f x) \cos (c+d x)}{b^2 d}+\frac{f \sin ^2(c+d x)}{4 b d^2}-\frac{(e+f x) \sin (c+d x) \cos (c+d x)}{2 b d}+\frac{e x}{2 b}+\frac{f x^2}{4 b}",1,"-1/8*(2*(2*a^2 + b^2)*(c + d*x)*(c*f - d*(2*e + f*x)) - 8*a*b*d*(e + f*x)*Cos[c + d*x] + b^2*f*Cos[2*(c + d*x)] + (8*a^3*d*(e + f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(-b + Sqrt[-a^2 + b^2] - a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] - (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2]))/(d*e - c*f + I*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1 + I*Tan[(c + d*x)/2]]) + 8*a*b*f*Sin[c + d*x] + 2*b^2*d*(e + f*x)*Sin[2*(c + d*x)])/(b^3*d^2)","A",0
231,1,97,107,0.2750413,"\int \frac{\sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Sin[c + d*x]^3/(a + b*Sin[c + d*x]),x]","\frac{2 \left(2 a^2+b^2\right) (c+d x)-\frac{8 a^3 \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 a b \cos (c+d x)-b^2 \sin (2 (c+d x))}{4 b^3 d}","\frac{x \left(2 a^2+b^2\right)}{2 b^3}-\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^3 d \sqrt{a^2-b^2}}+\frac{a \cos (c+d x)}{b^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}",1,"(2*(2*a^2 + b^2)*(c + d*x) - (8*a^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^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
232,1,894,732,2.6376311,"\int \frac{(e+f x)^3 \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{-2 d^3 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) (e+f x)^3+\frac{b \left(3 d^2 f \text{Li}_2\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right) (e+f x)^2+i \left(2 i e^3 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) d^3+f^3 x^3 \log \left(\frac{i e^{i (c+d x)} b}{\sqrt{a^2-b^2}-a}+1\right) d^3+3 e f^2 x^2 \log \left(\frac{i e^{i (c+d x)} b}{\sqrt{a^2-b^2}-a}+1\right) d^3+3 e^2 f x \log \left(\frac{i e^{i (c+d x)} b}{\sqrt{a^2-b^2}-a}+1\right) d^3-f^3 x^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d^3-3 e f^2 x^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d^3-3 e^2 f x \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d^3+3 i f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d^2+6 f^2 (e+f x) \text{Li}_3\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right) d-6 e f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d-6 f^3 x \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d+6 i f^3 \text{Li}_4\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)-6 i f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)\right)\right)}{\sqrt{a^2-b^2}}+3 i f \left(-2 \text{Li}_4(-\cos (c+d x)-i \sin (c+d x)) f^2+2 i d (e+f x) \text{Li}_3(-\cos (c+d x)-i \sin (c+d x)) f+d^2 (e+f x)^2 \text{Li}_2(-\cos (c+d x)-i \sin (c+d x))\right)-3 i f \left(-2 \text{Li}_4(\cos (c+d x)+i \sin (c+d x)) f^2+2 i d (e+f x) \text{Li}_3(\cos (c+d x)+i \sin (c+d x)) f+d^2 (e+f x)^2 \text{Li}_2(\cos (c+d x)+i \sin (c+d x))\right)}{a d^4}","-\frac{6 b f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a d^4 \sqrt{a^2-b^2}}+\frac{6 b f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a d^4 \sqrt{a^2-b^2}}+\frac{6 i b f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a d^3 \sqrt{a^2-b^2}}-\frac{6 i b f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a d^3 \sqrt{a^2-b^2}}+\frac{3 b f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a d^2 \sqrt{a^2-b^2}}-\frac{3 b f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a d^2 \sqrt{a^2-b^2}}+\frac{i b (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a d \sqrt{a^2-b^2}}-\frac{i b (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a d \sqrt{a^2-b^2}}-\frac{6 i f^3 \text{Li}_4\left(-e^{i (c+d x)}\right)}{a d^4}+\frac{6 i f^3 \text{Li}_4\left(e^{i (c+d x)}\right)}{a d^4}-\frac{6 f^2 (e+f x) \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}+\frac{3 i f (e+f x)^2 \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}-\frac{2 (e+f x)^3 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}",1,"(-2*d^3*(e + f*x)^3*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] + (b*(3*d^2*f*(e + f*x)^2*PolyLog[2, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + I*((2*I)*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 3*d^3*e^2*f*x*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + 3*d^3*e*f^2*x^2*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + d^3*f^3*x^3*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - 3*d^3*e^2*f*x*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] - 3*d^3*e*f^2*x^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] - d^3*f^3*x^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + (3*I)*d^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + 6*d*f^2*(e + f*x)*PolyLog[3, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - 6*d*e*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] - 6*d*f^3*x*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + (6*I)*f^3*PolyLog[4, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - (6*I)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])))/Sqrt[a^2 - b^2] + (3*I)*f*(d^2*(e + f*x)^2*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)*PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] - 2*f^2*PolyLog[4, -Cos[c + d*x] - I*Sin[c + d*x]]) - (3*I)*f*(d^2*(e + f*x)^2*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)*PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]] - 2*f^2*PolyLog[4, Cos[c + d*x] + I*Sin[c + d*x]]))/(a*d^4)","A",0
233,1,573,528,1.5776328,"\int \frac{(e+f x)^2 \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\frac{b \left(i \left(2 i d^2 e^2 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right)+2 d^2 e f x \log \left(1+\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)-2 d^2 e f x \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)+d^2 f^2 x^2 \log \left(1+\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)-d^2 f^2 x^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)+2 i d f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)+2 f^2 \text{Li}_3\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)-2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)\right)+2 d f (e+f x) \text{Li}_2\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)\right)}{\sqrt{a^2-b^2}}+d^2 (e+f x)^2 \log \left(1-e^{i (c+d x)}\right)-d^2 (e+f x)^2 \log \left(1+e^{i (c+d x)}\right)+2 i d f (e+f x) \text{Li}_2\left(-e^{i (c+d x)}\right)-2 i d f (e+f x) \text{Li}_2\left(e^{i (c+d x)}\right)-2 f^2 \text{Li}_3\left(-e^{i (c+d x)}\right)+2 f^2 \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}","\frac{2 i b f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a d^3 \sqrt{a^2-b^2}}-\frac{2 i b f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a d^3 \sqrt{a^2-b^2}}+\frac{2 b f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a d^2 \sqrt{a^2-b^2}}-\frac{2 b f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a d^2 \sqrt{a^2-b^2}}+\frac{i b (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a d \sqrt{a^2-b^2}}-\frac{i b (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a d \sqrt{a^2-b^2}}-\frac{2 f^2 \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^3}+\frac{2 f^2 \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}+\frac{2 i f (e+f x) \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}-\frac{2 i f (e+f x) \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}-\frac{2 (e+f x)^2 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}",1,"(d^2*(e + f*x)^2*Log[1 - E^(I*(c + d*x))] - d^2*(e + f*x)^2*Log[1 + E^(I*(c + d*x))] + (2*I)*d*f*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))] - (2*I)*d*f*(e + f*x)*PolyLog[2, E^(I*(c + d*x))] - 2*f^2*PolyLog[3, -E^(I*(c + d*x))] + 2*f^2*PolyLog[3, E^(I*(c + d*x))] + (b*(2*d*f*(e + f*x)*PolyLog[2, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + I*((2*I)*d^2*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 2*d^2*e*f*x*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + d^2*f^2*x^2*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - 2*d^2*e*f*x*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] - d^2*f^2*x^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + (2*I)*d*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + 2*f^2*PolyLog[3, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - 2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])))/Sqrt[a^2 - b^2])/(a*d^3)","A",1
234,1,764,325,6.3736199,"\int \frac{(e+f x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{-\frac{b d (e+f x) \left(\frac{2 (d e-c f) \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{i f \left(\text{Li}_2\left(\frac{a \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a+i \left(b+\sqrt{b^2-a^2}\right)}\right)+\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{b^2-a^2}-i a+b}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\text{Li}_2\left(\frac{a \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)}{a-i \left(b+\sqrt{b^2-a^2}\right)}\right)+\log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{b^2-a^2}+i a+b}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\text{Li}_2\left(\frac{a \left(\tan \left(\frac{1}{2} (c+d x)\right)+i\right)}{i a-b+\sqrt{b^2-a^2}}\right)+\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}-a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{b^2-a^2}+i a-b}\right)\right)}{\sqrt{b^2-a^2}}-\frac{i f \left(\text{Li}_2\left(\frac{i \tan \left(\frac{1}{2} (c+d x)\right) a+a}{a+i \left(\sqrt{b^2-a^2}-b\right)}\right)+\log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{-\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{-\sqrt{b^2-a^2}+i a+b}\right)\right)}{\sqrt{b^2-a^2}}\right)}{i f \log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)-i f \log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right)-c f+d e}+d e \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)+f \left(i \left(\text{Li}_2\left(-e^{i (c+d x)}\right)-\text{Li}_2\left(e^{i (c+d x)}\right)\right)+(c+d x) \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)\right)\right)-c f \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)}{a d^2}","\frac{b f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a d^2 \sqrt{a^2-b^2}}-\frac{b f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a d^2 \sqrt{a^2-b^2}}+\frac{i b (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a d \sqrt{a^2-b^2}}-\frac{i b (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a d \sqrt{a^2-b^2}}+\frac{i f \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}-\frac{i f \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}-\frac{2 (e+f x) \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}",1,"(d*e*Log[Tan[(c + d*x)/2]] - c*f*Log[Tan[(c + d*x)/2]] + f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])) - (b*d*(e + f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(-b + Sqrt[-a^2 + b^2] - a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] - (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2]))/(d*e - c*f + I*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1 + I*Tan[(c + d*x)/2]]))/(a*d^2)","B",0
235,1,77,67,0.0726488,"\int \frac{\csc (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]/(a + b*Sin[c + d*x]),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}}+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a d}","-\frac{2 b \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{a d \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",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*d)","A",1
236,1,1680,882,48.2063242,"\int \frac{(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\left(2 \sqrt{b^2-a^2} e^3 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) d^3+\sqrt{a^2-b^2} f^3 x^3 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e^2 f x \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3-\sqrt{a^2-b^2} f^3 x^3 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e^2 f x \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^2+3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d^2+6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d+6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d-6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d-6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d+6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right) b^2}{a^2 \sqrt{-\left(a^2-b^2\right)^2} d^4}+\frac{-b d^3 x^3 \log \left(1-e^{-i (c+d x)}\right) f^3+b d^3 x^3 \log \left(1+e^{-i (c+d x)}\right) f^3+3 b \left(i d^2 \text{Li}_2\left(-e^{-i (c+d x)}\right) x^2+2 d \text{Li}_3\left(-e^{-i (c+d x)}\right) x-2 i \text{Li}_4\left(-e^{-i (c+d x)}\right)\right) f^3-3 i b \left(d^2 \text{Li}_2\left(e^{-i (c+d x)}\right) x^2-2 i d \text{Li}_3\left(e^{-i (c+d x)}\right) x-2 \text{Li}_4\left(e^{-i (c+d x)}\right)\right) f^3-3 d^2 (b d e-a f) x^2 \log \left(1-e^{-i (c+d x)}\right) f^2+3 d^2 (b d e+a f) x^2 \log \left(1+e^{-i (c+d x)}\right) f^2+6 (b d e+a f) \left(i d x \text{Li}_2\left(-e^{-i (c+d x)}\right)+\text{Li}_3\left(-e^{-i (c+d x)}\right)\right) f^2+6 (a f-b d e) \left(i d x \text{Li}_2\left(e^{-i (c+d x)}\right)+\text{Li}_3\left(e^{-i (c+d x)}\right)\right) f^2-3 d^2 e (b d e-2 a f) x \log \left(1-e^{-i (c+d x)}\right) f+3 d^2 e (b d e+2 a f) x \log \left(1+e^{-i (c+d x)}\right) f+3 i d e (b d e+2 a f) \text{Li}_2\left(-e^{-i (c+d x)}\right) f-3 i d e (b d e-2 a f) \text{Li}_2\left(e^{-i (c+d x)}\right) f-\frac{2 i a d^3 (e+f x)^3}{-1+e^{2 i c}}+i d^2 e^2 (b d e-3 a f) \left(d x+i \log \left(1-e^{i (c+d x)}\right)\right)+d^2 e^2 (b d e+3 a f) \left(\log \left(1+e^{i (c+d x)}\right)-i d x\right)}{a^2 d^4}+\frac{\csc \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\sin \left(\frac{d x}{2}\right) e^3+3 f x \sin \left(\frac{d x}{2}\right) e^2+3 f^2 x^2 \sin \left(\frac{d x}{2}\right) e+f^3 x^3 \sin \left(\frac{d x}{2}\right)\right)}{2 a d}+\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\sin \left(\frac{d x}{2}\right) e^3+3 f x \sin \left(\frac{d x}{2}\right) e^2+3 f^2 x^2 \sin \left(\frac{d x}{2}\right) e+f^3 x^3 \sin \left(\frac{d x}{2}\right)\right)}{2 a d}","\frac{3 \text{Li}_3\left(e^{2 i (c+d x)}\right) f^3}{2 a d^4}+\frac{6 i b \text{Li}_4\left(-e^{i (c+d x)}\right) f^3}{a^2 d^4}-\frac{6 i b \text{Li}_4\left(e^{i (c+d x)}\right) f^3}{a^2 d^4}+\frac{6 b^2 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^3}{a^2 \sqrt{a^2-b^2} d^4}-\frac{6 b^2 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^3}{a^2 \sqrt{a^2-b^2} d^4}-\frac{3 i (e+f x) \text{Li}_2\left(e^{2 i (c+d x)}\right) f^2}{a d^3}+\frac{6 b (e+f x) \text{Li}_3\left(-e^{i (c+d x)}\right) f^2}{a^2 d^3}-\frac{6 b (e+f x) \text{Li}_3\left(e^{i (c+d x)}\right) f^2}{a^2 d^3}-\frac{6 i b^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^2}{a^2 \sqrt{a^2-b^2} d^3}+\frac{6 i b^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^2}{a^2 \sqrt{a^2-b^2} d^3}+\frac{3 (e+f x)^2 \log \left(1-e^{2 i (c+d x)}\right) f}{a d^2}-\frac{3 i b (e+f x)^2 \text{Li}_2\left(-e^{i (c+d x)}\right) f}{a^2 d^2}+\frac{3 i b (e+f x)^2 \text{Li}_2\left(e^{i (c+d x)}\right) f}{a^2 d^2}-\frac{3 b^2 (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f}{a^2 \sqrt{a^2-b^2} d^2}+\frac{3 b^2 (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f}{a^2 \sqrt{a^2-b^2} d^2}-\frac{i (e+f x)^3}{a d}+\frac{2 b (e+f x)^3 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a^2 d}-\frac{(e+f x)^3 \cot (c+d x)}{a d}-\frac{i b^2 (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 \sqrt{a^2-b^2} d}+\frac{i b^2 (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 \sqrt{a^2-b^2} d}",1,"(((-2*I)*a*d^3*(e + f*x)^3)/(-1 + E^((2*I)*c)) - 3*d^2*e*f*(b*d*e - 2*a*f)*x*Log[1 - E^((-I)*(c + d*x))] - 3*d^2*f^2*(b*d*e - a*f)*x^2*Log[1 - E^((-I)*(c + d*x))] - b*d^3*f^3*x^3*Log[1 - E^((-I)*(c + d*x))] + 3*d^2*e*f*(b*d*e + 2*a*f)*x*Log[1 + E^((-I)*(c + d*x))] + 3*d^2*f^2*(b*d*e + a*f)*x^2*Log[1 + E^((-I)*(c + d*x))] + b*d^3*f^3*x^3*Log[1 + E^((-I)*(c + d*x))] + I*d^2*e^2*(b*d*e - 3*a*f)*(d*x + I*Log[1 - E^(I*(c + d*x))]) + d^2*e^2*(b*d*e + 3*a*f)*((-I)*d*x + Log[1 + E^(I*(c + d*x))]) + (3*I)*d*e*f*(b*d*e + 2*a*f)*PolyLog[2, -E^((-I)*(c + d*x))] - (3*I)*d*e*f*(b*d*e - 2*a*f)*PolyLog[2, E^((-I)*(c + d*x))] + 6*f^2*(b*d*e + a*f)*(I*d*x*PolyLog[2, -E^((-I)*(c + d*x))] + PolyLog[3, -E^((-I)*(c + d*x))]) + 6*f^2*(-(b*d*e) + a*f)*(I*d*x*PolyLog[2, E^((-I)*(c + d*x))] + PolyLog[3, E^((-I)*(c + d*x))]) + 3*b*f^3*(I*d^2*x^2*PolyLog[2, -E^((-I)*(c + d*x))] + 2*d*x*PolyLog[3, -E^((-I)*(c + d*x))] - (2*I)*PolyLog[4, -E^((-I)*(c + d*x))]) - (3*I)*b*f^3*(d^2*x^2*PolyLog[2, E^((-I)*(c + d*x))] - (2*I)*d*x*PolyLog[3, E^((-I)*(c + d*x))] - 2*PolyLog[4, E^((-I)*(c + d*x))]))/(a^2*d^4) + (b^2*(2*Sqrt[-a^2 + b^2]*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))]))/(a^2*Sqrt[-(a^2 - b^2)^2]*d^4) + (Csc[c/2]*Csc[c/2 + (d*x)/2]*(e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2]))/(2*a*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2]))/(2*a*d)","A",0
237,1,911,639,12.1933903,"\int \frac{(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{i \left(-2 \sqrt{a^2-b^2} d f (e+f x) \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)+2 \sqrt{a^2-b^2} d f (e+f x) \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)-i \left(\left(2 \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) e^2+\sqrt{a^2-b^2} f x (2 e+f x) \left(\log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-\log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right)\right)\right) d^2+2 \sqrt{a^2-b^2} f^2 \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-2 \sqrt{a^2-b^2} f^2 \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)\right) b^2}{a^2 \sqrt{-\left(a^2-b^2\right)^2} d^3}+\frac{-b d^2 x^2 \log \left(1-e^{-i (c+d x)}\right) f^2+b d^2 x^2 \log \left(1+e^{-i (c+d x)}\right) f^2+2 b \left(i d x \text{Li}_2\left(-e^{-i (c+d x)}\right)+\text{Li}_3\left(-e^{-i (c+d x)}\right)\right) f^2-2 i b \left(d x \text{Li}_2\left(e^{-i (c+d x)}\right)-i \text{Li}_3\left(e^{-i (c+d x)}\right)\right) f^2-2 d (b d e-a f) x \log \left(1-e^{-i (c+d x)}\right) f+2 d (b d e+a f) x \log \left(1+e^{-i (c+d x)}\right) f+2 i (b d e+a f) \text{Li}_2\left(-e^{-i (c+d x)}\right) f+2 i (a f-b d e) \text{Li}_2\left(e^{-i (c+d x)}\right) f-\frac{2 i a d^2 (e+f x)^2}{-1+e^{2 i c}}+i d e (b d e-2 a f) \left(d x+i \log \left(1-e^{i (c+d x)}\right)\right)+d e (b d e+2 a f) \left(\log \left(1+e^{i (c+d x)}\right)-i d x\right)}{a^2 d^3}+\frac{\csc \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\sin \left(\frac{d x}{2}\right) e^2+2 f x \sin \left(\frac{d x}{2}\right) e+f^2 x^2 \sin \left(\frac{d x}{2}\right)\right)}{2 a d}+\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\sin \left(\frac{d x}{2}\right) e^2+2 f x \sin \left(\frac{d x}{2}\right) e+f^2 x^2 \sin \left(\frac{d x}{2}\right)\right)}{2 a d}","-\frac{2 i b^2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 d^3 \sqrt{a^2-b^2}}+\frac{2 i b^2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 d^3 \sqrt{a^2-b^2}}-\frac{2 b^2 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 d^2 \sqrt{a^2-b^2}}+\frac{2 b^2 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 d^2 \sqrt{a^2-b^2}}-\frac{i b^2 (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 d \sqrt{a^2-b^2}}+\frac{i b^2 (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a^2 d \sqrt{a^2-b^2}}+\frac{2 b f^2 \text{Li}_3\left(-e^{i (c+d x)}\right)}{a^2 d^3}-\frac{2 b f^2 \text{Li}_3\left(e^{i (c+d x)}\right)}{a^2 d^3}-\frac{2 i b f (e+f x) \text{Li}_2\left(-e^{i (c+d x)}\right)}{a^2 d^2}+\frac{2 i b f (e+f x) \text{Li}_2\left(e^{i (c+d x)}\right)}{a^2 d^2}+\frac{2 b (e+f x)^2 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a^2 d}-\frac{i f^2 \text{Li}_2\left(e^{2 i (c+d x)}\right)}{a d^3}+\frac{2 f (e+f x) \log \left(1-e^{2 i (c+d x)}\right)}{a d^2}-\frac{(e+f x)^2 \cot (c+d x)}{a d}-\frac{i (e+f x)^2}{a d}",1,"(((-2*I)*a*d^2*(e + f*x)^2)/(-1 + E^((2*I)*c)) - 2*d*f*(b*d*e - a*f)*x*Log[1 - E^((-I)*(c + d*x))] - b*d^2*f^2*x^2*Log[1 - E^((-I)*(c + d*x))] + 2*d*f*(b*d*e + a*f)*x*Log[1 + E^((-I)*(c + d*x))] + b*d^2*f^2*x^2*Log[1 + E^((-I)*(c + d*x))] + I*d*e*(b*d*e - 2*a*f)*(d*x + I*Log[1 - E^(I*(c + d*x))]) + d*e*(b*d*e + 2*a*f)*((-I)*d*x + Log[1 + E^(I*(c + d*x))]) + (2*I)*f*(b*d*e + a*f)*PolyLog[2, -E^((-I)*(c + d*x))] + (2*I)*f*(-(b*d*e) + a*f)*PolyLog[2, E^((-I)*(c + d*x))] + 2*b*f^2*(I*d*x*PolyLog[2, -E^((-I)*(c + d*x))] + PolyLog[3, -E^((-I)*(c + d*x))]) - (2*I)*b*f^2*(d*x*PolyLog[2, E^((-I)*(c + d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))]))/(a^2*d^3) + (I*b^2*(-2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - I*(d^2*(2*Sqrt[-a^2 + b^2]*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(2*e + f*x)*(Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])])) + 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))])))/(a^2*Sqrt[-(a^2 - b^2)^2]*d^3) + (Csc[c/2]*Csc[c/2 + (d*x)/2]*(e^2*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2]))/(2*a*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(e^2*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2]))/(2*a*d)","A",0
238,1,933,370,11.3724924,"\int \frac{(e+f x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{(d e+d f x) \left(\frac{2 (d e-c f) \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{i f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{-i a+b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a+i \left(b+\sqrt{b^2-a^2}\right)}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{i a+b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)}{a-i \left(b+\sqrt{b^2-a^2}\right)}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(-\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)-\sqrt{b^2-a^2}}{i a-b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(\tan \left(\frac{1}{2} (c+d x)\right)+i\right)}{i a-b+\sqrt{b^2-a^2}}\right)\right)}{\sqrt{b^2-a^2}}-\frac{i f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)-\sqrt{b^2-a^2}}{i a+b-\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{i \tan \left(\frac{1}{2} (c+d x)\right) a+a}{a+i \left(\sqrt{b^2-a^2}-b\right)}\right)\right)}{\sqrt{b^2-a^2}}\right) b^2}{a^2 d^2 \left(d e-c f+i f \log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)-i f \log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}-\frac{e \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right) b}{a^2 d}+\frac{c f \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right) b}{a^2 d^2}-\frac{f \left((c+d x) \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)\right)+i \left(\text{Li}_2\left(-e^{i (c+d x)}\right)-\text{Li}_2\left(e^{i (c+d x)}\right)\right)\right) b}{a^2 d^2}+\frac{\left(-d e \cos \left(\frac{1}{2} (c+d x)\right)+c f \cos \left(\frac{1}{2} (c+d x)\right)-f (c+d x) \cos \left(\frac{1}{2} (c+d x)\right)\right) \csc \left(\frac{1}{2} (c+d x)\right)}{2 a d^2}+\frac{f \log (\sin (c+d x))}{a d^2}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(d e \sin \left(\frac{1}{2} (c+d x)\right)-c f \sin \left(\frac{1}{2} (c+d x)\right)+f (c+d x) \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a d^2}","-\frac{b^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 d^2 \sqrt{a^2-b^2}}+\frac{b^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 d^2 \sqrt{a^2-b^2}}-\frac{i b^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 d \sqrt{a^2-b^2}}+\frac{i b^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a^2 d \sqrt{a^2-b^2}}-\frac{i b f \text{Li}_2\left(-e^{i (c+d x)}\right)}{a^2 d^2}+\frac{i b f \text{Li}_2\left(e^{i (c+d x)}\right)}{a^2 d^2}+\frac{2 b (e+f x) \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a^2 d}+\frac{f \log (\sin (c+d x))}{a d^2}-\frac{(e+f x) \cot (c+d x)}{a d}",1,"((-(d*e*Cos[(c + d*x)/2]) + c*f*Cos[(c + d*x)/2] - f*(c + d*x)*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(2*a*d^2) + (f*Log[Sin[c + d*x]])/(a*d^2) - (b*e*Log[Tan[(c + d*x)/2]])/(a^2*d) + (b*c*f*Log[Tan[(c + d*x)/2]])/(a^2*d^2) - (b*f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/(a^2*d^2) + (b^2*(d*e + d*f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] - (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2]))/(a^2*d^2*(d*e - c*f + I*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1 + I*Tan[(c + d*x)/2]])) + (Sec[(c + d*x)/2]*(d*e*Sin[(c + d*x)/2] - c*f*Sin[(c + d*x)/2] + f*(c + d*x)*Sin[(c + d*x)/2]))/(2*a*d^2)","B",0
239,1,111,83,0.4597815,"\int \frac{\csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Sin[c + d*x]),x]","\frac{\frac{4 b^2 \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}}+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 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 \sqrt{a^2-b^2}}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}",1,"((4*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] + 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
240,0,0,31,8.2396072,"\int \frac{(e+f x)^m \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx","\text{Int}\left(\frac{\sin ^2(c+d x) (e+f x)^m}{a+b \sin (c+d x)},x\right)",0,"Integrate[((e + f*x)^m*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]), x]","A",-1
241,0,0,29,0.8344861,"\int \frac{(e+f x)^m \sin (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \sin (c+d x)}{a+b \sin (c+d x)} \, dx","\text{Int}\left(\frac{\sin (c+d x) (e+f x)^m}{a+b \sin (c+d x)},x\right)",0,"Integrate[((e + f*x)^m*Sin[c + d*x])/(a + b*Sin[c + d*x]), x]","A",-1
242,0,0,23,0.3140006,"\int \frac{(e+f x)^m}{a+b \sin (c+d x)} \, dx","Integrate[(e + f*x)^m/(a + b*Sin[c + d*x]),x]","\int \frac{(e+f x)^m}{a+b \sin (c+d x)} \, dx","\text{Int}\left(\frac{(e+f x)^m}{a+b \sin (c+d x)},x\right)",0,"Integrate[(e + f*x)^m/(a + b*Sin[c + d*x]), x]","A",-1
243,0,0,29,23.7304549,"\int \frac{(e+f x)^m \csc (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Csc[c + d*x])/(a + b*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \csc (c+d x)}{a+b \sin (c+d x)} \, dx","\text{Int}\left(\frac{\csc (c+d x) (e+f x)^m}{a+b \sin (c+d x)},x\right)",0,"Integrate[((e + f*x)^m*Csc[c + d*x])/(a + b*Sin[c + d*x]), x]","A",-1
244,0,0,31,42.8590678,"\int \frac{(e+f x)^m \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx","\text{Int}\left(\frac{\csc ^2(c+d x) (e+f x)^m}{a+b \sin (c+d x)},x\right)",0,"Integrate[((e + f*x)^m*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]), x]","A",-1
245,1,2141,574,15.6517737,"\int \frac{(e+f x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[((e + f*x)*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\text{Result too large to show}","\frac{a^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^2 \left(a^2-b^2\right)^{3/2}}-\frac{a^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^2 \left(a^2-b^2\right)^{3/2}}-\frac{f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^2 \sqrt{a^2-b^2}}+\frac{f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^2 \sqrt{a^2-b^2}}+\frac{a f \log (a+b \sin (c+d x))}{b d^2 \left(a^2-b^2\right)}+\frac{i a^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d \left(a^2-b^2\right)^{3/2}}-\frac{i a^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d \left(a^2-b^2\right)^{3/2}}-\frac{i (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d \sqrt{a^2-b^2}}+\frac{i (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d \sqrt{a^2-b^2}}-\frac{a (e+f x) \cos (c+d x)}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}",1,"(-(a*d*e*Cos[c + d*x]) + a*c*f*Cos[c + d*x] - a*f*(c + d*x)*Cos[c + d*x])/((a - b)*(a + b)*d^2*(a + b*Sin[c + d*x])) + (((2*a*f*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (2*(-(b*d*e) + a*f + b*c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (a*f*Log[Sec[(c + d*x)/2]^2])/b - (a*f*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])])/b - (I*b*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*b*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*b*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] - (I*b*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2])*(-((b*e)/((a^2 - b^2)*(a + b*Sin[c + d*x]))) + (b*c*f)/((a^2 - b^2)*d*(a + b*Sin[c + d*x])) - (b*f*(c + d*x))/((a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (a*f*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))))/(d*((a*f*Tan[(c + d*x)/2])/b - (a*f*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(b*(a + b*Sin[c + d*x])) + (a^2*f*Sec[(c + d*x)/2]^2)/((a^2 - b^2)*(1 + (b + a*Tan[(c + d*x)/2])^2/(a^2 - b^2))) - (a*(-(b*d*e) + a*f + b*c*f)*Sec[(c + d*x)/2]^2)/((a^2 - b^2)*(1 + (b + a*Tan[(c + d*x)/2])^2/(a^2 - b^2))) + (I*b*f*(((-1/2*I)*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (Log[1 - (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(2*(I + Tan[(c + d*x)/2])) + (a*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(2*(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]))))/Sqrt[-a^2 + b^2] - (I*b*f*(((I/2)*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - ((I/2)*a*Log[1 - (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(a + I*a*Tan[(c + d*x)/2]) + (a*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(2*(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]))))/Sqrt[-a^2 + b^2] - (I*b*f*(((I/2)*Log[1 - (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - ((I/2)*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) + (a*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(2*(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]))))/Sqrt[-a^2 + b^2] + (I*b*f*(((-1/2*I)*Log[1 - (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + ((I/2)*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + (a*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(2*(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]))))/Sqrt[-a^2 + b^2]))","B",0
246,1,3759,1106,25.4256015,"\int \frac{(e+f x)^2 \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[((e + f*x)^2*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\text{Result too large to show}","\frac{i (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^{3/2} d}-\frac{i (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^{3/2} d}+\frac{2 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^{3/2} d^2}-\frac{2 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^{3/2} d^2}+\frac{2 i f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^{3/2} d^3}-\frac{2 i f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^{3/2} d^3}-\frac{i (e+f x)^2 a}{b \left(a^2-b^2\right) d}+\frac{2 f (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right) d^2}+\frac{2 f (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right) d^2}-\frac{2 i f^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right) d^3}-\frac{2 i f^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right) d^3}-\frac{(e+f x)^2 \cos (c+d x) a}{\left(a^2-b^2\right) d (a+b \sin (c+d x))}-\frac{i (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2} d}+\frac{i (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2} d}-\frac{2 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2} d^2}+\frac{2 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2} d^2}-\frac{2 i f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2} d^3}+\frac{2 i f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2} d^3}",1,"(2*b*e*f*((Pi*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (2*(-c + Pi/2 - d*x)*ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]] - 2*(-c + ArcCos[-(a/b)])*ArcTanh[((-a + b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]] + (ArcCos[-(a/b)] - (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]] - ArcTanh[((-a + b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]]))*Log[Sqrt[-a^2 + b^2]/(Sqrt[2]*Sqrt[b]*E^((I/2)*(-c + Pi/2 - d*x))*Sqrt[a + b*Sin[c + d*x]])] + (ArcCos[-(a/b)] + (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]] - ArcTanh[((-a + b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]]))*Log[(Sqrt[-a^2 + b^2]*E^((I/2)*(-c + Pi/2 - d*x)))/(Sqrt[2]*Sqrt[b]*Sqrt[a + b*Sin[c + d*x]])] - (ArcCos[-(a/b)] + (2*I)*ArcTanh[((-a + b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]])*Log[1 - ((a - I*Sqrt[-a^2 + b^2])*(a + b - Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))] + (-ArcCos[-(a/b)] + (2*I)*ArcTanh[((-a + b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]])*Log[1 - ((a + I*Sqrt[-a^2 + b^2])*(a + b - Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))] + I*(PolyLog[2, ((a - I*Sqrt[-a^2 + b^2])*(a + b - Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))] - PolyLog[2, ((a + I*Sqrt[-a^2 + b^2])*(a + b - Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))]))/Sqrt[-a^2 + b^2]))/((-a^2 + b^2)*d^2) + (2*a^2*f^2*Cot[c]*((Pi*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (2*(-c + Pi/2 - d*x)*ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]] - 2*(-c + ArcCos[-(a/b)])*ArcTanh[((-a + b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]] + (ArcCos[-(a/b)] - (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]] - ArcTanh[((-a + b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]]))*Log[Sqrt[-a^2 + b^2]/(Sqrt[2]*Sqrt[b]*E^((I/2)*(-c + Pi/2 - d*x))*Sqrt[a + b*Sin[c + d*x]])] + (ArcCos[-(a/b)] + (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]] - ArcTanh[((-a + b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]]))*Log[(Sqrt[-a^2 + b^2]*E^((I/2)*(-c + Pi/2 - d*x)))/(Sqrt[2]*Sqrt[b]*Sqrt[a + b*Sin[c + d*x]])] - (ArcCos[-(a/b)] + (2*I)*ArcTanh[((-a + b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]])*Log[1 - ((a - I*Sqrt[-a^2 + b^2])*(a + b - Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))] + (-ArcCos[-(a/b)] + (2*I)*ArcTanh[((-a + b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[-a^2 + b^2]])*Log[1 - ((a + I*Sqrt[-a^2 + b^2])*(a + b - Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))] + I*(PolyLog[2, ((a - I*Sqrt[-a^2 + b^2])*(a + b - Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))] - PolyLog[2, ((a + I*Sqrt[-a^2 + b^2])*(a + b - Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(-c + Pi/2 - d*x)/2]))]))/Sqrt[-a^2 + b^2]))/(b*(-a^2 + b^2)*d^3) + (b*E^(I*c)*f^2*(d^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - d^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (2*I)*d*x*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (2*I)*d*x*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/((-a^2 + b^2)*d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]) + ((2*I)*b*e^2*ArcTan[(I*b*Cos[c] - I*(-a + b*Sin[c])*Tan[(d*x)/2])/Sqrt[-a^2 + b^2*Cos[c]^2 + b^2*Sin[c]^2]])/((-a^2 + b^2)*d*Sqrt[-a^2 + b^2*Cos[c]^2 + b^2*Sin[c]^2]) + ((4*I)*a^2*e*f*ArcTan[(I*b*Cos[c] - I*(-a + b*Sin[c])*Tan[(d*x)/2])/Sqrt[-a^2 + b^2*Cos[c]^2 + b^2*Sin[c]^2]]*Cot[c])/(b*(-a^2 + b^2)*d^2*Sqrt[-a^2 + b^2*Cos[c]^2 + b^2*Sin[c]^2]) + (2*a*f^2*Csc[c]*(-1/2*(x^2*Cos[c])/b + (x*(d*x*Cos[c] - (2*a*ArcTan[(Sec[(d*x)/2]*(Cos[c] - I*Sin[c])*(b*Cos[c + (d*x)/2] + a*Sin[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*Cos[c]*(Cos[c] - I*Sin[c]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2]) - Log[a + b*Sin[c + d*x]]*Sin[c]))/(b*d) + (-((a*Cos[c]*((-I)*d*x*(Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])]) - PolyLog[2, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])]))/(Sqrt[a^2 - b^2]*d)) + (2*a*x*ArcTan[(Sec[(d*x)/2]*(Cos[c] - I*Sin[c])*(b*Cos[c + (d*x)/2] + a*Sin[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*Cos[c]*(Cos[c] - I*Sin[c]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2]) + ((c + d*x)*Log[a + b*Sin[c + d*x]]*Sin[c])/d - (b*(((c + d*x)*Log[a + b*Sin[c + d*x]])/b - ((-1/2*I)*(-c + Pi/2 - d*x)^2 + (4*I)*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]]*ArcTan[((a - b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[a^2 - b^2]] + (-c + Pi/2 - d*x + 2*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + ((a - Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] + (-c + Pi/2 - d*x - 2*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + ((a + Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] - (-c + Pi/2 - d*x)*Log[a + b*Sin[c + d*x]] - I*(PolyLog[2, ((-a - Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] + PolyLog[2, ((-a + Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b]))/b)*Sin[c])/d)/(b*d)))/((-a^2 + b^2)*d) - (2*a*e*f*Csc[c]*(-(b*d*x*Cos[c]) + b*Log[a + b*Cos[d*x]*Sin[c] + b*Cos[c]*Sin[d*x]]*Sin[c] + ((2*I)*a*b*ArcTan[(I*b*Cos[c] - I*(-a + b*Sin[c])*Tan[(d*x)/2])/Sqrt[-a^2 + b^2*Cos[c]^2 + b^2*Sin[c]^2]]*Cos[c])/Sqrt[-a^2 + b^2*Cos[c]^2 + b^2*Sin[c]^2]))/((-a^2 + b^2)*d^2*(b^2*Cos[c]^2 + b^2*Sin[c]^2)) + (Csc[c/2]*Sec[c/2]*(a^2*e^2*Cos[c] + 2*a^2*e*f*x*Cos[c] + a^2*f^2*x^2*Cos[c] + a*b*e^2*Sin[d*x] + 2*a*b*e*f*x*Sin[d*x] + a*b*f^2*x^2*Sin[d*x]))/(2*(a - b)*b*(a + b)*d*(a + b*Sin[c + d*x]))","B",0
247,1,5446,1512,22.3185497,"\int \frac{(e+f x)^3 \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[((e + f*x)^3*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\text{Result too large to show}","\frac{6 a \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^3}{b \left(a^2-b^2\right) d^4}+\frac{6 a \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^3}{b \left(a^2-b^2\right) d^4}+\frac{6 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^3}{b \sqrt{a^2-b^2} d^4}-\frac{6 a^2 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^3}{b \left(a^2-b^2\right)^{3/2} d^4}-\frac{6 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^3}{b \sqrt{a^2-b^2} d^4}+\frac{6 a^2 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^3}{b \left(a^2-b^2\right)^{3/2} d^4}-\frac{6 i a (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^2}{b \left(a^2-b^2\right) d^3}-\frac{6 i a (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^2}{b \left(a^2-b^2\right) d^3}-\frac{6 i (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^2}{b \sqrt{a^2-b^2} d^3}+\frac{6 i a^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^2}{b \left(a^2-b^2\right)^{3/2} d^3}+\frac{6 i (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^2}{b \sqrt{a^2-b^2} d^3}-\frac{6 i a^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^2}{b \left(a^2-b^2\right)^{3/2} d^3}+\frac{3 a (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f}{b \left(a^2-b^2\right) d^2}+\frac{3 a (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f}{b \left(a^2-b^2\right) d^2}-\frac{3 (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f}{b \sqrt{a^2-b^2} d^2}+\frac{3 a^2 (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f}{b \left(a^2-b^2\right)^{3/2} d^2}+\frac{3 (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f}{b \sqrt{a^2-b^2} d^2}-\frac{3 a^2 (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f}{b \left(a^2-b^2\right)^{3/2} d^2}-\frac{i a (e+f x)^3}{b \left(a^2-b^2\right) d}-\frac{i (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2} d}+\frac{i a^2 (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right)^{3/2} d}+\frac{i (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2} d}-\frac{i a^2 (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right)^{3/2} d}-\frac{a (e+f x)^3 \cos (c+d x)}{\left(a^2-b^2\right) d (a+b \sin (c+d x))}",1,"Result too large to show","B",0
248,1,2408,751,16.0046126,"\int \frac{(e+f x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[((e + f*x)*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\text{Result too large to show}","-\frac{3 a f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{2 b d^2 \left(a^2-b^2\right)^{3/2}}+\frac{3 a f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{2 b d^2 \left(a^2-b^2\right)^{3/2}}-\frac{a f}{2 b d^2 \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 a^2 f \log (a+b \sin (c+d x))}{2 b d^2 \left(a^2-b^2\right)^2}-\frac{f \log (a+b \sin (c+d x))}{b d^2 \left(a^2-b^2\right)}-\frac{3 i a (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{2 b d \left(a^2-b^2\right)^{3/2}}+\frac{3 i a (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{2 b d \left(a^2-b^2\right)^{3/2}}-\frac{3 a^2 (e+f x) \cos (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{a (e+f x) \cos (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}+\frac{(e+f x) \cos (c+d x)}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 a^3 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{2 b d^2 \left(a^2-b^2\right)^{5/2}}-\frac{3 a^3 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{2 b d^2 \left(a^2-b^2\right)^{5/2}}+\frac{3 i a^3 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{2 b d \left(a^2-b^2\right)^{5/2}}-\frac{3 i a^3 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{2 b d \left(a^2-b^2\right)^{5/2}}",1,"(-(a*d*e*Cos[c + d*x]) + a*c*f*Cos[c + d*x] - a*f*(c + d*x)*Cos[c + d*x])/(2*(a - b)*(a + b)*d^2*(a + b*Sin[c + d*x])^2) + (-(a^3*f) + a*b^2*f - a^2*b*d*e*Cos[c + d*x] - 2*b^3*d*e*Cos[c + d*x] + a^2*b*c*f*Cos[c + d*x] + 2*b^3*c*f*Cos[c + d*x] - a^2*b*f*(c + d*x)*Cos[c + d*x] - 2*b^3*f*(c + d*x)*Cos[c + d*x])/(2*(a - b)^2*b*(a + b)^2*d^2*(a + b*Sin[c + d*x])) + (((-2*(a^2 + 2*b^2)*f*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (2*(a^2*f + 2*b^2*f + a*b*(-3*d*e + 3*c*f))*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - ((a^2 + 2*b^2)*f*Log[Sec[(c + d*x)/2]^2])/b + ((a^2 + 2*b^2)*f*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])])/b + ((3*I)*a*b*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] - ((3*I)*a*b*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] - ((3*I)*a*b*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] + ((3*I)*a*b*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2])*((-3*a*b*e)/(2*(a^2 - b^2)^2*(a + b*Sin[c + d*x])) + (3*a*b*c*f)/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) - (3*a*b*f*(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + (a^2*f*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + (b^2*f*Cos[c + d*x])/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))))/(d*(-(((a^2 + 2*b^2)*f*Tan[(c + d*x)/2])/b) + ((a^2 + 2*b^2)*f*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(b*(a + b*Sin[c + d*x])) - (a*(a^2 + 2*b^2)*f*Sec[(c + d*x)/2]^2)/((a^2 - b^2)*(1 + (b + a*Tan[(c + d*x)/2])^2/(a^2 - b^2))) + (a*(a^2*f + 2*b^2*f + a*b*(-3*d*e + 3*c*f))*Sec[(c + d*x)/2]^2)/((a^2 - b^2)*(1 + (b + a*Tan[(c + d*x)/2])^2/(a^2 - b^2))) - ((3*I)*a*b*f*(((-1/2*I)*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (Log[1 - (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(2*(I + Tan[(c + d*x)/2])) + (a*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(2*(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]))))/Sqrt[-a^2 + b^2] + ((3*I)*a*b*f*(((I/2)*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - ((I/2)*a*Log[1 - (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(a + I*a*Tan[(c + d*x)/2]) + (a*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(2*(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]))))/Sqrt[-a^2 + b^2] + ((3*I)*a*b*f*(((I/2)*Log[1 - (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - ((I/2)*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) + (a*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(2*(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]))))/Sqrt[-a^2 + b^2] - ((3*I)*a*b*f*(((-1/2*I)*Log[1 - (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + ((I/2)*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + (a*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(2*(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]))))/Sqrt[-a^2 + b^2]))","B",0
249,1,13567,1584,26.0011211,"\int \frac{(e+f x)^2 \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[((e + f*x)^2*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\text{Result too large to show}","\frac{3 i (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^3}{2 b \left(a^2-b^2\right)^{5/2} d}-\frac{3 i (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^3}{2 b \left(a^2-b^2\right)^{5/2} d}+\frac{3 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^3}{b \left(a^2-b^2\right)^{5/2} d^2}-\frac{3 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^3}{b \left(a^2-b^2\right)^{5/2} d^2}+\frac{3 i f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^3}{b \left(a^2-b^2\right)^{5/2} d^3}-\frac{3 i f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^3}{b \left(a^2-b^2\right)^{5/2} d^3}-\frac{3 i (e+f x)^2 a^2}{2 b \left(a^2-b^2\right)^2 d}+\frac{3 f (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^2 d^2}+\frac{3 f (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^2 d^2}-\frac{3 i f^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^2 d^3}-\frac{3 i f^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^2 d^3}-\frac{3 (e+f x)^2 \cos (c+d x) a^2}{2 \left(a^2-b^2\right)^2 d (a+b \sin (c+d x))}+\frac{2 f^2 \tan ^{-1}\left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^3}-\frac{3 i (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a}{2 b \left(a^2-b^2\right)^{3/2} d}+\frac{3 i (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a}{2 b \left(a^2-b^2\right)^{3/2} d}-\frac{3 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^2}+\frac{3 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^2}-\frac{3 i f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^3}+\frac{3 i f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^3}-\frac{f (e+f x) a}{b \left(a^2-b^2\right) d^2 (a+b \sin (c+d x))}-\frac{(e+f x)^2 \cos (c+d x) a}{2 \left(a^2-b^2\right) d (a+b \sin (c+d x))^2}+\frac{i (e+f x)^2}{b \left(a^2-b^2\right) d}-\frac{2 f (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right) d^2}-\frac{2 f (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right) d^2}+\frac{2 i f^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right) d^3}+\frac{2 i f^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right) d^3}+\frac{(e+f x)^2 \cos (c+d x)}{\left(a^2-b^2\right) d (a+b \sin (c+d x))}",1,"Result too large to show","B",0
250,1,11208,2348,22.5995958,"\int \frac{(e+f x)^3 \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[((e + f*x)^3*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\text{Result too large to show}","\frac{3 i (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^3}{2 b \left(a^2-b^2\right)^{5/2} d}-\frac{3 i (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^3}{2 b \left(a^2-b^2\right)^{5/2} d}+\frac{9 f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^3}{2 b \left(a^2-b^2\right)^{5/2} d^2}-\frac{9 f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^3}{2 b \left(a^2-b^2\right)^{5/2} d^2}+\frac{9 i f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^3}{b \left(a^2-b^2\right)^{5/2} d^3}-\frac{9 i f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^3}{b \left(a^2-b^2\right)^{5/2} d^3}-\frac{9 f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^3}{b \left(a^2-b^2\right)^{5/2} d^4}+\frac{9 f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^3}{b \left(a^2-b^2\right)^{5/2} d^4}-\frac{3 i (e+f x)^3 a^2}{2 b \left(a^2-b^2\right)^2 d}+\frac{9 f (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^2}{2 b \left(a^2-b^2\right)^2 d^2}+\frac{9 f (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^2}{2 b \left(a^2-b^2\right)^2 d^2}-\frac{9 i f^2 (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^2 d^3}-\frac{9 i f^2 (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^2 d^3}+\frac{9 f^3 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^2 d^4}+\frac{9 f^3 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a^2}{b \left(a^2-b^2\right)^2 d^4}-\frac{3 (e+f x)^3 \cos (c+d x) a^2}{2 \left(a^2-b^2\right)^2 d (a+b \sin (c+d x))}-\frac{3 i (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a}{2 b \left(a^2-b^2\right)^{3/2} d}-\frac{3 i f^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^3}+\frac{3 i (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a}{2 b \left(a^2-b^2\right)^{3/2} d}+\frac{3 i f^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^3}-\frac{3 f^3 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^4}-\frac{9 f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a}{2 b \left(a^2-b^2\right)^{3/2} d^2}+\frac{3 f^3 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^4}+\frac{9 f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a}{2 b \left(a^2-b^2\right)^{3/2} d^2}-\frac{9 i f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^3}+\frac{9 i f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^3}+\frac{9 f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^4}-\frac{9 f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) a}{b \left(a^2-b^2\right)^{3/2} d^4}-\frac{3 f (e+f x)^2 a}{2 b \left(a^2-b^2\right) d^2 (a+b \sin (c+d x))}-\frac{(e+f x)^3 \cos (c+d x) a}{2 \left(a^2-b^2\right) d (a+b \sin (c+d x))^2}+\frac{i (e+f x)^3}{b \left(a^2-b^2\right) d}-\frac{3 f (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right) d^2}-\frac{3 f (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right) d^2}+\frac{6 i f^2 (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right) d^3}+\frac{6 i f^2 (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right) d^3}-\frac{6 f^3 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right) d^4}-\frac{6 f^3 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right) d^4}+\frac{(e+f x)^3 \cos (c+d x)}{\left(a^2-b^2\right) d (a+b \sin (c+d x))}",1,"Result too large to show","B",0
251,1,276,151,1.4954397,"\int \frac{(e+f x)^3 \cos (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{x \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right)}{4 a \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}-\frac{2 (\cos (c)+i \sin (c)) \left(\frac{3 f (\cos (c)-i \sin (c)) (\sin (c)-i \cos (c)+1) \left(d^2 (e+f x)^2 \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-2 i d f (e+f x) \text{Li}_3(-i \cos (c+d x)-\sin (c+d x))-2 f^2 \text{Li}_4(-i \cos (c+d x)-\sin (c+d x))\right)}{d^4}-\frac{(\sin (c)+i \cos (c)+1) (e+f x)^3 \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^4}{4 f}\right)}{a (\cos (c)+i (\sin (c)+1))}","\frac{12 i f^3 \text{Li}_4\left(i e^{i (c+d x)}\right)}{a d^4}+\frac{12 f^2 (e+f x) \text{Li}_3\left(i e^{i (c+d x)}\right)}{a d^3}-\frac{6 i f (e+f x)^2 \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^2}+\frac{2 (e+f x)^3 \log \left(1-i e^{i (c+d x)}\right)}{a d}-\frac{i (e+f x)^4}{4 a f}",1,"(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*(Cos[c/2] - Sin[c/2]))/(4*a*(Cos[c/2] + Sin[c/2])) - (2*(Cos[c] + I*Sin[c])*(((e + f*x)^4*(Cos[c] - I*Sin[c]))/(4*f) + (3*f*(d^2*(e + f*x)^2*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - (2*I)*d*f*(e + f*x)*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]] - 2*f^2*PolyLog[4, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*Sin[c])*(1 - I*Cos[c] + Sin[c]))/d^4 - ((e + f*x)^3*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d))/(a*(Cos[c] + I*(1 + Sin[c])))","A",1
252,1,221,114,1.05182,"\int \frac{(e+f x)^2 \cos (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{x \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(3 e^2+3 e f x+f^2 x^2\right)}{3 a \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}-\frac{2 (\cos (c)+i \sin (c)) \left(\frac{2 f (\cos (c)-i (\sin (c)+1)) (d (e+f x) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-i f \text{Li}_3(-i \cos (c+d x)-\sin (c+d x)))}{d^3}-\frac{(\sin (c)+i \cos (c)+1) (e+f x)^2 \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^3}{3 f}\right)}{a (\cos (c)+i (\sin (c)+1))}","\frac{4 f^2 \text{Li}_3\left(i e^{i (c+d x)}\right)}{a d^3}-\frac{4 i f (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^2}+\frac{2 (e+f x)^2 \log \left(1-i e^{i (c+d x)}\right)}{a d}-\frac{i (e+f x)^3}{3 a f}",1,"(x*(3*e^2 + 3*e*f*x + f^2*x^2)*(Cos[c/2] - Sin[c/2]))/(3*a*(Cos[c/2] + Sin[c/2])) - (2*(Cos[c] + I*Sin[c])*(((e + f*x)^3*(Cos[c] - I*Sin[c]))/(3*f) - ((e + f*x)^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (2*f*(d*(e + f*x)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*f*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*(1 + Sin[c])))/d^3))/(a*(Cos[c] + I*(1 + Sin[c])))","A",1
253,1,246,79,0.5738625,"\int \frac{(e+f x) \cos (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x])/(a + a*Sin[c + d*x]),x]","\frac{-i c^2 f+4 d e \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 i f \text{Li}_2\left(i e^{i (c+d x)}\right)-2 i c d f x+4 c f \log \left(1-i e^{i (c+d x)}\right)+4 \pi  f \log \left(1+e^{-i (c+d x)}\right)+4 d f x \log \left(1-i e^{i (c+d x)}\right)+2 \pi  f \log \left(1-i e^{i (c+d x)}\right)-2 \pi  f \log \left(\sin \left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)-4 \pi  f \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 c f \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+i \pi  c f-i d^2 f x^2+i \pi  d f x}{2 a d^2}","-\frac{2 i f \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^2}+\frac{2 (e+f x) \log \left(1-i e^{i (c+d x)}\right)}{a d}-\frac{i (e+f x)^2}{2 a f}",1,"((-I)*c^2*f + I*c*f*Pi - (2*I)*c*d*f*x + I*d*f*Pi*x - I*d^2*f*x^2 + 4*f*Pi*Log[1 + E^((-I)*(c + d*x))] + 4*c*f*Log[1 - I*E^(I*(c + d*x))] + 2*f*Pi*Log[1 - I*E^(I*(c + d*x))] + 4*d*f*x*Log[1 - I*E^(I*(c + d*x))] - 4*f*Pi*Log[Cos[(c + d*x)/2]] + 4*d*e*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 4*c*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2*f*Pi*Log[Sin[(2*c + Pi + 2*d*x)/4]] - (4*I)*f*PolyLog[2, I*E^(I*(c + d*x))])/(2*a*d^2)","B",1
254,1,16,16,0.0111317,"\int \frac{\cos (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]/(a + a*Sin[c + d*x]),x]","\frac{\log (\sin (c+d x)+1)}{a d}","\frac{\log (\sin (c+d x)+1)}{a d}",1,"Log[1 + Sin[c + d*x]]/(a*d)","A",1
255,0,0,29,3.2507426,"\int \frac{\cos (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Cos[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])),x]","\int \frac{\cos (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\cos (c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right)",0,"Integrate[Cos[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]","A",-1
256,0,0,29,4.444488,"\int \frac{\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Cos[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\int \frac{\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\cos (c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right)",0,"Integrate[Cos[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x]","A",-1
257,1,102,99,0.7639522,"\int \frac{(e+f x)^3 \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{-12 f \sin (c+d x) \left(d^2 (e+f x)^2-2 f^2\right)+4 d (e+f x) \cos (c+d x) \left(d^2 (e+f x)^2-6 f^2\right)+d^4 x \left(4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right)}{4 a d^4}","\frac{6 f^3 \sin (c+d x)}{a d^4}-\frac{6 f^2 (e+f x) \cos (c+d x)}{a d^3}-\frac{3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac{(e+f x)^3 \cos (c+d x)}{a d}+\frac{(e+f x)^4}{4 a f}",1,"(d^4*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) + 4*d*(e + f*x)*(-6*f^2 + d^2*(e + f*x)^2)*Cos[c + d*x] - 12*f*(-2*f^2 + d^2*(e + f*x)^2)*Sin[c + d*x])/(4*a*d^4)","A",1
258,1,74,75,0.4681903,"\int \frac{(e+f x)^2 \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{3 \cos (c+d x) \left(d^2 (e+f x)^2-2 f^2\right)-6 d f (e+f x) \sin (c+d x)+d^3 x \left(3 e^2+3 e f x+f^2 x^2\right)}{3 a d^3}","-\frac{2 f^2 \cos (c+d x)}{a d^3}-\frac{2 f (e+f x) \sin (c+d x)}{a d^2}+\frac{(e+f x)^2 \cos (c+d x)}{a d}+\frac{(e+f x)^3}{3 a f}",1,"(d^3*x*(3*e^2 + 3*e*f*x + f^2*x^2) + 3*(-2*f^2 + d^2*(e + f*x)^2)*Cos[c + d*x] - 6*d*f*(e + f*x)*Sin[c + d*x])/(3*a*d^3)","A",1
259,1,53,51,0.5376979,"\int \frac{(e+f x) \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","-\frac{(c+d x) (c f-2 d e-d f x)-2 d (e+f x) \cos (c+d x)+2 f \sin (c+d x)}{2 a d^2}","-\frac{f \sin (c+d x)}{a d^2}+\frac{(e+f x) \cos (c+d x)}{a d}+\frac{e x}{a}+\frac{f x^2}{2 a}",1,"-1/2*((c + d*x)*(-2*d*e + c*f - d*f*x) - 2*d*(e + f*x)*Cos[c + d*x] + 2*f*Sin[c + d*x])/(a*d^2)","A",1
260,1,97,19,0.148983,"\int \frac{\cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sin[c + d*x]),x]","-\frac{\left(2 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+(\sin (c+d x)-1) \sqrt{\sin (c+d x)+1}\right) \cos ^3(c+d x)}{a d (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^{3/2}}","\frac{\cos (c+d x)}{a d}+\frac{x}{a}",1,"-((Cos[c + d*x]^3*(2*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + (-1 + Sin[c + d*x])*Sqrt[1 + Sin[c + d*x]]))/(a*d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(3/2)))","B",1
261,1,58,72,0.3120044,"\int \frac{\cos ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Cos[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])),x]","\frac{-\sin \left(c-\frac{d e}{f}\right) \text{Ci}\left(d \left(\frac{e}{f}+x\right)\right)-\cos \left(c-\frac{d e}{f}\right) \text{Si}\left(d \left(\frac{e}{f}+x\right)\right)+\log (e+f x)}{a f}","-\frac{\sin \left(c-\frac{d e}{f}\right) \text{Ci}\left(\frac{d e}{f}+d x\right)}{a f}-\frac{\cos \left(c-\frac{d e}{f}\right) \text{Si}\left(\frac{d e}{f}+d x\right)}{a f}+\frac{\log (e+f x)}{a f}",1,"(Log[e + f*x] - CosIntegral[d*(e/f + x)]*Sin[c - (d*e)/f] - Cos[c - (d*e)/f]*SinIntegral[d*(e/f + x)])/(a*f)","A",1
262,1,80,95,0.441703,"\int \frac{\cos ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Cos[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\frac{-d (e+f x) \cos \left(c-\frac{d e}{f}\right) \text{Ci}\left(d \left(\frac{e}{f}+x\right)\right)+d (e+f x) \sin \left(c-\frac{d e}{f}\right) \text{Si}\left(d \left(\frac{e}{f}+x\right)\right)+f (\sin (c+d x)-1)}{a f^2 (e+f x)}","-\frac{d \cos \left(c-\frac{d e}{f}\right) \text{Ci}\left(\frac{d e}{f}+d x\right)}{a f^2}+\frac{d \sin \left(c-\frac{d e}{f}\right) \text{Si}\left(\frac{d e}{f}+d x\right)}{a f^2}+\frac{\sin (c+d x)}{a f (e+f x)}-\frac{1}{a f (e+f x)}",1,"(-(d*(e + f*x)*Cos[c - (d*e)/f]*CosIntegral[d*(e/f + x)]) + f*(-1 + Sin[c + d*x]) + d*(e + f*x)*Sin[c - (d*e)/f]*SinIntegral[d*(e/f + x)])/(a*f^2*(e + f*x))","A",1
263,1,132,219,1.364762,"\int \frac{(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{96 f \cos (c+d x) \left(d^2 (e+f x)^2-2 f^2\right)+4 d (e+f x) \cos (2 (c+d x)) \left(2 d^2 (e+f x)^2-3 f^2\right)+4 \sin (c+d x) \left(8 d (e+f x) \left(d^2 (e+f x)^2-6 f^2\right)-3 f \cos (c+d x) \left(2 d^2 (e+f x)^2-f^2\right)\right)}{32 a d^4}","-\frac{6 f^3 \cos (c+d x)}{a d^4}+\frac{3 f^3 \sin (c+d x) \cos (c+d x)}{8 a d^4}+\frac{3 f^2 (e+f x) \sin ^2(c+d x)}{4 a d^3}-\frac{6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac{3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac{3 f (e+f x)^2 \sin (c+d x) \cos (c+d x)}{4 a d^2}-\frac{(e+f x)^3 \sin ^2(c+d x)}{2 a d}+\frac{(e+f x)^3 \sin (c+d x)}{a d}-\frac{3 f^3 x}{8 a d^3}+\frac{(e+f x)^3}{4 a d}",1,"(96*f*(-2*f^2 + d^2*(e + f*x)^2)*Cos[c + d*x] + 4*d*(e + f*x)*(-3*f^2 + 2*d^2*(e + f*x)^2)*Cos[2*(c + d*x)] + 4*(8*d*(e + f*x)*(-6*f^2 + d^2*(e + f*x)^2) - 3*f*(-f^2 + 2*d^2*(e + f*x)^2)*Cos[c + d*x])*Sin[c + d*x])/(32*a*d^4)","A",1
264,1,95,161,1.0882671,"\int \frac{(e+f x)^2 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{\cos (2 (c+d x)) \left(2 d^2 (e+f x)^2-f^2\right)-4 \sin (c+d x) \left(d f (e+f x) \cos (c+d x)-2 \left(d^2 (e+f x)^2-2 f^2\right)\right)+16 d f (e+f x) \cos (c+d x)}{8 a d^3}","\frac{f^2 \sin ^2(c+d x)}{4 a d^3}-\frac{2 f^2 \sin (c+d x)}{a d^3}+\frac{2 f (e+f x) \cos (c+d x)}{a d^2}-\frac{f (e+f x) \sin (c+d x) \cos (c+d x)}{2 a d^2}-\frac{(e+f x)^2 \sin ^2(c+d x)}{2 a d}+\frac{(e+f x)^2 \sin (c+d x)}{a d}+\frac{e f x}{2 a d}+\frac{f^2 x^2}{4 a d}",1,"(16*d*f*(e + f*x)*Cos[c + d*x] + (-f^2 + 2*d^2*(e + f*x)^2)*Cos[2*(c + d*x)] - 4*(-2*(-2*f^2 + d^2*(e + f*x)^2) + d*f*(e + f*x)*Cos[c + d*x])*Sin[c + d*x])/(8*a*d^3)","A",1
265,1,52,91,0.9501947,"\int \frac{(e+f x) \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{d (e+f x) (4 \sin (c+d x)+\cos (2 (c+d x)))-f (\sin (c+d x)-4) \cos (c+d x)}{4 a d^2}","\frac{f \cos (c+d x)}{a d^2}-\frac{f \sin (c+d x) \cos (c+d x)}{4 a d^2}-\frac{(e+f x) \sin ^2(c+d x)}{2 a d}+\frac{(e+f x) \sin (c+d x)}{a d}+\frac{f x}{4 a d}",1,"(-(f*Cos[c + d*x]*(-4 + Sin[c + d*x])) + d*(e + f*x)*(Cos[2*(c + d*x)] + 4*Sin[c + d*x]))/(4*a*d^2)","A",1
266,1,24,32,0.0438196,"\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
267,1,105,128,0.3991771,"\int \frac{\cos ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Cos[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])),x]","-\frac{\sin \left(2 c-\frac{2 d e}{f}\right) \text{Ci}\left(\frac{2 d (e+f x)}{f}\right)-2 \cos \left(c-\frac{d e}{f}\right) \text{Ci}\left(d \left(\frac{e}{f}+x\right)\right)+2 \sin \left(c-\frac{d e}{f}\right) \text{Si}\left(d \left(\frac{e}{f}+x\right)\right)+\cos \left(2 c-\frac{2 d e}{f}\right) \text{Si}\left(\frac{2 d (e+f x)}{f}\right)}{2 a f}","-\frac{\sin \left(2 c-\frac{2 d e}{f}\right) \text{Ci}\left(\frac{2 d e}{f}+2 d x\right)}{2 a f}+\frac{\cos \left(c-\frac{d e}{f}\right) \text{Ci}\left(\frac{d e}{f}+d x\right)}{a f}-\frac{\sin \left(c-\frac{d e}{f}\right) \text{Si}\left(\frac{d e}{f}+d x\right)}{a f}-\frac{\cos \left(2 c-\frac{2 d e}{f}\right) \text{Si}\left(\frac{2 d e}{f}+2 d x\right)}{2 a f}",1,"-1/2*(-2*Cos[c - (d*e)/f]*CosIntegral[d*(e/f + x)] + CosIntegral[(2*d*(e + f*x))/f]*Sin[2*c - (2*d*e)/f] + 2*Sin[c - (d*e)/f]*SinIntegral[d*(e/f + x)] + Cos[2*c - (2*d*e)/f]*SinIntegral[(2*d*(e + f*x))/f])/(a*f)","A",1
268,1,203,175,0.6247604,"\int \frac{\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Cos[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\frac{-2 d (e+f x) \sin \left(c-\frac{d e}{f}\right) \text{Ci}\left(d \left(\frac{e}{f}+x\right)\right)-2 d (e+f x) \cos \left(2 c-\frac{2 d e}{f}\right) \text{Ci}\left(\frac{2 d (e+f x)}{f}\right)+2 d e \sin \left(2 c-\frac{2 d e}{f}\right) \text{Si}\left(\frac{2 d (e+f x)}{f}\right)+2 d f x \sin \left(2 c-\frac{2 d e}{f}\right) \text{Si}\left(\frac{2 d (e+f x)}{f}\right)-2 d e \cos \left(c-\frac{d e}{f}\right) \text{Si}\left(d \left(\frac{e}{f}+x\right)\right)-2 d f x \cos \left(c-\frac{d e}{f}\right) \text{Si}\left(d \left(\frac{e}{f}+x\right)\right)+f \sin (2 (c+d x))-2 f \cos (c+d x)}{2 a f^2 (e+f x)}","-\frac{d \sin \left(c-\frac{d e}{f}\right) \text{Ci}\left(\frac{d e}{f}+d x\right)}{a f^2}-\frac{d \cos \left(2 c-\frac{2 d e}{f}\right) \text{Ci}\left(\frac{2 d e}{f}+2 d x\right)}{a f^2}+\frac{d \sin \left(2 c-\frac{2 d e}{f}\right) \text{Si}\left(\frac{2 d e}{f}+2 d x\right)}{a f^2}-\frac{d \cos \left(c-\frac{d e}{f}\right) \text{Si}\left(\frac{d e}{f}+d x\right)}{a f^2}+\frac{\sin (2 c+2 d x)}{2 a f (e+f x)}-\frac{\cos (c+d x)}{a f (e+f x)}",1,"(-2*f*Cos[c + d*x] - 2*d*(e + f*x)*Cos[2*c - (2*d*e)/f]*CosIntegral[(2*d*(e + f*x))/f] - 2*d*(e + f*x)*CosIntegral[d*(e/f + x)]*Sin[c - (d*e)/f] + f*Sin[2*(c + d*x)] - 2*d*e*Cos[c - (d*e)/f]*SinIntegral[d*(e/f + x)] - 2*d*f*x*Cos[c - (d*e)/f]*SinIntegral[d*(e/f + x)] + 2*d*e*Sin[2*c - (2*d*e)/f]*SinIntegral[(2*d*(e + f*x))/f] + 2*d*f*x*Sin[2*c - (2*d*e)/f]*SinIntegral[(2*d*(e + f*x))/f])/(2*a*f^2*(e + f*x))","A",1
269,1,865,502,8.975388,"\int \frac{(e+f x)^3 \sec (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Sec[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{(e+f x)^3}{2 a d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}-\frac{(\cos (c)+i \sin (c)) \left(\frac{(\cos (c)-i \sin (c)) (e+f x)^4}{4 f}+\frac{\log (-i \cos (c+d x)-\sin (c+d x)+1) (-i \cos (c)-\sin (c)+1) (e+f x)^3}{d}+\frac{3 f \left(-2 \text{Li}_4(i \cos (c+d x)+\sin (c+d x)) f^2-2 i d (e+f x) \text{Li}_3(i \cos (c+d x)+\sin (c+d x)) f+d^2 (e+f x)^2 \text{Li}_2(i \cos (c+d x)+\sin (c+d x))\right) (\cos (c)+i (\sin (c)-1)) (i \cos (c)+\sin (c))}{d^4}\right)}{2 a (\cos (c)+i (\sin (c)-1))}-\frac{12 \left(d^2 \text{Li}_2(-i \cos (c+d x)-\sin (c+d x)) x^2-2 i d \text{Li}_3(-i \cos (c+d x)-\sin (c+d x)) x-2 \text{Li}_4(-i \cos (c+d x)-\sin (c+d x))\right) (-i \cos (c)+\sin (c)+1) f^4-4 d^3 x^3 \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)+i (\sin (c)+1)) f^4+24 d e (d x \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-i \text{Li}_3(-i \cos (c+d x)-\sin (c+d x))) (-i \cos (c)+\sin (c)+1) f^3-12 d^3 e x^2 \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)+i (\sin (c)+1)) f^3+12 \left(d^2 e^2+4 f^2\right) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x)) (-i \cos (c)+\sin (c)+1) f^2-12 d \left(d^2 e^2+4 f^2\right) x \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)+i (\sin (c)+1)) f^2+4 i d e \left(d^2 e^2+12 f^2\right) (d x+i \log (\cos (c+d x)+i (\sin (c+d x)+1))) (\cos (c)+i (\sin (c)+1)) f+\left(12 f^2+d^2 (e+f x)^2\right)^2}{8 a d^4 f (\cos (c)+i (\sin (c)+1))}+\frac{x \left(4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right)}{8 a \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right)}+\frac{3 \left(x^2 \sin \left(\frac{d x}{2}\right) f^3+2 e x \sin \left(\frac{d x}{2}\right) f^2+e^2 \sin \left(\frac{d x}{2}\right) f\right)}{a d^2 \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}","\frac{3 i f^3 \text{Li}_2\left(-i e^{i (c+d x)}\right)}{a d^4}-\frac{3 i f^3 \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^4}-\frac{3 i f^3 \text{Li}_2\left(-e^{2 i (c+d x)}\right)}{2 a d^4}-\frac{3 i f^3 \text{Li}_4\left(-i e^{i (c+d x)}\right)}{a d^4}+\frac{3 i f^3 \text{Li}_4\left(i e^{i (c+d x)}\right)}{a d^4}-\frac{3 f^2 (e+f x) \text{Li}_3\left(-i e^{i (c+d x)}\right)}{a d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left(i e^{i (c+d x)}\right)}{a d^3}+\frac{3 f^2 (e+f x) \log \left(1+e^{2 i (c+d x)}\right)}{a d^3}-\frac{6 i f^2 (e+f x) \tan ^{-1}\left(e^{i (c+d x)}\right)}{a d^3}+\frac{3 i f (e+f x)^2 \text{Li}_2\left(-i e^{i (c+d x)}\right)}{2 a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left(i e^{i (c+d x)}\right)}{2 a d^2}+\frac{3 f (e+f x)^2 \tan (c+d x)}{2 a d^2}-\frac{3 f (e+f x)^2 \sec (c+d x)}{2 a d^2}-\frac{i (e+f x)^3 \tan ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{(e+f x)^3 \sec ^2(c+d x)}{2 a d}+\frac{(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{3 i f (e+f x)^2}{2 a d^2}",1,"(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))/(8*a*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])) - ((Cos[c] + I*Sin[c])*(((e + f*x)^3*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(1 - I*Cos[c] - Sin[c]))/d + ((e + f*x)^4*(Cos[c] - I*Sin[c]))/(4*f) + (3*f*(d^2*(e + f*x)^2*PolyLog[2, I*Cos[c + d*x] + Sin[c + d*x]] - (2*I)*d*f*(e + f*x)*PolyLog[3, I*Cos[c + d*x] + Sin[c + d*x]] - 2*f^2*PolyLog[4, I*Cos[c + d*x] + Sin[c + d*x]])*(Cos[c] + I*(-1 + Sin[c]))*(I*Cos[c] + Sin[c]))/d^4))/(2*a*(Cos[c] + I*(-1 + Sin[c]))) - ((12*f^2 + d^2*(e + f*x)^2)^2 + 12*f^2*(d^2*e^2 + 4*f^2)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(1 - I*Cos[c] + Sin[c]) + 24*d*e*f^3*(d*x*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(1 - I*Cos[c] + Sin[c]) + 12*f^4*(d^2*x^2*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - (2*I)*d*x*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]] - 2*PolyLog[4, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(1 - I*Cos[c] + Sin[c]) - 12*d*f^2*(d^2*e^2 + 4*f^2)*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] + I*(1 + Sin[c])) - 12*d^3*e*f^3*x^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] + I*(1 + Sin[c])) - 4*d^3*f^4*x^3*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] + I*(1 + Sin[c])) + (4*I)*d*e*f*(d^2*e^2 + 12*f^2)*(d*x + I*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])])*(Cos[c] + I*(1 + Sin[c])))/(8*a*d^4*f*(Cos[c] + I*(1 + Sin[c]))) - (e + f*x)^3/(2*a*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (3*(e^2*f*Sin[(d*x)/2] + 2*e*f^2*x*Sin[(d*x)/2] + f^3*x^2*Sin[(d*x)/2]))/(a*d^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","A",0
270,1,670,278,8.2074211,"\int \frac{(e+f x)^2 \sec (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Sec[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{\frac{6 f \left(i d (e+f x) \text{Li}_2\left(i e^{-i (c+d x)}\right)+f \text{Li}_3\left(i e^{-i (c+d x)}\right)\right)}{d^3}+\frac{3 (e+f x)^2 \log \left(1-i e^{-i (c+d x)}\right)}{d}+\frac{(e+f x)^3}{\left(e^{i c}-i\right) f}}{6 a}-\frac{(\cos (c)+i \sin (c)) \left(x (\cos (c)-i \sin (c)) \left(d^2 e^2+4 f^2\right)+\frac{(\sin (c)+i \cos (c)) (\cos (c)+i (\sin (c)+1)) \left(d^2 e^2+4 f^2\right) (d x+i \log (\cos (c+d x)+i (\sin (c+d x)+1)))}{d}-i d^2 e f x^2 \sin (c)+d^2 e f x^2 \cos (c)+\frac{1}{3} d^2 f^2 x^3 (\cos (c)-i \sin (c))+2 e f (\cos (c)-i (\sin (c)+1)) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-2 d e f x (\cos (c)-i \sin (c)) (\cos (c)+i (\sin (c)+1)) \log (\sin (c+d x)+i \cos (c+d x)+1)+\frac{2 f^2 (\cos (c)-i \sin (c)) (\sin (c)-i \cos (c)+1) (d x \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-i \text{Li}_3(-i \cos (c+d x)-\sin (c+d x)))}{d}-d f^2 x^2 (\cos (c)-i \sin (c)) (\cos (c)+i (\sin (c)+1)) \log (\sin (c+d x)+i \cos (c+d x)+1)\right)}{2 a d^2 (\cos (c)+i (\sin (c)+1))}+\frac{2 \left(e f \sin \left(\frac{d x}{2}\right)+f^2 x \sin \left(\frac{d x}{2}\right)\right)}{a d^2 \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}-\frac{(e+f x)^2}{2 a d \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{x \left(3 e^2+3 e f x+f^2 x^2\right)}{6 a \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}","-\frac{f^2 \text{Li}_3\left(-i e^{i (c+d x)}\right)}{a d^3}+\frac{f^2 \text{Li}_3\left(i e^{i (c+d x)}\right)}{a d^3}+\frac{f^2 \tanh ^{-1}(\sin (c+d x))}{a d^3}+\frac{f^2 \log (\cos (c+d x))}{a d^3}+\frac{i f (e+f x) \text{Li}_2\left(-i e^{i (c+d x)}\right)}{a d^2}-\frac{i f (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^2}+\frac{f (e+f x) \tan (c+d x)}{a d^2}-\frac{f (e+f x) \sec (c+d x)}{a d^2}-\frac{i (e+f x)^2 \tan ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{(e+f x)^2 \sec ^2(c+d x)}{2 a d}+\frac{(e+f x)^2 \tan (c+d x) \sec (c+d x)}{2 a d}",1,"-1/6*((e + f*x)^3/((-I + E^(I*c))*f) + (3*(e + f*x)^2*Log[1 - I/E^(I*(c + d*x))])/d + (6*f*(I*d*(e + f*x)*PolyLog[2, I/E^(I*(c + d*x))] + f*PolyLog[3, I/E^(I*(c + d*x))]))/d^3)/a + (x*(3*e^2 + 3*e*f*x + f^2*x^2))/(6*a*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])) - ((Cos[c] + I*Sin[c])*(d^2*e*f*x^2*Cos[c] + (d^2*e^2 + 4*f^2)*x*(Cos[c] - I*Sin[c]) + (d^2*f^2*x^3*(Cos[c] - I*Sin[c]))/3 - I*d^2*e*f*x^2*Sin[c] + (2*f^2*(d*x*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*Sin[c])*(1 - I*Cos[c] + Sin[c]))/d + 2*e*f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])) - 2*d*e*f*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])) - d*f^2*x^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])) + ((d^2*e^2 + 4*f^2)*(d*x + I*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])])*(I*Cos[c] + Sin[c])*(Cos[c] + I*(1 + Sin[c])))/d))/(2*a*d^2*(Cos[c] + I*(1 + Sin[c]))) - (e + f*x)^2/(2*a*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (2*(e*f*Sin[(d*x)/2] + f^2*x*Sin[(d*x)/2]))/(a*d^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
271,1,655,172,3.0730012,"\int \frac{(e+f x) \sec (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Sec[c + d*x])/(a + a*Sin[c + d*x]),x]","-\frac{(c+d x) (c f-d (2 e+f x)) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+d e \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)+d e \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)-\frac{f \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left((-1)^{3/4} (c+d x)^2+\frac{4 i \text{Li}_2\left(-i e^{i (c+d x)}\right)-3 i \pi  (c+d x)-4 \pi  \log \left(1+e^{-i (c+d x)}\right)+2 (-2 c-2 d x+\pi ) \log \left(1+i e^{i (c+d x)}\right)-2 \pi  \log \left(\sin \left(\frac{1}{4} (2 c+2 d x-\pi )\right)\right)+4 \pi  \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{2}}\right)}{\sqrt{2}}+\frac{f \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(\sqrt[4]{-1} (c+d x)^2+\frac{4 i \text{Li}_2\left(i e^{i (c+d x)}\right)-i \pi  (c+d x)-4 \pi  \log \left(1+e^{-i (c+d x)}\right)-2 (2 c+2 d x+\pi ) \log \left(1-i e^{i (c+d x)}\right)+2 \pi  \log \left(\sin \left(\frac{1}{4} (2 c+2 d x+\pi )\right)\right)+4 \pi  \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{2}}\right)}{\sqrt{2}}-4 f \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)-c f \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)-c f \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+c+d x\right)+2 d (e+f x)}{4 a d^2 (\sin (c+d x)+1)}","\frac{i f \text{Li}_2\left(-i e^{i (c+d x)}\right)}{2 a d^2}-\frac{i f \text{Li}_2\left(i e^{i (c+d x)}\right)}{2 a d^2}+\frac{f \tan (c+d x)}{2 a d^2}-\frac{f \sec (c+d x)}{2 a d^2}-\frac{i (e+f x) \tan ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{(e+f x) \sec ^2(c+d x)}{2 a d}+\frac{(e+f x) \tan (c+d x) \sec (c+d x)}{2 a d}",1,"-1/4*(2*d*(e + f*x) - 4*f*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (c + d*x)*(c*f - d*(2*e + f*x))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + d*e*(c + d*x + 2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - c*f*(c + d*x + 2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + d*e*(c + d*x - 2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - c*f*(c + d*x - 2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (f*((-1)^(3/4)*(c + d*x)^2 + ((-3*I)*Pi*(c + d*x) - 4*Pi*Log[1 + E^((-I)*(c + d*x))] + 2*(-2*c + Pi - 2*d*x)*Log[1 + I*E^(I*(c + d*x))] + 4*Pi*Log[Cos[(c + d*x)/2]] - 2*Pi*Log[Sin[(2*c - Pi + 2*d*x)/4]] + (4*I)*PolyLog[2, (-I)*E^(I*(c + d*x))])/Sqrt[2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/Sqrt[2] + (f*((-1)^(1/4)*(c + d*x)^2 + ((-I)*Pi*(c + d*x) - 4*Pi*Log[1 + E^((-I)*(c + d*x))] - 2*(2*c + Pi + 2*d*x)*Log[1 - I*E^(I*(c + d*x))] + 4*Pi*Log[Cos[(c + d*x)/2]] + 2*Pi*Log[Sin[(2*c + Pi + 2*d*x)/4]] + (4*I)*PolyLog[2, I*E^(I*(c + d*x))])/Sqrt[2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/Sqrt[2])/(a*d^2*(1 + Sin[c + d*x]))","B",1
272,1,30,37,0.0391362,"\int \frac{\sec (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + a*Sin[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))-\frac{1}{\sin (c+d x)+1}}{2 a d}","\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{1}{2 d (a \sin (c+d x)+a)}",1,"(ArcTanh[Sin[c + d*x]] - (1 + Sin[c + d*x])^(-1))/(2*a*d)","A",1
273,0,0,29,14.2886825,"\int \frac{\sec (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Sec[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])),x]","\int \frac{\sec (c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sec (c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sec[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]","A",-1
274,0,0,29,23.2743402,"\int \frac{\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Sec[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\int \frac{\sec (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sec (c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sec[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x]","A",-1
275,1,1117,475,8.9255322,"\int \frac{(e+f x)^3 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\frac{d^3 (e+f x)^3}{-i+e^{i c}}+3 d^2 f \log \left(1-i e^{-i (c+d x)}\right) (e+f x)^2+6 f^2 \left(i d (e+f x) \text{Li}_2\left(i e^{-i (c+d x)}\right)+f \text{Li}_3\left(i e^{-i (c+d x)}\right)\right)}{2 a d^4}-\frac{f (\cos (c)+i \sin (c)) \left(\frac{5}{3} d^2 f^2 (\cos (c)-i \sin (c)) x^3+5 d^2 e f \cos (c) x^2-5 i d^2 e f \sin (c) x^2-5 d f^2 \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)-i \sin (c)) (\cos (c)+i (\sin (c)+1)) x^2+\left(5 d^2 e^2+4 f^2\right) (\cos (c)-i \sin (c)) x-10 d e f \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)-i \sin (c)) (\cos (c)+i (\sin (c)+1)) x+\frac{10 f^2 (d x \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-i \text{Li}_3(-i \cos (c+d x)-\sin (c+d x))) (\cos (c)-i \sin (c)) (-i \cos (c)+\sin (c)+1)}{d}+10 e f \text{Li}_2(-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (\sin (c)+1))+\frac{\left(5 d^2 e^2+4 f^2\right) (d x+i \log (\cos (c+d x)+i (\sin (c+d x)+1))) (i \cos (c)+\sin (c)) (\cos (c)+i (\sin (c)+1))}{d}\right)}{2 a d^3 (\cos (c)+i (\sin (c)+1))}+\frac{\sin \left(\frac{d x}{2}\right) e^3+3 f x \sin \left(\frac{d x}{2}\right) e^2+3 f^2 x^2 \sin \left(\frac{d x}{2}\right) e+f^3 x^3 \sin \left(\frac{d x}{2}\right)}{2 a d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{5 d^2 \sin \left(\frac{d x}{2}\right) e^3+15 d^2 f x \sin \left(\frac{d x}{2}\right) e^2+12 f^2 \sin \left(\frac{d x}{2}\right) e+15 d^2 f^2 x^2 \sin \left(\frac{d x}{2}\right) e+5 d^2 f^3 x^3 \sin \left(\frac{d x}{2}\right)+12 f^3 x \sin \left(\frac{d x}{2}\right)}{6 a d^3 \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{-d \cos \left(\frac{c}{2}\right) e^3+d \sin \left(\frac{c}{2}\right) e^3-3 f \cos \left(\frac{c}{2}\right) e^2-3 d f x \cos \left(\frac{c}{2}\right) e^2-3 f \sin \left(\frac{c}{2}\right) e^2+3 d f x \sin \left(\frac{c}{2}\right) e^2-3 d f^2 x^2 \cos \left(\frac{c}{2}\right) e-6 f^2 x \cos \left(\frac{c}{2}\right) e+3 d f^2 x^2 \sin \left(\frac{c}{2}\right) e-6 f^2 x \sin \left(\frac{c}{2}\right) e-d f^3 x^3 \cos \left(\frac{c}{2}\right)-3 f^3 x^2 \cos \left(\frac{c}{2}\right)+d f^3 x^3 \sin \left(\frac{c}{2}\right)-3 f^3 x^2 \sin \left(\frac{c}{2}\right)}{6 a d^2 \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\sin \left(\frac{d x}{2}\right) e^3+3 f x \sin \left(\frac{d x}{2}\right) e^2+3 f^2 x^2 \sin \left(\frac{d x}{2}\right) e+f^3 x^3 \sin \left(\frac{d x}{2}\right)}{3 a d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","-\frac{f^3 \text{Li}_3\left(-i e^{i (c+d x)}\right)}{a d^4}+\frac{f^3 \text{Li}_3\left(i e^{i (c+d x)}\right)}{a d^4}+\frac{f^3 \text{Li}_3\left(-e^{2 i (c+d x)}\right)}{a d^4}+\frac{f^3 \tanh ^{-1}(\sin (c+d x))}{a d^4}+\frac{f^3 \log (\cos (c+d x))}{a d^4}+\frac{i f^2 (e+f x) \text{Li}_2\left(-i e^{i (c+d x)}\right)}{a d^3}-\frac{i f^2 (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right)}{a d^3}-\frac{2 i f^2 (e+f x) \text{Li}_2\left(-e^{2 i (c+d x)}\right)}{a d^3}+\frac{f^2 (e+f x) \tan (c+d x)}{a d^3}-\frac{f^2 (e+f x) \sec (c+d x)}{a d^3}+\frac{2 f (e+f x)^2 \log \left(1+e^{2 i (c+d x)}\right)}{a d^2}-\frac{i f (e+f x)^2 \tan ^{-1}\left(e^{i (c+d x)}\right)}{a d^2}-\frac{f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}+\frac{f (e+f x)^2 \tan (c+d x) \sec (c+d x)}{2 a d^2}+\frac{2 (e+f x)^3 \tan (c+d x)}{3 a d}-\frac{(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac{(e+f x)^3 \tan (c+d x) \sec ^2(c+d x)}{3 a d}-\frac{2 i (e+f x)^3}{3 a d}",1,"((d^3*(e + f*x)^3)/(-I + E^(I*c)) + 3*d^2*f*(e + f*x)^2*Log[1 - I/E^(I*(c + d*x))] + 6*f^2*(I*d*(e + f*x)*PolyLog[2, I/E^(I*(c + d*x))] + f*PolyLog[3, I/E^(I*(c + d*x))]))/(2*a*d^4) - (f*(Cos[c] + I*Sin[c])*(5*d^2*e*f*x^2*Cos[c] + (5*d^2*e^2 + 4*f^2)*x*(Cos[c] - I*Sin[c]) + (5*d^2*f^2*x^3*(Cos[c] - I*Sin[c]))/3 - (5*I)*d^2*e*f*x^2*Sin[c] + (10*f^2*(d*x*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*Sin[c])*(1 - I*Cos[c] + Sin[c]))/d + 10*e*f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])) - 10*d*e*f*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])) - 5*d*f^2*x^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])) + ((5*d^2*e^2 + 4*f^2)*(d*x + I*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])])*(I*Cos[c] + Sin[c])*(Cos[c] + I*(1 + Sin[c])))/d))/(2*a*d^3*(Cos[c] + I*(1 + Sin[c]))) + (e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2])/(2*a*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2])/(3*a*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (-(d*e^3*Cos[c/2]) - 3*e^2*f*Cos[c/2] - 3*d*e^2*f*x*Cos[c/2] - 6*e*f^2*x*Cos[c/2] - 3*d*e*f^2*x^2*Cos[c/2] - 3*f^3*x^2*Cos[c/2] - d*f^3*x^3*Cos[c/2] + d*e^3*Sin[c/2] - 3*e^2*f*Sin[c/2] + 3*d*e^2*f*x*Sin[c/2] - 6*e*f^2*x*Sin[c/2] + 3*d*e*f^2*x^2*Sin[c/2] - 3*f^3*x^2*Sin[c/2] + d*f^3*x^3*Sin[c/2])/(6*a*d^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (5*d^2*e^3*Sin[(d*x)/2] + 12*e*f^2*Sin[(d*x)/2] + 15*d^2*e^2*f*x*Sin[(d*x)/2] + 12*f^3*x*Sin[(d*x)/2] + 15*d^2*e*f^2*x^2*Sin[(d*x)/2] + 5*d^2*f^3*x^3*Sin[(d*x)/2])/(6*a*d^3*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
276,1,637,343,7.0438786,"\int \frac{(e+f x)^2 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\frac{d^2 e^2 \sin (2 (c+d x))+2 d^2 e^2 \cos (c+d x)-4 d^2 e^2 \cos (c+2 d x)+2 d^2 e f x \sin (2 (c+d x))+4 d^2 e f x \cos (c+d x)-8 d^2 e f x \cos (c+2 d x)+d^2 f^2 x^2 \sin (2 (c+d x))+2 d^2 f^2 x^2 \cos (c+d x)-4 d^2 f^2 x^2 \cos (c+2 d x)-2 d e f \cos (2 c+d x)+2 f^2 \sin (2 (c+d x))-2 f^2 \sin (2 c+d x)-2 d f^2 x \cos (2 c+d x)+4 f^2 \cos (c+d x)-2 f^2 \cos (c+2 d x)-2 f^2 \cos (c)+8 d^2 e^2 \sin (d x)+16 d^2 e f x \sin (d x)+8 d^2 f^2 x^2 \sin (d x)-2 d f \cos (d x) (e+f x)+2 f^2 \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \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) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{12 d^2 f (\cos (c)+i \sin (c)) \left(\frac{f (\cos (c)-i (\sin (c)-1)) \text{Li}_2(i \cos (c+d x)+\sin (c+d x))}{d^2}+\frac{(-\sin (c)-i \cos (c)+1) (e+f x) \log (-\sin (c+d x)-i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^2}{2 f}\right)}{\cos (c)+i (\sin (c)-1)}-\frac{20 d^2 f (\cos (c)+i \sin (c)) \left(\frac{f (\cos (c)-i (\sin (c)+1)) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))}{d^2}-\frac{(\sin (c)+i \cos (c)+1) (e+f x) \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac{(\cos (c)-i \sin (c)) (e+f x)^2}{2 f}\right)}{\cos (c)+i (\sin (c)+1)}}{12 a d^3}","\frac{i f^2 \text{Li}_2\left(-i e^{i (c+d x)}\right)}{3 a d^3}-\frac{i f^2 \text{Li}_2\left(i e^{i (c+d x)}\right)}{3 a d^3}-\frac{2 i f^2 \text{Li}_2\left(-e^{2 i (c+d x)}\right)}{3 a d^3}+\frac{f^2 \tan (c+d x)}{3 a d^3}-\frac{f^2 \sec (c+d x)}{3 a d^3}+\frac{4 f (e+f x) \log \left(1+e^{2 i (c+d x)}\right)}{3 a d^2}-\frac{2 i f (e+f x) \tan ^{-1}\left(e^{i (c+d x)}\right)}{3 a d^2}-\frac{f (e+f x) \sec ^2(c+d x)}{3 a d^2}+\frac{f (e+f x) \tan (c+d x) \sec (c+d x)}{3 a d^2}+\frac{2 (e+f x)^2 \tan (c+d x)}{3 a d}-\frac{(e+f x)^2 \sec ^3(c+d x)}{3 a d}+\frac{(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 a d}-\frac{2 i (e+f x)^2}{3 a d}",1,"((12*d^2*f*((f*PolyLog[2, I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*(-1 + Sin[c])))/d^2 + ((e + f*x)*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(1 - I*Cos[c] - Sin[c]))/d + ((e + f*x)^2*(Cos[c] - I*Sin[c]))/(2*f))*(Cos[c] + I*Sin[c]))/(Cos[c] + I*(-1 + Sin[c])) - (20*d^2*f*(Cos[c] + I*Sin[c])*(((e + f*x)^2*(Cos[c] - I*Sin[c]))/(2*f) - ((e + f*x)*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2))/(Cos[c] + I*(1 + Sin[c])) + (-2*f^2*Cos[c] - 2*d*f*(e + f*x)*Cos[d*x] + 2*d^2*e^2*Cos[c + d*x] + 4*f^2*Cos[c + d*x] + 4*d^2*e*f*x*Cos[c + d*x] + 2*d^2*f^2*x^2*Cos[c + d*x] - 2*d*e*f*Cos[2*c + d*x] - 2*d*f^2*x*Cos[2*c + d*x] - 4*d^2*e^2*Cos[c + 2*d*x] - 2*f^2*Cos[c + 2*d*x] - 8*d^2*e*f*x*Cos[c + 2*d*x] - 4*d^2*f^2*x^2*Cos[c + 2*d*x] + 8*d^2*e^2*Sin[d*x] + 2*f^2*Sin[d*x] + 16*d^2*e*f*x*Sin[d*x] + 8*d^2*f^2*x^2*Sin[d*x] + d^2*e^2*Sin[2*(c + d*x)] + 2*f^2*Sin[2*(c + d*x)] + 2*d^2*e*f*x*Sin[2*(c + d*x)] + d^2*f^2*x^2*Sin[2*(c + d*x)] - 2*f^2*Sin[2*c + d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3))/(12*a*d^3)","A",0
277,1,231,152,1.1104529,"\int \frac{(e+f x) \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{\cos (c+d x) \left(\sin (c+d x) \left(3 f \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+5 f \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-c f+d e\right)+3 f \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+5 f \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-c f+d e-f\right)-2 d (e+f x) (\cos (2 (c+d x))-2 \sin (c+d x))}{6 a d^2 (\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{f \sec ^2(c+d x)}{6 a d^2}+\frac{f \tanh ^{-1}(\sin (c+d x))}{6 a d^2}+\frac{2 f \log (\cos (c+d x))}{3 a d^2}+\frac{f \tan (c+d x) \sec (c+d x)}{6 a d^2}+\frac{2 (e+f x) \tan (c+d x)}{3 a d}-\frac{(e+f x) \sec ^3(c+d x)}{3 a d}+\frac{(e+f x) \tan (c+d x) \sec ^2(c+d x)}{3 a d}",1,"(-2*d*(e + f*x)*(Cos[2*(c + d*x)] - 2*Sin[c + d*x]) + Cos[c + d*x]*(d*e - f - c*f + 3*f*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 5*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (d*e - c*f + 3*f*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 5*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sin[c + d*x]))/(6*a*d^2*(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
278,1,45,42,0.0568971,"\int \frac{\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sin[c + d*x]),x]","\frac{2 \tan (c+d x)-\cos (2 (c+d x)) \sec (c+d x)}{3 a d (\sin (c+d x)+1)}","\frac{2 \tan (c+d x)}{3 a d}-\frac{\sec (c+d x)}{3 d (a \sin (c+d x)+a)}",1,"(-(Cos[2*(c + d*x)]*Sec[c + d*x]) + 2*Tan[c + d*x])/(3*a*d*(1 + Sin[c + d*x]))","A",1
279,0,0,31,20.5626966,"\int \frac{\sec ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Sec[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])),x]","\int \frac{\sec ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sec ^2(c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sec[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])), x]","A",-1
280,0,0,31,26.6562341,"\int \frac{\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Sec[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\int \frac{\sec ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sec ^2(c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sec[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])), x]","A",-1
281,1,1901,698,10.3627026,"\int \frac{(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Sec[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{-e^3-3 f x e^2-3 f^2 x^2 e-f^3 x^3}{8 a d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}-\frac{-36 f^3 x^3 \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)+i (\sin (c)+1)) d^3-108 e f^2 x^2 \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)+i (\sin (c)+1)) d^3+216 e f^2 (d x \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-i \text{Li}_3(-i \cos (c+d x)-\sin (c+d x))) (-i \cos (c)+\sin (c)+1) d-12 f \left(9 d^2 e^2+28 f^2\right) x \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)+i (\sin (c)+1)) d+12 i e \left(3 d^2 e^2+28 f^2\right) (d x+i \log (\cos (c+d x)+i (\sin (c+d x)+1))) (\cos (c)+i (\sin (c)+1)) d+\frac{\left(28 f^2+3 d^2 (e+f x)^2\right)^2}{f}+12 f \left(9 d^2 e^2+28 f^2\right) \text{Li}_2(-i \cos (c+d x)-\sin (c+d x)) (-i \cos (c)+\sin (c)+1)+108 f^3 \left(d^2 \text{Li}_2(-i \cos (c+d x)-\sin (c+d x)) x^2-2 i d \text{Li}_3(-i \cos (c+d x)-\sin (c+d x)) x-2 \text{Li}_4(-i \cos (c+d x)-\sin (c+d x))\right) (-i \cos (c)+\sin (c)+1)}{96 a d^4 (\cos (c)+i (\sin (c)+1))}-\frac{3 \left(d^2 f^3 x^4+4 d^2 e f^2 x^3+4 d f^3 \log (-i \cos (c+d x)-\sin (c+d x)+1) (\cos (c)+i (\sin (c)-1)) x^3+8 f^3 x^2+6 d^2 e^2 f x^2+12 d e f^2 \log (-i \cos (c+d x)-\sin (c+d x)+1) (\cos (c)+i (\sin (c)-1)) x^2+4 d^2 e^3 x+16 e f^2 x+\frac{4 f \left(3 d^2 e^2+4 f^2\right) \log (-i \cos (c+d x)-\sin (c+d x)+1) (\cos (c)+i (\sin (c)-1)) x}{d}+\frac{4 e \left(d^2 e^2+4 f^2\right) (\log (-\cos (c+d x)-i (\sin (c+d x)-1))-i d x) (\cos (c)+i (\sin (c)-1))}{d}+\frac{24 e f^2 (i d x \text{Li}_2(i \cos (c+d x)+\sin (c+d x))+\text{Li}_3(i \cos (c+d x)+\sin (c+d x))) (\cos (c)+i (\sin (c)-1))}{d}+\frac{12 f^3 \left(i d^2 \text{Li}_2(i \cos (c+d x)+\sin (c+d x)) x^2+2 d \text{Li}_3(i \cos (c+d x)+\sin (c+d x)) x-2 i \text{Li}_4(i \cos (c+d x)+\sin (c+d x))\right) (\cos (c)+i (\sin (c)-1))}{d^2}+\frac{4 f \left(3 d^2 e^2+4 f^2\right) \text{Li}_2(i \cos (c+d x)+\sin (c+d x)) (i \cos (c)-\sin (c)+1)}{d^2}\right)}{32 a d^2 (\cos (c)+i (\sin (c)-1))}+\frac{\frac{3 x \cos (c) e^3}{4 a}+\frac{3 i x \sin (c) e^3}{4 a}}{\cos (2 c)+i \sin (2 c)+1}+\frac{\frac{9 e^2 f \cos (c) x^2}{8 a}+\frac{9 i e^2 f \sin (c) x^2}{8 a}}{\cos (2 c)+i \sin (2 c)+1}+\frac{\frac{3 e f^2 \cos (c) x^3}{4 a}+\frac{3 i e f^2 \sin (c) x^3}{4 a}}{\cos (2 c)+i \sin (2 c)+1}+\frac{\frac{3 f^3 \cos (c) x^4}{16 a}+\frac{3 i f^3 \sin (c) x^4}{16 a}}{\cos (2 c)+i \sin (2 c)+1}-\frac{3 \left(x^2 \sin \left(\frac{d x}{2}\right) f^3+2 e x \sin \left(\frac{d x}{2}\right) f^2+e^2 \sin \left(\frac{d x}{2}\right) f\right)}{4 a d^2 \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{7 d^2 x^2 \sin \left(\frac{d x}{2}\right) f^3+2 \sin \left(\frac{d x}{2}\right) f^3+14 d^2 e x \sin \left(\frac{d x}{2}\right) f^2+7 d^2 e^2 \sin \left(\frac{d x}{2}\right) f}{4 a d^4 \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{e^3+3 f x e^2+3 f^2 x^2 e+f^3 x^3}{8 a d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{-2 d^2 \cos \left(\frac{c}{2}\right) e^3-2 d^2 \sin \left(\frac{c}{2}\right) e^3-d f \cos \left(\frac{c}{2}\right) e^2-6 d^2 f x \cos \left(\frac{c}{2}\right) e^2+d f \sin \left(\frac{c}{2}\right) e^2-6 d^2 f x \sin \left(\frac{c}{2}\right) e^2-2 f^2 \cos \left(\frac{c}{2}\right) e-6 d^2 f^2 x^2 \cos \left(\frac{c}{2}\right) e-2 d f^2 x \cos \left(\frac{c}{2}\right) e-2 f^2 \sin \left(\frac{c}{2}\right) e-6 d^2 f^2 x^2 \sin \left(\frac{c}{2}\right) e+2 d f^2 x \sin \left(\frac{c}{2}\right) e-2 d^2 f^3 x^3 \cos \left(\frac{c}{2}\right)-d f^3 x^2 \cos \left(\frac{c}{2}\right)-2 f^3 x \cos \left(\frac{c}{2}\right)-2 d^2 f^3 x^3 \sin \left(\frac{c}{2}\right)+d f^3 x^2 \sin \left(\frac{c}{2}\right)-2 f^3 x \sin \left(\frac{c}{2}\right)}{8 a d^3 \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{x^2 \sin \left(\frac{d x}{2}\right) f^3+2 e x \sin \left(\frac{d x}{2}\right) f^2+e^2 \sin \left(\frac{d x}{2}\right) f}{4 a d^2 \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","\frac{5 i f^3 \text{Li}_2\left(-i e^{i (c+d x)}\right)}{2 a d^4}-\frac{5 i f^3 \text{Li}_2\left(i e^{i (c+d x)}\right)}{2 a d^4}-\frac{i f^3 \text{Li}_2\left(-e^{2 i (c+d x)}\right)}{2 a d^4}-\frac{9 i f^3 \text{Li}_4\left(-i e^{i (c+d x)}\right)}{4 a d^4}+\frac{9 i f^3 \text{Li}_4\left(i e^{i (c+d x)}\right)}{4 a d^4}+\frac{f^3 \tan (c+d x)}{4 a d^4}-\frac{f^3 \sec (c+d x)}{4 a d^4}-\frac{9 f^2 (e+f x) \text{Li}_3\left(-i e^{i (c+d x)}\right)}{4 a d^3}+\frac{9 f^2 (e+f x) \text{Li}_3\left(i e^{i (c+d x)}\right)}{4 a d^3}+\frac{f^2 (e+f x) \log \left(1+e^{2 i (c+d x)}\right)}{a d^3}-\frac{5 i f^2 (e+f x) \tan ^{-1}\left(e^{i (c+d x)}\right)}{a d^3}-\frac{f^2 (e+f x) \sec ^2(c+d x)}{4 a d^3}+\frac{f^2 (e+f x) \tan (c+d x) \sec (c+d x)}{4 a d^3}+\frac{9 i f (e+f x)^2 \text{Li}_2\left(-i e^{i (c+d x)}\right)}{8 a d^2}-\frac{9 i f (e+f x)^2 \text{Li}_2\left(i e^{i (c+d x)}\right)}{8 a d^2}+\frac{f (e+f x)^2 \tan (c+d x)}{2 a d^2}-\frac{f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac{9 f (e+f x)^2 \sec (c+d x)}{8 a d^2}+\frac{f (e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{4 a d^2}-\frac{3 i (e+f x)^3 \tan ^{-1}\left(e^{i (c+d x)}\right)}{4 a d}-\frac{(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac{(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac{3 (e+f x)^3 \tan (c+d x) \sec (c+d x)}{8 a d}-\frac{i f (e+f x)^2}{2 a d^2}",1,"(-3*(4*d^2*e^3*x + 16*e*f^2*x + 6*d^2*e^2*f*x^2 + 8*f^3*x^2 + 4*d^2*e*f^2*x^3 + d^2*f^3*x^4 + (4*e*(d^2*e^2 + 4*f^2)*((-I)*d*x + Log[-Cos[c + d*x] - I*(-1 + Sin[c + d*x])])*(Cos[c] + I*(-1 + Sin[c])))/d + (4*f*(3*d^2*e^2 + 4*f^2)*x*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*(-1 + Sin[c])))/d + 12*d*e*f^2*x^2*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*(-1 + Sin[c])) + 4*d*f^3*x^3*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*(-1 + Sin[c])) + (24*e*f^2*(I*d*x*PolyLog[2, I*Cos[c + d*x] + Sin[c + d*x]] + PolyLog[3, I*Cos[c + d*x] + Sin[c + d*x]])*(Cos[c] + I*(-1 + Sin[c])))/d + (12*f^3*(I*d^2*x^2*PolyLog[2, I*Cos[c + d*x] + Sin[c + d*x]] + 2*d*x*PolyLog[3, I*Cos[c + d*x] + Sin[c + d*x]] - (2*I)*PolyLog[4, I*Cos[c + d*x] + Sin[c + d*x]])*(Cos[c] + I*(-1 + Sin[c])))/d^2 + (4*f*(3*d^2*e^2 + 4*f^2)*PolyLog[2, I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] - Sin[c]))/d^2))/(32*a*d^2*(Cos[c] + I*(-1 + Sin[c]))) - ((28*f^2 + 3*d^2*(e + f*x)^2)^2/f + 12*f*(9*d^2*e^2 + 28*f^2)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(1 - I*Cos[c] + Sin[c]) + 216*d*e*f^2*(d*x*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(1 - I*Cos[c] + Sin[c]) + 108*f^3*(d^2*x^2*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - (2*I)*d*x*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]] - 2*PolyLog[4, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(1 - I*Cos[c] + Sin[c]) - 12*d*f*(9*d^2*e^2 + 28*f^2)*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] + I*(1 + Sin[c])) - 108*d^3*e*f^2*x^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] + I*(1 + Sin[c])) - 36*d^3*f^3*x^3*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] + I*(1 + Sin[c])) + (12*I)*d*e*(3*d^2*e^2 + 28*f^2)*(d*x + I*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])])*(Cos[c] + I*(1 + Sin[c])))/(96*a*d^4*(Cos[c] + I*(1 + Sin[c]))) + ((3*e^3*x*Cos[c])/(4*a) + (((3*I)/4)*e^3*x*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) + ((9*e^2*f*x^2*Cos[c])/(8*a) + (((9*I)/8)*e^2*f*x^2*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) + ((3*e*f^2*x^3*Cos[c])/(4*a) + (((3*I)/4)*e*f^2*x^3*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) + ((3*f^3*x^4*Cos[c])/(16*a) + (((3*I)/16)*f^3*x^4*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) + (e^3 + 3*e^2*f*x + 3*e*f^2*x^2 + f^3*x^3)/(8*a*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) - (3*(e^2*f*Sin[(d*x)/2] + 2*e*f^2*x*Sin[(d*x)/2] + f^3*x^2*Sin[(d*x)/2]))/(4*a*d^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (-e^3 - 3*e^2*f*x - 3*e*f^2*x^2 - f^3*x^3)/(8*a*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + (e^2*f*Sin[(d*x)/2] + 2*e*f^2*x*Sin[(d*x)/2] + f^3*x^2*Sin[(d*x)/2])/(4*a*d^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (-2*d^2*e^3*Cos[c/2] - d*e^2*f*Cos[c/2] - 2*e*f^2*Cos[c/2] - 6*d^2*e^2*f*x*Cos[c/2] - 2*d*e*f^2*x*Cos[c/2] - 2*f^3*x*Cos[c/2] - 6*d^2*e*f^2*x^2*Cos[c/2] - d*f^3*x^2*Cos[c/2] - 2*d^2*f^3*x^3*Cos[c/2] - 2*d^2*e^3*Sin[c/2] + d*e^2*f*Sin[c/2] - 2*e*f^2*Sin[c/2] - 6*d^2*e^2*f*x*Sin[c/2] + 2*d*e*f^2*x*Sin[c/2] - 2*f^3*x*Sin[c/2] - 6*d^2*e*f^2*x^2*Sin[c/2] + d*f^3*x^2*Sin[c/2] - 2*d^2*f^3*x^3*Sin[c/2])/(8*a*d^3*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (7*d^2*e^2*f*Sin[(d*x)/2] + 2*f^3*Sin[(d*x)/2] + 14*d^2*e*f^2*x*Sin[(d*x)/2] + 7*d^2*f^3*x^2*Sin[(d*x)/2])/(4*a*d^4*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
282,1,1468,431,9.0183939,"\int \frac{(e+f x)^2 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Sec[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{-e^2-2 f x e-f^2 x^2}{8 a d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}-\frac{(\cos (c)+i \sin (c)) \left(d^2 f^2 (\cos (c)-i \sin (c)) x^3+3 d^2 e f \cos (c) x^2+3 d f^2 \log (-i \cos (c+d x)-\sin (c+d x)+1) (\cos (c)+i (\sin (c)-1)) (\cos (c)-i \sin (c)) x^2-3 i d^2 e f \sin (c) x^2+\left(3 d^2 e^2+4 f^2\right) (\cos (c)-i \sin (c)) x+6 d e f \log (-i \cos (c+d x)-\sin (c+d x)+1) (\cos (c)+i (\sin (c)-1)) (\cos (c)-i \sin (c)) x+6 e f \text{Li}_2(i \cos (c+d x)+\sin (c+d x)) (\cos (c)-i (\sin (c)-1))+\frac{6 f^2 (i d x \text{Li}_2(i \cos (c+d x)+\sin (c+d x))+\text{Li}_3(i \cos (c+d x)+\sin (c+d x))) (\cos (c)+i (\sin (c)-1)) (\cos (c)-i \sin (c))}{d}+\frac{\left(3 d^2 e^2+4 f^2\right) (d x+i \log (-\cos (c+d x)-i (\sin (c+d x)-1))) (\cos (c)-i \sin (c)) (-i \cos (c)+\sin (c)-1)}{d}\right)}{8 a d^2 (\cos (c)+i (\sin (c)-1))}-\frac{(\cos (c)+i \sin (c)) \left(3 d^2 f^2 \cos (c) x^3-3 i d^2 f^2 \sin (c) x^3+9 d^2 e f \cos (c) x^2-9 i d^2 e f \sin (c) x^2-9 d f^2 \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)-i \sin (c)) (\cos (c)+i (\sin (c)+1)) x^2+\left(9 d^2 e^2+28 f^2\right) (\cos (c)-i \sin (c)) x-18 d e f \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)-i \sin (c)) (\cos (c)+i (\sin (c)+1)) x+\frac{18 f^2 (d x \text{Li}_2(-i \cos (c+d x)-\sin (c+d x))-i \text{Li}_3(-i \cos (c+d x)-\sin (c+d x))) (\cos (c)-i \sin (c)) (-i \cos (c)+\sin (c)+1)}{d}+18 e f \text{Li}_2(-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (\sin (c)+1))+\frac{\left(9 d^2 e^2+28 f^2\right) (d x+i \log (\cos (c+d x)+i (\sin (c+d x)+1))) (i \cos (c)+\sin (c)) (\cos (c)+i (\sin (c)+1))}{d}\right)}{24 a d^2 (\cos (c)+i (\sin (c)+1))}+\frac{\frac{3 x \cos (c) e^2}{4 a}+\frac{3 i x \sin (c) e^2}{4 a}}{\cos (2 c)+i \sin (2 c)+1}+\frac{\frac{3 e f \cos (c) x^2}{4 a}+\frac{3 i e f \sin (c) x^2}{4 a}}{\cos (2 c)+i \sin (2 c)+1}+\frac{\frac{f^2 \cos (c) x^3}{4 a}+\frac{i f^2 \sin (c) x^3}{4 a}}{\cos (2 c)+i \sin (2 c)+1}+\frac{-x \sin \left(\frac{d x}{2}\right) f^2-e \sin \left(\frac{d x}{2}\right) f}{2 a d^2 \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{7 \left(x \sin \left(\frac{d x}{2}\right) f^2+e \sin \left(\frac{d x}{2}\right) f\right)}{6 a d^2 \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{e^2+2 f x e+f^2 x^2}{8 a d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{-3 e^2 \cos \left(\frac{c}{2}\right) d^2-3 f^2 x^2 \cos \left(\frac{c}{2}\right) d^2-6 e f x \cos \left(\frac{c}{2}\right) d^2-3 e^2 \sin \left(\frac{c}{2}\right) d^2-3 f^2 x^2 \sin \left(\frac{c}{2}\right) d^2-6 e f x \sin \left(\frac{c}{2}\right) d^2-e f \cos \left(\frac{c}{2}\right) d-f^2 x \cos \left(\frac{c}{2}\right) d+e f \sin \left(\frac{c}{2}\right) d+f^2 x \sin \left(\frac{c}{2}\right) d-f^2 \cos \left(\frac{c}{2}\right)-f^2 \sin \left(\frac{c}{2}\right)}{12 a d^3 \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{x \sin \left(\frac{d x}{2}\right) f^2+e \sin \left(\frac{d x}{2}\right) f}{6 a d^2 \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","-\frac{3 f^2 \text{Li}_3\left(-i e^{i (c+d x)}\right)}{4 a d^3}+\frac{3 f^2 \text{Li}_3\left(i e^{i (c+d x)}\right)}{4 a d^3}-\frac{f^2 \sec ^2(c+d x)}{12 a d^3}+\frac{5 f^2 \tanh ^{-1}(\sin (c+d x))}{6 a d^3}+\frac{f^2 \log (\cos (c+d x))}{3 a d^3}+\frac{f^2 \tan (c+d x) \sec (c+d x)}{12 a d^3}+\frac{3 i f (e+f x) \text{Li}_2\left(-i e^{i (c+d x)}\right)}{4 a d^2}-\frac{3 i f (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right)}{4 a d^2}+\frac{f (e+f x) \tan (c+d x)}{3 a d^2}-\frac{f (e+f x) \sec ^3(c+d x)}{6 a d^2}-\frac{3 f (e+f x) \sec (c+d x)}{4 a d^2}+\frac{f (e+f x) \tan (c+d x) \sec ^2(c+d x)}{6 a d^2}-\frac{3 i (e+f x)^2 \tan ^{-1}\left(e^{i (c+d x)}\right)}{4 a d}-\frac{(e+f x)^2 \sec ^4(c+d x)}{4 a d}+\frac{(e+f x)^2 \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac{3 (e+f x)^2 \tan (c+d x) \sec (c+d x)}{8 a d}",1,"-1/8*((Cos[c] + I*Sin[c])*(3*d^2*e*f*x^2*Cos[c] + 6*e*f*PolyLog[2, I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*(-1 + Sin[c])) + (3*d^2*e^2 + 4*f^2)*x*(Cos[c] - I*Sin[c]) + d^2*f^2*x^3*(Cos[c] - I*Sin[c]) + 6*d*e*f*x*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*(-1 + Sin[c]))*(Cos[c] - I*Sin[c]) + 3*d*f^2*x^2*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*(-1 + Sin[c]))*(Cos[c] - I*Sin[c]) + (6*f^2*(I*d*x*PolyLog[2, I*Cos[c + d*x] + Sin[c + d*x]] + PolyLog[3, I*Cos[c + d*x] + Sin[c + d*x]])*(Cos[c] + I*(-1 + Sin[c]))*(Cos[c] - I*Sin[c]))/d - (3*I)*d^2*e*f*x^2*Sin[c] + ((3*d^2*e^2 + 4*f^2)*(d*x + I*Log[-Cos[c + d*x] - I*(-1 + Sin[c + d*x])])*(Cos[c] - I*Sin[c])*(-1 - I*Cos[c] + Sin[c]))/d))/(a*d^2*(Cos[c] + I*(-1 + Sin[c]))) - ((Cos[c] + I*Sin[c])*(9*d^2*e*f*x^2*Cos[c] + 3*d^2*f^2*x^3*Cos[c] + (9*d^2*e^2 + 28*f^2)*x*(Cos[c] - I*Sin[c]) - (9*I)*d^2*e*f*x^2*Sin[c] - (3*I)*d^2*f^2*x^3*Sin[c] + (18*f^2*(d*x*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*Sin[c])*(1 - I*Cos[c] + Sin[c]))/d + 18*e*f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])) - 18*d*e*f*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])) - 9*d*f^2*x^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])) + ((9*d^2*e^2 + 28*f^2)*(d*x + I*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])])*(I*Cos[c] + Sin[c])*(Cos[c] + I*(1 + Sin[c])))/d))/(24*a*d^2*(Cos[c] + I*(1 + Sin[c]))) + ((3*e^2*x*Cos[c])/(4*a) + (((3*I)/4)*e^2*x*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) + ((3*e*f*x^2*Cos[c])/(4*a) + (((3*I)/4)*e*f*x^2*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) + ((f^2*x^3*Cos[c])/(4*a) + ((I/4)*f^2*x^3*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) + (e^2 + 2*e*f*x + f^2*x^2)/(8*a*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (-(e*f*Sin[(d*x)/2]) - f^2*x*Sin[(d*x)/2])/(2*a*d^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (-e^2 - 2*e*f*x - f^2*x^2)/(8*a*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + (e*f*Sin[(d*x)/2] + f^2*x*Sin[(d*x)/2])/(6*a*d^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (-3*d^2*e^2*Cos[c/2] - d*e*f*Cos[c/2] - f^2*Cos[c/2] - 6*d^2*e*f*x*Cos[c/2] - d*f^2*x*Cos[c/2] - 3*d^2*f^2*x^2*Cos[c/2] - 3*d^2*e^2*Sin[c/2] + d*e*f*Sin[c/2] - f^2*Sin[c/2] - 6*d^2*e*f*x*Sin[c/2] + d*f^2*x*Sin[c/2] - 3*d^2*f^2*x^2*Sin[c/2])/(12*a*d^3*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (7*(e*f*Sin[(d*x)/2] + f^2*x*Sin[(d*x)/2]))/(6*a*d^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
283,1,1171,241,6.5843646,"\int \frac{(e+f x) \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)*Sec[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{3 (c+d x) (2 d e-2 c f+f (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}{16 d^2 (\sin (c+d x) a+a)}+\frac{3 e \left(\frac{1}{2} (-c-d x)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}{8 d (\sin (c+d x) a+a)}-\frac{3 c f \left(\frac{1}{2} (-c-d x)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}{8 d^2 (\sin (c+d x) a+a)}-\frac{3 e \left(\frac{1}{2} (c+d x)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}{8 d (\sin (c+d x) a+a)}+\frac{3 c f \left(\frac{1}{2} (c+d x)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}{8 d^2 (\sin (c+d x) a+a)}-\frac{3 f \left(\frac{1}{4} e^{-\frac{i \pi }{4}} (c+d x)^2-\frac{-\frac{3}{4} i \pi  (c+d x)-\pi  \log \left(1+e^{-i (c+d x)}\right)-2 \left(\frac{1}{2} (c+d x)-\frac{\pi }{4}\right) \log \left(1-e^{2 i \left(\frac{1}{2} (c+d x)-\frac{\pi }{4}\right)}\right)+\pi  \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{1}{2} \pi  \log \left(-\sin \left(\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right)\right)+i \text{Li}_2\left(e^{2 i \left(\frac{1}{2} (c+d x)-\frac{\pi }{4}\right)}\right)}{\sqrt{2}}\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}{4 \sqrt{2} d^2 (\sin (c+d x) a+a)}-\frac{3 f \left(\frac{1}{4} e^{\frac{i \pi }{4}} (c+d x)^2+\frac{-\frac{1}{4} i \pi  (c+d x)-\pi  \log \left(1+e^{-i (c+d x)}\right)-2 \left(\frac{1}{2} (c+d x)+\frac{\pi }{4}\right) \log \left(1-e^{2 i \left(\frac{1}{2} (c+d x)+\frac{\pi }{4}\right)}\right)+\pi  \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{1}{2} \pi  \log \left(\sin \left(\frac{1}{2} (c+d x)+\frac{\pi }{4}\right)\right)+i \text{Li}_2\left(e^{2 i \left(\frac{1}{2} (c+d x)+\frac{\pi }{4}\right)}\right)}{\sqrt{2}}\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}{4 \sqrt{2} d^2 (\sin (c+d x) a+a)}-\frac{f \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)^2}{4 d^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (\sin (c+d x) a+a)}+\frac{(d e-c f+f (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}{8 d^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (\sin (c+d x) a+a)}+\frac{7 f \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)}{12 d^2 (\sin (c+d x) a+a)}+\frac{-6 d e+6 c f-f-6 f (c+d x)}{24 d^2 (\sin (c+d x) a+a)}+\frac{f \sin \left(\frac{1}{2} (c+d x)\right)}{12 d^2 (\sin (c+d x) a+a) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{-d e+c f-f (c+d x)}{8 d^2 (\sin (c+d x) a+a) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}","\frac{3 i f \text{Li}_2\left(-i e^{i (c+d x)}\right)}{8 a d^2}-\frac{3 i f \text{Li}_2\left(i e^{i (c+d x)}\right)}{8 a d^2}+\frac{f \tan ^3(c+d x)}{12 a d^2}+\frac{f \tan (c+d x)}{4 a d^2}-\frac{f \sec ^3(c+d x)}{12 a d^2}-\frac{3 f \sec (c+d x)}{8 a d^2}-\frac{3 i (e+f x) \tan ^{-1}\left(e^{i (c+d x)}\right)}{4 a d}-\frac{(e+f x) \sec ^4(c+d x)}{4 a d}+\frac{(e+f x) \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac{3 (e+f x) \tan (c+d x) \sec (c+d x)}{8 a d}",1,"(-6*d*e - f + 6*c*f - 6*f*(c + d*x))/(24*d^2*(a + a*Sin[c + d*x])) + (-(d*e) + c*f - f*(c + d*x))/(8*d^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a + a*Sin[c + d*x])) + (f*Sin[(c + d*x)/2])/(12*d^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a + a*Sin[c + d*x])) + (7*f*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(12*d^2*(a + a*Sin[c + d*x])) + (3*(c + d*x)*(2*d*e - 2*c*f + f*(c + d*x))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(16*d^2*(a + a*Sin[c + d*x])) + (3*e*((-c - d*x)/2 - Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(8*d*(a + a*Sin[c + d*x])) - (3*c*f*((-c - d*x)/2 - Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(8*d^2*(a + a*Sin[c + d*x])) - (3*e*((c + d*x)/2 - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(8*d*(a + a*Sin[c + d*x])) + (3*c*f*((c + d*x)/2 - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(8*d^2*(a + a*Sin[c + d*x])) - (3*f*((c + d*x)^2/(4*E^((I/4)*Pi)) - (((-3*I)/4)*Pi*(c + d*x) - Pi*Log[1 + E^((-I)*(c + d*x))] - 2*(-1/4*Pi + (c + d*x)/2)*Log[1 - E^((2*I)*(-1/4*Pi + (c + d*x)/2))] + Pi*Log[Cos[(c + d*x)/2]] - (Pi*Log[-Sin[Pi/4 + (-c - d*x)/2]])/2 + I*PolyLog[2, E^((2*I)*(-1/4*Pi + (c + d*x)/2))])/Sqrt[2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(4*Sqrt[2]*d^2*(a + a*Sin[c + d*x])) - (3*f*((E^((I/4)*Pi)*(c + d*x)^2)/4 + ((-1/4*I)*Pi*(c + d*x) - Pi*Log[1 + E^((-I)*(c + d*x))] - 2*(Pi/4 + (c + d*x)/2)*Log[1 - E^((2*I)*(Pi/4 + (c + d*x)/2))] + Pi*Log[Cos[(c + d*x)/2]] + (Pi*Log[Sin[Pi/4 + (c + d*x)/2]])/2 + I*PolyLog[2, E^((2*I)*(Pi/4 + (c + d*x)/2))])/Sqrt[2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(4*Sqrt[2]*d^2*(a + a*Sin[c + d*x])) + ((d*e - c*f + f*(c + d*x))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(8*d^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a + a*Sin[c + d*x])) - (f*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(4*d^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a + a*Sin[c + d*x]))","B",1
284,1,75,77,0.111161,"\int \frac{\sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^2(c+d x) \left(-3 \sin ^2(c+d x)-3 \sin (c+d x)+3 (\sin (c+d x)-1) (\sin (c+d x)+1)^2 \tanh ^{-1}(\sin (c+d x))+2\right)}{8 a d (\sin (c+d x)+1)}","-\frac{a}{8 d (a \sin (c+d x)+a)^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{1}{4 d (a \sin (c+d x)+a)}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}",1,"-1/8*(Sec[c + d*x]^2*(2 - 3*Sin[c + d*x] - 3*Sin[c + d*x]^2 + 3*ArcTanh[Sin[c + d*x]]*(-1 + Sin[c + d*x])*(1 + Sin[c + d*x])^2))/(a*d*(1 + Sin[c + d*x]))","A",1
285,0,0,31,36.0849644,"\int \frac{\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","Integrate[Sec[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])),x]","\int \frac{\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sec ^3(c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sec[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x]","A",-1
286,0,0,31,54.477907,"\int \frac{\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","Integrate[Sec[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])),x]","\int \frac{\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx","\text{Int}\left(\frac{\sec ^3(c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right)",0,"Integrate[Sec[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x]","A",-1
287,1,405,449,4.7916693,"\int \frac{(e+f x)^m \cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Cos[c + d*x]^4)/(a + a*Sin[c + d*x]),x]","\frac{i (e+f x)^m \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(-3 i e^{i \left(c-\frac{d e}{f}\right)} \left(-\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,-\frac{i d (e+f x)}{f}\right)-3\ 2^{-m} e^{2 i \left(c-\frac{d e}{f}\right)} \left(-\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,-\frac{2 i d (e+f x)}{f}\right)-i 3^{-m} e^{3 i \left(c-\frac{d e}{f}\right)} \left(-\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,-\frac{3 i d (e+f x)}{f}\right)-3 i e^{-i \left(c-\frac{d e}{f}\right)} \left(\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,\frac{i d (e+f x)}{f}\right)+3\ 2^{-m} e^{-2 i \left(c-\frac{d e}{f}\right)} \left(\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,\frac{2 i d (e+f x)}{f}\right)-i 3^{-m} e^{-3 i \left(c-\frac{d e}{f}\right)} \left(\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,\frac{3 i d (e+f x)}{f}\right)-\frac{12 i d (e+f x)}{f (m+1)}\right)}{24 a d (\sin (c+d x)+1)}","\frac{e^{i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(-\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,-\frac{i d (e+f x)}{f}\right)}{8 a d}-\frac{i 2^{-m-3} e^{2 i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(-\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,-\frac{2 i d (e+f x)}{f}\right)}{a d}+\frac{3^{-m-1} e^{3 i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(-\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,-\frac{3 i d (e+f x)}{f}\right)}{8 a d}+\frac{e^{-i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,\frac{i d (e+f x)}{f}\right)}{8 a d}+\frac{i 2^{-m-3} e^{-2 i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,\frac{2 i d (e+f x)}{f}\right)}{a d}+\frac{3^{-m-1} e^{-3 i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,\frac{3 i d (e+f x)}{f}\right)}{8 a d}+\frac{(e+f x)^{m+1}}{2 a f (m+1)}",1,"((I/24)*(e + f*x)^m*(((-12*I)*d*(e + f*x))/(f*(1 + m)) - ((3*I)*E^(I*(c - (d*e)/f))*Gamma[1 + m, ((-I)*d*(e + f*x))/f])/(((-I)*d*(e + f*x))/f)^m - ((3*I)*Gamma[1 + m, (I*d*(e + f*x))/f])/(E^(I*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m) - (3*E^((2*I)*(c - (d*e)/f))*Gamma[1 + m, ((-2*I)*d*(e + f*x))/f])/(2^m*(((-I)*d*(e + f*x))/f)^m) + (3*Gamma[1 + m, ((2*I)*d*(e + f*x))/f])/(2^m*E^((2*I)*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m) - (I*E^((3*I)*(c - (d*e)/f))*Gamma[1 + m, ((-3*I)*d*(e + f*x))/f])/(3^m*(((-I)*d*(e + f*x))/f)^m) - (I*Gamma[1 + m, ((3*I)*d*(e + f*x))/f])/(3^m*E^((3*I)*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(a*d*(1 + Sin[c + d*x]))","A",1
288,1,253,277,2.5546154,"\int \frac{(e+f x)^m \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Cos[c + d*x]^3)/(a + a*Sin[c + d*x]),x]","\frac{2^{-m-3} e^{-\frac{2 i (c f+d e)}{f}} (e+f x)^m \left(\frac{d^2 (e+f x)^2}{f^2}\right)^{-m} \left(i 2^{m+2} e^{i \left(c+\frac{3 d e}{f}\right)} \left(-\frac{i d (e+f x)}{f}\right)^m \Gamma \left(m+1,\frac{i d (e+f x)}{f}\right)-i 2^{m+2} e^{i \left(3 c+\frac{d e}{f}\right)} \left(\frac{i d (e+f x)}{f}\right)^m \Gamma \left(m+1,-\frac{i d (e+f x)}{f}\right)+e^{4 i c} \left(\frac{i d (e+f x)}{f}\right)^m \Gamma \left(m+1,-\frac{2 i d (e+f x)}{f}\right)+e^{\frac{4 i d e}{f}} \left(-\frac{i d (e+f x)}{f}\right)^m \Gamma \left(m+1,\frac{2 i d (e+f x)}{f}\right)\right)}{a d}","-\frac{i e^{i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(-\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,-\frac{i d (e+f x)}{f}\right)}{2 a d}+\frac{2^{-m-3} e^{2 i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(-\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,-\frac{2 i d (e+f x)}{f}\right)}{a d}+\frac{i e^{-i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,\frac{i d (e+f x)}{f}\right)}{2 a d}+\frac{2^{-m-3} e^{-2 i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,\frac{2 i d (e+f x)}{f}\right)}{a d}",1,"(2^(-3 - m)*(e + f*x)^m*((-I)*2^(2 + m)*E^(I*(3*c + (d*e)/f))*((I*d*(e + f*x))/f)^m*Gamma[1 + m, ((-I)*d*(e + f*x))/f] + I*2^(2 + m)*E^(I*(c + (3*d*e)/f))*(((-I)*d*(e + f*x))/f)^m*Gamma[1 + m, (I*d*(e + f*x))/f] + E^((4*I)*c)*((I*d*(e + f*x))/f)^m*Gamma[1 + m, ((-2*I)*d*(e + f*x))/f] + E^(((4*I)*d*e)/f)*(((-I)*d*(e + f*x))/f)^m*Gamma[1 + m, ((2*I)*d*(e + f*x))/f]))/(a*d*E^(((2*I)*(d*e + c*f))/f)*((d^2*(e + f*x)^2)/f^2)^m)","A",1
289,1,220,154,0.979182,"\int \frac{(e+f x)^m \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Cos[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\frac{e^{i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(\frac{d^2 (e+f x)^2}{f^2}\right)^{-m} \left(2 d (e+f x) e^{-i \left(c-\frac{d e}{f}\right)} \left(\frac{d^2 (e+f x)^2}{f^2}\right)^m+f (m+1) e^{-2 i \left(c-\frac{d e}{f}\right)} \left(-\frac{i d (e+f x)}{f}\right)^m \Gamma \left(m+1,\frac{i d (e+f x)}{f}\right)+f (m+1) \left(\frac{i d (e+f x)}{f}\right)^m \Gamma \left(m+1,-\frac{i d (e+f x)}{f}\right)\right)}{2 a d f (m+1) (\sin (c+d x)+1)}","\frac{e^{i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(-\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,-\frac{i d (e+f x)}{f}\right)}{2 a d}+\frac{e^{-i \left(c-\frac{d e}{f}\right)} (e+f x)^m \left(\frac{i d (e+f x)}{f}\right)^{-m} \Gamma \left(m+1,\frac{i d (e+f x)}{f}\right)}{2 a d}+\frac{(e+f x)^{m+1}}{a f (m+1)}",1,"(E^(I*(c - (d*e)/f))*(e + f*x)^m*((2*d*(e + f*x)*((d^2*(e + f*x)^2)/f^2)^m)/E^(I*(c - (d*e)/f)) + f*(1 + m)*((I*d*(e + f*x))/f)^m*Gamma[1 + m, ((-I)*d*(e + f*x))/f] + (f*(1 + m)*(((-I)*d*(e + f*x))/f)^m*Gamma[1 + m, (I*d*(e + f*x))/f])/E^((2*I)*(c - (d*e)/f)))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(2*a*d*f*(1 + m)*((d^2*(e + f*x)^2)/f^2)^m*(1 + Sin[c + d*x]))","A",1
290,0,0,29,8.3048935,"\int \frac{(e+f x)^m \cos (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Cos[c + d*x])/(a + a*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \cos (c+d x)}{a+a \sin (c+d x)} \, dx","\text{Int}\left(\frac{\cos (c+d x) (e+f x)^m}{a \sin (c+d x)+a},x\right)",0,"Integrate[((e + f*x)^m*Cos[c + d*x])/(a + a*Sin[c + d*x]), x]","A",-1
291,0,0,23,0.3480115,"\int \frac{(e+f x)^m}{a+a \sin (c+d x)} \, dx","Integrate[(e + f*x)^m/(a + a*Sin[c + d*x]),x]","\int \frac{(e+f x)^m}{a+a \sin (c+d x)} \, dx","\text{Int}\left(\frac{(e+f x)^m}{a \sin (c+d x)+a},x\right)",0,"Integrate[(e + f*x)^m/(a + a*Sin[c + d*x]), x]","A",-1
292,0,0,29,164.9263158,"\int \frac{(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Sec[c + d*x])/(a + a*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx","\text{Int}\left(\frac{\sec (c+d x) (e+f x)^m}{a \sin (c+d x)+a},x\right)",0,"Integrate[((e + f*x)^m*Sec[c + d*x])/(a + a*Sin[c + d*x]), x]","A",-1
293,0,0,31,27.3290686,"\int \frac{(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","\text{Int}\left(\frac{\sec ^2(c+d x) (e+f x)^m}{a \sin (c+d x)+a},x\right)",0,"Integrate[((e + f*x)^m*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]), x]","A",-1
294,1,410,432,0.1906591,"\int \frac{(e+f x)^3 \cos (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\frac{12 f \left(2 f \left(d (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)+i f \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)\right)-i d^2 (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)\right)}{d^4}+\frac{12 f \left(2 f \left(d (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)+i f \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)\right)-i d^2 (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)\right)}{d^4}+\frac{4 (e+f x)^3 \log \left(1+\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)}{d}+\frac{4 (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{d}-\frac{i (e+f x)^4}{f}}{4 b}","\frac{6 i f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^4}+\frac{6 i f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^4}+\frac{6 f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^3}-\frac{3 i f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^2}+\frac{(e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d}+\frac{(e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d}-\frac{i (e+f x)^4}{4 b f}",1,"(((-I)*(e + f*x)^4)/f + (4*(e + f*x)^3*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])])/d + (4*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/d + (12*f*((-I)*d^2*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])] + 2*f*(d*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])] + I*f*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])))/d^4 + (12*f*((-I)*d^2*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + 2*f*(d*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + I*f*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])))/d^4)/(4*b)","A",1
295,1,302,320,0.1643362,"\int \frac{(e+f x)^2 \cos (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\frac{6 f \left(f \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)-i d (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)\right)}{d^3}+\frac{6 f \left(f \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)-i d (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)\right)}{d^3}+\frac{3 (e+f x)^2 \log \left(1+\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)}{d}+\frac{3 (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{d}-\frac{i (e+f x)^3}{f}}{3 b}","\frac{2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^3}+\frac{2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^3}-\frac{2 i f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^2}-\frac{2 i f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^2}+\frac{(e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d}+\frac{(e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d}-\frac{i (e+f x)^3}{3 b f}",1,"(((-I)*(e + f*x)^3)/f + (3*(e + f*x)^2*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])])/d + (3*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/d + (6*f*((-I)*d*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])] + f*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])]))/d^3 + (6*f*((-I)*d*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + f*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])]))/d^3)/(3*b)","A",1
296,1,197,212,0.0502571,"\int \frac{(e+f x) \cos (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{i \left(d (e+f x) \left(2 i f \log \left(1+\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)+2 i f \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)+d e+d f x\right)+2 f^2 \text{Li}_2\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)+2 f^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)\right)}{2 b d^2 f}","-\frac{i f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^2}-\frac{i f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^2}+\frac{(e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d}+\frac{(e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d}-\frac{i (e+f x)^2}{2 b f}",1,"((-1/2*I)*(d*(e + f*x)*(d*e + d*f*x + (2*I)*f*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + (2*I)*f*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])]) + 2*f^2*PolyLog[2, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + 2*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])]))/(b*d^2*f)","A",1
297,1,18,18,0.0081667,"\int \frac{\cos (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{\log (a+b \sin (c+d x))}{b d}","\frac{\log (a+b \sin (c+d x))}{b d}",1,"Log[a + b*Sin[c + d*x]]/(b*d)","A",1
298,1,1025,618,3.5099861,"\int \frac{(e+f x)^3 \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{a x \left(4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right) d^4+4 b (e+f x) \left(d^2 (e+f x)^2-6 f^2\right) \cos (c+d x) d+\frac{4 \left(b^2-a^2\right) \left(2 \sqrt{b^2-a^2} e^3 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) d^3+\sqrt{a^2-b^2} f^3 x^3 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e^2 f x \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3-\sqrt{a^2-b^2} f^3 x^3 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e^2 f x \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^2+3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d^2+6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d+6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d-6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d-6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d+6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{\sqrt{-\left(a^2-b^2\right)^2}}-12 b f \left(d^2 (e+f x)^2-2 f^2\right) \sin (c+d x)}{4 b^2 d^4}","-\frac{6 f^3 \sqrt{a^2-b^2} \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^4}+\frac{6 f^3 \sqrt{a^2-b^2} \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^4}+\frac{6 i f^2 \sqrt{a^2-b^2} (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^3}-\frac{6 i f^2 \sqrt{a^2-b^2} (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^3}+\frac{3 f \sqrt{a^2-b^2} (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^2}-\frac{3 f \sqrt{a^2-b^2} (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^2}+\frac{i \sqrt{a^2-b^2} (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d}-\frac{i \sqrt{a^2-b^2} (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b^2 d}+\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^3 \sin (c+d x)}{b d^4}-\frac{6 f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac{3 f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac{(e+f x)^3 \cos (c+d x)}{b d}",1,"(a*d^4*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) + 4*b*d*(e + f*x)*(-6*f^2 + d^2*(e + f*x)^2)*Cos[c + d*x] + (4*(-a^2 + b^2)*(2*Sqrt[-a^2 + b^2]*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))]))/Sqrt[-(a^2 - b^2)^2] - 12*b*f*(-2*f^2 + d^2*(e + f*x)^2)*Sin[c + d*x])/(4*b^2*d^4)","A",0
299,1,536,460,2.6403768,"\int \frac{(e+f x)^2 \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\frac{3 i \left(b^2-a^2\right) \left(-i \left(d^2 \left(2 e^2 \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right)+f x \sqrt{a^2-b^2} (2 e+f x) \left(\log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-\log \left(1+\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}+i a}\right)\right)\right)+2 f^2 \sqrt{a^2-b^2} \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-2 f^2 \sqrt{a^2-b^2} \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)-2 d f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)+2 d f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{d^3 \sqrt{-\left(a^2-b^2\right)^2}}+a x \left(3 e^2+3 e f x+f^2 x^2\right)+\frac{3 b \cos (d x) \left(\cos (c) \left(d^2 (e+f x)^2-2 f^2\right)-2 d f \sin (c) (e+f x)\right)}{d^3}-\frac{3 b \sin (d x) \left(\sin (c) \left(d^2 (e+f x)^2-2 f^2\right)+2 d f \cos (c) (e+f x)\right)}{d^3}}{3 b^2}","\frac{2 i f^2 \sqrt{a^2-b^2} \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^3}-\frac{2 i f^2 \sqrt{a^2-b^2} \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^3}+\frac{2 f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^2}-\frac{2 f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^2}+\frac{i \sqrt{a^2-b^2} (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d}-\frac{i \sqrt{a^2-b^2} (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b^2 d}+\frac{a (e+f x)^3}{3 b^2 f}-\frac{2 f^2 \cos (c+d x)}{b d^3}-\frac{2 f (e+f x) \sin (c+d x)}{b d^2}+\frac{(e+f x)^2 \cos (c+d x)}{b d}",1,"(a*x*(3*e^2 + 3*e*f*x + f^2*x^2) + ((3*I)*(-a^2 + b^2)*(-2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - I*(d^2*(2*Sqrt[-a^2 + b^2]*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(2*e + f*x)*(Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])])) + 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))])))/(Sqrt[-(a^2 - b^2)^2]*d^3) + (3*b*Cos[d*x]*((-2*f^2 + d^2*(e + f*x)^2)*Cos[c] - 2*d*f*(e + f*x)*Sin[c]))/d^3 - (3*b*(2*d*f*(e + f*x)*Cos[c] + (-2*f^2 + d^2*(e + f*x)^2)*Sin[c])*Sin[d*x])/d^3)/(3*b^2)","A",0
300,1,716,298,7.1702905,"\int \frac{(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\frac{2 d \left(b^2-a^2\right) (e+f x) \left(\frac{2 (d e-c f) \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{i f \left(\text{Li}_2\left(\frac{a \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a+i \left(b+\sqrt{b^2-a^2}\right)}\right)+\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{b^2-a^2}-i a+b}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\text{Li}_2\left(\frac{a \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)}{a-i \left(b+\sqrt{b^2-a^2}\right)}\right)+\log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{b^2-a^2}+i a+b}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\text{Li}_2\left(\frac{a \left(\tan \left(\frac{1}{2} (c+d x)\right)+i\right)}{i a-b+\sqrt{b^2-a^2}}\right)+\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{b^2-a^2}-a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{b^2-a^2}+i a-b}\right)\right)}{\sqrt{b^2-a^2}}-\frac{i f \left(\text{Li}_2\left(\frac{i \tan \left(\frac{1}{2} (c+d x)\right) a+a}{a+i \left(\sqrt{b^2-a^2}-b\right)}\right)+\log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{-\sqrt{b^2-a^2}+a \tan \left(\frac{1}{2} (c+d x)\right)+b}{-\sqrt{b^2-a^2}+i a+b}\right)\right)}{\sqrt{b^2-a^2}}\right)}{i f \log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)-i f \log \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right)-c f+d e}-a (c+d x) (c f-d (2 e+f x))+2 b d (e+f x) \cos (c+d x)-2 b f \sin (c+d x)}{2 b^2 d^2}","\frac{f \sqrt{a^2-b^2} \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d^2}-\frac{f \sqrt{a^2-b^2} \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^2 d^2}+\frac{i \sqrt{a^2-b^2} (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^2 d}-\frac{i \sqrt{a^2-b^2} (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b^2 d}+\frac{a e x}{b^2}+\frac{a f x^2}{2 b^2}-\frac{f \sin (c+d x)}{b d^2}+\frac{(e+f x) \cos (c+d x)}{b d}",1,"(-(a*(c + d*x)*(c*f - d*(2*e + f*x))) + 2*b*d*(e + f*x)*Cos[c + d*x] + (2*(-a^2 + b^2)*d*(e + f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(-b + Sqrt[-a^2 + b^2] - a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] - (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2]))/(d*e - c*f + I*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1 + I*Tan[(c + d*x)/2]]) - 2*b*f*Sin[c + d*x])/(2*b^2*d^2)","B",0
301,1,361,70,1.4017881,"\int \frac{\cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x]),x]","\frac{\cos (c+d x) \left(2 (a-b) \sqrt{1-\sin (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}{\sqrt{a+b} \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}}}\right)+\sqrt{a+b} \left(\sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \left(\sqrt{a-b} \sqrt{1-\sin (c+d x)} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}}+2 \sqrt{b} \sinh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}{\sqrt{2} \sqrt{b}}\right)\right)-2 \sqrt{a-b} \sqrt{1-\sin (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{\frac{b (\sin (c+d x)+1)}{b-a}}}{\sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}}}\right)\right)\right)}{b d \sqrt{a-b} \sqrt{a+b} \sqrt{1-\sin (c+d x)} \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}","-\frac{2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d}+\frac{a x}{b^2}+\frac{\cos (c+d x)}{b d}",1,"(Cos[c + d*x]*(2*(a - b)*ArcTanh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[a + b]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))])]*Sqrt[1 - Sin[c + d*x]] + Sqrt[a + b]*(-2*Sqrt[a - b]*ArcTanh[Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]/Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*(2*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[2]*Sqrt[b])] + Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]))))/(Sqrt[a - b]*b*Sqrt[a + b]*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])","B",1
302,1,2452,737,10.7057496,"\int \frac{(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","-\frac{6 i f^3 \left(a^2-b^2\right) \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d^4}-\frac{6 i f^3 \left(a^2-b^2\right) \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^3 d^4}-\frac{6 f^2 \left(a^2-b^2\right) (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d^3}-\frac{6 f^2 \left(a^2-b^2\right) (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^3 d^3}+\frac{3 i f \left(a^2-b^2\right) (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d^2}+\frac{3 i f \left(a^2-b^2\right) (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^3 d^2}-\frac{\left(a^2-b^2\right) (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d}-\frac{\left(a^2-b^2\right) (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b^3 d}+\frac{i \left(a^2-b^2\right) (e+f x)^4}{4 b^3 f}-\frac{6 a f^3 \cos (c+d x)}{b^2 d^4}-\frac{6 a f^2 (e+f x) \sin (c+d x)}{b^2 d^3}+\frac{3 a f (e+f x)^2 \cos (c+d x)}{b^2 d^2}+\frac{a (e+f x)^3 \sin (c+d x)}{b^2 d}+\frac{3 f^3 \sin (c+d x) \cos (c+d x)}{8 b d^4}+\frac{3 f^2 (e+f x) \sin ^2(c+d x)}{4 b d^3}-\frac{3 f (e+f x)^2 \sin (c+d x) \cos (c+d x)}{4 b d^2}-\frac{(e+f x)^3 \sin ^2(c+d x)}{2 b d}-\frac{3 f^3 x}{8 b d^3}+\frac{(e+f x)^3}{4 b d}",1,"(-32*(a^2 - b^2)*e^3*x*Cot[c] - 48*(a^2 - b^2)*e^2*f*x^2*Cot[c] - 32*(a^2 - b^2)*e*f^2*x^3*Cot[c] - 8*(a^2 - b^2)*f^3*x^4*Cot[c] + (16*(a^2 - b^2)*((4*I)*d^4*e^3*E^((2*I)*c)*x + (6*I)*d^4*e^2*E^((2*I)*c)*f*x^2 + (4*I)*d^4*e*E^((2*I)*c)*f^2*x^3 + I*d^4*E^((2*I)*c)*f^3*x^4 + (2*I)*d^3*e^3*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] - (2*I)*d^3*e^3*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + d^3*e^3*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] - d^3*e^3*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] + 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*e*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*e*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (12*I)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12*I)*E^((2*I)*c)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (12*I)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (12*I)*E^((2*I)*c)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(d^4*(-1 + E^((2*I)*c))) + (16*a*b*(-6*f^3 - (6*I)*d*f^2*(e + f*x) + 3*d^2*f*(e + f*x)^2 + I*d^3*(e + f*x)^3)*(Cos[c + d*x] - I*Sin[c + d*x]))/d^4 + (16*a*b*(-6*f^3 + (6*I)*d*f^2*(e + f*x) + 3*d^2*f*(e + f*x)^2 - I*d^3*(e + f*x)^3)*(Cos[c + d*x] + I*Sin[c + d*x]))/d^4 + (b^2*((3*I)*f^3 - 6*d*f^2*(e + f*x) - (6*I)*d^2*f*(e + f*x)^2 + 4*d^3*(e + f*x)^3)*(Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)]))/d^4 + (b^2*((-3*I)*f^3 - 6*d*f^2*(e + f*x) + (6*I)*d^2*f*(e + f*x)^2 + 4*d^3*(e + f*x)^3)*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]))/d^4)/(32*b^3)","B",1
303,1,2397,548,5.9698008,"\int \frac{(e+f x)^2 \cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","-\frac{2 f^2 \left(a^2-b^2\right) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d^3}-\frac{2 f^2 \left(a^2-b^2\right) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^3 d^3}+\frac{2 i f \left(a^2-b^2\right) (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d^2}+\frac{2 i f \left(a^2-b^2\right) (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^3 d^2}-\frac{\left(a^2-b^2\right) (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d}-\frac{\left(a^2-b^2\right) (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b^3 d}+\frac{i \left(a^2-b^2\right) (e+f x)^3}{3 b^3 f}-\frac{2 a f^2 \sin (c+d x)}{b^2 d^3}+\frac{2 a f (e+f x) \cos (c+d x)}{b^2 d^2}+\frac{a (e+f x)^2 \sin (c+d x)}{b^2 d}+\frac{f^2 \sin ^2(c+d x)}{4 b d^3}-\frac{f (e+f x) \sin (c+d x) \cos (c+d x)}{2 b d^2}-\frac{(e+f x)^2 \sin ^2(c+d x)}{2 b d}+\frac{e f x}{2 b d}+\frac{f^2 x^2}{4 b d}",1,"((48*I)*a^2*d^3*e^2*E^((2*I)*c)*x - (48*I)*b^2*d^3*e^2*E^((2*I)*c)*x + (48*I)*a^2*d^3*e*E^((2*I)*c)*f*x^2 - (48*I)*b^2*d^3*e*E^((2*I)*c)*f*x^2 + (16*I)*a^2*d^3*E^((2*I)*c)*f^2*x^3 - (16*I)*b^2*d^3*E^((2*I)*c)*f^2*x^3 - (48*I)*a^2*d^2*e^2*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + (48*I)*b^2*d^2*e^2*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + (24*I)*a*b*d^2*e^2*E^(I*c)*Cos[d*x] - (24*I)*a*b*d^2*e^2*E^((3*I)*c)*Cos[d*x] + 48*a*b*d*e*E^(I*c)*f*Cos[d*x] + 48*a*b*d*e*E^((3*I)*c)*f*Cos[d*x] - (48*I)*a*b*E^(I*c)*f^2*Cos[d*x] + (48*I)*a*b*E^((3*I)*c)*f^2*Cos[d*x] + (48*I)*a*b*d^2*e*E^(I*c)*f*x*Cos[d*x] - (48*I)*a*b*d^2*e*E^((3*I)*c)*f*x*Cos[d*x] + 48*a*b*d*E^(I*c)*f^2*x*Cos[d*x] + 48*a*b*d*E^((3*I)*c)*f^2*x*Cos[d*x] + (24*I)*a*b*d^2*E^(I*c)*f^2*x^2*Cos[d*x] - (24*I)*a*b*d^2*E^((3*I)*c)*f^2*x^2*Cos[d*x] + 6*b^2*d^2*e^2*Cos[2*d*x] + 6*b^2*d^2*e^2*E^((4*I)*c)*Cos[2*d*x] - (6*I)*b^2*d*e*f*Cos[2*d*x] + (6*I)*b^2*d*e*E^((4*I)*c)*f*Cos[2*d*x] - 3*b^2*f^2*Cos[2*d*x] - 3*b^2*E^((4*I)*c)*f^2*Cos[2*d*x] + 12*b^2*d^2*e*f*x*Cos[2*d*x] + 12*b^2*d^2*e*E^((4*I)*c)*f*x*Cos[2*d*x] - (6*I)*b^2*d*f^2*x*Cos[2*d*x] + (6*I)*b^2*d*E^((4*I)*c)*f^2*x*Cos[2*d*x] + 6*b^2*d^2*f^2*x^2*Cos[2*d*x] + 6*b^2*d^2*E^((4*I)*c)*f^2*x^2*Cos[2*d*x] - 24*a^2*d^2*e^2*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] + 24*b^2*d^2*e^2*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] - 96*a^2*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 96*b^2*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 48*a^2*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 48*b^2*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 96*a^2*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 96*b^2*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 48*a^2*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 48*b^2*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (96*I)*(a^2 - b^2)*d*E^((2*I)*c)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (96*I)*(a^2 - b^2)*d*E^((2*I)*c)*f*(e + f*x)*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 96*a^2*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 96*b^2*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 96*a^2*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 96*b^2*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 24*a*b*d^2*e^2*E^(I*c)*Sin[d*x] + 24*a*b*d^2*e^2*E^((3*I)*c)*Sin[d*x] - (48*I)*a*b*d*e*E^(I*c)*f*Sin[d*x] + (48*I)*a*b*d*e*E^((3*I)*c)*f*Sin[d*x] - 48*a*b*E^(I*c)*f^2*Sin[d*x] - 48*a*b*E^((3*I)*c)*f^2*Sin[d*x] + 48*a*b*d^2*e*E^(I*c)*f*x*Sin[d*x] + 48*a*b*d^2*e*E^((3*I)*c)*f*x*Sin[d*x] - (48*I)*a*b*d*E^(I*c)*f^2*x*Sin[d*x] + (48*I)*a*b*d*E^((3*I)*c)*f^2*x*Sin[d*x] + 24*a*b*d^2*E^(I*c)*f^2*x^2*Sin[d*x] + 24*a*b*d^2*E^((3*I)*c)*f^2*x^2*Sin[d*x] - (6*I)*b^2*d^2*e^2*Sin[2*d*x] + (6*I)*b^2*d^2*e^2*E^((4*I)*c)*Sin[2*d*x] - 6*b^2*d*e*f*Sin[2*d*x] - 6*b^2*d*e*E^((4*I)*c)*f*Sin[2*d*x] + (3*I)*b^2*f^2*Sin[2*d*x] - (3*I)*b^2*E^((4*I)*c)*f^2*Sin[2*d*x] - (12*I)*b^2*d^2*e*f*x*Sin[2*d*x] + (12*I)*b^2*d^2*e*E^((4*I)*c)*f*x*Sin[2*d*x] - 6*b^2*d*f^2*x*Sin[2*d*x] - 6*b^2*d*E^((4*I)*c)*f^2*x*Sin[2*d*x] - (6*I)*b^2*d^2*f^2*x^2*Sin[2*d*x] + (6*I)*b^2*d^2*E^((4*I)*c)*f^2*x^2*Sin[2*d*x])/(48*b^3*d^3*E^((2*I)*c))","B",1
304,1,2165,351,14.6648192,"\int \frac{(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x]^3)/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","\frac{i f \left(a^2-b^2\right) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d^2}+\frac{i f \left(a^2-b^2\right) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b^3 d^2}-\frac{\left(a^2-b^2\right) (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b^3 d}-\frac{\left(a^2-b^2\right) (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b^3 d}+\frac{i \left(a^2-b^2\right) (e+f x)^2}{2 b^3 f}+\frac{a f \cos (c+d x)}{b^2 d^2}+\frac{a (e+f x) \sin (c+d x)}{b^2 d}-\frac{f \sin (c+d x) \cos (c+d x)}{4 b d^2}-\frac{(e+f x) \sin ^2(c+d x)}{2 b d}+\frac{f x}{4 b d}",1,"(a*f*Cos[c + d*x])/(b^2*d^2) + ((d*e - c*f + f*(c + d*x))*Cos[2*(c + d*x)])/(4*b*d^2) + (a*(d*e - c*f + f*(c + d*x))*Sin[c + d*x])/(b^2*d^2) - (f*Sin[2*(c + d*x)])/(8*b*d^2) + ((f*(c + d*x)^2 + (2*I)*d*e*Log[Sec[(c + d*x)/2]^2] - (2*I)*c*f*Log[Sec[(c + d*x)/2]^2] - (2*I)*d*e*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] + (2*I)*c*f*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] - (4*I)*f*(c + d*x)*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + 4*f*PolyLog[2, -Cos[c + d*x] + I*Sin[c + d*x]] + 2*f*PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))] - 2*f*PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))] + 2*f*PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])] - 2*f*PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))])*((e*Cos[c + d*x])/(a + b*Sin[c + d*x]) - (a^2*e*Cos[c + d*x])/(b^2*(a + b*Sin[c + d*x])) - (c*f*Cos[c + d*x])/(d*(a + b*Sin[c + d*x])) + (a^2*c*f*Cos[c + d*x])/(b^2*d*(a + b*Sin[c + d*x])) + (f*(c + d*x)*Cos[c + d*x])/(d*(a + b*Sin[c + d*x])) - (a^2*f*(c + d*x)*Cos[c + d*x])/(b^2*d*(a + b*Sin[c + d*x]))))/(d*(2*f*(c + d*x) - (4*I)*f*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - (4*f*Log[1 + Cos[c + d*x] - I*Sin[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x]))/(-Cos[c + d*x] + I*Sin[c + d*x]) + (I*f*Log[1 - (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) + (I*f*Log[1 - (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - (I*f*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + (2*I)*d*e*Tan[(c + d*x)/2] - (2*I)*c*f*Tan[(c + d*x)/2] + ((2*I)*f*(c + d*x)*Sec[(c + d*x)/2]^2)/(-I + Tan[(c + d*x)/2]) - (f*Log[1 - (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(I + Tan[(c + d*x)/2]) + (I*a*f*Log[1 - (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(a + I*a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - ((2*I)*d*e*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x]) + ((2*I)*c*f*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x])))","B",0
305,1,54,61,0.0763633,"\int \frac{\cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(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)-\frac{1}{2} b^2 \sin ^2(c+d x)}{b^3 d}","-\frac{\left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^3 d}+\frac{a \sin (c+d x)}{b^2 d}-\frac{\sin ^2(c+d x)}{2 b d}",1,"(-((a^2 - b^2)*Log[a + b*Sin[c + d*x]]) + a*b*Sin[c + d*x] - (b^2*Sin[c + d*x]^2)/2)/(b^3*d)","A",1
306,1,2496,937,10.0338311,"\int \frac{(e+f x)^3 \sec (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Sec[c + d*x])/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","-\frac{6 i a \text{Li}_4\left(-i e^{i (c+d x)}\right) f^3}{\left(a^2-b^2\right) d^4}+\frac{6 i a \text{Li}_4\left(i e^{i (c+d x)}\right) f^3}{\left(a^2-b^2\right) d^4}-\frac{6 i b \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^3}{\left(a^2-b^2\right) d^4}-\frac{6 i b \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^3}{\left(a^2-b^2\right) d^4}+\frac{3 i b \text{Li}_4\left(-e^{2 i (c+d x)}\right) f^3}{4 \left(a^2-b^2\right) d^4}-\frac{6 a (e+f x) \text{Li}_3\left(-i e^{i (c+d x)}\right) f^2}{\left(a^2-b^2\right) d^3}+\frac{6 a (e+f x) \text{Li}_3\left(i e^{i (c+d x)}\right) f^2}{\left(a^2-b^2\right) d^3}-\frac{6 b (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^2}{\left(a^2-b^2\right) d^3}-\frac{6 b (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^2}{\left(a^2-b^2\right) d^3}+\frac{3 b (e+f x) \text{Li}_3\left(-e^{2 i (c+d x)}\right) f^2}{2 \left(a^2-b^2\right) d^3}+\frac{3 i a (e+f x)^2 \text{Li}_2\left(-i e^{i (c+d x)}\right) f}{\left(a^2-b^2\right) d^2}-\frac{3 i a (e+f x)^2 \text{Li}_2\left(i e^{i (c+d x)}\right) f}{\left(a^2-b^2\right) d^2}+\frac{3 i b (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f}{\left(a^2-b^2\right) d^2}+\frac{3 i b (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f}{\left(a^2-b^2\right) d^2}-\frac{3 i b (e+f x)^2 \text{Li}_2\left(-e^{2 i (c+d x)}\right) f}{2 \left(a^2-b^2\right) d^2}-\frac{2 i a (e+f x)^3 \tan ^{-1}\left(e^{i (c+d x)}\right)}{\left(a^2-b^2\right) d}-\frac{b (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right) d}-\frac{b (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right) d}+\frac{b (e+f x)^3 \log \left(1+e^{2 i (c+d x)}\right)}{\left(a^2-b^2\right) d}",1,"((4*((I*b*(e + f*x)^4)/f - (2*(a - b)*(1 + E^((2*I)*c))*(e + f*x)^3*Log[1 - I/E^(I*(c + d*x))])/d + (2*(a + b)*(1 + E^((2*I)*c))*(e + f*x)^3*Log[1 + I/E^(I*(c + d*x))])/d + (6*(a + b)*(1 + E^((2*I)*c))*f*(I*d^2*(e + f*x)^2*PolyLog[2, (-I)/E^(I*(c + d*x))] + 2*f*(d*(e + f*x)*PolyLog[3, (-I)/E^(I*(c + d*x))] - I*f*PolyLog[4, (-I)/E^(I*(c + d*x))])))/d^4 - ((6*I)*(a - b)*(1 + E^((2*I)*c))*f*(d^2*(e + f*x)^2*PolyLog[2, I/E^(I*(c + d*x))] - (2*I)*d*f*(e + f*x)*PolyLog[3, I/E^(I*(c + d*x))] - 2*f^2*PolyLog[4, I/E^(I*(c + d*x))]))/d^4))/((a^2 - b^2)*(1 + E^((2*I)*c))) + (4*b*((-4*I)*d^4*e^3*E^((2*I)*c)*x - (6*I)*d^4*e^2*E^((2*I)*c)*f*x^2 - (4*I)*d^4*e*E^((2*I)*c)*f^2*x^3 - I*d^4*E^((2*I)*c)*f^3*x^4 - (2*I)*d^3*e^3*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + (2*I)*d^3*e^3*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] - d^3*e^3*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] + d^3*e^3*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] - 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*e*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*e*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (12*I)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (12*I)*E^((2*I)*c)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12*I)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (12*I)*E^((2*I)*c)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/((-a^2 + b^2)*d^4*(-1 + E^((2*I)*c))) - (8*b*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*Csc[c]^3)/((a - b)*(a + b)*(Csc[c/2] - Sec[c/2])*(Csc[c/2] + Sec[c/2])))/8","B",1
307,1,1561,667,5.8599844,"\int \frac{(e+f x)^2 \sec (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Sec[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{1}{6} \left(-\frac{8 b x \left(3 e^2+3 f x e+f^2 x^2\right) \csc ^3(c)}{(a-b) (a+b) \left(\csc \left(\frac{c}{2}\right)-\sec \left(\frac{c}{2}\right)\right) \left(\csc \left(\frac{c}{2}\right)+\sec \left(\frac{c}{2}\right)\right)}+\frac{2 \left(\frac{2 i b (e+f x)^3}{f}-\frac{3 (a-b) \left(1+e^{2 i c}\right) \log \left(1-i e^{-i (c+d x)}\right) (e+f x)^2}{d}+\frac{3 (a+b) \left(1+e^{2 i c}\right) \log \left(1+i e^{-i (c+d x)}\right) (e+f x)^2}{d}+\frac{6 (a+b) \left(1+e^{2 i c}\right) f \left(i d (e+f x) \text{Li}_2\left(-i e^{-i (c+d x)}\right)+f \text{Li}_3\left(-i e^{-i (c+d x)}\right)\right)}{d^3}-\frac{6 i (a-b) \left(1+e^{2 i c}\right) f \left(d (e+f x) \text{Li}_2\left(i e^{-i (c+d x)}\right)-i f \text{Li}_3\left(i e^{-i (c+d x)}\right)\right)}{d^3}\right)}{\left(a^2-b^2\right) \left(1+e^{2 i c}\right)}+\frac{b \left(-4 i e^{2 i c} f^2 x^3 d^3-12 i e e^{2 i c} f x^2 d^3-12 i e^2 e^{2 i c} x d^3+6 i e^2 e^{2 i c} \tan ^{-1}\left(\frac{2 a e^{i (c+d x)}}{b \left(-1+e^{2 i (c+d x)}\right)}\right) d^2-6 i e^2 \tan ^{-1}\left(\frac{2 a e^{i (c+d x)}}{b \left(-1+e^{2 i (c+d x)}\right)}\right) d^2+3 e^2 e^{2 i c} \log \left(4 e^{2 i (c+d x)} a^2+b^2 \left(-1+e^{2 i (c+d x)}\right)^2\right) d^2-3 e^2 \log \left(4 e^{2 i (c+d x)} a^2+b^2 \left(-1+e^{2 i (c+d x)}\right)^2\right) d^2+6 e^{2 i c} f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-6 f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+12 e e^{2 i c} f x \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-12 e f x \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+6 e^{2 i c} f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-6 f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+12 e e^{2 i c} f x \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-12 e f x \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-12 i \left(-1+e^{2 i c}\right) f (e+f x) \text{Li}_2\left(\frac{i b e^{i (2 c+d x)}}{e^{i c} a+i \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d-12 i \left(-1+e^{2 i c}\right) f (e+f x) \text{Li}_2\left(-\frac{b e^{i (2 c+d x)}}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d+12 e^{2 i c} f^2 \text{Li}_3\left(\frac{i b e^{i (2 c+d x)}}{e^{i c} a+i \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-12 f^2 \text{Li}_3\left(\frac{i b e^{i (2 c+d x)}}{e^{i c} a+i \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+12 e^{2 i c} f^2 \text{Li}_3\left(-\frac{b e^{i (2 c+d x)}}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-12 f^2 \text{Li}_3\left(-\frac{b e^{i (2 c+d x)}}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)\right)}{\left(b^2-a^2\right) d^3 \left(-1+e^{2 i c}\right)}\right)","-\frac{2 a f^2 \text{Li}_3\left(-i e^{i (c+d x)}\right)}{d^3 \left(a^2-b^2\right)}+\frac{2 a f^2 \text{Li}_3\left(i e^{i (c+d x)}\right)}{d^3 \left(a^2-b^2\right)}-\frac{2 b f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d^3 \left(a^2-b^2\right)}-\frac{2 b f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{d^3 \left(a^2-b^2\right)}+\frac{b f^2 \text{Li}_3\left(-e^{2 i (c+d x)}\right)}{2 d^3 \left(a^2-b^2\right)}+\frac{2 i a f (e+f x) \text{Li}_2\left(-i e^{i (c+d x)}\right)}{d^2 \left(a^2-b^2\right)}-\frac{2 i a f (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right)}{d^2 \left(a^2-b^2\right)}+\frac{2 i b f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d^2 \left(a^2-b^2\right)}+\frac{2 i b f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{d^2 \left(a^2-b^2\right)}-\frac{i b f (e+f x) \text{Li}_2\left(-e^{2 i (c+d x)}\right)}{d^2 \left(a^2-b^2\right)}-\frac{b (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)}-\frac{b (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{d \left(a^2-b^2\right)}+\frac{b (e+f x)^2 \log \left(1+e^{2 i (c+d x)}\right)}{d \left(a^2-b^2\right)}-\frac{2 i a (e+f x)^2 \tan ^{-1}\left(e^{i (c+d x)}\right)}{d \left(a^2-b^2\right)}",1,"((2*(((2*I)*b*(e + f*x)^3)/f - (3*(a - b)*(1 + E^((2*I)*c))*(e + f*x)^2*Log[1 - I/E^(I*(c + d*x))])/d + (3*(a + b)*(1 + E^((2*I)*c))*(e + f*x)^2*Log[1 + I/E^(I*(c + d*x))])/d + (6*(a + b)*(1 + E^((2*I)*c))*f*(I*d*(e + f*x)*PolyLog[2, (-I)/E^(I*(c + d*x))] + f*PolyLog[3, (-I)/E^(I*(c + d*x))]))/d^3 - ((6*I)*(a - b)*(1 + E^((2*I)*c))*f*(d*(e + f*x)*PolyLog[2, I/E^(I*(c + d*x))] - I*f*PolyLog[3, I/E^(I*(c + d*x))]))/d^3))/((a^2 - b^2)*(1 + E^((2*I)*c))) + (b*((-12*I)*d^3*e^2*E^((2*I)*c)*x - (12*I)*d^3*e*E^((2*I)*c)*f*x^2 - (4*I)*d^3*E^((2*I)*c)*f^2*x^3 - (6*I)*d^2*e^2*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + (6*I)*d^2*e^2*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] - 3*d^2*e^2*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] + 3*d^2*e^2*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] - 12*d^2*e*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^2*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d^2*e*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^2*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12*I)*d*(-1 + E^((2*I)*c))*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12*I)*d*(-1 + E^((2*I)*c))*f*(e + f*x)*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/((-a^2 + b^2)*d^3*(-1 + E^((2*I)*c))) - (8*b*x*(3*e^2 + 3*e*f*x + f^2*x^2)*Csc[c]^3)/((a - b)*(a + b)*(Csc[c/2] - Sec[c/2])*(Csc[c/2] + Sec[c/2])))/6","B",1
308,1,2743,413,16.8286499,"\int \frac{(e+f x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Sec[c + d*x])/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","\frac{i a f \text{Li}_2\left(-i e^{i (c+d x)}\right)}{d^2 \left(a^2-b^2\right)}-\frac{i a f \text{Li}_2\left(i e^{i (c+d x)}\right)}{d^2 \left(a^2-b^2\right)}+\frac{i b f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d^2 \left(a^2-b^2\right)}+\frac{i b f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{d^2 \left(a^2-b^2\right)}-\frac{i b f \text{Li}_2\left(-e^{2 i (c+d x)}\right)}{2 d^2 \left(a^2-b^2\right)}-\frac{b (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)}-\frac{b (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{d \left(a^2-b^2\right)}+\frac{b (e+f x) \log \left(1+e^{2 i (c+d x)}\right)}{d \left(a^2-b^2\right)}-\frac{2 i a (e+f x) \tan ^{-1}\left(e^{i (c+d x)}\right)}{d \left(a^2-b^2\right)}",1,"((d*e + d*f*x)*(((-I)*b*(d*e + d*f*x)^2)/f + 2*(a - b)*(d*e - c*f)*Log[1 - Tan[(c + d*x)/2]] - 4*b*(d*e + d*f*x)*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - 2*(a + b)*(d*e - c*f)*Log[1 + Tan[(c + d*x)/2]] - (4*I)*b*f*PolyLog[2, -Cos[c + d*x] + I*Sin[c + d*x]] + (2*I)*(a + b)*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(1/2 - I/2)*(1 + Tan[(c + d*x)/2])] + PolyLog[2, ((1 + I) - (1 - I)*Tan[(c + d*x)/2])/2]) - (2*I)*(a + b)*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(1/2 + I/2)*(1 + Tan[(c + d*x)/2])] + PolyLog[2, (-1/2 - I/2)*(I + Tan[(c + d*x)/2])]) + (2*I)*(a - b)*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(-1/2 + I/2)*(-1 + Tan[(c + d*x)/2])] + PolyLog[2, ((1 + I) + (1 - I)*Tan[(c + d*x)/2])/2]) - (2*I)*(a - b)*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(-1/2 - I/2)*(-1 + Tan[(c + d*x)/2])] + PolyLog[2, ((1 - I) + (1 + I)*Tan[(c + d*x)/2])/2]))*(a - b*Sin[c + d*x]))/((a^2 - b^2)*d^2*(-2*a*d*e + 2*a*c*f - (2*I)*a*f*Log[1 - I*Tan[(c + d*x)/2]] + (2*I)*a*f*Log[1 + I*Tan[(c + d*x)/2]] + 4*b*f*Cos[c + d*x]*(Log[1 + Cos[c + d*x] - I*Sin[c + d*x]] - Log[(-2*I)/(-I + Tan[(c + d*x)/2])]) + b*(d*e - c*f + f*(c + d*x))*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + b*d*e*Tan[(c + d*x)/2] - b*c*f*Tan[(c + d*x)/2] - b*f*(c + d*x)*Tan[(c + d*x)/2] + (2*I)*b*f*Log[1 - I*Tan[(c + d*x)/2]]*Tan[(c + d*x)/2] - (2*I)*b*f*Log[1 + I*Tan[(c + d*x)/2]]*Tan[(c + d*x)/2])) + ((f*(c + d*x)^2 + (2*I)*d*e*Log[Sec[(c + d*x)/2]^2] - (2*I)*c*f*Log[Sec[(c + d*x)/2]^2] - (2*I)*d*e*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] + (2*I)*c*f*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] - (4*I)*f*(c + d*x)*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + 4*f*PolyLog[2, -Cos[c + d*x] + I*Sin[c + d*x]] + 2*f*PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))] - 2*f*PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))] + 2*f*PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])] - 2*f*PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))])*(-((b^2*e*Cos[c + d*x])/((a^2 - b^2)*(a + b*Sin[c + d*x]))) + (b^2*c*f*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x])) - (b^2*f*(c + d*x)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))))/(d*(2*f*(c + d*x) - (4*I)*f*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - (4*f*Log[1 + Cos[c + d*x] - I*Sin[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x]))/(-Cos[c + d*x] + I*Sin[c + d*x]) + (I*f*Log[1 - (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) + (I*f*Log[1 - (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - (I*f*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + (2*I)*d*e*Tan[(c + d*x)/2] - (2*I)*c*f*Tan[(c + d*x)/2] + ((2*I)*f*(c + d*x)*Sec[(c + d*x)/2]^2)/(-I + Tan[(c + d*x)/2]) - (f*Log[1 - (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(I + Tan[(c + d*x)/2]) + (I*a*f*Log[1 - (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(a + I*a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - ((2*I)*d*e*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x]) + ((2*I)*c*f*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x])))","B",0
309,1,64,75,0.0645511,"\int \frac{\sec (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{(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 d (a-b) (a+b)}","-\frac{b \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[1 - Sin[c + d*x]] + (a + b)*Log[1 + Sin[c + d*x]] - 2*b*Log[a + b*Sin[c + d*x]])/(2*(a - b)*(a + b)*d)","A",1
310,1,1438,923,9.3379262,"\int \frac{(e+f x)^3 \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{b \sec (c) (e+f x)^3}{\left(b^2-a^2\right) d}+\frac{f \left(\frac{2 i a (e+f x)^3}{f}+\frac{3 (a-b) \left(1+e^{2 i c}\right) \log \left(1-i e^{-i (c+d x)}\right) (e+f x)^2}{d}+\frac{3 (a+b) \left(1+e^{2 i c}\right) \log \left(1+i e^{-i (c+d x)}\right) (e+f x)^2}{d}+\frac{6 (a+b) \left(1+e^{2 i c}\right) f \left(i d (e+f x) \text{Li}_2\left(-i e^{-i (c+d x)}\right)+f \text{Li}_3\left(-i e^{-i (c+d x)}\right)\right)}{d^3}+\frac{6 (a-b) \left(1+e^{2 i c}\right) f \left(i d (e+f x) \text{Li}_2\left(i e^{-i (c+d x)}\right)+f \text{Li}_3\left(i e^{-i (c+d x)}\right)\right)}{d^3}\right)}{\left(a^2-b^2\right) d \left(1+e^{2 i c}\right)}+\frac{b^2 \left(2 \sqrt{b^2-a^2} e^3 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) d^3+\sqrt{a^2-b^2} f^3 x^3 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e^2 f x \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3-\sqrt{a^2-b^2} f^3 x^3 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e^2 f x \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^2+3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d^2+6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d+6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d-6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d-6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d+6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{\sqrt{-\left(a^2-b^2\right)^2} \left(b^2-a^2\right) d^4}+\frac{\sin \left(\frac{d x}{2}\right) e^3+3 f x \sin \left(\frac{d x}{2}\right) e^2+3 f^2 x^2 \sin \left(\frac{d x}{2}\right) e+f^3 x^3 \sin \left(\frac{d x}{2}\right)}{(a+b) d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\sin \left(\frac{d x}{2}\right) e^3+3 f x \sin \left(\frac{d x}{2}\right) e^2+3 f^2 x^2 \sin \left(\frac{d x}{2}\right) e+f^3 x^3 \sin \left(\frac{d x}{2}\right)}{(a-b) d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}","-\frac{6 b \text{Li}_3\left(-i e^{i (c+d x)}\right) f^3}{\left(a^2-b^2\right) d^4}+\frac{6 b \text{Li}_3\left(i e^{i (c+d x)}\right) f^3}{\left(a^2-b^2\right) d^4}+\frac{3 a \text{Li}_3\left(-e^{2 i (c+d x)}\right) f^3}{2 \left(a^2-b^2\right) d^4}-\frac{6 b^2 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^3}{\left(a^2-b^2\right)^{3/2} d^4}+\frac{6 b^2 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^3}{\left(a^2-b^2\right)^{3/2} d^4}+\frac{6 i b (e+f x) \text{Li}_2\left(-i e^{i (c+d x)}\right) f^2}{\left(a^2-b^2\right) d^3}-\frac{6 i b (e+f x) \text{Li}_2\left(i e^{i (c+d x)}\right) f^2}{\left(a^2-b^2\right) d^3}-\frac{3 i a (e+f x) \text{Li}_2\left(-e^{2 i (c+d x)}\right) f^2}{\left(a^2-b^2\right) d^3}+\frac{6 i b^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f^2}{\left(a^2-b^2\right)^{3/2} d^3}-\frac{6 i b^2 (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f^2}{\left(a^2-b^2\right)^{3/2} d^3}-\frac{6 i b (e+f x)^2 \tan ^{-1}\left(e^{i (c+d x)}\right) f}{\left(a^2-b^2\right) d^2}+\frac{3 a (e+f x)^2 \log \left(1+e^{2 i (c+d x)}\right) f}{\left(a^2-b^2\right) d^2}+\frac{3 b^2 (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) f}{\left(a^2-b^2\right)^{3/2} d^2}-\frac{3 b^2 (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) f}{\left(a^2-b^2\right)^{3/2} d^2}-\frac{i a (e+f x)^3}{\left(a^2-b^2\right) d}+\frac{i b^2 (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2} d}-\frac{i b^2 (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2} d}-\frac{b (e+f x)^3 \sec (c+d x)}{\left(a^2-b^2\right) d}+\frac{a (e+f x)^3 \tan (c+d x)}{\left(a^2-b^2\right) d}",1,"(f*(((2*I)*a*(e + f*x)^3)/f + (3*(a - b)*(1 + E^((2*I)*c))*(e + f*x)^2*Log[1 - I/E^(I*(c + d*x))])/d + (3*(a + b)*(1 + E^((2*I)*c))*(e + f*x)^2*Log[1 + I/E^(I*(c + d*x))])/d + (6*(a + b)*(1 + E^((2*I)*c))*f*(I*d*(e + f*x)*PolyLog[2, (-I)/E^(I*(c + d*x))] + f*PolyLog[3, (-I)/E^(I*(c + d*x))]))/d^3 + (6*(a - b)*(1 + E^((2*I)*c))*f*(I*d*(e + f*x)*PolyLog[2, I/E^(I*(c + d*x))] + f*PolyLog[3, I/E^(I*(c + d*x))]))/d^3))/((a^2 - b^2)*d*(1 + E^((2*I)*c))) + (b^2*(2*Sqrt[-a^2 + b^2]*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))]))/(Sqrt[-(a^2 - b^2)^2]*(-a^2 + b^2)*d^4) + (b*(e + f*x)^3*Sec[c])/((-a^2 + b^2)*d) + (e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2])/((a + b)*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2])/((a - b)*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","A",0
311,1,1122,659,7.9291443,"\int \frac{(e+f x)^2 \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{i \left(-2 \sqrt{a^2-b^2} d f (e+f x) \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)+2 \sqrt{a^2-b^2} d f (e+f x) \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)-i \left(\left(2 \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) e^2+\sqrt{a^2-b^2} f x (2 e+f x) \left(\log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-\log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right)\right)\right) d^2+2 \sqrt{a^2-b^2} f^2 \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-2 \sqrt{a^2-b^2} f^2 \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)\right) b^2}{\sqrt{-\left(a^2-b^2\right)^2} \left(b^2-a^2\right) d^3}+\frac{(e+f x)^2 \sec (c) b}{\left(b^2-a^2\right) d}+\frac{2 f^2 \left(\frac{2 \tan ^{-1}(\cot (c)) \tanh ^{-1}\left(\frac{\sin (c)+\cos (c) \tan \left(\frac{d x}{2}\right)}{\sqrt{\cos ^2(c)+\sin ^2(c)}}\right)}{\sqrt{\cos ^2(c)+\sin ^2(c)}}-\frac{\csc (c) \left(\left(d x-\tan ^{-1}(\cot (c))\right) \left(\log \left(1-e^{i \left(d x-\tan ^{-1}(\cot (c))\right)}\right)-\log \left(1+e^{i \left(d x-\tan ^{-1}(\cot (c))\right)}\right)\right)+i \left(\text{Li}_2\left(-e^{i \left(d x-\tan ^{-1}(\cot (c))\right)}\right)-\text{Li}_2\left(e^{i \left(d x-\tan ^{-1}(\cot (c))\right)}\right)\right)\right)}{\sqrt{\cot ^2(c)+1}}\right) b}{\left(a^2-b^2\right) d^3}+\frac{4 i e f \tan ^{-1}\left(\frac{-i \sin (c)-i \cos (c) \tan \left(\frac{d x}{2}\right)}{\sqrt{\cos ^2(c)+\sin ^2(c)}}\right) b}{\left(a^2-b^2\right) d^2 \sqrt{\cos ^2(c)+\sin ^2(c)}}+\frac{a f^2 \csc (c) \left(d^2 e^{-i \tan ^{-1}(\cot (c))} x^2-\frac{\cot (c) \left(i d x \left(-2 \tan ^{-1}(\cot (c))-\pi \right)-\pi  \log \left(1+e^{-2 i d x}\right)-2 \left(d x-\tan ^{-1}(\cot (c))\right) \log \left(1-e^{2 i \left(d x-\tan ^{-1}(\cot (c))\right)}\right)+\pi  \log (\cos (d x))-2 \tan ^{-1}(\cot (c)) \log \left(\sin \left(d x-\tan ^{-1}(\cot (c))\right)\right)+i \text{Li}_2\left(e^{2 i \left(d x-\tan ^{-1}(\cot (c))\right)}\right)\right)}{\sqrt{\cot ^2(c)+1}}\right) \sec (c)}{\left(a^2-b^2\right) d^3 \sqrt{\csc ^2(c) \left(\cos ^2(c)+\sin ^2(c)\right)}}+\frac{2 a e f \sec (c) (\cos (c) \log (\cos (c) \cos (d x)-\sin (c) \sin (d x))+d x \sin (c))}{\left(a^2-b^2\right) d^2 \left(\cos ^2(c)+\sin ^2(c)\right)}+\frac{\sin \left(\frac{d x}{2}\right) e^2+2 f x \sin \left(\frac{d x}{2}\right) e+f^2 x^2 \sin \left(\frac{d x}{2}\right)}{(a+b) d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\sin \left(\frac{d x}{2}\right) e^2+2 f x \sin \left(\frac{d x}{2}\right) e+f^2 x^2 \sin \left(\frac{d x}{2}\right)}{(a-b) d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}","\frac{2 i b^2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d^3 \left(a^2-b^2\right)^{3/2}}-\frac{2 i b^2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{d^3 \left(a^2-b^2\right)^{3/2}}+\frac{2 i b f^2 \text{Li}_2\left(-i e^{i (c+d x)}\right)}{d^3 \left(a^2-b^2\right)}-\frac{2 i b f^2 \text{Li}_2\left(i e^{i (c+d x)}\right)}{d^3 \left(a^2-b^2\right)}-\frac{i a f^2 \text{Li}_2\left(-e^{2 i (c+d x)}\right)}{d^3 \left(a^2-b^2\right)}+\frac{2 b^2 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d^2 \left(a^2-b^2\right)^{3/2}}-\frac{2 b^2 f (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{d^2 \left(a^2-b^2\right)^{3/2}}+\frac{2 a f (e+f x) \log \left(1+e^{2 i (c+d x)}\right)}{d^2 \left(a^2-b^2\right)}-\frac{4 i b f (e+f x) \tan ^{-1}\left(e^{i (c+d x)}\right)}{d^2 \left(a^2-b^2\right)}+\frac{i b^2 (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}-\frac{i b^2 (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{a (e+f x)^2 \tan (c+d x)}{d \left(a^2-b^2\right)}-\frac{b (e+f x)^2 \sec (c+d x)}{d \left(a^2-b^2\right)}-\frac{i a (e+f x)^2}{d \left(a^2-b^2\right)}",1,"(I*b^2*(-2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - I*(d^2*(2*Sqrt[-a^2 + b^2]*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(2*e + f*x)*(Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])])) + 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))])))/(Sqrt[-(a^2 - b^2)^2]*(-a^2 + b^2)*d^3) + (b*(e + f*x)^2*Sec[c])/((-a^2 + b^2)*d) + (2*a*e*f*Sec[c]*(Cos[c]*Log[Cos[c]*Cos[d*x] - Sin[c]*Sin[d*x]] + d*x*Sin[c]))/((a^2 - b^2)*d^2*(Cos[c]^2 + Sin[c]^2)) + ((4*I)*b*e*f*ArcTan[((-I)*Sin[c] - I*Cos[c]*Tan[(d*x)/2])/Sqrt[Cos[c]^2 + Sin[c]^2]])/((a^2 - b^2)*d^2*Sqrt[Cos[c]^2 + Sin[c]^2]) + (a*f^2*Csc[c]*((d^2*x^2)/E^(I*ArcTan[Cot[c]]) - (Cot[c]*(I*d*x*(-Pi - 2*ArcTan[Cot[c]]) - Pi*Log[1 + E^((-2*I)*d*x)] - 2*(d*x - ArcTan[Cot[c]])*Log[1 - E^((2*I)*(d*x - ArcTan[Cot[c]]))] + Pi*Log[Cos[d*x]] - 2*ArcTan[Cot[c]]*Log[Sin[d*x - ArcTan[Cot[c]]]] + I*PolyLog[2, E^((2*I)*(d*x - ArcTan[Cot[c]]))]))/Sqrt[1 + Cot[c]^2])*Sec[c])/((a^2 - b^2)*d^3*Sqrt[Csc[c]^2*(Cos[c]^2 + Sin[c]^2)]) + (2*b*f^2*(-((Csc[c]*((d*x - ArcTan[Cot[c]])*(Log[1 - E^(I*(d*x - ArcTan[Cot[c]]))] - Log[1 + E^(I*(d*x - ArcTan[Cot[c]]))]) + I*(PolyLog[2, -E^(I*(d*x - ArcTan[Cot[c]]))] - PolyLog[2, E^(I*(d*x - ArcTan[Cot[c]]))])))/Sqrt[1 + Cot[c]^2]) + (2*ArcTan[Cot[c]]*ArcTanh[(Sin[c] + Cos[c]*Tan[(d*x)/2])/Sqrt[Cos[c]^2 + Sin[c]^2]])/Sqrt[Cos[c]^2 + Sin[c]^2]))/((a^2 - b^2)*d^3) + (e^2*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2])/((a + b)*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (e^2*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2])/((a - b)*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","A",0
312,1,842,349,9.6221223,"\int \frac{(e+f x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\frac{d (e+f x) \left(\frac{2 (d e-c f) \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{i f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{-i a+b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a+i \left(b+\sqrt{b^2-a^2}\right)}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{i a+b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)}{a-i \left(b+\sqrt{b^2-a^2}\right)}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{-b-a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{i a-b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(\tan \left(\frac{1}{2} (c+d x)\right)+i\right)}{i a-b+\sqrt{b^2-a^2}}\right)\right)}{\sqrt{b^2-a^2}}-\frac{i f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)-\sqrt{b^2-a^2}}{i a+b-\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{i \tan \left(\frac{1}{2} (c+d x)\right) a+a}{a+i \left(\sqrt{b^2-a^2}-b\right)}\right)\right)}{\sqrt{b^2-a^2}}\right) b^2}{\left(b^2-a^2\right) \left(d e-c f+i f \log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)-i f \log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}+\frac{d (e+f x) b}{b^2-a^2}+\frac{f \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a+b}+\frac{f \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a-b}+\frac{d (e+f x) \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{d (e+f x) \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)}}{d^2}","\frac{b^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d^2 \left(a^2-b^2\right)^{3/2}}-\frac{b^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{d^2 \left(a^2-b^2\right)^{3/2}}+\frac{b f \tanh ^{-1}(\sin (c+d x))}{d^2 \left(a^2-b^2\right)}+\frac{a f \log (\cos (c+d x))}{d^2 \left(a^2-b^2\right)}+\frac{i b^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}-\frac{i b^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{a (e+f x) \tan (c+d x)}{d \left(a^2-b^2\right)}-\frac{b (e+f x) \sec (c+d x)}{d \left(a^2-b^2\right)}",1,"((b*d*(e + f*x))/(-a^2 + b^2) + (f*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a + b) + (f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a - b) + (b^2*d*(e + f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(-b + Sqrt[-a^2 + b^2] - a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] - (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2]))/((-a^2 + b^2)*(d*e - c*f + I*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1 + I*Tan[(c + d*x)/2]])) + (d*(e + f*x)*Sin[(c + d*x)/2])/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (d*(e + f*x)*Sin[(c + d*x)/2])/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/d^2","B",0
313,1,152,84,0.258929,"\int \frac{\sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Sec[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 b^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 (b-a) (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 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)^{3/2}}-\frac{\sec (c+d x) (b-a \sin (c+d x))}{d \left(a^2-b^2\right)}",1,"(2*b^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
314,0,0,31,6.1363829,"\int \frac{(e+f x)^m \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Cos[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx","\text{Int}\left(\frac{\cos ^2(c+d x) (e+f x)^m}{a+b \sin (c+d x)},x\right)",0,"Integrate[((e + f*x)^m*Cos[c + d*x]^2)/(a + b*Sin[c + d*x]), x]","A",-1
315,0,0,29,3.4575933,"\int \frac{(e+f x)^m \cos (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Cos[c + d*x])/(a + b*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \cos (c+d x)}{a+b \sin (c+d x)} \, dx","\text{Int}\left(\frac{\cos (c+d x) (e+f x)^m}{a+b \sin (c+d x)},x\right)",0,"Integrate[((e + f*x)^m*Cos[c + d*x])/(a + b*Sin[c + d*x]), x]","A",-1
316,0,0,23,0.1081859,"\int \frac{(e+f x)^m}{a+b \sin (c+d x)} \, dx","Integrate[(e + f*x)^m/(a + b*Sin[c + d*x]),x]","\int \frac{(e+f x)^m}{a+b \sin (c+d x)} \, dx","\text{Int}\left(\frac{(e+f x)^m}{a+b \sin (c+d x)},x\right)",0,"Integrate[(e + f*x)^m/(a + b*Sin[c + d*x]), x]","A",-1
317,0,0,29,148.6017158,"\int \frac{(e+f x)^m \sec (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Sec[c + d*x])/(a + b*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \sec (c+d x)}{a+b \sin (c+d x)} \, dx","\text{Int}\left(\frac{\sec (c+d x) (e+f x)^m}{a+b \sin (c+d x)},x\right)",0,"Integrate[((e + f*x)^m*Sec[c + d*x])/(a + b*Sin[c + d*x]), x]","A",-1
318,0,0,31,21.7388517,"\int \frac{(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^m*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\int \frac{(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","\text{Int}\left(\frac{\sec ^2(c+d x) (e+f x)^m}{a+b \sin (c+d x)},x\right)",0,"Integrate[((e + f*x)^m*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]), x]","A",-1
319,1,73,77,0.4372859,"\int \frac{(e+f x) \cos (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[((e + f*x)*Cos[c + d*x])/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{2 f \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{d (e+f x)}{a+b \sin (c+d x)}}{b d^2}","\frac{2 f \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d^2 \sqrt{a^2-b^2}}-\frac{e+f x}{b d (a+b \sin (c+d x))}",1,"((2*f*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (d*(e + f*x))/(a + b*Sin[c + d*x]))/(b*d^2)","A",1
320,1,311,280,3.214338,"\int \frac{(e+f x)^2 \cos (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x])/(a + b*Sin[c + d*x])^2,x]","-\frac{(e+f x)^2}{b d (a+b \sin (c+d x))}+\frac{2 i f \left(-i d \left(2 e \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right)+f x \sqrt{a^2-b^2} \left(\log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-\log \left(1+\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}+i a}\right)\right)\right)-f \sqrt{a^2-b^2} \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)+f \sqrt{a^2-b^2} \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{b d^3 \sqrt{-\left(a^2-b^2\right)^2}}","-\frac{2 f^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^3 \sqrt{a^2-b^2}}+\frac{2 f^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^3 \sqrt{a^2-b^2}}-\frac{2 i f (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^2 \sqrt{a^2-b^2}}+\frac{2 i f (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d^2 \sqrt{a^2-b^2}}-\frac{(e+f x)^2}{b d (a+b \sin (c+d x))}",1,"((2*I)*f*((-I)*d*(2*Sqrt[-a^2 + b^2]*e*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])])) - Sqrt[a^2 - b^2]*f*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*f*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))]))/(b*Sqrt[-(a^2 - b^2)^2]*d^3) - (e + f*x)^2/(b*d*(a + b*Sin[c + d*x]))","A",0
321,1,446,418,2.4569245,"\int \frac{(e+f x)^3 \cos (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x])/(a + b*Sin[c + d*x])^2,x]","-\frac{(e+f x)^3}{b d (a+b \sin (c+d x))}+\frac{3 i f \left(-i \left(d^2 \left(2 e^2 \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right)+f x \sqrt{a^2-b^2} (2 e+f x) \left(\log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-\log \left(1+\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}+i a}\right)\right)\right)+2 f^2 \sqrt{a^2-b^2} \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-2 f^2 \sqrt{a^2-b^2} \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)-2 d f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)+2 d f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{b d^4 \sqrt{-\left(a^2-b^2\right)^2}}","-\frac{6 i f^3 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^4 \sqrt{a^2-b^2}}+\frac{6 i f^3 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^4 \sqrt{a^2-b^2}}-\frac{6 f^2 (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^3 \sqrt{a^2-b^2}}+\frac{6 f^2 (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^3 \sqrt{a^2-b^2}}-\frac{3 i f (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^2 \sqrt{a^2-b^2}}+\frac{3 i f (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d^2 \sqrt{a^2-b^2}}-\frac{(e+f x)^3}{b d (a+b \sin (c+d x))}",1,"((3*I)*f*(-2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - I*(d^2*(2*Sqrt[-a^2 + b^2]*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(2*e + f*x)*(Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])])) + 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))])))/(b*Sqrt[-(a^2 - b^2)^2]*d^4) - (e + f*x)^3/(b*d*(a + b*Sin[c + d*x]))","A",0
322,1,112,116,1.1651801,"\int \frac{(e+f x) \cos (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[((e + f*x)*Cos[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{2 a f \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right)^{3/2}}+\frac{\frac{f \cos (c+d x) (a+b \sin (c+d x))}{(a-b) (a+b)}-\frac{d (e+f x)}{b}}{(a+b \sin (c+d x))^2}}{2 d^2}","\frac{a f \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b d^2 \left(a^2-b^2\right)^{3/2}}+\frac{f \cos (c+d x)}{2 d^2 \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{e+f x}{2 b d (a+b \sin (c+d x))^2}",1,"((2*a*f*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(3/2)) + (-((d*(e + f*x))/b) + (f*Cos[c + d*x]*(a + b*Sin[c + d*x]))/((a - b)*(a + b)))/(a + b*Sin[c + d*x])^2)/(2*d^2)","A",1
323,1,1104,357,15.3437175,"\int \frac{(e+f x)^2 \cos (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\frac{x \cot (c) f^2}{b \left(b^2-a^2\right) d^2}-\frac{x \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) f^2}{2 b (b-a) (a+b) d^2}-\frac{i e^{i c} \left(4 e^{i c} f x-\frac{2 i a e^{2 i c} f \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) x}{\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+\frac{2 i a f \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) x}{\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+\frac{2 i a e^{2 i c} f \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) x}{\sqrt{\left(b^2-a^2\right) e^{2 i c}}}-\frac{2 i a f \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) x}{\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+\frac{4 i a e e^{-i c} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{4 i a e e^{i c} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{2 e^{-i c} f \tan ^{-1}\left(\frac{2 a e^{i (c+d x)}}{b \left(-1+e^{2 i (c+d x)}\right)}\right)}{d}-\frac{2 e^{i c} f \tan ^{-1}\left(\frac{2 a e^{i (c+d x)}}{b \left(-1+e^{2 i (c+d x)}\right)}\right)}{d}-\frac{i e^{-i c} f \log \left(4 e^{2 i (c+d x)} a^2+b^2 \left(-1+e^{2 i (c+d x)}\right)^2\right)}{d}+\frac{i e^{i c} f \log \left(4 e^{2 i (c+d x)} a^2+b^2 \left(-1+e^{2 i (c+d x)}\right)^2\right)}{d}-\frac{2 a \left(-1+e^{2 i c}\right) f \text{Li}_2\left(\frac{i b e^{i (2 c+d x)}}{e^{i c} a+i \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)}{d \sqrt{\left(b^2-a^2\right) e^{2 i c}}}+\frac{2 a \left(-1+e^{2 i c}\right) f \text{Li}_2\left(-\frac{b e^{i (2 c+d x)}}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)}{d \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) f}{2 b \left(b^2-a^2\right) d^2 \left(-1+e^{2 i c}\right)}+\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(-a x \cos (c) f^2-b x \sin (d x) f^2-a e \cos (c) f-b e \sin (d x) f\right)}{2 (a-b) b (a+b) d^2 (a+b \sin (c+d x))}-\frac{(e+f x)^2}{2 b d (a+b \sin (c+d x))^2}","-\frac{a f^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^3 \left(a^2-b^2\right)^{3/2}}+\frac{a f^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^3 \left(a^2-b^2\right)^{3/2}}-\frac{f^2 \log (a+b \sin (c+d x))}{b d^3 \left(a^2-b^2\right)}-\frac{i a f (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^2 \left(a^2-b^2\right)^{3/2}}+\frac{i a f (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d^2 \left(a^2-b^2\right)^{3/2}}+\frac{f (e+f x) \cos (c+d x)}{d^2 \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{(e+f x)^2}{2 b d (a+b \sin (c+d x))^2}",1,"(f^2*x*Cot[c])/(b*(-a^2 + b^2)*d^2) - ((I/2)*E^(I*c)*f*(4*E^(I*c)*f*x + ((4*I)*a*e*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*E^(I*c)) - ((4*I)*a*e*E^(I*c)*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (2*f*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))])/(d*E^(I*c)) - (2*E^(I*c)*f*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))])/d - (I*f*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2])/(d*E^(I*c)) + (I*E^(I*c)*f*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2])/d + ((2*I)*a*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - ((2*I)*a*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - ((2*I)*a*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] + ((2*I)*a*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - (2*a*(-1 + E^((2*I)*c))*f*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/(d*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]) + (2*a*(-1 + E^((2*I)*c))*f*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])/(d*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])))/(b*(-a^2 + b^2)*d^2*(-1 + E^((2*I)*c))) - (f^2*x*Csc[c/2]*Sec[c/2]*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))/(2*b*(-a + b)*(a + b)*d^2) - (e + f*x)^2/(2*b*d*(a + b*Sin[c + d*x])^2) + (Csc[c/2]*Sec[c/2]*(-(a*e*f*Cos[c]) - a*f^2*x*Cos[c] - b*e*f*Sin[d*x] - b*f^2*x*Sin[d*x]))/(2*(a - b)*b*(a + b)*d^2*(a + b*Sin[c + d*x]))","B",1
324,1,2311,753,20.0170408,"\int \frac{(e+f x)^3 \cos (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x])/(a + b*Sin[c + d*x])^3,x]","\text{Result too large to show}","\frac{3 i f^3 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^4 \left(a^2-b^2\right)}+\frac{3 i f^3 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^4 \left(a^2-b^2\right)}-\frac{3 i a f^3 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^4 \left(a^2-b^2\right)^{3/2}}+\frac{3 i a f^3 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^4 \left(a^2-b^2\right)^{3/2}}-\frac{3 a f^2 (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^3 \left(a^2-b^2\right)^{3/2}}+\frac{3 a f^2 (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{b d^3 \left(a^2-b^2\right)^{3/2}}-\frac{3 f^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{b d^3 \left(a^2-b^2\right)}-\frac{3 f^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{b d^3 \left(a^2-b^2\right)}-\frac{3 i a f (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{2 b d^2 \left(a^2-b^2\right)^{3/2}}+\frac{3 i a f (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{2 b d^2 \left(a^2-b^2\right)^{3/2}}+\frac{3 f (e+f x)^2 \cos (c+d x)}{2 d^2 \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 i f (e+f x)^2}{2 b d^2 \left(a^2-b^2\right)}-\frac{(e+f x)^3}{2 b d (a+b \sin (c+d x))^2}",1,"((-3*I)*E^(I*c)*f*(2*e*E^(I*c)*f*x + E^(I*c)*f^2*x^2 + (I*a*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*E^(I*c)) - (I*a*e^2*E^(I*c)*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (2*a*e*f*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d*E^(I*c)) + (e*f*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))])/(d*E^(I*c)) - (e*E^(I*c)*f*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))])/d + ((2*I)*a*e*f*ArcTanh[(-a + I*b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d*E^(I*c)) - ((I/2)*e*f*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2])/(d*E^(I*c)) + ((I/2)*e*E^(I*c)*f*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2])/d + (I*a*e*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - (I*a*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - (I*f^2*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/(d*E^(I*c)) + (I*E^(I*c)*f^2*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/d + ((I/2)*a*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - ((I/2)*a*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - (I*a*e*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] + (I*a*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - (I*f^2*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/(d*E^(I*c)) + (I*E^(I*c)*f^2*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/d - ((I/2)*a*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] + ((I/2)*a*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - ((-1 + E^((2*I)*c))*f*(-(Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*f) + a*d*E^(I*c)*(e + f*x))*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/(d^2*E^(I*c)*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]) + ((-1 + E^((2*I)*c))*f*(Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*f + a*d*E^(I*c)*(e + f*x))*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])/(d^2*E^(I*c)*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]) + (I*a*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/(d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]) - (I*a*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/(d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]) - (I*a*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])/(d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]) + (I*a*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])/(d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])))/(b*(-a^2 + b^2)*d^2*(-1 + E^((2*I)*c))) - (e + f*x)^3/(2*b*d*(a + b*Sin[c + d*x])^2) - (3*Csc[c/2]*Sec[c/2]*(a*e^2*f*Cos[c] + 2*a*e*f^2*x*Cos[c] + a*f^3*x^2*Cos[c] + b*e^2*f*Sin[d*x] + 2*b*e*f^2*x*Sin[d*x] + b*f^3*x^2*Sin[d*x]))/(4*(a - b)*b*(a + b)*d^2*(a + b*Sin[c + d*x]))","B",0
325,1,1194,765,1.9266383,"\int \frac{(e+f x)^3 \cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{x \left(4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right)}{4 b}+\frac{\left(a^2-b^2\right) \left(2 \sqrt{b^2-a^2} e^3 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) d^3+\sqrt{a^2-b^2} f^3 x^3 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3+3 \sqrt{a^2-b^2} e^2 f x \log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^3-\sqrt{a^2-b^2} f^3 x^3 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e f^2 x^2 \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 \sqrt{a^2-b^2} e^2 f x \log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right) d^3-3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d^2+3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d^2+6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d+6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right) d-6 \sqrt{a^2-b^2} e f^2 \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d-6 \sqrt{a^2-b^2} f^3 x \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right) d+6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-6 i \sqrt{a^2-b^2} f^3 \text{Li}_4\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{a b \sqrt{-\left(a^2-b^2\right)^2} d^4}+\frac{i \left(2 i \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) (e+f x)^3+\frac{3 f \left(-2 \text{Li}_4(-\cos (c+d x)-i \sin (c+d x)) f^2+2 i d (e+f x) \text{Li}_3(-\cos (c+d x)-i \sin (c+d x)) f+d^2 (e+f x)^2 \text{Li}_2(-\cos (c+d x)-i \sin (c+d x))\right)}{d^3}-\frac{3 f \left(-2 \text{Li}_4(\cos (c+d x)+i \sin (c+d x)) f^2+2 i d (e+f x) \text{Li}_3(\cos (c+d x)+i \sin (c+d x)) f+d^2 (e+f x)^2 \text{Li}_2(\cos (c+d x)+i \sin (c+d x))\right)}{d^3}\right)}{a d}","\frac{6 f^3 \sqrt{a^2-b^2} \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b d^4}-\frac{6 f^3 \sqrt{a^2-b^2} \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b d^4}-\frac{6 i f^2 \sqrt{a^2-b^2} (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b d^3}+\frac{6 i f^2 \sqrt{a^2-b^2} (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b d^3}-\frac{3 f \sqrt{a^2-b^2} (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b d^2}+\frac{3 f \sqrt{a^2-b^2} (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b d^2}-\frac{i \sqrt{a^2-b^2} (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b d}+\frac{i \sqrt{a^2-b^2} (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a b d}-\frac{6 i f^3 \text{Li}_4\left(-e^{i (c+d x)}\right)}{a d^4}+\frac{6 i f^3 \text{Li}_4\left(e^{i (c+d x)}\right)}{a d^4}-\frac{6 f^2 (e+f x) \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}+\frac{3 i f (e+f x)^2 \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}-\frac{2 (e+f x)^3 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{(e+f x)^4}{4 b f}",1,"-1/4*(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))/b + ((a^2 - b^2)*(2*Sqrt[-a^2 + b^2]*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))]))/(a*b*Sqrt[-(a^2 - b^2)^2]*d^4) + (I*((2*I)*(e + f*x)^3*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] + (3*f*(d^2*(e + f*x)^2*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)*PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] - 2*f^2*PolyLog[4, -Cos[c + d*x] - I*Sin[c + d*x]]))/d^3 - (3*f*(d^2*(e + f*x)^2*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)*PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]] - 2*f^2*PolyLog[4, Cos[c + d*x] + I*Sin[c + d*x]]))/d^3))/(a*d)","A",0
326,1,607,557,1.6710726,"\int \frac{(e+f x)^2 \cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{i \left(a^2-b^2\right) \left(-i \left(d^2 \left(2 e^2 \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right)+f x \sqrt{a^2-b^2} (2 e+f x) \left(\log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-\log \left(1+\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}+i a}\right)\right)\right)+2 f^2 \sqrt{a^2-b^2} \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-2 f^2 \sqrt{a^2-b^2} \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)-2 d f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)+2 d f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)}{a b d^3 \sqrt{-\left(a^2-b^2\right)^2}}+\frac{\frac{2 i f \left(d (e+f x) \text{Li}_2\left(-e^{i (c+d x)}\right)+i f \text{Li}_3\left(-e^{i (c+d x)}\right)\right)}{d^2}+\frac{2 f \left(f \text{Li}_3\left(e^{i (c+d x)}\right)-i d (e+f x) \text{Li}_2\left(e^{i (c+d x)}\right)\right)}{d^2}+(e+f x)^2 \log \left(1-e^{i (c+d x)}\right)-(e+f x)^2 \log \left(1+e^{i (c+d x)}\right)}{a d}-\frac{x \left(3 e^2+3 e f x+f^2 x^2\right)}{3 b}","-\frac{2 i f^2 \sqrt{a^2-b^2} \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b d^3}+\frac{2 i f^2 \sqrt{a^2-b^2} \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b d^3}-\frac{2 f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b d^2}+\frac{2 f \sqrt{a^2-b^2} (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b d^2}-\frac{i \sqrt{a^2-b^2} (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b d}+\frac{i \sqrt{a^2-b^2} (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a b d}-\frac{2 f^2 \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^3}+\frac{2 f^2 \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}+\frac{2 i f (e+f x) \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}-\frac{2 i f (e+f x) \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}-\frac{2 (e+f x)^2 \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{(e+f x)^3}{3 b f}",1,"-1/3*(x*(3*e^2 + 3*e*f*x + f^2*x^2))/b + ((e + f*x)^2*Log[1 - E^(I*(c + d*x))] - (e + f*x)^2*Log[1 + E^(I*(c + d*x))] + ((2*I)*f*(d*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))] + I*f*PolyLog[3, -E^(I*(c + d*x))]))/d^2 + (2*f*((-I)*d*(e + f*x)*PolyLog[2, E^(I*(c + d*x))] + f*PolyLog[3, E^(I*(c + d*x))]))/d^2)/(a*d) + (I*(a^2 - b^2)*(-2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - I*(d^2*(2*Sqrt[-a^2 + b^2]*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(2*e + f*x)*(Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])])) + 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))])))/(a*b*Sqrt[-(a^2 - b^2)^2]*d^3)","A",1
327,1,812,351,6.7739211,"\int \frac{(e+f x) \cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\frac{(c+d x) (c f-d (2 e+f x))}{b}+\frac{2 d e \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)}{a}-\frac{2 c f \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)}{a}+\frac{2 f \left((c+d x) \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)\right)+i \left(\text{Li}_2\left(-e^{i (c+d x)}\right)-\text{Li}_2\left(e^{i (c+d x)}\right)\right)\right)}{a}+\frac{2 \left(a^2-b^2\right) d (e+f x) \left(\frac{2 (d e-c f) \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{i f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{-i a+b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a+i \left(b+\sqrt{b^2-a^2}\right)}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{i a+b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)}{a-i \left(b+\sqrt{b^2-a^2}\right)}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{-b-a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{i a-b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(\tan \left(\frac{1}{2} (c+d x)\right)+i\right)}{i a-b+\sqrt{b^2-a^2}}\right)\right)}{\sqrt{b^2-a^2}}-\frac{i f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)-\sqrt{b^2-a^2}}{i a+b-\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{i \tan \left(\frac{1}{2} (c+d x)\right) a+a}{a+i \left(\sqrt{b^2-a^2}-b\right)}\right)\right)}{\sqrt{b^2-a^2}}\right)}{a b \left(d e-c f+i f \log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)-i f \log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}}{2 d^2}","-\frac{f \sqrt{a^2-b^2} \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b d^2}+\frac{f \sqrt{a^2-b^2} \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b d^2}-\frac{i \sqrt{a^2-b^2} (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b d}+\frac{i \sqrt{a^2-b^2} (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a b d}+\frac{i f \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}-\frac{i f \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}-\frac{2 (e+f x) \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{e x}{b}-\frac{f x^2}{2 b}",1,"(((c + d*x)*(c*f - d*(2*e + f*x)))/b + (2*d*e*Log[Tan[(c + d*x)/2]])/a - (2*c*f*Log[Tan[(c + d*x)/2]])/a + (2*f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/a + (2*(a^2 - b^2)*d*(e + f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(-b + Sqrt[-a^2 + b^2] - a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] - (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2]))/(a*b*(d*e - c*f + I*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1 + I*Tan[(c + d*x)/2]])))/(2*d^2)","B",0
328,1,90,75,0.1256379,"\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
329,1,4014,763,10.7530353,"\int \frac{(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x]^2*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","\frac{6 i f^3 \left(a^2-b^2\right) \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^2 d^4}+\frac{6 i f^3 \left(a^2-b^2\right) \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b^2 d^4}+\frac{6 f^2 \left(a^2-b^2\right) (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^2 d^3}+\frac{6 f^2 \left(a^2-b^2\right) (e+f x) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b^2 d^3}-\frac{3 i f \left(a^2-b^2\right) (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^2 d^2}-\frac{3 i f \left(a^2-b^2\right) (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b^2 d^2}+\frac{\left(a^2-b^2\right) (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^2 d}+\frac{\left(a^2-b^2\right) (e+f x)^3 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a b^2 d}-\frac{i \left(a^2-b^2\right) (e+f x)^4}{4 a b^2 f}+\frac{3 i f^3 \text{Li}_4\left(e^{2 i (c+d x)}\right)}{4 a d^4}+\frac{3 f^2 (e+f x) \text{Li}_3\left(e^{2 i (c+d x)}\right)}{2 a d^3}-\frac{3 i f (e+f x)^2 \text{Li}_2\left(e^{2 i (c+d x)}\right)}{2 a d^2}+\frac{(e+f x)^3 \log \left(1-e^{2 i (c+d x)}\right)}{a d}-\frac{i (e+f x)^4}{4 a f}+\frac{6 f^3 \cos (c+d x)}{b d^4}+\frac{6 f^2 (e+f x) \sin (c+d x)}{b d^3}-\frac{3 f (e+f x)^2 \cos (c+d x)}{b d^2}-\frac{(e+f x)^3 \sin (c+d x)}{b d}",1,"-1/2*(e*E^(I*c)*f^2*Csc[c]*((2*d^3*x^3)/E^((2*I)*c) + (3*I)*d^2*(1 - E^((-2*I)*c))*x^2*Log[1 - E^((-I)*(c + d*x))] + (3*I)*d^2*(1 - E^((-2*I)*c))*x^2*Log[1 + E^((-I)*(c + d*x))] - (6*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, -E^((-I)*(c + d*x))] - I*PolyLog[3, -E^((-I)*(c + d*x))]))/E^((2*I)*c) - (6*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, E^((-I)*(c + d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))]))/E^((2*I)*c)))/(a*d^3) - (E^(I*c)*f^3*Csc[c]*((d^4*x^4)/E^((2*I)*c) + (2*I)*d^3*(1 - E^((-2*I)*c))*x^3*Log[1 - E^((-I)*(c + d*x))] + (2*I)*d^3*(1 - E^((-2*I)*c))*x^3*Log[1 + E^((-I)*(c + d*x))] - (6*(-1 + E^((2*I)*c))*(d^2*x^2*PolyLog[2, -E^((-I)*(c + d*x))] - (2*I)*d*x*PolyLog[3, -E^((-I)*(c + d*x))] - 2*PolyLog[4, -E^((-I)*(c + d*x))]))/E^((2*I)*c) - (6*(-1 + E^((2*I)*c))*(d^2*x^2*PolyLog[2, E^((-I)*(c + d*x))] - (2*I)*d*x*PolyLog[3, E^((-I)*(c + d*x))] - 2*PolyLog[4, E^((-I)*(c + d*x))]))/E^((2*I)*c)))/(4*a*d^4) + ((a^2 - b^2)*((-4*I)*d^4*e^3*E^((2*I)*c)*x - (6*I)*d^4*e^2*E^((2*I)*c)*f*x^2 - (4*I)*d^4*e*E^((2*I)*c)*f^2*x^3 - I*d^4*E^((2*I)*c)*f^3*x^4 - (2*I)*d^3*e^3*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + (2*I)*d^3*e^3*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] - d^3*e^3*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] + d^3*e^3*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] - 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*e*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*e*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (12*I)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (12*I)*E^((2*I)*c)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12*I)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (12*I)*E^((2*I)*c)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(2*a*b^2*d^4*(-1 + E^((2*I)*c))) + (e^3*Csc[c]*(-(d*x*Cos[c]) + Log[Cos[d*x]*Sin[c] + Cos[c]*Sin[d*x]]*Sin[c]))/(a*d*(Cos[c]^2 + Sin[c]^2)) + Csc[c]*(Cos[c + d*x]/(8*b^2*d^4) - ((I/8)*Sin[c + d*x])/(b^2*d^4))*(4*a*d^4*e^3*x*Cos[d*x] + 6*a*d^4*e^2*f*x^2*Cos[d*x] + 4*a*d^4*e*f^2*x^3*Cos[d*x] + a*d^4*f^3*x^4*Cos[d*x] + 4*a*d^4*e^3*x*Cos[2*c + d*x] + 6*a*d^4*e^2*f*x^2*Cos[2*c + d*x] + 4*a*d^4*e*f^2*x^3*Cos[2*c + d*x] + a*d^4*f^3*x^4*Cos[2*c + d*x] - 2*b*d^3*e^3*Cos[c + 2*d*x] - (6*I)*b*d^2*e^2*f*Cos[c + 2*d*x] + 12*b*d*e*f^2*Cos[c + 2*d*x] + (12*I)*b*f^3*Cos[c + 2*d*x] - 6*b*d^3*e^2*f*x*Cos[c + 2*d*x] - (12*I)*b*d^2*e*f^2*x*Cos[c + 2*d*x] + 12*b*d*f^3*x*Cos[c + 2*d*x] - 6*b*d^3*e*f^2*x^2*Cos[c + 2*d*x] - (6*I)*b*d^2*f^3*x^2*Cos[c + 2*d*x] - 2*b*d^3*f^3*x^3*Cos[c + 2*d*x] + 2*b*d^3*e^3*Cos[3*c + 2*d*x] + (6*I)*b*d^2*e^2*f*Cos[3*c + 2*d*x] - 12*b*d*e*f^2*Cos[3*c + 2*d*x] - (12*I)*b*f^3*Cos[3*c + 2*d*x] + 6*b*d^3*e^2*f*x*Cos[3*c + 2*d*x] + (12*I)*b*d^2*e*f^2*x*Cos[3*c + 2*d*x] - 12*b*d*f^3*x*Cos[3*c + 2*d*x] + 6*b*d^3*e*f^2*x^2*Cos[3*c + 2*d*x] + (6*I)*b*d^2*f^3*x^2*Cos[3*c + 2*d*x] + 2*b*d^3*f^3*x^3*Cos[3*c + 2*d*x] - (4*I)*b*d^3*e^3*Sin[c] - 12*b*d^2*e^2*f*Sin[c] + (24*I)*b*d*e*f^2*Sin[c] + 24*b*f^3*Sin[c] - (12*I)*b*d^3*e^2*f*x*Sin[c] - 24*b*d^2*e*f^2*x*Sin[c] + (24*I)*b*d*f^3*x*Sin[c] - (12*I)*b*d^3*e*f^2*x^2*Sin[c] - 12*b*d^2*f^3*x^2*Sin[c] - (4*I)*b*d^3*f^3*x^3*Sin[c] + (4*I)*a*d^4*e^3*x*Sin[d*x] + (6*I)*a*d^4*e^2*f*x^2*Sin[d*x] + (4*I)*a*d^4*e*f^2*x^3*Sin[d*x] + I*a*d^4*f^3*x^4*Sin[d*x] + (4*I)*a*d^4*e^3*x*Sin[2*c + d*x] + (6*I)*a*d^4*e^2*f*x^2*Sin[2*c + d*x] + (4*I)*a*d^4*e*f^2*x^3*Sin[2*c + d*x] + I*a*d^4*f^3*x^4*Sin[2*c + d*x] - (2*I)*b*d^3*e^3*Sin[c + 2*d*x] + 6*b*d^2*e^2*f*Sin[c + 2*d*x] + (12*I)*b*d*e*f^2*Sin[c + 2*d*x] - 12*b*f^3*Sin[c + 2*d*x] - (6*I)*b*d^3*e^2*f*x*Sin[c + 2*d*x] + 12*b*d^2*e*f^2*x*Sin[c + 2*d*x] + (12*I)*b*d*f^3*x*Sin[c + 2*d*x] - (6*I)*b*d^3*e*f^2*x^2*Sin[c + 2*d*x] + 6*b*d^2*f^3*x^2*Sin[c + 2*d*x] - (2*I)*b*d^3*f^3*x^3*Sin[c + 2*d*x] + (2*I)*b*d^3*e^3*Sin[3*c + 2*d*x] - 6*b*d^2*e^2*f*Sin[3*c + 2*d*x] - (12*I)*b*d*e*f^2*Sin[3*c + 2*d*x] + 12*b*f^3*Sin[3*c + 2*d*x] + (6*I)*b*d^3*e^2*f*x*Sin[3*c + 2*d*x] - 12*b*d^2*e*f^2*x*Sin[3*c + 2*d*x] - (12*I)*b*d*f^3*x*Sin[3*c + 2*d*x] + (6*I)*b*d^3*e*f^2*x^2*Sin[3*c + 2*d*x] - 6*b*d^2*f^3*x^2*Sin[3*c + 2*d*x] + (2*I)*b*d^3*f^3*x^3*Sin[3*c + 2*d*x]) - (3*e^2*f*Csc[c]*Sec[c]*(d^2*E^(I*ArcTan[Tan[c]])*x^2 + ((I*d*x*(-Pi + 2*ArcTan[Tan[c]]) - Pi*Log[1 + E^((-2*I)*d*x)] - 2*(d*x + ArcTan[Tan[c]])*Log[1 - E^((2*I)*(d*x + ArcTan[Tan[c]]))] + Pi*Log[Cos[d*x]] + 2*ArcTan[Tan[c]]*Log[Sin[d*x + ArcTan[Tan[c]]]] + I*PolyLog[2, E^((2*I)*(d*x + ArcTan[Tan[c]]))])*Tan[c])/Sqrt[1 + Tan[c]^2]))/(2*a*d^2*Sqrt[Sec[c]^2*(Cos[c]^2 + Sin[c]^2)])","B",0
330,1,1834,566,9.5541807,"\int \frac{(e+f x)^2 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x]^2*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{\csc (c) (\log (\cos (d x) \sin (c)+\cos (c) \sin (d x)) \sin (c)-d x \cos (c)) e^2}{a d \left(\cos ^2(c)+\sin ^2(c)\right)}-\frac{f \csc (c) \sec (c) \left(d^2 e^{i \tan ^{-1}(\tan (c))} x^2+\frac{\left(i d x \left(2 \tan ^{-1}(\tan (c))-\pi \right)-\pi  \log \left(1+e^{-2 i d x}\right)-2 \left(d x+\tan ^{-1}(\tan (c))\right) \log \left(1-e^{2 i \left(d x+\tan ^{-1}(\tan (c))\right)}\right)+\pi  \log (\cos (d x))+2 \tan ^{-1}(\tan (c)) \log \left(\sin \left(d x+\tan ^{-1}(\tan (c))\right)\right)+i \text{Li}_2\left(e^{2 i \left(d x+\tan ^{-1}(\tan (c))\right)}\right)\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}\right) e}{a d^2 \sqrt{\sec ^2(c) \left(\cos ^2(c)+\sin ^2(c)\right)}}-\frac{e^{i c} f^2 \csc (c) \left(2 d^3 e^{-2 i c} x^3+3 i d^2 \left(1-e^{-2 i c}\right) \log \left(1-e^{-i (c+d x)}\right) x^2+3 i d^2 \left(1-e^{-2 i c}\right) \log \left(1+e^{-i (c+d x)}\right) x^2-6 e^{-2 i c} \left(-1+e^{2 i c}\right) \left(d x \text{Li}_2\left(-e^{-i (c+d x)}\right)-i \text{Li}_3\left(-e^{-i (c+d x)}\right)\right)-6 e^{-2 i c} \left(-1+e^{2 i c}\right) \left(d x \text{Li}_2\left(e^{-i (c+d x)}\right)-i \text{Li}_3\left(e^{-i (c+d x)}\right)\right)\right)}{6 a d^3}+\frac{\left(a^2-b^2\right) \left(-4 i e^{2 i c} f^2 x^3 d^3-12 i e e^{2 i c} f x^2 d^3-12 i e^2 e^{2 i c} x d^3+6 i e^2 e^{2 i c} \tan ^{-1}\left(\frac{2 a e^{i (c+d x)}}{b \left(-1+e^{2 i (c+d x)}\right)}\right) d^2-6 i e^2 \tan ^{-1}\left(\frac{2 a e^{i (c+d x)}}{b \left(-1+e^{2 i (c+d x)}\right)}\right) d^2+3 e^2 e^{2 i c} \log \left(4 e^{2 i (c+d x)} a^2+b^2 \left(-1+e^{2 i (c+d x)}\right)^2\right) d^2-3 e^2 \log \left(4 e^{2 i (c+d x)} a^2+b^2 \left(-1+e^{2 i (c+d x)}\right)^2\right) d^2+6 e^{2 i c} f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-6 f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+12 e e^{2 i c} f x \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-12 e f x \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+6 e^{2 i c} f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-6 f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+12 e e^{2 i c} f x \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-12 e f x \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-12 i \left(-1+e^{2 i c}\right) f (e+f x) \text{Li}_2\left(\frac{i b e^{i (2 c+d x)}}{e^{i c} a+i \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d-12 i \left(-1+e^{2 i c}\right) f (e+f x) \text{Li}_2\left(-\frac{b e^{i (2 c+d x)}}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d+12 e^{2 i c} f^2 \text{Li}_3\left(\frac{i b e^{i (2 c+d x)}}{e^{i c} a+i \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-12 f^2 \text{Li}_3\left(\frac{i b e^{i (2 c+d x)}}{e^{i c} a+i \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+12 e^{2 i c} f^2 \text{Li}_3\left(-\frac{b e^{i (2 c+d x)}}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-12 f^2 \text{Li}_3\left(-\frac{b e^{i (2 c+d x)}}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)\right)}{6 a b^2 d^3 \left(-1+e^{2 i c}\right)}+\frac{a x \left(3 e^2+3 f x e+f^2 x^2\right) \cos (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{6 b^2}-\frac{\cos (d x) \left(e^2 \sin (c) d^2+f^2 x^2 \sin (c) d^2+2 e f x \sin (c) d^2+2 e f \cos (c) d+2 f^2 x \cos (c) d-2 f^2 \sin (c)\right)}{b d^3}-\frac{\left(e^2 \cos (c) d^2+f^2 x^2 \cos (c) d^2+2 e f x \cos (c) d^2-2 e f \sin (c) d-2 f^2 x \sin (c) d-2 f^2 \cos (c)\right) \sin (d x)}{b d^3}","\frac{2 f^2 \left(a^2-b^2\right) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^2 d^3}+\frac{2 f^2 \left(a^2-b^2\right) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b^2 d^3}-\frac{2 i f \left(a^2-b^2\right) (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^2 d^2}-\frac{2 i f \left(a^2-b^2\right) (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b^2 d^2}+\frac{\left(a^2-b^2\right) (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^2 d}+\frac{\left(a^2-b^2\right) (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a b^2 d}-\frac{i \left(a^2-b^2\right) (e+f x)^3}{3 a b^2 f}+\frac{f^2 \text{Li}_3\left(e^{2 i (c+d x)}\right)}{2 a d^3}-\frac{i f (e+f x) \text{Li}_2\left(e^{2 i (c+d x)}\right)}{a d^2}+\frac{(e+f x)^2 \log \left(1-e^{2 i (c+d x)}\right)}{a d}-\frac{i (e+f x)^3}{3 a f}+\frac{2 f^2 \sin (c+d x)}{b d^3}-\frac{2 f (e+f x) \cos (c+d x)}{b d^2}-\frac{(e+f x)^2 \sin (c+d x)}{b d}",1,"-1/6*(E^(I*c)*f^2*Csc[c]*((2*d^3*x^3)/E^((2*I)*c) + (3*I)*d^2*(1 - E^((-2*I)*c))*x^2*Log[1 - E^((-I)*(c + d*x))] + (3*I)*d^2*(1 - E^((-2*I)*c))*x^2*Log[1 + E^((-I)*(c + d*x))] - (6*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, -E^((-I)*(c + d*x))] - I*PolyLog[3, -E^((-I)*(c + d*x))]))/E^((2*I)*c) - (6*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, E^((-I)*(c + d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))]))/E^((2*I)*c)))/(a*d^3) + ((a^2 - b^2)*((-12*I)*d^3*e^2*E^((2*I)*c)*x - (12*I)*d^3*e*E^((2*I)*c)*f*x^2 - (4*I)*d^3*E^((2*I)*c)*f^2*x^3 - (6*I)*d^2*e^2*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + (6*I)*d^2*e^2*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] - 3*d^2*e^2*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] + 3*d^2*e^2*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] - 12*d^2*e*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^2*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d^2*e*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^2*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12*I)*d*(-1 + E^((2*I)*c))*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12*I)*d*(-1 + E^((2*I)*c))*f*(e + f*x)*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(6*a*b^2*d^3*(-1 + E^((2*I)*c))) + (a*x*(3*e^2 + 3*e*f*x + f^2*x^2)*Cos[c]*Csc[c/2]*Sec[c/2])/(6*b^2) - (Cos[d*x]*(2*d*e*f*Cos[c] + 2*d*f^2*x*Cos[c] + d^2*e^2*Sin[c] - 2*f^2*Sin[c] + 2*d^2*e*f*x*Sin[c] + d^2*f^2*x^2*Sin[c]))/(b*d^3) + (e^2*Csc[c]*(-(d*x*Cos[c]) + Log[Cos[d*x]*Sin[c] + Cos[c]*Sin[d*x]]*Sin[c]))/(a*d*(Cos[c]^2 + Sin[c]^2)) - ((d^2*e^2*Cos[c] - 2*f^2*Cos[c] + 2*d^2*e*f*x*Cos[c] + d^2*f^2*x^2*Cos[c] - 2*d*e*f*Sin[c] - 2*d*f^2*x*Sin[c])*Sin[d*x])/(b*d^3) - (e*f*Csc[c]*Sec[c]*(d^2*E^(I*ArcTan[Tan[c]])*x^2 + ((I*d*x*(-Pi + 2*ArcTan[Tan[c]]) - Pi*Log[1 + E^((-2*I)*d*x)] - 2*(d*x + ArcTan[Tan[c]])*Log[1 - E^((2*I)*(d*x + ArcTan[Tan[c]]))] + Pi*Log[Cos[d*x]] + 2*ArcTan[Tan[c]]*Log[Sin[d*x + ArcTan[Tan[c]]]] + I*PolyLog[2, E^((2*I)*(d*x + ArcTan[Tan[c]]))])*Tan[c])/Sqrt[1 + Tan[c]^2]))/(a*d^2*Sqrt[Sec[c]^2*(Cos[c]^2 + Sin[c]^2)])","B",0
331,1,2209,379,14.9424847,"\int \frac{(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x]^2*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","-\frac{i f \left(a^2-b^2\right) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^2 d^2}-\frac{i f \left(a^2-b^2\right) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b^2 d^2}+\frac{\left(a^2-b^2\right) (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^2 d}+\frac{\left(a^2-b^2\right) (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a b^2 d}-\frac{i \left(a^2-b^2\right) (e+f x)^2}{2 a b^2 f}-\frac{i f \text{Li}_2\left(e^{2 i (c+d x)}\right)}{2 a d^2}+\frac{(e+f x) \log \left(1-e^{2 i (c+d x)}\right)}{a d}-\frac{i (e+f x)^2}{2 a f}-\frac{f \cos (c+d x)}{b d^2}-\frac{(e+f x) \sin (c+d x)}{b d}",1,"-((f*Cos[c + d*x])/(b*d^2)) + (e*Log[Sin[c + d*x]])/(a*d) - (c*f*Log[Sin[c + d*x]])/(a*d^2) + (f*((c + d*x)*Log[1 - E^((2*I)*(c + d*x))] - (I/2)*((c + d*x)^2 + PolyLog[2, E^((2*I)*(c + d*x))])))/(a*d^2) - ((d*e - c*f + f*(c + d*x))*Sin[c + d*x])/(b*d^2) + ((f*(c + d*x)^2 + (2*I)*d*e*Log[Sec[(c + d*x)/2]^2] - (2*I)*c*f*Log[Sec[(c + d*x)/2]^2] - (2*I)*d*e*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] + (2*I)*c*f*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] - (4*I)*f*(c + d*x)*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + 4*f*PolyLog[2, -Cos[c + d*x] + I*Sin[c + d*x]] + 2*f*PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))] - 2*f*PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))] + 2*f*PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])] - 2*f*PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))])*((a*e*Cos[c + d*x])/(b*(a + b*Sin[c + d*x])) - (b*e*Cos[c + d*x])/(a*(a + b*Sin[c + d*x])) - (a*c*f*Cos[c + d*x])/(b*d*(a + b*Sin[c + d*x])) + (b*c*f*Cos[c + d*x])/(a*d*(a + b*Sin[c + d*x])) + (a*f*(c + d*x)*Cos[c + d*x])/(b*d*(a + b*Sin[c + d*x])) - (b*f*(c + d*x)*Cos[c + d*x])/(a*d*(a + b*Sin[c + d*x]))))/(d*(2*f*(c + d*x) - (4*I)*f*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - (4*f*Log[1 + Cos[c + d*x] - I*Sin[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x]))/(-Cos[c + d*x] + I*Sin[c + d*x]) + (I*f*Log[1 - (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) + (I*f*Log[1 - (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - (I*f*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + (2*I)*d*e*Tan[(c + d*x)/2] - (2*I)*c*f*Tan[(c + d*x)/2] + ((2*I)*f*(c + d*x)*Sec[(c + d*x)/2]^2)/(-I + Tan[(c + d*x)/2]) - (f*Log[1 - (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(I + Tan[(c + d*x)/2]) + (I*a*f*Log[1 - (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(a + I*a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - ((2*I)*d*e*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x]) + ((2*I)*c*f*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x])))","B",0
332,1,53,59,0.0738507,"\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
333,1,1181,1138,6.7339551,"\int \frac{(e+f x)^3 \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","\frac{2 a \left(2 a^2-3 b^2\right) f^3 x^4 d^4+8 a \left(2 a^2-3 b^2\right) e f^2 x^3 d^4+12 a \left(2 a^2-3 b^2\right) e^2 f x^2 d^4+8 a \left(2 a^2-3 b^2\right) e^3 x d^4-32 b^3 (e+f x)^3 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) d^3+16 a^2 b (e+f x)^3 \cos (c+d x) d^3-4 a b^2 (e+f x)^3 \sin (2 (c+d x)) d^3-6 a b^2 f (e+f x)^2 \cos (2 (c+d x)) d^2+48 \left(a^2-b^2\right)^{3/2} f (e+f x)^2 \text{Li}_2\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right) d^2-48 a^2 b f (e+f x)^2 \sin (c+d x) d^2-96 a^2 b f^2 (e+f x) \cos (c+d x) d+6 a b^2 f^2 (e+f x) \sin (2 (c+d x)) d+3 a b^2 f^3 \cos (2 (c+d x))+16 i \left(a^2-b^2\right)^{3/2} \left(2 i e^3 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) d^3+f^3 x^3 \log \left(\frac{i e^{i (c+d x)} b}{\sqrt{a^2-b^2}-a}+1\right) d^3+3 e f^2 x^2 \log \left(\frac{i e^{i (c+d x)} b}{\sqrt{a^2-b^2}-a}+1\right) d^3+3 e^2 f x \log \left(\frac{i e^{i (c+d x)} b}{\sqrt{a^2-b^2}-a}+1\right) d^3-f^3 x^3 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d^3-3 e f^2 x^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d^3-3 e^2 f x \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d^3+3 i f (e+f x)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d^2+6 f^2 (e+f x) \text{Li}_3\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right) d-6 e f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d-6 f^3 x \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) d+6 i f^3 \text{Li}_4\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)-6 i f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)\right)+48 i b^3 f \left(-2 \text{Li}_4(-\cos (c+d x)-i \sin (c+d x)) f^2+2 i d (e+f x) \text{Li}_3(-\cos (c+d x)-i \sin (c+d x)) f+d^2 (e+f x)^2 \text{Li}_2(-\cos (c+d x)-i \sin (c+d x))\right)-48 i b^3 f \left(-2 \text{Li}_4(\cos (c+d x)+i \sin (c+d x)) f^2+2 i d (e+f x) \text{Li}_3(\cos (c+d x)+i \sin (c+d x)) f+d^2 (e+f x)^2 \text{Li}_2(\cos (c+d x)+i \sin (c+d x))\right)+96 a^2 b f^3 \sin (c+d x)}{16 a b^3 d^4}","\frac{\left(a^2-b^2\right) (e+f x)^4}{4 b^3 f}-\frac{(e+f x)^4}{8 b f}-\frac{2 \tanh ^{-1}\left(e^{i (c+d x)}\right) (e+f x)^3}{a d}+\frac{\left(a^2-b^2\right) \cos (c+d x) (e+f x)^3}{a b^2 d}+\frac{\cos (c+d x) (e+f x)^3}{a d}+\frac{i \left(a^2-b^2\right)^{3/2} \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^3}{a b^3 d}-\frac{i \left(a^2-b^2\right)^{3/2} \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^3}{a b^3 d}-\frac{\cos (c+d x) \sin (c+d x) (e+f x)^3}{2 b d}-\frac{3 f \cos ^2(c+d x) (e+f x)^2}{4 b d^2}+\frac{3 i f \text{Li}_2\left(-e^{i (c+d x)}\right) (e+f x)^2}{a d^2}-\frac{3 i f \text{Li}_2\left(e^{i (c+d x)}\right) (e+f x)^2}{a d^2}+\frac{3 \left(a^2-b^2\right)^{3/2} f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^2}{a b^3 d^2}-\frac{3 \left(a^2-b^2\right)^{3/2} f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^2}{a b^3 d^2}-\frac{3 \left(a^2-b^2\right) f \sin (c+d x) (e+f x)^2}{a b^2 d^2}-\frac{3 f \sin (c+d x) (e+f x)^2}{a d^2}-\frac{6 \left(a^2-b^2\right) f^2 \cos (c+d x) (e+f x)}{a b^2 d^3}-\frac{6 f^2 \cos (c+d x) (e+f x)}{a d^3}-\frac{6 f^2 \text{Li}_3\left(-e^{i (c+d x)}\right) (e+f x)}{a d^3}+\frac{6 f^2 \text{Li}_3\left(e^{i (c+d x)}\right) (e+f x)}{a d^3}+\frac{6 i \left(a^2-b^2\right)^{3/2} f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)}{a b^3 d^3}-\frac{6 i \left(a^2-b^2\right)^{3/2} f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)}{a b^3 d^3}+\frac{3 f^2 \cos (c+d x) \sin (c+d x) (e+f x)}{4 b d^3}+\frac{3 f^3 x^2}{8 b d^2}+\frac{3 f^3 \cos ^2(c+d x)}{8 b d^4}+\frac{3 e f^2 x}{4 b d^2}-\frac{6 i f^3 \text{Li}_4\left(-e^{i (c+d x)}\right)}{a d^4}+\frac{6 i f^3 \text{Li}_4\left(e^{i (c+d x)}\right)}{a d^4}-\frac{6 \left(a^2-b^2\right)^{3/2} f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^3 d^4}+\frac{6 \left(a^2-b^2\right)^{3/2} f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b^3 d^4}+\frac{6 \left(a^2-b^2\right) f^3 \sin (c+d x)}{a b^2 d^4}+\frac{6 f^3 \sin (c+d x)}{a d^4}",1,"(8*a*(2*a^2 - 3*b^2)*d^4*e^3*x + 12*a*(2*a^2 - 3*b^2)*d^4*e^2*f*x^2 + 8*a*(2*a^2 - 3*b^2)*d^4*e*f^2*x^3 + 2*a*(2*a^2 - 3*b^2)*d^4*f^3*x^4 - 32*b^3*d^3*(e + f*x)^3*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - 96*a^2*b*d*f^2*(e + f*x)*Cos[c + d*x] + 16*a^2*b*d^3*(e + f*x)^3*Cos[c + d*x] + 3*a*b^2*f^3*Cos[2*(c + d*x)] - 6*a*b^2*d^2*f*(e + f*x)^2*Cos[2*(c + d*x)] + 48*(a^2 - b^2)^(3/2)*d^2*f*(e + f*x)^2*PolyLog[2, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + (16*I)*(a^2 - b^2)^(3/2)*((2*I)*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 3*d^3*e^2*f*x*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + 3*d^3*e*f^2*x^2*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + d^3*f^3*x^3*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - 3*d^3*e^2*f*x*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] - 3*d^3*e*f^2*x^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] - d^3*f^3*x^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + (3*I)*d^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + 6*d*f^2*(e + f*x)*PolyLog[3, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - 6*d*e*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] - 6*d*f^3*x*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + (6*I)*f^3*PolyLog[4, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - (6*I)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])]) + (48*I)*b^3*f*(d^2*(e + f*x)^2*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)*PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] - 2*f^2*PolyLog[4, -Cos[c + d*x] - I*Sin[c + d*x]]) - (48*I)*b^3*f*(d^2*(e + f*x)^2*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)*PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]] - 2*f^2*PolyLog[4, Cos[c + d*x] + I*Sin[c + d*x]]) + 96*a^2*b*f^3*Sin[c + d*x] - 48*a^2*b*d^2*f*(e + f*x)^2*Sin[c + d*x] + 6*a*b^2*d*f^2*(e + f*x)*Sin[2*(c + d*x)] - 4*a*b^2*d^3*(e + f*x)^3*Sin[2*(c + d*x)])/(16*a*b^3*d^4)","A",0
334,1,1254,825,5.0781042,"\int \frac{(e+f x)^2 \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{-24 d^2 e^2 \log \left(1-e^{i (c+d x)}\right) b^3-24 d^2 f^2 x^2 \log \left(1-e^{i (c+d x)}\right) b^3-48 d^2 e f x \log \left(1-e^{i (c+d x)}\right) b^3+24 d^2 e^2 \log \left(1+e^{i (c+d x)}\right) b^3+24 d^2 f^2 x^2 \log \left(1+e^{i (c+d x)}\right) b^3+48 d^2 e f x \log \left(1+e^{i (c+d x)}\right) b^3-48 i d e f \text{Li}_2\left(-e^{i (c+d x)}\right) b^3-48 i d f^2 x \text{Li}_2\left(-e^{i (c+d x)}\right) b^3+48 i d e f \text{Li}_2\left(e^{i (c+d x)}\right) b^3+48 i d f^2 x \text{Li}_2\left(e^{i (c+d x)}\right) b^3+48 f^2 \text{Li}_3\left(-e^{i (c+d x)}\right) b^3-48 f^2 \text{Li}_3\left(e^{i (c+d x)}\right) b^3+12 a d^3 f^2 x^3 b^2+36 a d^3 e f x^2 b^2+36 a d^3 e^2 x b^2+6 a d e f \cos (2 (c+d x)) b^2+6 a d f^2 x \cos (2 (c+d x)) b^2+6 a d^2 e^2 \sin (2 (c+d x)) b^2-3 a f^2 \sin (2 (c+d x)) b^2+6 a d^2 f^2 x^2 \sin (2 (c+d x)) b^2+12 a d^2 e f x \sin (2 (c+d x)) b^2-24 a^2 d^2 e^2 \cos (c+d x) b+48 a^2 f^2 \cos (c+d x) b-24 a^2 d^2 f^2 x^2 \cos (c+d x) b-48 a^2 d^2 e f x \cos (c+d x) b+48 a^2 d e f \sin (c+d x) b+48 a^2 d f^2 x \sin (c+d x) b-8 a^3 d^3 f^2 x^3-24 a^3 d^3 e f x^2-24 a^3 d^3 e^2 x+48 \left(a^2-b^2\right)^{3/2} d^2 e^2 \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right)-24 i \left(a^2-b^2\right)^{3/2} d^2 f^2 x^2 \log \left(\frac{i e^{i (c+d x)} b}{\sqrt{a^2-b^2}-a}+1\right)-48 i \left(a^2-b^2\right)^{3/2} d^2 e f x \log \left(\frac{i e^{i (c+d x)} b}{\sqrt{a^2-b^2}-a}+1\right)+24 i \left(a^2-b^2\right)^{3/2} d^2 f^2 x^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)+48 i \left(a^2-b^2\right)^{3/2} d^2 e f x \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)-48 \left(a^2-b^2\right)^{3/2} d e f \text{Li}_2\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)-48 \left(a^2-b^2\right)^{3/2} d f^2 x \text{Li}_2\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)+48 \left(a^2-b^2\right)^{3/2} d e f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)+48 \left(a^2-b^2\right)^{3/2} d f^2 x \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)-48 i \left(a^2-b^2\right)^{3/2} f^2 \text{Li}_3\left(-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right)+48 i \left(a^2-b^2\right)^{3/2} f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{24 a b^3 d^3}","\frac{\left(a^2-b^2\right) (e+f x)^3}{3 b^3 f}-\frac{(e+f x)^3}{6 b f}-\frac{2 \tanh ^{-1}\left(e^{i (c+d x)}\right) (e+f x)^2}{a d}+\frac{\left(a^2-b^2\right) \cos (c+d x) (e+f x)^2}{a b^2 d}+\frac{\cos (c+d x) (e+f x)^2}{a d}+\frac{i \left(a^2-b^2\right)^{3/2} \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^2}{a b^3 d}-\frac{i \left(a^2-b^2\right)^{3/2} \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^2}{a b^3 d}-\frac{\cos (c+d x) \sin (c+d x) (e+f x)^2}{2 b d}-\frac{f \cos ^2(c+d x) (e+f x)}{2 b d^2}+\frac{2 i f \text{Li}_2\left(-e^{i (c+d x)}\right) (e+f x)}{a d^2}-\frac{2 i f \text{Li}_2\left(e^{i (c+d x)}\right) (e+f x)}{a d^2}+\frac{2 \left(a^2-b^2\right)^{3/2} f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)}{a b^3 d^2}-\frac{2 \left(a^2-b^2\right)^{3/2} f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)}{a b^3 d^2}-\frac{2 \left(a^2-b^2\right) f \sin (c+d x) (e+f x)}{a b^2 d^2}-\frac{2 f \sin (c+d x) (e+f x)}{a d^2}+\frac{f^2 x}{4 b d^2}-\frac{2 \left(a^2-b^2\right) f^2 \cos (c+d x)}{a b^2 d^3}-\frac{2 f^2 \cos (c+d x)}{a d^3}-\frac{2 f^2 \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^3}+\frac{2 f^2 \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^3}+\frac{2 i \left(a^2-b^2\right)^{3/2} f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^3 d^3}-\frac{2 i \left(a^2-b^2\right)^{3/2} f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b^3 d^3}+\frac{f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}",1,"-1/24*(-24*a^3*d^3*e^2*x + 36*a*b^2*d^3*e^2*x - 24*a^3*d^3*e*f*x^2 + 36*a*b^2*d^3*e*f*x^2 - 8*a^3*d^3*f^2*x^3 + 12*a*b^2*d^3*f^2*x^3 + 48*(a^2 - b^2)^(3/2)*d^2*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] - 24*a^2*b*d^2*e^2*Cos[c + d*x] + 48*a^2*b*f^2*Cos[c + d*x] - 48*a^2*b*d^2*e*f*x*Cos[c + d*x] - 24*a^2*b*d^2*f^2*x^2*Cos[c + d*x] + 6*a*b^2*d*e*f*Cos[2*(c + d*x)] + 6*a*b^2*d*f^2*x*Cos[2*(c + d*x)] - 24*b^3*d^2*e^2*Log[1 - E^(I*(c + d*x))] - 48*b^3*d^2*e*f*x*Log[1 - E^(I*(c + d*x))] - 24*b^3*d^2*f^2*x^2*Log[1 - E^(I*(c + d*x))] + 24*b^3*d^2*e^2*Log[1 + E^(I*(c + d*x))] + 48*b^3*d^2*e*f*x*Log[1 + E^(I*(c + d*x))] + 24*b^3*d^2*f^2*x^2*Log[1 + E^(I*(c + d*x))] - (48*I)*(a^2 - b^2)^(3/2)*d^2*e*f*x*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - (24*I)*(a^2 - b^2)^(3/2)*d^2*f^2*x^2*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + (48*I)*(a^2 - b^2)^(3/2)*d^2*e*f*x*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + (24*I)*(a^2 - b^2)^(3/2)*d^2*f^2*x^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] - (48*I)*b^3*d*e*f*PolyLog[2, -E^(I*(c + d*x))] - (48*I)*b^3*d*f^2*x*PolyLog[2, -E^(I*(c + d*x))] + (48*I)*b^3*d*e*f*PolyLog[2, E^(I*(c + d*x))] + (48*I)*b^3*d*f^2*x*PolyLog[2, E^(I*(c + d*x))] - 48*(a^2 - b^2)^(3/2)*d*e*f*PolyLog[2, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - 48*(a^2 - b^2)^(3/2)*d*f^2*x*PolyLog[2, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + 48*(a^2 - b^2)^(3/2)*d*e*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + 48*(a^2 - b^2)^(3/2)*d*f^2*x*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + 48*b^3*f^2*PolyLog[3, -E^(I*(c + d*x))] - 48*b^3*f^2*PolyLog[3, E^(I*(c + d*x))] - (48*I)*(a^2 - b^2)^(3/2)*f^2*PolyLog[3, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + (48*I)*(a^2 - b^2)^(3/2)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + 48*a^2*b*d*e*f*Sin[c + d*x] + 48*a^2*b*d*f^2*x*Sin[c + d*x] + 6*a*b^2*d^2*e^2*Sin[2*(c + d*x)] - 3*a*b^2*f^2*Sin[2*(c + d*x)] + 12*a*b^2*d^2*e*f*x*Sin[2*(c + d*x)] + 6*a*b^2*d^2*f^2*x^2*Sin[2*(c + d*x)])/(a*b^3*d^3)","A",1
335,1,934,524,11.7643479,"\int \frac{(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]","-\frac{(d e+d f x) \left(\frac{2 (d e-c f) \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{i f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{-i a+b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a+i \left(b+\sqrt{b^2-a^2}\right)}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{i a+b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)}{a-i \left(b+\sqrt{b^2-a^2}\right)}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(-\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)-\sqrt{b^2-a^2}}{i a-b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(\tan \left(\frac{1}{2} (c+d x)\right)+i\right)}{i a-b+\sqrt{b^2-a^2}}\right)\right)}{\sqrt{b^2-a^2}}-\frac{i f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)-\sqrt{b^2-a^2}}{i a+b-\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{i \tan \left(\frac{1}{2} (c+d x)\right) a+a}{a+i \left(\sqrt{b^2-a^2}-b\right)}\right)\right)}{\sqrt{b^2-a^2}}\right) \left(a^2-b^2\right)^2}{a b^3 d^2 \left(d e-c f+i f \log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)-i f \log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}-\frac{\left(3 b^2-2 a^2\right) (c+d x) (2 d e-2 c f+f (c+d x))}{4 b^3 d^2}+\frac{a (d e-c f+f (c+d x)) \cos (c+d x)}{b^2 d^2}-\frac{f \cos (2 (c+d x))}{8 b d^2}+\frac{e \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)}{a d}-\frac{c f \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)}{a d^2}+\frac{f \left((c+d x) \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)\right)+i \left(\text{Li}_2\left(-e^{i (c+d x)}\right)-\text{Li}_2\left(e^{i (c+d x)}\right)\right)\right)}{a d^2}-\frac{a f \sin (c+d x)}{b^2 d^2}-\frac{(d e-c f+f (c+d x)) \sin (2 (c+d x))}{4 b d^2}","-\frac{f \left(a^2-b^2\right) \sin (c+d x)}{a b^2 d^2}+\frac{\left(a^2-b^2\right) (e+f x) \cos (c+d x)}{a b^2 d}+\frac{f \left(a^2-b^2\right)^{3/2} \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^3 d^2}-\frac{f \left(a^2-b^2\right)^{3/2} \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a b^3 d^2}+\frac{i \left(a^2-b^2\right)^{3/2} (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a b^3 d}-\frac{i \left(a^2-b^2\right)^{3/2} (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a b^3 d}+\frac{e x \left(a^2-b^2\right)}{b^3}+\frac{f x^2 \left(a^2-b^2\right)}{2 b^3}+\frac{i f \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^2}-\frac{i f \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^2}-\frac{f \sin (c+d x)}{a d^2}+\frac{(e+f x) \cos (c+d x)}{a d}-\frac{2 (e+f x) \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d}-\frac{f \cos ^2(c+d x)}{4 b d^2}-\frac{(e+f x) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac{e x}{2 b}-\frac{f x^2}{4 b}",1,"-1/4*((-2*a^2 + 3*b^2)*(c + d*x)*(2*d*e - 2*c*f + f*(c + d*x)))/(b^3*d^2) + (a*(d*e - c*f + f*(c + d*x))*Cos[c + d*x])/(b^2*d^2) - (f*Cos[2*(c + d*x)])/(8*b*d^2) + (e*Log[Tan[(c + d*x)/2]])/(a*d) - (c*f*Log[Tan[(c + d*x)/2]])/(a*d^2) + (f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/(a*d^2) - ((a^2 - b^2)^2*(d*e + d*f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] - (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2]))/(a*b^3*d^2*(d*e - c*f + I*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1 + I*Tan[(c + d*x)/2]])) - (a*f*Sin[c + d*x])/(b^2*d^2) - ((d*e - c*f + f*(c + d*x))*Sin[2*(c + d*x)])/(4*b*d^2)","A",0
336,1,143,124,0.271855,"\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
337,1,2974,852,49.182381,"\int \frac{(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","\frac{i b (e+f x)^4}{4 a^2 f}+\frac{i \left(a^2-b^2\right) (e+f x)^4}{4 a^2 b f}-\frac{\csc (c+d x) (e+f x)^3}{a d}-\frac{\left(a^2-b^2\right) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^3}{a^2 b d}-\frac{\left(a^2-b^2\right) \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^3}{a^2 b d}-\frac{b \log \left(1-e^{2 i (c+d x)}\right) (e+f x)^3}{a^2 d}-\frac{6 f \tanh ^{-1}\left(e^{i (c+d x)}\right) (e+f x)^2}{a d^2}+\frac{3 i \left(a^2-b^2\right) f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^2}{a^2 b d^2}+\frac{3 i \left(a^2-b^2\right) f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^2}{a^2 b d^2}+\frac{3 i b f \text{Li}_2\left(e^{2 i (c+d x)}\right) (e+f x)^2}{2 a^2 d^2}+\frac{6 i f^2 \text{Li}_2\left(-e^{i (c+d x)}\right) (e+f x)}{a d^3}-\frac{6 i f^2 \text{Li}_2\left(e^{i (c+d x)}\right) (e+f x)}{a d^3}-\frac{6 \left(a^2-b^2\right) f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)}{a^2 b d^3}-\frac{6 \left(a^2-b^2\right) f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)}{a^2 b d^3}-\frac{3 b f^2 \text{Li}_3\left(e^{2 i (c+d x)}\right) (e+f x)}{2 a^2 d^3}-\frac{6 f^3 \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^4}+\frac{6 f^3 \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^4}-\frac{6 i \left(a^2-b^2\right) f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b d^4}-\frac{6 i \left(a^2-b^2\right) f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 b d^4}-\frac{3 i b f^3 \text{Li}_4\left(e^{2 i (c+d x)}\right)}{4 a^2 d^4}",1,"-1/2*(((-I)*b*(e + f*x)^4)/((-1 + E^((2*I)*c))*f) + (6*e*f*(b*d*e - 2*a*f)*x*Log[1 - E^((-I)*(c + d*x))])/d^2 + (6*f^2*(b*d*e - a*f)*x^2*Log[1 - E^((-I)*(c + d*x))])/d^2 + (2*b*f^3*x^3*Log[1 - E^((-I)*(c + d*x))])/d + (6*e*f*(b*d*e + 2*a*f)*x*Log[1 + E^((-I)*(c + d*x))])/d^2 + (6*f^2*(b*d*e + a*f)*x^2*Log[1 + E^((-I)*(c + d*x))])/d^2 + (2*b*f^3*x^3*Log[1 + E^((-I)*(c + d*x))])/d + (2*e^2*(b*d*e - 3*a*f)*((-I)*d*x + Log[1 - E^(I*(c + d*x))]))/d^2 + (2*e^2*(b*d*e + 3*a*f)*((-I)*d*x + Log[1 + E^(I*(c + d*x))]))/d^2 + ((6*I)*e*f*(b*d*e + 2*a*f)*PolyLog[2, -E^((-I)*(c + d*x))])/d^3 + ((6*I)*e*f*(b*d*e - 2*a*f)*PolyLog[2, E^((-I)*(c + d*x))])/d^3 + (12*f^2*(b*d*e + a*f)*(I*d*x*PolyLog[2, -E^((-I)*(c + d*x))] + PolyLog[3, -E^((-I)*(c + d*x))]))/d^4 + (12*f^2*(b*d*e - a*f)*(I*d*x*PolyLog[2, E^((-I)*(c + d*x))] + PolyLog[3, E^((-I)*(c + d*x))]))/d^4 + (6*b*f^3*(I*d^2*x^2*PolyLog[2, -E^((-I)*(c + d*x))] + 2*d*x*PolyLog[3, -E^((-I)*(c + d*x))] - (2*I)*PolyLog[4, -E^((-I)*(c + d*x))]))/d^4 + (6*b*f^3*(I*d^2*x^2*PolyLog[2, E^((-I)*(c + d*x))] + 2*d*x*PolyLog[3, E^((-I)*(c + d*x))] - (2*I)*PolyLog[4, E^((-I)*(c + d*x))]))/d^4)/a^2 + ((a^2 - b^2)*((4*I)*d^4*e^3*E^((2*I)*c)*x + (6*I)*d^4*e^2*E^((2*I)*c)*f*x^2 + (4*I)*d^4*e*E^((2*I)*c)*f^2*x^3 + I*d^4*E^((2*I)*c)*f^3*x^4 + (2*I)*d^3*e^3*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] - (2*I)*d^3*e^3*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + d^3*e^3*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] - d^3*e^3*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] + 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*e*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*e*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (12*I)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12*I)*E^((2*I)*c)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (12*I)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (12*I)*E^((2*I)*c)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(2*a^2*b*d^4*(-1 + E^((2*I)*c))) + ((-4*b*e^3 - 12*b*e^2*f*x - 12*b*e*f^2*x^2 - 4*b*f^3*x^3 - 4*a*d*e^3*x*Cos[c] - 6*a*d*e^2*f*x^2*Cos[c] - 4*a*d*e*f^2*x^3*Cos[c] - a*d*f^3*x^4*Cos[c])*Csc[c/2]*Sec[c/2])/(8*a*b*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(e^3*Sin[(d*x)/2]) - 3*e^2*f*x*Sin[(d*x)/2] - 3*e*f^2*x^2*Sin[(d*x)/2] - f^3*x^3*Sin[(d*x)/2]))/(2*a*d) + (Csc[c/2]*Csc[c/2 + (d*x)/2]*(e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2]))/(2*a*d)","B",1
338,1,1833,616,14.228811,"\int \frac{(e+f x)^2 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{\frac{2 i b (e+f x)^3}{\left(-1+e^{2 i c}\right) f}-\frac{3 b f^2 x^2 \log \left(1-e^{-i (c+d x)}\right)}{d}+\frac{6 f (a f-b d e) x \log \left(1-e^{-i (c+d x)}\right)}{d^2}-\frac{3 b f^2 x^2 \log \left(1+e^{-i (c+d x)}\right)}{d}-\frac{6 f (b d e+a f) x \log \left(1+e^{-i (c+d x)}\right)}{d^2}+\frac{3 i e (b d e-2 a f) \left(d x+i \log \left(1-e^{i (c+d x)}\right)\right)}{d^2}+\frac{3 i e (b d e+2 a f) \left(d x+i \log \left(1+e^{i (c+d x)}\right)\right)}{d^2}-\frac{6 i f (b d e+a f) \text{Li}_2\left(-e^{-i (c+d x)}\right)}{d^3}+\frac{6 i f (a f-b d e) \text{Li}_2\left(e^{-i (c+d x)}\right)}{d^3}-\frac{6 i b f^2 \left(d x \text{Li}_2\left(-e^{-i (c+d x)}\right)-i \text{Li}_3\left(-e^{-i (c+d x)}\right)\right)}{d^3}-\frac{6 i b f^2 \left(d x \text{Li}_2\left(e^{-i (c+d x)}\right)-i \text{Li}_3\left(e^{-i (c+d x)}\right)\right)}{d^3}}{3 a^2}+\frac{\left(a^2-b^2\right) \left(4 i e^{2 i c} f^2 x^3 d^3+12 i e e^{2 i c} f x^2 d^3+12 i e^2 e^{2 i c} x d^3-6 i e^2 e^{2 i c} \tan ^{-1}\left(\frac{2 a e^{i (c+d x)}}{b \left(-1+e^{2 i (c+d x)}\right)}\right) d^2+6 i e^2 \tan ^{-1}\left(\frac{2 a e^{i (c+d x)}}{b \left(-1+e^{2 i (c+d x)}\right)}\right) d^2-3 e^2 e^{2 i c} \log \left(4 e^{2 i (c+d x)} a^2+b^2 \left(-1+e^{2 i (c+d x)}\right)^2\right) d^2+3 e^2 \log \left(4 e^{2 i (c+d x)} a^2+b^2 \left(-1+e^{2 i (c+d x)}\right)^2\right) d^2-6 e^{2 i c} f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+6 f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-12 e e^{2 i c} f x \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+12 e f x \log \left(\frac{e^{i (2 c+d x)} b}{i a e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-6 e^{2 i c} f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+6 f^2 x^2 \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-12 e e^{2 i c} f x \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+12 e f x \log \left(\frac{e^{i (2 c+d x)} b}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+12 i \left(-1+e^{2 i c}\right) f (e+f x) \text{Li}_2\left(\frac{i b e^{i (2 c+d x)}}{e^{i c} a+i \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d+12 i \left(-1+e^{2 i c}\right) f (e+f x) \text{Li}_2\left(-\frac{b e^{i (2 c+d x)}}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d-12 e^{2 i c} f^2 \text{Li}_3\left(\frac{i b e^{i (2 c+d x)}}{e^{i c} a+i \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+12 f^2 \text{Li}_3\left(\frac{i b e^{i (2 c+d x)}}{e^{i c} a+i \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-12 e^{2 i c} f^2 \text{Li}_3\left(-\frac{b e^{i (2 c+d x)}}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+12 f^2 \text{Li}_3\left(-\frac{b e^{i (2 c+d x)}}{i e^{i c} a+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)\right)}{6 a^2 b d^3 \left(-1+e^{2 i c}\right)}+\frac{\left(-a d f^2 \cos (c) x^3-3 b f^2 x^2-3 a d e f \cos (c) x^2-6 b e f x-3 a d e^2 \cos (c) x-3 b e^2\right) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{6 a b d}+\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(-\sin \left(\frac{d x}{2}\right) e^2-2 f x \sin \left(\frac{d x}{2}\right) e-f^2 x^2 \sin \left(\frac{d x}{2}\right)\right)}{2 a d}+\frac{\csc \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\sin \left(\frac{d x}{2}\right) e^2+2 f x \sin \left(\frac{d x}{2}\right) e+f^2 x^2 \sin \left(\frac{d x}{2}\right)\right)}{2 a d}","-\frac{2 f^2 \left(a^2-b^2\right) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b d^3}-\frac{2 f^2 \left(a^2-b^2\right) \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 b d^3}+\frac{2 i f \left(a^2-b^2\right) (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b d^2}+\frac{2 i f \left(a^2-b^2\right) (e+f x) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 b d^2}-\frac{\left(a^2-b^2\right) (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b d}-\frac{\left(a^2-b^2\right) (e+f x)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a^2 b d}+\frac{i \left(a^2-b^2\right) (e+f x)^3}{3 a^2 b f}-\frac{b f^2 \text{Li}_3\left(e^{2 i (c+d x)}\right)}{2 a^2 d^3}+\frac{i b f (e+f x) \text{Li}_2\left(e^{2 i (c+d x)}\right)}{a^2 d^2}-\frac{b (e+f x)^2 \log \left(1-e^{2 i (c+d x)}\right)}{a^2 d}+\frac{i b (e+f x)^3}{3 a^2 f}+\frac{2 i f^2 \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^3}-\frac{2 i f^2 \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^3}-\frac{4 f (e+f x) \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a d^2}-\frac{(e+f x)^2 \csc (c+d x)}{a d}",1,"(((2*I)*b*(e + f*x)^3)/((-1 + E^((2*I)*c))*f) + (6*f*(-(b*d*e) + a*f)*x*Log[1 - E^((-I)*(c + d*x))])/d^2 - (3*b*f^2*x^2*Log[1 - E^((-I)*(c + d*x))])/d - (6*f*(b*d*e + a*f)*x*Log[1 + E^((-I)*(c + d*x))])/d^2 - (3*b*f^2*x^2*Log[1 + E^((-I)*(c + d*x))])/d + ((3*I)*e*(b*d*e - 2*a*f)*(d*x + I*Log[1 - E^(I*(c + d*x))]))/d^2 + ((3*I)*e*(b*d*e + 2*a*f)*(d*x + I*Log[1 + E^(I*(c + d*x))]))/d^2 - ((6*I)*f*(b*d*e + a*f)*PolyLog[2, -E^((-I)*(c + d*x))])/d^3 + ((6*I)*f*(-(b*d*e) + a*f)*PolyLog[2, E^((-I)*(c + d*x))])/d^3 - ((6*I)*b*f^2*(d*x*PolyLog[2, -E^((-I)*(c + d*x))] - I*PolyLog[3, -E^((-I)*(c + d*x))]))/d^3 - ((6*I)*b*f^2*(d*x*PolyLog[2, E^((-I)*(c + d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))]))/d^3)/(3*a^2) + ((a^2 - b^2)*((12*I)*d^3*e^2*E^((2*I)*c)*x + (12*I)*d^3*e*E^((2*I)*c)*f*x^2 + (4*I)*d^3*E^((2*I)*c)*f^2*x^3 + (6*I)*d^2*e^2*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] - (6*I)*d^2*e^2*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + 3*d^2*e^2*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] - 3*d^2*e^2*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] + 12*d^2*e*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^2*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d^2*e*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d^2*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^2*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^2*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (12*I)*d*(-1 + E^((2*I)*c))*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (12*I)*d*(-1 + E^((2*I)*c))*f*(e + f*x)*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(6*a^2*b*d^3*(-1 + E^((2*I)*c))) + ((-3*b*e^2 - 6*b*e*f*x - 3*b*f^2*x^2 - 3*a*d*e^2*x*Cos[c] - 3*a*d*e*f*x^2*Cos[c] - a*d*f^2*x^3*Cos[c])*Csc[c/2]*Sec[c/2])/(6*a*b*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(e^2*Sin[(d*x)/2]) - 2*e*f*x*Sin[(d*x)/2] - f^2*x^2*Sin[(d*x)/2]))/(2*a*d) + (Csc[c/2]*Csc[c/2 + (d*x)/2]*(e^2*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2]))/(2*a*d)","B",1
339,1,2314,386,14.8954742,"\int \frac{(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","\frac{i f \left(a^2-b^2\right) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b d^2}+\frac{i f \left(a^2-b^2\right) \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 b d^2}-\frac{\left(a^2-b^2\right) (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b d}-\frac{\left(a^2-b^2\right) (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a^2 b d}+\frac{i \left(a^2-b^2\right) (e+f x)^2}{2 a^2 b f}+\frac{i b f \text{Li}_2\left(e^{2 i (c+d x)}\right)}{2 a^2 d^2}-\frac{b (e+f x) \log \left(1-e^{2 i (c+d x)}\right)}{a^2 d}+\frac{i b (e+f x)^2}{2 a^2 f}-\frac{f \tanh ^{-1}(\cos (c+d x))}{a d^2}-\frac{(e+f x) \csc (c+d x)}{a d}",1,"((-(d*e*Cos[(c + d*x)/2]) + c*f*Cos[(c + d*x)/2] - f*(c + d*x)*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(2*a*d^2) - (b*e*Log[Sin[c + d*x]])/(a^2*d) + (b*c*f*Log[Sin[c + d*x]])/(a^2*d^2) + (f*Log[Tan[(c + d*x)/2]])/(a*d^2) - (b*f*((c + d*x)*Log[1 - E^((2*I)*(c + d*x))] - (I/2)*((c + d*x)^2 + PolyLog[2, E^((2*I)*(c + d*x))])))/(a^2*d^2) + (Sec[(c + d*x)/2]*(-(d*e*Sin[(c + d*x)/2]) + c*f*Sin[(c + d*x)/2] - f*(c + d*x)*Sin[(c + d*x)/2]))/(2*a*d^2) + ((f*(c + d*x)^2 + (2*I)*d*e*Log[Sec[(c + d*x)/2]^2] - (2*I)*c*f*Log[Sec[(c + d*x)/2]^2] - (2*I)*d*e*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] + (2*I)*c*f*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] - (4*I)*f*(c + d*x)*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + 4*f*PolyLog[2, -Cos[c + d*x] + I*Sin[c + d*x]] + 2*f*PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))] - 2*f*PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))] + 2*f*PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])] - 2*f*PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))])*(-((e*Cos[c + d*x])/(a + b*Sin[c + d*x])) + (b^2*e*Cos[c + d*x])/(a^2*(a + b*Sin[c + d*x])) + (c*f*Cos[c + d*x])/(d*(a + b*Sin[c + d*x])) - (b^2*c*f*Cos[c + d*x])/(a^2*d*(a + b*Sin[c + d*x])) - (f*(c + d*x)*Cos[c + d*x])/(d*(a + b*Sin[c + d*x])) + (b^2*f*(c + d*x)*Cos[c + d*x])/(a^2*d*(a + b*Sin[c + d*x]))))/(d*(2*f*(c + d*x) - (4*I)*f*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - (4*f*Log[1 + Cos[c + d*x] - I*Sin[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x]))/(-Cos[c + d*x] + I*Sin[c + d*x]) + (I*f*Log[1 - (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) + (I*f*Log[1 - (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - (I*f*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + (2*I)*d*e*Tan[(c + d*x)/2] - (2*I)*c*f*Tan[(c + d*x)/2] + ((2*I)*f*(c + d*x)*Sec[(c + d*x)/2]^2)/(-I + Tan[(c + d*x)/2]) - (f*Log[1 - (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(I + Tan[(c + d*x)/2]) + (I*a*f*Log[1 - (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(a + I*a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - ((2*I)*d*e*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x]) + ((2*I)*c*f*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x])))","B",0
340,1,54,60,0.0985852,"\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
341,1,3860,1144,45.9154197,"\int \frac{(e+f x)^3 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","-\frac{\left(a^2-b^2\right) (e+f x)^4}{4 a b^2 f}-\frac{(e+f x)^4}{4 a f}+\frac{2 b \tanh ^{-1}\left(e^{i (c+d x)}\right) (e+f x)^3}{a^2 d}-\frac{b \cos (c+d x) (e+f x)^3}{a^2 d}-\frac{\left(a^2-b^2\right) \cos (c+d x) (e+f x)^3}{a^2 b d}-\frac{\cot (c+d x) (e+f x)^3}{a d}-\frac{i \left(a^2-b^2\right)^{3/2} \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^3}{a^2 b^2 d}+\frac{i \left(a^2-b^2\right)^{3/2} \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^3}{a^2 b^2 d}-\frac{i (e+f x)^3}{a d}+\frac{3 f \log \left(1-e^{2 i (c+d x)}\right) (e+f x)^2}{a d^2}-\frac{3 i b f \text{Li}_2\left(-e^{i (c+d x)}\right) (e+f x)^2}{a^2 d^2}+\frac{3 i b f \text{Li}_2\left(e^{i (c+d x)}\right) (e+f x)^2}{a^2 d^2}-\frac{3 \left(a^2-b^2\right)^{3/2} f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^2}{a^2 b^2 d^2}+\frac{3 \left(a^2-b^2\right)^{3/2} f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^2}{a^2 b^2 d^2}+\frac{3 b f \sin (c+d x) (e+f x)^2}{a^2 d^2}+\frac{3 \left(a^2-b^2\right) f \sin (c+d x) (e+f x)^2}{a^2 b d^2}+\frac{6 b f^2 \cos (c+d x) (e+f x)}{a^2 d^3}+\frac{6 \left(a^2-b^2\right) f^2 \cos (c+d x) (e+f x)}{a^2 b d^3}-\frac{3 i f^2 \text{Li}_2\left(e^{2 i (c+d x)}\right) (e+f x)}{a d^3}+\frac{6 b f^2 \text{Li}_3\left(-e^{i (c+d x)}\right) (e+f x)}{a^2 d^3}-\frac{6 b f^2 \text{Li}_3\left(e^{i (c+d x)}\right) (e+f x)}{a^2 d^3}-\frac{6 i \left(a^2-b^2\right)^{3/2} f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)}{a^2 b^2 d^3}+\frac{6 i \left(a^2-b^2\right)^{3/2} f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)}{a^2 b^2 d^3}+\frac{3 f^3 \text{Li}_3\left(e^{2 i (c+d x)}\right)}{2 a d^4}+\frac{6 i b f^3 \text{Li}_4\left(-e^{i (c+d x)}\right)}{a^2 d^4}-\frac{6 i b f^3 \text{Li}_4\left(e^{i (c+d x)}\right)}{a^2 d^4}+\frac{6 \left(a^2-b^2\right)^{3/2} f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b^2 d^4}-\frac{6 \left(a^2-b^2\right)^{3/2} f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 b^2 d^4}-\frac{6 b f^3 \sin (c+d x)}{a^2 d^4}-\frac{6 \left(a^2-b^2\right) f^3 \sin (c+d x)}{a^2 b d^4}",1,"(((-2*I)*a*d^3*(e + f*x)^3)/(-1 + E^((2*I)*c)) - 3*d^2*e*f*(b*d*e - 2*a*f)*x*Log[1 - E^((-I)*(c + d*x))] - 3*d^2*f^2*(b*d*e - a*f)*x^2*Log[1 - E^((-I)*(c + d*x))] - b*d^3*f^3*x^3*Log[1 - E^((-I)*(c + d*x))] + 3*d^2*e*f*(b*d*e + 2*a*f)*x*Log[1 + E^((-I)*(c + d*x))] + 3*d^2*f^2*(b*d*e + a*f)*x^2*Log[1 + E^((-I)*(c + d*x))] + b*d^3*f^3*x^3*Log[1 + E^((-I)*(c + d*x))] + I*d^2*e^2*(b*d*e - 3*a*f)*(d*x + I*Log[1 - E^(I*(c + d*x))]) + d^2*e^2*(b*d*e + 3*a*f)*((-I)*d*x + Log[1 + E^(I*(c + d*x))]) + (3*I)*d*e*f*(b*d*e + 2*a*f)*PolyLog[2, -E^((-I)*(c + d*x))] - (3*I)*d*e*f*(b*d*e - 2*a*f)*PolyLog[2, E^((-I)*(c + d*x))] + 6*f^2*(b*d*e + a*f)*(I*d*x*PolyLog[2, -E^((-I)*(c + d*x))] + PolyLog[3, -E^((-I)*(c + d*x))]) + 6*f^2*(-(b*d*e) + a*f)*(I*d*x*PolyLog[2, E^((-I)*(c + d*x))] + PolyLog[3, E^((-I)*(c + d*x))]) + 3*b*f^3*(I*d^2*x^2*PolyLog[2, -E^((-I)*(c + d*x))] + 2*d*x*PolyLog[3, -E^((-I)*(c + d*x))] - (2*I)*PolyLog[4, -E^((-I)*(c + d*x))]) - (3*I)*b*f^3*(d^2*x^2*PolyLog[2, E^((-I)*(c + d*x))] - (2*I)*d*x*PolyLog[3, E^((-I)*(c + d*x))] - 2*PolyLog[4, E^((-I)*(c + d*x))]))/(a^2*d^4) + (Sqrt[-(a^2 - b^2)^2]*(-2*Sqrt[-a^2 + b^2]*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] - 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] + 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] + (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))]))/(a^2*b^2*d^4) + Csc[c]*Csc[c + d*x]*(Cos[c + d*x]/(16*a*b^2*d^4) - ((I/16)*Sin[c + d*x])/(a*b^2*d^4))*((8*I)*b^2*d^3*e^3*Cos[c] + (24*I)*b^2*d^3*e^2*f*x*Cos[c] + (24*I)*b^2*d^3*e*f^2*x^2*Cos[c] + (8*I)*b^2*d^3*f^3*x^3*Cos[c] - 2*a*b*d^3*e^3*Cos[d*x] + (18*I)*a*b*d^2*e^2*f*Cos[d*x] + 12*a*b*d*e*f^2*Cos[d*x] - (36*I)*a*b*f^3*Cos[d*x] - 6*a*b*d^3*e^2*f*x*Cos[d*x] + (36*I)*a*b*d^2*e*f^2*x*Cos[d*x] + 12*a*b*d*f^3*x*Cos[d*x] - 6*a*b*d^3*e*f^2*x^2*Cos[d*x] + (18*I)*a*b*d^2*f^3*x^2*Cos[d*x] - 2*a*b*d^3*f^3*x^3*Cos[d*x] + 2*a*b*d^3*e^3*Cos[2*c + d*x] - (18*I)*a*b*d^2*e^2*f*Cos[2*c + d*x] - 12*a*b*d*e*f^2*Cos[2*c + d*x] + (36*I)*a*b*f^3*Cos[2*c + d*x] + 6*a*b*d^3*e^2*f*x*Cos[2*c + d*x] - (36*I)*a*b*d^2*e*f^2*x*Cos[2*c + d*x] - 12*a*b*d*f^3*x*Cos[2*c + d*x] + 6*a*b*d^3*e*f^2*x^2*Cos[2*c + d*x] - (18*I)*a*b*d^2*f^3*x^2*Cos[2*c + d*x] + 2*a*b*d^3*f^3*x^3*Cos[2*c + d*x] - (8*I)*b^2*d^3*e^3*Cos[c + 2*d*x] - 4*a^2*d^4*e^3*x*Cos[c + 2*d*x] - (24*I)*b^2*d^3*e^2*f*x*Cos[c + 2*d*x] - 6*a^2*d^4*e^2*f*x^2*Cos[c + 2*d*x] - (24*I)*b^2*d^3*e*f^2*x^2*Cos[c + 2*d*x] - 4*a^2*d^4*e*f^2*x^3*Cos[c + 2*d*x] - (8*I)*b^2*d^3*f^3*x^3*Cos[c + 2*d*x] - a^2*d^4*f^3*x^4*Cos[c + 2*d*x] + 4*a^2*d^4*e^3*x*Cos[3*c + 2*d*x] + 6*a^2*d^4*e^2*f*x^2*Cos[3*c + 2*d*x] + 4*a^2*d^4*e*f^2*x^3*Cos[3*c + 2*d*x] + a^2*d^4*f^3*x^4*Cos[3*c + 2*d*x] - 2*a*b*d^3*e^3*Cos[2*c + 3*d*x] - (6*I)*a*b*d^2*e^2*f*Cos[2*c + 3*d*x] + 12*a*b*d*e*f^2*Cos[2*c + 3*d*x] + (12*I)*a*b*f^3*Cos[2*c + 3*d*x] - 6*a*b*d^3*e^2*f*x*Cos[2*c + 3*d*x] - (12*I)*a*b*d^2*e*f^2*x*Cos[2*c + 3*d*x] + 12*a*b*d*f^3*x*Cos[2*c + 3*d*x] - 6*a*b*d^3*e*f^2*x^2*Cos[2*c + 3*d*x] - (6*I)*a*b*d^2*f^3*x^2*Cos[2*c + 3*d*x] - 2*a*b*d^3*f^3*x^3*Cos[2*c + 3*d*x] + 2*a*b*d^3*e^3*Cos[4*c + 3*d*x] + (6*I)*a*b*d^2*e^2*f*Cos[4*c + 3*d*x] - 12*a*b*d*e*f^2*Cos[4*c + 3*d*x] - (12*I)*a*b*f^3*Cos[4*c + 3*d*x] + 6*a*b*d^3*e^2*f*x*Cos[4*c + 3*d*x] + (12*I)*a*b*d^2*e*f^2*x*Cos[4*c + 3*d*x] - 12*a*b*d*f^3*x*Cos[4*c + 3*d*x] + 6*a*b*d^3*e*f^2*x^2*Cos[4*c + 3*d*x] + (6*I)*a*b*d^2*f^3*x^2*Cos[4*c + 3*d*x] + 2*a*b*d^3*f^3*x^3*Cos[4*c + 3*d*x] - 8*b^2*d^3*e^3*Sin[c] - (8*I)*a^2*d^4*e^3*x*Sin[c] - 24*b^2*d^3*e^2*f*x*Sin[c] - (12*I)*a^2*d^4*e^2*f*x^2*Sin[c] - 24*b^2*d^3*e*f^2*x^2*Sin[c] - (8*I)*a^2*d^4*e*f^2*x^3*Sin[c] - 8*b^2*d^3*f^3*x^3*Sin[c] - (2*I)*a^2*d^4*f^3*x^4*Sin[c] + (2*I)*a*b*d^3*e^3*Sin[d*x] - 6*a*b*d^2*e^2*f*Sin[d*x] - (12*I)*a*b*d*e*f^2*Sin[d*x] + 12*a*b*f^3*Sin[d*x] + (6*I)*a*b*d^3*e^2*f*x*Sin[d*x] - 12*a*b*d^2*e*f^2*x*Sin[d*x] - (12*I)*a*b*d*f^3*x*Sin[d*x] + (6*I)*a*b*d^3*e*f^2*x^2*Sin[d*x] - 6*a*b*d^2*f^3*x^2*Sin[d*x] + (2*I)*a*b*d^3*f^3*x^3*Sin[d*x] - (2*I)*a*b*d^3*e^3*Sin[2*c + d*x] + 6*a*b*d^2*e^2*f*Sin[2*c + d*x] + (12*I)*a*b*d*e*f^2*Sin[2*c + d*x] - 12*a*b*f^3*Sin[2*c + d*x] - (6*I)*a*b*d^3*e^2*f*x*Sin[2*c + d*x] + 12*a*b*d^2*e*f^2*x*Sin[2*c + d*x] + (12*I)*a*b*d*f^3*x*Sin[2*c + d*x] - (6*I)*a*b*d^3*e*f^2*x^2*Sin[2*c + d*x] + 6*a*b*d^2*f^3*x^2*Sin[2*c + d*x] - (2*I)*a*b*d^3*f^3*x^3*Sin[2*c + d*x] + 8*b^2*d^3*e^3*Sin[c + 2*d*x] - (4*I)*a^2*d^4*e^3*x*Sin[c + 2*d*x] + 24*b^2*d^3*e^2*f*x*Sin[c + 2*d*x] - (6*I)*a^2*d^4*e^2*f*x^2*Sin[c + 2*d*x] + 24*b^2*d^3*e*f^2*x^2*Sin[c + 2*d*x] - (4*I)*a^2*d^4*e*f^2*x^3*Sin[c + 2*d*x] + 8*b^2*d^3*f^3*x^3*Sin[c + 2*d*x] - I*a^2*d^4*f^3*x^4*Sin[c + 2*d*x] + (4*I)*a^2*d^4*e^3*x*Sin[3*c + 2*d*x] + (6*I)*a^2*d^4*e^2*f*x^2*Sin[3*c + 2*d*x] + (4*I)*a^2*d^4*e*f^2*x^3*Sin[3*c + 2*d*x] + I*a^2*d^4*f^3*x^4*Sin[3*c + 2*d*x] - (2*I)*a*b*d^3*e^3*Sin[2*c + 3*d*x] + 6*a*b*d^2*e^2*f*Sin[2*c + 3*d*x] + (12*I)*a*b*d*e*f^2*Sin[2*c + 3*d*x] - 12*a*b*f^3*Sin[2*c + 3*d*x] - (6*I)*a*b*d^3*e^2*f*x*Sin[2*c + 3*d*x] + 12*a*b*d^2*e*f^2*x*Sin[2*c + 3*d*x] + (12*I)*a*b*d*f^3*x*Sin[2*c + 3*d*x] - (6*I)*a*b*d^3*e*f^2*x^2*Sin[2*c + 3*d*x] + 6*a*b*d^2*f^3*x^2*Sin[2*c + 3*d*x] - (2*I)*a*b*d^3*f^3*x^3*Sin[2*c + 3*d*x] + (2*I)*a*b*d^3*e^3*Sin[4*c + 3*d*x] - 6*a*b*d^2*e^2*f*Sin[4*c + 3*d*x] - (12*I)*a*b*d*e*f^2*Sin[4*c + 3*d*x] + 12*a*b*f^3*Sin[4*c + 3*d*x] + (6*I)*a*b*d^3*e^2*f*x*Sin[4*c + 3*d*x] - 12*a*b*d^2*e*f^2*x*Sin[4*c + 3*d*x] - (12*I)*a*b*d*f^3*x*Sin[4*c + 3*d*x] + (6*I)*a*b*d^3*e*f^2*x^2*Sin[4*c + 3*d*x] - 6*a*b*d^2*f^3*x^2*Sin[4*c + 3*d*x] + (2*I)*a*b*d^3*f^3*x^3*Sin[4*c + 3*d*x])","B",0
342,1,951,840,10.8508543,"\int \frac{(e+f x)^2 \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{12 \left(-b d^2 x^2 \log \left(1-e^{-i (c+d x)}\right) f^2+b d^2 x^2 \log \left(1+e^{-i (c+d x)}\right) f^2+2 b \left(i d x \text{Li}_2\left(-e^{-i (c+d x)}\right)+\text{Li}_3\left(-e^{-i (c+d x)}\right)\right) f^2-2 i b \left(d x \text{Li}_2\left(e^{-i (c+d x)}\right)-i \text{Li}_3\left(e^{-i (c+d x)}\right)\right) f^2-2 d (b d e-a f) x \log \left(1-e^{-i (c+d x)}\right) f+2 d (b d e+a f) x \log \left(1+e^{-i (c+d x)}\right) f+2 i (b d e+a f) \text{Li}_2\left(-e^{-i (c+d x)}\right) f+2 i (a f-b d e) \text{Li}_2\left(e^{-i (c+d x)}\right) f-\frac{2 i a d^2 (e+f x)^2}{-1+e^{2 i c}}+i d e (b d e-2 a f) \left(d x+i \log \left(1-e^{i (c+d x)}\right)\right)+d e (b d e+2 a f) \left(\log \left(1+e^{i (c+d x)}\right)-i d x\right)\right)-\frac{12 i \sqrt{-\left(a^2-b^2\right)^2} \left(-2 \sqrt{a^2-b^2} d f (e+f x) \text{Li}_2\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)+2 \sqrt{a^2-b^2} d f (e+f x) \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)-i \left(\left(2 \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right) e^2+\sqrt{a^2-b^2} f x (2 e+f x) \left(\log \left(1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-\log \left(\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right)\right)\right) d^2+2 \sqrt{a^2-b^2} f^2 \text{Li}_3\left(\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right)-2 \sqrt{a^2-b^2} f^2 \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right)\right)\right)}{b^2}+\frac{a \csc (c) \csc (c+d x) \left(-2 a^2 x \left(3 e^2+3 f x e+f^2 x^2\right) \cos (d x) d^3+2 a^2 x \left(3 e^2+3 f x e+f^2 x^2\right) \cos (2 c+d x) d^3+3 b \left(-a \left(d^2 (e+f x)^2-2 f^2\right) \cos (c+2 d x)+a \left(d^2 (e+f x)^2-2 f^2\right) \cos (3 c+2 d x)+2 d (e+f x) \left(4 a f \sin (c) \sin ^2(c+d x)+2 b d (e+f x) \sin (d x)\right)\right)\right)}{b^2}}{12 a^2 d^3}","-\frac{\left(a^2-b^2\right) (e+f x)^3}{3 a b^2 f}-\frac{(e+f x)^3}{3 a f}+\frac{2 b \tanh ^{-1}\left(e^{i (c+d x)}\right) (e+f x)^2}{a^2 d}-\frac{b \cos (c+d x) (e+f x)^2}{a^2 d}-\frac{\left(a^2-b^2\right) \cos (c+d x) (e+f x)^2}{a^2 b d}-\frac{\cot (c+d x) (e+f x)^2}{a d}-\frac{i \left(a^2-b^2\right)^{3/2} \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^2}{a^2 b^2 d}+\frac{i \left(a^2-b^2\right)^{3/2} \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^2}{a^2 b^2 d}-\frac{i (e+f x)^2}{a d}+\frac{2 f \log \left(1-e^{2 i (c+d x)}\right) (e+f x)}{a d^2}-\frac{2 i b f \text{Li}_2\left(-e^{i (c+d x)}\right) (e+f x)}{a^2 d^2}+\frac{2 i b f \text{Li}_2\left(e^{i (c+d x)}\right) (e+f x)}{a^2 d^2}-\frac{2 \left(a^2-b^2\right)^{3/2} f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)}{a^2 b^2 d^2}+\frac{2 \left(a^2-b^2\right)^{3/2} f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)}{a^2 b^2 d^2}+\frac{2 b f \sin (c+d x) (e+f x)}{a^2 d^2}+\frac{2 \left(a^2-b^2\right) f \sin (c+d x) (e+f x)}{a^2 b d^2}+\frac{2 b f^2 \cos (c+d x)}{a^2 d^3}+\frac{2 \left(a^2-b^2\right) f^2 \cos (c+d x)}{a^2 b d^3}-\frac{i f^2 \text{Li}_2\left(e^{2 i (c+d x)}\right)}{a d^3}+\frac{2 b f^2 \text{Li}_3\left(-e^{i (c+d x)}\right)}{a^2 d^3}-\frac{2 b f^2 \text{Li}_3\left(e^{i (c+d x)}\right)}{a^2 d^3}-\frac{2 i \left(a^2-b^2\right)^{3/2} f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b^2 d^3}+\frac{2 i \left(a^2-b^2\right)^{3/2} f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 b^2 d^3}",1,"(12*(((-2*I)*a*d^2*(e + f*x)^2)/(-1 + E^((2*I)*c)) - 2*d*f*(b*d*e - a*f)*x*Log[1 - E^((-I)*(c + d*x))] - b*d^2*f^2*x^2*Log[1 - E^((-I)*(c + d*x))] + 2*d*f*(b*d*e + a*f)*x*Log[1 + E^((-I)*(c + d*x))] + b*d^2*f^2*x^2*Log[1 + E^((-I)*(c + d*x))] + I*d*e*(b*d*e - 2*a*f)*(d*x + I*Log[1 - E^(I*(c + d*x))]) + d*e*(b*d*e + 2*a*f)*((-I)*d*x + Log[1 + E^(I*(c + d*x))]) + (2*I)*f*(b*d*e + a*f)*PolyLog[2, -E^((-I)*(c + d*x))] + (2*I)*f*(-(b*d*e) + a*f)*PolyLog[2, E^((-I)*(c + d*x))] + 2*b*f^2*(I*d*x*PolyLog[2, -E^((-I)*(c + d*x))] + PolyLog[3, -E^((-I)*(c + d*x))]) - (2*I)*b*f^2*(d*x*PolyLog[2, E^((-I)*(c + d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))])) - ((12*I)*Sqrt[-(a^2 - b^2)^2]*(-2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - I*(d^2*(2*Sqrt[-a^2 + b^2]*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(2*e + f*x)*(Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])])) + 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))])))/b^2 + (a*Csc[c]*Csc[c + d*x]*(-2*a^2*d^3*x*(3*e^2 + 3*e*f*x + f^2*x^2)*Cos[d*x] + 2*a^2*d^3*x*(3*e^2 + 3*e*f*x + f^2*x^2)*Cos[2*c + d*x] + 3*b*(-(a*(-2*f^2 + d^2*(e + f*x)^2)*Cos[c + 2*d*x]) + a*(-2*f^2 + d^2*(e + f*x)^2)*Cos[3*c + 2*d*x] + 2*d*(e + f*x)*(2*b*d*(e + f*x)*Sin[d*x] + 4*a*f*Sin[c]*Sin[c + d*x]^2))))/b^2)/(12*a^2*d^3)","A",0
343,1,1019,517,11.8860775,"\int \frac{(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\frac{(d e+d f x) \left(\frac{2 (d e-c f) \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{i f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{-i a+b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a+i \left(b+\sqrt{b^2-a^2}\right)}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{b^2-a^2}}{i a+b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)}{a-i \left(b+\sqrt{b^2-a^2}\right)}\right)\right)}{\sqrt{b^2-a^2}}+\frac{i f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(-\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)-\sqrt{b^2-a^2}}{i a-b+\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{a \left(\tan \left(\frac{1}{2} (c+d x)\right)+i\right)}{i a-b+\sqrt{b^2-a^2}}\right)\right)}{\sqrt{b^2-a^2}}-\frac{i f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{b+a \tan \left(\frac{1}{2} (c+d x)\right)-\sqrt{b^2-a^2}}{i a+b-\sqrt{b^2-a^2}}\right)+\text{Li}_2\left(\frac{i \tan \left(\frac{1}{2} (c+d x)\right) a+a}{a+i \left(\sqrt{b^2-a^2}-b\right)}\right)\right)}{\sqrt{b^2-a^2}}\right) \left(a^2-b^2\right)^2}{a^2 b^2 d^2 \left(d e-c f+i f \log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)-i f \log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)\right)}-\frac{a (c+d x) (2 d e-2 c f+f (c+d x))}{2 b^2 d^2}-\frac{(d e-c f+f (c+d x)) \cos (c+d x)}{b d^2}+\frac{\left(-d e \cos \left(\frac{1}{2} (c+d x)\right)+c f \cos \left(\frac{1}{2} (c+d x)\right)-f (c+d x) \cos \left(\frac{1}{2} (c+d x)\right)\right) \csc \left(\frac{1}{2} (c+d x)\right)}{2 a d^2}+\frac{f \log (\sin (c+d x))}{a d^2}-\frac{b e \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)}{a^2 d}+\frac{b c f \log \left(\tan \left(\frac{1}{2} (c+d x)\right)\right)}{a^2 d^2}-\frac{b f \left((c+d x) \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)\right)+i \left(\text{Li}_2\left(-e^{i (c+d x)}\right)-\text{Li}_2\left(e^{i (c+d x)}\right)\right)\right)}{a^2 d^2}+\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(d e \sin \left(\frac{1}{2} (c+d x)\right)-c f \sin \left(\frac{1}{2} (c+d x)\right)+f (c+d x) \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a d^2}+\frac{f \sin (c+d x)}{b d^2}","-\frac{f \left(a^2-b^2\right)^{3/2} \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b^2 d^2}+\frac{f \left(a^2-b^2\right)^{3/2} \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 b^2 d^2}+\frac{f \left(a^2-b^2\right) \sin (c+d x)}{a^2 b d^2}-\frac{i \left(a^2-b^2\right)^{3/2} (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b^2 d}+\frac{i \left(a^2-b^2\right)^{3/2} (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a^2 b^2 d}-\frac{\left(a^2-b^2\right) (e+f x) \cos (c+d x)}{a^2 b d}+\frac{e x \left(1-\frac{a^2}{b^2}\right)}{a}+\frac{f x^2 \left(1-\frac{a^2}{b^2}\right)}{2 a}-\frac{i b f \text{Li}_2\left(-e^{i (c+d x)}\right)}{a^2 d^2}+\frac{i b f \text{Li}_2\left(e^{i (c+d x)}\right)}{a^2 d^2}+\frac{b f \sin (c+d x)}{a^2 d^2}-\frac{b (e+f x) \cos (c+d x)}{a^2 d}+\frac{2 b (e+f x) \tanh ^{-1}\left(e^{i (c+d x)}\right)}{a^2 d}+\frac{f \log (\sin (c+d x))}{a d^2}-\frac{(e+f x) \cot (c+d x)}{a d}-\frac{e x}{a}-\frac{f x^2}{2 a}",1,"-1/2*(a*(c + d*x)*(2*d*e - 2*c*f + f*(c + d*x)))/(b^2*d^2) - ((d*e - c*f + f*(c + d*x))*Cos[c + d*x])/(b*d^2) + ((-(d*e*Cos[(c + d*x)/2]) + c*f*Cos[(c + d*x)/2] - f*(c + d*x)*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(2*a*d^2) + (f*Log[Sin[c + d*x]])/(a*d^2) - (b*e*Log[Tan[(c + d*x)/2]])/(a^2*d) + (b*c*f*Log[Tan[(c + d*x)/2]])/(a^2*d^2) - (b*f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/(a^2*d^2) + ((a^2 - b^2)^2*(d*e + d*f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] - (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2]))/(a^2*b^2*d^2*(d*e - c*f + I*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1 + I*Tan[(c + d*x)/2]])) + (Sec[(c + d*x)/2]*(d*e*Sin[(c + d*x)/2] - c*f*Sin[(c + d*x)/2] + f*(c + d*x)*Sin[(c + d*x)/2]))/(2*a*d^2) + (f*Sin[c + d*x])/(b*d^2)","A",0
344,1,146,104,0.8226022,"\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
345,1,3944,1432,49.2265713,"\int \frac{(e+f x)^3 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^3*Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","-\frac{i \left(a^2-b^2\right)^2 (e+f x)^4}{4 a^2 b^3 f}+\frac{i b (e+f x)^4}{4 a^2 f}+\frac{b \sin ^2(c+d x) (e+f x)^3}{2 a^2 d}+\frac{\left(a^2-b^2\right) \sin ^2(c+d x) (e+f x)^3}{2 a^2 b d}-\frac{\csc (c+d x) (e+f x)^3}{a d}+\frac{\left(a^2-b^2\right)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^3}{a^2 b^3 d}+\frac{\left(a^2-b^2\right)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^3}{a^2 b^3 d}-\frac{b \log \left(1-e^{2 i (c+d x)}\right) (e+f x)^3}{a^2 d}-\frac{\left(a^2-b^2\right) \sin (c+d x) (e+f x)^3}{a b^2 d}-\frac{\sin (c+d x) (e+f x)^3}{a d}-\frac{b (e+f x)^3}{4 a^2 d}-\frac{\left(a^2-b^2\right) (e+f x)^3}{4 a^2 b d}-\frac{6 f \tanh ^{-1}\left(e^{i (c+d x)}\right) (e+f x)^2}{a d^2}-\frac{3 \left(a^2-b^2\right) f \cos (c+d x) (e+f x)^2}{a b^2 d^2}-\frac{3 f \cos (c+d x) (e+f x)^2}{a d^2}-\frac{3 i \left(a^2-b^2\right)^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^2}{a^2 b^3 d^2}-\frac{3 i \left(a^2-b^2\right)^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^2}{a^2 b^3 d^2}+\frac{3 i b f \text{Li}_2\left(e^{2 i (c+d x)}\right) (e+f x)^2}{2 a^2 d^2}+\frac{3 b f \cos (c+d x) \sin (c+d x) (e+f x)^2}{4 a^2 d^2}+\frac{3 \left(a^2-b^2\right) f \cos (c+d x) \sin (c+d x) (e+f x)^2}{4 a^2 b d^2}-\frac{3 b f^2 \sin ^2(c+d x) (e+f x)}{4 a^2 d^3}-\frac{3 \left(a^2-b^2\right) f^2 \sin ^2(c+d x) (e+f x)}{4 a^2 b d^3}+\frac{6 i f^2 \text{Li}_2\left(-e^{i (c+d x)}\right) (e+f x)}{a d^3}-\frac{6 i f^2 \text{Li}_2\left(e^{i (c+d x)}\right) (e+f x)}{a d^3}+\frac{6 \left(a^2-b^2\right)^2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)}{a^2 b^3 d^3}+\frac{6 \left(a^2-b^2\right)^2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)}{a^2 b^3 d^3}-\frac{3 b f^2 \text{Li}_3\left(e^{2 i (c+d x)}\right) (e+f x)}{2 a^2 d^3}+\frac{6 \left(a^2-b^2\right) f^2 \sin (c+d x) (e+f x)}{a b^2 d^3}+\frac{6 f^2 \sin (c+d x) (e+f x)}{a d^3}+\frac{3 b f^3 x}{8 a^2 d^3}+\frac{3 \left(a^2-b^2\right) f^3 x}{8 a^2 b d^3}+\frac{6 \left(a^2-b^2\right) f^3 \cos (c+d x)}{a b^2 d^4}+\frac{6 f^3 \cos (c+d x)}{a d^4}-\frac{6 f^3 \text{Li}_3\left(-e^{i (c+d x)}\right)}{a d^4}+\frac{6 f^3 \text{Li}_3\left(e^{i (c+d x)}\right)}{a d^4}+\frac{6 i \left(a^2-b^2\right)^2 f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b^3 d^4}+\frac{6 i \left(a^2-b^2\right)^2 f^3 \text{Li}_4\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 b^3 d^4}-\frac{3 i b f^3 \text{Li}_4\left(e^{2 i (c+d x)}\right)}{4 a^2 d^4}-\frac{3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac{3 \left(a^2-b^2\right) f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 b d^4}",1,"((-e^3 - 3*e^2*f*x - 3*e*f^2*x^2 - f^3*x^3)*Csc[c + d*x])/(a*d) - (((-I)*b*(e + f*x)^4)/((-1 + E^((2*I)*c))*f) + (6*e*f*(b*d*e - 2*a*f)*x*Log[1 - E^((-I)*(c + d*x))])/d^2 + (6*f^2*(b*d*e - a*f)*x^2*Log[1 - E^((-I)*(c + d*x))])/d^2 + (2*b*f^3*x^3*Log[1 - E^((-I)*(c + d*x))])/d + (6*e*f*(b*d*e + 2*a*f)*x*Log[1 + E^((-I)*(c + d*x))])/d^2 + (6*f^2*(b*d*e + a*f)*x^2*Log[1 + E^((-I)*(c + d*x))])/d^2 + (2*b*f^3*x^3*Log[1 + E^((-I)*(c + d*x))])/d + (2*e^2*(b*d*e - 3*a*f)*((-I)*d*x + Log[1 - E^(I*(c + d*x))]))/d^2 + (2*e^2*(b*d*e + 3*a*f)*((-I)*d*x + Log[1 + E^(I*(c + d*x))]))/d^2 + ((6*I)*e*f*(b*d*e + 2*a*f)*PolyLog[2, -E^((-I)*(c + d*x))])/d^3 + ((6*I)*e*f*(b*d*e - 2*a*f)*PolyLog[2, E^((-I)*(c + d*x))])/d^3 + (12*f^2*(b*d*e + a*f)*(I*d*x*PolyLog[2, -E^((-I)*(c + d*x))] + PolyLog[3, -E^((-I)*(c + d*x))]))/d^4 + (12*f^2*(b*d*e - a*f)*(I*d*x*PolyLog[2, E^((-I)*(c + d*x))] + PolyLog[3, E^((-I)*(c + d*x))]))/d^4 + (6*b*f^3*(I*d^2*x^2*PolyLog[2, -E^((-I)*(c + d*x))] + 2*d*x*PolyLog[3, -E^((-I)*(c + d*x))] - (2*I)*PolyLog[4, -E^((-I)*(c + d*x))]))/d^4 + (6*b*f^3*(I*d^2*x^2*PolyLog[2, E^((-I)*(c + d*x))] + 2*d*x*PolyLog[3, E^((-I)*(c + d*x))] - (2*I)*PolyLog[4, E^((-I)*(c + d*x))]))/d^4)/(2*a^2) + ((a^2 - b^2)^2*((-4*I)*d^4*e^3*E^((2*I)*c)*x - (6*I)*d^4*e^2*E^((2*I)*c)*f*x^2 - (4*I)*d^4*e*E^((2*I)*c)*f^2*x^3 - I*d^4*E^((2*I)*c)*f^3*x^4 - (2*I)*d^3*e^3*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + (2*I)*d^3*e^3*E^((2*I)*c)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] - d^3*e^3*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] + d^3*e^3*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2] - 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*e*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 12*d*e*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (12*I)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (12*I)*E^((2*I)*c)*f^3*PolyLog[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12*I)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (12*I)*E^((2*I)*c)*f^3*PolyLog[4, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(2*a^2*b^3*d^4*(-1 + E^((2*I)*c))) - (I*(-a^2 + 2*b^2)*e^3*x*(1 + Cos[2*c] + I*Sin[2*c]))/(b^3*(-1 + Cos[2*c] + I*Sin[2*c])) - (((3*I)/2)*(-a^2 + 2*b^2)*e^2*f*x^2*(1 + Cos[2*c] + I*Sin[2*c]))/(b^3*(-1 + Cos[2*c] + I*Sin[2*c])) - (I*(-a^2 + 2*b^2)*e*f^2*x^3*(1 + Cos[2*c] + I*Sin[2*c]))/(b^3*(-1 + Cos[2*c] + I*Sin[2*c])) - ((I/4)*(-a^2 + 2*b^2)*f^3*x^4*(1 + Cos[2*c] + I*Sin[2*c]))/(b^3*(-1 + Cos[2*c] + I*Sin[2*c])) + (((-1/2*I)*a*f^3*x^3*Cos[c])/(b^2*d) - (a*f^3*x^3*Sin[c])/(2*b^2*d) + ((-I)*d^3*e^3 - 3*d^2*e^2*f + (6*I)*d*e*f^2 + 6*f^3)*((a*Cos[c])/(2*b^2*d^4) - ((I/2)*a*Sin[c])/(b^2*d^4)) + (a*d^2*e^2*f - (2*I)*a*d*e*f^2 - 2*a*f^3)*((((-3*I)/2)*x*Cos[c])/(b^2*d^3) - (3*x*Sin[c])/(2*b^2*d^3)) + (a*d*e*f^2 - I*a*f^3)*((((-3*I)/2)*x^2*Cos[c])/(b^2*d^2) - (3*x^2*Sin[c])/(2*b^2*d^2)))*(Cos[d*x] - I*Sin[d*x]) + (((I/2)*a*f^3*x^3*Cos[c])/(b^2*d) - (a*f^3*x^3*Sin[c])/(2*b^2*d) + (I*d^3*e^3 - 3*d^2*e^2*f - (6*I)*d*e*f^2 + 6*f^3)*((a*Cos[c])/(2*b^2*d^4) + ((I/2)*a*Sin[c])/(b^2*d^4)) + (((3*I)/2)*x^2*(a*d*e*f^2*Cos[c] + I*a*f^3*Cos[c] + I*a*d*e*f^2*Sin[c] - a*f^3*Sin[c]))/(b^2*d^2) + (((3*I)/2)*x*(a*d^2*e^2*f*Cos[c] + (2*I)*a*d*e*f^2*Cos[c] - 2*a*f^3*Cos[c] + I*a*d^2*e^2*f*Sin[c] - 2*a*d*e*f^2*Sin[c] - (2*I)*a*f^3*Sin[c]))/(b^2*d^3))*(Cos[d*x] + I*Sin[d*x]) + (-1/8*(f^3*x^3*Cos[2*c])/(b*d) + ((I/8)*f^3*x^3*Sin[2*c])/(b*d) + (4*d^3*e^3 - (6*I)*d^2*e^2*f - 6*d*e*f^2 + (3*I)*f^3)*(-1/32*Cos[2*c]/(b*d^4) + ((I/32)*Sin[2*c])/(b*d^4)) + ((2*I)*d^2*e^2*f + 2*d*e*f^2 - I*f^3)*((((3*I)/16)*x*Cos[2*c])/(b*d^3) + (3*x*Sin[2*c])/(16*b*d^3)) + ((2*I)*d*e*f^2 + f^3)*((((3*I)/16)*x^2*Cos[2*c])/(b*d^2) + (3*x^2*Sin[2*c])/(16*b*d^2)))*(Cos[2*d*x] - I*Sin[2*d*x]) + (-1/8*(f^3*x^3*Cos[2*c])/(b*d) - ((I/8)*f^3*x^3*Sin[2*c])/(b*d) + (4*d^3*e^3 + (6*I)*d^2*e^2*f - 6*d*e*f^2 - (3*I)*f^3)*(-1/32*Cos[2*c]/(b*d^4) - ((I/32)*Sin[2*c])/(b*d^4)) - (((3*I)/16)*x*((-2*I)*d^2*e^2*f*Cos[2*c] + 2*d*e*f^2*Cos[2*c] + I*f^3*Cos[2*c] + 2*d^2*e^2*f*Sin[2*c] + (2*I)*d*e*f^2*Sin[2*c] - f^3*Sin[2*c]))/(b*d^3) - (((3*I)/16)*x^2*((-2*I)*d*e*f^2*Cos[2*c] + f^3*Cos[2*c] + 2*d*e*f^2*Sin[2*c] + I*f^3*Sin[2*c]))/(b*d^2))*(Cos[2*d*x] + I*Sin[2*d*x])","B",1
346,1,5156,1051,13.9792738,"\int \frac{(e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)^2*Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","-\frac{i \left(a^2-b^2\right)^2 (e+f x)^3}{3 a^2 b^3 f}+\frac{i b (e+f x)^3}{3 a^2 f}+\frac{b \sin ^2(c+d x) (e+f x)^2}{2 a^2 d}+\frac{\left(a^2-b^2\right) \sin ^2(c+d x) (e+f x)^2}{2 a^2 b d}-\frac{\csc (c+d x) (e+f x)^2}{a d}+\frac{\left(a^2-b^2\right)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)^2}{a^2 b^3 d}+\frac{\left(a^2-b^2\right)^2 \log \left(1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)^2}{a^2 b^3 d}-\frac{b \log \left(1-e^{2 i (c+d x)}\right) (e+f x)^2}{a^2 d}-\frac{\left(a^2-b^2\right) \sin (c+d x) (e+f x)^2}{a b^2 d}-\frac{\sin (c+d x) (e+f x)^2}{a d}-\frac{4 f \tanh ^{-1}\left(e^{i (c+d x)}\right) (e+f x)}{a d^2}-\frac{2 \left(a^2-b^2\right) f \cos (c+d x) (e+f x)}{a b^2 d^2}-\frac{2 f \cos (c+d x) (e+f x)}{a d^2}-\frac{2 i \left(a^2-b^2\right)^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right) (e+f x)}{a^2 b^3 d^2}-\frac{2 i \left(a^2-b^2\right)^2 f \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right) (e+f x)}{a^2 b^3 d^2}+\frac{i b f \text{Li}_2\left(e^{2 i (c+d x)}\right) (e+f x)}{a^2 d^2}+\frac{b f \cos (c+d x) \sin (c+d x) (e+f x)}{2 a^2 d^2}+\frac{\left(a^2-b^2\right) f \cos (c+d x) \sin (c+d x) (e+f x)}{2 a^2 b d^2}-\frac{b f^2 x^2}{4 a^2 d}-\frac{\left(a^2-b^2\right) f^2 x^2}{4 a^2 b d}-\frac{b f^2 \sin ^2(c+d x)}{4 a^2 d^3}-\frac{\left(a^2-b^2\right) f^2 \sin ^2(c+d x)}{4 a^2 b d^3}-\frac{b e f x}{2 a^2 d}-\frac{\left(a^2-b^2\right) e f x}{2 a^2 b d}+\frac{2 i f^2 \text{Li}_2\left(-e^{i (c+d x)}\right)}{a d^3}-\frac{2 i f^2 \text{Li}_2\left(e^{i (c+d x)}\right)}{a d^3}+\frac{2 \left(a^2-b^2\right)^2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b^3 d^3}+\frac{2 \left(a^2-b^2\right)^2 f^2 \text{Li}_3\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 b^3 d^3}-\frac{b f^2 \text{Li}_3\left(e^{2 i (c+d x)}\right)}{2 a^2 d^3}+\frac{2 \left(a^2-b^2\right) f^2 \sin (c+d x)}{a b^2 d^3}+\frac{2 f^2 \sin (c+d x)}{a d^3}",1,"Result too large to show","B",1
347,1,2504,641,15.4101253,"\int \frac{(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[((e + f*x)*Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","-\frac{f \left(a^2-b^2\right) \cos (c+d x)}{a b^2 d^2}+\frac{f \left(a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{4 a^2 b d^2}+\frac{\left(a^2-b^2\right) (e+f x) \sin ^2(c+d x)}{2 a^2 b d}-\frac{\left(a^2-b^2\right) (e+f x) \sin (c+d x)}{a b^2 d}-\frac{f x \left(a^2-b^2\right)}{4 a^2 b d}-\frac{i f \left(a^2-b^2\right)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b^3 d^2}-\frac{i f \left(a^2-b^2\right)^2 \text{Li}_2\left(\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{a^2 b^3 d^2}+\frac{\left(a^2-b^2\right)^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{a^2 b^3 d}+\frac{\left(a^2-b^2\right)^2 (e+f x) \log \left(1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{a^2 b^3 d}-\frac{i \left(a^2-b^2\right)^2 (e+f x)^2}{2 a^2 b^3 f}+\frac{i b f \text{Li}_2\left(e^{2 i (c+d x)}\right)}{2 a^2 d^2}+\frac{b f \sin (c+d x) \cos (c+d x)}{4 a^2 d^2}-\frac{b (e+f x) \log \left(1-e^{2 i (c+d x)}\right)}{a^2 d}+\frac{b (e+f x) \sin ^2(c+d x)}{2 a^2 d}-\frac{b f x}{4 a^2 d}+\frac{i b (e+f x)^2}{2 a^2 f}-\frac{f \cos (c+d x)}{a d^2}-\frac{f \tanh ^{-1}(\cos (c+d x))}{a d^2}-\frac{(e+f x) \sin (c+d x)}{a d}-\frac{(e+f x) \csc (c+d x)}{a d}",1,"-((a*f*Cos[c + d*x])/(b^2*d^2)) - ((d*e - c*f + f*(c + d*x))*Cos[2*(c + d*x)])/(4*b*d^2) + ((-(d*e*Cos[(c + d*x)/2]) + c*f*Cos[(c + d*x)/2] - f*(c + d*x)*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(2*a*d^2) - (b*e*Log[Sin[c + d*x]])/(a^2*d) + (b*c*f*Log[Sin[c + d*x]])/(a^2*d^2) + (f*Log[Tan[(c + d*x)/2]])/(a*d^2) - (b*f*((c + d*x)*Log[1 - E^((2*I)*(c + d*x))] - (I/2)*((c + d*x)^2 + PolyLog[2, E^((2*I)*(c + d*x))])))/(a^2*d^2) + (Sec[(c + d*x)/2]*(-(d*e*Sin[(c + d*x)/2]) + c*f*Sin[(c + d*x)/2] - f*(c + d*x)*Sin[(c + d*x)/2]))/(2*a*d^2) - (a*(d*e - c*f + f*(c + d*x))*Sin[c + d*x])/(b^2*d^2) + (f*Sin[2*(c + d*x)])/(8*b*d^2) + ((f*(c + d*x)^2 + (2*I)*d*e*Log[Sec[(c + d*x)/2]^2] - (2*I)*c*f*Log[Sec[(c + d*x)/2]^2] - (2*I)*d*e*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] + (2*I)*c*f*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] - (4*I)*f*(c + d*x)*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])] + 4*f*PolyLog[2, -Cos[c + d*x] + I*Sin[c + d*x]] + 2*f*PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))] - 2*f*PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))] + 2*f*PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])] - 2*f*PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))])*((-2*e*Cos[c + d*x])/(a + b*Sin[c + d*x]) + (a^2*e*Cos[c + d*x])/(b^2*(a + b*Sin[c + d*x])) + (b^2*e*Cos[c + d*x])/(a^2*(a + b*Sin[c + d*x])) + (2*c*f*Cos[c + d*x])/(d*(a + b*Sin[c + d*x])) - (a^2*c*f*Cos[c + d*x])/(b^2*d*(a + b*Sin[c + d*x])) - (b^2*c*f*Cos[c + d*x])/(a^2*d*(a + b*Sin[c + d*x])) - (2*f*(c + d*x)*Cos[c + d*x])/(d*(a + b*Sin[c + d*x])) + (a^2*f*(c + d*x)*Cos[c + d*x])/(b^2*d*(a + b*Sin[c + d*x])) + (b^2*f*(c + d*x)*Cos[c + d*x])/(a^2*d*(a + b*Sin[c + d*x]))))/(d*(2*f*(c + d*x) - (4*I)*f*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - (4*f*Log[1 + Cos[c + d*x] - I*Sin[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x]))/(-Cos[c + d*x] + I*Sin[c + d*x]) + (I*f*Log[1 - (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) + (I*f*Log[1 - (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - (I*f*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + (2*I)*d*e*Tan[(c + d*x)/2] - (2*I)*c*f*Tan[(c + d*x)/2] + ((2*I)*f*(c + d*x)*Sec[(c + d*x)/2]^2)/(-I + Tan[(c + d*x)/2]) - (f*Log[1 - (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(I + Tan[(c + d*x)/2]) + (I*a*f*Log[1 - (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(a + I*a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - ((2*I)*d*e*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x]) + ((2*I)*c*f*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x])))","B",0
348,1,86,96,0.2078395,"\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