1,1,76,91,0.3458828,"\int (c+d x)^4 \cos (a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x],x]","\frac{4 b d (c+d x) \cos (a+b x) \left(b^2 (c+d x)^2-6 d^2\right)+\sin (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 \sin (a+b x)}{b^5}-\frac{24 d^3 (c+d x) \cos (a+b x)}{b^4}-\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}+\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}+\frac{(c+d x)^4 \sin (a+b x)}{b}",1,"(4*b*d*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + (24*d^4 - 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Sin[a + b*x])/b^5","A",1
2,1,61,70,0.2314359,"\int (c+d x)^3 \cos (a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x],x]","\frac{b (c+d x) \sin (a+b x) \left(b^2 (c+d x)^2-6 d^2\right)+3 d \cos (a+b x) \left(b^2 (c+d x)^2-2 d^2\right)}{b^4}","-\frac{6 d^3 \cos (a+b x)}{b^4}-\frac{6 d^2 (c+d x) \sin (a+b x)}{b^3}+\frac{3 d (c+d x)^2 \cos (a+b x)}{b^2}+\frac{(c+d x)^3 \sin (a+b x)}{b}",1,"(3*d*(-2*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + b*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Sin[a + b*x])/b^4","A",1
3,1,44,49,0.1771424,"\int (c+d x)^2 \cos (a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x],x]","\frac{\sin (a+b x) \left(b^2 (c+d x)^2-2 d^2\right)+2 b d (c+d x) \cos (a+b x)}{b^3}","-\frac{2 d^2 \sin (a+b x)}{b^3}+\frac{2 d (c+d x) \cos (a+b x)}{b^2}+\frac{(c+d x)^2 \sin (a+b x)}{b}",1,"(2*b*d*(c + d*x)*Cos[a + b*x] + (-2*d^2 + b^2*(c + d*x)^2)*Sin[a + b*x])/b^3","A",1
4,1,26,27,0.0608944,"\int (c+d x) \cos (a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x],x]","\frac{b (c+d x) \sin (a+b x)+d \cos (a+b x)}{b^2}","\frac{d \cos (a+b x)}{b^2}+\frac{(c+d x) \sin (a+b x)}{b}",1,"(d*Cos[a + b*x] + b*(c + d*x)*Sin[a + b*x])/b^2","A",1
5,1,50,52,0.1009762,"\int \frac{\cos (a+b x)}{c+d x} \, dx","Integrate[Cos[a + b*x]/(c + d*x),x]","\frac{\cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)-\sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d}","\frac{\cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d}-\frac{\sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d}",1,"(Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x] - Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d","A",1
6,1,65,73,0.402281,"\int \frac{\cos (a+b x)}{(c+d x)^2} \, dx","Integrate[Cos[a + b*x]/(c + d*x)^2,x]","-\frac{b \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+b \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+\frac{d \cos (a+b x)}{c+d x}}{d^2}","-\frac{b \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d^2}-\frac{b \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d^2}-\frac{\cos (a+b x)}{d (c+d x)}",1,"-(((d*Cos[a + b*x])/(c + d*x) + b*CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + b*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)])/d^2)","A",1
7,1,89,104,0.6134029,"\int \frac{\cos (a+b x)}{(c+d x)^3} \, dx","Integrate[Cos[a + b*x]/(c + d*x)^3,x]","\frac{b^2 \left(-\cos \left(a-\frac{b c}{d}\right)\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+b^2 \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+\frac{d (b (c+d x) \sin (a+b x)-d \cos (a+b x))}{(c+d x)^2}}{2 d^3}","-\frac{b^2 \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{2 d^3}+\frac{b^2 \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{2 d^3}+\frac{b \sin (a+b x)}{2 d^2 (c+d x)}-\frac{\cos (a+b x)}{2 d (c+d x)^2}",1,"(-(b^2*Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)]) + (d*(-(d*Cos[a + b*x]) + b*(c + d*x)*Sin[a + b*x]))/(c + d*x)^2 + b^2*Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)])/(2*d^3)","A",1
8,1,144,127,0.5667661,"\int \frac{\cos (a+b x)}{(c+d x)^4} \, dx","Integrate[Cos[a + b*x]/(c + d*x)^4,x]","\frac{b^3 (c+d x)^3 \left(\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)\right)+d \cos (b x) \left(\cos (a) \left(b^2 (c+d x)^2-2 d^2\right)+b d \sin (a) (c+d x)\right)+d \sin (b x) \left(b d \cos (a) (c+d x)-\sin (a) \left(b^2 (c+d x)^2-2 d^2\right)\right)}{6 d^4 (c+d x)^3}","\frac{b^3 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{6 d^4}+\frac{b^3 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{6 d^4}+\frac{b^2 \cos (a+b x)}{6 d^3 (c+d x)}+\frac{b \sin (a+b x)}{6 d^2 (c+d x)^2}-\frac{\cos (a+b x)}{3 d (c+d x)^3}",1,"(d*Cos[b*x]*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] + b*d*(c + d*x)*Sin[a]) + d*(b*d*(c + d*x)*Cos[a] - (-2*d^2 + b^2*(c + d*x)^2)*Sin[a])*Sin[b*x] + b^3*(c + d*x)^3*(CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)]))/(6*d^4*(c + d*x)^3)","A",1
9,1,132,161,0.6798937,"\int (c+d x)^4 \cos ^2(a+b x) \, dx","Integrate[(c + d*x)^4*Cos[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) \cos ^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 \cos ^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
10,1,106,134,0.4452582,"\int (c+d x)^3 \cos ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Cos[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 \cos ^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 \cos ^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
11,1,77,95,0.3058063,"\int (c+d x)^2 \cos ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Cos[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) \cos ^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
12,1,50,55,0.1629207,"\int (c+d x) \cos ^2(a+b x) \, dx","Integrate[(c + d*x)*Cos[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 \cos ^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
13,1,65,78,0.1310059,"\int \frac{\cos ^2(a+b x)}{c+d x} \, dx","Integrate[Cos[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
14,1,75,83,0.6229761,"\int \frac{\cos ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[Cos[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 \cos ^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{\cos ^2(a+b x)}{d (c+d x)}",1,"-(((d*Cos[a + b*x]^2)/(c + d*x) + b*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] + b*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/d^2)","A",1
15,1,102,112,0.9239615,"\int \frac{\cos ^2(a+b x)}{(c+d x)^3} \, dx","Integrate[Cos[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 \cos ^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{\cos ^2(a+b x)}{2 d (c+d x)^2}",1,"(-2*b^2*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] + (d*(-(d*Cos[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])/(2*d^3)","A",1
16,1,385,225,0.9861173,"\int (c+d x)^4 \cos ^3(a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]^3,x]","\frac{243 b^4 c^4 \sin (a+b x)+27 b^4 c^4 \sin (3 (a+b x))+972 b^4 c^3 d x \sin (a+b x)+108 b^4 c^3 d x \sin (3 (a+b x))+1458 b^4 c^2 d^2 x^2 \sin (a+b x)+162 b^4 c^2 d^2 x^2 \sin (3 (a+b x))+972 b^4 c d^3 x^3 \sin (a+b x)+108 b^4 c d^3 x^3 \sin (3 (a+b x))+243 b^4 d^4 x^4 \sin (a+b x)+27 b^4 d^4 x^4 \sin (3 (a+b x))-2916 b^2 c^2 d^2 \sin (a+b x)-36 b^2 c^2 d^2 \sin (3 (a+b x))-5832 b^2 c d^3 x \sin (a+b x)-72 b^2 c d^3 x \sin (3 (a+b x))+972 b d (c+d x) \cos (a+b x) \left(b^2 (c+d x)^2-6 d^2\right)+12 b d (c+d x) \cos (3 (a+b x)) \left(3 b^2 (c+d x)^2-2 d^2\right)-2916 b^2 d^4 x^2 \sin (a+b x)-36 b^2 d^4 x^2 \sin (3 (a+b x))+5832 d^4 \sin (a+b x)+8 d^4 \sin (3 (a+b x))}{324 b^5}","-\frac{8 d^4 \sin ^3(a+b x)}{81 b^5}+\frac{488 d^4 \sin (a+b x)}{27 b^5}-\frac{8 d^3 (c+d x) \cos ^3(a+b x)}{27 b^4}-\frac{160 d^3 (c+d x) \cos (a+b x)}{9 b^4}-\frac{80 d^2 (c+d x)^2 \sin (a+b x)}{9 b^3}-\frac{4 d^2 (c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{9 b^3}+\frac{4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac{8 d (c+d x)^3 \cos (a+b x)}{3 b^2}+\frac{2 (c+d x)^4 \sin (a+b x)}{3 b}+\frac{(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}",1,"(972*b*d*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + 12*b*d*(c + d*x)*(-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] + 243*b^4*c^4*Sin[a + b*x] - 2916*b^2*c^2*d^2*Sin[a + b*x] + 5832*d^4*Sin[a + b*x] + 972*b^4*c^3*d*x*Sin[a + b*x] - 5832*b^2*c*d^3*x*Sin[a + b*x] + 1458*b^4*c^2*d^2*x^2*Sin[a + b*x] - 2916*b^2*d^4*x^2*Sin[a + b*x] + 972*b^4*c*d^3*x^3*Sin[a + b*x] + 243*b^4*d^4*x^4*Sin[a + b*x] + 27*b^4*c^4*Sin[3*(a + b*x)] - 36*b^2*c^2*d^2*Sin[3*(a + b*x)] + 8*d^4*Sin[3*(a + b*x)] + 108*b^4*c^3*d*x*Sin[3*(a + b*x)] - 72*b^2*c*d^3*x*Sin[3*(a + b*x)] + 162*b^4*c^2*d^2*x^2*Sin[3*(a + b*x)] - 36*b^2*d^4*x^2*Sin[3*(a + b*x)] + 108*b^4*c*d^3*x^3*Sin[3*(a + b*x)] + 27*b^4*d^4*x^4*Sin[3*(a + b*x)])/(324*b^5)","A",1
17,1,121,175,0.9585399,"\int (c+d x)^3 \cos ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]^3,x]","\frac{243 d \cos (a+b x) \left(b^2 (c+d x)^2-2 d^2\right)+d \cos (3 (a+b x)) \left(9 b^2 (c+d x)^2-2 d^2\right)+6 b (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)+15 b^2 (c+d x)^2-82 d^2\right)}{108 b^4}","-\frac{2 d^3 \cos ^3(a+b x)}{27 b^4}-\frac{40 d^3 \cos (a+b x)}{9 b^4}-\frac{40 d^2 (c+d x) \sin (a+b x)}{9 b^3}-\frac{2 d^2 (c+d x) \sin (a+b x) \cos ^2(a+b x)}{9 b^3}+\frac{d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac{2 d (c+d x)^2 \cos (a+b x)}{b^2}+\frac{2 (c+d x)^3 \sin (a+b x)}{3 b}+\frac{(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}",1,"(243*d*(-2*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + d*(-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] + 6*b*(c + d*x)*(-82*d^2 + 15*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])/(108*b^4)","A",1
18,1,93,123,0.6163333,"\int (c+d x)^2 \cos ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]^3,x]","\frac{2 \sin (a+b x) \left(\cos (2 (a+b x)) \left(9 b^2 (c+d x)^2-2 d^2\right)+45 b^2 (c+d x)^2-82 d^2\right)+162 b d (c+d x) \cos (a+b x)+6 b d (c+d x) \cos (3 (a+b x))}{108 b^3}","\frac{2 d^2 \sin ^3(a+b x)}{27 b^3}-\frac{14 d^2 \sin (a+b x)}{9 b^3}+\frac{2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac{4 d (c+d x) \cos (a+b x)}{3 b^2}+\frac{2 (c+d x)^2 \sin (a+b x)}{3 b}+\frac{(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b}",1,"(162*b*d*(c + d*x)*Cos[a + b*x] + 6*b*d*(c + d*x)*Cos[3*(a + b*x)] + 2*(-82*d^2 + 45*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])/(108*b^3)","A",1
19,1,52,75,0.172176,"\int (c+d x) \cos ^3(a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]^3,x]","\frac{3 b (c+d x) (9 \sin (a+b x)+\sin (3 (a+b x)))+27 d \cos (a+b x)+d \cos (3 (a+b x))}{36 b^2}","\frac{d \cos ^3(a+b x)}{9 b^2}+\frac{2 d \cos (a+b x)}{3 b^2}+\frac{2 (c+d x) \sin (a+b x)}{3 b}+\frac{(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}",1,"(27*d*Cos[a + b*x] + d*Cos[3*(a + b*x)] + 3*b*(c + d*x)*(9*Sin[a + b*x] + Sin[3*(a + b*x)]))/(36*b^2)","A",1
20,1,103,121,0.2533164,"\int \frac{\cos ^3(a+b x)}{c+d x} \, dx","Integrate[Cos[a + b*x]^3/(c + d*x),x]","\frac{3 \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+\cos \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{Si}\left(b \left(\frac{c}{d}+x\right)\right)-\sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)}{4 d}","\frac{3 \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{4 d}+\frac{\cos \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{Si}\left(\frac{b c}{d}+b x\right)}{4 d}-\frac{\sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{4 d}",1,"(3*Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] + Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] - 3*Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)] - Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/(4*d)","A",1
21,1,200,145,0.7010725,"\int \frac{\cos ^3(a+b x)}{(c+d x)^2} \, dx","Integrate[Cos[a + b*x]^3/(c + d*x)^2,x]","-\frac{3 b (c+d x) \sin \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{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+3 b c \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+3 b d x \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+3 b c \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)+3 b d x \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)+3 d \cos (a+b x)+d \cos (3 (a+b x))}{4 d^2 (c+d x)}","-\frac{3 b \sin \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{Ci}\left(\frac{b c}{d}+b x\right)}{4 d^2}-\frac{3 b \cos \left(a-\frac{b c}{d}\right) \text{Si}\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{Si}\left(\frac{3 b c}{d}+3 b x\right)}{4 d^2}-\frac{\cos ^3(a+b x)}{d (c+d x)}",1,"-1/4*(3*d*Cos[a + b*x] + d*Cos[3*(a + b*x)] + 3*b*(c + d*x)*CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] + 3*b*(c + d*x)*CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + 3*b*c*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + 3*b*d*x*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + 3*b*c*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d] + 3*b*d*x*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/(d^2*(c + d*x))","A",1
22,1,221,184,0.8259076,"\int \frac{\cos ^3(a+b x)}{(c+d x)^3} \, dx","Integrate[Cos[a + b*x]^3/(c + d*x)^3,x]","\frac{-6 b^2 (c+d x)^2 \left(\cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+3 \cos \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{Si}\left(b \left(\frac{c}{d}+x\right)\right)-3 \sin \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 \sin (a) (c+d x)-d \cos (a))+2 d \cos (3 b x) (3 b \sin (3 a) (c+d x)-d \cos (3 a))+6 d \sin (b x) (b \cos (a) (c+d x)+d \sin (a))+2 d \sin (3 b x) (3 b \cos (3 a) (c+d x)+d \sin (3 a))}{16 d^3 (c+d x)^2}","-\frac{3 b^2 \cos \left(a-\frac{b c}{d}\right) \text{Ci}\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{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{Si}\left(\frac{b c}{d}+b x\right)}{8 d^3}+\frac{9 b^2 \sin \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 (a+b x) \cos ^2(a+b x)}{2 d^2 (c+d x)}-\frac{\cos ^3(a+b x)}{2 d (c+d x)^2}",1,"(6*d*Cos[b*x]*(-(d*Cos[a]) + b*(c + d*x)*Sin[a]) + 2*d*Cos[3*b*x]*(-(d*Cos[3*a]) + 3*b*(c + d*x)*Sin[3*a]) + 6*d*(b*(c + d*x)*Cos[a] + d*Sin[a])*Sin[b*x] + 2*d*(3*b*(c + d*x)*Cos[3*a] + d*Sin[3*a])*Sin[3*b*x] - 6*b^2*(c + d*x)^2*(Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] + 3*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] - Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)] - 3*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d]))/(16*d^3*(c + d*x)^2)","A",1
23,1,100,172,0.418898,"\int x^3 \cos ^4(a+b x) \, dx","Integrate[x^3*Cos[a + b*x]^4,x]","\frac{192 \left(2 b^2 x^2-1\right) \cos (2 (a+b x))+3 \left(8 b^2 x^2-1\right) \cos (4 (a+b x))+4 b x \left(32 \left(2 b^2 x^2-3\right) \sin (2 (a+b x))+\left(8 b^2 x^2-3\right) \sin (4 (a+b x))+24 b^3 x^3\right)}{1024 b^4}","-\frac{3 \cos ^4(a+b x)}{128 b^4}-\frac{45 \cos ^2(a+b x)}{128 b^4}-\frac{3 x \sin (a+b x) \cos ^3(a+b x)}{32 b^3}-\frac{45 x \sin (a+b x) \cos (a+b x)}{64 b^3}+\frac{3 x^2 \cos ^4(a+b x)}{16 b^2}+\frac{9 x^2 \cos ^2(a+b x)}{16 b^2}+\frac{x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{3 x^3 \sin (a+b x) \cos (a+b x)}{8 b}-\frac{45 x^2}{128 b^2}+\frac{3 x^4}{32}",1,"(192*(-1 + 2*b^2*x^2)*Cos[2*(a + b*x)] + 3*(-1 + 8*b^2*x^2)*Cos[4*(a + b*x)] + 4*b*x*(24*b^3*x^3 + 32*(-3 + 2*b^2*x^2)*Sin[2*(a + b*x)] + (-3 + 8*b^2*x^2)*Sin[4*(a + b*x)]))/(1024*b^4)","A",1
24,1,92,134,0.17791,"\int x^2 \cos ^4(a+b x) \, dx","Integrate[x^2*Cos[a + b*x]^4,x]","\frac{64 b^2 x^2 \sin (2 (a+b x))+8 b^2 x^2 \sin (4 (a+b x))-32 \sin (2 (a+b x))-\sin (4 (a+b x))+64 b x \cos (2 (a+b x))+4 b x \cos (4 (a+b x))+32 b^3 x^3}{256 b^3}","-\frac{\sin (a+b x) \cos ^3(a+b x)}{32 b^3}-\frac{15 \sin (a+b x) \cos (a+b x)}{64 b^3}+\frac{x \cos ^4(a+b x)}{8 b^2}+\frac{3 x \cos ^2(a+b x)}{8 b^2}+\frac{x^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{3 x^2 \sin (a+b x) \cos (a+b x)}{8 b}-\frac{15 x}{64 b^2}+\frac{x^3}{8}",1,"(32*b^3*x^3 + 64*b*x*Cos[2*(a + b*x)] + 4*b*x*Cos[4*(a + b*x)] - 32*Sin[2*(a + b*x)] + 64*b^2*x^2*Sin[2*(a + b*x)] - Sin[4*(a + b*x)] + 8*b^2*x^2*Sin[4*(a + b*x)])/(256*b^3)","A",1
25,1,53,80,0.1253364,"\int x \cos ^4(a+b x) \, dx","Integrate[x*Cos[a + b*x]^4,x]","\frac{4 b x (8 \sin (2 (a+b x))+\sin (4 (a+b x))+6 b x)+16 \cos (2 (a+b x))+\cos (4 (a+b x))}{128 b^2}","\frac{\cos ^4(a+b x)}{16 b^2}+\frac{3 \cos ^2(a+b x)}{16 b^2}+\frac{x \sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{3 x \sin (a+b x) \cos (a+b x)}{8 b}+\frac{3 x^2}{16}",1,"(16*Cos[2*(a + b*x)] + Cos[4*(a + b*x)] + 4*b*x*(6*b*x + 8*Sin[2*(a + b*x)] + Sin[4*(a + b*x)]))/(128*b^2)","A",1
26,1,52,59,0.0995261,"\int \frac{\cos ^4(a+b x)}{x} \, dx","Integrate[Cos[a + b*x]^4/x,x]","\frac{1}{8} (4 \cos (2 a) \text{Ci}(2 b x)+\cos (4 a) \text{Ci}(4 b x)-4 \sin (2 a) \text{Si}(2 b x)-\sin (4 a) \text{Si}(4 b x)+3 \log (x))","\frac{1}{2} \cos (2 a) \text{Ci}(2 b x)+\frac{1}{8} \cos (4 a) \text{Ci}(4 b x)-\frac{1}{2} \sin (2 a) \text{Si}(2 b x)-\frac{1}{8} \sin (4 a) \text{Si}(4 b x)+\frac{3 \log (x)}{8}",1,"(4*Cos[2*a]*CosIntegral[2*b*x] + Cos[4*a]*CosIntegral[4*b*x] + 3*Log[x] - 4*Sin[2*a]*SinIntegral[2*b*x] - Sin[4*a]*SinIntegral[4*b*x])/8","A",1
27,1,79,66,0.2201304,"\int \frac{\cos ^4(a+b x)}{x^2} \, dx","Integrate[Cos[a + b*x]^4/x^2,x]","-\frac{8 b x \sin (2 a) \text{Ci}(2 b x)+4 b x \sin (4 a) \text{Ci}(4 b x)+8 b x \cos (2 a) \text{Si}(2 b x)+4 b x \cos (4 a) \text{Si}(4 b x)+4 \cos (2 (a+b x))+\cos (4 (a+b x))+3}{8 x}","-b \sin (2 a) \text{Ci}(2 b x)-\frac{1}{2} b \sin (4 a) \text{Ci}(4 b x)-b \cos (2 a) \text{Si}(2 b x)-\frac{1}{2} b \cos (4 a) \text{Si}(4 b x)-\frac{\cos ^4(a+b x)}{x}",1,"-1/8*(3 + 4*Cos[2*(a + b*x)] + Cos[4*(a + b*x)] + 8*b*x*CosIntegral[2*b*x]*Sin[2*a] + 4*b*x*CosIntegral[4*b*x]*Sin[4*a] + 8*b*x*Cos[2*a]*SinIntegral[2*b*x] + 4*b*x*Cos[4*a]*SinIntegral[4*b*x])/x","A",1
28,1,119,90,0.3040573,"\int \frac{\cos ^4(a+b x)}{x^3} \, dx","Integrate[Cos[a + b*x]^4/x^3,x]","-\frac{16 b^2 x^2 \cos (2 a) \text{Ci}(2 b x)+16 b^2 x^2 \cos (4 a) \text{Ci}(4 b x)-16 b^2 x^2 \sin (2 a) \text{Si}(2 b x)-16 b^2 x^2 \sin (4 a) \text{Si}(4 b x)-8 b x \sin (2 (a+b x))-4 b x \sin (4 (a+b x))+4 \cos (2 (a+b x))+\cos (4 (a+b x))+3}{16 x^2}","-b^2 \cos (2 a) \text{Ci}(2 b x)-b^2 \cos (4 a) \text{Ci}(4 b x)+b^2 \sin (2 a) \text{Si}(2 b x)+b^2 \sin (4 a) \text{Si}(4 b x)-\frac{\cos ^4(a+b x)}{2 x^2}+\frac{2 b \sin (a+b x) \cos ^3(a+b x)}{x}",1,"-1/16*(3 + 4*Cos[2*(a + b*x)] + Cos[4*(a + b*x)] + 16*b^2*x^2*Cos[2*a]*CosIntegral[2*b*x] + 16*b^2*x^2*Cos[4*a]*CosIntegral[4*b*x] - 8*b*x*Sin[2*(a + b*x)] - 4*b*x*Sin[4*(a + b*x)] - 16*b^2*x^2*Sin[2*a]*SinIntegral[2*b*x] - 16*b^2*x^2*Sin[4*a]*SinIntegral[4*b*x])/x^2","A",1
29,1,196,205,0.1905303,"\int (c+d x)^3 \sec (a+b x) \, dx","Integrate[(c + d*x)^3*Sec[a + b*x],x]","-\frac{i \left(2 b^3 (c+d x)^3 \tan ^{-1}\left(e^{i (a+b x)}\right)-3 d \left(b^2 (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_3\left(-i e^{i (a+b x)}\right)-2 d^2 \text{Li}_4\left(-i e^{i (a+b x)}\right)\right)+3 d \left(b^2 (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_3\left(i e^{i (a+b x)}\right)-2 d^2 \text{Li}_4\left(i e^{i (a+b x)}\right)\right)\right)}{b^4}","-\frac{6 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)}{b^4}+\frac{6 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)}{b^4}-\frac{6 d^2 (c+d x) \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{6 d^2 (c+d x) \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{2 i (c+d x)^3 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"((-I)*(2*b^3*(c + d*x)^3*ArcTan[E^(I*(a + b*x))] - 3*d*(b^2*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))] - 2*d^2*PolyLog[4, (-I)*E^(I*(a + b*x))]) + 3*d*(b^2*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))] - 2*d^2*PolyLog[4, I*E^(I*(a + b*x))])))/b^4","A",1
30,1,130,137,0.0981447,"\int (c+d x)^2 \sec (a+b x) \, dx","Integrate[(c + d*x)^2*Sec[a + b*x],x]","-\frac{2 i \left(b^2 (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)-d \left(b (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)+i d \text{Li}_3\left(-i e^{i (a+b x)}\right)\right)+d \left(b (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)+i d \text{Li}_3\left(i e^{i (a+b x)}\right)\right)\right)}{b^3}","-\frac{2 d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{2 d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}+\frac{2 i d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{2 i d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{2 i (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"((-2*I)*(b^2*(c + d*x)^2*ArcTan[E^(I*(a + b*x))] - d*(b*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] + I*d*PolyLog[3, (-I)*E^(I*(a + b*x))]) + d*(b*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] + I*d*PolyLog[3, I*E^(I*(a + b*x))])))/b^3","A",1
31,1,87,75,0.0053521,"\int (c+d x) \sec (a+b x) \, dx","Integrate[(c + d*x)*Sec[a + b*x],x]","\frac{i d \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{i d \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}+\frac{c \tanh ^{-1}(\sin (a+b x))}{b}-\frac{2 i d x \tan ^{-1}\left(e^{i a+i b x}\right)}{b}","\frac{i d \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{i d \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{2 i (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"((-2*I)*d*x*ArcTan[E^(I*a + I*b*x)])/b + (c*ArcTanh[Sin[a + b*x]])/b + (I*d*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (I*d*PolyLog[2, I*E^(I*(a + b*x))])/b^2","A",0
32,0,0,17,4.5488799,"\int \frac{\sec (a+b x)}{c+d x} \, dx","Integrate[Sec[a + b*x]/(c + d*x),x]","\int \frac{\sec (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\sec (a+b x)}{c+d x},x\right)",0,"Integrate[Sec[a + b*x]/(c + d*x), x]","A",-1
33,1,109,114,0.5069298,"\int (c+d x)^3 \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Sec[a + b*x]^2,x]","\frac{2 b^2 (c+d x)^2 \left(b (c+d x) \tan (a+b x)+3 d \log \left(1+e^{2 i (a+b x)}\right)-i b (c+d x)\right)-6 i b d^2 (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)+3 d^3 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^4}","\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 \tan (a+b x)}{b}-\frac{i (c+d x)^3}{b}",1,"((-6*I)*b*d^2*(c + d*x)*PolyLog[2, -E^((2*I)*(a + b*x))] + 3*d^3*PolyLog[3, -E^((2*I)*(a + b*x))] + 2*b^2*(c + d*x)^2*((-I)*b*(c + d*x) + 3*d*Log[1 + E^((2*I)*(a + b*x))] + b*(c + d*x)*Tan[a + b*x]))/(2*b^4)","A",1
34,1,75,82,0.2520052,"\int (c+d x)^2 \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Sec[a + b*x]^2,x]","\frac{b (c+d x) \left(b (c+d x) \tan (a+b x)+2 d \log \left(1+e^{2 i (a+b x)}\right)-i b (c+d x)\right)-i d^2 \text{Li}_2\left(-e^{2 i (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 \tan (a+b x)}{b}-\frac{i (c+d x)^2}{b}",1,"((-I)*d^2*PolyLog[2, -E^((2*I)*(a + b*x))] + b*(c + d*x)*((-I)*b*(c + d*x) + 2*d*Log[1 + E^((2*I)*(a + b*x))] + b*(c + d*x)*Tan[a + b*x]))/b^3","A",1
35,1,36,28,0.0117558,"\int (c+d x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)*Sec[a + b*x]^2,x]","\frac{d \log (\cos (a+b x))}{b^2}+\frac{c \tan (a+b x)}{b}+\frac{d x \tan (a+b x)}{b}","\frac{d \log (\cos (a+b x))}{b^2}+\frac{(c+d x) \tan (a+b x)}{b}",1,"(d*Log[Cos[a + b*x]])/b^2 + (c*Tan[a + b*x])/b + (d*x*Tan[a + b*x])/b","A",1
36,0,0,19,5.6231403,"\int \frac{\sec ^2(a+b x)}{c+d x} \, dx","Integrate[Sec[a + b*x]^2/(c + d*x),x]","\int \frac{\sec ^2(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\sec ^2(a+b x)}{c+d x},x\right)",0,"Integrate[Sec[a + b*x]^2/(c + d*x), x]","A",-1
37,1,311,337,2.9923424,"\int (c+d x)^3 \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Sec[a + b*x]^3,x]","\frac{-2 i b^3 (c+d x)^3 \tan ^{-1}\left(e^{i (a+b x)}\right)+b^3 (c+d x)^3 \tan (a+b x) \sec (a+b x)+3 i d \left(b^2 (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_3\left(-i e^{i (a+b x)}\right)-2 d^2 \text{Li}_4\left(-i e^{i (a+b x)}\right)\right)-3 i d \left(b^2 (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_3\left(i e^{i (a+b x)}\right)-2 d^2 \text{Li}_4\left(i e^{i (a+b x)}\right)\right)-3 b^2 d (c+d x)^2 \sec (a+b x)-6 i d^2 \left(2 b (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)-d \text{Li}_2\left(-i e^{i (a+b x)}\right)+d \text{Li}_2\left(i e^{i (a+b x)}\right)\right)}{2 b^4}","\frac{3 i d^3 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^4}-\frac{3 i d^3 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^4}-\frac{3 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)}{b^4}+\frac{3 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)}{b^4}-\frac{3 d^2 (c+d x) \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{3 d^2 (c+d x) \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}-\frac{6 i d^2 (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{2 b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{2 b^2}-\frac{3 d (c+d x)^2 \sec (a+b x)}{2 b^2}-\frac{i (c+d x)^3 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}+\frac{(c+d x)^3 \tan (a+b x) \sec (a+b x)}{2 b}",1,"((-2*I)*b^3*(c + d*x)^3*ArcTan[E^(I*(a + b*x))] - (6*I)*d^2*(2*b*(c + d*x)*ArcTan[E^(I*(a + b*x))] - d*PolyLog[2, (-I)*E^(I*(a + b*x))] + d*PolyLog[2, I*E^(I*(a + b*x))]) + (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))] - 2*d^2*PolyLog[4, (-I)*E^(I*(a + b*x))]) - (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))] - 2*d^2*PolyLog[4, I*E^(I*(a + b*x))]) - 3*b^2*d*(c + d*x)^2*Sec[a + b*x] + b^3*(c + d*x)^3*Sec[a + b*x]*Tan[a + b*x])/(2*b^4)","A",1
38,1,184,193,0.9224246,"\int (c+d x)^2 \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Sec[a + b*x]^3,x]","\frac{-2 i b^2 (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)+b^2 (c+d x)^2 \tan (a+b x) \sec (a+b x)+2 i b d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)-2 i b d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)-2 b d (c+d x) \sec (a+b x)-2 d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)+2 d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)+2 d^2 \tanh ^{-1}(\sin (a+b x))}{2 b^3}","-\frac{d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}+\frac{d^2 \tanh ^{-1}(\sin (a+b x))}{b^3}+\frac{i d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{i d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{d (c+d x) \sec (a+b x)}{b^2}-\frac{i (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}+\frac{(c+d x)^2 \tan (a+b x) \sec (a+b x)}{2 b}",1,"((-2*I)*b^2*(c + d*x)^2*ArcTan[E^(I*(a + b*x))] + 2*d^2*ArcTanh[Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] - 2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] + 2*d^2*PolyLog[3, I*E^(I*(a + b*x))] - 2*b*d*(c + d*x)*Sec[a + b*x] + b^2*(c + d*x)^2*Sec[a + b*x]*Tan[a + b*x])/(2*b^3)","A",1
39,1,389,117,3.682146,"\int (c+d x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)*Sec[a + b*x]^3,x]","\frac{d \left(i \left(\text{Li}_2\left(-e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)-\text{Li}_2\left(e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)\right)+\left(-a-b x+\frac{\pi }{2}\right) \left(\log \left(1-e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)-\log \left(1+e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)\right)-\left(\frac{\pi }{2}-a\right) \log \left(\tan \left(\frac{1}{2} \left(-a-b x+\frac{\pi }{2}\right)\right)\right)\right)}{2 b^2}-\frac{d \sin \left(\frac{b x}{2}\right)}{2 b^2 \left(\cos \left(\frac{a}{2}\right)-\sin \left(\frac{a}{2}\right)\right) \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)-\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}+\frac{d \sin \left(\frac{b x}{2}\right)}{2 b^2 \left(\sin \left(\frac{a}{2}\right)+\cos \left(\frac{a}{2}\right)\right) \left(\sin \left(\frac{a}{2}+\frac{b x}{2}\right)+\cos \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}+\frac{c \tanh ^{-1}(\sin (a+b x))}{2 b}+\frac{c \tan (a+b x) \sec (a+b x)}{2 b}+\frac{d x}{4 b \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)-\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)^2}-\frac{d x}{4 b \left(\sin \left(\frac{a}{2}+\frac{b x}{2}\right)+\cos \left(\frac{a}{2}+\frac{b x}{2}\right)\right)^2}","\frac{i d \text{Li}_2\left(-i e^{i (a+b x)}\right)}{2 b^2}-\frac{i d \text{Li}_2\left(i e^{i (a+b x)}\right)}{2 b^2}-\frac{d \sec (a+b x)}{2 b^2}-\frac{i (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}+\frac{(c+d x) \tan (a+b x) \sec (a+b x)}{2 b}",1,"(c*ArcTanh[Sin[a + b*x]])/(2*b) + (d*((-a + Pi/2 - b*x)*(Log[1 - E^(I*(-a + Pi/2 - b*x))] - Log[1 + E^(I*(-a + Pi/2 - b*x))]) - (-a + Pi/2)*Log[Tan[(-a + Pi/2 - b*x)/2]] + I*(PolyLog[2, -E^(I*(-a + Pi/2 - b*x))] - PolyLog[2, E^(I*(-a + Pi/2 - b*x))])))/(2*b^2) + (d*x)/(4*b*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])^2) - (d*Sin[(b*x)/2])/(2*b^2*(Cos[a/2] - Sin[a/2])*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])) - (d*x)/(4*b*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2])^2) + (d*Sin[(b*x)/2])/(2*b^2*(Cos[a/2] + Sin[a/2])*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2])) + (c*Sec[a + b*x]*Tan[a + b*x])/(2*b)","B",1
40,0,0,19,1.4219301,"\int \frac{\sec ^2(a+b x)}{c+d x} \, dx","Integrate[Sec[a + b*x]^2/(c + d*x),x]","\int \frac{\sec ^2(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\sec ^2(a+b x)}{c+d x},x\right)",0,"Integrate[Sec[a + b*x]^2/(c + d*x), x]","A",-1
41,1,124,194,0.0591565,"\int (c+d x)^{5/2} \cos (a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x],x]","-\frac{d^3 e^{-\frac{i (a d+b c)}{d}} \left(e^{2 i a} \sqrt{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{7}{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{7}{2},\frac{i b (c+d x)}{d}\right)\right)}{2 b^4 \sqrt{c+d x}}","\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{7/2}}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{4 b^3}+\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{2 b^2}+\frac{(c+d x)^{5/2} \sin (a+b x)}{b}",1,"-1/2*(d^3*(E^((2*I)*a)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[7/2, ((-I)*b*(c + d*x))/d] + E^(((2*I)*b*c)/d)*Sqrt[(I*b*(c + d*x))/d]*Gamma[7/2, (I*b*(c + d*x))/d]))/(b^4*E^((I*(b*c + a*d))/d)*Sqrt[c + d*x])","C",1
42,1,122,169,0.1002309,"\int (c+d x)^{3/2} \cos (a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x],x]","\frac{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} \cos \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} \sin \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} \cos (a+b x)}{2 b^2}+\frac{(c+d x)^{3/2} \sin (a+b x)}{b}",1,"(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]))/(2*b^2*E^((I*(b*c + a*d))/d))","C",1
43,1,124,142,0.0917795,"\int \sqrt{c+d x} \cos (a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x],x]","-\frac{i \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} \sin \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} \cos \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} \sin (a+b x)}{b}",1,"((-1/2*I)*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]))/(b*E^((I*(b*c + a*d))/d))","C",1
44,1,124,118,0.0582443,"\int \frac{\cos (a+b x)}{\sqrt{c+d x}} \, dx","Integrate[Cos[a + b*x]/Sqrt[c + d*x],x]","\frac{i e^{-\frac{i (a d+b c)}{d}} \left(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)-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)\right)}{2 b \sqrt{c+d x}}","\frac{\sqrt{2 \pi } \cos \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 } \sin \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,"((I/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
45,1,147,139,0.3193175,"\int \frac{\cos (a+b x)}{(c+d x)^{3/2}} \, dx","Integrate[Cos[a + b*x]/(c + d*x)^(3/2),x]","\frac{e^{-i a} \left(e^{2 i a-\frac{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)+e^{-i 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)\right)}{d \sqrt{c+d x}}","-\frac{2 \sqrt{2 \pi } \sqrt{b} \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^{3/2}}-\frac{2 \sqrt{2 \pi } \sqrt{b} \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^{3/2}}-\frac{2 \cos (a+b x)}{d \sqrt{c+d x}}",1,"(E^((2*I)*a - (I*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*b*x))/(d*E^(I*a)*Sqrt[c + d*x])","C",1
46,1,190,168,0.2942626,"\int \frac{\cos (a+b x)}{(c+d x)^{5/2}} \, dx","Integrate[Cos[a + b*x]/(c + d*x)^(5/2),x]","\frac{e^{-i a} \left(e^{-i 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)-2 i e^{2 i a-\frac{i b c}{d}} \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)\right)}{6 d^2 (c+d x)^{3/2}}","-\frac{4 \sqrt{2 \pi } b^{3/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{3 d^{5/2}}+\frac{4 \sqrt{2 \pi } b^{3/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{3 d^{5/2}}+\frac{4 b \sin (a+b x)}{3 d^2 \sqrt{c+d x}}-\frac{2 \cos (a+b x)}{3 d (c+d x)^{3/2}}",1,"((-2*I)*E^((2*I)*a - (I*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]) + (-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*b*x))/(6*d^2*E^(I*a)*(c + d*x)^(3/2))","C",1
47,1,228,193,0.3440789,"\int \frac{\cos (a+b x)}{(c+d x)^{7/2}} \, dx","Integrate[Cos[a + b*x]/(c + d*x)^(7/2),x]","\frac{e^{-i a} \left(2 e^{2 i a} \left(2 b e^{-\frac{i b c}{d}} (c+d x) \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)-3 d^2 e^{i b x}\right)+e^{-i b x} \left(8 b^2 (c+d x)^2+8 d^2 e^{\frac{i b (c+d x)}{d}} \left(\frac{i b (c+d x)}{d}\right)^{5/2} \Gamma \left(\frac{1}{2},\frac{i b (c+d x)}{d}\right)+4 i b d (c+d x)-6 d^2\right)\right)}{30 d^3 (c+d x)^{5/2}}","\frac{8 \sqrt{2 \pi } b^{5/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{15 d^{7/2}}+\frac{8 \sqrt{2 \pi } b^{5/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{15 d^{7/2}}+\frac{8 b^2 \cos (a+b x)}{15 d^3 \sqrt{c+d x}}+\frac{4 b \sin (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac{2 \cos (a+b x)}{5 d (c+d x)^{5/2}}",1,"(2*E^((2*I)*a)*(-3*d^2*E^(I*b*x) + (2*b*(c + d*x)*(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]))/E^((I*b*c)/d)) + (-6*d^2 + (4*I)*b*d*(c + d*x) + 8*b^2*(c + d*x)^2 + 8*d^2*E^((I*b*(c + d*x))/d)*((I*b*(c + d*x))/d)^(5/2)*Gamma[1/2, (I*b*(c + d*x))/d])/E^(I*b*x))/(30*d^3*E^(I*a)*(c + d*x)^(5/2))","C",1
48,1,194,231,2.1427272,"\int (c+d x)^{5/2} \cos ^2(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[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} \cos ^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
49,1,175,203,1.6660626,"\int (c+d x)^{3/2} \cos ^2(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[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) (5 d \sin (2 (a+b x))+4 b (c+d 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} \cos ^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
50,1,146,158,0.5313423,"\int \sqrt{c+d x} \cos ^2(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^2,x]","\frac{1}{48} \sqrt{c+d x} \left(-\frac{3 i \sqrt{2} e^{2 i \left(a-\frac{b c}{d}\right)} \Gamma \left(\frac{3}{2},-\frac{2 i b (c+d x)}{d}\right)}{b \sqrt{-\frac{i b (c+d x)}{d}}}+\frac{3 i \sqrt{2} e^{-2 i \left(a-\frac{b c}{d}\right)} \Gamma \left(\frac{3}{2},\frac{2 i b (c+d x)}{d}\right)}{b \sqrt{\frac{i b (c+d x)}{d}}}+\frac{16 (c+d x)}{d}\right)","-\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,"(Sqrt[c + d*x]*((16*(c + d*x))/d - ((3*I)*Sqrt[2]*E^((2*I)*(a - (b*c)/d))*Gamma[3/2, ((-2*I)*b*(c + d*x))/d])/(b*Sqrt[((-I)*b*(c + d*x))/d]) + ((3*I)*Sqrt[2]*Gamma[3/2, ((2*I)*b*(c + d*x))/d])/(b*E^((2*I)*(a - (b*c)/d))*Sqrt[(I*b*(c + d*x))/d])))/48","C",1
51,1,145,130,0.2229471,"\int \frac{\cos ^2(a+b x)}{\sqrt{c+d x}} \, dx","Integrate[Cos[a + b*x]^2/Sqrt[c + d*x],x]","\frac{-\frac{i \sqrt{2} e^{2 i \left(a-\frac{b c}{d}\right)} \sqrt{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{1}{2},-\frac{2 i b (c+d x)}{d}\right)}{b}+\frac{i \sqrt{2} e^{-2 i \left(a-\frac{b c}{d}\right)} \sqrt{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{1}{2},\frac{2 i b (c+d x)}{d}\right)}{b}+8 \left(\frac{c}{d}+x\right)}{8 \sqrt{c+d x}}","\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,"(8*(c/d + x) - (I*Sqrt[2]*E^((2*I)*(a - (b*c)/d))*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[1/2, ((-2*I)*b*(c + d*x))/d])/b + (I*Sqrt[2]*Sqrt[(I*b*(c + d*x))/d]*Gamma[1/2, ((2*I)*b*(c + d*x))/d])/(b*E^((2*I)*(a - (b*c)/d))))/(8*Sqrt[c + d*x])","C",1
52,1,133,135,0.6625959,"\int \frac{\cos ^2(a+b x)}{(c+d x)^{3/2}} \, dx","Integrate[Cos[a + b*x]^2/(c + d*x)^(3/2),x]","\frac{2 \left(-\sqrt{\pi } \sqrt{\frac{b}{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)-\sqrt{\pi } \sqrt{\frac{b}{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)-\frac{\cos ^2(a+b x)}{\sqrt{c+d x}}\right)}{d}","-\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 \cos ^2(a+b x)}{d \sqrt{c+d x}}",1,"(2*(-(Cos[a + b*x]^2/Sqrt[c + d*x]) - Sqrt[b/d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - Sqrt[b/d]*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d]))/d","A",1
53,1,181,170,1.3893194,"\int \frac{\cos ^2(a+b x)}{(c+d x)^{5/2}} \, dx","Integrate[Cos[a + b*x]^2/(c + d*x)^(5/2),x]","\frac{e^{-\frac{2 i (a d+b c)}{d}} \left(-2 \sqrt{2} e^{4 i a} d \left(-\frac{i b (c+d x)}{d}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{2 i b (c+d x)}{d}\right)+2 e^{\frac{2 i (a d+b c)}{d}} \left(2 b (c+d x) \sin (2 (a+b x))-d \cos ^2(a+b x)\right)-2 \sqrt{2} d e^{\frac{4 i b c}{d}} \left(\frac{i b (c+d x)}{d}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{2 i b (c+d x)}{d}\right)\right)}{3 d^2 (c+d x)^{3/2}}","-\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 \cos ^2(a+b x)}{3 d (c+d x)^{3/2}}",1,"(-2*Sqrt[2]*d*E^((4*I)*a)*(((-I)*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((-2*I)*b*(c + d*x))/d] - 2*Sqrt[2]*d*E^(((4*I)*b*c)/d)*((I*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((2*I)*b*(c + d*x))/d] + 2*E^(((2*I)*(b*c + a*d))/d)*(-(d*Cos[a + b*x]^2) + 2*b*(c + d*x)*Sin[2*(a + b*x)]))/(3*d^2*E^(((2*I)*(b*c + a*d))/d)*(c + d*x)^(3/2))","C",1
54,1,244,216,1.2675225,"\int \frac{\cos ^2(a+b x)}{(c+d x)^{7/2}} \, dx","Integrate[Cos[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 \cos ^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 \cos ^2(a+b x)}{5 d (c+d x)^{5/2}}-\frac{16 b^2}{15 d^3 \sqrt{c+d x}}",1,"(-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)])/(15*d^3*(c + d*x)^(5/2))","A",1
55,1,237,247,0.8630061,"\int \frac{\cos ^2(a+b x)}{(c+d x)^{9/2}} \, dx","Integrate[Cos[a + b*x]^2/(c + d*x)^(9/2),x]","\frac{2 \left(-32 b^3 (c+d x)^3 \sin (2 (a+b x))+16 b^2 d (c+d x)^2 \cos ^2(a+b x)+16 \sqrt{2} b^2 d (c+d x)^2 e^{2 i \left(a-\frac{b c}{d}\right)} \left(-\frac{i b (c+d x)}{d}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{2 i b (c+d x)}{d}\right)+16 \sqrt{2} b^2 d (c+d x)^2 e^{-2 i \left(a-\frac{b c}{d}\right)} \left(\frac{i b (c+d x)}{d}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{2 i b (c+d x)}{d}\right)+6 b d^2 (c+d x) \sin (2 (a+b x))-15 d^3 \cos ^2(a+b x)-8 b^2 d (c+d x)^2\right)}{105 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 \cos ^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 \cos ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac{16 b^2}{105 d^3 (c+d x)^{3/2}}",1,"(2*(-8*b^2*d*(c + d*x)^2 - 15*d^3*Cos[a + b*x]^2 + 16*b^2*d*(c + d*x)^2*Cos[a + b*x]^2 + 16*Sqrt[2]*b^2*d*E^((2*I)*(a - (b*c)/d))*(c + d*x)^2*(((-I)*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((-2*I)*b*(c + d*x))/d] + (16*Sqrt[2]*b^2*d*(c + d*x)^2*((I*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((2*I)*b*(c + d*x))/d])/E^((2*I)*(a - (b*c)/d)) + 6*b*d^2*(c + d*x)*Sin[2*(a + b*x)] - 32*b^3*(c + d*x)^3*Sin[2*(a + b*x)]))/(105*d^4*(c + d*x)^(7/2))","C",1
56,1,542,410,3.0988028,"\int (c+d x)^{5/2} \cos ^3(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^3,x]","\frac{648 b^3 c^2 \sqrt{c+d x} \sin (a+b x)+72 b^3 c^2 \sqrt{c+d x} \sin (3 (a+b x))+648 b^3 d^2 x^2 \sqrt{c+d x} \sin (a+b x)+72 b^3 d^2 x^2 \sqrt{c+d x} \sin (3 (a+b x))+1296 b^3 c d x \sqrt{c+d x} \sin (a+b x)+144 b^3 c d x \sqrt{c+d x} \sin (3 (a+b x))+1620 b^2 d^2 x \sqrt{c+d x} \cos (a+b x)+60 b^2 d^2 x \sqrt{c+d x} \cos (3 (a+b x))+1620 b^2 c d \sqrt{c+d x} \cos (a+b x)+60 b^2 c d \sqrt{c+d x} \cos (3 (a+b x))+5 \sqrt{6 \pi } d^3 \sqrt{\frac{b}{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)+1215 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left(a-\frac{b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+1215 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left(a-\frac{b c}{d}\right) S\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) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-2430 b d^2 \sqrt{c+d x} \sin (a+b x)-30 b d^2 \sqrt{c+d x} \sin (3 (a+b x))}{864 b^4}","\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \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)}{144 b^{7/2}}+\frac{45 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}+\frac{45 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}+\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \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)}{144 b^{7/2}}-\frac{45 d^2 \sqrt{c+d x} \sin (a+b x)}{16 b^3}-\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \cos ^3(a+b x)}{18 b^2}+\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{3 b^2}+\frac{2 (c+d x)^{5/2} \sin (a+b x)}{3 b}+\frac{(c+d x)^{5/2} \sin (a+b x) \cos ^2(a+b x)}{3 b}",1,"(1620*b^2*c*d*Sqrt[c + d*x]*Cos[a + b*x] + 1620*b^2*d^2*x*Sqrt[c + d*x]*Cos[a + b*x] + 60*b^2*c*d*Sqrt[c + d*x]*Cos[3*(a + b*x)] + 60*b^2*d^2*x*Sqrt[c + d*x]*Cos[3*(a + b*x)] + 1215*Sqrt[b/d]*d^3*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelS[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]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + 5*Sqrt[b/d]*d^3*Sqrt[6*Pi]*FresnelC[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]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[a - (b*c)/d] + 648*b^3*c^2*Sqrt[c + d*x]*Sin[a + b*x] - 2430*b*d^2*Sqrt[c + d*x]*Sin[a + b*x] + 1296*b^3*c*d*x*Sqrt[c + d*x]*Sin[a + b*x] + 648*b^3*d^2*x^2*Sqrt[c + d*x]*Sin[a + b*x] + 72*b^3*c^2*Sqrt[c + d*x]*Sin[3*(a + b*x)] - 30*b*d^2*Sqrt[c + d*x]*Sin[3*(a + b*x)] + 144*b^3*c*d*x*Sqrt[c + d*x]*Sin[3*(a + b*x)] + 72*b^3*d^2*x^2*Sqrt[c + d*x]*Sin[3*(a + b*x)])/(864*b^4)","A",1
57,1,390,354,1.6003031,"\int (c+d x)^{3/2} \cos ^3(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^3,x]","\frac{-81 \sqrt{2 \pi } d \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 } 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)+\sqrt{6 \pi } 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)+81 \sqrt{2 \pi } d \sin \left(a-\frac{b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+108 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (a+b x)+108 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (a+b x)+12 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))+12 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))+162 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (a+b x)+6 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))}{144 b^2 \sqrt{\frac{b}{d}}}","-\frac{9 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \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)}{24 b^{5/2}}+\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \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)}{24 b^{5/2}}+\frac{9 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}+\frac{d \sqrt{c+d x} \cos ^3(a+b x)}{6 b^2}+\frac{d \sqrt{c+d x} \cos (a+b x)}{b^2}+\frac{2 (c+d x)^{3/2} \sin (a+b x)}{3 b}+\frac{(c+d x)^{3/2} \sin (a+b x) \cos ^2(a+b x)}{3 b}",1,"(162*Sqrt[b/d]*d*Sqrt[c + d*x]*Cos[a + b*x] + 6*Sqrt[b/d]*d*Sqrt[c + d*x]*Cos[3*(a + b*x)] - 81*d*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - d*Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + d*Sqrt[6*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 81*d*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[a - (b*c)/d] + 108*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Sin[a + b*x] + 108*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Sin[a + b*x] + 12*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)] + 12*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Sin[3*(a + b*x)])/(144*b^2*Sqrt[b/d])","A",1
58,1,254,304,0.4238786,"\int \sqrt{c+d x} \cos ^3(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^3,x]","\frac{i \sqrt{c+d x} e^{-\frac{3 i (a d+b c)}{d}} \left(-27 e^{2 i \left(2 a+\frac{b c}{d}\right)} \sqrt{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)+27 e^{2 i a+\frac{4 i b c}{d}} \sqrt{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)+\sqrt{3} \left(e^{\frac{6 i b c}{d}} \sqrt{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{3}{2},\frac{3 i b (c+d x)}{d}\right)-e^{6 i a} \sqrt{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{3}{2},-\frac{3 i b (c+d x)}{d}\right)\right)\right)}{72 b \sqrt{\frac{b^2 (c+d x)^2}{d^2}}}","-\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \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)}{12 b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{d} \sin \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{3 \sqrt{\frac{\pi }{2}} \sqrt{d} \cos \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{\sqrt{\frac{\pi }{6}} \sqrt{d} \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)}{12 b^{3/2}}+\frac{3 \sqrt{c+d x} \sin (a+b x)}{4 b}+\frac{\sqrt{c+d x} \sin (3 a+3 b x)}{12 b}",1,"((I/72)*Sqrt[c + d*x]*(-27*E^((2*I)*(2*a + (b*c)/d))*Sqrt[(I*b*(c + d*x))/d]*Gamma[3/2, ((-I)*b*(c + d*x))/d] + 27*E^((2*I)*a + ((4*I)*b*c)/d)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[3/2, (I*b*(c + d*x))/d] + Sqrt[3]*(-(E^((6*I)*a)*Sqrt[(I*b*(c + d*x))/d]*Gamma[3/2, ((-3*I)*b*(c + d*x))/d]) + E^(((6*I)*b*c)/d)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[3/2, ((3*I)*b*(c + d*x))/d])))/(b*E^(((3*I)*(b*c + a*d))/d)*Sqrt[(b^2*(c + d*x)^2)/d^2])","C",1
59,1,236,257,0.3557402,"\int \frac{\cos ^3(a+b x)}{\sqrt{c+d x}} \, dx","Integrate[Cos[a + b*x]^3/Sqrt[c + d*x],x]","\frac{i e^{-\frac{3 i (a d+b c)}{d}} \left(-9 e^{2 i \left(2 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)+9 e^{2 i a+\frac{4 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)+\sqrt{3} \left(e^{\frac{6 i b c}{d}} \sqrt{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{1}{2},\frac{3 i b (c+d x)}{d}\right)-e^{6 i a} \sqrt{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{1}{2},-\frac{3 i b (c+d x)}{d}\right)\right)\right)}{24 b \sqrt{c+d x}}","\frac{3 \sqrt{\frac{\pi }{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)}{2 \sqrt{b} \sqrt{d}}+\frac{\sqrt{\frac{\pi }{6}} \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)}{2 \sqrt{b} \sqrt{d}}-\frac{\sqrt{\frac{\pi }{6}} \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)}{2 \sqrt{b} \sqrt{d}}-\frac{3 \sqrt{\frac{\pi }{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)}{2 \sqrt{b} \sqrt{d}}",1,"((I/24)*(-9*E^((2*I)*(2*a + (b*c)/d))*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[1/2, ((-I)*b*(c + d*x))/d] + 9*E^((2*I)*a + ((4*I)*b*c)/d)*Sqrt[(I*b*(c + d*x))/d]*Gamma[1/2, (I*b*(c + d*x))/d] + Sqrt[3]*(-(E^((6*I)*a)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[1/2, ((-3*I)*b*(c + d*x))/d]) + E^(((6*I)*b*c)/d)*Sqrt[(I*b*(c + d*x))/d]*Gamma[1/2, ((3*I)*b*(c + d*x))/d])))/(b*E^(((3*I)*(b*c + a*d))/d)*Sqrt[c + d*x])","C",1
60,1,299,271,1.563423,"\int \frac{\cos ^3(a+b x)}{(c+d x)^{3/2}} \, dx","Integrate[Cos[a + b*x]^3/(c + d*x)^(3/2),x]","-\frac{\sqrt{6 \pi } \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)+3 \sqrt{2 \pi } \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)+3 \sqrt{2 \pi } \sqrt{\frac{b}{d}} \sqrt{c+d x} \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 } \sqrt{\frac{b}{d}} \sqrt{c+d x} \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)+3 \cos (a+b x)+\cos (3 (a+b x))}{2 d \sqrt{c+d x}}","-\frac{\sqrt{\frac{3 \pi }{2}} \sqrt{b} \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^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{b} \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^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{b} \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^{3/2}}-\frac{\sqrt{\frac{3 \pi }{2}} \sqrt{b} \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^{3/2}}-\frac{2 \cos ^3(a+b x)}{d \sqrt{c+d x}}",1,"-1/2*(3*Cos[a + b*x] + Cos[3*(a + b*x)] + 3*Sqrt[b/d]*Sqrt[2*Pi]*Sqrt[c + d*x]*Cos[a - (b*c)/d]*FresnelS[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]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + 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] + 3*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])/(d*Sqrt[c + d*x])","A",1
61,1,268,292,2.0066305,"\int \frac{\cos ^3(a+b x)}{(c+d x)^{5/2}} \, dx","Integrate[Cos[a + b*x]^3/(c + d*x)^(5/2),x]","\frac{-3 d e^{i \left(a-\frac{b c}{d}\right)} \left(-\frac{i b (c+d x)}{d}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{i b (c+d x)}{d}\right)-3 d e^{-i \left(a-\frac{b c}{d}\right)} \left(\frac{i b (c+d x)}{d}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{i b (c+d x)}{d}\right)-3 \sqrt{3} d e^{3 i \left(a-\frac{b c}{d}\right)} \left(-\frac{i b (c+d x)}{d}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{3 i b (c+d x)}{d}\right)-3 \sqrt{3} d e^{-3 i \left(a-\frac{b c}{d}\right)} \left(\frac{i b (c+d x)}{d}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{3 i b (c+d x)}{d}\right)+24 b (c+d x) \sin (a+b x) \cos ^2(a+b x)-4 d \cos ^3(a+b x)}{6 d^2 (c+d x)^{3/2}}","-\frac{\sqrt{2 \pi } b^{3/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{5/2}}-\frac{\sqrt{6 \pi } b^{3/2} \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)}{d^{5/2}}+\frac{\sqrt{6 \pi } b^{3/2} \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)}{d^{5/2}}+\frac{\sqrt{2 \pi } b^{3/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{d^{5/2}}+\frac{4 b \sin (a+b x) \cos ^2(a+b x)}{d^2 \sqrt{c+d x}}-\frac{2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}",1,"(-4*d*Cos[a + b*x]^3 - 3*d*E^(I*(a - (b*c)/d))*(((-I)*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((-I)*b*(c + d*x))/d] - (3*d*((I*b*(c + d*x))/d)^(3/2)*Gamma[1/2, (I*b*(c + d*x))/d])/E^(I*(a - (b*c)/d)) - 3*Sqrt[3]*d*E^((3*I)*(a - (b*c)/d))*(((-I)*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((-3*I)*b*(c + d*x))/d] - (3*Sqrt[3]*d*((I*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((3*I)*b*(c + d*x))/d])/E^((3*I)*(a - (b*c)/d)) + 24*b*(c + d*x)*Cos[a + b*x]^2*Sin[a + b*x])/(6*d^2*(c + d*x)^(3/2))","C",1
62,1,1429,356,6.3092487,"\int \frac{\cos ^3(a+b x)}{(c+d x)^{7/2}} \, dx","Integrate[Cos[a + b*x]^3/(c + d*x)^(7/2),x]","\frac{3}{4} \left(\cos (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)-\sin (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)\right)+\frac{1}{4} \left(\cos (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)-\sin (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)\right)","\frac{6 \sqrt{6 \pi } b^{5/2} \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)}{5 d^{7/2}}+\frac{2 \sqrt{2 \pi } b^{5/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{5 d^{7/2}}+\frac{2 \sqrt{2 \pi } b^{5/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{5 d^{7/2}}+\frac{6 \sqrt{6 \pi } b^{5/2} \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)}{5 d^{7/2}}+\frac{24 b^2 \cos ^3(a+b x)}{5 d^3 \sqrt{c+d x}}-\frac{16 b^2 \cos (a+b x)}{5 d^3 \sqrt{c+d x}}+\frac{4 b \sin (a+b x) \cos ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac{2 \cos ^3(a+b x)}{5 d (c+d x)^{5/2}}",1,"(3*(-(Sin[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))) + Cos[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 + (-(Sin[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))) + Cos[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
63,1,55,49,0.010672,"\int x^{3/2} \cos (x) \, dx","Integrate[x^(3/2)*Cos[x],x]","\frac{\sqrt{x} \Gamma \left(\frac{5}{2},-i x\right)}{2 \sqrt{-i x}}+\frac{\sqrt{x} \Gamma \left(\frac{5}{2},i x\right)}{2 \sqrt{i x}}","-\frac{3}{2} \sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{x}\right)+x^{3/2} \sin (x)+\frac{3}{2} \sqrt{x} \cos (x)",1,"(Sqrt[x]*Gamma[5/2, (-I)*x])/(2*Sqrt[(-I)*x]) + (Sqrt[x]*Gamma[5/2, I*x])/(2*Sqrt[I*x])","C",1
64,1,48,36,0.0058557,"\int \sqrt{x} \cos (x) \, dx","Integrate[Sqrt[x]*Cos[x],x]","\frac{\sqrt{-i x} \Gamma \left(\frac{3}{2},-i x\right)+\sqrt{i x} \Gamma \left(\frac{3}{2},i x\right)}{2 \sqrt{x}}","\sqrt{x} \sin (x)-\sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }} \sqrt{x}\right)",1,"(Sqrt[(-I)*x]*Gamma[3/2, (-I)*x] + Sqrt[I*x]*Gamma[3/2, I*x])/(2*Sqrt[x])","C",1
65,1,51,24,0.0064169,"\int \frac{\cos (x)}{\sqrt{x}} \, dx","Integrate[Cos[x]/Sqrt[x],x]","-\frac{i \left(\sqrt{-i x} \Gamma \left(\frac{1}{2},-i x\right)-\sqrt{i x} \Gamma \left(\frac{1}{2},i x\right)\right)}{2 \sqrt{x}}","\sqrt{2 \pi } C\left(\sqrt{\frac{2}{\pi }} \sqrt{x}\right)",1,"((-1/2*I)*(Sqrt[(-I)*x]*Gamma[1/2, (-I)*x] - Sqrt[I*x]*Gamma[1/2, I*x]))/Sqrt[x]","C",1
66,1,63,35,0.0432051,"\int \frac{\cos (x)}{x^{3/2}} \, dx","Integrate[Cos[x]/x^(3/2),x]","\frac{-e^{-i x} \left(1+e^{2 i x}\right)+\sqrt{-i x} \Gamma \left(\frac{1}{2},-i x\right)+\sqrt{i x} \Gamma \left(\frac{1}{2},i x\right)}{\sqrt{x}}","-2 \sqrt{2 \pi } S\left(\sqrt{\frac{2}{\pi }} \sqrt{x}\right)-\frac{2 \cos (x)}{\sqrt{x}}",1,"(-((1 + E^((2*I)*x))/E^(I*x)) + Sqrt[(-I)*x]*Gamma[1/2, (-I)*x] + Sqrt[I*x]*Gamma[1/2, I*x])/Sqrt[x]","C",1
67,1,122,183,0.1079867,"\int (c+d x)^{4/3} \cos (a+b x) \, dx","Integrate[(c + d*x)^(4/3)*Cos[a + b*x],x]","\frac{d \sqrt[3]{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(\frac{e^{2 i a} \Gamma \left(\frac{7}{3},-\frac{i b (c+d x)}{d}\right)}{\sqrt[3]{-\frac{i b (c+d x)}{d}}}+\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{7}{3},\frac{i b (c+d x)}{d}\right)}{\sqrt[3]{\frac{i b (c+d x)}{d}}}\right)}{2 b^2}","\frac{2 i d^2 e^{i \left(a-\frac{b c}{d}\right)} \left(-\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(\frac{1}{3},-\frac{i b (c+d x)}{d}\right)}{9 b^3 (c+d x)^{2/3}}-\frac{2 i d^2 e^{-i \left(a-\frac{b c}{d}\right)} \left(\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(\frac{1}{3},\frac{i b (c+d x)}{d}\right)}{9 b^3 (c+d x)^{2/3}}+\frac{4 d \sqrt[3]{c+d x} \cos (a+b x)}{3 b^2}+\frac{(c+d x)^{4/3} \sin (a+b x)}{b}",1,"(d*(c + d*x)^(1/3)*((E^((2*I)*a)*Gamma[7/3, ((-I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^(1/3) + (E^(((2*I)*b*c)/d)*Gamma[7/3, (I*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^(1/3)))/(2*b^2*E^((I*(b*c + a*d))/d))","A",1
68,1,124,152,0.1124758,"\int (c+d x)^{2/3} \cos (a+b x) \, dx","Integrate[(c + d*x)^(2/3)*Cos[a + b*x],x]","-\frac{i (c+d x)^{2/3} e^{-\frac{i (a d+b c)}{d}} \left(\frac{e^{2 i a} \Gamma \left(\frac{5}{3},-\frac{i b (c+d x)}{d}\right)}{\left(-\frac{i b (c+d x)}{d}\right)^{2/3}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{5}{3},\frac{i b (c+d x)}{d}\right)}{\left(\frac{i b (c+d x)}{d}\right)^{2/3}}\right)}{2 b}","\frac{d e^{i \left(a-\frac{b c}{d}\right)} \sqrt[3]{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{2}{3},-\frac{i b (c+d x)}{d}\right)}{3 b^2 \sqrt[3]{c+d x}}+\frac{d e^{-i \left(a-\frac{b c}{d}\right)} \sqrt[3]{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{2}{3},\frac{i b (c+d x)}{d}\right)}{3 b^2 \sqrt[3]{c+d x}}+\frac{(c+d x)^{2/3} \sin (a+b x)}{b}",1,"((-1/2*I)*(c + d*x)^(2/3)*((E^((2*I)*a)*Gamma[5/3, ((-I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^(2/3) - (E^(((2*I)*b*c)/d)*Gamma[5/3, (I*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^(2/3)))/(b*E^((I*(b*c + a*d))/d))","A",1
69,1,124,152,0.0938332,"\int \sqrt[3]{c+d x} \cos (a+b x) \, dx","Integrate[(c + d*x)^(1/3)*Cos[a + b*x],x]","-\frac{i \sqrt[3]{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(\frac{e^{2 i a} \Gamma \left(\frac{4}{3},-\frac{i b (c+d x)}{d}\right)}{\sqrt[3]{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{4}{3},\frac{i b (c+d x)}{d}\right)}{\sqrt[3]{\frac{i b (c+d x)}{d}}}\right)}{2 b}","\frac{d e^{i \left(a-\frac{b c}{d}\right)} \left(-\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(\frac{1}{3},-\frac{i b (c+d x)}{d}\right)}{6 b^2 (c+d x)^{2/3}}+\frac{d e^{-i \left(a-\frac{b c}{d}\right)} \left(\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(\frac{1}{3},\frac{i b (c+d x)}{d}\right)}{6 b^2 (c+d x)^{2/3}}+\frac{\sqrt[3]{c+d x} \sin (a+b x)}{b}",1,"((-1/2*I)*(c + d*x)^(1/3)*((E^((2*I)*a)*Gamma[4/3, ((-I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^(1/3) - (E^(((2*I)*b*c)/d)*Gamma[4/3, (I*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^(1/3)))/(b*E^((I*(b*c + a*d))/d))","A",1
70,1,124,135,0.0585179,"\int \frac{\cos (a+b x)}{\sqrt[3]{c+d x}} \, dx","Integrate[Cos[a + b*x]/(c + d*x)^(1/3),x]","\frac{i e^{-\frac{i (a d+b c)}{d}} \left(e^{\frac{2 i b c}{d}} \sqrt[3]{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{2}{3},\frac{i b (c+d x)}{d}\right)-e^{2 i a} \sqrt[3]{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{2}{3},-\frac{i b (c+d x)}{d}\right)\right)}{2 b \sqrt[3]{c+d x}}","\frac{i e^{-i \left(a-\frac{b c}{d}\right)} \sqrt[3]{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{2}{3},\frac{i b (c+d x)}{d}\right)}{2 b \sqrt[3]{c+d x}}-\frac{i e^{i \left(a-\frac{b c}{d}\right)} \sqrt[3]{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{2}{3},-\frac{i b (c+d x)}{d}\right)}{2 b \sqrt[3]{c+d x}}",1,"((I/2)*(-(E^((2*I)*a)*(((-I)*b*(c + d*x))/d)^(1/3)*Gamma[2/3, ((-I)*b*(c + d*x))/d]) + E^(((2*I)*b*c)/d)*((I*b*(c + d*x))/d)^(1/3)*Gamma[2/3, (I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)*(c + d*x)^(1/3))","A",1
71,1,124,135,0.0607587,"\int \frac{\cos (a+b x)}{(c+d x)^{2/3}} \, dx","Integrate[Cos[a + b*x]/(c + d*x)^(2/3),x]","\frac{i e^{-\frac{i (a d+b c)}{d}} \left(e^{\frac{2 i b c}{d}} \left(\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(\frac{1}{3},\frac{i b (c+d x)}{d}\right)-e^{2 i a} \left(-\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(\frac{1}{3},-\frac{i b (c+d x)}{d}\right)\right)}{2 b (c+d x)^{2/3}}","\frac{i e^{-i \left(a-\frac{b c}{d}\right)} \left(\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(\frac{1}{3},\frac{i b (c+d x)}{d}\right)}{2 b (c+d x)^{2/3}}-\frac{i e^{i \left(a-\frac{b c}{d}\right)} \left(-\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(\frac{1}{3},-\frac{i b (c+d x)}{d}\right)}{2 b (c+d x)^{2/3}}",1,"((I/2)*(-(E^((2*I)*a)*(((-I)*b*(c + d*x))/d)^(2/3)*Gamma[1/3, ((-I)*b*(c + d*x))/d]) + E^(((2*I)*b*c)/d)*((I*b*(c + d*x))/d)^(2/3)*Gamma[1/3, (I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)*(c + d*x)^(2/3))","A",1
72,1,121,151,0.0555036,"\int \frac{\cos (a+b x)}{(c+d x)^{4/3}} \, dx","Integrate[Cos[a + b*x]/(c + d*x)^(4/3),x]","-\frac{e^{-\frac{i (a d+b c)}{d}} \left(e^{2 i a} \sqrt[3]{-\frac{i b (c+d x)}{d}} \Gamma \left(-\frac{1}{3},-\frac{i b (c+d x)}{d}\right)+e^{\frac{2 i b c}{d}} \sqrt[3]{\frac{i b (c+d x)}{d}} \Gamma \left(-\frac{1}{3},\frac{i b (c+d x)}{d}\right)\right)}{2 d \sqrt[3]{c+d x}}","-\frac{3 \cos (a+b x)}{d \sqrt[3]{c+d x}}+\frac{3 e^{i \left(a-\frac{b c}{d}\right)} \sqrt[3]{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{2}{3},-\frac{i b (c+d x)}{d}\right)}{2 d \sqrt[3]{c+d x}}+\frac{3 e^{-i \left(a-\frac{b c}{d}\right)} \sqrt[3]{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{2}{3},\frac{i b (c+d x)}{d}\right)}{2 d \sqrt[3]{c+d x}}",1,"-1/2*(E^((2*I)*a)*(((-I)*b*(c + d*x))/d)^(1/3)*Gamma[-1/3, ((-I)*b*(c + d*x))/d] + E^(((2*I)*b*c)/d)*((I*b*(c + d*x))/d)^(1/3)*Gamma[-1/3, (I*b*(c + d*x))/d])/(d*E^((I*(b*c + a*d))/d)*(c + d*x)^(1/3))","A",1
73,1,121,153,0.0533511,"\int \frac{\cos (a+b x)}{(c+d x)^{5/3}} \, dx","Integrate[Cos[a + b*x]/(c + d*x)^(5/3),x]","-\frac{e^{-\frac{i (a d+b c)}{d}} \left(e^{2 i a} \left(-\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(-\frac{2}{3},-\frac{i b (c+d x)}{d}\right)+e^{\frac{2 i b c}{d}} \left(\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(-\frac{2}{3},\frac{i b (c+d x)}{d}\right)\right)}{2 d (c+d x)^{2/3}}","-\frac{3 \cos (a+b x)}{2 d (c+d x)^{2/3}}+\frac{3 e^{i \left(a-\frac{b c}{d}\right)} \left(-\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(\frac{1}{3},-\frac{i b (c+d x)}{d}\right)}{4 d (c+d x)^{2/3}}+\frac{3 e^{-i \left(a-\frac{b c}{d}\right)} \left(\frac{i b (c+d x)}{d}\right)^{2/3} \Gamma \left(\frac{1}{3},\frac{i b (c+d x)}{d}\right)}{4 d (c+d x)^{2/3}}",1,"-1/2*(E^((2*I)*a)*(((-I)*b*(c + d*x))/d)^(2/3)*Gamma[-2/3, ((-I)*b*(c + d*x))/d] + E^(((2*I)*b*c)/d)*((I*b*(c + d*x))/d)^(2/3)*Gamma[-2/3, (I*b*(c + d*x))/d])/(d*E^((I*(b*c + a*d))/d)*(c + d*x)^(2/3))","A",1
74,1,125,182,0.0523563,"\int \frac{\cos (a+b x)}{(c+d x)^{7/3}} \, dx","Integrate[Cos[a + b*x]/(c + d*x)^(7/3),x]","\frac{i b e^{-\frac{i (a d+b c)}{d}} \left(e^{2 i a} \sqrt[3]{-\frac{i b (c+d x)}{d}} \Gamma \left(-\frac{4}{3},-\frac{i b (c+d x)}{d}\right)-e^{\frac{2 i b c}{d}} \sqrt[3]{\frac{i b (c+d x)}{d}} \Gamma \left(-\frac{4}{3},\frac{i b (c+d x)}{d}\right)\right)}{2 d^2 \sqrt[3]{c+d x}}","\frac{9 b \sin (a+b x)}{4 d^2 \sqrt[3]{c+d x}}+\frac{9 i b e^{i \left(a-\frac{b c}{d}\right)} \sqrt[3]{-\frac{i b (c+d x)}{d}} \Gamma \left(\frac{2}{3},-\frac{i b (c+d x)}{d}\right)}{8 d^2 \sqrt[3]{c+d x}}-\frac{9 i b e^{-i \left(a-\frac{b c}{d}\right)} \sqrt[3]{\frac{i b (c+d x)}{d}} \Gamma \left(\frac{2}{3},\frac{i b (c+d x)}{d}\right)}{8 d^2 \sqrt[3]{c+d x}}-\frac{3 \cos (a+b x)}{4 d (c+d x)^{4/3}}",1,"((I/2)*b*(E^((2*I)*a)*(((-I)*b*(c + d*x))/d)^(1/3)*Gamma[-4/3, ((-I)*b*(c + d*x))/d] - E^(((2*I)*b*c)/d)*((I*b*(c + d*x))/d)^(1/3)*Gamma[-4/3, (I*b*(c + d*x))/d]))/(d^2*E^((I*(b*c + a*d))/d)*(c + d*x)^(1/3))","A",1
75,-1,0,15,0,"\int x \sqrt{\cos (a+b x)} \, dx","Integrate[x*Sqrt[Cos[a + b*x]],x]","\text{\$Aborted}","\text{Int}\left(x \sqrt{\cos (a+b x)},x\right)",0,"$Aborted","F",-1
76,1,16,16,0.0135142,"\int \sqrt{\cos (a+b x)} \, dx","Integrate[Sqrt[Cos[a + b*x]],x]","\frac{2 E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b}","\frac{2 E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b}",1,"(2*EllipticE[(a + b*x)/2, 2])/b","A",1
77,0,0,17,0.6568817,"\int \frac{\sqrt{\cos (a+b x)}}{x} \, dx","Integrate[Sqrt[Cos[a + b*x]]/x,x]","\int \frac{\sqrt{\cos (a+b x)}}{x} \, dx","\text{Int}\left(\frac{\sqrt{\cos (a+b x)}}{x},x\right)",0,"Integrate[Sqrt[Cos[a + b*x]]/x, x]","A",-1
78,0,0,61,1.9399849,"\int x \cos ^{\frac{3}{2}}(a+b x) \, dx","Integrate[x*Cos[a + b*x]^(3/2),x]","\int x \cos ^{\frac{3}{2}}(a+b x) \, dx","\frac{1}{3} \text{Int}\left(\frac{x}{\sqrt{\cos (a+b x)}},x\right)+\frac{4 \cos ^{\frac{3}{2}}(a+b x)}{9 b^2}+\frac{2 x \sin (a+b x) \sqrt{\cos (a+b x)}}{3 b}",0,"Integrate[x*Cos[a + b*x]^(3/2), x]","A",-1
79,1,36,42,0.0235905,"\int \cos ^{\frac{3}{2}}(a+b x) \, dx","Integrate[Cos[a + b*x]^(3/2),x]","\frac{2 \left(F\left(\left.\frac{1}{2} (a+b x)\right|2\right)+\sin (a+b x) \sqrt{\cos (a+b x)}\right)}{3 b}","\frac{2 F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{3 b}+\frac{2 \sin (a+b x) \sqrt{\cos (a+b x)}}{3 b}",1,"(2*(EllipticF[(a + b*x)/2, 2] + Sqrt[Cos[a + b*x]]*Sin[a + b*x]))/(3*b)","A",1
80,0,0,17,15.4810201,"\int \frac{\cos ^{\frac{3}{2}}(a+b x)}{x} \, dx","Integrate[Cos[a + b*x]^(3/2)/x,x]","\int \frac{\cos ^{\frac{3}{2}}(a+b x)}{x} \, dx","\text{Int}\left(\frac{\cos ^{\frac{3}{2}}(a+b x)}{x},x\right)",0,"Integrate[Cos[a + b*x]^(3/2)/x, x]","A",-1
81,1,40,42,0.4330266,"\int \left(-\frac{x}{3 \sqrt{\cos (a+b x)}}+x \cos ^{\frac{3}{2}}(a+b x)\right) \, dx","Integrate[-1/3*x/Sqrt[Cos[a + b*x]] + x*Cos[a + b*x]^(3/2),x]","\frac{\sqrt{\cos (a+b x)} \left(4 x \sin (a+b x)+\frac{8 \cos (a+b x)}{3 b}\right)}{6 b}","\frac{4 \cos ^{\frac{3}{2}}(a+b x)}{9 b^2}+\frac{2 x \sin (a+b x) \sqrt{\cos (a+b x)}}{3 b}",1,"(Sqrt[Cos[a + b*x]]*((8*Cos[a + b*x])/(3*b) + 4*x*Sin[a + b*x]))/(6*b)","A",1
82,0,0,63,4.9721259,"\int \frac{\cos ^{\frac{3}{2}}(x)}{x^3} \, dx","Integrate[Cos[x]^(3/2)/x^3,x]","\int \frac{\cos ^{\frac{3}{2}}(x)}{x^3} \, dx","-\frac{9}{8} \text{Int}\left(\frac{\cos ^{\frac{3}{2}}(x)}{x},x\right)+\frac{3}{8} \text{Int}\left(\frac{1}{x \sqrt{\cos (x)}},x\right)-\frac{\cos ^{\frac{3}{2}}(x)}{2 x^2}+\frac{3 \sin (x) \sqrt{\cos (x)}}{4 x}",0,"Integrate[Cos[x]^(3/2)/x^3, x]","A",-1
83,0,0,15,0.2969801,"\int \frac{x}{\sqrt{\cos (a+b x)}} \, dx","Integrate[x/Sqrt[Cos[a + b*x]],x]","\int \frac{x}{\sqrt{\cos (a+b x)}} \, dx","\text{Int}\left(\frac{x}{\sqrt{\cos (a+b x)}},x\right)",0,"Integrate[x/Sqrt[Cos[a + b*x]], x]","A",-1
84,1,16,16,0.0156971,"\int \frac{1}{\sqrt{\cos (a+b x)}} \, dx","Integrate[1/Sqrt[Cos[a + b*x]],x]","\frac{2 F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b}","\frac{2 F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b}",1,"(2*EllipticF[(a + b*x)/2, 2])/b","A",1
85,0,0,17,0.1975808,"\int \frac{1}{x \sqrt{\cos (a+b x)}} \, dx","Integrate[1/(x*Sqrt[Cos[a + b*x]]),x]","\int \frac{1}{x \sqrt{\cos (a+b x)}} \, dx","\text{Int}\left(\frac{1}{x \sqrt{\cos (a+b x)}},x\right)",0,"Integrate[1/(x*Sqrt[Cos[a + b*x]]), x]","A",-1
86,-1,0,55,0,"\int \frac{x}{\cos ^{\frac{3}{2}}(a+b x)} \, dx","Integrate[x/Cos[a + b*x]^(3/2),x]","\text{\$Aborted}","-\text{Int}\left(x \sqrt{\cos (a+b x)},x\right)+\frac{4 \sqrt{\cos (a+b x)}}{b^2}+\frac{2 x \sin (a+b x)}{b \sqrt{\cos (a+b x)}}",0,"$Aborted","F",-1
87,1,38,38,0.0487315,"\int \frac{1}{\cos ^{\frac{3}{2}}(a+b x)} \, dx","Integrate[Cos[a + b*x]^(-3/2),x]","\frac{2 \sin (a+b x)}{b \sqrt{\cos (a+b x)}}-\frac{2 E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b}","\frac{2 \sin (a+b x)}{b \sqrt{\cos (a+b x)}}-\frac{2 E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b}",1,"(-2*EllipticE[(a + b*x)/2, 2])/b + (2*Sin[a + b*x])/(b*Sqrt[Cos[a + b*x]])","A",1
88,0,0,17,14.9105798,"\int \frac{1}{x \cos ^{\frac{3}{2}}(a+b x)} \, dx","Integrate[1/(x*Cos[a + b*x]^(3/2)),x]","\int \frac{1}{x \cos ^{\frac{3}{2}}(a+b x)} \, dx","\text{Int}\left(\frac{1}{x \cos ^{\frac{3}{2}}(a+b x)},x\right)",0,"Integrate[1/(x*Cos[a + b*x]^(3/2)), x]","A",-1
89,1,33,38,0.3976222,"\int \left(\frac{x}{\cos ^{\frac{3}{2}}(a+b x)}+x \sqrt{\cos (a+b x)}\right) \, dx","Integrate[x/Cos[a + b*x]^(3/2) + x*Sqrt[Cos[a + b*x]],x]","\frac{2 (b x \sin (a+b x)+2 \cos (a+b x))}{b^2 \sqrt{\cos (a+b x)}}","\frac{4 \sqrt{\cos (a+b x)}}{b^2}+\frac{2 x \sin (a+b x)}{b \sqrt{\cos (a+b x)}}",1,"(2*(2*Cos[a + b*x] + b*x*Sin[a + b*x]))/(b^2*Sqrt[Cos[a + b*x]])","A",1
90,1,17,20,0.0866452,"\int \left(\frac{x}{\cos ^{\frac{3}{2}}(x)}+x \sqrt{\cos (x)}\right) \, dx","Integrate[x/Cos[x]^(3/2) + x*Sqrt[Cos[x]],x]","\frac{2 (x \sin (x)+2 \cos (x))}{\sqrt{\cos (x)}}","4 \sqrt{\cos (x)}+\frac{2 x \sin (x)}{\sqrt{\cos (x)}}",1,"(2*(2*Cos[x] + x*Sin[x]))/Sqrt[Cos[x]]","A",1
91,1,17,24,0.0799311,"\int \left(\frac{x}{\cos ^{\frac{5}{2}}(x)}-\frac{x}{3 \sqrt{\cos (x)}}\right) \, dx","Integrate[x/Cos[x]^(5/2) - x/(3*Sqrt[Cos[x]]),x]","-\frac{8-4 x \tan (x)}{6 \sqrt{\cos (x)}}","\frac{2 x \sin (x)}{3 \cos ^{\frac{3}{2}}(x)}-\frac{4}{3 \sqrt{\cos (x)}}",1,"-1/6*(8 - 4*x*Tan[x])/Sqrt[Cos[x]]","A",1
92,1,33,47,0.1396928,"\int \left(\frac{x}{\cos ^{\frac{7}{2}}(x)}+\frac{3}{5} x \sqrt{\cos (x)}\right) \, dx","Integrate[x/Cos[x]^(7/2) + (3*x*Sqrt[Cos[x]])/5,x]","\frac{21 x \sin (x)+9 x \sin (3 x)+46 \cos (x)+18 \cos (3 x)}{30 \cos ^{\frac{5}{2}}(x)}","-\frac{4}{15 \cos ^{\frac{3}{2}}(x)}+\frac{12 \sqrt{\cos (x)}}{5}+\frac{2 x \sin (x)}{5 \cos ^{\frac{5}{2}}(x)}+\frac{6 x \sin (x)}{5 \sqrt{\cos (x)}}",1,"(46*Cos[x] + 18*Cos[3*x] + 21*x*Sin[x] + 9*x*Sin[3*x])/(30*Cos[x]^(5/2))","A",1
93,1,29,32,0.1501344,"\int \left(\frac{x^2}{\cos ^{\frac{3}{2}}(x)}+x^2 \sqrt{\cos (x)}\right) \, dx","Integrate[x^2/Cos[x]^(3/2) + x^2*Sqrt[Cos[x]],x]","2 \left(\frac{x (x \sin (x)+4 \cos (x))}{\sqrt{\cos (x)}}-8 E\left(\left.\frac{x}{2}\right|2\right)\right)","\frac{2 x^2 \sin (x)}{\sqrt{\cos (x)}}+8 x \sqrt{\cos (x)}-16 E\left(\left.\frac{x}{2}\right|2\right)",1,"2*(-8*EllipticE[x/2, 2] + (x*(4*Cos[x] + x*Sin[x]))/Sqrt[Cos[x]])","A",1
94,1,17,24,0.0947237,"\int \left(\frac{x}{\sec ^{\frac{3}{2}}(x)}-\frac{1}{3} x \sqrt{\sec (x)}\right) \, dx","Integrate[x/Sec[x]^(3/2) - (x*Sqrt[Sec[x]])/3,x]","\frac{2 (3 x \tan (x)+2)}{9 \sec ^{\frac{3}{2}}(x)}","\frac{4}{9 \sec ^{\frac{3}{2}}(x)}+\frac{2 x \sin (x)}{3 \sqrt{\sec (x)}}",1,"(2*(2 + 3*x*Tan[x]))/(9*Sec[x]^(3/2))","A",1
95,1,17,24,0.1438331,"\int \left(\frac{x}{\sec ^{\frac{5}{2}}(x)}-\frac{3 x}{5 \sqrt{\sec (x)}}\right) \, dx","Integrate[x/Sec[x]^(5/2) - (3*x)/(5*Sqrt[Sec[x]]),x]","\frac{2 (5 x \tan (x)+2)}{25 \sec ^{\frac{5}{2}}(x)}","\frac{4}{25 \sec ^{\frac{5}{2}}(x)}+\frac{2 x \sin (x)}{5 \sec ^{\frac{3}{2}}(x)}",1,"(2*(2 + 5*x*Tan[x]))/(25*Sec[x]^(5/2))","A",1
96,1,45,47,0.1097153,"\int \left(\frac{x}{\sec ^{\frac{7}{2}}(x)}-\frac{5}{21} x \sqrt{\sec (x)}\right) \, dx","Integrate[x/Sec[x]^(7/2) - (5*x*Sqrt[Sec[x]])/21,x]","\sqrt{\sec (x)} \left(\frac{13}{42} x \sin (2 x)+\frac{1}{28} x \sin (4 x)+\frac{88}{441} \cos (2 x)+\frac{1}{98} \cos (4 x)+\frac{167}{882}\right)","\frac{20}{63 \sec ^{\frac{3}{2}}(x)}+\frac{4}{49 \sec ^{\frac{7}{2}}(x)}+\frac{2 x \sin (x)}{7 \sec ^{\frac{5}{2}}(x)}+\frac{10 x \sin (x)}{21 \sqrt{\sec (x)}}",1,"Sqrt[Sec[x]]*(167/882 + (88*Cos[2*x])/441 + Cos[4*x]/98 + (13*x*Sin[2*x])/42 + (x*Sin[4*x])/28)","A",1
97,1,51,62,0.1091312,"\int \left(\frac{x^2}{\sec ^{\frac{3}{2}}(x)}-\frac{1}{3} x^2 \sqrt{\sec (x)}\right) \, dx","Integrate[x^2/Sec[x]^(3/2) - (x^2*Sqrt[Sec[x]])/3,x]","\frac{1}{27} \sqrt{\sec (x)} \left(9 x^2 \sin (2 x)+12 x-8 \sin (2 x)+12 x \cos (2 x)-16 \sqrt{\cos (x)} F\left(\left.\frac{x}{2}\right|2\right)\right)","\frac{2 x^2 \sin (x)}{3 \sqrt{\sec (x)}}+\frac{8 x}{9 \sec ^{\frac{3}{2}}(x)}-\frac{16 \sin (x)}{27 \sqrt{\sec (x)}}-\frac{16}{27} \sqrt{\cos (x)} \sqrt{\sec (x)} F\left(\left.\frac{x}{2}\right|2\right)",1,"(Sqrt[Sec[x]]*(12*x + 12*x*Cos[2*x] - 16*Sqrt[Cos[x]]*EllipticF[x/2, 2] - 8*Sin[2*x] + 9*x^2*Sin[2*x]))/27","A",1
98,0,0,21,0.8231368,"\int (c+d x)^m (b \cos (e+f x))^n \, dx","Integrate[(c + d*x)^m*(b*Cos[e + f*x])^n,x]","\int (c+d x)^m (b \cos (e+f x))^n \, dx","\text{Int}\left((c+d x)^m (b \cos (e+f x))^n,x\right)",0,"Integrate[(c + d*x)^m*(b*Cos[e + f*x])^n, x]","A",-1
99,1,253,275,0.1672375,"\int (c+d x)^m \cos ^3(a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]^3,x]","\frac{i 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 i 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{i 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 i 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{i 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,"((I/8)*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]))/(b*E^(((3*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^m)","A",1
100,1,150,162,0.1955599,"\int (c+d x)^m \cos ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]^2,x]","\frac{1}{8} (c+d x)^m \left(-\frac{i 2^{-m} e^{2 i \left(a-\frac{b c}{d}\right)} \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} e^{-2 i \left(a-\frac{b c}{d}\right)} \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{4 c+4 d x}{d m+d}\right)","-\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,"((c + d*x)^m*((4*c + 4*d*x)/(d + d*m) - (I*E^((2*I)*(a - (b*c)/d))*Gamma[1 + m, ((-2*I)*b*(c + d*x))/d])/(2^m*b*(((-I)*b*(c + d*x))/d)^m) + (I*Gamma[1 + m, ((2*I)*b*(c + d*x))/d])/(2^m*b*E^((2*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m)))/8","A",1
101,1,122,131,0.0452414,"\int (c+d x)^m \cos (a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x],x]","-\frac{i 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{i 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{i 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,"((-1/2*I)*(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))/(b*E^((I*(b*c + a*d))/d))","A",1
102,0,0,17,5.4362935,"\int (c+d x)^m \sec (a+b x) \, dx","Integrate[(c + d*x)^m*Sec[a + b*x],x]","\int (c+d x)^m \sec (a+b x) \, dx","\text{Int}\left(\sec (a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Sec[a + b*x], x]","A",-1
103,0,0,19,0.8036385,"\int (c+d x)^m \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Sec[a + b*x]^2,x]","\int (c+d x)^m \sec ^2(a+b x) \, dx","\text{Int}\left(\sec ^2(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Sec[a + b*x]^2, x]","A",-1
104,1,75,75,0.019937,"\int x^{3+m} \cos (a+b x) \, dx","Integrate[x^(3 + m)*Cos[a + b*x],x]","-\frac{e^{i a} x^m (-i b x)^{-m} \Gamma (m+4,-i b x)}{2 b^4}-\frac{e^{-i a} x^m (i b x)^{-m} \Gamma (m+4,i b x)}{2 b^4}","-\frac{e^{i a} x^m (-i b x)^{-m} \Gamma (m+4,-i b x)}{2 b^4}-\frac{e^{-i a} x^m (i b x)^{-m} \Gamma (m+4,i b x)}{2 b^4}",1,"-1/2*(E^(I*a)*x^m*Gamma[4 + m, (-I)*b*x])/(b^4*((-I)*b*x)^m) - (x^m*Gamma[4 + m, I*b*x])/(2*b^4*E^(I*a)*(I*b*x)^m)","A",1
105,1,79,79,0.0170553,"\int x^{2+m} \cos (a+b x) \, dx","Integrate[x^(2 + m)*Cos[a + b*x],x]","\frac{i e^{i a} x^m (-i b x)^{-m} \Gamma (m+3,-i b x)}{2 b^3}-\frac{i e^{-i a} x^m (i b x)^{-m} \Gamma (m+3,i b x)}{2 b^3}","\frac{i e^{i a} x^m (-i b x)^{-m} \Gamma (m+3,-i b x)}{2 b^3}-\frac{i e^{-i a} x^m (i b x)^{-m} \Gamma (m+3,i b x)}{2 b^3}",1,"((I/2)*E^(I*a)*x^m*Gamma[3 + m, (-I)*b*x])/(b^3*((-I)*b*x)^m) - ((I/2)*x^m*Gamma[3 + m, I*b*x])/(b^3*E^(I*a)*(I*b*x)^m)","A",1
106,1,75,75,0.0156876,"\int x^{1+m} \cos (a+b x) \, dx","Integrate[x^(1 + m)*Cos[a + b*x],x]","\frac{e^{i a} x^m (-i b x)^{-m} \Gamma (m+2,-i b x)}{2 b^2}+\frac{e^{-i a} x^m (i b x)^{-m} \Gamma (m+2,i b x)}{2 b^2}","\frac{e^{i a} x^m (-i b x)^{-m} \Gamma (m+2,-i b x)}{2 b^2}+\frac{e^{-i a} x^m (i b x)^{-m} \Gamma (m+2,i b x)}{2 b^2}",1,"(E^(I*a)*x^m*Gamma[2 + m, (-I)*b*x])/(2*b^2*((-I)*b*x)^m) + (x^m*Gamma[2 + m, I*b*x])/(2*b^2*E^(I*a)*(I*b*x)^m)","A",1
107,1,79,79,0.0145913,"\int x^m \cos (a+b x) \, dx","Integrate[x^m*Cos[a + b*x],x]","\frac{i e^{-i a} x^m (i b x)^{-m} \Gamma (m+1,i b x)}{2 b}-\frac{i e^{i a} x^m (-i b x)^{-m} \Gamma (m+1,-i b x)}{2 b}","\frac{i e^{-i a} x^m (i b x)^{-m} \Gamma (m+1,i b x)}{2 b}-\frac{i e^{i a} x^m (-i b x)^{-m} \Gamma (m+1,-i b x)}{2 b}",1,"((-1/2*I)*E^(I*a)*x^m*Gamma[1 + m, (-I)*b*x])/(b*((-I)*b*x)^m) + ((I/2)*x^m*Gamma[1 + m, I*b*x])/(b*E^(I*a)*(I*b*x)^m)","A",1
108,1,62,65,0.0225281,"\int x^{-1+m} \cos (a+b x) \, dx","Integrate[x^(-1 + m)*Cos[a + b*x],x]","\frac{1}{2} 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} e^{i a} x^m (-i b x)^{-m} \Gamma (m,-i b x)-\frac{1}{2} e^{-i a} x^m (i b x)^{-m} \Gamma (m,i b x)",1,"(x^m*(-((E^((2*I)*a)*Gamma[m, (-I)*b*x])/((-I)*b*x)^m) - Gamma[m, I*b*x]/(I*b*x)^m))/(2*E^(I*a))","A",1
109,1,75,75,0.0170213,"\int x^{-2+m} \cos (a+b x) \, dx","Integrate[x^(-2 + m)*Cos[a + b*x],x]","\frac{1}{2} i e^{i a} b x^m (-i b x)^{-m} \Gamma (m-1,-i b x)-\frac{1}{2} i e^{-i a} b x^m (i b x)^{-m} \Gamma (m-1,i b x)","\frac{1}{2} i e^{i a} b x^m (-i b x)^{-m} \Gamma (m-1,-i b x)-\frac{1}{2} i e^{-i a} b x^m (i b x)^{-m} \Gamma (m-1,i b x)",1,"((I/2)*b*E^(I*a)*x^m*Gamma[-1 + m, (-I)*b*x])/((-I)*b*x)^m - ((I/2)*b*x^m*Gamma[-1 + m, I*b*x])/(E^(I*a)*(I*b*x)^m)","A",1
110,1,75,75,0.0158594,"\int x^{-3+m} \cos (a+b x) \, dx","Integrate[x^(-3 + m)*Cos[a + b*x],x]","\frac{1}{2} e^{i a} b^2 x^m (-i b x)^{-m} \Gamma (m-2,-i b x)+\frac{1}{2} e^{-i a} b^2 x^m (i b x)^{-m} \Gamma (m-2,i b x)","\frac{1}{2} e^{i a} b^2 x^m (-i b x)^{-m} \Gamma (m-2,-i b x)+\frac{1}{2} e^{-i a} b^2 x^m (i b x)^{-m} \Gamma (m-2,i b x)",1,"(b^2*E^(I*a)*x^m*Gamma[-2 + m, (-I)*b*x])/(2*((-I)*b*x)^m) + (b^2*x^m*Gamma[-2 + m, I*b*x])/(2*E^(I*a)*(I*b*x)^m)","A",1
111,1,92,99,0.1110734,"\int x^{3+m} \cos ^2(a+b x) \, dx","Integrate[x^(3 + m)*Cos[a + b*x]^2,x]","\frac{1}{64} x^m \left(-\frac{e^{2 i a} 2^{-m} (-i b x)^{-m} \Gamma (m+4,-2 i b x)}{b^4}-\frac{e^{-2 i a} 2^{-m} (i b x)^{-m} \Gamma (m+4,2 i b x)}{b^4}+\frac{32 x^4}{m+4}\right)","-\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,"(x^m*((32*x^4)/(4 + m) - (E^((2*I)*a)*Gamma[4 + m, (-2*I)*b*x])/(2^m*b^4*((-I)*b*x)^m) - Gamma[4 + m, (2*I)*b*x]/(2^m*b^4*E^((2*I)*a)*(I*b*x)^m)))/64","A",1
112,1,96,103,0.1045442,"\int x^{2+m} \cos ^2(a+b x) \, dx","Integrate[x^(2 + m)*Cos[a + b*x]^2,x]","\frac{1}{32} x^m \left(\frac{i e^{2 i a} 2^{-m} (-i b x)^{-m} \Gamma (m+3,-2 i b x)}{b^3}-\frac{i e^{-2 i a} 2^{-m} (i b x)^{-m} \Gamma (m+3,2 i b x)}{b^3}+\frac{16 x^3}{m+3}\right)","\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,"(x^m*((16*x^3)/(3 + m) + (I*E^((2*I)*a)*Gamma[3 + m, (-2*I)*b*x])/(2^m*b^3*((-I)*b*x)^m) - (I*Gamma[3 + m, (2*I)*b*x])/(2^m*b^3*E^((2*I)*a)*(I*b*x)^m)))/32","A",1
113,1,90,97,0.1005358,"\int x^{1+m} \cos ^2(a+b x) \, dx","Integrate[x^(1 + m)*Cos[a + b*x]^2,x]","\frac{1}{16} x^m \left(\frac{e^{2 i a} 2^{-m} (-i b x)^{-m} \Gamma (m+2,-2 i b x)}{b^2}+\frac{e^{-2 i a} 2^{-m} (i b x)^{-m} \Gamma (m+2,2 i b x)}{b^2}+\frac{8 x^2}{m+2}\right)","\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,"(x^m*((8*x^2)/(2 + m) + (E^((2*I)*a)*Gamma[2 + m, (-2*I)*b*x])/(2^m*b^2*((-I)*b*x)^m) + Gamma[2 + m, (2*I)*b*x]/(2^m*b^2*E^((2*I)*a)*(I*b*x)^m)))/16","A",1
114,1,90,103,0.115293,"\int x^m \cos ^2(a+b x) \, dx","Integrate[x^m*Cos[a + b*x]^2,x]","\frac{1}{8} x^m \left(e^{2 i a} \left(-2^{-m}\right) x (-i b x)^{-m-1} \Gamma (m+1,-2 i b x)-e^{-2 i a} 2^{-m} x (i b x)^{-m-1} \Gamma (m+1,2 i b x)+\frac{4 x}{m+1}\right)","-\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,"(x^m*((4*x)/(1 + m) - (E^((2*I)*a)*x*((-I)*b*x)^(-1 - m)*Gamma[1 + m, (-2*I)*b*x])/2^m - (x*(I*b*x)^(-1 - m)*Gamma[1 + m, (2*I)*b*x])/(2^m*E^((2*I)*a))))/8","A",1
115,1,77,85,0.0589036,"\int x^{-1+m} \cos ^2(a+b x) \, dx","Integrate[x^(-1 + m)*Cos[a + b*x]^2,x]","\frac{1}{4} x^m \left(e^{2 i a} \left(-2^{-m}\right) (-i b x)^{-m} \Gamma (m,-2 i b x)-e^{-2 i a} 2^{-m} (i b x)^{-m} \Gamma (m,2 i b x)+\frac{2}{m}\right)","e^{2 i a} \left(-2^{-m-2}\right) 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,"(x^m*(2/m - (E^((2*I)*a)*Gamma[m, (-2*I)*b*x])/(2^m*((-I)*b*x)^m) - Gamma[m, (2*I)*b*x]/(2^m*E^((2*I)*a)*(I*b*x)^m)))/4","A",1
116,1,91,101,0.1040686,"\int x^{-2+m} \cos ^2(a+b x) \, dx","Integrate[x^(-2 + m)*Cos[a + b*x]^2,x]","\frac{1}{2} x^m \left(i e^{2 i a} b 2^{-m} (-i b x)^{-m} \Gamma (m-1,-2 i b x)-i e^{-2 i a} b 2^{-m} (i b x)^{-m} \Gamma (m-1,2 i b x)+\frac{1}{(m-1) x}\right)","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,"(x^m*(1/((-1 + m)*x) + (I*b*E^((2*I)*a)*Gamma[-1 + m, (-2*I)*b*x])/(2^m*((-I)*b*x)^m) - (I*b*Gamma[-1 + m, (2*I)*b*x])/(2^m*E^((2*I)*a)*(I*b*x)^m)))/2","A",1
117,1,95,95,0.0926882,"\int x^{-3+m} \cos ^2(a+b x) \, dx","Integrate[x^(-3 + m)*Cos[a + b*x]^2,x]","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)}","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,"-1/2*x^(-2 + m)/(2 - m) + (b^2*E^((2*I)*a)*x^m*Gamma[-2 + m, (-2*I)*b*x])/(2^m*((-I)*b*x)^m) + (b^2*x^m*Gamma[-2 + m, (2*I)*b*x])/(2^m*E^((2*I)*a)*(I*b*x)^m)","A",1
118,1,122,89,0.5882332,"\int (c+d x)^3 (a+a \cos (e+f x)) \, dx","Integrate[(c + d*x)^3*(a + a*Cos[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) \cos (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) \sin (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) \sin (e+f x)}{f^3}+\frac{3 a d (c+d x)^2 \cos (e+f x)}{f^2}+\frac{a (c+d x)^3 \sin (e+f x)}{f}+\frac{a (c+d x)^4}{4 d}-\frac{6 a d^3 \cos (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 + (3*d*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Cos[e + f*x])/f^4 + ((c + d*x)*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-6 + f^2*x^2))*Sin[e + f*x])/f^3)","A",1
119,1,80,67,0.3534174,"\int (c+d x)^2 (a+a \cos (e+f x)) \, dx","Integrate[(c + d*x)^2*(a + a*Cos[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) \sin (e+f x)}{f^3}+c^2 x+\frac{2 d (c+d x) \cos (e+f x)}{f^2}+c d x^2+\frac{d^2 x^3}{3}\right)","\frac{2 a d (c+d x) \cos (e+f x)}{f^2}+\frac{a (c+d x)^2 \sin (e+f x)}{f}+\frac{a (c+d x)^3}{3 d}-\frac{2 a d^2 \sin (e+f x)}{f^3}",1,"a*(c^2*x + c*d*x^2 + (d^2*x^3)/3 + (2*d*(c + d*x)*Cos[e + f*x])/f^2 + ((c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + f^2*x^2))*Sin[e + f*x])/f^3)","A",1
120,1,52,44,0.2239659,"\int (c+d x) (a+a \cos (e+f x)) \, dx","Integrate[(c + d*x)*(a + a*Cos[e + f*x]),x]","\frac{a (-2 (e+f x) (-2 c f+d e-d f x)+4 f (c+d x) \sin (e+f x)+4 d \cos (e+f x))}{4 f^2}","\frac{a (c+d x) \sin (e+f x)}{f}+\frac{a (c+d x)^2}{2 d}+\frac{a d \cos (e+f x)}{f^2}",1,"(a*(-2*(e + f*x)*(d*e - 2*c*f - d*f*x) + 4*d*Cos[e + f*x] + 4*f*(c + d*x)*Sin[e + f*x]))/(4*f^2)","A",1
121,1,55,65,0.1326635,"\int \frac{a+a \cos (e+f x)}{c+d x} \, dx","Integrate[(a + a*Cos[e + f*x])/(c + d*x),x]","\frac{a \left(\text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \cos \left(e-\frac{c f}{d}\right)-\sin \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) \cos \left(e-\frac{c f}{d}\right)}{d}-\frac{a \sin \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*(Cos[e - (c*f)/d]*CosIntegral[f*(c/d + x)] + Log[c + d*x] - Sin[e - (c*f)/d]*SinIntegral[f*(c/d + x)]))/d","A",1
122,1,78,89,0.330909,"\int \frac{a+a \cos (e+f x)}{(c+d x)^2} \, dx","Integrate[(a + a*Cos[e + f*x])/(c + d*x)^2,x]","-\frac{a \left(f (c+d x) \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \sin \left(e-\frac{c f}{d}\right)+f (c+d x) \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)+d (\cos (e+f x)+1)\right)}{d^2 (c+d x)}","-\frac{a f \text{Ci}\left(x f+\frac{c f}{d}\right) \sin \left(e-\frac{c f}{d}\right)}{d^2}-\frac{a f \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{d^2}-\frac{a \cos (e+f x)}{d (c+d x)}-\frac{a}{d (c+d x)}",1,"-((a*(d*(1 + Cos[e + f*x]) + f*(c + d*x)*CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + f*(c + d*x)*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)]))/(d^2*(c + d*x)))","A",1
123,1,217,237,1.4433873,"\int (c+d x)^3 (a+a \cos (e+f x))^2 \, dx","Integrate[(c + d*x)^3*(a + a*Cos[e + f*x])^2,x]","\frac{a^2 \left(96 d \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)+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))+2 f \left(16 (c+d x) \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2-6\right)\right) \sin (e+f x)+(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 f^3 x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)\right)\right)}{16 f^4}","-\frac{12 a^2 d^2 (c+d x) \sin (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 \cos ^2(e+f x)}{4 f^2}+\frac{6 a^2 d (c+d x)^2 \cos (e+f x)}{f^2}+\frac{2 a^2 (c+d x)^3 \sin (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 \cos ^2(e+f x)}{8 f^4}-\frac{12 a^2 d^3 \cos (e+f x)}{f^4}-\frac{3 a^2 d^3 x^2}{8 f^2}",1,"(a^2*(96*d*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-2 + 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)] + 2*f*(3*f^3*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + 16*(c + d*x)*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-6 + f^2*x^2))*Sin[e + f*x] + (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
124,1,193,168,0.6474648,"\int (c+d x)^2 (a+a \cos (e+f x))^2 \, dx","Integrate[(c + d*x)^2*(a + a*Cos[e + f*x])^2,x]","\frac{a^2 \left(16 c^2 f^2 \sin (e+f x)+2 c^2 f^2 \sin (2 (e+f x))+12 c^2 f^3 x+32 c d f^2 x \sin (e+f x)+4 c d f^2 x \sin (2 (e+f x))+32 d f (c+d x) \cos (e+f x)+2 d f (c+d x) \cos (2 (e+f x))+12 c d f^3 x^2+16 d^2 f^2 x^2 \sin (e+f x)+2 d^2 f^2 x^2 \sin (2 (e+f x))-32 d^2 \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) \cos ^2(e+f x)}{2 f^2}+\frac{4 a^2 d (c+d x) \cos (e+f x)}{f^2}+\frac{2 a^2 (c+d x)^2 \sin (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 \sin (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 + 32*d*f*(c + d*x)*Cos[e + f*x] + 2*d*f*(c + d*x)*Cos[2*(e + f*x)] - 32*d^2*Sin[e + f*x] + 16*c^2*f^2*Sin[e + f*x] + 32*c*d*f^2*x*Sin[e + f*x] + 16*d^2*f^2*x^2*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
125,1,80,118,0.4957786,"\int (c+d x) (a+a \cos (e+f x))^2 \, dx","Integrate[(c + d*x)*(a + a*Cos[e + f*x])^2,x]","\frac{a^2 (-6 (e+f x) (d (e-f x)-2 c f)+16 f (c+d x) \sin (e+f x)+2 f (c+d x) \sin (2 (e+f x))+16 d \cos (e+f x)+d \cos (2 (e+f x)))}{8 f^2}","\frac{2 a^2 (c+d x) \sin (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 \cos ^2(e+f x)}{4 f^2}+\frac{2 a^2 d \cos (e+f x)}{f^2}+\frac{1}{4} a^2 d x^2",1,"(a^2*(-6*(e + f*x)*(-2*c*f + d*(e - f*x)) + 16*d*Cos[e + f*x] + d*Cos[2*(e + f*x)] + 16*f*(c + d*x)*Sin[e + f*x] + 2*f*(c + d*x)*Sin[2*(e + f*x)]))/(8*f^2)","A",1
126,1,114,145,0.227366,"\int \frac{(a+a \cos (e+f x))^2}{c+d x} \, dx","Integrate[(a + a*Cos[e + f*x])^2/(c + d*x),x]","\frac{a^2 \left(4 \text{Ci}\left(f \left(\frac{c}{d}+x\right)\right) \cos \left(e-\frac{c f}{d}\right)+\text{Ci}\left(\frac{2 f (c+d x)}{d}\right) \cos \left(2 e-\frac{2 c f}{d}\right)-4 \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\right)-\sin \left(2 e-\frac{2 c f}{d}\right) \text{Si}\left(\frac{2 f (c+d x)}{d}\right)+3 \log (c+d x)\right)}{2 d}","\frac{2 a^2 \text{Ci}\left(x f+\frac{c f}{d}\right) \cos \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{2 a^2 \sin \left(e-\frac{c f}{d}\right) \text{Si}\left(x f+\frac{c f}{d}\right)}{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{3 a^2 \log (c+d x)}{2 d}",1,"(a^2*(4*Cos[e - (c*f)/d]*CosIntegral[f*(c/d + x)] + Cos[2*e - (2*c*f)/d]*CosIntegral[(2*f*(c + d*x))/d] + 3*Log[c + d*x] - 4*Sin[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
127,1,206,159,0.4998148,"\int \frac{(a+a \cos (e+f x))^2}{(c+d x)^2} \, dx","Integrate[(a + a*Cos[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) \sin \left(e-\frac{c f}{d}\right)+4 d f x \cos \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)+4 c f \cos \left(e-\frac{c f}{d}\right) \text{Si}\left(f \left(\frac{c}{d}+x\right)\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 \cos (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) \sin \left(e-\frac{c f}{d}\right)}{d^2}-\frac{2 a^2 f \cos \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 \cos ^4\left(\frac{e}{2}+\frac{f x}{2}\right)}{d (c+d x)}",1,"-1/2*(a^2*(3*d + 4*d*Cos[e + f*x] + d*Cos[2*(e + f*x)] + 2*f*(c + d*x)*CosIntegral[(2*f*(c + d*x))/d]*Sin[2*e - (2*c*f)/d] + 4*f*(c + d*x)*CosIntegral[f*(c/d + x)]*Sin[e - (c*f)/d] + 4*c*f*Cos[e - (c*f)/d]*SinIntegral[f*(c/d + x)] + 4*d*f*x*Cos[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]))/(d^2*(c + d*x))","A",1
128,1,151,134,0.3281366,"\int \frac{(c+d x)^3}{a+a \cos (e+f x)} \, dx","Integrate[(c + d*x)^3/(a + a*Cos[e + f*x]),x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \left((c+d x)^3 \sin \left(\frac{1}{2} (e+f x)\right)-\frac{i \cos \left(\frac{1}{2} (e+f x)\right) \left(12 d^2 f (c+d x) \text{Li}_2\left(-e^{i (e+f x)}\right)+f^2 (c+d x)^2 \left(f (c+d x)+6 i d \log \left(1+e^{i (e+f x)}\right)\right)+12 i d^3 \text{Li}_3\left(-e^{i (e+f x)}\right)\right)}{f^3}\right)}{a f (\cos (e+f x)+1)}","-\frac{12 i d^2 (c+d x) \text{Li}_2\left(-e^{i (e+f x)}\right)}{a f^3}+\frac{6 d (c+d x)^2 \log \left(1+e^{i (e+f x)}\right)}{a f^2}+\frac{(c+d x)^3 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{a f}-\frac{i (c+d x)^3}{a f}+\frac{12 d^3 \text{Li}_3\left(-e^{i (e+f x)}\right)}{a f^4}",1,"(2*Cos[(e + f*x)/2]*(((-I)*Cos[(e + f*x)/2]*(f^2*(c + d*x)^2*(f*(c + d*x) + (6*I)*d*Log[1 + E^(I*(e + f*x))]) + 12*d^2*f*(c + d*x)*PolyLog[2, -E^(I*(e + f*x))] + (12*I)*d^3*PolyLog[3, -E^(I*(e + f*x))]))/f^3 + (c + d*x)^3*Sin[(e + f*x)/2]))/(a*f*(1 + Cos[e + f*x]))","A",1
129,1,125,101,0.3364389,"\int \frac{(c+d x)^2}{a+a \cos (e+f x)} \, dx","Integrate[(c + d*x)^2/(a + a*Cos[e + f*x]),x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \left(f (c+d x) \left(f (c+d x) \sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right) \left(4 d \log \left(1+e^{i (e+f x)}\right)-i f (c+d x)\right)\right)-4 i d^2 \text{Li}_2\left(-e^{i (e+f x)}\right) \cos \left(\frac{1}{2} (e+f x)\right)\right)}{a f^3 (\cos (e+f x)+1)}","\frac{4 d (c+d x) \log \left(1+e^{i (e+f x)}\right)}{a f^2}+\frac{(c+d x)^2 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{a f}-\frac{i (c+d x)^2}{a f}-\frac{4 i d^2 \text{Li}_2\left(-e^{i (e+f x)}\right)}{a f^3}",1,"(2*Cos[(e + f*x)/2]*((-4*I)*d^2*Cos[(e + f*x)/2]*PolyLog[2, -E^(I*(e + f*x))] + f*(c + d*x)*(Cos[(e + f*x)/2]*((-I)*f*(c + d*x) + 4*d*Log[1 + E^(I*(e + f*x))]) + f*(c + d*x)*Sin[(e + f*x)/2])))/(a*f^3*(1 + Cos[e + f*x]))","A",1
130,1,70,49,0.080148,"\int \frac{c+d x}{a+a \cos (e+f x)} \, dx","Integrate[(c + d*x)/(a + a*Cos[e + f*x]),x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \left(f (c+d x) \sin \left(\frac{1}{2} (e+f x)\right)+2 d \cos \left(\frac{1}{2} (e+f x)\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)}{a f^2 (\cos (e+f x)+1)}","\frac{(c+d x) \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{a f}+\frac{2 d \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{a f^2}",1,"(2*Cos[(e + f*x)/2]*(2*d*Cos[(e + f*x)/2]*Log[Cos[(e + f*x)/2]] + f*(c + d*x)*Sin[(e + f*x)/2]))/(a*f^2*(1 + Cos[e + f*x]))","A",1
131,0,0,23,2.8032203,"\int \frac{1}{(c+d x) (a+a \cos (e+f x))} \, dx","Integrate[1/((c + d*x)*(a + a*Cos[e + f*x])),x]","\int \frac{1}{(c+d x) (a+a \cos (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a \cos (e+f x)+a)},x\right)",0,"Integrate[1/((c + d*x)*(a + a*Cos[e + f*x])), x]","A",-1
132,0,0,23,2.6894521,"\int \frac{1}{(c+d x)^2 (a+a \cos (e+f x))} \, dx","Integrate[1/((c + d*x)^2*(a + a*Cos[e + f*x])),x]","\int \frac{1}{(c+d x)^2 (a+a \cos (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a \cos (e+f x)+a)},x\right)",0,"Integrate[1/((c + d*x)^2*(a + a*Cos[e + f*x])), x]","A",-1
133,1,250,271,1.0476628,"\int \frac{(c+d x)^3}{(a+a \cos (e+f x))^2} \, dx","Integrate[(c + d*x)^3/(a + a*Cos[e + f*x])^2,x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \left(-2 \cos ^3\left(\frac{1}{2} (e+f x)\right) \left(-6 d \left(f^2 (c+d x)^2 \log \left(1+e^{i (e+f x)}\right)-2 i d f (c+d x) \text{Li}_2\left(-e^{i (e+f x)}\right)+2 d^2 \text{Li}_3\left(-e^{i (e+f x)}\right)\right)-f^3 (c+d x)^3 \tan \left(\frac{1}{2} (e+f x)\right)+i f^3 (c+d x)^3\right)+12 d^2 \cos ^3\left(\frac{1}{2} (e+f x)\right) \left(f (c+d x) \tan \left(\frac{1}{2} (e+f x)\right)+2 d \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+f^3 (c+d x)^3 \sin \left(\frac{1}{2} (e+f x)\right)-3 d f^2 (c+d x)^2 \cos \left(\frac{1}{2} (e+f x)\right)\right)}{3 a^2 f^4 (\cos (e+f x)+1)^2}","-\frac{4 i d^2 (c+d x) \text{Li}_2\left(-e^{i (e+f x)}\right)}{a^2 f^3}+\frac{2 d^2 (c+d x) \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{a^2 f^3}+\frac{2 d (c+d x)^2 \log \left(1+e^{i (e+f x)}\right)}{a^2 f^2}-\frac{d (c+d x)^2 \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{2 a^2 f^2}+\frac{(c+d x)^3 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{3 a^2 f}+\frac{(c+d x)^3 \tan \left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{6 a^2 f}-\frac{i (c+d x)^3}{3 a^2 f}+\frac{4 d^3 \text{Li}_3\left(-e^{i (e+f x)}\right)}{a^2 f^4}+\frac{4 d^3 \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{a^2 f^4}",1,"(2*Cos[(e + f*x)/2]*(-3*d*f^2*(c + d*x)^2*Cos[(e + f*x)/2] + f^3*(c + d*x)^3*Sin[(e + f*x)/2] + 12*d^2*Cos[(e + f*x)/2]^3*(2*d*Log[Cos[(e + f*x)/2]] + f*(c + d*x)*Tan[(e + f*x)/2]) - 2*Cos[(e + f*x)/2]^3*(I*f^3*(c + d*x)^3 - 6*d*(f^2*(c + d*x)^2*Log[1 + E^(I*(e + f*x))] - (2*I)*d*f*(c + d*x)*PolyLog[2, -E^(I*(e + f*x))] + 2*d^2*PolyLog[3, -E^(I*(e + f*x))]) - f^3*(c + d*x)^3*Tan[(e + f*x)/2])))/(3*a^2*f^4*(1 + Cos[e + f*x])^2)","A",1
134,1,212,212,1.0810208,"\int \frac{(c+d x)^2}{(a+a \cos (e+f x))^2} \, dx","Integrate[(c + d*x)^2/(a + a*Cos[e + f*x])^2,x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right) \left(\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 \left(c^2 f^2+2 c d f^2 x+d^2 \left(f^2 x^2+1\right)\right)\right)-2 d f (c+d x) \cos \left(\frac{1}{2} (e+f x)\right)-2 i f (c+d x) \cos ^3\left(\frac{1}{2} (e+f x)\right) \left(f (c+d x)+4 i d \log \left(1+e^{i (e+f x)}\right)\right)-8 i d^2 \text{Li}_2\left(-e^{i (e+f x)}\right) \cos ^3\left(\frac{1}{2} (e+f x)\right)\right)}{3 a^2 f^3 (\cos (e+f x)+1)^2}","\frac{4 d (c+d x) \log \left(1+e^{i (e+f x)}\right)}{3 a^2 f^2}-\frac{d (c+d x) \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{3 a^2 f^2}+\frac{(c+d x)^2 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{3 a^2 f}+\frac{(c+d x)^2 \tan \left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{6 a^2 f}-\frac{i (c+d x)^2}{3 a^2 f}-\frac{4 i d^2 \text{Li}_2\left(-e^{i (e+f x)}\right)}{3 a^2 f^3}+\frac{2 d^2 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{3 a^2 f^3}",1,"(2*Cos[(e + f*x)/2]*(-2*d*f*(c + d*x)*Cos[(e + f*x)/2] - (2*I)*f*(c + d*x)*Cos[(e + f*x)/2]^3*(f*(c + d*x) + (4*I)*d*Log[1 + E^(I*(e + f*x))]) - (8*I)*d^2*Cos[(e + f*x)/2]^3*PolyLog[2, -E^(I*(e + f*x))] + (2*(c^2*f^2 + 2*c*d*f^2*x + d^2*(1 + f^2*x^2)) + (c^2*f^2 + 2*c*d*f^2*x + d^2*(2 + f^2*x^2))*Cos[e + f*x])*Sin[(e + f*x)/2]))/(3*a^2*f^3*(1 + Cos[e + f*x])^2)","A",1
135,1,113,123,0.5203094,"\int \frac{c+d x}{(a+a \cos (e+f x))^2} \, dx","Integrate[(c + d*x)/(a + a*Cos[e + f*x])^2,x]","\frac{\cos \left(\frac{1}{2} (e+f x)\right) \left(f (c+d x) \left(3 \sin \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{3}{2} (e+f x)\right)\right)+2 d \cos \left(\frac{3}{2} (e+f x)\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)+2 d \cos \left(\frac{1}{2} (e+f x)\right) \left(3 \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)-1\right)\right)}{3 a^2 f^2 (\cos (e+f x)+1)^2}","\frac{(c+d x) \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{3 a^2 f}+\frac{(c+d x) \tan \left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{6 a^2 f}-\frac{d \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{6 a^2 f^2}+\frac{2 d \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{3 a^2 f^2}",1,"(Cos[(e + f*x)/2]*(2*d*Cos[(3*(e + f*x))/2]*Log[Cos[(e + f*x)/2]] + 2*d*Cos[(e + f*x)/2]*(-1 + 3*Log[Cos[(e + f*x)/2]]) + f*(c + d*x)*(3*Sin[(e + f*x)/2] + Sin[(3*(e + f*x))/2])))/(3*a^2*f^2*(1 + Cos[e + f*x])^2)","A",1
136,0,0,23,12.1713522,"\int \frac{1}{(c+d x) (a+a \cos (e+f x))^2} \, dx","Integrate[1/((c + d*x)*(a + a*Cos[e + f*x])^2),x]","\int \frac{1}{(c+d x) (a+a \cos (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a \cos (e+f x)+a)^2},x\right)",0,"Integrate[1/((c + d*x)*(a + a*Cos[e + f*x])^2), x]","A",-1
137,0,0,23,12.7829829,"\int \frac{1}{(c+d x)^2 (a+a \cos (e+f x))^2} \, dx","Integrate[1/((c + d*x)^2*(a + a*Cos[e + f*x])^2),x]","\int \frac{1}{(c+d x)^2 (a+a \cos (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a \cos (e+f x)+a)^2},x\right)",0,"Integrate[1/((c + d*x)^2*(a + a*Cos[e + f*x])^2), x]","A",-1
138,1,164,133,1.2502775,"\int \frac{(c+d x)^3}{a-a \cos (e+f x)} \, dx","Integrate[(c + d*x)^3/(a - a*Cos[e + f*x]),x]","\frac{2 \sin \left(\frac{1}{2} (e+f x)\right) \left(f^3 \csc \left(\frac{e}{2}\right) (c+d x)^3 \sin \left(\frac{f x}{2}\right)+2 \sin \left(\frac{1}{2} (e+f x)\right) \left(6 i d^2 f (c+d x) \text{Li}_2\left(e^{-i (e+f x)}\right)-\frac{i f^3 (c+d x)^3}{-1+e^{i e}}+3 d f^2 (c+d x)^2 \log \left(1-e^{-i (e+f x)}\right)+6 d^3 \text{Li}_3\left(e^{-i (e+f x)}\right)\right)\right)}{f^4 (a-a \cos (e+f x))}","-\frac{12 i d^2 (c+d x) \text{Li}_2\left(e^{i (e+f x)}\right)}{a f^3}+\frac{6 d (c+d x)^2 \log \left(1-e^{i (e+f x)}\right)}{a f^2}-\frac{(c+d x)^3 \cot \left(\frac{e}{2}+\frac{f x}{2}\right)}{a f}-\frac{i (c+d x)^3}{a f}+\frac{12 d^3 \text{Li}_3\left(e^{i (e+f x)}\right)}{a f^4}",1,"(2*Sin[(e + f*x)/2]*(f^3*(c + d*x)^3*Csc[e/2]*Sin[(f*x)/2] + 2*(((-I)*f^3*(c + d*x)^3)/(-1 + E^(I*e)) + 3*d*f^2*(c + d*x)^2*Log[1 - E^((-I)*(e + f*x))] + (6*I)*d^2*f*(c + d*x)*PolyLog[2, E^((-I)*(e + f*x))] + 6*d^3*PolyLog[3, E^((-I)*(e + f*x))])*Sin[(e + f*x)/2]))/(f^4*(a - a*Cos[e + f*x]))","A",1
139,1,292,102,5.5279158,"\int \frac{(c+d x)^2}{a-a \cos (e+f x)} \, dx","Integrate[(c + d*x)^2/(a - a*Cos[e + f*x]),x]","\frac{2 \csc \left(\frac{e}{2}\right) \sin \left(\frac{1}{2} (e+f x)\right) \left(-2 c d f \sin \left(\frac{1}{2} (e+f x)\right) \left(f x \cos \left(\frac{e}{2}\right)-2 \sin \left(\frac{e}{2}\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)+f^2 (c+d x)^2 \sin \left(\frac{f x}{2}\right)+d^2 \sin \left(\frac{1}{2} (e+f x)\right) \left(f^2 x^2 \cos \left(\frac{e}{2}\right) \left(-e^{i \tan ^{-1}\left(\tan \left(\frac{e}{2}\right)\right)}\right) \sqrt{\sec ^2\left(\frac{e}{2}\right)}-4 \sin \left(\frac{e}{2}\right) \left(i \text{Li}_2\left(e^{i \left(f x+2 \tan ^{-1}\left(\tan \left(\frac{e}{2}\right)\right)\right)}\right)-\frac{1}{2} i f x \left(\pi -2 \tan ^{-1}\left(\tan \left(\frac{e}{2}\right)\right)\right)-\left(2 \tan ^{-1}\left(\tan \left(\frac{e}{2}\right)\right)+f x\right) \log \left(1-e^{i \left(2 \tan ^{-1}\left(\tan \left(\frac{e}{2}\right)\right)+f x\right)}\right)+2 \tan ^{-1}\left(\tan \left(\frac{e}{2}\right)\right) \log \left(\sin \left(\tan ^{-1}\left(\tan \left(\frac{e}{2}\right)\right)+\frac{f x}{2}\right)\right)-\pi  \log \left(1+e^{-i f x}\right)+\pi  \log \left(\cos \left(\frac{f x}{2}\right)\right)\right)\right)\right)}{f^3 (a-a \cos (e+f x))}","\frac{4 d (c+d x) \log \left(1-e^{i (e+f x)}\right)}{a f^2}-\frac{(c+d x)^2 \cot \left(\frac{e}{2}+\frac{f x}{2}\right)}{a f}-\frac{i (c+d x)^2}{a f}-\frac{4 i d^2 \text{Li}_2\left(e^{i (e+f x)}\right)}{a f^3}",1,"(2*Csc[e/2]*Sin[(e + f*x)/2]*(f^2*(c + d*x)^2*Sin[(f*x)/2] - 2*c*d*f*(f*x*Cos[e/2] - 2*Log[Sin[(e + f*x)/2]]*Sin[e/2])*Sin[(e + f*x)/2] + d^2*(-(E^(I*ArcTan[Tan[e/2]])*f^2*x^2*Cos[e/2]*Sqrt[Sec[e/2]^2]) - 4*((-1/2*I)*f*x*(Pi - 2*ArcTan[Tan[e/2]]) - Pi*Log[1 + E^((-I)*f*x)] - (f*x + 2*ArcTan[Tan[e/2]])*Log[1 - E^(I*(f*x + 2*ArcTan[Tan[e/2]]))] + Pi*Log[Cos[(f*x)/2]] + 2*ArcTan[Tan[e/2]]*Log[Sin[(f*x)/2 + ArcTan[Tan[e/2]]]] + I*PolyLog[2, E^(I*(f*x + 2*ArcTan[Tan[e/2]]))])*Sin[e/2])*Sin[(e + f*x)/2]))/(f^3*(a - a*Cos[e + f*x]))","B",0
140,1,57,50,0.2458547,"\int \frac{c+d x}{a-a \cos (e+f x)} \, dx","Integrate[(c + d*x)/(a - a*Cos[e + f*x]),x]","\frac{f (c+d x) \sin (e+f x)-4 d \sin ^2\left(\frac{1}{2} (e+f x)\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)\right)}{a f^2 (\cos (e+f x)-1)}","\frac{2 d \log \left(\sin \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{a f^2}-\frac{(c+d x) \cot \left(\frac{e}{2}+\frac{f x}{2}\right)}{a f}",1,"(-4*d*Log[Sin[(e + f*x)/2]]*Sin[(e + f*x)/2]^2 + f*(c + d*x)*Sin[e + f*x])/(a*f^2*(-1 + Cos[e + f*x]))","A",1
141,0,0,24,2.3981117,"\int \frac{1}{(c+d x) (a-a \cos (e+f x))} \, dx","Integrate[1/((c + d*x)*(a - a*Cos[e + f*x])),x]","\int \frac{1}{(c+d x) (a-a \cos (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a-a \cos (e+f x))},x\right)",0,"Integrate[1/((c + d*x)*(a - a*Cos[e + f*x])), x]","A",-1
142,0,0,24,2.3314797,"\int \frac{1}{(c+d x)^2 (a-a \cos (e+f x))} \, dx","Integrate[1/((c + d*x)^2*(a - a*Cos[e + f*x])),x]","\int \frac{1}{(c+d x)^2 (a-a \cos (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a-a \cos (e+f x))},x\right)",0,"Integrate[1/((c + d*x)^2*(a - a*Cos[e + f*x])), x]","A",-1
143,1,53,110,0.2243499,"\int x^3 \sqrt{a+a \cos (c+d x)} \, dx","Integrate[x^3*Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \left(d x \left(d^2 x^2-24\right) \tan \left(\frac{1}{2} (c+d x)\right)+6 \left(d^2 x^2-8\right)\right) \sqrt{a (\cos (c+d x)+1)}}{d^4}","-\frac{96 \sqrt{a \cos (c+d x)+a}}{d^4}-\frac{48 x \tan \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}}{d^3}+\frac{12 x^2 \sqrt{a \cos (c+d x)+a}}{d^2}+\frac{2 x^3 \tan \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}}{d}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(6*(-8 + d^2*x^2) + d*x*(-24 + d^2*x^2)*Tan[(c + d*x)/2]))/d^4","A",1
144,1,44,88,0.1614773,"\int x^2 \sqrt{a+a \cos (c+d x)} \, dx","Integrate[x^2*Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \left(\left(d^2 x^2-8\right) \tan \left(\frac{1}{2} (c+d x)\right)+4 d x\right) \sqrt{a (\cos (c+d x)+1)}}{d^3}","-\frac{16 \tan \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}}{d^3}+\frac{8 x \sqrt{a \cos (c+d x)+a}}{d^2}+\frac{2 x^2 \tan \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}}{d}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(4*d*x + (-8 + d^2*x^2)*Tan[(c + d*x)/2]))/d^3","A",1
145,1,34,53,0.1272311,"\int x \sqrt{a+a \cos (c+d x)} \, dx","Integrate[x*Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \left(d x \tan \left(\frac{1}{2} (c+d x)\right)+2\right) \sqrt{a (\cos (c+d x)+1)}}{d^2}","\frac{4 \sqrt{a \cos (c+d x)+a}}{d^2}+\frac{2 x \tan \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}}{d}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(2 + d*x*Tan[(c + d*x)/2]))/d^2","A",1
146,1,29,26,0.0326689,"\int \sqrt{a+a \cos (c+d x)} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{d}","\frac{2 a \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*Tan[(c + d*x)/2])/d","A",1
147,1,55,84,0.0888183,"\int \frac{\sqrt{a+a \cos (c+d x)}}{x} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]/x,x]","\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\cos \left(\frac{c}{2}\right) \text{Ci}\left(\frac{d x}{2}\right)-\sin \left(\frac{c}{2}\right) \text{Si}\left(\frac{d x}{2}\right)\right)","\cos \left(\frac{c}{2}\right) \text{Ci}\left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}-\sin \left(\frac{c}{2}\right) \text{Si}\left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}",1,"Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Cos[c/2]*CosIntegral[(d*x)/2] - Sin[c/2]*SinIntegral[(d*x)/2])","A",1
148,1,75,110,0.1566194,"\int \frac{\sqrt{a+a \cos (c+d x)}}{x^2} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]/x^2,x]","-\frac{\sqrt{a (\cos (c+d x)+1)} \left(d x \sin \left(\frac{c}{2}\right) \text{Ci}\left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+d x \cos \left(\frac{c}{2}\right) \text{Si}\left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+2\right)}{2 x}","-\frac{1}{2} d \sin \left(\frac{c}{2}\right) \text{Ci}\left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}-\frac{1}{2} d \cos \left(\frac{c}{2}\right) \text{Si}\left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}-\frac{\sqrt{a \cos (c+d x)+a}}{x}",1,"-1/2*(Sqrt[a*(1 + Cos[c + d*x])]*(2 + d*x*CosIntegral[(d*x)/2]*Sec[(c + d*x)/2]*Sin[c/2] + d*x*Cos[c/2]*Sec[(c + d*x)/2]*SinIntegral[(d*x)/2]))/x","A",1
149,1,98,151,0.2662942,"\int \frac{\sqrt{a+a \cos (c+d x)}}{x^3} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]/x^3,x]","\frac{\sqrt{a (\cos (c+d x)+1)} \left(-d^2 x^2 \cos \left(\frac{c}{2}\right) \text{Ci}\left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+d^2 x^2 \sin \left(\frac{c}{2}\right) \text{Si}\left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+2 d x \tan \left(\frac{1}{2} (c+d x)\right)-4\right)}{8 x^2}","-\frac{1}{8} d^2 \cos \left(\frac{c}{2}\right) \text{Ci}\left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}+\frac{1}{8} d^2 \sin \left(\frac{c}{2}\right) \text{Si}\left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}-\frac{\sqrt{a \cos (c+d x)+a}}{2 x^2}+\frac{d \tan \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a \cos (c+d x)+a}}{4 x}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(-4 - d^2*x^2*Cos[c/2]*CosIntegral[(d*x)/2]*Sec[(c + d*x)/2] + d^2*x^2*Sec[(c + d*x)/2]*Sin[c/2]*SinIntegral[(d*x)/2] + 2*d*x*Tan[(c + d*x)/2]))/(8*x^2)","A",1
150,1,33,68,0.0547792,"\int x^3 \sqrt{a+a \cos (x)} \, dx","Integrate[x^3*Sqrt[a + a*Cos[x]],x]","2 \left(6 \left(x^2-8\right)+x \left(x^2-24\right) \tan \left(\frac{x}{2}\right)\right) \sqrt{a (\cos (x)+1)}","2 x^3 \tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+12 x^2 \sqrt{a \cos (x)+a}-96 \sqrt{a \cos (x)+a}-48 x \tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}",1,"2*Sqrt[a*(1 + Cos[x])]*(6*(-8 + x^2) + x*(-24 + x^2)*Tan[x/2])","A",1
151,1,29,53,0.043656,"\int x^2 \sqrt{a+a \cos (x)} \, dx","Integrate[x^2*Sqrt[a + a*Cos[x]],x]","8 \left(\frac{1}{4} \left(x^2-8\right) \tan \left(\frac{x}{2}\right)+x\right) \sqrt{a (\cos (x)+1)}","2 x^2 \tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+8 x \sqrt{a \cos (x)+a}-16 \tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}",1,"8*Sqrt[a*(1 + Cos[x])]*(x + ((-8 + x^2)*Tan[x/2])/4)","A",1
152,1,22,32,0.0206636,"\int x \sqrt{a+a \cos (x)} \, dx","Integrate[x*Sqrt[a + a*Cos[x]],x]","2 \left(x \tan \left(\frac{x}{2}\right)+2\right) \sqrt{a (\cos (x)+1)}","4 \sqrt{a \cos (x)+a}+2 x \tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}",1,"2*Sqrt[a*(1 + Cos[x])]*(2 + x*Tan[x/2])","A",1
153,1,18,15,0.0074555,"\int \sqrt{a+a \cos (x)} \, dx","Integrate[Sqrt[a + a*Cos[x]],x]","2 \tan \left(\frac{x}{2}\right) \sqrt{a (\cos (x)+1)}","\frac{2 a \sin (x)}{\sqrt{a \cos (x)+a}}",1,"2*Sqrt[a*(1 + Cos[x])]*Tan[x/2]","A",1
154,1,23,23,0.0064348,"\int \frac{\sqrt{a+a \cos (x)}}{x} \, dx","Integrate[Sqrt[a + a*Cos[x]]/x,x]","\text{Ci}\left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) \sqrt{a (\cos (x)+1)}","\text{Ci}\left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}",1,"Sqrt[a*(1 + Cos[x])]*CosIntegral[x/2]*Sec[x/2]","A",1
155,1,33,42,0.0590008,"\int \frac{\sqrt{a+a \cos (x)}}{x^2} \, dx","Integrate[Sqrt[a + a*Cos[x]]/x^2,x]","-\frac{\sqrt{a (\cos (x)+1)} \left(x \text{Si}\left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right)+2\right)}{2 x}","-\frac{1}{2} \text{Si}\left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}-\frac{\sqrt{a \cos (x)+a}}{x}",1,"-1/2*(Sqrt[a*(1 + Cos[x])]*(2 + x*Sec[x/2]*SinIntegral[x/2]))/x","A",1
156,1,44,67,0.0742053,"\int \frac{\sqrt{a+a \cos (x)}}{x^3} \, dx","Integrate[Sqrt[a + a*Cos[x]]/x^3,x]","-\frac{\sqrt{a (\cos (x)+1)} \left(x^2 \text{Ci}\left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right)-2 x \tan \left(\frac{x}{2}\right)+4\right)}{8 x^2}","-\frac{1}{8} \text{Ci}\left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}-\frac{\sqrt{a \cos (x)+a}}{2 x^2}+\frac{\tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}}{4 x}",1,"-1/8*(Sqrt[a*(1 + Cos[x])]*(4 + x^2*CosIntegral[x/2]*Sec[x/2] - 2*x*Tan[x/2]))/x^2","A",1
157,1,34,72,0.051534,"\int x^3 \sqrt{a-a \cos (x)} \, dx","Integrate[x^3*Sqrt[a - a*Cos[x]],x]","-2 \left(x \left(x^2-24\right) \cot \left(\frac{x}{2}\right)-6 \left(x^2-8\right)\right) \sqrt{a-a \cos (x)}","-2 x^3 \cot \left(\frac{x}{2}\right) \sqrt{a-a \cos (x)}+12 x^2 \sqrt{a-a \cos (x)}-96 \sqrt{a-a \cos (x)}+48 x \cot \left(\frac{x}{2}\right) \sqrt{a-a \cos (x)}",1,"-2*Sqrt[a - a*Cos[x]]*(-6*(-8 + x^2) + x*(-24 + x^2)*Cot[x/2])","A",1
158,1,30,56,0.0423358,"\int x^2 \sqrt{a-a \cos (x)} \, dx","Integrate[x^2*Sqrt[a - a*Cos[x]],x]","8 \left(x-\frac{1}{4} \left(x^2-8\right) \cot \left(\frac{x}{2}\right)\right) \sqrt{a-a \cos (x)}","-2 x^2 \cot \left(\frac{x}{2}\right) \sqrt{a-a \cos (x)}+8 x \sqrt{a-a \cos (x)}+16 \cot \left(\frac{x}{2}\right) \sqrt{a-a \cos (x)}",1,"8*Sqrt[a - a*Cos[x]]*(x - ((-8 + x^2)*Cot[x/2])/4)","A",1
159,1,23,34,0.0226898,"\int x \sqrt{a-a \cos (x)} \, dx","Integrate[x*Sqrt[a - a*Cos[x]],x]","-2 \left(x \cot \left(\frac{x}{2}\right)-2\right) \sqrt{a-a \cos (x)}","4 \sqrt{a-a \cos (x)}-2 x \cot \left(\frac{x}{2}\right) \sqrt{a-a \cos (x)}",1,"-2*Sqrt[a - a*Cos[x]]*(-2 + x*Cot[x/2])","A",1
160,1,19,16,0.0067965,"\int \sqrt{a-a \cos (x)} \, dx","Integrate[Sqrt[a - a*Cos[x]],x]","-2 \cot \left(\frac{x}{2}\right) \sqrt{a-a \cos (x)}","-\frac{2 a \sin (x)}{\sqrt{a-a \cos (x)}}",1,"-2*Sqrt[a - a*Cos[x]]*Cot[x/2]","A",1
161,1,24,24,0.0214108,"\int \frac{\sqrt{a-a \cos (x)}}{x} \, dx","Integrate[Sqrt[a - a*Cos[x]]/x,x]","\text{Si}\left(\frac{x}{2}\right) \csc \left(\frac{x}{2}\right) \sqrt{a-a \cos (x)}","\text{Si}\left(\frac{x}{2}\right) \csc \left(\frac{x}{2}\right) \sqrt{a-a \cos (x)}",1,"Sqrt[a - a*Cos[x]]*Csc[x/2]*SinIntegral[x/2]","A",1
162,1,34,44,0.0271957,"\int \frac{\sqrt{a-a \cos (x)}}{x^2} \, dx","Integrate[Sqrt[a - a*Cos[x]]/x^2,x]","\frac{\sqrt{a-a \cos (x)} \left(x \text{Ci}\left(\frac{x}{2}\right) \csc \left(\frac{x}{2}\right)-2\right)}{2 x}","\frac{1}{2} \text{Ci}\left(\frac{x}{2}\right) \csc \left(\frac{x}{2}\right) \sqrt{a-a \cos (x)}-\frac{\sqrt{a-a \cos (x)}}{x}",1,"(Sqrt[a - a*Cos[x]]*(-2 + x*CosIntegral[x/2]*Csc[x/2]))/(2*x)","A",1
163,1,45,70,0.0572827,"\int \frac{\sqrt{a-a \cos (x)}}{x^3} \, dx","Integrate[Sqrt[a - a*Cos[x]]/x^3,x]","-\frac{\sqrt{a-a \cos (x)} \left(x^2 \text{Si}\left(\frac{x}{2}\right) \csc \left(\frac{x}{2}\right)+2 x \cot \left(\frac{x}{2}\right)+4\right)}{8 x^2}","-\frac{1}{8} \text{Si}\left(\frac{x}{2}\right) \csc \left(\frac{x}{2}\right) \sqrt{a-a \cos (x)}-\frac{\sqrt{a-a \cos (x)}}{2 x^2}-\frac{\cot \left(\frac{x}{2}\right) \sqrt{a-a \cos (x)}}{4 x}",1,"-1/8*(Sqrt[a - a*Cos[x]]*(4 + 2*x*Cot[x/2] + x^2*Csc[x/2]*SinIntegral[x/2]))/x^2","A",1
164,1,67,185,0.296541,"\int x^3 (a+a \cos (x))^{3/2} \, dx","Integrate[x^3*(a + a*Cos[x])^(3/2),x]","\frac{2}{27} a \sqrt{a (\cos (x)+1)} \left(234 x^2+3 \left(15 x^2-328\right) x \tan \left(\frac{x}{2}\right)+\cos (x) \left(2 \left(9 x^2-8\right)+3 x \left(3 x^2-8\right) \tan \left(\frac{x}{2}\right)\right)-1936\right)","\frac{4}{3} a x^3 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+\frac{8}{3} a x^3 \tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+\frac{8}{3} a x^2 \cos ^2\left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+16 a x^2 \sqrt{a \cos (x)+a}-\frac{64}{27} a \cos ^2\left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}-\frac{1280}{9} a \sqrt{a \cos (x)+a}-\frac{32}{9} a x \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}-\frac{640}{9} a x \tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}",1,"(2*a*Sqrt[a*(1 + Cos[x])]*(-1936 + 234*x^2 + 3*x*(-328 + 15*x^2)*Tan[x/2] + Cos[x]*(2*(-8 + 9*x^2) + 3*x*(-8 + 3*x^2)*Tan[x/2])))/27","A",1
165,1,54,145,0.2193735,"\int x^2 (a+a \cos (x))^{3/2} \, dx","Integrate[x^2*(a + a*Cos[x])^(3/2),x]","\frac{2}{27} a \sqrt{a (\cos (x)+1)} \left(\left(45 x^2-328\right) \tan \left(\frac{x}{2}\right)+\cos (x) \left(\left(9 x^2-8\right) \tan \left(\frac{x}{2}\right)+12 x\right)+156 x\right)","\frac{4}{3} a x^2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+\frac{8}{3} a x^2 \tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+\frac{16}{9} a x \cos ^2\left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+\frac{32}{3} a x \sqrt{a \cos (x)+a}-\frac{224}{9} a \tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+\frac{32}{27} a \sin ^2\left(\frac{x}{2}\right) \tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}",1,"(2*a*Sqrt[a*(1 + Cos[x])]*(156*x + (-328 + 45*x^2)*Tan[x/2] + Cos[x]*(12*x + (-8 + 9*x^2)*Tan[x/2])))/27","A",1
166,1,45,89,0.0682503,"\int x (a+a \cos (x))^{3/2} \, dx","Integrate[x*(a + a*Cos[x])^(3/2),x]","\frac{1}{9} a \sqrt{a (\cos (x)+1)} \left(4 \cos (x)+27 x \tan \left(\frac{x}{2}\right)+3 x \sin \left(\frac{3 x}{2}\right) \sec \left(\frac{x}{2}\right)+52\right)","\frac{8}{9} a \cos ^2\left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+\frac{16}{3} a \sqrt{a \cos (x)+a}+\frac{4}{3} a x \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+\frac{8}{3} a x \tan \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}",1,"(a*Sqrt[a*(1 + Cos[x])]*(52 + 4*Cos[x] + 3*x*Sec[x/2]*Sin[(3*x)/2] + 27*x*Tan[x/2]))/9","A",1
167,1,36,55,0.0201196,"\int \frac{(a+a \cos (x))^{3/2}}{x} \, dx","Integrate[(a + a*Cos[x])^(3/2)/x,x]","\frac{1}{2} a \left(3 \text{Ci}\left(\frac{x}{2}\right)+\text{Ci}\left(\frac{3 x}{2}\right)\right) \sec \left(\frac{x}{2}\right) \sqrt{a (\cos (x)+1)}","\frac{3}{2} a \text{Ci}\left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}+\frac{1}{2} a \text{Ci}\left(\frac{3 x}{2}\right) \sec \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}",1,"(a*Sqrt[a*(1 + Cos[x])]*(3*CosIntegral[x/2] + CosIntegral[(3*x)/2])*Sec[x/2])/2","A",1
168,1,53,79,0.0848767,"\int \frac{(a+a \cos (x))^{3/2}}{x^2} \, dx","Integrate[(a + a*Cos[x])^(3/2)/x^2,x]","-\frac{a \sec \left(\frac{x}{2}\right) \sqrt{a (\cos (x)+1)} \left(3 x \text{Si}\left(\frac{x}{2}\right)+3 x \text{Si}\left(\frac{3 x}{2}\right)+8 \cos ^3\left(\frac{x}{2}\right)\right)}{4 x}","-\frac{3}{4} a \text{Si}\left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}-\frac{3}{4} a \text{Si}\left(\frac{3 x}{2}\right) \sec \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}-\frac{2 a \cos ^2\left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}}{x}",1,"-1/4*(a*Sqrt[a*(1 + Cos[x])]*Sec[x/2]*(8*Cos[x/2]^3 + 3*x*SinIntegral[x/2] + 3*x*SinIntegral[(3*x)/2]))/x","A",1
169,1,66,109,0.0530355,"\int \frac{(a+a \cos (x))^{3/2}}{x^3} \, dx","Integrate[(a + a*Cos[x])^(3/2)/x^3,x]","-\frac{(a (\cos (x)+1))^{3/2} \left(3 x^2 \text{Ci}\left(\frac{x}{2}\right) \sec ^3\left(\frac{x}{2}\right)+9 x^2 \text{Ci}\left(\frac{3 x}{2}\right) \sec ^3\left(\frac{x}{2}\right)-24 x \tan \left(\frac{x}{2}\right)+16\right)}{32 x^2}","-\frac{3}{16} a \text{Ci}\left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}-\frac{9}{16} a \text{Ci}\left(\frac{3 x}{2}\right) \sec \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}-\frac{a \cos ^2\left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}}{x^2}+\frac{3 a \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) \sqrt{a \cos (x)+a}}{2 x}",1,"-1/32*((a*(1 + Cos[x]))^(3/2)*(16 + 3*x^2*CosIntegral[x/2]*Sec[x/2]^3 + 9*x^2*CosIntegral[(3*x)/2]*Sec[x/2]^3 - 24*x*Tan[x/2]))/x^2","A",1
170,1,199,374,0.1256069,"\int \frac{x^3}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[x^3/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{4 i \cos \left(\frac{1}{2} (c+d x)\right) \left(d^3 x^3 \tan ^{-1}\left(e^{\frac{1}{2} i (c+d x)}\right)-3 d^2 x^2 \text{Li}_2\left(-i e^{\frac{1}{2} i (c+d x)}\right)+3 d^2 x^2 \text{Li}_2\left(i e^{\frac{1}{2} i (c+d x)}\right)-12 i d x \text{Li}_3\left(-i e^{\frac{1}{2} i (c+d x)}\right)+12 i d x \text{Li}_3\left(i e^{\frac{1}{2} i (c+d x)}\right)+24 \text{Li}_4\left(-i e^{\frac{1}{2} i (c+d x)}\right)-24 \text{Li}_4\left(i e^{\frac{1}{2} i (c+d x)}\right)\right)}{d^4 \sqrt{a (\cos (c+d x)+1)}}","-\frac{96 i \text{Li}_4\left(-i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^4 \sqrt{a \cos (c+d x)+a}}+\frac{96 i \text{Li}_4\left(i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^4 \sqrt{a \cos (c+d x)+a}}-\frac{48 x \text{Li}_3\left(-i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^3 \sqrt{a \cos (c+d x)+a}}+\frac{48 x \text{Li}_3\left(i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^3 \sqrt{a \cos (c+d x)+a}}+\frac{12 i x^2 \text{Li}_2\left(-i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^2 \sqrt{a \cos (c+d x)+a}}-\frac{12 i x^2 \text{Li}_2\left(i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^2 \sqrt{a \cos (c+d x)+a}}-\frac{4 i x^3 \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \tan ^{-1}\left(e^{\frac{1}{2} i (c+d x)}\right)}{d \sqrt{a \cos (c+d x)+a}}",1,"((-4*I)*Cos[(c + d*x)/2]*(d^3*x^3*ArcTan[E^((I/2)*(c + d*x))] - 3*d^2*x^2*PolyLog[2, (-I)*E^((I/2)*(c + d*x))] + 3*d^2*x^2*PolyLog[2, I*E^((I/2)*(c + d*x))] - (12*I)*d*x*PolyLog[3, (-I)*E^((I/2)*(c + d*x))] + (12*I)*d*x*PolyLog[3, I*E^((I/2)*(c + d*x))] + 24*PolyLog[4, (-I)*E^((I/2)*(c + d*x))] - 24*PolyLog[4, I*E^((I/2)*(c + d*x))]))/(d^4*Sqrt[a*(1 + Cos[c + d*x])])","A",1
171,1,146,262,0.0754813,"\int \frac{x^2}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[x^2/Sqrt[a + a*Cos[c + d*x]],x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(-i d^2 x^2 \tan ^{-1}\left(e^{\frac{1}{2} i (c+d x)}\right)+2 i d x \text{Li}_2\left(-i e^{\frac{1}{2} i (c+d x)}\right)-2 i d x \text{Li}_2\left(i e^{\frac{1}{2} i (c+d x)}\right)-4 \text{Li}_3\left(-i e^{\frac{1}{2} i (c+d x)}\right)+4 \text{Li}_3\left(i e^{\frac{1}{2} i (c+d x)}\right)\right)}{d^3 \sqrt{a (\cos (c+d x)+1)}}","-\frac{16 \text{Li}_3\left(-i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^3 \sqrt{a \cos (c+d x)+a}}+\frac{16 \text{Li}_3\left(i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^3 \sqrt{a \cos (c+d x)+a}}+\frac{8 i x \text{Li}_2\left(-i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^2 \sqrt{a \cos (c+d x)+a}}-\frac{8 i x \text{Li}_2\left(i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^2 \sqrt{a \cos (c+d x)+a}}-\frac{4 i x^2 \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \tan ^{-1}\left(e^{\frac{1}{2} i (c+d x)}\right)}{d \sqrt{a \cos (c+d x)+a}}",1,"(4*Cos[(c + d*x)/2]*((-I)*d^2*x^2*ArcTan[E^((I/2)*(c + d*x))] + (2*I)*d*x*PolyLog[2, (-I)*E^((I/2)*(c + d*x))] - (2*I)*d*x*PolyLog[2, I*E^((I/2)*(c + d*x))] - 4*PolyLog[3, (-I)*E^((I/2)*(c + d*x))] + 4*PolyLog[3, I*E^((I/2)*(c + d*x))]))/(d^3*Sqrt[a*(1 + Cos[c + d*x])])","A",1
172,1,89,156,0.046918,"\int \frac{x}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[x/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{4 i \cos \left(\frac{1}{2} (c+d x)\right) \left(-\text{Li}_2\left(-i e^{\frac{1}{2} i (c+d x)}\right)+\text{Li}_2\left(i e^{\frac{1}{2} i (c+d x)}\right)+d x \tan ^{-1}\left(e^{\frac{1}{2} i (c+d x)}\right)\right)}{d^2 \sqrt{a (\cos (c+d x)+1)}}","\frac{4 i \text{Li}_2\left(-i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^2 \sqrt{a \cos (c+d x)+a}}-\frac{4 i \text{Li}_2\left(i e^{\frac{1}{2} i (c+d x)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d^2 \sqrt{a \cos (c+d x)+a}}-\frac{4 i x \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \tan ^{-1}\left(e^{\frac{1}{2} i (c+d x)}\right)}{d \sqrt{a \cos (c+d x)+a}}",1,"((-4*I)*Cos[(c + d*x)/2]*(d*x*ArcTan[E^((I/2)*(c + d*x))] - PolyLog[2, (-I)*E^((I/2)*(c + d*x))] + PolyLog[2, I*E^((I/2)*(c + d*x))]))/(d^2*Sqrt[a*(1 + Cos[c + d*x])])","A",1
173,1,40,46,0.015413,"\int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[1/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
174,0,0,21,0.8930548,"\int \frac{1}{x \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[1/(x*Sqrt[a + a*Cos[c + d*x]]),x]","\int \frac{1}{x \sqrt{a+a \cos (c+d x)}} \, dx","\text{Int}\left(\frac{1}{x \sqrt{a \cos (c+d x)+a}},x\right)",0,"Integrate[1/(x*Sqrt[a + a*Cos[c + d*x]]), x]","A",-1
175,1,170,235,0.0998899,"\int \frac{x^3}{\sqrt{a-a \cos (x)}} \, dx","Integrate[x^3/Sqrt[a - a*Cos[x]],x]","-\frac{i \sin \left(\frac{x}{2}\right) \left(-48 x^2 \text{Li}_2\left(e^{-\frac{i x}{2}}\right)-48 x^2 \text{Li}_2\left(-e^{\frac{i x}{2}}\right)+192 i x \text{Li}_3\left(e^{-\frac{i x}{2}}\right)-192 i x \text{Li}_3\left(-e^{\frac{i x}{2}}\right)+384 \text{Li}_4\left(e^{-\frac{i x}{2}}\right)+384 \text{Li}_4\left(-e^{\frac{i x}{2}}\right)-x^4+8 i x^3 \log \left(1-e^{-\frac{i x}{2}}\right)-8 i x^3 \log \left(1+e^{\frac{i x}{2}}\right)+8 \pi ^4\right)}{4 \sqrt{a-a \cos (x)}}","\frac{12 i x^2 \text{Li}_2\left(-e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}-\frac{12 i x^2 \text{Li}_2\left(e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}-\frac{48 x \text{Li}_3\left(-e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}+\frac{48 x \text{Li}_3\left(e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}-\frac{96 i \text{Li}_4\left(-e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}+\frac{96 i \text{Li}_4\left(e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}-\frac{4 x^3 \sin \left(\frac{x}{2}\right) \tanh ^{-1}\left(e^{\frac{i x}{2}}\right)}{\sqrt{a-a \cos (x)}}",1,"((-1/4*I)*(8*Pi^4 - x^4 + (8*I)*x^3*Log[1 - E^((-1/2*I)*x)] - (8*I)*x^3*Log[1 + E^((I/2)*x)] - 48*x^2*PolyLog[2, E^((-1/2*I)*x)] - 48*x^2*PolyLog[2, -E^((I/2)*x)] + (192*I)*x*PolyLog[3, E^((-1/2*I)*x)] - (192*I)*x*PolyLog[3, -E^((I/2)*x)] + 384*PolyLog[4, E^((-1/2*I)*x)] + 384*PolyLog[4, -E^((I/2)*x)])*Sin[x/2])/Sqrt[a - a*Cos[x]]","A",1
176,1,117,163,0.0488879,"\int \frac{x^2}{\sqrt{a-a \cos (x)}} \, dx","Integrate[x^2/Sqrt[a - a*Cos[x]],x]","\frac{2 \sin \left(\frac{x}{2}\right) \left(4 i x \text{Li}_2\left(-e^{\frac{i x}{2}}\right)-4 i x \text{Li}_2\left(e^{\frac{i x}{2}}\right)-8 \text{Li}_3\left(-e^{\frac{i x}{2}}\right)+8 \text{Li}_3\left(e^{\frac{i x}{2}}\right)+x^2 \log \left(1-e^{\frac{i x}{2}}\right)-x^2 \log \left(1+e^{\frac{i x}{2}}\right)\right)}{\sqrt{a-a \cos (x)}}","\frac{8 i x \text{Li}_2\left(-e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}-\frac{8 i x \text{Li}_2\left(e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}-\frac{16 \text{Li}_3\left(-e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}+\frac{16 \text{Li}_3\left(e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}-\frac{4 x^2 \sin \left(\frac{x}{2}\right) \tanh ^{-1}\left(e^{\frac{i x}{2}}\right)}{\sqrt{a-a \cos (x)}}",1,"(2*(x^2*Log[1 - E^((I/2)*x)] - x^2*Log[1 + E^((I/2)*x)] + (4*I)*x*PolyLog[2, -E^((I/2)*x)] - (4*I)*x*PolyLog[2, E^((I/2)*x)] - 8*PolyLog[3, -E^((I/2)*x)] + 8*PolyLog[3, E^((I/2)*x)])*Sin[x/2])/Sqrt[a - a*Cos[x]]","A",1
177,1,83,97,0.0331485,"\int \frac{x}{\sqrt{a-a \cos (x)}} \, dx","Integrate[x/Sqrt[a - a*Cos[x]],x]","\frac{2 \sin \left(\frac{x}{2}\right) \left(2 i \text{Li}_2\left(-e^{\frac{i x}{2}}\right)-2 i \text{Li}_2\left(e^{\frac{i x}{2}}\right)+x \left(\log \left(1-e^{\frac{i x}{2}}\right)-\log \left(1+e^{\frac{i x}{2}}\right)\right)\right)}{\sqrt{a-a \cos (x)}}","\frac{4 i \text{Li}_2\left(-e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}-\frac{4 i \text{Li}_2\left(e^{\frac{i x}{2}}\right) \sin \left(\frac{x}{2}\right)}{\sqrt{a-a \cos (x)}}-\frac{4 x \sin \left(\frac{x}{2}\right) \tanh ^{-1}\left(e^{\frac{i x}{2}}\right)}{\sqrt{a-a \cos (x)}}",1,"(2*(x*(Log[1 - E^((I/2)*x)] - Log[1 + E^((I/2)*x)]) + (2*I)*PolyLog[2, -E^((I/2)*x)] - (2*I)*PolyLog[2, E^((I/2)*x)])*Sin[x/2])/Sqrt[a - a*Cos[x]]","A",1
178,1,36,37,0.014775,"\int \frac{1}{\sqrt{a-a \cos (x)}} \, dx","Integrate[1/Sqrt[a - a*Cos[x]],x]","\frac{2 \sin \left(\frac{x}{2}\right) \left(\log \left(\sin \left(\frac{x}{4}\right)\right)-\log \left(\cos \left(\frac{x}{4}\right)\right)\right)}{\sqrt{a-a \cos (x)}}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{a-a \cos (x)}}\right)}{\sqrt{a}}",1,"(2*(-Log[Cos[x/4]] + Log[Sin[x/4]])*Sin[x/2])/Sqrt[a - a*Cos[x]]","A",1
179,0,0,18,2.9562406,"\int \frac{1}{x \sqrt{a-a \cos (x)}} \, dx","Integrate[1/(x*Sqrt[a - a*Cos[x]]),x]","\int \frac{1}{x \sqrt{a-a \cos (x)}} \, dx","\text{Int}\left(\frac{1}{x \sqrt{a-a \cos (x)}},x\right)",0,"Integrate[1/(x*Sqrt[a - a*Cos[x]]), x]","A",-1
180,1,257,423,0.411021,"\int \frac{x^3}{(a+a \cos (x))^{3/2}} \, dx","Integrate[x^3/(a + a*Cos[x])^(3/2),x]","-\frac{i \cos \left(\frac{x}{2}\right) \left(-6 \left(x^2+8\right) \text{Li}_2\left(-i e^{\frac{i x}{2}}\right) \cos ^2\left(\frac{x}{2}\right)+6 \left(x^2+8\right) \text{Li}_2\left(i e^{\frac{i x}{2}}\right) \cos ^2\left(\frac{x}{2}\right)-24 i x \text{Li}_3\left(-i e^{\frac{i x}{2}}\right) \cos ^2\left(\frac{x}{2}\right)+24 i x \text{Li}_3\left(i e^{\frac{i x}{2}}\right) \cos ^2\left(\frac{x}{2}\right)+48 \text{Li}_4\left(-i e^{\frac{i x}{2}}\right) \cos ^2\left(\frac{x}{2}\right)-48 \text{Li}_4\left(i e^{\frac{i x}{2}}\right) \cos ^2\left(\frac{x}{2}\right)+i x^3 \sin \left(\frac{x}{2}\right)+2 x^3 \cos ^2\left(\frac{x}{2}\right) \tan ^{-1}\left(e^{\frac{i x}{2}}\right)-6 i x^2 \cos \left(\frac{x}{2}\right)+48 x \cos ^2\left(\frac{x}{2}\right) \tan ^{-1}\left(e^{\frac{i x}{2}}\right)\right)}{(a (\cos (x)+1))^{3/2}}","\frac{3 i x^2 \text{Li}_2\left(-i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}-\frac{3 i x^2 \text{Li}_2\left(i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}-\frac{12 x \text{Li}_3\left(-i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}+\frac{12 x \text{Li}_3\left(i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}+\frac{24 i \text{Li}_2\left(-i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}-\frac{24 i \text{Li}_2\left(i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}-\frac{24 i \text{Li}_4\left(-i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}+\frac{24 i \text{Li}_4\left(i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}-\frac{i x^3 \cos \left(\frac{x}{2}\right) \tan ^{-1}\left(e^{\frac{i x}{2}}\right)}{a \sqrt{a \cos (x)+a}}+\frac{x^3 \tan \left(\frac{x}{2}\right)}{2 a \sqrt{a \cos (x)+a}}-\frac{3 x^2}{a \sqrt{a \cos (x)+a}}-\frac{24 i x \cos \left(\frac{x}{2}\right) \tan ^{-1}\left(e^{\frac{i x}{2}}\right)}{a \sqrt{a \cos (x)+a}}",1,"((-I)*Cos[x/2]*((-6*I)*x^2*Cos[x/2] + 48*x*ArcTan[E^((I/2)*x)]*Cos[x/2]^2 + 2*x^3*ArcTan[E^((I/2)*x)]*Cos[x/2]^2 - 6*(8 + x^2)*Cos[x/2]^2*PolyLog[2, (-I)*E^((I/2)*x)] + 6*(8 + x^2)*Cos[x/2]^2*PolyLog[2, I*E^((I/2)*x)] - (24*I)*x*Cos[x/2]^2*PolyLog[3, (-I)*E^((I/2)*x)] + (24*I)*x*Cos[x/2]^2*PolyLog[3, I*E^((I/2)*x)] + 48*Cos[x/2]^2*PolyLog[4, (-I)*E^((I/2)*x)] - 48*Cos[x/2]^2*PolyLog[4, I*E^((I/2)*x)] + I*x^3*Sin[x/2]))/(a*(1 + Cos[x]))^(3/2)","A",1
181,1,185,257,0.1189719,"\int \frac{x^2}{(a+a \cos (x))^{3/2}} \, dx","Integrate[x^2/(a + a*Cos[x])^(3/2),x]","\frac{\cos \left(\frac{x}{2}\right) \left(4 i x \text{Li}_2\left(-i e^{\frac{i x}{2}}\right) \cos ^2\left(\frac{x}{2}\right)-4 i x \text{Li}_2\left(i e^{\frac{i x}{2}}\right) \cos ^2\left(\frac{x}{2}\right)-8 \text{Li}_3\left(-i e^{\frac{i x}{2}}\right) \cos ^2\left(\frac{x}{2}\right)+8 \text{Li}_3\left(i e^{\frac{i x}{2}}\right) \cos ^2\left(\frac{x}{2}\right)+x^2 \sin \left(\frac{x}{2}\right)-2 i x^2 \cos ^2\left(\frac{x}{2}\right) \tan ^{-1}\left(e^{\frac{i x}{2}}\right)-4 x \cos \left(\frac{x}{2}\right)+8 \cos ^2\left(\frac{x}{2}\right) \tanh ^{-1}\left(\sin \left(\frac{x}{2}\right)\right)\right)}{(a (\cos (x)+1))^{3/2}}","\frac{2 i x \text{Li}_2\left(-i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}-\frac{2 i x \text{Li}_2\left(i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}-\frac{4 \text{Li}_3\left(-i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}+\frac{4 \text{Li}_3\left(i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}-\frac{i x^2 \cos \left(\frac{x}{2}\right) \tan ^{-1}\left(e^{\frac{i x}{2}}\right)}{a \sqrt{a \cos (x)+a}}+\frac{x^2 \tan \left(\frac{x}{2}\right)}{2 a \sqrt{a \cos (x)+a}}-\frac{2 x}{a \sqrt{a \cos (x)+a}}+\frac{4 \cos \left(\frac{x}{2}\right) \tanh ^{-1}\left(\sin \left(\frac{x}{2}\right)\right)}{a \sqrt{a \cos (x)+a}}",1,"(Cos[x/2]*(-4*x*Cos[x/2] - (2*I)*x^2*ArcTan[E^((I/2)*x)]*Cos[x/2]^2 + 8*ArcTanh[Sin[x/2]]*Cos[x/2]^2 + (4*I)*x*Cos[x/2]^2*PolyLog[2, (-I)*E^((I/2)*x)] - (4*I)*x*Cos[x/2]^2*PolyLog[2, I*E^((I/2)*x)] - 8*Cos[x/2]^2*PolyLog[3, (-I)*E^((I/2)*x)] + 8*Cos[x/2]^2*PolyLog[3, I*E^((I/2)*x)] + x^2*Sin[x/2]))/(a*(1 + Cos[x]))^(3/2)","A",1
182,1,165,150,0.1770641,"\int \frac{x}{(a+a \cos (x))^{3/2}} \, dx","Integrate[x/(a + a*Cos[x])^(3/2),x]","\frac{\sec \left(\frac{x}{2}\right) \left(2 i \text{Li}_2\left(-i e^{\frac{i x}{2}}\right) (\cos (x)+1)-2 i \text{Li}_2\left(i e^{\frac{i x}{2}}\right) (\cos (x)+1)+x \log \left(1-i e^{\frac{i x}{2}}\right)-x \log \left(1+i e^{\frac{i x}{2}}\right)+2 x \sin \left(\frac{x}{2}\right)-4 \cos \left(\frac{x}{2}\right)+x \log \left(1-i e^{\frac{i x}{2}}\right) \cos (x)-x \log \left(1+i e^{\frac{i x}{2}}\right) \cos (x)\right)}{4 a \sqrt{a (\cos (x)+1)}}","\frac{i \text{Li}_2\left(-i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}-\frac{i \text{Li}_2\left(i e^{\frac{i x}{2}}\right) \cos \left(\frac{x}{2}\right)}{a \sqrt{a \cos (x)+a}}-\frac{1}{a \sqrt{a \cos (x)+a}}-\frac{i x \cos \left(\frac{x}{2}\right) \tan ^{-1}\left(e^{\frac{i x}{2}}\right)}{a \sqrt{a \cos (x)+a}}+\frac{x \tan \left(\frac{x}{2}\right)}{2 a \sqrt{a \cos (x)+a}}",1,"(Sec[x/2]*(-4*Cos[x/2] + x*Log[1 - I*E^((I/2)*x)] + x*Cos[x]*Log[1 - I*E^((I/2)*x)] - x*Log[1 + I*E^((I/2)*x)] - x*Cos[x]*Log[1 + I*E^((I/2)*x)] + (2*I)*(1 + Cos[x])*PolyLog[2, (-I)*E^((I/2)*x)] - (2*I)*(1 + Cos[x])*PolyLog[2, I*E^((I/2)*x)] + 2*x*Sin[x/2]))/(4*a*Sqrt[a*(1 + Cos[x])])","A",1
183,0,0,17,11.1389937,"\int \frac{1}{x (a+a \cos (x))^{3/2}} \, dx","Integrate[1/(x*(a + a*Cos[x])^(3/2)),x]","\int \frac{1}{x (a+a \cos (x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{x (a \cos (x)+a)^{3/2}},x\right)",0,"Integrate[1/(x*(a + a*Cos[x])^(3/2)), x]","A",-1
184,0,0,21,1.5073135,"\int \frac{\sqrt[3]{a+a \cos (c+d x)}}{x} \, dx","Integrate[(a + a*Cos[c + d*x])^(1/3)/x,x]","\int \frac{\sqrt[3]{a+a \cos (c+d x)}}{x} \, dx","\text{Int}\left(\frac{\sqrt[3]{a \cos (c+d x)+a}}{x},x\right)",0,"Integrate[(a + a*Cos[c + d*x])^(1/3)/x, x]","A",-1
185,1,290,383,0.9167555,"\int \frac{x^3}{a+b \cos (x)} \, dx","Integrate[x^3/(a + b*Cos[x]),x]","\frac{-3 x^2 \text{Li}_2\left(\frac{b e^{i x}}{\sqrt{a^2-b^2}-a}\right)+3 x^2 \text{Li}_2\left(-\frac{b e^{i x}}{a+\sqrt{a^2-b^2}}\right)-6 i x \text{Li}_3\left(\frac{b e^{i x}}{\sqrt{a^2-b^2}-a}\right)+6 i x \text{Li}_3\left(-\frac{b e^{i x}}{a+\sqrt{a^2-b^2}}\right)+6 \text{Li}_4\left(\frac{b e^{i x}}{\sqrt{a^2-b^2}-a}\right)-6 \text{Li}_4\left(-\frac{b e^{i x}}{a+\sqrt{a^2-b^2}}\right)-i x^3 \log \left(1+\frac{b e^{i x}}{a-\sqrt{a^2-b^2}}\right)+i x^3 \log \left(1+\frac{b e^{i x}}{\sqrt{a^2-b^2}+a}\right)}{\sqrt{a^2-b^2}}","-\frac{3 x^2 \text{Li}_2\left(-\frac{b e^{i x}}{a-\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{3 x^2 \text{Li}_2\left(-\frac{b e^{i x}}{a+\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{6 i x \text{Li}_3\left(-\frac{b e^{i x}}{a-\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{6 i x \text{Li}_3\left(-\frac{b e^{i x}}{a+\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{6 \text{Li}_4\left(-\frac{b e^{i x}}{a-\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{6 \text{Li}_4\left(-\frac{b e^{i x}}{a+\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{i x^3 \log \left(1+\frac{b e^{i x}}{a-\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{i x^3 \log \left(1+\frac{b e^{i x}}{\sqrt{a^2-b^2}+a}\right)}{\sqrt{a^2-b^2}}",1,"((-I)*x^3*Log[1 + (b*E^(I*x))/(a - Sqrt[a^2 - b^2])] + I*x^3*Log[1 + (b*E^(I*x))/(a + Sqrt[a^2 - b^2])] - 3*x^2*PolyLog[2, (b*E^(I*x))/(-a + Sqrt[a^2 - b^2])] + 3*x^2*PolyLog[2, -((b*E^(I*x))/(a + Sqrt[a^2 - b^2]))] - (6*I)*x*PolyLog[3, (b*E^(I*x))/(-a + Sqrt[a^2 - b^2])] + (6*I)*x*PolyLog[3, -((b*E^(I*x))/(a + Sqrt[a^2 - b^2]))] + 6*PolyLog[4, (b*E^(I*x))/(-a + Sqrt[a^2 - b^2])] - 6*PolyLog[4, -((b*E^(I*x))/(a + Sqrt[a^2 - b^2]))])/Sqrt[a^2 - b^2]","A",1
186,1,379,329,0.7158919,"\int \frac{x^2}{a+b \cos (c+d x)} \, dx","Integrate[x^2/(a + b*Cos[c + d*x]),x]","\frac{e^{i c} \left(-i \left(d^2 x^2 \log \left(1+\frac{b e^{i (2 c+d x)}}{a e^{i c}-\sqrt{e^{2 i c} \left(a^2-b^2\right)}}\right)-d^2 x^2 \log \left(1+\frac{b e^{i (2 c+d x)}}{\sqrt{e^{2 i c} \left(a^2-b^2\right)}+a e^{i c}}\right)+2 i d x \text{Li}_2\left(-\frac{b e^{i (2 c+d x)}}{e^{i c} a+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+2 \text{Li}_3\left(-\frac{b e^{i (2 c+d x)}}{a e^{i c}-\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-2 \text{Li}_3\left(-\frac{b e^{i (2 c+d x)}}{e^{i c} a+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)\right)-2 d x \text{Li}_2\left(-\frac{b e^{i (2 c+d x)}}{a e^{i c}-\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)\right)}{d^3 \sqrt{e^{2 i c} \left(a^2-b^2\right)}}","-\frac{2 i \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d^3 \sqrt{a^2-b^2}}+\frac{2 i \text{Li}_3\left(-\frac{b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{d^3 \sqrt{a^2-b^2}}-\frac{2 x \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d^2 \sqrt{a^2-b^2}}+\frac{2 x \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{d^2 \sqrt{a^2-b^2}}-\frac{i x^2 \log \left(1+\frac{b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d \sqrt{a^2-b^2}}+\frac{i x^2 \log \left(1+\frac{b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{d \sqrt{a^2-b^2}}",1,"(E^(I*c)*(-2*d*x*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(a*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - I*(d^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(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)))/(a*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*d*x*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(a*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(a*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(a*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))])))/(d^3*Sqrt[(a^2 - b^2)*E^((2*I)*c)])","A",1
187,1,756,214,0.829473,"\int \frac{x}{a+b \cos (c+d x)} \, dx","Integrate[x/(a + b*Cos[c + d*x]),x]","\frac{i \left(\text{Li}_2\left(\frac{\left(a-i \sqrt{b^2-a^2}\right) \left(a+b-\sqrt{b^2-a^2} \tan \left(\frac{1}{2} (c+d x)\right)\right)}{b \left(a+b+\sqrt{b^2-a^2} \tan \left(\frac{1}{2} (c+d x)\right)\right)}\right)-\text{Li}_2\left(\frac{\left(a+i \sqrt{b^2-a^2}\right) \left(a+b-\sqrt{b^2-a^2} \tan \left(\frac{1}{2} (c+d x)\right)\right)}{b \left(a+b+\sqrt{b^2-a^2} \tan \left(\frac{1}{2} (c+d x)\right)\right)}\right)\right)+2 (c+d x) \tanh ^{-1}\left(\frac{(a+b) \cot \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)-2 \left(\cos ^{-1}\left(-\frac{a}{b}\right)+c\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)-\log \left(\frac{(a+b) \left(-i \sqrt{b^2-a^2}-a+b\right) \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{b \left(\sqrt{b^2-a^2} \tan \left(\frac{1}{2} (c+d x)\right)+a+b\right)}\right) \left(\cos ^{-1}\left(-\frac{a}{b}\right)-2 i \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)\right)-\log \left(\frac{(a+b) \left(\sqrt{b^2-a^2}+i a-i b\right) \left(\tan \left(\frac{1}{2} (c+d x)\right)+i\right)}{b \left(\sqrt{b^2-a^2} \tan \left(\frac{1}{2} (c+d x)\right)+a+b\right)}\right) \left(\cos ^{-1}\left(-\frac{a}{b}\right)+2 i \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)\right)+\log \left(\frac{\sqrt{b^2-a^2} e^{-\frac{1}{2} i (c+d x)}}{\sqrt{2} \sqrt{b} \sqrt{a+b \cos (c+d x)}}\right) \left(2 i \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)-2 i \tanh ^{-1}\left(\frac{(a+b) \cot \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)+\cos ^{-1}\left(-\frac{a}{b}\right)\right)+\log \left(\frac{\sqrt{b^2-a^2} e^{\frac{1}{2} i (c+d x)}}{\sqrt{2} \sqrt{b} \sqrt{a+b \cos (c+d x)}}\right) \left(\cos ^{-1}\left(-\frac{a}{b}\right)+2 i \left(\tanh ^{-1}\left(\frac{(a+b) \cot \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)-\tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)\right)\right)}{d^2 \sqrt{b^2-a^2}}","-\frac{\text{Li}_2\left(-\frac{b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d^2 \sqrt{a^2-b^2}}+\frac{\text{Li}_2\left(-\frac{b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{d^2 \sqrt{a^2-b^2}}-\frac{i x \log \left(1+\frac{b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d \sqrt{a^2-b^2}}+\frac{i x \log \left(1+\frac{b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{d \sqrt{a^2-b^2}}",1,"(2*(c + d*x)*ArcTanh[((a + b)*Cot[(c + d*x)/2])/Sqrt[-a^2 + b^2]] - 2*(c + ArcCos[-(a/b)])*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + (ArcCos[-(a/b)] - (2*I)*ArcTanh[((a + b)*Cot[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + (2*I)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])*Log[Sqrt[-a^2 + b^2]/(Sqrt[2]*Sqrt[b]*E^((I/2)*(c + d*x))*Sqrt[a + b*Cos[c + d*x]])] + (ArcCos[-(a/b)] + (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x)/2])/Sqrt[-a^2 + b^2]] - ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]))*Log[(Sqrt[-a^2 + b^2]*E^((I/2)*(c + d*x)))/(Sqrt[2]*Sqrt[b]*Sqrt[a + b*Cos[c + d*x]])] - (ArcCos[-(a/b)] - (2*I)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])*Log[((a + b)*(-a + b - I*Sqrt[-a^2 + b^2])*(1 + I*Tan[(c + d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(c + d*x)/2]))] - (ArcCos[-(a/b)] + (2*I)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])*Log[((a + b)*(I*a - I*b + Sqrt[-a^2 + b^2])*(I + Tan[(c + d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(c + d*x)/2]))] + I*(PolyLog[2, ((a - I*Sqrt[-a^2 + b^2])*(a + b - Sqrt[-a^2 + b^2]*Tan[(c + d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(c + d*x)/2]))] - PolyLog[2, ((a + I*Sqrt[-a^2 + b^2])*(a + b - Sqrt[-a^2 + b^2]*Tan[(c + d*x)/2]))/(b*(a + b + Sqrt[-a^2 + b^2]*Tan[(c + d*x)/2]))]))/(Sqrt[-a^2 + b^2]*d^2)","B",1
188,0,0,15,0.837234,"\int \frac{1}{x (a+b \cos (x))} \, dx","Integrate[1/(x*(a + b*Cos[x])),x]","\int \frac{1}{x (a+b \cos (x))} \, dx","\text{Int}\left(\frac{1}{x (a+b \cos (x))},x\right)",0,"Integrate[1/(x*(a + b*Cos[x])), x]","A",-1
189,1,933,296,9.7507678,"\int \frac{e+f x}{(a+b \cos (c+d x))^2} \, dx","Integrate[(e + f*x)/(a + b*Cos[c + d*x])^2,x]","\frac{\left(\frac{2 a (d e-c f) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}+f \log \left(\sec ^2\left(\frac{1}{2} (c+d x)\right)\right)-f \log \left((a+b \cos (c+d x)) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)-\frac{i a f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{\sqrt{a+b}-\sqrt{b-a} \tan \left(\frac{1}{2} (c+d x)\right)}{i \sqrt{b-a}+\sqrt{a+b}}\right)+\text{Li}_2\left(\frac{\sqrt{b-a} \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{b-a}-i \sqrt{a+b}}\right)\right)}{\sqrt{b-a} \sqrt{a+b}}+\frac{i a f \left(\log \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \log \left(\frac{i \left(\sqrt{b-a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{a+b}\right)}{\sqrt{b-a}+i \sqrt{a+b}}\right)+\text{Li}_2\left(\frac{\sqrt{b-a} \left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{b-a}+i \sqrt{a+b}}\right)\right)}{\sqrt{b-a} \sqrt{a+b}}-\frac{i a f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{\sqrt{b-a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{a+b}}{i \sqrt{b-a}+\sqrt{a+b}}\right)+\text{Li}_2\left(\frac{\sqrt{b-a} \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)}{\sqrt{b-a}-i \sqrt{a+b}}\right)\right)}{\sqrt{b-a} \sqrt{a+b}}+\frac{i a f \left(\log \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right) \log \left(\frac{i \left(\sqrt{a+b}-\sqrt{b-a} \tan \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{b-a}+i \sqrt{a+b}}\right)+\text{Li}_2\left(\frac{\sqrt{b-a} \left(i \tan \left(\frac{1}{2} (c+d x)\right)+1\right)}{\sqrt{b-a}+i \sqrt{a+b}}\right)\right)}{\sqrt{b-a} \sqrt{a+b}}\right) (a d e+a d f x+b f \sin (c+d x)) \left(\sqrt{a+b}-\sqrt{b-a} \tan \left(\frac{1}{2} (c+d x)\right)\right) \left(\sqrt{b-a} \tan \left(\frac{1}{2} (c+d x)\right)+\sqrt{a+b}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)}{\left(a^2-b^2\right) d^2 (a+b \cos (c+d x)) \left(a \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)+b f \sin (c+d x)\right)}+\frac{-b d e \sin (c+d x)+b c f \sin (c+d x)-b f (c+d x) \sin (c+d x)}{(a-b) (a+b) d^2 (a+b \cos (c+d x))}","-\frac{a f \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d^2 \left(a^2-b^2\right)^{3/2}}+\frac{a f \text{Li}_2\left(-\frac{b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right)}{d^2 \left(a^2-b^2\right)^{3/2}}-\frac{f \log (a+b \cos (c+d x))}{d^2 \left(a^2-b^2\right)}-\frac{i a (e+f x) \log \left(1+\frac{b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{i a (e+f x) \log \left(1+\frac{b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right)}{d \left(a^2-b^2\right)^{3/2}}-\frac{b (e+f x) \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(-(b*d*e*Sin[c + d*x]) + b*c*f*Sin[c + d*x] - b*f*(c + d*x)*Sin[c + d*x])/((a - b)*(a + b)*d^2*(a + b*Cos[c + d*x])) + (Cos[(c + d*x)/2]^2*((2*a*(d*e - c*f)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) + f*Log[Sec[(c + d*x)/2]^2] - f*Log[(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2] - (I*a*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(Sqrt[a + b] - Sqrt[-a + b]*Tan[(c + d*x)/2])/(I*Sqrt[-a + b] + Sqrt[a + b])] + PolyLog[2, (Sqrt[-a + b]*(1 - I*Tan[(c + d*x)/2]))/(Sqrt[-a + b] - I*Sqrt[a + b])]))/(Sqrt[-a + b]*Sqrt[a + b]) + (I*a*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(I*(Sqrt[a + b] + Sqrt[-a + b]*Tan[(c + d*x)/2]))/(Sqrt[-a + b] + I*Sqrt[a + b])] + PolyLog[2, (Sqrt[-a + b]*(1 - I*Tan[(c + d*x)/2]))/(Sqrt[-a + b] + I*Sqrt[a + b])]))/(Sqrt[-a + b]*Sqrt[a + b]) - (I*a*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(Sqrt[a + b] + Sqrt[-a + b]*Tan[(c + d*x)/2])/(I*Sqrt[-a + b] + Sqrt[a + b])] + PolyLog[2, (Sqrt[-a + b]*(1 + I*Tan[(c + d*x)/2]))/(Sqrt[-a + b] - I*Sqrt[a + b])]))/(Sqrt[-a + b]*Sqrt[a + b]) + (I*a*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(I*(Sqrt[a + b] - Sqrt[-a + b]*Tan[(c + d*x)/2]))/(Sqrt[-a + b] + I*Sqrt[a + b])] + PolyLog[2, (Sqrt[-a + b]*(1 + I*Tan[(c + d*x)/2]))/(Sqrt[-a + b] + I*Sqrt[a + b])]))/(Sqrt[-a + b]*Sqrt[a + b]))*(a*d*e + a*d*f*x + b*f*Sin[c + d*x])*(Sqrt[a + b] - Sqrt[-a + b]*Tan[(c + d*x)/2])*(Sqrt[a + b] + Sqrt[-a + b]*Tan[(c + d*x)/2]))/((a^2 - b^2)*d^2*(a + b*Cos[c + d*x])*(a*(d*e - c*f + I*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1 + I*Tan[(c + d*x)/2]]) + b*f*Sin[c + d*x]))","B",0