1,1,82,126,0.1686682,"\int x^3 (a+b x) \sin (c+d x) \, dx","Integrate[x^3*(a + b*x)*Sin[c + d*x],x]","\frac{d \left(3 a \left(d^2 x^2-2\right)+4 b x \left(d^2 x^2-6\right)\right) \sin (c+d x)-\left(a d^2 x \left(d^2 x^2-6\right)+b \left(d^4 x^4-12 d^2 x^2+24\right)\right) \cos (c+d x)}{d^5}","-\frac{6 a \sin (c+d x)}{d^4}+\frac{6 a x \cos (c+d x)}{d^3}+\frac{3 a x^2 \sin (c+d x)}{d^2}-\frac{a x^3 \cos (c+d x)}{d}-\frac{24 b \cos (c+d x)}{d^5}-\frac{24 b x \sin (c+d x)}{d^4}+\frac{12 b x^2 \cos (c+d x)}{d^3}+\frac{4 b x^3 \sin (c+d x)}{d^2}-\frac{b x^4 \cos (c+d x)}{d}",1,"(-((a*d^2*x*(-6 + d^2*x^2) + b*(24 - 12*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + d*(4*b*x*(-6 + d^2*x^2) + 3*a*(-2 + d^2*x^2))*Sin[c + d*x])/d^5","A",1
2,1,65,96,0.1469889,"\int x^2 (a+b x) \sin (c+d x) \, dx","Integrate[x^2*(a + b*x)*Sin[c + d*x],x]","\frac{\left(2 a d^2 x+3 b \left(d^2 x^2-2\right)\right) \sin (c+d x)-d \left(a \left(d^2 x^2-2\right)+b x \left(d^2 x^2-6\right)\right) \cos (c+d x)}{d^4}","\frac{2 a \cos (c+d x)}{d^3}+\frac{2 a x \sin (c+d x)}{d^2}-\frac{a x^2 \cos (c+d x)}{d}-\frac{6 b \sin (c+d x)}{d^4}+\frac{6 b x \cos (c+d x)}{d^3}+\frac{3 b x^2 \sin (c+d x)}{d^2}-\frac{b x^3 \cos (c+d x)}{d}",1,"(-(d*(b*x*(-6 + d^2*x^2) + a*(-2 + d^2*x^2))*Cos[c + d*x]) + (2*a*d^2*x + 3*b*(-2 + d^2*x^2))*Sin[c + d*x])/d^4","A",1
3,1,45,65,0.1044408,"\int x (a+b x) \sin (c+d x) \, dx","Integrate[x*(a + b*x)*Sin[c + d*x],x]","\frac{d (a+2 b x) \sin (c+d x)-\left(a d^2 x+b \left(d^2 x^2-2\right)\right) \cos (c+d x)}{d^3}","\frac{a \sin (c+d x)}{d^2}-\frac{a x \cos (c+d x)}{d}+\frac{2 b \cos (c+d x)}{d^3}+\frac{2 b x \sin (c+d x)}{d^2}-\frac{b x^2 \cos (c+d x)}{d}",1,"(-((a*d^2*x + b*(-2 + d^2*x^2))*Cos[c + d*x]) + d*(a + 2*b*x)*Sin[c + d*x])/d^3","A",1
4,1,27,28,0.0766166,"\int (a+b x) \sin (c+d x) \, dx","Integrate[(a + b*x)*Sin[c + d*x],x]","\frac{b \sin (c+d x)-d (a+b x) \cos (c+d x)}{d^2}","\frac{b \sin (c+d x)}{d^2}-\frac{(a+b x) \cos (c+d x)}{d}",1,"(-(d*(a + b*x)*Cos[c + d*x]) + b*Sin[c + d*x])/d^2","A",1
5,1,40,29,0.037682,"\int \frac{(a+b x) \sin (c+d x)}{x} \, dx","Integrate[((a + b*x)*Sin[c + d*x])/x,x]","a \sin (c) \text{Ci}(d x)+a \cos (c) \text{Si}(d x)+\frac{b \sin (c) \sin (d x)}{d}-\frac{b \cos (c) \cos (d x)}{d}","a \sin (c) \text{Ci}(d x)+a \cos (c) \text{Si}(d x)-\frac{b \cos (c+d x)}{d}",1,"-((b*Cos[c]*Cos[d*x])/d) + a*CosIntegral[d*x]*Sin[c] + (b*Sin[c]*Sin[d*x])/d + a*Cos[c]*SinIntegral[d*x]","A",1
6,1,60,48,0.1429502,"\int \frac{(a+b x) \sin (c+d x)}{x^2} \, dx","Integrate[((a + b*x)*Sin[c + d*x])/x^2,x]","a d (\cos (c) \text{Ci}(d x)-\sin (c) \text{Si}(d x))-\frac{a \sin (c) \cos (d x)}{x}-\frac{a \cos (c) \sin (d x)}{x}+b \sin (c) \text{Ci}(d x)+b \cos (c) \text{Si}(d x)","a d \cos (c) \text{Ci}(d x)-a d \sin (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{x}+b \sin (c) \text{Ci}(d x)+b \cos (c) \text{Si}(d x)",1,"-((a*Cos[d*x]*Sin[c])/x) + b*CosIntegral[d*x]*Sin[c] - (a*Cos[c]*Sin[d*x])/x + b*Cos[c]*SinIntegral[d*x] + a*d*(Cos[c]*CosIntegral[d*x] - Sin[c]*SinIntegral[d*x])","A",1
7,1,76,89,0.2690256,"\int \frac{(a+b x) \sin (c+d x)}{x^3} \, dx","Integrate[((a + b*x)*Sin[c + d*x])/x^3,x]","-\frac{d x^2 \text{Ci}(d x) (a d \sin (c)-2 b \cos (c))+d x^2 \text{Si}(d x) (a d \cos (c)+2 b \sin (c))+a \sin (c+d x)+a d x \cos (c+d x)+2 b x \sin (c+d x)}{2 x^2}","-\frac{1}{2} a d^2 \sin (c) \text{Ci}(d x)-\frac{1}{2} a d^2 \cos (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{2 x^2}-\frac{a d \cos (c+d x)}{2 x}+b d \cos (c) \text{Ci}(d x)-b d \sin (c) \text{Si}(d x)-\frac{b \sin (c+d x)}{x}",1,"-1/2*(a*d*x*Cos[c + d*x] + d*x^2*CosIntegral[d*x]*(-2*b*Cos[c] + a*d*Sin[c]) + a*Sin[c + d*x] + 2*b*x*Sin[c + d*x] + d*x^2*(a*d*Cos[c] + 2*b*Sin[c])*SinIntegral[d*x])/x^2","A",1
8,1,110,132,0.3424058,"\int \frac{(a+b x) \sin (c+d x)}{x^4} \, dx","Integrate[((a + b*x)*Sin[c + d*x])/x^4,x]","-\frac{d^2 x^3 \text{Ci}(d x) (a d \cos (c)+3 b \sin (c))+d^2 x^3 \text{Si}(d x) (3 b \cos (c)-a d \sin (c))-a d^2 x^2 \sin (c+d x)+2 a \sin (c+d x)+a d x \cos (c+d x)+3 b d x^2 \cos (c+d x)+3 b x \sin (c+d x)}{6 x^3}","-\frac{1}{6} a d^3 \cos (c) \text{Ci}(d x)+\frac{1}{6} a d^3 \sin (c) \text{Si}(d x)+\frac{a d^2 \sin (c+d x)}{6 x}-\frac{a \sin (c+d x)}{3 x^3}-\frac{a d \cos (c+d x)}{6 x^2}-\frac{1}{2} b d^2 \sin (c) \text{Ci}(d x)-\frac{1}{2} b d^2 \cos (c) \text{Si}(d x)-\frac{b \sin (c+d x)}{2 x^2}-\frac{b d \cos (c+d x)}{2 x}",1,"-1/6*(a*d*x*Cos[c + d*x] + 3*b*d*x^2*Cos[c + d*x] + d^2*x^3*CosIntegral[d*x]*(a*d*Cos[c] + 3*b*Sin[c]) + 2*a*Sin[c + d*x] + 3*b*x*Sin[c + d*x] - a*d^2*x^2*Sin[c + d*x] + d^2*x^3*(3*b*Cos[c] - a*d*Sin[c])*SinIntegral[d*x])/x^3","A",1
9,1,138,166,0.2894205,"\int \frac{(a+b x) \sin (c+d x)}{x^5} \, dx","Integrate[((a + b*x)*Sin[c + d*x])/x^5,x]","\frac{d^3 x^4 \text{Ci}(d x) (a d \sin (c)-4 b \cos (c))+d^3 x^4 \text{Si}(d x) (a d \cos (c)+4 b \sin (c))+a d^3 x^3 \cos (c+d x)+a d^2 x^2 \sin (c+d x)-6 a \sin (c+d x)-2 a d x \cos (c+d x)+4 b d^2 x^3 \sin (c+d x)-4 b d x^2 \cos (c+d x)-8 b x \sin (c+d x)}{24 x^4}","\frac{1}{24} a d^4 \sin (c) \text{Ci}(d x)+\frac{1}{24} a d^4 \cos (c) \text{Si}(d x)+\frac{a d^3 \cos (c+d x)}{24 x}+\frac{a d^2 \sin (c+d x)}{24 x^2}-\frac{a \sin (c+d x)}{4 x^4}-\frac{a d \cos (c+d x)}{12 x^3}-\frac{1}{6} b d^3 \cos (c) \text{Ci}(d x)+\frac{1}{6} b d^3 \sin (c) \text{Si}(d x)+\frac{b d^2 \sin (c+d x)}{6 x}-\frac{b \sin (c+d x)}{3 x^3}-\frac{b d \cos (c+d x)}{6 x^2}",1,"(-2*a*d*x*Cos[c + d*x] - 4*b*d*x^2*Cos[c + d*x] + a*d^3*x^3*Cos[c + d*x] + d^3*x^4*CosIntegral[d*x]*(-4*b*Cos[c] + a*d*Sin[c]) - 6*a*Sin[c + d*x] - 8*b*x*Sin[c + d*x] + a*d^2*x^2*Sin[c + d*x] + 4*b*d^2*x^3*Sin[c + d*x] + d^3*x^4*(a*d*Cos[c] + 4*b*Sin[c])*SinIntegral[d*x])/(24*x^4)","A",1
10,1,101,186,0.2931325,"\int x^2 (a+b x)^2 \sin (c+d x) \, dx","Integrate[x^2*(a + b*x)^2*Sin[c + d*x],x]","\frac{2 d (a+2 b x) \left(a d^2 x+b \left(d^2 x^2-6\right)\right) \sin (c+d x)-\left(a^2 d^2 \left(d^2 x^2-2\right)+2 a b d^2 x \left(d^2 x^2-6\right)+b^2 \left(d^4 x^4-12 d^2 x^2+24\right)\right) \cos (c+d x)}{d^5}","\frac{2 a^2 \cos (c+d x)}{d^3}+\frac{2 a^2 x \sin (c+d x)}{d^2}-\frac{a^2 x^2 \cos (c+d x)}{d}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{12 a b x \cos (c+d x)}{d^3}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{24 b^2 \cos (c+d x)}{d^5}-\frac{24 b^2 x \sin (c+d x)}{d^4}+\frac{12 b^2 x^2 \cos (c+d x)}{d^3}+\frac{4 b^2 x^3 \sin (c+d x)}{d^2}-\frac{b^2 x^4 \cos (c+d x)}{d}",1,"(-((2*a*b*d^2*x*(-6 + d^2*x^2) + a^2*d^2*(-2 + d^2*x^2) + b^2*(24 - 12*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + 2*d*(a + 2*b*x)*(a*d^2*x + b*(-6 + d^2*x^2))*Sin[c + d*x])/d^5","A",1
11,1,87,135,0.2240768,"\int x (a+b x)^2 \sin (c+d x) \, dx","Integrate[x*(a + b*x)^2*Sin[c + d*x],x]","\frac{\left(a^2 d^2+4 a b d^2 x+3 b^2 \left(d^2 x^2-2\right)\right) \sin (c+d x)-d \left(a^2 d^2 x+2 a b \left(d^2 x^2-2\right)+b^2 x \left(d^2 x^2-6\right)\right) \cos (c+d x)}{d^4}","\frac{a^2 \sin (c+d x)}{d^2}-\frac{a^2 x \cos (c+d x)}{d}+\frac{4 a b \cos (c+d x)}{d^3}+\frac{4 a b x \sin (c+d x)}{d^2}-\frac{2 a b x^2 \cos (c+d x)}{d}-\frac{6 b^2 \sin (c+d x)}{d^4}+\frac{6 b^2 x \cos (c+d x)}{d^3}+\frac{3 b^2 x^2 \sin (c+d x)}{d^2}-\frac{b^2 x^3 \cos (c+d x)}{d}",1,"(-(d*(a^2*d^2*x + b^2*x*(-6 + d^2*x^2) + 2*a*b*(-2 + d^2*x^2))*Cos[c + d*x]) + (a^2*d^2 + 4*a*b*d^2*x + 3*b^2*(-2 + d^2*x^2))*Sin[c + d*x])/d^4","A",1
12,1,57,50,0.182777,"\int (a+b x)^2 \sin (c+d x) \, dx","Integrate[(a + b*x)^2*Sin[c + d*x],x]","\frac{2 b d (a+b x) \sin (c+d x)-\left(a^2 d^2+2 a b d^2 x+b^2 \left(d^2 x^2-2\right)\right) \cos (c+d x)}{d^3}","\frac{2 b (a+b x) \sin (c+d x)}{d^2}-\frac{(a+b x)^2 \cos (c+d x)}{d}+\frac{2 b^2 \cos (c+d x)}{d^3}",1,"(-((a^2*d^2 + 2*a*b*d^2*x + b^2*(-2 + d^2*x^2))*Cos[c + d*x]) + 2*b*d*(a + b*x)*Sin[c + d*x])/d^3","A",1
13,1,51,62,0.3100912,"\int \frac{(a+b x)^2 \sin (c+d x)}{x} \, dx","Integrate[((a + b*x)^2*Sin[c + d*x])/x,x]","a^2 \sin (c) \text{Ci}(d x)+a^2 \cos (c) \text{Si}(d x)+\frac{b (b \sin (c+d x)-d (2 a+b x) \cos (c+d x))}{d^2}","a^2 \sin (c) \text{Ci}(d x)+a^2 \cos (c) \text{Si}(d x)-\frac{2 a b \cos (c+d x)}{d}+\frac{b^2 \sin (c+d x)}{d^2}-\frac{b^2 x \cos (c+d x)}{d}",1,"a^2*CosIntegral[d*x]*Sin[c] + (b*(-(d*(2*a + b*x)*Cos[c + d*x]) + b*Sin[c + d*x]))/d^2 + a^2*Cos[c]*SinIntegral[d*x]","A",1
14,1,64,72,0.2644213,"\int \frac{(a+b x)^2 \sin (c+d x)}{x^2} \, dx","Integrate[((a + b*x)^2*Sin[c + d*x])/x^2,x]","-\frac{a^2 \sin (c+d x)}{x}+a \text{Ci}(d x) (a d \cos (c)+2 b \sin (c))-a \text{Si}(d x) (a d \sin (c)-2 b \cos (c))-\frac{b^2 \cos (c+d x)}{d}","a^2 d \cos (c) \text{Ci}(d x)-a^2 d \sin (c) \text{Si}(d x)-\frac{a^2 \sin (c+d x)}{x}+2 a b \sin (c) \text{Ci}(d x)+2 a b \cos (c) \text{Si}(d x)-\frac{b^2 \cos (c+d x)}{d}",1,"-((b^2*Cos[c + d*x])/d) + a*CosIntegral[d*x]*(a*d*Cos[c] + 2*b*Sin[c]) - (a^2*Sin[c + d*x])/x - a*(-2*b*Cos[c] + a*d*Sin[c])*SinIntegral[d*x]","A",1
15,1,95,121,0.4377696,"\int \frac{(a+b x)^2 \sin (c+d x)}{x^3} \, dx","Integrate[((a + b*x)^2*Sin[c + d*x])/x^3,x]","\frac{1}{2} \left(\text{Ci}(d x) \left(\sin (c) \left(2 b^2-a^2 d^2\right)+4 a b d \cos (c)\right)+\text{Si}(d x) \left(\cos (c) \left(2 b^2-a^2 d^2\right)-4 a b d \sin (c)\right)-\frac{a ((a+4 b x) \sin (c+d x)+a d x \cos (c+d x))}{x^2}\right)","-\frac{1}{2} a^2 d^2 \sin (c) \text{Ci}(d x)-\frac{1}{2} a^2 d^2 \cos (c) \text{Si}(d x)-\frac{a^2 \sin (c+d x)}{2 x^2}-\frac{a^2 d \cos (c+d x)}{2 x}+2 a b d \cos (c) \text{Ci}(d x)-2 a b d \sin (c) \text{Si}(d x)-\frac{2 a b \sin (c+d x)}{x}+b^2 \sin (c) \text{Ci}(d x)+b^2 \cos (c) \text{Si}(d x)",1,"(CosIntegral[d*x]*(4*a*b*d*Cos[c] + (2*b^2 - a^2*d^2)*Sin[c]) - (a*(a*d*x*Cos[c + d*x] + (a + 4*b*x)*Sin[c + d*x]))/x^2 + ((2*b^2 - a^2*d^2)*Cos[c] - 4*a*b*d*Sin[c])*SinIntegral[d*x])/2","A",1
16,1,154,175,0.5652868,"\int \frac{(a+b x)^2 \sin (c+d x)}{x^4} \, dx","Integrate[((a + b*x)^2*Sin[c + d*x])/x^4,x]","-\frac{d x^3 \text{Ci}(d x) \left(\cos (c) \left(a^2 d^2-6 b^2\right)+6 a b d \sin (c)\right)+d x^3 \text{Si}(d x) \left(-a^2 d^2 \sin (c)+6 a b d \cos (c)+6 b^2 \sin (c)\right)-a^2 d^2 x^2 \sin (c+d x)+2 a^2 \sin (c+d x)+a^2 d x \cos (c+d x)+6 a b d x^2 \cos (c+d x)+6 a b x \sin (c+d x)+6 b^2 x^2 \sin (c+d x)}{6 x^3}","-\frac{1}{6} a^2 d^3 \cos (c) \text{Ci}(d x)+\frac{1}{6} a^2 d^3 \sin (c) \text{Si}(d x)+\frac{a^2 d^2 \sin (c+d x)}{6 x}-\frac{a^2 \sin (c+d x)}{3 x^3}-\frac{a^2 d \cos (c+d x)}{6 x^2}-a b d^2 \sin (c) \text{Ci}(d x)-a b d^2 \cos (c) \text{Si}(d x)-\frac{a b \sin (c+d x)}{x^2}-\frac{a b d \cos (c+d x)}{x}+b^2 d \cos (c) \text{Ci}(d x)-b^2 d \sin (c) \text{Si}(d x)-\frac{b^2 \sin (c+d x)}{x}",1,"-1/6*(a^2*d*x*Cos[c + d*x] + 6*a*b*d*x^2*Cos[c + d*x] + d*x^3*CosIntegral[d*x]*((-6*b^2 + a^2*d^2)*Cos[c] + 6*a*b*d*Sin[c]) + 2*a^2*Sin[c + d*x] + 6*a*b*x*Sin[c + d*x] + 6*b^2*x^2*Sin[c + d*x] - a^2*d^2*x^2*Sin[c + d*x] + d*x^3*(6*a*b*d*Cos[c] + 6*b^2*Sin[c] - a^2*d^2*Sin[c])*SinIntegral[d*x])/x^3","A",1
17,1,204,248,0.4850047,"\int \frac{(a+b x)^2 \sin (c+d x)}{x^5} \, dx","Integrate[((a + b*x)^2*Sin[c + d*x])/x^5,x]","\frac{d^2 x^4 \text{Ci}(d x) \left(\sin (c) \left(a^2 d^2-12 b^2\right)-8 a b d \cos (c)\right)+d^2 x^4 \text{Si}(d x) \left(a^2 d^2 \cos (c)+8 a b d \sin (c)-12 b^2 \cos (c)\right)+a^2 d^3 x^3 \cos (c+d x)+a^2 d^2 x^2 \sin (c+d x)-6 a^2 \sin (c+d x)-2 a^2 d x \cos (c+d x)+8 a b d^2 x^3 \sin (c+d x)-8 a b d x^2 \cos (c+d x)-16 a b x \sin (c+d x)-12 b^2 d x^3 \cos (c+d x)-12 b^2 x^2 \sin (c+d x)}{24 x^4}","\frac{1}{24} a^2 d^4 \sin (c) \text{Ci}(d x)+\frac{1}{24} a^2 d^4 \cos (c) \text{Si}(d x)+\frac{a^2 d^3 \cos (c+d x)}{24 x}+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a^2 d \cos (c+d x)}{12 x^3}-\frac{1}{3} a b d^3 \cos (c) \text{Ci}(d x)+\frac{1}{3} a b d^3 \sin (c) \text{Si}(d x)+\frac{a b d^2 \sin (c+d x)}{3 x}-\frac{2 a b \sin (c+d x)}{3 x^3}-\frac{a b d \cos (c+d x)}{3 x^2}-\frac{1}{2} b^2 d^2 \sin (c) \text{Ci}(d x)-\frac{1}{2} b^2 d^2 \cos (c) \text{Si}(d x)-\frac{b^2 \sin (c+d x)}{2 x^2}-\frac{b^2 d \cos (c+d x)}{2 x}",1,"(-2*a^2*d*x*Cos[c + d*x] - 8*a*b*d*x^2*Cos[c + d*x] - 12*b^2*d*x^3*Cos[c + d*x] + a^2*d^3*x^3*Cos[c + d*x] + d^2*x^4*CosIntegral[d*x]*(-8*a*b*d*Cos[c] + (-12*b^2 + a^2*d^2)*Sin[c]) - 6*a^2*Sin[c + d*x] - 16*a*b*x*Sin[c + d*x] - 12*b^2*x^2*Sin[c + d*x] + a^2*d^2*x^2*Sin[c + d*x] + 8*a*b*d^2*x^3*Sin[c + d*x] + d^2*x^4*(-12*b^2*Cos[c] + a^2*d^2*Cos[c] + 8*a*b*d*Sin[c])*SinIntegral[d*x])/(24*x^4)","A",1
18,1,158,218,0.6874917,"\int \frac{x^4 \sin (c+d x)}{a+b x} \, dx","Integrate[(x^4*Sin[c + d*x])/(a + b*x),x]","\frac{a^4 d^4 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)+a^4 d^4 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+b \left(b \left(a^2 d^2-2 a b d^2 x+3 b^2 \left(d^2 x^2-2\right)\right) \sin (c+d x)+d \left(a^3 d^2-a^2 b d^2 x+a b^2 \left(d^2 x^2-2\right)+b^3 x \left(6-d^2 x^2\right)\right) \cos (c+d x)\right)}{b^5 d^4}","\frac{a^4 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^5}+\frac{a^4 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^5}+\frac{a^3 \cos (c+d x)}{b^4 d}+\frac{a^2 \sin (c+d x)}{b^3 d^2}-\frac{a^2 x \cos (c+d x)}{b^3 d}-\frac{2 a \cos (c+d x)}{b^2 d^3}-\frac{2 a x \sin (c+d x)}{b^2 d^2}+\frac{a x^2 \cos (c+d x)}{b^2 d}-\frac{6 \sin (c+d x)}{b d^4}+\frac{6 x \cos (c+d x)}{b d^3}+\frac{3 x^2 \sin (c+d x)}{b d^2}-\frac{x^3 \cos (c+d x)}{b d}",1,"(a^4*d^4*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + b*(d*(a^3*d^2 - a^2*b*d^2*x + b^3*x*(6 - d^2*x^2) + a*b^2*(-2 + d^2*x^2))*Cos[c + d*x] + b*(a^2*d^2 - 2*a*b*d^2*x + 3*b^2*(-2 + d^2*x^2))*Sin[c + d*x]) + a^4*d^4*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(b^5*d^4)","A",1
19,1,117,152,0.5772625,"\int \frac{x^3 \sin (c+d x)}{a+b x} \, dx","Integrate[(x^3*Sin[c + d*x])/(a + b*x),x]","-\frac{a^3 d^3 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)+a^3 d^3 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+b \left(\left(a^2 d^2-a b d^2 x+b^2 \left(d^2 x^2-2\right)\right) \cos (c+d x)+b d (a-2 b x) \sin (c+d x)\right)}{b^4 d^3}","-\frac{a^3 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^4}-\frac{a^3 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^4}-\frac{a^2 \cos (c+d x)}{b^3 d}-\frac{a \sin (c+d x)}{b^2 d^2}+\frac{a x \cos (c+d x)}{b^2 d}+\frac{2 \cos (c+d x)}{b d^3}+\frac{2 x \sin (c+d x)}{b d^2}-\frac{x^2 \cos (c+d x)}{b d}",1,"-((a^3*d^3*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + b*((a^2*d^2 - a*b*d^2*x + b^2*(-2 + d^2*x^2))*Cos[c + d*x] + b*d*(a - 2*b*x)*Sin[c + d*x]) + a^3*d^3*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(b^4*d^3))","A",1
20,1,87,99,0.3177327,"\int \frac{x^2 \sin (c+d x)}{a+b x} \, dx","Integrate[(x^2*Sin[c + d*x])/(a + b*x),x]","\frac{a^2 d^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)+a^2 d^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+b (d (a-b x) \cos (c+d x)+b \sin (c+d x))}{b^3 d^2}","\frac{a^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^3}+\frac{a^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^3}+\frac{a \cos (c+d x)}{b^2 d}+\frac{\sin (c+d x)}{b d^2}-\frac{x \cos (c+d x)}{b d}",1,"(a^2*d^2*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + b*(d*(a - b*x)*Cos[c + d*x] + b*Sin[c + d*x]) + a^2*d^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(b^3*d^2)","A",1
21,1,63,69,0.1992422,"\int \frac{x \sin (c+d x)}{a+b x} \, dx","Integrate[(x*Sin[c + d*x])/(a + b*x),x]","-\frac{a d \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)+a d \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+b \cos (c+d x)}{b^2 d}","-\frac{a \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^2}-\frac{a \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^2}-\frac{\cos (c+d x)}{b d}",1,"-((b*Cos[c + d*x] + a*d*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + a*d*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(b^2*d))","A",1
22,1,49,51,0.0717032,"\int \frac{\sin (c+d x)}{a+b x} \, dx","Integrate[Sin[c + d*x]/(a + b*x),x]","\frac{\sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)+\cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b}","\frac{\sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b}+\frac{\cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b}",1,"(CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b] + Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b","A",1
23,1,63,73,0.1673549,"\int \frac{\sin (c+d x)}{x (a+b x)} \, dx","Integrate[Sin[c + d*x]/(x*(a + b*x)),x]","\frac{-\sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)-\cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+\sin (c) \text{Ci}(d x)+\cos (c) \text{Si}(d x)}{a}","-\frac{\sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a}-\frac{\cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a}+\frac{\sin (c) \text{Ci}(d x)}{a}+\frac{\cos (c) \text{Si}(d x)}{a}",1,"(CosIntegral[d*x]*Sin[c] - CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + Cos[c]*SinIntegral[d*x] - Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/a","A",1
24,1,101,114,0.4167618,"\int \frac{\sin (c+d x)}{x^2 (a+b x)} \, dx","Integrate[Sin[c + d*x]/(x^2*(a + b*x)),x]","\frac{b x \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)+x \text{Ci}(d x) (a d \cos (c)-b \sin (c))+b x \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)-a d x \sin (c) \text{Si}(d x)-a \sin (c+d x)-b x \cos (c) \text{Si}(d x)}{a^2 x}","-\frac{b \sin (c) \text{Ci}(d x)}{a^2}+\frac{b \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a^2}-\frac{b \cos (c) \text{Si}(d x)}{a^2}+\frac{b \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a^2}+\frac{d \cos (c) \text{Ci}(d x)}{a}-\frac{d \sin (c) \text{Si}(d x)}{a}-\frac{\sin (c+d x)}{a x}",1,"(x*CosIntegral[d*x]*(a*d*Cos[c] - b*Sin[c]) + b*x*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] - a*Sin[c + d*x] - b*x*Cos[c]*SinIntegral[d*x] - a*d*x*Sin[c]*SinIntegral[d*x] + b*x*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(a^2*x)","A",1
25,1,176,189,0.6725586,"\int \frac{\sin (c+d x)}{x^3 (a+b x)} \, dx","Integrate[Sin[c + d*x]/(x^3*(a + b*x)),x]","-\frac{x^2 \text{Ci}(d x) \left(\sin (c) \left(a^2 d^2-2 b^2\right)+2 a b d \cos (c)\right)+a^2 d^2 x^2 \cos (c) \text{Si}(d x)+a^2 \sin (c+d x)+a^2 d x \cos (c+d x)+2 b^2 x^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)+2 b^2 x^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)-2 a b d x^2 \sin (c) \text{Si}(d x)-2 a b x \sin (c+d x)-2 b^2 x^2 \cos (c) \text{Si}(d x)}{2 a^3 x^2}","\frac{b^2 \sin (c) \text{Ci}(d x)}{a^3}-\frac{b^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a^3}+\frac{b^2 \cos (c) \text{Si}(d x)}{a^3}-\frac{b^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a^3}-\frac{b d \cos (c) \text{Ci}(d x)}{a^2}+\frac{b d \sin (c) \text{Si}(d x)}{a^2}+\frac{b \sin (c+d x)}{a^2 x}-\frac{d^2 \sin (c) \text{Ci}(d x)}{2 a}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a}-\frac{\sin (c+d x)}{2 a x^2}-\frac{d \cos (c+d x)}{2 a x}",1,"-1/2*(a^2*d*x*Cos[c + d*x] + x^2*CosIntegral[d*x]*(2*a*b*d*Cos[c] + (-2*b^2 + a^2*d^2)*Sin[c]) + 2*b^2*x^2*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + a^2*Sin[c + d*x] - 2*a*b*x*Sin[c + d*x] - 2*b^2*x^2*Cos[c]*SinIntegral[d*x] + a^2*d^2*x^2*Cos[c]*SinIntegral[d*x] - 2*a*b*d*x^2*Sin[c]*SinIntegral[d*x] + 2*b^2*x^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(a^3*x^2)","A",1
26,1,177,233,1.0637733,"\int \frac{x^4 \sin (c+d x)}{(a+b x)^2} \, dx","Integrate[(x^4*Sin[c + d*x])/(a + b*x)^2,x]","\frac{a^3 \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right) \left(a d \cos \left(c-\frac{a d}{b}\right)-4 b \sin \left(c-\frac{a d}{b}\right)\right)-a^3 \text{Si}\left(d \left(\frac{a}{b}+x\right)\right) \left(a d \sin \left(c-\frac{a d}{b}\right)+4 b \cos \left(c-\frac{a d}{b}\right)\right)-\frac{b \left(b (a+b x) \left(3 a^2 d^2-2 a b d^2 x+b^2 \left(d^2 x^2-2\right)\right) \cos (c+d x)+d \left(a^4 d^2+2 a^2 b^2-2 b^4 x^2\right) \sin (c+d x)\right)}{d^3 (a+b x)}}{b^6}","\frac{a^4 d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^6}-\frac{a^4 d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^6}-\frac{a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac{4 a^3 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^5}-\frac{4 a^3 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^5}-\frac{3 a^2 \cos (c+d x)}{b^4 d}-\frac{2 a \sin (c+d x)}{b^3 d^2}+\frac{2 a x \cos (c+d x)}{b^3 d}+\frac{2 \cos (c+d x)}{b^2 d^3}+\frac{2 x \sin (c+d x)}{b^2 d^2}-\frac{x^2 \cos (c+d x)}{b^2 d}",1,"(a^3*CosIntegral[d*(a/b + x)]*(a*d*Cos[c - (a*d)/b] - 4*b*Sin[c - (a*d)/b]) - (b*(b*(a + b*x)*(3*a^2*d^2 - 2*a*b*d^2*x + b^2*(-2 + d^2*x^2))*Cos[c + d*x] + d*(2*a^2*b^2 + a^4*d^2 - 2*b^4*x^2)*Sin[c + d*x]))/(d^3*(a + b*x)) - a^3*(4*b*Cos[c - (a*d)/b] + a*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^6","A",1
27,1,153,181,0.9134031,"\int \frac{x^3 \sin (c+d x)}{(a+b x)^2} \, dx","Integrate[(x^3*Sin[c + d*x])/(a + b*x)^2,x]","\frac{a^2 \left(-\text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)\right) \left(a d \cos \left(c-\frac{a d}{b}\right)-3 b \sin \left(c-\frac{a d}{b}\right)\right)+a^2 \text{Si}\left(d \left(\frac{a}{b}+x\right)\right) \left(a d \sin \left(c-\frac{a d}{b}\right)+3 b \cos \left(c-\frac{a d}{b}\right)\right)+\frac{b \left(\left(a^3 d^2+a b^2+b^3 x\right) \sin (c+d x)+b d \left(2 a^2+a b x-b^2 x^2\right) \cos (c+d x)\right)}{d^2 (a+b x)}}{b^5}","-\frac{a^3 d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^5}+\frac{a^3 d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^5}+\frac{a^3 \sin (c+d x)}{b^4 (a+b x)}+\frac{3 a^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^4}+\frac{3 a^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^4}+\frac{2 a \cos (c+d x)}{b^3 d}+\frac{\sin (c+d x)}{b^2 d^2}-\frac{x \cos (c+d x)}{b^2 d}",1,"(-(a^2*CosIntegral[d*(a/b + x)]*(a*d*Cos[c - (a*d)/b] - 3*b*Sin[c - (a*d)/b])) + (b*(b*d*(2*a^2 + a*b*x - b^2*x^2)*Cos[c + d*x] + (a*b^2 + a^3*d^2 + b^3*x)*Sin[c + d*x]))/(d^2*(a + b*x)) + a^2*(3*b*Cos[c - (a*d)/b] + a*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^5","A",1
28,1,117,149,0.8364962,"\int \frac{x^2 \sin (c+d x)}{(a+b x)^2} \, dx","Integrate[(x^2*Sin[c + d*x])/(a + b*x)^2,x]","\frac{b \left(-\frac{a^2 \sin (c+d x)}{a+b x}-\frac{b \cos (c+d x)}{d}\right)+a \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right) \left(a d \cos \left(c-\frac{a d}{b}\right)-2 b \sin \left(c-\frac{a d}{b}\right)\right)-a \text{Si}\left(d \left(\frac{a}{b}+x\right)\right) \left(a d \sin \left(c-\frac{a d}{b}\right)+2 b \cos \left(c-\frac{a d}{b}\right)\right)}{b^4}","\frac{a^2 d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^4}-\frac{a^2 d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^4}-\frac{a^2 \sin (c+d x)}{b^3 (a+b x)}-\frac{2 a \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^3}-\frac{2 a \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^3}-\frac{\cos (c+d x)}{b^2 d}",1,"(a*CosIntegral[d*(a/b + x)]*(a*d*Cos[c - (a*d)/b] - 2*b*Sin[c - (a*d)/b]) + b*(-((b*Cos[c + d*x])/d) - (a^2*Sin[c + d*x])/(a + b*x)) - a*(2*b*Cos[c - (a*d)/b] + a*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^4","A",1
29,1,96,124,0.4813678,"\int \frac{x \sin (c+d x)}{(a+b x)^2} \, dx","Integrate[(x*Sin[c + d*x])/(a + b*x)^2,x]","\frac{\text{Ci}\left(d \left(\frac{a}{b}+x\right)\right) \left(b \sin \left(c-\frac{a d}{b}\right)-a d \cos \left(c-\frac{a d}{b}\right)\right)+\text{Si}\left(d \left(\frac{a}{b}+x\right)\right) \left(a d \sin \left(c-\frac{a d}{b}\right)+b \cos \left(c-\frac{a d}{b}\right)\right)+\frac{a b \sin (c+d x)}{a+b x}}{b^3}","-\frac{a d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^3}+\frac{a d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^3}+\frac{\sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^2}+\frac{\cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^2}+\frac{a \sin (c+d x)}{b^2 (a+b x)}",1,"(CosIntegral[d*(a/b + x)]*(-(a*d*Cos[c - (a*d)/b]) + b*Sin[c - (a*d)/b]) + (a*b*Sin[c + d*x])/(a + b*x) + (b*Cos[c - (a*d)/b] + a*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^3","A",1
30,1,66,72,0.2250068,"\int \frac{\sin (c+d x)}{(a+b x)^2} \, dx","Integrate[Sin[c + d*x]/(a + b*x)^2,x]","\frac{d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)-d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)-\frac{b \sin (c+d x)}{a+b x}}{b^2}","\frac{d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^2}-\frac{d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^2}-\frac{\sin (c+d x)}{b (a+b x)}",1,"(d*Cos[c - (a*d)/b]*CosIntegral[d*(a/b + x)] - (b*Sin[c + d*x])/(a + b*x) - d*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/b^2","A",1
31,1,138,149,1.0693423,"\int \frac{\sin (c+d x)}{x (a+b x)^2} \, dx","Integrate[Sin[c + d*x]/(x*(a + b*x)^2),x]","\frac{-\frac{\text{Ci}\left(d \left(\frac{a}{b}+x\right)\right) \left(b \sin \left(c-\frac{a d}{b}\right)+a d \cos \left(c-\frac{a d}{b}\right)\right)}{b}+\frac{a d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)}{b}-\cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+\frac{a \sin (c) \cos (d x)}{a+b x}+\frac{a \cos (c) \sin (d x)}{a+b x}+\sin (c) \text{Ci}(d x)+\cos (c) \text{Si}(d x)}{a^2}","-\frac{\sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a^2}-\frac{\cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a^2}+\frac{\sin (c) \text{Ci}(d x)}{a^2}+\frac{\cos (c) \text{Si}(d x)}{a^2}-\frac{d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a b}+\frac{d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a b}+\frac{\sin (c+d x)}{a (a+b x)}",1,"((a*Cos[d*x]*Sin[c])/(a + b*x) + CosIntegral[d*x]*Sin[c] - (CosIntegral[d*(a/b + x)]*(a*d*Cos[c - (a*d)/b] + b*Sin[c - (a*d)/b]))/b + (a*Cos[c]*Sin[d*x])/(a + b*x) + Cos[c]*SinIntegral[d*x] - Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + (a*d*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/b)/a^2","A",1
32,1,184,188,2.0490856,"\int \frac{\sin (c+d x)}{x^2 (a+b x)^2} \, dx","Integrate[Sin[c + d*x]/(x^2*(a + b*x)^2),x]","-\frac{-2 b \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)-a d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)+a d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)-2 b \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+\frac{a \sin (c) (a+2 b x) \cos (d x)}{x (a+b x)}+\frac{a \cos (c) (a+2 b x) \sin (d x)}{x (a+b x)}-a d \cos (c) \text{Ci}(d x)+a d \sin (c) \text{Si}(d x)+2 b \sin (c) \text{Ci}(d x)+2 b \cos (c) \text{Si}(d x)}{a^3}","-\frac{2 b \sin (c) \text{Ci}(d x)}{a^3}+\frac{2 b \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a^3}-\frac{2 b \cos (c) \text{Si}(d x)}{a^3}+\frac{2 b \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a^3}+\frac{d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a^2}-\frac{d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a^2}-\frac{b \sin (c+d x)}{a^2 (a+b x)}+\frac{d \cos (c) \text{Ci}(d x)}{a^2}-\frac{d \sin (c) \text{Si}(d x)}{a^2}-\frac{\sin (c+d x)}{a^2 x}",1,"-((-(a*d*Cos[c]*CosIntegral[d*x]) - a*d*Cos[c - (a*d)/b]*CosIntegral[d*(a/b + x)] + (a*(a + 2*b*x)*Cos[d*x]*Sin[c])/(x*(a + b*x)) + 2*b*CosIntegral[d*x]*Sin[c] - 2*b*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + (a*(a + 2*b*x)*Cos[c]*Sin[d*x])/(x*(a + b*x)) + 2*b*Cos[c]*SinIntegral[d*x] + a*d*Sin[c]*SinIntegral[d*x] - 2*b*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + a*d*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/a^3)","A",1
33,1,235,265,1.1177431,"\int \frac{x^3 \sin (c+d x)}{(a+b x)^3} \, dx","Integrate[(x^3*Sin[c + d*x])/(a + b*x)^3,x]","-\frac{-a d (a+b x)^2 \left(\text{Ci}\left(d \left(\frac{a}{b}+x\right)\right) \left(\left(a^2 d^2-6 b^2\right) \sin \left(c-\frac{a d}{b}\right)+6 a b d \cos \left(c-\frac{a d}{b}\right)\right)+\text{Si}\left(d \left(\frac{a}{b}+x\right)\right) \left(\left(a^2 d^2-6 b^2\right) \cos \left(c-\frac{a d}{b}\right)-6 a b d \sin \left(c-\frac{a d}{b}\right)\right)\right)+b \cos (d x) \left(a^2 b d \sin (c) (5 a+6 b x)-\cos (c) (a+b x) \left(a^3 d^2-2 a b^2-2 b^3 x\right)\right)+b \sin (d x) \left(\sin (c) (a+b x) \left(a^3 d^2-2 a b^2-2 b^3 x\right)+a^2 b d \cos (c) (5 a+6 b x)\right)}{2 b^6 d (a+b x)^2}","\frac{a^3 d^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{2 b^6}+\frac{a^3 d^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{2 b^6}+\frac{a^3 d \cos (c+d x)}{2 b^5 (a+b x)}+\frac{a^3 \sin (c+d x)}{2 b^4 (a+b x)^2}+\frac{3 a^2 d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^5}-\frac{3 a^2 d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^5}-\frac{3 a^2 \sin (c+d x)}{b^4 (a+b x)}-\frac{3 a \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^4}-\frac{3 a \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^4}-\frac{\cos (c+d x)}{b^3 d}",1,"-1/2*(b*Cos[d*x]*(-((a + b*x)*(-2*a*b^2 + a^3*d^2 - 2*b^3*x)*Cos[c]) + a^2*b*d*(5*a + 6*b*x)*Sin[c]) + b*(a^2*b*d*(5*a + 6*b*x)*Cos[c] + (a + b*x)*(-2*a*b^2 + a^3*d^2 - 2*b^3*x)*Sin[c])*Sin[d*x] - a*d*(a + b*x)^2*(CosIntegral[d*(a/b + x)]*(6*a*b*d*Cos[c - (a*d)/b] + (-6*b^2 + a^2*d^2)*Sin[c - (a*d)/b]) + ((-6*b^2 + a^2*d^2)*Cos[c - (a*d)/b] - 6*a*b*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)]))/(b^6*d*(a + b*x)^2)","A",1
34,1,154,241,1.2475713,"\int \frac{x^2 \sin (c+d x)}{(a+b x)^3} \, dx","Integrate[(x^2*Sin[c + d*x])/(a + b*x)^3,x]","-\frac{-\text{Ci}\left(d \left(\frac{a}{b}+x\right)\right) \left(\left(2 b^2-a^2 d^2\right) \sin \left(c-\frac{a d}{b}\right)-4 a b d \cos \left(c-\frac{a d}{b}\right)\right)+\text{Si}\left(d \left(\frac{a}{b}+x\right)\right) \left(\left(a^2 d^2-2 b^2\right) \cos \left(c-\frac{a d}{b}\right)-4 a b d \sin \left(c-\frac{a d}{b}\right)\right)+\frac{a b (a d (a+b x) \cos (c+d x)-b (3 a+4 b x) \sin (c+d x))}{(a+b x)^2}}{2 b^5}","-\frac{a^2 d^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{2 b^5}-\frac{a^2 d^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{2 b^5}-\frac{a^2 d \cos (c+d x)}{2 b^4 (a+b x)}-\frac{a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}-\frac{2 a d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^4}+\frac{2 a d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^4}+\frac{\sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^3}+\frac{\cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^3}+\frac{2 a \sin (c+d x)}{b^3 (a+b x)}",1,"-1/2*(-(CosIntegral[d*(a/b + x)]*(-4*a*b*d*Cos[c - (a*d)/b] + (2*b^2 - a^2*d^2)*Sin[c - (a*d)/b])) + (a*b*(a*d*(a + b*x)*Cos[c + d*x] - b*(3*a + 4*b*x)*Sin[c + d*x]))/(a + b*x)^2 + ((-2*b^2 + a^2*d^2)*Cos[c - (a*d)/b] - 4*a*b*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^5","A",1
35,1,157,179,0.6062874,"\int \frac{x \sin (c+d x)}{(a+b x)^3} \, dx","Integrate[(x*Sin[c + d*x])/(a + b*x)^3,x]","\frac{d (a+b x)^2 \left(\text{Ci}\left(d \left(\frac{a}{b}+x\right)\right) \left(a d \sin \left(c-\frac{a d}{b}\right)+2 b \cos \left(c-\frac{a d}{b}\right)\right)+\text{Si}\left(d \left(\frac{a}{b}+x\right)\right) \left(a d \cos \left(c-\frac{a d}{b}\right)-2 b \sin \left(c-\frac{a d}{b}\right)\right)\right)+b \cos (d x) (a d \cos (c) (a+b x)-b \sin (c) (a+2 b x))-b \sin (d x) (a d \sin (c) (a+b x)+b \cos (c) (a+2 b x))}{2 b^4 (a+b x)^2}","\frac{a d^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{2 b^4}+\frac{a d^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{2 b^4}+\frac{d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{b^3}-\frac{d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{b^3}+\frac{a d \cos (c+d x)}{2 b^3 (a+b x)}-\frac{\sin (c+d x)}{b^2 (a+b x)}+\frac{a \sin (c+d x)}{2 b^2 (a+b x)^2}",1,"(b*Cos[d*x]*(a*d*(a + b*x)*Cos[c] - b*(a + 2*b*x)*Sin[c]) - b*(b*(a + 2*b*x)*Cos[c] + a*d*(a + b*x)*Sin[c])*Sin[d*x] + d*(a + b*x)^2*(CosIntegral[d*(a/b + x)]*(2*b*Cos[c - (a*d)/b] + a*d*Sin[c - (a*d)/b]) + (a*d*Cos[c - (a*d)/b] - 2*b*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)]))/(2*b^4*(a + b*x)^2)","A",1
36,1,87,104,0.735448,"\int \frac{\sin (c+d x)}{(a+b x)^3} \, dx","Integrate[Sin[c + d*x]/(a + b*x)^3,x]","-\frac{d^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right)+d^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+\frac{b (d (a+b x) \cos (c+d x)+b \sin (c+d x))}{(a+b x)^2}}{2 b^3}","-\frac{d^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{2 b^3}-\frac{d^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{2 b^3}-\frac{d \cos (c+d x)}{2 b^2 (a+b x)}-\frac{\sin (c+d x)}{2 b (a+b x)^2}",1,"-1/2*(d^2*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + (b*(d*(a + b*x)*Cos[c + d*x] + b*Sin[c + d*x]))/(a + b*x)^2 + d^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/b^3","A",1
37,1,449,261,1.0487003,"\int \frac{\sin (c+d x)}{x (a+b x)^3} \, dx","Integrate[Sin[c + d*x]/(x*(a + b*x)^3),x]","\frac{a^4 d^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+2 a^3 b d^2 x \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+2 a^3 b d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+a^3 b d \cos (c+d x)+(a+b x)^2 \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right) \left(\left(a^2 d^2-2 b^2\right) \sin \left(c-\frac{a d}{b}\right)-2 a b d \cos \left(c-\frac{a d}{b}\right)\right)+a^2 b^2 d^2 x^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+4 a^2 b^2 d x \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+2 a^2 b^2 \cos (c) \text{Si}(d x)-2 a^2 b^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+3 a^2 b^2 \sin (c+d x)+a^2 b^2 d x \cos (c+d x)-2 b^4 x^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+2 a b^3 d x^2 \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+4 a b^3 x \cos (c) \text{Si}(d x)-4 a b^3 x \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+2 a b^3 x \sin (c+d x)+2 b^2 \sin (c) (a+b x)^2 \text{Ci}(d x)+2 b^4 x^2 \cos (c) \text{Si}(d x)}{2 a^3 b^2 (a+b x)^2}","-\frac{\sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a^3}-\frac{\cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a^3}+\frac{\sin (c) \text{Ci}(d x)}{a^3}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a^2 b}+\frac{d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a^2 b}+\frac{\sin (c+d x)}{a^2 (a+b x)}+\frac{d^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{2 a b^2}+\frac{d^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{2 a b^2}+\frac{\sin (c+d x)}{2 a (a+b x)^2}+\frac{d \cos (c+d x)}{2 a b (a+b x)}",1,"(a^3*b*d*Cos[c + d*x] + a^2*b^2*d*x*Cos[c + d*x] + 2*b^2*(a + b*x)^2*CosIntegral[d*x]*Sin[c] + (a + b*x)^2*CosIntegral[d*(a/b + x)]*(-2*a*b*d*Cos[c - (a*d)/b] + (-2*b^2 + a^2*d^2)*Sin[c - (a*d)/b]) + 3*a^2*b^2*Sin[c + d*x] + 2*a*b^3*x*Sin[c + d*x] + 2*a^2*b^2*Cos[c]*SinIntegral[d*x] + 4*a*b^3*x*Cos[c]*SinIntegral[d*x] + 2*b^4*x^2*Cos[c]*SinIntegral[d*x] - 2*a^2*b^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + a^4*d^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] - 4*a*b^3*x*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 2*a^3*b*d^2*x*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] - 2*b^4*x^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + a^2*b^2*d^2*x^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 2*a^3*b*d*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 4*a^2*b^2*d*x*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 2*a*b^3*d*x^2*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(2*a^3*b^2*(a + b*x)^2)","A",1
38,1,540,299,2.0741833,"\int \frac{\sin (c+d x)}{x^2 (a+b x)^3} \, dx","Integrate[Sin[c + d*x]/(x^2*(a + b*x)^3),x]","-\frac{a^4 d^2 x \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+2 a^3 b d^2 x^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+2 a^3 b d x \sin (c) \text{Si}(d x)+4 a^3 b d x \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+2 a^3 b \sin (c+d x)+a^3 b d x \cos (c+d x)+x (a+b x)^2 \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right) \left(\left(a^2 d^2-6 b^2\right) \sin \left(c-\frac{a d}{b}\right)-4 a b d \cos \left(c-\frac{a d}{b}\right)\right)+a^2 b^2 d^2 x^3 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+4 a^2 b^2 d x^2 \sin (c) \text{Si}(d x)+8 a^2 b^2 d x^2 \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+6 a^2 b^2 x \cos (c) \text{Si}(d x)-6 a^2 b^2 x \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+a^2 b^2 d x^2 \cos (c+d x)+9 a^2 b^2 x \sin (c+d x)-6 b^4 x^3 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+2 a b^3 d x^3 \sin (c) \text{Si}(d x)+4 a b^3 d x^3 \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+12 a b^3 x^2 \cos (c) \text{Si}(d x)-12 a b^3 x^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+6 a b^3 x^2 \sin (c+d x)+2 b x (a+b x)^2 \text{Ci}(d x) (3 b \sin (c)-a d \cos (c))+6 b^4 x^3 \cos (c) \text{Si}(d x)}{2 a^4 b x (a+b x)^2}","-\frac{3 b \sin (c) \text{Ci}(d x)}{a^4}+\frac{3 b \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a^4}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}+\frac{3 b \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a^4}+\frac{2 d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a^3}-\frac{2 d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a^3}-\frac{2 b \sin (c+d x)}{a^3 (a+b x)}+\frac{d \cos (c) \text{Ci}(d x)}{a^3}-\frac{d \sin (c) \text{Si}(d x)}{a^3}-\frac{\sin (c+d x)}{a^3 x}-\frac{d^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{2 a^2 b}-\frac{d^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{2 a^2 b}-\frac{b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac{d \cos (c+d x)}{2 a^2 (a+b x)}",1,"-1/2*(a^3*b*d*x*Cos[c + d*x] + a^2*b^2*d*x^2*Cos[c + d*x] + 2*b*x*(a + b*x)^2*CosIntegral[d*x]*(-(a*d*Cos[c]) + 3*b*Sin[c]) + x*(a + b*x)^2*CosIntegral[d*(a/b + x)]*(-4*a*b*d*Cos[c - (a*d)/b] + (-6*b^2 + a^2*d^2)*Sin[c - (a*d)/b]) + 2*a^3*b*Sin[c + d*x] + 9*a^2*b^2*x*Sin[c + d*x] + 6*a*b^3*x^2*Sin[c + d*x] + 6*a^2*b^2*x*Cos[c]*SinIntegral[d*x] + 12*a*b^3*x^2*Cos[c]*SinIntegral[d*x] + 6*b^4*x^3*Cos[c]*SinIntegral[d*x] + 2*a^3*b*d*x*Sin[c]*SinIntegral[d*x] + 4*a^2*b^2*d*x^2*Sin[c]*SinIntegral[d*x] + 2*a*b^3*d*x^3*Sin[c]*SinIntegral[d*x] - 6*a^2*b^2*x*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + a^4*d^2*x*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] - 12*a*b^3*x^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 2*a^3*b*d^2*x^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] - 6*b^4*x^3*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + a^2*b^2*d^2*x^3*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 4*a^3*b*d*x*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 8*a^2*b^2*d*x^2*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 4*a*b^3*d*x^3*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(a^4*b*x*(a + b*x)^2)","A",1
39,1,630,377,2.0747684,"\int \frac{\sin (c+d x)}{x^3 (a+b x)^3} \, dx","Integrate[Sin[c + d*x]/(x^3*(a + b*x)^3),x]","\frac{a^4 d^2 x^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)-a^4 d^2 x^2 \cos (c) \text{Si}(d x)-a^4 \sin (c+d x)+a^4 (-d) x \cos (c+d x)-2 a^3 b d^2 x^3 \cos (c) \text{Si}(d x)+2 a^3 b d^2 x^3 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+6 a^3 b d x^2 \sin (c) \text{Si}(d x)+6 a^3 b d x^2 \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)-a^3 b d x^2 \cos (c+d x)+4 a^3 b x \sin (c+d x)-x^2 (a+b x)^2 \text{Ci}(d x) \left(\sin (c) \left(a^2 d^2-12 b^2\right)+6 a b d \cos (c)\right)+x^2 (a+b x)^2 \text{Ci}\left(d \left(\frac{a}{b}+x\right)\right) \left(\left(a^2 d^2-12 b^2\right) \sin \left(c-\frac{a d}{b}\right)-6 a b d \cos \left(c-\frac{a d}{b}\right)\right)-a^2 b^2 d^2 x^4 \cos (c) \text{Si}(d x)+a^2 b^2 d^2 x^4 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+12 a^2 b^2 d x^3 \sin (c) \text{Si}(d x)+12 a^2 b^2 d x^3 \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+12 a^2 b^2 x^2 \cos (c) \text{Si}(d x)-12 a^2 b^2 x^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+18 a^2 b^2 x^2 \sin (c+d x)-12 b^4 x^4 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+6 a b^3 d x^4 \sin (c) \text{Si}(d x)+6 a b^3 d x^4 \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+24 a b^3 x^3 \cos (c) \text{Si}(d x)-24 a b^3 x^3 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(d \left(\frac{a}{b}+x\right)\right)+12 a b^3 x^3 \sin (c+d x)+12 b^4 x^4 \cos (c) \text{Si}(d x)}{2 a^5 x^2 (a+b x)^2}","\frac{6 b^2 \sin (c) \text{Ci}(d x)}{a^5}-\frac{6 b^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a^5}+\frac{6 b^2 \cos (c) \text{Si}(d x)}{a^5}-\frac{6 b^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a^5}+\frac{3 b^2 \sin (c+d x)}{a^4 (a+b x)}-\frac{3 b d \cos (c) \text{Ci}(d x)}{a^4}-\frac{3 b d \cos \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{a^4}+\frac{3 b d \sin (c) \text{Si}(d x)}{a^4}+\frac{3 b d \sin \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{a^4}+\frac{3 b \sin (c+d x)}{a^4 x}+\frac{b^2 \sin (c+d x)}{2 a^3 (a+b x)^2}+\frac{d^2 \sin \left(c-\frac{a d}{b}\right) \text{Ci}\left(x d+\frac{a d}{b}\right)}{2 a^3}+\frac{d^2 \cos \left(c-\frac{a d}{b}\right) \text{Si}\left(x d+\frac{a d}{b}\right)}{2 a^3}+\frac{b d \cos (c+d x)}{2 a^3 (a+b x)}-\frac{d^2 \sin (c) \text{Ci}(d x)}{2 a^3}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{d \cos (c+d x)}{2 a^3 x}",1,"(-(a^4*d*x*Cos[c + d*x]) - a^3*b*d*x^2*Cos[c + d*x] - x^2*(a + b*x)^2*CosIntegral[d*x]*(6*a*b*d*Cos[c] + (-12*b^2 + a^2*d^2)*Sin[c]) + x^2*(a + b*x)^2*CosIntegral[d*(a/b + x)]*(-6*a*b*d*Cos[c - (a*d)/b] + (-12*b^2 + a^2*d^2)*Sin[c - (a*d)/b]) - a^4*Sin[c + d*x] + 4*a^3*b*x*Sin[c + d*x] + 18*a^2*b^2*x^2*Sin[c + d*x] + 12*a*b^3*x^3*Sin[c + d*x] + 12*a^2*b^2*x^2*Cos[c]*SinIntegral[d*x] - a^4*d^2*x^2*Cos[c]*SinIntegral[d*x] + 24*a*b^3*x^3*Cos[c]*SinIntegral[d*x] - 2*a^3*b*d^2*x^3*Cos[c]*SinIntegral[d*x] + 12*b^4*x^4*Cos[c]*SinIntegral[d*x] - a^2*b^2*d^2*x^4*Cos[c]*SinIntegral[d*x] + 6*a^3*b*d*x^2*Sin[c]*SinIntegral[d*x] + 12*a^2*b^2*d*x^3*Sin[c]*SinIntegral[d*x] + 6*a*b^3*d*x^4*Sin[c]*SinIntegral[d*x] - 12*a^2*b^2*x^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + a^4*d^2*x^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] - 24*a*b^3*x^3*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 2*a^3*b*d^2*x^3*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] - 12*b^4*x^4*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + a^2*b^2*d^2*x^4*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 6*a^3*b*d*x^2*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 12*a^2*b^2*d*x^3*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)] + 6*a*b^3*d*x^4*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(2*a^5*x^2*(a + b*x)^2)","A",1
40,1,92,141,0.1829433,"\int x^3 \left(a+b x^2\right) \sin (c+d x) \, dx","Integrate[x^3*(a + b*x^2)*Sin[c + d*x],x]","\frac{\left(3 a d^2 \left(d^2 x^2-2\right)+5 b \left(d^4 x^4-12 d^2 x^2+24\right)\right) \sin (c+d x)-d x \left(a d^2 \left(d^2 x^2-6\right)+b \left(d^4 x^4-20 d^2 x^2+120\right)\right) \cos (c+d x)}{d^6}","-\frac{6 a \sin (c+d x)}{d^4}+\frac{6 a x \cos (c+d x)}{d^3}+\frac{3 a x^2 \sin (c+d x)}{d^2}-\frac{a x^3 \cos (c+d x)}{d}+\frac{120 b \sin (c+d x)}{d^6}-\frac{120 b x \cos (c+d x)}{d^5}-\frac{60 b x^2 \sin (c+d x)}{d^4}+\frac{20 b x^3 \cos (c+d x)}{d^3}+\frac{5 b x^4 \sin (c+d x)}{d^2}-\frac{b x^5 \cos (c+d x)}{d}",1,"(-(d*x*(a*d^2*(-6 + d^2*x^2) + b*(120 - 20*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + (3*a*d^2*(-2 + d^2*x^2) + 5*b*(24 - 12*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^6","A",1
41,1,75,111,0.1452874,"\int x^2 \left(a+b x^2\right) \sin (c+d x) \, dx","Integrate[x^2*(a + b*x^2)*Sin[c + d*x],x]","\frac{2 d x \left(a d^2+2 b \left(d^2 x^2-6\right)\right) \sin (c+d x)-\left(a d^2 \left(d^2 x^2-2\right)+b \left(d^4 x^4-12 d^2 x^2+24\right)\right) \cos (c+d x)}{d^5}","\frac{2 a \cos (c+d x)}{d^3}+\frac{2 a x \sin (c+d x)}{d^2}-\frac{a x^2 \cos (c+d x)}{d}-\frac{24 b \cos (c+d x)}{d^5}-\frac{24 b x \sin (c+d x)}{d^4}+\frac{12 b x^2 \cos (c+d x)}{d^3}+\frac{4 b x^3 \sin (c+d x)}{d^2}-\frac{b x^4 \cos (c+d x)}{d}",1,"(-((a*d^2*(-2 + d^2*x^2) + b*(24 - 12*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + 2*d*x*(a*d^2 + 2*b*(-6 + d^2*x^2))*Sin[c + d*x])/d^5","A",1
42,1,57,80,0.1264359,"\int x \left(a+b x^2\right) \sin (c+d x) \, dx","Integrate[x*(a + b*x^2)*Sin[c + d*x],x]","\frac{\left(a d^2+3 b \left(d^2 x^2-2\right)\right) \sin (c+d x)-d x \left(a d^2+b \left(d^2 x^2-6\right)\right) \cos (c+d x)}{d^4}","\frac{a \sin (c+d x)}{d^2}-\frac{a x \cos (c+d x)}{d}-\frac{6 b \sin (c+d x)}{d^4}+\frac{6 b x \cos (c+d x)}{d^3}+\frac{3 b x^2 \sin (c+d x)}{d^2}-\frac{b x^3 \cos (c+d x)}{d}",1,"(-(d*x*(a*d^2 + b*(-6 + d^2*x^2))*Cos[c + d*x]) + (a*d^2 + 3*b*(-2 + d^2*x^2))*Sin[c + d*x])/d^4","A",1
43,1,41,53,0.0876091,"\int \left(a+b x^2\right) \sin (c+d x) \, dx","Integrate[(a + b*x^2)*Sin[c + d*x],x]","\frac{2 b d x \sin (c+d x)-\left(a d^2+b \left(d^2 x^2-2\right)\right) \cos (c+d x)}{d^3}","-\frac{a \cos (c+d x)}{d}+\frac{2 b \cos (c+d x)}{d^3}+\frac{2 b x \sin (c+d x)}{d^2}-\frac{b x^2 \cos (c+d x)}{d}",1,"(-((a*d^2 + b*(-2 + d^2*x^2))*Cos[c + d*x]) + 2*b*d*x*Sin[c + d*x])/d^3","A",1
44,1,54,41,0.1368717,"\int \frac{\left(a+b x^2\right) \sin (c+d x)}{x} \, dx","Integrate[((a + b*x^2)*Sin[c + d*x])/x,x]","a \sin (c) \text{Ci}(d x)+a \cos (c) \text{Si}(d x)-\frac{b \cos (d x) (d x \cos (c)-\sin (c))}{d^2}+\frac{b \sin (d x) (d x \sin (c)+\cos (c))}{d^2}","a \sin (c) \text{Ci}(d x)+a \cos (c) \text{Si}(d x)+\frac{b \sin (c+d x)}{d^2}-\frac{b x \cos (c+d x)}{d}",1,"-((b*Cos[d*x]*(d*x*Cos[c] - Sin[c]))/d^2) + a*CosIntegral[d*x]*Sin[c] + (b*(Cos[c] + d*x*Sin[c])*Sin[d*x])/d^2 + a*Cos[c]*SinIntegral[d*x]","A",1
45,1,44,44,0.0971917,"\int \frac{\left(a+b x^2\right) \sin (c+d x)}{x^2} \, dx","Integrate[((a + b*x^2)*Sin[c + d*x])/x^2,x]","a d \cos (c) \text{Ci}(d x)-a d \sin (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{x}-\frac{b \cos (c+d x)}{d}","a d \cos (c) \text{Ci}(d x)-a d \sin (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{x}-\frac{b \cos (c+d x)}{d}",1,"-((b*Cos[c + d*x])/d) + a*d*Cos[c]*CosIntegral[d*x] - (a*Sin[c + d*x])/x - a*d*Sin[c]*SinIntegral[d*x]","A",1
46,1,82,74,0.1935677,"\int \frac{\left(a+b x^2\right) \sin (c+d x)}{x^3} \, dx","Integrate[((a + b*x^2)*Sin[c + d*x])/x^3,x]","-\frac{1}{2} a d^2 (\sin (c) \text{Ci}(d x)+\cos (c) \text{Si}(d x))-\frac{a \cos (d x) (d x \cos (c)+\sin (c))}{2 x^2}+\frac{a \sin (d x) (d x \sin (c)-\cos (c))}{2 x^2}+b \sin (c) \text{Ci}(d x)+b \cos (c) \text{Si}(d x)","-\frac{1}{2} a d^2 \sin (c) \text{Ci}(d x)-\frac{1}{2} a d^2 \cos (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{2 x^2}-\frac{a d \cos (c+d x)}{2 x}+b \sin (c) \text{Ci}(d x)+b \cos (c) \text{Si}(d x)",1,"b*CosIntegral[d*x]*Sin[c] - (a*Cos[d*x]*(d*x*Cos[c] + Sin[c]))/(2*x^2) + (a*(-Cos[c] + d*x*Sin[c])*Sin[d*x])/(2*x^2) + b*Cos[c]*SinIntegral[d*x] - (a*d^2*(CosIntegral[d*x]*Sin[c] + Cos[c]*SinIntegral[d*x]))/2","A",1
47,1,95,106,0.2014803,"\int \frac{\left(a+b x^2\right) \sin (c+d x)}{x^4} \, dx","Integrate[((a + b*x^2)*Sin[c + d*x])/x^4,x]","\frac{d x^3 \cos (c) \left(6 b-a d^2\right) \text{Ci}(d x)+d x^3 \sin (c) \left(a d^2-6 b\right) \text{Si}(d x)+a d^2 x^2 \sin (c+d x)-2 a \sin (c+d x)-a d x \cos (c+d x)-6 b x^2 \sin (c+d x)}{6 x^3}","-\frac{1}{6} a d^3 \cos (c) \text{Ci}(d x)+\frac{1}{6} a d^3 \sin (c) \text{Si}(d x)+\frac{a d^2 \sin (c+d x)}{6 x}-\frac{a \sin (c+d x)}{3 x^3}-\frac{a d \cos (c+d x)}{6 x^2}+b d \cos (c) \text{Ci}(d x)-b d \sin (c) \text{Si}(d x)-\frac{b \sin (c+d x)}{x}",1,"(-(a*d*x*Cos[c + d*x]) + d*(6*b - a*d^2)*x^3*Cos[c]*CosIntegral[d*x] - 2*a*Sin[c + d*x] - 6*b*x^2*Sin[c + d*x] + a*d^2*x^2*Sin[c + d*x] + d*(-6*b + a*d^2)*x^3*Sin[c]*SinIntegral[d*x])/(6*x^3)","A",1
48,1,125,149,0.2490686,"\int \frac{\left(a+b x^2\right) \sin (c+d x)}{x^5} \, dx","Integrate[((a + b*x^2)*Sin[c + d*x])/x^5,x]","\frac{d^2 x^4 \sin (c) \left(a d^2-12 b\right) \text{Ci}(d x)+d^2 x^4 \cos (c) \left(a d^2-12 b\right) \text{Si}(d x)+a d^3 x^3 \cos (c+d x)+a d^2 x^2 \sin (c+d x)-6 a \sin (c+d x)-2 a d x \cos (c+d x)-12 b d x^3 \cos (c+d x)-12 b x^2 \sin (c+d x)}{24 x^4}","\frac{1}{24} a d^4 \sin (c) \text{Ci}(d x)+\frac{1}{24} a d^4 \cos (c) \text{Si}(d x)+\frac{a d^3 \cos (c+d x)}{24 x}+\frac{a d^2 \sin (c+d x)}{24 x^2}-\frac{a \sin (c+d x)}{4 x^4}-\frac{a d \cos (c+d x)}{12 x^3}-\frac{1}{2} b d^2 \sin (c) \text{Ci}(d x)-\frac{1}{2} b d^2 \cos (c) \text{Si}(d x)-\frac{b \sin (c+d x)}{2 x^2}-\frac{b d \cos (c+d x)}{2 x}",1,"(-2*a*d*x*Cos[c + d*x] - 12*b*d*x^3*Cos[c + d*x] + a*d^3*x^3*Cos[c + d*x] + d^2*(-12*b + a*d^2)*x^4*CosIntegral[d*x]*Sin[c] - 6*a*Sin[c + d*x] - 12*b*x^2*Sin[c + d*x] + a*d^2*x^2*Sin[c + d*x] + d^2*(-12*b + a*d^2)*x^4*Cos[c]*SinIntegral[d*x])/(24*x^4)","A",1
49,1,139,236,0.4162462,"\int x^2 \left(a+b x^2\right)^2 \sin (c+d x) \, dx","Integrate[x^2*(a + b*x^2)^2*Sin[c + d*x],x]","\frac{2 d x \left(a^2 d^4+4 a b d^2 \left(d^2 x^2-6\right)+3 b^2 \left(d^4 x^4-20 d^2 x^2+120\right)\right) \sin (c+d x)-\left(a^2 d^4 \left(d^2 x^2-2\right)+2 a b d^2 \left(d^4 x^4-12 d^2 x^2+24\right)+b^2 \left(d^6 x^6-30 d^4 x^4+360 d^2 x^2-720\right)\right) \cos (c+d x)}{d^7}","\frac{2 a^2 \cos (c+d x)}{d^3}+\frac{2 a^2 x \sin (c+d x)}{d^2}-\frac{a^2 x^2 \cos (c+d x)}{d}-\frac{48 a b \cos (c+d x)}{d^5}-\frac{48 a b x \sin (c+d x)}{d^4}+\frac{24 a b x^2 \cos (c+d x)}{d^3}+\frac{8 a b x^3 \sin (c+d x)}{d^2}-\frac{2 a b x^4 \cos (c+d x)}{d}+\frac{720 b^2 \cos (c+d x)}{d^7}+\frac{720 b^2 x \sin (c+d x)}{d^6}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{b^2 x^6 \cos (c+d x)}{d}",1,"(-((a^2*d^4*(-2 + d^2*x^2) + 2*a*b*d^2*(24 - 12*d^2*x^2 + d^4*x^4) + b^2*(-720 + 360*d^2*x^2 - 30*d^4*x^4 + d^6*x^6))*Cos[c + d*x]) + 2*d*x*(a^2*d^4 + 4*a*b*d^2*(-6 + d^2*x^2) + 3*b^2*(120 - 20*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^7","A",1
50,1,113,185,0.2753451,"\int x \left(a+b x^2\right)^2 \sin (c+d x) \, dx","Integrate[x*(a + b*x^2)^2*Sin[c + d*x],x]","\frac{\left(a^2 d^4+6 a b d^2 \left(d^2 x^2-2\right)+5 b^2 \left(d^4 x^4-12 d^2 x^2+24\right)\right) \sin (c+d x)-d x \left(a^2 d^4+2 a b d^2 \left(d^2 x^2-6\right)+b^2 \left(d^4 x^4-20 d^2 x^2+120\right)\right) \cos (c+d x)}{d^6}","\frac{a^2 \sin (c+d x)}{d^2}-\frac{a^2 x \cos (c+d x)}{d}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{12 a b x \cos (c+d x)}{d^3}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{120 b^2 \sin (c+d x)}{d^6}-\frac{120 b^2 x \cos (c+d x)}{d^5}-\frac{60 b^2 x^2 \sin (c+d x)}{d^4}+\frac{20 b^2 x^3 \cos (c+d x)}{d^3}+\frac{5 b^2 x^4 \sin (c+d x)}{d^2}-\frac{b^2 x^5 \cos (c+d x)}{d}",1,"(-(d*x*(a^2*d^4 + 2*a*b*d^2*(-6 + d^2*x^2) + b^2*(120 - 20*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + (a^2*d^4 + 6*a*b*d^2*(-2 + d^2*x^2) + 5*b^2*(24 - 12*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^6","A",1
51,1,86,138,0.2055829,"\int \left(a+b x^2\right)^2 \sin (c+d x) \, dx","Integrate[(a + b*x^2)^2*Sin[c + d*x],x]","\frac{4 b d x \left(a d^2+b \left(d^2 x^2-6\right)\right) \sin (c+d x)-\left(a^2 d^4+2 a b d^2 \left(d^2 x^2-2\right)+b^2 \left(d^4 x^4-12 d^2 x^2+24\right)\right) \cos (c+d x)}{d^5}","-\frac{a^2 \cos (c+d x)}{d}+\frac{4 a b \cos (c+d x)}{d^3}+\frac{4 a b x \sin (c+d x)}{d^2}-\frac{2 a b x^2 \cos (c+d x)}{d}-\frac{24 b^2 \cos (c+d x)}{d^5}-\frac{24 b^2 x \sin (c+d x)}{d^4}+\frac{12 b^2 x^2 \cos (c+d x)}{d^3}+\frac{4 b^2 x^3 \sin (c+d x)}{d^2}-\frac{b^2 x^4 \cos (c+d x)}{d}",1,"(-((a^2*d^4 + 2*a*b*d^2*(-2 + d^2*x^2) + b^2*(24 - 12*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + 4*b*d*x*(a*d^2 + b*(-6 + d^2*x^2))*Sin[c + d*x])/d^5","A",1
52,1,82,111,0.4167461,"\int \frac{\left(a+b x^2\right)^2 \sin (c+d x)}{x} \, dx","Integrate[((a + b*x^2)^2*Sin[c + d*x])/x,x]","a^2 \sin (c) \text{Ci}(d x)+a^2 \cos (c) \text{Si}(d x)+\frac{b \left(2 a d^2+3 b \left(d^2 x^2-2\right)\right) \sin (c+d x)}{d^4}-\frac{b x \left(2 a d^2+b \left(d^2 x^2-6\right)\right) \cos (c+d x)}{d^3}","a^2 \sin (c) \text{Ci}(d x)+a^2 \cos (c) \text{Si}(d x)+\frac{2 a b \sin (c+d x)}{d^2}-\frac{2 a b x \cos (c+d x)}{d}-\frac{6 b^2 \sin (c+d x)}{d^4}+\frac{6 b^2 x \cos (c+d x)}{d^3}+\frac{3 b^2 x^2 \sin (c+d x)}{d^2}-\frac{b^2 x^3 \cos (c+d x)}{d}",1,"-((b*x*(2*a*d^2 + b*(-6 + d^2*x^2))*Cos[c + d*x])/d^3) + a^2*CosIntegral[d*x]*Sin[c] + (b*(2*a*d^2 + 3*b*(-2 + d^2*x^2))*Sin[c + d*x])/d^4 + a^2*Cos[c]*SinIntegral[d*x]","A",1
53,1,97,97,0.2837108,"\int \frac{\left(a+b x^2\right)^2 \sin (c+d x)}{x^2} \, dx","Integrate[((a + b*x^2)^2*Sin[c + d*x])/x^2,x]","a^2 d \cos (c) \text{Ci}(d x)-a^2 d \sin (c) \text{Si}(d x)-\frac{a^2 \sin (c+d x)}{x}-\frac{2 a b \cos (c+d x)}{d}+\frac{2 b^2 \cos (c+d x)}{d^3}+\frac{2 b^2 x \sin (c+d x)}{d^2}-\frac{b^2 x^2 \cos (c+d x)}{d}","a^2 d \cos (c) \text{Ci}(d x)-a^2 d \sin (c) \text{Si}(d x)-\frac{a^2 \sin (c+d x)}{x}-\frac{2 a b \cos (c+d x)}{d}+\frac{2 b^2 \cos (c+d x)}{d^3}+\frac{2 b^2 x \sin (c+d x)}{d^2}-\frac{b^2 x^2 \cos (c+d x)}{d}",1,"(2*b^2*Cos[c + d*x])/d^3 - (2*a*b*Cos[c + d*x])/d - (b^2*x^2*Cos[c + d*x])/d + a^2*d*Cos[c]*CosIntegral[d*x] - (a^2*Sin[c + d*x])/x + (2*b^2*x*Sin[c + d*x])/d^2 - a^2*d*Sin[c]*SinIntegral[d*x]","A",1
54,1,99,114,0.4429864,"\int \frac{\left(a+b x^2\right)^2 \sin (c+d x)}{x^3} \, dx","Integrate[((a + b*x^2)^2*Sin[c + d*x])/x^3,x]","\frac{1}{2} \left(-\frac{a^2 \sin (c+d x)}{x^2}-\frac{a^2 d \cos (c+d x)}{x}+a \sin (c) \left(4 b-a d^2\right) \text{Ci}(d x)+a \cos (c) \left(4 b-a d^2\right) \text{Si}(d x)+\frac{2 b^2 \sin (c+d x)}{d^2}-\frac{2 b^2 x \cos (c+d x)}{d}\right)","-\frac{1}{2} a^2 d^2 \sin (c) \text{Ci}(d x)-\frac{1}{2} a^2 d^2 \cos (c) \text{Si}(d x)-\frac{a^2 \sin (c+d x)}{2 x^2}-\frac{a^2 d \cos (c+d x)}{2 x}+2 a b \sin (c) \text{Ci}(d x)+2 a b \cos (c) \text{Si}(d x)+\frac{b^2 \sin (c+d x)}{d^2}-\frac{b^2 x \cos (c+d x)}{d}",1,"(-((a^2*d*Cos[c + d*x])/x) - (2*b^2*x*Cos[c + d*x])/d + a*(4*b - a*d^2)*CosIntegral[d*x]*Sin[c] + (2*b^2*Sin[c + d*x])/d^2 - (a^2*Sin[c + d*x])/x^2 + a*(4*b - a*d^2)*Cos[c]*SinIntegral[d*x])/2","A",1
55,1,114,134,0.4479954,"\int \frac{\left(a+b x^2\right)^2 \sin (c+d x)}{x^4} \, dx","Integrate[((a + b*x^2)^2*Sin[c + d*x])/x^4,x]","\frac{1}{6} \left(\frac{a^2 d^2 \sin (c+d x)}{x}-\frac{2 a^2 \sin (c+d x)}{x^3}-\frac{a^2 d \cos (c+d x)}{x^2}-a d \cos (c) \left(a d^2-12 b\right) \text{Ci}(d x)+a d \sin (c) \left(a d^2-12 b\right) \text{Si}(d x)-\frac{12 a b \sin (c+d x)}{x}-\frac{6 b^2 \cos (c+d x)}{d}\right)","-\frac{1}{6} a^2 d^3 \cos (c) \text{Ci}(d x)+\frac{1}{6} a^2 d^3 \sin (c) \text{Si}(d x)+\frac{a^2 d^2 \sin (c+d x)}{6 x}-\frac{a^2 \sin (c+d x)}{3 x^3}-\frac{a^2 d \cos (c+d x)}{6 x^2}+2 a b d \cos (c) \text{Ci}(d x)-2 a b d \sin (c) \text{Si}(d x)-\frac{2 a b \sin (c+d x)}{x}-\frac{b^2 \cos (c+d x)}{d}",1,"((-6*b^2*Cos[c + d*x])/d - (a^2*d*Cos[c + d*x])/x^2 - a*d*(-12*b + a*d^2)*Cos[c]*CosIntegral[d*x] - (2*a^2*Sin[c + d*x])/x^3 - (12*a*b*Sin[c + d*x])/x + (a^2*d^2*Sin[c + d*x])/x + a*d*(-12*b + a*d^2)*Sin[c]*SinIntegral[d*x])/6","A",1
56,1,122,177,0.4700259,"\int \frac{\left(a+b x^2\right)^2 \sin (c+d x)}{x^5} \, dx","Integrate[((a + b*x^2)^2*Sin[c + d*x])/x^5,x]","\frac{x^4 \sin (c) \left(a^2 d^4-24 a b d^2+24 b^2\right) \text{Ci}(d x)+x^4 \cos (c) \left(a^2 d^4-24 a b d^2+24 b^2\right) \text{Si}(d x)+a \left(a \left(d^2 x^2-6\right)-24 b x^2\right) \sin (c+d x)+a d x \left(a \left(d^2 x^2-2\right)-24 b x^2\right) \cos (c+d x)}{24 x^4}","\frac{1}{24} a^2 d^4 \sin (c) \text{Ci}(d x)+\frac{1}{24} a^2 d^4 \cos (c) \text{Si}(d x)+\frac{a^2 d^3 \cos (c+d x)}{24 x}+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a^2 d \cos (c+d x)}{12 x^3}-a b d^2 \sin (c) \text{Ci}(d x)-a b d^2 \cos (c) \text{Si}(d x)-\frac{a b \sin (c+d x)}{x^2}-\frac{a b d \cos (c+d x)}{x}+b^2 \sin (c) \text{Ci}(d x)+b^2 \cos (c) \text{Si}(d x)",1,"(a*d*x*(-24*b*x^2 + a*(-2 + d^2*x^2))*Cos[c + d*x] + (24*b^2 - 24*a*b*d^2 + a^2*d^4)*x^4*CosIntegral[d*x]*Sin[c] + a*(-24*b*x^2 + a*(-6 + d^2*x^2))*Sin[c + d*x] + (24*b^2 - 24*a*b*d^2 + a^2*d^4)*x^4*Cos[c]*SinIntegral[d*x])/(24*x^4)","A",1
57,1,275,273,0.4979676,"\int \frac{x^4 \sin (c+d x)}{a+b x^2} \, dx","Integrate[(x^4*Sin[c + d*x])/(a + b*x^2),x]","\frac{i a^{3/2} d^3 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-i a^{3/2} d^3 \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i a^{3/2} d^3 \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i a^{3/2} d^3 \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+2 a \sqrt{b} d^2 \cos (c+d x)-2 b^{3/2} d^2 x^2 \cos (c+d x)+4 b^{3/2} d x \sin (c+d x)+4 b^{3/2} \cos (c+d x)}{2 b^{5/2} d^3}","-\frac{(-a)^{3/2} \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 b^{5/2}}+\frac{(-a)^{3/2} \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 b^{5/2}}-\frac{(-a)^{3/2} \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 b^{5/2}}-\frac{(-a)^{3/2} \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 b^{5/2}}+\frac{a \cos (c+d x)}{b^2 d}+\frac{2 \cos (c+d x)}{b d^3}+\frac{2 x \sin (c+d x)}{b d^2}-\frac{x^2 \cos (c+d x)}{b d}",1,"(4*b^(3/2)*Cos[c + d*x] + 2*a*Sqrt[b]*d^2*Cos[c + d*x] - 2*b^(3/2)*d^2*x^2*Cos[c + d*x] + I*a^(3/2)*d^3*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] - I*a^(3/2)*d^3*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] + 4*b^(3/2)*d*x*Sin[c + d*x] + I*a^(3/2)*d^3*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*a^(3/2)*d^3*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(2*b^(5/2)*d^3)","C",0
58,1,202,209,0.4267741,"\int \frac{x^3 \sin (c+d x)}{a+b x^2} \, dx","Integrate[(x^3*Sin[c + d*x])/(a + b*x^2),x]","-\frac{a d^2 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+a d^2 \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+a d^2 \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-a d^2 \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-2 b \sin (c+d x)+2 b d x \cos (c+d x)}{2 b^2 d^2}","-\frac{a \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 b^2}-\frac{a \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 b^2}+\frac{a \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 b^2}-\frac{a \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 b^2}+\frac{\sin (c+d x)}{b d^2}-\frac{x \cos (c+d x)}{b d}",1,"-1/2*(2*b*d*x*Cos[c + d*x] + a*d^2*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] + a*d^2*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] - 2*b*Sin[c + d*x] + a*d^2*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - a*d^2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(b^2*d^2)","C",0
59,1,216,227,0.3563608,"\int \frac{x^2 \sin (c+d x)}{a+b x^2} \, dx","Integrate[(x^2*Sin[c + d*x])/(a + b*x^2),x]","-\frac{i \sqrt{a} d \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-i \sqrt{a} d \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \sqrt{a} d \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \sqrt{a} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+2 \sqrt{b} \cos (c+d x)}{2 b^{3/2} d}","-\frac{\sqrt{-a} \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 b^{3/2}}+\frac{\sqrt{-a} \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 b^{3/2}}-\frac{\sqrt{-a} \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 b^{3/2}}-\frac{\sqrt{-a} \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 b^{3/2}}-\frac{\cos (c+d x)}{b d}",1,"-1/2*(2*Sqrt[b]*Cos[c + d*x] + I*Sqrt[a]*d*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] - I*Sqrt[a]*d*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] + I*Sqrt[a]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sqrt[a]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(b^(3/2)*d)","C",0
60,1,163,177,0.2163815,"\int \frac{x \sin (c+d x)}{a+b x^2} \, dx","Integrate[(x*Sin[c + d*x])/(a + b*x^2),x]","\frac{\sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{2 b}","\frac{\sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 b}+\frac{\sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 b}-\frac{\cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 b}+\frac{\cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 b}",1,"(CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] + CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] + Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(2*b)","C",0
61,1,172,213,0.2102998,"\int \frac{\sin (c+d x)}{a+b x^2} \, dx","Integrate[Sin[c + d*x]/(a + b*x^2),x]","\frac{i \left(\sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)\right)}{2 \sqrt{a} \sqrt{b}}","-\frac{\sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 \sqrt{-a} \sqrt{b}}+\frac{\sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 \sqrt{-a} \sqrt{b}}-\frac{\cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 \sqrt{-a} \sqrt{b}}-\frac{\cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 \sqrt{-a} \sqrt{b}}",1,"((I/2)*(CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] - CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] + Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x]))/(Sqrt[a]*Sqrt[b])","C",0
62,1,179,197,0.3722739,"\int \frac{\sin (c+d x)}{x \left(a+b x^2\right)} \, dx","Integrate[Sin[c + d*x]/(x*(a + b*x^2)),x]","-\frac{\sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-2 \sin (c) \text{Ci}(d x)-2 \cos (c) \text{Si}(d x)}{2 a}","-\frac{\sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 a}-\frac{\sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 a}+\frac{\cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 a}-\frac{\cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 a}+\frac{\sin (c) \text{Ci}(d x)}{a}+\frac{\cos (c) \text{Si}(d x)}{a}",1,"-1/2*(-2*CosIntegral[d*x]*Sin[c] + CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] + CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] - 2*Cos[c]*SinIntegral[d*x] + Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/a","C",0
63,1,238,250,0.5171143,"\int \frac{\sin (c+d x)}{x^2 \left(a+b x^2\right)} \, dx","Integrate[Sin[c + d*x]/(x^2*(a + b*x^2)),x]","\frac{d \cos (c) \text{Ci}(d x)}{a}-\frac{i \left(\sqrt{b} x \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\sqrt{b} x \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sqrt{b} x \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sqrt{b} x \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-2 i \sqrt{a} d x \sin (c) \text{Si}(d x)-2 i \sqrt{a} \sin (c+d x)\right)}{2 a^{3/2} x}","-\frac{\sqrt{b} \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 (-a)^{3/2}}+\frac{\sqrt{b} \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 (-a)^{3/2}}-\frac{\sqrt{b} \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 (-a)^{3/2}}-\frac{\sqrt{b} \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 (-a)^{3/2}}+\frac{d \cos (c) \text{Ci}(d x)}{a}-\frac{d \sin (c) \text{Si}(d x)}{a}-\frac{\sin (c+d x)}{a x}",1,"(d*Cos[c]*CosIntegral[d*x])/a - ((I/2)*(Sqrt[b]*x*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] - Sqrt[b]*x*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] - (2*I)*Sqrt[a]*Sin[c + d*x] - (2*I)*Sqrt[a]*d*x*Sin[c]*SinIntegral[d*x] + Sqrt[b]*x*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + Sqrt[b]*x*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x]))/(a^(3/2)*x)","C",0
64,1,247,270,0.6929126,"\int \frac{\sin (c+d x)}{x^3 \left(a+b x^2\right)} \, dx","Integrate[Sin[c + d*x]/(x^3*(a + b*x^2)),x]","-\frac{x^2 \sin (c) \left(a d^2+2 b\right) \text{Ci}(d x)-b x^2 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-b x^2 \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-b x^2 \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+b x^2 \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+a d^2 x^2 \cos (c) \text{Si}(d x)+a \sin (c+d x)+a d x \cos (c+d x)+2 b x^2 \cos (c) \text{Si}(d x)}{2 a^2 x^2}","-\frac{b \sin (c) \text{Ci}(d x)}{a^2}+\frac{b \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 a^2}+\frac{b \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 a^2}-\frac{b \cos (c) \text{Si}(d x)}{a^2}-\frac{b \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 a^2}+\frac{b \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 a^2}-\frac{d^2 \sin (c) \text{Ci}(d x)}{2 a}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a}-\frac{\sin (c+d x)}{2 a x^2}-\frac{d \cos (c+d x)}{2 a x}",1,"-1/2*(a*d*x*Cos[c + d*x] + (2*b + a*d^2)*x^2*CosIntegral[d*x]*Sin[c] - b*x^2*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] - b*x^2*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]] + a*Sin[c + d*x] + 2*b*x^2*Cos[c]*SinIntegral[d*x] + a*d^2*x^2*Cos[c]*SinIntegral[d*x] - b*x^2*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + b*x^2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(a^2*x^2)","C",0
65,1,632,450,1.1835389,"\int \frac{x^4 \sin (c+d x)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^4*Sin[c + d*x])/(a + b*x^2)^2,x]","-\frac{3 i a^{3/2} \sqrt{b} d \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+3 i a^{3/2} \sqrt{b} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-a^2 d^2 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+a^2 d^2 \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+3 i \sqrt{a} b^{3/2} d x^2 \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+3 i \sqrt{a} b^{3/2} d x^2 \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+\sqrt{a} d \left(a+b x^2\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(3 i \sqrt{b} \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)+\sqrt{a} d \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)+\sqrt{a} d \left(a+b x^2\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(\sqrt{a} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)-3 i \sqrt{b} \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)-a b d^2 x^2 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+a b d^2 x^2 \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-2 a b d x \sin (c+d x)+4 a b \cos (c+d x)+4 b^2 x^2 \cos (c+d x)}{4 b^3 d \left(a+b x^2\right)}","-\frac{3 \sqrt{-a} \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 b^{5/2}}+\frac{3 \sqrt{-a} \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 b^{5/2}}-\frac{3 \sqrt{-a} \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 b^{5/2}}-\frac{3 \sqrt{-a} \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 b^{5/2}}-\frac{a d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 b^3}-\frac{a d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 b^3}-\frac{a d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 b^3}+\frac{a d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 b^3}-\frac{x^3 \sin (c+d x)}{2 b \left(a+b x^2\right)}+\frac{x \sin (c+d x)}{2 b^2}-\frac{\cos (c+d x)}{b^2 d}",1,"-1/4*(4*a*b*Cos[c + d*x] + 4*b^2*x^2*Cos[c + d*x] + Sqrt[a]*d*(a + b*x^2)*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] + (3*I)*Sqrt[b]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]) + Sqrt[a]*d*(a + b*x^2)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] - (3*I)*Sqrt[b]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) - 2*a*b*d*x*Sin[c + d*x] + (3*I)*a^(3/2)*Sqrt[b]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + (3*I)*Sqrt[a]*b^(3/2)*d*x^2*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - a^2*d^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - a*b*d^2*x^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + (3*I)*a^(3/2)*Sqrt[b]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + (3*I)*Sqrt[a]*b^(3/2)*d*x^2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + a^2*d^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + a*b*d^2*x^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(b^3*d*(a + b*x^2))","C",0
66,1,583,431,0.8920974,"\int \frac{x^3 \sin (c+d x)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^3*Sin[c + d*x])/(a + b*x^2)^2,x]","\frac{i a^{3/2} d \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i a^{3/2} d \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+2 b^{3/2} x^2 \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-2 b^{3/2} x^2 \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+\left(a+b x^2\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(2 \sqrt{b} \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)-i \sqrt{a} d \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)+\left(a+b x^2\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(2 \sqrt{b} \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)+i \sqrt{a} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)+i \sqrt{a} b d x^2 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \sqrt{a} b d x^2 \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+2 a \sqrt{b} \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-2 a \sqrt{b} \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+2 a \sqrt{b} \sin (c+d x)}{4 b^{5/2} \left(a+b x^2\right)}","\frac{\sqrt{-a} d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 b^{5/2}}-\frac{\sqrt{-a} d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 b^{5/2}}+\frac{\sqrt{-a} d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 b^{5/2}}+\frac{\sqrt{-a} d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 b^{5/2}}+\frac{\sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 b^2}+\frac{\sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 b^2}-\frac{\cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 b^2}+\frac{\cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 b^2}-\frac{x^2 \sin (c+d x)}{2 b \left(a+b x^2\right)}+\frac{\sin (c+d x)}{2 b^2}",1,"((a + b*x^2)*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*((-I)*Sqrt[a]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] + 2*Sqrt[b]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]) + (a + b*x^2)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(I*Sqrt[a]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] + 2*Sqrt[b]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) + 2*a*Sqrt[b]*Sin[c + d*x] + 2*a*Sqrt[b]*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 2*b^(3/2)*x^2*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*a^(3/2)*d*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sqrt[a]*b*d*x^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - 2*a*Sqrt[b]*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - 2*b^(3/2)*x^2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*a^(3/2)*d*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*Sqrt[a]*b*d*x^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)*(a + b*x^2))","C",0
67,1,583,416,0.8625497,"\int \frac{x^2 \sin (c+d x)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x^2*Sin[c + d*x])/(a + b*x^2)^2,x]","\frac{-a^{3/2} d \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+a^{3/2} d \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+i b^{3/2} x^2 \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i b^{3/2} x^2 \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+\left(a+b x^2\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(i \sqrt{b} \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)+\sqrt{a} d \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)+\left(a+b x^2\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(\sqrt{a} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)-i \sqrt{b} \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)-\sqrt{a} b d x^2 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sqrt{a} b d x^2 \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+i a \sqrt{b} \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i a \sqrt{b} \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-2 \sqrt{a} b x \sin (c+d x)}{4 \sqrt{a} b^2 \left(a+b x^2\right)}","\frac{\sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 \sqrt{-a} b^{3/2}}-\frac{\sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 \sqrt{-a} b^{3/2}}-\frac{\cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 \sqrt{-a} b^{3/2}}-\frac{\cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 \sqrt{-a} b^{3/2}}+\frac{d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 b^2}+\frac{d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 b^2}+\frac{d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 b^2}-\frac{d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 b^2}-\frac{x \sin (c+d x)}{2 b \left(a+b x^2\right)}",1,"((a + b*x^2)*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] + I*Sqrt[b]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]) + (a + b*x^2)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] - I*Sqrt[b]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) - 2*Sqrt[a]*b*x*Sin[c + d*x] + I*a*Sqrt[b]*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*b^(3/2)*x^2*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - a^(3/2)*d*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - Sqrt[a]*b*d*x^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*a*Sqrt[b]*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*b^(3/2)*x^2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + a^(3/2)*d*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + Sqrt[a]*b*d*x^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(4*Sqrt[a]*b^2*(a + b*x^2))","C",0
68,1,309,239,0.3989061,"\int \frac{x \sin (c+d x)}{\left(a+b x^2\right)^2} \, dx","Integrate[(x*Sin[c + d*x])/(a + b*x^2)^2,x]","-\frac{i \left(d \left(a+b x^2\right) \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-d \left(a+b x^2\right) \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+b d x^2 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+b d x^2 \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+a d \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+a d \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-2 i \sqrt{a} \sqrt{b} \sin (c+d x)\right)}{4 \sqrt{a} b^{3/2} \left(a+b x^2\right)}","\frac{d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 \sqrt{-a} b^{3/2}}-\frac{d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 \sqrt{-a} b^{3/2}}+\frac{d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 \sqrt{-a} b^{3/2}}+\frac{d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 \sqrt{-a} b^{3/2}}-\frac{\sin (c+d x)}{2 b \left(a+b x^2\right)}",1,"((-1/4*I)*(d*(a + b*x^2)*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] - d*(a + b*x^2)*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (2*I)*Sqrt[a]*Sqrt[b]*Sin[c + d*x] + a*d*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + b*d*x^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + a*d*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + b*d*x^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x]))/(Sqrt[a]*b^(3/2)*(a + b*x^2))","C",0
69,1,585,476,0.6483675,"\int \frac{\sin (c+d x)}{\left(a+b x^2\right)^2} \, dx","Integrate[Sin[c + d*x]/(a + b*x^2)^2,x]","\frac{a^{3/2} d \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-a^{3/2} d \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+i b^{3/2} x^2 \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i b^{3/2} x^2 \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-\left(a+b x^2\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(\sqrt{a} d \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)-i \sqrt{b} \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)-\left(a+b x^2\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(i \sqrt{b} \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)+\sqrt{a} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)+\sqrt{a} b d x^2 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\sqrt{a} b d x^2 \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+i a \sqrt{b} \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i a \sqrt{b} \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+2 \sqrt{a} b x \sin (c+d x)}{4 a^{3/2} b \left(a+b x^2\right)}","-\frac{\sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 (-a)^{3/2} \sqrt{b}}+\frac{\sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 (-a)^{3/2} \sqrt{b}}-\frac{d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 a b}-\frac{d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 a b}-\frac{d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 a b}+\frac{d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 a b}+\frac{\cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 (-a)^{3/2} \sqrt{b}}+\frac{\cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 (-a)^{3/2} \sqrt{b}}-\frac{\sin (c+d x)}{4 a \sqrt{b} \left(\sqrt{-a}-\sqrt{b} x\right)}+\frac{\sin (c+d x)}{4 a \sqrt{b} \left(\sqrt{-a}+\sqrt{b} x\right)}",1,"(-((a + b*x^2)*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] - I*Sqrt[b]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]])) - (a + b*x^2)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] + I*Sqrt[b]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) + 2*Sqrt[a]*b*x*Sin[c + d*x] + I*a*Sqrt[b]*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*b^(3/2)*x^2*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + a^(3/2)*d*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + Sqrt[a]*b*d*x^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*a*Sqrt[b]*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*b^(3/2)*x^2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - a^(3/2)*d*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - Sqrt[a]*b*d*x^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(4*a^(3/2)*b*(a + b*x^2))","C",0
70,1,650,435,0.982875,"\int \frac{\sin (c+d x)}{x \left(a+b x^2\right)^2} \, dx","Integrate[Sin[c + d*x]/(x*(a + b*x^2)^2),x]","\frac{i a^{3/2} d \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i a^{3/2} d \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-2 b^{3/2} x^2 \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+2 b^{3/2} x^2 \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-i \left(a+b x^2\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(\sqrt{a} d \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)-2 i \sqrt{b} \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)+i \left(a+b x^2\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(2 i \sqrt{b} \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)+\sqrt{a} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)+4 a \sqrt{b} \sin (c) \text{Ci}(d x)+i \sqrt{a} b d x^2 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \sqrt{a} b d x^2 \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+4 a \sqrt{b} \cos (c) \text{Si}(d x)-2 a \sqrt{b} \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+2 a \sqrt{b} \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+2 a \sqrt{b} \sin (c+d x)+4 b^{3/2} x^2 \sin (c) \text{Ci}(d x)+4 b^{3/2} x^2 \cos (c) \text{Si}(d x)}{4 a^2 \sqrt{b} \left(a+b x^2\right)}","-\frac{\sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 a^2}-\frac{\sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 a^2}+\frac{\cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 a^2}-\frac{\cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 a^2}+\frac{\sin (c) \text{Ci}(d x)}{a^2}+\frac{\cos (c) \text{Si}(d x)}{a^2}+\frac{d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 (-a)^{3/2} \sqrt{b}}-\frac{d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 (-a)^{3/2} \sqrt{b}}+\frac{d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 (-a)^{3/2} \sqrt{b}}+\frac{d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 (-a)^{3/2} \sqrt{b}}+\frac{\sin (c+d x)}{2 a \left(a+b x^2\right)}",1,"(4*a*Sqrt[b]*CosIntegral[d*x]*Sin[c] + 4*b^(3/2)*x^2*CosIntegral[d*x]*Sin[c] - I*(a + b*x^2)*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] - (2*I)*Sqrt[b]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]) + I*(a + b*x^2)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] + (2*I)*Sqrt[b]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) + 2*a*Sqrt[b]*Sin[c + d*x] + 4*a*Sqrt[b]*Cos[c]*SinIntegral[d*x] + 4*b^(3/2)*x^2*Cos[c]*SinIntegral[d*x] - 2*a*Sqrt[b]*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - 2*b^(3/2)*x^2*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*a^(3/2)*d*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sqrt[a]*b*d*x^2*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 2*a*Sqrt[b]*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + 2*b^(3/2)*x^2*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*a^(3/2)*d*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + I*Sqrt[a]*b*d*x^2*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(4*a^2*Sqrt[b]*(a + b*x^2))","C",0
71,1,768,501,1.0976452,"\int \frac{\sin (c+d x)}{x^2 \left(a+b x^2\right)^2} \, dx","Integrate[Sin[c + d*x]/(x^2*(a + b*x^2)^2),x]","\frac{a^{3/2} d x \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-a^{3/2} d x \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+a^{3/2} d x \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-4 a^{3/2} d x \sin (c) \text{Si}(d x)-4 a^{3/2} \sin (c+d x)-3 i b^{3/2} x^3 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-3 i b^{3/2} x^3 \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-3 i b^{3/2} x^3 \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+\sqrt{a} b d x^3 \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+4 \sqrt{a} d x \cos (c) \left(a+b x^2\right) \text{Ci}(d x)+x \left(a+b x^2\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(3 i \sqrt{b} \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)+\sqrt{a} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)-3 i a \sqrt{b} x \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-4 \sqrt{a} b d x^3 \sin (c) \text{Si}(d x)-\sqrt{a} b d x^3 \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sqrt{a} b d x^3 \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-3 i a \sqrt{b} x \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-3 i a \sqrt{b} x \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-6 \sqrt{a} b x^2 \sin (c+d x)}{4 a^{5/2} x \left(a+b x^2\right)}","\frac{d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 a^2}+\frac{d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 a^2}+\frac{d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 a^2}-\frac{d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 a^2}+\frac{\sqrt{b} \sin (c+d x)}{4 a^2 \left(\sqrt{-a}-\sqrt{b} x\right)}-\frac{\sqrt{b} \sin (c+d x)}{4 a^2 \left(\sqrt{-a}+\sqrt{b} x\right)}+\frac{d \cos (c) \text{Ci}(d x)}{a^2}-\frac{d \sin (c) \text{Si}(d x)}{a^2}-\frac{\sin (c+d x)}{a^2 x}+\frac{3 \sqrt{b} \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 (-a)^{5/2}}-\frac{3 \sqrt{b} \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 (-a)^{5/2}}+\frac{3 \sqrt{b} \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{4 (-a)^{5/2}}+\frac{3 \sqrt{b} \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{4 (-a)^{5/2}}",1,"(4*Sqrt[a]*d*x*(a + b*x^2)*Cos[c]*CosIntegral[d*x] + a^(3/2)*d*x*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + Sqrt[a]*b*d*x^3*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (3*I)*a*Sqrt[b]*x*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] - (3*I)*b^(3/2)*x^3*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]] + x*(a + b*x^2)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(Sqrt[a]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] + (3*I)*Sqrt[b]*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]) - 4*a^(3/2)*Sin[c + d*x] - 6*Sqrt[a]*b*x^2*Sin[c + d*x] - 4*a^(3/2)*d*x*Sin[c]*SinIntegral[d*x] - 4*Sqrt[a]*b*d*x^3*Sin[c]*SinIntegral[d*x] - (3*I)*a*Sqrt[b]*x*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (3*I)*b^(3/2)*x^3*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - a^(3/2)*d*x*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - Sqrt[a]*b*d*x^3*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (3*I)*a*Sqrt[b]*x*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - (3*I)*b^(3/2)*x^3*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + a^(3/2)*d*x*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + Sqrt[a]*b*d*x^3*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(4*a^(5/2)*x*(a + b*x^2))","C",0
72,1,647,476,1.9942004,"\int \frac{x^3 \sin (c+d x)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^3*Sin[c + d*x])/(a + b*x^2)^3,x]","\frac{\frac{d^2 \cos (c) \left(-i \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)\right)\right)}{b}-\frac{d^2 \sin (c) \left(\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)\right)\right)}{b}+\frac{3 d \cos (c) \left(-i \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)\right)\right)}{\sqrt{a} \sqrt{b}}-\frac{3 d \sin (c) \left(\sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)\right)\right)}{\sqrt{a} \sqrt{b}}-\frac{2 \cos (d x) \left(d x \cos (c) \left(a+b x^2\right)+2 \sin (c) \left(a+2 b x^2\right)\right)}{\left(a+b x^2\right)^2}+\frac{2 \sin (d x) \left(d x \sin (c) \left(a+b x^2\right)-2 \cos (c) \left(a+2 b x^2\right)\right)}{\left(a+b x^2\right)^2}}{16 b^2}","\frac{3 d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 \sqrt{-a} b^{5/2}}-\frac{3 d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 \sqrt{-a} b^{5/2}}+\frac{3 d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 \sqrt{-a} b^{5/2}}+\frac{3 d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 b^3}-\frac{d^2 \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 b^3}+\frac{d^2 \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 b^3}-\frac{d^2 \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 b^3}-\frac{\sin (c+d x)}{4 b^2 \left(a+b x^2\right)}-\frac{d x \cos (c+d x)}{8 b^2 \left(a+b x^2\right)}-\frac{x^2 \sin (c+d x)}{4 b \left(a+b x^2\right)^2}",1,"((-2*Cos[d*x]*(d*x*(a + b*x^2)*Cos[c] + 2*(a + 2*b*x^2)*Sin[c]))/(a + b*x^2)^2 + (2*(-2*(a + 2*b*x^2)*Cos[c] + d*x*(a + b*x^2)*Sin[c])*Sin[d*x])/(a + b*x^2)^2 + (d^2*Cos[c]*((-I)*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + I*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + Cosh[(Sqrt[a]*d)/Sqrt[b]]*(-SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/b + (3*d*Cos[c]*((-I)*Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + I*Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + Sinh[(Sqrt[a]*d)/Sqrt[b]]*(-SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/(Sqrt[a]*Sqrt[b]) - (3*d*Sin[c]*(CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + I*Cosh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/(Sqrt[a]*Sqrt[b]) - (d^2*Sin[c]*(Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sinh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/b)/(16*b^2)","C",0
73,1,927,746,2.7724167,"\int \frac{x^2 \sin (c+d x)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x^2*Sin[c + d*x])/(a + b*x^2)^3,x]","\frac{\frac{2 \sqrt{a} b^2 \cos (d x) \sin (c) x^3}{\left(b x^2+a\right)^2}+\frac{2 \sqrt{a} b^2 \cos (c) \sin (d x) x^3}{\left(b x^2+a\right)^2}-\frac{2 a^{3/2} b d \cos (c) \cos (d x) x^2}{\left(b x^2+a\right)^2}+\frac{2 a^{3/2} b d \sin (c) \sin (d x) x^2}{\left(b x^2+a\right)^2}-\frac{2 a^{3/2} b \cos (d x) \sin (c) x}{\left(b x^2+a\right)^2}-\frac{2 a^{3/2} b \cos (c) \sin (d x) x}{\left(b x^2+a\right)^2}-\frac{2 a^{5/2} d \cos (c) \cos (d x)}{\left(b x^2+a\right)^2}+\frac{\text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(i \left(b-a d^2\right) \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)-\sqrt{a} \sqrt{b} d \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)}{\sqrt{b}}+\frac{i \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(i \sqrt{a} \sqrt{b} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)+\left(a d^2-b\right) \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)}{\sqrt{b}}+\frac{2 a^{5/2} d \sin (c) \sin (d x)}{\left(b x^2+a\right)^2}-\frac{i a d^2 \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)}{\sqrt{b}}+i \sqrt{b} \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sqrt{a} d \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \sin (c) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-i \sqrt{a} d \cos (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\frac{a d^2 \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)}{\sqrt{b}}-\sqrt{b} \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\frac{i a d^2 \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{\sqrt{b}}+i \sqrt{b} \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-\sqrt{a} d \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \sin (c) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-i \sqrt{a} d \cos (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-\frac{a d^2 \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{\sqrt{b}}+\sqrt{b} \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{16 a^{3/2} b^2}","\frac{d^2 \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 \sqrt{-a} b^{5/2}}-\frac{d^2 \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 \sqrt{-a} b^{5/2}}+\frac{\sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{3/2} b^{3/2}}-\frac{\sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 (-a)^{3/2} b^{3/2}}+\frac{d^2 \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 \sqrt{-a} b^{5/2}}+\frac{d^2 \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 \sqrt{-a} b^{5/2}}+\frac{\cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 (-a)^{3/2} b^{3/2}}+\frac{\cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{3/2} b^{3/2}}-\frac{\sin (c+d x)}{16 a b^{3/2} \left(\sqrt{-a}-\sqrt{b} x\right)}+\frac{\sin (c+d x)}{16 a b^{3/2} \left(\sqrt{-a}+\sqrt{b} x\right)}-\frac{d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 a b^2}-\frac{d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 a b^2}-\frac{d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 a b^2}+\frac{d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 a b^2}-\frac{d \cos (c+d x)}{8 b^2 \left(a+b x^2\right)}-\frac{x \sin (c+d x)}{4 b \left(a+b x^2\right)^2}",1,"((-2*a^(5/2)*d*Cos[c]*Cos[d*x])/(a + b*x^2)^2 - (2*a^(3/2)*b*d*x^2*Cos[c]*Cos[d*x])/(a + b*x^2)^2 - (2*a^(3/2)*b*x*Cos[d*x]*Sin[c])/(a + b*x^2)^2 + (2*Sqrt[a]*b^2*x^3*Cos[d*x]*Sin[c])/(a + b*x^2)^2 + (CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*(-(Sqrt[a]*Sqrt[b]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]]) + I*(b - a*d^2)*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]))/Sqrt[b] + (I*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(I*Sqrt[a]*Sqrt[b]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] + (-b + a*d^2)*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]))/Sqrt[b] - (2*a^(3/2)*b*x*Cos[c]*Sin[d*x])/(a + b*x^2)^2 + (2*Sqrt[a]*b^2*x^3*Cos[c]*Sin[d*x])/(a + b*x^2)^2 + (2*a^(5/2)*d*Sin[c]*Sin[d*x])/(a + b*x^2)^2 + (2*a^(3/2)*b*d*x^2*Sin[c]*Sin[d*x])/(a + b*x^2)^2 + I*Sqrt[b]*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (I*a*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[b] + Sqrt[a]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - I*Sqrt[a]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - Sqrt[b]*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + (a*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[b] + I*Sqrt[b]*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - (I*a*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[b] - Sqrt[a]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - I*Sqrt[a]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + Sqrt[b]*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - (a*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[b])/(16*a^(3/2)*b^2)","C",0
74,1,634,512,1.8240682,"\int \frac{x \sin (c+d x)}{\left(a+b x^2\right)^3} \, dx","Integrate[(x*Sin[c + d*x])/(a + b*x^2)^3,x]","\frac{\frac{d^2 \cos (c) \left(i \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-i \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)\right)\right)}{b}+\frac{d^2 \sin (c) \left(\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)\right)\right)}{b}+\frac{d \cos (c) \left(-i \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)\right)\right)}{\sqrt{a} \sqrt{b}}-\frac{d \sin (c) \left(\sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)\right)\right)}{\sqrt{a} \sqrt{b}}+\frac{2 \cos (d x) \left(d x \cos (c) \left(a+b x^2\right)-2 a \sin (c)\right)}{\left(a+b x^2\right)^2}-\frac{2 \sin (d x) \left(d x \sin (c) \left(a+b x^2\right)+2 a \cos (c)\right)}{\left(a+b x^2\right)^2}}{16 a b}","-\frac{d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 (-a)^{3/2} b^{3/2}}+\frac{d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{3/2} b^{3/2}}-\frac{d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 (-a)^{3/2} b^{3/2}}-\frac{d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{3/2} b^{3/2}}-\frac{d \cos (c+d x)}{16 a b^{3/2} \left(\sqrt{-a}-\sqrt{b} x\right)}+\frac{d \cos (c+d x)}{16 a b^{3/2} \left(\sqrt{-a}+\sqrt{b} x\right)}+\frac{d^2 \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 a b^2}+\frac{d^2 \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 a b^2}-\frac{d^2 \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 a b^2}+\frac{d^2 \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 a b^2}-\frac{\sin (c+d x)}{4 b \left(a+b x^2\right)^2}",1,"((2*Cos[d*x]*(d*x*(a + b*x^2)*Cos[c] - 2*a*Sin[c]))/(a + b*x^2)^2 - (2*(2*a*Cos[c] + d*x*(a + b*x^2)*Sin[c])*Sin[d*x])/(a + b*x^2)^2 + (d^2*Cos[c]*(I*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] - I*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + Cosh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/b + (d*Cos[c]*((-I)*Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + I*Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + Sinh[(Sqrt[a]*d)/Sqrt[b]]*(-SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/(Sqrt[a]*Sqrt[b]) - (d*Sin[c]*(CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + I*Cosh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/(Sqrt[a]*Sqrt[b]) + (d^2*Sin[c]*(Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sinh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/b)/(16*a*b)","C",0
75,1,932,856,2.52148,"\int \frac{\sin (c+d x)}{\left(a+b x^2\right)^3} \, dx","Integrate[Sin[c + d*x]/(a + b*x^2)^3,x]","\frac{\frac{6 b^{5/2} \cos (d x) \sin (c) x^3}{\left(b x^2+a\right)^2}+\frac{6 b^{5/2} \cos (c) \sin (d x) x^3}{\left(b x^2+a\right)^2}+\frac{2 a b^{3/2} d \cos (c) \cos (d x) x^2}{\left(b x^2+a\right)^2}-\frac{2 a b^{3/2} d \sin (c) \sin (d x) x^2}{\left(b x^2+a\right)^2}+\frac{10 a b^{3/2} \cos (d x) \sin (c) x}{\left(b x^2+a\right)^2}+\frac{10 a b^{3/2} \cos (c) \sin (d x) x}{\left(b x^2+a\right)^2}+\frac{2 a^2 \sqrt{b} d \cos (c) \cos (d x)}{\left(b x^2+a\right)^2}+\frac{i \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(3 i \sqrt{a} \sqrt{b} d \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)+\left(a d^2+3 b\right) \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)}{\sqrt{a}}-\frac{i \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(\left(a d^2+3 b\right) \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)-3 i \sqrt{a} \sqrt{b} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)}{\sqrt{a}}-\frac{2 a^2 \sqrt{b} d \sin (c) \sin (d x)}{\left(b x^2+a\right)^2}+i \sqrt{a} d^2 \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\frac{3 i b \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)}{\sqrt{a}}+3 \sqrt{b} d \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \sin (c) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-3 i \sqrt{b} d \cos (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\sqrt{a} d^2 \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\frac{3 b \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)}{\sqrt{a}}+i \sqrt{a} d^2 \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+\frac{3 i b \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{\sqrt{a}}-3 \sqrt{b} d \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \sin (c) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-3 i \sqrt{b} d \cos (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+\sqrt{a} d^2 \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+\frac{3 b \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{\sqrt{a}}}{16 a^2 b^{3/2}}","\frac{\text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) d^2}{16 (-a)^{3/2} b^{3/2}}-\frac{\text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right) \sin \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac{\cos \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac{\cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac{\cos (c+d x) d}{16 (-a)^{3/2} b \left(\sqrt{-a}-\sqrt{b} x\right)}+\frac{\cos (c+d x) d}{16 (-a)^{3/2} b \left(\sqrt{b} x+\sqrt{-a}\right)}-\frac{3 \cos \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right) d}{16 a^2 b}-\frac{3 \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right) d}{16 a^2 b}-\frac{3 \sin \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right) d}{16 a^2 b}+\frac{3 \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right) d}{16 a^2 b}-\frac{3 \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{5/2} \sqrt{b}}+\frac{3 \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right) \sin \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{5/2} \sqrt{b}}-\frac{3 \sin (c+d x)}{16 a^2 \sqrt{b} \left(\sqrt{-a}-\sqrt{b} x\right)}+\frac{3 \sin (c+d x)}{16 a^2 \sqrt{b} \left(\sqrt{b} x+\sqrt{-a}\right)}-\frac{\sin (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left(\sqrt{-a}-\sqrt{b} x\right)^2}+\frac{\sin (c+d x)}{16 (-a)^{3/2} \sqrt{b} \left(\sqrt{b} x+\sqrt{-a}\right)^2}-\frac{3 \cos \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 (-a)^{5/2} \sqrt{b}}-\frac{3 \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{5/2} \sqrt{b}}",1,"((2*a^2*Sqrt[b]*d*Cos[c]*Cos[d*x])/(a + b*x^2)^2 + (2*a*b^(3/2)*d*x^2*Cos[c]*Cos[d*x])/(a + b*x^2)^2 + (10*a*b^(3/2)*x*Cos[d*x]*Sin[c])/(a + b*x^2)^2 + (6*b^(5/2)*x^3*Cos[d*x]*Sin[c])/(a + b*x^2)^2 + (I*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*((3*I)*Sqrt[a]*Sqrt[b]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] + (3*b + a*d^2)*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]))/Sqrt[a] - (I*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*((-3*I)*Sqrt[a]*Sqrt[b]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] + (3*b + a*d^2)*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]))/Sqrt[a] + (10*a*b^(3/2)*x*Cos[c]*Sin[d*x])/(a + b*x^2)^2 + (6*b^(5/2)*x^3*Cos[c]*Sin[d*x])/(a + b*x^2)^2 - (2*a^2*Sqrt[b]*d*Sin[c]*Sin[d*x])/(a + b*x^2)^2 - (2*a*b^(3/2)*d*x^2*Sin[c]*Sin[d*x])/(a + b*x^2)^2 + ((3*I)*b*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[a] + I*Sqrt[a]*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 3*Sqrt[b]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (3*I)*Sqrt[b]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (3*b*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[a] - Sqrt[a]*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + ((3*I)*b*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[a] + I*Sqrt[a]*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - 3*Sqrt[b]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - (3*I)*Sqrt[b]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + (3*b*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[a] + Sqrt[a]*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/(16*a^2*b^(3/2))","C",0
76,1,924,730,2.8688468,"\int \frac{\sin (c+d x)}{x \left(a+b x^2\right)^3} \, dx","Integrate[Sin[c + d*x]/(x*(a + b*x^2)^3),x]","\frac{\frac{16 b^2 \text{Ci}(d x) \sin (c) x^4}{\left(b x^2+a\right)^2}+\frac{16 b^2 \cos (c) \text{Si}(d x) x^4}{\left(b x^2+a\right)^2}-\frac{2 a b d \cos (c+d x) x^3}{\left(b x^2+a\right)^2}+\frac{32 a b \text{Ci}(d x) \sin (c) x^2}{\left(b x^2+a\right)^2}+\frac{8 a b \sin (c+d x) x^2}{\left(b x^2+a\right)^2}+\frac{32 a b \cos (c) \text{Si}(d x) x^2}{\left(b x^2+a\right)^2}-\frac{2 a^2 d \cos (c+d x) x}{\left(b x^2+a\right)^2}+\frac{16 a^2 \text{Ci}(d x) \sin (c)}{\left(b x^2+a\right)^2}-\frac{\text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(5 i \sqrt{a} \sqrt{b} d \cos \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)+\left(a d^2+8 b\right) \sin \left(c-\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)}{b}-\frac{\text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(\left(a d^2+8 b\right) \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)-5 i \sqrt{a} \sqrt{b} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)}{b}+\frac{12 a^2 \sin (c+d x)}{\left(b x^2+a\right)^2}+\frac{16 a^2 \cos (c) \text{Si}(d x)}{\left(b x^2+a\right)^2}-\frac{a d^2 \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)}{b}-8 \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\frac{5 i \sqrt{a} d \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \sin (c) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)}{\sqrt{b}}+\frac{5 \sqrt{a} d \cos (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)}{\sqrt{b}}-\frac{i a d^2 \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)}{b}-8 i \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\frac{a d^2 \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{b}+8 \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+\frac{5 i \sqrt{a} d \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \sin (c) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{\sqrt{b}}-\frac{5 \sqrt{a} d \cos (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{\sqrt{b}}-\frac{i a d^2 \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{b}-8 i \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{16 a^3}","-\frac{\sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 a^3}-\frac{\sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 a^3}+\frac{\cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 a^3}-\frac{\cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 a^3}+\frac{\sin (c) \text{Ci}(d x)}{a^3}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{d^2 \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 a^2 b}-\frac{d^2 \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 a^2 b}+\frac{d^2 \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 a^2 b}-\frac{d^2 \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 a^2 b}+\frac{\sin (c+d x)}{2 a^2 \left(a+b x^2\right)}+\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left(\sqrt{-a}-\sqrt{b} x\right)}-\frac{d \cos (c+d x)}{16 a^2 \sqrt{b} \left(\sqrt{-a}+\sqrt{b} x\right)}-\frac{5 d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 (-a)^{5/2} \sqrt{b}}+\frac{5 d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{5/2} \sqrt{b}}-\frac{5 d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 (-a)^{5/2} \sqrt{b}}-\frac{5 d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{5/2} \sqrt{b}}+\frac{\sin (c+d x)}{4 a \left(a+b x^2\right)^2}",1,"((-2*a^2*d*x*Cos[c + d*x])/(a + b*x^2)^2 - (2*a*b*d*x^3*Cos[c + d*x])/(a + b*x^2)^2 + (16*a^2*CosIntegral[d*x]*Sin[c])/(a + b*x^2)^2 + (32*a*b*x^2*CosIntegral[d*x]*Sin[c])/(a + b*x^2)^2 + (16*b^2*x^4*CosIntegral[d*x]*Sin[c])/(a + b*x^2)^2 - (CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*((5*I)*Sqrt[a]*Sqrt[b]*d*Cos[c - (I*Sqrt[a]*d)/Sqrt[b]] + (8*b + a*d^2)*Sin[c - (I*Sqrt[a]*d)/Sqrt[b]]))/b - (CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*((-5*I)*Sqrt[a]*Sqrt[b]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] + (8*b + a*d^2)*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]))/b + (12*a^2*Sin[c + d*x])/(a + b*x^2)^2 + (8*a*b*x^2*Sin[c + d*x])/(a + b*x^2)^2 + (16*a^2*Cos[c]*SinIntegral[d*x])/(a + b*x^2)^2 + (32*a*b*x^2*Cos[c]*SinIntegral[d*x])/(a + b*x^2)^2 + (16*b^2*x^4*Cos[c]*SinIntegral[d*x])/(a + b*x^2)^2 - 8*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (a*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/b + ((5*I)*Sqrt[a]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[b] + (5*Sqrt[a]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[b] - (8*I)*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (I*a*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/b + 8*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + (a*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/b + ((5*I)*Sqrt[a]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[b] - (5*Sqrt[a]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[b] - (8*I)*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - (I*a*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/b)/(16*a^3)","C",0
77,1,1177,875,3.0375214,"\int \frac{\sin (c+d x)}{x^2 \left(a+b x^2\right)^3} \, dx","Integrate[Sin[c + d*x]/(x^2*(a + b*x^2)^3),x]","\frac{-\frac{30 \sqrt{a} b^2 \cos (d x) \sin (c) x^3}{\left(b x^2+a\right)^2}-\frac{30 \sqrt{a} b^2 \cos (c) \sin (d x) x^3}{\left(b x^2+a\right)^2}-\frac{2 a^{3/2} b d \cos (c) \cos (d x) x^2}{\left(b x^2+a\right)^2}+\frac{2 a^{3/2} b d \sin (c) \sin (d x) x^2}{\left(b x^2+a\right)^2}-\frac{50 a^{3/2} b \cos (d x) \sin (c) x}{\left(b x^2+a\right)^2}-\frac{50 a^{3/2} b \cos (c) \sin (d x) x}{\left(b x^2+a\right)^2}-\frac{2 a^{5/2} d \cos (c) \cos (d x)}{\left(b x^2+a\right)^2}+16 \sqrt{a} d \cos (c) \text{Ci}(d x)+7 \sqrt{a} d \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\frac{i a d^2 \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \sin (c)}{\sqrt{b}}-15 i \sqrt{b} \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \sin (c)+\frac{\text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \left(7 \sqrt{a} \sqrt{b} d \cos \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)+i \left(a d^2+15 b\right) \sin \left(c+\frac{i \sqrt{a} d}{\sqrt{b}}\right)\right)}{\sqrt{b}}+\frac{2 a^{5/2} d \sin (c) \sin (d x)}{\left(b x^2+a\right)^2}-\frac{a d^2 \cos (c) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right)}{\sqrt{b}}-15 \sqrt{b} \cos (c) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right)+7 i \sqrt{a} d \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right)-16 \sqrt{a} d \sin (c) \text{Si}(d x)-\frac{i a d^2 \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)}{\sqrt{b}}-15 i \sqrt{b} \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-7 \sqrt{a} d \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \sin (c) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+7 i \sqrt{a} d \cos (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\frac{a d^2 \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)}{\sqrt{b}}+15 \sqrt{b} \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\frac{i a d^2 \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{\sqrt{b}}-15 i \sqrt{b} \cos (c) \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+7 \sqrt{a} d \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \sin (c) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)+7 i \sqrt{a} d \cos (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-\frac{a d^2 \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)}{\sqrt{b}}-15 \sqrt{b} \sin (c) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-\frac{16 a^{5/2} \cos (d x) \sin (c)}{\left(b x^2+a\right)^2 x}-\frac{16 a^{5/2} \cos (c) \sin (d x)}{\left(b x^2+a\right)^2 x}}{16 a^{7/2}}","\frac{\text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) d^2}{16 (-a)^{5/2} \sqrt{b}}-\frac{\text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right) \sin \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right) d^2}{16 (-a)^{5/2} \sqrt{b}}+\frac{\cos \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right) d^2}{16 (-a)^{5/2} \sqrt{b}}+\frac{\cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right) d^2}{16 (-a)^{5/2} \sqrt{b}}+\frac{\cos (c+d x) d}{16 (-a)^{5/2} \left(\sqrt{-a}-\sqrt{b} x\right)}+\frac{\cos (c+d x) d}{16 (-a)^{5/2} \left(\sqrt{b} x+\sqrt{-a}\right)}+\frac{\cos (c) \text{Ci}(d x) d}{a^3}+\frac{7 \cos \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right) d}{16 a^3}+\frac{7 \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right) d}{16 a^3}-\frac{\sin (c) \text{Si}(d x) d}{a^3}+\frac{7 \sin \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right) d}{16 a^3}-\frac{7 \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right) d}{16 a^3}-\frac{15 \sqrt{b} \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{7/2}}+\frac{15 \sqrt{b} \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right) \sin \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{7/2}}-\frac{\sin (c+d x)}{a^3 x}+\frac{7 \sqrt{b} \sin (c+d x)}{16 a^3 \left(\sqrt{-a}-\sqrt{b} x\right)}-\frac{7 \sqrt{b} \sin (c+d x)}{16 a^3 \left(\sqrt{b} x+\sqrt{-a}\right)}-\frac{\sqrt{b} \sin (c+d x)}{16 (-a)^{5/2} \left(\sqrt{-a}-\sqrt{b} x\right)^2}+\frac{\sqrt{b} \sin (c+d x)}{16 (-a)^{5/2} \left(\sqrt{b} x+\sqrt{-a}\right)^2}-\frac{15 \sqrt{b} \cos \left(c+\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 (-a)^{7/2}}-\frac{15 \sqrt{b} \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{7/2}}",1,"((-2*a^(5/2)*d*Cos[c]*Cos[d*x])/(a + b*x^2)^2 - (2*a^(3/2)*b*d*x^2*Cos[c]*Cos[d*x])/(a + b*x^2)^2 + 16*Sqrt[a]*d*Cos[c]*CosIntegral[d*x] + 7*Sqrt[a]*d*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (16*a^(5/2)*Cos[d*x]*Sin[c])/(x*(a + b*x^2)^2) - (50*a^(3/2)*b*x*Cos[d*x]*Sin[c])/(a + b*x^2)^2 - (30*Sqrt[a]*b^2*x^3*Cos[d*x]*Sin[c])/(a + b*x^2)^2 - (15*I)*Sqrt[b]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c] - (I*a*d^2*Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c])/Sqrt[b] + (CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*(7*Sqrt[a]*Sqrt[b]*d*Cos[c + (I*Sqrt[a]*d)/Sqrt[b]] + I*(15*b + a*d^2)*Sin[c + (I*Sqrt[a]*d)/Sqrt[b]]))/Sqrt[b] - (16*a^(5/2)*Cos[c]*Sin[d*x])/(x*(a + b*x^2)^2) - (50*a^(3/2)*b*x*Cos[c]*Sin[d*x])/(a + b*x^2)^2 - (30*Sqrt[a]*b^2*x^3*Cos[c]*Sin[d*x])/(a + b*x^2)^2 + (2*a^(5/2)*d*Sin[c]*Sin[d*x])/(a + b*x^2)^2 + (2*a^(3/2)*b*d*x^2*Sin[c]*Sin[d*x])/(a + b*x^2)^2 - 15*Sqrt[b]*Cos[c]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] - (a*d^2*Cos[c]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]])/Sqrt[b] + (7*I)*Sqrt[a]*d*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]] - 16*Sqrt[a]*d*Sin[c]*SinIntegral[d*x] - (15*I)*Sqrt[b]*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - (I*a*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[b] - 7*Sqrt[a]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + (7*I)*Sqrt[a]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + 15*Sqrt[b]*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + (a*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)])/Sqrt[b] - (15*I)*Sqrt[b]*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - (I*a*d^2*Cos[c]*Cosh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[b] + 7*Sqrt[a]*d*Cosh[(Sqrt[a]*d)/Sqrt[b]]*Sin[c]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] + (7*I)*Sqrt[a]*d*Cos[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - 15*Sqrt[b]*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x] - (a*d^2*Sin[c]*Sinh[(Sqrt[a]*d)/Sqrt[b]]*SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])/Sqrt[b])/(16*a^(7/2))","C",0
78,1,995,791,2.870842,"\int \frac{\sin (c+d x)}{x^3 \left(a+b x^2\right)^3} \, dx","Integrate[Sin[c + d*x]/(x^3*(a + b*x^2)^3),x]","\frac{a \cos (c) \left(i \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right)-i \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right)+\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)\right)\right) d^2+a \sin (c) \left(\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)\right)\right) d^2+9 \sqrt{a} \sqrt{b} \cos (c) \left(-i \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)-\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)\right)\right) d-9 \sqrt{a} \sqrt{b} \sin (c) \left(\text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right)+\text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right)+i \cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)\right)\right) d-\frac{2 a \cos (d x) \left(d x \left(3 b^2 x^4+7 a b x^2+4 a^2\right) \cos (c)+2 \left(6 b^2 x^4+9 a b x^2+2 a^2\right) \sin (c)\right)}{x^2 \left(b x^2+a\right)^2}+\frac{2 a \left(d x \left(3 b^2 x^4+7 a b x^2+4 a^2\right) \sin (c)-2 \left(6 b^2 x^4+9 a b x^2+2 a^2\right) \cos (c)\right) \sin (d x)}{x^2 \left(b x^2+a\right)^2}-8 \left(a d^2+6 b\right) (\text{Ci}(d x) \sin (c)+\cos (c) \text{Si}(d x))+24 b \cos (c) \left(i \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right)-i \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right) \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right)+\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)-\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)\right)\right)+24 b \sin (c) \left(\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x-\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\cosh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \text{Ci}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+i \sinh \left(\frac{\sqrt{a} d}{\sqrt{b}}\right) \left(\text{Si}\left(d \left(x+\frac{i \sqrt{a}}{\sqrt{b}}\right)\right)+\text{Si}\left(\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right)\right)\right)}{16 a^4}","-\frac{3 b \sin (c) \text{Ci}(d x)}{a^4}+\frac{3 b \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 a^4}+\frac{3 b \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 a^4}-\frac{3 b \cos (c) \text{Si}(d x)}{a^4}-\frac{3 b \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{2 a^4}+\frac{3 b \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{2 a^4}+\frac{d^2 \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 a^3}+\frac{d^2 \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 a^3}-\frac{d^2 \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 a^3}+\frac{d^2 \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 a^3}-\frac{b \sin (c+d x)}{a^3 \left(a+b x^2\right)}-\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left(\sqrt{-a}-\sqrt{b} x\right)}+\frac{\sqrt{b} d \cos (c+d x)}{16 a^3 \left(\sqrt{-a}+\sqrt{b} x\right)}-\frac{d^2 \sin (c) \text{Ci}(d x)}{2 a^3}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^3}-\frac{\sin (c+d x)}{2 a^3 x^2}-\frac{d \cos (c+d x)}{2 a^3 x}-\frac{b \sin (c+d x)}{4 a^2 \left(a+b x^2\right)^2}-\frac{9 \sqrt{b} d \cos \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Ci}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 (-a)^{7/2}}+\frac{9 \sqrt{b} d \cos \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Ci}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{7/2}}-\frac{9 \sqrt{b} d \sin \left(\frac{\sqrt{-a} d}{\sqrt{b}}+c\right) \text{Si}\left(\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right)}{16 (-a)^{7/2}}-\frac{9 \sqrt{b} d \sin \left(c-\frac{\sqrt{-a} d}{\sqrt{b}}\right) \text{Si}\left(x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right)}{16 (-a)^{7/2}}",1,"((-2*a*Cos[d*x]*(d*x*(4*a^2 + 7*a*b*x^2 + 3*b^2*x^4)*Cos[c] + 2*(2*a^2 + 9*a*b*x^2 + 6*b^2*x^4)*Sin[c]))/(x^2*(a + b*x^2)^2) + (2*a*(-2*(2*a^2 + 9*a*b*x^2 + 6*b^2*x^4)*Cos[c] + d*x*(4*a^2 + 7*a*b*x^2 + 3*b^2*x^4)*Sin[c])*Sin[d*x])/(x^2*(a + b*x^2)^2) - 8*(6*b + a*d^2)*(CosIntegral[d*x]*Sin[c] + Cos[c]*SinIntegral[d*x]) + 24*b*Cos[c]*(I*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] - I*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + Cosh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])) + a*d^2*Cos[c]*(I*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] - I*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + Cosh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] - SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])) + 9*Sqrt[a]*Sqrt[b]*d*Cos[c]*((-I)*Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + I*Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + Sinh[(Sqrt[a]*d)/Sqrt[b]]*(-SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])) - 9*Sqrt[a]*Sqrt[b]*d*Sin[c]*(CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)]*Sinh[(Sqrt[a]*d)/Sqrt[b]] + I*Cosh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])) + 24*b*Sin[c]*(Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sinh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])) + a*d^2*Sin[c]*(Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*(((-I)*Sqrt[a])/Sqrt[b] + x)] + Cosh[(Sqrt[a]*d)/Sqrt[b]]*CosIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + I*Sinh[(Sqrt[a]*d)/Sqrt[b]]*(SinIntegral[d*((I*Sqrt[a])/Sqrt[b] + x)] + SinIntegral[(I*Sqrt[a]*d)/Sqrt[b] - d*x])))/(16*a^4)","C",0
79,1,101,156,0.1989089,"\int x^3 \left(a+b x^3\right) \sin (c+d x) \, dx","Integrate[x^3*(a + b*x^3)*Sin[c + d*x],x]","\frac{3 d \left(a d^2 \left(d^2 x^2-2\right)+2 b x \left(d^4 x^4-20 d^2 x^2+120\right)\right) \sin (c+d x)-\left(a d^4 x \left(d^2 x^2-6\right)+b \left(d^6 x^6-30 d^4 x^4+360 d^2 x^2-720\right)\right) \cos (c+d x)}{d^7}","-\frac{6 a \sin (c+d x)}{d^4}+\frac{6 a x \cos (c+d x)}{d^3}+\frac{3 a x^2 \sin (c+d x)}{d^2}-\frac{a x^3 \cos (c+d x)}{d}+\frac{720 b \cos (c+d x)}{d^7}+\frac{720 b x \sin (c+d x)}{d^6}-\frac{360 b x^2 \cos (c+d x)}{d^5}-\frac{120 b x^3 \sin (c+d x)}{d^4}+\frac{30 b x^4 \cos (c+d x)}{d^3}+\frac{6 b x^5 \sin (c+d x)}{d^2}-\frac{b x^6 \cos (c+d x)}{d}",1,"(-((a*d^4*x*(-6 + d^2*x^2) + b*(-720 + 360*d^2*x^2 - 30*d^4*x^4 + d^6*x^6))*Cos[c + d*x]) + 3*d*(a*d^2*(-2 + d^2*x^2) + 2*b*x*(120 - 20*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^7","A",1
80,1,84,126,0.1631708,"\int x^2 \left(a+b x^3\right) \sin (c+d x) \, dx","Integrate[x^2*(a + b*x^3)*Sin[c + d*x],x]","\frac{\left(2 a d^4 x+5 b \left(d^4 x^4-12 d^2 x^2+24\right)\right) \sin (c+d x)-d \left(a d^2 \left(d^2 x^2-2\right)+b x \left(d^4 x^4-20 d^2 x^2+120\right)\right) \cos (c+d x)}{d^6}","\frac{2 a \cos (c+d x)}{d^3}+\frac{2 a x \sin (c+d x)}{d^2}-\frac{a x^2 \cos (c+d x)}{d}+\frac{120 b \sin (c+d x)}{d^6}-\frac{120 b x \cos (c+d x)}{d^5}-\frac{60 b x^2 \sin (c+d x)}{d^4}+\frac{20 b x^3 \cos (c+d x)}{d^3}+\frac{5 b x^4 \sin (c+d x)}{d^2}-\frac{b x^5 \cos (c+d x)}{d}",1,"(-(d*(a*d^2*(-2 + d^2*x^2) + b*x*(120 - 20*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + (2*a*d^4*x + 5*b*(24 - 12*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^6","A",1
81,1,66,95,0.1312257,"\int x \left(a+b x^3\right) \sin (c+d x) \, dx","Integrate[x*(a + b*x^3)*Sin[c + d*x],x]","\frac{d \left(a d^2+4 b x \left(d^2 x^2-6\right)\right) \sin (c+d x)-\left(a d^4 x+b \left(d^4 x^4-12 d^2 x^2+24\right)\right) \cos (c+d x)}{d^5}","\frac{a \sin (c+d x)}{d^2}-\frac{a x \cos (c+d x)}{d}-\frac{24 b \cos (c+d x)}{d^5}-\frac{24 b x \sin (c+d x)}{d^4}+\frac{12 b x^2 \cos (c+d x)}{d^3}+\frac{4 b x^3 \sin (c+d x)}{d^2}-\frac{b x^4 \cos (c+d x)}{d}",1,"(-((a*d^4*x + b*(24 - 12*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + d*(a*d^2 + 4*b*x*(-6 + d^2*x^2))*Sin[c + d*x])/d^5","A",1
82,1,50,68,0.092064,"\int \left(a+b x^3\right) \sin (c+d x) \, dx","Integrate[(a + b*x^3)*Sin[c + d*x],x]","\frac{3 b \left(d^2 x^2-2\right) \sin (c+d x)-d \left(a d^2+b x \left(d^2 x^2-6\right)\right) \cos (c+d x)}{d^4}","-\frac{a \cos (c+d x)}{d}-\frac{6 b \sin (c+d x)}{d^4}+\frac{6 b x \cos (c+d x)}{d^3}+\frac{3 b x^2 \sin (c+d x)}{d^2}-\frac{b x^3 \cos (c+d x)}{d}",1,"(-(d*(a*d^2 + b*x*(-6 + d^2*x^2))*Cos[c + d*x]) + 3*b*(-2 + d^2*x^2)*Sin[c + d*x])/d^4","A",1
83,1,50,57,0.2034306,"\int \frac{\left(a+b x^3\right) \sin (c+d x)}{x} \, dx","Integrate[((a + b*x^3)*Sin[c + d*x])/x,x]","a \sin (c) \text{Ci}(d x)+a \cos (c) \text{Si}(d x)+\frac{b \left(\left(2-d^2 x^2\right) \cos (c+d x)+2 d x \sin (c+d x)\right)}{d^3}","a \sin (c) \text{Ci}(d x)+a \cos (c) \text{Si}(d x)+\frac{2 b \cos (c+d x)}{d^3}+\frac{2 b x \sin (c+d x)}{d^2}-\frac{b x^2 \cos (c+d x)}{d}",1,"a*CosIntegral[d*x]*Sin[c] + (b*((2 - d^2*x^2)*Cos[c + d*x] + 2*d*x*Sin[c + d*x]))/d^3 + a*Cos[c]*SinIntegral[d*x]","A",1
84,1,56,56,0.1389729,"\int \frac{\left(a+b x^3\right) \sin (c+d x)}{x^2} \, dx","Integrate[((a + b*x^3)*Sin[c + d*x])/x^2,x]","a d \cos (c) \text{Ci}(d x)-a d \sin (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{x}+\frac{b \sin (c+d x)}{d^2}-\frac{b x \cos (c+d x)}{d}","a d \cos (c) \text{Ci}(d x)-a d \sin (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{x}+\frac{b \sin (c+d x)}{d^2}-\frac{b x \cos (c+d x)}{d}",1,"-((b*x*Cos[c + d*x])/d) + a*d*Cos[c]*CosIntegral[d*x] + (b*Sin[c + d*x])/d^2 - (a*Sin[c + d*x])/x - a*d*Sin[c]*SinIntegral[d*x]","A",1
85,1,66,70,0.163528,"\int \frac{\left(a+b x^3\right) \sin (c+d x)}{x^3} \, dx","Integrate[((a + b*x^3)*Sin[c + d*x])/x^3,x]","\frac{1}{2} \left(-a d^2 \sin (c) \text{Ci}(d x)-a d^2 \cos (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{x^2}-\frac{a d \cos (c+d x)}{x}-\frac{2 b \cos (c+d x)}{d}\right)","-\frac{1}{2} a d^2 \sin (c) \text{Ci}(d x)-\frac{1}{2} a d^2 \cos (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{2 x^2}-\frac{a d \cos (c+d x)}{2 x}-\frac{b \cos (c+d x)}{d}",1,"((-2*b*Cos[c + d*x])/d - (a*d*Cos[c + d*x])/x - a*d^2*CosIntegral[d*x]*Sin[c] - (a*Sin[c + d*x])/x^2 - a*d^2*Cos[c]*SinIntegral[d*x])/2","A",1
86,1,104,91,0.2290972,"\int \frac{\left(a+b x^3\right) \sin (c+d x)}{x^4} \, dx","Integrate[((a + b*x^3)*Sin[c + d*x])/x^4,x]","-\frac{1}{6} a d^3 (\cos (c) \text{Ci}(d x)-\sin (c) \text{Si}(d x))+\frac{a \cos (d x) \left(d^2 x^2 \sin (c)-d x \cos (c)-2 \sin (c)\right)}{6 x^3}+\frac{a \sin (d x) \left(d^2 x^2 \cos (c)+d x \sin (c)-2 \cos (c)\right)}{6 x^3}+b \sin (c) \text{Ci}(d x)+b \cos (c) \text{Si}(d x)","-\frac{1}{6} a d^3 \cos (c) \text{Ci}(d x)+\frac{1}{6} a d^3 \sin (c) \text{Si}(d x)+\frac{a d^2 \sin (c+d x)}{6 x}-\frac{a \sin (c+d x)}{3 x^3}-\frac{a d \cos (c+d x)}{6 x^2}+b \sin (c) \text{Ci}(d x)+b \cos (c) \text{Si}(d x)",1,"b*CosIntegral[d*x]*Sin[c] + (a*Cos[d*x]*(-(d*x*Cos[c]) - 2*Sin[c] + d^2*x^2*Sin[c]))/(6*x^3) + (a*(-2*Cos[c] + d^2*x^2*Cos[c] + d*x*Sin[c])*Sin[d*x])/(6*x^3) + b*Cos[c]*SinIntegral[d*x] - (a*d^3*(Cos[c]*CosIntegral[d*x] - Sin[c]*SinIntegral[d*x]))/6","A",1
87,1,139,235,0.4056898,"\int x \left(a+b x^3\right)^2 \sin (c+d x) \, dx","Integrate[x*(a + b*x^3)^2*Sin[c + d*x],x]","\frac{\left(a^2 d^6+8 a b d^4 x \left(d^2 x^2-6\right)+7 b^2 \left(d^6 x^6-30 d^4 x^4+360 d^2 x^2-720\right)\right) \sin (c+d x)-d \left(a^2 d^6 x+2 a b d^2 \left(d^4 x^4-12 d^2 x^2+24\right)+b^2 x \left(d^6 x^6-42 d^4 x^4+840 d^2 x^2-5040\right)\right) \cos (c+d x)}{d^8}","\frac{a^2 \sin (c+d x)}{d^2}-\frac{a^2 x \cos (c+d x)}{d}-\frac{48 a b \cos (c+d x)}{d^5}-\frac{48 a b x \sin (c+d x)}{d^4}+\frac{24 a b x^2 \cos (c+d x)}{d^3}+\frac{8 a b x^3 \sin (c+d x)}{d^2}-\frac{2 a b x^4 \cos (c+d x)}{d}-\frac{5040 b^2 \sin (c+d x)}{d^8}+\frac{5040 b^2 x \cos (c+d x)}{d^7}+\frac{2520 b^2 x^2 \sin (c+d x)}{d^6}-\frac{840 b^2 x^3 \cos (c+d x)}{d^5}-\frac{210 b^2 x^4 \sin (c+d x)}{d^4}+\frac{42 b^2 x^5 \cos (c+d x)}{d^3}+\frac{7 b^2 x^6 \sin (c+d x)}{d^2}-\frac{b^2 x^7 \cos (c+d x)}{d}",1,"(-(d*(a^2*d^6*x + 2*a*b*d^2*(24 - 12*d^2*x^2 + d^4*x^4) + b^2*x*(-5040 + 840*d^2*x^2 - 42*d^4*x^4 + d^6*x^6))*Cos[c + d*x]) + (a^2*d^6 + 8*a*b*d^4*x*(-6 + d^2*x^2) + 7*b^2*(-720 + 360*d^2*x^2 - 30*d^4*x^4 + d^6*x^6))*Sin[c + d*x])/d^8","A",1
88,1,112,188,0.3265126,"\int \left(a+b x^3\right)^2 \sin (c+d x) \, dx","Integrate[(a + b*x^3)^2*Sin[c + d*x],x]","\frac{6 b d \left(a d^2 \left(d^2 x^2-2\right)+b x \left(d^4 x^4-20 d^2 x^2+120\right)\right) \sin (c+d x)-\left(a^2 d^6+2 a b d^4 x \left(d^2 x^2-6\right)+b^2 \left(d^6 x^6-30 d^4 x^4+360 d^2 x^2-720\right)\right) \cos (c+d x)}{d^7}","-\frac{a^2 \cos (c+d x)}{d}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{12 a b x \cos (c+d x)}{d^3}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{720 b^2 \cos (c+d x)}{d^7}+\frac{720 b^2 x \sin (c+d x)}{d^6}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{b^2 x^6 \cos (c+d x)}{d}",1,"(-((a^2*d^6 + 2*a*b*d^4*x*(-6 + d^2*x^2) + b^2*(-720 + 360*d^2*x^2 - 30*d^4*x^4 + d^6*x^6))*Cos[c + d*x]) + 6*b*d*(a*d^2*(-2 + d^2*x^2) + b*x*(120 - 20*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^7","A",1
89,1,108,161,0.5278663,"\int \frac{\left(a+b x^3\right)^2 \sin (c+d x)}{x} \, dx","Integrate[((a + b*x^3)^2*Sin[c + d*x])/x,x]","a^2 \sin (c) \text{Ci}(d x)+a^2 \cos (c) \text{Si}(d x)+\frac{b \left(4 a d^4 x+5 b \left(d^4 x^4-12 d^2 x^2+24\right)\right) \sin (c+d x)}{d^6}-\frac{b \left(2 a d^2 \left(d^2 x^2-2\right)+b x \left(d^4 x^4-20 d^2 x^2+120\right)\right) \cos (c+d x)}{d^5}","a^2 \sin (c) \text{Ci}(d x)+a^2 \cos (c) \text{Si}(d x)+\frac{4 a b \cos (c+d x)}{d^3}+\frac{4 a b x \sin (c+d x)}{d^2}-\frac{2 a b x^2 \cos (c+d x)}{d}+\frac{120 b^2 \sin (c+d x)}{d^6}-\frac{120 b^2 x \cos (c+d x)}{d^5}-\frac{60 b^2 x^2 \sin (c+d x)}{d^4}+\frac{20 b^2 x^3 \cos (c+d x)}{d^3}+\frac{5 b^2 x^4 \sin (c+d x)}{d^2}-\frac{b^2 x^5 \cos (c+d x)}{d}",1,"-((b*(2*a*d^2*(-2 + d^2*x^2) + b*x*(120 - 20*d^2*x^2 + d^4*x^4))*Cos[c + d*x])/d^5) + a^2*CosIntegral[d*x]*Sin[c] + (b*(4*a*d^4*x + 5*b*(24 - 12*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^6 + a^2*Cos[c]*SinIntegral[d*x]","A",1
90,1,145,145,0.3942035,"\int \frac{\left(a+b x^3\right)^2 \sin (c+d x)}{x^2} \, dx","Integrate[((a + b*x^3)^2*Sin[c + d*x])/x^2,x]","a^2 d \cos (c) \text{Ci}(d x)-a^2 d \sin (c) \text{Si}(d x)-\frac{a^2 \sin (c+d x)}{x}+\frac{2 a b \sin (c+d x)}{d^2}-\frac{2 a b x \cos (c+d x)}{d}-\frac{24 b^2 \cos (c+d x)}{d^5}-\frac{24 b^2 x \sin (c+d x)}{d^4}+\frac{12 b^2 x^2 \cos (c+d x)}{d^3}+\frac{4 b^2 x^3 \sin (c+d x)}{d^2}-\frac{b^2 x^4 \cos (c+d x)}{d}","a^2 d \cos (c) \text{Ci}(d x)-a^2 d \sin (c) \text{Si}(d x)-\frac{a^2 \sin (c+d x)}{x}+\frac{2 a b \sin (c+d x)}{d^2}-\frac{2 a b x \cos (c+d x)}{d}-\frac{24 b^2 \cos (c+d x)}{d^5}-\frac{24 b^2 x \sin (c+d x)}{d^4}+\frac{12 b^2 x^2 \cos (c+d x)}{d^3}+\frac{4 b^2 x^3 \sin (c+d x)}{d^2}-\frac{b^2 x^4 \cos (c+d x)}{d}",1,"(-24*b^2*Cos[c + d*x])/d^5 - (2*a*b*x*Cos[c + d*x])/d + (12*b^2*x^2*Cos[c + d*x])/d^3 - (b^2*x^4*Cos[c + d*x])/d + a^2*d*Cos[c]*CosIntegral[d*x] + (2*a*b*Sin[c + d*x])/d^2 - (a^2*Sin[c + d*x])/x - (24*b^2*x*Sin[c + d*x])/d^4 + (4*b^2*x^3*Sin[c + d*x])/d^2 - a^2*d*Sin[c]*SinIntegral[d*x]","A",1
91,1,138,142,0.4066933,"\int \frac{\left(a+b x^3\right)^2 \sin (c+d x)}{x^3} \, dx","Integrate[((a + b*x^3)^2*Sin[c + d*x])/x^3,x]","\frac{1}{2} \left(-a^2 d^2 \sin (c) \text{Ci}(d x)-a^2 d^2 \cos (c) \text{Si}(d x)-\frac{a^2 \sin (c+d x)}{x^2}-\frac{a^2 d \cos (c+d x)}{x}-\frac{4 a b \cos (c+d x)}{d}-\frac{12 b^2 \sin (c+d x)}{d^4}+\frac{12 b^2 x \cos (c+d x)}{d^3}+\frac{6 b^2 x^2 \sin (c+d x)}{d^2}-\frac{2 b^2 x^3 \cos (c+d x)}{d}\right)","-\frac{1}{2} a^2 d^2 \sin (c) \text{Ci}(d x)-\frac{1}{2} a^2 d^2 \cos (c) \text{Si}(d x)-\frac{a^2 \sin (c+d x)}{2 x^2}-\frac{a^2 d \cos (c+d x)}{2 x}-\frac{2 a b \cos (c+d x)}{d}-\frac{6 b^2 \sin (c+d x)}{d^4}+\frac{6 b^2 x \cos (c+d x)}{d^3}+\frac{3 b^2 x^2 \sin (c+d x)}{d^2}-\frac{b^2 x^3 \cos (c+d x)}{d}",1,"((-4*a*b*Cos[c + d*x])/d - (a^2*d*Cos[c + d*x])/x + (12*b^2*x*Cos[c + d*x])/d^3 - (2*b^2*x^3*Cos[c + d*x])/d - a^2*d^2*CosIntegral[d*x]*Sin[c] - (12*b^2*Sin[c + d*x])/d^4 - (a^2*Sin[c + d*x])/x^2 + (6*b^2*x^2*Sin[c + d*x])/d^2 - a^2*d^2*Cos[c]*SinIntegral[d*x])/2","A",1
92,1,135,151,0.6260057,"\int \frac{\left(a+b x^3\right)^2 \sin (c+d x)}{x^4} \, dx","Integrate[((a + b*x^3)^2*Sin[c + d*x])/x^4,x]","\frac{1}{6} \left(\frac{a^2 d^2 \sin (c+d x)}{x}-\frac{2 a^2 \sin (c+d x)}{x^3}-\frac{a^2 d \cos (c+d x)}{x^2}-a \text{Ci}(d x) \left(a d^3 \cos (c)-12 b \sin (c)\right)+a \text{Si}(d x) \left(a d^3 \sin (c)+12 b \cos (c)\right)+\frac{12 b^2 \cos (c+d x)}{d^3}+\frac{12 b^2 x \sin (c+d x)}{d^2}-\frac{6 b^2 x^2 \cos (c+d x)}{d}\right)","-\frac{1}{6} a^2 d^3 \cos (c) \text{Ci}(d x)+\frac{1}{6} a^2 d^3 \sin (c) \text{Si}(d x)+\frac{a^2 d^2 \sin (c+d x)}{6 x}-\frac{a^2 \sin (c+d x)}{3 x^3}-\frac{a^2 d \cos (c+d x)}{6 x^2}+2 a b \sin (c) \text{Ci}(d x)+2 a b \cos (c) \text{Si}(d x)+\frac{2 b^2 \cos (c+d x)}{d^3}+\frac{2 b^2 x \sin (c+d x)}{d^2}-\frac{b^2 x^2 \cos (c+d x)}{d}",1,"((12*b^2*Cos[c + d*x])/d^3 - (a^2*d*Cos[c + d*x])/x^2 - (6*b^2*x^2*Cos[c + d*x])/d - a*CosIntegral[d*x]*(a*d^3*Cos[c] - 12*b*Sin[c]) - (2*a^2*Sin[c + d*x])/x^3 + (a^2*d^2*Sin[c + d*x])/x + (12*b^2*x*Sin[c + d*x])/d^2 + a*(12*b*Cos[c] + a*d^3*Sin[c])*SinIntegral[d*x])/6","A",1
93,1,148,167,0.6057486,"\int \frac{\left(a+b x^3\right)^2 \sin (c+d x)}{x^5} \, dx","Integrate[((a + b*x^3)^2*Sin[c + d*x])/x^5,x]","\frac{1}{24} \left(\frac{a^2 d^3 \cos (c+d x)}{x}+\frac{a^2 d^2 \sin (c+d x)}{x^2}-\frac{6 a^2 \sin (c+d x)}{x^4}-\frac{2 a^2 d \cos (c+d x)}{x^3}+a d \text{Ci}(d x) \left(a d^3 \sin (c)+48 b \cos (c)\right)+a d \text{Si}(d x) \left(a d^3 \cos (c)-48 b \sin (c)\right)-\frac{48 a b \sin (c+d x)}{x}+\frac{24 b^2 \sin (c+d x)}{d^2}-\frac{24 b^2 x \cos (c+d x)}{d}\right)","\frac{1}{24} a^2 d^4 \sin (c) \text{Ci}(d x)+\frac{1}{24} a^2 d^4 \cos (c) \text{Si}(d x)+\frac{a^2 d^3 \cos (c+d x)}{24 x}+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a^2 d \cos (c+d x)}{12 x^3}+2 a b d \cos (c) \text{Ci}(d x)-2 a b d \sin (c) \text{Si}(d x)-\frac{2 a b \sin (c+d x)}{x}+\frac{b^2 \sin (c+d x)}{d^2}-\frac{b^2 x \cos (c+d x)}{d}",1,"((-2*a^2*d*Cos[c + d*x])/x^3 + (a^2*d^3*Cos[c + d*x])/x - (24*b^2*x*Cos[c + d*x])/d + a*d*CosIntegral[d*x]*(48*b*Cos[c] + a*d^3*Sin[c]) + (24*b^2*Sin[c + d*x])/d^2 - (6*a^2*Sin[c + d*x])/x^4 + (a^2*d^2*Sin[c + d*x])/x^2 - (48*a*b*Sin[c + d*x])/x + a*d*(a*d^3*Cos[c] - 48*b*Sin[c])*SinIntegral[d*x])/24","A",1
94,1,231,371,0.5725401,"\int \frac{x^4 \sin (c+d x)}{a+b x^3} \, dx","Integrate[(x^4*Sin[c + d*x])/(a + b*x^3),x]","\frac{-i a d^2 \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\&\right]+i a d^2 \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\&\right]+6 b (\sin (c+d x)-d x \cos (c+d x))}{6 b^2 d^2}","\frac{a^{2/3} \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b^{5/3}}+\frac{(-1)^{2/3} a^{2/3} \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 b^{5/3}}-\frac{\sqrt[3]{-1} a^{2/3} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b^{5/3}}-\frac{(-1)^{2/3} a^{2/3} \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 b^{5/3}}+\frac{a^{2/3} \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b^{5/3}}-\frac{\sqrt[3]{-1} a^{2/3} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b^{5/3}}+\frac{\sin (c+d x)}{b d^2}-\frac{x \cos (c+d x)}{b d}",1,"((-I)*a*d^2*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1 & ] + I*a*d^2*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1 & ] + 6*b*(-(d*x*Cos[c + d*x]) + Sin[c + d*x]))/(6*b^2*d^2)","C",0
95,1,216,357,0.3551339,"\int \frac{x^3 \sin (c+d x)}{a+b x^3} \, dx","Integrate[(x^3*Sin[c + d*x])/(a + b*x^3),x]","-\frac{i a d \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]-i a d \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]+6 b \cos (c+d x)}{6 b^2 d}","-\frac{\sqrt[3]{a} \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b^{4/3}}+\frac{\sqrt[3]{-1} \sqrt[3]{a} \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 b^{4/3}}-\frac{(-1)^{2/3} \sqrt[3]{a} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b^{4/3}}-\frac{\sqrt[3]{-1} \sqrt[3]{a} \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 b^{4/3}}-\frac{\sqrt[3]{a} \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b^{4/3}}-\frac{(-1)^{2/3} \sqrt[3]{a} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b^{4/3}}-\frac{\cos (c+d x)}{b d}",1,"-1/6*(6*b*Cos[c + d*x] + I*a*d*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - I*a*d*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ])/(b^2*d)","C",0
96,1,186,281,0.3146364,"\int \frac{x^2 \sin (c+d x)}{a+b x^3} \, dx","Integrate[(x^2*Sin[c + d*x])/(a + b*x^3),x]","\frac{i \left(\text{RootSum}\left[\text{$\#$1}^3 b+a\&,-i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))\&\right]-\text{RootSum}\left[\text{$\#$1}^3 b+a\&,i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))\&\right]\right)}{6 b}","\frac{\sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b}+\frac{\sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 b}+\frac{\sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b}-\frac{\cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 b}+\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b}+\frac{\cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 b}",1,"((I/6)*(RootSum[a + b*#1^3 & , Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)] & ] - RootSum[a + b*#1^3 & , Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)] & ]))/b","C",0
97,1,196,343,0.3033139,"\int \frac{x \sin (c+d x)}{a+b x^3} \, dx","Integrate[(x*Sin[c + d*x])/(a + b*x^3),x]","\frac{i \left(\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\&\right]-\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\&\right]\right)}{6 b}","-\frac{\sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 \sqrt[3]{a} b^{2/3}}-\frac{(-1)^{2/3} \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 \sqrt[3]{a} b^{2/3}}+\frac{\sqrt[3]{-1} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 \sqrt[3]{a} b^{2/3}}+\frac{(-1)^{2/3} \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 \sqrt[3]{a} b^{2/3}}-\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 \sqrt[3]{a} b^{2/3}}+\frac{\sqrt[3]{-1} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 \sqrt[3]{a} b^{2/3}}",1,"((I/6)*(RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1 & ] - RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1 & ]))/b","C",0
98,1,196,343,0.2068732,"\int \frac{\sin (c+d x)}{a+b x^3} \, dx","Integrate[Sin[c + d*x]/(a + b*x^3),x]","\frac{i \left(\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]-\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]\right)}{6 b}","\frac{\sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{2/3} \sqrt[3]{b}}-\frac{\sqrt[3]{-1} \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 a^{2/3} \sqrt[3]{b}}+\frac{(-1)^{2/3} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{2/3} \sqrt[3]{b}}+\frac{\sqrt[3]{-1} \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 a^{2/3} \sqrt[3]{b}}+\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{2/3} \sqrt[3]{b}}+\frac{(-1)^{2/3} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{2/3} \sqrt[3]{b}}",1,"((I/6)*(RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ]))/b","C",0
99,1,206,301,0.3819656,"\int \frac{\sin (c+d x)}{x \left(a+b x^3\right)} \, dx","Integrate[Sin[c + d*x]/(x*(a + b*x^3)),x]","\frac{-i \text{RootSum}\left[\text{$\#$1}^3 b+a\&,-i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))\&\right]+i \text{RootSum}\left[\text{$\#$1}^3 b+a\&,i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))\&\right]+6 \sin (c) \text{Ci}(d x)+6 \cos (c) \text{Si}(d x)}{6 a}","-\frac{\sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a}-\frac{\sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 a}-\frac{\sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a}+\frac{\cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 a}-\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a}-\frac{\cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a}+\frac{\sin (c) \text{Ci}(d x)}{a}+\frac{\cos (c) \text{Si}(d x)}{a}",1,"((-I)*RootSum[a + b*#1^3 & , Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)] & ] + I*RootSum[a + b*#1^3 & , Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)] & ] + 6*CosIntegral[d*x]*Sin[c] + 6*Cos[c]*SinIntegral[d*x])/(6*a)","C",0
100,1,233,380,0.4970774,"\int \frac{\sin (c+d x)}{x^2 \left(a+b x^3\right)} \, dx","Integrate[Sin[c + d*x]/(x^2*(a + b*x^3)),x]","\frac{-i x \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\&\right]+i x \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\&\right]+6 d x \cos (c) \text{Ci}(d x)-6 d x \sin (c) \text{Si}(d x)-6 \sin (c+d x)}{6 a x}","\frac{\sqrt[3]{b} \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{4/3}}+\frac{(-1)^{2/3} \sqrt[3]{b} \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 a^{4/3}}-\frac{\sqrt[3]{-1} \sqrt[3]{b} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{4/3}}-\frac{(-1)^{2/3} \sqrt[3]{b} \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 a^{4/3}}+\frac{\sqrt[3]{b} \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{4/3}}-\frac{\sqrt[3]{-1} \sqrt[3]{b} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{4/3}}+\frac{d \cos (c) \text{Ci}(d x)}{a}-\frac{d \sin (c) \text{Si}(d x)}{a}-\frac{\sin (c+d x)}{a x}",1,"(6*d*x*Cos[c]*CosIntegral[d*x] - I*x*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1 & ] + I*x*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1 & ] - 6*Sin[c + d*x] - 6*d*x*Sin[c]*SinIntegral[d*x])/(6*a*x)","C",0
101,1,253,408,0.5031281,"\int \frac{\sin (c+d x)}{x^3 \left(a+b x^3\right)} \, dx","Integrate[Sin[c + d*x]/(x^3*(a + b*x^3)),x]","\frac{-i x^2 \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]+i x^2 \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]-3 \left(d^2 x^2 \sin (c) \text{Ci}(d x)+d^2 x^2 \cos (c) \text{Si}(d x)+\sin (c+d x)+d x \cos (c+d x)\right)}{6 a x^2}","-\frac{b^{2/3} \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{5/3}}+\frac{\sqrt[3]{-1} b^{2/3} \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{5/3}}-\frac{\sqrt[3]{-1} b^{2/3} \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 a^{5/3}}-\frac{b^{2/3} \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^{5/3}}-\frac{d^2 \sin (c) \text{Ci}(d x)}{2 a}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a}-\frac{\sin (c+d x)}{2 a x^2}-\frac{d \cos (c+d x)}{2 a x}",1,"((-I)*x^2*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] + I*x^2*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - 3*(d*x*Cos[c + d*x] + d^2*x^2*CosIntegral[d*x]*Sin[c] + Sin[c + d*x] + d^2*x^2*Cos[c]*SinIntegral[d*x]))/(6*a*x^2)","C",0
102,1,383,714,0.484473,"\int \frac{x^3 \sin (c+d x)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^3*Sin[c + d*x])/(a + b*x^3)^2,x]","\frac{\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{\sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]+\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{\sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]-\frac{6 b x \sin (c+d x)}{a+b x^3}}{18 b^2}","-\frac{\sqrt[3]{-1} \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{2/3} b^{4/3}}+\frac{\sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{2/3} b^{4/3}}+\frac{(-1)^{2/3} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{2/3} b^{4/3}}+\frac{\sqrt[3]{-1} \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{2/3} b^{4/3}}+\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{2/3} b^{4/3}}+\frac{(-1)^{2/3} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{2/3} b^{4/3}}-\frac{(-1)^{2/3} d \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 \sqrt[3]{a} b^{5/3}}-\frac{d \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 \sqrt[3]{a} b^{5/3}}+\frac{\sqrt[3]{-1} d \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 \sqrt[3]{a} b^{5/3}}-\frac{(-1)^{2/3} d \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 \sqrt[3]{a} b^{5/3}}+\frac{d \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 \sqrt[3]{a} b^{5/3}}-\frac{\sqrt[3]{-1} d \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 \sqrt[3]{a} b^{5/3}}-\frac{x \sin (c+d x)}{3 b \left(a+b x^3\right)}",1,"(RootSum[a + b*#1^3 & , (I*Cos[c + d*#1]*CosIntegral[d*(x - #1)] + CosIntegral[d*(x - #1)]*Sin[c + d*#1] + Cos[c + d*#1]*SinIntegral[d*(x - #1)] - I*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] + RootSum[a + b*#1^3 & , ((-I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] + CosIntegral[d*(x - #1)]*Sin[c + d*#1] + Cos[c + d*#1]*SinIntegral[d*(x - #1)] + I*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] - (6*b*x*Sin[c + d*x])/(a + b*x^3))/(18*b^2)","C",0
103,1,214,371,0.1880775,"\int \frac{x^2 \sin (c+d x)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x^2*Sin[c + d*x])/(a + b*x^3)^2,x]","\frac{d \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]+d \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]-\frac{6 b \sin (c+d x)}{a+b x^3}}{18 b^2}","-\frac{\sqrt[3]{-1} d \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{2/3} b^{4/3}}+\frac{d \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{2/3} b^{4/3}}+\frac{(-1)^{2/3} d \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{2/3} b^{4/3}}-\frac{\sqrt[3]{-1} d \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{2/3} b^{4/3}}-\frac{d \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{2/3} b^{4/3}}-\frac{(-1)^{2/3} d \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{2/3} b^{4/3}}-\frac{\sin (c+d x)}{3 b \left(a+b x^3\right)}",1,"(d*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] + d*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - (6*b*Sin[c + d*x])/(a + b*x^3))/(18*b^2)","C",0
104,1,408,691,0.2224449,"\int \frac{x \sin (c+d x)}{\left(a+b x^3\right)^2} \, dx","Integrate[(x*Sin[c + d*x])/(a + b*x^3)^2,x]","-\frac{\left(a+b x^3\right) \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-\sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\&\right]+\left(a+b x^3\right) \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-\sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\&\right]-6 b x^2 \sin (c+d x)}{18 a b \left(a+b x^3\right)}","-\frac{(-1)^{2/3} \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{4/3} b^{2/3}}-\frac{\sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{4/3} b^{2/3}}+\frac{\sqrt[3]{-1} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{4/3} b^{2/3}}+\frac{(-1)^{2/3} \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{4/3} b^{2/3}}-\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{4/3} b^{2/3}}+\frac{\sqrt[3]{-1} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{4/3} b^{2/3}}-\frac{d \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a b}-\frac{d \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a b}-\frac{d \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a b}-\frac{d \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a b}+\frac{d \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a b}+\frac{d \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a b}-\frac{\sin (c+d x)}{3 b x \left(a+b x^3\right)}+\frac{\sin (c+d x)}{3 a b x}",1,"-1/18*((a + b*x^3)*RootSum[a + b*#1^3 & , ((-I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - CosIntegral[d*(x - #1)]*Sin[c + d*#1] - Cos[c + d*#1]*SinIntegral[d*(x - #1)] + I*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1 & ] + (a + b*x^3)*RootSum[a + b*#1^3 & , (I*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - CosIntegral[d*(x - #1)]*Sin[c + d*#1] - Cos[c + d*#1]*SinIntegral[d*(x - #1)] - I*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1 & ] - 6*b*x^2*Sin[c + d*x])/(a*b*(a + b*x^3))","C",0
105,1,406,735,0.2269771,"\int \frac{\sin (c+d x)}{\left(a+b x^3\right)^2} \, dx","Integrate[Sin[c + d*x]/(a + b*x^3)^2,x]","-\frac{\left(a+b x^3\right) \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-2 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-2 i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+2 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]+\left(a+b x^3\right) \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-2 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+2 i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-2 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]-6 b x \sin (c+d x)}{18 a b \left(a+b x^3\right)}","\frac{(-1)^{2/3} d \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{4/3} b^{2/3}}+\frac{d \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{4/3} b^{2/3}}-\frac{\sqrt[3]{-1} d \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{4/3} b^{2/3}}+\frac{(-1)^{2/3} d \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{4/3} b^{2/3}}-\frac{d \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{4/3} b^{2/3}}+\frac{\sqrt[3]{-1} d \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{4/3} b^{2/3}}-\frac{2 \sqrt[3]{-1} \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{5/3} \sqrt[3]{b}}+\frac{2 \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{5/3} \sqrt[3]{b}}+\frac{2 (-1)^{2/3} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{5/3} \sqrt[3]{b}}+\frac{2 \sqrt[3]{-1} \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{5/3} \sqrt[3]{b}}+\frac{2 \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{5/3} \sqrt[3]{b}}+\frac{2 (-1)^{2/3} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{5/3} \sqrt[3]{b}}+\frac{\sin (c+d x)}{3 a b x^2}-\frac{\sin (c+d x)}{3 b x^2 \left(a+b x^3\right)}",1,"-1/18*((a + b*x^3)*RootSum[a + b*#1^3 & , ((-2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] + (a + b*x^3)*RootSum[a + b*#1^3 & , ((2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] - 6*b*x*Sin[c + d*x])/(a*b*(a + b*x^3))","C",0
106,1,446,693,0.8908931,"\int \frac{\sin (c+d x)}{x \left(a+b x^3\right)^2} \, dx","Integrate[Sin[c + d*x]/(x*(a + b*x^3)^2),x]","\frac{-\frac{1}{2} i \text{RootSum}\left[\text{$\#$1}^3 b+a\&,-i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))\&\right]+\frac{1}{2} i \text{RootSum}\left[\text{$\#$1}^3 b+a\&,i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))\&\right]-\frac{a d \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]}{6 b}-\frac{a d \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]}{6 b}+\frac{a \sin (c) \cos (d x)}{a+b x^3}+\frac{a \cos (c) \sin (d x)}{a+b x^3}+3 \sin (c) \text{Ci}(d x)+3 \cos (c) \text{Si}(d x)}{3 a^2}","\frac{\sqrt[3]{-1} d \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{5/3} \sqrt[3]{b}}-\frac{d \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{5/3} \sqrt[3]{b}}-\frac{(-1)^{2/3} d \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{5/3} \sqrt[3]{b}}+\frac{\sqrt[3]{-1} d \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{5/3} \sqrt[3]{b}}+\frac{d \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{5/3} \sqrt[3]{b}}+\frac{(-1)^{2/3} d \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{5/3} \sqrt[3]{b}}-\frac{\sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 a^2}-\frac{\sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^2}-\frac{\sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^2}+\frac{\cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 a^2}-\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^2}-\frac{\cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^2}+\frac{\sin (c) \text{Ci}(d x)}{a^2}+\frac{\cos (c) \text{Si}(d x)}{a^2}-\frac{\sin (c+d x)}{3 b x^3 \left(a+b x^3\right)}+\frac{\sin (c+d x)}{3 a b x^3}",1,"((-1/2*I)*RootSum[a + b*#1^3 & , Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)] & ] + (I/2)*RootSum[a + b*#1^3 & , Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)] & ] - (a*d*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ])/(6*b) - (a*d*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ])/(6*b) + (a*Cos[d*x]*Sin[c])/(a + b*x^3) + 3*CosIntegral[d*x]*Sin[c] + (a*Cos[c]*Sin[d*x])/(a + b*x^3) + 3*Cos[c]*SinIntegral[d*x])/(3*a^2)","C",0
107,1,445,712,1.1623823,"\int \frac{\sin (c+d x)}{x^2 \left(a+b x^3\right)^2} \, dx","Integrate[Sin[c + d*x]/(x^2*(a + b*x^3)^2),x]","-\frac{-\frac{1}{6} x \left(a+b x^3\right) \left(\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-4 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-4 i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+4 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\&\right]+\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-4 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+4 i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-4 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\&\right]+18 d \cos (c) \text{Ci}(d x)-18 d \sin (c) \text{Si}(d x)\right)+\sin (c) \left(3 a+4 b x^3\right) \cos (d x)+\cos (c) \left(3 a+4 b x^3\right) \sin (d x)}{3 a^2 x \left(a+b x^3\right)}","\frac{4 \sqrt[3]{b} \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{7/3}}+\frac{4 (-1)^{2/3} \sqrt[3]{b} \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{7/3}}-\frac{4 \sqrt[3]{-1} \sqrt[3]{b} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{7/3}}-\frac{4 (-1)^{2/3} \sqrt[3]{b} \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{7/3}}+\frac{4 \sqrt[3]{b} \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{7/3}}-\frac{4 \sqrt[3]{-1} \sqrt[3]{b} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{7/3}}+\frac{d \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^2}+\frac{d \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^2}+\frac{d \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^2}+\frac{d \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^2}-\frac{d \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^2}-\frac{d \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^2}+\frac{d \cos (c) \text{Ci}(d x)}{a^2}-\frac{d \sin (c) \text{Si}(d x)}{a^2}-\frac{4 \sin (c+d x)}{3 a^2 x}+\frac{\sin (c+d x)}{3 a b x^4}-\frac{\sin (c+d x)}{3 b x^4 \left(a+b x^3\right)}",1,"-1/3*((3*a + 4*b*x^3)*Cos[d*x]*Sin[c] + (3*a + 4*b*x^3)*Cos[c]*Sin[d*x] - (x*(a + b*x^3)*(18*d*Cos[c]*CosIntegral[d*x] + RootSum[a + b*#1^3 & , ((-4*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 4*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 4*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (4*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1 & ] + RootSum[a + b*#1^3 & , ((4*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 4*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 4*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - (4*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1 & ] - 18*d*Sin[c]*SinIntegral[d*x]))/6)/(a^2*x*(a + b*x^3))","C",0
108,1,470,800,1.1952432,"\int \frac{\sin (c+d x)}{x^3 \left(a+b x^3\right)^2} \, dx","Integrate[Sin[c + d*x]/(x^3*(a + b*x^3)^2),x]","\frac{\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-5 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-5 i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+5 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-5 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]+\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-5 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+5 i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-5 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-5 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]-\frac{3 \left(3 d^2 x^2 \sin (c) \left(a+b x^3\right) \text{Ci}(d x)+3 d^2 x^2 \cos (c) \left(a+b x^3\right) \text{Si}(d x)+3 a \sin (c+d x)+3 a d x \cos (c+d x)+3 b d x^4 \cos (c+d x)+5 b x^3 \sin (c+d x)\right)}{x^2 \left(a+b x^3\right)}}{18 a^2}","-\frac{\text{Ci}(d x) \sin (c) d^2}{2 a^2}-\frac{\cos (c) \text{Si}(d x) d^2}{2 a^2}-\frac{\cos (c+d x) d}{2 a^2 x}-\frac{(-1)^{2/3} \sqrt[3]{b} \cos \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) d}{9 a^{7/3}}-\frac{\sqrt[3]{b} \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{9 a^{7/3}}+\frac{\sqrt[3]{-1} \sqrt[3]{b} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{9 a^{7/3}}-\frac{(-1)^{2/3} \sqrt[3]{b} \sin \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) d}{9 a^{7/3}}+\frac{\sqrt[3]{b} \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{9 a^{7/3}}-\frac{\sqrt[3]{-1} \sqrt[3]{b} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{9 a^{7/3}}-\frac{5 b^{2/3} \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{8/3}}+\frac{5 \sqrt[3]{-1} b^{2/3} \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) \sin \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{8/3}}-\frac{5 (-1)^{2/3} b^{2/3} \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{8/3}}-\frac{\sin (c+d x)}{3 b x^5 \left(b x^3+a\right)}-\frac{5 \sin (c+d x)}{6 a^2 x^2}+\frac{\sin (c+d x)}{3 a b x^5}-\frac{5 \sqrt[3]{-1} b^{2/3} \cos \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{9 a^{8/3}}-\frac{5 b^{2/3} \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{8/3}}-\frac{5 (-1)^{2/3} b^{2/3} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{9 a^{8/3}}",1,"(RootSum[a + b*#1^3 & , ((-5*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 5*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 5*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (5*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] + RootSum[a + b*#1^3 & , ((5*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 5*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 5*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - (5*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] - (3*(3*a*d*x*Cos[c + d*x] + 3*b*d*x^4*Cos[c + d*x] + 3*d^2*x^2*(a + b*x^3)*CosIntegral[d*x]*Sin[c] + 3*a*Sin[c + d*x] + 5*b*x^3*Sin[c + d*x] + 3*d^2*x^2*(a + b*x^3)*Cos[c]*SinIntegral[d*x]))/(x^2*(a + b*x^3)))/(18*a^2)","C",0
109,1,457,772,0.6649197,"\int \frac{x^3 \sin (c+d x)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^3*Sin[c + d*x])/(a + b*x^3)^3,x]","\frac{i \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-i \text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-2 i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+2 \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-2 i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]-i \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{i \text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+2 i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+2 \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+2 i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]+\frac{6 b x \left(\left(b x^3-2 a\right) \sin (c+d x)+d x \left(a+b x^3\right) \cos (c+d x)\right)}{\left(a+b x^3\right)^2}}{108 a b^2}","\frac{\sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{5/3} b^{4/3}}-\frac{\sqrt[3]{-1} \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{5/3} b^{4/3}}+\frac{\sqrt[3]{-1} \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{27 a^{5/3} b^{4/3}}+\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{5/3} b^{4/3}}+\frac{d^2 \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{54 a b^2}+\frac{d^2 \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{54 a b^2}+\frac{d^2 \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{54 a b^2}-\frac{d^2 \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{54 a b^2}+\frac{d^2 \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{54 a b^2}+\frac{d^2 \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{54 a b^2}-\frac{d \cos (c+d x)}{18 b^2 x \left(a+b x^3\right)}+\frac{\sin (c+d x)}{18 a b^2 x^2}-\frac{\sin (c+d x)}{18 b^2 x^2 \left(a+b x^3\right)}+\frac{d \cos (c+d x)}{18 a b^2 x}-\frac{x \sin (c+d x)}{6 b \left(a+b x^3\right)^2}",1,"(I*RootSum[a + b*#1^3 & , (2*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - (2*I)*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - (2*I)*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - 2*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1^2 - I*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 - I*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 - d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ] - I*RootSum[a + b*#1^3 & , (2*Cos[c + d*#1]*CosIntegral[d*(x - #1)] + (2*I)*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + (2*I)*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - 2*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1^2 + I*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 + I*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 - d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ] + (6*b*x*(d*x*(a + b*x^3)*Cos[c + d*x] + (-2*a + b*x^3)*Sin[c + d*x]))/(a + b*x^3)^2)/(108*a*b^2)","C",0
110,1,449,777,0.4373603,"\int \frac{x^2 \sin (c+d x)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x^2*Sin[c + d*x])/(a + b*x^3)^3,x]","\frac{i d \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-2 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-2 i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+2 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]-i d \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-2 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+2 i \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-2 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]+\frac{6 b \cos (d x) \left(d x \cos (c) \left(a+b x^3\right)-3 a \sin (c)\right)}{\left(a+b x^3\right)^2}-\frac{6 b \sin (d x) \left(d x \sin (c) \left(a+b x^3\right)+3 a \cos (c)\right)}{\left(a+b x^3\right)^2}}{108 a b^2}","-\frac{d^2 \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{54 a^{4/3} b^{5/3}}-\frac{(-1)^{2/3} d^2 \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{54 a^{4/3} b^{5/3}}+\frac{\sqrt[3]{-1} d^2 \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{54 a^{4/3} b^{5/3}}-\frac{\sqrt[3]{-1} d \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{27 a^{5/3} b^{4/3}}+\frac{d \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} d \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} d^2 \cos \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{54 a^{4/3} b^{5/3}}-\frac{d^2 \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{54 a^{4/3} b^{5/3}}+\frac{\sqrt[3]{-1} d^2 \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{54 a^{4/3} b^{5/3}}-\frac{\sqrt[3]{-1} d \sin \left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}+c\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{27 a^{5/3} b^{4/3}}-\frac{d \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{5/3} b^{4/3}}-\frac{(-1)^{2/3} d \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{5/3} b^{4/3}}+\frac{d \cos (c+d x)}{18 a b^2 x^2}-\frac{d \cos (c+d x)}{18 b^2 x^2 \left(a+b x^3\right)}-\frac{\sin (c+d x)}{6 b \left(a+b x^3\right)^2}",1,"(I*d*RootSum[a + b*#1^3 & , ((-2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] - I*d*RootSum[a + b*#1^3 & , ((2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] + (6*b*Cos[d*x]*(d*x*(a + b*x^3)*Cos[c] - 3*a*Sin[c]))/(a + b*x^3)^2 - (6*b*(3*a*Cos[c] + d*x*(a + b*x^3)*Sin[c])*Sin[d*x])/(a + b*x^3)^2)/(108*a*b^2)","C",0
111,1,698,1141,0.5855673,"\int \frac{x \sin (c+d x)}{\left(a+b x^3\right)^3} \, dx","Integrate[(x*Sin[c + d*x])/(a + b*x^3)^3,x]","-\frac{\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-4 i \text{$\#$1}^2 b d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+4 \text{$\#$1}^2 b d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-4 \text{$\#$1}^2 b d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 i \text{$\#$1}^2 b d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-a d^2 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i a d^2 \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i a d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-a d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-4 i \text{$\#$1} b \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+4 i \text{$\#$1} b \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]+\text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{4 i \text{$\#$1}^2 b d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+4 \text{$\#$1}^2 b d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-4 \text{$\#$1}^2 b d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+4 i \text{$\#$1}^2 b d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-a d^2 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+i a d^2 \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-i a d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-a d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+4 i \text{$\#$1} b \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-4 i \text{$\#$1} b \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]-\frac{6 b \cos (d x) \left(a d \cos (c) \left(a+b x^3\right)+b x^2 \sin (c) \left(7 a+4 b x^3\right)\right)}{\left(a+b x^3\right)^2}-\frac{6 b \sin (d x) \left(b x^2 \cos (c) \left(7 a+4 b x^3\right)-a d \sin (c) \left(a+b x^3\right)\right)}{\left(a+b x^3\right)^2}}{108 a^2 b^2}","\frac{\text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^{5/3} b^{4/3}}-\frac{\sqrt[3]{-1} \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) \sin \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^{5/3} b^{4/3}}+\frac{\sqrt[3]{-1} \cos \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) d^2}{54 a^{5/3} b^{4/3}}+\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^{5/3} b^{4/3}}-\frac{\cos (c+d x) d}{18 b^2 x^3 \left(b x^3+a\right)}+\frac{\cos (c+d x) d}{18 a b^2 x^3}-\frac{2 \cos \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) d}{27 a^2 b}-\frac{2 \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{27 a^2 b}-\frac{2 \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{27 a^2 b}-\frac{2 \sin \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) d}{27 a^2 b}+\frac{2 \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{27 a^2 b}+\frac{2 \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{27 a^2 b}-\frac{2 \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{7/3} b^{2/3}}-\frac{2 (-1)^{2/3} \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) \sin \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{7/3} b^{2/3}}+\frac{2 \sqrt[3]{-1} \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{7/3} b^{2/3}}+\frac{2 \sin (c+d x)}{9 a^2 b x}+\frac{\sin (c+d x)}{18 b^2 x^4 \left(b x^3+a\right)}-\frac{\sin (c+d x)}{6 b x \left(b x^3+a\right)^2}-\frac{\sin (c+d x)}{18 a b^2 x^4}+\frac{2 (-1)^{2/3} \cos \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{27 a^{7/3} b^{2/3}}-\frac{2 \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{7/3} b^{2/3}}+\frac{2 \sqrt[3]{-1} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{7/3} b^{2/3}}",1,"-1/108*(RootSum[a + b*#1^3 & , ((-I)*a*d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - a*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - a*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + I*a*d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)] - (4*I)*b*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - 4*b*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 4*b*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + (4*I)*b*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + 4*b*d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1^2 - (4*I)*b*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 - (4*I)*b*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 - 4*b*d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ] + RootSum[a + b*#1^3 & , (I*a*d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - a*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - a*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - I*a*d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + (4*I)*b*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - 4*b*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 4*b*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - (4*I)*b*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + 4*b*d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1^2 + (4*I)*b*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 + (4*I)*b*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 - 4*b*d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ] - (6*b*Cos[d*x]*(a*d*(a + b*x^3)*Cos[c] + b*x^2*(7*a + 4*b*x^3)*Sin[c]))/(a + b*x^3)^2 - (6*b*(b*x^2*(7*a + 4*b*x^3)*Cos[c] - a*d*(a + b*x^3)*Sin[c])*Sin[d*x])/(a + b*x^3)^2)/(a^2*b^2)","C",0
112,1,675,1161,0.4324134,"\int \frac{\sin (c+d x)}{\left(a+b x^3\right)^3} \, dx","Integrate[Sin[c + d*x]/(a + b*x^3)^3,x]","\frac{-\frac{i \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{-i \text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-6 \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+10 i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-6 i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-10 \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+6 i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+10 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-6 \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+10 i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]}{b}+\frac{i \text{RootSum}\left[\text{$\#$1}^3 b+a\&,\frac{i \text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-\text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-6 \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-10 i \sin (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))+6 i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-10 \cos (\text{$\#$1} d+c) \text{Ci}(d (x-\text{$\#$1}))-6 i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+10 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-6 \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-10 i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\&\right]}{b}-\frac{6 x \cos (d x) \left(d x \cos (c) \left(a+b x^3\right)-\sin (c) \left(8 a+5 b x^3\right)\right)}{\left(a+b x^3\right)^2}+\frac{6 x \sin (d x) \left(d x \sin (c) \left(a+b x^3\right)+\cos (c) \left(8 a+5 b x^3\right)\right)}{\left(a+b x^3\right)^2}}{108 a^2}","-\frac{\text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^2 b}-\frac{\text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) \sin \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^2 b}-\frac{\text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^2 b}+\frac{\cos \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) d^2}{54 a^2 b}-\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^2 b}-\frac{\cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^2 b}-\frac{\cos (c+d x) d}{18 a^2 b x}-\frac{\cos (c+d x) d}{18 b^2 x^4 \left(b x^3+a\right)}+\frac{\cos (c+d x) d}{18 a b^2 x^4}+\frac{(-1)^{2/3} \cos \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) d}{9 a^{7/3} b^{2/3}}+\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{9 a^{7/3} b^{2/3}}-\frac{\sqrt[3]{-1} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{9 a^{7/3} b^{2/3}}+\frac{(-1)^{2/3} \sin \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) d}{9 a^{7/3} b^{2/3}}-\frac{\sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{9 a^{7/3} b^{2/3}}+\frac{\sqrt[3]{-1} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{9 a^{7/3} b^{2/3}}+\frac{5 \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{8/3} \sqrt[3]{b}}-\frac{5 \sqrt[3]{-1} \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) \sin \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{8/3} \sqrt[3]{b}}+\frac{5 (-1)^{2/3} \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{8/3} \sqrt[3]{b}}+\frac{\sin (c+d x)}{9 b^2 x^5 \left(b x^3+a\right)}+\frac{5 \sin (c+d x)}{18 a^2 b x^2}-\frac{\sin (c+d x)}{6 b x^2 \left(b x^3+a\right)^2}-\frac{\sin (c+d x)}{9 a b^2 x^5}+\frac{5 \sqrt[3]{-1} \cos \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{27 a^{8/3} \sqrt[3]{b}}+\frac{5 \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{8/3} \sqrt[3]{b}}+\frac{5 (-1)^{2/3} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{27 a^{8/3} \sqrt[3]{b}}",1,"(((-I)*RootSum[a + b*#1^3 & , (-10*Cos[c + d*#1]*CosIntegral[d*(x - #1)] + (10*I)*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + (10*I)*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + 10*Sin[c + d*#1]*SinIntegral[d*(x - #1)] - (6*I)*d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - 6*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 6*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + (6*I)*d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1^2 - I*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 - I*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 - d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ])/b + (I*RootSum[a + b*#1^3 & , (-10*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - (10*I)*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - (10*I)*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + 10*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + (6*I)*d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - 6*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 6*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - (6*I)*d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1^2 + I*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 + I*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 - d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ])/b - (6*x*Cos[d*x]*(d*x*(a + b*x^3)*Cos[c] - (8*a + 5*b*x^3)*Sin[c]))/(a + b*x^3)^2 + (6*x*((8*a + 5*b*x^3)*Cos[c] + d*x*(a + b*x^3)*Sin[c])*Sin[d*x])/(a + b*x^3)^2)/(108*a^2)","C",0
113,1,2109,1163,1.1482144,"\int \frac{\sin (c+d x)}{x \left(a+b x^3\right)^3} \, dx","Integrate[Sin[c + d*x]/(x*(a + b*x^3)^3),x]","\text{Result too large to show}","\frac{\text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^{7/3} b^{2/3}}+\frac{(-1)^{2/3} \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) \sin \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^{7/3} b^{2/3}}-\frac{\sqrt[3]{-1} \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^{7/3} b^{2/3}}-\frac{(-1)^{2/3} \cos \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) d^2}{54 a^{7/3} b^{2/3}}+\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^{7/3} b^{2/3}}-\frac{\sqrt[3]{-1} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d^2}{54 a^{7/3} b^{2/3}}-\frac{\cos (c+d x) d}{18 b^2 x^5 \left(b x^3+a\right)}-\frac{\cos (c+d x) d}{18 a^2 b x^2}+\frac{\cos (c+d x) d}{18 a b^2 x^5}+\frac{4 \sqrt[3]{-1} \cos \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) d}{27 a^{8/3} \sqrt[3]{b}}-\frac{4 \cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{27 a^{8/3} \sqrt[3]{b}}-\frac{4 (-1)^{2/3} \cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{27 a^{8/3} \sqrt[3]{b}}+\frac{4 \sqrt[3]{-1} \sin \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) d}{27 a^{8/3} \sqrt[3]{b}}+\frac{4 \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{27 a^{8/3} \sqrt[3]{b}}+\frac{4 (-1)^{2/3} \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) d}{27 a^{8/3} \sqrt[3]{b}}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\text{Ci}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^3}-\frac{\text{Ci}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right) \sin \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^3}-\frac{\text{Ci}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \sin \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^3}+\frac{\sin (c+d x)}{6 b^2 x^6 \left(b x^3+a\right)}-\frac{\sin (c+d x)}{6 b x^3 \left(b x^3+a\right)^2}+\frac{\sin (c+d x)}{3 a^2 b x^3}-\frac{\sin (c+d x)}{6 a b^2 x^6}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{\cos \left(c+\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(\frac{\sqrt[3]{-1} \sqrt[3]{a} d}{\sqrt[3]{b}}-d x\right)}{3 a^3}-\frac{\cos \left(c-\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{\sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^3}-\frac{\cos \left(c-\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right) \text{Si}\left(x d+\frac{(-1)^{2/3} \sqrt[3]{a} d}{\sqrt[3]{b}}\right)}{3 a^3}",1,"(-6*a^2*b*d*x*Cos[c + d*x] - 6*a*b^2*d*x^4*Cos[c + d*x] - (18*I)*b*(a + b*x^3)^2*RootSum[a + b*#1^3 & , Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)] & ] + (18*I)*b*(a + b*x^3)^2*RootSum[a + b*#1^3 & , Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)] & ] - 6*a^3*d*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - 12*a^2*b*d*x^3*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - 6*a*b^2*d*x^6*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - 6*a^3*d*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - 12*a^2*b*d*x^3*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - 6*a*b^2*d*x^6*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - I*a^3*d*RootSum[a + b*#1^3 & , ((-2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] - (2*I)*a^2*b*d*x^3*RootSum[a + b*#1^3 & , ((-2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] - I*a*b^2*d*x^6*RootSum[a + b*#1^3 & , ((-2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] + I*a^3*d*RootSum[a + b*#1^3 & , ((2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] + (2*I)*a^2*b*d*x^3*RootSum[a + b*#1^3 & , ((2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] + I*a*b^2*d*x^6*RootSum[a + b*#1^3 & , ((2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] + 108*a^2*b*CosIntegral[d*x]*Sin[c] + 216*a*b^2*x^3*CosIntegral[d*x]*Sin[c] + 108*b^3*x^6*CosIntegral[d*x]*Sin[c] + 54*a^2*b*Sin[c + d*x] + 36*a*b^2*x^3*Sin[c + d*x] + 108*a^2*b*Cos[c]*SinIntegral[d*x] + 216*a*b^2*x^3*Cos[c]*SinIntegral[d*x] + 108*b^3*x^6*Cos[c]*SinIntegral[d*x])/(108*a^3*b*(a + b*x^3)^2)","C",0