1,1,138,137,0.0823265,"\int (c+d x)^m \cos (a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]*Sin[a + b*x],x]","-\frac{2^{-m-3} e^{-\frac{2 i (a d+b c)}{d}} (c+d x)^m \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(e^{4 i a} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)+e^{\frac{4 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)\right)}{b}","-\frac{2^{-m-3} e^{2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)}{b}-\frac{2^{-m-3} e^{-2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)}{b}",1,"-((2^(-3 - m)*(c + d*x)^m*(E^((4*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-2*I)*b*(c + d*x))/d] + E^(((4*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((2*I)*b*(c + d*x))/d]))/(b*E^(((2*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^m))","A",1
2,1,86,156,0.4790578,"\int (c+d x)^4 \cos (a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]*Sin[a + b*x],x]","\frac{4 b d (c+d x) \sin (2 (a+b x)) \left(2 b^2 (c+d x)^2-3 d^2\right)-2 \cos (2 (a+b x)) \left(2 b^4 (c+d x)^4-6 b^2 d^2 (c+d x)^2+3 d^4\right)}{16 b^5}","\frac{3 d^4 \sin ^2(a+b x)}{4 b^5}-\frac{3 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^4}-\frac{3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}+\frac{d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{b^2}+\frac{(c+d x)^4 \sin ^2(a+b x)}{2 b}+\frac{3 c d^3 x}{2 b^3}+\frac{3 d^4 x^2}{4 b^3}-\frac{(c+d x)^4}{4 b}",1,"(-2*(3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Cos[2*(a + b*x)] + 4*b*d*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Sin[2*(a + b*x)])/(16*b^5)","A",1
3,1,71,120,0.2983256,"\int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x],x]","\frac{3 d \sin (2 (a+b x)) \left(2 b^2 (c+d x)^2-d^2\right)-2 b (c+d x) \cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-3 d^2\right)}{16 b^4}","-\frac{3 d^3 \sin (a+b x) \cos (a+b x)}{8 b^4}-\frac{3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac{3 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{4 b^2}+\frac{(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac{3 d^3 x}{8 b^3}-\frac{(c+d x)^3}{4 b}",1,"(-2*b*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + 3*d*(-d^2 + 2*b^2*(c + d*x)^2)*Sin[2*(a + b*x)])/(16*b^4)","A",1
4,1,50,89,0.2286477,"\int (c+d x)^2 \cos (a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x],x]","\frac{\cos (2 (a+b x)) \left(d^2-2 b^2 (c+d x)^2\right)+2 b d (c+d x) \sin (2 (a+b x))}{8 b^3}","-\frac{d^2 \sin ^2(a+b x)}{4 b^3}+\frac{d (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^2}+\frac{(c+d x)^2 \sin ^2(a+b x)}{2 b}-\frac{c d x}{2 b}-\frac{d^2 x^2}{4 b}",1,"((d^2 - 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + 2*b*d*(c + d*x)*Sin[2*(a + b*x)])/(8*b^3)","A",1
5,1,34,50,0.1000549,"\int (c+d x) \cos (a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]*Sin[a + b*x],x]","\frac{d \sin (2 (a+b x))-2 b (c+d x) \cos (2 (a+b x))}{8 b^2}","\frac{d \sin (a+b x) \cos (a+b x)}{4 b^2}+\frac{(c+d x) \sin ^2(a+b x)}{2 b}-\frac{d x}{4 b}",1,"(-2*b*(c + d*x)*Cos[2*(a + b*x)] + d*Sin[2*(a + b*x)])/(8*b^2)","A",1
6,1,60,65,0.1256453,"\int \frac{\cos (a+b x) \sin (a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x])/(c + d*x),x]","\frac{\sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)+\cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{2 d}","\frac{\sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{2 d}+\frac{\cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{2 d}",1,"(CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d] + Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(2*d)","A",1
7,1,80,85,0.3082443,"\int \frac{\cos (a+b x) \sin (a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x])/(c + d*x)^2,x]","\frac{2 b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-2 b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)-\frac{d \sin (2 (a+b x))}{c+d x}}{2 d^2}","\frac{b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}-\frac{b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}-\frac{\sin (2 a+2 b x)}{2 d (c+d x)}",1,"(2*b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] - (d*Sin[2*(a + b*x)])/(c + d*x) - 2*b*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/(2*d^2)","A",1
8,1,102,114,1.0841562,"\int \frac{\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x])/(c + d*x)^3,x]","-\frac{4 b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)+4 b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\frac{d (2 b (c+d x) \cos (2 (a+b x))+d \sin (2 (a+b x)))}{(c+d x)^2}}{4 d^3}","-\frac{b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d^3}-\frac{b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d^3}-\frac{b \cos (2 a+2 b x)}{2 d^2 (c+d x)}-\frac{\sin (2 a+2 b x)}{4 d (c+d x)^2}",1,"-1/4*(4*b^2*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] + (d*(2*b*(c + d*x)*Cos[2*(a + b*x)] + d*Sin[2*(a + b*x)]))/(c + d*x)^2 + 4*b^2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/d^3","A",1
9,1,164,144,0.6667268,"\int \frac{\cos (a+b x) \sin (a+b x)}{(c+d x)^4} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x])/(c + d*x)^4,x]","\frac{-4 b^3 (c+d x)^3 \left(\cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-\sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)\right)-d \cos (2 b x) \left(\sin (2 a) \left(d^2-2 b^2 (c+d x)^2\right)+b d \cos (2 a) (c+d x)\right)+d \sin (2 b x) \left(\cos (2 a) \left(2 b^2 (c+d x)^2-d^2\right)+b d \sin (2 a) (c+d x)\right)}{6 d^4 (c+d x)^3}","-\frac{2 b^3 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{3 d^4}+\frac{2 b^3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{3 d^4}+\frac{b^2 \sin (2 a+2 b x)}{3 d^3 (c+d x)}-\frac{b \cos (2 a+2 b x)}{6 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{6 d (c+d x)^3}",1,"(-(d*Cos[2*b*x]*(b*d*(c + d*x)*Cos[2*a] + (d^2 - 2*b^2*(c + d*x)^2)*Sin[2*a])) + d*((-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*a] + b*d*(c + d*x)*Sin[2*a])*Sin[2*b*x] - 4*b^3*(c + d*x)^3*(Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] - Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d]))/(6*d^4*(c + d*x)^3)","A",1
10,1,8,8,0.005694,"\int \frac{\cos (x) \sin (x)}{x} \, dx","Integrate[(Cos[x]*Sin[x])/x,x]","\frac{\text{Si}(2 x)}{2}","\frac{\text{Si}(2 x)}{2}",1,"SinIntegral[2*x]/2","A",1
11,1,16,16,0.0058853,"\int \frac{\cos (x) \sin (x)}{x^2} \, dx","Integrate[(Cos[x]*Sin[x])/x^2,x]","\text{Ci}(2 x)-\frac{\sin (2 x)}{2 x}","\text{Ci}(2 x)-\frac{\sin (2 x)}{2 x}",1,"CosIntegral[2*x] - Sin[2*x]/(2*x)","A",1
12,1,29,29,0.0076605,"\int \frac{\cos (x) \sin (x)}{x^3} \, dx","Integrate[(Cos[x]*Sin[x])/x^3,x]","-\text{Si}(2 x)-\frac{\sin (2 x)}{4 x^2}-\frac{\cos (2 x)}{2 x}","-\text{Si}(2 x)-\frac{\sin (2 x)}{4 x^2}-\frac{\cos (2 x)}{2 x}",1,"-1/2*Cos[2*x]/x - Sin[2*x]/(4*x^2) - SinIntegral[2*x]","A",1
13,1,237,275,0.6923526,"\int (c+d x)^m \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]*Sin[a + b*x]^2,x]","-\frac{i e^{-\frac{3 i (a d+b c)}{d}} (c+d x)^m \left(\left(\frac{i b (c+d x)}{d}\right)^{-m} \left(3^{-m} \left(e^{\frac{6 i b c}{d}} \Gamma \left(m+1,\frac{3 i b (c+d x)}{d}\right)-e^{6 i a} \left(\frac{i b (c+d x)}{d}\right)^{2 m} \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \Gamma \left(m+1,-\frac{3 i b (c+d x)}{d}\right)\right)-3 e^{2 i a+\frac{4 i b c}{d}} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)\right)+3 e^{2 i \left(2 a+\frac{b c}{d}\right)} \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)\right)}{24 b}","-\frac{i e^{i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)}{8 b}+\frac{i 3^{-m-1} e^{3 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{3 i b (c+d x)}{d}\right)}{8 b}+\frac{i e^{-i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)}{8 b}-\frac{i 3^{-m-1} e^{-3 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{3 i b (c+d x)}{d}\right)}{8 b}",1,"((-1/24*I)*(c + d*x)^m*((3*E^((2*I)*(2*a + (b*c)/d))*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^m + (-3*E^((2*I)*a + ((4*I)*b*c)/d)*Gamma[1 + m, (I*b*(c + d*x))/d] + (-((E^((6*I)*a)*((I*b*(c + d*x))/d)^(2*m)*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d])/((b^2*(c + d*x)^2)/d^2)^m) + E^(((6*I)*b*c)/d)*Gamma[1 + m, ((3*I)*b*(c + d*x))/d])/3^m)/((I*b*(c + d*x))/d)^m))/(b*E^(((3*I)*(b*c + a*d))/d))","A",1
14,1,385,205,1.4493846,"\int (c+d x)^4 \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]*Sin[a + b*x]^2,x]","\frac{81 b^4 c^4 \sin (a+b x)-27 b^4 c^4 \sin (3 (a+b x))+324 b^4 c^3 d x \sin (a+b x)-108 b^4 c^3 d x \sin (3 (a+b x))+486 b^4 c^2 d^2 x^2 \sin (a+b x)-162 b^4 c^2 d^2 x^2 \sin (3 (a+b x))+324 b^4 c d^3 x^3 \sin (a+b x)-108 b^4 c d^3 x^3 \sin (3 (a+b x))+81 b^4 d^4 x^4 \sin (a+b x)-27 b^4 d^4 x^4 \sin (3 (a+b x))-972 b^2 c^2 d^2 \sin (a+b x)+36 b^2 c^2 d^2 \sin (3 (a+b x))-1944 b^2 c d^3 x \sin (a+b x)+72 b^2 c d^3 x \sin (3 (a+b x))+324 b d (c+d x) \cos (a+b x) \left(b^2 (c+d x)^2-6 d^2\right)-12 b d (c+d x) \cos (3 (a+b x)) \left(3 b^2 (c+d x)^2-2 d^2\right)-972 b^2 d^4 x^2 \sin (a+b x)+36 b^2 d^4 x^2 \sin (3 (a+b x))+1944 d^4 \sin (a+b x)-8 d^4 \sin (3 (a+b x))}{324 b^5}","\frac{8 d^4 \sin ^3(a+b x)}{81 b^5}+\frac{160 d^4 \sin (a+b x)}{27 b^5}-\frac{160 d^3 (c+d x) \cos (a+b x)}{27 b^4}-\frac{8 d^3 (c+d x) \sin ^2(a+b x) \cos (a+b x)}{27 b^4}-\frac{4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}-\frac{8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}+\frac{8 d (c+d x)^3 \cos (a+b x)}{9 b^2}+\frac{4 d (c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{9 b^2}+\frac{(c+d x)^4 \sin ^3(a+b x)}{3 b}",1,"(324*b*d*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] - 12*b*d*(c + d*x)*(-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] + 81*b^4*c^4*Sin[a + b*x] - 972*b^2*c^2*d^2*Sin[a + b*x] + 1944*d^4*Sin[a + b*x] + 324*b^4*c^3*d*x*Sin[a + b*x] - 1944*b^2*c*d^3*x*Sin[a + b*x] + 486*b^4*c^2*d^2*x^2*Sin[a + b*x] - 972*b^2*d^4*x^2*Sin[a + b*x] + 324*b^4*c*d^3*x^3*Sin[a + b*x] + 81*b^4*d^4*x^4*Sin[a + b*x] - 27*b^4*c^4*Sin[3*(a + b*x)] + 36*b^2*c^2*d^2*Sin[3*(a + b*x)] - 8*d^4*Sin[3*(a + b*x)] - 108*b^4*c^3*d*x*Sin[3*(a + b*x)] + 72*b^2*c*d^3*x*Sin[3*(a + b*x)] - 162*b^4*c^2*d^2*x^2*Sin[3*(a + b*x)] + 36*b^2*d^4*x^2*Sin[3*(a + b*x)] - 108*b^4*c*d^3*x^3*Sin[3*(a + b*x)] - 27*b^4*d^4*x^4*Sin[3*(a + b*x)])/(324*b^5)","A",1
15,1,121,151,0.9408627,"\int (c+d x)^3 \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x]^2,x]","-\frac{-81 d \cos (a+b x) \left(b^2 (c+d x)^2-2 d^2\right)+d \cos (3 (a+b x)) \left(9 b^2 (c+d x)^2-2 d^2\right)+6 b (c+d x) \sin (a+b x) \left(\cos (2 (a+b x)) \left(3 b^2 (c+d x)^2-2 d^2\right)-3 b^2 (c+d x)^2+26 d^2\right)}{108 b^4}","\frac{2 d^3 \cos ^3(a+b x)}{27 b^4}-\frac{14 d^3 \cos (a+b x)}{9 b^4}-\frac{2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}-\frac{4 d^2 (c+d x) \sin (a+b x)}{3 b^3}+\frac{2 d (c+d x)^2 \cos (a+b x)}{3 b^2}+\frac{d (c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b^2}+\frac{(c+d x)^3 \sin ^3(a+b x)}{3 b}",1,"-1/108*(-81*d*(-2*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + d*(-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] + 6*b*(c + d*x)*(26*d^2 - 3*b^2*(c + d*x)^2 + (-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[a + b*x])/b^4","A",1
16,1,93,103,0.5799682,"\int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x]^2,x]","\frac{-2 \sin (a+b x) \left(\cos (2 (a+b x)) \left(9 b^2 (c+d x)^2-2 d^2\right)-9 b^2 (c+d x)^2+26 d^2\right)+54 b d (c+d x) \cos (a+b x)-6 b d (c+d x) \cos (3 (a+b x))}{108 b^3}","-\frac{2 d^2 \sin ^3(a+b x)}{27 b^3}-\frac{4 d^2 \sin (a+b x)}{9 b^3}+\frac{4 d (c+d x) \cos (a+b x)}{9 b^2}+\frac{2 d (c+d x) \sin ^2(a+b x) \cos (a+b x)}{9 b^2}+\frac{(c+d x)^2 \sin ^3(a+b x)}{3 b}",1,"(54*b*d*(c + d*x)*Cos[a + b*x] - 6*b*d*(c + d*x)*Cos[3*(a + b*x)] - 2*(26*d^2 - 9*b^2*(c + d*x)^2 + (-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[a + b*x])/(108*b^3)","A",1
17,1,44,51,0.1694837,"\int (c+d x) \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^2,x]","\frac{12 b (c+d x) \sin ^3(a+b x)+9 d \cos (a+b x)-d \cos (3 (a+b x))}{36 b^2}","-\frac{d \cos ^3(a+b x)}{9 b^2}+\frac{d \cos (a+b x)}{3 b^2}+\frac{(c+d x) \sin ^3(a+b x)}{3 b}",1,"(9*d*Cos[a + b*x] - d*Cos[3*(a + b*x)] + 12*b*(c + d*x)*Sin[a + b*x]^3)/(36*b^2)","A",1
18,1,102,121,0.3159881,"\int \frac{\cos (a+b x) \sin ^2(a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x]^2)/(c + d*x),x]","\frac{\cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)-\cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)-\sin \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+\sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)}{4 d}","\frac{\cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{4 d}-\frac{\cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{4 d}-\frac{\sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{4 d}+\frac{\sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{4 d}",1,"(Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] - Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] - Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/(4*d)","A",1
19,1,139,168,1.3700464,"\int \frac{\cos (a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x]^2)/(c + d*x)^2,x]","-\frac{-3 b \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)+b \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+b \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)-3 b \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)+\frac{d \cos (a+b x)}{c+d x}-\frac{d \cos (3 (a+b x))}{c+d x}}{4 d^2}","\frac{3 b \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{4 d^2}-\frac{b \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{4 d^2}-\frac{b \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{4 d^2}+\frac{3 b \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{4 d^2}-\frac{\cos (a+b x)}{4 d (c+d x)}+\frac{\cos (3 a+3 b x)}{4 d (c+d x)}",1,"-1/4*((d*Cos[a + b*x])/(c + d*x) - (d*Cos[3*(a + b*x)])/(c + d*x) - 3*b*CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] + b*CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + b*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] - 3*b*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/d^2","A",1
20,1,183,221,2.1558692,"\int \frac{\cos (a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x]^2)/(c + d*x)^3,x]","\frac{b^2 \left(-\cos \left(a-\frac{b c}{d}\right)\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+9 b^2 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)+b^2 \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)-9 b^2 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)+\frac{d (b (c+d x) \sin (a+b x)-d \cos (a+b x))}{(c+d x)^2}+\frac{d (d \cos (3 (a+b x))-3 b (c+d x) \sin (3 (a+b x)))}{(c+d x)^2}}{8 d^3}","-\frac{b^2 \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{8 d^3}+\frac{9 b^2 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{8 d^3}+\frac{b^2 \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{8 d^3}-\frac{9 b^2 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{8 d^3}+\frac{b \sin (a+b x)}{8 d^2 (c+d x)}-\frac{3 b \sin (3 a+3 b x)}{8 d^2 (c+d x)}-\frac{\cos (a+b x)}{8 d (c+d x)^2}+\frac{\cos (3 a+3 b x)}{8 d (c+d x)^2}",1,"(-(b^2*Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)]) + 9*b^2*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] + (d*(-(d*Cos[a + b*x]) + b*(c + d*x)*Sin[a + b*x]))/(c + d*x)^2 + (d*(d*Cos[3*(a + b*x)] - 3*b*(c + d*x)*Sin[3*(a + b*x)]))/(c + d*x)^2 + b^2*Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)] - 9*b^2*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/(8*d^3)","A",1
21,1,298,270,1.6817414,"\int \frac{\cos (a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x]^2)/(c + d*x)^4,x]","\frac{b^3 (c+d x)^3 \left(\sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+\cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)\right)-27 b^3 (c+d x)^3 \left(\sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)+\cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)\right)+d \cos (b x) \left(\cos (a) \left(b^2 (c+d x)^2-2 d^2\right)+b d \sin (a) (c+d x)\right)-d \cos (3 b x) \left(\cos (3 a) \left(9 b^2 (c+d x)^2-2 d^2\right)+3 b d \sin (3 a) (c+d x)\right)+d \sin (b x) \left(b d \cos (a) (c+d x)-\sin (a) \left(b^2 (c+d x)^2-2 d^2\right)\right)-d \sin (3 b x) \left(3 b d \cos (3 a) (c+d x)-\sin (3 a) \left(9 b^2 (c+d x)^2-2 d^2\right)\right)}{24 d^4 (c+d x)^3}","-\frac{9 b^3 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{8 d^4}+\frac{b^3 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{24 d^4}+\frac{b^3 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{24 d^4}-\frac{9 b^3 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{8 d^4}+\frac{b^2 \cos (a+b x)}{24 d^3 (c+d x)}-\frac{3 b^2 \cos (3 a+3 b x)}{8 d^3 (c+d x)}+\frac{b \sin (a+b x)}{24 d^2 (c+d x)^2}-\frac{b \sin (3 a+3 b x)}{8 d^2 (c+d x)^2}-\frac{\cos (a+b x)}{12 d (c+d x)^3}+\frac{\cos (3 a+3 b x)}{12 d (c+d x)^3}",1,"(d*Cos[b*x]*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] + b*d*(c + d*x)*Sin[a]) - d*Cos[3*b*x]*((-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*a] + 3*b*d*(c + d*x)*Sin[3*a]) + d*(b*d*(c + d*x)*Cos[a] - (-2*d^2 + b^2*(c + d*x)^2)*Sin[a])*Sin[b*x] - d*(3*b*d*(c + d*x)*Cos[3*a] - (-2*d^2 + 9*b^2*(c + d*x)^2)*Sin[3*a])*Sin[3*b*x] + b^3*(c + d*x)^3*(CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)]) - 27*b^3*(c + d*x)^3*(CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] + Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d]))/(24*d^4*(c + d*x)^3)","A",1
22,1,246,271,0.2896452,"\int (c+d x)^m \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{4^{-m-3} e^{-\frac{4 i (a d+b c)}{d}} (c+d x)^m \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(-2^{m+2} e^{2 i \left(a+\frac{3 b c}{d}\right)} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)-2^{m+2} e^{2 i \left(3 a+\frac{b c}{d}\right)} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)+e^{8 i a} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{4 i b (c+d x)}{d}\right)+e^{\frac{8 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{4 i b (c+d x)}{d}\right)\right)}{b}","-\frac{2^{-m-4} e^{2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)}{b}+\frac{2^{-2 (m+3)} e^{4 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{4 i b (c+d x)}{d}\right)}{b}-\frac{2^{-m-4} e^{-2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)}{b}+\frac{2^{-2 (m+3)} e^{-4 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{4 i b (c+d x)}{d}\right)}{b}",1,"(4^(-3 - m)*(c + d*x)^m*(-(2^(2 + m)*E^((2*I)*(3*a + (b*c)/d))*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-2*I)*b*(c + d*x))/d]) - 2^(2 + m)*E^((2*I)*(a + (3*b*c)/d))*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((2*I)*b*(c + d*x))/d] + E^((8*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-4*I)*b*(c + d*x))/d] + E^(((8*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((4*I)*b*(c + d*x))/d]))/(b*E^(((4*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^m)","A",1
23,1,158,260,1.6986681,"\int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{-8 b d (c+d x) \sin (2 (a+b x)) \left(\cos (2 (a+b x)) \left(8 b^2 (c+d x)^2-3 d^2\right)-16 \left(2 b^2 (c+d x)^2-3 d^2\right)\right)-64 \cos (2 (a+b x)) \left(2 b^4 (c+d x)^4-6 b^2 d^2 (c+d x)^2+3 d^4\right)+\cos (4 (a+b x)) \left(32 b^4 (c+d x)^4-24 b^2 d^2 (c+d x)^2+3 d^4\right)}{1024 b^5}","\frac{3 d^4 \sin ^4(a+b x)}{128 b^5}+\frac{45 d^4 \sin ^2(a+b x)}{128 b^5}-\frac{3 d^3 (c+d x) \sin ^3(a+b x) \cos (a+b x)}{32 b^4}-\frac{45 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{64 b^4}-\frac{3 d^2 (c+d x)^2 \sin ^4(a+b x)}{16 b^3}-\frac{9 d^2 (c+d x)^2 \sin ^2(a+b x)}{16 b^3}+\frac{d (c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b^2}+\frac{3 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{8 b^2}+\frac{(c+d x)^4 \sin ^4(a+b x)}{4 b}+\frac{45 c d^3 x}{64 b^3}+\frac{45 d^4 x^2}{128 b^3}-\frac{3 (c+d x)^4}{32 b}",1,"(-64*(3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Cos[2*(a + b*x)] + (3*d^4 - 24*b^2*d^2*(c + d*x)^2 + 32*b^4*(c + d*x)^4)*Cos[4*(a + b*x)] - 8*b*d*(c + d*x)*(-16*(-3*d^2 + 2*b^2*(c + d*x)^2) + (-3*d^2 + 8*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[2*(a + b*x)])/(1024*b^5)","A",1
24,1,135,196,0.8927353,"\int (c+d x)^3 \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{-64 b (c+d x) \cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-3 d^2\right)+4 b (c+d x) \cos (4 (a+b x)) \left(8 b^2 (c+d x)^2-3 d^2\right)-6 d \sin (2 (a+b x)) \left(\cos (2 (a+b x)) \left(8 b^2 (c+d x)^2-d^2\right)-16 \left(2 b^2 (c+d x)^2-d^2\right)\right)}{1024 b^4}","-\frac{3 d^3 \sin ^3(a+b x) \cos (a+b x)}{128 b^4}-\frac{45 d^3 \sin (a+b x) \cos (a+b x)}{256 b^4}-\frac{3 d^2 (c+d x) \sin ^4(a+b x)}{32 b^3}-\frac{9 d^2 (c+d x) \sin ^2(a+b x)}{32 b^3}+\frac{3 d (c+d x)^2 \sin ^3(a+b x) \cos (a+b x)}{16 b^2}+\frac{9 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{32 b^2}+\frac{(c+d x)^3 \sin ^4(a+b x)}{4 b}+\frac{45 d^3 x}{256 b^3}-\frac{3 (c+d x)^3}{32 b}",1,"(-64*b*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + 4*b*(c + d*x)*(-3*d^2 + 8*b^2*(c + d*x)^2)*Cos[4*(a + b*x)] - 6*d*(-16*(-d^2 + 2*b^2*(c + d*x)^2) + (-d^2 + 8*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[2*(a + b*x)])/(1024*b^4)","A",1
25,1,91,134,0.4913858,"\int (c+d x)^2 \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{-16 \cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-d^2\right)+\cos (4 (a+b x)) \left(8 b^2 (c+d x)^2-d^2\right)-4 b d (c+d x) (\sin (4 (a+b x))-8 \sin (2 (a+b x)))}{256 b^3}","-\frac{d^2 \sin ^4(a+b x)}{32 b^3}-\frac{3 d^2 \sin ^2(a+b x)}{32 b^3}+\frac{d (c+d x) \sin ^3(a+b x) \cos (a+b x)}{8 b^2}+\frac{3 d (c+d x) \sin (a+b x) \cos (a+b x)}{16 b^2}+\frac{(c+d x)^2 \sin ^4(a+b x)}{4 b}-\frac{3 c d x}{16 b}-\frac{3 d^2 x^2}{32 b}",1,"(-16*(-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + (-d^2 + 8*b^2*(c + d*x)^2)*Cos[4*(a + b*x)] - 4*b*d*(c + d*x)*(-8*Sin[2*(a + b*x)] + Sin[4*(a + b*x)]))/(256*b^3)","A",1
26,1,75,72,0.108595,"\int (c+d x) \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{d (\sin (2 (a+b x))-2 b x \cos (2 (a+b x)))}{16 b^2}-\frac{d (\sin (4 (a+b x))-4 b x \cos (4 (a+b x)))}{128 b^2}+\frac{c \sin ^4(a+b x)}{4 b}","\frac{d \sin ^3(a+b x) \cos (a+b x)}{16 b^2}+\frac{3 d \sin (a+b x) \cos (a+b x)}{32 b^2}+\frac{(c+d x) \sin ^4(a+b x)}{4 b}-\frac{3 d x}{32 b}",1,"(c*Sin[a + b*x]^4)/(4*b) + (d*(-2*b*x*Cos[2*(a + b*x)] + Sin[2*(a + b*x)]))/(16*b^2) - (d*(-4*b*x*Cos[4*(a + b*x)] + Sin[4*(a + b*x)]))/(128*b^2)","A",1
27,1,110,129,0.391777,"\int \frac{\cos (a+b x) \sin ^3(a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x]^3)/(c + d*x),x]","-\frac{\sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)-2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)}{8 d}","-\frac{\sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{8 d}+\frac{\sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{4 d}+\frac{\cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{4 d}-\frac{\cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{8 d}",1,"-1/8*(CosIntegral[(4*b*(c + d*x))/d]*Sin[4*a - (4*b*c)/d] - 2*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] - 2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/d","A",1
28,1,151,179,1.2285841,"\int \frac{\cos (a+b x) \sin ^3(a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x]^3)/(c + d*x)^2,x]","\frac{4 b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-4 b \cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)-4 b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+4 b \sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)-\frac{2 d \sin (2 (a+b x))}{c+d x}+\frac{d \sin (4 (a+b x))}{c+d x}}{8 d^2}","\frac{b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{2 d^2}-\frac{b \cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{2 d^2}-\frac{b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{2 d^2}+\frac{b \sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{2 d^2}-\frac{\sin (2 a+2 b x)}{4 d (c+d x)}+\frac{\sin (4 a+4 b x)}{8 d (c+d x)}",1,"(4*b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] - 4*b*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*(c + d*x))/d] - (2*d*Sin[2*(a + b*x)])/(c + d*x) + (d*Sin[4*(a + b*x)])/(c + d*x) - 4*b*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + 4*b*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/(8*d^2)","A",1
29,1,199,229,2.7596677,"\int \frac{\cos (a+b x) \sin ^3(a+b x)}{(c+d x)^3} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x]^3)/(c + d*x)^3,x]","\frac{-2 \left(4 b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)+4 b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\frac{d (2 b (c+d x) \cos (2 (a+b x))+d \sin (2 (a+b x)))}{(c+d x)^2}\right)+16 b^2 \sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)+16 b^2 \cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)+\frac{d (4 b (c+d x) \cos (4 (a+b x))+d \sin (4 (a+b x)))}{(c+d x)^2}}{16 d^3}","\frac{b^2 \sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{d^3}-\frac{b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{2 d^3}-\frac{b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{2 d^3}+\frac{b^2 \cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{d^3}-\frac{b \cos (2 a+2 b x)}{4 d^2 (c+d x)}+\frac{b \cos (4 a+4 b x)}{4 d^2 (c+d x)}-\frac{\sin (2 a+2 b x)}{8 d (c+d x)^2}+\frac{\sin (4 a+4 b x)}{16 d (c+d x)^2}",1,"(16*b^2*CosIntegral[(4*b*(c + d*x))/d]*Sin[4*a - (4*b*c)/d] + (d*(4*b*(c + d*x)*Cos[4*(a + b*x)] + d*Sin[4*(a + b*x)]))/(c + d*x)^2 - 2*(4*b^2*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] + (d*(2*b*(c + d*x)*Cos[2*(a + b*x)] + d*Sin[2*(a + b*x)]))/(c + d*x)^2 + 4*b^2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d]) + 16*b^2*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/(16*d^3)","A",1
30,1,316,287,2.2483627,"\int \frac{\cos (a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx","Integrate[(Cos[a + b*x]*Sin[a + b*x]^3)/(c + d*x)^4,x]","\frac{-8 b^3 (c+d x)^3 \left(\cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-\sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)\right)+32 b^3 (c+d x)^3 \left(\cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)-\sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)\right)-2 d \cos (2 b x) \left(\sin (2 a) \left(d^2-2 b^2 (c+d x)^2\right)+b d \cos (2 a) (c+d x)\right)+d \cos (4 b x) \left(\sin (4 a) \left(d^2-8 b^2 (c+d x)^2\right)+2 b d \cos (4 a) (c+d x)\right)+2 d \sin (2 b x) \left(\cos (2 a) \left(2 b^2 (c+d x)^2-d^2\right)+b d \sin (2 a) (c+d x)\right)-d \sin (4 b x) \left(\cos (4 a) \left(8 b^2 (c+d x)^2-d^2\right)+2 b d \sin (4 a) (c+d x)\right)}{24 d^4 (c+d x)^3}","-\frac{b^3 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{3 d^4}+\frac{4 b^3 \cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{3 d^4}+\frac{b^3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{3 d^4}-\frac{4 b^3 \sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{3 d^4}+\frac{b^2 \sin (2 a+2 b x)}{6 d^3 (c+d x)}-\frac{b^2 \sin (4 a+4 b x)}{3 d^3 (c+d x)}-\frac{b \cos (2 a+2 b x)}{12 d^2 (c+d x)^2}+\frac{b \cos (4 a+4 b x)}{12 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{12 d (c+d x)^3}+\frac{\sin (4 a+4 b x)}{24 d (c+d x)^3}",1,"(-2*d*Cos[2*b*x]*(b*d*(c + d*x)*Cos[2*a] + (d^2 - 2*b^2*(c + d*x)^2)*Sin[2*a]) + d*Cos[4*b*x]*(2*b*d*(c + d*x)*Cos[4*a] + (d^2 - 8*b^2*(c + d*x)^2)*Sin[4*a]) + 2*d*((-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*a] + b*d*(c + d*x)*Sin[2*a])*Sin[2*b*x] - d*((-d^2 + 8*b^2*(c + d*x)^2)*Cos[4*a] + 2*b*d*(c + d*x)*Sin[4*a])*Sin[4*b*x] - 8*b^3*(c + d*x)^3*(Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] - Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d]) + 32*b^3*(c + d*x)^3*(Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*(c + d*x))/d] - Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d]))/(24*d^4*(c + d*x)^3)","A",1
31,0,0,17,2.5750289,"\int (c+d x)^m \cot (a+b x) \, dx","Integrate[(c + d*x)^m*Cot[a + b*x],x]","\int (c+d x)^m \cot (a+b x) \, dx","\text{Int}\left(\cot (a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Cot[a + b*x], x]","A",-1
32,1,799,151,6.1147448,"\int (c+d x)^4 \cot (a+b x) \, dx","Integrate[(c + d*x)^4*Cot[a + b*x],x]","\frac{1}{5} i d^4 x^5+i c d^3 x^4+\frac{d^4 \log \left(1-e^{-i (a+b x)}\right) x^4}{b}+\frac{d^4 \log \left(1+e^{-i (a+b x)}\right) x^4}{b}+2 i c^2 d^2 x^3+\frac{4 c d^3 \log \left(1-e^{-i (a+b x)}\right) x^3}{b}+\frac{4 c d^3 \log \left(1+e^{-i (a+b x)}\right) x^3}{b}+2 c^3 d \cot (a) x^2+\frac{6 c^2 d^2 \log \left(1-e^{-i (a+b x)}\right) x^2}{b}+\frac{6 c^2 d^2 \log \left(1+e^{-i (a+b x)}\right) x^2}{b}+\frac{12 d^4 \text{Li}_3\left(-e^{-i (a+b x)}\right) x^2}{b^3}+\frac{12 d^4 \text{Li}_3\left(e^{-i (a+b x)}\right) x^2}{b^3}-2 c^3 d e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt{\sec ^2(a)} x^2-\frac{4 i c^3 d \tan ^{-1}(\tan (a)) x}{b}+\frac{4 c^3 d \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right) x}{b}+\frac{4 i d^2 \left(3 c^2+3 d x c+d^2 x^2\right) \text{Li}_2\left(-e^{-i (a+b x)}\right) x}{b^2}+\frac{4 i d^2 \left(3 c^2+3 d x c+d^2 x^2\right) \text{Li}_2\left(e^{-i (a+b x)}\right) x}{b^2}+\frac{24 c d^3 \text{Li}_3\left(-e^{-i (a+b x)}\right) x}{b^3}+\frac{24 c d^3 \text{Li}_3\left(e^{-i (a+b x)}\right) x}{b^3}-\frac{24 i d^4 \text{Li}_4\left(-e^{-i (a+b x)}\right) x}{b^4}-\frac{24 i d^4 \text{Li}_4\left(e^{-i (a+b x)}\right) x}{b^4}+\frac{2 i c^3 d \pi  x}{b}+\frac{2 c^3 d \pi  \log \left(1+e^{-2 i b x}\right)}{b^2}+\frac{4 c^3 d \tan ^{-1}(\tan (a)) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)}{b^2}-\frac{2 c^3 d \pi  \log (\cos (b x))}{b^2}+\frac{c^4 \log (\sin (a+b x))}{b}-\frac{4 c^3 d \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)}{b^2}-\frac{2 i c^3 d \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)}{b^2}+\frac{12 c^2 d^2 \text{Li}_3\left(-e^{-i (a+b x)}\right)}{b^3}+\frac{12 c^2 d^2 \text{Li}_3\left(e^{-i (a+b x)}\right)}{b^3}-\frac{24 i c d^3 \text{Li}_4\left(-e^{-i (a+b x)}\right)}{b^4}-\frac{24 i c d^3 \text{Li}_4\left(e^{-i (a+b x)}\right)}{b^4}-\frac{24 d^4 \text{Li}_5\left(-e^{-i (a+b x)}\right)}{b^5}-\frac{24 d^4 \text{Li}_5\left(e^{-i (a+b x)}\right)}{b^5}","-\frac{3 d^4 \text{Li}_5\left(e^{2 i (a+b x)}\right)}{2 b^5}+\frac{3 i d^3 (c+d x) \text{Li}_4\left(e^{2 i (a+b x)}\right)}{b^4}+\frac{3 d^2 (c+d x)^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{b^3}-\frac{2 i d (c+d x)^3 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}+\frac{(c+d x)^4 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{i (c+d x)^5}{5 d}",1,"((2*I)*c^3*d*Pi*x)/b + (2*I)*c^2*d^2*x^3 + I*c*d^3*x^4 + (I/5)*d^4*x^5 - ((4*I)*c^3*d*x*ArcTan[Tan[a]])/b + 2*c^3*d*x^2*Cot[a] + (2*c^3*d*Pi*Log[1 + E^((-2*I)*b*x)])/b^2 + (6*c^2*d^2*x^2*Log[1 - E^((-I)*(a + b*x))])/b + (4*c*d^3*x^3*Log[1 - E^((-I)*(a + b*x))])/b + (d^4*x^4*Log[1 - E^((-I)*(a + b*x))])/b + (6*c^2*d^2*x^2*Log[1 + E^((-I)*(a + b*x))])/b + (4*c*d^3*x^3*Log[1 + E^((-I)*(a + b*x))])/b + (d^4*x^4*Log[1 + E^((-I)*(a + b*x))])/b + (4*c^3*d*x*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))])/b + (4*c^3*d*ArcTan[Tan[a]]*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))])/b^2 - (2*c^3*d*Pi*Log[Cos[b*x]])/b^2 + (c^4*Log[Sin[a + b*x]])/b - (4*c^3*d*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]])/b^2 + ((4*I)*d^2*x*(3*c^2 + 3*c*d*x + d^2*x^2)*PolyLog[2, -E^((-I)*(a + b*x))])/b^2 + ((4*I)*d^2*x*(3*c^2 + 3*c*d*x + d^2*x^2)*PolyLog[2, E^((-I)*(a + b*x))])/b^2 - ((2*I)*c^3*d*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])/b^2 + (12*c^2*d^2*PolyLog[3, -E^((-I)*(a + b*x))])/b^3 + (24*c*d^3*x*PolyLog[3, -E^((-I)*(a + b*x))])/b^3 + (12*d^4*x^2*PolyLog[3, -E^((-I)*(a + b*x))])/b^3 + (12*c^2*d^2*PolyLog[3, E^((-I)*(a + b*x))])/b^3 + (24*c*d^3*x*PolyLog[3, E^((-I)*(a + b*x))])/b^3 + (12*d^4*x^2*PolyLog[3, E^((-I)*(a + b*x))])/b^3 - ((24*I)*c*d^3*PolyLog[4, -E^((-I)*(a + b*x))])/b^4 - ((24*I)*d^4*x*PolyLog[4, -E^((-I)*(a + b*x))])/b^4 - ((24*I)*c*d^3*PolyLog[4, E^((-I)*(a + b*x))])/b^4 - ((24*I)*d^4*x*PolyLog[4, E^((-I)*(a + b*x))])/b^4 - (24*d^4*PolyLog[5, -E^((-I)*(a + b*x))])/b^5 - (24*d^4*PolyLog[5, E^((-I)*(a + b*x))])/b^5 - 2*c^3*d*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2]","B",0
33,1,560,127,2.6667239,"\int (c+d x)^3 \cot (a+b x) \, dx","Integrate[(c + d*x)^3*Cot[a + b*x],x]","\frac{6 b^4 c^2 d x^2 \cot (a)-6 b^4 c^2 d x^2 e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt{\sec ^2(a)}+4 b^3 c^3 \log (\sin (a+b x))-12 i b^3 c^2 d x \tan ^{-1}(\tan (a))+12 b^3 c^2 d x \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+12 b^3 c d^2 x^2 \log \left(1-e^{-i (a+b x)}\right)+12 b^3 c d^2 x^2 \log \left(1+e^{-i (a+b x)}\right)+4 b^3 d^3 x^3 \log \left(1-e^{-i (a+b x)}\right)+4 b^3 d^3 x^3 \log \left(1+e^{-i (a+b x)}\right)-6 i b^2 c^2 d \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+12 b^2 c^2 d \tan ^{-1}(\tan (a)) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)-12 b^2 c^2 d \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)+12 i b^2 d^2 x (2 c+d x) \text{Li}_2\left(-e^{-i (a+b x)}\right)+12 i b^2 d^2 x (2 c+d x) \text{Li}_2\left(e^{-i (a+b x)}\right)+24 b c d^2 \text{Li}_3\left(-e^{-i (a+b x)}\right)+24 b c d^2 \text{Li}_3\left(e^{-i (a+b x)}\right)+24 b d^3 x \text{Li}_3\left(-e^{-i (a+b x)}\right)+24 b d^3 x \text{Li}_3\left(e^{-i (a+b x)}\right)-24 i d^3 \text{Li}_4\left(-e^{-i (a+b x)}\right)-24 i d^3 \text{Li}_4\left(e^{-i (a+b x)}\right)+4 i b^4 c d^2 x^3+i b^4 d^3 x^4+6 i \pi  b^3 c^2 d x+6 \pi  b^2 c^2 d \log \left(1+e^{-2 i b x}\right)-6 \pi  b^2 c^2 d \log (\cos (b x))}{4 b^4}","\frac{3 i d^3 \text{Li}_4\left(e^{2 i (a+b x)}\right)}{4 b^4}+\frac{3 d^2 (c+d x) \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}+\frac{(c+d x)^3 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{i (c+d x)^4}{4 d}",1,"((6*I)*b^3*c^2*d*Pi*x + (4*I)*b^4*c*d^2*x^3 + I*b^4*d^3*x^4 - (12*I)*b^3*c^2*d*x*ArcTan[Tan[a]] + 6*b^4*c^2*d*x^2*Cot[a] + 6*b^2*c^2*d*Pi*Log[1 + E^((-2*I)*b*x)] + 12*b^3*c*d^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 4*b^3*d^3*x^3*Log[1 - E^((-I)*(a + b*x))] + 12*b^3*c*d^2*x^2*Log[1 + E^((-I)*(a + b*x))] + 4*b^3*d^3*x^3*Log[1 + E^((-I)*(a + b*x))] + 12*b^3*c^2*d*x*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + 12*b^2*c^2*d*ArcTan[Tan[a]]*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] - 6*b^2*c^2*d*Pi*Log[Cos[b*x]] + 4*b^3*c^3*Log[Sin[a + b*x]] - 12*b^2*c^2*d*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + (12*I)*b^2*d^2*x*(2*c + d*x)*PolyLog[2, -E^((-I)*(a + b*x))] + (12*I)*b^2*d^2*x*(2*c + d*x)*PolyLog[2, E^((-I)*(a + b*x))] - (6*I)*b^2*c^2*d*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))] + 24*b*c*d^2*PolyLog[3, -E^((-I)*(a + b*x))] + 24*b*d^3*x*PolyLog[3, -E^((-I)*(a + b*x))] + 24*b*c*d^2*PolyLog[3, E^((-I)*(a + b*x))] + 24*b*d^3*x*PolyLog[3, E^((-I)*(a + b*x))] - (24*I)*d^3*PolyLog[4, -E^((-I)*(a + b*x))] - (24*I)*d^3*PolyLog[4, E^((-I)*(a + b*x))] - 6*b^4*c^2*d*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2])/(4*b^4)","B",0
34,1,356,93,1.4070179,"\int (c+d x)^2 \cot (a+b x) \, dx","Integrate[(c + d*x)^2*Cot[a + b*x],x]","\frac{3 b^3 c d x^2 \cot (a)-3 b^3 c d x^2 e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt{\sec ^2(a)}+3 b^2 c^2 \log (\sin (a+b x))-6 i b^2 c d x \tan ^{-1}(\tan (a))+6 b^2 c d x \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+3 b^2 d^2 x^2 \log \left(1-e^{-i (a+b x)}\right)+3 b^2 d^2 x^2 \log \left(1+e^{-i (a+b x)}\right)-3 i b c d \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+6 b c d \tan ^{-1}(\tan (a)) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)-6 b c d \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)+6 i b d^2 x \text{Li}_2\left(-e^{-i (a+b x)}\right)+6 i b d^2 x \text{Li}_2\left(e^{-i (a+b x)}\right)+6 d^2 \text{Li}_3\left(-e^{-i (a+b x)}\right)+6 d^2 \text{Li}_3\left(e^{-i (a+b x)}\right)+i b^3 d^2 x^3+3 i \pi  b^2 c d x+3 \pi  b c d \log \left(1+e^{-2 i b x}\right)-3 \pi  b c d \log (\cos (b x))}{3 b^3}","\frac{d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}-\frac{i d (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}+\frac{(c+d x)^2 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{i (c+d x)^3}{3 d}",1,"((3*I)*b^2*c*d*Pi*x + I*b^3*d^2*x^3 - (6*I)*b^2*c*d*x*ArcTan[Tan[a]] + 3*b^3*c*d*x^2*Cot[a] + 3*b*c*d*Pi*Log[1 + E^((-2*I)*b*x)] + 3*b^2*d^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 3*b^2*d^2*x^2*Log[1 + E^((-I)*(a + b*x))] + 6*b^2*c*d*x*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + 6*b*c*d*ArcTan[Tan[a]]*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] - 3*b*c*d*Pi*Log[Cos[b*x]] + 3*b^2*c^2*Log[Sin[a + b*x]] - 6*b*c*d*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + (6*I)*b*d^2*x*PolyLog[2, -E^((-I)*(a + b*x))] + (6*I)*b*d^2*x*PolyLog[2, E^((-I)*(a + b*x))] - (3*I)*b*c*d*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))] + 6*d^2*PolyLog[3, -E^((-I)*(a + b*x))] + 6*d^2*PolyLog[3, E^((-I)*(a + b*x))] - 3*b^3*c*d*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2])/(3*b^3)","B",0
35,1,188,65,5.1117412,"\int (c+d x) \cot (a+b x) \, dx","Integrate[(c + d*x)*Cot[a + b*x],x]","-\frac{d \csc (a) \sec (a) \left(b^2 x^2 e^{i \tan ^{-1}(\tan (a))}+\frac{\tan (a) \left(i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\tan ^2(a)+1}}\right)}{2 b^2 \sqrt{\sec ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}+\frac{c (\log (\tan (a+b x))+\log (\cos (a+b x)))}{b}+\frac{1}{2} d x^2 \cot (a)","-\frac{i d \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}+\frac{(c+d x) \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{i (c+d x)^2}{2 d}",1,"(d*x^2*Cot[a])/2 + (c*(Log[Cos[a + b*x]] + Log[Tan[a + b*x]]))/b - (d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(2*b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
36,0,0,17,3.6617052,"\int \frac{\cot (a+b x)}{c+d x} \, dx","Integrate[Cot[a + b*x]/(c + d*x),x]","\int \frac{\cot (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\cot (a+b x)}{c+d x},x\right)",0,"Integrate[Cot[a + b*x]/(c + d*x), x]","A",-1
37,0,0,17,6.9830974,"\int \frac{\cot (a+b x)}{(c+d x)^2} \, dx","Integrate[Cot[a + b*x]/(c + d*x)^2,x]","\int \frac{\cot (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\cot (a+b x)}{(c+d x)^2},x\right)",0,"Integrate[Cot[a + b*x]/(c + d*x)^2, x]","A",-1
38,0,0,23,2.9299522,"\int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx","Integrate[(c + d*x)^m*Cot[a + b*x]*Csc[a + b*x],x]","\int (c+d x)^m \cot (a+b x) \csc (a+b x) \, dx","\text{Int}\left(\cot (a+b x) \csc (a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Cot[a + b*x]*Csc[a + b*x], x]","A",-1
39,1,308,208,1.3575522,"\int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx","Integrate[(c + d*x)^4*Cot[a + b*x]*Csc[a + b*x],x]","\frac{8 i d \left(\frac{3 d \left(b^2 (c+d x)^2 \text{Li}_2(-\cos (a+b x)-i \sin (a+b x))+2 i b d (c+d x) \text{Li}_3(-\cos (a+b x)-i \sin (a+b x))-2 d^2 \text{Li}_4(-\cos (a+b x)-i \sin (a+b x))\right)}{b^3}-\frac{3 d \left(b^2 (c+d x)^2 \text{Li}_2(\cos (a+b x)+i \sin (a+b x))+2 i b d (c+d x) \text{Li}_3(\cos (a+b x)+i \sin (a+b x))-2 d^2 \text{Li}_4(\cos (a+b x)+i \sin (a+b x))\right)}{b^3}+2 i (c+d x)^3 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x))\right)-2 b \csc (a) (c+d x)^4+b \csc \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) (c+d x)^4 \csc \left(\frac{1}{2} (a+b x)\right)-b \sec \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) (c+d x)^4 \sec \left(\frac{1}{2} (a+b x)\right)}{2 b^2}","-\frac{24 i d^4 \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^5}+\frac{24 i d^4 \text{Li}_4\left(e^{i (a+b x)}\right)}{b^5}-\frac{24 d^3 (c+d x) \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^4}+\frac{24 d^3 (c+d x) \text{Li}_3\left(e^{i (a+b x)}\right)}{b^4}+\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^3}-\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^3}-\frac{8 d (c+d x)^3 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{(c+d x)^4 \csc (a+b x)}{b}",1,"(-2*b*(c + d*x)^4*Csc[a] + (8*I)*d*((2*I)*(c + d*x)^3*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]] + (3*d*(b^2*(c + d*x)^2*PolyLog[2, -Cos[a + b*x] - I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]] - 2*d^2*PolyLog[4, -Cos[a + b*x] - I*Sin[a + b*x]]))/b^3 - (3*d*(b^2*(c + d*x)^2*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]] - 2*d^2*PolyLog[4, Cos[a + b*x] + I*Sin[a + b*x]]))/b^3) + b*(c + d*x)^4*Csc[a/2]*Csc[(a + b*x)/2]*Sin[(b*x)/2] - b*(c + d*x)^4*Sec[a/2]*Sec[(a + b*x)/2]*Sin[(b*x)/2])/(2*b^2)","A",0
40,1,311,146,1.1682564,"\int (c+d x)^3 \cot (a+b x) \csc (a+b x) \, dx","Integrate[(c + d*x)^3*Cot[a + b*x]*Csc[a + b*x],x]","-\frac{b^3 c^3 \csc (a+b x)+3 b^3 c^2 d x \csc (a+b x)+3 b^3 c d^2 x^2 \csc (a+b x)+b^3 d^3 x^3 \csc (a+b x)-3 b^2 c^2 d \log \left(1-e^{i (a+b x)}\right)+3 b^2 c^2 d \log \left(1+e^{i (a+b x)}\right)-6 b^2 c d^2 x \log \left(1-e^{i (a+b x)}\right)+6 b^2 c d^2 x \log \left(1+e^{i (a+b x)}\right)-3 b^2 d^3 x^2 \log \left(1-e^{i (a+b x)}\right)+3 b^2 d^3 x^2 \log \left(1+e^{i (a+b x)}\right)-6 i b d^2 (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)+6 i b d^2 (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)+6 d^3 \text{Li}_3\left(-e^{i (a+b x)}\right)-6 d^3 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^4}","-\frac{6 d^3 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^4}+\frac{6 d^3 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^4}+\frac{6 i d^2 (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^3}-\frac{6 i d^2 (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^3}-\frac{6 d (c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{(c+d x)^3 \csc (a+b x)}{b}",1,"-((b^3*c^3*Csc[a + b*x] + 3*b^3*c^2*d*x*Csc[a + b*x] + 3*b^3*c*d^2*x^2*Csc[a + b*x] + b^3*d^3*x^3*Csc[a + b*x] - 3*b^2*c^2*d*Log[1 - E^(I*(a + b*x))] - 6*b^2*c*d^2*x*Log[1 - E^(I*(a + b*x))] - 3*b^2*d^3*x^2*Log[1 - E^(I*(a + b*x))] + 3*b^2*c^2*d*Log[1 + E^(I*(a + b*x))] + 6*b^2*c*d^2*x*Log[1 + E^(I*(a + b*x))] + 3*b^2*d^3*x^2*Log[1 + E^(I*(a + b*x))] - (6*I)*b*d^2*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] + (6*I)*b*d^2*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] + 6*d^3*PolyLog[3, -E^(I*(a + b*x))] - 6*d^3*PolyLog[3, E^(I*(a + b*x))])/b^4)","B",1
41,1,234,90,2.0366739,"\int (c+d x)^2 \cot (a+b x) \csc (a+b x) \, dx","Integrate[(c + d*x)^2*Cot[a + b*x]*Csc[a + b*x],x]","\frac{-2 b^2 \csc (a) (c+d x)^2+b^2 \csc \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) (c+d x)^2 \csc \left(\frac{1}{2} (a+b x)\right)-b^2 \sec \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) (c+d x)^2 \sec \left(\frac{1}{2} (a+b x)\right)-8 b c d \tanh ^{-1}\left(\cos (a)-\sin (a) \tan \left(\frac{b x}{2}\right)\right)+4 d^2 \left(2 \tan ^{-1}(\tan (a)) \tanh ^{-1}\left(\cos (a)-\sin (a) \tan \left(\frac{b x}{2}\right)\right)+\frac{\sec (a) \left(i \text{Li}_2\left(-e^{i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)-i \text{Li}_2\left(e^{i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\left(\tan ^{-1}(\tan (a))+b x\right) \left(\log \left(1-e^{i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)-\log \left(1+e^{i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)\right)\right)}{\sqrt{\sec ^2(a)}}\right)}{2 b^3}","\frac{2 i d^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^3}-\frac{2 i d^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^3}-\frac{4 d (c+d x) \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{(c+d x)^2 \csc (a+b x)}{b}",1,"(-8*b*c*d*ArcTanh[Cos[a] - Sin[a]*Tan[(b*x)/2]] - 2*b^2*(c + d*x)^2*Csc[a] + 4*d^2*(2*ArcTan[Tan[a]]*ArcTanh[Cos[a] - Sin[a]*Tan[(b*x)/2]] + (((b*x + ArcTan[Tan[a]])*(Log[1 - E^(I*(b*x + ArcTan[Tan[a]]))] - Log[1 + E^(I*(b*x + ArcTan[Tan[a]]))]) + I*PolyLog[2, -E^(I*(b*x + ArcTan[Tan[a]]))] - I*PolyLog[2, E^(I*(b*x + ArcTan[Tan[a]]))])*Sec[a])/Sqrt[Sec[a]^2]) + b^2*(c + d*x)^2*Csc[a/2]*Csc[(a + b*x)/2]*Sin[(b*x)/2] - b^2*(c + d*x)^2*Sec[a/2]*Sec[(a + b*x)/2]*Sin[(b*x)/2])/(2*b^3)","B",0
42,1,131,30,0.0552609,"\int (c+d x) \cot (a+b x) \csc (a+b x) \, dx","Integrate[(c + d*x)*Cot[a + b*x]*Csc[a + b*x],x]","\frac{d \log \left(\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}{b^2}-\frac{d \log \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}{b^2}-\frac{c \csc (a+b x)}{b}-\frac{d x \csc (a)}{b}+\frac{d x \csc \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) \csc \left(\frac{a}{2}+\frac{b x}{2}\right)}{2 b}-\frac{d x \sec \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) \sec \left(\frac{a}{2}+\frac{b x}{2}\right)}{2 b}","-\frac{d \tanh ^{-1}(\cos (a+b x))}{b^2}-\frac{(c+d x) \csc (a+b x)}{b}",1,"-((d*x*Csc[a])/b) - (c*Csc[a + b*x])/b - (d*Log[Cos[a/2 + (b*x)/2]])/b^2 + (d*Log[Sin[a/2 + (b*x)/2]])/b^2 + (d*x*Csc[a/2]*Csc[a/2 + (b*x)/2]*Sin[(b*x)/2])/(2*b) - (d*x*Sec[a/2]*Sec[a/2 + (b*x)/2]*Sin[(b*x)/2])/(2*b)","B",1
43,0,0,23,16.7592241,"\int \frac{\cot (a+b x) \csc (a+b x)}{c+d x} \, dx","Integrate[(Cot[a + b*x]*Csc[a + b*x])/(c + d*x),x]","\int \frac{\cot (a+b x) \csc (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\cot (a+b x) \csc (a+b x)}{c+d x},x\right)",0,"Integrate[(Cot[a + b*x]*Csc[a + b*x])/(c + d*x), x]","A",-1
44,0,0,23,20.304161,"\int \frac{\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx","Integrate[(Cot[a + b*x]*Csc[a + b*x])/(c + d*x)^2,x]","\int \frac{\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\cot (a+b x) \csc (a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Cot[a + b*x]*Csc[a + b*x])/(c + d*x)^2, x]","A",-1
45,0,0,25,6.3140497,"\int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Cot[a + b*x]*Csc[a + b*x]^2,x]","\int (c+d x)^m \cot (a+b x) \csc ^2(a+b x) \, dx","\text{Int}\left(\cot (a+b x) \csc ^2(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Cot[a + b*x]*Csc[a + b*x]^2, x]","A",-1
46,1,504,137,6.6084654,"\int (c+d x)^4 \cot (a+b x) \csc ^2(a+b x) \, dx","Integrate[(c + d*x)^4*Cot[a + b*x]*Csc[a + b*x]^2,x]","\frac{6 c^2 d^2 \csc (a) (\sin (a) \log (\sin (a) \cos (b x)+\cos (a) \sin (b x))-b x \cos (a))}{b^3 \left(\sin ^2(a)+\cos ^2(a)\right)}+\frac{2 \csc (a) \csc (a+b x) \left(c^3 d \sin (b x)+3 c^2 d^2 x \sin (b x)+3 c d^3 x^2 \sin (b x)+d^4 x^3 \sin (b x)\right)}{b^2}-\frac{6 c d^3 \csc (a) \sec (a) \left(b^2 x^2 e^{i \tan ^{-1}(\tan (a))}+\frac{\tan (a) \left(i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\tan ^2(a)+1}}\right)}{b^4 \sqrt{\sec ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}-\frac{e^{i a} d^4 \csc (a) \left(2 e^{-2 i a} b^3 x^3+3 i \left(1-e^{-2 i a}\right) b^2 x^2 \log \left(1-e^{-i (a+b x)}\right)+3 i \left(1-e^{-2 i a}\right) b^2 x^2 \log \left(1+e^{-i (a+b x)}\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right)}{b^5}-\frac{(c+d x)^4 \csc ^2(a+b x)}{2 b}","\frac{3 d^4 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{b^5}-\frac{6 i d^3 (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^4}+\frac{6 d^2 (c+d x)^2 \log \left(1-e^{2 i (a+b x)}\right)}{b^3}-\frac{2 d (c+d x)^3 \cot (a+b x)}{b^2}-\frac{(c+d x)^4 \csc ^2(a+b x)}{2 b}-\frac{2 i d (c+d x)^3}{b^2}",1,"-1/2*((c + d*x)^4*Csc[a + b*x]^2)/b - (d^4*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^5 + (6*c^2*d^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (2*Csc[a]*Csc[a + b*x]*(c^3*d*Sin[b*x] + 3*c^2*d^2*x*Sin[b*x] + 3*c*d^3*x^2*Sin[b*x] + d^4*x^3*Sin[b*x]))/b^2 - (6*c*d^3*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^4*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
47,1,277,115,6.413808,"\int (c+d x)^3 \cot (a+b x) \csc ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Cot[a + b*x]*Csc[a + b*x]^2,x]","\frac{3 c d^2 \csc (a) (\sin (a) \log (\sin (a) \cos (b x)+\cos (a) \sin (b x))-b x \cos (a))}{b^3 \left(\sin ^2(a)+\cos ^2(a)\right)}+\frac{3 \csc (a) \csc (a+b x) \left(c^2 d \sin (b x)+2 c d^2 x \sin (b x)+d^3 x^2 \sin (b x)\right)}{2 b^2}-\frac{3 d^3 \csc (a) \sec (a) \left(b^2 x^2 e^{i \tan ^{-1}(\tan (a))}+\frac{\tan (a) \left(i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\tan ^2(a)+1}}\right)}{2 b^4 \sqrt{\sec ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}-\frac{(c+d x)^3 \csc ^2(a+b x)}{2 b}","-\frac{3 i d^3 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^4}+\frac{3 d^2 (c+d x) \log \left(1-e^{2 i (a+b x)}\right)}{b^3}-\frac{3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac{(c+d x)^3 \csc ^2(a+b x)}{2 b}-\frac{3 i d (c+d x)^2}{2 b^2}",1,"-1/2*((c + d*x)^3*Csc[a + b*x]^2)/b + (3*c*d^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (3*Csc[a]*Csc[a + b*x]*(c^2*d*Sin[b*x] + 2*c*d^2*x*Sin[b*x] + d^3*x^2*Sin[b*x]))/(2*b^2) - (3*d^3*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(2*b^4*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
48,1,94,54,0.8918601,"\int (c+d x)^2 \cot (a+b x) \csc ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Cot[a + b*x]*Csc[a + b*x]^2,x]","\frac{-b^2 (c+d x)^2 \csc ^2(a+b x)+2 b d \csc (a) \sin (b x) (c+d x) \csc (a+b x)-2 i d^2 \tan ^{-1}(\tan (a+b x))-2 b d^2 x \cot (a)+d^2 \log \left(\sin ^2(a+b x)\right)+2 i b d^2 x}{2 b^3}","\frac{d^2 \log (\sin (a+b x))}{b^3}-\frac{d (c+d x) \cot (a+b x)}{b^2}-\frac{(c+d x)^2 \csc ^2(a+b x)}{2 b}",1,"((2*I)*b*d^2*x - (2*I)*d^2*ArcTan[Tan[a + b*x]] - 2*b*d^2*x*Cot[a] - b^2*(c + d*x)^2*Csc[a + b*x]^2 + d^2*Log[Sin[a + b*x]^2] + 2*b*d*(c + d*x)*Csc[a]*Csc[a + b*x]*Sin[b*x])/(2*b^3)","C",1
49,1,48,35,0.0726576,"\int (c+d x) \cot (a+b x) \csc ^2(a+b x) \, dx","Integrate[(c + d*x)*Cot[a + b*x]*Csc[a + b*x]^2,x]","-\frac{d \cot (a+b x)}{2 b^2}-\frac{c \csc ^2(a+b x)}{2 b}-\frac{d x \csc ^2(a+b x)}{2 b}","-\frac{d \cot (a+b x)}{2 b^2}-\frac{(c+d x) \csc ^2(a+b x)}{2 b}",1,"-1/2*(d*Cot[a + b*x])/b^2 - (c*Csc[a + b*x]^2)/(2*b) - (d*x*Csc[a + b*x]^2)/(2*b)","A",1
50,0,0,25,11.3287026,"\int \frac{\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx","Integrate[(Cot[a + b*x]*Csc[a + b*x]^2)/(c + d*x),x]","\int \frac{\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\cot (a+b x) \csc ^2(a+b x)}{c+d x},x\right)",0,"Integrate[(Cot[a + b*x]*Csc[a + b*x]^2)/(c + d*x), x]","A",-1
51,0,0,25,10.6413363,"\int \frac{\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[(Cot[a + b*x]*Csc[a + b*x]^2)/(c + d*x)^2,x]","\int \frac{\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Cot[a + b*x]*Csc[a + b*x]^2)/(c + d*x)^2, x]","A",-1
52,1,179,196,2.2386374,"\int (c+d x)^{5/2} \cos (a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]*Sin[a + b*x],x]","\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x} \left(20 b d (c+d x) \sin (2 (a+b x))-\cos (2 (a+b x)) \left(16 b^2 (c+d x)^2-15 d^2\right)\right)-15 \sqrt{\pi } d^2 \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+15 \sqrt{\pi } d^2 \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)}{128 b^3 \sqrt{\frac{b}{d}}}","-\frac{15 \sqrt{\pi } d^{5/2} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{128 b^{7/2}}+\frac{15 \sqrt{\pi } d^{5/2} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{128 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{64 b^3}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{16 b^2}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{4 b}",1,"(-15*d^2*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + 15*d^2*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 2*Sqrt[b/d]*Sqrt[c + d*x]*(-((-15*d^2 + 16*b^2*(c + d*x)^2)*Cos[2*(a + b*x)]) + 20*b*d*(c + d*x)*Sin[2*(a + b*x)]))/(128*b^3*Sqrt[b/d])","A",1
53,1,157,168,0.8419977,"\int (c+d x)^{3/2} \cos (a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]*Sin[a + b*x],x]","\frac{-3 \sqrt{\pi } d \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-3 \sqrt{\pi } d \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-2 \sqrt{\frac{b}{d}} \sqrt{c+d x} (4 b (c+d x) \cos (2 (a+b x))-3 d \sin (2 (a+b x)))}{32 d^2 \left(\frac{b}{d}\right)^{5/2}}","-\frac{3 \sqrt{\pi } d^{3/2} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{32 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{32 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (2 a+2 b x)}{16 b^2}-\frac{(c+d x)^{3/2} \cos (2 a+2 b x)}{4 b}",1,"(-3*d*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 3*d*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] - 2*Sqrt[b/d]*Sqrt[c + d*x]*(4*b*(c + d*x)*Cos[2*(a + b*x)] - 3*d*Sin[2*(a + b*x)]))/(32*(b/d)^(5/2)*d^2)","A",1
54,1,134,142,0.2576622,"\int \sqrt{c+d x} \cos (a+b x) \sin (a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]*Sin[a + b*x],x]","\frac{\sqrt{\pi } \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-\sqrt{\pi } \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-2 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))}{8 b \sqrt{\frac{b}{d}}}","\frac{\sqrt{\pi } \sqrt{d} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{8 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{8 b^{3/2}}-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{4 b}",1,"(-2*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] + Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d])/(8*b*Sqrt[b/d])","A",1
55,1,134,142,0.023758,"\int \sqrt{c+d x} \cos (a+b x) \sin (a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]*Sin[a + b*x],x]","\frac{\sqrt{\pi } \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-\sqrt{\pi } \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-2 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))}{8 b \sqrt{\frac{b}{d}}}","\frac{\sqrt{\pi } \sqrt{d} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{8 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{8 b^{3/2}}-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{4 b}",1,"(-2*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] + Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d])/(8*b*Sqrt[b/d])","A",1
56,1,157,168,0.0395031,"\int (c+d x)^{3/2} \cos (a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]*Sin[a + b*x],x]","\frac{-3 \sqrt{\pi } d \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-3 \sqrt{\pi } d \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-2 \sqrt{\frac{b}{d}} \sqrt{c+d x} (4 b (c+d x) \cos (2 (a+b x))-3 d \sin (2 (a+b x)))}{32 d^2 \left(\frac{b}{d}\right)^{5/2}}","-\frac{3 \sqrt{\pi } d^{3/2} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{32 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{32 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (2 a+2 b x)}{16 b^2}-\frac{(c+d x)^{3/2} \cos (2 a+2 b x)}{4 b}",1,"(-3*d*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 3*d*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] - 2*Sqrt[b/d]*Sqrt[c + d*x]*(4*b*(c + d*x)*Cos[2*(a + b*x)] - 3*d*Sin[2*(a + b*x)]))/(32*(b/d)^(5/2)*d^2)","A",1
57,1,179,196,1.0564881,"\int (c+d x)^{5/2} \cos (a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]*Sin[a + b*x],x]","\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x} \left(20 b d (c+d x) \sin (2 (a+b x))-\cos (2 (a+b x)) \left(16 b^2 (c+d x)^2-15 d^2\right)\right)-15 \sqrt{\pi } d^2 \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+15 \sqrt{\pi } d^2 \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)}{128 b^3 \sqrt{\frac{b}{d}}}","-\frac{15 \sqrt{\pi } d^{5/2} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{128 b^{7/2}}+\frac{15 \sqrt{\pi } d^{5/2} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{128 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{64 b^3}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{16 b^2}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{4 b}",1,"(-15*d^2*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + 15*d^2*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 2*Sqrt[b/d]*Sqrt[c + d*x]*(-((-15*d^2 + 16*b^2*(c + d*x)^2)*Cos[2*(a + b*x)]) + 20*b*d*(c + d*x)*Sin[2*(a + b*x)]))/(128*b^3*Sqrt[b/d])","A",1
58,1,1171,406,15.0709202,"\int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]*Sin[a + b*x]^2,x]","-\frac{i e^{-\frac{i (b c+a d)}{d}} \sqrt{c+d x} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right) c^2}{8 b}-\frac{\left(-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \sin \left(3 a-\frac{3 b c}{d}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))\right) c^2}{24 \sqrt{3} b \sqrt{\frac{b}{d}}}+\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(a-\frac{b c}{d}\right)-3 d \cos \left(a-\frac{b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(a-\frac{b c}{d}\right)+3 d \sin \left(a-\frac{b c}{d}\right)\right)+2 b \sqrt{c+d x} (3 \cos (a+b x)+2 b x \sin (a+b x))\right) c}{8 b^3}-\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(3 a-\frac{3 b c}{d}\right)-d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(3 a-\frac{3 b c}{d}\right)+d \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} b \sqrt{c+d x} (\cos (3 (a+b x))+2 b x \sin (3 (a+b x)))\right) c}{24 \sqrt{3} b^3}+\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \cos \left(a-\frac{b c}{d}\right)+12 b c d \sin \left(a-\frac{b c}{d}\right)\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \sin \left(a-\frac{b c}{d}\right)-12 b c d \cos \left(a-\frac{b c}{d}\right)\right)+2 \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(4 b^2 x^2-15\right) \sin (a+b x)-2 b (c-5 d x) \cos (a+b x)\right)\right)}{32 b^5}-\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \cos \left(3 a-\frac{3 b c}{d}\right)+12 b c d \sin \left(3 a-\frac{3 b c}{d}\right)\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \sin \left(3 a-\frac{3 b c}{d}\right)-12 b c d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(12 b^2 x^2-5\right) \sin (3 (a+b x))-2 b (c-5 d x) \cos (3 (a+b x))\right)\right)}{288 \sqrt{3} b^5}","-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{144 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{144 b^{7/2}}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{16 b^3}+\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac{(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}",1,"((-1/8*I)*c^2*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) + (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[a - (b*c)/d] + 2*b*c*Sin[a - (b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*b*Sqrt[c + d*x]*(3*Cos[a + b*x] + 2*b*x*Sin[a + b*x])))/(8*b^3) + ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*((4*b^2*c^2 - 15*d^2)*Cos[a - (b*c)/d] + 12*b*c*d*Sin[a - (b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[a - (b*c)/d] + (4*b^2*c^2 - 15*d^2)*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[a + b*x] + d*(-15 + 4*b^2*x^2)*Sin[a + b*x])))/(32*b^5) - (c^2*(-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)]))/(24*Sqrt[3]*b*Sqrt[b/d]) - (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-(d*Cos[3*a - (3*b*c)/d]) + 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*b*Sqrt[c + d*x]*(Cos[3*(a + b*x)] + 2*b*x*Sin[3*(a + b*x)])))/(24*Sqrt[3]*b^3) - ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*((12*b^2*c^2 - 5*d^2)*Cos[3*a - (3*b*c)/d] + 12*b*c*d*Sin[3*a - (3*b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[3*a - (3*b*c)/d] + (12*b^2*c^2 - 5*d^2)*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[3*(a + b*x)] + d*(-5 + 12*b^2*x^2)*Sin[3*(a + b*x)])))/(288*Sqrt[3]*b^5)","C",0
59,1,677,353,9.1404243,"\int (c+d x)^{3/2} \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]*Sin[a + b*x]^2,x]","\frac{d \left(\sqrt{2 \pi } \sqrt{\frac{b}{d}} C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(a-\frac{b c}{d}\right)-3 d \cos \left(a-\frac{b c}{d}\right)\right)+\sqrt{2 \pi } \sqrt{\frac{b}{d}} S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(3 d \sin \left(a-\frac{b c}{d}\right)+2 b c \cos \left(a-\frac{b c}{d}\right)\right)+2 b \sqrt{c+d x} (2 b x \sin (a+b x)+3 \cos (a+b x))\right)}{16 b^3}-\frac{d \left(\sqrt{2 \pi } \sqrt{\frac{b}{d}} C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(3 a-\frac{3 b c}{d}\right)-d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{2 \pi } \sqrt{\frac{b}{d}} S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(d \sin \left(3 a-\frac{3 b c}{d}\right)+2 b c \cos \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} b \sqrt{c+d x} (2 b x \sin (3 (a+b x))+\cos (3 (a+b x)))\right)}{48 \sqrt{3} b^3}-\frac{c \left(-\sqrt{2 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))\right)}{24 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{i c \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{8 b}","-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}+\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{24 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{24 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}+\frac{3 d \sqrt{c+d x} \cos (a+b x)}{8 b^2}-\frac{d \sqrt{c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac{(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}",1,"((-1/8*I)*c*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) + (d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[a - (b*c)/d] + 2*b*c*Sin[a - (b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*b*Sqrt[c + d*x]*(3*Cos[a + b*x] + 2*b*x*Sin[a + b*x])))/(16*b^3) - (c*(-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)]))/(24*Sqrt[3]*b*Sqrt[b/d]) - (d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-(d*Cos[3*a - (3*b*c)/d]) + 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*b*Sqrt[c + d*x]*(Cos[3*(a + b*x)] + 2*b*x*Sin[3*(a + b*x)])))/(48*Sqrt[3]*b^3)","C",1
60,1,264,304,5.363931,"\int \sqrt{c+d x} \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]*Sin[a + b*x]^2,x]","\frac{\frac{\sqrt{6 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{6 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-6 \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))}{\sqrt{\frac{b}{d}}}-9 i \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{72 b}","\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{12 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{12 b^{3/2}}+\frac{\sqrt{c+d x} \sin (a+b x)}{4 b}-\frac{\sqrt{c+d x} \sin (3 a+3 b x)}{12 b}",1,"(((-9*I)*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/E^((I*(b*c + a*d))/d) + (Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[6*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] - 6*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)])/Sqrt[b/d])/(72*b)","C",1
61,1,264,304,3.0494103,"\int \sqrt{c+d x} \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]*Sin[a + b*x]^2,x]","\frac{\frac{\sqrt{6 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{6 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-6 \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))}{\sqrt{\frac{b}{d}}}-9 i \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{72 b}","\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{12 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{12 b^{3/2}}+\frac{\sqrt{c+d x} \sin (a+b x)}{4 b}-\frac{\sqrt{c+d x} \sin (3 a+3 b x)}{12 b}",1,"(((-9*I)*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/E^((I*(b*c + a*d))/d) + (Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[6*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] - 6*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)])/Sqrt[b/d])/(72*b)","C",1
62,1,677,353,8.9492338,"\int (c+d x)^{3/2} \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]*Sin[a + b*x]^2,x]","\frac{d \left(\sqrt{2 \pi } \sqrt{\frac{b}{d}} C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(a-\frac{b c}{d}\right)-3 d \cos \left(a-\frac{b c}{d}\right)\right)+\sqrt{2 \pi } \sqrt{\frac{b}{d}} S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(3 d \sin \left(a-\frac{b c}{d}\right)+2 b c \cos \left(a-\frac{b c}{d}\right)\right)+2 b \sqrt{c+d x} (2 b x \sin (a+b x)+3 \cos (a+b x))\right)}{16 b^3}-\frac{d \left(\sqrt{2 \pi } \sqrt{\frac{b}{d}} C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(3 a-\frac{3 b c}{d}\right)-d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{2 \pi } \sqrt{\frac{b}{d}} S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(d \sin \left(3 a-\frac{3 b c}{d}\right)+2 b c \cos \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} b \sqrt{c+d x} (2 b x \sin (3 (a+b x))+\cos (3 (a+b x)))\right)}{48 \sqrt{3} b^3}-\frac{c \left(-\sqrt{2 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))\right)}{24 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{i c \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{8 b}","-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}+\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{24 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{24 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}+\frac{3 d \sqrt{c+d x} \cos (a+b x)}{8 b^2}-\frac{d \sqrt{c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac{(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}",1,"((-1/8*I)*c*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) + (d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[a - (b*c)/d] + 2*b*c*Sin[a - (b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*b*Sqrt[c + d*x]*(3*Cos[a + b*x] + 2*b*x*Sin[a + b*x])))/(16*b^3) - (c*(-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)]))/(24*Sqrt[3]*b*Sqrt[b/d]) - (d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-(d*Cos[3*a - (3*b*c)/d]) + 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*b*Sqrt[c + d*x]*(Cos[3*(a + b*x)] + 2*b*x*Sin[3*(a + b*x)])))/(48*Sqrt[3]*b^3)","C",1
63,1,1171,406,13.6277807,"\int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]*Sin[a + b*x]^2,x]","-\frac{i e^{-\frac{i (b c+a d)}{d}} \sqrt{c+d x} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right) c^2}{8 b}-\frac{\left(-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \sin \left(3 a-\frac{3 b c}{d}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))\right) c^2}{24 \sqrt{3} b \sqrt{\frac{b}{d}}}+\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(a-\frac{b c}{d}\right)-3 d \cos \left(a-\frac{b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(a-\frac{b c}{d}\right)+3 d \sin \left(a-\frac{b c}{d}\right)\right)+2 b \sqrt{c+d x} (3 \cos (a+b x)+2 b x \sin (a+b x))\right) c}{8 b^3}-\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(3 a-\frac{3 b c}{d}\right)-d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(3 a-\frac{3 b c}{d}\right)+d \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} b \sqrt{c+d x} (\cos (3 (a+b x))+2 b x \sin (3 (a+b x)))\right) c}{24 \sqrt{3} b^3}+\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \cos \left(a-\frac{b c}{d}\right)+12 b c d \sin \left(a-\frac{b c}{d}\right)\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \sin \left(a-\frac{b c}{d}\right)-12 b c d \cos \left(a-\frac{b c}{d}\right)\right)+2 \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(4 b^2 x^2-15\right) \sin (a+b x)-2 b (c-5 d x) \cos (a+b x)\right)\right)}{32 b^5}-\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \cos \left(3 a-\frac{3 b c}{d}\right)+12 b c d \sin \left(3 a-\frac{3 b c}{d}\right)\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \sin \left(3 a-\frac{3 b c}{d}\right)-12 b c d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(12 b^2 x^2-5\right) \sin (3 (a+b x))-2 b (c-5 d x) \cos (3 (a+b x))\right)\right)}{288 \sqrt{3} b^5}","-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{144 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{144 b^{7/2}}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{16 b^3}+\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac{(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}",1,"((-1/8*I)*c^2*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) + (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[a - (b*c)/d] + 2*b*c*Sin[a - (b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*b*Sqrt[c + d*x]*(3*Cos[a + b*x] + 2*b*x*Sin[a + b*x])))/(8*b^3) + ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*((4*b^2*c^2 - 15*d^2)*Cos[a - (b*c)/d] + 12*b*c*d*Sin[a - (b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[a - (b*c)/d] + (4*b^2*c^2 - 15*d^2)*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[a + b*x] + d*(-15 + 4*b^2*x^2)*Sin[a + b*x])))/(32*b^5) - (c^2*(-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)]))/(24*Sqrt[3]*b*Sqrt[b/d]) - (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-(d*Cos[3*a - (3*b*c)/d]) + 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*b*Sqrt[c + d*x]*(Cos[3*(a + b*x)] + 2*b*x*Sin[3*(a + b*x)])))/(24*Sqrt[3]*b^3) - ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*((12*b^2*c^2 - 5*d^2)*Cos[3*a - (3*b*c)/d] + 12*b*c*d*Sin[3*a - (3*b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[3*a - (3*b*c)/d] + (12*b^2*c^2 - 5*d^2)*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[3*(a + b*x)] + d*(-5 + 12*b^2*x^2)*Sin[3*(a + b*x)])))/(288*Sqrt[3]*b^5)","C",0
64,1,550,407,14.246312,"\int (c+d x)^{5/2} \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{-1024 b^3 c^2 \sqrt{c+d x} \cos (2 (a+b x))+256 b^3 c^2 \sqrt{c+d x} \cos (4 (a+b x))-1024 b^3 d^2 x^2 \sqrt{c+d x} \cos (2 (a+b x))+256 b^3 d^2 x^2 \sqrt{c+d x} \cos (4 (a+b x))-2048 b^3 c d x \sqrt{c+d x} \cos (2 (a+b x))+512 b^3 c d x \sqrt{c+d x} \cos (4 (a+b x))+1280 b^2 d^2 x \sqrt{c+d x} \sin (2 (a+b x))-160 b^2 d^2 x \sqrt{c+d x} \sin (4 (a+b x))+1280 b^2 c d \sqrt{c+d x} \sin (2 (a+b x))-160 b^2 c d \sqrt{c+d x} \sin (4 (a+b x))+15 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-480 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-15 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+480 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+960 b d^2 \sqrt{c+d x} \cos (2 (a+b x))-60 b d^2 \sqrt{c+d x} \cos (4 (a+b x))}{8192 b^4}","\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}-\frac{15 \sqrt{\pi } d^{5/2} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{256 b^{7/2}}-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}+\frac{15 \sqrt{\pi } d^{5/2} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{256 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac{15 d^2 \sqrt{c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}-\frac{5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}+\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}",1,"(-1024*b^3*c^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 960*b*d^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 2048*b^3*c*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 1024*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 256*b^3*c^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 60*b*d^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 512*b^3*c*d*x*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 256*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 15*Sqrt[b/d]*d^3*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 480*Sqrt[b/d]*d^3*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 15*Sqrt[b/d]*d^3*Sqrt[2*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] + 480*Sqrt[b/d]*d^3*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 1280*b^2*c*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 1280*b^2*d^2*x*Sqrt[c + d*x]*Sin[2*(a + b*x)] - 160*b^2*c*d*Sqrt[c + d*x]*Sin[4*(a + b*x)] - 160*b^2*d^2*x*Sqrt[c + d*x]*Sin[4*(a + b*x)])/(8192*b^4)","A",1
65,1,393,351,3.0758725,"\int (c+d x)^{3/2} \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{3 \sqrt{2 \pi } d \sin \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-48 \sqrt{\pi } d \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+3 \sqrt{2 \pi } d \cos \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-48 \sqrt{\pi } d \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+96 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (2 (a+b x))-12 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (4 (a+b x))-128 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-128 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))+32 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))+32 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))}{1024 b^2 \sqrt{\frac{b}{d}}}","\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{64 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{64 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac{3 d \sqrt{c+d x} \sin (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac{(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}",1,"(-128*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 128*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 32*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 32*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 3*d*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 48*d*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + 3*d*Sqrt[2*Pi]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] - 48*d*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 96*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] - 12*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[4*(a + b*x)])/(1024*b^2*Sqrt[b/d])","A",1
66,1,264,299,0.8110785,"\int \sqrt{c+d x} \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{-\sqrt{2 \pi } \cos \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+8 \sqrt{\pi } \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+\sqrt{2 \pi } \sin \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-8 \sqrt{\pi } \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-16 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))+4 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))}{128 b \sqrt{\frac{b}{d}}}","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}+\frac{\sqrt{\pi } \sqrt{d} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{16 b^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{16 b^{3/2}}-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{8 b}+\frac{\sqrt{c+d x} \cos (4 a+4 b x)}{32 b}",1,"(-16*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 4*Sqrt[b/d]*Sqrt[c + d*x]*Cos[4*(a + b*x)] - Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] + 8*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + Sqrt[2*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] - 8*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d])/(128*b*Sqrt[b/d])","A",1
67,1,264,299,0.2874138,"\int \sqrt{c+d x} \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{-\sqrt{2 \pi } \cos \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+8 \sqrt{\pi } \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+\sqrt{2 \pi } \sin \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-8 \sqrt{\pi } \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-16 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))+4 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))}{128 b \sqrt{\frac{b}{d}}}","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}+\frac{\sqrt{\pi } \sqrt{d} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{16 b^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{16 b^{3/2}}-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{8 b}+\frac{\sqrt{c+d x} \cos (4 a+4 b x)}{32 b}",1,"(-16*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 4*Sqrt[b/d]*Sqrt[c + d*x]*Cos[4*(a + b*x)] - Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] + 8*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + Sqrt[2*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] - 8*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d])/(128*b*Sqrt[b/d])","A",1
68,1,393,351,2.3671327,"\int (c+d x)^{3/2} \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{3 \sqrt{2 \pi } d \sin \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-48 \sqrt{\pi } d \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+3 \sqrt{2 \pi } d \cos \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-48 \sqrt{\pi } d \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+96 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (2 (a+b x))-12 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (4 (a+b x))-128 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-128 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))+32 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))+32 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))}{1024 b^2 \sqrt{\frac{b}{d}}}","\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{64 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{64 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac{3 d \sqrt{c+d x} \sin (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac{(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}",1,"(-128*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 128*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 32*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 32*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 3*d*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 48*d*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + 3*d*Sqrt[2*Pi]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] - 48*d*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 96*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] - 12*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[4*(a + b*x)])/(1024*b^2*Sqrt[b/d])","A",1
69,1,550,407,9.9662156,"\int (c+d x)^{5/2} \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{-1024 b^3 c^2 \sqrt{c+d x} \cos (2 (a+b x))+256 b^3 c^2 \sqrt{c+d x} \cos (4 (a+b x))-1024 b^3 d^2 x^2 \sqrt{c+d x} \cos (2 (a+b x))+256 b^3 d^2 x^2 \sqrt{c+d x} \cos (4 (a+b x))-2048 b^3 c d x \sqrt{c+d x} \cos (2 (a+b x))+512 b^3 c d x \sqrt{c+d x} \cos (4 (a+b x))+1280 b^2 d^2 x \sqrt{c+d x} \sin (2 (a+b x))-160 b^2 d^2 x \sqrt{c+d x} \sin (4 (a+b x))+1280 b^2 c d \sqrt{c+d x} \sin (2 (a+b x))-160 b^2 c d \sqrt{c+d x} \sin (4 (a+b x))+15 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-480 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-15 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+480 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+960 b d^2 \sqrt{c+d x} \cos (2 (a+b x))-60 b d^2 \sqrt{c+d x} \cos (4 (a+b x))}{8192 b^4}","\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}-\frac{15 \sqrt{\pi } d^{5/2} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{256 b^{7/2}}-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}+\frac{15 \sqrt{\pi } d^{5/2} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{256 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{128 b^3}-\frac{15 d^2 \sqrt{c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}-\frac{5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}+\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}",1,"(-1024*b^3*c^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 960*b*d^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 2048*b^3*c*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 1024*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 256*b^3*c^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 60*b*d^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 512*b^3*c*d*x*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 256*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 15*Sqrt[b/d]*d^3*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 480*Sqrt[b/d]*d^3*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 15*Sqrt[b/d]*d^3*Sqrt[2*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] + 480*Sqrt[b/d]*d^3*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 1280*b^2*c*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 1280*b^2*d^2*x*Sqrt[c + d*x]*Sin[2*(a + b*x)] - 160*b^2*c*d*Sqrt[c + d*x]*Sin[4*(a + b*x)] - 160*b^2*d^2*x*Sqrt[c + d*x]*Sin[4*(a + b*x)])/(8192*b^4)","A",1
70,1,250,267,0.455077,"\int (c+d x)^m \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]^2*Sin[a + b*x],x]","\frac{e^{-\frac{3 i (a d+b c)}{d}} (c+d x)^m \left(3 e^{\frac{2 i (a d+b c)}{d}} \left(-e^{2 i a} \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)-e^{\frac{2 i b c}{d}} \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)\right)-3^{-m} \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(e^{6 i a} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{3 i b (c+d x)}{d}\right)+e^{\frac{6 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{3 i b (c+d x)}{d}\right)\right)\right)}{24 b}","-\frac{e^{i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)}{8 b}-\frac{3^{-m-1} e^{3 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{3 i b (c+d x)}{d}\right)}{8 b}-\frac{e^{-i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)}{8 b}-\frac{3^{-m-1} e^{-3 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{3 i b (c+d x)}{d}\right)}{8 b}",1,"((c + d*x)^m*(3*E^(((2*I)*(b*c + a*d))/d)*(-((E^((2*I)*a)*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^m) - (E^(((2*I)*b*c)/d)*Gamma[1 + m, (I*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^m) - (E^((6*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d] + E^(((6*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((3*I)*b*(c + d*x))/d])/(3^m*((b^2*(c + d*x)^2)/d^2)^m)))/(24*b*E^(((3*I)*(b*c + a*d))/d))","A",1
71,1,150,205,1.5504337,"\int (c+d x)^4 \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]^2*Sin[a + b*x],x]","-\frac{-24 b d (c+d x) \sin (a+b x) \left(\cos (2 (a+b x)) \left(3 b^2 (c+d x)^2-2 d^2\right)+15 b^2 (c+d x)^2-82 d^2\right)+81 \cos (a+b x) \left(b^4 (c+d x)^4-12 b^2 d^2 (c+d x)^2+24 d^4\right)+\cos (3 (a+b x)) \left(27 b^4 (c+d x)^4-36 b^2 d^2 (c+d x)^2+8 d^4\right)}{324 b^5}","-\frac{8 d^4 \cos ^3(a+b x)}{81 b^5}-\frac{160 d^4 \cos (a+b x)}{27 b^5}-\frac{160 d^3 (c+d x) \sin (a+b x)}{27 b^4}-\frac{8 d^3 (c+d x) \sin (a+b x) \cos ^2(a+b x)}{27 b^4}+\frac{4 d^2 (c+d x)^2 \cos ^3(a+b x)}{9 b^3}+\frac{8 d^2 (c+d x)^2 \cos (a+b x)}{3 b^3}+\frac{8 d (c+d x)^3 \sin (a+b x)}{9 b^2}+\frac{4 d (c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{9 b^2}-\frac{(c+d x)^4 \cos ^3(a+b x)}{3 b}",1,"-1/324*(81*(24*d^4 - 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Cos[a + b*x] + (8*d^4 - 36*b^2*d^2*(c + d*x)^2 + 27*b^4*(c + d*x)^4)*Cos[3*(a + b*x)] - 24*b*d*(c + d*x)*(-82*d^2 + 15*b^2*(c + d*x)^2 + (-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[a + b*x])/b^5","A",1
72,1,127,151,0.8903174,"\int (c+d x)^3 \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]^2*Sin[a + b*x],x]","\frac{-27 b (c+d x) \cos (a+b x) \left(b^2 (c+d x)^2-6 d^2\right)-3 b (c+d x) \cos (3 (a+b x)) \left(3 b^2 (c+d x)^2-2 d^2\right)+2 d \sin (a+b x) \left(\cos (2 (a+b x)) \left(9 b^2 (c+d x)^2-2 d^2\right)+45 b^2 (c+d x)^2-82 d^2\right)}{108 b^4}","\frac{2 d^3 \sin ^3(a+b x)}{27 b^4}-\frac{14 d^3 \sin (a+b x)}{9 b^4}+\frac{2 d^2 (c+d x) \cos ^3(a+b x)}{9 b^3}+\frac{4 d^2 (c+d x) \cos (a+b x)}{3 b^3}+\frac{2 d (c+d x)^2 \sin (a+b x)}{3 b^2}+\frac{d (c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b^2}-\frac{(c+d x)^3 \cos ^3(a+b x)}{3 b}",1,"(-27*b*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] - 3*b*(c + d*x)*(-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] + 2*d*(-82*d^2 + 45*b^2*(c + d*x)^2 + (-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[a + b*x])/(108*b^4)","A",1
73,1,86,103,0.4901808,"\int (c+d x)^2 \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]^2*Sin[a + b*x],x]","-\frac{27 \cos (a+b x) \left(b^2 (c+d x)^2-2 d^2\right)+\cos (3 (a+b x)) \left(9 b^2 (c+d x)^2-2 d^2\right)-6 b d (c+d x) (9 \sin (a+b x)+\sin (3 (a+b x)))}{108 b^3}","\frac{2 d^2 \cos ^3(a+b x)}{27 b^3}+\frac{4 d^2 \cos (a+b x)}{9 b^3}+\frac{4 d (c+d x) \sin (a+b x)}{9 b^2}+\frac{2 d (c+d x) \sin (a+b x) \cos ^2(a+b x)}{9 b^2}-\frac{(c+d x)^2 \cos ^3(a+b x)}{3 b}",1,"-1/108*(27*(-2*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + (-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] - 6*b*d*(c + d*x)*(9*Sin[a + b*x] + Sin[3*(a + b*x)]))/b^3","A",1
74,1,71,51,0.1454665,"\int (c+d x) \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]^2*Sin[a + b*x],x]","\frac{d (\sin (a+b x)-b x \cos (a+b x))}{4 b^2}+\frac{d (\sin (3 (a+b x))-3 b x \cos (3 (a+b x)))}{36 b^2}-\frac{c \cos ^3(a+b x)}{3 b}","-\frac{d \sin ^3(a+b x)}{9 b^2}+\frac{d \sin (a+b x)}{3 b^2}-\frac{(c+d x) \cos ^3(a+b x)}{3 b}",1,"-1/3*(c*Cos[a + b*x]^3)/b + (d*(-(b*x*Cos[a + b*x]) + Sin[a + b*x]))/(4*b^2) + (d*(-3*b*x*Cos[3*(a + b*x)] + Sin[3*(a + b*x)]))/(36*b^2)","A",1
75,1,100,121,0.2964467,"\int \frac{\cos ^2(a+b x) \sin (a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x),x]","\frac{\sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)+\sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+\cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+\cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)}{4 d}","\frac{\sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{4 d}+\frac{\sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{4 d}+\frac{\cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{4 d}+\frac{\cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{4 d}",1,"(CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] + CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/(4*d)","A",1
76,1,139,168,1.0872668,"\int \frac{\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x)^2,x]","-\frac{-b \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)-3 b \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)+b \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+3 b \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)+\frac{d \sin (a+b x)}{c+d x}+\frac{d \sin (3 (a+b x))}{c+d x}}{4 d^2}","\frac{b \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{4 d^2}+\frac{3 b \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{4 d^2}-\frac{b \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{4 d^2}-\frac{3 b \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{4 d^2}-\frac{\sin (a+b x)}{4 d (c+d x)}-\frac{\sin (3 a+3 b x)}{4 d (c+d x)}",1,"-1/4*(-(b*Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)]) - 3*b*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] + (d*Sin[a + b*x])/(c + d*x) + (d*Sin[3*(a + b*x)])/(c + d*x) + b*Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + 3*b*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/d^2","A",1
77,1,181,221,2.5583592,"\int \frac{\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^3} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x)^3,x]","-\frac{9 b^2 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)+b^2 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+b^2 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+9 b^2 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)+\frac{d (b (c+d x) \cos (a+b x)+d \sin (a+b x))}{(c+d x)^2}+\frac{d (3 b (c+d x) \cos (3 (a+b x))+d \sin (3 (a+b x)))}{(c+d x)^2}}{8 d^3}","-\frac{9 b^2 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{8 d^3}-\frac{b^2 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{8 d^3}-\frac{b^2 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{8 d^3}-\frac{9 b^2 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{8 d^3}-\frac{b \cos (a+b x)}{8 d^2 (c+d x)}-\frac{3 b \cos (3 a+3 b x)}{8 d^2 (c+d x)}-\frac{\sin (a+b x)}{8 d (c+d x)^2}-\frac{\sin (3 a+3 b x)}{8 d (c+d x)^2}",1,"-1/8*(9*b^2*CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] + b^2*CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + (d*(b*(c + d*x)*Cos[a + b*x] + d*Sin[a + b*x]))/(c + d*x)^2 + (d*(3*b*(c + d*x)*Cos[3*(a + b*x)] + d*Sin[3*(a + b*x)]))/(c + d*x)^2 + b^2*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + 9*b^2*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/d^3","A",1
78,1,300,270,1.8234819,"\int \frac{\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^4} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x)^4,x]","-\frac{b^3 (c+d x)^3 \left(\cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)-\sin \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)\right)+27 b^3 (c+d x)^3 \left(\cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)-\sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)\right)+d \cos (b x) \left(b d \cos (a) (c+d x)-\sin (a) \left(b^2 (c+d x)^2-2 d^2\right)\right)+d \cos (3 b x) \left(3 b d \cos (3 a) (c+d x)-\sin (3 a) \left(9 b^2 (c+d x)^2-2 d^2\right)\right)-d \sin (b x) \left(\cos (a) \left(b^2 (c+d x)^2-2 d^2\right)+b d \sin (a) (c+d x)\right)-d \sin (3 b x) \left(\cos (3 a) \left(9 b^2 (c+d x)^2-2 d^2\right)+3 b d \sin (3 a) (c+d x)\right)}{24 d^4 (c+d x)^3}","-\frac{b^3 \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{24 d^4}-\frac{9 b^3 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{8 d^4}+\frac{b^3 \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{24 d^4}+\frac{9 b^3 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{8 d^4}+\frac{b^2 \sin (a+b x)}{24 d^3 (c+d x)}+\frac{3 b^2 \sin (3 a+3 b x)}{8 d^3 (c+d x)}-\frac{b \cos (a+b x)}{24 d^2 (c+d x)^2}-\frac{b \cos (3 a+3 b x)}{8 d^2 (c+d x)^2}-\frac{\sin (a+b x)}{12 d (c+d x)^3}-\frac{\sin (3 a+3 b x)}{12 d (c+d x)^3}",1,"-1/24*(d*Cos[b*x]*(b*d*(c + d*x)*Cos[a] - (-2*d^2 + b^2*(c + d*x)^2)*Sin[a]) + d*Cos[3*b*x]*(3*b*d*(c + d*x)*Cos[3*a] - (-2*d^2 + 9*b^2*(c + d*x)^2)*Sin[3*a]) - d*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] + b*d*(c + d*x)*Sin[a])*Sin[b*x] - d*((-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*a] + 3*b*d*(c + d*x)*Sin[3*a])*Sin[3*b*x] + b^3*(c + d*x)^3*(Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] - Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)]) + 27*b^3*(c + d*x)^3*(Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] - Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d]))/(d^4*(c + d*x)^3)","A",1
79,1,213,162,1.038301,"\int (c+d x)^m \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]^2*Sin[a + b*x]^2,x]","\frac{4^{-m-3} (c+d x)^m \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(-i d (m+1) \left(-\frac{i b (c+d x)}{d}\right)^m \left(\cos \left(4 a-\frac{4 b c}{d}\right)-i \sin \left(4 a-\frac{4 b c}{d}\right)\right) \Gamma \left(m+1,\frac{4 i b (c+d x)}{d}\right)+i d (m+1) \left(\frac{i b (c+d x)}{d}\right)^m \left(\cos \left(4 a-\frac{4 b c}{d}\right)+i \sin \left(4 a-\frac{4 b c}{d}\right)\right) \Gamma \left(m+1,-\frac{4 i b (c+d x)}{d}\right)+b 2^{2 m+3} (c+d x) \left(\frac{b^2 (c+d x)^2}{d^2}\right)^m\right)}{b d (m+1)}","\frac{i 2^{-2 (m+3)} e^{4 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{4 i b (c+d x)}{d}\right)}{b}-\frac{i 2^{-2 (m+3)} e^{-4 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{4 i b (c+d x)}{d}\right)}{b}+\frac{(c+d x)^{m+1}}{8 d (m+1)}",1,"(4^(-3 - m)*(c + d*x)^m*(2^(3 + 2*m)*b*(c + d*x)*((b^2*(c + d*x)^2)/d^2)^m - I*d*(1 + m)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((4*I)*b*(c + d*x))/d]*(Cos[4*a - (4*b*c)/d] - I*Sin[4*a - (4*b*c)/d]) + I*d*(1 + m)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-4*I)*b*(c + d*x))/d]*(Cos[4*a - (4*b*c)/d] + I*Sin[4*a - (4*b*c)/d])))/(b*d*(1 + m)*((b^2*(c + d*x)^2)/d^2)^m)","A",1
80,1,132,131,1.2696583,"\int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]^2*Sin[a + b*x]^2,x]","\frac{20 b d (c+d x) \cos (4 (a+b x)) \left(3 d^2-8 b^2 (c+d x)^2\right)-5 \sin (4 (a+b x)) \left(32 b^4 (c+d x)^4-24 b^2 d^2 (c+d x)^2+3 d^4\right)+128 b^5 x \left(5 c^4+10 c^3 d x+10 c^2 d^2 x^2+5 c d^3 x^3+d^4 x^4\right)}{5120 b^5}","-\frac{3 d^4 \sin (4 a+4 b x)}{1024 b^5}+\frac{3 d^3 (c+d x) \cos (4 a+4 b x)}{256 b^4}+\frac{3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}-\frac{d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}-\frac{(c+d x)^4 \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^5}{40 d}",1,"(128*b^5*x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4) + 20*b*d*(c + d*x)*(3*d^2 - 8*b^2*(c + d*x)^2)*Cos[4*(a + b*x)] - 5*(3*d^4 - 24*b^2*d^2*(c + d*x)^2 + 32*b^4*(c + d*x)^4)*Sin[4*(a + b*x)])/(5120*b^5)","A",1
81,1,106,105,0.6551861,"\int (c+d x)^3 \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]^2*Sin[a + b*x]^2,x]","\frac{-4 b (c+d x) \sin (4 (a+b x)) \left(8 b^2 (c+d x)^2-3 d^2\right)-3 d \cos (4 (a+b x)) \left(8 b^2 (c+d x)^2-d^2\right)+32 b^4 x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)}{1024 b^4}","\frac{3 d^3 \cos (4 a+4 b x)}{1024 b^4}+\frac{3 d^2 (c+d x) \sin (4 a+4 b x)}{256 b^3}-\frac{3 d (c+d x)^2 \cos (4 a+4 b x)}{128 b^2}-\frac{(c+d x)^3 \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^4}{32 d}",1,"(32*b^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) - 3*d*(-d^2 + 8*b^2*(c + d*x)^2)*Cos[4*(a + b*x)] - 4*b*(c + d*x)*(-3*d^2 + 8*b^2*(c + d*x)^2)*Sin[4*(a + b*x)])/(1024*b^4)","A",1
82,1,77,79,0.4245387,"\int (c+d x)^2 \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]^2*Sin[a + b*x]^2,x]","\frac{-3 \sin (4 (a+b x)) \left(8 b^2 (c+d x)^2-d^2\right)-12 b d (c+d x) \cos (4 (a+b x))+32 b^3 x \left(3 c^2+3 c d x+d^2 x^2\right)}{768 b^3}","\frac{d^2 \sin (4 a+4 b x)}{256 b^3}-\frac{d (c+d x) \cos (4 a+4 b x)}{64 b^2}-\frac{(c+d x)^2 \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^3}{24 d}",1,"(32*b^3*x*(3*c^2 + 3*c*d*x + d^2*x^2) - 12*b*d*(c + d*x)*Cos[4*(a + b*x)] - 3*(-d^2 + 8*b^2*(c + d*x)^2)*Sin[4*(a + b*x)])/(768*b^3)","A",1
83,1,54,53,0.2860968,"\int (c+d x) \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]^2*Sin[a + b*x]^2,x]","-\frac{8 (a+b x) (a d-2 b c-b d x)+4 b (c+d x) \sin (4 (a+b x))+d \cos (4 (a+b x))}{128 b^2}","-\frac{d \cos (4 a+4 b x)}{128 b^2}-\frac{(c+d x) \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^2}{16 d}",1,"-1/128*(8*(a + b*x)*(-2*b*c + a*d - b*d*x) + d*Cos[4*(a + b*x)] + 4*b*(c + d*x)*Sin[4*(a + b*x)])/b^2","A",1
84,1,65,78,0.15952,"\int \frac{\cos ^2(a+b x) \sin ^2(a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x),x]","\frac{-\cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)+\sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)+\log (c+d x)}{8 d}","-\frac{\cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{8 d}+\frac{\sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{8 d}+\frac{\log (c+d x)}{8 d}",1,"(-(Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*(c + d*x))/d]) + Log[c + d*x] + Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/(8*d)","A",1
85,1,81,104,0.4320765,"\int \frac{\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^2,x]","\frac{4 b \sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)+4 b \cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)+\frac{d (\cos (4 (a+b x))-1)}{c+d x}}{8 d^2}","\frac{b \sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{2 d^2}+\frac{b \cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{2 d^2}+\frac{\cos (4 a+4 b x)}{8 d (c+d x)}-\frac{1}{8 d (c+d x)}",1,"((d*(-1 + Cos[4*(a + b*x)]))/(c + d*x) + 4*b*CosIntegral[(4*b*(c + d*x))/d]*Sin[4*a - (4*b*c)/d] + 4*b*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/(8*d^2)","A",1
86,1,105,127,0.8463085,"\int \frac{\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^3,x]","\frac{16 b^2 \cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)-16 b^2 \sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)+\frac{d (-4 b (c+d x) \sin (4 (a+b x))+d \cos (4 (a+b x))-d)}{(c+d x)^2}}{16 d^3}","\frac{b^2 \cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{d^3}-\frac{b^2 \sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{d^3}-\frac{b \sin (4 a+4 b x)}{4 d^2 (c+d x)}+\frac{\cos (4 a+4 b x)}{16 d (c+d x)^2}-\frac{1}{16 d (c+d x)^2}",1,"(16*b^2*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*(c + d*x))/d] + (d*(-d + d*Cos[4*(a + b*x)] - 4*b*(c + d*x)*Sin[4*(a + b*x)]))/(c + d*x)^2 - 16*b^2*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/(16*d^3)","A",1
87,1,123,158,1.6514804,"\int \frac{\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^4,x]","-\frac{32 b^3 \sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)+32 b^3 \cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)+\frac{d \left(\cos (4 (a+b x)) \left(8 b^2 (c+d x)^2-d^2\right)+d (2 b (c+d x) \sin (4 (a+b x))+d)\right)}{(c+d x)^3}}{24 d^4}","-\frac{4 b^3 \sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{3 d^4}-\frac{4 b^3 \cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{3 d^4}-\frac{b^2 \cos (4 a+4 b x)}{3 d^3 (c+d x)}-\frac{b \sin (4 a+4 b x)}{12 d^2 (c+d x)^2}+\frac{\cos (4 a+4 b x)}{24 d (c+d x)^3}-\frac{1}{24 d (c+d x)^3}",1,"-1/24*(32*b^3*CosIntegral[(4*b*(c + d*x))/d]*Sin[4*a - (4*b*c)/d] + (d*((-d^2 + 8*b^2*(c + d*x)^2)*Cos[4*(a + b*x)] + d*(d + 2*b*(c + d*x)*Sin[4*(a + b*x)])))/(c + d*x)^3 + 32*b^3*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/d^4","A",1
88,1,376,407,0.6282521,"\int (c+d x)^m \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\frac{e^{-\frac{5 i (a d+b c)}{d}} (c+d x)^m \left(-5\ 3^{-m} e^{\frac{2 i (a d+b c)}{d}} \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(e^{6 i a} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{3 i b (c+d x)}{d}\right)+e^{\frac{6 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{3 i b (c+d x)}{d}\right)\right)+3\ 5^{-m} \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(e^{10 i a} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{5 i b (c+d x)}{d}\right)+e^{\frac{10 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{5 i b (c+d x)}{d}\right)\right)+30 e^{\frac{4 i (a d+b c)}{d}} \left(-e^{2 i a} \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)-e^{\frac{2 i b c}{d}} \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)\right)\right)}{480 b}","-\frac{e^{i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)}{16 b}-\frac{3^{-m-1} e^{3 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{3 i b (c+d x)}{d}\right)}{32 b}+\frac{5^{-m-1} e^{5 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{5 i b (c+d x)}{d}\right)}{32 b}-\frac{e^{-i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)}{16 b}-\frac{3^{-m-1} e^{-3 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{3 i b (c+d x)}{d}\right)}{32 b}+\frac{5^{-m-1} e^{-5 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{5 i b (c+d x)}{d}\right)}{32 b}",1,"((c + d*x)^m*(30*E^(((4*I)*(b*c + a*d))/d)*(-((E^((2*I)*a)*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^m) - (E^(((2*I)*b*c)/d)*Gamma[1 + m, (I*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^m) - (5*E^(((2*I)*(b*c + a*d))/d)*(E^((6*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d] + E^(((6*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((3*I)*b*(c + d*x))/d]))/(3^m*((b^2*(c + d*x)^2)/d^2)^m) + (3*(E^((10*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-5*I)*b*(c + d*x))/d] + E^(((10*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((5*I)*b*(c + d*x))/d]))/(5^m*((b^2*(c + d*x)^2)/d^2)^m)))/(480*b*E^(((5*I)*(b*c + a*d))/d))","A",1
89,1,238,330,3.1720138,"\int (c+d x)^4 \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\frac{120 b d (c+d x) \sin (a+b x) \left(16 \cos (2 (a+b x)) \left(75 b^2 (c+d x)^2-68 d^2\right)-27 \cos (4 (a+b x)) \left(25 b^2 (c+d x)^2-6 d^2\right)+17475 b^2 c^2+34950 b^2 c d x+17475 b^2 d^2 x^2-101794 d^2\right)-506250 \cos (a+b x) \left(b^4 (c+d x)^4-12 b^2 d^2 (c+d x)^2+24 d^4\right)-3125 \cos (3 (a+b x)) \left(27 b^4 (c+d x)^4-36 b^2 d^2 (c+d x)^2+8 d^4\right)+81 \cos (5 (a+b x)) \left(625 b^4 (c+d x)^4-300 b^2 d^2 (c+d x)^2+24 d^4\right)}{4050000 b^5}","-\frac{3 d^4 \cos (a+b x)}{b^5}-\frac{d^4 \cos (3 a+3 b x)}{162 b^5}+\frac{3 d^4 \cos (5 a+5 b x)}{6250 b^5}-\frac{3 d^3 (c+d x) \sin (a+b x)}{b^4}-\frac{d^3 (c+d x) \sin (3 a+3 b x)}{54 b^4}+\frac{3 d^3 (c+d x) \sin (5 a+5 b x)}{1250 b^4}+\frac{3 d^2 (c+d x)^2 \cos (a+b x)}{2 b^3}+\frac{d^2 (c+d x)^2 \cos (3 a+3 b x)}{36 b^3}-\frac{3 d^2 (c+d x)^2 \cos (5 a+5 b x)}{500 b^3}+\frac{d (c+d x)^3 \sin (a+b x)}{2 b^2}+\frac{d (c+d x)^3 \sin (3 a+3 b x)}{36 b^2}-\frac{d (c+d x)^3 \sin (5 a+5 b x)}{100 b^2}-\frac{(c+d x)^4 \cos (a+b x)}{8 b}-\frac{(c+d x)^4 \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^4 \cos (5 a+5 b x)}{80 b}",1,"(-506250*(24*d^4 - 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Cos[a + b*x] - 3125*(8*d^4 - 36*b^2*d^2*(c + d*x)^2 + 27*b^4*(c + d*x)^4)*Cos[3*(a + b*x)] + 81*(24*d^4 - 300*b^2*d^2*(c + d*x)^2 + 625*b^4*(c + d*x)^4)*Cos[5*(a + b*x)] + 120*b*d*(c + d*x)*(17475*b^2*c^2 - 101794*d^2 + 34950*b^2*c*d*x + 17475*b^2*d^2*x^2 + 16*(-68*d^2 + 75*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] - 27*(-6*d^2 + 25*b^2*(c + d*x)^2)*Cos[4*(a + b*x)])*Sin[a + b*x])/(4050000*b^5)","A",1
90,1,369,259,1.4527962,"\int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\frac{3375 b^3 c^3 \cos (5 (a+b x))+10125 b^3 c^2 d x \cos (5 (a+b x))+10125 b^3 c d^2 x^2 \cos (5 (a+b x))+3375 b^3 d^3 x^3 \cos (5 (a+b x))+101250 b^2 c^2 d \sin (a+b x)+5625 b^2 c^2 d \sin (3 (a+b x))-2025 b^2 c^2 d \sin (5 (a+b x))+202500 b^2 c d^2 x \sin (a+b x)+11250 b^2 c d^2 x \sin (3 (a+b x))-4050 b^2 c d^2 x \sin (5 (a+b x))-33750 b (c+d x) \cos (a+b x) \left(b^2 (c+d x)^2-6 d^2\right)-1875 b (c+d x) \cos (3 (a+b x)) \left(3 b^2 (c+d x)^2-2 d^2\right)+101250 b^2 d^3 x^2 \sin (a+b x)+5625 b^2 d^3 x^2 \sin (3 (a+b x))-2025 b^2 d^3 x^2 \sin (5 (a+b x))-810 b c d^2 \cos (5 (a+b x))-202500 d^3 \sin (a+b x)-1250 d^3 \sin (3 (a+b x))+162 d^3 \sin (5 (a+b x))-810 b d^3 x \cos (5 (a+b x))}{270000 b^4}","-\frac{3 d^3 \sin (a+b x)}{4 b^4}-\frac{d^3 \sin (3 a+3 b x)}{216 b^4}+\frac{3 d^3 \sin (5 a+5 b x)}{5000 b^4}+\frac{3 d^2 (c+d x) \cos (a+b x)}{4 b^3}+\frac{d^2 (c+d x) \cos (3 a+3 b x)}{72 b^3}-\frac{3 d^2 (c+d x) \cos (5 a+5 b x)}{1000 b^3}+\frac{3 d (c+d x)^2 \sin (a+b x)}{8 b^2}+\frac{d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}-\frac{3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2}-\frac{(c+d x)^3 \cos (a+b x)}{8 b}-\frac{(c+d x)^3 \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^3 \cos (5 a+5 b x)}{80 b}",1,"(-33750*b*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] - 1875*b*(c + d*x)*(-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] + 3375*b^3*c^3*Cos[5*(a + b*x)] - 810*b*c*d^2*Cos[5*(a + b*x)] + 10125*b^3*c^2*d*x*Cos[5*(a + b*x)] - 810*b*d^3*x*Cos[5*(a + b*x)] + 10125*b^3*c*d^2*x^2*Cos[5*(a + b*x)] + 3375*b^3*d^3*x^3*Cos[5*(a + b*x)] + 101250*b^2*c^2*d*Sin[a + b*x] - 202500*d^3*Sin[a + b*x] + 202500*b^2*c*d^2*x*Sin[a + b*x] + 101250*b^2*d^3*x^2*Sin[a + b*x] + 5625*b^2*c^2*d*Sin[3*(a + b*x)] - 1250*d^3*Sin[3*(a + b*x)] + 11250*b^2*c*d^2*x*Sin[3*(a + b*x)] + 5625*b^2*d^3*x^2*Sin[3*(a + b*x)] - 2025*b^2*c^2*d*Sin[5*(a + b*x)] + 162*d^3*Sin[5*(a + b*x)] - 4050*b^2*c*d^2*x*Sin[5*(a + b*x)] - 2025*b^2*d^3*x^2*Sin[5*(a + b*x)])/(270000*b^4)","A",1
91,1,127,184,0.8794798,"\int (c+d x)^2 \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\frac{-6750 \cos (a+b x) \left(b^2 (c+d x)^2-2 d^2\right)-125 \cos (3 (a+b x)) \left(9 b^2 (c+d x)^2-2 d^2\right)+27 \cos (5 (a+b x)) \left(25 b^2 (c+d x)^2-2 d^2\right)+30 b d (c+d x) (450 \sin (a+b x)+25 \sin (3 (a+b x))-9 \sin (5 (a+b x)))}{54000 b^3}","\frac{d^2 \cos (a+b x)}{4 b^3}+\frac{d^2 \cos (3 a+3 b x)}{216 b^3}-\frac{d^2 \cos (5 a+5 b x)}{1000 b^3}+\frac{d (c+d x) \sin (a+b x)}{4 b^2}+\frac{d (c+d x) \sin (3 a+3 b x)}{72 b^2}-\frac{d (c+d x) \sin (5 a+5 b x)}{200 b^2}-\frac{(c+d x)^2 \cos (a+b x)}{8 b}-\frac{(c+d x)^2 \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^2 \cos (5 a+5 b x)}{80 b}",1,"(-6750*(-2*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] - 125*(-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] + 27*(-2*d^2 + 25*b^2*(c + d*x)^2)*Cos[5*(a + b*x)] + 30*b*d*(c + d*x)*(450*Sin[a + b*x] + 25*Sin[3*(a + b*x)] - 9*Sin[5*(a + b*x)]))/(54000*b^3)","A",1
92,1,94,109,0.2893004,"\int (c+d x) \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\frac{-450 b (c+d x) \cos (a+b x)-75 b (c+d x) \cos (3 (a+b x))+45 b c \cos (5 (a+b x))+450 d \sin (a+b x)+25 d \sin (3 (a+b x))-9 d \sin (5 (a+b x))+45 b d x \cos (5 (a+b x))}{3600 b^2}","\frac{d \sin (a+b x)}{8 b^2}+\frac{d \sin (3 a+3 b x)}{144 b^2}-\frac{d \sin (5 a+5 b x)}{400 b^2}-\frac{(c+d x) \cos (a+b x)}{8 b}-\frac{(c+d x) \cos (3 a+3 b x)}{48 b}+\frac{(c+d x) \cos (5 a+5 b x)}{80 b}",1,"(-450*b*(c + d*x)*Cos[a + b*x] - 75*b*(c + d*x)*Cos[3*(a + b*x)] + 45*b*c*Cos[5*(a + b*x)] + 45*b*d*x*Cos[5*(a + b*x)] + 450*d*Sin[a + b*x] + 25*d*Sin[3*(a + b*x)] - 9*d*Sin[5*(a + b*x)])/(3600*b^2)","A",1
93,1,154,185,0.4897939,"\int \frac{\cos ^2(a+b x) \sin ^3(a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^3)/(c + d*x),x]","\frac{\sin \left(5 a-\frac{5 b c}{d}\right) \left(-\text{Ci}\left(\frac{5 b (c+d x)}{d}\right)\right)+\sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)+2 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+2 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+\cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)-\cos \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b (c+d x)}{d}\right)}{16 d}","-\frac{\sin \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b c}{d}+5 b x\right)}{16 d}+\frac{\sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{16 d}+\frac{\sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{8 d}+\frac{\cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{8 d}+\frac{\cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{16 d}-\frac{\cos \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b c}{d}+5 b x\right)}{16 d}",1,"(-(CosIntegral[(5*b*(c + d*x))/d]*Sin[5*a - (5*b*c)/d]) + CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] + 2*CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + 2*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d] - Cos[5*a - (5*b*c)/d]*SinIntegral[(5*b*(c + d*x))/d])/(16*d)","A",1
94,1,213,257,1.4681062,"\int \frac{\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^3)/(c + d*x)^2,x]","\frac{2 b \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+3 b \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)-5 b \cos \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b (c+d x)}{d}\right)-2 b \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)-3 b \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)+5 b \sin \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b (c+d x)}{d}\right)-\frac{2 d \sin (a+b x)}{c+d x}-\frac{d \sin (3 (a+b x))}{c+d x}+\frac{d \sin (5 (a+b x))}{c+d x}}{16 d^2}","\frac{b \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{8 d^2}+\frac{3 b \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{16 d^2}-\frac{5 b \cos \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b c}{d}+5 b x\right)}{16 d^2}-\frac{b \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{8 d^2}-\frac{3 b \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{16 d^2}+\frac{5 b \sin \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b c}{d}+5 b x\right)}{16 d^2}-\frac{\sin (a+b x)}{8 d (c+d x)}-\frac{\sin (3 a+3 b x)}{16 d (c+d x)}+\frac{\sin (5 a+5 b x)}{16 d (c+d x)}",1,"(2*b*Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] + 3*b*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] - 5*b*Cos[5*a - (5*b*c)/d]*CosIntegral[(5*b*(c + d*x))/d] - (2*d*Sin[a + b*x])/(c + d*x) - (d*Sin[3*(a + b*x)])/(c + d*x) + (d*Sin[5*(a + b*x)])/(c + d*x) - 2*b*Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)] - 3*b*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d] + 5*b*Sin[5*a - (5*b*c)/d]*SinIntegral[(5*b*(c + d*x))/d])/(16*d^2)","A",1
95,1,279,338,3.9049271,"\int \frac{\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^3} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^3)/(c + d*x)^3,x]","\frac{-2 \left(b^2 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+b^2 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+\frac{d (b (c+d x) \cos (a+b x)+d \sin (a+b x))}{(c+d x)^2}\right)+25 b^2 \sin \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b (c+d x)}{d}\right)-9 b^2 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)-9 b^2 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)+25 b^2 \cos \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b (c+d x)}{d}\right)-\frac{d (3 b (c+d x) \cos (3 (a+b x))+d \sin (3 (a+b x)))}{(c+d x)^2}+\frac{d (5 b (c+d x) \cos (5 (a+b x))+d \sin (5 (a+b x)))}{(c+d x)^2}}{32 d^3}","\frac{25 b^2 \sin \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b c}{d}+5 b x\right)}{32 d^3}-\frac{9 b^2 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{32 d^3}-\frac{b^2 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{16 d^3}-\frac{b^2 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{16 d^3}-\frac{9 b^2 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{32 d^3}+\frac{25 b^2 \cos \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b c}{d}+5 b x\right)}{32 d^3}-\frac{b \cos (a+b x)}{16 d^2 (c+d x)}-\frac{3 b \cos (3 a+3 b x)}{32 d^2 (c+d x)}+\frac{5 b \cos (5 a+5 b x)}{32 d^2 (c+d x)}-\frac{\sin (a+b x)}{16 d (c+d x)^2}-\frac{\sin (3 a+3 b x)}{32 d (c+d x)^2}+\frac{\sin (5 a+5 b x)}{32 d (c+d x)^2}",1,"(25*b^2*CosIntegral[(5*b*(c + d*x))/d]*Sin[5*a - (5*b*c)/d] - 9*b^2*CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] - (d*(3*b*(c + d*x)*Cos[3*(a + b*x)] + d*Sin[3*(a + b*x)]))/(c + d*x)^2 + (d*(5*b*(c + d*x)*Cos[5*(a + b*x)] + d*Sin[5*(a + b*x)]))/(c + d*x)^2 - 2*(b^2*CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + (d*(b*(c + d*x)*Cos[a + b*x] + d*Sin[a + b*x]))/(c + d*x)^2 + b^2*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)]) - 9*b^2*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d] + 25*b^2*Cos[5*a - (5*b*c)/d]*SinIntegral[(5*b*(c + d*x))/d])/(32*d^3)","A",1
96,1,457,413,2.8956845,"\int \frac{\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx","Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^3)/(c + d*x)^4,x]","\frac{-27 b^3 (c+d x)^3 \left(\cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)-\sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)\right)+125 b^3 (c+d x)^3 \left(\cos \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b (c+d x)}{d}\right)-\sin \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b (c+d x)}{d}\right)\right)-d \cos (3 b x) \left(3 b d \cos (3 a) (c+d x)-\sin (3 a) \left(9 b^2 (c+d x)^2-2 d^2\right)\right)+d \cos (5 b x) \left(5 b d \cos (5 a) (c+d x)-\sin (5 a) \left(25 b^2 (c+d x)^2-2 d^2\right)\right)+d \sin (3 b x) \left(\cos (3 a) \left(9 b^2 (c+d x)^2-2 d^2\right)+3 b d \sin (3 a) (c+d x)\right)-d \sin (5 b x) \left(\cos (5 a) \left(25 b^2 (c+d x)^2-2 d^2\right)+5 b d \sin (5 a) (c+d x)\right)-2 \left(b^3 (c+d x)^3 \left(\cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)-\sin \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)\right)+d \cos (b x) \left(b d \cos (a) (c+d x)-\sin (a) \left(b^2 (c+d x)^2-2 d^2\right)\right)-d \sin (b x) \left(\cos (a) \left(b^2 (c+d x)^2-2 d^2\right)+b d \sin (a) (c+d x)\right)\right)}{96 d^4 (c+d x)^3}","-\frac{b^3 \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{48 d^4}-\frac{9 b^3 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{32 d^4}+\frac{125 b^3 \cos \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b c}{d}+5 b x\right)}{96 d^4}+\frac{b^3 \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{48 d^4}+\frac{9 b^3 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{32 d^4}-\frac{125 b^3 \sin \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b c}{d}+5 b x\right)}{96 d^4}+\frac{b^2 \sin (a+b x)}{48 d^3 (c+d x)}+\frac{3 b^2 \sin (3 a+3 b x)}{32 d^3 (c+d x)}-\frac{25 b^2 \sin (5 a+5 b x)}{96 d^3 (c+d x)}-\frac{b \cos (a+b x)}{48 d^2 (c+d x)^2}-\frac{b \cos (3 a+3 b x)}{32 d^2 (c+d x)^2}+\frac{5 b \cos (5 a+5 b x)}{96 d^2 (c+d x)^2}-\frac{\sin (a+b x)}{24 d (c+d x)^3}-\frac{\sin (3 a+3 b x)}{48 d (c+d x)^3}+\frac{\sin (5 a+5 b x)}{48 d (c+d x)^3}",1,"(-(d*Cos[3*b*x]*(3*b*d*(c + d*x)*Cos[3*a] - (-2*d^2 + 9*b^2*(c + d*x)^2)*Sin[3*a])) + d*Cos[5*b*x]*(5*b*d*(c + d*x)*Cos[5*a] - (-2*d^2 + 25*b^2*(c + d*x)^2)*Sin[5*a]) + d*((-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*a] + 3*b*d*(c + d*x)*Sin[3*a])*Sin[3*b*x] - d*((-2*d^2 + 25*b^2*(c + d*x)^2)*Cos[5*a] + 5*b*d*(c + d*x)*Sin[5*a])*Sin[5*b*x] - 2*(d*Cos[b*x]*(b*d*(c + d*x)*Cos[a] - (-2*d^2 + b^2*(c + d*x)^2)*Sin[a]) - d*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] + b*d*(c + d*x)*Sin[a])*Sin[b*x] + b^3*(c + d*x)^3*(Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] - Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)])) - 27*b^3*(c + d*x)^3*(Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] - Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d]) + 125*b^3*(c + d*x)^3*(Cos[5*a - (5*b*c)/d]*CosIntegral[(5*b*(c + d*x))/d] - Sin[5*a - (5*b*c)/d]*SinIntegral[(5*b*(c + d*x))/d]))/(96*d^4*(c + d*x)^3)","A",1
97,0,0,144,6.3833248,"\int (c+d x)^m \cos (a+b x) \cot (a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]*Cot[a + b*x],x]","\int (c+d x)^m \cos (a+b x) \cot (a+b x) \, dx","\text{Int}\left(\csc (a+b x) (c+d x)^m,x\right)+\frac{e^{i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)}{2 b}+\frac{e^{-i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)}{2 b}",0,"Integrate[(c + d*x)^m*Cos[a + b*x]*Cot[a + b*x], x]","A",-1
98,1,837,333,1.3247472,"\int (c+d x)^4 \cos (a+b x) \cot (a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]*Cot[a + b*x],x]","\frac{c^4 \cos (a+b x) b^4+d^4 x^4 \cos (a+b x) b^4+4 c d^3 x^3 \cos (a+b x) b^4+6 c^2 d^2 x^2 \cos (a+b x) b^4+4 c^3 d x \cos (a+b x) b^4+c^4 \log \left(1-e^{i (a+b x)}\right) b^4+d^4 x^4 \log \left(1-e^{i (a+b x)}\right) b^4+4 c d^3 x^3 \log \left(1-e^{i (a+b x)}\right) b^4+6 c^2 d^2 x^2 \log \left(1-e^{i (a+b x)}\right) b^4+4 c^3 d x \log \left(1-e^{i (a+b x)}\right) b^4-c^4 \log \left(1+e^{i (a+b x)}\right) b^4-d^4 x^4 \log \left(1+e^{i (a+b x)}\right) b^4-4 c d^3 x^3 \log \left(1+e^{i (a+b x)}\right) b^4-6 c^2 d^2 x^2 \log \left(1+e^{i (a+b x)}\right) b^4-4 c^3 d x \log \left(1+e^{i (a+b x)}\right) b^4+4 i d (c+d x)^3 \text{Li}_2\left(-e^{i (a+b x)}\right) b^3-4 i d (c+d x)^3 \text{Li}_2\left(e^{i (a+b x)}\right) b^3-4 d^4 x^3 \sin (a+b x) b^3-12 c d^3 x^2 \sin (a+b x) b^3-4 c^3 d \sin (a+b x) b^3-12 c^2 d^2 x \sin (a+b x) b^3-12 c^2 d^2 \cos (a+b x) b^2-12 d^4 x^2 \cos (a+b x) b^2-24 c d^3 x \cos (a+b x) b^2-12 c^2 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right) b^2-12 d^4 x^2 \text{Li}_3\left(-e^{i (a+b x)}\right) b^2-24 c d^3 x \text{Li}_3\left(-e^{i (a+b x)}\right) b^2+12 c^2 d^2 \text{Li}_3\left(e^{i (a+b x)}\right) b^2+12 d^4 x^2 \text{Li}_3\left(e^{i (a+b x)}\right) b^2+24 c d^3 x \text{Li}_3\left(e^{i (a+b x)}\right) b^2-24 i c d^3 \text{Li}_4\left(-e^{i (a+b x)}\right) b-24 i d^4 x \text{Li}_4\left(-e^{i (a+b x)}\right) b+24 i c d^3 \text{Li}_4\left(e^{i (a+b x)}\right) b+24 i d^4 x \text{Li}_4\left(e^{i (a+b x)}\right) b+24 c d^3 \sin (a+b x) b+24 d^4 x \sin (a+b x) b+24 d^4 \cos (a+b x)+24 d^4 \text{Li}_5\left(-e^{i (a+b x)}\right)-24 d^4 \text{Li}_5\left(e^{i (a+b x)}\right)}{b^5}","\frac{24 d^4 \text{Li}_5\left(-e^{i (a+b x)}\right)}{b^5}-\frac{24 d^4 \text{Li}_5\left(e^{i (a+b x)}\right)}{b^5}+\frac{24 d^4 \cos (a+b x)}{b^5}-\frac{24 i d^3 (c+d x) \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^4}+\frac{24 i d^3 (c+d x) \text{Li}_4\left(e^{i (a+b x)}\right)}{b^4}+\frac{24 d^3 (c+d x) \sin (a+b x)}{b^4}-\frac{12 d^2 (c+d x)^2 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{12 d^2 (c+d x)^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{12 d^2 (c+d x)^2 \cos (a+b x)}{b^3}+\frac{4 i d (c+d x)^3 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{4 i d (c+d x)^3 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{4 d (c+d x)^3 \sin (a+b x)}{b^2}+\frac{(c+d x)^4 \cos (a+b x)}{b}-\frac{2 (c+d x)^4 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(b^4*c^4*Cos[a + b*x] - 12*b^2*c^2*d^2*Cos[a + b*x] + 24*d^4*Cos[a + b*x] + 4*b^4*c^3*d*x*Cos[a + b*x] - 24*b^2*c*d^3*x*Cos[a + b*x] + 6*b^4*c^2*d^2*x^2*Cos[a + b*x] - 12*b^2*d^4*x^2*Cos[a + b*x] + 4*b^4*c*d^3*x^3*Cos[a + b*x] + b^4*d^4*x^4*Cos[a + b*x] + b^4*c^4*Log[1 - E^(I*(a + b*x))] + 4*b^4*c^3*d*x*Log[1 - E^(I*(a + b*x))] + 6*b^4*c^2*d^2*x^2*Log[1 - E^(I*(a + b*x))] + 4*b^4*c*d^3*x^3*Log[1 - E^(I*(a + b*x))] + b^4*d^4*x^4*Log[1 - E^(I*(a + b*x))] - b^4*c^4*Log[1 + E^(I*(a + b*x))] - 4*b^4*c^3*d*x*Log[1 + E^(I*(a + b*x))] - 6*b^4*c^2*d^2*x^2*Log[1 + E^(I*(a + b*x))] - 4*b^4*c*d^3*x^3*Log[1 + E^(I*(a + b*x))] - b^4*d^4*x^4*Log[1 + E^(I*(a + b*x))] + (4*I)*b^3*d*(c + d*x)^3*PolyLog[2, -E^(I*(a + b*x))] - (4*I)*b^3*d*(c + d*x)^3*PolyLog[2, E^(I*(a + b*x))] - 12*b^2*c^2*d^2*PolyLog[3, -E^(I*(a + b*x))] - 24*b^2*c*d^3*x*PolyLog[3, -E^(I*(a + b*x))] - 12*b^2*d^4*x^2*PolyLog[3, -E^(I*(a + b*x))] + 12*b^2*c^2*d^2*PolyLog[3, E^(I*(a + b*x))] + 24*b^2*c*d^3*x*PolyLog[3, E^(I*(a + b*x))] + 12*b^2*d^4*x^2*PolyLog[3, E^(I*(a + b*x))] - (24*I)*b*c*d^3*PolyLog[4, -E^(I*(a + b*x))] - (24*I)*b*d^4*x*PolyLog[4, -E^(I*(a + b*x))] + (24*I)*b*c*d^3*PolyLog[4, E^(I*(a + b*x))] + (24*I)*b*d^4*x*PolyLog[4, E^(I*(a + b*x))] + 24*d^4*PolyLog[5, -E^(I*(a + b*x))] - 24*d^4*PolyLog[5, E^(I*(a + b*x))] - 4*b^3*c^3*d*Sin[a + b*x] + 24*b*c*d^3*Sin[a + b*x] - 12*b^3*c^2*d^2*x*Sin[a + b*x] + 24*b*d^4*x*Sin[a + b*x] - 12*b^3*c*d^3*x^2*Sin[a + b*x] - 4*b^3*d^4*x^3*Sin[a + b*x])/b^5","B",1
99,1,330,254,0.9379576,"\int (c+d x)^3 \cos (a+b x) \cot (a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]*Cot[a + b*x],x]","\frac{-2 b^3 (c+d x)^3 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x))+3 i d \left(b^2 (c+d x)^2 \text{Li}_2(-\cos (a+b x)-i \sin (a+b x))+2 i b d (c+d x) \text{Li}_3(-\cos (a+b x)-i \sin (a+b x))-2 d^2 \text{Li}_4(-\cos (a+b x)-i \sin (a+b x))\right)-3 i d \left(b^2 (c+d x)^2 \text{Li}_2(\cos (a+b x)+i \sin (a+b x))+2 i b d (c+d x) \text{Li}_3(\cos (a+b x)+i \sin (a+b x))-2 d^2 \text{Li}_4(\cos (a+b x)+i \sin (a+b x))\right)+\cos (b x) \left(b \cos (a) (c+d x) \left(b^2 (c+d x)^2-6 d^2\right)-3 d \sin (a) \left(b^2 (c+d x)^2-2 d^2\right)\right)-\sin (b x) \left(b \sin (a) (c+d x) \left(b^2 (c+d x)^2-6 d^2\right)+3 d \cos (a) \left(b^2 (c+d x)^2-2 d^2\right)\right)}{b^4}","-\frac{6 i d^3 \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^4}+\frac{6 i d^3 \text{Li}_4\left(e^{i (a+b x)}\right)}{b^4}+\frac{6 d^3 \sin (a+b x)}{b^4}-\frac{6 d^2 (c+d x) \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{6 d^2 (c+d x) \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{3 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac{(c+d x)^3 \cos (a+b x)}{b}-\frac{2 (c+d x)^3 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(-2*b^3*(c + d*x)^3*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]] + (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, -Cos[a + b*x] - I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]] - 2*d^2*PolyLog[4, -Cos[a + b*x] - I*Sin[a + b*x]]) - (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]] - 2*d^2*PolyLog[4, Cos[a + b*x] + I*Sin[a + b*x]]) + Cos[b*x]*(b*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a] - 3*d*(-2*d^2 + b^2*(c + d*x)^2)*Sin[a]) - (3*d*(-2*d^2 + b^2*(c + d*x)^2)*Cos[a] + b*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Sin[a])*Sin[b*x])/b^4","A",0
100,1,221,171,0.8467448,"\int (c+d x)^2 \cos (a+b x) \cot (a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]*Cot[a + b*x],x]","\frac{\cos (b x) \left(\cos (a) \left(b^2 (c+d x)^2-2 d^2\right)-2 b d \sin (a) (c+d x)\right)-\sin (b x) \left(\sin (a) \left(b^2 (c+d x)^2-2 d^2\right)+2 b d \cos (a) (c+d x)\right)+b^2 (c+d x)^2 \log \left(1-e^{i (a+b x)}\right)-b^2 (c+d x)^2 \log \left(1+e^{i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)-2 i b d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)-2 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)+2 d^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}","-\frac{2 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{2 d^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{2 d^2 \cos (a+b x)}{b^3}+\frac{2 i d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{2 i d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{2 d (c+d x) \sin (a+b x)}{b^2}+\frac{(c+d x)^2 \cos (a+b x)}{b}-\frac{2 (c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(b^2*(c + d*x)^2*Log[1 - E^(I*(a + b*x))] - b^2*(c + d*x)^2*Log[1 + E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] - 2*d^2*PolyLog[3, -E^(I*(a + b*x))] + 2*d^2*PolyLog[3, E^(I*(a + b*x))] + Cos[b*x]*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] - 2*b*d*(c + d*x)*Sin[a]) - (2*b*d*(c + d*x)*Cos[a] + (-2*d^2 + b^2*(c + d*x)^2)*Sin[a])*Sin[b*x])/b^3","A",1
101,1,176,94,0.1705737,"\int (c+d x) \cos (a+b x) \cot (a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]*Cot[a + b*x],x]","\frac{d \left(i \left(\text{Li}_2\left(-e^{i (a+b x)}\right)-\text{Li}_2\left(e^{i (a+b x)}\right)\right)+(a+b x) \left(\log \left(1-e^{i (a+b x)}\right)-\log \left(1+e^{i (a+b x)}\right)\right)-a \log \left(\tan \left(\frac{1}{2} (a+b x)\right)\right)\right)}{b^2}+\frac{d \cos (b x) (b x \cos (a)-\sin (a))}{b^2}-\frac{d \sin (b x) (b x \sin (a)+\cos (a))}{b^2}+\frac{c \cos (a+b x)}{b}+\frac{c \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{c \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}","\frac{i d \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{i d \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{d \sin (a+b x)}{b^2}+\frac{(c+d x) \cos (a+b x)}{b}-\frac{2 (c+d x) \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(c*Cos[a + b*x])/b - (c*Log[Cos[(a + b*x)/2]])/b + (c*Log[Sin[(a + b*x)/2]])/b + (d*((a + b*x)*(Log[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(a + b*x))]) - a*Log[Tan[(a + b*x)/2]] + I*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^(I*(a + b*x))])))/b^2 + (d*Cos[b*x]*(b*x*Cos[a] - Sin[a]))/b^2 - (d*(Cos[a] + b*x*Sin[a])*Sin[b*x])/b^2","A",1
102,0,0,70,8.3541979,"\int \frac{\cos (a+b x) \cot (a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]*Cot[a + b*x])/(c + d*x),x]","\int \frac{\cos (a+b x) \cot (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\csc (a+b x)}{c+d x},x\right)-\frac{\sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d}-\frac{\cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d}",0,"Integrate[(Cos[a + b*x]*Cot[a + b*x])/(c + d*x), x]","A",-1
103,0,0,88,4.0196588,"\int \frac{\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]*Cot[a + b*x])/(c + d*x)^2,x]","\int \frac{\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\csc (a+b x)}{(c+d x)^2},x\right)-\frac{b \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d^2}+\frac{b \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d^2}+\frac{\sin (a+b x)}{d (c+d x)}",0,"Integrate[(Cos[a + b*x]*Cot[a + b*x])/(c + d*x)^2, x]","A",-1
104,0,0,19,1.1984355,"\int (c+d x)^m \cot ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Cot[a + b*x]^2,x]","\int (c+d x)^m \cot ^2(a+b x) \, dx","\text{Int}\left(\cot ^2(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Cot[a + b*x]^2, x]","A",-1
105,1,795,155,6.7199815,"\int (c+d x)^4 \cot ^2(a+b x) \, dx","Integrate[(c + d*x)^4*Cot[a + b*x]^2,x]","-\frac{e^{i a} \csc (a) \left(b^4 e^{-2 i a} x^4+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^3+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^3-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(-e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(-e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(e^{-i (a+b x)}\right)\right)\right) d^4}{b^5}-\frac{2 c e^{i a} \csc (a) \left(2 b^3 e^{-2 i a} x^3+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^2+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^2-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right) d^3}{b^4}-\frac{6 c^2 \csc (a) \sec (a) \left(b^2 e^{i \tan ^{-1}(\tan (a))} x^2+\frac{\left(i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x+\tan ^{-1}(\tan (a))\right) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\pi  \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right) \tan (a)}{\sqrt{\tan ^2(a)+1}}\right) d^2}{b^3 \sqrt{\sec ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}+\frac{4 c^3 \csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a)) d}{b^2 \left(\cos ^2(a)+\sin ^2(a)\right)}-\frac{1}{5} x \left(5 c^4+10 d x c^3+10 d^2 x^2 c^2+5 d^3 x^3 c+d^4 x^4\right)+\frac{\csc (a) \csc (a+b x) \left(\sin (b x) c^4+4 d x \sin (b x) c^3+6 d^2 x^2 \sin (b x) c^2+4 d^3 x^3 \sin (b x) c+d^4 x^4 \sin (b x)\right)}{b}","\frac{3 i d^4 \text{Li}_4\left(e^{2 i (a+b x)}\right)}{b^5}+\frac{6 d^3 (c+d x) \text{Li}_3\left(e^{2 i (a+b x)}\right)}{b^4}-\frac{6 i d^2 (c+d x)^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^3}+\frac{4 d (c+d x)^3 \log \left(1-e^{2 i (a+b x)}\right)}{b^2}-\frac{(c+d x)^4 \cot (a+b x)}{b}-\frac{i (c+d x)^4}{b}-\frac{(c+d x)^5}{5 d}",1,"-1/5*(x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4)) - (2*c*d^3*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^4 - (d^4*E^(I*a)*Csc[a]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-I)*(a + b*x))] - 2*PolyLog[4, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, E^((-I)*(a + b*x))] - 2*PolyLog[4, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^5 + (4*c^3*d*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^2*(Cos[a]^2 + Sin[a]^2)) + (Csc[a]*Csc[a + b*x]*(c^4*Sin[b*x] + 4*c^3*d*x*Sin[b*x] + 6*c^2*d^2*x^2*Sin[b*x] + 4*c*d^3*x^3*Sin[b*x] + d^4*x^4*Sin[b*x]))/b - (6*c^2*d^2*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^3*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
106,1,374,127,6.1535085,"\int (c+d x)^3 \cot ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Cot[a + b*x]^2,x]","-\frac{3 c^2 d (b x \cot (a)-\log (\sin (a+b x)))}{b^2}+\frac{3 c d^2 \left(-b^2 x^2 e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt{\sec ^2(a)}-i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+i b x \left(\pi -2 \tan ^{-1}(\tan (a))\right)+2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)-2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)+\pi  \log \left(1+e^{-2 i b x}\right)-\pi  \log (\cos (b x))\right)}{b^3}+\frac{e^{-i a} d^3 \sin (a) (\cot (a)+i) \left(-b^3 x^3 \cot (a)+3 b^2 x^2 \log \left(1-e^{-i (a+b x)}\right)+3 b^2 x^2 \log \left(1+e^{-i (a+b x)}\right)+6 i b x \text{Li}_2\left(-e^{-i (a+b x)}\right)+6 i b x \text{Li}_2\left(e^{-i (a+b x)}\right)+6 \text{Li}_3\left(-e^{-i (a+b x)}\right)+6 \text{Li}_3\left(e^{-i (a+b x)}\right)+i b^3 x^3\right)}{b^4}+\frac{\csc (a) \sin (b x) (c+d x)^3 \csc (a+b x)}{b}-\frac{1}{4} x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)","\frac{3 d^3 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 i d^2 (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^3}+\frac{3 d (c+d x)^2 \log \left(1-e^{2 i (a+b x)}\right)}{b^2}-\frac{(c+d x)^3 \cot (a+b x)}{b}-\frac{i (c+d x)^3}{b}-\frac{(c+d x)^4}{4 d}",1,"-1/4*(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)) - (3*c^2*d*(b*x*Cot[a] - Log[Sin[a + b*x]]))/b^2 + (3*c*d^2*(I*b*x*(Pi - 2*ArcTan[Tan[a]]) + Pi*Log[1 + E^((-2*I)*b*x)] + 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] - Pi*Log[Cos[b*x]] - 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] - I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))] - b^2*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2]))/b^3 + (d^3*(I + Cot[a])*(I*b^3*x^3 - b^3*x^3*Cot[a] + 3*b^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 3*b^2*x^2*Log[1 + E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, -E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, E^((-I)*(a + b*x))] + 6*PolyLog[3, -E^((-I)*(a + b*x))] + 6*PolyLog[3, E^((-I)*(a + b*x))])*Sin[a])/(b^4*E^(I*a)) + ((c + d*x)^3*Csc[a]*Csc[a + b*x]*Sin[b*x])/b","B",0
107,1,198,97,6.0255172,"\int (c+d x)^2 \cot ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Cot[a + b*x]^2,x]","-\frac{2 c d (b x \cot (a)-\log (\sin (a+b x)))}{b^2}+\frac{d^2 \left(-b^2 x^2 e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt{\sec ^2(a)}-i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+i b x \left(\pi -2 \tan ^{-1}(\tan (a))\right)+2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)-2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)+\pi  \log \left(1+e^{-2 i b x}\right)-\pi  \log (\cos (b x))\right)}{b^3}+\frac{\csc (a) \sin (b x) (c+d x)^2 \csc (a+b x)}{b}-\frac{1}{3} x \left(3 c^2+3 c d x+d^2 x^2\right)","-\frac{i d^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^3}+\frac{2 d (c+d x) \log \left(1-e^{2 i (a+b x)}\right)}{b^2}-\frac{(c+d x)^2 \cot (a+b x)}{b}-\frac{i (c+d x)^2}{b}-\frac{(c+d x)^3}{3 d}",1,"-1/3*(x*(3*c^2 + 3*c*d*x + d^2*x^2)) - (2*c*d*(b*x*Cot[a] - Log[Sin[a + b*x]]))/b^2 + (d^2*(I*b*x*(Pi - 2*ArcTan[Tan[a]]) + Pi*Log[1 + E^((-2*I)*b*x)] + 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] - Pi*Log[Cos[b*x]] - 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] - I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))] - b^2*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2]))/b^3 + ((c + d*x)^2*Csc[a]*Csc[a + b*x]*Sin[b*x])/b","B",0
108,1,82,41,0.4788958,"\int (c+d x) \cot ^2(a+b x) \, dx","Integrate[(c + d*x)*Cot[a + b*x]^2,x]","\frac{d \log (\sin (a+b x))}{b^2}-\frac{c \cot (a+b x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(a+b x)\right)}{b}+\frac{d x \csc (a) \sin (b x) \csc (a+b x)}{b}-\frac{d x \csc (a) (b x \sin (a)+2 \cos (a))}{2 b}","\frac{d \log (\sin (a+b x))}{b^2}-\frac{(c+d x) \cot (a+b x)}{b}-c x-\frac{d x^2}{2}",1,"-((c*Cot[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[a + b*x]^2])/b) + (d*Log[Sin[a + b*x]])/b^2 - (d*x*Csc[a]*(2*Cos[a] + b*x*Sin[a]))/(2*b) + (d*x*Csc[a]*Csc[a + b*x]*Sin[b*x])/b","C",1
109,0,0,19,4.5208306,"\int \frac{\cot ^2(a+b x)}{c+d x} \, dx","Integrate[Cot[a + b*x]^2/(c + d*x),x]","\int \frac{\cot ^2(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\cot ^2(a+b x)}{c+d x},x\right)",0,"Integrate[Cot[a + b*x]^2/(c + d*x), x]","A",-1
110,0,0,19,2.3488481,"\int \frac{\cot ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[Cot[a + b*x]^2/(c + d*x)^2,x]","\int \frac{\cot ^2(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\cot ^2(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[Cot[a + b*x]^2/(c + d*x)^2, x]","A",-1
111,0,0,39,38.5671792,"\int (c+d x)^m \cot ^2(a+b x) \csc (a+b x) \, dx","Integrate[(c + d*x)^m*Cot[a + b*x]^2*Csc[a + b*x],x]","\int (c+d x)^m \cot ^2(a+b x) \csc (a+b x) \, dx","\text{Int}\left(\csc ^3(a+b x) (c+d x)^m,x\right)-\text{Int}\left(\csc (a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Cot[a + b*x]^2*Csc[a + b*x], x]","A",-1
112,1,966,416,8.2953919,"\int (c+d x)^4 \cot ^2(a+b x) \csc (a+b x) \, dx","Integrate[(c + d*x)^4*Cot[a + b*x]^2*Csc[a + b*x],x]","\frac{-c^4 \log \left(1-e^{i (a+b x)}\right) b^4-d^4 x^4 \log \left(1-e^{i (a+b x)}\right) b^4-4 c d^3 x^3 \log \left(1-e^{i (a+b x)}\right) b^4-6 c^2 d^2 x^2 \log \left(1-e^{i (a+b x)}\right) b^4-4 c^3 d x \log \left(1-e^{i (a+b x)}\right) b^4+c^4 \log \left(1+e^{i (a+b x)}\right) b^4+d^4 x^4 \log \left(1+e^{i (a+b x)}\right) b^4+4 c d^3 x^3 \log \left(1+e^{i (a+b x)}\right) b^4+6 c^2 d^2 x^2 \log \left(1+e^{i (a+b x)}\right) b^4+4 c^3 d x \log \left(1+e^{i (a+b x)}\right) b^4+12 c^2 d^2 \log \left(1-e^{i (a+b x)}\right) b^2+12 d^4 x^2 \log \left(1-e^{i (a+b x)}\right) b^2+24 c d^3 x \log \left(1-e^{i (a+b x)}\right) b^2-12 c^2 d^2 \log \left(1+e^{i (a+b x)}\right) b^2-12 d^4 x^2 \log \left(1+e^{i (a+b x)}\right) b^2-24 c d^3 x \log \left(1+e^{i (a+b x)}\right) b^2+12 c^2 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right) b^2+12 d^4 x^2 \text{Li}_3\left(-e^{i (a+b x)}\right) b^2+24 c d^3 x \text{Li}_3\left(-e^{i (a+b x)}\right) b^2-12 c^2 d^2 \text{Li}_3\left(e^{i (a+b x)}\right) b^2-12 d^4 x^2 \text{Li}_3\left(e^{i (a+b x)}\right) b^2-24 c d^3 x \text{Li}_3\left(e^{i (a+b x)}\right) b^2-4 i d (c+d x) \left(b^2 (c+d x)^2-6 d^2\right) \text{Li}_2\left(-e^{i (a+b x)}\right) b+4 i d (c+d x) \left(b^2 (c+d x)^2-6 d^2\right) \text{Li}_2\left(e^{i (a+b x)}\right) b+24 i c d^3 \text{Li}_4\left(-e^{i (a+b x)}\right) b+24 i d^4 x \text{Li}_4\left(-e^{i (a+b x)}\right) b-24 i c d^3 \text{Li}_4\left(e^{i (a+b x)}\right) b-24 i d^4 x \text{Li}_4\left(e^{i (a+b x)}\right) b-24 d^4 \text{Li}_3\left(-e^{i (a+b x)}\right)+24 d^4 \text{Li}_3\left(e^{i (a+b x)}\right)-24 d^4 \text{Li}_5\left(-e^{i (a+b x)}\right)+24 d^4 \text{Li}_5\left(e^{i (a+b x)}\right)}{2 b^5}-\frac{\csc ^2(a+b x) \left(b \cos (a+b x) c^4+4 b d x \cos (a+b x) c^3+4 d \sin (a+b x) c^3+6 b d^2 x^2 \cos (a+b x) c^2+12 d^2 x \sin (a+b x) c^2+4 b d^3 x^3 \cos (a+b x) c+12 d^3 x^2 \sin (a+b x) c+b d^4 x^4 \cos (a+b x)+4 d^4 x^3 \sin (a+b x)\right)}{2 b^2}","-\frac{12 d^4 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^5}+\frac{12 d^4 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^5}-\frac{12 d^4 \text{Li}_5\left(-e^{i (a+b x)}\right)}{b^5}+\frac{12 d^4 \text{Li}_5\left(e^{i (a+b x)}\right)}{b^5}+\frac{12 i d^3 (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^4}-\frac{12 i d^3 (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^4}+\frac{12 i d^3 (c+d x) \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^4}-\frac{12 i d^3 (c+d x) \text{Li}_4\left(e^{i (a+b x)}\right)}{b^4}+\frac{6 d^2 (c+d x)^2 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}-\frac{6 d^2 (c+d x)^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{12 d^2 (c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^3}-\frac{2 i d (c+d x)^3 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}+\frac{2 i d (c+d x)^3 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{2 d (c+d x)^3 \csc (a+b x)}{b^2}+\frac{(c+d x)^4 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{(c+d x)^4 \cot (a+b x) \csc (a+b x)}{2 b}",1,"(-(b^4*c^4*Log[1 - E^(I*(a + b*x))]) + 12*b^2*c^2*d^2*Log[1 - E^(I*(a + b*x))] - 4*b^4*c^3*d*x*Log[1 - E^(I*(a + b*x))] + 24*b^2*c*d^3*x*Log[1 - E^(I*(a + b*x))] - 6*b^4*c^2*d^2*x^2*Log[1 - E^(I*(a + b*x))] + 12*b^2*d^4*x^2*Log[1 - E^(I*(a + b*x))] - 4*b^4*c*d^3*x^3*Log[1 - E^(I*(a + b*x))] - b^4*d^4*x^4*Log[1 - E^(I*(a + b*x))] + b^4*c^4*Log[1 + E^(I*(a + b*x))] - 12*b^2*c^2*d^2*Log[1 + E^(I*(a + b*x))] + 4*b^4*c^3*d*x*Log[1 + E^(I*(a + b*x))] - 24*b^2*c*d^3*x*Log[1 + E^(I*(a + b*x))] + 6*b^4*c^2*d^2*x^2*Log[1 + E^(I*(a + b*x))] - 12*b^2*d^4*x^2*Log[1 + E^(I*(a + b*x))] + 4*b^4*c*d^3*x^3*Log[1 + E^(I*(a + b*x))] + b^4*d^4*x^4*Log[1 + E^(I*(a + b*x))] - (4*I)*b*d*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*PolyLog[2, -E^(I*(a + b*x))] + (4*I)*b*d*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*PolyLog[2, E^(I*(a + b*x))] + 12*b^2*c^2*d^2*PolyLog[3, -E^(I*(a + b*x))] - 24*d^4*PolyLog[3, -E^(I*(a + b*x))] + 24*b^2*c*d^3*x*PolyLog[3, -E^(I*(a + b*x))] + 12*b^2*d^4*x^2*PolyLog[3, -E^(I*(a + b*x))] - 12*b^2*c^2*d^2*PolyLog[3, E^(I*(a + b*x))] + 24*d^4*PolyLog[3, E^(I*(a + b*x))] - 24*b^2*c*d^3*x*PolyLog[3, E^(I*(a + b*x))] - 12*b^2*d^4*x^2*PolyLog[3, E^(I*(a + b*x))] + (24*I)*b*c*d^3*PolyLog[4, -E^(I*(a + b*x))] + (24*I)*b*d^4*x*PolyLog[4, -E^(I*(a + b*x))] - (24*I)*b*c*d^3*PolyLog[4, E^(I*(a + b*x))] - (24*I)*b*d^4*x*PolyLog[4, E^(I*(a + b*x))] - 24*d^4*PolyLog[5, -E^(I*(a + b*x))] + 24*d^4*PolyLog[5, E^(I*(a + b*x))])/(2*b^5) - (Csc[a + b*x]^2*(b*c^4*Cos[a + b*x] + 4*b*c^3*d*x*Cos[a + b*x] + 6*b*c^2*d^2*x^2*Cos[a + b*x] + 4*b*c*d^3*x^3*Cos[a + b*x] + b*d^4*x^4*Cos[a + b*x] + 4*c^3*d*Sin[a + b*x] + 12*c^2*d^2*x*Sin[a + b*x] + 12*c*d^3*x^2*Sin[a + b*x] + 4*d^4*x^3*Sin[a + b*x]))/(2*b^2)","B",1
113,1,528,308,4.7444744,"\int (c+d x)^3 \cot ^2(a+b x) \csc (a+b x) \, dx","Integrate[(c + d*x)^3*Cot[a + b*x]^2*Csc[a + b*x],x]","-\frac{b^3 c^3 \log \left(1-e^{i (a+b x)}\right)-b^3 c^3 \log \left(1+e^{i (a+b x)}\right)+3 b^3 c^2 d x \log \left(1-e^{i (a+b x)}\right)-3 b^3 c^2 d x \log \left(1+e^{i (a+b x)}\right)+3 b^3 c d^2 x^2 \log \left(1-e^{i (a+b x)}\right)-3 b^3 c d^2 x^2 \log \left(1+e^{i (a+b x)}\right)+b^3 d^3 x^3 \log \left(1-e^{i (a+b x)}\right)-b^3 d^3 x^3 \log \left(1+e^{i (a+b x)}\right)+3 i d \text{Li}_2\left(-e^{i (a+b x)}\right) \left(b^2 (c+d x)^2-2 d^2\right)-3 i d \text{Li}_2\left(e^{i (a+b x)}\right) \left(b^2 (c+d x)^2-2 d^2\right)+b^2 (c+d x)^2 \csc (a+b x) (b (c+d x) \cot (a+b x)+3 d)-6 b c d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)+6 b c d^2 \text{Li}_3\left(e^{i (a+b x)}\right)-6 b c d^2 \log \left(1-e^{i (a+b x)}\right)+6 b c d^2 \log \left(1+e^{i (a+b x)}\right)-6 b d^3 x \text{Li}_3\left(-e^{i (a+b x)}\right)+6 b d^3 x \text{Li}_3\left(e^{i (a+b x)}\right)-6 i d^3 \text{Li}_4\left(-e^{i (a+b x)}\right)+6 i d^3 \text{Li}_4\left(e^{i (a+b x)}\right)-6 b d^3 x \log \left(1-e^{i (a+b x)}\right)+6 b d^3 x \log \left(1+e^{i (a+b x)}\right)}{2 b^4}","\frac{3 i d^3 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^4}-\frac{3 i d^3 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^4}+\frac{3 i d^3 \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^4}-\frac{3 i d^3 \text{Li}_4\left(e^{i (a+b x)}\right)}{b^4}+\frac{3 d^2 (c+d x) \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}-\frac{3 d^2 (c+d x) \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{6 d^2 (c+d x) \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^3}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{2 b^2}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{2 b^2}-\frac{3 d (c+d x)^2 \csc (a+b x)}{2 b^2}+\frac{(c+d x)^3 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b}",1,"-1/2*(b^2*(c + d*x)^2*(3*d + b*(c + d*x)*Cot[a + b*x])*Csc[a + b*x] + b^3*c^3*Log[1 - E^(I*(a + b*x))] - 6*b*c*d^2*Log[1 - E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 - E^(I*(a + b*x))] - 6*b*d^3*x*Log[1 - E^(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 - E^(I*(a + b*x))] + b^3*d^3*x^3*Log[1 - E^(I*(a + b*x))] - b^3*c^3*Log[1 + E^(I*(a + b*x))] + 6*b*c*d^2*Log[1 + E^(I*(a + b*x))] - 3*b^3*c^2*d*x*Log[1 + E^(I*(a + b*x))] + 6*b*d^3*x*Log[1 + E^(I*(a + b*x))] - 3*b^3*c*d^2*x^2*Log[1 + E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 + E^(I*(a + b*x))] + (3*I)*d*(-2*d^2 + b^2*(c + d*x)^2)*PolyLog[2, -E^(I*(a + b*x))] - (3*I)*d*(-2*d^2 + b^2*(c + d*x)^2)*PolyLog[2, E^(I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, -E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, -E^(I*(a + b*x))] + 6*b*c*d^2*PolyLog[3, E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, E^(I*(a + b*x))] - (6*I)*d^3*PolyLog[4, -E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4","A",1
114,1,471,179,7.577551,"\int (c+d x)^2 \cot ^2(a+b x) \csc (a+b x) \, dx","Integrate[(c + d*x)^2*Cot[a + b*x]^2*Csc[a + b*x],x]","\frac{\csc \left(\frac{a}{2}\right) \csc \left(\frac{a}{2}+\frac{b x}{2}\right) \left(c d \sin \left(\frac{b x}{2}\right)+d^2 x \sin \left(\frac{b x}{2}\right)\right)}{2 b^2}+\frac{\sec \left(\frac{a}{2}\right) \sec \left(\frac{a}{2}+\frac{b x}{2}\right) \left(d^2 (-x) \sin \left(\frac{b x}{2}\right)-c d \sin \left(\frac{b x}{2}\right)\right)}{2 b^2}-\frac{d \csc (a) (c+d x)}{b^2}+\frac{b^2 \left(-c^2\right) \log \left(1-e^{i (a+b x)}\right)+b^2 c^2 \log \left(1+e^{i (a+b x)}\right)-2 b^2 c d x \log \left(1-e^{i (a+b x)}\right)+2 b^2 c d x \log \left(1+e^{i (a+b x)}\right)-b^2 d^2 x^2 \log \left(1-e^{i (a+b x)}\right)+b^2 d^2 x^2 \log \left(1+e^{i (a+b x)}\right)-2 i b d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)+2 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)-2 d^2 \text{Li}_3\left(e^{i (a+b x)}\right)+2 d^2 \log \left(1-e^{i (a+b x)}\right)-2 d^2 \log \left(1+e^{i (a+b x)}\right)}{2 b^3}+\frac{\left(-c^2-2 c d x-d^2 x^2\right) \csc ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}+\frac{\left(c^2+2 c d x+d^2 x^2\right) \sec ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}","\frac{d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}-\frac{d^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{d^2 \tanh ^{-1}(\cos (a+b x))}{b^3}-\frac{i d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}+\frac{i d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{d (c+d x) \csc (a+b x)}{b^2}+\frac{(c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{(c+d x)^2 \cot (a+b x) \csc (a+b x)}{2 b}",1,"-((d*(c + d*x)*Csc[a])/b^2) + ((-c^2 - 2*c*d*x - d^2*x^2)*Csc[a/2 + (b*x)/2]^2)/(8*b) + (-(b^2*c^2*Log[1 - E^(I*(a + b*x))]) + 2*d^2*Log[1 - E^(I*(a + b*x))] - 2*b^2*c*d*x*Log[1 - E^(I*(a + b*x))] - b^2*d^2*x^2*Log[1 - E^(I*(a + b*x))] + b^2*c^2*Log[1 + E^(I*(a + b*x))] - 2*d^2*Log[1 + E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 + E^(I*(a + b*x))] + b^2*d^2*x^2*Log[1 + E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] + 2*d^2*PolyLog[3, -E^(I*(a + b*x))] - 2*d^2*PolyLog[3, E^(I*(a + b*x))])/(2*b^3) + ((c^2 + 2*c*d*x + d^2*x^2)*Sec[a/2 + (b*x)/2]^2)/(8*b) + (Sec[a/2]*Sec[a/2 + (b*x)/2]*(-(c*d*Sin[(b*x)/2]) - d^2*x*Sin[(b*x)/2]))/(2*b^2) + (Csc[a/2]*Csc[a/2 + (b*x)/2]*(c*d*Sin[(b*x)/2] + d^2*x*Sin[(b*x)/2]))/(2*b^2)","B",1
115,1,260,108,1.592944,"\int (c+d x) \cot ^2(a+b x) \csc (a+b x) \, dx","Integrate[(c + d*x)*Cot[a + b*x]^2*Csc[a + b*x],x]","-\frac{d \left(i \left(\text{Li}_2\left(-e^{i (a+b x)}\right)-\text{Li}_2\left(e^{i (a+b x)}\right)\right)+(a+b x) \left(\log \left(1-e^{i (a+b x)}\right)-\log \left(1+e^{i (a+b x)}\right)\right)\right)}{2 b^2}-\frac{d \tan \left(\frac{1}{2} (a+b x)\right)}{4 b^2}-\frac{d \cot \left(\frac{1}{2} (a+b x)\right)}{4 b^2}+\frac{a d \log \left(\tan \left(\frac{1}{2} (a+b x)\right)\right)}{2 b^2}-\frac{c \csc ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{c \sec ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}-\frac{c \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}+\frac{c \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}-\frac{d x \csc ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{d x \sec ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}","-\frac{i d \text{Li}_2\left(-e^{i (a+b x)}\right)}{2 b^2}+\frac{i d \text{Li}_2\left(e^{i (a+b x)}\right)}{2 b^2}-\frac{d \csc (a+b x)}{2 b^2}+\frac{(c+d x) \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{(c+d x) \cot (a+b x) \csc (a+b x)}{2 b}",1,"-1/4*(d*Cot[(a + b*x)/2])/b^2 - (c*Csc[(a + b*x)/2]^2)/(8*b) - (d*x*Csc[(a + b*x)/2]^2)/(8*b) + (c*Log[Cos[(a + b*x)/2]])/(2*b) - (c*Log[Sin[(a + b*x)/2]])/(2*b) + (a*d*Log[Tan[(a + b*x)/2]])/(2*b^2) - (d*((a + b*x)*(Log[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(a + b*x))]) + I*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^(I*(a + b*x))])))/(2*b^2) + (c*Sec[(a + b*x)/2]^2)/(8*b) + (d*x*Sec[(a + b*x)/2]^2)/(8*b) - (d*Tan[(a + b*x)/2])/(4*b^2)","B",1
116,0,0,39,35.8867674,"\int \frac{\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx","Integrate[(Cot[a + b*x]^2*Csc[a + b*x])/(c + d*x),x]","\int \frac{\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\csc ^3(a+b x)}{c+d x},x\right)-\text{Int}\left(\frac{\csc (a+b x)}{c+d x},x\right)",0,"Integrate[(Cot[a + b*x]^2*Csc[a + b*x])/(c + d*x), x]","A",-1
117,0,0,39,41.7212031,"\int \frac{\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx","Integrate[(Cot[a + b*x]^2*Csc[a + b*x])/(c + d*x)^2,x]","\int \frac{\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\csc ^3(a+b x)}{(c+d x)^2},x\right)-\text{Int}\left(\frac{\csc (a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Cot[a + b*x]^2*Csc[a + b*x])/(c + d*x)^2, x]","A",-1
118,1,1168,406,15.9457762,"\int (c+d x)^{5/2} \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x],x]","\frac{e^{-\frac{i (b c+a d)}{d}} \sqrt{c+d x} \left(-\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right) c^2}{8 b}-\frac{\left(2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \sin \left(3 a-\frac{3 b c}{d}\right)\right) c^2}{24 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{\sqrt{\frac{b}{d}} d \left(\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(3 d \cos \left(a-\frac{b c}{d}\right)-2 b c \sin \left(a-\frac{b c}{d}\right)\right)+\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(a-\frac{b c}{d}\right)+3 d \sin \left(a-\frac{b c}{d}\right)\right)+2 \sqrt{\frac{b}{d}} d \sqrt{c+d x} (2 b x \cos (a+b x)-3 \sin (a+b x))\right) c}{8 b^3}-\frac{\sqrt{\frac{b}{d}} d \left(\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(d \cos \left(3 a-\frac{3 b c}{d}\right)-2 b c \sin \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(3 a-\frac{3 b c}{d}\right)+d \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} d \sqrt{c+d x} (2 b x \cos (3 (a+b x))-\sin (3 (a+b x)))\right) c}{24 \sqrt{3} b^3}+\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \cos \left(a-\frac{b c}{d}\right)+12 b c d \sin \left(a-\frac{b c}{d}\right)\right)-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \sin \left(a-\frac{b c}{d}\right)-12 b c d \cos \left(a-\frac{b c}{d}\right)\right)-2 \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(4 b^2 x^2-15\right) \cos (a+b x)+2 b (c-5 d x) \sin (a+b x)\right)\right)}{32 b^5}+\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \cos \left(3 a-\frac{3 b c}{d}\right)+12 b c d \sin \left(3 a-\frac{3 b c}{d}\right)\right)-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \sin \left(3 a-\frac{3 b c}{d}\right)-12 b c d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(5-12 b^2 x^2\right) \cos (3 (a+b x))-2 b (c-5 d x) \sin (3 (a+b x))\right)\right)}{288 \sqrt{3} b^5}","-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{144 b^{7/2}}+\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{144 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (a+b x)}{16 b^3}+\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{8 b^2}+\frac{5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{72 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x)}{4 b}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{12 b}",1,"(c^2*Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(8*b*E^((I*(b*c + a*d))/d)) - (c^2*(2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d]))/(24*Sqrt[3]*b*Sqrt[b/d]) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(3*d*Cos[a - (b*c)/d] - 2*b*c*Sin[a - (b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[a + b*x] - 3*Sin[a + b*x])))/(8*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*((4*b^2*c^2 - 15*d^2)*Cos[a - (b*c)/d] + 12*b*c*d*Sin[a - (b*c)/d]) - Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[a - (b*c)/d] + (4*b^2*c^2 - 15*d^2)*Sin[a - (b*c)/d]) - 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(d*(-15 + 4*b^2*x^2)*Cos[a + b*x] + 2*b*(c - 5*d*x)*Sin[a + b*x])))/(32*b^5) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(d*Cos[3*a - (3*b*c)/d] - 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[3*(a + b*x)] - Sin[3*(a + b*x)])))/(24*Sqrt[3]*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*((12*b^2*c^2 - 5*d^2)*Cos[3*a - (3*b*c)/d] + 12*b*c*d*Sin[3*a - (3*b*c)/d]) - Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[3*a - (3*b*c)/d] + (12*b^2*c^2 - 5*d^2)*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(d*(5 - 12*b^2*x^2)*Cos[3*(a + b*x)] - 2*b*(c - 5*d*x)*Sin[3*(a + b*x)])))/(288*Sqrt[3]*b^5)","C",1
119,1,676,353,9.0356963,"\int (c+d x)^{3/2} \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x],x]","-\frac{d \sqrt{\frac{b}{d}} \left(\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(3 d \sin \left(a-\frac{b c}{d}\right)+2 b c \cos \left(a-\frac{b c}{d}\right)\right)+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(3 d \cos \left(a-\frac{b c}{d}\right)-2 b c \sin \left(a-\frac{b c}{d}\right)\right)+2 d \sqrt{\frac{b}{d}} \sqrt{c+d x} (2 b x \cos (a+b x)-3 \sin (a+b x))\right)}{16 b^3}-\frac{d \sqrt{\frac{b}{d}} \left(\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(d \sin \left(3 a-\frac{3 b c}{d}\right)+2 b c \cos \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(d \cos \left(3 a-\frac{3 b c}{d}\right)-2 b c \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} d \sqrt{\frac{b}{d}} \sqrt{c+d x} (2 b x \cos (3 (a+b x))-\sin (3 (a+b x)))\right)}{48 \sqrt{3} b^3}-\frac{c \left(-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))\right)}{24 \sqrt{3} b \sqrt{\frac{b}{d}}}+\frac{c \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(-\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{8 b}","-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{24 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{24 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (a+b x)}{8 b^2}+\frac{d \sqrt{c+d x} \sin (3 a+3 b x)}{24 b^2}-\frac{(c+d x)^{3/2} \cos (a+b x)}{4 b}-\frac{(c+d x)^{3/2} \cos (3 a+3 b x)}{12 b}",1,"(c*Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(8*b*E^((I*(b*c + a*d))/d)) - (c*(2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d]))/(24*Sqrt[3]*b*Sqrt[b/d]) - (Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(3*d*Cos[a - (b*c)/d] - 2*b*c*Sin[a - (b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[a + b*x] - 3*Sin[a + b*x])))/(16*b^3) - (Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(d*Cos[3*a - (3*b*c)/d] - 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[3*(a + b*x)] - Sin[3*(a + b*x)])))/(48*Sqrt[3]*b^3)","C",1
120,1,278,304,6.6159151,"\int \sqrt{c+d x} \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^2*Sin[a + b*x],x]","-\frac{-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))}{24 \sqrt{3} b \sqrt{\frac{b}{d}}}+\frac{\sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(-\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{8 b}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{12 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{12 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{3/2}}-\frac{\sqrt{c+d x} \cos (a+b x)}{4 b}-\frac{\sqrt{c+d x} \cos (3 a+3 b x)}{12 b}",1,"(Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(8*b*E^((I*(b*c + a*d))/d)) - (2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d])/(24*Sqrt[3]*b*Sqrt[b/d])","C",1
121,1,264,304,6.3893641,"\int \sqrt{c+d x} \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^2*Sin[a + b*x],x]","\frac{\frac{\sqrt{6 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-\sqrt{6 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-6 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))}{\sqrt{\frac{b}{d}}}+9 \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(-\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{72 b}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{12 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{12 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4 b^{3/2}}-\frac{\sqrt{c+d x} \cos (a+b x)}{4 b}-\frac{\sqrt{c+d x} \cos (3 a+3 b x)}{12 b}",1,"((9*Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/E^((I*(b*c + a*d))/d) + (-6*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] + Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] - Sqrt[6*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d])/Sqrt[b/d])/(72*b)","C",1
122,1,676,353,8.9288982,"\int (c+d x)^{3/2} \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x],x]","-\frac{d \sqrt{\frac{b}{d}} \left(\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(3 d \sin \left(a-\frac{b c}{d}\right)+2 b c \cos \left(a-\frac{b c}{d}\right)\right)+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(3 d \cos \left(a-\frac{b c}{d}\right)-2 b c \sin \left(a-\frac{b c}{d}\right)\right)+2 d \sqrt{\frac{b}{d}} \sqrt{c+d x} (2 b x \cos (a+b x)-3 \sin (a+b x))\right)}{16 b^3}-\frac{d \sqrt{\frac{b}{d}} \left(\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(d \sin \left(3 a-\frac{3 b c}{d}\right)+2 b c \cos \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(d \cos \left(3 a-\frac{3 b c}{d}\right)-2 b c \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} d \sqrt{\frac{b}{d}} \sqrt{c+d x} (2 b x \cos (3 (a+b x))-\sin (3 (a+b x)))\right)}{48 \sqrt{3} b^3}-\frac{c \left(-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))\right)}{24 \sqrt{3} b \sqrt{\frac{b}{d}}}+\frac{c \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(-\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{8 b}","-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{24 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{24 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (a+b x)}{8 b^2}+\frac{d \sqrt{c+d x} \sin (3 a+3 b x)}{24 b^2}-\frac{(c+d x)^{3/2} \cos (a+b x)}{4 b}-\frac{(c+d x)^{3/2} \cos (3 a+3 b x)}{12 b}",1,"(c*Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(8*b*E^((I*(b*c + a*d))/d)) - (c*(2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d]))/(24*Sqrt[3]*b*Sqrt[b/d]) - (Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(3*d*Cos[a - (b*c)/d] - 2*b*c*Sin[a - (b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[a + b*x] - 3*Sin[a + b*x])))/(16*b^3) - (Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(d*Cos[3*a - (3*b*c)/d] - 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[3*(a + b*x)] - Sin[3*(a + b*x)])))/(48*Sqrt[3]*b^3)","C",1
123,1,1168,406,15.3130761,"\int (c+d x)^{5/2} \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x],x]","\frac{e^{-\frac{i (b c+a d)}{d}} \sqrt{c+d x} \left(-\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right) c^2}{8 b}-\frac{\left(2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \sin \left(3 a-\frac{3 b c}{d}\right)\right) c^2}{24 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{\sqrt{\frac{b}{d}} d \left(\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(3 d \cos \left(a-\frac{b c}{d}\right)-2 b c \sin \left(a-\frac{b c}{d}\right)\right)+\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(a-\frac{b c}{d}\right)+3 d \sin \left(a-\frac{b c}{d}\right)\right)+2 \sqrt{\frac{b}{d}} d \sqrt{c+d x} (2 b x \cos (a+b x)-3 \sin (a+b x))\right) c}{8 b^3}-\frac{\sqrt{\frac{b}{d}} d \left(\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(d \cos \left(3 a-\frac{3 b c}{d}\right)-2 b c \sin \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(3 a-\frac{3 b c}{d}\right)+d \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} d \sqrt{c+d x} (2 b x \cos (3 (a+b x))-\sin (3 (a+b x)))\right) c}{24 \sqrt{3} b^3}+\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \cos \left(a-\frac{b c}{d}\right)+12 b c d \sin \left(a-\frac{b c}{d}\right)\right)-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \sin \left(a-\frac{b c}{d}\right)-12 b c d \cos \left(a-\frac{b c}{d}\right)\right)-2 \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(4 b^2 x^2-15\right) \cos (a+b x)+2 b (c-5 d x) \sin (a+b x)\right)\right)}{32 b^5}+\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \cos \left(3 a-\frac{3 b c}{d}\right)+12 b c d \sin \left(3 a-\frac{3 b c}{d}\right)\right)-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \sin \left(3 a-\frac{3 b c}{d}\right)-12 b c d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(5-12 b^2 x^2\right) \cos (3 (a+b x))-2 b (c-5 d x) \sin (3 (a+b x))\right)\right)}{288 \sqrt{3} b^5}","-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{144 b^{7/2}}+\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{144 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (a+b x)}{16 b^3}+\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{8 b^2}+\frac{5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{72 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x)}{4 b}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{12 b}",1,"(c^2*Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(8*b*E^((I*(b*c + a*d))/d)) - (c^2*(2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d]))/(24*Sqrt[3]*b*Sqrt[b/d]) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(3*d*Cos[a - (b*c)/d] - 2*b*c*Sin[a - (b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[a + b*x] - 3*Sin[a + b*x])))/(8*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*((4*b^2*c^2 - 15*d^2)*Cos[a - (b*c)/d] + 12*b*c*d*Sin[a - (b*c)/d]) - Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[a - (b*c)/d] + (4*b^2*c^2 - 15*d^2)*Sin[a - (b*c)/d]) - 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(d*(-15 + 4*b^2*x^2)*Cos[a + b*x] + 2*b*(c - 5*d*x)*Sin[a + b*x])))/(32*b^5) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(d*Cos[3*a - (3*b*c)/d] - 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[3*(a + b*x)] - Sin[3*(a + b*x)])))/(24*Sqrt[3]*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*((12*b^2*c^2 - 5*d^2)*Cos[3*a - (3*b*c)/d] + 12*b*c*d*Sin[3*a - (3*b*c)/d]) - Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[3*a - (3*b*c)/d] + (12*b^2*c^2 - 5*d^2)*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(d*(5 - 12*b^2*x^2)*Cos[3*(a + b*x)] - 2*b*(c - 5*d*x)*Sin[3*(a + b*x)])))/(288*Sqrt[3]*b^5)","C",1
124,1,206,228,3.4192977,"\int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x]^2,x]","\frac{\sqrt{\frac{b}{d}} \left(4 \sqrt{\frac{b}{d}} \sqrt{c+d x} \left(-7 d \sin (4 (a+b x)) \left(64 b^2 (c+d x)^2-15 d^2\right)-280 b d^2 (c+d x) \cos (4 (a+b x))+512 b^3 (c+d x)^3\right)-105 \sqrt{2 \pi } d^3 \sin \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-105 \sqrt{2 \pi } d^3 \cos \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)\right)}{57344 b^4}","-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac{5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^{7/2}}{28 d}",1,"(Sqrt[b/d]*(-105*d^3*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 105*d^3*Sqrt[2*Pi]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] + 4*Sqrt[b/d]*Sqrt[c + d*x]*(512*b^3*(c + d*x)^3 - 280*b*d^2*(c + d*x)*Cos[4*(a + b*x)] - 7*d*(-15*d^2 + 64*b^2*(c + d*x)^2)*Sin[4*(a + b*x)])))/(57344*b^4)","A",1
125,1,187,200,2.7968721,"\int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x]^2,x]","\frac{\sqrt{\frac{b}{d}} \left(15 \sqrt{2 \pi } d^2 \cos \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-15 \sqrt{2 \pi } d^2 \sin \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+4 \sqrt{\frac{b}{d}} \sqrt{c+d x} \left(8 b (c+d x) (8 b (c+d x)-5 d \sin (4 (a+b x)))-15 d^2 \cos (4 (a+b x))\right)\right)}{5120 b^3}","\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 d \sqrt{c+d x} \cos (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{3/2} \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^{5/2}}{20 d}",1,"(Sqrt[b/d]*(15*d^2*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 15*d^2*Sqrt[2*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] + 4*Sqrt[b/d]*Sqrt[c + d*x]*(-15*d^2*Cos[4*(a + b*x)] + 8*b*(c + d*x)*(8*b*(c + d*x) - 5*d*Sin[4*(a + b*x)]))))/(5120*b^3)","A",1
126,1,161,174,0.8221888,"\int \sqrt{c+d x} \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^2*Sin[a + b*x]^2,x]","\frac{3 \sqrt{2 \pi } d \sin \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+3 \sqrt{2 \pi } d \cos \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+4 \sqrt{\frac{b}{d}} \sqrt{c+d x} (8 b (c+d x)-3 d \sin (4 (a+b x)))}{384 d^2 \left(\frac{b}{d}\right)^{3/2}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}-\frac{\sqrt{c+d x} \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^{3/2}}{12 d}",1,"(3*d*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] + 3*d*Sqrt[2*Pi]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] + 4*Sqrt[b/d]*Sqrt[c + d*x]*(8*b*(c + d*x) - 3*d*Sin[4*(a + b*x)]))/(384*(b/d)^(3/2)*d^2)","A",1
127,1,161,174,0.1122836,"\int \sqrt{c+d x} \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^2*Sin[a + b*x]^2,x]","\frac{3 \sqrt{2 \pi } d \sin \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+3 \sqrt{2 \pi } d \cos \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+4 \sqrt{\frac{b}{d}} \sqrt{c+d x} (8 b (c+d x)-3 d \sin (4 (a+b x)))}{384 d^2 \left(\frac{b}{d}\right)^{3/2}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}-\frac{\sqrt{c+d x} \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^{3/2}}{12 d}",1,"(3*d*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] + 3*d*Sqrt[2*Pi]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] + 4*Sqrt[b/d]*Sqrt[c + d*x]*(8*b*(c + d*x) - 3*d*Sin[4*(a + b*x)]))/(384*(b/d)^(3/2)*d^2)","A",1
128,1,187,200,1.2683486,"\int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x]^2,x]","\frac{\sqrt{\frac{b}{d}} \left(15 \sqrt{2 \pi } d^2 \cos \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-15 \sqrt{2 \pi } d^2 \sin \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+4 \sqrt{\frac{b}{d}} \sqrt{c+d x} \left(8 b (c+d x) (8 b (c+d x)-5 d \sin (4 (a+b x)))-15 d^2 \cos (4 (a+b x))\right)\right)}{5120 b^3}","\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 d \sqrt{c+d x} \cos (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{3/2} \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^{5/2}}{20 d}",1,"(Sqrt[b/d]*(15*d^2*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 15*d^2*Sqrt[2*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] + 4*Sqrt[b/d]*Sqrt[c + d*x]*(-15*d^2*Cos[4*(a + b*x)] + 8*b*(c + d*x)*(8*b*(c + d*x) - 5*d*Sin[4*(a + b*x)]))))/(5120*b^3)","A",1
129,1,206,228,2.3246603,"\int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x]^2,x]","\frac{\sqrt{\frac{b}{d}} \left(4 \sqrt{\frac{b}{d}} \sqrt{c+d x} \left(-7 d \sin (4 (a+b x)) \left(64 b^2 (c+d x)^2-15 d^2\right)-280 b d^2 (c+d x) \cos (4 (a+b x))+512 b^3 (c+d x)^3\right)-105 \sqrt{2 \pi } d^3 \sin \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-105 \sqrt{2 \pi } d^3 \cos \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)\right)}{57344 b^4}","-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac{5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^{7/2}}{28 d}",1,"(Sqrt[b/d]*(-105*d^3*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 105*d^3*Sqrt[2*Pi]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] + 4*Sqrt[b/d]*Sqrt[c + d*x]*(512*b^3*(c + d*x)^3 - 280*b*d^2*(c + d*x)*Cos[4*(a + b*x)] - 7*d*(-15*d^2 + 64*b^2*(c + d*x)^2)*Sin[4*(a + b*x)])))/(57344*b^4)","A",1
130,1,3348,615,24.0975893,"\int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\text{Result too large to show}","-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{32 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{576 b^{7/2}}+\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \cos \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1600 b^{7/2}}-\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \sin \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1600 b^{7/2}}+\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{576 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{32 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (a+b x)}{32 b^3}+\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac{3 d^2 \sqrt{c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac{5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}",1,"(c^2*Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(16*b*E^((I*(b*c + a*d))/d)) + (c^2*(2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[5*(a + b*x)] - Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d]))/(160*Sqrt[5]*b*Sqrt[b/d]) - (c^2*(2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(3*d*Cos[a - (b*c)/d] - 2*b*c*Sin[a - (b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[a + b*x] - 3*Sin[a + b*x])))/(16*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*((4*b^2*c^2 - 15*d^2)*Cos[a - (b*c)/d] + 12*b*c*d*Sin[a - (b*c)/d]) - Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[a - (b*c)/d] + (4*b^2*c^2 - 15*d^2)*Sin[a - (b*c)/d]) - 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(d*(-15 + 4*b^2*x^2)*Cos[a + b*x] + 2*b*(c - 5*d*x)*Sin[a + b*x])))/(64*b^5) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(d*Cos[3*a - (3*b*c)/d] - 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[3*(a + b*x)] - Sin[3*(a + b*x)])))/(96*Sqrt[3]*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*((12*b^2*c^2 - 5*d^2)*Cos[3*a - (3*b*c)/d] + 12*b*c*d*Sin[3*a - (3*b*c)/d]) - Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[3*a - (3*b*c)/d] + (12*b^2*c^2 - 5*d^2)*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(d*(5 - 12*b^2*x^2)*Cos[3*(a + b*x)] - 2*b*(c - 5*d*x)*Sin[3*(a + b*x)])))/(1152*Sqrt[3]*b^5) + (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(3*d*Cos[5*a - (5*b*c)/d] - 10*b*c*Sin[5*a - (5*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(10*b*c*Cos[5*a - (5*b*c)/d] + 3*d*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*Sqrt[b/d]*d*Sqrt[c + d*x]*(10*b*x*Cos[5*(a + b*x)] - 3*Sin[5*(a + b*x)])))/(800*Sqrt[5]*b^3) - (d^2*(Sin[5*a]*((c^2*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]])*Sin[(5*b*c)/d])/(5*Sqrt[5]*(b/d)^(3/2)*d^3) + (c^2*Cos[(5*b*c)/d]*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/(5*Sqrt[5]*(b/d)^(3/2)*d^3) - (2*c*Cos[(5*b*c)/d]*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) - (2*c*Sin[(5*b*c)/d]*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) + (Sin[(5*b*c)/d]*(-25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Cos[(5*b*(c + d*x))/d] + (5*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3) + (Cos[(5*b*c)/d]*(25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Sin[(5*b*(c + d*x))/d] - (5*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3)) + Cos[5*a]*((c^2*Cos[(5*b*c)/d]*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/(5*Sqrt[5]*(b/d)^(3/2)*d^3) - (c^2*Sin[(5*b*c)/d]*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/(5*Sqrt[5]*(b/d)^(3/2)*d^3) + (2*c*Sin[(5*b*c)/d]*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) - (2*c*Cos[(5*b*c)/d]*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) + (Cos[(5*b*c)/d]*(-25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Cos[(5*b*(c + d*x))/d] + (5*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3) - (Sin[(5*b*c)/d]*(25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Sin[(5*b*(c + d*x))/d] - (5*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3))))/16","C",0
131,1,1041,534,11.683534,"\int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\frac{c e^{-\frac{i (b c+a d)}{d}} \sqrt{c+d x} \left(-\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{16 b}+\frac{c \left(2 \sqrt{5} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (5 (a+b x))-\sqrt{2 \pi } \cos \left(5 a-\frac{5 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \sin \left(5 a-\frac{5 b c}{d}\right)\right)}{160 \sqrt{5} b \sqrt{\frac{b}{d}}}-\frac{c \left(2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \sin \left(3 a-\frac{3 b c}{d}\right)\right)}{96 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{\sqrt{\frac{b}{d}} d \left(\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(3 d \cos \left(a-\frac{b c}{d}\right)-2 b c \sin \left(a-\frac{b c}{d}\right)\right)+\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(a-\frac{b c}{d}\right)+3 d \sin \left(a-\frac{b c}{d}\right)\right)+2 \sqrt{\frac{b}{d}} d \sqrt{c+d x} (2 b x \cos (a+b x)-3 \sin (a+b x))\right)}{32 b^3}-\frac{\sqrt{\frac{b}{d}} d \left(\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(d \cos \left(3 a-\frac{3 b c}{d}\right)-2 b c \sin \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(3 a-\frac{3 b c}{d}\right)+d \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} d \sqrt{c+d x} (2 b x \cos (3 (a+b x))-\sin (3 (a+b x)))\right)}{192 \sqrt{3} b^3}+\frac{\sqrt{\frac{b}{d}} d \left(\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(3 d \cos \left(5 a-\frac{5 b c}{d}\right)-10 b c \sin \left(5 a-\frac{5 b c}{d}\right)\right)+\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(10 b c \cos \left(5 a-\frac{5 b c}{d}\right)+3 d \sin \left(5 a-\frac{5 b c}{d}\right)\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} d \sqrt{c+d x} (10 b x \cos (5 (a+b x))-3 \sin (5 (a+b x)))\right)}{1600 \sqrt{5} b^3}","\frac{3 \sqrt{\frac{\pi }{10}} d^{3/2} \sin \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{800 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{96 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{96 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{10}} d^{3/2} \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{800 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (a+b x)}{16 b^2}+\frac{d \sqrt{c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac{3 d \sqrt{c+d x} \sin (5 a+5 b x)}{800 b^2}-\frac{(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac{(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}",1,"(c*Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(16*b*E^((I*(b*c + a*d))/d)) + (c*(2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[5*(a + b*x)] - Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d]))/(160*Sqrt[5]*b*Sqrt[b/d]) - (c*(2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(3*d*Cos[a - (b*c)/d] - 2*b*c*Sin[a - (b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[a + b*x] - 3*Sin[a + b*x])))/(32*b^3) - (Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(d*Cos[3*a - (3*b*c)/d] - 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[3*(a + b*x)] - Sin[3*(a + b*x)])))/(192*Sqrt[3]*b^3) + (Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(3*d*Cos[5*a - (5*b*c)/d] - 10*b*c*Sin[5*a - (5*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(10*b*c*Cos[5*a - (5*b*c)/d] + 3*d*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*Sqrt[b/d]*d*Sqrt[c + d*x]*(10*b*x*Cos[5*(a + b*x)] - 3*Sin[5*(a + b*x)])))/(1600*Sqrt[5]*b^3)","C",0
132,1,432,459,7.3377732,"\int \sqrt{c+d x} \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\frac{-\sqrt{2 \pi } \cos \left(5 a-\frac{5 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } \sin \left(5 a-\frac{5 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (5 (a+b x))}{160 \sqrt{5} b \sqrt{\frac{b}{d}}}-\frac{-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))}{96 \sqrt{3} b \sqrt{\frac{b}{d}}}+\frac{\sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(-\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{16 b}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{48 b^{3/2}}-\frac{\sqrt{\frac{\pi }{10}} \sqrt{d} \cos \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{80 b^{3/2}}+\frac{\sqrt{\frac{\pi }{10}} \sqrt{d} \sin \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{80 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{48 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{3/2}}-\frac{\sqrt{c+d x} \cos (a+b x)}{8 b}-\frac{\sqrt{c+d x} \cos (3 a+3 b x)}{48 b}+\frac{\sqrt{c+d x} \cos (5 a+5 b x)}{80 b}",1,"(Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(16*b*E^((I*(b*c + a*d))/d)) + (2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[5*(a + b*x)] - Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d])/(160*Sqrt[5]*b*Sqrt[b/d]) - (2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d])/(96*Sqrt[3]*b*Sqrt[b/d])","C",1
133,1,432,459,7.2594661,"\int \sqrt{c+d x} \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\frac{-\sqrt{2 \pi } \cos \left(5 a-\frac{5 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } \sin \left(5 a-\frac{5 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (5 (a+b x))}{160 \sqrt{5} b \sqrt{\frac{b}{d}}}-\frac{-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))}{96 \sqrt{3} b \sqrt{\frac{b}{d}}}+\frac{\sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(-\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{16 b}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{48 b^{3/2}}-\frac{\sqrt{\frac{\pi }{10}} \sqrt{d} \cos \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{80 b^{3/2}}+\frac{\sqrt{\frac{\pi }{10}} \sqrt{d} \sin \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{80 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{48 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{3/2}}-\frac{\sqrt{c+d x} \cos (a+b x)}{8 b}-\frac{\sqrt{c+d x} \cos (3 a+3 b x)}{48 b}+\frac{\sqrt{c+d x} \cos (5 a+5 b x)}{80 b}",1,"(Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(16*b*E^((I*(b*c + a*d))/d)) + (2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[5*(a + b*x)] - Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d])/(160*Sqrt[5]*b*Sqrt[b/d]) - (2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d])/(96*Sqrt[3]*b*Sqrt[b/d])","C",1
134,1,1041,534,11.4545219,"\int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\frac{c e^{-\frac{i (b c+a d)}{d}} \sqrt{c+d x} \left(-\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{16 b}+\frac{c \left(2 \sqrt{5} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (5 (a+b x))-\sqrt{2 \pi } \cos \left(5 a-\frac{5 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \sin \left(5 a-\frac{5 b c}{d}\right)\right)}{160 \sqrt{5} b \sqrt{\frac{b}{d}}}-\frac{c \left(2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (3 (a+b x))-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \sin \left(3 a-\frac{3 b c}{d}\right)\right)}{96 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{\sqrt{\frac{b}{d}} d \left(\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(3 d \cos \left(a-\frac{b c}{d}\right)-2 b c \sin \left(a-\frac{b c}{d}\right)\right)+\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(a-\frac{b c}{d}\right)+3 d \sin \left(a-\frac{b c}{d}\right)\right)+2 \sqrt{\frac{b}{d}} d \sqrt{c+d x} (2 b x \cos (a+b x)-3 \sin (a+b x))\right)}{32 b^3}-\frac{\sqrt{\frac{b}{d}} d \left(\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(d \cos \left(3 a-\frac{3 b c}{d}\right)-2 b c \sin \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(3 a-\frac{3 b c}{d}\right)+d \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} d \sqrt{c+d x} (2 b x \cos (3 (a+b x))-\sin (3 (a+b x)))\right)}{192 \sqrt{3} b^3}+\frac{\sqrt{\frac{b}{d}} d \left(\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(3 d \cos \left(5 a-\frac{5 b c}{d}\right)-10 b c \sin \left(5 a-\frac{5 b c}{d}\right)\right)+\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(10 b c \cos \left(5 a-\frac{5 b c}{d}\right)+3 d \sin \left(5 a-\frac{5 b c}{d}\right)\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} d \sqrt{c+d x} (10 b x \cos (5 (a+b x))-3 \sin (5 (a+b x)))\right)}{1600 \sqrt{5} b^3}","\frac{3 \sqrt{\frac{\pi }{10}} d^{3/2} \sin \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{800 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{96 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{96 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{10}} d^{3/2} \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{800 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (a+b x)}{16 b^2}+\frac{d \sqrt{c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac{3 d \sqrt{c+d x} \sin (5 a+5 b x)}{800 b^2}-\frac{(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac{(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}",1,"(c*Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(16*b*E^((I*(b*c + a*d))/d)) + (c*(2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[5*(a + b*x)] - Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d]))/(160*Sqrt[5]*b*Sqrt[b/d]) - (c*(2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(3*d*Cos[a - (b*c)/d] - 2*b*c*Sin[a - (b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[a + b*x] - 3*Sin[a + b*x])))/(32*b^3) - (Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(d*Cos[3*a - (3*b*c)/d] - 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[3*(a + b*x)] - Sin[3*(a + b*x)])))/(192*Sqrt[3]*b^3) + (Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(3*d*Cos[5*a - (5*b*c)/d] - 10*b*c*Sin[5*a - (5*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(10*b*c*Cos[5*a - (5*b*c)/d] + 3*d*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*Sqrt[b/d]*d*Sqrt[c + d*x]*(10*b*x*Cos[5*(a + b*x)] - 3*Sin[5*(a + b*x)])))/(1600*Sqrt[5]*b^3)","C",0
135,1,3348,615,22.9659822,"\int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\text{Result too large to show}","-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{32 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{576 b^{7/2}}+\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \cos \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1600 b^{7/2}}-\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \sin \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1600 b^{7/2}}+\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{576 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{32 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (a+b x)}{32 b^3}+\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac{3 d^2 \sqrt{c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac{5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}",1,"(c^2*Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(16*b*E^((I*(b*c + a*d))/d)) + (c^2*(2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[5*(a + b*x)] - Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d]))/(160*Sqrt[5]*b*Sqrt[b/d]) - (c^2*(2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(3*d*Cos[a - (b*c)/d] - 2*b*c*Sin[a - (b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[a + b*x] - 3*Sin[a + b*x])))/(16*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*((4*b^2*c^2 - 15*d^2)*Cos[a - (b*c)/d] + 12*b*c*d*Sin[a - (b*c)/d]) - Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[a - (b*c)/d] + (4*b^2*c^2 - 15*d^2)*Sin[a - (b*c)/d]) - 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(d*(-15 + 4*b^2*x^2)*Cos[a + b*x] + 2*b*(c - 5*d*x)*Sin[a + b*x])))/(64*b^5) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(d*Cos[3*a - (3*b*c)/d] - 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[3*(a + b*x)] - Sin[3*(a + b*x)])))/(96*Sqrt[3]*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*((12*b^2*c^2 - 5*d^2)*Cos[3*a - (3*b*c)/d] + 12*b*c*d*Sin[3*a - (3*b*c)/d]) - Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[3*a - (3*b*c)/d] + (12*b^2*c^2 - 5*d^2)*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(d*(5 - 12*b^2*x^2)*Cos[3*(a + b*x)] - 2*b*(c - 5*d*x)*Sin[3*(a + b*x)])))/(1152*Sqrt[3]*b^5) + (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(3*d*Cos[5*a - (5*b*c)/d] - 10*b*c*Sin[5*a - (5*b*c)/d]) + Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(10*b*c*Cos[5*a - (5*b*c)/d] + 3*d*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*Sqrt[b/d]*d*Sqrt[c + d*x]*(10*b*x*Cos[5*(a + b*x)] - 3*Sin[5*(a + b*x)])))/(800*Sqrt[5]*b^3) - (d^2*(Sin[5*a]*((c^2*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]])*Sin[(5*b*c)/d])/(5*Sqrt[5]*(b/d)^(3/2)*d^3) + (c^2*Cos[(5*b*c)/d]*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/(5*Sqrt[5]*(b/d)^(3/2)*d^3) - (2*c*Cos[(5*b*c)/d]*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) - (2*c*Sin[(5*b*c)/d]*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) + (Sin[(5*b*c)/d]*(-25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Cos[(5*b*(c + d*x))/d] + (5*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3) + (Cos[(5*b*c)/d]*(25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Sin[(5*b*(c + d*x))/d] - (5*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3)) + Cos[5*a]*((c^2*Cos[(5*b*c)/d]*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/(5*Sqrt[5]*(b/d)^(3/2)*d^3) - (c^2*Sin[(5*b*c)/d]*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/(5*Sqrt[5]*(b/d)^(3/2)*d^3) + (2*c*Sin[(5*b*c)/d]*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) - (2*c*Cos[(5*b*c)/d]*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) + (Cos[(5*b*c)/d]*(-25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Cos[(5*b*(c + d*x))/d] + (5*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3) - (Sin[(5*b*c)/d]*(25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Sin[(5*b*(c + d*x))/d] - (5*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3))))/16","C",0
136,1,245,273,0.2368544,"\int (c+d x)^m \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]^3*Sin[a + b*x],x]","-\frac{4^{-m-3} e^{-\frac{4 i (a d+b c)}{d}} (c+d x)^m \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(2^{m+2} e^{2 i \left(a+\frac{3 b c}{d}\right)} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)+2^{m+2} e^{2 i \left(3 a+\frac{b c}{d}\right)} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)+e^{8 i a} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{4 i b (c+d x)}{d}\right)+e^{\frac{8 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{4 i b (c+d x)}{d}\right)\right)}{b}","-\frac{2^{-m-4} e^{2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)}{b}-\frac{2^{-2 (m+3)} e^{4 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{4 i b (c+d x)}{d}\right)}{b}-\frac{2^{-m-4} e^{-2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)}{b}-\frac{2^{-2 (m+3)} e^{-4 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{4 i b (c+d x)}{d}\right)}{b}",1,"-((4^(-3 - m)*(c + d*x)^m*(2^(2 + m)*E^((2*I)*(3*a + (b*c)/d))*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-2*I)*b*(c + d*x))/d] + 2^(2 + m)*E^((2*I)*(a + (3*b*c)/d))*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((2*I)*b*(c + d*x))/d] + E^((8*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-4*I)*b*(c + d*x))/d] + E^(((8*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((4*I)*b*(c + d*x))/d]))/(b*E^(((4*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^m))","A",1
137,1,158,260,1.8493693,"\int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]^3*Sin[a + b*x],x]","-\frac{-8 b d (c+d x) \sin (2 (a+b x)) \left(\cos (2 (a+b x)) \left(8 b^2 (c+d x)^2-3 d^2\right)+16 \left(2 b^2 (c+d x)^2-3 d^2\right)\right)+64 \cos (2 (a+b x)) \left(2 b^4 (c+d x)^4-6 b^2 d^2 (c+d x)^2+3 d^4\right)+\cos (4 (a+b x)) \left(32 b^4 (c+d x)^4-24 b^2 d^2 (c+d x)^2+3 d^4\right)}{1024 b^5}","-\frac{3 d^4 \cos ^4(a+b x)}{128 b^5}-\frac{45 d^4 \cos ^2(a+b x)}{128 b^5}-\frac{3 d^3 (c+d x) \sin (a+b x) \cos ^3(a+b x)}{32 b^4}-\frac{45 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{64 b^4}+\frac{3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}+\frac{9 d^2 (c+d x)^2 \cos ^2(a+b x)}{16 b^3}+\frac{d (c+d x)^3 \sin (a+b x) \cos ^3(a+b x)}{4 b^2}+\frac{3 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{8 b^2}-\frac{(c+d x)^4 \cos ^4(a+b x)}{4 b}-\frac{45 c d^3 x}{64 b^3}-\frac{45 d^4 x^2}{128 b^3}+\frac{3 (c+d x)^4}{32 b}",1,"-1/1024*(64*(3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Cos[2*(a + b*x)] + (3*d^4 - 24*b^2*d^2*(c + d*x)^2 + 32*b^4*(c + d*x)^4)*Cos[4*(a + b*x)] - 8*b*d*(c + d*x)*(16*(-3*d^2 + 2*b^2*(c + d*x)^2) + (-3*d^2 + 8*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[2*(a + b*x)])/b^5","A",1
138,1,135,196,0.9249087,"\int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x],x]","\frac{-64 b (c+d x) \cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-3 d^2\right)-4 b (c+d x) \cos (4 (a+b x)) \left(8 b^2 (c+d x)^2-3 d^2\right)+6 d \sin (2 (a+b x)) \left(\cos (2 (a+b x)) \left(8 b^2 (c+d x)^2-d^2\right)+16 \left(2 b^2 (c+d x)^2-d^2\right)\right)}{1024 b^4}","-\frac{3 d^3 \sin (a+b x) \cos ^3(a+b x)}{128 b^4}-\frac{45 d^3 \sin (a+b x) \cos (a+b x)}{256 b^4}+\frac{3 d^2 (c+d x) \cos ^4(a+b x)}{32 b^3}+\frac{9 d^2 (c+d x) \cos ^2(a+b x)}{32 b^3}+\frac{3 d (c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{16 b^2}+\frac{9 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{32 b^2}-\frac{(c+d x)^3 \cos ^4(a+b x)}{4 b}-\frac{45 d^3 x}{256 b^3}+\frac{3 (c+d x)^3}{32 b}",1,"(-64*b*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] - 4*b*(c + d*x)*(-3*d^2 + 8*b^2*(c + d*x)^2)*Cos[4*(a + b*x)] + 6*d*(16*(-d^2 + 2*b^2*(c + d*x)^2) + (-d^2 + 8*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[2*(a + b*x)])/(1024*b^4)","A",1
139,1,89,134,0.4620967,"\int (c+d x)^2 \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]^3*Sin[a + b*x],x]","\frac{-16 \cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-d^2\right)+\cos (4 (a+b x)) \left(d^2-8 b^2 (c+d x)^2\right)+4 b d (c+d x) (8 \sin (2 (a+b x))+\sin (4 (a+b x)))}{256 b^3}","\frac{d^2 \cos ^4(a+b x)}{32 b^3}+\frac{3 d^2 \cos ^2(a+b x)}{32 b^3}+\frac{d (c+d x) \sin (a+b x) \cos ^3(a+b x)}{8 b^2}+\frac{3 d (c+d x) \sin (a+b x) \cos (a+b x)}{16 b^2}-\frac{(c+d x)^2 \cos ^4(a+b x)}{4 b}+\frac{3 c d x}{16 b}+\frac{3 d^2 x^2}{32 b}",1,"(-16*(-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + (d^2 - 8*b^2*(c + d*x)^2)*Cos[4*(a + b*x)] + 4*b*d*(c + d*x)*(8*Sin[2*(a + b*x)] + Sin[4*(a + b*x)]))/(256*b^3)","A",1
140,1,75,72,0.1375041,"\int (c+d x) \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]^3*Sin[a + b*x],x]","\frac{d (\sin (2 (a+b x))-2 b x \cos (2 (a+b x)))}{16 b^2}+\frac{d (\sin (4 (a+b x))-4 b x \cos (4 (a+b x)))}{128 b^2}-\frac{c \cos ^4(a+b x)}{4 b}","\frac{d \sin (a+b x) \cos ^3(a+b x)}{16 b^2}+\frac{3 d \sin (a+b x) \cos (a+b x)}{32 b^2}-\frac{(c+d x) \cos ^4(a+b x)}{4 b}+\frac{3 d x}{32 b}",1,"-1/4*(c*Cos[a + b*x]^4)/b + (d*(-2*b*x*Cos[2*(a + b*x)] + Sin[2*(a + b*x)]))/(16*b^2) + (d*(-4*b*x*Cos[4*(a + b*x)] + Sin[4*(a + b*x)]))/(128*b^2)","A",1
141,1,110,129,0.3410741,"\int \frac{\cos ^3(a+b x) \sin (a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x])/(c + d*x),x]","\frac{\sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)+2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)+2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)}{8 d}","\frac{\sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{8 d}+\frac{\sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{4 d}+\frac{\cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{4 d}+\frac{\cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{8 d}",1,"(CosIntegral[(4*b*(c + d*x))/d]*Sin[4*a - (4*b*c)/d] + 2*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] + 2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/(8*d)","A",1
142,1,151,179,1.6309429,"\int \frac{\cos ^3(a+b x) \sin (a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x])/(c + d*x)^2,x]","-\frac{-4 b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-4 b \cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)+4 b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+4 b \sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)+\frac{2 d \sin (2 (a+b x))}{c+d x}+\frac{d \sin (4 (a+b x))}{c+d x}}{8 d^2}","\frac{b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{2 d^2}+\frac{b \cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{2 d^2}-\frac{b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{2 d^2}-\frac{b \sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{2 d^2}-\frac{\sin (2 a+2 b x)}{4 d (c+d x)}-\frac{\sin (4 a+4 b x)}{8 d (c+d x)}",1,"-1/8*(-4*b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] - 4*b*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*(c + d*x))/d] + (2*d*Sin[2*(a + b*x)])/(c + d*x) + (d*Sin[4*(a + b*x)])/(c + d*x) + 4*b*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + 4*b*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/d^2","A",1
143,1,197,231,3.7535343,"\int \frac{\cos ^3(a+b x) \sin (a+b x)}{(c+d x)^3} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x])/(c + d*x)^3,x]","-\frac{16 b^2 \sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)+8 b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)+8 b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+16 b^2 \cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)+\frac{2 d (2 b (c+d x) \cos (2 (a+b x))+d \sin (2 (a+b x)))}{(c+d x)^2}+\frac{d (4 b (c+d x) \cos (4 (a+b x))+d \sin (4 (a+b x)))}{(c+d x)^2}}{16 d^3}","-\frac{b^2 \sin \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{d^3}-\frac{b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{2 d^3}-\frac{b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{2 d^3}-\frac{b^2 \cos \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{d^3}-\frac{b \cos (2 a+2 b x)}{4 d^2 (c+d x)}-\frac{b \cos (4 a+4 b x)}{4 d^2 (c+d x)}-\frac{\sin (2 a+2 b x)}{8 d (c+d x)^2}-\frac{\sin (4 a+4 b x)}{16 d (c+d x)^2}",1,"-1/16*(16*b^2*CosIntegral[(4*b*(c + d*x))/d]*Sin[4*a - (4*b*c)/d] + 8*b^2*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] + (2*d*(2*b*(c + d*x)*Cos[2*(a + b*x)] + d*Sin[2*(a + b*x)]))/(c + d*x)^2 + (d*(4*b*(c + d*x)*Cos[4*(a + b*x)] + d*Sin[4*(a + b*x)]))/(c + d*x)^2 + 8*b^2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + 16*b^2*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/d^3","A",1
144,1,316,287,2.4897001,"\int \frac{\cos ^3(a+b x) \sin (a+b x)}{(c+d x)^4} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x])/(c + d*x)^4,x]","-\frac{8 b^3 (c+d x)^3 \left(\cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-\sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)\right)+32 b^3 (c+d x)^3 \left(\cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b (c+d x)}{d}\right)-\sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b (c+d x)}{d}\right)\right)+2 d \cos (2 b x) \left(\sin (2 a) \left(d^2-2 b^2 (c+d x)^2\right)+b d \cos (2 a) (c+d x)\right)+d \cos (4 b x) \left(\sin (4 a) \left(d^2-8 b^2 (c+d x)^2\right)+2 b d \cos (4 a) (c+d x)\right)-2 d \sin (2 b x) \left(\cos (2 a) \left(2 b^2 (c+d x)^2-d^2\right)+b d \sin (2 a) (c+d x)\right)-d \sin (4 b x) \left(\cos (4 a) \left(8 b^2 (c+d x)^2-d^2\right)+2 b d \sin (4 a) (c+d x)\right)}{24 d^4 (c+d x)^3}","-\frac{b^3 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{3 d^4}-\frac{4 b^3 \cos \left(4 a-\frac{4 b c}{d}\right) \text{Ci}\left(\frac{4 b c}{d}+4 b x\right)}{3 d^4}+\frac{b^3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{3 d^4}+\frac{4 b^3 \sin \left(4 a-\frac{4 b c}{d}\right) \text{Si}\left(\frac{4 b c}{d}+4 b x\right)}{3 d^4}+\frac{b^2 \sin (2 a+2 b x)}{6 d^3 (c+d x)}+\frac{b^2 \sin (4 a+4 b x)}{3 d^3 (c+d x)}-\frac{b \cos (2 a+2 b x)}{12 d^2 (c+d x)^2}-\frac{b \cos (4 a+4 b x)}{12 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{12 d (c+d x)^3}-\frac{\sin (4 a+4 b x)}{24 d (c+d x)^3}",1,"-1/24*(2*d*Cos[2*b*x]*(b*d*(c + d*x)*Cos[2*a] + (d^2 - 2*b^2*(c + d*x)^2)*Sin[2*a]) + d*Cos[4*b*x]*(2*b*d*(c + d*x)*Cos[4*a] + (d^2 - 8*b^2*(c + d*x)^2)*Sin[4*a]) - 2*d*((-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*a] + b*d*(c + d*x)*Sin[2*a])*Sin[2*b*x] - d*((-d^2 + 8*b^2*(c + d*x)^2)*Cos[4*a] + 2*b*d*(c + d*x)*Sin[4*a])*Sin[4*b*x] + 8*b^3*(c + d*x)^3*(Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] - Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d]) + 32*b^3*(c + d*x)^3*(Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*(c + d*x))/d] - Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d]))/(d^4*(c + d*x)^3)","A",1
145,1,409,419,0.5727299,"\int (c+d x)^m \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]^3*Sin[a + b*x]^2,x]","-\frac{i 3^{-m-1} e^{-\frac{3 i (a d+b c)}{d}} (c+d x)^m \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(e^{\frac{6 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{3 i b (c+d x)}{d}\right)-e^{6 i a} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{3 i b (c+d x)}{d}\right)\right)}{32 b}-\frac{i 5^{-m-1} e^{-\frac{5 i (a d+b c)}{d}} (c+d x)^m \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(e^{\frac{10 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{5 i b (c+d x)}{d}\right)-e^{10 i a} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{5 i b (c+d x)}{d}\right)\right)}{32 b}-\frac{i e^{-\frac{i (a d+b c)}{d}} (c+d x)^m \left(e^{2 i a} \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)-e^{\frac{2 i b c}{d}} \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)\right)}{16 b}","-\frac{i e^{i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)}{16 b}+\frac{i 3^{-m-1} e^{3 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{3 i b (c+d x)}{d}\right)}{32 b}+\frac{i 5^{-m-1} e^{5 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{5 i b (c+d x)}{d}\right)}{32 b}+\frac{i e^{-i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)}{16 b}-\frac{i 3^{-m-1} e^{-3 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{3 i b (c+d x)}{d}\right)}{32 b}-\frac{i 5^{-m-1} e^{-5 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{5 i b (c+d x)}{d}\right)}{32 b}",1,"((-1/16*I)*(c + d*x)^m*((E^((2*I)*a)*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^m - (E^(((2*I)*b*c)/d)*Gamma[1 + m, (I*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^m))/(b*E^((I*(b*c + a*d))/d)) - ((I/32)*3^(-1 - m)*(c + d*x)^m*(-(E^((6*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d]) + E^(((6*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((3*I)*b*(c + d*x))/d]))/(b*E^(((3*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^m) - ((I/32)*5^(-1 - m)*(c + d*x)^m*(-(E^((10*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-5*I)*b*(c + d*x))/d]) + E^(((10*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((5*I)*b*(c + d*x))/d]))/(b*E^(((5*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^m)","A",1
146,1,563,330,3.4801057,"\int (c+d x)^4 \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]^3*Sin[a + b*x]^2,x]","-\frac{-506250 b^4 c^4 \sin (a+b x)+84375 b^4 c^4 \sin (3 (a+b x))+50625 b^4 c^4 \sin (5 (a+b x))-2025000 b^3 c^3 d (b x \sin (a+b x)+\cos (a+b x))+112500 b^3 c^3 d (3 b x \sin (3 (a+b x))+\cos (3 (a+b x)))+40500 b^3 c^3 d (5 b x \sin (5 (a+b x))+\cos (5 (a+b x)))-3037500 b^2 c^2 d^2 \left(\left(b^2 x^2-2\right) \sin (a+b x)+2 b x \cos (a+b x)\right)+56250 b^2 c^2 d^2 \left(\left(9 b^2 x^2-2\right) \sin (3 (a+b x))+6 b x \cos (3 (a+b x))\right)+12150 b^2 c^2 d^2 \left(\left(25 b^2 x^2-2\right) \sin (5 (a+b x))+10 b x \cos (5 (a+b x))\right)-2025000 b c d^3 \left(b x \left(b^2 x^2-6\right) \sin (a+b x)+3 \left(b^2 x^2-2\right) \cos (a+b x)\right)+37500 b c d^3 \left(3 b x \left(3 b^2 x^2-2\right) \sin (3 (a+b x))+\left(9 b^2 x^2-2\right) \cos (3 (a+b x))\right)+1620 b c d^3 \left(5 b x \left(25 b^2 x^2-6\right) \sin (5 (a+b x))+\left(75 b^2 x^2-6\right) \cos (5 (a+b x))\right)-506250 d^4 \left(4 b x \left(b^2 x^2-6\right) \cos (a+b x)+\left(b^4 x^4-12 b^2 x^2+24\right) \sin (a+b x)\right)+3125 d^4 \left(12 b x \left(3 b^2 x^2-2\right) \cos (3 (a+b x))+\left(27 b^4 x^4-36 b^2 x^2+8\right) \sin (3 (a+b x))\right)+81 d^4 \left(20 b x \left(25 b^2 x^2-6\right) \cos (5 (a+b x))+\left(625 b^4 x^4-300 b^2 x^2+24\right) \sin (5 (a+b x))\right)}{4050000 b^5}","\frac{3 d^4 \sin (a+b x)}{b^5}-\frac{d^4 \sin (3 a+3 b x)}{162 b^5}-\frac{3 d^4 \sin (5 a+5 b x)}{6250 b^5}-\frac{3 d^3 (c+d x) \cos (a+b x)}{b^4}+\frac{d^3 (c+d x) \cos (3 a+3 b x)}{54 b^4}+\frac{3 d^3 (c+d x) \cos (5 a+5 b x)}{1250 b^4}-\frac{3 d^2 (c+d x)^2 \sin (a+b x)}{2 b^3}+\frac{d^2 (c+d x)^2 \sin (3 a+3 b x)}{36 b^3}+\frac{3 d^2 (c+d x)^2 \sin (5 a+5 b x)}{500 b^3}+\frac{d (c+d x)^3 \cos (a+b x)}{2 b^2}-\frac{d (c+d x)^3 \cos (3 a+3 b x)}{36 b^2}-\frac{d (c+d x)^3 \cos (5 a+5 b x)}{100 b^2}+\frac{(c+d x)^4 \sin (a+b x)}{8 b}-\frac{(c+d x)^4 \sin (3 a+3 b x)}{48 b}-\frac{(c+d x)^4 \sin (5 a+5 b x)}{80 b}",1,"-1/4050000*(-506250*b^4*c^4*Sin[a + b*x] - 2025000*b^3*c^3*d*(Cos[a + b*x] + b*x*Sin[a + b*x]) - 2025000*b*c*d^3*(3*(-2 + b^2*x^2)*Cos[a + b*x] + b*x*(-6 + b^2*x^2)*Sin[a + b*x]) - 3037500*b^2*c^2*d^2*(2*b*x*Cos[a + b*x] + (-2 + b^2*x^2)*Sin[a + b*x]) - 506250*d^4*(4*b*x*(-6 + b^2*x^2)*Cos[a + b*x] + (24 - 12*b^2*x^2 + b^4*x^4)*Sin[a + b*x]) + 84375*b^4*c^4*Sin[3*(a + b*x)] + 112500*b^3*c^3*d*(Cos[3*(a + b*x)] + 3*b*x*Sin[3*(a + b*x)]) + 37500*b*c*d^3*((-2 + 9*b^2*x^2)*Cos[3*(a + b*x)] + 3*b*x*(-2 + 3*b^2*x^2)*Sin[3*(a + b*x)]) + 56250*b^2*c^2*d^2*(6*b*x*Cos[3*(a + b*x)] + (-2 + 9*b^2*x^2)*Sin[3*(a + b*x)]) + 3125*d^4*(12*b*x*(-2 + 3*b^2*x^2)*Cos[3*(a + b*x)] + (8 - 36*b^2*x^2 + 27*b^4*x^4)*Sin[3*(a + b*x)]) + 50625*b^4*c^4*Sin[5*(a + b*x)] + 40500*b^3*c^3*d*(Cos[5*(a + b*x)] + 5*b*x*Sin[5*(a + b*x)]) + 1620*b*c*d^3*((-6 + 75*b^2*x^2)*Cos[5*(a + b*x)] + 5*b*x*(-6 + 25*b^2*x^2)*Sin[5*(a + b*x)]) + 12150*b^2*c^2*d^2*(10*b*x*Cos[5*(a + b*x)] + (-2 + 25*b^2*x^2)*Sin[5*(a + b*x)]) + 81*d^4*(20*b*x*(-6 + 25*b^2*x^2)*Cos[5*(a + b*x)] + (24 - 300*b^2*x^2 + 625*b^4*x^4)*Sin[5*(a + b*x)]))/b^5","A",1
147,1,195,259,2.1808856,"\int (c+d x)^3 \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x]^2,x]","-\frac{30 b (c+d x) \sin (a+b x) \left(8 \cos (2 (a+b x)) \left(75 b^2 (c+d x)^2-38 d^2\right)+9 \cos (4 (a+b x)) \left(25 b^2 (c+d x)^2-6 d^2\right)-825 b^2 c^2-1650 b^2 c d x-825 b^2 d^2 x^2+6598 d^2\right)-101250 d \cos (a+b x) \left(b^2 (c+d x)^2-2 d^2\right)+625 d \cos (3 (a+b x)) \left(9 b^2 (c+d x)^2-2 d^2\right)+81 d \cos (5 (a+b x)) \left(25 b^2 (c+d x)^2-2 d^2\right)}{270000 b^4}","-\frac{3 d^3 \cos (a+b x)}{4 b^4}+\frac{d^3 \cos (3 a+3 b x)}{216 b^4}+\frac{3 d^3 \cos (5 a+5 b x)}{5000 b^4}-\frac{3 d^2 (c+d x) \sin (a+b x)}{4 b^3}+\frac{d^2 (c+d x) \sin (3 a+3 b x)}{72 b^3}+\frac{3 d^2 (c+d x) \sin (5 a+5 b x)}{1000 b^3}+\frac{3 d (c+d x)^2 \cos (a+b x)}{8 b^2}-\frac{d (c+d x)^2 \cos (3 a+3 b x)}{48 b^2}-\frac{3 d (c+d x)^2 \cos (5 a+5 b x)}{400 b^2}+\frac{(c+d x)^3 \sin (a+b x)}{8 b}-\frac{(c+d x)^3 \sin (3 a+3 b x)}{48 b}-\frac{(c+d x)^3 \sin (5 a+5 b x)}{80 b}",1,"-1/270000*(-101250*d*(-2*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + 625*d*(-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] + 81*d*(-2*d^2 + 25*b^2*(c + d*x)^2)*Cos[5*(a + b*x)] + 30*b*(c + d*x)*(-825*b^2*c^2 + 6598*d^2 - 1650*b^2*c*d*x - 825*b^2*d^2*x^2 + 8*(-38*d^2 + 75*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + 9*(-6*d^2 + 25*b^2*(c + d*x)^2)*Cos[4*(a + b*x)])*Sin[a + b*x])/b^4","A",1
148,1,252,184,0.9603893,"\int (c+d x)^2 \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]^3*Sin[a + b*x]^2,x]","-\frac{-6750 b^2 c^2 \sin (a+b x)+1125 b^2 c^2 \sin (3 (a+b x))+675 b^2 c^2 \sin (5 (a+b x))-13500 b^2 c d x \sin (a+b x)+2250 b^2 c d x \sin (3 (a+b x))+1350 b^2 c d x \sin (5 (a+b x))-6750 b^2 d^2 x^2 \sin (a+b x)+1125 b^2 d^2 x^2 \sin (3 (a+b x))+675 b^2 d^2 x^2 \sin (5 (a+b x))-13500 b d (c+d x) \cos (a+b x)+750 b d (c+d x) \cos (3 (a+b x))+270 b c d \cos (5 (a+b x))+13500 d^2 \sin (a+b x)-250 d^2 \sin (3 (a+b x))-54 d^2 \sin (5 (a+b x))+270 b d^2 x \cos (5 (a+b x))}{54000 b^3}","-\frac{d^2 \sin (a+b x)}{4 b^3}+\frac{d^2 \sin (3 a+3 b x)}{216 b^3}+\frac{d^2 \sin (5 a+5 b x)}{1000 b^3}+\frac{d (c+d x) \cos (a+b x)}{4 b^2}-\frac{d (c+d x) \cos (3 a+3 b x)}{72 b^2}-\frac{d (c+d x) \cos (5 a+5 b x)}{200 b^2}+\frac{(c+d x)^2 \sin (a+b x)}{8 b}-\frac{(c+d x)^2 \sin (3 a+3 b x)}{48 b}-\frac{(c+d x)^2 \sin (5 a+5 b x)}{80 b}",1,"-1/54000*(-13500*b*d*(c + d*x)*Cos[a + b*x] + 750*b*d*(c + d*x)*Cos[3*(a + b*x)] + 270*b*c*d*Cos[5*(a + b*x)] + 270*b*d^2*x*Cos[5*(a + b*x)] - 6750*b^2*c^2*Sin[a + b*x] + 13500*d^2*Sin[a + b*x] - 13500*b^2*c*d*x*Sin[a + b*x] - 6750*b^2*d^2*x^2*Sin[a + b*x] + 1125*b^2*c^2*Sin[3*(a + b*x)] - 250*d^2*Sin[3*(a + b*x)] + 2250*b^2*c*d*x*Sin[3*(a + b*x)] + 1125*b^2*d^2*x^2*Sin[3*(a + b*x)] + 675*b^2*c^2*Sin[5*(a + b*x)] - 54*d^2*Sin[5*(a + b*x)] + 1350*b^2*c*d*x*Sin[5*(a + b*x)] + 675*b^2*d^2*x^2*Sin[5*(a + b*x)])/b^3","A",1
149,1,110,109,0.3568214,"\int (c+d x) \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]^3*Sin[a + b*x]^2,x]","-\frac{-450 b c \sin (a+b x)+75 b c \sin (3 (a+b x))+45 b c \sin (5 (a+b x))-450 b d x \sin (a+b x)+75 b d x \sin (3 (a+b x))+45 b d x \sin (5 (a+b x))-450 d \cos (a+b x)+25 d \cos (3 (a+b x))+9 d \cos (5 (a+b x))}{3600 b^2}","\frac{d \cos (a+b x)}{8 b^2}-\frac{d \cos (3 a+3 b x)}{144 b^2}-\frac{d \cos (5 a+5 b x)}{400 b^2}+\frac{(c+d x) \sin (a+b x)}{8 b}-\frac{(c+d x) \sin (3 a+3 b x)}{48 b}-\frac{(c+d x) \sin (5 a+5 b x)}{80 b}",1,"-1/3600*(-450*d*Cos[a + b*x] + 25*d*Cos[3*(a + b*x)] + 9*d*Cos[5*(a + b*x)] - 450*b*c*Sin[a + b*x] - 450*b*d*x*Sin[a + b*x] + 75*b*c*Sin[3*(a + b*x)] + 75*b*d*x*Sin[3*(a + b*x)] + 45*b*c*Sin[5*(a + b*x)] + 45*b*d*x*Sin[5*(a + b*x)])/b^2","A",1
150,1,154,185,0.5071445,"\int \frac{\cos ^3(a+b x) \sin ^2(a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x]^2)/(c + d*x),x]","\frac{2 \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)-\cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)-\cos \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b (c+d x)}{d}\right)-2 \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+\sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)+\sin \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b (c+d x)}{d}\right)}{16 d}","\frac{\cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{8 d}-\frac{\cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{16 d}-\frac{\cos \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b c}{d}+5 b x\right)}{16 d}-\frac{\sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{8 d}+\frac{\sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{16 d}+\frac{\sin \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b c}{d}+5 b x\right)}{16 d}",1,"(2*Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] - Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] - Cos[5*a - (5*b*c)/d]*CosIntegral[(5*b*(c + d*x))/d] - 2*Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d] + Sin[5*a - (5*b*c)/d]*SinIntegral[(5*b*(c + d*x))/d])/(16*d)","A",1
151,1,212,257,2.1053606,"\int \frac{\cos ^3(a+b x) \sin ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x]^2)/(c + d*x)^2,x]","\frac{-2 \left(b \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+b \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)+\frac{d \cos (a+b x)}{c+d x}\right)+5 b \sin \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b (c+d x)}{d}\right)+3 b \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)+3 b \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)+5 b \cos \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b (c+d x)}{d}\right)+\frac{d \cos (3 (a+b x))}{c+d x}+\frac{d \cos (5 (a+b x))}{c+d x}}{16 d^2}","\frac{5 b \sin \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b c}{d}+5 b x\right)}{16 d^2}+\frac{3 b \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{16 d^2}-\frac{b \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{8 d^2}-\frac{b \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{8 d^2}+\frac{3 b \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{16 d^2}+\frac{5 b \cos \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b c}{d}+5 b x\right)}{16 d^2}-\frac{\cos (a+b x)}{8 d (c+d x)}+\frac{\cos (3 a+3 b x)}{16 d (c+d x)}+\frac{\cos (5 a+5 b x)}{16 d (c+d x)}",1,"((d*Cos[3*(a + b*x)])/(c + d*x) + (d*Cos[5*(a + b*x)])/(c + d*x) + 5*b*CosIntegral[(5*b*(c + d*x))/d]*Sin[5*a - (5*b*c)/d] + 3*b*CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] - 2*((d*Cos[a + b*x])/(c + d*x) + b*CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + b*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)]) + 3*b*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d] + 5*b*Cos[5*a - (5*b*c)/d]*SinIntegral[(5*b*(c + d*x))/d])/(16*d^2)","A",1
152,1,283,338,3.3731958,"\int \frac{\cos ^3(a+b x) \sin ^2(a+b x)}{(c+d x)^3} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x]^2)/(c + d*x)^3,x]","\frac{-2 b^2 \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+9 b^2 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)+25 b^2 \cos \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b (c+d x)}{d}\right)+2 b^2 \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)-9 b^2 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)-25 b^2 \sin \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b (c+d x)}{d}\right)+\frac{d^2 \cos (3 (a+b x))}{(c+d x)^2}+\frac{d^2 \cos (5 (a+b x))}{(c+d x)^2}-\frac{3 b d \sin (3 (a+b x))}{c+d x}-\frac{5 b d \sin (5 (a+b x))}{c+d x}+\frac{2 d (b (c+d x) \sin (a+b x)-d \cos (a+b x))}{(c+d x)^2}}{32 d^3}","-\frac{b^2 \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{16 d^3}+\frac{9 b^2 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{32 d^3}+\frac{25 b^2 \cos \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b c}{d}+5 b x\right)}{32 d^3}+\frac{b^2 \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{16 d^3}-\frac{9 b^2 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{32 d^3}-\frac{25 b^2 \sin \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b c}{d}+5 b x\right)}{32 d^3}+\frac{b \sin (a+b x)}{16 d^2 (c+d x)}-\frac{3 b \sin (3 a+3 b x)}{32 d^2 (c+d x)}-\frac{5 b \sin (5 a+5 b x)}{32 d^2 (c+d x)}-\frac{\cos (a+b x)}{16 d (c+d x)^2}+\frac{\cos (3 a+3 b x)}{32 d (c+d x)^2}+\frac{\cos (5 a+5 b x)}{32 d (c+d x)^2}",1,"((d^2*Cos[3*(a + b*x)])/(c + d*x)^2 + (d^2*Cos[5*(a + b*x)])/(c + d*x)^2 - 2*b^2*Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] + 9*b^2*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] + 25*b^2*Cos[5*a - (5*b*c)/d]*CosIntegral[(5*b*(c + d*x))/d] + (2*d*(-(d*Cos[a + b*x]) + b*(c + d*x)*Sin[a + b*x]))/(c + d*x)^2 - (3*b*d*Sin[3*(a + b*x)])/(c + d*x) - (5*b*d*Sin[5*(a + b*x)])/(c + d*x) + 2*b^2*Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)] - 9*b^2*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d] - 25*b^2*Sin[5*a - (5*b*c)/d]*SinIntegral[(5*b*(c + d*x))/d])/(32*d^3)","A",1
153,1,451,413,3.4018206,"\int \frac{\cos ^3(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x]^2)/(c + d*x)^4,x]","-\frac{27 b^3 (c+d x)^3 \left(\sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b (c+d x)}{d}\right)+\cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b (c+d x)}{d}\right)\right)+125 b^3 (c+d x)^3 \left(\sin \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b (c+d x)}{d}\right)+\cos \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b (c+d x)}{d}\right)\right)+d \cos (3 b x) \left(\cos (3 a) \left(9 b^2 (c+d x)^2-2 d^2\right)+3 b d \sin (3 a) (c+d x)\right)+d \cos (5 b x) \left(\cos (5 a) \left(25 b^2 (c+d x)^2-2 d^2\right)+5 b d \sin (5 a) (c+d x)\right)+d \sin (3 b x) \left(3 b d \cos (3 a) (c+d x)-\sin (3 a) \left(9 b^2 (c+d x)^2-2 d^2\right)\right)+d \sin (5 b x) \left(5 b d \cos (5 a) (c+d x)-\sin (5 a) \left(25 b^2 (c+d x)^2-2 d^2\right)\right)-2 \left(b^3 (c+d x)^3 \left(\sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(b \left(\frac{c}{d}+x\right)\right)+\cos \left(a-\frac{b c}{d}\right) \text{Si}\left(b \left(\frac{c}{d}+x\right)\right)\right)+d \cos (b x) \left(\cos (a) \left(b^2 (c+d x)^2-2 d^2\right)+b d \sin (a) (c+d x)\right)+d \sin (b x) \left(b d \cos (a) (c+d x)-\sin (a) \left(b^2 (c+d x)^2-2 d^2\right)\right)\right)}{96 d^4 (c+d x)^3}","-\frac{125 b^3 \sin \left(5 a-\frac{5 b c}{d}\right) \text{Ci}\left(\frac{5 b c}{d}+5 b x\right)}{96 d^4}-\frac{9 b^3 \sin \left(3 a-\frac{3 b c}{d}\right) \text{Ci}\left(\frac{3 b c}{d}+3 b x\right)}{32 d^4}+\frac{b^3 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{48 d^4}+\frac{b^3 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{48 d^4}-\frac{9 b^3 \cos \left(3 a-\frac{3 b c}{d}\right) \text{Si}\left(\frac{3 b c}{d}+3 b x\right)}{32 d^4}-\frac{125 b^3 \cos \left(5 a-\frac{5 b c}{d}\right) \text{Si}\left(\frac{5 b c}{d}+5 b x\right)}{96 d^4}+\frac{b^2 \cos (a+b x)}{48 d^3 (c+d x)}-\frac{3 b^2 \cos (3 a+3 b x)}{32 d^3 (c+d x)}-\frac{25 b^2 \cos (5 a+5 b x)}{96 d^3 (c+d x)}+\frac{b \sin (a+b x)}{48 d^2 (c+d x)^2}-\frac{b \sin (3 a+3 b x)}{32 d^2 (c+d x)^2}-\frac{5 b \sin (5 a+5 b x)}{96 d^2 (c+d x)^2}-\frac{\cos (a+b x)}{24 d (c+d x)^3}+\frac{\cos (3 a+3 b x)}{48 d (c+d x)^3}+\frac{\cos (5 a+5 b x)}{48 d (c+d x)^3}",1,"-1/96*(d*Cos[3*b*x]*((-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*a] + 3*b*d*(c + d*x)*Sin[3*a]) + d*Cos[5*b*x]*((-2*d^2 + 25*b^2*(c + d*x)^2)*Cos[5*a] + 5*b*d*(c + d*x)*Sin[5*a]) + d*(3*b*d*(c + d*x)*Cos[3*a] - (-2*d^2 + 9*b^2*(c + d*x)^2)*Sin[3*a])*Sin[3*b*x] + d*(5*b*d*(c + d*x)*Cos[5*a] - (-2*d^2 + 25*b^2*(c + d*x)^2)*Sin[5*a])*Sin[5*b*x] - 2*(d*Cos[b*x]*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] + b*d*(c + d*x)*Sin[a]) + d*(b*d*(c + d*x)*Cos[a] - (-2*d^2 + b^2*(c + d*x)^2)*Sin[a])*Sin[b*x] + b^3*(c + d*x)^3*(CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)])) + 27*b^3*(c + d*x)^3*(CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] + Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d]) + 125*b^3*(c + d*x)^3*(CosIntegral[(5*b*(c + d*x))/d]*Sin[5*a - (5*b*c)/d] + Cos[5*a - (5*b*c)/d]*SinIntegral[(5*b*(c + d*x))/d]))/(d^4*(c + d*x)^3)","A",1
154,1,255,285,3.3383533,"\int (c+d x)^m \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{2^{-m-7} 3^{-m-1} e^{-\frac{6 i (a d+b c)}{d}} (c+d x)^m \left(\frac{b^2 (c+d x)^2}{d^2}\right)^{-m} \left(-3^{m+2} e^{4 i a+\frac{8 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)-3^{m+2} e^{4 i \left(2 a+\frac{b c}{d}\right)} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)+e^{12 i a} \left(\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,-\frac{6 i b (c+d x)}{d}\right)+e^{\frac{12 i b c}{d}} \left(-\frac{i b (c+d x)}{d}\right)^m \Gamma \left(m+1,\frac{6 i b (c+d x)}{d}\right)\right)}{b}","-\frac{3\ 2^{-m-7} e^{2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)}{b}+\frac{2^{-m-7} 3^{-m-1} e^{6 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{6 i b (c+d x)}{d}\right)}{b}-\frac{3\ 2^{-m-7} e^{-2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)}{b}+\frac{2^{-m-7} 3^{-m-1} e^{-6 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{6 i b (c+d x)}{d}\right)}{b}",1,"(2^(-7 - m)*3^(-1 - m)*(c + d*x)^m*(-(3^(2 + m)*E^((4*I)*(2*a + (b*c)/d))*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-2*I)*b*(c + d*x))/d]) - 3^(2 + m)*E^((4*I)*a + ((8*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((2*I)*b*(c + d*x))/d] + E^((12*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-6*I)*b*(c + d*x))/d] + E^(((12*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 + m, ((6*I)*b*(c + d*x))/d]))/(b*E^(((6*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^m)","A",1
155,1,153,233,1.5366054,"\int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{-12 b d (c+d x) \sin (2 (a+b x)) \left(\cos (4 (a+b x)) \left(6 b^2 (c+d x)^2-d^2\right)-78 b^2 (c+d x)^2+121 d^2\right)-243 \cos (2 (a+b x)) \left(2 b^4 (c+d x)^4-6 b^2 d^2 (c+d x)^2+3 d^4\right)+\cos (6 (a+b x)) \left(54 b^4 (c+d x)^4-18 b^2 d^2 (c+d x)^2+d^4\right)}{10368 b^5}","-\frac{9 d^4 \cos (2 a+2 b x)}{128 b^5}+\frac{d^4 \cos (6 a+6 b x)}{10368 b^5}-\frac{9 d^3 (c+d x) \sin (2 a+2 b x)}{64 b^4}+\frac{d^3 (c+d x) \sin (6 a+6 b x)}{1728 b^4}+\frac{9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac{d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac{3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}-\frac{d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}-\frac{3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^4 \cos (6 a+6 b x)}{192 b}",1,"(-243*(3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Cos[2*(a + b*x)] + (d^4 - 18*b^2*d^2*(c + d*x)^2 + 54*b^4*(c + d*x)^4)*Cos[6*(a + b*x)] - 12*b*d*(c + d*x)*(121*d^2 - 78*b^2*(c + d*x)^2 + (-d^2 + 6*b^2*(c + d*x)^2)*Cos[4*(a + b*x)])*Sin[2*(a + b*x)])/(10368*b^5)","A",1
156,1,132,181,2.3131175,"\int (c+d x)^3 \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{-324 b (c+d x) \cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-3 d^2\right)+12 b (c+d x) \cos (6 (a+b x)) \left(6 b^2 (c+d x)^2-d^2\right)-4 d \sin (2 (a+b x)) \left(\cos (4 (a+b x)) \left(18 b^2 (c+d x)^2-d^2\right)-234 b^2 (c+d x)^2+121 d^2\right)}{13824 b^4}","-\frac{9 d^3 \sin (2 a+2 b x)}{256 b^4}+\frac{d^3 \sin (6 a+6 b x)}{6912 b^4}+\frac{9 d^2 (c+d x) \cos (2 a+2 b x)}{128 b^3}-\frac{d^2 (c+d x) \cos (6 a+6 b x)}{1152 b^3}+\frac{9 d (c+d x)^2 \sin (2 a+2 b x)}{128 b^2}-\frac{d (c+d x)^2 \sin (6 a+6 b x)}{384 b^2}-\frac{3 (c+d x)^3 \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^3 \cos (6 a+6 b x)}{192 b}",1,"(-324*b*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + 12*b*(c + d*x)*(-d^2 + 6*b^2*(c + d*x)^2)*Cos[6*(a + b*x)] - 4*d*(121*d^2 - 234*b^2*(c + d*x)^2 + (-d^2 + 18*b^2*(c + d*x)^2)*Cos[4*(a + b*x)])*Sin[2*(a + b*x)])/(13824*b^4)","A",1
157,1,91,129,0.5626437,"\int (c+d x)^2 \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{-81 \cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-d^2\right)+\cos (6 (a+b x)) \left(18 b^2 (c+d x)^2-d^2\right)-6 b d (c+d x) (\sin (6 (a+b x))-27 \sin (2 (a+b x)))}{3456 b^3}","\frac{3 d^2 \cos (2 a+2 b x)}{128 b^3}-\frac{d^2 \cos (6 a+6 b x)}{3456 b^3}+\frac{3 d (c+d x) \sin (2 a+2 b x)}{64 b^2}-\frac{d (c+d x) \sin (6 a+6 b x)}{576 b^2}-\frac{3 (c+d x)^2 \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^2 \cos (6 a+6 b x)}{192 b}",1,"(-81*(-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + (-d^2 + 18*b^2*(c + d*x)^2)*Cos[6*(a + b*x)] - 6*b*d*(c + d*x)*(-27*Sin[2*(a + b*x)] + Sin[6*(a + b*x)]))/(3456*b^3)","A",1
158,1,63,77,0.2191625,"\int (c+d x) \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{-54 b (c+d x) \cos (2 (a+b x))+6 b (c+d x) \cos (6 (a+b x))+d (27 \sin (2 (a+b x))-\sin (6 (a+b x)))}{1152 b^2}","\frac{3 d \sin (2 a+2 b x)}{128 b^2}-\frac{d \sin (6 a+6 b x)}{1152 b^2}-\frac{3 (c+d x) \cos (2 a+2 b x)}{64 b}+\frac{(c+d x) \cos (6 a+6 b x)}{192 b}",1,"(-54*b*(c + d*x)*Cos[2*(a + b*x)] + 6*b*(c + d*x)*Cos[6*(a + b*x)] + d*(27*Sin[2*(a + b*x)] - Sin[6*(a + b*x)]))/(1152*b^2)","A",1
159,1,110,129,0.3080745,"\int \frac{\cos ^3(a+b x) \sin ^3(a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x]^3)/(c + d*x),x]","-\frac{\sin \left(6 a-\frac{6 b c}{d}\right) \text{Ci}\left(\frac{6 b (c+d x)}{d}\right)-3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-3 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\cos \left(6 a-\frac{6 b c}{d}\right) \text{Si}\left(\frac{6 b (c+d x)}{d}\right)}{32 d}","-\frac{\sin \left(6 a-\frac{6 b c}{d}\right) \text{Ci}\left(\frac{6 b c}{d}+6 b x\right)}{32 d}+\frac{3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{32 d}+\frac{3 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{32 d}-\frac{\cos \left(6 a-\frac{6 b c}{d}\right) \text{Si}\left(\frac{6 b c}{d}+6 b x\right)}{32 d}",1,"-1/32*(CosIntegral[(6*b*(c + d*x))/d]*Sin[6*a - (6*b*c)/d] - 3*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] - 3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + Cos[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d])/d","A",1
160,1,189,179,0.9551661,"\int \frac{\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x]^3)/(c + d*x)^2,x]","\frac{6 b (c+d x) \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-6 b (c+d x) \cos \left(6 a-\frac{6 b c}{d}\right) \text{Ci}\left(\frac{6 b (c+d x)}{d}\right)-6 b (c+d x) \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+6 b (c+d x) \sin \left(6 a-\frac{6 b c}{d}\right) \text{Si}\left(\frac{6 b (c+d x)}{d}\right)-3 d \sin (2 a) \cos (2 b x)+d \sin (6 a) \cos (6 b x)-3 d \cos (2 a) \sin (2 b x)+d \cos (6 a) \sin (6 b x)}{32 d^2 (c+d x)}","\frac{3 b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{16 d^2}-\frac{3 b \cos \left(6 a-\frac{6 b c}{d}\right) \text{Ci}\left(\frac{6 b c}{d}+6 b x\right)}{16 d^2}-\frac{3 b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{16 d^2}+\frac{3 b \sin \left(6 a-\frac{6 b c}{d}\right) \text{Si}\left(\frac{6 b c}{d}+6 b x\right)}{16 d^2}-\frac{3 \sin (2 a+2 b x)}{32 d (c+d x)}+\frac{\sin (6 a+6 b x)}{32 d (c+d x)}",1,"(6*b*(c + d*x)*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] - 6*b*(c + d*x)*Cos[6*a - (6*b*c)/d]*CosIntegral[(6*b*(c + d*x))/d] - 3*d*Cos[2*b*x]*Sin[2*a] + d*Cos[6*b*x]*Sin[6*a] - 3*d*Cos[2*a]*Sin[2*b*x] + d*Cos[6*a]*Sin[6*b*x] - 6*b*(c + d*x)*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + 6*b*(c + d*x)*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d])/(32*d^2*(c + d*x))","A",1
161,1,239,235,1.0393474,"\int \frac{\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^3} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x]^3)/(c + d*x)^3,x]","\frac{6 b^2 (c+d x)^2 \left(6 \sin \left(6 a-\frac{6 b c}{d}\right) \text{Ci}\left(\frac{6 b (c+d x)}{d}\right)-2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+6 \cos \left(6 a-\frac{6 b c}{d}\right) \text{Si}\left(\frac{6 b (c+d x)}{d}\right)\right)-3 d \cos (2 b x) (2 b \cos (2 a) (c+d x)+d \sin (2 a))+d \cos (6 b x) (6 b \cos (6 a) (c+d x)+d \sin (6 a))+3 d \sin (2 b x) (2 b \sin (2 a) (c+d x)-d \cos (2 a))+d \sin (6 b x) (d \cos (6 a)-6 b \sin (6 a) (c+d x))}{64 d^3 (c+d x)^2}","\frac{9 b^2 \sin \left(6 a-\frac{6 b c}{d}\right) \text{Ci}\left(\frac{6 b c}{d}+6 b x\right)}{16 d^3}-\frac{3 b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{16 d^3}-\frac{3 b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{16 d^3}+\frac{9 b^2 \cos \left(6 a-\frac{6 b c}{d}\right) \text{Si}\left(\frac{6 b c}{d}+6 b x\right)}{16 d^3}-\frac{3 b \cos (2 a+2 b x)}{32 d^2 (c+d x)}+\frac{3 b \cos (6 a+6 b x)}{32 d^2 (c+d x)}-\frac{3 \sin (2 a+2 b x)}{64 d (c+d x)^2}+\frac{\sin (6 a+6 b x)}{64 d (c+d x)^2}",1,"(-3*d*Cos[2*b*x]*(2*b*(c + d*x)*Cos[2*a] + d*Sin[2*a]) + d*Cos[6*b*x]*(6*b*(c + d*x)*Cos[6*a] + d*Sin[6*a]) + 3*d*(-(d*Cos[2*a]) + 2*b*(c + d*x)*Sin[2*a])*Sin[2*b*x] + d*(d*Cos[6*a] - 6*b*(c + d*x)*Sin[6*a])*Sin[6*b*x] + 6*b^2*(c + d*x)^2*(6*CosIntegral[(6*b*(c + d*x))/d]*Sin[6*a - (6*b*c)/d] - 2*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] - 2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + 6*Cos[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d]))/(64*d^3*(c + d*x)^2)","A",1
162,1,554,287,4.9435845,"\int \frac{\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx","Integrate[(Cos[a + b*x]^3*Sin[a + b*x]^3)/(c + d*x)^4,x]","\frac{12 b^3 c^3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)-108 b^3 c^3 \sin \left(6 a-\frac{6 b c}{d}\right) \text{Si}\left(\frac{6 b (c+d x)}{d}\right)+36 b^3 c^2 d x \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)-324 b^3 c^2 d x \sin \left(6 a-\frac{6 b c}{d}\right) \text{Si}\left(\frac{6 b (c+d x)}{d}\right)-12 b^3 (c+d x)^3 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)+108 b^3 (c+d x)^3 \cos \left(6 a-\frac{6 b c}{d}\right) \text{Ci}\left(\frac{6 b (c+d x)}{d}\right)+12 b^3 d^3 x^3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)-108 b^3 d^3 x^3 \sin \left(6 a-\frac{6 b c}{d}\right) \text{Si}\left(\frac{6 b (c+d x)}{d}\right)+36 b^3 c d^2 x^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)-324 b^3 c d^2 x^2 \sin \left(6 a-\frac{6 b c}{d}\right) \text{Si}\left(\frac{6 b (c+d x)}{d}\right)+6 b^2 c^2 d \sin (2 (a+b x))-18 b^2 c^2 d \sin (6 (a+b x))+12 b^2 c d^2 x \sin (2 (a+b x))-36 b^2 c d^2 x \sin (6 (a+b x))+6 b^2 d^3 x^2 \sin (2 (a+b x))-18 b^2 d^3 x^2 \sin (6 (a+b x))-3 b c d^2 \cos (2 (a+b x))+3 b c d^2 \cos (6 (a+b x))-3 d^3 \sin (2 (a+b x))+d^3 \sin (6 (a+b x))-3 b d^3 x \cos (2 (a+b x))+3 b d^3 x \cos (6 (a+b x))}{96 d^4 (c+d x)^3}","-\frac{b^3 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{8 d^4}+\frac{9 b^3 \cos \left(6 a-\frac{6 b c}{d}\right) \text{Ci}\left(\frac{6 b c}{d}+6 b x\right)}{8 d^4}+\frac{b^3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{8 d^4}-\frac{9 b^3 \sin \left(6 a-\frac{6 b c}{d}\right) \text{Si}\left(\frac{6 b c}{d}+6 b x\right)}{8 d^4}+\frac{b^2 \sin (2 a+2 b x)}{16 d^3 (c+d x)}-\frac{3 b^2 \sin (6 a+6 b x)}{16 d^3 (c+d x)}-\frac{b \cos (2 a+2 b x)}{32 d^2 (c+d x)^2}+\frac{b \cos (6 a+6 b x)}{32 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{32 d (c+d x)^3}+\frac{\sin (6 a+6 b x)}{96 d (c+d x)^3}",1,"(-3*b*c*d^2*Cos[2*(a + b*x)] - 3*b*d^3*x*Cos[2*(a + b*x)] + 3*b*c*d^2*Cos[6*(a + b*x)] + 3*b*d^3*x*Cos[6*(a + b*x)] - 12*b^3*(c + d*x)^3*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] + 108*b^3*(c + d*x)^3*Cos[6*a - (6*b*c)/d]*CosIntegral[(6*b*(c + d*x))/d] + 6*b^2*c^2*d*Sin[2*(a + b*x)] - 3*d^3*Sin[2*(a + b*x)] + 12*b^2*c*d^2*x*Sin[2*(a + b*x)] + 6*b^2*d^3*x^2*Sin[2*(a + b*x)] - 18*b^2*c^2*d*Sin[6*(a + b*x)] + d^3*Sin[6*(a + b*x)] - 36*b^2*c*d^2*x*Sin[6*(a + b*x)] - 18*b^2*d^3*x^2*Sin[6*(a + b*x)] + 12*b^3*c^3*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + 36*b^3*c^2*d*x*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + 36*b^3*c*d^2*x^2*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + 12*b^3*d^3*x^3*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] - 108*b^3*c^3*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d] - 324*b^3*c^2*d*x*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d] - 324*b^3*c*d^2*x^2*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d] - 108*b^3*d^3*x^3*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d])/(96*d^4*(c + d*x)^3)","A",1
163,0,0,152,7.7735129,"\int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]^2*Cot[a + b*x],x]","\int (c+d x)^m \cos ^2(a+b x) \cot (a+b x) \, dx","\text{Int}\left(\cot (a+b x) (c+d x)^m,x\right)+\frac{2^{-m-3} e^{2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)}{b}+\frac{2^{-m-3} e^{-2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)}{b}",0,"Integrate[(c + d*x)^m*Cos[a + b*x]^2*Cot[a + b*x], x]","A",-1
164,1,2828,307,6.5206608,"\int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]^2*Cot[a + b*x],x]","\text{Result too large to show}","-\frac{3 d^4 \text{Li}_5\left(e^{2 i (a+b x)}\right)}{2 b^5}-\frac{3 d^4 \sin ^2(a+b x)}{4 b^5}+\frac{3 i d^3 (c+d x) \text{Li}_4\left(e^{2 i (a+b x)}\right)}{b^4}+\frac{3 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^4}+\frac{3 d^2 (c+d x)^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{b^3}+\frac{3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}-\frac{2 i d (c+d x)^3 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{b^2}+\frac{(c+d x)^4 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac{3 c d^3 x}{2 b^3}-\frac{3 d^4 x^2}{4 b^3}+\frac{(c+d x)^4}{4 b}-\frac{i (c+d x)^5}{5 d}",1,"-((c^2*d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^3) - (c*d^3*E^(I*a)*Csc[a]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-I)*(a + b*x))] - 2*PolyLog[4, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, E^((-I)*(a + b*x))] - 2*PolyLog[4, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^4 - (d^4*E^(I*a)*Csc[a]*((2*b^5*x^5)/E^((2*I)*a) + (5*I)*b^4*(1 - E^((-2*I)*a))*x^4*Log[1 - E^((-I)*(a + b*x))] + (5*I)*b^4*(1 - E^((-2*I)*a))*x^4*Log[1 + E^((-I)*(a + b*x))] - (20*(-1 + E^((2*I)*a))*(b^3*x^3*PolyLog[2, -E^((-I)*(a + b*x))] - (3*I)*b^2*x^2*PolyLog[3, -E^((-I)*(a + b*x))] - 6*b*x*PolyLog[4, -E^((-I)*(a + b*x))] + (6*I)*PolyLog[5, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (20*(-1 + E^((2*I)*a))*(b^3*x^3*PolyLog[2, E^((-I)*(a + b*x))] - (3*I)*b^2*x^2*PolyLog[3, E^((-I)*(a + b*x))] - 6*b*x*PolyLog[4, E^((-I)*(a + b*x))] + (6*I)*PolyLog[5, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(10*b^5) + (c^4*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + Csc[a]*(Cos[2*a + 2*b*x]/(160*b^5) - ((I/160)*Sin[2*a + 2*b*x])/b^5)*(80*b^5*c^4*x*Cos[a + 2*b*x] + 160*b^5*c^3*d*x^2*Cos[a + 2*b*x] + 160*b^5*c^2*d^2*x^3*Cos[a + 2*b*x] + 80*b^5*c*d^3*x^4*Cos[a + 2*b*x] + 16*b^5*d^4*x^5*Cos[a + 2*b*x] + 80*b^5*c^4*x*Cos[3*a + 2*b*x] + 160*b^5*c^3*d*x^2*Cos[3*a + 2*b*x] + 160*b^5*c^2*d^2*x^3*Cos[3*a + 2*b*x] + 80*b^5*c*d^3*x^4*Cos[3*a + 2*b*x] + 16*b^5*d^4*x^5*Cos[3*a + 2*b*x] + (10*I)*b^4*c^4*Cos[3*a + 4*b*x] - 20*b^3*c^3*d*Cos[3*a + 4*b*x] - (30*I)*b^2*c^2*d^2*Cos[3*a + 4*b*x] + 30*b*c*d^3*Cos[3*a + 4*b*x] + (15*I)*d^4*Cos[3*a + 4*b*x] + (40*I)*b^4*c^3*d*x*Cos[3*a + 4*b*x] - 60*b^3*c^2*d^2*x*Cos[3*a + 4*b*x] - (60*I)*b^2*c*d^3*x*Cos[3*a + 4*b*x] + 30*b*d^4*x*Cos[3*a + 4*b*x] + (60*I)*b^4*c^2*d^2*x^2*Cos[3*a + 4*b*x] - 60*b^3*c*d^3*x^2*Cos[3*a + 4*b*x] - (30*I)*b^2*d^4*x^2*Cos[3*a + 4*b*x] + (40*I)*b^4*c*d^3*x^3*Cos[3*a + 4*b*x] - 20*b^3*d^4*x^3*Cos[3*a + 4*b*x] + (10*I)*b^4*d^4*x^4*Cos[3*a + 4*b*x] - (10*I)*b^4*c^4*Cos[5*a + 4*b*x] + 20*b^3*c^3*d*Cos[5*a + 4*b*x] + (30*I)*b^2*c^2*d^2*Cos[5*a + 4*b*x] - 30*b*c*d^3*Cos[5*a + 4*b*x] - (15*I)*d^4*Cos[5*a + 4*b*x] - (40*I)*b^4*c^3*d*x*Cos[5*a + 4*b*x] + 60*b^3*c^2*d^2*x*Cos[5*a + 4*b*x] + (60*I)*b^2*c*d^3*x*Cos[5*a + 4*b*x] - 30*b*d^4*x*Cos[5*a + 4*b*x] - (60*I)*b^4*c^2*d^2*x^2*Cos[5*a + 4*b*x] + 60*b^3*c*d^3*x^2*Cos[5*a + 4*b*x] + (30*I)*b^2*d^4*x^2*Cos[5*a + 4*b*x] - (40*I)*b^4*c*d^3*x^3*Cos[5*a + 4*b*x] + 20*b^3*d^4*x^3*Cos[5*a + 4*b*x] - (10*I)*b^4*d^4*x^4*Cos[5*a + 4*b*x] + 20*b^4*c^4*Sin[a] - (40*I)*b^3*c^3*d*Sin[a] - 60*b^2*c^2*d^2*Sin[a] + (60*I)*b*c*d^3*Sin[a] + 30*d^4*Sin[a] + 80*b^4*c^3*d*x*Sin[a] - (120*I)*b^3*c^2*d^2*x*Sin[a] - 120*b^2*c*d^3*x*Sin[a] + (60*I)*b*d^4*x*Sin[a] + 120*b^4*c^2*d^2*x^2*Sin[a] - (120*I)*b^3*c*d^3*x^2*Sin[a] - 60*b^2*d^4*x^2*Sin[a] + 80*b^4*c*d^3*x^3*Sin[a] - (40*I)*b^3*d^4*x^3*Sin[a] + 20*b^4*d^4*x^4*Sin[a] + (80*I)*b^5*c^4*x*Sin[a + 2*b*x] + (160*I)*b^5*c^3*d*x^2*Sin[a + 2*b*x] + (160*I)*b^5*c^2*d^2*x^3*Sin[a + 2*b*x] + (80*I)*b^5*c*d^3*x^4*Sin[a + 2*b*x] + (16*I)*b^5*d^4*x^5*Sin[a + 2*b*x] + (80*I)*b^5*c^4*x*Sin[3*a + 2*b*x] + (160*I)*b^5*c^3*d*x^2*Sin[3*a + 2*b*x] + (160*I)*b^5*c^2*d^2*x^3*Sin[3*a + 2*b*x] + (80*I)*b^5*c*d^3*x^4*Sin[3*a + 2*b*x] + (16*I)*b^5*d^4*x^5*Sin[3*a + 2*b*x] - 10*b^4*c^4*Sin[3*a + 4*b*x] - (20*I)*b^3*c^3*d*Sin[3*a + 4*b*x] + 30*b^2*c^2*d^2*Sin[3*a + 4*b*x] + (30*I)*b*c*d^3*Sin[3*a + 4*b*x] - 15*d^4*Sin[3*a + 4*b*x] - 40*b^4*c^3*d*x*Sin[3*a + 4*b*x] - (60*I)*b^3*c^2*d^2*x*Sin[3*a + 4*b*x] + 60*b^2*c*d^3*x*Sin[3*a + 4*b*x] + (30*I)*b*d^4*x*Sin[3*a + 4*b*x] - 60*b^4*c^2*d^2*x^2*Sin[3*a + 4*b*x] - (60*I)*b^3*c*d^3*x^2*Sin[3*a + 4*b*x] + 30*b^2*d^4*x^2*Sin[3*a + 4*b*x] - 40*b^4*c*d^3*x^3*Sin[3*a + 4*b*x] - (20*I)*b^3*d^4*x^3*Sin[3*a + 4*b*x] - 10*b^4*d^4*x^4*Sin[3*a + 4*b*x] + 10*b^4*c^4*Sin[5*a + 4*b*x] + (20*I)*b^3*c^3*d*Sin[5*a + 4*b*x] - 30*b^2*c^2*d^2*Sin[5*a + 4*b*x] - (30*I)*b*c*d^3*Sin[5*a + 4*b*x] + 15*d^4*Sin[5*a + 4*b*x] + 40*b^4*c^3*d*x*Sin[5*a + 4*b*x] + (60*I)*b^3*c^2*d^2*x*Sin[5*a + 4*b*x] - 60*b^2*c*d^3*x*Sin[5*a + 4*b*x] - (30*I)*b*d^4*x*Sin[5*a + 4*b*x] + 60*b^4*c^2*d^2*x^2*Sin[5*a + 4*b*x] + (60*I)*b^3*c*d^3*x^2*Sin[5*a + 4*b*x] - 30*b^2*d^4*x^2*Sin[5*a + 4*b*x] + 40*b^4*c*d^3*x^3*Sin[5*a + 4*b*x] + (20*I)*b^3*d^4*x^3*Sin[5*a + 4*b*x] + 10*b^4*d^4*x^4*Sin[5*a + 4*b*x]) - (2*c^3*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
165,1,1918,246,6.4058551,"\int (c+d x)^3 \cos ^2(a+b x) \cot (a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]^2*Cot[a + b*x],x]","\frac{\csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a)) c^3}{b \left(\cos ^2(a)+\sin ^2(a)\right)}-\frac{3 d \csc (a) \sec (a) \left(b^2 e^{i \tan ^{-1}(\tan (a))} x^2+\frac{\left(i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x+\tan ^{-1}(\tan (a))\right) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\pi  \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right) \tan (a)}{\sqrt{\tan ^2(a)+1}}\right) c^2}{2 b^2 \sqrt{\sec ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}-\frac{d^2 e^{i a} \csc (a) \left(2 b^3 e^{-2 i a} x^3+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^2+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^2-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right) c}{2 b^3}-\frac{d^3 e^{i a} \csc (a) \left(b^4 e^{-2 i a} x^4+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^3+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^3-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(-e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(-e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(e^{-i (a+b x)}\right)\right)\right)}{4 b^4}+\csc (a) \left(\frac{\cos (2 a+2 b x)}{64 b^4}-\frac{i \sin (2 a+2 b x)}{64 b^4}\right) \left(8 d^3 x^4 \cos (a+2 b x) b^4+32 c d^2 x^3 \cos (a+2 b x) b^4+48 c^2 d x^2 \cos (a+2 b x) b^4+32 c^3 x \cos (a+2 b x) b^4+8 d^3 x^4 \cos (3 a+2 b x) b^4+32 c d^2 x^3 \cos (3 a+2 b x) b^4+48 c^2 d x^2 \cos (3 a+2 b x) b^4+32 c^3 x \cos (3 a+2 b x) b^4+8 i d^3 x^4 \sin (a+2 b x) b^4+32 i c d^2 x^3 \sin (a+2 b x) b^4+48 i c^2 d x^2 \sin (a+2 b x) b^4+32 i c^3 x \sin (a+2 b x) b^4+8 i d^3 x^4 \sin (3 a+2 b x) b^4+32 i c d^2 x^3 \sin (3 a+2 b x) b^4+48 i c^2 d x^2 \sin (3 a+2 b x) b^4+32 i c^3 x \sin (3 a+2 b x) b^4+4 i c^3 \cos (3 a+4 b x) b^3+4 i d^3 x^3 \cos (3 a+4 b x) b^3+12 i c d^2 x^2 \cos (3 a+4 b x) b^3+12 i c^2 d x \cos (3 a+4 b x) b^3-4 i c^3 \cos (5 a+4 b x) b^3-4 i d^3 x^3 \cos (5 a+4 b x) b^3-12 i c d^2 x^2 \cos (5 a+4 b x) b^3-12 i c^2 d x \cos (5 a+4 b x) b^3+8 c^3 \sin (a) b^3+8 d^3 x^3 \sin (a) b^3+24 c d^2 x^2 \sin (a) b^3+24 c^2 d x \sin (a) b^3-4 c^3 \sin (3 a+4 b x) b^3-4 d^3 x^3 \sin (3 a+4 b x) b^3-12 c d^2 x^2 \sin (3 a+4 b x) b^3-12 c^2 d x \sin (3 a+4 b x) b^3+4 c^3 \sin (5 a+4 b x) b^3+4 d^3 x^3 \sin (5 a+4 b x) b^3+12 c d^2 x^2 \sin (5 a+4 b x) b^3+12 c^2 d x \sin (5 a+4 b x) b^3-6 d^3 x^2 \cos (3 a+4 b x) b^2-6 c^2 d \cos (3 a+4 b x) b^2-12 c d^2 x \cos (3 a+4 b x) b^2+6 d^3 x^2 \cos (5 a+4 b x) b^2+6 c^2 d \cos (5 a+4 b x) b^2+12 c d^2 x \cos (5 a+4 b x) b^2-12 i d^3 x^2 \sin (a) b^2-12 i c^2 d \sin (a) b^2-24 i c d^2 x \sin (a) b^2-6 i d^3 x^2 \sin (3 a+4 b x) b^2-6 i c^2 d \sin (3 a+4 b x) b^2-12 i c d^2 x \sin (3 a+4 b x) b^2+6 i d^3 x^2 \sin (5 a+4 b x) b^2+6 i c^2 d \sin (5 a+4 b x) b^2+12 i c d^2 x \sin (5 a+4 b x) b^2-6 i c d^2 \cos (3 a+4 b x) b-6 i d^3 x \cos (3 a+4 b x) b+6 i c d^2 \cos (5 a+4 b x) b+6 i d^3 x \cos (5 a+4 b x) b-12 c d^2 \sin (a) b-12 d^3 x \sin (a) b+6 c d^2 \sin (3 a+4 b x) b+6 d^3 x \sin (3 a+4 b x) b-6 c d^2 \sin (5 a+4 b x) b-6 d^3 x \sin (5 a+4 b x) b+3 d^3 \cos (3 a+4 b x)-3 d^3 \cos (5 a+4 b x)+6 i d^3 \sin (a)+3 i d^3 \sin (3 a+4 b x)-3 i d^3 \sin (5 a+4 b x)\right)","\frac{3 i d^3 \text{Li}_4\left(e^{2 i (a+b x)}\right)}{4 b^4}+\frac{3 d^3 \sin (a+b x) \cos (a+b x)}{8 b^4}+\frac{3 d^2 (c+d x) \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}+\frac{3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{3 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{4 b^2}+\frac{(c+d x)^3 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac{3 d^3 x}{8 b^3}+\frac{(c+d x)^3}{4 b}-\frac{i (c+d x)^4}{4 d}",1,"-1/2*(c*d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^3 - (d^3*E^(I*a)*Csc[a]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-I)*(a + b*x))] - 2*PolyLog[4, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, E^((-I)*(a + b*x))] - 2*PolyLog[4, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(4*b^4) + (c^3*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + Csc[a]*(Cos[2*a + 2*b*x]/(64*b^4) - ((I/64)*Sin[2*a + 2*b*x])/b^4)*(32*b^4*c^3*x*Cos[a + 2*b*x] + 48*b^4*c^2*d*x^2*Cos[a + 2*b*x] + 32*b^4*c*d^2*x^3*Cos[a + 2*b*x] + 8*b^4*d^3*x^4*Cos[a + 2*b*x] + 32*b^4*c^3*x*Cos[3*a + 2*b*x] + 48*b^4*c^2*d*x^2*Cos[3*a + 2*b*x] + 32*b^4*c*d^2*x^3*Cos[3*a + 2*b*x] + 8*b^4*d^3*x^4*Cos[3*a + 2*b*x] + (4*I)*b^3*c^3*Cos[3*a + 4*b*x] - 6*b^2*c^2*d*Cos[3*a + 4*b*x] - (6*I)*b*c*d^2*Cos[3*a + 4*b*x] + 3*d^3*Cos[3*a + 4*b*x] + (12*I)*b^3*c^2*d*x*Cos[3*a + 4*b*x] - 12*b^2*c*d^2*x*Cos[3*a + 4*b*x] - (6*I)*b*d^3*x*Cos[3*a + 4*b*x] + (12*I)*b^3*c*d^2*x^2*Cos[3*a + 4*b*x] - 6*b^2*d^3*x^2*Cos[3*a + 4*b*x] + (4*I)*b^3*d^3*x^3*Cos[3*a + 4*b*x] - (4*I)*b^3*c^3*Cos[5*a + 4*b*x] + 6*b^2*c^2*d*Cos[5*a + 4*b*x] + (6*I)*b*c*d^2*Cos[5*a + 4*b*x] - 3*d^3*Cos[5*a + 4*b*x] - (12*I)*b^3*c^2*d*x*Cos[5*a + 4*b*x] + 12*b^2*c*d^2*x*Cos[5*a + 4*b*x] + (6*I)*b*d^3*x*Cos[5*a + 4*b*x] - (12*I)*b^3*c*d^2*x^2*Cos[5*a + 4*b*x] + 6*b^2*d^3*x^2*Cos[5*a + 4*b*x] - (4*I)*b^3*d^3*x^3*Cos[5*a + 4*b*x] + 8*b^3*c^3*Sin[a] - (12*I)*b^2*c^2*d*Sin[a] - 12*b*c*d^2*Sin[a] + (6*I)*d^3*Sin[a] + 24*b^3*c^2*d*x*Sin[a] - (24*I)*b^2*c*d^2*x*Sin[a] - 12*b*d^3*x*Sin[a] + 24*b^3*c*d^2*x^2*Sin[a] - (12*I)*b^2*d^3*x^2*Sin[a] + 8*b^3*d^3*x^3*Sin[a] + (32*I)*b^4*c^3*x*Sin[a + 2*b*x] + (48*I)*b^4*c^2*d*x^2*Sin[a + 2*b*x] + (32*I)*b^4*c*d^2*x^3*Sin[a + 2*b*x] + (8*I)*b^4*d^3*x^4*Sin[a + 2*b*x] + (32*I)*b^4*c^3*x*Sin[3*a + 2*b*x] + (48*I)*b^4*c^2*d*x^2*Sin[3*a + 2*b*x] + (32*I)*b^4*c*d^2*x^3*Sin[3*a + 2*b*x] + (8*I)*b^4*d^3*x^4*Sin[3*a + 2*b*x] - 4*b^3*c^3*Sin[3*a + 4*b*x] - (6*I)*b^2*c^2*d*Sin[3*a + 4*b*x] + 6*b*c*d^2*Sin[3*a + 4*b*x] + (3*I)*d^3*Sin[3*a + 4*b*x] - 12*b^3*c^2*d*x*Sin[3*a + 4*b*x] - (12*I)*b^2*c*d^2*x*Sin[3*a + 4*b*x] + 6*b*d^3*x*Sin[3*a + 4*b*x] - 12*b^3*c*d^2*x^2*Sin[3*a + 4*b*x] - (6*I)*b^2*d^3*x^2*Sin[3*a + 4*b*x] - 4*b^3*d^3*x^3*Sin[3*a + 4*b*x] + 4*b^3*c^3*Sin[5*a + 4*b*x] + (6*I)*b^2*c^2*d*Sin[5*a + 4*b*x] - 6*b*c*d^2*Sin[5*a + 4*b*x] - (3*I)*d^3*Sin[5*a + 4*b*x] + 12*b^3*c^2*d*x*Sin[5*a + 4*b*x] + (12*I)*b^2*c*d^2*x*Sin[5*a + 4*b*x] - 6*b*d^3*x*Sin[5*a + 4*b*x] + 12*b^3*c*d^2*x^2*Sin[5*a + 4*b*x] + (6*I)*b^2*d^3*x^2*Sin[5*a + 4*b*x] + 4*b^3*d^3*x^3*Sin[5*a + 4*b*x]) - (3*c^2*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(2*b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
166,1,564,181,2.9039449,"\int (c+d x)^2 \cos ^2(a+b x) \cot (a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]^2*Cot[a + b*x],x]","\frac{48 b^3 c d x^2 \cot (a)-48 b^3 c d x^2 e^{i \tan ^{-1}(\tan (a))} \cot (a) \sqrt{\sec ^2(a)}+48 b^2 c^2 \log (\sin (a+b x))-6 b^2 c^2 \csc (a) \sin (a+2 b x)+6 b^2 c^2 \csc (a) \sin (3 a+2 b x)-96 i b^2 c d x \tan ^{-1}(\tan (a))+96 b^2 c d x \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)-12 b^2 c d x \csc (a) \sin (a+2 b x)+12 b^2 c d x \csc (a) \sin (3 a+2 b x)+48 b^2 d^2 x^2 \log \left(1-e^{-i (a+b x)}\right)+48 b^2 d^2 x^2 \log \left(1+e^{-i (a+b x)}\right)-6 b^2 d^2 x^2 \csc (a) \sin (a+2 b x)+6 b^2 d^2 x^2 \csc (a) \sin (3 a+2 b x)-48 i b c d \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+96 b c d \tan ^{-1}(\tan (a)) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)-6 b c d \csc (a) \cos (a+2 b x)+6 b c d \csc (a) \cos (3 a+2 b x)-96 b c d \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)+96 i b d^2 x \text{Li}_2\left(-e^{-i (a+b x)}\right)+96 i b d^2 x \text{Li}_2\left(e^{-i (a+b x)}\right)+96 d^2 \text{Li}_3\left(-e^{-i (a+b x)}\right)+96 d^2 \text{Li}_3\left(e^{-i (a+b x)}\right)-6 b d^2 x \csc (a) \cos (a+2 b x)+6 b d^2 x \csc (a) \cos (3 a+2 b x)+3 d^2 \csc (a) \sin (a+2 b x)-3 d^2 \csc (a) \sin (3 a+2 b x)+16 i b^3 d^2 x^3+48 i \pi  b^2 c d x+48 \pi  b c d \log \left(1+e^{-2 i b x}\right)-48 \pi  b c d \log (\cos (b x))}{48 b^3}","\frac{d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}+\frac{d^2 \sin ^2(a+b x)}{4 b^3}-\frac{i d (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{d (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^2}+\frac{(c+d x)^2 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{(c+d x)^2 \sin ^2(a+b x)}{2 b}+\frac{c d x}{2 b}+\frac{d^2 x^2}{4 b}-\frac{i (c+d x)^3}{3 d}",1,"((48*I)*b^2*c*d*Pi*x + (16*I)*b^3*d^2*x^3 - (96*I)*b^2*c*d*x*ArcTan[Tan[a]] + 48*b^3*c*d*x^2*Cot[a] - 6*b*c*d*Cos[a + 2*b*x]*Csc[a] - 6*b*d^2*x*Cos[a + 2*b*x]*Csc[a] + 6*b*c*d*Cos[3*a + 2*b*x]*Csc[a] + 6*b*d^2*x*Cos[3*a + 2*b*x]*Csc[a] + 48*b*c*d*Pi*Log[1 + E^((-2*I)*b*x)] + 48*b^2*d^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 48*b^2*d^2*x^2*Log[1 + E^((-I)*(a + b*x))] + 96*b^2*c*d*x*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + 96*b*c*d*ArcTan[Tan[a]]*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] - 48*b*c*d*Pi*Log[Cos[b*x]] + 48*b^2*c^2*Log[Sin[a + b*x]] - 96*b*c*d*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + (96*I)*b*d^2*x*PolyLog[2, -E^((-I)*(a + b*x))] + (96*I)*b*d^2*x*PolyLog[2, E^((-I)*(a + b*x))] - (48*I)*b*c*d*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))] + 96*d^2*PolyLog[3, -E^((-I)*(a + b*x))] + 96*d^2*PolyLog[3, E^((-I)*(a + b*x))] - 48*b^3*c*d*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2] - 6*b^2*c^2*Csc[a]*Sin[a + 2*b*x] + 3*d^2*Csc[a]*Sin[a + 2*b*x] - 12*b^2*c*d*x*Csc[a]*Sin[a + 2*b*x] - 6*b^2*d^2*x^2*Csc[a]*Sin[a + 2*b*x] + 6*b^2*c^2*Csc[a]*Sin[3*a + 2*b*x] - 3*d^2*Csc[a]*Sin[3*a + 2*b*x] + 12*b^2*c*d*x*Csc[a]*Sin[3*a + 2*b*x] + 6*b^2*d^2*x^2*Csc[a]*Sin[3*a + 2*b*x])/(48*b^3)","B",0
167,1,131,114,0.3458991,"\int (c+d x) \cos ^2(a+b x) \cot (a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]^2*Cot[a + b*x],x]","\frac{d \left((a+b x) \log \left(1-e^{2 i (a+b x)}\right)-\frac{1}{2} i \left((a+b x)^2+\text{Li}_2\left(e^{2 i (a+b x)}\right)\right)\right)}{b^2}-\frac{d \sin (2 (a+b x))}{8 b^2}-\frac{a d \log (\sin (a+b x))}{b^2}-\frac{c \sin ^2(a+b x)}{2 b}+\frac{c \log (\sin (a+b x))}{b}+\frac{d x \cos (2 (a+b x))}{4 b}","-\frac{i d \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{d \sin (a+b x) \cos (a+b x)}{4 b^2}+\frac{(c+d x) \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{(c+d x) \sin ^2(a+b x)}{2 b}+\frac{d x}{4 b}-\frac{i (c+d x)^2}{2 d}",1,"(d*x*Cos[2*(a + b*x)])/(4*b) + (c*Log[Sin[a + b*x]])/b - (a*d*Log[Sin[a + b*x]])/b^2 + (d*((a + b*x)*Log[1 - E^((2*I)*(a + b*x))] - (I/2)*((a + b*x)^2 + PolyLog[2, E^((2*I)*(a + b*x))])))/b^2 - (c*Sin[a + b*x]^2)/(2*b) - (d*Sin[2*(a + b*x)])/(8*b^2)","A",1
168,0,0,82,0.8129631,"\int \frac{\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]^2*Cot[a + b*x])/(c + d*x),x]","\int \frac{\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\cot (a+b x)}{c+d x},x\right)-\frac{\sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{2 d}-\frac{\cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{2 d}",0,"Integrate[(Cos[a + b*x]^2*Cot[a + b*x])/(c + d*x), x]","A",-1
169,0,0,102,2.4930994,"\int \frac{\cos ^2(a+b x) \cot (a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]^2*Cot[a + b*x])/(c + d*x)^2,x]","\int \frac{\cos ^2(a+b x) \cot (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\cot (a+b x)}{(c+d x)^2},x\right)-\frac{b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}+\frac{b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}+\frac{\sin (2 a+2 b x)}{2 d (c+d x)}",0,"Integrate[(Cos[a + b*x]^2*Cot[a + b*x])/(c + d*x)^2, x]","A",-1
170,0,0,154,9.8925999,"\int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Cos[a + b*x]*Cot[a + b*x]^2,x]","\int (c+d x)^m \cos (a+b x) \cot ^2(a+b x) \, dx","\text{Int}\left(\cot (a+b x) \csc (a+b x) (c+d x)^m,x\right)+\frac{i e^{i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)}{2 b}-\frac{i e^{-i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)}{2 b}",0,"Integrate[(c + d*x)^m*Cos[a + b*x]*Cot[a + b*x]^2, x]","A",-1
171,1,798,299,1.7128765,"\int (c+d x)^4 \cos (a+b x) \cot ^2(a+b x) \, dx","Integrate[(c + d*x)^4*Cos[a + b*x]*Cot[a + b*x]^2,x]","\frac{\csc (a+b x) \left(-3 c^4 b^4-3 d^4 x^4 b^4-12 c d^3 x^3 b^4-18 c^2 d^2 x^2 b^4-12 c^3 d x b^4+c^4 \cos (2 (a+b x)) b^4+d^4 x^4 \cos (2 (a+b x)) b^4+4 c d^3 x^3 \cos (2 (a+b x)) b^4+6 c^2 d^2 x^2 \cos (2 (a+b x)) b^4+4 c^3 d x \cos (2 (a+b x)) b^4-16 d^4 x^3 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-48 c d^3 x^2 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-16 c^3 d \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-48 c^2 d^2 x \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-4 d^4 x^3 \sin (2 (a+b x)) b^3-12 c d^3 x^2 \sin (2 (a+b x)) b^3-4 c^3 d \sin (2 (a+b x)) b^3-12 c^2 d^2 x \sin (2 (a+b x)) b^3+12 c^2 d^2 b^2+12 d^4 x^2 b^2+24 c d^3 x b^2-12 c^2 d^2 \cos (2 (a+b x)) b^2-12 d^4 x^2 \cos (2 (a+b x)) b^2-24 c d^3 x \cos (2 (a+b x)) b^2+24 i d^2 (c+d x)^2 \text{Li}_2(-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x) b^2-24 i d^2 (c+d x)^2 \text{Li}_2(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^2-48 c d^3 \text{Li}_3(-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x) b-48 d^4 x \text{Li}_3(-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x) b+48 c d^3 \text{Li}_3(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b+48 d^4 x \text{Li}_3(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b+24 c d^3 \sin (2 (a+b x)) b+24 d^4 x \sin (2 (a+b x)) b-24 d^4+24 d^4 \cos (2 (a+b x))-48 i d^4 \text{Li}_4(-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x)+48 i d^4 \text{Li}_4(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x)\right)}{2 b^5}","-\frac{24 i d^4 \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^5}+\frac{24 i d^4 \text{Li}_4\left(e^{i (a+b x)}\right)}{b^5}-\frac{24 d^4 \sin (a+b x)}{b^5}-\frac{24 d^3 (c+d x) \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^4}+\frac{24 d^3 (c+d x) \text{Li}_3\left(e^{i (a+b x)}\right)}{b^4}+\frac{24 d^3 (c+d x) \cos (a+b x)}{b^4}+\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^3}-\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^3}+\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac{8 d (c+d x)^3 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{(c+d x)^4 \sin (a+b x)}{b}-\frac{(c+d x)^4 \csc (a+b x)}{b}",1,"(Csc[a + b*x]*(-3*b^4*c^4 + 12*b^2*c^2*d^2 - 24*d^4 - 12*b^4*c^3*d*x + 24*b^2*c*d^3*x - 18*b^4*c^2*d^2*x^2 + 12*b^2*d^4*x^2 - 12*b^4*c*d^3*x^3 - 3*b^4*d^4*x^4 + b^4*c^4*Cos[2*(a + b*x)] - 12*b^2*c^2*d^2*Cos[2*(a + b*x)] + 24*d^4*Cos[2*(a + b*x)] + 4*b^4*c^3*d*x*Cos[2*(a + b*x)] - 24*b^2*c*d^3*x*Cos[2*(a + b*x)] + 6*b^4*c^2*d^2*x^2*Cos[2*(a + b*x)] - 12*b^2*d^4*x^2*Cos[2*(a + b*x)] + 4*b^4*c*d^3*x^3*Cos[2*(a + b*x)] + b^4*d^4*x^4*Cos[2*(a + b*x)] - 16*b^3*c^3*d*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - 48*b^3*c^2*d^2*x*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - 48*b^3*c*d^3*x^2*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - 16*b^3*d^4*x^3*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] + (24*I)*b^2*d^2*(c + d*x)^2*PolyLog[2, -Cos[a + b*x] - I*Sin[a + b*x]]*Sin[a + b*x] - (24*I)*b^2*d^2*(c + d*x)^2*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - 48*b*c*d^3*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]]*Sin[a + b*x] - 48*b*d^4*x*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]]*Sin[a + b*x] + 48*b*c*d^3*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] + 48*b*d^4*x*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - (48*I)*d^4*PolyLog[4, -Cos[a + b*x] - I*Sin[a + b*x]]*Sin[a + b*x] + (48*I)*d^4*PolyLog[4, Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - 4*b^3*c^3*d*Sin[2*(a + b*x)] + 24*b*c*d^3*Sin[2*(a + b*x)] - 12*b^3*c^2*d^2*x*Sin[2*(a + b*x)] + 24*b*d^4*x*Sin[2*(a + b*x)] - 12*b^3*c*d^3*x^2*Sin[2*(a + b*x)] - 4*b^3*d^4*x^3*Sin[2*(a + b*x)]))/(2*b^5)","B",0
172,1,539,216,1.4243051,"\int (c+d x)^3 \cos (a+b x) \cot ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Cos[a + b*x]*Cot[a + b*x]^2,x]","\frac{\csc (a+b x) \left(b^3 c^3 \cos (2 (a+b x))+3 b^3 c^2 d x \cos (2 (a+b x))+3 b^3 c d^2 x^2 \cos (2 (a+b x))+b^3 d^3 x^3 \cos (2 (a+b x))-3 b^2 c^2 d \sin (2 (a+b x))+6 b^2 c^2 d \log \left(1-e^{i (a+b x)}\right) \sin (a+b x)-6 b^2 c^2 d \log \left(1+e^{i (a+b x)}\right) \sin (a+b x)-6 b^2 c d^2 x \sin (2 (a+b x))+12 b^2 c d^2 x \log \left(1-e^{i (a+b x)}\right) \sin (a+b x)-12 b^2 c d^2 x \log \left(1+e^{i (a+b x)}\right) \sin (a+b x)-3 b^2 d^3 x^2 \sin (2 (a+b x))+6 b^2 d^3 x^2 \log \left(1-e^{i (a+b x)}\right) \sin (a+b x)-6 b^2 d^3 x^2 \log \left(1+e^{i (a+b x)}\right) \sin (a+b x)+12 i b d^2 (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right) \sin (a+b x)-12 i b d^2 (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right) \sin (a+b x)-6 b c d^2 \cos (2 (a+b x))-12 d^3 \text{Li}_3\left(-e^{i (a+b x)}\right) \sin (a+b x)+12 d^3 \text{Li}_3\left(e^{i (a+b x)}\right) \sin (a+b x)+6 d^3 \sin (2 (a+b x))-6 b d^3 x \cos (2 (a+b x))-3 b^3 c^3-9 b^3 c^2 d x-9 b^3 c d^2 x^2-3 b^3 d^3 x^3+6 b c d^2+6 b d^3 x\right)}{2 b^4}","-\frac{6 d^3 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^4}+\frac{6 d^3 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^4}+\frac{6 d^3 \cos (a+b x)}{b^4}+\frac{6 i d^2 (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^3}-\frac{6 i d^2 (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^3}+\frac{6 d^2 (c+d x) \sin (a+b x)}{b^3}-\frac{3 d (c+d x)^2 \cos (a+b x)}{b^2}-\frac{6 d (c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{(c+d x)^3 \sin (a+b x)}{b}-\frac{(c+d x)^3 \csc (a+b x)}{b}",1,"(Csc[a + b*x]*(-3*b^3*c^3 + 6*b*c*d^2 - 9*b^3*c^2*d*x + 6*b*d^3*x - 9*b^3*c*d^2*x^2 - 3*b^3*d^3*x^3 + b^3*c^3*Cos[2*(a + b*x)] - 6*b*c*d^2*Cos[2*(a + b*x)] + 3*b^3*c^2*d*x*Cos[2*(a + b*x)] - 6*b*d^3*x*Cos[2*(a + b*x)] + 3*b^3*c*d^2*x^2*Cos[2*(a + b*x)] + b^3*d^3*x^3*Cos[2*(a + b*x)] + 6*b^2*c^2*d*Log[1 - E^(I*(a + b*x))]*Sin[a + b*x] + 12*b^2*c*d^2*x*Log[1 - E^(I*(a + b*x))]*Sin[a + b*x] + 6*b^2*d^3*x^2*Log[1 - E^(I*(a + b*x))]*Sin[a + b*x] - 6*b^2*c^2*d*Log[1 + E^(I*(a + b*x))]*Sin[a + b*x] - 12*b^2*c*d^2*x*Log[1 + E^(I*(a + b*x))]*Sin[a + b*x] - 6*b^2*d^3*x^2*Log[1 + E^(I*(a + b*x))]*Sin[a + b*x] + (12*I)*b*d^2*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))]*Sin[a + b*x] - (12*I)*b*d^2*(c + d*x)*PolyLog[2, E^(I*(a + b*x))]*Sin[a + b*x] - 12*d^3*PolyLog[3, -E^(I*(a + b*x))]*Sin[a + b*x] + 12*d^3*PolyLog[3, E^(I*(a + b*x))]*Sin[a + b*x] - 3*b^2*c^2*d*Sin[2*(a + b*x)] + 6*d^3*Sin[2*(a + b*x)] - 6*b^2*c*d^2*x*Sin[2*(a + b*x)] - 3*b^2*d^3*x^2*Sin[2*(a + b*x)]))/(2*b^4)","B",1
173,1,310,139,3.9153176,"\int (c+d x)^2 \cos (a+b x) \cot ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Cos[a + b*x]*Cot[a + b*x]^2,x]","-\frac{2 \cos (b x) \left(\sin (a) \left(b^2 (c+d x)^2-2 d^2\right)+2 b d \cos (a) (c+d x)\right)+2 \sin (b x) \left(\cos (a) \left(b^2 (c+d x)^2-2 d^2\right)-2 b d \sin (a) (c+d x)\right)+2 b^2 \csc (a) (c+d x)^2-b^2 \csc \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) (c+d x)^2 \csc \left(\frac{1}{2} (a+b x)\right)+b^2 \sec \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) (c+d x)^2 \sec \left(\frac{1}{2} (a+b x)\right)+8 b c d \tanh ^{-1}\left(\cos (a)-\sin (a) \tan \left(\frac{b x}{2}\right)\right)-4 d^2 \left(2 \tan ^{-1}(\tan (a)) \tanh ^{-1}\left(\cos (a)-\sin (a) \tan \left(\frac{b x}{2}\right)\right)+\frac{\sec (a) \left(i \text{Li}_2\left(-e^{i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)-i \text{Li}_2\left(e^{i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\left(\tan ^{-1}(\tan (a))+b x\right) \left(\log \left(1-e^{i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)-\log \left(1+e^{i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)\right)\right)}{\sqrt{\sec ^2(a)}}\right)}{2 b^3}","\frac{2 i d^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^3}-\frac{2 i d^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^3}+\frac{2 d^2 \sin (a+b x)}{b^3}-\frac{2 d (c+d x) \cos (a+b x)}{b^2}-\frac{4 d (c+d x) \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{(c+d x)^2 \sin (a+b x)}{b}-\frac{(c+d x)^2 \csc (a+b x)}{b}",1,"-1/2*(8*b*c*d*ArcTanh[Cos[a] - Sin[a]*Tan[(b*x)/2]] + 2*b^2*(c + d*x)^2*Csc[a] - 4*d^2*(2*ArcTan[Tan[a]]*ArcTanh[Cos[a] - Sin[a]*Tan[(b*x)/2]] + (((b*x + ArcTan[Tan[a]])*(Log[1 - E^(I*(b*x + ArcTan[Tan[a]]))] - Log[1 + E^(I*(b*x + ArcTan[Tan[a]]))]) + I*PolyLog[2, -E^(I*(b*x + ArcTan[Tan[a]]))] - I*PolyLog[2, E^(I*(b*x + ArcTan[Tan[a]]))])*Sec[a])/Sqrt[Sec[a]^2]) + 2*Cos[b*x]*(2*b*d*(c + d*x)*Cos[a] + (-2*d^2 + b^2*(c + d*x)^2)*Sin[a]) - b^2*(c + d*x)^2*Csc[a/2]*Csc[(a + b*x)/2]*Sin[(b*x)/2] + b^2*(c + d*x)^2*Sec[a/2]*Sec[(a + b*x)/2]*Sin[(b*x)/2] + 2*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] - 2*b*d*(c + d*x)*Sin[a])*Sin[b*x])/b^3","B",0
174,1,104,58,0.6793853,"\int (c+d x) \cos (a+b x) \cot ^2(a+b x) \, dx","Integrate[(c + d*x)*Cos[a + b*x]*Cot[a + b*x]^2,x]","-\frac{2 b c \sin (a+b x)+2 b c \csc (a+b x)+2 b d x \sin (a+b x)+2 d \cos (a+b x)+b d x \tan \left(\frac{1}{2} (a+b x)\right)+b d x \cot \left(\frac{1}{2} (a+b x)\right)-2 d \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+2 d \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{2 b^2}","-\frac{d \cos (a+b x)}{b^2}-\frac{d \tanh ^{-1}(\cos (a+b x))}{b^2}-\frac{(c+d x) \sin (a+b x)}{b}-\frac{(c+d x) \csc (a+b x)}{b}",1,"-1/2*(2*d*Cos[a + b*x] + b*d*x*Cot[(a + b*x)/2] + 2*b*c*Csc[a + b*x] + 2*d*Log[Cos[(a + b*x)/2]] - 2*d*Log[Sin[(a + b*x)/2]] + 2*b*c*Sin[a + b*x] + 2*b*d*x*Sin[a + b*x] + b*d*x*Tan[(a + b*x)/2])/b^2","A",1
175,0,0,75,3.7170084,"\int \frac{\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx","Integrate[(Cos[a + b*x]*Cot[a + b*x]^2)/(c + d*x),x]","\int \frac{\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\cot (a+b x) \csc (a+b x)}{c+d x},x\right)-\frac{\cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d}+\frac{\sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d}",0,"Integrate[(Cos[a + b*x]*Cot[a + b*x]^2)/(c + d*x), x]","A",-1
176,0,0,93,4.0408976,"\int \frac{\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[(Cos[a + b*x]*Cot[a + b*x]^2)/(c + d*x)^2,x]","\int \frac{\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\cot (a+b x) \csc (a+b x)}{(c+d x)^2},x\right)+\frac{b \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d^2}+\frac{b \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d^2}+\frac{\cos (a+b x)}{d (c+d x)}",0,"Integrate[(Cos[a + b*x]*Cot[a + b*x]^2)/(c + d*x)^2, x]","A",-1
177,0,0,19,11.4813366,"\int (c+d x)^m \cot ^3(a+b x) \, dx","Integrate[(c + d*x)^m*Cot[a + b*x]^3,x]","\int (c+d x)^m \cot ^3(a+b x) \, dx","\text{Int}\left(\cot ^3(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Cot[a + b*x]^3, x]","A",-1
178,1,1534,302,7.1200087,"\int (c+d x)^4 \cot ^3(a+b x) \, dx","Integrate[(c + d*x)^4*Cot[a + b*x]^3,x]","-\frac{\csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a)) c^4}{b \left(\cos ^2(a)+\sin ^2(a)\right)}+\frac{2 d \csc (a) \sec (a) \left(b^2 e^{i \tan ^{-1}(\tan (a))} x^2+\frac{\left(i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x+\tan ^{-1}(\tan (a))\right) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\pi  \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right) \tan (a)}{\sqrt{\tan ^2(a)+1}}\right) c^3}{b^2 \sqrt{\sec ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}+\frac{d^2 e^{i a} \csc (a) \left(2 b^3 e^{-2 i a} x^3+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^2+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^2-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right) c^2}{b^3}+\frac{6 d^2 \csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a)) c^2}{b^3 \left(\cos ^2(a)+\sin ^2(a)\right)}+\frac{d^3 e^{i a} \csc (a) \left(b^4 e^{-2 i a} x^4+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^3+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^3-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(-e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(-e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(e^{-i (a+b x)}\right)\right)\right) c}{b^4}-\frac{6 d^3 \csc (a) \sec (a) \left(b^2 e^{i \tan ^{-1}(\tan (a))} x^2+\frac{\left(i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x+\tan ^{-1}(\tan (a))\right) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\pi  \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right) \tan (a)}{\sqrt{\tan ^2(a)+1}}\right) c}{b^4 \sqrt{\sec ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}-\frac{(c+d x)^4 \csc ^2(a+b x)}{2 b}-\frac{1}{5} x \left(5 c^4+10 d x c^3+10 d^2 x^2 c^2+5 d^3 x^3 c+d^4 x^4\right) \cot (a)-\frac{d^4 e^{i a} \csc (a) \left(2 b^3 e^{-2 i a} x^3+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^2+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^2-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right)}{b^5}+\frac{d^4 e^{i a} \csc (a) \left(2 b^5 e^{-2 i a} x^5+5 i b^4 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^4+5 i b^4 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^4-20 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^3 \text{Li}_2\left(-e^{-i (a+b x)}\right) x^3-3 i b^2 \text{Li}_3\left(-e^{-i (a+b x)}\right) x^2-6 b \text{Li}_4\left(-e^{-i (a+b x)}\right) x+6 i \text{Li}_5\left(-e^{-i (a+b x)}\right)\right)-20 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^3 \text{Li}_2\left(e^{-i (a+b x)}\right) x^3-3 i b^2 \text{Li}_3\left(e^{-i (a+b x)}\right) x^2-6 b \text{Li}_4\left(e^{-i (a+b x)}\right) x+6 i \text{Li}_5\left(e^{-i (a+b x)}\right)\right)\right)}{10 b^5}+\frac{2 \csc (a) \csc (a+b x) \left(x^3 \sin (b x) d^4+3 c x^2 \sin (b x) d^3+3 c^2 x \sin (b x) d^2+c^3 \sin (b x) d\right)}{b^2}","\frac{3 d^4 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{b^5}+\frac{3 d^4 \text{Li}_5\left(e^{2 i (a+b x)}\right)}{2 b^5}-\frac{6 i d^3 (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^4}-\frac{3 i d^3 (c+d x) \text{Li}_4\left(e^{2 i (a+b x)}\right)}{b^4}-\frac{3 d^2 (c+d x)^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{b^3}+\frac{6 d^2 (c+d x)^2 \log \left(1-e^{2 i (a+b x)}\right)}{b^3}+\frac{2 i d (c+d x)^3 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{2 d (c+d x)^3 \cot (a+b x)}{b^2}-\frac{(c+d x)^4 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{(c+d x)^4 \cot ^2(a+b x)}{2 b}-\frac{2 i d (c+d x)^3}{b^2}-\frac{(c+d x)^4}{2 b}+\frac{i (c+d x)^5}{5 d}",1,"-1/5*(x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4)*Cot[a]) - ((c + d*x)^4*Csc[a + b*x]^2)/(2*b) + (c^2*d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^3 - (d^4*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^5 + (c*d^3*E^(I*a)*Csc[a]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-I)*(a + b*x))] - 2*PolyLog[4, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, E^((-I)*(a + b*x))] - 2*PolyLog[4, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^4 + (d^4*E^(I*a)*Csc[a]*((2*b^5*x^5)/E^((2*I)*a) + (5*I)*b^4*(1 - E^((-2*I)*a))*x^4*Log[1 - E^((-I)*(a + b*x))] + (5*I)*b^4*(1 - E^((-2*I)*a))*x^4*Log[1 + E^((-I)*(a + b*x))] - (20*(-1 + E^((2*I)*a))*(b^3*x^3*PolyLog[2, -E^((-I)*(a + b*x))] - (3*I)*b^2*x^2*PolyLog[3, -E^((-I)*(a + b*x))] - 6*b*x*PolyLog[4, -E^((-I)*(a + b*x))] + (6*I)*PolyLog[5, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (20*(-1 + E^((2*I)*a))*(b^3*x^3*PolyLog[2, E^((-I)*(a + b*x))] - (3*I)*b^2*x^2*PolyLog[3, E^((-I)*(a + b*x))] - 6*b*x*PolyLog[4, E^((-I)*(a + b*x))] + (6*I)*PolyLog[5, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(10*b^5) - (c^4*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + (6*c^2*d^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (2*Csc[a]*Csc[a + b*x]*(c^3*d*Sin[b*x] + 3*c^2*d^2*x*Sin[b*x] + 3*c*d^3*x^2*Sin[b*x] + d^4*x^3*Sin[b*x]))/b^2 + (2*c^3*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (6*c*d^3*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^4*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
179,1,994,256,6.8928159,"\int (c+d x)^3 \cot ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Cot[a + b*x]^3,x]","-\frac{\csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a)) c^3}{b \left(\cos ^2(a)+\sin ^2(a)\right)}+\frac{3 d \csc (a) \sec (a) \left(b^2 e^{i \tan ^{-1}(\tan (a))} x^2+\frac{\left(i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x+\tan ^{-1}(\tan (a))\right) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\pi  \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right) \tan (a)}{\sqrt{\tan ^2(a)+1}}\right) c^2}{2 b^2 \sqrt{\sec ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}+\frac{d^2 e^{i a} \csc (a) \left(2 b^3 e^{-2 i a} x^3+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^2+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^2-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right) c}{2 b^3}+\frac{3 d^2 \csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a)) c}{b^3 \left(\cos ^2(a)+\sin ^2(a)\right)}-\frac{(c+d x)^3 \csc ^2(a+b x)}{2 b}-\frac{1}{4} x \left(4 c^3+6 d x c^2+4 d^2 x^2 c+d^3 x^3\right) \cot (a)+\frac{d^3 e^{i a} \csc (a) \left(b^4 e^{-2 i a} x^4+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^3+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^3-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(-e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(-e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(e^{-i (a+b x)}\right)\right)\right)}{4 b^4}+\frac{3 \csc (a) \csc (a+b x) \left(x^2 \sin (b x) d^3+2 c x \sin (b x) d^2+c^2 \sin (b x) d\right)}{2 b^2}-\frac{3 d^3 \csc (a) \sec (a) \left(b^2 e^{i \tan ^{-1}(\tan (a))} x^2+\frac{\left(i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x+\tan ^{-1}(\tan (a))\right) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\pi  \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right) \tan (a)}{\sqrt{\tan ^2(a)+1}}\right)}{2 b^4 \sqrt{\sec ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}","-\frac{3 i d^3 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 i d^3 \text{Li}_4\left(e^{2 i (a+b x)}\right)}{4 b^4}-\frac{3 d^2 (c+d x) \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}+\frac{3 d^2 (c+d x) \log \left(1-e^{2 i (a+b x)}\right)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac{(c+d x)^3 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{(c+d x)^3 \cot ^2(a+b x)}{2 b}-\frac{3 i d (c+d x)^2}{2 b^2}-\frac{(c+d x)^3}{2 b}+\frac{i (c+d x)^4}{4 d}",1,"-1/4*(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Cot[a]) - ((c + d*x)^3*Csc[a + b*x]^2)/(2*b) + (c*d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(2*b^3) + (d^3*E^(I*a)*Csc[a]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-I)*(a + b*x))] - 2*PolyLog[4, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, E^((-I)*(a + b*x))] - 2*PolyLog[4, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(4*b^4) - (c^3*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + (3*c*d^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (3*Csc[a]*Csc[a + b*x]*(c^2*d*Sin[b*x] + 2*c*d^2*x*Sin[b*x] + d^3*x^2*Sin[b*x]))/(2*b^2) + (3*c^2*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(2*b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (3*d^3*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(2*b^4*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
180,1,540,168,6.6813394,"\int (c+d x)^2 \cot ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Cot[a + b*x]^3,x]","\frac{d^2 \csc (a) (\sin (a) \log (\sin (a) \cos (b x)+\cos (a) \sin (b x))-b x \cos (a))}{b^3 \left(\sin ^2(a)+\cos ^2(a)\right)}+\frac{\csc (a) \csc (a+b x) \left(c d \sin (b x)+d^2 x \sin (b x)\right)}{b^2}+\frac{c d \csc (a) \sec (a) \left(b^2 x^2 e^{i \tan ^{-1}(\tan (a))}+\frac{\tan (a) \left(i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\tan ^2(a)+1}}\right)}{b^2 \sqrt{\sec ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}+\frac{e^{i a} d^2 \csc (a) \left(2 e^{-2 i a} b^3 x^3+3 i \left(1-e^{-2 i a}\right) b^2 x^2 \log \left(1-e^{-i (a+b x)}\right)+3 i \left(1-e^{-2 i a}\right) b^2 x^2 \log \left(1+e^{-i (a+b x)}\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right)}{6 b^3}-\frac{c^2 \csc (a) (\sin (a) \log (\sin (a) \cos (b x)+\cos (a) \sin (b x))-b x \cos (a))}{b \left(\sin ^2(a)+\cos ^2(a)\right)}-\frac{(c+d x)^2 \csc ^2(a+b x)}{2 b}-\frac{1}{3} x \cot (a) \left(3 c^2+3 c d x+d^2 x^2\right)","-\frac{d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}+\frac{d^2 \log (\sin (a+b x))}{b^3}+\frac{i d (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{d (c+d x) \cot (a+b x)}{b^2}-\frac{(c+d x)^2 \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{(c+d x)^2 \cot ^2(a+b x)}{2 b}-\frac{c d x}{b}-\frac{d^2 x^2}{2 b}+\frac{i (c+d x)^3}{3 d}",1,"-1/3*(x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cot[a]) - ((c + d*x)^2*Csc[a + b*x]^2)/(2*b) + (d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(6*b^3) - (c^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + (d^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (Csc[a]*Csc[a + b*x]*(c*d*Sin[b*x] + d^2*x*Sin[b*x]))/b^2 + (c*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
181,1,240,109,6.1662672,"\int (c+d x) \cot ^3(a+b x) \, dx","Integrate[(c + d*x)*Cot[a + b*x]^3,x]","\frac{d \csc (a) \sec (a) \left(b^2 x^2 e^{i \tan ^{-1}(\tan (a))}+\frac{\tan (a) \left(i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-2 \left(\tan ^{-1}(\tan (a))+b x\right) \log \left(1-e^{2 i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(\tan ^{-1}(\tan (a))+b x\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\tan ^2(a)+1}}\right)}{2 b^2 \sqrt{\sec ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}+\frac{d \csc (a) \sin (b x) \csc (a+b x)}{2 b^2}-\frac{c \left(\cot ^2(a+b x)+2 \log (\tan (a+b x))+2 \log (\cos (a+b x))\right)}{2 b}-\frac{d x \csc ^2(a+b x)}{2 b}-\frac{1}{2} d x^2 \cot (a)","\frac{i d \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{d \cot (a+b x)}{2 b^2}-\frac{(c+d x) \log \left(1-e^{2 i (a+b x)}\right)}{b}-\frac{(c+d x) \cot ^2(a+b x)}{2 b}-\frac{d x}{2 b}+\frac{i (c+d x)^2}{2 d}",1,"-1/2*(d*x^2*Cot[a]) - (d*x*Csc[a + b*x]^2)/(2*b) - (c*(Cot[a + b*x]^2 + 2*Log[Cos[a + b*x]] + 2*Log[Tan[a + b*x]]))/(2*b) + (d*Csc[a]*Csc[a + b*x]*Sin[b*x])/(2*b^2) + (d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(2*b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
182,0,0,19,8.0804232,"\int \frac{\cot ^3(a+b x)}{c+d x} \, dx","Integrate[Cot[a + b*x]^3/(c + d*x),x]","\int \frac{\cot ^3(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\cot ^3(a+b x)}{c+d x},x\right)",0,"Integrate[Cot[a + b*x]^3/(c + d*x), x]","A",-1
183,0,0,19,9.2887314,"\int \frac{\cot ^3(a+b x)}{(c+d x)^2} \, dx","Integrate[Cot[a + b*x]^3/(c + d*x)^2,x]","\int \frac{\cot ^3(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\cot ^3(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[Cot[a + b*x]^3/(c + d*x)^2, x]","A",-1
184,1,550,407,12.14809,"\int (c+d x)^{5/2} \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x],x]","\frac{-1024 b^3 c^2 \sqrt{c+d x} \cos (2 (a+b x))-256 b^3 c^2 \sqrt{c+d x} \cos (4 (a+b x))-1024 b^3 d^2 x^2 \sqrt{c+d x} \cos (2 (a+b x))-256 b^3 d^2 x^2 \sqrt{c+d x} \cos (4 (a+b x))-2048 b^3 c d x \sqrt{c+d x} \cos (2 (a+b x))-512 b^3 c d x \sqrt{c+d x} \cos (4 (a+b x))+1280 b^2 d^2 x \sqrt{c+d x} \sin (2 (a+b x))+160 b^2 d^2 x \sqrt{c+d x} \sin (4 (a+b x))+1280 b^2 c d \sqrt{c+d x} \sin (2 (a+b x))+160 b^2 c d \sqrt{c+d x} \sin (4 (a+b x))-15 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-480 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+15 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+480 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+960 b d^2 \sqrt{c+d x} \cos (2 (a+b x))+60 b d^2 \sqrt{c+d x} \cos (4 (a+b x))}{8192 b^4}","-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}-\frac{15 \sqrt{\pi } d^{5/2} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{256 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}+\frac{15 \sqrt{\pi } d^{5/2} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{256 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{128 b^3}+\frac{15 d^2 \sqrt{c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}+\frac{5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}",1,"(-1024*b^3*c^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 960*b*d^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 2048*b^3*c*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 1024*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 256*b^3*c^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 60*b*d^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 512*b^3*c*d*x*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 256*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 15*Sqrt[b/d]*d^3*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 480*Sqrt[b/d]*d^3*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + 15*Sqrt[b/d]*d^3*Sqrt[2*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] + 480*Sqrt[b/d]*d^3*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 1280*b^2*c*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 1280*b^2*d^2*x*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 160*b^2*c*d*Sqrt[c + d*x]*Sin[4*(a + b*x)] + 160*b^2*d^2*x*Sqrt[c + d*x]*Sin[4*(a + b*x)])/(8192*b^4)","A",1
185,1,393,351,2.9827589,"\int (c+d x)^{3/2} \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x],x]","\frac{-3 \sqrt{2 \pi } d \sin \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-48 \sqrt{\pi } d \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-3 \sqrt{2 \pi } d \cos \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-48 \sqrt{\pi } d \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+96 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (2 (a+b x))+12 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (4 (a+b x))-128 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-128 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-32 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))-32 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))}{1024 b^2 \sqrt{\frac{b}{d}}}","-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{64 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{64 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (2 a+2 b x)}{32 b^2}+\frac{3 d \sqrt{c+d x} \sin (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}-\frac{(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}",1,"(-128*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 128*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 32*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 32*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 3*d*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 48*d*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 3*d*Sqrt[2*Pi]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] - 48*d*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 96*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 12*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[4*(a + b*x)])/(1024*b^2*Sqrt[b/d])","A",1
186,1,264,299,0.5901064,"\int \sqrt{c+d x} \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^3*Sin[a + b*x],x]","\frac{\sqrt{2 \pi } \cos \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+8 \sqrt{\pi } \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-\sqrt{2 \pi } \sin \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-8 \sqrt{\pi } \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-16 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-4 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))}{128 b \sqrt{\frac{b}{d}}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}+\frac{\sqrt{\pi } \sqrt{d} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{16 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{16 b^{3/2}}-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{8 b}-\frac{\sqrt{c+d x} \cos (4 a+4 b x)}{32 b}",1,"(-16*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 4*Sqrt[b/d]*Sqrt[c + d*x]*Cos[4*(a + b*x)] + Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] + 8*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - Sqrt[2*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] - 8*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d])/(128*b*Sqrt[b/d])","A",1
187,1,264,299,0.1589278,"\int \sqrt{c+d x} \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^3*Sin[a + b*x],x]","\frac{\sqrt{2 \pi } \cos \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+8 \sqrt{\pi } \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-\sqrt{2 \pi } \sin \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-8 \sqrt{\pi } \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-16 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-4 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))}{128 b \sqrt{\frac{b}{d}}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}+\frac{\sqrt{\pi } \sqrt{d} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{16 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{64 b^{3/2}}-\frac{\sqrt{\pi } \sqrt{d} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{16 b^{3/2}}-\frac{\sqrt{c+d x} \cos (2 a+2 b x)}{8 b}-\frac{\sqrt{c+d x} \cos (4 a+4 b x)}{32 b}",1,"(-16*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 4*Sqrt[b/d]*Sqrt[c + d*x]*Cos[4*(a + b*x)] + Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] + 8*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - Sqrt[2*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] - 8*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d])/(128*b*Sqrt[b/d])","A",1
188,1,393,351,1.1988973,"\int (c+d x)^{3/2} \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x],x]","\frac{-3 \sqrt{2 \pi } d \sin \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-48 \sqrt{\pi } d \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-3 \sqrt{2 \pi } d \cos \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-48 \sqrt{\pi } d \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+96 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (2 (a+b x))+12 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (4 (a+b x))-128 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-128 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-32 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))-32 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (4 (a+b x))}{1024 b^2 \sqrt{\frac{b}{d}}}","-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{64 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{512 b^{5/2}}-\frac{3 \sqrt{\pi } d^{3/2} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{64 b^{5/2}}+\frac{3 d \sqrt{c+d x} \sin (2 a+2 b x)}{32 b^2}+\frac{3 d \sqrt{c+d x} \sin (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}-\frac{(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}",1,"(-128*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 128*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 32*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 32*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 3*d*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 48*d*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 3*d*Sqrt[2*Pi]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] - 48*d*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 96*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 12*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[4*(a + b*x)])/(1024*b^2*Sqrt[b/d])","A",1
189,1,550,407,8.3945835,"\int (c+d x)^{5/2} \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x],x]","\frac{-1024 b^3 c^2 \sqrt{c+d x} \cos (2 (a+b x))-256 b^3 c^2 \sqrt{c+d x} \cos (4 (a+b x))-1024 b^3 d^2 x^2 \sqrt{c+d x} \cos (2 (a+b x))-256 b^3 d^2 x^2 \sqrt{c+d x} \cos (4 (a+b x))-2048 b^3 c d x \sqrt{c+d x} \cos (2 (a+b x))-512 b^3 c d x \sqrt{c+d x} \cos (4 (a+b x))+1280 b^2 d^2 x \sqrt{c+d x} \sin (2 (a+b x))+160 b^2 d^2 x \sqrt{c+d x} \sin (4 (a+b x))+1280 b^2 c d \sqrt{c+d x} \sin (2 (a+b x))+160 b^2 c d \sqrt{c+d x} \sin (4 (a+b x))-15 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)-480 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+15 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right)+480 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+960 b d^2 \sqrt{c+d x} \cos (2 (a+b x))+60 b d^2 \sqrt{c+d x} \cos (4 (a+b x))}{8192 b^4}","-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(4 a-\frac{4 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}-\frac{15 \sqrt{\pi } d^{5/2} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{256 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(4 a-\frac{4 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{4096 b^{7/2}}+\frac{15 \sqrt{\pi } d^{5/2} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{256 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{128 b^3}+\frac{15 d^2 \sqrt{c+d x} \cos (4 a+4 b x)}{2048 b^3}+\frac{5 d (c+d x)^{3/2} \sin (2 a+2 b x)}{32 b^2}+\frac{5 d (c+d x)^{3/2} \sin (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{5/2} \cos (2 a+2 b x)}{8 b}-\frac{(c+d x)^{5/2} \cos (4 a+4 b x)}{32 b}",1,"(-1024*b^3*c^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 960*b*d^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 2048*b^3*c*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 1024*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 256*b^3*c^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] + 60*b*d^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 512*b^3*c*d*x*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 256*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[4*(a + b*x)] - 15*Sqrt[b/d]*d^3*Sqrt[2*Pi]*Cos[4*a - (4*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]] - 480*Sqrt[b/d]*d^3*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + 15*Sqrt[b/d]*d^3*Sqrt[2*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[4*a - (4*b*c)/d] + 480*Sqrt[b/d]*d^3*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 1280*b^2*c*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 1280*b^2*d^2*x*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 160*b^2*c*d*Sqrt[c + d*x]*Sin[4*(a + b*x)] + 160*b^2*d^2*x*Sqrt[c + d*x]*Sin[4*(a + b*x)])/(8192*b^4)","A",1
190,1,1795,615,22.420874,"\int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x]^2,x]","-\frac{i e^{-\frac{i (b c+a d)}{d}} \sqrt{c+d x} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right) c^2}{16 b}-\frac{\left(-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \sin \left(3 a-\frac{3 b c}{d}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))\right) c^2}{96 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{\left(-\sqrt{2 \pi } \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \sin \left(5 a-\frac{5 b c}{d}\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (5 (a+b x))\right) c^2}{160 \sqrt{5} b \sqrt{\frac{b}{d}}}+\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(a-\frac{b c}{d}\right)-3 d \cos \left(a-\frac{b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(a-\frac{b c}{d}\right)+3 d \sin \left(a-\frac{b c}{d}\right)\right)+2 b \sqrt{c+d x} (3 \cos (a+b x)+2 b x \sin (a+b x))\right) c}{16 b^3}-\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(3 a-\frac{3 b c}{d}\right)-d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(3 a-\frac{3 b c}{d}\right)+d \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} b \sqrt{c+d x} (\cos (3 (a+b x))+2 b x \sin (3 (a+b x)))\right) c}{96 \sqrt{3} b^3}-\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(10 b c \sin \left(5 a-\frac{5 b c}{d}\right)-3 d \cos \left(5 a-\frac{5 b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(10 b c \cos \left(5 a-\frac{5 b c}{d}\right)+3 d \sin \left(5 a-\frac{5 b c}{d}\right)\right)+2 \sqrt{5} b \sqrt{c+d x} (3 \cos (5 (a+b x))+10 b x \sin (5 (a+b x)))\right) c}{800 \sqrt{5} b^3}+\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \cos \left(a-\frac{b c}{d}\right)+12 b c d \sin \left(a-\frac{b c}{d}\right)\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \sin \left(a-\frac{b c}{d}\right)-12 b c d \cos \left(a-\frac{b c}{d}\right)\right)+2 \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(4 b^2 x^2-15\right) \sin (a+b x)-2 b (c-5 d x) \cos (a+b x)\right)\right)}{64 b^5}-\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \cos \left(3 a-\frac{3 b c}{d}\right)+12 b c d \sin \left(3 a-\frac{3 b c}{d}\right)\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \sin \left(3 a-\frac{3 b c}{d}\right)-12 b c d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(12 b^2 x^2-5\right) \sin (3 (a+b x))-2 b (c-5 d x) \cos (3 (a+b x))\right)\right)}{1152 \sqrt{3} b^5}-\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(\left(20 b^2 c^2-3 d^2\right) \cos \left(5 a-\frac{5 b c}{d}\right)+12 b c d \sin \left(5 a-\frac{5 b c}{d}\right)\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(\left(20 b^2 c^2-3 d^2\right) \sin \left(5 a-\frac{5 b c}{d}\right)-12 b c d \cos \left(5 a-\frac{5 b c}{d}\right)\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(20 b^2 x^2-3\right) \sin (5 (a+b x))-2 b (c-5 d x) \cos (5 (a+b x))\right)\right)}{3200 \sqrt{5} b^5}","-\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \sin \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1600 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{576 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{32 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{32 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{576 b^{7/2}}-\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1600 b^{7/2}}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{32 b^3}+\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{576 b^3}+\frac{3 d^2 \sqrt{c+d x} \sin (5 a+5 b x)}{1600 b^3}+\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{16 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \cos (5 a+5 b x)}{160 b^2}+\frac{(c+d x)^{5/2} \sin (a+b x)}{8 b}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{48 b}-\frac{(c+d x)^{5/2} \sin (5 a+5 b x)}{80 b}",1,"((-1/16*I)*c^2*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) + (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[a - (b*c)/d] + 2*b*c*Sin[a - (b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*b*Sqrt[c + d*x]*(3*Cos[a + b*x] + 2*b*x*Sin[a + b*x])))/(16*b^3) + ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*((4*b^2*c^2 - 15*d^2)*Cos[a - (b*c)/d] + 12*b*c*d*Sin[a - (b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[a - (b*c)/d] + (4*b^2*c^2 - 15*d^2)*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[a + b*x] + d*(-15 + 4*b^2*x^2)*Sin[a + b*x])))/(64*b^5) - (c^2*(-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-(d*Cos[3*a - (3*b*c)/d]) + 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*b*Sqrt[c + d*x]*(Cos[3*(a + b*x)] + 2*b*x*Sin[3*(a + b*x)])))/(96*Sqrt[3]*b^3) - ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*((12*b^2*c^2 - 5*d^2)*Cos[3*a - (3*b*c)/d] + 12*b*c*d*Sin[3*a - (3*b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[3*a - (3*b*c)/d] + (12*b^2*c^2 - 5*d^2)*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[3*(a + b*x)] + d*(-5 + 12*b^2*x^2)*Sin[3*(a + b*x)])))/(1152*Sqrt[3]*b^5) - (c^2*(-(Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d] + 2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[5*(a + b*x)]))/(160*Sqrt[5]*b*Sqrt[b/d]) - (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[5*a - (5*b*c)/d] + 10*b*c*Sin[5*a - (5*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(10*b*c*Cos[5*a - (5*b*c)/d] + 3*d*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*b*Sqrt[c + d*x]*(3*Cos[5*(a + b*x)] + 10*b*x*Sin[5*(a + b*x)])))/(800*Sqrt[5]*b^3) - ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*((20*b^2*c^2 - 3*d^2)*Cos[5*a - (5*b*c)/d] + 12*b*c*d*Sin[5*a - (5*b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[5*a - (5*b*c)/d] + (20*b^2*c^2 - 3*d^2)*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[5*(a + b*x)] + d*(-3 + 20*b^2*x^2)*Sin[5*(a + b*x)])))/(3200*Sqrt[5]*b^5)","C",0
191,1,1043,534,12.0023799,"\int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^2,x]","-\frac{i c e^{-\frac{i (b c+a d)}{d}} \sqrt{c+d x} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{16 b}+\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(a-\frac{b c}{d}\right)-3 d \cos \left(a-\frac{b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(a-\frac{b c}{d}\right)+3 d \sin \left(a-\frac{b c}{d}\right)\right)+2 b \sqrt{c+d x} (3 \cos (a+b x)+2 b x \sin (a+b x))\right)}{32 b^3}-\frac{c \left(-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \sin \left(3 a-\frac{3 b c}{d}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))\right)}{96 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(3 a-\frac{3 b c}{d}\right)-d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(3 a-\frac{3 b c}{d}\right)+d \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} b \sqrt{c+d x} (\cos (3 (a+b x))+2 b x \sin (3 (a+b x)))\right)}{192 \sqrt{3} b^3}-\frac{c \left(-\sqrt{2 \pi } \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \sin \left(5 a-\frac{5 b c}{d}\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (5 (a+b x))\right)}{160 \sqrt{5} b \sqrt{\frac{b}{d}}}-\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(10 b c \sin \left(5 a-\frac{5 b c}{d}\right)-3 d \cos \left(5 a-\frac{5 b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(10 b c \cos \left(5 a-\frac{5 b c}{d}\right)+3 d \sin \left(5 a-\frac{5 b c}{d}\right)\right)+2 \sqrt{5} b \sqrt{c+d x} (3 \cos (5 (a+b x))+10 b x \sin (5 (a+b x)))\right)}{1600 \sqrt{5} b^3}","-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{5/2}}+\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{96 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{10}} d^{3/2} \cos \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{800 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{10}} d^{3/2} \sin \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{800 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{96 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{5/2}}+\frac{3 d \sqrt{c+d x} \cos (a+b x)}{16 b^2}-\frac{d \sqrt{c+d x} \cos (3 a+3 b x)}{96 b^2}-\frac{3 d \sqrt{c+d x} \cos (5 a+5 b x)}{800 b^2}+\frac{(c+d x)^{3/2} \sin (a+b x)}{8 b}-\frac{(c+d x)^{3/2} \sin (3 a+3 b x)}{48 b}-\frac{(c+d x)^{3/2} \sin (5 a+5 b x)}{80 b}",1,"((-1/16*I)*c*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) + (d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[a - (b*c)/d] + 2*b*c*Sin[a - (b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*b*Sqrt[c + d*x]*(3*Cos[a + b*x] + 2*b*x*Sin[a + b*x])))/(32*b^3) - (c*(-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-(d*Cos[3*a - (3*b*c)/d]) + 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*b*Sqrt[c + d*x]*(Cos[3*(a + b*x)] + 2*b*x*Sin[3*(a + b*x)])))/(192*Sqrt[3]*b^3) - (c*(-(Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d] + 2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[5*(a + b*x)]))/(160*Sqrt[5]*b*Sqrt[b/d]) - (d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[5*a - (5*b*c)/d] + 10*b*c*Sin[5*a - (5*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(10*b*c*Cos[5*a - (5*b*c)/d] + 3*d*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*b*Sqrt[c + d*x]*(3*Cos[5*(a + b*x)] + 10*b*x*Sin[5*(a + b*x)])))/(1600*Sqrt[5]*b^3)","C",1
192,1,435,459,6.9424824,"\int \sqrt{c+d x} \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^3*Sin[a + b*x]^2,x]","-\frac{-\sqrt{2 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))}{96 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{-\sqrt{2 \pi } \sin \left(5 a-\frac{5 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (5 (a+b x))}{160 \sqrt{5} b \sqrt{\frac{b}{d}}}-\frac{i \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{16 b}","\frac{\sqrt{\frac{\pi }{10}} \sqrt{d} \sin \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{80 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{48 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{48 b^{3/2}}+\frac{\sqrt{\frac{\pi }{10}} \sqrt{d} \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{80 b^{3/2}}+\frac{\sqrt{c+d x} \sin (a+b x)}{8 b}-\frac{\sqrt{c+d x} \sin (3 a+3 b x)}{48 b}-\frac{\sqrt{c+d x} \sin (5 a+5 b x)}{80 b}",1,"((-1/16*I)*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) - (-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)])/(96*Sqrt[3]*b*Sqrt[b/d]) - (-(Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d] + 2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[5*(a + b*x)])/(160*Sqrt[5]*b*Sqrt[b/d])","C",1
193,1,435,459,6.8741728,"\int \sqrt{c+d x} \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^3*Sin[a + b*x]^2,x]","-\frac{-\sqrt{2 \pi } \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))}{96 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{-\sqrt{2 \pi } \sin \left(5 a-\frac{5 b c}{d}\right) C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (5 (a+b x))}{160 \sqrt{5} b \sqrt{\frac{b}{d}}}-\frac{i \sqrt{c+d x} e^{-\frac{i (a d+b c)}{d}} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{16 b}","\frac{\sqrt{\frac{\pi }{10}} \sqrt{d} \sin \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{80 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{48 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{8 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{d} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{48 b^{3/2}}+\frac{\sqrt{\frac{\pi }{10}} \sqrt{d} \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{80 b^{3/2}}+\frac{\sqrt{c+d x} \sin (a+b x)}{8 b}-\frac{\sqrt{c+d x} \sin (3 a+3 b x)}{48 b}-\frac{\sqrt{c+d x} \sin (5 a+5 b x)}{80 b}",1,"((-1/16*I)*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) - (-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)])/(96*Sqrt[3]*b*Sqrt[b/d]) - (-(Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d] + 2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[5*(a + b*x)])/(160*Sqrt[5]*b*Sqrt[b/d])","C",1
194,1,1043,534,11.7098116,"\int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^2,x]","-\frac{i c e^{-\frac{i (b c+a d)}{d}} \sqrt{c+d x} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right)}{16 b}+\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(a-\frac{b c}{d}\right)-3 d \cos \left(a-\frac{b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(a-\frac{b c}{d}\right)+3 d \sin \left(a-\frac{b c}{d}\right)\right)+2 b \sqrt{c+d x} (3 \cos (a+b x)+2 b x \sin (a+b x))\right)}{32 b^3}-\frac{c \left(-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \sin \left(3 a-\frac{3 b c}{d}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))\right)}{96 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(3 a-\frac{3 b c}{d}\right)-d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(3 a-\frac{3 b c}{d}\right)+d \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} b \sqrt{c+d x} (\cos (3 (a+b x))+2 b x \sin (3 (a+b x)))\right)}{192 \sqrt{3} b^3}-\frac{c \left(-\sqrt{2 \pi } \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \sin \left(5 a-\frac{5 b c}{d}\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (5 (a+b x))\right)}{160 \sqrt{5} b \sqrt{\frac{b}{d}}}-\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(10 b c \sin \left(5 a-\frac{5 b c}{d}\right)-3 d \cos \left(5 a-\frac{5 b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(10 b c \cos \left(5 a-\frac{5 b c}{d}\right)+3 d \sin \left(5 a-\frac{5 b c}{d}\right)\right)+2 \sqrt{5} b \sqrt{c+d x} (3 \cos (5 (a+b x))+10 b x \sin (5 (a+b x)))\right)}{1600 \sqrt{5} b^3}","-\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \cos \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{5/2}}+\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \cos \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{96 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{10}} d^{3/2} \cos \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{800 b^{5/2}}-\frac{3 \sqrt{\frac{\pi }{10}} d^{3/2} \sin \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{800 b^{5/2}}-\frac{\sqrt{\frac{\pi }{6}} d^{3/2} \sin \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{96 b^{5/2}}+\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} \sin \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{16 b^{5/2}}+\frac{3 d \sqrt{c+d x} \cos (a+b x)}{16 b^2}-\frac{d \sqrt{c+d x} \cos (3 a+3 b x)}{96 b^2}-\frac{3 d \sqrt{c+d x} \cos (5 a+5 b x)}{800 b^2}+\frac{(c+d x)^{3/2} \sin (a+b x)}{8 b}-\frac{(c+d x)^{3/2} \sin (3 a+3 b x)}{48 b}-\frac{(c+d x)^{3/2} \sin (5 a+5 b x)}{80 b}",1,"((-1/16*I)*c*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) + (d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[a - (b*c)/d] + 2*b*c*Sin[a - (b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*b*Sqrt[c + d*x]*(3*Cos[a + b*x] + 2*b*x*Sin[a + b*x])))/(32*b^3) - (c*(-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-(d*Cos[3*a - (3*b*c)/d]) + 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*b*Sqrt[c + d*x]*(Cos[3*(a + b*x)] + 2*b*x*Sin[3*(a + b*x)])))/(192*Sqrt[3]*b^3) - (c*(-(Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d] + 2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[5*(a + b*x)]))/(160*Sqrt[5]*b*Sqrt[b/d]) - (d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[5*a - (5*b*c)/d] + 10*b*c*Sin[5*a - (5*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(10*b*c*Cos[5*a - (5*b*c)/d] + 3*d*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*b*Sqrt[c + d*x]*(3*Cos[5*(a + b*x)] + 10*b*x*Sin[5*(a + b*x)])))/(1600*Sqrt[5]*b^3)","C",1
195,1,1795,615,21.7476483,"\int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x]^2,x]","-\frac{i e^{-\frac{i (b c+a d)}{d}} \sqrt{c+d x} \left(\frac{e^{2 i a} \Gamma \left(\frac{3}{2},-\frac{i b (c+d x)}{d}\right)}{\sqrt{-\frac{i b (c+d x)}{d}}}-\frac{e^{\frac{2 i b c}{d}} \Gamma \left(\frac{3}{2},\frac{i b (c+d x)}{d}\right)}{\sqrt{\frac{i b (c+d x)}{d}}}\right) c^2}{16 b}-\frac{\left(-\sqrt{2 \pi } \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \sin \left(3 a-\frac{3 b c}{d}\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))\right) c^2}{96 \sqrt{3} b \sqrt{\frac{b}{d}}}-\frac{\left(-\sqrt{2 \pi } \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \sin \left(5 a-\frac{5 b c}{d}\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (5 (a+b x))\right) c^2}{160 \sqrt{5} b \sqrt{\frac{b}{d}}}+\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(a-\frac{b c}{d}\right)-3 d \cos \left(a-\frac{b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(a-\frac{b c}{d}\right)+3 d \sin \left(a-\frac{b c}{d}\right)\right)+2 b \sqrt{c+d x} (3 \cos (a+b x)+2 b x \sin (a+b x))\right) c}{16 b^3}-\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \sin \left(3 a-\frac{3 b c}{d}\right)-d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(2 b c \cos \left(3 a-\frac{3 b c}{d}\right)+d \sin \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} b \sqrt{c+d x} (\cos (3 (a+b x))+2 b x \sin (3 (a+b x)))\right) c}{96 \sqrt{3} b^3}-\frac{d \left(\sqrt{\frac{b}{d}} \sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(10 b c \sin \left(5 a-\frac{5 b c}{d}\right)-3 d \cos \left(5 a-\frac{5 b c}{d}\right)\right)+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(10 b c \cos \left(5 a-\frac{5 b c}{d}\right)+3 d \sin \left(5 a-\frac{5 b c}{d}\right)\right)+2 \sqrt{5} b \sqrt{c+d x} (3 \cos (5 (a+b x))+10 b x \sin (5 (a+b x)))\right) c}{800 \sqrt{5} b^3}+\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \cos \left(a-\frac{b c}{d}\right)+12 b c d \sin \left(a-\frac{b c}{d}\right)\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right) \left(\left(4 b^2 c^2-15 d^2\right) \sin \left(a-\frac{b c}{d}\right)-12 b c d \cos \left(a-\frac{b c}{d}\right)\right)+2 \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(4 b^2 x^2-15\right) \sin (a+b x)-2 b (c-5 d x) \cos (a+b x)\right)\right)}{64 b^5}-\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \cos \left(3 a-\frac{3 b c}{d}\right)+12 b c d \sin \left(3 a-\frac{3 b c}{d}\right)\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right) \left(\left(12 b^2 c^2-5 d^2\right) \sin \left(3 a-\frac{3 b c}{d}\right)-12 b c d \cos \left(3 a-\frac{3 b c}{d}\right)\right)+2 \sqrt{3} \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(12 b^2 x^2-5\right) \sin (3 (a+b x))-2 b (c-5 d x) \cos (3 (a+b x))\right)\right)}{1152 \sqrt{3} b^5}-\frac{\left(\frac{b}{d}\right)^{3/2} d^2 \left(-\sqrt{2 \pi } S\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(\left(20 b^2 c^2-3 d^2\right) \cos \left(5 a-\frac{5 b c}{d}\right)+12 b c d \sin \left(5 a-\frac{5 b c}{d}\right)\right)-\sqrt{2 \pi } C\left(\sqrt{\frac{b}{d}} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}\right) \left(\left(20 b^2 c^2-3 d^2\right) \sin \left(5 a-\frac{5 b c}{d}\right)-12 b c d \cos \left(5 a-\frac{5 b c}{d}\right)\right)+2 \sqrt{5} \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left(d \left(20 b^2 x^2-3\right) \sin (5 (a+b x))-2 b (c-5 d x) \cos (5 (a+b x))\right)\right)}{3200 \sqrt{5} b^5}","-\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \sin \left(5 a-\frac{5 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1600 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left(3 a-\frac{3 b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{576 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left(a-\frac{b c}{d}\right) C\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{32 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left(a-\frac{b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{32 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left(3 a-\frac{3 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{576 b^{7/2}}-\frac{3 \sqrt{\frac{\pi }{10}} d^{5/2} \cos \left(5 a-\frac{5 b c}{d}\right) S\left(\frac{\sqrt{b} \sqrt{\frac{10}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1600 b^{7/2}}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{32 b^3}+\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{576 b^3}+\frac{3 d^2 \sqrt{c+d x} \sin (5 a+5 b x)}{1600 b^3}+\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{16 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{288 b^2}-\frac{d (c+d x)^{3/2} \cos (5 a+5 b x)}{160 b^2}+\frac{(c+d x)^{5/2} \sin (a+b x)}{8 b}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{48 b}-\frac{(c+d x)^{5/2} \sin (5 a+5 b x)}{80 b}",1,"((-1/16*I)*c^2*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) + (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[a - (b*c)/d] + 2*b*c*Sin[a - (b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*c)/d]) + 2*b*Sqrt[c + d*x]*(3*Cos[a + b*x] + 2*b*x*Sin[a + b*x])))/(16*b^3) + ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*((4*b^2*c^2 - 15*d^2)*Cos[a - (b*c)/d] + 12*b*c*d*Sin[a - (b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[a - (b*c)/d] + (4*b^2*c^2 - 15*d^2)*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[a + b*x] + d*(-15 + 4*b^2*x^2)*Sin[a + b*x])))/(64*b^5) - (c^2*(-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[3*(a + b*x)]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-(d*Cos[3*a - (3*b*c)/d]) + 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*b*Sqrt[c + d*x]*(Cos[3*(a + b*x)] + 2*b*x*Sin[3*(a + b*x)])))/(96*Sqrt[3]*b^3) - ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*((12*b^2*c^2 - 5*d^2)*Cos[3*a - (3*b*c)/d] + 12*b*c*d*Sin[3*a - (3*b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[3*a - (3*b*c)/d] + (12*b^2*c^2 - 5*d^2)*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[3*(a + b*x)] + d*(-5 + 12*b^2*x^2)*Sin[3*(a + b*x)])))/(1152*Sqrt[3]*b^5) - (c^2*(-(Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d] + 2*Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[5*(a + b*x)]))/(160*Sqrt[5]*b*Sqrt[b/d]) - (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[5*a - (5*b*c)/d] + 10*b*c*Sin[5*a - (5*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(10*b*c*Cos[5*a - (5*b*c)/d] + 3*d*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*b*Sqrt[c + d*x]*(3*Cos[5*(a + b*x)] + 10*b*x*Sin[5*(a + b*x)])))/(800*Sqrt[5]*b^3) - ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*((20*b^2*c^2 - 3*d^2)*Cos[5*a - (5*b*c)/d] + 12*b*c*d*Sin[5*a - (5*b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[5*a - (5*b*c)/d] + (20*b^2*c^2 - 3*d^2)*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[5*(a + b*x)] + d*(-3 + 20*b^2*x^2)*Sin[5*(a + b*x)])))/(3200*Sqrt[5]*b^5)","C",0
196,1,550,407,5.0732855,"\int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{-2592 b^3 c^2 \sqrt{c+d x} \cos (2 (a+b x))+288 b^3 c^2 \sqrt{c+d x} \cos (6 (a+b x))-2592 b^3 d^2 x^2 \sqrt{c+d x} \cos (2 (a+b x))+288 b^3 d^2 x^2 \sqrt{c+d x} \cos (6 (a+b x))-5184 b^3 c d x \sqrt{c+d x} \cos (2 (a+b x))+576 b^3 c d x \sqrt{c+d x} \cos (6 (a+b x))+3240 b^2 d^2 x \sqrt{c+d x} \sin (2 (a+b x))-120 b^2 d^2 x \sqrt{c+d x} \sin (6 (a+b x))+3240 b^2 c d \sqrt{c+d x} \sin (2 (a+b x))-120 b^2 c d \sqrt{c+d x} \sin (6 (a+b x))+5 \sqrt{3 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left(6 a-\frac{6 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)-1215 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-5 \sqrt{3 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left(6 a-\frac{6 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)+1215 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+2430 b d^2 \sqrt{c+d x} \cos (2 (a+b x))-30 b d^2 \sqrt{c+d x} \cos (6 (a+b x))}{55296 b^4}","\frac{5 \sqrt{\frac{\pi }{3}} d^{5/2} \cos \left(6 a-\frac{6 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{18432 b^{7/2}}-\frac{45 \sqrt{\pi } d^{5/2} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{2048 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{3}} d^{5/2} \sin \left(6 a-\frac{6 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{18432 b^{7/2}}+\frac{45 \sqrt{\pi } d^{5/2} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{2048 b^{7/2}}+\frac{45 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac{5 d^2 \sqrt{c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac{15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac{5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}-\frac{3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}",1,"(-2592*b^3*c^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 2430*b*d^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 5184*b^3*c*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 2592*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 288*b^3*c^2*Sqrt[c + d*x]*Cos[6*(a + b*x)] - 30*b*d^2*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 576*b^3*c*d*x*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 288*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 5*Sqrt[b/d]*d^3*Sqrt[3*Pi]*Cos[6*a - (6*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]] - 1215*Sqrt[b/d]*d^3*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 5*Sqrt[b/d]*d^3*Sqrt[3*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]]*Sin[6*a - (6*b*c)/d] + 1215*Sqrt[b/d]*d^3*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 3240*b^2*c*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 3240*b^2*d^2*x*Sqrt[c + d*x]*Sin[2*(a + b*x)] - 120*b^2*c*d*Sqrt[c + d*x]*Sin[6*(a + b*x)] - 120*b^2*d^2*x*Sqrt[c + d*x]*Sin[6*(a + b*x)])/(55296*b^4)","A",1
197,1,391,351,2.8313682,"\int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{\sqrt{3 \pi } d \sin \left(6 a-\frac{6 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)-81 \sqrt{\pi } d \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+\sqrt{3 \pi } d \cos \left(6 a-\frac{6 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)-81 \sqrt{\pi } d \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+162 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (2 (a+b x))-6 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (6 (a+b x))-216 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-216 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))+24 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (6 (a+b x))+24 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (6 (a+b x))}{4608 b^2 \sqrt{\frac{b}{d}}}","\frac{\sqrt{\frac{\pi }{3}} d^{3/2} \sin \left(6 a-\frac{6 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1536 b^{5/2}}-\frac{9 \sqrt{\pi } d^{3/2} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{512 b^{5/2}}+\frac{\sqrt{\frac{\pi }{3}} d^{3/2} \cos \left(6 a-\frac{6 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1536 b^{5/2}}-\frac{9 \sqrt{\pi } d^{3/2} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{512 b^{5/2}}+\frac{9 d \sqrt{c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac{d \sqrt{c+d x} \sin (6 a+6 b x)}{768 b^2}-\frac{3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}",1,"(-216*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 216*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 24*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 24*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[6*(a + b*x)] + d*Sqrt[3*Pi]*Cos[6*a - (6*b*c)/d]*FresnelS[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]] - 81*d*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + d*Sqrt[3*Pi]*FresnelC[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]]*Sin[6*a - (6*b*c)/d] - 81*d*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 162*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] - 6*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[6*(a + b*x)])/(4608*b^2*Sqrt[b/d])","A",1
198,1,264,299,1.2504359,"\int \sqrt{c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{-\sqrt{3 \pi } \cos \left(6 a-\frac{6 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)+27 \sqrt{\pi } \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+\sqrt{3 \pi } \sin \left(6 a-\frac{6 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)-27 \sqrt{\pi } \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-54 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))+6 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (6 (a+b x))}{1152 b \sqrt{\frac{b}{d}}}","-\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} \cos \left(6 a-\frac{6 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{384 b^{3/2}}+\frac{3 \sqrt{\pi } \sqrt{d} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{128 b^{3/2}}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} \sin \left(6 a-\frac{6 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{384 b^{3/2}}-\frac{3 \sqrt{\pi } \sqrt{d} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{128 b^{3/2}}-\frac{3 \sqrt{c+d x} \cos (2 a+2 b x)}{64 b}+\frac{\sqrt{c+d x} \cos (6 a+6 b x)}{192 b}",1,"(-54*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 6*Sqrt[b/d]*Sqrt[c + d*x]*Cos[6*(a + b*x)] - Sqrt[3*Pi]*Cos[6*a - (6*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]] + 27*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + Sqrt[3*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]]*Sin[6*a - (6*b*c)/d] - 27*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d])/(1152*b*Sqrt[b/d])","A",1
199,1,264,299,0.5229121,"\int \sqrt{c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[Sqrt[c + d*x]*Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{-\sqrt{3 \pi } \cos \left(6 a-\frac{6 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)+27 \sqrt{\pi } \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+\sqrt{3 \pi } \sin \left(6 a-\frac{6 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)-27 \sqrt{\pi } \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-54 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))+6 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (6 (a+b x))}{1152 b \sqrt{\frac{b}{d}}}","-\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} \cos \left(6 a-\frac{6 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{384 b^{3/2}}+\frac{3 \sqrt{\pi } \sqrt{d} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{128 b^{3/2}}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} \sin \left(6 a-\frac{6 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{384 b^{3/2}}-\frac{3 \sqrt{\pi } \sqrt{d} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{128 b^{3/2}}-\frac{3 \sqrt{c+d x} \cos (2 a+2 b x)}{64 b}+\frac{\sqrt{c+d x} \cos (6 a+6 b x)}{192 b}",1,"(-54*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 6*Sqrt[b/d]*Sqrt[c + d*x]*Cos[6*(a + b*x)] - Sqrt[3*Pi]*Cos[6*a - (6*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]] + 27*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + Sqrt[3*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]]*Sin[6*a - (6*b*c)/d] - 27*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d])/(1152*b*Sqrt[b/d])","A",1
200,1,391,351,0.2023272,"\int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{\sqrt{3 \pi } d \sin \left(6 a-\frac{6 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)-81 \sqrt{\pi } d \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+\sqrt{3 \pi } d \cos \left(6 a-\frac{6 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)-81 \sqrt{\pi } d \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+162 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (2 (a+b x))-6 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (6 (a+b x))-216 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-216 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))+24 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (6 (a+b x))+24 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (6 (a+b x))}{4608 b^2 \sqrt{\frac{b}{d}}}","\frac{\sqrt{\frac{\pi }{3}} d^{3/2} \sin \left(6 a-\frac{6 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1536 b^{5/2}}-\frac{9 \sqrt{\pi } d^{3/2} \sin \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{512 b^{5/2}}+\frac{\sqrt{\frac{\pi }{3}} d^{3/2} \cos \left(6 a-\frac{6 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{1536 b^{5/2}}-\frac{9 \sqrt{\pi } d^{3/2} \cos \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{512 b^{5/2}}+\frac{9 d \sqrt{c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac{d \sqrt{c+d x} \sin (6 a+6 b x)}{768 b^2}-\frac{3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}",1,"(-216*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 216*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 24*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 24*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[6*(a + b*x)] + d*Sqrt[3*Pi]*Cos[6*a - (6*b*c)/d]*FresnelS[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]] - 81*d*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + d*Sqrt[3*Pi]*FresnelC[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]]*Sin[6*a - (6*b*c)/d] - 81*d*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 162*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] - 6*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[6*(a + b*x)])/(4608*b^2*Sqrt[b/d])","A",1
201,1,550,407,2.7382415,"\int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{-2592 b^3 c^2 \sqrt{c+d x} \cos (2 (a+b x))+288 b^3 c^2 \sqrt{c+d x} \cos (6 (a+b x))-2592 b^3 d^2 x^2 \sqrt{c+d x} \cos (2 (a+b x))+288 b^3 d^2 x^2 \sqrt{c+d x} \cos (6 (a+b x))-5184 b^3 c d x \sqrt{c+d x} \cos (2 (a+b x))+576 b^3 c d x \sqrt{c+d x} \cos (6 (a+b x))+3240 b^2 d^2 x \sqrt{c+d x} \sin (2 (a+b x))-120 b^2 d^2 x \sqrt{c+d x} \sin (6 (a+b x))+3240 b^2 c d \sqrt{c+d x} \sin (2 (a+b x))-120 b^2 c d \sqrt{c+d x} \sin (6 (a+b x))+5 \sqrt{3 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left(6 a-\frac{6 b c}{d}\right) C\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)-1215 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)-5 \sqrt{3 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left(6 a-\frac{6 b c}{d}\right) S\left(2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right)+1215 \sqrt{\pi } d^3 \sqrt{\frac{b}{d}} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right)+2430 b d^2 \sqrt{c+d x} \cos (2 (a+b x))-30 b d^2 \sqrt{c+d x} \cos (6 (a+b x))}{55296 b^4}","\frac{5 \sqrt{\frac{\pi }{3}} d^{5/2} \cos \left(6 a-\frac{6 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{18432 b^{7/2}}-\frac{45 \sqrt{\pi } d^{5/2} \cos \left(2 a-\frac{2 b c}{d}\right) C\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{2048 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{3}} d^{5/2} \sin \left(6 a-\frac{6 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right)}{18432 b^{7/2}}+\frac{45 \sqrt{\pi } d^{5/2} \sin \left(2 a-\frac{2 b c}{d}\right) S\left(\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right)}{2048 b^{7/2}}+\frac{45 d^2 \sqrt{c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac{5 d^2 \sqrt{c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac{15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac{5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}-\frac{3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}",1,"(-2592*b^3*c^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 2430*b*d^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 5184*b^3*c*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 2592*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 288*b^3*c^2*Sqrt[c + d*x]*Cos[6*(a + b*x)] - 30*b*d^2*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 576*b^3*c*d*x*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 288*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 5*Sqrt[b/d]*d^3*Sqrt[3*Pi]*Cos[6*a - (6*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]] - 1215*Sqrt[b/d]*d^3*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] - 5*Sqrt[b/d]*d^3*Sqrt[3*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]]*Sin[6*a - (6*b*c)/d] + 1215*Sqrt[b/d]*d^3*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 3240*b^2*c*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 3240*b^2*d^2*x*Sqrt[c + d*x]*Sin[2*(a + b*x)] - 120*b^2*c*d*Sqrt[c + d*x]*Sin[6*(a + b*x)] - 120*b^2*d^2*x*Sqrt[c + d*x]*Sin[6*(a + b*x)])/(55296*b^4)","A",1
202,1,104,112,0.1710885,"\int x^3 \cos ^2(x) \cot ^2(x) \, dx","Integrate[x^3*Cos[x]^2*Cot[x]^2,x]","\frac{1}{16} \left(48 i x \text{Li}_2\left(e^{-2 i x}\right)+24 \text{Li}_3\left(e^{-2 i x}\right)-6 x^4+16 i x^3-4 x^3 \sin (2 x)-16 x^3 \cot (x)+48 x^2 \log \left(1-e^{-2 i x}\right)-6 x^2 \cos (2 x)+6 x \sin (2 x)+3 \cos (2 x)-2 i \pi ^3\right)","-3 i x \text{Li}_2\left(e^{2 i x}\right)+\frac{3}{2} \text{Li}_3\left(e^{2 i x}\right)-\frac{3 x^4}{8}-i x^3-x^3 \cot (x)-\frac{1}{2} x^3 \sin (x) \cos (x)+\frac{3 x^2}{8}+3 x^2 \log \left(1-e^{2 i x}\right)-\frac{3}{4} x^2 \cos ^2(x)+\frac{3 \cos ^2(x)}{8}+\frac{3}{4} x \sin (x) \cos (x)",1,"((-2*I)*Pi^3 + (16*I)*x^3 - 6*x^4 + 3*Cos[2*x] - 6*x^2*Cos[2*x] - 16*x^3*Cot[x] + 48*x^2*Log[1 - E^((-2*I)*x)] + (48*I)*x*PolyLog[2, E^((-2*I)*x)] + 24*PolyLog[3, E^((-2*I)*x)] + 6*x*Sin[2*x] - 4*x^3*Sin[2*x])/16","A",1
203,1,72,83,0.0996245,"\int x^2 \cos ^2(x) \cot ^2(x) \, dx","Integrate[x^2*Cos[x]^2*Cot[x]^2,x]","\frac{1}{8} \left(-8 i \text{Li}_2\left(e^{2 i x}\right)-4 x^3-8 i x^2-2 x^2 \sin (2 x)-8 x^2 \cot (x)+16 x \log \left(1-e^{2 i x}\right)+\sin (2 x)-2 x \cos (2 x)\right)","-i \text{Li}_2\left(e^{2 i x}\right)-\frac{x^3}{2}-i x^2-x^2 \cot (x)-\frac{1}{2} x^2 \sin (x) \cos (x)+\frac{x}{4}+2 x \log \left(1-e^{2 i x}\right)-\frac{1}{2} x \cos ^2(x)+\frac{1}{4} \sin (x) \cos (x)",1,"((-8*I)*x^2 - 4*x^3 - 2*x*Cos[2*x] - 8*x^2*Cot[x] + 16*x*Log[1 - E^((2*I)*x)] - (8*I)*PolyLog[2, E^((2*I)*x)] + Sin[2*x] - 2*x^2*Sin[2*x])/8","A",1
204,1,33,33,0.0236007,"\int x \cos ^2(x) \cot ^2(x) \, dx","Integrate[x*Cos[x]^2*Cot[x]^2,x]","-\frac{3 x^2}{4}-\frac{1}{4} x \sin (2 x)-\frac{1}{8} \cos (2 x)-x \cot (x)+\log (\sin (x))","-\frac{3 x^2}{4}-\frac{\cos ^2(x)}{4}-x \cot (x)+\log (\sin (x))-\frac{1}{2} x \sin (x) \cos (x)",1,"(-3*x^2)/4 - Cos[2*x]/8 - x*Cot[x] + Log[Sin[x]] - (x*Sin[2*x])/4","A",1
205,1,159,180,0.4077576,"\int x^3 \cos ^2(x) \cot ^3(x) \, dx","Integrate[x^3*Cos[x]^2*Cot[x]^3,x]","\frac{1}{32} \left(-96 i x^2 \text{Li}_2\left(e^{-2 i x}\right)-96 x \text{Li}_3\left(e^{-2 i x}\right)-48 i \text{Li}_2\left(e^{2 i x}\right)+48 i \text{Li}_4\left(e^{-2 i x}\right)-16 i x^4-64 x^3 \log \left(1-e^{-2 i x}\right)-8 x^3 \cos (2 x)-16 x^3 \csc ^2(x)-48 i x^2+12 x^2 \sin (2 x)-48 x^2 \cot (x)+96 x \log \left(1-e^{2 i x}\right)-6 \sin (2 x)+12 x \cos (2 x)+i \pi ^4\right)","3 i x^2 \text{Li}_2\left(e^{2 i x}\right)-3 x \text{Li}_3\left(e^{2 i x}\right)-\frac{3}{2} i \text{Li}_2\left(e^{2 i x}\right)-\frac{3}{2} i \text{Li}_4\left(e^{2 i x}\right)+\frac{i x^4}{2}-\frac{3 x^3}{4}-2 x^3 \log \left(1-e^{2 i x}\right)+\frac{1}{2} x^3 \sin ^2(x)-\frac{1}{2} x^3 \cot ^2(x)-\frac{3 i x^2}{2}-\frac{3}{2} x^2 \cot (x)+\frac{3}{4} x^2 \sin (x) \cos (x)+\frac{3 x}{8}+3 x \log \left(1-e^{2 i x}\right)-\frac{3}{4} x \sin ^2(x)-\frac{3}{8} \sin (x) \cos (x)",1,"(I*Pi^4 - (48*I)*x^2 - (16*I)*x^4 + 12*x*Cos[2*x] - 8*x^3*Cos[2*x] - 48*x^2*Cot[x] - 16*x^3*Csc[x]^2 - 64*x^3*Log[1 - E^((-2*I)*x)] + 96*x*Log[1 - E^((2*I)*x)] - (96*I)*x^2*PolyLog[2, E^((-2*I)*x)] - (48*I)*PolyLog[2, E^((2*I)*x)] - 96*x*PolyLog[3, E^((-2*I)*x)] + (48*I)*PolyLog[4, E^((-2*I)*x)] - 6*Sin[2*x] + 12*x^2*Sin[2*x])/32","A",1
206,1,108,106,0.3315343,"\int x^2 \cos ^2(x) \cot ^3(x) \, dx","Integrate[x^2*Cos[x]^2*Cot[x]^3,x]","-2 i x \text{Li}_2\left(e^{-2 i x}\right)-\text{Li}_3\left(e^{-2 i x}\right)-\frac{2 i x^3}{3}-2 x^2 \log \left(1-e^{-2 i x}\right)-\frac{1}{4} x^2 \cos (2 x)-\frac{1}{2} x^2 \csc ^2(x)+\frac{1}{4} x \sin (2 x)+\frac{1}{8} \cos (2 x)-x \cot (x)+\log (\sin (x))+\frac{i \pi ^3}{12}","2 i x \text{Li}_2\left(e^{2 i x}\right)-\text{Li}_3\left(e^{2 i x}\right)+\frac{2 i x^3}{3}-\frac{3 x^2}{4}-2 x^2 \log \left(1-e^{2 i x}\right)+\frac{1}{2} x^2 \sin ^2(x)-\frac{1}{2} x^2 \cot ^2(x)-\frac{\sin ^2(x)}{4}-x \cot (x)+\log (\sin (x))+\frac{1}{2} x \sin (x) \cos (x)",1,"(I/12)*Pi^3 - ((2*I)/3)*x^3 + Cos[2*x]/8 - (x^2*Cos[2*x])/4 - x*Cot[x] - (x^2*Csc[x]^2)/2 - 2*x^2*Log[1 - E^((-2*I)*x)] + Log[Sin[x]] - (2*I)*x*PolyLog[2, E^((-2*I)*x)] - PolyLog[3, E^((-2*I)*x)] + (x*Sin[2*x])/4","A",1
207,1,62,73,0.111481,"\int x \cos ^2(x) \cot ^3(x) \, dx","Integrate[x*Cos[x]^2*Cot[x]^3,x]","\frac{1}{8} \left(8 i \text{Li}_2\left(e^{2 i x}\right)+8 i x^2-16 x \log \left(1-e^{2 i x}\right)+\sin (2 x)-2 x \cos (2 x)-4 \cot (x)-4 x \csc ^2(x)\right)","i \text{Li}_2\left(e^{2 i x}\right)+i x^2-\frac{3 x}{4}-2 x \log \left(1-e^{2 i x}\right)+\frac{1}{2} x \sin ^2(x)-\frac{1}{2} x \cot ^2(x)-\frac{\cot (x)}{2}+\frac{1}{4} \sin (x) \cos (x)",1,"((8*I)*x^2 - 2*x*Cos[2*x] - 4*Cot[x] - 4*x*Csc[x]^2 - 16*x*Log[1 - E^((2*I)*x)] + (8*I)*PolyLog[2, E^((2*I)*x)] + Sin[2*x])/8","A",1
208,0,0,17,2.5541156,"\int (c+d x)^m \tan (a+b x) \, dx","Integrate[(c + d*x)^m*Tan[a + b*x],x]","\int (c+d x)^m \tan (a+b x) \, dx","\text{Int}\left(\tan (a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Tan[a + b*x], x]","A",-1
209,1,157,158,0.0681306,"\int (c+d x)^4 \tan (a+b x) \, dx","Integrate[(c + d*x)^4*Tan[a + b*x],x]","\frac{2 i d (c+d x)^3 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{3 d^2 \left(2 b^2 (c+d x)^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)+d \left(2 i b (c+d x) \text{Li}_4\left(-e^{2 i (a+b x)}\right)-d \text{Li}_5\left(-e^{2 i (a+b x)}\right)\right)\right)}{2 b^5}-\frac{(c+d x)^4 \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{i (c+d x)^5}{5 d}","\frac{3 d^4 \text{Li}_5\left(-e^{2 i (a+b x)}\right)}{2 b^5}-\frac{3 i d^3 (c+d x) \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{b^4}-\frac{3 d^2 (c+d x)^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{b^3}+\frac{2 i d (c+d x)^3 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{(c+d x)^4 \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{i (c+d x)^5}{5 d}",1,"((I/5)*(c + d*x)^5)/d - ((c + d*x)^4*Log[1 + E^((2*I)*(a + b*x))])/b + ((2*I)*d*(c + d*x)^3*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2 - (3*d^2*(2*b^2*(c + d*x)^2*PolyLog[3, -E^((2*I)*(a + b*x))] + d*((2*I)*b*(c + d*x)*PolyLog[4, -E^((2*I)*(a + b*x))] - d*PolyLog[5, -E^((2*I)*(a + b*x))])))/(2*b^5)","A",1
210,1,126,132,0.0846889,"\int (c+d x)^3 \tan (a+b x) \, dx","Integrate[(c + d*x)^3*Tan[a + b*x],x]","\frac{1}{4} i \left(\frac{3 d \left(2 b^2 (c+d x)^2 \text{Li}_2\left(-e^{2 i (a+b x)}\right)+d \left(2 i b (c+d x) \text{Li}_3\left(-e^{2 i (a+b x)}\right)-d \text{Li}_4\left(-e^{2 i (a+b x)}\right)\right)\right)}{b^4}+\frac{4 i (c+d x)^3 \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{(c+d x)^4}{d}\right)","-\frac{3 i d^3 \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{4 b^4}-\frac{3 d^2 (c+d x) \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{(c+d x)^3 \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{i (c+d x)^4}{4 d}",1,"(I/4)*((c + d*x)^4/d + ((4*I)*(c + d*x)^3*Log[1 + E^((2*I)*(a + b*x))])/b + (3*d*(2*b^2*(c + d*x)^2*PolyLog[2, -E^((2*I)*(a + b*x))] + d*((2*I)*b*(c + d*x)*PolyLog[3, -E^((2*I)*(a + b*x))] - d*PolyLog[4, -E^((2*I)*(a + b*x))])))/b^4)","A",1
211,1,100,96,0.0414193,"\int (c+d x)^2 \tan (a+b x) \, dx","Integrate[(c + d*x)^2*Tan[a + b*x],x]","\frac{2 i b^2 (c+d x)^2 \left(b (c+d x)+3 i d \log \left(1+e^{2 i (a+b x)}\right)\right)+6 i b d^2 (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)-3 d^3 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{6 b^3 d}","-\frac{d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}+\frac{i d (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{(c+d x)^2 \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{i (c+d x)^3}{3 d}",1,"((2*I)*b^2*(c + d*x)^2*(b*(c + d*x) + (3*I)*d*Log[1 + E^((2*I)*(a + b*x))]) + (6*I)*b*d^2*(c + d*x)*PolyLog[2, -E^((2*I)*(a + b*x))] - 3*d^3*PolyLog[3, -E^((2*I)*(a + b*x))])/(6*b^3*d)","A",1
212,1,70,66,0.0135624,"\int (c+d x) \tan (a+b x) \, dx","Integrate[(c + d*x)*Tan[a + b*x],x]","\frac{i d \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{c \log (\cos (a+b x))}{b}-\frac{d x \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{1}{2} i d x^2","\frac{i d \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{(c+d x) \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{i (c+d x)^2}{2 d}",1,"(I/2)*d*x^2 - (d*x*Log[1 + E^((2*I)*(a + b*x))])/b - (c*Log[Cos[a + b*x]])/b + ((I/2)*d*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2","A",1
213,0,0,17,3.5322884,"\int \frac{\tan (a+b x)}{c+d x} \, dx","Integrate[Tan[a + b*x]/(c + d*x),x]","\int \frac{\tan (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\tan (a+b x)}{c+d x},x\right)",0,"Integrate[Tan[a + b*x]/(c + d*x), x]","A",-1
214,0,0,17,5.2987085,"\int \frac{\tan (a+b x)}{(c+d x)^2} \, dx","Integrate[Tan[a + b*x]/(c + d*x)^2,x]","\int \frac{\tan (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\tan (a+b x)}{(c+d x)^2},x\right)",0,"Integrate[Tan[a + b*x]/(c + d*x)^2, x]","A",-1
215,0,0,148,6.6167819,"\int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^m*Sin[a + b*x]*Tan[a + b*x],x]","\int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx","\text{Int}\left(\sec (a+b x) (c+d x)^m,x\right)+\frac{i e^{i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)}{2 b}-\frac{i e^{-i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)}{2 b}",0,"Integrate[(c + d*x)^m*Sin[a + b*x]*Tan[a + b*x], x]","A",-1
216,1,557,275,1.4703248,"\int (c+d x)^3 \sin (a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^3*Sin[a + b*x]*Tan[a + b*x],x]","-\frac{b^3 c^3 \sin (a+b x)+2 i b^3 c^3 \tan ^{-1}\left(e^{i (a+b x)}\right)-3 b^3 c^2 d x \log \left(1-i e^{i (a+b x)}\right)+3 b^3 c^2 d x \log \left(1+i e^{i (a+b x)}\right)+3 b^3 c^2 d x \sin (a+b x)-3 b^3 c d^2 x^2 \log \left(1-i e^{i (a+b x)}\right)+3 b^3 c d^2 x^2 \log \left(1+i e^{i (a+b x)}\right)+3 b^3 c d^2 x^2 \sin (a+b x)-b^3 d^3 x^3 \log \left(1-i e^{i (a+b x)}\right)+b^3 d^3 x^3 \log \left(1+i e^{i (a+b x)}\right)+b^3 d^3 x^3 \sin (a+b x)+3 b^2 c^2 d \cos (a+b x)+6 b^2 c d^2 x \cos (a+b x)-3 i b^2 d (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)+3 i b^2 d (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)+3 b^2 d^3 x^2 \cos (a+b x)+6 b c d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)-6 b c d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)-6 b c d^2 \sin (a+b x)+6 b d^3 x \text{Li}_3\left(-i e^{i (a+b x)}\right)-6 b d^3 x \text{Li}_3\left(i e^{i (a+b x)}\right)+6 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)-6 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)-6 b d^3 x \sin (a+b x)-6 d^3 \cos (a+b x)}{b^4}","-\frac{6 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)}{b^4}+\frac{6 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)}{b^4}+\frac{6 d^3 \cos (a+b x)}{b^4}-\frac{6 d^2 (c+d x) \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{6 d^2 (c+d x) \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}+\frac{6 d^2 (c+d x) \sin (a+b x)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{3 d (c+d x)^2 \cos (a+b x)}{b^2}-\frac{(c+d x)^3 \sin (a+b x)}{b}-\frac{2 i (c+d x)^3 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"-(((2*I)*b^3*c^3*ArcTan[E^(I*(a + b*x))] + 3*b^2*c^2*d*Cos[a + b*x] - 6*d^3*Cos[a + b*x] + 6*b^2*c*d^2*x*Cos[a + b*x] + 3*b^2*d^3*x^2*Cos[a + b*x] - 3*b^3*c^2*d*x*Log[1 - I*E^(I*(a + b*x))] - 3*b^3*c*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 - I*E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 + I*E^(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] + b^3*d^3*x^3*Log[1 + I*E^(I*(a + b*x))] - (3*I)*b^2*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))] + (3*I)*b^2*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))] + 6*b*c*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, (-I)*E^(I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, I*E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, I*E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))] - (6*I)*d^3*PolyLog[4, I*E^(I*(a + b*x))] + b^3*c^3*Sin[a + b*x] - 6*b*c*d^2*Sin[a + b*x] + 3*b^3*c^2*d*x*Sin[a + b*x] - 6*b*d^3*x*Sin[a + b*x] + 3*b^3*c*d^2*x^2*Sin[a + b*x] + b^3*d^3*x^3*Sin[a + b*x])/b^4)","B",1
217,1,315,186,0.8318777,"\int (c+d x)^2 \sin (a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^2*Sin[a + b*x]*Tan[a + b*x],x]","-\frac{b^2 c^2 \sin (a+b x)+2 i b^2 c^2 \tan ^{-1}\left(e^{i (a+b x)}\right)-2 b^2 c d x \log \left(1-i e^{i (a+b x)}\right)+2 b^2 c d x \log \left(1+i e^{i (a+b x)}\right)+2 b^2 c d x \sin (a+b x)-b^2 d^2 x^2 \log \left(1-i e^{i (a+b x)}\right)+b^2 d^2 x^2 \log \left(1+i e^{i (a+b x)}\right)+b^2 d^2 x^2 \sin (a+b x)-2 i b d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)+2 b c d \cos (a+b x)+2 d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)-2 d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)-2 d^2 \sin (a+b x)+2 b d^2 x \cos (a+b x)}{b^3}","-\frac{2 d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{2 d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}+\frac{2 d^2 \sin (a+b x)}{b^3}+\frac{2 i d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{2 i d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{2 d (c+d x) \cos (a+b x)}{b^2}-\frac{(c+d x)^2 \sin (a+b x)}{b}-\frac{2 i (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"-(((2*I)*b^2*c^2*ArcTan[E^(I*(a + b*x))] + 2*b*c*d*Cos[a + b*x] + 2*b*d^2*x*Cos[a + b*x] - 2*b^2*c*d*x*Log[1 - I*E^(I*(a + b*x))] - b^2*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 + I*E^(I*(a + b*x))] + b^2*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] + 2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] - 2*d^2*PolyLog[3, I*E^(I*(a + b*x))] + b^2*c^2*Sin[a + b*x] - 2*d^2*Sin[a + b*x] + 2*b^2*c*d*x*Sin[a + b*x] + b^2*d^2*x^2*Sin[a + b*x])/b^3)","A",1
218,1,213,103,0.4018327,"\int (c+d x) \sin (a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)*Sin[a + b*x]*Tan[a + b*x],x]","\frac{d \left(i \left(\text{Li}_2\left(-e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)-\text{Li}_2\left(e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)\right)+\left(-a-b x+\frac{\pi }{2}\right) \left(\log \left(1-e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)-\log \left(1+e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)\right)-\left(\frac{\pi }{2}-a\right) \log \left(\tan \left(\frac{1}{2} \left(-a-b x+\frac{\pi }{2}\right)\right)\right)\right)}{b^2}-\frac{d \cos (b x) (b x \sin (a)+\cos (a))}{b^2}-\frac{d \sin (b x) (b x \cos (a)-\sin (a))}{b^2}-\frac{c \sin (a+b x)}{b}+\frac{c \tanh ^{-1}(\sin (a+b x))}{b}","\frac{i d \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{i d \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{d \cos (a+b x)}{b^2}-\frac{(c+d x) \sin (a+b x)}{b}-\frac{2 i (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(c*ArcTanh[Sin[a + b*x]])/b + (d*((-a + Pi/2 - b*x)*(Log[1 - E^(I*(-a + Pi/2 - b*x))] - Log[1 + E^(I*(-a + Pi/2 - b*x))]) - (-a + Pi/2)*Log[Tan[(-a + Pi/2 - b*x)/2]] + I*(PolyLog[2, -E^(I*(-a + Pi/2 - b*x))] - PolyLog[2, E^(I*(-a + Pi/2 - b*x))])))/b^2 - (d*Cos[b*x]*(Cos[a] + b*x*Sin[a]))/b^2 - (d*(b*x*Cos[a] - Sin[a])*Sin[b*x])/b^2 - (c*Sin[a + b*x])/b","B",1
219,0,0,69,6.0382584,"\int \frac{\sin (a+b x) \tan (a+b x)}{c+d x} \, dx","Integrate[(Sin[a + b*x]*Tan[a + b*x])/(c + d*x),x]","\int \frac{\sin (a+b x) \tan (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\sec (a+b x)}{c+d x},x\right)-\frac{\cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d}+\frac{\sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d}",0,"Integrate[(Sin[a + b*x]*Tan[a + b*x])/(c + d*x), x]","A",-1
220,0,0,87,7.0799636,"\int \frac{\sin (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx","Integrate[(Sin[a + b*x]*Tan[a + b*x])/(c + d*x)^2,x]","\int \frac{\sin (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\sec (a+b x)}{(c+d x)^2},x\right)+\frac{b \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d^2}+\frac{b \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d^2}+\frac{\cos (a+b x)}{d (c+d x)}",0,"Integrate[(Sin[a + b*x]*Tan[a + b*x])/(c + d*x)^2, x]","A",-1
221,0,0,152,7.8095595,"\int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^m*Sin[a + b*x]^2*Tan[a + b*x],x]","\int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx","\text{Int}\left(\tan (a+b x) (c+d x)^m,x\right)+\frac{2^{-m-3} e^{2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{2 i b (c+d x)}{d}\right)}{b}+\frac{2^{-m-3} e^{-2 i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{2 i b (c+d x)}{d}\right)}{b}",0,"Integrate[(c + d*x)^m*Sin[a + b*x]^2*Tan[a + b*x], x]","A",-1
222,1,1720,251,7.1196782,"\int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^3*Sin[a + b*x]^2*Tan[a + b*x],x]","-\frac{\sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a)) c^3}{b \left(\cos ^2(a)+\sin ^2(a)\right)}-\frac{3 d \csc (a) \left(b^2 e^{-i \tan ^{-1}(\cot (a))} x^2-\frac{\cot (a) \left(i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\pi  \log (\cos (b x))-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)}{\sqrt{\cot ^2(a)+1}}\right) \sec (a) c^2}{2 b^2 \sqrt{\csc ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}-\frac{i d^2 e^{-i a} \left(2 b^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right) x^2+6 b \left(1+e^{2 i a}\right) \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right) \sec (a) c}{4 b^3}-\frac{1}{8} i d^3 e^{i a} \left(2 e^{-2 i a} x^4-\frac{4 i \left(1+e^{-2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right) x^3}{b}+\frac{3 e^{-2 i a} \left(1+e^{2 i a}\right) \left(2 b^2 \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(-e^{-2 i (a+b x)}\right) x-\text{Li}_4\left(-e^{-2 i (a+b x)}\right)\right)}{b^4}\right) \sec (a)+\sec (a) \left(\frac{\cos (2 a+2 b x)}{64 b^4}-\frac{i \sin (2 a+2 b x)}{64 b^4}\right) \left(8 i d^3 x^4 \cos (a+2 b x) b^4+32 i c d^2 x^3 \cos (a+2 b x) b^4+48 i c^2 d x^2 \cos (a+2 b x) b^4+32 i c^3 x \cos (a+2 b x) b^4-8 i d^3 x^4 \cos (3 a+2 b x) b^4-32 i c d^2 x^3 \cos (3 a+2 b x) b^4-48 i c^2 d x^2 \cos (3 a+2 b x) b^4-32 i c^3 x \cos (3 a+2 b x) b^4-8 d^3 x^4 \sin (a+2 b x) b^4-32 c d^2 x^3 \sin (a+2 b x) b^4-48 c^2 d x^2 \sin (a+2 b x) b^4-32 c^3 x \sin (a+2 b x) b^4+8 d^3 x^4 \sin (3 a+2 b x) b^4+32 c d^2 x^3 \sin (3 a+2 b x) b^4+48 c^2 d x^2 \sin (3 a+2 b x) b^4+32 c^3 x \sin (3 a+2 b x) b^4+8 c^3 \cos (a) b^3+8 d^3 x^3 \cos (a) b^3+24 c d^2 x^2 \cos (a) b^3+24 c^2 d x \cos (a) b^3+4 c^3 \cos (3 a+4 b x) b^3+4 d^3 x^3 \cos (3 a+4 b x) b^3+12 c d^2 x^2 \cos (3 a+4 b x) b^3+12 c^2 d x \cos (3 a+4 b x) b^3+4 c^3 \cos (5 a+4 b x) b^3+4 d^3 x^3 \cos (5 a+4 b x) b^3+12 c d^2 x^2 \cos (5 a+4 b x) b^3+12 c^2 d x \cos (5 a+4 b x) b^3+4 i c^3 \sin (3 a+4 b x) b^3+4 i d^3 x^3 \sin (3 a+4 b x) b^3+12 i c d^2 x^2 \sin (3 a+4 b x) b^3+12 i c^2 d x \sin (3 a+4 b x) b^3+4 i c^3 \sin (5 a+4 b x) b^3+4 i d^3 x^3 \sin (5 a+4 b x) b^3+12 i c d^2 x^2 \sin (5 a+4 b x) b^3+12 i c^2 d x \sin (5 a+4 b x) b^3-12 i d^3 x^2 \cos (a) b^2-12 i c^2 d \cos (a) b^2-24 i c d^2 x \cos (a) b^2+6 i d^3 x^2 \cos (3 a+4 b x) b^2+6 i c^2 d \cos (3 a+4 b x) b^2+12 i c d^2 x \cos (3 a+4 b x) b^2+6 i d^3 x^2 \cos (5 a+4 b x) b^2+6 i c^2 d \cos (5 a+4 b x) b^2+12 i c d^2 x \cos (5 a+4 b x) b^2-6 d^3 x^2 \sin (3 a+4 b x) b^2-6 c^2 d \sin (3 a+4 b x) b^2-12 c d^2 x \sin (3 a+4 b x) b^2-6 d^3 x^2 \sin (5 a+4 b x) b^2-6 c^2 d \sin (5 a+4 b x) b^2-12 c d^2 x \sin (5 a+4 b x) b^2-12 c d^2 \cos (a) b-12 d^3 x \cos (a) b-6 c d^2 \cos (3 a+4 b x) b-6 d^3 x \cos (3 a+4 b x) b-6 c d^2 \cos (5 a+4 b x) b-6 d^3 x \cos (5 a+4 b x) b-6 i c d^2 \sin (3 a+4 b x) b-6 i d^3 x \sin (3 a+4 b x) b-6 i c d^2 \sin (5 a+4 b x) b-6 i d^3 x \sin (5 a+4 b x) b+6 i d^3 \cos (a)-3 i d^3 \cos (3 a+4 b x)-3 i d^3 \cos (5 a+4 b x)+3 d^3 \sin (3 a+4 b x)+3 d^3 \sin (5 a+4 b x)\right)","-\frac{3 i d^3 \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{4 b^4}+\frac{3 d^3 \sin (a+b x) \cos (a+b x)}{8 b^4}-\frac{3 d^2 (c+d x) \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}+\frac{3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{3 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{4 b^2}-\frac{(c+d x)^3 \log \left(1+e^{2 i (a+b x)}\right)}{b}-\frac{(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac{3 d^3 x}{8 b^3}+\frac{(c+d x)^3}{4 b}+\frac{i (c+d x)^4}{4 d}",1,"((-1/4*I)*c*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) - (I/8)*d^3*E^(I*a)*((2*x^4)/E^((2*I)*a) - ((4*I)*(1 + E^((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x))])/b + (3*(1 + E^((2*I)*a))*(2*b^2*x^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-2*I)*(a + b*x))] - PolyLog[4, -E^((-2*I)*(a + b*x))]))/(b^4*E^((2*I)*a)))*Sec[a] - (c^3*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (3*c^2*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(2*b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + Sec[a]*(Cos[2*a + 2*b*x]/(64*b^4) - ((I/64)*Sin[2*a + 2*b*x])/b^4)*(8*b^3*c^3*Cos[a] - (12*I)*b^2*c^2*d*Cos[a] - 12*b*c*d^2*Cos[a] + (6*I)*d^3*Cos[a] + 24*b^3*c^2*d*x*Cos[a] - (24*I)*b^2*c*d^2*x*Cos[a] - 12*b*d^3*x*Cos[a] + 24*b^3*c*d^2*x^2*Cos[a] - (12*I)*b^2*d^3*x^2*Cos[a] + 8*b^3*d^3*x^3*Cos[a] + (32*I)*b^4*c^3*x*Cos[a + 2*b*x] + (48*I)*b^4*c^2*d*x^2*Cos[a + 2*b*x] + (32*I)*b^4*c*d^2*x^3*Cos[a + 2*b*x] + (8*I)*b^4*d^3*x^4*Cos[a + 2*b*x] - (32*I)*b^4*c^3*x*Cos[3*a + 2*b*x] - (48*I)*b^4*c^2*d*x^2*Cos[3*a + 2*b*x] - (32*I)*b^4*c*d^2*x^3*Cos[3*a + 2*b*x] - (8*I)*b^4*d^3*x^4*Cos[3*a + 2*b*x] + 4*b^3*c^3*Cos[3*a + 4*b*x] + (6*I)*b^2*c^2*d*Cos[3*a + 4*b*x] - 6*b*c*d^2*Cos[3*a + 4*b*x] - (3*I)*d^3*Cos[3*a + 4*b*x] + 12*b^3*c^2*d*x*Cos[3*a + 4*b*x] + (12*I)*b^2*c*d^2*x*Cos[3*a + 4*b*x] - 6*b*d^3*x*Cos[3*a + 4*b*x] + 12*b^3*c*d^2*x^2*Cos[3*a + 4*b*x] + (6*I)*b^2*d^3*x^2*Cos[3*a + 4*b*x] + 4*b^3*d^3*x^3*Cos[3*a + 4*b*x] + 4*b^3*c^3*Cos[5*a + 4*b*x] + (6*I)*b^2*c^2*d*Cos[5*a + 4*b*x] - 6*b*c*d^2*Cos[5*a + 4*b*x] - (3*I)*d^3*Cos[5*a + 4*b*x] + 12*b^3*c^2*d*x*Cos[5*a + 4*b*x] + (12*I)*b^2*c*d^2*x*Cos[5*a + 4*b*x] - 6*b*d^3*x*Cos[5*a + 4*b*x] + 12*b^3*c*d^2*x^2*Cos[5*a + 4*b*x] + (6*I)*b^2*d^3*x^2*Cos[5*a + 4*b*x] + 4*b^3*d^3*x^3*Cos[5*a + 4*b*x] - 32*b^4*c^3*x*Sin[a + 2*b*x] - 48*b^4*c^2*d*x^2*Sin[a + 2*b*x] - 32*b^4*c*d^2*x^3*Sin[a + 2*b*x] - 8*b^4*d^3*x^4*Sin[a + 2*b*x] + 32*b^4*c^3*x*Sin[3*a + 2*b*x] + 48*b^4*c^2*d*x^2*Sin[3*a + 2*b*x] + 32*b^4*c*d^2*x^3*Sin[3*a + 2*b*x] + 8*b^4*d^3*x^4*Sin[3*a + 2*b*x] + (4*I)*b^3*c^3*Sin[3*a + 4*b*x] - 6*b^2*c^2*d*Sin[3*a + 4*b*x] - (6*I)*b*c*d^2*Sin[3*a + 4*b*x] + 3*d^3*Sin[3*a + 4*b*x] + (12*I)*b^3*c^2*d*x*Sin[3*a + 4*b*x] - 12*b^2*c*d^2*x*Sin[3*a + 4*b*x] - (6*I)*b*d^3*x*Sin[3*a + 4*b*x] + (12*I)*b^3*c*d^2*x^2*Sin[3*a + 4*b*x] - 6*b^2*d^3*x^2*Sin[3*a + 4*b*x] + (4*I)*b^3*d^3*x^3*Sin[3*a + 4*b*x] + (4*I)*b^3*c^3*Sin[5*a + 4*b*x] - 6*b^2*c^2*d*Sin[5*a + 4*b*x] - (6*I)*b*c*d^2*Sin[5*a + 4*b*x] + 3*d^3*Sin[5*a + 4*b*x] + (12*I)*b^3*c^2*d*x*Sin[5*a + 4*b*x] - 12*b^2*c*d^2*x*Sin[5*a + 4*b*x] - (6*I)*b*d^3*x*Sin[5*a + 4*b*x] + (12*I)*b^3*c*d^2*x^2*Sin[5*a + 4*b*x] - 6*b^2*d^3*x^2*Sin[5*a + 4*b*x] + (4*I)*b^3*d^3*x^3*Sin[5*a + 4*b*x])","B",0
223,1,518,184,6.4689522,"\int (c+d x)^2 \sin ^2(a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^2*Sin[a + b*x]^2*Tan[a + b*x],x]","-\frac{c d \csc (a) \sec (a) \left(b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left(i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\cot ^2(a)+1}}\right)}{b^2 \sqrt{\csc ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}+\frac{\cos (2 b x) \left(2 b^2 c^2 \cos (2 a)+4 b^2 c d x \cos (2 a)+2 b^2 d^2 x^2 \cos (2 a)-2 b c d \sin (2 a)-2 b d^2 x \sin (2 a)-d^2 \cos (2 a)\right)}{8 b^3}-\frac{\sin (2 b x) \left(2 b^2 c^2 \sin (2 a)+4 b^2 c d x \sin (2 a)+2 b^2 d^2 x^2 \sin (2 a)+2 b c d \cos (2 a)+2 b d^2 x \cos (2 a)-d^2 \sin (2 a)\right)}{8 b^3}-\frac{i e^{-i a} d^2 \sec (a) \left(2 b^2 x^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right)+6 \left(1+e^{2 i a}\right) b x \text{Li}_2\left(-e^{-2 i (a+b x)}\right)-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right)}{12 b^3}-\frac{c^2 \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b \left(\sin ^2(a)+\cos ^2(a)\right)}+\frac{1}{3} x \tan (a) \left(3 c^2+3 c d x+d^2 x^2\right)","-\frac{d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}+\frac{d^2 \sin ^2(a+b x)}{4 b^3}+\frac{i d (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{d (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^2}-\frac{(c+d x)^2 \log \left(1+e^{2 i (a+b x)}\right)}{b}-\frac{(c+d x)^2 \sin ^2(a+b x)}{2 b}+\frac{c d x}{2 b}+\frac{d^2 x^2}{4 b}+\frac{i (c+d x)^3}{3 d}",1,"((-1/12*I)*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) - (c^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (c*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + (Cos[2*b*x]*(2*b^2*c^2*Cos[2*a] - d^2*Cos[2*a] + 4*b^2*c*d*x*Cos[2*a] + 2*b^2*d^2*x^2*Cos[2*a] - 2*b*c*d*Sin[2*a] - 2*b*d^2*x*Sin[2*a]))/(8*b^3) - ((2*b*c*d*Cos[2*a] + 2*b*d^2*x*Cos[2*a] + 2*b^2*c^2*Sin[2*a] - d^2*Sin[2*a] + 4*b^2*c*d*x*Sin[2*a] + 2*b^2*d^2*x^2*Sin[2*a])*Sin[2*b*x])/(8*b^3) + (x*(3*c^2 + 3*c*d*x + d^2*x^2)*Tan[a])/3","B",0
224,1,134,115,0.2999783,"\int (c+d x) \sin ^2(a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)*Sin[a + b*x]^2*Tan[a + b*x],x]","\frac{d \left(\frac{1}{2} i \text{Li}_2\left(-e^{2 i (a+b x)}\right)+\frac{1}{2} i (a+b x)^2-(a+b x) \log \left(1+e^{2 i (a+b x)}\right)\right)}{b^2}-\frac{d \sin (2 (a+b x))}{8 b^2}+\frac{a d \log (\cos (a+b x))}{b^2}-\frac{c \left(\log (\cos (a+b x))-\frac{1}{2} \cos ^2(a+b x)\right)}{b}+\frac{d x \cos (2 (a+b x))}{4 b}","\frac{i d \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{d \sin (a+b x) \cos (a+b x)}{4 b^2}-\frac{(c+d x) \log \left(1+e^{2 i (a+b x)}\right)}{b}-\frac{(c+d x) \sin ^2(a+b x)}{2 b}+\frac{d x}{4 b}+\frac{i (c+d x)^2}{2 d}",1,"(d*x*Cos[2*(a + b*x)])/(4*b) + (a*d*Log[Cos[a + b*x]])/b^2 - (c*(-1/2*Cos[a + b*x]^2 + Log[Cos[a + b*x]]))/b + (d*((I/2)*(a + b*x)^2 - (a + b*x)*Log[1 + E^((2*I)*(a + b*x))] + (I/2)*PolyLog[2, -E^((2*I)*(a + b*x))]))/b^2 - (d*Sin[2*(a + b*x)])/(8*b^2)","A",1
225,0,0,82,0.7634772,"\int \frac{\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx","Integrate[(Sin[a + b*x]^2*Tan[a + b*x])/(c + d*x),x]","\int \frac{\sin ^2(a+b x) \tan (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\tan (a+b x)}{c+d x},x\right)-\frac{\sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{2 d}-\frac{\cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{2 d}",0,"Integrate[(Sin[a + b*x]^2*Tan[a + b*x])/(c + d*x), x]","A",-1
226,0,0,102,2.4010565,"\int \frac{\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx","Integrate[(Sin[a + b*x]^2*Tan[a + b*x])/(c + d*x)^2,x]","\int \frac{\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\tan (a+b x)}{(c+d x)^2},x\right)-\frac{b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}+\frac{b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}+\frac{\sin (2 a+2 b x)}{2 d (c+d x)}",0,"Integrate[(Sin[a + b*x]^2*Tan[a + b*x])/(c + d*x)^2, x]","A",-1
227,0,0,23,5.8919254,"\int (c+d x)^m \csc (a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)^m*Csc[a + b*x]*Sec[a + b*x],x]","\int (c+d x)^m \csc (a+b x) \sec (a+b x) \, dx","\text{Int}\left(\csc (a+b x) \sec (a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Csc[a + b*x]*Sec[a + b*x], x]","A",-1
228,1,578,247,1.2851556,"\int (c+d x)^4 \csc (a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x],x]","\frac{-4 b^4 c^4 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)+8 b^4 c^3 d x \log \left(1-e^{2 i (a+b x)}\right)-8 b^4 c^3 d x \log \left(1+e^{2 i (a+b x)}\right)+12 b^4 c^2 d^2 x^2 \log \left(1-e^{2 i (a+b x)}\right)-12 b^4 c^2 d^2 x^2 \log \left(1+e^{2 i (a+b x)}\right)+8 b^4 c d^3 x^3 \log \left(1-e^{2 i (a+b x)}\right)-8 b^4 c d^3 x^3 \log \left(1+e^{2 i (a+b x)}\right)+2 b^4 d^4 x^4 \log \left(1-e^{2 i (a+b x)}\right)-2 b^4 d^4 x^4 \log \left(1+e^{2 i (a+b x)}\right)+4 i b^3 d (c+d x)^3 \text{Li}_2\left(-e^{2 i (a+b x)}\right)-4 i b^3 d (c+d x)^3 \text{Li}_2\left(e^{2 i (a+b x)}\right)-6 b^2 c^2 d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)+6 b^2 c^2 d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)-12 b^2 c d^3 x \text{Li}_3\left(-e^{2 i (a+b x)}\right)+12 b^2 c d^3 x \text{Li}_3\left(e^{2 i (a+b x)}\right)-6 b^2 d^4 x^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)+6 b^2 d^4 x^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)-6 i b c d^3 \text{Li}_4\left(-e^{2 i (a+b x)}\right)+6 i b c d^3 \text{Li}_4\left(e^{2 i (a+b x)}\right)-6 i b d^4 x \text{Li}_4\left(-e^{2 i (a+b x)}\right)+6 i b d^4 x \text{Li}_4\left(e^{2 i (a+b x)}\right)+3 d^4 \text{Li}_5\left(-e^{2 i (a+b x)}\right)-3 d^4 \text{Li}_5\left(e^{2 i (a+b x)}\right)}{2 b^5}","\frac{3 d^4 \text{Li}_5\left(-e^{2 i (a+b x)}\right)}{2 b^5}-\frac{3 d^4 \text{Li}_5\left(e^{2 i (a+b x)}\right)}{2 b^5}-\frac{3 i d^3 (c+d x) \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{b^4}+\frac{3 i d^3 (c+d x) \text{Li}_4\left(e^{2 i (a+b x)}\right)}{b^4}-\frac{3 d^2 (c+d x)^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{b^3}+\frac{3 d^2 (c+d x)^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{b^3}+\frac{2 i d (c+d x)^3 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{2 i d (c+d x)^3 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{2 (c+d x)^4 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}",1,"(-4*b^4*c^4*ArcTanh[E^((2*I)*(a + b*x))] + 8*b^4*c^3*d*x*Log[1 - E^((2*I)*(a + b*x))] + 12*b^4*c^2*d^2*x^2*Log[1 - E^((2*I)*(a + b*x))] + 8*b^4*c*d^3*x^3*Log[1 - E^((2*I)*(a + b*x))] + 2*b^4*d^4*x^4*Log[1 - E^((2*I)*(a + b*x))] - 8*b^4*c^3*d*x*Log[1 + E^((2*I)*(a + b*x))] - 12*b^4*c^2*d^2*x^2*Log[1 + E^((2*I)*(a + b*x))] - 8*b^4*c*d^3*x^3*Log[1 + E^((2*I)*(a + b*x))] - 2*b^4*d^4*x^4*Log[1 + E^((2*I)*(a + b*x))] + (4*I)*b^3*d*(c + d*x)^3*PolyLog[2, -E^((2*I)*(a + b*x))] - (4*I)*b^3*d*(c + d*x)^3*PolyLog[2, E^((2*I)*(a + b*x))] - 6*b^2*c^2*d^2*PolyLog[3, -E^((2*I)*(a + b*x))] - 12*b^2*c*d^3*x*PolyLog[3, -E^((2*I)*(a + b*x))] - 6*b^2*d^4*x^2*PolyLog[3, -E^((2*I)*(a + b*x))] + 6*b^2*c^2*d^2*PolyLog[3, E^((2*I)*(a + b*x))] + 12*b^2*c*d^3*x*PolyLog[3, E^((2*I)*(a + b*x))] + 6*b^2*d^4*x^2*PolyLog[3, E^((2*I)*(a + b*x))] - (6*I)*b*c*d^3*PolyLog[4, -E^((2*I)*(a + b*x))] - (6*I)*b*d^4*x*PolyLog[4, -E^((2*I)*(a + b*x))] + (6*I)*b*c*d^3*PolyLog[4, E^((2*I)*(a + b*x))] + (6*I)*b*d^4*x*PolyLog[4, E^((2*I)*(a + b*x))] + 3*d^4*PolyLog[5, -E^((2*I)*(a + b*x))] - 3*d^4*PolyLog[5, E^((2*I)*(a + b*x))])/(2*b^5)","B",1
229,1,350,197,0.9457198,"\int (c+d x)^3 \csc (a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]*Sec[a + b*x],x]","\frac{-8 b^3 c^3 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)+12 b^3 c^2 d x \log \left(1-e^{2 i (a+b x)}\right)-12 b^3 c^2 d x \log \left(1+e^{2 i (a+b x)}\right)+12 b^3 c d^2 x^2 \log \left(1-e^{2 i (a+b x)}\right)-12 b^3 c d^2 x^2 \log \left(1+e^{2 i (a+b x)}\right)+4 b^3 d^3 x^3 \log \left(1-e^{2 i (a+b x)}\right)-4 b^3 d^3 x^3 \log \left(1+e^{2 i (a+b x)}\right)+6 i b^2 d (c+d x)^2 \text{Li}_2\left(-e^{2 i (a+b x)}\right)-6 i b^2 d (c+d x)^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)-6 b d^2 (c+d x) \text{Li}_3\left(-e^{2 i (a+b x)}\right)+6 b c d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)+6 b d^3 x \text{Li}_3\left(e^{2 i (a+b x)}\right)-3 i d^3 \text{Li}_4\left(-e^{2 i (a+b x)}\right)+3 i d^3 \text{Li}_4\left(e^{2 i (a+b x)}\right)}{4 b^4}","-\frac{3 i d^3 \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{4 b^4}+\frac{3 i d^3 \text{Li}_4\left(e^{2 i (a+b x)}\right)}{4 b^4}-\frac{3 d^2 (c+d x) \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}+\frac{3 d^2 (c+d x) \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{2 (c+d x)^3 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}",1,"(-8*b^3*c^3*ArcTanh[E^((2*I)*(a + b*x))] + 12*b^3*c^2*d*x*Log[1 - E^((2*I)*(a + b*x))] + 12*b^3*c*d^2*x^2*Log[1 - E^((2*I)*(a + b*x))] + 4*b^3*d^3*x^3*Log[1 - E^((2*I)*(a + b*x))] - 12*b^3*c^2*d*x*Log[1 + E^((2*I)*(a + b*x))] - 12*b^3*c*d^2*x^2*Log[1 + E^((2*I)*(a + b*x))] - 4*b^3*d^3*x^3*Log[1 + E^((2*I)*(a + b*x))] + (6*I)*b^2*d*(c + d*x)^2*PolyLog[2, -E^((2*I)*(a + b*x))] - (6*I)*b^2*d*(c + d*x)^2*PolyLog[2, E^((2*I)*(a + b*x))] - 6*b*d^2*(c + d*x)*PolyLog[3, -E^((2*I)*(a + b*x))] + 6*b*c*d^2*PolyLog[3, E^((2*I)*(a + b*x))] + 6*b*d^3*x*PolyLog[3, E^((2*I)*(a + b*x))] - (3*I)*d^3*PolyLog[4, -E^((2*I)*(a + b*x))] + (3*I)*d^3*PolyLog[4, E^((2*I)*(a + b*x))])/(4*b^4)","A",1
230,1,213,127,0.5892512,"\int (c+d x)^2 \csc (a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]*Sec[a + b*x],x]","\frac{-4 b^2 c^2 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)+4 b^2 c d x \log \left(1-e^{2 i (a+b x)}\right)-4 b^2 c d x \log \left(1+e^{2 i (a+b x)}\right)+2 b^2 d^2 x^2 \log \left(1-e^{2 i (a+b x)}\right)-2 b^2 d^2 x^2 \log \left(1+e^{2 i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)-2 i b d (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)-d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)+d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}","-\frac{d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}+\frac{d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}+\frac{i d (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{i d (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{2 (c+d x)^2 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}",1,"(-4*b^2*c^2*ArcTanh[E^((2*I)*(a + b*x))] + 4*b^2*c*d*x*Log[1 - E^((2*I)*(a + b*x))] + 2*b^2*d^2*x^2*Log[1 - E^((2*I)*(a + b*x))] - 4*b^2*c*d*x*Log[1 + E^((2*I)*(a + b*x))] - 2*b^2*d^2*x^2*Log[1 + E^((2*I)*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, -E^((2*I)*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, E^((2*I)*(a + b*x))] - d^2*PolyLog[3, -E^((2*I)*(a + b*x))] + d^2*PolyLog[3, E^((2*I)*(a + b*x))])/(2*b^3)","A",1
231,1,141,71,0.1149133,"\int (c+d x) \csc (a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]*Sec[a + b*x],x]","\frac{d \left(i \left(\text{Li}_2\left(-e^{i (2 a+2 b x)}\right)-\text{Li}_2\left(e^{i (2 a+2 b x)}\right)\right)+(2 a+2 b x) \left(\log \left(1-e^{i (2 a+2 b x)}\right)-\log \left(1+e^{i (2 a+2 b x)}\right)\right)-2 a \log \left(\tan \left(\frac{1}{2} (2 a+2 b x)\right)\right)\right)}{2 b^2}+\frac{c \log (\sin (a+b x))}{b}-\frac{c \log (\cos (a+b x))}{b}","\frac{i d \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{i d \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{2 (c+d x) \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}",1,"-((c*Log[Cos[a + b*x]])/b) + (c*Log[Sin[a + b*x]])/b + (d*((2*a + 2*b*x)*(Log[1 - E^(I*(2*a + 2*b*x))] - Log[1 + E^(I*(2*a + 2*b*x))]) - 2*a*Log[Tan[(2*a + 2*b*x)/2]] + I*(PolyLog[2, -E^(I*(2*a + 2*b*x))] - PolyLog[2, E^(I*(2*a + 2*b*x))])))/(2*b^2)","A",1
232,0,0,22,4.5333895,"\int \frac{\csc (a+b x) \sec (a+b x)}{c+d x} \, dx","Integrate[(Csc[a + b*x]*Sec[a + b*x])/(c + d*x),x]","\int \frac{\csc (a+b x) \sec (a+b x)}{c+d x} \, dx","2 \text{Int}\left(\frac{\csc (2 a+2 b x)}{c+d x},x\right)",0,"Integrate[(Csc[a + b*x]*Sec[a + b*x])/(c + d*x), x]","A",-1
233,0,0,22,5.7153687,"\int \frac{\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx","Integrate[(Csc[a + b*x]*Sec[a + b*x])/(c + d*x)^2,x]","\int \frac{\csc (a+b x) \sec (a+b x)}{(c+d x)^2} \, dx","2 \text{Int}\left(\frac{\csc (2 a+2 b x)}{(c+d x)^2},x\right)",0,"Integrate[(Csc[a + b*x]*Sec[a + b*x])/(c + d*x)^2, x]","A",-1
234,0,0,25,17.7858693,"\int (c+d x)^m \csc ^2(a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)^m*Csc[a + b*x]^2*Sec[a + b*x],x]","\int (c+d x)^m \csc ^2(a+b x) \sec (a+b x) \, dx","\text{Int}\left(\csc ^2(a+b x) \sec (a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Csc[a + b*x]^2*Sec[a + b*x], x]","A",-1
235,1,739,350,6.4102749,"\int (c+d x)^3 \csc ^2(a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x],x]","\frac{3 d \left(\frac{2 i d \left(b (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)+i d \text{Li}_3\left(-e^{i (a+b x)}\right)\right)}{b^2}+\frac{2 d \left(d \text{Li}_3\left(e^{i (a+b x)}\right)-i b (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)\right)}{b^2}+(c+d x)^2 \log \left(1-e^{i (a+b x)}\right)-(c+d x)^2 \log \left(1+e^{i (a+b x)}\right)\right)}{b^2}+\frac{-2 i b^3 c^3 \tan ^{-1}\left(e^{i (a+b x)}\right)+3 b^3 c^2 d x \log \left(1-i e^{i (a+b x)}\right)-3 b^3 c^2 d x \log \left(1+i e^{i (a+b x)}\right)+3 b^3 c d^2 x^2 \log \left(1-i e^{i (a+b x)}\right)-3 b^3 c d^2 x^2 \log \left(1+i e^{i (a+b x)}\right)+b^3 d^3 x^3 \log \left(1-i e^{i (a+b x)}\right)-b^3 d^3 x^3 \log \left(1+i e^{i (a+b x)}\right)+3 i b^2 d (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)-3 i b^2 d (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)-6 b c d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)+6 b c d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)-6 b d^3 x \text{Li}_3\left(-i e^{i (a+b x)}\right)+6 b d^3 x \text{Li}_3\left(i e^{i (a+b x)}\right)-6 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)+6 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)}{b^4}+\frac{\csc \left(\frac{a}{2}\right) \csc \left(\frac{a}{2}+\frac{b x}{2}\right) \left(c^3 \sin \left(\frac{b x}{2}\right)+3 c^2 d x \sin \left(\frac{b x}{2}\right)+3 c d^2 x^2 \sin \left(\frac{b x}{2}\right)+d^3 x^3 \sin \left(\frac{b x}{2}\right)\right)}{2 b}+\frac{\sec \left(\frac{a}{2}\right) \sec \left(\frac{a}{2}+\frac{b x}{2}\right) \left(c^3 \left(-\sin \left(\frac{b x}{2}\right)\right)-3 c^2 d x \sin \left(\frac{b x}{2}\right)-3 c d^2 x^2 \sin \left(\frac{b x}{2}\right)-d^3 x^3 \sin \left(\frac{b x}{2}\right)\right)}{2 b}-\frac{\csc (a) (c+d x)^3}{b}","-\frac{6 d^3 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^4}+\frac{6 d^3 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^4}-\frac{6 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)}{b^4}+\frac{6 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)}{b^4}+\frac{6 i d^2 (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^3}-\frac{6 i d^2 (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^3}-\frac{6 d^2 (c+d x) \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{6 d^2 (c+d x) \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{6 d (c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{2 i (c+d x)^3 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{(c+d x)^3 \csc (a+b x)}{b}",1,"-(((c + d*x)^3*Csc[a])/b) + (3*d*((c + d*x)^2*Log[1 - E^(I*(a + b*x))] - (c + d*x)^2*Log[1 + E^(I*(a + b*x))] + ((2*I)*d*(b*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] + I*d*PolyLog[3, -E^(I*(a + b*x))]))/b^2 + (2*d*((-I)*b*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] + d*PolyLog[3, E^(I*(a + b*x))]))/b^2))/b^2 + ((-2*I)*b^3*c^3*ArcTan[E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 - I*E^(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] + b^3*d^3*x^3*Log[1 - I*E^(I*(a + b*x))] - 3*b^3*c^2*d*x*Log[1 + I*E^(I*(a + b*x))] - 3*b^3*c*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 + I*E^(I*(a + b*x))] + (3*I)*b^2*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))] - (3*I)*b^2*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, (-I)*E^(I*(a + b*x))] + 6*b*c*d^2*PolyLog[3, I*E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, I*E^(I*(a + b*x))] - (6*I)*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, I*E^(I*(a + b*x))])/b^4 + (Sec[a/2]*Sec[a/2 + (b*x)/2]*(-(c^3*Sin[(b*x)/2]) - 3*c^2*d*x*Sin[(b*x)/2] - 3*c*d^2*x^2*Sin[(b*x)/2] - d^3*x^3*Sin[(b*x)/2]))/(2*b) + (Csc[a/2]*Csc[a/2 + (b*x)/2]*(c^3*Sin[(b*x)/2] + 3*c^2*d*x*Sin[(b*x)/2] + 3*c*d^2*x^2*Sin[(b*x)/2] + d^3*x^3*Sin[(b*x)/2]))/(2*b)","B",1
236,1,593,226,6.2749278,"\int (c+d x)^2 \csc ^2(a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]^2*Sec[a + b*x],x]","\frac{2 d^2 \left(-\frac{2 \tan ^{-1}(\tan (a)) \tanh ^{-1}\left(\frac{\sin (a) \tan \left(\frac{b x}{2}\right)-\cos (a)}{\sqrt{\sin ^2(a)+\cos ^2(a)}}\right)}{\sqrt{\sin ^2(a)+\cos ^2(a)}}+\frac{\sec (a) \left(i \left(\text{Li}_2\left(-e^{i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)-\text{Li}_2\left(e^{i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right)+\left(\tan ^{-1}(\tan (a))+b x\right) \left(\log \left(1-e^{i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)-\log \left(1+e^{i \left(\tan ^{-1}(\tan (a))+b x\right)}\right)\right)\right)}{\sqrt{\tan ^2(a)+1}}\right)}{b^3}+\frac{4 i c d \tan ^{-1}\left(\frac{i \cos (a)-i \sin (a) \tan \left(\frac{b x}{2}\right)}{\sqrt{\sin ^2(a)+\cos ^2(a)}}\right)}{b^2 \sqrt{\sin ^2(a)+\cos ^2(a)}}+\frac{-2 i b^2 c^2 \tan ^{-1}\left(e^{i (a+b x)}\right)+2 b^2 c d x \log \left(1-i e^{i (a+b x)}\right)-2 b^2 c d x \log \left(1+i e^{i (a+b x)}\right)+b^2 d^2 x^2 \log \left(1-i e^{i (a+b x)}\right)-b^2 d^2 x^2 \log \left(1+i e^{i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)-2 i b d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)-2 d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)+2 d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}+\frac{\csc \left(\frac{a}{2}\right) \csc \left(\frac{a}{2}+\frac{b x}{2}\right) \left(c^2 \sin \left(\frac{b x}{2}\right)+2 c d x \sin \left(\frac{b x}{2}\right)+d^2 x^2 \sin \left(\frac{b x}{2}\right)\right)}{2 b}+\frac{\sec \left(\frac{a}{2}\right) \sec \left(\frac{a}{2}+\frac{b x}{2}\right) \left(c^2 \left(-\sin \left(\frac{b x}{2}\right)\right)-2 c d x \sin \left(\frac{b x}{2}\right)-d^2 x^2 \sin \left(\frac{b x}{2}\right)\right)}{2 b}-\frac{\csc (a) (c+d x)^2}{b}","\frac{2 i d^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^3}-\frac{2 i d^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^3}-\frac{2 d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{2 d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}+\frac{2 i d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{2 i d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{4 d (c+d x) \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{2 i (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{(c+d x)^2 \csc (a+b x)}{b}",1,"-(((c + d*x)^2*Csc[a])/b) + ((-2*I)*b^2*c^2*ArcTan[E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 - I*E^(I*(a + b*x))] + b^2*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] - 2*b^2*c*d*x*Log[1 + I*E^(I*(a + b*x))] - b^2*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] - 2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] + 2*d^2*PolyLog[3, I*E^(I*(a + b*x))])/b^3 + ((4*I)*c*d*ArcTan[(I*Cos[a] - I*Sin[a]*Tan[(b*x)/2])/Sqrt[Cos[a]^2 + Sin[a]^2]])/(b^2*Sqrt[Cos[a]^2 + Sin[a]^2]) + (Sec[a/2]*Sec[a/2 + (b*x)/2]*(-(c^2*Sin[(b*x)/2]) - 2*c*d*x*Sin[(b*x)/2] - d^2*x^2*Sin[(b*x)/2]))/(2*b) + (Csc[a/2]*Csc[a/2 + (b*x)/2]*(c^2*Sin[(b*x)/2] + 2*c*d*x*Sin[(b*x)/2] + d^2*x^2*Sin[(b*x)/2]))/(2*b) + (2*d^2*((-2*ArcTan[Tan[a]]*ArcTanh[(-Cos[a] + Sin[a]*Tan[(b*x)/2])/Sqrt[Cos[a]^2 + Sin[a]^2]])/Sqrt[Cos[a]^2 + Sin[a]^2] + (((b*x + ArcTan[Tan[a]])*(Log[1 - E^(I*(b*x + ArcTan[Tan[a]]))] - Log[1 + E^(I*(b*x + ArcTan[Tan[a]]))]) + I*(PolyLog[2, -E^(I*(b*x + ArcTan[Tan[a]]))] - PolyLog[2, E^(I*(b*x + ArcTan[Tan[a]]))]))*Sec[a])/Sqrt[1 + Tan[a]^2]))/b^3","B",0
237,1,517,131,2.9082815,"\int (c+d x) \csc ^2(a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]^2*Sec[a + b*x],x]","\frac{d \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b^2}-\frac{d \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b^2}+\frac{d \left(a \cos \left(\frac{1}{2} (a+b x)\right)-(a+b x) \cos \left(\frac{1}{2} (a+b x)\right)\right) \csc \left(\frac{1}{2} (a+b x)\right)}{2 b^2}+\frac{d \left(a \sin \left(\frac{1}{2} (a+b x)\right)-(a+b x) \sin \left(\frac{1}{2} (a+b x)\right)\right) \sec \left(\frac{1}{2} (a+b x)\right)}{2 b^2}-\frac{c \csc (a+b x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sin ^2(a+b x)\right)}{b}-\frac{d x \left(-i \left(\text{Li}_2\left(\frac{1}{2} \left((1+i)-(1-i) \tan \left(\frac{1}{2} (a+b x)\right)\right)\right)+\log \left(1+i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(\frac{1}{2}-\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)+1\right)\right)\right)+i \left(\text{Li}_2\left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)+i\right)\right)+\log \left(1-i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)+1\right)\right)\right)-i \left(\text{Li}_2\left(\frac{1}{2} \left((1-i) \tan \left(\frac{1}{2} (a+b x)\right)+(1+i)\right)\right)+\log \left(1-i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(-\frac{1}{2}+\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)-1\right)\right)\right)+i \left(\text{Li}_2\left(\frac{1}{2} \left((1+i) \tan \left(\frac{1}{2} (a+b x)\right)+(1-i)\right)\right)+\log \left(1+i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)-1\right)\right)\right)+a \log \left(1-\tan \left(\frac{1}{2} (a+b x)\right)\right)-a \log \left(\tan \left(\frac{1}{2} (a+b x)\right)+1\right)\right)}{b \left(-i \log \left(1-i \tan \left(\frac{1}{2} (a+b x)\right)\right)+i \log \left(1+i \tan \left(\frac{1}{2} (a+b x)\right)\right)+a\right)}","\frac{i d \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{i d \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{d \tanh ^{-1}(\cos (a+b x))}{b^2}-\frac{(c+d x) \csc (a+b x)}{b}+\frac{(c+d x) \tanh ^{-1}(\sin (a+b x))}{b}-\frac{2 i d x \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{d x \tanh ^{-1}(\sin (a+b x))}{b}",1,"(d*(a*Cos[(a + b*x)/2] - (a + b*x)*Cos[(a + b*x)/2])*Csc[(a + b*x)/2])/(2*b^2) - (c*Csc[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[a + b*x]^2])/b - (d*Log[Cos[(a + b*x)/2]])/b^2 + (d*Log[Sin[(a + b*x)/2]])/b^2 - (d*x*(a*Log[1 - Tan[(a + b*x)/2]] - a*Log[1 + Tan[(a + b*x)/2]] - I*(Log[1 + I*Tan[(a + b*x)/2]]*Log[(1/2 - I/2)*(1 + Tan[(a + b*x)/2])] + PolyLog[2, ((1 + I) - (1 - I)*Tan[(a + b*x)/2])/2]) + I*(Log[1 - I*Tan[(a + b*x)/2]]*Log[(1/2 + I/2)*(1 + Tan[(a + b*x)/2])] + PolyLog[2, (-1/2 - I/2)*(I + Tan[(a + b*x)/2])]) - I*(Log[1 - I*Tan[(a + b*x)/2]]*Log[(-1/2 + I/2)*(-1 + Tan[(a + b*x)/2])] + PolyLog[2, ((1 + I) + (1 - I)*Tan[(a + b*x)/2])/2]) + I*(Log[1 + I*Tan[(a + b*x)/2]]*Log[(-1/2 - I/2)*(-1 + Tan[(a + b*x)/2])] + PolyLog[2, ((1 - I) + (1 + I)*Tan[(a + b*x)/2])/2])))/(b*(a - I*Log[1 - I*Tan[(a + b*x)/2]] + I*Log[1 + I*Tan[(a + b*x)/2]])) + (d*Sec[(a + b*x)/2]*(a*Sin[(a + b*x)/2] - (a + b*x)*Sin[(a + b*x)/2]))/(2*b^2)","C",0
238,0,0,25,11.5457276,"\int \frac{\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx","Integrate[(Csc[a + b*x]^2*Sec[a + b*x])/(c + d*x),x]","\int \frac{\csc ^2(a+b x) \sec (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\csc ^2(a+b x) \sec (a+b x)}{c+d x},x\right)",0,"Integrate[(Csc[a + b*x]^2*Sec[a + b*x])/(c + d*x), x]","A",-1
239,0,0,25,11.0771857,"\int \frac{\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx","Integrate[(Csc[a + b*x]^2*Sec[a + b*x])/(c + d*x)^2,x]","\int \frac{\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\csc ^2(a+b x) \sec (a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Csc[a + b*x]^2*Sec[a + b*x])/(c + d*x)^2, x]","A",-1
240,0,0,25,18.2877293,"\int (c+d x)^m \csc ^3(a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x],x]","\int (c+d x)^m \csc ^3(a+b x) \sec (a+b x) \, dx","\text{Int}\left(\csc ^3(a+b x) \sec (a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x], x]","A",-1
241,1,1477,325,6.9747163,"\int (c+d x)^3 \csc ^3(a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]^3*Sec[a + b*x],x]","-\frac{\sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a)) c^3}{b \left(\cos ^2(a)+\sin ^2(a)\right)}+\frac{\csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a)) c^3}{b \left(\cos ^2(a)+\sin ^2(a)\right)}-\frac{3 d \csc (a) \sec (a) \left(b^2 e^{i \tan ^{-1}(\tan (a))} x^2+\frac{\left(i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x+\tan ^{-1}(\tan (a))\right) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\pi  \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right) \tan (a)}{\sqrt{\tan ^2(a)+1}}\right) c^2}{2 b^2 \sqrt{\sec ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}-\frac{3 d \csc (a) \left(b^2 e^{-i \tan ^{-1}(\cot (a))} x^2-\frac{\cot (a) \left(i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\pi  \log (\cos (b x))-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)}{\sqrt{\cot ^2(a)+1}}\right) \sec (a) c^2}{2 b^2 \sqrt{\csc ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}-\frac{d^2 e^{i a} \csc (a) \left(2 b^3 e^{-2 i a} x^3+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^2+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^2-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right) c}{2 b^3}-\frac{i d^2 e^{-i a} \left(2 b^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right) x^2+6 b \left(1+e^{2 i a}\right) \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right) \sec (a) c}{4 b^3}+\frac{3 d^2 \csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a)) c}{b^3 \left(\cos ^2(a)+\sin ^2(a)\right)}-\frac{(c+d x)^3 \csc ^2(a+b x)}{2 b}-\frac{d^3 e^{i a} \csc (a) \left(b^4 e^{-2 i a} x^4+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^3+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^3-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(-e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(-e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(e^{-i (a+b x)}\right)\right)\right)}{4 b^4}+\frac{1}{4} x \left(4 c^3+6 d x c^2+4 d^2 x^2 c+d^3 x^3\right) \csc (a) \sec (a)-\frac{1}{8} i d^3 e^{i a} \left(2 e^{-2 i a} x^4-\frac{4 i \left(1+e^{-2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right) x^3}{b}+\frac{3 e^{-2 i a} \left(1+e^{2 i a}\right) \left(2 b^2 \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(-e^{-2 i (a+b x)}\right) x-\text{Li}_4\left(-e^{-2 i (a+b x)}\right)\right)}{b^4}\right) \sec (a)+\frac{3 \csc (a) \csc (a+b x) \left(x^2 \sin (b x) d^3+2 c x \sin (b x) d^2+c^2 \sin (b x) d\right)}{2 b^2}-\frac{3 d^3 \csc (a) \sec (a) \left(b^2 e^{i \tan ^{-1}(\tan (a))} x^2+\frac{\left(i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x+\tan ^{-1}(\tan (a))\right) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\pi  \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right) \tan (a)}{\sqrt{\tan ^2(a)+1}}\right)}{2 b^4 \sqrt{\sec ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}","-\frac{3 i d^3 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 i d^3 \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{4 b^4}+\frac{3 i d^3 \text{Li}_4\left(e^{2 i (a+b x)}\right)}{4 b^4}-\frac{3 d^2 (c+d x) \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}+\frac{3 d^2 (c+d x) \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}+\frac{3 d^2 (c+d x) \log \left(1-e^{2 i (a+b x)}\right)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac{(c+d x)^3 \cot ^2(a+b x)}{2 b}-\frac{2 (c+d x)^3 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}-\frac{3 i d (c+d x)^2}{2 b^2}-\frac{(c+d x)^3}{2 b}",1,"-1/2*((c + d*x)^3*Csc[a + b*x]^2)/b - (c*d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(2*b^3) - (d^3*E^(I*a)*Csc[a]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-I)*(a + b*x))] - 2*PolyLog[4, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, E^((-I)*(a + b*x))] - 2*PolyLog[4, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(4*b^4) + (x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Csc[a]*Sec[a])/4 - ((I/4)*c*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) - (I/8)*d^3*E^(I*a)*((2*x^4)/E^((2*I)*a) - ((4*I)*(1 + E^((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x))])/b + (3*(1 + E^((2*I)*a))*(2*b^2*x^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-2*I)*(a + b*x))] - PolyLog[4, -E^((-2*I)*(a + b*x))]))/(b^4*E^((2*I)*a)))*Sec[a] - (c^3*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + (c^3*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + (3*c*d^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) - (3*c^2*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(2*b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + (3*Csc[a]*Csc[a + b*x]*(c^2*d*Sin[b*x] + 2*c*d^2*x*Sin[b*x] + d^3*x^2*Sin[b*x]))/(2*b^2) - (3*c^2*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(2*b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (3*d^3*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(2*b^4*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
242,1,872,201,6.757022,"\int (c+d x)^2 \csc ^3(a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]^3*Sec[a + b*x],x]","-\frac{\sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a)) c^2}{b \left(\cos ^2(a)+\sin ^2(a)\right)}+\frac{\csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a)) c^2}{b \left(\cos ^2(a)+\sin ^2(a)\right)}-\frac{d \csc (a) \sec (a) \left(b^2 e^{i \tan ^{-1}(\tan (a))} x^2+\frac{\left(i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x+\tan ^{-1}(\tan (a))\right) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\pi  \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right) \tan (a)}{\sqrt{\tan ^2(a)+1}}\right) c}{b^2 \sqrt{\sec ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}-\frac{d \csc (a) \left(b^2 e^{-i \tan ^{-1}(\cot (a))} x^2-\frac{\cot (a) \left(i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\pi  \log (\cos (b x))-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)}{\sqrt{\cot ^2(a)+1}}\right) \sec (a) c}{b^2 \sqrt{\csc ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}-\frac{(c+d x)^2 \csc ^2(a+b x)}{2 b}-\frac{d^2 e^{i a} \csc (a) \left(2 b^3 e^{-2 i a} x^3+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^2+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^2-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right)}{6 b^3}+\frac{1}{3} x \left(3 c^2+3 d x c+d^2 x^2\right) \csc (a) \sec (a)-\frac{i d^2 e^{-i a} \left(2 b^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right) x^2+6 b \left(1+e^{2 i a}\right) \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right) \sec (a)}{12 b^3}+\frac{\csc (a) \csc (a+b x) \left(x \sin (b x) d^2+c \sin (b x) d\right)}{b^2}+\frac{d^2 \csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a))}{b^3 \left(\cos ^2(a)+\sin ^2(a)\right)}","-\frac{d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}+\frac{d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}+\frac{d^2 \log (\sin (a+b x))}{b^3}+\frac{i d (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{i d (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{d (c+d x) \cot (a+b x)}{b^2}-\frac{(c+d x)^2 \cot ^2(a+b x)}{2 b}-\frac{2 (c+d x)^2 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}-\frac{c d x}{b}-\frac{d^2 x^2}{2 b}",1,"-1/2*((c + d*x)^2*Csc[a + b*x]^2)/b - (d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(6*b^3) + (x*(3*c^2 + 3*c*d*x + d^2*x^2)*Csc[a]*Sec[a])/3 - ((I/12)*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) - (c^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + (c^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + (d^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) - (c*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + (Csc[a]*Csc[a + b*x]*(c*d*Sin[b*x] + d^2*x*Sin[b*x]))/b^2 - (c*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
243,1,210,141,0.8976262,"\int (c+d x) \csc ^3(a+b x) \sec (a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]^3*Sec[a + b*x],x]","\frac{d \left(\frac{1}{2} i \text{Li}_2\left(-e^{2 i (a+b x)}\right)+\frac{1}{2} i (a+b x)^2-(a+b x) \log \left(1+e^{2 i (a+b x)}\right)\right)}{b^2}+\frac{d \left((a+b x) \log \left(1-e^{2 i (a+b x)}\right)-\frac{1}{2} i \left((a+b x)^2+\text{Li}_2\left(e^{2 i (a+b x)}\right)\right)\right)}{b^2}-\frac{d \cot (a+b x)}{2 b^2}-\frac{a d \log (\sin (a+b x))}{b^2}+\frac{a d \log (\cos (a+b x))}{b^2}-\frac{c \left(\csc ^2(a+b x)-2 \log (\sin (a+b x))+2 \log (\cos (a+b x))\right)}{2 b}-\frac{d x \csc ^2(a+b x)}{2 b}","\frac{i d \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{i d \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{d \cot (a+b x)}{2 b^2}-\frac{(c+d x) \cot ^2(a+b x)}{2 b}+\frac{(c+d x) \log (\tan (a+b x))}{b}-\frac{2 d x \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}-\frac{d x \log (\tan (a+b x))}{b}-\frac{d x}{2 b}",1,"-1/2*(d*Cot[a + b*x])/b^2 - (d*x*Csc[a + b*x]^2)/(2*b) + (a*d*Log[Cos[a + b*x]])/b^2 - (c*(Csc[a + b*x]^2 + 2*Log[Cos[a + b*x]] - 2*Log[Sin[a + b*x]]))/(2*b) - (a*d*Log[Sin[a + b*x]])/b^2 + (d*((I/2)*(a + b*x)^2 - (a + b*x)*Log[1 + E^((2*I)*(a + b*x))] + (I/2)*PolyLog[2, -E^((2*I)*(a + b*x))]))/b^2 + (d*((a + b*x)*Log[1 - E^((2*I)*(a + b*x))] - (I/2)*((a + b*x)^2 + PolyLog[2, E^((2*I)*(a + b*x))])))/b^2","A",1
244,0,0,25,14.0680626,"\int \frac{\csc ^3(a+b x) \sec (a+b x)}{c+d x} \, dx","Integrate[(Csc[a + b*x]^3*Sec[a + b*x])/(c + d*x),x]","\int \frac{\csc ^3(a+b x) \sec (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\csc ^3(a+b x) \sec (a+b x)}{c+d x},x\right)",0,"Integrate[(Csc[a + b*x]^3*Sec[a + b*x])/(c + d*x), x]","A",-1
245,0,0,25,16.2305933,"\int \frac{\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx","Integrate[(Csc[a + b*x]^3*Sec[a + b*x])/(c + d*x)^2,x]","\int \frac{\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\csc ^3(a+b x) \sec (a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Csc[a + b*x]^3*Sec[a + b*x])/(c + d*x)^2, x]","A",-1
246,0,0,23,2.7004892,"\int (c+d x)^m \sec (a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x],x]","\int (c+d x)^m \sec (a+b x) \tan (a+b x) \, dx","\text{Int}\left(\tan (a+b x) \sec (a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x], x]","A",-1
247,1,428,227,1.1680908,"\int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^4*Sec[a + b*x]*Tan[a + b*x],x]","\frac{(c+d x)^4 \sec (a+b x)}{b}-\frac{4 d \left(-2 i b^3 c^3 \tan ^{-1}\left(e^{i (a+b x)}\right)+3 b^3 c^2 d x \log \left(1-i e^{i (a+b x)}\right)-3 b^3 c^2 d x \log \left(1+i e^{i (a+b x)}\right)+3 b^3 c d^2 x^2 \log \left(1-i e^{i (a+b x)}\right)-3 b^3 c d^2 x^2 \log \left(1+i e^{i (a+b x)}\right)+b^3 d^3 x^3 \log \left(1-i e^{i (a+b x)}\right)-b^3 d^3 x^3 \log \left(1+i e^{i (a+b x)}\right)+3 i b^2 d (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)-3 i b^2 d (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)-6 b c d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)+6 b c d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)-6 b d^3 x \text{Li}_3\left(-i e^{i (a+b x)}\right)+6 b d^3 x \text{Li}_3\left(i e^{i (a+b x)}\right)-6 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)+6 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)\right)}{b^5}","\frac{24 i d^4 \text{Li}_4\left(-i e^{i (a+b x)}\right)}{b^5}-\frac{24 i d^4 \text{Li}_4\left(i e^{i (a+b x)}\right)}{b^5}+\frac{24 d^3 (c+d x) \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^4}-\frac{24 d^3 (c+d x) \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^4}-\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}+\frac{8 i d (c+d x)^3 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}+\frac{(c+d x)^4 \sec (a+b x)}{b}",1,"(-4*d*((-2*I)*b^3*c^3*ArcTan[E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 - I*E^(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] + b^3*d^3*x^3*Log[1 - I*E^(I*(a + b*x))] - 3*b^3*c^2*d*x*Log[1 + I*E^(I*(a + b*x))] - 3*b^3*c*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 + I*E^(I*(a + b*x))] + (3*I)*b^2*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))] - (3*I)*b^2*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, (-I)*E^(I*(a + b*x))] + 6*b*c*d^2*PolyLog[3, I*E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, I*E^(I*(a + b*x))] - (6*I)*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, I*E^(I*(a + b*x))]))/b^5 + ((c + d*x)^4*Sec[a + b*x])/b","A",1
248,1,256,159,0.8072178,"\int (c+d x)^3 \sec (a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^3*Sec[a + b*x]*Tan[a + b*x],x]","\frac{(c+d x)^3 \sec (a+b x)}{b}-\frac{3 d \left(-2 i b^2 c^2 \tan ^{-1}\left(e^{i (a+b x)}\right)+2 b^2 c d x \log \left(1-i e^{i (a+b x)}\right)-2 b^2 c d x \log \left(1+i e^{i (a+b x)}\right)+b^2 d^2 x^2 \log \left(1-i e^{i (a+b x)}\right)-b^2 d^2 x^2 \log \left(1+i e^{i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)-2 i b d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)-2 d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)+2 d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)\right)}{b^4}","\frac{6 d^3 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^4}-\frac{6 d^3 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^4}-\frac{6 i d^2 (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{6 i d^2 (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}+\frac{6 i d (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}+\frac{(c+d x)^3 \sec (a+b x)}{b}",1,"(-3*d*((-2*I)*b^2*c^2*ArcTan[E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 - I*E^(I*(a + b*x))] + b^2*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] - 2*b^2*c*d*x*Log[1 + I*E^(I*(a + b*x))] - b^2*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] - 2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] + 2*d^2*PolyLog[3, I*E^(I*(a + b*x))]))/b^4 + ((c + d*x)^3*Sec[a + b*x])/b","A",1
249,1,174,97,1.6554953,"\int (c+d x)^2 \sec (a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^2*Sec[a + b*x]*Tan[a + b*x],x]","\frac{b^2 (c+d x)^2 \sec (a+b x)-4 b c d \tanh ^{-1}\left(\cos (a) \tan \left(\frac{b x}{2}\right)+\sin (a)\right)+\frac{2 d^2 \csc (a) \left(i \text{Li}_2\left(-e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-i \text{Li}_2\left(e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\left(b x-\tan ^{-1}(\cot (a))\right) \left(\log \left(1-e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-\log \left(1+e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)\right)}{\sqrt{\csc ^2(a)}}-4 d^2 \tan ^{-1}(\cot (a)) \tanh ^{-1}\left(\cos (a) \tan \left(\frac{b x}{2}\right)+\sin (a)\right)}{b^3}","-\frac{2 i d^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{2 i d^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}+\frac{4 i d (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}+\frac{(c+d x)^2 \sec (a+b x)}{b}",1,"(-4*b*c*d*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] - 4*d^2*ArcTan[Cot[a]]*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] + (2*d^2*Csc[a]*((b*x - ArcTan[Cot[a]])*(Log[1 - E^(I*(b*x - ArcTan[Cot[a]]))] - Log[1 + E^(I*(b*x - ArcTan[Cot[a]]))]) + I*PolyLog[2, -E^(I*(b*x - ArcTan[Cot[a]]))] - I*PolyLog[2, E^(I*(b*x - ArcTan[Cot[a]]))]))/Sqrt[Csc[a]^2] + b^2*(c + d*x)^2*Sec[a + b*x])/b^3","A",0
250,1,93,29,0.0448432,"\int (c+d x) \sec (a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)*Sec[a + b*x]*Tan[a + b*x],x]","\frac{d \log \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)-\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}{b^2}-\frac{d \log \left(\sin \left(\frac{a}{2}+\frac{b x}{2}\right)+\cos \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}{b^2}+\frac{c \sec (a+b x)}{b}+\frac{d x \sec (a+b x)}{b}","\frac{(c+d x) \sec (a+b x)}{b}-\frac{d \tanh ^{-1}(\sin (a+b x))}{b^2}",1,"(d*Log[Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2]])/b^2 - (d*Log[Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2]])/b^2 + (c*Sec[a + b*x])/b + (d*x*Sec[a + b*x])/b","B",1
251,0,0,23,11.0495917,"\int \frac{\sec (a+b x) \tan (a+b x)}{c+d x} \, dx","Integrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x),x]","\int \frac{\sec (a+b x) \tan (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\tan (a+b x) \sec (a+b x)}{c+d x},x\right)",0,"Integrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x), x]","A",-1
252,0,0,23,19.1244818,"\int \frac{\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx","Integrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x)^2,x]","\int \frac{\sec (a+b x) \tan (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\tan (a+b x) \sec (a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x)^2, x]","A",-1
253,0,0,19,2.8769524,"\int (c+d x)^m \tan ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Tan[a + b*x]^2,x]","\int (c+d x)^m \tan ^2(a+b x) \, dx","\text{Int}\left(\tan ^2(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Tan[a + b*x]^2, x]","A",-1
254,1,424,128,6.5452945,"\int (c+d x)^3 \tan ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Tan[a + b*x]^2,x]","\frac{3 c^2 d \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b^2 \left(\sin ^2(a)+\cos ^2(a)\right)}+\frac{i e^{-i a} d^3 \sec (a) \left(2 b^2 x^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right)+6 \left(1+e^{2 i a}\right) b x \text{Li}_2\left(-e^{-2 i (a+b x)}\right)-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right)}{4 b^4}+\frac{3 c d^2 \csc (a) \sec (a) \left(b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left(i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\cot ^2(a)+1}}\right)}{b^3 \sqrt{\csc ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}+\frac{\sec (a) \sec (a+b x) \left(c^3 \sin (b x)+3 c^2 d x \sin (b x)+3 c d^2 x^2 \sin (b x)+d^3 x^3 \sin (b x)\right)}{b}-\frac{1}{4} x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)","\frac{3 d^3 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 i d^2 (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^3}+\frac{3 d (c+d x)^2 \log \left(1+e^{2 i (a+b x)}\right)}{b^2}+\frac{(c+d x)^3 \tan (a+b x)}{b}-\frac{i (c+d x)^3}{b}-\frac{(c+d x)^4}{4 d}",1,"-1/4*(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)) + ((I/4)*d^3*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^4*E^(I*a)) + (3*c^2*d*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b^2*(Cos[a]^2 + Sin[a]^2)) + (3*c*d^2*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^3*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + (Sec[a]*Sec[a + b*x]*(c^3*Sin[b*x] + 3*c^2*d*x*Sin[b*x] + 3*c*d^2*x^2*Sin[b*x] + d^3*x^3*Sin[b*x]))/b","B",0
255,1,276,96,6.3675938,"\int (c+d x)^2 \tan ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Tan[a + b*x]^2,x]","\frac{2 c d \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b^2 \left(\sin ^2(a)+\cos ^2(a)\right)}+\frac{d^2 \csc (a) \sec (a) \left(b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left(i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\cot ^2(a)+1}}\right)}{b^3 \sqrt{\csc ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}+\frac{\sec (a) \sec (a+b x) \left(c^2 \sin (b x)+2 c d x \sin (b x)+d^2 x^2 \sin (b x)\right)}{b}-\frac{1}{3} x \left(3 c^2+3 c d x+d^2 x^2\right)","-\frac{i d^2 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^3}+\frac{2 d (c+d x) \log \left(1+e^{2 i (a+b x)}\right)}{b^2}+\frac{(c+d x)^2 \tan (a+b x)}{b}-\frac{i (c+d x)^2}{b}-\frac{(c+d x)^3}{3 d}",1,"-1/3*(x*(3*c^2 + 3*c*d*x + d^2*x^2)) + (2*c*d*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b^2*(Cos[a]^2 + Sin[a]^2)) + (d^2*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^3*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + (Sec[a]*Sec[a + b*x]*(c^2*Sin[b*x] + 2*c*d*x*Sin[b*x] + d^2*x^2*Sin[b*x]))/b","B",0
256,1,76,40,0.2762641,"\int (c+d x) \tan ^2(a+b x) \, dx","Integrate[(c + d*x)*Tan[a + b*x]^2,x]","\frac{d \log (\cos (a+b x))}{b^2}-\frac{c \tan ^{-1}(\tan (a+b x))}{b}+\frac{c \tan (a+b x)}{b}+\frac{d x \sec (a) \sin (b x) \sec (a+b x)}{b}-\frac{d x \sec (a) (b x \cos (a)-2 \sin (a))}{2 b}","\frac{d \log (\cos (a+b x))}{b^2}+\frac{(c+d x) \tan (a+b x)}{b}-c x-\frac{d x^2}{2}",1,"-((c*ArcTan[Tan[a + b*x]])/b) + (d*Log[Cos[a + b*x]])/b^2 - (d*x*Sec[a]*(b*x*Cos[a] - 2*Sin[a]))/(2*b) + (d*x*Sec[a]*Sec[a + b*x]*Sin[b*x])/b + (c*Tan[a + b*x])/b","A",1
257,0,0,19,3.832934,"\int \frac{\tan ^2(a+b x)}{c+d x} \, dx","Integrate[Tan[a + b*x]^2/(c + d*x),x]","\int \frac{\tan ^2(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\tan ^2(a+b x)}{c+d x},x\right)",0,"Integrate[Tan[a + b*x]^2/(c + d*x), x]","A",-1
258,0,0,19,5.0711933,"\int \frac{\tan ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[Tan[a + b*x]^2/(c + d*x)^2,x]","\int \frac{\tan ^2(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\tan ^2(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[Tan[a + b*x]^2/(c + d*x)^2, x]","A",-1
259,0,0,150,23.6973757,"\int (c+d x)^m \sin (a+b x) \tan ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Sin[a + b*x]*Tan[a + b*x]^2,x]","\int (c+d x)^m \sin (a+b x) \tan ^2(a+b x) \, dx","\text{Int}\left(\tan (a+b x) \sec (a+b x) (c+d x)^m,x\right)+\frac{e^{i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(-\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,-\frac{i b (c+d x)}{d}\right)}{2 b}+\frac{e^{-i \left(a-\frac{b c}{d}\right)} (c+d x)^m \left(\frac{i b (c+d x)}{d}\right)^{-m} \Gamma \left(m+1,\frac{i b (c+d x)}{d}\right)}{2 b}",0,"Integrate[(c + d*x)^m*Sin[a + b*x]*Tan[a + b*x]^2, x]","A",-1
260,1,532,228,1.5449782,"\int (c+d x)^3 \sin (a+b x) \tan ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Sin[a + b*x]*Tan[a + b*x]^2,x]","\frac{\sec (a+b x) \left(b^3 c^3 \cos (2 (a+b x))+3 b^3 c^2 d x \cos (2 (a+b x))+3 b^3 c d^2 x^2 \cos (2 (a+b x))+b^3 d^3 x^3 \cos (2 (a+b x))-3 b^2 c^2 d \sin (2 (a+b x))+12 i b^2 c^2 d \cos (a+b x) \tan ^{-1}\left(e^{i (a+b x)}\right)-6 b^2 c d^2 x \sin (2 (a+b x))-12 b^2 c d^2 x \log \left(1-i e^{i (a+b x)}\right) \cos (a+b x)+12 b^2 c d^2 x \log \left(1+i e^{i (a+b x)}\right) \cos (a+b x)-3 b^2 d^3 x^2 \sin (2 (a+b x))-6 b^2 d^3 x^2 \log \left(1-i e^{i (a+b x)}\right) \cos (a+b x)+6 b^2 d^3 x^2 \log \left(1+i e^{i (a+b x)}\right) \cos (a+b x)-12 i b d^2 (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right) \cos (a+b x)+12 i b d^2 (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right) \cos (a+b x)-6 b c d^2 \cos (2 (a+b x))+12 d^3 \text{Li}_3\left(-i e^{i (a+b x)}\right) \cos (a+b x)-12 d^3 \text{Li}_3\left(i e^{i (a+b x)}\right) \cos (a+b x)+6 d^3 \sin (2 (a+b x))-6 b d^3 x \cos (2 (a+b x))+3 b^3 c^3+9 b^3 c^2 d x+9 b^3 c d^2 x^2+3 b^3 d^3 x^3-6 b c d^2-6 b d^3 x\right)}{2 b^4}","\frac{6 d^3 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^4}-\frac{6 d^3 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^4}+\frac{6 d^3 \sin (a+b x)}{b^4}-\frac{6 i d^2 (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{6 i d^2 (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}-\frac{6 d^2 (c+d x) \cos (a+b x)}{b^3}-\frac{3 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac{6 i d (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}+\frac{(c+d x)^3 \cos (a+b x)}{b}+\frac{(c+d x)^3 \sec (a+b x)}{b}",1,"(Sec[a + b*x]*(3*b^3*c^3 - 6*b*c*d^2 + 9*b^3*c^2*d*x - 6*b*d^3*x + 9*b^3*c*d^2*x^2 + 3*b^3*d^3*x^3 + (12*I)*b^2*c^2*d*ArcTan[E^(I*(a + b*x))]*Cos[a + b*x] + b^3*c^3*Cos[2*(a + b*x)] - 6*b*c*d^2*Cos[2*(a + b*x)] + 3*b^3*c^2*d*x*Cos[2*(a + b*x)] - 6*b*d^3*x*Cos[2*(a + b*x)] + 3*b^3*c*d^2*x^2*Cos[2*(a + b*x)] + b^3*d^3*x^3*Cos[2*(a + b*x)] - 12*b^2*c*d^2*x*Cos[a + b*x]*Log[1 - I*E^(I*(a + b*x))] - 6*b^2*d^3*x^2*Cos[a + b*x]*Log[1 - I*E^(I*(a + b*x))] + 12*b^2*c*d^2*x*Cos[a + b*x]*Log[1 + I*E^(I*(a + b*x))] + 6*b^2*d^3*x^2*Cos[a + b*x]*Log[1 + I*E^(I*(a + b*x))] - (12*I)*b*d^2*(c + d*x)*Cos[a + b*x]*PolyLog[2, (-I)*E^(I*(a + b*x))] + (12*I)*b*d^2*(c + d*x)*Cos[a + b*x]*PolyLog[2, I*E^(I*(a + b*x))] + 12*d^3*Cos[a + b*x]*PolyLog[3, (-I)*E^(I*(a + b*x))] - 12*d^3*Cos[a + b*x]*PolyLog[3, I*E^(I*(a + b*x))] - 3*b^2*c^2*d*Sin[2*(a + b*x)] + 6*d^3*Sin[2*(a + b*x)] - 6*b^2*c*d^2*x*Sin[2*(a + b*x)] - 3*b^2*d^3*x^2*Sin[2*(a + b*x)]))/(2*b^4)","B",1
261,1,362,145,3.1182363,"\int (c+d x)^2 \sin (a+b x) \tan ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Sin[a + b*x]*Tan[a + b*x]^2,x]","\frac{\cos (b x) \left(\cos (a) \left(b^2 (c+d x)^2-2 d^2\right)-2 b d \sin (a) (c+d x)\right)-\sin (b x) \left(\sin (a) \left(b^2 (c+d x)^2-2 d^2\right)+2 b d \cos (a) (c+d x)\right)+b^2 \sec (a) (c+d x)^2+\frac{b^2 \sin \left(\frac{b x}{2}\right) (c+d x)^2}{\left(\cos \left(\frac{a}{2}\right)-\sin \left(\frac{a}{2}\right)\right) \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)}-\frac{b^2 \sin \left(\frac{b x}{2}\right) (c+d x)^2}{\left(\sin \left(\frac{a}{2}\right)+\cos \left(\frac{a}{2}\right)\right) \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)}-4 b c d \tanh ^{-1}\left(\cos (a) \tan \left(\frac{b x}{2}\right)+\sin (a)\right)+\frac{2 d^2 \csc (a) \left(i \text{Li}_2\left(-e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-i \text{Li}_2\left(e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\left(b x-\tan ^{-1}(\cot (a))\right) \left(\log \left(1-e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-\log \left(1+e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)\right)}{\sqrt{\csc ^2(a)}}-4 d^2 \tan ^{-1}(\cot (a)) \tanh ^{-1}\left(\cos (a) \tan \left(\frac{b x}{2}\right)+\sin (a)\right)}{b^3}","-\frac{2 i d^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{2 i d^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}-\frac{2 d^2 \cos (a+b x)}{b^3}-\frac{2 d (c+d x) \sin (a+b x)}{b^2}+\frac{4 i d (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}+\frac{(c+d x)^2 \cos (a+b x)}{b}+\frac{(c+d x)^2 \sec (a+b x)}{b}",1,"(-4*b*c*d*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] - 4*d^2*ArcTan[Cot[a]]*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] + (2*d^2*Csc[a]*((b*x - ArcTan[Cot[a]])*(Log[1 - E^(I*(b*x - ArcTan[Cot[a]]))] - Log[1 + E^(I*(b*x - ArcTan[Cot[a]]))]) + I*PolyLog[2, -E^(I*(b*x - ArcTan[Cot[a]]))] - I*PolyLog[2, E^(I*(b*x - ArcTan[Cot[a]]))]))/Sqrt[Csc[a]^2] + b^2*(c + d*x)^2*Sec[a] + Cos[b*x]*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] - 2*b*d*(c + d*x)*Sin[a]) - (2*b*d*(c + d*x)*Cos[a] + (-2*d^2 + b^2*(c + d*x)^2)*Sin[a])*Sin[b*x] + (b^2*(c + d*x)^2*Sin[(b*x)/2])/((Cos[a/2] - Sin[a/2])*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])) - (b^2*(c + d*x)^2*Sin[(b*x)/2])/((Cos[a/2] + Sin[a/2])*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])))/b^3","B",0
262,1,107,56,0.2992428,"\int (c+d x) \sin (a+b x) \tan ^2(a+b x) \, dx","Integrate[(c + d*x)*Sin[a + b*x]*Tan[a + b*x]^2,x]","\frac{\sec (a+b x) \left(b (c+d x) \cos (2 (a+b x))-d \sin (2 (a+b x))+2 d \cos (a+b x) \left(\log \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)\right)+3 b c+3 b d x\right)}{2 b^2}","-\frac{d \sin (a+b x)}{b^2}-\frac{d \tanh ^{-1}(\sin (a+b x))}{b^2}+\frac{(c+d x) \cos (a+b x)}{b}+\frac{(c+d x) \sec (a+b x)}{b}",1,"(Sec[a + b*x]*(3*b*c + 3*b*d*x + b*(c + d*x)*Cos[2*(a + b*x)] + 2*d*Cos[a + b*x]*(Log[Cos[(a + b*x)/2] - Sin[(a + b*x)/2]] - Log[Cos[(a + b*x)/2] + Sin[(a + b*x)/2]]) - d*Sin[2*(a + b*x)]))/(2*b^2)","A",1
263,0,0,76,3.7778153,"\int \frac{\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx","Integrate[(Sin[a + b*x]*Tan[a + b*x]^2)/(c + d*x),x]","\int \frac{\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\tan (a+b x) \sec (a+b x)}{c+d x},x\right)-\frac{\sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d}-\frac{\cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d}",0,"Integrate[(Sin[a + b*x]*Tan[a + b*x]^2)/(c + d*x), x]","A",-1
264,0,0,94,4.0693844,"\int \frac{\sin (a+b x) \tan ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[(Sin[a + b*x]*Tan[a + b*x]^2)/(c + d*x)^2,x]","\int \frac{\sin (a+b x) \tan ^2(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\tan (a+b x) \sec (a+b x)}{(c+d x)^2},x\right)-\frac{b \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d^2}+\frac{b \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d^2}+\frac{\sin (a+b x)}{d (c+d x)}",0,"Integrate[(Sin[a + b*x]*Tan[a + b*x]^2)/(c + d*x)^2, x]","A",-1
265,0,0,25,24.5686057,"\int (c+d x)^m \csc (a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Csc[a + b*x]*Sec[a + b*x]^2,x]","\int (c+d x)^m \csc (a+b x) \sec ^2(a+b x) \, dx","\text{Int}\left(\csc (a+b x) \sec ^2(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Csc[a + b*x]*Sec[a + b*x]^2, x]","A",-1
266,1,694,469,3.365469,"\int (c+d x)^4 \csc (a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x]^2,x]","\frac{b^4 (c+d x)^4 \log \left(1-e^{i (a+b x)}\right)-b^4 (c+d x)^4 \log \left(1+e^{i (a+b x)}\right)+b^4 (c+d x)^4 \sec (a+b x)-4 d \left(-2 i b^3 c^3 \tan ^{-1}\left(e^{i (a+b x)}\right)+3 b^3 c^2 d x \log \left(1-i e^{i (a+b x)}\right)-3 b^3 c^2 d x \log \left(1+i e^{i (a+b x)}\right)+3 b^3 c d^2 x^2 \log \left(1-i e^{i (a+b x)}\right)-3 b^3 c d^2 x^2 \log \left(1+i e^{i (a+b x)}\right)+b^3 d^3 x^3 \log \left(1-i e^{i (a+b x)}\right)-b^3 d^3 x^3 \log \left(1+i e^{i (a+b x)}\right)+3 i b^2 d (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)-3 i b^2 d (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)-6 b c d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)+6 b c d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)-6 b d^3 x \text{Li}_3\left(-i e^{i (a+b x)}\right)+6 b d^3 x \text{Li}_3\left(i e^{i (a+b x)}\right)-6 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)+6 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)\right)+4 i d \left(b^3 (c+d x)^3 \text{Li}_2\left(-e^{i (a+b x)}\right)+3 i b^2 d (c+d x)^2 \text{Li}_3\left(-e^{i (a+b x)}\right)-6 d^2 \left(b (c+d x) \text{Li}_4\left(-e^{i (a+b x)}\right)+i d \text{Li}_5\left(-e^{i (a+b x)}\right)\right)\right)-4 i d \left(b^3 (c+d x)^3 \text{Li}_2\left(e^{i (a+b x)}\right)+3 i b^2 d (c+d x)^2 \text{Li}_3\left(e^{i (a+b x)}\right)-6 d^2 \left(b (c+d x) \text{Li}_4\left(e^{i (a+b x)}\right)+i d \text{Li}_5\left(e^{i (a+b x)}\right)\right)\right)}{b^5}","\frac{24 i d^4 \text{Li}_4\left(-i e^{i (a+b x)}\right)}{b^5}-\frac{24 i d^4 \text{Li}_4\left(i e^{i (a+b x)}\right)}{b^5}+\frac{24 d^4 \text{Li}_5\left(-e^{i (a+b x)}\right)}{b^5}-\frac{24 d^4 \text{Li}_5\left(e^{i (a+b x)}\right)}{b^5}+\frac{24 d^3 (c+d x) \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^4}-\frac{24 d^3 (c+d x) \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^4}-\frac{24 i d^3 (c+d x) \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^4}+\frac{24 i d^3 (c+d x) \text{Li}_4\left(e^{i (a+b x)}\right)}{b^4}-\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}-\frac{12 d^2 (c+d x)^2 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{12 d^2 (c+d x)^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}+\frac{4 i d (c+d x)^3 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{4 i d (c+d x)^3 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}+\frac{8 i d (c+d x)^3 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}+\frac{(c+d x)^4 \sec (a+b x)}{b}-\frac{2 (c+d x)^4 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(b^4*(c + d*x)^4*Log[1 - E^(I*(a + b*x))] - b^4*(c + d*x)^4*Log[1 + E^(I*(a + b*x))] - 4*d*((-2*I)*b^3*c^3*ArcTan[E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 - I*E^(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] + b^3*d^3*x^3*Log[1 - I*E^(I*(a + b*x))] - 3*b^3*c^2*d*x*Log[1 + I*E^(I*(a + b*x))] - 3*b^3*c*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 + I*E^(I*(a + b*x))] + (3*I)*b^2*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))] - (3*I)*b^2*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, (-I)*E^(I*(a + b*x))] + 6*b*c*d^2*PolyLog[3, I*E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, I*E^(I*(a + b*x))] - (6*I)*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, I*E^(I*(a + b*x))]) + (4*I)*d*(b^3*(c + d*x)^3*PolyLog[2, -E^(I*(a + b*x))] + (3*I)*b^2*d*(c + d*x)^2*PolyLog[3, -E^(I*(a + b*x))] - 6*d^2*(b*(c + d*x)*PolyLog[4, -E^(I*(a + b*x))] + I*d*PolyLog[5, -E^(I*(a + b*x))])) - (4*I)*d*(b^3*(c + d*x)^3*PolyLog[2, E^(I*(a + b*x))] + (3*I)*b^2*d*(c + d*x)^2*PolyLog[3, E^(I*(a + b*x))] - 6*d^2*(b*(c + d*x)*PolyLog[4, E^(I*(a + b*x))] + I*d*PolyLog[5, E^(I*(a + b*x))])) + b^4*(c + d*x)^4*Sec[a + b*x])/b^5","A",1
267,1,473,343,1.3422259,"\int (c+d x)^3 \csc (a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]*Sec[a + b*x]^2,x]","\frac{b^3 (c+d x)^3 \sec (a+b x)-2 b^3 (c+d x)^3 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x))-3 d \left(-2 i b^2 c^2 \tan ^{-1}\left(e^{i (a+b x)}\right)+2 b^2 c d x \log \left(1-i e^{i (a+b x)}\right)-2 b^2 c d x \log \left(1+i e^{i (a+b x)}\right)+b^2 d^2 x^2 \log \left(1-i e^{i (a+b x)}\right)-b^2 d^2 x^2 \log \left(1+i e^{i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)-2 i b d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)-2 d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)+2 d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)\right)+3 i d \left(b^2 (c+d x)^2 \text{Li}_2(-\cos (a+b x)-i \sin (a+b x))+2 i b d (c+d x) \text{Li}_3(-\cos (a+b x)-i \sin (a+b x))-2 d^2 \text{Li}_4(-\cos (a+b x)-i \sin (a+b x))\right)-3 i d \left(b^2 (c+d x)^2 \text{Li}_2(\cos (a+b x)+i \sin (a+b x))+2 i b d (c+d x) \text{Li}_3(\cos (a+b x)+i \sin (a+b x))-2 d^2 \text{Li}_4(\cos (a+b x)+i \sin (a+b x))\right)}{b^4}","\frac{6 d^3 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^4}-\frac{6 d^3 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^4}-\frac{6 i d^3 \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^4}+\frac{6 i d^3 \text{Li}_4\left(e^{i (a+b x)}\right)}{b^4}-\frac{6 i d^2 (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{6 i d^2 (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}-\frac{6 d^2 (c+d x) \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{6 d^2 (c+d x) \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}+\frac{6 i d (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}+\frac{(c+d x)^3 \sec (a+b x)}{b}-\frac{2 (c+d x)^3 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(-2*b^3*(c + d*x)^3*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]] - 3*d*((-2*I)*b^2*c^2*ArcTan[E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 - I*E^(I*(a + b*x))] + b^2*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] - 2*b^2*c*d*x*Log[1 + I*E^(I*(a + b*x))] - b^2*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] - 2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] + 2*d^2*PolyLog[3, I*E^(I*(a + b*x))]) + (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, -Cos[a + b*x] - I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]] - 2*d^2*PolyLog[4, -Cos[a + b*x] - I*Sin[a + b*x]]) - (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]] - 2*d^2*PolyLog[4, Cos[a + b*x] + I*Sin[a + b*x]]) + b^3*(c + d*x)^3*Sec[a + b*x])/b^4","A",0
268,1,317,219,2.4723955,"\int (c+d x)^2 \csc (a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]*Sec[a + b*x]^2,x]","\frac{b^2 (c+d x)^2 \log \left(1-e^{i (a+b x)}\right)-b^2 (c+d x)^2 \log \left(1+e^{i (a+b x)}\right)+b^2 (c+d x)^2 \sec (a+b x)+2 i d \left(b (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)+i d \text{Li}_3\left(-e^{i (a+b x)}\right)\right)+2 d \left(d \text{Li}_3\left(e^{i (a+b x)}\right)-i b (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)\right)-4 b c d \tanh ^{-1}\left(\cos (a) \tan \left(\frac{b x}{2}\right)+\sin (a)\right)+\frac{2 d^2 \csc (a) \left(i \text{Li}_2\left(-e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-i \text{Li}_2\left(e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\left(b x-\tan ^{-1}(\cot (a))\right) \left(\log \left(1-e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-\log \left(1+e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)\right)}{\sqrt{\csc ^2(a)}}-4 d^2 \tan ^{-1}(\cot (a)) \tanh ^{-1}\left(\cos (a) \tan \left(\frac{b x}{2}\right)+\sin (a)\right)}{b^3}","-\frac{2 i d^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{2 i d^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}-\frac{2 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{2 d^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}+\frac{2 i d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{2 i d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}+\frac{4 i d (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}+\frac{(c+d x)^2 \sec (a+b x)}{b}-\frac{2 (c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(-4*b*c*d*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] - 4*d^2*ArcTan[Cot[a]]*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] + b^2*(c + d*x)^2*Log[1 - E^(I*(a + b*x))] - b^2*(c + d*x)^2*Log[1 + E^(I*(a + b*x))] + (2*d^2*Csc[a]*((b*x - ArcTan[Cot[a]])*(Log[1 - E^(I*(b*x - ArcTan[Cot[a]]))] - Log[1 + E^(I*(b*x - ArcTan[Cot[a]]))]) + I*PolyLog[2, -E^(I*(b*x - ArcTan[Cot[a]]))] - I*PolyLog[2, E^(I*(b*x - ArcTan[Cot[a]]))]))/Sqrt[Csc[a]^2] + (2*I)*d*(b*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] + I*d*PolyLog[3, -E^(I*(a + b*x))]) + 2*d*((-I)*b*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] + d*PolyLog[3, E^(I*(a + b*x))]) + b^2*(c + d*x)^2*Sec[a + b*x])/b^3","A",0
269,1,212,113,0.4850535,"\int (c+d x) \csc (a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]*Sec[a + b*x]^2,x]","\frac{d \left(i \left(\text{Li}_2\left(-e^{i (a+b x)}\right)-\text{Li}_2\left(e^{i (a+b x)}\right)\right)+(a+b x) \left(\log \left(1-e^{i (a+b x)}\right)-\log \left(1+e^{i (a+b x)}\right)\right)\right)}{b^2}-\frac{a d \log \left(\tan \left(\frac{1}{2} (a+b x)\right)\right)}{b^2}+\frac{d \log \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b^2}-\frac{d \log \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b^2}+\frac{c \sec (a+b x)}{b}+\frac{c \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{c \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}+\frac{d x \sec (a+b x)}{b}","\frac{i d \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{i d \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{d \tanh ^{-1}(\sin (a+b x))}{b^2}+\frac{c \sec (a+b x)}{b}-\frac{c \tanh ^{-1}(\cos (a+b x))}{b}+\frac{d x \sec (a+b x)}{b}-\frac{2 d x \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"-((c*Log[Cos[(a + b*x)/2]])/b) + (d*Log[Cos[(a + b*x)/2] - Sin[(a + b*x)/2]])/b^2 + (c*Log[Sin[(a + b*x)/2]])/b - (d*Log[Cos[(a + b*x)/2] + Sin[(a + b*x)/2]])/b^2 - (a*d*Log[Tan[(a + b*x)/2]])/b^2 + (d*((a + b*x)*(Log[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(a + b*x))]) + I*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^(I*(a + b*x))])))/b^2 + (c*Sec[a + b*x])/b + (d*x*Sec[a + b*x])/b","A",1
270,0,0,25,9.6515492,"\int \frac{\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx","Integrate[(Csc[a + b*x]*Sec[a + b*x]^2)/(c + d*x),x]","\int \frac{\csc (a+b x) \sec ^2(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\csc (a+b x) \sec ^2(a+b x)}{c+d x},x\right)",0,"Integrate[(Csc[a + b*x]*Sec[a + b*x]^2)/(c + d*x), x]","A",-1
271,0,0,25,9.7577328,"\int \frac{\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[(Csc[a + b*x]*Sec[a + b*x]^2)/(c + d*x)^2,x]","\int \frac{\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\csc (a+b x) \sec ^2(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Csc[a + b*x]*Sec[a + b*x]^2)/(c + d*x)^2, x]","A",-1
272,0,0,27,2.6332404,"\int (c+d x)^m \csc ^2(a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Csc[a + b*x]^2*Sec[a + b*x]^2,x]","\int (c+d x)^m \csc ^2(a+b x) \sec ^2(a+b x) \, dx","\text{Int}\left(\csc ^2(a+b x) \sec ^2(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Csc[a + b*x]^2*Sec[a + b*x]^2, x]","A",-1
273,1,285,118,2.1306185,"\int (c+d x)^3 \csc ^2(a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x]^2,x]","\frac{-\frac{8 i b^3 (c+d x)^3}{-1+e^{4 i a}}+4 b^3 \csc (2 a) \sin (2 b x) (c+d x)^3 \csc (2 (a+b x))+6 b^2 d (c+d x)^2 \log \left(1-e^{-i (a+b x)}\right)+6 b^2 d (c+d x)^2 \log \left(1+e^{-i (a+b x)}\right)+6 b^2 d (c+d x)^2 \log \left(1+e^{-2 i (a+b x)}\right)+12 i b d^2 (c+d x) \text{Li}_2\left(-e^{-i (a+b x)}\right)+12 i b d^2 (c+d x) \text{Li}_2\left(e^{-i (a+b x)}\right)+6 i b d^2 (c+d x) \text{Li}_2\left(-e^{-2 i (a+b x)}\right)+12 d^3 \text{Li}_3\left(-e^{-i (a+b x)}\right)+12 d^3 \text{Li}_3\left(e^{-i (a+b x)}\right)+3 d^3 \text{Li}_3\left(-e^{-2 i (a+b x)}\right)}{2 b^4}","\frac{3 d^3 \text{Li}_3\left(e^{4 i (a+b x)}\right)}{8 b^4}-\frac{3 i d^2 (c+d x) \text{Li}_2\left(e^{4 i (a+b x)}\right)}{2 b^3}+\frac{3 d (c+d x)^2 \log \left(1-e^{4 i (a+b x)}\right)}{b^2}-\frac{2 (c+d x)^3 \cot (2 a+2 b x)}{b}-\frac{2 i (c+d x)^3}{b}",1,"(((-8*I)*b^3*(c + d*x)^3)/(-1 + E^((4*I)*a)) + 6*b^2*d*(c + d*x)^2*Log[1 - E^((-I)*(a + b*x))] + 6*b^2*d*(c + d*x)^2*Log[1 + E^((-I)*(a + b*x))] + 6*b^2*d*(c + d*x)^2*Log[1 + E^((-2*I)*(a + b*x))] + (12*I)*b*d^2*(c + d*x)*PolyLog[2, -E^((-I)*(a + b*x))] + (12*I)*b*d^2*(c + d*x)*PolyLog[2, E^((-I)*(a + b*x))] + (6*I)*b*d^2*(c + d*x)*PolyLog[2, -E^((-2*I)*(a + b*x))] + 12*d^3*PolyLog[3, -E^((-I)*(a + b*x))] + 12*d^3*PolyLog[3, E^((-I)*(a + b*x))] + 3*d^3*PolyLog[3, -E^((-2*I)*(a + b*x))] + 4*b^3*(c + d*x)^3*Csc[2*a]*Csc[2*(a + b*x)]*Sin[2*b*x])/(2*b^4)","B",1
274,1,277,88,1.71606,"\int (c+d x)^2 \csc ^2(a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]^2*Sec[a + b*x]^2,x]","\frac{2 b^2 \csc (2 a) \sin (2 b x) (c+d x)^2 \csc (2 (a+b x))-\frac{i e^{4 i a} \left(4 e^{-4 i a} b^2 (c+d x)^2+2 i \left(1-e^{-4 i a}\right) b d (c+d x) \log \left(1-e^{-i (a+b x)}\right)+2 i \left(1-e^{-4 i a}\right) b d (c+d x) \log \left(1+e^{-i (a+b x)}\right)+2 i \left(1-e^{-4 i a}\right) b d (c+d x) \log \left(1+e^{-2 i (a+b x)}\right)-2 \left(1-e^{-4 i a}\right) d^2 \text{Li}_2\left(-e^{-i (a+b x)}\right)-2 \left(1-e^{-4 i a}\right) d^2 \text{Li}_2\left(e^{-i (a+b x)}\right)-\left(1-e^{-4 i a}\right) d^2 \text{Li}_2\left(-e^{-2 i (a+b x)}\right)\right)}{-1+e^{4 i a}}}{b^3}","-\frac{i d^2 \text{Li}_2\left(e^{4 i (a+b x)}\right)}{2 b^3}+\frac{2 d (c+d x) \log \left(1-e^{4 i (a+b x)}\right)}{b^2}-\frac{2 (c+d x)^2 \cot (2 a+2 b x)}{b}-\frac{2 i (c+d x)^2}{b}",1,"(((-I)*E^((4*I)*a)*((4*b^2*(c + d*x)^2)/E^((4*I)*a) + (2*I)*b*d*(1 - E^((-4*I)*a))*(c + d*x)*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b*d*(1 - E^((-4*I)*a))*(c + d*x)*Log[1 + E^((-I)*(a + b*x))] + (2*I)*b*d*(1 - E^((-4*I)*a))*(c + d*x)*Log[1 + E^((-2*I)*(a + b*x))] - 2*d^2*(1 - E^((-4*I)*a))*PolyLog[2, -E^((-I)*(a + b*x))] - 2*d^2*(1 - E^((-4*I)*a))*PolyLog[2, E^((-I)*(a + b*x))] - d^2*(1 - E^((-4*I)*a))*PolyLog[2, -E^((-2*I)*(a + b*x))]))/(-1 + E^((4*I)*a)) + 2*b^2*(c + d*x)^2*Csc[2*a]*Csc[2*(a + b*x)]*Sin[2*b*x])/b^3","B",1
275,1,32,35,0.2029541,"\int (c+d x) \csc ^2(a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]^2*Sec[a + b*x]^2,x]","\frac{d \log (\sin (2 (a+b x)))-2 b (c+d x) \cot (2 (a+b x))}{b^2}","\frac{d \log (\sin (2 a+2 b x))}{b^2}-\frac{2 (c+d x) \cot (2 a+2 b x)}{b}",1,"(-2*b*(c + d*x)*Cot[2*(a + b*x)] + d*Log[Sin[2*(a + b*x)]])/b^2","A",1
276,0,0,24,7.0649383,"\int \frac{\csc ^2(a+b x) \sec ^2(a+b x)}{c+d x} \, dx","Integrate[(Csc[a + b*x]^2*Sec[a + b*x]^2)/(c + d*x),x]","\int \frac{\csc ^2(a+b x) \sec ^2(a+b x)}{c+d x} \, dx","4 \text{Int}\left(\frac{\csc ^2(2 a+2 b x)}{c+d x},x\right)",0,"Integrate[(Csc[a + b*x]^2*Sec[a + b*x]^2)/(c + d*x), x]","A",-1
277,0,0,24,7.3263692,"\int \frac{\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[(Csc[a + b*x]^2*Sec[a + b*x]^2)/(c + d*x)^2,x]","\int \frac{\csc ^2(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx","4 \text{Int}\left(\frac{\csc ^2(2 a+2 b x)}{(c+d x)^2},x\right)",0,"Integrate[(Csc[a + b*x]^2*Sec[a + b*x]^2)/(c + d*x)^2, x]","A",-1
278,0,0,27,25.5144987,"\int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x]^2,x]","\int (c+d x)^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx","\text{Int}\left(\csc ^3(a+b x) \sec ^2(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x]^2, x]","A",-1
279,1,907,601,8.3460061,"\int (c+d x)^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]^3*Sec[a + b*x]^2,x]","-\frac{\sec (a+b x) \left(-b c^3+3 b \cos (2 a+2 b x) c^3-3 b d x c^2+9 b d x \cos (2 a+2 b x) c^2+3 d \sin (2 a+2 b x) c^2-3 b d^2 x^2 c+9 b d^2 x^2 \cos (2 a+2 b x) c+6 d^2 x \sin (2 a+2 b x) c-b d^3 x^3+3 b d^3 x^3 \cos (2 a+2 b x)+3 d^3 x^2 \sin (2 a+2 b x)\right) \csc ^2(a+b x)}{4 b^2}-\frac{3 d \left(-2 i c^2 \tan ^{-1}\left(e^{i (a+b x)}\right) b^2+d^2 x^2 \log \left(1-i e^{i (a+b x)}\right) b^2+2 c d x \log \left(1-i e^{i (a+b x)}\right) b^2-d^2 x^2 \log \left(1+i e^{i (a+b x)}\right) b^2-2 c d x \log \left(1+i e^{i (a+b x)}\right) b^2+2 i d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right) b-2 i d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right) b-2 d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)+2 d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)\right)}{b^4}+\frac{3 \left(c^3 \log \left(1-e^{i (a+b x)}\right) b^3+d^3 x^3 \log \left(1-e^{i (a+b x)}\right) b^3+3 c d^2 x^2 \log \left(1-e^{i (a+b x)}\right) b^3+3 c^2 d x \log \left(1-e^{i (a+b x)}\right) b^3-c^3 \log \left(1+e^{i (a+b x)}\right) b^3-d^3 x^3 \log \left(1+e^{i (a+b x)}\right) b^3-3 c d^2 x^2 \log \left(1+e^{i (a+b x)}\right) b^3-3 c^2 d x \log \left(1+e^{i (a+b x)}\right) b^3+2 c d^2 \log \left(1-e^{i (a+b x)}\right) b+2 d^3 x \log \left(1-e^{i (a+b x)}\right) b-2 c d^2 \log \left(1+e^{i (a+b x)}\right) b-2 d^3 x \log \left(1+e^{i (a+b x)}\right) b-6 c d^2 \text{Li}_3\left(-e^{i (a+b x)}\right) b-6 d^3 x \text{Li}_3\left(-e^{i (a+b x)}\right) b+6 c d^2 \text{Li}_3\left(e^{i (a+b x)}\right) b+6 d^3 x \text{Li}_3\left(e^{i (a+b x)}\right) b+i d \left(2 d^2+3 b^2 (c+d x)^2\right) \text{Li}_2\left(-e^{i (a+b x)}\right)-i d \left(2 d^2+3 b^2 (c+d x)^2\right) \text{Li}_2\left(e^{i (a+b x)}\right)-6 i d^3 \text{Li}_4\left(-e^{i (a+b x)}\right)+6 i d^3 \text{Li}_4\left(e^{i (a+b x)}\right)\right)}{2 b^4}","\frac{3 i d^3 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^4}-\frac{3 i d^3 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^4}+\frac{6 d^3 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^4}-\frac{6 d^3 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^4}-\frac{9 i d^3 \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^4}+\frac{9 i d^3 \text{Li}_4\left(e^{i (a+b x)}\right)}{b^4}-\frac{6 i c d^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{6 i c d^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}-\frac{9 d^2 (c+d x) \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{9 d^2 (c+d x) \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{3 c d^2 \tanh ^{-1}(\cos (a+b x))}{b^3}-\frac{6 i d^3 x \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{6 i d^3 x \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}-\frac{6 d^3 x \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^3}-\frac{3 c^2 d \csc (a+b x)}{2 b^2}-\frac{3 c^2 d \tanh ^{-1}(\sin (a+b x))}{b^2}+\frac{12 i c d^2 x \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{3 c d^2 x \csc (a+b x)}{b^2}+\frac{9 i d (c+d x)^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{2 b^2}-\frac{9 i d (c+d x)^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{2 b^2}+\frac{6 i d^3 x^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{3 d^3 x^2 \csc (a+b x)}{2 b^2}+\frac{3 (c+d x)^3 \sec (a+b x)}{2 b}-\frac{3 (c+d x)^3 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{(c+d x)^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}",1,"(-3*d*((-2*I)*b^2*c^2*ArcTan[E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 - I*E^(I*(a + b*x))] + b^2*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] - 2*b^2*c*d*x*Log[1 + I*E^(I*(a + b*x))] - b^2*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] - 2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] + 2*d^2*PolyLog[3, I*E^(I*(a + b*x))]))/b^4 + (3*(b^3*c^3*Log[1 - E^(I*(a + b*x))] + 2*b*c*d^2*Log[1 - E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 - E^(I*(a + b*x))] + 2*b*d^3*x*Log[1 - E^(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 - E^(I*(a + b*x))] + b^3*d^3*x^3*Log[1 - E^(I*(a + b*x))] - b^3*c^3*Log[1 + E^(I*(a + b*x))] - 2*b*c*d^2*Log[1 + E^(I*(a + b*x))] - 3*b^3*c^2*d*x*Log[1 + E^(I*(a + b*x))] - 2*b*d^3*x*Log[1 + E^(I*(a + b*x))] - 3*b^3*c*d^2*x^2*Log[1 + E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 + E^(I*(a + b*x))] + I*d*(2*d^2 + 3*b^2*(c + d*x)^2)*PolyLog[2, -E^(I*(a + b*x))] - I*d*(2*d^2 + 3*b^2*(c + d*x)^2)*PolyLog[2, E^(I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, -E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, -E^(I*(a + b*x))] + 6*b*c*d^2*PolyLog[3, E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, E^(I*(a + b*x))] - (6*I)*d^3*PolyLog[4, -E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, E^(I*(a + b*x))]))/(2*b^4) - (Csc[a + b*x]^2*Sec[a + b*x]*(-(b*c^3) - 3*b*c^2*d*x - 3*b*c*d^2*x^2 - b*d^3*x^3 + 3*b*c^3*Cos[2*a + 2*b*x] + 9*b*c^2*d*x*Cos[2*a + 2*b*x] + 9*b*c*d^2*x^2*Cos[2*a + 2*b*x] + 3*b*d^3*x^3*Cos[2*a + 2*b*x] + 3*c^2*d*Sin[2*a + 2*b*x] + 6*c*d^2*x*Sin[2*a + 2*b*x] + 3*d^3*x^2*Sin[2*a + 2*b*x]))/(4*b^2)","A",1
280,1,889,305,7.9712248,"\int (c+d x)^2 \csc ^3(a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]^3*Sec[a + b*x]^2,x]","-\frac{2 \left(\frac{2 \tan ^{-1}(\cot (a)) \tanh ^{-1}\left(\frac{\sin (a)+\cos (a) \tan \left(\frac{b x}{2}\right)}{\sqrt{\cos ^2(a)+\sin ^2(a)}}\right)}{\sqrt{\cos ^2(a)+\sin ^2(a)}}-\frac{\csc (a) \left(\left(b x-\tan ^{-1}(\cot (a))\right) \left(\log \left(1-e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-\log \left(1+e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)+i \left(\text{Li}_2\left(-e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-\text{Li}_2\left(e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)\right)}{\sqrt{\cot ^2(a)+1}}\right) d^2}{b^3}-\frac{4 i c \tan ^{-1}\left(\frac{-i \sin (a)-i \cos (a) \tan \left(\frac{b x}{2}\right)}{\sqrt{\cos ^2(a)+\sin ^2(a)}}\right) d}{b^2 \sqrt{\cos ^2(a)+\sin ^2(a)}}+\frac{\left(-c^2-2 d x c-d^2 x^2\right) \csc ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}+\frac{\left(c^2+2 d x c+d^2 x^2\right) \sec ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}+\frac{3 c^2 \log \left(1-e^{i (a+b x)}\right) b^2+3 d^2 x^2 \log \left(1-e^{i (a+b x)}\right) b^2+6 c d x \log \left(1-e^{i (a+b x)}\right) b^2-3 c^2 \log \left(1+e^{i (a+b x)}\right) b^2-3 d^2 x^2 \log \left(1+e^{i (a+b x)}\right) b^2-6 c d x \log \left(1+e^{i (a+b x)}\right) b^2+6 i d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right) b-6 i d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right) b+2 d^2 \log \left(1-e^{i (a+b x)}\right)-2 d^2 \log \left(1+e^{i (a+b x)}\right)-6 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)+6 d^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{2 b^3}+\frac{(c+d x) \csc (a) \sec (a) (-d \cos (a)+b c \sin (a)+b d x \sin (a))}{b^2}+\frac{\sec \left(\frac{a}{2}\right) \sec \left(\frac{a}{2}+\frac{b x}{2}\right) \left(-x \sin \left(\frac{b x}{2}\right) d^2-c \sin \left(\frac{b x}{2}\right) d\right)}{2 b^2}+\frac{\csc \left(\frac{a}{2}\right) \csc \left(\frac{a}{2}+\frac{b x}{2}\right) \left(x \sin \left(\frac{b x}{2}\right) d^2+c \sin \left(\frac{b x}{2}\right) d\right)}{2 b^2}+\frac{\sin \left(\frac{b x}{2}\right) c^2+2 d x \sin \left(\frac{b x}{2}\right) c+d^2 x^2 \sin \left(\frac{b x}{2}\right)}{b \left(\cos \left(\frac{a}{2}\right)-\sin \left(\frac{a}{2}\right)\right) \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)-\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}+\frac{-\sin \left(\frac{b x}{2}\right) c^2-2 d x \sin \left(\frac{b x}{2}\right) c-d^2 x^2 \sin \left(\frac{b x}{2}\right)}{b \left(\cos \left(\frac{a}{2}\right)+\sin \left(\frac{a}{2}\right)\right) \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)+\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}","-\frac{2 i d^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{2 i d^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}-\frac{3 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{3 d^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{d^2 \tanh ^{-1}(\cos (a+b x))}{b^3}+\frac{3 i d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{3 i d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{c d \csc (a+b x)}{b^2}-\frac{2 c d \tanh ^{-1}(\sin (a+b x))}{b^2}+\frac{4 i d^2 x \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{d^2 x \csc (a+b x)}{b^2}+\frac{3 (c+d x)^2 \sec (a+b x)}{2 b}-\frac{3 (c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{(c+d x)^2 \csc ^2(a+b x) \sec (a+b x)}{2 b}",1,"((-c^2 - 2*c*d*x - d^2*x^2)*Csc[a/2 + (b*x)/2]^2)/(8*b) + (3*b^2*c^2*Log[1 - E^(I*(a + b*x))] + 2*d^2*Log[1 - E^(I*(a + b*x))] + 6*b^2*c*d*x*Log[1 - E^(I*(a + b*x))] + 3*b^2*d^2*x^2*Log[1 - E^(I*(a + b*x))] - 3*b^2*c^2*Log[1 + E^(I*(a + b*x))] - 2*d^2*Log[1 + E^(I*(a + b*x))] - 6*b^2*c*d*x*Log[1 + E^(I*(a + b*x))] - 3*b^2*d^2*x^2*Log[1 + E^(I*(a + b*x))] + (6*I)*b*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] - (6*I)*b*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] - 6*d^2*PolyLog[3, -E^(I*(a + b*x))] + 6*d^2*PolyLog[3, E^(I*(a + b*x))])/(2*b^3) + ((c^2 + 2*c*d*x + d^2*x^2)*Sec[a/2 + (b*x)/2]^2)/(8*b) + ((c + d*x)*Csc[a]*Sec[a]*(-(d*Cos[a]) + b*c*Sin[a] + b*d*x*Sin[a]))/b^2 - ((4*I)*c*d*ArcTan[((-I)*Sin[a] - I*Cos[a]*Tan[(b*x)/2])/Sqrt[Cos[a]^2 + Sin[a]^2]])/(b^2*Sqrt[Cos[a]^2 + Sin[a]^2]) - (2*d^2*(-((Csc[a]*((b*x - ArcTan[Cot[a]])*(Log[1 - E^(I*(b*x - ArcTan[Cot[a]]))] - Log[1 + E^(I*(b*x - ArcTan[Cot[a]]))]) + I*(PolyLog[2, -E^(I*(b*x - ArcTan[Cot[a]]))] - PolyLog[2, E^(I*(b*x - ArcTan[Cot[a]]))])))/Sqrt[1 + Cot[a]^2]) + (2*ArcTan[Cot[a]]*ArcTanh[(Sin[a] + Cos[a]*Tan[(b*x)/2])/Sqrt[Cos[a]^2 + Sin[a]^2]])/Sqrt[Cos[a]^2 + Sin[a]^2]))/b^3 + (Sec[a/2]*Sec[a/2 + (b*x)/2]*(-(c*d*Sin[(b*x)/2]) - d^2*x*Sin[(b*x)/2]))/(2*b^2) + (Csc[a/2]*Csc[a/2 + (b*x)/2]*(c*d*Sin[(b*x)/2] + d^2*x*Sin[(b*x)/2]))/(2*b^2) + (c^2*Sin[(b*x)/2] + 2*c*d*x*Sin[(b*x)/2] + d^2*x^2*Sin[(b*x)/2])/(b*(Cos[a/2] - Sin[a/2])*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])) + (-(c^2*Sin[(b*x)/2]) - 2*c*d*x*Sin[(b*x)/2] - d^2*x^2*Sin[(b*x)/2])/(b*(Cos[a/2] + Sin[a/2])*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2]))","B",0
281,1,520,154,4.9230339,"\int (c+d x) \csc ^3(a+b x) \sec ^2(a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]^3*Sec[a + b*x]^2,x]","\frac{3 d \left(i \left(\text{Li}_2\left(-e^{i (a+b x)}\right)-\text{Li}_2\left(e^{i (a+b x)}\right)\right)+(a+b x) \left(\log \left(1-e^{i (a+b x)}\right)-\log \left(1+e^{i (a+b x)}\right)\right)\right)}{2 b^2}-\frac{d \tan \left(\frac{1}{2} (a+b x)\right)}{4 b^2}-\frac{d \cot \left(\frac{1}{2} (a+b x)\right)}{4 b^2}-\frac{3 a d \log \left(\tan \left(\frac{1}{2} (a+b x)\right)\right)}{2 b^2}+\frac{d \left(a \sin \left(\frac{1}{2} (a+b x)\right)-(a+b x) \sin \left(\frac{1}{2} (a+b x)\right)\right)}{b^2 \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)}+\frac{d \left((a+b x) \sin \left(\frac{1}{2} (a+b x)\right)-a \sin \left(\frac{1}{2} (a+b x)\right)\right)}{b^2 \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)}+\frac{d \log \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b^2}-\frac{d \log \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b^2}-\frac{c \csc ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{c \sec ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{3 c \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}-\frac{3 c \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}+\frac{c \sin \left(\frac{1}{2} (a+b x)\right)}{b \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)}-\frac{c \sin \left(\frac{1}{2} (a+b x)\right)}{b \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)}-\frac{d x \csc ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{d x \sec ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{d x}{b}","\frac{3 i d \text{Li}_2\left(-e^{i (a+b x)}\right)}{2 b^2}-\frac{3 i d \text{Li}_2\left(e^{i (a+b x)}\right)}{2 b^2}-\frac{d \csc (a+b x)}{2 b^2}-\frac{d \tanh ^{-1}(\sin (a+b x))}{b^2}+\frac{3 (c+d x) \sec (a+b x)}{2 b}-\frac{(c+d x) \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac{3 c \tanh ^{-1}(\cos (a+b x))}{2 b}-\frac{3 d x \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(d*x)/b - (d*Cot[(a + b*x)/2])/(4*b^2) - (c*Csc[(a + b*x)/2]^2)/(8*b) - (d*x*Csc[(a + b*x)/2]^2)/(8*b) - (3*c*Log[Cos[(a + b*x)/2]])/(2*b) + (d*Log[Cos[(a + b*x)/2] - Sin[(a + b*x)/2]])/b^2 + (3*c*Log[Sin[(a + b*x)/2]])/(2*b) - (d*Log[Cos[(a + b*x)/2] + Sin[(a + b*x)/2]])/b^2 - (3*a*d*Log[Tan[(a + b*x)/2]])/(2*b^2) + (3*d*((a + b*x)*(Log[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(a + b*x))]) + I*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^(I*(a + b*x))])))/(2*b^2) + (c*Sec[(a + b*x)/2]^2)/(8*b) + (d*x*Sec[(a + b*x)/2]^2)/(8*b) + (c*Sin[(a + b*x)/2])/(b*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])) - (c*Sin[(a + b*x)/2])/(b*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])) + (d*(a*Sin[(a + b*x)/2] - (a + b*x)*Sin[(a + b*x)/2]))/(b^2*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])) + (d*(-(a*Sin[(a + b*x)/2]) + (a + b*x)*Sin[(a + b*x)/2]))/(b^2*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])) - (d*Tan[(a + b*x)/2])/(4*b^2)","B",1
282,0,0,27,22.2192414,"\int \frac{\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx","Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/(c + d*x),x]","\int \frac{\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\csc ^3(a+b x) \sec ^2(a+b x)}{c+d x},x\right)",0,"Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/(c + d*x), x]","A",-1
283,0,0,27,30.3700171,"\int \frac{\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/(c + d*x)^2,x]","\int \frac{\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\csc ^3(a+b x) \sec ^2(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/(c + d*x)^2, x]","A",-1
284,0,0,23,39.6513465,"\int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx","Integrate[x^m*Csc[a + b*x]^3*Sec[a + b*x]^2,x]","\int x^m \csc ^3(a+b x) \sec ^2(a+b x) \, dx","\text{Int}\left(x^m \csc ^3(a+b x) \sec ^2(a+b x),x\right)",0,"Integrate[x^m*Csc[a + b*x]^3*Sec[a + b*x]^2, x]","A",-1
285,1,672,387,7.2096962,"\int x^3 \csc ^3(a+b x) \sec ^2(a+b x) \, dx","Integrate[x^3*Csc[a + b*x]^3*Sec[a + b*x]^2,x]","\frac{3 x^2 \csc \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) \csc \left(\frac{a}{2}+\frac{b x}{2}\right)}{4 b^2}-\frac{3 x^2 \sec \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) \sec \left(\frac{a}{2}+\frac{b x}{2}\right)}{4 b^2}+\frac{x^2 \csc (a) \sec (a) (2 b x \sin (a)-3 \cos (a))}{2 b^2}+\frac{6 \left(i b^2 x^2 \tan ^{-1}(\cos (a+b x)+i \sin (a+b x))+i b x \text{Li}_2(i \cos (a+b x)-\sin (a+b x))-i b x \text{Li}_2(\sin (a+b x)-i \cos (a+b x))-\text{Li}_3(i \cos (a+b x)-\sin (a+b x))+\text{Li}_3(\sin (a+b x)-i \cos (a+b x))\right)}{b^4}+\frac{3 \left(b^3 x^3 \log (-i \sin (a+b x)-\cos (a+b x)+1)-b^3 x^3 \log (i \sin (a+b x)+\cos (a+b x)+1)+i \left(3 b^2 x^2+2\right) \text{Li}_2(-\cos (a+b x)-i \sin (a+b x))-i \left(3 b^2 x^2+2\right) \text{Li}_2(\cos (a+b x)+i \sin (a+b x))-6 b x \text{Li}_3(-\cos (a+b x)-i \sin (a+b x))+6 b x \text{Li}_3(\cos (a+b x)+i \sin (a+b x))-6 i \text{Li}_4(-\cos (a+b x)-i \sin (a+b x))+6 i \text{Li}_4(\cos (a+b x)+i \sin (a+b x))+2 b x \log (-i \sin (a+b x)-\cos (a+b x)+1)-2 b x \log (i \sin (a+b x)+\cos (a+b x)+1)\right)}{2 b^4}-\frac{x^3 \csc ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}+\frac{x^3 \sec ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}+\frac{x^3 \sin \left(\frac{b x}{2}\right)}{b \left(\cos \left(\frac{a}{2}\right)-\sin \left(\frac{a}{2}\right)\right) \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)-\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}-\frac{x^3 \sin \left(\frac{b x}{2}\right)}{b \left(\sin \left(\frac{a}{2}\right)+\cos \left(\frac{a}{2}\right)\right) \left(\sin \left(\frac{a}{2}+\frac{b x}{2}\right)+\cos \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}","\frac{3 i \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^4}-\frac{3 i \text{Li}_2\left(e^{i (a+b x)}\right)}{b^4}+\frac{6 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^4}-\frac{6 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^4}-\frac{9 i \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^4}+\frac{9 i \text{Li}_4\left(e^{i (a+b x)}\right)}{b^4}-\frac{6 i x \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{6 i x \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}-\frac{9 x \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{9 x \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{6 x \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^3}+\frac{9 i x^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{2 b^2}-\frac{9 i x^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{2 b^2}+\frac{6 i x^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{3 x^2 \csc (a+b x)}{2 b^2}+\frac{3 x^3 \sec (a+b x)}{2 b}-\frac{3 x^3 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{x^3 \csc ^2(a+b x) \sec (a+b x)}{2 b}",1,"-1/8*(x^3*Csc[a/2 + (b*x)/2]^2)/b + (6*(I*b^2*x^2*ArcTan[Cos[a + b*x] + I*Sin[a + b*x]] + I*b*x*PolyLog[2, I*Cos[a + b*x] - Sin[a + b*x]] - I*b*x*PolyLog[2, (-I)*Cos[a + b*x] + Sin[a + b*x]] - PolyLog[3, I*Cos[a + b*x] - Sin[a + b*x]] + PolyLog[3, (-I)*Cos[a + b*x] + Sin[a + b*x]]))/b^4 + (3*(2*b*x*Log[1 - Cos[a + b*x] - I*Sin[a + b*x]] + b^3*x^3*Log[1 - Cos[a + b*x] - I*Sin[a + b*x]] - 2*b*x*Log[1 + Cos[a + b*x] + I*Sin[a + b*x]] - b^3*x^3*Log[1 + Cos[a + b*x] + I*Sin[a + b*x]] + I*(2 + 3*b^2*x^2)*PolyLog[2, -Cos[a + b*x] - I*Sin[a + b*x]] - I*(2 + 3*b^2*x^2)*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]] - 6*b*x*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]] + 6*b*x*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]] - (6*I)*PolyLog[4, -Cos[a + b*x] - I*Sin[a + b*x]] + (6*I)*PolyLog[4, Cos[a + b*x] + I*Sin[a + b*x]]))/(2*b^4) + (x^3*Sec[a/2 + (b*x)/2]^2)/(8*b) + (x^2*Csc[a]*Sec[a]*(-3*Cos[a] + 2*b*x*Sin[a]))/(2*b^2) + (3*x^2*Csc[a/2]*Csc[a/2 + (b*x)/2]*Sin[(b*x)/2])/(4*b^2) - (3*x^2*Sec[a/2]*Sec[a/2 + (b*x)/2]*Sin[(b*x)/2])/(4*b^2) + (x^3*Sin[(b*x)/2])/(b*(Cos[a/2] - Sin[a/2])*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])) - (x^3*Sin[(b*x)/2])/(b*(Cos[a/2] + Sin[a/2])*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2]))","A",0
286,1,613,235,6.6233749,"\int x^2 \csc ^3(a+b x) \sec ^2(a+b x) \, dx","Integrate[x^2*Csc[a + b*x]^3*Sec[a + b*x]^2,x]","-\frac{2 \left(i \left(\text{Li}_2\left(-e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)-\text{Li}_2\left(e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)\right)+\left(-a-b x+\frac{\pi }{2}\right) \left(\log \left(1-e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)-\log \left(1+e^{i \left(-a-b x+\frac{\pi }{2}\right)}\right)\right)-\left(\frac{\pi }{2}-a\right) \log \left(\tan \left(\frac{1}{2} \left(-a-b x+\frac{\pi }{2}\right)\right)\right)\right)}{b^3}+\frac{x \csc \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) \csc \left(\frac{a}{2}+\frac{b x}{2}\right)}{2 b^2}-\frac{x \sec \left(\frac{a}{2}\right) \sin \left(\frac{b x}{2}\right) \sec \left(\frac{a}{2}+\frac{b x}{2}\right)}{2 b^2}+\frac{x \csc (a) \sec (a) (b x \sin (a)-\cos (a))}{b^2}+\frac{3 b^2 x^2 \log (-i \sin (a+b x)-\cos (a+b x)+1)-3 b^2 x^2 \log (i \sin (a+b x)+\cos (a+b x)+1)+6 i b x \text{Li}_2(-\cos (a+b x)-i \sin (a+b x))-6 i b x \text{Li}_2(\cos (a+b x)+i \sin (a+b x))-6 \text{Li}_3(-\cos (a+b x)-i \sin (a+b x))+6 \text{Li}_3(\cos (a+b x)+i \sin (a+b x))+2 \log (-i \sin (a+b x)-\cos (a+b x)+1)-2 \log (i \sin (a+b x)+\cos (a+b x)+1)}{2 b^3}-\frac{x^2 \csc ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}+\frac{x^2 \sec ^2\left(\frac{a}{2}+\frac{b x}{2}\right)}{8 b}+\frac{x^2 \sin \left(\frac{b x}{2}\right)}{b \left(\cos \left(\frac{a}{2}\right)-\sin \left(\frac{a}{2}\right)\right) \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)-\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}-\frac{x^2 \sin \left(\frac{b x}{2}\right)}{b \left(\sin \left(\frac{a}{2}\right)+\cos \left(\frac{a}{2}\right)\right) \left(\sin \left(\frac{a}{2}+\frac{b x}{2}\right)+\cos \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}","-\frac{2 i \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{2 i \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}-\frac{3 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{3 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{\tanh ^{-1}(\cos (a+b x))}{b^3}+\frac{3 i x \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{3 i x \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}+\frac{4 i x \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{x \csc (a+b x)}{b^2}+\frac{3 x^2 \sec (a+b x)}{2 b}-\frac{3 x^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{x^2 \csc ^2(a+b x) \sec (a+b x)}{2 b}",1,"-1/8*(x^2*Csc[a/2 + (b*x)/2]^2)/b - (2*((-a + Pi/2 - b*x)*(Log[1 - E^(I*(-a + Pi/2 - b*x))] - Log[1 + E^(I*(-a + Pi/2 - b*x))]) - (-a + Pi/2)*Log[Tan[(-a + Pi/2 - b*x)/2]] + I*(PolyLog[2, -E^(I*(-a + Pi/2 - b*x))] - PolyLog[2, E^(I*(-a + Pi/2 - b*x))])))/b^3 + (2*Log[1 - Cos[a + b*x] - I*Sin[a + b*x]] + 3*b^2*x^2*Log[1 - Cos[a + b*x] - I*Sin[a + b*x]] - 2*Log[1 + Cos[a + b*x] + I*Sin[a + b*x]] - 3*b^2*x^2*Log[1 + Cos[a + b*x] + I*Sin[a + b*x]] + (6*I)*b*x*PolyLog[2, -Cos[a + b*x] - I*Sin[a + b*x]] - (6*I)*b*x*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]] - 6*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]] + 6*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]])/(2*b^3) + (x^2*Sec[a/2 + (b*x)/2]^2)/(8*b) + (x*Csc[a]*Sec[a]*(-Cos[a] + b*x*Sin[a]))/b^2 + (x*Csc[a/2]*Csc[a/2 + (b*x)/2]*Sin[(b*x)/2])/(2*b^2) - (x*Sec[a/2]*Sec[a/2 + (b*x)/2]*Sin[(b*x)/2])/(2*b^2) + (x^2*Sin[(b*x)/2])/(b*(Cos[a/2] - Sin[a/2])*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])) - (x^2*Sin[(b*x)/2])/(b*(Cos[a/2] + Sin[a/2])*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2]))","B",1
287,1,282,126,2.4094369,"\int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx","Integrate[x*Csc[a + b*x]^3*Sec[a + b*x]^2,x]","\frac{12 i \left(\text{Li}_2\left(-e^{i (a+b x)}\right)-\text{Li}_2\left(e^{i (a+b x)}\right)\right)+12 (a+b x) \left(\log \left(1-e^{i (a+b x)}\right)-\log \left(1+e^{i (a+b x)}\right)\right)-2 \tan \left(\frac{1}{2} (a+b x)\right)-2 \cot \left(\frac{1}{2} (a+b x)\right)-b x \csc ^2\left(\frac{1}{2} (a+b x)\right)+b x \sec ^2\left(\frac{1}{2} (a+b x)\right)-12 a \log \left(\tan \left(\frac{1}{2} (a+b x)\right)\right)+\frac{8 b x \sin \left(\frac{1}{2} (a+b x)\right)}{\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)}-\frac{8 b x \sin \left(\frac{1}{2} (a+b x)\right)}{\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)}+8 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)-8 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)+8 b x}{8 b^2}","\frac{3 i \text{Li}_2\left(-e^{i (a+b x)}\right)}{2 b^2}-\frac{3 i \text{Li}_2\left(e^{i (a+b x)}\right)}{2 b^2}-\frac{\csc (a+b x)}{2 b^2}-\frac{\tanh ^{-1}(\sin (a+b x))}{b^2}+\frac{3 x \sec (a+b x)}{2 b}-\frac{3 x \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{x \csc ^2(a+b x) \sec (a+b x)}{2 b}",1,"(8*b*x - 2*Cot[(a + b*x)/2] - b*x*Csc[(a + b*x)/2]^2 + 12*(a + b*x)*(Log[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(a + b*x))]) + 8*Log[Cos[(a + b*x)/2] - Sin[(a + b*x)/2]] - 8*Log[Cos[(a + b*x)/2] + Sin[(a + b*x)/2]] - 12*a*Log[Tan[(a + b*x)/2]] + (12*I)*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^(I*(a + b*x))]) + b*x*Sec[(a + b*x)/2]^2 + (8*b*x*Sin[(a + b*x)/2])/(Cos[(a + b*x)/2] - Sin[(a + b*x)/2]) - (8*b*x*Sin[(a + b*x)/2])/(Cos[(a + b*x)/2] + Sin[(a + b*x)/2]) - 2*Tan[(a + b*x)/2])/(8*b^2)","B",1
288,0,0,23,47.0088859,"\int \frac{\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx","Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/x,x]","\int \frac{\csc ^3(a+b x) \sec ^2(a+b x)}{x} \, dx","\text{Int}\left(\frac{\csc ^3(a+b x) \sec ^2(a+b x)}{x},x\right)",0,"Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/x, x]","A",-1
289,0,0,23,19.7837863,"\int \frac{\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx","Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/x^2,x]","\int \frac{\csc ^3(a+b x) \sec ^2(a+b x)}{x^2} \, dx","\text{Int}\left(\frac{\csc ^3(a+b x) \sec ^2(a+b x)}{x^2},x\right)",0,"Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/x^2, x]","A",-1
290,0,0,25,5.4053581,"\int (c+d x)^m \sec ^2(a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^m*Sec[a + b*x]^2*Tan[a + b*x],x]","\int (c+d x)^m \sec ^2(a+b x) \tan (a+b x) \, dx","\text{Int}\left(\tan (a+b x) \sec ^2(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Sec[a + b*x]^2*Tan[a + b*x], x]","A",-1
291,1,418,139,6.5608939,"\int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^4*Sec[a + b*x]^2*Tan[a + b*x],x]","-\frac{6 c^2 d^2 \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b^3 \left(\sin ^2(a)+\cos ^2(a)\right)}-\frac{2 \sec (a) \sec (a+b x) \left(c^3 d \sin (b x)+3 c^2 d^2 x \sin (b x)+3 c d^3 x^2 \sin (b x)+d^4 x^3 \sin (b x)\right)}{b^2}-\frac{i e^{-i a} d^4 \sec (a) \left(2 b^2 x^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right)+6 \left(1+e^{2 i a}\right) b x \text{Li}_2\left(-e^{-2 i (a+b x)}\right)-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right)}{2 b^5}-\frac{6 c d^3 \csc (a) \sec (a) \left(b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left(i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\cot ^2(a)+1}}\right)}{b^4 \sqrt{\csc ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}+\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b}","-\frac{3 d^4 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{b^5}+\frac{6 i d^3 (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^4}-\frac{6 d^2 (c+d x)^2 \log \left(1+e^{2 i (a+b x)}\right)}{b^3}-\frac{2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac{(c+d x)^4 \sec ^2(a+b x)}{2 b}+\frac{2 i d (c+d x)^3}{b^2}",1,"((-1/2*I)*d^4*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^5*E^(I*a)) + ((c + d*x)^4*Sec[a + b*x]^2)/(2*b) - (6*c^2*d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) - (6*c*d^3*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^4*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (2*Sec[a]*Sec[a + b*x]*(c^3*d*Sin[b*x] + 3*c^2*d^2*x*Sin[b*x] + 3*c*d^3*x^2*Sin[b*x] + d^4*x^3*Sin[b*x]))/b^2","B",0
292,1,286,115,6.3778408,"\int (c+d x)^3 \sec ^2(a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^3*Sec[a + b*x]^2*Tan[a + b*x],x]","-\frac{3 c d^2 \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b^3 \left(\sin ^2(a)+\cos ^2(a)\right)}-\frac{3 \sec (a) \sec (a+b x) \left(c^2 d \sin (b x)+2 c d^2 x \sin (b x)+d^3 x^2 \sin (b x)\right)}{2 b^2}-\frac{3 d^3 \csc (a) \sec (a) \left(b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left(i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\cot ^2(a)+1}}\right)}{2 b^4 \sqrt{\csc ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}+\frac{(c+d x)^3 \sec ^2(a+b x)}{2 b}","\frac{3 i d^3 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 d^2 (c+d x) \log \left(1+e^{2 i (a+b x)}\right)}{b^3}-\frac{3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac{(c+d x)^3 \sec ^2(a+b x)}{2 b}+\frac{3 i d (c+d x)^2}{2 b^2}",1,"((c + d*x)^3*Sec[a + b*x]^2)/(2*b) - (3*c*d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) - (3*d^3*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(2*b^4*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (3*Sec[a]*Sec[a + b*x]*(c^2*d*Sin[b*x] + 2*c*d^2*x*Sin[b*x] + d^3*x^2*Sin[b*x]))/(2*b^2)","B",0
293,1,66,55,0.5103065,"\int (c+d x)^2 \sec ^2(a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)^2*Sec[a + b*x]^2*Tan[a + b*x],x]","\frac{b^2 (c+d x)^2 \sec ^2(a+b x)-2 b d \sec (a) \sin (b x) (c+d x) \sec (a+b x)-2 d^2 (b x \tan (a)+\log (\cos (a+b x)))}{2 b^3}","-\frac{d^2 \log (\cos (a+b x))}{b^3}-\frac{d (c+d x) \tan (a+b x)}{b^2}+\frac{(c+d x)^2 \sec ^2(a+b x)}{2 b}",1,"(b^2*(c + d*x)^2*Sec[a + b*x]^2 - 2*b*d*(c + d*x)*Sec[a]*Sec[a + b*x]*Sin[b*x] - 2*d^2*(Log[Cos[a + b*x]] + b*x*Tan[a]))/(2*b^3)","A",1
294,1,48,35,0.0611154,"\int (c+d x) \sec ^2(a+b x) \tan (a+b x) \, dx","Integrate[(c + d*x)*Sec[a + b*x]^2*Tan[a + b*x],x]","-\frac{d \tan (a+b x)}{2 b^2}+\frac{c \sec ^2(a+b x)}{2 b}+\frac{d x \sec ^2(a+b x)}{2 b}","\frac{(c+d x) \sec ^2(a+b x)}{2 b}-\frac{d \tan (a+b x)}{2 b^2}",1,"(c*Sec[a + b*x]^2)/(2*b) + (d*x*Sec[a + b*x]^2)/(2*b) - (d*Tan[a + b*x])/(2*b^2)","A",1
295,0,0,25,7.2437807,"\int \frac{\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx","Integrate[(Sec[a + b*x]^2*Tan[a + b*x])/(c + d*x),x]","\int \frac{\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\tan (a+b x) \sec ^2(a+b x)}{c+d x},x\right)",0,"Integrate[(Sec[a + b*x]^2*Tan[a + b*x])/(c + d*x), x]","A",-1
296,0,0,25,10.3853089,"\int \frac{\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx","Integrate[(Sec[a + b*x]^2*Tan[a + b*x])/(c + d*x)^2,x]","\int \frac{\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\tan (a+b x) \sec ^2(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Sec[a + b*x]^2*Tan[a + b*x])/(c + d*x)^2, x]","A",-1
297,-1,0,39,180.0175136,"\int (c+d x)^m \sec (a+b x) \tan ^2(a+b x) \, dx","Integrate[(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x]^2,x]","\text{\$Aborted}","\text{Int}\left(\sec ^3(a+b x) (c+d x)^m,x\right)-\text{Int}\left(\sec (a+b x) (c+d x)^m,x\right)",0,"$Aborted","F",-1
298,1,530,337,3.400416,"\int (c+d x)^3 \sec (a+b x) \tan ^2(a+b x) \, dx","Integrate[(c + d*x)^3*Sec[a + b*x]*Tan[a + b*x]^2,x]","\frac{2 i b^3 c^3 \tan ^{-1}\left(e^{i (a+b x)}\right)-3 b^3 c^2 d x \log \left(1-i e^{i (a+b x)}\right)+3 b^3 c^2 d x \log \left(1+i e^{i (a+b x)}\right)-3 b^3 c d^2 x^2 \log \left(1-i e^{i (a+b x)}\right)+3 b^3 c d^2 x^2 \log \left(1+i e^{i (a+b x)}\right)-b^3 d^3 x^3 \log \left(1-i e^{i (a+b x)}\right)+b^3 d^3 x^3 \log \left(1+i e^{i (a+b x)}\right)-3 i d \text{Li}_2\left(-i e^{i (a+b x)}\right) \left(b^2 (c+d x)^2-2 d^2\right)+3 i d \text{Li}_2\left(i e^{i (a+b x)}\right) \left(b^2 (c+d x)^2-2 d^2\right)+b^2 (c+d x)^2 \sec (a+b x) (b (c+d x) \tan (a+b x)-3 d)+6 b c d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)-6 b c d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)-12 i b c d^2 \tan ^{-1}\left(e^{i (a+b x)}\right)+6 b d^3 x \text{Li}_3\left(-i e^{i (a+b x)}\right)-6 b d^3 x \text{Li}_3\left(i e^{i (a+b x)}\right)+6 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)-6 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)+6 b d^3 x \log \left(1-i e^{i (a+b x)}\right)-6 b d^3 x \log \left(1+i e^{i (a+b x)}\right)}{2 b^4}","\frac{3 i d^3 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^4}-\frac{3 i d^3 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^4}+\frac{3 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)}{b^4}-\frac{3 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)}{b^4}+\frac{3 d^2 (c+d x) \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}-\frac{3 d^2 (c+d x) \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}-\frac{6 i d^2 (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^3}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{2 b^2}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{2 b^2}-\frac{3 d (c+d x)^2 \sec (a+b x)}{2 b^2}+\frac{i (c+d x)^3 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}+\frac{(c+d x)^3 \tan (a+b x) \sec (a+b x)}{2 b}",1,"((2*I)*b^3*c^3*ArcTan[E^(I*(a + b*x))] - (12*I)*b*c*d^2*ArcTan[E^(I*(a + b*x))] - 3*b^3*c^2*d*x*Log[1 - I*E^(I*(a + b*x))] + 6*b*d^3*x*Log[1 - I*E^(I*(a + b*x))] - 3*b^3*c*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 - I*E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 + I*E^(I*(a + b*x))] - 6*b*d^3*x*Log[1 + I*E^(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] + b^3*d^3*x^3*Log[1 + I*E^(I*(a + b*x))] - (3*I)*d*(-2*d^2 + b^2*(c + d*x)^2)*PolyLog[2, (-I)*E^(I*(a + b*x))] + (3*I)*d*(-2*d^2 + b^2*(c + d*x)^2)*PolyLog[2, I*E^(I*(a + b*x))] + 6*b*c*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, (-I)*E^(I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, I*E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, I*E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))] - (6*I)*d^3*PolyLog[4, I*E^(I*(a + b*x))] + b^2*(c + d*x)^2*Sec[a + b*x]*(-3*d + b*(c + d*x)*Tan[a + b*x]))/(2*b^4)","A",1
299,1,526,193,7.1830689,"\int (c+d x)^2 \sec (a+b x) \tan ^2(a+b x) \, dx","Integrate[(c + d*x)^2*Sec[a + b*x]*Tan[a + b*x]^2,x]","\frac{i b c^2 \tan ^{-1}\left(e^{i (a+b x)}\right)-i d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)+i d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)-b c d x \log \left(1-i e^{i (a+b x)}\right)+b c d x \log \left(1+i e^{i (a+b x)}\right)+\frac{d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b}-\frac{d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b}-\frac{1}{2} b d^2 x^2 \log \left(1-i e^{i (a+b x)}\right)+\frac{1}{2} b d^2 x^2 \log \left(1+i e^{i (a+b x)}\right)-\frac{2 i d^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}}{b^2}+\frac{d^2 (-x) \sin \left(\frac{b x}{2}\right)-c d \sin \left(\frac{b x}{2}\right)}{b^2 \left(\cos \left(\frac{a}{2}\right)-\sin \left(\frac{a}{2}\right)\right) \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)-\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}+\frac{c d \sin \left(\frac{b x}{2}\right)+d^2 x \sin \left(\frac{b x}{2}\right)}{b^2 \left(\sin \left(\frac{a}{2}\right)+\cos \left(\frac{a}{2}\right)\right) \left(\sin \left(\frac{a}{2}+\frac{b x}{2}\right)+\cos \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}-\frac{d \sec (a) (c+d x)}{b^2}+\frac{-c^2-2 c d x-d^2 x^2}{4 b \left(\sin \left(\frac{a}{2}+\frac{b x}{2}\right)+\cos \left(\frac{a}{2}+\frac{b x}{2}\right)\right)^2}+\frac{c^2+2 c d x+d^2 x^2}{4 b \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)-\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)^2}","\frac{d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}-\frac{d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}+\frac{d^2 \tanh ^{-1}(\sin (a+b x))}{b^3}-\frac{i d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}+\frac{i d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{d (c+d x) \sec (a+b x)}{b^2}+\frac{i (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}+\frac{(c+d x)^2 \tan (a+b x) \sec (a+b x)}{2 b}",1,"(I*b*c^2*ArcTan[E^(I*(a + b*x))] - ((2*I)*d^2*ArcTan[E^(I*(a + b*x))])/b - b*c*d*x*Log[1 - I*E^(I*(a + b*x))] - (b*d^2*x^2*Log[1 - I*E^(I*(a + b*x))])/2 + b*c*d*x*Log[1 + I*E^(I*(a + b*x))] + (b*d^2*x^2*Log[1 + I*E^(I*(a + b*x))])/2 - I*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] + I*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] + (d^2*PolyLog[3, (-I)*E^(I*(a + b*x))])/b - (d^2*PolyLog[3, I*E^(I*(a + b*x))])/b)/b^2 - (d*(c + d*x)*Sec[a])/b^2 + (c^2 + 2*c*d*x + d^2*x^2)/(4*b*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])^2) + (-(c*d*Sin[(b*x)/2]) - d^2*x*Sin[(b*x)/2])/(b^2*(Cos[a/2] - Sin[a/2])*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])) + (-c^2 - 2*c*d*x - d^2*x^2)/(4*b*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2])^2) + (c*d*Sin[(b*x)/2] + d^2*x*Sin[(b*x)/2])/(b^2*(Cos[a/2] + Sin[a/2])*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2]))","B",1
300,1,555,117,6.5036712,"\int (c+d x) \sec (a+b x) \tan ^2(a+b x) \, dx","Integrate[(c + d*x)*Sec[a + b*x]*Tan[a + b*x]^2,x]","-\frac{d \sin \left(\frac{1}{2} (a+b x)\right)}{2 b^2 \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)}+\frac{d \sin \left(\frac{1}{2} (a+b x)\right)}{2 b^2 \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)}-\frac{c \tanh ^{-1}(\sin (a+b x))}{2 b}+\frac{c \tan (a+b x) \sec (a+b x)}{2 b}+\frac{d x \left(-i \text{Li}_2\left(\frac{1}{2} \left((1+i)-(1-i) \tan \left(\frac{1}{2} (a+b x)\right)\right)\right)+i \text{Li}_2\left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)+i\right)\right)-i \text{Li}_2\left(\frac{1}{2} \left((1-i) \tan \left(\frac{1}{2} (a+b x)\right)+(1+i)\right)\right)+i \text{Li}_2\left(\frac{1}{2} \left((1+i) \tan \left(\frac{1}{2} (a+b x)\right)+(1-i)\right)\right)+a \log \left(1-\tan \left(\frac{1}{2} (a+b x)\right)\right)+i \log \left(1+i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)-1\right)\right)-i \log \left(1-i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(-\frac{1}{2}+\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)-1\right)\right)-i \log \left(1+i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(\frac{1}{2}-\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)+1\right)\right)+i \log \left(1-i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)+1\right)\right)-a \log \left(\tan \left(\frac{1}{2} (a+b x)\right)+1\right)\right)}{2 b \left(-i \log \left(1-i \tan \left(\frac{1}{2} (a+b x)\right)\right)+i \log \left(1+i \tan \left(\frac{1}{2} (a+b x)\right)\right)+a\right)}+\frac{d x}{4 b \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)^2}-\frac{d x}{4 b \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)^2}","-\frac{i d \text{Li}_2\left(-i e^{i (a+b x)}\right)}{2 b^2}+\frac{i d \text{Li}_2\left(i e^{i (a+b x)}\right)}{2 b^2}-\frac{d \sec (a+b x)}{2 b^2}+\frac{i (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}+\frac{(c+d x) \tan (a+b x) \sec (a+b x)}{2 b}",1,"-1/2*(c*ArcTanh[Sin[a + b*x]])/b + (d*x*(a*Log[1 - Tan[(a + b*x)/2]] + I*Log[1 + I*Tan[(a + b*x)/2]]*Log[(-1/2 - I/2)*(-1 + Tan[(a + b*x)/2])] - I*Log[1 - I*Tan[(a + b*x)/2]]*Log[(-1/2 + I/2)*(-1 + Tan[(a + b*x)/2])] - I*Log[1 + I*Tan[(a + b*x)/2]]*Log[(1/2 - I/2)*(1 + Tan[(a + b*x)/2])] + I*Log[1 - I*Tan[(a + b*x)/2]]*Log[(1/2 + I/2)*(1 + Tan[(a + b*x)/2])] - a*Log[1 + Tan[(a + b*x)/2]] - I*PolyLog[2, ((1 + I) - (1 - I)*Tan[(a + b*x)/2])/2] + I*PolyLog[2, (-1/2 - I/2)*(I + Tan[(a + b*x)/2])] - I*PolyLog[2, ((1 + I) + (1 - I)*Tan[(a + b*x)/2])/2] + I*PolyLog[2, ((1 - I) + (1 + I)*Tan[(a + b*x)/2])/2]))/(2*b*(a - I*Log[1 - I*Tan[(a + b*x)/2]] + I*Log[1 + I*Tan[(a + b*x)/2]])) + (d*x)/(4*b*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])^2) - (d*Sin[(a + b*x)/2])/(2*b^2*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])) - (d*x)/(4*b*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])^2) + (d*Sin[(a + b*x)/2])/(2*b^2*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])) + (c*Sec[a + b*x]*Tan[a + b*x])/(2*b)","B",0
301,0,0,39,26.3901579,"\int \frac{\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx","Integrate[(Sec[a + b*x]*Tan[a + b*x]^2)/(c + d*x),x]","\int \frac{\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\sec ^3(a+b x)}{c+d x},x\right)-\text{Int}\left(\frac{\sec (a+b x)}{c+d x},x\right)",0,"Integrate[(Sec[a + b*x]*Tan[a + b*x]^2)/(c + d*x), x]","A",-1
302,0,0,39,29.1197277,"\int \frac{\sec (a+b x) \tan ^2(a+b x)}{(c+d x)^2} \, dx","Integrate[(Sec[a + b*x]*Tan[a + b*x]^2)/(c + d*x)^2,x]","\int \frac{\sec (a+b x) \tan ^2(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\sec ^3(a+b x)}{(c+d x)^2},x\right)-\text{Int}\left(\frac{\sec (a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Sec[a + b*x]*Tan[a + b*x]^2)/(c + d*x)^2, x]","A",-1
303,0,0,19,9.7039924,"\int (c+d x)^m \tan ^3(a+b x) \, dx","Integrate[(c + d*x)^m*Tan[a + b*x]^3,x]","\int (c+d x)^m \tan ^3(a+b x) \, dx","\text{Int}\left(\tan ^3(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Tan[a + b*x]^3, x]","A",-1
304,1,803,259,6.8863859,"\int (c+d x)^3 \tan ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Tan[a + b*x]^3,x]","\frac{\sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a)) c^3}{b \left(\cos ^2(a)+\sin ^2(a)\right)}+\frac{3 d \csc (a) \left(b^2 e^{-i \tan ^{-1}(\cot (a))} x^2-\frac{\cot (a) \left(i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\pi  \log (\cos (b x))-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)}{\sqrt{\cot ^2(a)+1}}\right) \sec (a) c^2}{2 b^2 \sqrt{\csc ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}+\frac{i d^2 e^{-i a} \left(2 b^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right) x^2+6 b \left(1+e^{2 i a}\right) \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right) \sec (a) c}{4 b^3}-\frac{3 d^2 \sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a)) c}{b^3 \left(\cos ^2(a)+\sin ^2(a)\right)}+\frac{(c+d x)^3 \sec ^2(a+b x)}{2 b}+\frac{1}{8} i d^3 e^{i a} \left(2 e^{-2 i a} x^4-\frac{4 i \left(1+e^{-2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right) x^3}{b}+\frac{3 e^{-2 i a} \left(1+e^{2 i a}\right) \left(2 b^2 \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(-e^{-2 i (a+b x)}\right) x-\text{Li}_4\left(-e^{-2 i (a+b x)}\right)\right)}{b^4}\right) \sec (a)-\frac{3 \sec (a) \sec (a+b x) \left(x^2 \sin (b x) d^3+2 c x \sin (b x) d^2+c^2 \sin (b x) d\right)}{2 b^2}-\frac{1}{4} x \left(4 c^3+6 d x c^2+4 d^2 x^2 c+d^3 x^3\right) \tan (a)-\frac{3 d^3 \csc (a) \left(b^2 e^{-i \tan ^{-1}(\cot (a))} x^2-\frac{\cot (a) \left(i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\pi  \log (\cos (b x))-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)}{\sqrt{\cot ^2(a)+1}}\right) \sec (a)}{2 b^4 \sqrt{\csc ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}","\frac{3 i d^3 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^4}+\frac{3 i d^3 \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{4 b^4}+\frac{3 d^2 (c+d x) \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}-\frac{3 d^2 (c+d x) \log \left(1+e^{2 i (a+b x)}\right)}{b^3}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac{(c+d x)^3 \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{(c+d x)^3 \tan ^2(a+b x)}{2 b}+\frac{3 i d (c+d x)^2}{2 b^2}+\frac{(c+d x)^3}{2 b}-\frac{i (c+d x)^4}{4 d}",1,"((I/4)*c*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) + (I/8)*d^3*E^(I*a)*((2*x^4)/E^((2*I)*a) - ((4*I)*(1 + E^((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x))])/b + (3*(1 + E^((2*I)*a))*(2*b^2*x^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-2*I)*(a + b*x))] - PolyLog[4, -E^((-2*I)*(a + b*x))]))/(b^4*E^((2*I)*a)))*Sec[a] + ((c + d*x)^3*Sec[a + b*x]^2)/(2*b) + (c^3*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (3*c*d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (3*c^2*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(2*b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (3*d^3*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(2*b^4*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (3*Sec[a]*Sec[a + b*x]*(c^2*d*Sin[b*x] + 2*c*d^2*x*Sin[b*x] + d^3*x^2*Sin[b*x]))/(2*b^2) - (x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Tan[a])/4","B",0
305,1,454,169,6.6563245,"\int (c+d x)^2 \tan ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Tan[a + b*x]^3,x]","-\frac{d^2 \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b^3 \left(\sin ^2(a)+\cos ^2(a)\right)}+\frac{\sec (a) \sec (a+b x) \left(d^2 (-x) \sin (b x)-c d \sin (b x)\right)}{b^2}+\frac{c d \csc (a) \sec (a) \left(b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left(i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\cot ^2(a)+1}}\right)}{b^2 \sqrt{\csc ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}+\frac{i e^{-i a} d^2 \sec (a) \left(2 b^2 x^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right)+6 \left(1+e^{2 i a}\right) b x \text{Li}_2\left(-e^{-2 i (a+b x)}\right)-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right)}{12 b^3}+\frac{c^2 \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b \left(\sin ^2(a)+\cos ^2(a)\right)}+\frac{(c+d x)^2 \sec ^2(a+b x)}{2 b}-\frac{1}{3} x \tan (a) \left(3 c^2+3 c d x+d^2 x^2\right)","\frac{d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}-\frac{d^2 \log (\cos (a+b x))}{b^3}-\frac{i d (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{d (c+d x) \tan (a+b x)}{b^2}+\frac{(c+d x)^2 \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{(c+d x)^2 \tan ^2(a+b x)}{2 b}+\frac{c d x}{b}+\frac{d^2 x^2}{2 b}-\frac{i (c+d x)^3}{3 d}",1,"((I/12)*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) + ((c + d*x)^2*Sec[a + b*x]^2)/(2*b) + (c^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (c*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + (Sec[a]*Sec[a + b*x]*(-(c*d*Sin[b*x]) - d^2*x*Sin[b*x]))/b^2 - (x*(3*c^2 + 3*c*d*x + d^2*x^2)*Tan[a])/3","B",0
306,1,240,108,6.1531183,"\int (c+d x) \tan ^3(a+b x) \, dx","Integrate[(c + d*x)*Tan[a + b*x]^3,x]","\frac{d \csc (a) \sec (a) \left(b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left(i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\cot ^2(a)+1}}\right)}{2 b^2 \sqrt{\csc ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}-\frac{d \sec (a) \sin (b x) \sec (a+b x)}{2 b^2}+\frac{c \left(\tan ^2(a+b x)+2 \log (\cos (a+b x))\right)}{2 b}+\frac{d x \sec ^2(a+b x)}{2 b}-\frac{1}{2} d x^2 \tan (a)","-\frac{i d \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{d \tan (a+b x)}{2 b^2}+\frac{(c+d x) \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{(c+d x) \tan ^2(a+b x)}{2 b}+\frac{d x}{2 b}-\frac{i (c+d x)^2}{2 d}",1,"(d*x*Sec[a + b*x]^2)/(2*b) + (d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(2*b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (d*Sec[a]*Sec[a + b*x]*Sin[b*x])/(2*b^2) - (d*x^2*Tan[a])/2 + (c*(2*Log[Cos[a + b*x]] + Tan[a + b*x]^2))/(2*b)","B",0
307,0,0,19,6.530619,"\int \frac{\tan ^3(a+b x)}{c+d x} \, dx","Integrate[Tan[a + b*x]^3/(c + d*x),x]","\int \frac{\tan ^3(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\tan ^3(a+b x)}{c+d x},x\right)",0,"Integrate[Tan[a + b*x]^3/(c + d*x), x]","A",-1
308,0,0,19,6.9612515,"\int \frac{\tan ^3(a+b x)}{(c+d x)^2} \, dx","Integrate[Tan[a + b*x]^3/(c + d*x)^2,x]","\int \frac{\tan ^3(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\tan ^3(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[Tan[a + b*x]^3/(c + d*x)^2, x]","A",-1
309,0,0,25,13.3680261,"\int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^m*Csc[a + b*x]*Sec[a + b*x]^3,x]","\int (c+d x)^m \csc (a+b x) \sec ^3(a+b x) \, dx","\text{Int}\left(\csc (a+b x) \sec ^3(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Csc[a + b*x]*Sec[a + b*x]^3, x]","A",-1
310,1,2090,399,7.4701335,"\int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x]^3,x]","\text{Result too large to show}","-\frac{3 d^4 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{b^5}+\frac{3 d^4 \text{Li}_5\left(-e^{2 i (a+b x)}\right)}{2 b^5}-\frac{3 d^4 \text{Li}_5\left(e^{2 i (a+b x)}\right)}{2 b^5}+\frac{6 i d^3 (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^4}-\frac{3 i d^3 (c+d x) \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{b^4}+\frac{3 i d^3 (c+d x) \text{Li}_4\left(e^{2 i (a+b x)}\right)}{b^4}-\frac{3 d^2 (c+d x)^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{b^3}+\frac{3 d^2 (c+d x)^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{b^3}-\frac{6 d^2 (c+d x)^2 \log \left(1+e^{2 i (a+b x)}\right)}{b^3}+\frac{2 i d (c+d x)^3 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{2 i d (c+d x)^3 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac{(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac{2 (c+d x)^4 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}+\frac{2 i d (c+d x)^3}{b^2}+\frac{(c+d x)^4}{2 b}",1,"-((c^2*d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^3) - (c*d^3*E^(I*a)*Csc[a]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-I)*(a + b*x))] - 2*PolyLog[4, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, E^((-I)*(a + b*x))] - 2*PolyLog[4, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^4 - (d^4*E^(I*a)*Csc[a]*((2*b^5*x^5)/E^((2*I)*a) + (5*I)*b^4*(1 - E^((-2*I)*a))*x^4*Log[1 - E^((-I)*(a + b*x))] + (5*I)*b^4*(1 - E^((-2*I)*a))*x^4*Log[1 + E^((-I)*(a + b*x))] - (20*(-1 + E^((2*I)*a))*(b^3*x^3*PolyLog[2, -E^((-I)*(a + b*x))] - (3*I)*b^2*x^2*PolyLog[3, -E^((-I)*(a + b*x))] - 6*b*x*PolyLog[4, -E^((-I)*(a + b*x))] + (6*I)*PolyLog[5, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (20*(-1 + E^((2*I)*a))*(b^3*x^3*PolyLog[2, E^((-I)*(a + b*x))] - (3*I)*b^2*x^2*PolyLog[3, E^((-I)*(a + b*x))] - 6*b*x*PolyLog[4, E^((-I)*(a + b*x))] + (6*I)*PolyLog[5, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(10*b^5) + (x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4)*Csc[a]*Sec[a])/5 - ((I/2)*c^2*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) - ((I/2)*d^4*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^5*E^(I*a)) - (I/2)*c*d^3*E^(I*a)*((2*x^4)/E^((2*I)*a) - ((4*I)*(1 + E^((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x))])/b + (3*(1 + E^((2*I)*a))*(2*b^2*x^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-2*I)*(a + b*x))] - PolyLog[4, -E^((-2*I)*(a + b*x))]))/(b^4*E^((2*I)*a)))*Sec[a] + (d^4*((-4*I)*x^5 - (10*(1 + E^((2*I)*a))*x^4*Log[1 + E^((-2*I)*(a + b*x))])/b + (5*(1 + E^((2*I)*a))*((-4*I)*b^3*x^3*PolyLog[2, -E^((-2*I)*(a + b*x))] - 6*b^2*x^2*PolyLog[3, -E^((-2*I)*(a + b*x))] + (6*I)*b*x*PolyLog[4, -E^((-2*I)*(a + b*x))] + 3*PolyLog[5, -E^((-2*I)*(a + b*x))]))/b^5)*Sec[a])/(20*E^(I*a)) + ((c + d*x)^4*Sec[a + b*x]^2)/(2*b) - (c^4*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (6*c^2*d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (c^4*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (2*c^3*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (6*c*d^3*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^4*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (2*Sec[a]*Sec[a + b*x]*(c^3*d*Sin[b*x] + 3*c^2*d^2*x*Sin[b*x] + 3*c*d^3*x^2*Sin[b*x] + d^4*x^3*Sin[b*x]))/b^2 - (2*c^3*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
311,1,1486,325,6.9088239,"\int (c+d x)^3 \csc (a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]*Sec[a + b*x]^3,x]","-\frac{\sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a)) c^3}{b \left(\cos ^2(a)+\sin ^2(a)\right)}+\frac{\csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a)) c^3}{b \left(\cos ^2(a)+\sin ^2(a)\right)}-\frac{3 d \csc (a) \sec (a) \left(b^2 e^{i \tan ^{-1}(\tan (a))} x^2+\frac{\left(i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x+\tan ^{-1}(\tan (a))\right) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\pi  \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right) \tan (a)}{\sqrt{\tan ^2(a)+1}}\right) c^2}{2 b^2 \sqrt{\sec ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}-\frac{3 d \csc (a) \left(b^2 e^{-i \tan ^{-1}(\cot (a))} x^2-\frac{\cot (a) \left(i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\pi  \log (\cos (b x))-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)}{\sqrt{\cot ^2(a)+1}}\right) \sec (a) c^2}{2 b^2 \sqrt{\csc ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}-\frac{d^2 e^{i a} \csc (a) \left(2 b^3 e^{-2 i a} x^3+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^2+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^2-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right) c}{2 b^3}-\frac{i d^2 e^{-i a} \left(2 b^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right) x^2+6 b \left(1+e^{2 i a}\right) \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right) \sec (a) c}{4 b^3}-\frac{3 d^2 \sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a)) c}{b^3 \left(\cos ^2(a)+\sin ^2(a)\right)}+\frac{(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac{d^3 e^{i a} \csc (a) \left(b^4 e^{-2 i a} x^4+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^3+2 i b^3 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^3-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(-e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(-e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b^2 \text{Li}_2\left(e^{-i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(e^{-i (a+b x)}\right) x-2 \text{Li}_4\left(e^{-i (a+b x)}\right)\right)\right)}{4 b^4}+\frac{1}{4} x \left(4 c^3+6 d x c^2+4 d^2 x^2 c+d^3 x^3\right) \csc (a) \sec (a)-\frac{1}{8} i d^3 e^{i a} \left(2 e^{-2 i a} x^4-\frac{4 i \left(1+e^{-2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right) x^3}{b}+\frac{3 e^{-2 i a} \left(1+e^{2 i a}\right) \left(2 b^2 \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(-e^{-2 i (a+b x)}\right) x-\text{Li}_4\left(-e^{-2 i (a+b x)}\right)\right)}{b^4}\right) \sec (a)-\frac{3 \sec (a) \sec (a+b x) \left(x^2 \sin (b x) d^3+2 c x \sin (b x) d^2+c^2 \sin (b x) d\right)}{2 b^2}-\frac{3 d^3 \csc (a) \left(b^2 e^{-i \tan ^{-1}(\cot (a))} x^2-\frac{\cot (a) \left(i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\pi  \log (\cos (b x))-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)}{\sqrt{\cot ^2(a)+1}}\right) \sec (a)}{2 b^4 \sqrt{\csc ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}","\frac{3 i d^3 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 i d^3 \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{4 b^4}+\frac{3 i d^3 \text{Li}_4\left(e^{2 i (a+b x)}\right)}{4 b^4}-\frac{3 d^2 (c+d x) \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}+\frac{3 d^2 (c+d x) \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}-\frac{3 d^2 (c+d x) \log \left(1+e^{2 i (a+b x)}\right)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac{(c+d x)^3 \tan ^2(a+b x)}{2 b}-\frac{2 (c+d x)^3 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}+\frac{3 i d (c+d x)^2}{2 b^2}+\frac{(c+d x)^3}{2 b}",1,"-1/2*(c*d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^3 - (d^3*E^(I*a)*Csc[a]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-I)*(a + b*x))] - 2*PolyLog[4, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, E^((-I)*(a + b*x))] - 2*PolyLog[4, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(4*b^4) + (x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Csc[a]*Sec[a])/4 - ((I/4)*c*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) - (I/8)*d^3*E^(I*a)*((2*x^4)/E^((2*I)*a) - ((4*I)*(1 + E^((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x))])/b + (3*(1 + E^((2*I)*a))*(2*b^2*x^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-2*I)*(a + b*x))] - PolyLog[4, -E^((-2*I)*(a + b*x))]))/(b^4*E^((2*I)*a)))*Sec[a] + ((c + d*x)^3*Sec[a + b*x]^2)/(2*b) - (c^3*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (3*c*d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (c^3*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (3*c^2*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(2*b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (3*d^3*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(2*b^4*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (3*Sec[a]*Sec[a + b*x]*(c^2*d*Sin[b*x] + 2*c*d^2*x*Sin[b*x] + d^3*x^2*Sin[b*x]))/(2*b^2) - (3*c^2*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(2*b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
312,1,875,201,6.7574205,"\int (c+d x)^2 \csc (a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]*Sec[a + b*x]^3,x]","-\frac{\sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a)) c^2}{b \left(\cos ^2(a)+\sin ^2(a)\right)}+\frac{\csc (a) (\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a)-b x \cos (a)) c^2}{b \left(\cos ^2(a)+\sin ^2(a)\right)}-\frac{d \csc (a) \sec (a) \left(b^2 e^{i \tan ^{-1}(\tan (a))} x^2+\frac{\left(i b x \left(2 \tan ^{-1}(\tan (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x+\tan ^{-1}(\tan (a))\right) \log \left(1-e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)+\pi  \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left(\sin \left(b x+\tan ^{-1}(\tan (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right) \tan (a)}{\sqrt{\tan ^2(a)+1}}\right) c}{b^2 \sqrt{\sec ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}-\frac{d \csc (a) \left(b^2 e^{-i \tan ^{-1}(\cot (a))} x^2-\frac{\cot (a) \left(i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\pi  \log (\cos (b x))-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)}{\sqrt{\cot ^2(a)+1}}\right) \sec (a) c}{b^2 \sqrt{\csc ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}+\frac{(c+d x)^2 \sec ^2(a+b x)}{2 b}-\frac{d^2 e^{i a} \csc (a) \left(2 b^3 e^{-2 i a} x^3+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1-e^{-i (a+b x)}\right) x^2+3 i b^2 \left(1-e^{-2 i a}\right) \log \left(1+e^{-i (a+b x)}\right) x^2-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(-e^{-i (a+b x)}\right)-i \text{Li}_3\left(-e^{-i (a+b x)}\right)\right)-6 e^{-2 i a} \left(-1+e^{2 i a}\right) \left(b x \text{Li}_2\left(e^{-i (a+b x)}\right)-i \text{Li}_3\left(e^{-i (a+b x)}\right)\right)\right)}{6 b^3}+\frac{1}{3} x \left(3 c^2+3 d x c+d^2 x^2\right) \csc (a) \sec (a)-\frac{i d^2 e^{-i a} \left(2 b^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right) x^2+6 b \left(1+e^{2 i a}\right) \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right) \sec (a)}{12 b^3}+\frac{\sec (a) \sec (a+b x) \left(-x \sin (b x) d^2-c \sin (b x) d\right)}{b^2}-\frac{d^2 \sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a))}{b^3 \left(\cos ^2(a)+\sin ^2(a)\right)}","-\frac{d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}+\frac{d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{2 b^3}-\frac{d^2 \log (\cos (a+b x))}{b^3}+\frac{i d (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{i d (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{d (c+d x) \tan (a+b x)}{b^2}+\frac{(c+d x)^2 \tan ^2(a+b x)}{2 b}-\frac{2 (c+d x)^2 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}+\frac{c d x}{b}+\frac{d^2 x^2}{2 b}",1,"-1/6*(d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/b^3 + (x*(3*c^2 + 3*c*d*x + d^2*x^2)*Csc[a]*Sec[a])/3 - ((I/12)*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) + ((c + d*x)^2*Sec[a + b*x]^2)/(2*b) - (c^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (c^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (c*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + (Sec[a]*Sec[a + b*x]*(-(c*d*Sin[b*x]) - d^2*x*Sin[b*x]))/b^2 - (c*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])","B",0
313,1,212,139,0.5642286,"\int (c+d x) \csc (a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]*Sec[a + b*x]^3,x]","\frac{d \left(\frac{1}{2} i \text{Li}_2\left(-e^{2 i (a+b x)}\right)+\frac{1}{2} i (a+b x)^2-(a+b x) \log \left(1+e^{2 i (a+b x)}\right)\right)}{b^2}+\frac{d \left((a+b x) \log \left(1-e^{2 i (a+b x)}\right)-\frac{1}{2} i \left((a+b x)^2+\text{Li}_2\left(e^{2 i (a+b x)}\right)\right)\right)}{b^2}-\frac{d \tan (a+b x)}{2 b^2}-\frac{a d \log (\sin (a+b x))}{b^2}+\frac{a d \log (\cos (a+b x))}{b^2}-\frac{c \left(-\sec ^2(a+b x)-2 \log (\sin (a+b x))+2 \log (\cos (a+b x))\right)}{2 b}+\frac{d x \sec ^2(a+b x)}{2 b}","\frac{i d \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}-\frac{i d \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^2}-\frac{d \tan (a+b x)}{2 b^2}+\frac{c \tan ^2(a+b x)}{2 b}+\frac{c \log (\tan (a+b x))}{b}+\frac{d x \tan ^2(a+b x)}{2 b}-\frac{2 d x \tanh ^{-1}\left(e^{2 i a+2 i b x}\right)}{b}+\frac{d x}{2 b}",1,"(a*d*Log[Cos[a + b*x]])/b^2 - (a*d*Log[Sin[a + b*x]])/b^2 + (d*((I/2)*(a + b*x)^2 - (a + b*x)*Log[1 + E^((2*I)*(a + b*x))] + (I/2)*PolyLog[2, -E^((2*I)*(a + b*x))]))/b^2 + (d*((a + b*x)*Log[1 - E^((2*I)*(a + b*x))] - (I/2)*((a + b*x)^2 + PolyLog[2, E^((2*I)*(a + b*x))])))/b^2 + (d*x*Sec[a + b*x]^2)/(2*b) - (c*(2*Log[Cos[a + b*x]] - 2*Log[Sin[a + b*x]] - Sec[a + b*x]^2))/(2*b) - (d*Tan[a + b*x])/(2*b^2)","A",1
314,0,0,25,9.0760976,"\int \frac{\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx","Integrate[(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x),x]","\int \frac{\csc (a+b x) \sec ^3(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\csc (a+b x) \sec ^3(a+b x)}{c+d x},x\right)",0,"Integrate[(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x), x]","A",-1
315,0,0,25,6.6507969,"\int \frac{\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx","Integrate[(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x)^2,x]","\int \frac{\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\csc (a+b x) \sec ^3(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x)^2, x]","A",-1
316,0,0,27,30.5131005,"\int (c+d x)^m \csc ^2(a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^m*Csc[a + b*x]^2*Sec[a + b*x]^3,x]","\int (c+d x)^m \csc ^2(a+b x) \sec ^3(a+b x) \, dx","\text{Int}\left(\csc ^2(a+b x) \sec ^3(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Csc[a + b*x]^2*Sec[a + b*x]^3, x]","A",-1
317,1,819,486,7.8974038,"\int (c+d x)^3 \csc ^2(a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x]^3,x]","-\frac{\csc (a+b x) \left(b c^3+3 b \cos (2 a+2 b x) c^3+3 b d x c^2+9 b d x \cos (2 a+2 b x) c^2+3 d \sin (2 a+2 b x) c^2+3 b d^2 x^2 c+9 b d^2 x^2 \cos (2 a+2 b x) c+6 d^2 x \sin (2 a+2 b x) c+b d^3 x^3+3 b d^3 x^3 \cos (2 a+2 b x)+3 d^3 x^2 \sin (2 a+2 b x)\right) \sec ^2(a+b x)}{4 b^2}+\frac{3 d \left(\log \left(1-e^{i (a+b x)}\right) (c+d x)^2-\log \left(1+e^{i (a+b x)}\right) (c+d x)^2+\frac{2 i d \left(b (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)+i d \text{Li}_3\left(-e^{i (a+b x)}\right)\right)}{b^2}+\frac{2 d \left(d \text{Li}_3\left(e^{i (a+b x)}\right)-i b (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)\right)}{b^2}\right)}{b^2}-\frac{3 \left(2 i c^3 \tan ^{-1}\left(e^{i (a+b x)}\right) b^3-d^3 x^3 \log \left(1-i e^{i (a+b x)}\right) b^3-3 c d^2 x^2 \log \left(1-i e^{i (a+b x)}\right) b^3-3 c^2 d x \log \left(1-i e^{i (a+b x)}\right) b^3+d^3 x^3 \log \left(1+i e^{i (a+b x)}\right) b^3+3 c d^2 x^2 \log \left(1+i e^{i (a+b x)}\right) b^3+3 c^2 d x \log \left(1+i e^{i (a+b x)}\right) b^3+4 i c d^2 \tan ^{-1}\left(e^{i (a+b x)}\right) b-2 d^3 x \log \left(1-i e^{i (a+b x)}\right) b+2 d^3 x \log \left(1+i e^{i (a+b x)}\right) b+6 c d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right) b+6 d^3 x \text{Li}_3\left(-i e^{i (a+b x)}\right) b-6 c d^2 \text{Li}_3\left(i e^{i (a+b x)}\right) b-6 d^3 x \text{Li}_3\left(i e^{i (a+b x)}\right) b-i d \left(2 d^2+3 b^2 (c+d x)^2\right) \text{Li}_2\left(-i e^{i (a+b x)}\right)+i d \left(2 d^2+3 b^2 (c+d x)^2\right) \text{Li}_2\left(i e^{i (a+b x)}\right)+6 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)-6 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)\right)}{2 b^4}","\frac{3 i d^3 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^4}-\frac{3 i d^3 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^4}-\frac{6 d^3 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^4}+\frac{6 d^3 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^4}-\frac{9 i d^3 \text{Li}_4\left(-i e^{i (a+b x)}\right)}{b^4}+\frac{9 i d^3 \text{Li}_4\left(i e^{i (a+b x)}\right)}{b^4}+\frac{6 i d^2 (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^3}-\frac{6 i d^2 (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^3}-\frac{9 d^2 (c+d x) \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{9 d^2 (c+d x) \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}-\frac{6 i d^2 (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^3}+\frac{9 i d (c+d x)^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{2 b^2}-\frac{9 i d (c+d x)^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{2 b^2}-\frac{3 d (c+d x)^2 \sec (a+b x)}{2 b^2}-\frac{6 d (c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{3 i (c+d x)^3 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{3 (c+d x)^3 \csc (a+b x)}{2 b}+\frac{(c+d x)^3 \csc (a+b x) \sec ^2(a+b x)}{2 b}",1,"(3*d*((c + d*x)^2*Log[1 - E^(I*(a + b*x))] - (c + d*x)^2*Log[1 + E^(I*(a + b*x))] + ((2*I)*d*(b*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] + I*d*PolyLog[3, -E^(I*(a + b*x))]))/b^2 + (2*d*((-I)*b*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] + d*PolyLog[3, E^(I*(a + b*x))]))/b^2))/b^2 - (3*((2*I)*b^3*c^3*ArcTan[E^(I*(a + b*x))] + (4*I)*b*c*d^2*ArcTan[E^(I*(a + b*x))] - 3*b^3*c^2*d*x*Log[1 - I*E^(I*(a + b*x))] - 2*b*d^3*x*Log[1 - I*E^(I*(a + b*x))] - 3*b^3*c*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 - I*E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 + I*E^(I*(a + b*x))] + 2*b*d^3*x*Log[1 + I*E^(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] + b^3*d^3*x^3*Log[1 + I*E^(I*(a + b*x))] - I*d*(2*d^2 + 3*b^2*(c + d*x)^2)*PolyLog[2, (-I)*E^(I*(a + b*x))] + I*d*(2*d^2 + 3*b^2*(c + d*x)^2)*PolyLog[2, I*E^(I*(a + b*x))] + 6*b*c*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, (-I)*E^(I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, I*E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, I*E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))] - (6*I)*d^3*PolyLog[4, I*E^(I*(a + b*x))]))/(2*b^4) - (Csc[a + b*x]*Sec[a + b*x]^2*(b*c^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2 + b*d^3*x^3 + 3*b*c^3*Cos[2*a + 2*b*x] + 9*b*c^2*d*x*Cos[2*a + 2*b*x] + 9*b*c*d^2*x^2*Cos[2*a + 2*b*x] + 3*b*d^3*x^3*Cos[2*a + 2*b*x] + 3*c^2*d*Sin[2*a + 2*b*x] + 6*c*d^2*x*Sin[2*a + 2*b*x] + 3*d^3*x^2*Sin[2*a + 2*b*x]))/(4*b^2)","A",1
318,1,889,341,7.5304911,"\int (c+d x)^2 \csc ^2(a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]^2*Sec[a + b*x]^3,x]","\frac{2 \left(\frac{\left(\left(b x+\tan ^{-1}(\tan (a))\right) \left(\log \left(1-e^{i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)-\log \left(1+e^{i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right)+i \left(\text{Li}_2\left(-e^{i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)-\text{Li}_2\left(e^{i \left(b x+\tan ^{-1}(\tan (a))\right)}\right)\right)\right) \sec (a)}{\sqrt{\tan ^2(a)+1}}-\frac{2 \tan ^{-1}(\tan (a)) \tanh ^{-1}\left(\frac{\sin (a) \tan \left(\frac{b x}{2}\right)-\cos (a)}{\sqrt{\cos ^2(a)+\sin ^2(a)}}\right)}{\sqrt{\cos ^2(a)+\sin ^2(a)}}\right) d^2}{b^3}+\frac{4 i c \tan ^{-1}\left(\frac{i \cos (a)-i \sin (a) \tan \left(\frac{b x}{2}\right)}{\sqrt{\cos ^2(a)+\sin ^2(a)}}\right) d}{b^2 \sqrt{\cos ^2(a)+\sin ^2(a)}}-\frac{6 i c^2 \tan ^{-1}\left(e^{i (a+b x)}\right) b^2-3 d^2 x^2 \log \left(1-i e^{i (a+b x)}\right) b^2-6 c d x \log \left(1-i e^{i (a+b x)}\right) b^2+3 d^2 x^2 \log \left(1+i e^{i (a+b x)}\right) b^2+6 c d x \log \left(1+i e^{i (a+b x)}\right) b^2-6 i d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right) b+6 i d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right) b+4 i d^2 \tan ^{-1}\left(e^{i (a+b x)}\right)+6 d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)-6 d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)}{2 b^3}-\frac{(c+d x) \csc (a) \sec (a) (b c \cos (a)+b d x \cos (a)+d \sin (a))}{b^2}+\frac{\sec \left(\frac{a}{2}\right) \sec \left(\frac{a}{2}+\frac{b x}{2}\right) \left(-\sin \left(\frac{b x}{2}\right) c^2-2 d x \sin \left(\frac{b x}{2}\right) c-d^2 x^2 \sin \left(\frac{b x}{2}\right)\right)}{2 b}+\frac{\csc \left(\frac{a}{2}\right) \csc \left(\frac{a}{2}+\frac{b x}{2}\right) \left(\sin \left(\frac{b x}{2}\right) c^2+2 d x \sin \left(\frac{b x}{2}\right) c+d^2 x^2 \sin \left(\frac{b x}{2}\right)\right)}{2 b}+\frac{-x \sin \left(\frac{b x}{2}\right) d^2-c \sin \left(\frac{b x}{2}\right) d}{b^2 \left(\cos \left(\frac{a}{2}\right)-\sin \left(\frac{a}{2}\right)\right) \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)-\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}+\frac{x \sin \left(\frac{b x}{2}\right) d^2+c \sin \left(\frac{b x}{2}\right) d}{b^2 \left(\cos \left(\frac{a}{2}\right)+\sin \left(\frac{a}{2}\right)\right) \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)+\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}+\frac{c^2+2 d x c+d^2 x^2}{4 b \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)-\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)^2}+\frac{-c^2-2 d x c-d^2 x^2}{4 b \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)+\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)^2}","\frac{2 i d^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^3}-\frac{2 i d^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^3}-\frac{3 d^2 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^3}+\frac{3 d^2 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^3}+\frac{d^2 \tanh ^{-1}(\sin (a+b x))}{b^3}+\frac{3 i d (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^2}-\frac{3 i d (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^2}-\frac{d (c+d x) \sec (a+b x)}{b^2}-\frac{6 d (c+d x) \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^2}+\frac{d (c+d x) \tanh ^{-1}(\cos (a+b x))}{b^2}+\frac{2 d^2 x \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{d^2 x \tanh ^{-1}(\cos (a+b x))}{b^2}-\frac{3 i (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}-\frac{3 (c+d x)^2 \csc (a+b x)}{2 b}+\frac{(c+d x)^2 \csc (a+b x) \sec ^2(a+b x)}{2 b}",1,"-1/2*((6*I)*b^2*c^2*ArcTan[E^(I*(a + b*x))] + (4*I)*d^2*ArcTan[E^(I*(a + b*x))] - 6*b^2*c*d*x*Log[1 - I*E^(I*(a + b*x))] - 3*b^2*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] + 6*b^2*c*d*x*Log[1 + I*E^(I*(a + b*x))] + 3*b^2*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] - (6*I)*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))] + (6*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] + 6*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))] - 6*d^2*PolyLog[3, I*E^(I*(a + b*x))])/b^3 - ((c + d*x)*Csc[a]*Sec[a]*(b*c*Cos[a] + b*d*x*Cos[a] + d*Sin[a]))/b^2 + ((4*I)*c*d*ArcTan[(I*Cos[a] - I*Sin[a]*Tan[(b*x)/2])/Sqrt[Cos[a]^2 + Sin[a]^2]])/(b^2*Sqrt[Cos[a]^2 + Sin[a]^2]) + (Sec[a/2]*Sec[a/2 + (b*x)/2]*(-(c^2*Sin[(b*x)/2]) - 2*c*d*x*Sin[(b*x)/2] - d^2*x^2*Sin[(b*x)/2]))/(2*b) + (Csc[a/2]*Csc[a/2 + (b*x)/2]*(c^2*Sin[(b*x)/2] + 2*c*d*x*Sin[(b*x)/2] + d^2*x^2*Sin[(b*x)/2]))/(2*b) + (c^2 + 2*c*d*x + d^2*x^2)/(4*b*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])^2) + (-(c*d*Sin[(b*x)/2]) - d^2*x*Sin[(b*x)/2])/(b^2*(Cos[a/2] - Sin[a/2])*(Cos[a/2 + (b*x)/2] - Sin[a/2 + (b*x)/2])) + (-c^2 - 2*c*d*x - d^2*x^2)/(4*b*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2])^2) + (c*d*Sin[(b*x)/2] + d^2*x*Sin[(b*x)/2])/(b^2*(Cos[a/2] + Sin[a/2])*(Cos[a/2 + (b*x)/2] + Sin[a/2 + (b*x)/2])) + (2*d^2*((-2*ArcTan[Tan[a]]*ArcTanh[(-Cos[a] + Sin[a]*Tan[(b*x)/2])/Sqrt[Cos[a]^2 + Sin[a]^2]])/Sqrt[Cos[a]^2 + Sin[a]^2] + (((b*x + ArcTan[Tan[a]])*(Log[1 - E^(I*(b*x + ArcTan[Tan[a]]))] - Log[1 + E^(I*(b*x + ArcTan[Tan[a]]))]) + I*(PolyLog[2, -E^(I*(b*x + ArcTan[Tan[a]]))] - PolyLog[2, E^(I*(b*x + ArcTan[Tan[a]]))]))*Sec[a])/Sqrt[1 + Tan[a]^2]))/b^3","B",0
319,1,669,162,6.5878417,"\int (c+d x) \csc ^2(a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]^2*Sec[a + b*x]^3,x]","\frac{d \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b^2}-\frac{d \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b^2}-\frac{d \sin \left(\frac{1}{2} (a+b x)\right)}{2 b^2 \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)}+\frac{d \sin \left(\frac{1}{2} (a+b x)\right)}{2 b^2 \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)}+\frac{d \left(a \cos \left(\frac{1}{2} (a+b x)\right)-(a+b x) \cos \left(\frac{1}{2} (a+b x)\right)\right) \csc \left(\frac{1}{2} (a+b x)\right)}{2 b^2}+\frac{d \left(a \sin \left(\frac{1}{2} (a+b x)\right)-(a+b x) \sin \left(\frac{1}{2} (a+b x)\right)\right) \sec \left(\frac{1}{2} (a+b x)\right)}{2 b^2}-\frac{c \csc (a+b x) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\sin ^2(a+b x)\right)}{b}-\frac{3 d x \left(-i \left(\text{Li}_2\left(\frac{1}{2} \left((1+i)-(1-i) \tan \left(\frac{1}{2} (a+b x)\right)\right)\right)+\log \left(1+i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(\frac{1}{2}-\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)+1\right)\right)\right)+i \left(\text{Li}_2\left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)+i\right)\right)+\log \left(1-i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)+1\right)\right)\right)-i \left(\text{Li}_2\left(\frac{1}{2} \left((1-i) \tan \left(\frac{1}{2} (a+b x)\right)+(1+i)\right)\right)+\log \left(1-i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(-\frac{1}{2}+\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)-1\right)\right)\right)+i \left(\text{Li}_2\left(\frac{1}{2} \left((1+i) \tan \left(\frac{1}{2} (a+b x)\right)+(1-i)\right)\right)+\log \left(1+i \tan \left(\frac{1}{2} (a+b x)\right)\right) \log \left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(\tan \left(\frac{1}{2} (a+b x)\right)-1\right)\right)\right)+a \log \left(1-\tan \left(\frac{1}{2} (a+b x)\right)\right)-a \log \left(\tan \left(\frac{1}{2} (a+b x)\right)+1\right)\right)}{2 b \left(-i \log \left(1-i \tan \left(\frac{1}{2} (a+b x)\right)\right)+i \log \left(1+i \tan \left(\frac{1}{2} (a+b x)\right)\right)+a\right)}+\frac{d x}{4 b \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)^2}-\frac{d x}{4 b \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)^2}","\frac{3 i d \text{Li}_2\left(-i e^{i (a+b x)}\right)}{2 b^2}-\frac{3 i d \text{Li}_2\left(i e^{i (a+b x)}\right)}{2 b^2}-\frac{d \sec (a+b x)}{2 b^2}-\frac{d \tanh ^{-1}(\cos (a+b x))}{b^2}-\frac{3 (c+d x) \csc (a+b x)}{2 b}+\frac{(c+d x) \csc (a+b x) \sec ^2(a+b x)}{2 b}+\frac{3 c \tanh ^{-1}(\sin (a+b x))}{2 b}-\frac{3 i d x \tan ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(d*(a*Cos[(a + b*x)/2] - (a + b*x)*Cos[(a + b*x)/2])*Csc[(a + b*x)/2])/(2*b^2) - (c*Csc[a + b*x]*Hypergeometric2F1[-1/2, 2, 1/2, Sin[a + b*x]^2])/b - (d*Log[Cos[(a + b*x)/2]])/b^2 + (d*Log[Sin[(a + b*x)/2]])/b^2 - (3*d*x*(a*Log[1 - Tan[(a + b*x)/2]] - a*Log[1 + Tan[(a + b*x)/2]] - I*(Log[1 + I*Tan[(a + b*x)/2]]*Log[(1/2 - I/2)*(1 + Tan[(a + b*x)/2])] + PolyLog[2, ((1 + I) - (1 - I)*Tan[(a + b*x)/2])/2]) + I*(Log[1 - I*Tan[(a + b*x)/2]]*Log[(1/2 + I/2)*(1 + Tan[(a + b*x)/2])] + PolyLog[2, (-1/2 - I/2)*(I + Tan[(a + b*x)/2])]) - I*(Log[1 - I*Tan[(a + b*x)/2]]*Log[(-1/2 + I/2)*(-1 + Tan[(a + b*x)/2])] + PolyLog[2, ((1 + I) + (1 - I)*Tan[(a + b*x)/2])/2]) + I*(Log[1 + I*Tan[(a + b*x)/2]]*Log[(-1/2 - I/2)*(-1 + Tan[(a + b*x)/2])] + PolyLog[2, ((1 - I) + (1 + I)*Tan[(a + b*x)/2])/2])))/(2*b*(a - I*Log[1 - I*Tan[(a + b*x)/2]] + I*Log[1 + I*Tan[(a + b*x)/2]])) + (d*x)/(4*b*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])^2) - (d*Sin[(a + b*x)/2])/(2*b^2*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])) - (d*x)/(4*b*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])^2) + (d*Sin[(a + b*x)/2])/(2*b^2*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])) + (d*Sec[(a + b*x)/2]*(a*Sin[(a + b*x)/2] - (a + b*x)*Sin[(a + b*x)/2]))/(2*b^2)","C",0
320,0,0,27,20.7427288,"\int \frac{\csc ^2(a+b x) \sec ^3(a+b x)}{c+d x} \, dx","Integrate[(Csc[a + b*x]^2*Sec[a + b*x]^3)/(c + d*x),x]","\int \frac{\csc ^2(a+b x) \sec ^3(a+b x)}{c+d x} \, dx","\text{Int}\left(\frac{\csc ^2(a+b x) \sec ^3(a+b x)}{c+d x},x\right)",0,"Integrate[(Csc[a + b*x]^2*Sec[a + b*x]^3)/(c + d*x), x]","A",-1
321,0,0,27,25.8835339,"\int \frac{\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx","Integrate[(Csc[a + b*x]^2*Sec[a + b*x]^3)/(c + d*x)^2,x]","\int \frac{\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{\csc ^2(a+b x) \sec ^3(a+b x)}{(c+d x)^2},x\right)",0,"Integrate[(Csc[a + b*x]^2*Sec[a + b*x]^3)/(c + d*x)^2, x]","A",-1
322,0,0,27,32.3715466,"\int (c+d x)^m \csc ^3(a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x]^3,x]","\int (c+d x)^m \csc ^3(a+b x) \sec ^3(a+b x) \, dx","\text{Int}\left(\csc ^3(a+b x) \sec ^3(a+b x) (c+d x)^m,x\right)",0,"Integrate[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x]^3, x]","A",-1
323,1,483,318,8.6260031,"\int (c+d x)^3 \csc ^3(a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]^3*Sec[a + b*x]^3,x]","-\frac{8 b^3 c^3 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)-12 b^3 c^2 d x \log \left(1-e^{2 i (a+b x)}\right)+12 b^3 c^2 d x \log \left(1+e^{2 i (a+b x)}\right)-12 b^3 c d^2 x^2 \log \left(1-e^{2 i (a+b x)}\right)+12 b^3 c d^2 x^2 \log \left(1+e^{2 i (a+b x)}\right)-4 b^3 d^3 x^3 \log \left(1-e^{2 i (a+b x)}\right)+4 b^3 d^3 x^3 \log \left(1+e^{2 i (a+b x)}\right)-3 i d \text{Li}_2\left(-e^{2 i (a+b x)}\right) \left(2 b^2 (c+d x)^2+d^2\right)+3 i d \text{Li}_2\left(e^{2 i (a+b x)}\right) \left(2 b^2 (c+d x)^2+d^2\right)+2 b^2 (c+d x)^2 \csc (2 (a+b x)) (2 b (c+d x) \cot (2 (a+b x))+3 d)+6 b c d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)-6 b c d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)+12 b c d^2 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)+6 b d^3 x \text{Li}_3\left(-e^{2 i (a+b x)}\right)-6 b d^3 x \text{Li}_3\left(e^{2 i (a+b x)}\right)+3 i d^3 \text{Li}_4\left(-e^{2 i (a+b x)}\right)-3 i d^3 \text{Li}_4\left(e^{2 i (a+b x)}\right)-6 b d^3 x \log \left(1-e^{2 i (a+b x)}\right)+6 b d^3 x \log \left(1+e^{2 i (a+b x)}\right)}{2 b^4}","\frac{3 i d^3 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 i d^3 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 i d^3 \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{2 b^4}+\frac{3 i d^3 \text{Li}_4\left(e^{2 i (a+b x)}\right)}{2 b^4}-\frac{3 d^2 (c+d x) \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{b^3}+\frac{3 d^2 (c+d x) \text{Li}_3\left(e^{2 i (a+b x)}\right)}{b^3}-\frac{6 d^2 (c+d x) \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b^3}+\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{3 d (c+d x)^2 \csc (2 a+2 b x)}{b^2}-\frac{4 (c+d x)^3 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}-\frac{2 (c+d x)^3 \cot (2 a+2 b x) \csc (2 a+2 b x)}{b}",1,"-1/2*(8*b^3*c^3*ArcTanh[E^((2*I)*(a + b*x))] + 12*b*c*d^2*ArcTanh[E^((2*I)*(a + b*x))] + 2*b^2*(c + d*x)^2*(3*d + 2*b*(c + d*x)*Cot[2*(a + b*x)])*Csc[2*(a + b*x)] - 12*b^3*c^2*d*x*Log[1 - E^((2*I)*(a + b*x))] - 6*b*d^3*x*Log[1 - E^((2*I)*(a + b*x))] - 12*b^3*c*d^2*x^2*Log[1 - E^((2*I)*(a + b*x))] - 4*b^3*d^3*x^3*Log[1 - E^((2*I)*(a + b*x))] + 12*b^3*c^2*d*x*Log[1 + E^((2*I)*(a + b*x))] + 6*b*d^3*x*Log[1 + E^((2*I)*(a + b*x))] + 12*b^3*c*d^2*x^2*Log[1 + E^((2*I)*(a + b*x))] + 4*b^3*d^3*x^3*Log[1 + E^((2*I)*(a + b*x))] - (3*I)*d*(d^2 + 2*b^2*(c + d*x)^2)*PolyLog[2, -E^((2*I)*(a + b*x))] + (3*I)*d*(d^2 + 2*b^2*(c + d*x)^2)*PolyLog[2, E^((2*I)*(a + b*x))] + 6*b*c*d^2*PolyLog[3, -E^((2*I)*(a + b*x))] + 6*b*d^3*x*PolyLog[3, -E^((2*I)*(a + b*x))] - 6*b*c*d^2*PolyLog[3, E^((2*I)*(a + b*x))] - 6*b*d^3*x*PolyLog[3, E^((2*I)*(a + b*x))] + (3*I)*d^3*PolyLog[4, -E^((2*I)*(a + b*x))] - (3*I)*d^3*PolyLog[4, E^((2*I)*(a + b*x))])/b^4","A",1
324,1,381,190,8.1771778,"\int (c+d x)^2 \csc ^3(a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]^3*Sec[a + b*x]^3,x]","8 \left(\frac{\csc (a) \csc (a+b x) \left(c d \sin (b x)+d^2 x \sin (b x)\right)}{8 b^2}+\frac{\sec (a) \sec (a+b x) \left(d^2 (-x) \sin (b x)-c d \sin (b x)\right)}{8 b^2}-\frac{d \csc (2 a) (c+d x)}{4 b^2}-\frac{4 b^2 c^2 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)-4 b^2 c d x \log \left(1-e^{2 i (a+b x)}\right)+4 b^2 c d x \log \left(1+e^{2 i (a+b x)}\right)-2 b^2 d^2 x^2 \log \left(1-e^{2 i (a+b x)}\right)+2 b^2 d^2 x^2 \log \left(1+e^{2 i (a+b x)}\right)-2 i b d (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)+2 i b d (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)+d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)-d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)+2 d^2 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{8 b^3}+\frac{\left(-c^2-2 c d x-d^2 x^2\right) \csc ^2(a+b x)}{16 b}+\frac{\left(c^2+2 c d x+d^2 x^2\right) \sec ^2(a+b x)}{16 b}\right)","-\frac{d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{b^3}+\frac{d^2 \text{Li}_3\left(e^{2 i (a+b x)}\right)}{b^3}-\frac{d^2 \tanh ^{-1}(\cos (2 a+2 b x))}{b^3}+\frac{2 i d (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{2 i d (c+d x) \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{2 d (c+d x) \csc (2 a+2 b x)}{b^2}-\frac{4 (c+d x)^2 \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}-\frac{2 (c+d x)^2 \cot (2 a+2 b x) \csc (2 a+2 b x)}{b}",1,"8*(-1/4*(d*(c + d*x)*Csc[2*a])/b^2 + ((-c^2 - 2*c*d*x - d^2*x^2)*Csc[a + b*x]^2)/(16*b) - (4*b^2*c^2*ArcTanh[E^((2*I)*(a + b*x))] + 2*d^2*ArcTanh[E^((2*I)*(a + b*x))] - 4*b^2*c*d*x*Log[1 - E^((2*I)*(a + b*x))] - 2*b^2*d^2*x^2*Log[1 - E^((2*I)*(a + b*x))] + 4*b^2*c*d*x*Log[1 + E^((2*I)*(a + b*x))] + 2*b^2*d^2*x^2*Log[1 + E^((2*I)*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, -E^((2*I)*(a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, E^((2*I)*(a + b*x))] + d^2*PolyLog[3, -E^((2*I)*(a + b*x))] - d^2*PolyLog[3, E^((2*I)*(a + b*x))])/(8*b^3) + ((c^2 + 2*c*d*x + d^2*x^2)*Sec[a + b*x]^2)/(16*b) + (Sec[a]*Sec[a + b*x]*(-(c*d*Sin[b*x]) - d^2*x*Sin[b*x]))/(8*b^2) + (Csc[a]*Csc[a + b*x]*(c*d*Sin[b*x] + d^2*x*Sin[b*x]))/(8*b^2))","B",1
325,1,236,110,2.0884975,"\int (c+d x) \csc ^3(a+b x) \sec ^3(a+b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]^3*Sec[a + b*x]^3,x]","\frac{d \left(i \left(\text{Li}_2\left(-e^{2 i (a+b x)}\right)-\text{Li}_2\left(e^{2 i (a+b x)}\right)\right)+2 (a+b x) \left(\log \left(1-e^{2 i (a+b x)}\right)-\log \left(1+e^{2 i (a+b x)}\right)\right)\right)}{b^2}-\frac{d \tan (a+b x)}{2 b^2}-\frac{d \cot (a+b x)}{2 b^2}+\frac{d (2 a-2 (a+b x)) \csc ^2(a+b x)}{4 b^2}+\frac{d (2 (a+b x)-2 a) \sec ^2(a+b x)}{4 b^2}-\frac{2 a d \log (\tan (a+b x))}{b^2}-\frac{c \csc ^2(a+b x)}{2 b}+\frac{c \sec ^2(a+b x)}{2 b}+\frac{2 c \log (\sin (a+b x))}{b}-\frac{2 c \log (\cos (a+b x))}{b}","\frac{i d \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}-\frac{i d \text{Li}_2\left(e^{2 i (a+b x)}\right)}{b^2}-\frac{d \csc (2 a+2 b x)}{b^2}-\frac{4 (c+d x) \tanh ^{-1}\left(e^{2 i (a+b x)}\right)}{b}-\frac{2 (c+d x) \cot (2 a+2 b x) \csc (2 a+2 b x)}{b}",1,"-1/2*(d*Cot[a + b*x])/b^2 - (c*Csc[a + b*x]^2)/(2*b) + (d*(2*a - 2*(a + b*x))*Csc[a + b*x]^2)/(4*b^2) - (2*c*Log[Cos[a + b*x]])/b + (2*c*Log[Sin[a + b*x]])/b - (2*a*d*Log[Tan[a + b*x]])/b^2 + (d*(2*(a + b*x)*(Log[1 - E^((2*I)*(a + b*x))] - Log[1 + E^((2*I)*(a + b*x))]) + I*(PolyLog[2, -E^((2*I)*(a + b*x))] - PolyLog[2, E^((2*I)*(a + b*x))])))/b^2 + (c*Sec[a + b*x]^2)/(2*b) + (d*(-2*a + 2*(a + b*x))*Sec[a + b*x]^2)/(4*b^2) - (d*Tan[a + b*x])/(2*b^2)","B",1
326,0,0,24,28.7027376,"\int \frac{\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx","Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^3)/(c + d*x),x]","\int \frac{\csc ^3(a+b x) \sec ^3(a+b x)}{c+d x} \, dx","8 \text{Int}\left(\frac{\csc ^3(2 a+2 b x)}{c+d x},x\right)",0,"Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^3)/(c + d*x), x]","A",-1
327,0,0,24,32.4656433,"\int \frac{\csc ^3(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx","Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^3)/(c + d*x)^2,x]","\int \frac{\csc ^3(a+b x) \sec ^3(a+b x)}{(c+d x)^2} \, dx","8 \text{Int}\left(\frac{\csc ^3(2 a+2 b x)}{(c+d x)^2},x\right)",0,"Integrate[(Csc[a + b*x]^3*Sec[a + b*x]^3)/(c + d*x)^2, x]","A",-1
328,1,73,83,0.3301083,"\int x \cos ^{\frac{5}{2}}(a+b x) \sin (a+b x) \, dx","Integrate[x*Cos[a + b*x]^(5/2)*Sin[a + b*x],x]","\frac{40 F\left(\left.\frac{1}{2} (a+b x)\right|2\right)+\sqrt{\cos (a+b x)} (46 \sin (a+b x)+6 \sin (3 (a+b x))-63 b x \cos (a+b x)-21 b x \cos (3 (a+b x)))}{294 b^2}","\frac{20 F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{147 b^2}+\frac{4 \sin (a+b x) \cos ^{\frac{5}{2}}(a+b x)}{49 b^2}+\frac{20 \sin (a+b x) \sqrt{\cos (a+b x)}}{147 b^2}-\frac{2 x \cos ^{\frac{7}{2}}(a+b x)}{7 b}",1,"(40*EllipticF[(a + b*x)/2, 2] + Sqrt[Cos[a + b*x]]*(-63*b*x*Cos[a + b*x] - 21*b*x*Cos[3*(a + b*x)] + 46*Sin[a + b*x] + 6*Sin[3*(a + b*x)]))/(294*b^2)","A",1
329,1,51,60,0.3745522,"\int x \cos ^{\frac{3}{2}}(a+b x) \sin (a+b x) \, dx","Integrate[x*Cos[a + b*x]^(3/2)*Sin[a + b*x],x]","-\frac{2 \left(\cos ^{\frac{3}{2}}(a+b x) (5 b x \cos (a+b x)-2 \sin (a+b x))-6 E\left(\left.\frac{1}{2} (a+b x)\right|2\right)\right)}{25 b^2}","\frac{12 E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{25 b^2}+\frac{4 \sin (a+b x) \cos ^{\frac{3}{2}}(a+b x)}{25 b^2}-\frac{2 x \cos ^{\frac{5}{2}}(a+b x)}{5 b}",1,"(-2*(-6*EllipticE[(a + b*x)/2, 2] + Cos[a + b*x]^(3/2)*(5*b*x*Cos[a + b*x] - 2*Sin[a + b*x])))/(25*b^2)","A",1
330,1,52,60,0.1520215,"\int x \sqrt{\cos (a+b x)} \sin (a+b x) \, dx","Integrate[x*Sqrt[Cos[a + b*x]]*Sin[a + b*x],x]","\frac{4 F\left(\left.\frac{1}{2} (a+b x)\right|2\right)+2 \sqrt{\cos (a+b x)} (2 \sin (a+b x)-3 b x \cos (a+b x))}{9 b^2}","\frac{4 F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{9 b^2}+\frac{4 \sin (a+b x) \sqrt{\cos (a+b x)}}{9 b^2}-\frac{2 x \cos ^{\frac{3}{2}}(a+b x)}{3 b}",1,"(4*EllipticF[(a + b*x)/2, 2] + 2*Sqrt[Cos[a + b*x]]*(-3*b*x*Cos[a + b*x] + 2*Sin[a + b*x]))/(9*b^2)","A",1
331,1,181,33,1.7592788,"\int \frac{x \sin (a+b x)}{\sqrt{\cos (a+b x)}} \, dx","Integrate[(x*Sin[a + b*x])/Sqrt[Cos[a + b*x]],x]","\frac{4 \cos ^2\left(\frac{1}{2} (a+b x)\right)^{3/2} \sqrt{\frac{\cos (a+b x)}{(\cos (a+b x)+1)^2}} \sqrt{\frac{1}{\cos (a+b x)+1}} \left(\left(2 \tan \left(\frac{1}{2} (a+b x)\right)-b x\right) \sqrt{\cos (a+b x) \sec ^2\left(\frac{1}{2} (a+b x)\right)}-2 \sqrt{\sec ^2\left(\frac{1}{2} (a+b x)\right)} F\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (a+b x)\right)\right)\right|-1\right)+2 \sqrt{\sec ^2\left(\frac{1}{2} (a+b x)\right)} E\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (a+b x)\right)\right)\right|-1\right)\right)}{b^2 \sqrt{\frac{\cos (a+b x)}{\cos (a+b x)+1}}}","\frac{4 E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b^2}-\frac{2 x \sqrt{\cos (a+b x)}}{b}",1,"(4*(Cos[(a + b*x)/2]^2)^(3/2)*Sqrt[Cos[a + b*x]/(1 + Cos[a + b*x])^2]*Sqrt[(1 + Cos[a + b*x])^(-1)]*(2*EllipticE[ArcSin[Tan[(a + b*x)/2]], -1]*Sqrt[Sec[(a + b*x)/2]^2] - 2*EllipticF[ArcSin[Tan[(a + b*x)/2]], -1]*Sqrt[Sec[(a + b*x)/2]^2] + Sqrt[Cos[a + b*x]*Sec[(a + b*x)/2]^2]*(-(b*x) + 2*Tan[(a + b*x)/2])))/(b^2*Sqrt[Cos[a + b*x]/(1 + Cos[a + b*x])])","B",0
332,1,33,33,0.1595367,"\int \frac{x \sin (a+b x)}{\cos ^{\frac{3}{2}}(a+b x)} \, dx","Integrate[(x*Sin[a + b*x])/Cos[a + b*x]^(3/2),x]","\frac{2 x}{b \sqrt{\cos (a+b x)}}-\frac{4 F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b^2}","\frac{2 x}{b \sqrt{\cos (a+b x)}}-\frac{4 F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b^2}",1,"(2*x)/(b*Sqrt[Cos[a + b*x]]) - (4*EllipticF[(a + b*x)/2, 2])/b^2","A",1
333,1,54,60,0.1851865,"\int \frac{x \sin (a+b x)}{\cos ^{\frac{5}{2}}(a+b x)} \, dx","Integrate[(x*Sin[a + b*x])/Cos[a + b*x]^(5/2),x]","\frac{2 \left(-\sin (2 (a+b x))+2 \cos ^{\frac{3}{2}}(a+b x) E\left(\left.\frac{1}{2} (a+b x)\right|2\right)+b x\right)}{3 b^2 \cos ^{\frac{3}{2}}(a+b x)}","\frac{4 E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{3 b^2}-\frac{4 \sin (a+b x)}{3 b^2 \sqrt{\cos (a+b x)}}+\frac{2 x}{3 b \cos ^{\frac{3}{2}}(a+b x)}",1,"(2*(b*x + 2*Cos[a + b*x]^(3/2)*EllipticE[(a + b*x)/2, 2] - Sin[2*(a + b*x)]))/(3*b^2*Cos[a + b*x]^(3/2))","A",1
334,1,53,60,0.2295068,"\int \frac{x \sin (a+b x)}{\cos ^{\frac{7}{2}}(a+b x)} \, dx","Integrate[(x*Sin[a + b*x])/Cos[a + b*x]^(7/2),x]","-\frac{2 \left(\sin (2 (a+b x))+2 \cos ^{\frac{5}{2}}(a+b x) F\left(\left.\frac{1}{2} (a+b x)\right|2\right)-3 b x\right)}{15 b^2 \cos ^{\frac{5}{2}}(a+b x)}","-\frac{4 F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{15 b^2}-\frac{4 \sin (a+b x)}{15 b^2 \cos ^{\frac{3}{2}}(a+b x)}+\frac{2 x}{5 b \cos ^{\frac{5}{2}}(a+b x)}",1,"(-2*(-3*b*x + 2*Cos[a + b*x]^(5/2)*EllipticF[(a + b*x)/2, 2] + Sin[2*(a + b*x)]))/(15*b^2*Cos[a + b*x]^(5/2))","A",1
335,1,65,83,0.2561844,"\int \frac{x \sin (a+b x)}{\cos ^{\frac{9}{2}}(a+b x)} \, dx","Integrate[(x*Sin[a + b*x])/Cos[a + b*x]^(9/2),x]","\frac{-10 \sin (2 (a+b x))-3 \sin (4 (a+b x))+24 \cos ^{\frac{7}{2}}(a+b x) E\left(\left.\frac{1}{2} (a+b x)\right|2\right)+20 b x}{70 b^2 \cos ^{\frac{7}{2}}(a+b x)}","\frac{12 E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{35 b^2}-\frac{4 \sin (a+b x)}{35 b^2 \cos ^{\frac{5}{2}}(a+b x)}-\frac{12 \sin (a+b x)}{35 b^2 \sqrt{\cos (a+b x)}}+\frac{2 x}{7 b \cos ^{\frac{7}{2}}(a+b x)}",1,"(20*b*x + 24*Cos[a + b*x]^(7/2)*EllipticE[(a + b*x)/2, 2] - 10*Sin[2*(a + b*x)] - 3*Sin[4*(a + b*x)])/(70*b^2*Cos[a + b*x]^(7/2))","A",1
336,1,65,103,0.2799902,"\int x \sec ^{\frac{9}{2}}(a+b x) \sin (a+b x) \, dx","Integrate[x*Sec[a + b*x]^(9/2)*Sin[a + b*x],x]","\frac{\sec ^{\frac{7}{2}}(a+b x) \left(-10 \sin (2 (a+b x))-3 \sin (4 (a+b x))+24 \cos ^{\frac{7}{2}}(a+b x) E\left(\left.\frac{1}{2} (a+b x)\right|2\right)+20 b x\right)}{70 b^2}","-\frac{4 \sin (a+b x) \sec ^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{12 \sin (a+b x) \sqrt{\sec (a+b x)}}{35 b^2}+\frac{12 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{35 b^2}+\frac{2 x \sec ^{\frac{7}{2}}(a+b x)}{7 b}",1,"(Sec[a + b*x]^(7/2)*(20*b*x + 24*Cos[a + b*x]^(7/2)*EllipticE[(a + b*x)/2, 2] - 10*Sin[2*(a + b*x)] - 3*Sin[4*(a + b*x)]))/(70*b^2)","A",1
337,1,61,80,0.2311761,"\int x \sec ^{\frac{7}{2}}(a+b x) \sin (a+b x) \, dx","Integrate[x*Sec[a + b*x]^(7/2)*Sin[a + b*x],x]","\frac{2 \sqrt{\sec (a+b x)} \left(-2 \tan (a+b x)+3 b x \sec ^2(a+b x)-2 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)\right)}{15 b^2}","-\frac{4 \sin (a+b x) \sec ^{\frac{3}{2}}(a+b x)}{15 b^2}-\frac{4 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{15 b^2}+\frac{2 x \sec ^{\frac{5}{2}}(a+b x)}{5 b}",1,"(2*Sqrt[Sec[a + b*x]]*(-2*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2] + 3*b*x*Sec[a + b*x]^2 - 2*Tan[a + b*x]))/(15*b^2)","A",1
338,1,54,80,0.1842613,"\int x \sec ^{\frac{5}{2}}(a+b x) \sin (a+b x) \, dx","Integrate[x*Sec[a + b*x]^(5/2)*Sin[a + b*x],x]","\frac{2 \sec ^{\frac{3}{2}}(a+b x) \left(-\sin (2 (a+b x))+2 \cos ^{\frac{3}{2}}(a+b x) E\left(\left.\frac{1}{2} (a+b x)\right|2\right)+b x\right)}{3 b^2}","-\frac{4 \sin (a+b x) \sqrt{\sec (a+b x)}}{3 b^2}+\frac{4 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{3 b^2}+\frac{2 x \sec ^{\frac{3}{2}}(a+b x)}{3 b}",1,"(2*Sec[a + b*x]^(3/2)*(b*x + 2*Cos[a + b*x]^(3/2)*EllipticE[(a + b*x)/2, 2] - Sin[2*(a + b*x)]))/(3*b^2)","A",1
339,1,42,53,0.1269515,"\int x \sec ^{\frac{3}{2}}(a+b x) \sin (a+b x) \, dx","Integrate[x*Sec[a + b*x]^(3/2)*Sin[a + b*x],x]","\frac{2 \sqrt{\sec (a+b x)} \left(b x-2 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)\right)}{b^2}","\frac{2 x \sqrt{\sec (a+b x)}}{b}-\frac{4 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b^2}",1,"(2*(b*x - 2*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])*Sqrt[Sec[a + b*x]])/b^2","A",1
340,1,132,53,2.1535641,"\int x \sqrt{\sec (a+b x)} \sin (a+b x) \, dx","Integrate[x*Sqrt[Sec[a + b*x]]*Sin[a + b*x],x]","\frac{2 \left(2 \tan \left(\frac{1}{2} (a+b x)\right)-\frac{2 \sec ^2\left(\frac{1}{2} (a+b x)\right) F\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (a+b x)\right)\right)\right|-1\right)}{\sqrt{\cos (a+b x) \sec ^4\left(\frac{1}{2} (a+b x)\right)}}+\frac{2 \sec ^2\left(\frac{1}{2} (a+b x)\right) E\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (a+b x)\right)\right)\right|-1\right)}{\sqrt{\cos (a+b x) \sec ^4\left(\frac{1}{2} (a+b x)\right)}}-b x\right)}{b^2 \sqrt{\sec (a+b x)}}","\frac{4 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b^2}-\frac{2 x}{b \sqrt{\sec (a+b x)}}",1,"(2*(-(b*x) + (2*EllipticE[ArcSin[Tan[(a + b*x)/2]], -1]*Sec[(a + b*x)/2]^2)/Sqrt[Cos[a + b*x]*Sec[(a + b*x)/2]^4] - (2*EllipticF[ArcSin[Tan[(a + b*x)/2]], -1]*Sec[(a + b*x)/2]^2)/Sqrt[Cos[a + b*x]*Sec[(a + b*x)/2]^4] + 2*Tan[(a + b*x)/2]))/(b^2*Sqrt[Sec[a + b*x]])","B",1
341,1,63,80,0.2362568,"\int \frac{x \sin (a+b x)}{\sqrt{\sec (a+b x)}} \, dx","Integrate[(x*Sin[a + b*x])/Sqrt[Sec[a + b*x]],x]","\frac{\sqrt{\sec (a+b x)} \left(2 \sin (2 (a+b x))-6 b x \cos ^2(a+b x)+4 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)\right)}{9 b^2}","\frac{4 \sin (a+b x)}{9 b^2 \sqrt{\sec (a+b x)}}+\frac{4 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{9 b^2}-\frac{2 x}{3 b \sec ^{\frac{3}{2}}(a+b x)}",1,"(Sqrt[Sec[a + b*x]]*(-6*b*x*Cos[a + b*x]^2 + 4*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2] + 2*Sin[2*(a + b*x)]))/(9*b^2)","A",1
342,1,212,80,8.1041523,"\int \frac{x \sin (a+b x)}{\sec ^{\frac{3}{2}}(a+b x)} \, dx","Integrate[(x*Sin[a + b*x])/Sec[a + b*x]^(3/2),x]","\frac{\cos ^2\left(\frac{1}{2} (a+b x)\right) \sqrt{\sec (a+b x)} \left(\left(5 (a+b x)-12 \tan \left(\frac{1}{2} (a+b x)\right)-5 a\right) \left(\tan ^2\left(\frac{1}{2} (a+b x)\right)-1\right)-12 \sqrt{\cos (a+b x) \sec ^4\left(\frac{1}{2} (a+b x)\right)} F\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (a+b x)\right)\right)\right|-1\right)+12 \sqrt{\cos (a+b x) \sec ^4\left(\frac{1}{2} (a+b x)\right)} E\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (a+b x)\right)\right)\right|-1\right)\right)}{25 b^2}+\frac{\sqrt{\sec (a+b x)} \left(\frac{\sin (a+b x)}{25 b}+\frac{\sin (3 (a+b x))}{25 b}-\frac{1}{10} x \cos (a+b x)-\frac{1}{10} x \cos (3 (a+b x))\right)}{b}","\frac{4 \sin (a+b x)}{25 b^2 \sec ^{\frac{3}{2}}(a+b x)}+\frac{12 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{25 b^2}-\frac{2 x}{5 b \sec ^{\frac{5}{2}}(a+b x)}",1,"(Sqrt[Sec[a + b*x]]*(-1/10*(x*Cos[a + b*x]) - (x*Cos[3*(a + b*x)])/10 + Sin[a + b*x]/(25*b) + Sin[3*(a + b*x)]/(25*b)))/b + (Cos[(a + b*x)/2]^2*Sqrt[Sec[a + b*x]]*(12*EllipticE[ArcSin[Tan[(a + b*x)/2]], -1]*Sqrt[Cos[a + b*x]*Sec[(a + b*x)/2]^4] - 12*EllipticF[ArcSin[Tan[(a + b*x)/2]], -1]*Sqrt[Cos[a + b*x]*Sec[(a + b*x)/2]^4] + (-5*a + 5*(a + b*x) - 12*Tan[(a + b*x)/2])*(-1 + Tan[(a + b*x)/2]^2)))/(25*b^2)","B",1
343,1,89,103,0.3379422,"\int \frac{x \sin (a+b x)}{\sec ^{\frac{5}{2}}(a+b x)} \, dx","Integrate[(x*Sin[a + b*x])/Sec[a + b*x]^(5/2),x]","\frac{\sqrt{\sec (a+b x)} \left(52 \sin (2 (a+b x))+6 \sin (4 (a+b x))-84 b x \cos (2 (a+b x))-21 b x \cos (4 (a+b x))+80 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)-63 b x\right)}{588 b^2}","\frac{4 \sin (a+b x)}{49 b^2 \sec ^{\frac{5}{2}}(a+b x)}+\frac{20 \sin (a+b x)}{147 b^2 \sqrt{\sec (a+b x)}}+\frac{20 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{147 b^2}-\frac{2 x}{7 b \sec ^{\frac{7}{2}}(a+b x)}",1,"(Sqrt[Sec[a + b*x]]*(-63*b*x - 84*b*x*Cos[2*(a + b*x)] - 21*b*x*Cos[4*(a + b*x)] + 80*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2] + 52*Sin[2*(a + b*x)] + 6*Sin[4*(a + b*x)]))/(588*b^2)","A",1
344,1,67,88,0.5411846,"\int x \cos (a+b x) \sin ^{\frac{5}{2}}(a+b x) \, dx","Integrate[x*Cos[a + b*x]*Sin[a + b*x]^(5/2),x]","\frac{40 F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)+\sqrt{\sin (a+b x)} \left(84 b x \sin ^3(a+b x)+46 \cos (a+b x)-6 \cos (3 (a+b x))\right)}{294 b^2}","-\frac{20 F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{147 b^2}+\frac{4 \sin ^{\frac{5}{2}}(a+b x) \cos (a+b x)}{49 b^2}+\frac{20 \sqrt{\sin (a+b x)} \cos (a+b x)}{147 b^2}+\frac{2 x \sin ^{\frac{7}{2}}(a+b x)}{7 b}",1,"(40*EllipticF[(-2*a + Pi - 2*b*x)/4, 2] + Sqrt[Sin[a + b*x]]*(46*Cos[a + b*x] - 6*Cos[3*(a + b*x)] + 84*b*x*Sin[a + b*x]^3))/(294*b^2)","A",1
345,1,108,65,0.9034003,"\int x \cos (a+b x) \sin ^{\frac{3}{2}}(a+b x) \, dx","Integrate[x*Cos[a + b*x]*Sin[a + b*x]^(3/2),x]","\frac{\sqrt{\sin (a+b x)} \left(4 \tan \left(\frac{1}{2} (a+b x)\right) \sqrt{\sec ^2\left(\frac{1}{2} (a+b x)\right)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+2 \sin (2 (a+b x))-5 b x \cos (2 (a+b x))-12 \tan \left(\frac{1}{2} (a+b x)\right)+5 b x\right)}{25 b^2}","-\frac{12 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{25 b^2}+\frac{4 \sin ^{\frac{3}{2}}(a+b x) \cos (a+b x)}{25 b^2}+\frac{2 x \sin ^{\frac{5}{2}}(a+b x)}{5 b}",1,"(Sqrt[Sin[a + b*x]]*(5*b*x - 5*b*x*Cos[2*(a + b*x)] + 2*Sin[2*(a + b*x)] - 12*Tan[(a + b*x)/2] + 4*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[(a + b*x)/2]^2]*Sqrt[Sec[(a + b*x)/2]^2]*Tan[(a + b*x)/2]))/(25*b^2)","C",1
346,1,56,65,0.1891106,"\int x \cos (a+b x) \sqrt{\sin (a+b x)} \, dx","Integrate[x*Cos[a + b*x]*Sqrt[Sin[a + b*x]],x]","\frac{4 F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)+2 \sqrt{\sin (a+b x)} (3 b x \sin (a+b x)+2 \cos (a+b x))}{9 b^2}","-\frac{4 F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{9 b^2}+\frac{4 \sqrt{\sin (a+b x)} \cos (a+b x)}{9 b^2}+\frac{2 x \sin ^{\frac{3}{2}}(a+b x)}{3 b}",1,"(4*EllipticF[(-2*a + Pi - 2*b*x)/4, 2] + 2*Sqrt[Sin[a + b*x]]*(2*Cos[a + b*x] + 3*b*x*Sin[a + b*x]))/(9*b^2)","A",1
347,1,86,38,1.1588184,"\int \frac{x \cos (a+b x)}{\sqrt{\sin (a+b x)}} \, dx","Integrate[(x*Cos[a + b*x])/Sqrt[Sin[a + b*x]],x]","\frac{2 \sqrt{\sin (a+b x)} \left(2 \tan \left(\frac{1}{2} (a+b x)\right) \sqrt{\sec ^2\left(\frac{1}{2} (a+b x)\right)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-6 \tan \left(\frac{1}{2} (a+b x)\right)+3 b x\right)}{3 b^2}","\frac{2 x \sqrt{\sin (a+b x)}}{b}-\frac{4 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b^2}",1,"(2*Sqrt[Sin[a + b*x]]*(3*b*x - 6*Tan[(a + b*x)/2] + 2*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[(a + b*x)/2]^2]*Sqrt[Sec[(a + b*x)/2]^2]*Tan[(a + b*x)/2]))/(3*b^2)","C",1
348,1,37,38,0.1771462,"\int \frac{x \cos (a+b x)}{\sin ^{\frac{3}{2}}(a+b x)} \, dx","Integrate[(x*Cos[a + b*x])/Sin[a + b*x]^(3/2),x]","\frac{2 \left(-\frac{b x}{\sqrt{\sin (a+b x)}}-2 F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{b^2}","\frac{4 F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b^2}-\frac{2 x}{b \sqrt{\sin (a+b x)}}",1,"(2*(-2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2] - (b*x)/Sqrt[Sin[a + b*x]]))/b^2","A",1
349,1,56,65,0.1791203,"\int \frac{x \cos (a+b x)}{\sin ^{\frac{5}{2}}(a+b x)} \, dx","Integrate[(x*Cos[a + b*x])/Sin[a + b*x]^(5/2),x]","-\frac{2 \left(\sin (2 (a+b x))-2 \sin ^{\frac{3}{2}}(a+b x) E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)+b x\right)}{3 b^2 \sin ^{\frac{3}{2}}(a+b x)}","-\frac{4 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{3 b^2}-\frac{4 \cos (a+b x)}{3 b^2 \sqrt{\sin (a+b x)}}-\frac{2 x}{3 b \sin ^{\frac{3}{2}}(a+b x)}",1,"(-2*(b*x - 2*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(3/2) + Sin[2*(a + b*x)]))/(3*b^2*Sin[a + b*x]^(3/2))","A",1
350,1,57,65,0.2280152,"\int \frac{x \cos (a+b x)}{\sin ^{\frac{7}{2}}(a+b x)} \, dx","Integrate[(x*Cos[a + b*x])/Sin[a + b*x]^(7/2),x]","-\frac{2 \left(\sin (2 (a+b x))+2 \sin ^{\frac{5}{2}}(a+b x) F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)+3 b x\right)}{15 b^2 \sin ^{\frac{5}{2}}(a+b x)}","\frac{4 F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{15 b^2}-\frac{4 \cos (a+b x)}{15 b^2 \sin ^{\frac{3}{2}}(a+b x)}-\frac{2 x}{5 b \sin ^{\frac{5}{2}}(a+b x)}",1,"(-2*(3*b*x + 2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(5/2) + Sin[2*(a + b*x)]))/(15*b^2*Sin[a + b*x]^(5/2))","A",1
351,1,73,88,0.2883177,"\int \frac{x \cos (a+b x)}{\sin ^{\frac{9}{2}}(a+b x)} \, dx","Integrate[(x*Cos[a + b*x])/Sin[a + b*x]^(9/2),x]","-\frac{2 \left(\sin (2 (a+b x))+6 \sin ^3(a+b x) \cos (a+b x)-6 \sin ^{\frac{7}{2}}(a+b x) E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)+5 b x\right)}{35 b^2 \sin ^{\frac{7}{2}}(a+b x)}","-\frac{12 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{35 b^2}-\frac{4 \cos (a+b x)}{35 b^2 \sin ^{\frac{5}{2}}(a+b x)}-\frac{12 \cos (a+b x)}{35 b^2 \sqrt{\sin (a+b x)}}-\frac{2 x}{7 b \sin ^{\frac{7}{2}}(a+b x)}",1,"(-2*(5*b*x + 6*Cos[a + b*x]*Sin[a + b*x]^3 - 6*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(7/2) + Sin[2*(a + b*x)]))/(35*b^2*Sin[a + b*x]^(7/2))","A",1
352,1,73,108,0.265639,"\int x \cos (a+b x) \csc ^{\frac{9}{2}}(a+b x) \, dx","Integrate[x*Cos[a + b*x]*Csc[a + b*x]^(9/2),x]","-\frac{2 \csc ^{\frac{7}{2}}(a+b x) \left(\sin (2 (a+b x))+6 \sin ^3(a+b x) \cos (a+b x)-6 \sin ^{\frac{7}{2}}(a+b x) E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)+5 b x\right)}{35 b^2}","-\frac{4 \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x)}{35 b^2}-\frac{12 \cos (a+b x) \sqrt{\csc (a+b x)}}{35 b^2}-\frac{12 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{35 b^2}-\frac{2 x \csc ^{\frac{7}{2}}(a+b x)}{7 b}",1,"(-2*Csc[a + b*x]^(7/2)*(5*b*x + 6*Cos[a + b*x]*Sin[a + b*x]^3 - 6*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(7/2) + Sin[2*(a + b*x)]))/(35*b^2)","A",1
353,1,65,85,0.2920763,"\int x \cos (a+b x) \csc ^{\frac{7}{2}}(a+b x) \, dx","Integrate[x*Cos[a + b*x]*Csc[a + b*x]^(7/2),x]","-\frac{2 \sqrt{\csc (a+b x)} \left(2 \cot (a+b x)+3 b x \csc ^2(a+b x)+2 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{15 b^2}","-\frac{4 \cos (a+b x) \csc ^{\frac{3}{2}}(a+b x)}{15 b^2}+\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{15 b^2}-\frac{2 x \csc ^{\frac{5}{2}}(a+b x)}{5 b}",1,"(-2*Sqrt[Csc[a + b*x]]*(2*Cot[a + b*x] + 3*b*x*Csc[a + b*x]^2 + 2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]]))/(15*b^2)","A",1
354,1,56,85,0.1777053,"\int x \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x) \, dx","Integrate[x*Cos[a + b*x]*Csc[a + b*x]^(5/2),x]","-\frac{2 \csc ^{\frac{3}{2}}(a+b x) \left(\sin (2 (a+b x))-2 \sin ^{\frac{3}{2}}(a+b x) E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)+b x\right)}{3 b^2}","-\frac{4 \cos (a+b x) \sqrt{\csc (a+b x)}}{3 b^2}-\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{3 b^2}-\frac{2 x \csc ^{\frac{3}{2}}(a+b x)}{3 b}",1,"(-2*Csc[a + b*x]^(3/2)*(b*x - 2*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(3/2) + Sin[2*(a + b*x)]))/(3*b^2)","A",1
355,1,46,58,0.137444,"\int x \cos (a+b x) \csc ^{\frac{3}{2}}(a+b x) \, dx","Integrate[x*Cos[a + b*x]*Csc[a + b*x]^(3/2),x]","-\frac{2 \sqrt{\csc (a+b x)} \left(2 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)+b x\right)}{b^2}","\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b^2}-\frac{2 x \sqrt{\csc (a+b x)}}{b}",1,"(-2*Sqrt[Csc[a + b*x]]*(b*x + 2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]]))/b^2","A",1
356,1,106,58,0.7243858,"\int x \cos (a+b x) \sqrt{\csc (a+b x)} \, dx","Integrate[x*Cos[a + b*x]*Sqrt[Csc[a + b*x]],x]","\frac{4 \sin \left(\frac{1}{2} (a+b x)\right) \cos \left(\frac{1}{2} (a+b x)\right) \sqrt{\csc (a+b x)} \left(2 \tan \left(\frac{1}{2} (a+b x)\right) \sqrt{\sec ^2\left(\frac{1}{2} (a+b x)\right)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-6 \tan \left(\frac{1}{2} (a+b x)\right)+3 b x\right)}{3 b^2}","\frac{2 x}{b \sqrt{\csc (a+b x)}}-\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b^2}",1,"(4*Cos[(a + b*x)/2]*Sqrt[Csc[a + b*x]]*Sin[(a + b*x)/2]*(3*b*x - 6*Tan[(a + b*x)/2] + 2*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[(a + b*x)/2]^2]*Sqrt[Sec[(a + b*x)/2]^2]*Tan[(a + b*x)/2]))/(3*b^2)","C",1
357,1,65,85,0.225048,"\int \frac{x \cos (a+b x)}{\sqrt{\csc (a+b x)}} \, dx","Integrate[(x*Cos[a + b*x])/Sqrt[Csc[a + b*x]],x]","\frac{2 \sqrt{\csc (a+b x)} \left(3 b x \sin ^2(a+b x)+\sin (2 (a+b x))+2 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{9 b^2}","\frac{4 \cos (a+b x)}{9 b^2 \sqrt{\csc (a+b x)}}-\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{9 b^2}+\frac{2 x}{3 b \csc ^{\frac{3}{2}}(a+b x)}",1,"(2*Sqrt[Csc[a + b*x]]*(2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + 3*b*x*Sin[a + b*x]^2 + Sin[2*(a + b*x)]))/(9*b^2)","A",1
358,1,114,85,0.9830183,"\int \frac{x \cos (a+b x)}{\csc ^{\frac{3}{2}}(a+b x)} \, dx","Integrate[(x*Cos[a + b*x])/Csc[a + b*x]^(3/2),x]","\frac{\tan \left(\frac{1}{2} (a+b x)\right) \left(4 \sqrt{2} \sqrt{\frac{1}{\cos (a+b x)+1}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+10 b x \sin (a+b x)+5 b x \sin (2 (a+b x))+4 \cos (a+b x)+2 \cos (2 (a+b x))-10\right)}{25 b^2 \sqrt{\csc (a+b x)}}","\frac{4 \cos (a+b x)}{25 b^2 \csc ^{\frac{3}{2}}(a+b x)}-\frac{12 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{25 b^2}+\frac{2 x}{5 b \csc ^{\frac{5}{2}}(a+b x)}",1,"((-10 + 4*Cos[a + b*x] + 2*Cos[2*(a + b*x)] + 4*Sqrt[2]*Sqrt[(1 + Cos[a + b*x])^(-1)]*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[(a + b*x)/2]^2] + 10*b*x*Sin[a + b*x] + 5*b*x*Sin[2*(a + b*x)])*Tan[(a + b*x)/2])/(25*b^2*Sqrt[Csc[a + b*x]])","C",1
359,1,93,108,0.3941194,"\int \frac{x \cos (a+b x)}{\csc ^{\frac{5}{2}}(a+b x)} \, dx","Integrate[(x*Cos[a + b*x])/Csc[a + b*x]^(5/2),x]","\frac{\sqrt{\csc (a+b x)} \left(52 \sin (2 (a+b x))-6 \sin (4 (a+b x))-84 b x \cos (2 (a+b x))+21 b x \cos (4 (a+b x))+80 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)+63 b x\right)}{588 b^2}","\frac{4 \cos (a+b x)}{49 b^2 \csc ^{\frac{5}{2}}(a+b x)}+\frac{20 \cos (a+b x)}{147 b^2 \sqrt{\csc (a+b x)}}-\frac{20 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{147 b^2}+\frac{2 x}{7 b \csc ^{\frac{7}{2}}(a+b x)}",1,"(Sqrt[Csc[a + b*x]]*(63*b*x - 84*b*x*Cos[2*(a + b*x)] + 21*b*x*Cos[4*(a + b*x)] + 80*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + 52*Sin[2*(a + b*x)] - 6*Sin[4*(a + b*x)]))/(588*b^2)","A",1
360,1,22,31,0.0181187,"\int x \csc (x) \sin (3 x) \, dx","Integrate[x*Csc[x]*Sin[3*x],x]","\frac{x^2}{2}+x \sin (2 x)+\frac{1}{2} \cos (2 x)","\frac{x^2}{2}-\frac{\sin ^2(x)}{4}+\frac{3 \cos ^2(x)}{4}+2 x \sin (x) \cos (x)",1,"x^2/2 + Cos[2*x]/2 + x*Sin[2*x]","A",1
361,1,154,131,0.2240023,"\int (c+d x)^4 \csc (x) \sin (3 x) \, dx","Integrate[(c + d*x)^4*Csc[x]*Sin[3*x],x]","c^4 x+2 c^3 d x^2+2 c^2 d^2 x^3+d \cos (2 x) \left(2 c^3+6 c^2 d x+3 c d^2 \left(2 x^2-1\right)+d^3 x \left(2 x^2-3\right)\right)+\frac{1}{2} \sin (2 x) \left(2 c^4+8 c^3 d x+6 c^2 d^2 \left(2 x^2-1\right)+4 c d^3 x \left(2 x^2-3\right)+d^4 \left(2 x^4-6 x^2+3\right)\right)+c d^3 x^4+\frac{d^4 x^5}{5}","\frac{3}{2} d^3 \sin ^2(x) (c+d x)-\frac{9}{2} d^3 \cos ^2(x) (c+d x)-6 d^2 \sin (x) \cos (x) (c+d x)^2+\frac{(c+d x)^5}{5 d}-d (c+d x)^3-d \sin ^2(x) (c+d x)^3+3 d \cos ^2(x) (c+d x)^3+2 \sin (x) \cos (x) (c+d x)^4+\frac{3 d^4 x}{2}+3 d^4 \sin (x) \cos (x)",1,"c^4*x + 2*c^3*d*x^2 + 2*c^2*d^2*x^3 + c*d^3*x^4 + (d^4*x^5)/5 + d*(2*c^3 + 6*c^2*d*x + d^3*x*(-3 + 2*x^2) + 3*c*d^2*(-1 + 2*x^2))*Cos[2*x] + ((2*c^4 + 8*c^3*d*x + 4*c*d^3*x*(-3 + 2*x^2) + 6*c^2*d^2*(-1 + 2*x^2) + d^4*(3 - 6*x^2 + 2*x^4))*Sin[2*x])/2","A",1
362,1,109,115,0.1625682,"\int (c+d x)^3 \csc (x) \sin (3 x) \, dx","Integrate[(c + d*x)^3*Csc[x]*Sin[3*x],x]","\frac{1}{4} \left(3 d \cos (2 x) \left(2 c^2+4 c d x+d^2 \left(2 x^2-1\right)\right)+2 \sin (2 x) \left(2 c^3+6 c^2 d x+3 c d^2 \left(2 x^2-1\right)+d^3 x \left(2 x^2-3\right)\right)+x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)\right)","-\frac{3}{2} c d^2 x-3 d^2 \sin (x) \cos (x) (c+d x)+\frac{(c+d x)^4}{4 d}-\frac{3}{4} d \sin ^2(x) (c+d x)^2+\frac{9}{4} d \cos ^2(x) (c+d x)^2+2 \sin (x) \cos (x) (c+d x)^3-\frac{3 d^3 x^2}{4}+\frac{3}{8} d^3 \sin ^2(x)-\frac{9}{8} d^3 \cos ^2(x)",1,"(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + 3*d*(2*c^2 + 4*c*d*x + d^2*(-1 + 2*x^2))*Cos[2*x] + 2*(2*c^3 + 6*c^2*d*x + d^3*x*(-3 + 2*x^2) + 3*c*d^2*(-1 + 2*x^2))*Sin[2*x])/4","A",1
363,1,60,73,0.1092243,"\int (c+d x)^2 \csc (x) \sin (3 x) \, dx","Integrate[(c + d*x)^2*Csc[x]*Sin[3*x],x]","\sin (x) \cos (x) \left(2 c^2+4 c d x+d^2 \left(2 x^2-1\right)\right)+c^2 x+c d x^2+d \cos (2 x) (c+d x)+\frac{d^2 x^3}{3}","\frac{(c+d x)^3}{3 d}-\frac{1}{2} d \sin ^2(x) (c+d x)+\frac{3}{2} d \cos ^2(x) (c+d x)+2 \sin (x) \cos (x) (c+d x)^2-\frac{d^2 x}{2}-d^2 \sin (x) \cos (x)",1,"c^2*x + c*d*x^2 + (d^2*x^3)/3 + d*(c + d*x)*Cos[2*x] + (2*c^2 + 4*c*d*x + d^2*(-1 + 2*x^2))*Cos[x]*Sin[x]","A",1
364,1,34,41,0.0182347,"\int (c+d x) \csc (x) \sin (3 x) \, dx","Integrate[(c + d*x)*Csc[x]*Sin[3*x],x]","c x+c \sin (2 x)+\frac{d x^2}{2}+d x \sin (2 x)+\frac{1}{2} d \cos (2 x)","2 \sin (x) \cos (x) (c+d x)+c x+\frac{d x^2}{2}-\frac{1}{4} d \sin ^2(x)+\frac{3}{4} d \cos ^2(x)",1,"c*x + (d*x^2)/2 + (d*Cos[2*x])/2 + c*Sin[2*x] + d*x*Sin[2*x]","A",1
365,1,49,57,0.060766,"\int \frac{\csc (x) \sin (3 x)}{c+d x} \, dx","Integrate[(Csc[x]*Sin[3*x])/(c + d*x),x]","\frac{2 \cos \left(\frac{2 c}{d}\right) \text{Ci}\left(2 \left(\frac{c}{d}+x\right)\right)+2 \sin \left(\frac{2 c}{d}\right) \text{Si}\left(2 \left(\frac{c}{d}+x\right)\right)+\log (c+d x)}{d}","\frac{2 \cos \left(\frac{2 c}{d}\right) \text{Ci}\left(\frac{2 c}{d}+2 x\right)}{d}+\frac{2 \sin \left(\frac{2 c}{d}\right) \text{Si}\left(\frac{2 c}{d}+2 x\right)}{d}+\frac{\log (c+d x)}{d}",1,"(2*Cos[(2*c)/d]*CosIntegral[2*(c/d + x)] + Log[c + d*x] + 2*Sin[(2*c)/d]*SinIntegral[2*(c/d + x)])/d","A",1
366,1,61,78,0.1382257,"\int \frac{\csc (x) \sin (3 x)}{(c+d x)^2} \, dx","Integrate[(Csc[x]*Sin[3*x])/(c + d*x)^2,x]","\frac{4 \sin \left(\frac{2 c}{d}\right) \text{Ci}\left(2 \left(\frac{c}{d}+x\right)\right)-4 \cos \left(\frac{2 c}{d}\right) \text{Si}\left(2 \left(\frac{c}{d}+x\right)\right)-\frac{d (2 \cos (2 x)+1)}{c+d x}}{d^2}","\frac{4 \sin \left(\frac{2 c}{d}\right) \text{Ci}\left(\frac{2 c}{d}+2 x\right)}{d^2}-\frac{4 \cos \left(\frac{2 c}{d}\right) \text{Si}\left(\frac{2 c}{d}+2 x\right)}{d^2}+\frac{\sin ^2(x)}{d (c+d x)}-\frac{3 \cos ^2(x)}{d (c+d x)}",1,"(-((d*(1 + 2*Cos[2*x]))/(c + d*x)) + 4*CosIntegral[2*(c/d + x)]*Sin[(2*c)/d] - 4*Cos[(2*c)/d]*SinIntegral[2*(c/d + x)])/d^2","A",1
367,1,77,99,0.2428927,"\int \frac{\csc (x) \sin (3 x)}{(c+d x)^3} \, dx","Integrate[(Csc[x]*Sin[3*x])/(c + d*x)^3,x]","\frac{-8 \cos \left(\frac{2 c}{d}\right) \text{Ci}\left(2 \left(\frac{c}{d}+x\right)\right)-8 \sin \left(\frac{2 c}{d}\right) \text{Si}\left(2 \left(\frac{c}{d}+x\right)\right)+\frac{d (4 \sin (2 x) (c+d x)-2 d \cos (2 x)-d)}{(c+d x)^2}}{2 d^3}","-\frac{4 \cos \left(\frac{2 c}{d}\right) \text{Ci}\left(\frac{2 c}{d}+2 x\right)}{d^3}-\frac{4 \sin \left(\frac{2 c}{d}\right) \text{Si}\left(\frac{2 c}{d}+2 x\right)}{d^3}+\frac{4 \sin (x) \cos (x)}{d^2 (c+d x)}+\frac{\sin ^2(x)}{2 d (c+d x)^2}-\frac{3 \cos ^2(x)}{2 d (c+d x)^2}",1,"(-8*Cos[(2*c)/d]*CosIntegral[2*(c/d + x)] + (d*(-d - 2*d*Cos[2*x] + 4*(c + d*x)*Sin[2*x]))/(c + d*x)^2 - 8*Sin[(2*c)/d]*SinIntegral[2*(c/d + x)])/(2*d^3)","A",1
368,1,128,198,0.6827425,"\int (c+d x)^4 \csc (a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)^4*Csc[a + b*x]*Sin[3*a + 3*b*x],x]","\frac{d (c+d x) \cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-3 d^2\right)}{b^4}+\frac{\sin (2 (a+b x)) \left(2 b^4 (c+d x)^4-6 b^2 d^2 (c+d x)^2+3 d^4\right)}{2 b^5}+c^4 x+2 c^3 d x^2+2 c^2 d^2 x^3+c d^3 x^4+\frac{d^4 x^5}{5}","\frac{3 d^4 \sin (a+b x) \cos (a+b x)}{b^5}+\frac{3 d^3 (c+d x) \sin ^2(a+b x)}{2 b^4}-\frac{9 d^3 (c+d x) \cos ^2(a+b x)}{2 b^4}-\frac{6 d^2 (c+d x)^2 \sin (a+b x) \cos (a+b x)}{b^3}-\frac{d (c+d x)^3 \sin ^2(a+b x)}{b^2}+\frac{3 d (c+d x)^3 \cos ^2(a+b x)}{b^2}+\frac{2 (c+d x)^4 \sin (a+b x) \cos (a+b x)}{b}+\frac{3 d^4 x}{2 b^4}-\frac{d (c+d x)^3}{b^2}+\frac{(c+d x)^5}{5 d}",1,"c^4*x + 2*c^3*d*x^2 + 2*c^2*d^2*x^3 + c*d^3*x^4 + (d^4*x^5)/5 + (d*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])/b^4 + ((3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Sin[2*(a + b*x)])/(2*b^5)","A",1
369,1,105,171,0.4216416,"\int (c+d x)^3 \csc (a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]*Sin[3*a + 3*b*x],x]","\frac{2 b (c+d x) \sin (2 (a+b x)) \left(2 b^2 (c+d x)^2-3 d^2\right)+3 d \cos (2 (a+b x)) \left(2 b^2 (c+d x)^2-d^2\right)+b^4 x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)}{4 b^4}","\frac{3 d^3 \sin ^2(a+b x)}{8 b^4}-\frac{9 d^3 \cos ^2(a+b x)}{8 b^4}-\frac{3 d^2 (c+d x) \sin (a+b x) \cos (a+b x)}{b^3}-\frac{3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}+\frac{9 d (c+d x)^2 \cos ^2(a+b x)}{4 b^2}+\frac{2 (c+d x)^3 \sin (a+b x) \cos (a+b x)}{b}-\frac{3 c d^2 x}{2 b^2}-\frac{3 d^3 x^2}{4 b^2}+\frac{(c+d x)^4}{4 d}",1,"(b^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + 3*d*(-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + 2*b*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Sin[2*(a + b*x)])/(4*b^4)","A",1
370,1,73,112,0.3999106,"\int (c+d x)^2 \csc (a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]*Sin[3*a + 3*b*x],x]","\frac{d (c+d x) \cos (2 (a+b x))}{b^2}+\frac{\sin (2 (a+b x)) \left(2 b^2 (c+d x)^2-d^2\right)}{2 b^3}+c^2 x+c d x^2+\frac{d^2 x^3}{3}","-\frac{d^2 \sin (a+b x) \cos (a+b x)}{b^3}-\frac{d (c+d x) \sin ^2(a+b x)}{2 b^2}+\frac{3 d (c+d x) \cos ^2(a+b x)}{2 b^2}+\frac{2 (c+d x)^2 \sin (a+b x) \cos (a+b x)}{b}-\frac{d^2 x}{2 b^2}+\frac{(c+d x)^3}{3 d}",1,"c^2*x + c*d*x^2 + (d^2*x^3)/3 + (d*(c + d*x)*Cos[2*(a + b*x)])/b^2 + ((-d^2 + 2*b^2*(c + d*x)^2)*Sin[2*(a + b*x)])/(2*b^3)","A",1
371,1,46,66,0.142919,"\int (c+d x) \csc (a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]*Sin[3*a + 3*b*x],x]","\frac{b (2 (c+d x) \sin (2 (a+b x))+b x (2 c+d x))+d \cos (2 (a+b x))}{2 b^2}","-\frac{d \sin ^2(a+b x)}{4 b^2}+\frac{3 d \cos ^2(a+b x)}{4 b^2}+\frac{2 (c+d x) \sin (a+b x) \cos (a+b x)}{b}+c x+\frac{d x^2}{2}",1,"(d*Cos[2*(a + b*x)] + b*(b*x*(2*c + d*x) + 2*(c + d*x)*Sin[2*(a + b*x)]))/(2*b^2)","A",1
372,1,63,71,0.1482059,"\int \frac{\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx","Integrate[(Csc[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x),x]","\frac{2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\log (c+d x)}{d}","\frac{2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d}-\frac{2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d}+\frac{\log (c+d x)}{d}",1,"(2*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] + Log[c + d*x] - 2*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/d","A",1
373,1,81,102,0.531931,"\int \frac{\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx","Integrate[(Csc[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x)^2,x]","-\frac{4 b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)+4 b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\frac{d (2 \cos (2 (a+b x))+1)}{c+d x}}{d^2}","-\frac{4 b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}-\frac{4 b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}+\frac{\sin ^2(a+b x)}{d (c+d x)}-\frac{3 \cos ^2(a+b x)}{d (c+d x)}",1,"-(((d*(1 + 2*Cos[2*(a + b*x)]))/(c + d*x) + 4*b*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] + 4*b*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/d^2)","A",1
374,1,104,136,0.9886329,"\int \frac{\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx","Integrate[(Csc[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x)^3,x]","-\frac{8 b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)-8 b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\frac{d (-4 b (c+d x) \sin (2 (a+b x))+2 d \cos (2 (a+b x))+d)}{(c+d x)^2}}{2 d^3}","-\frac{4 b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d^3}+\frac{4 b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d^3}+\frac{4 b \sin (a+b x) \cos (a+b x)}{d^2 (c+d x)}+\frac{\sin ^2(a+b x)}{2 d (c+d x)^2}-\frac{3 \cos ^2(a+b x)}{2 d (c+d x)^2}",1,"-1/2*(8*b^2*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] + (d*(d + 2*d*Cos[2*(a + b*x)] - 4*b*(c + d*x)*Sin[2*(a + b*x)]))/(c + d*x)^2 - 8*b^2*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/d^3","A",1
375,1,125,205,1.1419944,"\int \frac{\csc (a+b x) \sin (3 a+3 b x)}{(c+d x)^4} \, dx","Integrate[(Csc[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x)^4,x]","\frac{8 b^3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b (c+d x)}{d}\right)+8 b^3 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b (c+d x)}{d}\right)+\frac{d \left(\cos (2 (a+b x)) \left(4 b^2 (c+d x)^2-2 d^2\right)+d (2 b (c+d x) \sin (2 (a+b x))-d)\right)}{(c+d x)^3}}{3 d^4}","\frac{8 b^3 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{3 d^4}+\frac{8 b^3 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{3 d^4}-\frac{2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}+\frac{2 b^2 \cos ^2(a+b x)}{d^3 (c+d x)}+\frac{4 b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}+\frac{\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac{\cos ^2(a+b x)}{d (c+d x)^3}-\frac{2 b^2}{3 d^3 (c+d x)}",1,"(8*b^3*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] + (d*((-2*d^2 + 4*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + d*(-d + 2*b*(c + d*x)*Sin[2*(a + b*x)])))/(c + d*x)^3 + 8*b^3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d])/(3*d^4)","A",1
376,1,459,255,1.5371456,"\int (c+d x)^3 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)^3*Csc[a + b*x]^2*Sin[3*a + 3*b*x],x]","\frac{4 \cos (b x) \left(b^3 c^3 \cos (a)+3 b^3 c^2 d x \cos (a)+3 b^3 c d^2 x^2 \cos (a)+b^3 d^3 x^3 \cos (a)-3 b^2 c^2 d \sin (a)-6 b^2 c d^2 x \sin (a)-3 b^2 d^3 x^2 \sin (a)-6 b c d^2 \cos (a)-6 b d^3 x \cos (a)+6 d^3 \sin (a)\right)}{b^4}-\frac{4 \sin (b x) \left(b^3 c^3 \sin (a)+3 b^3 c^2 d x \sin (a)+3 b^3 c d^2 x^2 \sin (a)+b^3 d^3 x^3 \sin (a)+3 b^2 c^2 d \cos (a)+6 b^2 c d^2 x \cos (a)+3 b^2 d^3 x^2 \cos (a)-6 b c d^2 \sin (a)-6 b d^3 x \sin (a)-6 d^3 \cos (a)\right)}{b^4}+\frac{3 \left(-2 b^3 (c+d x)^3 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x))+3 i d \left(b^2 (c+d x)^2 \text{Li}_2(-\cos (a+b x)-i \sin (a+b x))+2 i b d (c+d x) \text{Li}_3(-\cos (a+b x)-i \sin (a+b x))-2 d^2 \text{Li}_4(-\cos (a+b x)-i \sin (a+b x))\right)-3 i d \left(b^2 (c+d x)^2 \text{Li}_2(\cos (a+b x)+i \sin (a+b x))+2 i b d (c+d x) \text{Li}_3(\cos (a+b x)+i \sin (a+b x))-2 d^2 \text{Li}_4(\cos (a+b x)+i \sin (a+b x))\right)\right)}{b^4}","-\frac{18 i d^3 \text{Li}_4\left(-e^{i (a+b x)}\right)}{b^4}+\frac{18 i d^3 \text{Li}_4\left(e^{i (a+b x)}\right)}{b^4}+\frac{24 d^3 \sin (a+b x)}{b^4}-\frac{18 d^2 (c+d x) \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{18 d^2 (c+d x) \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{24 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac{9 i d (c+d x)^2 \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{9 i d (c+d x)^2 \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{12 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac{4 (c+d x)^3 \cos (a+b x)}{b}-\frac{6 (c+d x)^3 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(3*(-2*b^3*(c + d*x)^3*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]] + (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, -Cos[a + b*x] - I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]] - 2*d^2*PolyLog[4, -Cos[a + b*x] - I*Sin[a + b*x]]) - (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]] - 2*d^2*PolyLog[4, Cos[a + b*x] + I*Sin[a + b*x]])))/b^4 + (4*Cos[b*x]*(b^3*c^3*Cos[a] - 6*b*c*d^2*Cos[a] + 3*b^3*c^2*d*x*Cos[a] - 6*b*d^3*x*Cos[a] + 3*b^3*c*d^2*x^2*Cos[a] + b^3*d^3*x^3*Cos[a] - 3*b^2*c^2*d*Sin[a] + 6*d^3*Sin[a] - 6*b^2*c*d^2*x*Sin[a] - 3*b^2*d^3*x^2*Sin[a]))/b^4 - (4*(3*b^2*c^2*d*Cos[a] - 6*d^3*Cos[a] + 6*b^2*c*d^2*x*Cos[a] + 3*b^2*d^3*x^2*Cos[a] + b^3*c^3*Sin[a] - 6*b*c*d^2*Sin[a] + 3*b^3*c^2*d*x*Sin[a] - 6*b*d^3*x*Sin[a] + 3*b^3*c*d^2*x^2*Sin[a] + b^3*d^3*x^3*Sin[a])*Sin[b*x])/b^4","A",0
377,1,223,172,1.0958701,"\int (c+d x)^2 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)^2*Csc[a + b*x]^2*Sin[3*a + 3*b*x],x]","\frac{4 \cos (b x) \left(\cos (a) \left(b^2 (c+d x)^2-2 d^2\right)-2 b d \sin (a) (c+d x)\right)-4 \sin (b x) \left(\sin (a) \left(b^2 (c+d x)^2-2 d^2\right)+2 b d \cos (a) (c+d x)\right)+3 b^2 (c+d x)^2 \log \left(1-e^{i (a+b x)}\right)-3 b^2 (c+d x)^2 \log \left(1+e^{i (a+b x)}\right)+6 i b d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)-6 i b d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)-6 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)+6 d^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}","-\frac{6 d^2 \text{Li}_3\left(-e^{i (a+b x)}\right)}{b^3}+\frac{6 d^2 \text{Li}_3\left(e^{i (a+b x)}\right)}{b^3}-\frac{8 d^2 \cos (a+b x)}{b^3}+\frac{6 i d (c+d x) \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{6 i d (c+d x) \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{8 d (c+d x) \sin (a+b x)}{b^2}+\frac{4 (c+d x)^2 \cos (a+b x)}{b}-\frac{6 (c+d x)^2 \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(3*b^2*(c + d*x)^2*Log[1 - E^(I*(a + b*x))] - 3*b^2*(c + d*x)^2*Log[1 + E^(I*(a + b*x))] + (6*I)*b*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] - (6*I)*b*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] - 6*d^2*PolyLog[3, -E^(I*(a + b*x))] + 6*d^2*PolyLog[3, E^(I*(a + b*x))] + 4*Cos[b*x]*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] - 2*b*d*(c + d*x)*Sin[a]) - 4*(2*b*d*(c + d*x)*Cos[a] + (-2*d^2 + b^2*(c + d*x)^2)*Sin[a])*Sin[b*x])/b^3","A",1
378,1,171,95,0.3044783,"\int (c+d x) \csc ^2(a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)*Csc[a + b*x]^2*Sin[3*a + 3*b*x],x]","\frac{3 d \left(i \left(\text{Li}_2\left(-e^{i (a+b x)}\right)-\text{Li}_2\left(e^{i (a+b x)}\right)\right)+(a+b x) \left(\log \left(1-e^{i (a+b x)}\right)-\log \left(1+e^{i (a+b x)}\right)\right)\right)}{b^2}-\frac{4 d \sin (a+b x)}{b^2}-\frac{3 a d \log \left(\tan \left(\frac{1}{2} (a+b x)\right)\right)}{b^2}+\frac{4 c \cos (a+b x)}{b}+\frac{3 c \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{3 c \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}+\frac{4 d x \cos (a+b x)}{b}","\frac{3 i d \text{Li}_2\left(-e^{i (a+b x)}\right)}{b^2}-\frac{3 i d \text{Li}_2\left(e^{i (a+b x)}\right)}{b^2}-\frac{4 d \sin (a+b x)}{b^2}+\frac{4 (c+d x) \cos (a+b x)}{b}-\frac{6 (c+d x) \tanh ^{-1}\left(e^{i (a+b x)}\right)}{b}",1,"(4*c*Cos[a + b*x])/b + (4*d*x*Cos[a + b*x])/b - (3*c*Log[Cos[(a + b*x)/2]])/b + (3*c*Log[Sin[(a + b*x)/2]])/b - (3*a*d*Log[Tan[(a + b*x)/2]])/b^2 + (3*d*((a + b*x)*(Log[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(a + b*x))]) + I*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^(I*(a + b*x))])))/b^2 - (4*d*Sin[a + b*x])/b^2","A",1
379,0,0,72,6.2300725,"\int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx","Integrate[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x),x]","\int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx","3 \text{Int}\left(\frac{\csc (a+b x)}{c+d x},x\right)-\frac{4 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d}-\frac{4 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d}",0,"Integrate[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x), x]","A",-1
380,0,0,92,6.7143855,"\int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx","Integrate[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x)^2,x]","\int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx","3 \text{Int}\left(\frac{\csc (a+b x)}{(c+d x)^2},x\right)-\frac{4 b \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d^2}+\frac{4 b \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d^2}+\frac{4 \sin (a+b x)}{d (c+d x)}",0,"Integrate[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x)^2, x]","A",-1
381,0,0,115,7.1398107,"\int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx","Integrate[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x)^3,x]","\int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx","3 \text{Int}\left(\frac{\csc (a+b x)}{(c+d x)^3},x\right)+\frac{2 b^2 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d^3}+\frac{2 b^2 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d^3}+\frac{2 b \cos (a+b x)}{d^2 (c+d x)}+\frac{2 \sin (a+b x)}{d (c+d x)^2}",0,"Integrate[(Csc[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x)^3, x]","A",-1
382,1,2482,299,6.7460711,"\int (c+d x)^4 \sec (a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)^4*Sec[a + b*x]*Sin[3*a + 3*b*x],x]","\text{Result too large to show}","-\frac{3 d^4 \text{Li}_5\left(-e^{2 i (a+b x)}\right)}{2 b^5}+\frac{3 d^4 \sin ^2(a+b x)}{b^5}+\frac{3 i d^3 (c+d x) \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{b^4}-\frac{6 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{b^4}+\frac{3 d^2 (c+d x)^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{b^3}-\frac{6 d^2 (c+d x)^2 \sin ^2(a+b x)}{b^3}-\frac{2 i d (c+d x)^3 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}+\frac{4 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{b^2}+\frac{(c+d x)^4 \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{2 (c+d x)^4 \sin ^2(a+b x)}{b}+\frac{6 c d^3 x}{b^3}+\frac{3 d^4 x^2}{b^3}-\frac{(c+d x)^4}{b}-\frac{i (c+d x)^5}{5 d}",1,"((I/2)*c^2*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) + (I/2)*c*d^3*E^(I*a)*((2*x^4)/E^((2*I)*a) - ((4*I)*(1 + E^((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x))])/b + (3*(1 + E^((2*I)*a))*(2*b^2*x^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-2*I)*(a + b*x))] - PolyLog[4, -E^((-2*I)*(a + b*x))]))/(b^4*E^((2*I)*a)))*Sec[a] - (d^4*((-4*I)*x^5 - (10*(1 + E^((2*I)*a))*x^4*Log[1 + E^((-2*I)*(a + b*x))])/b + (5*(1 + E^((2*I)*a))*((-4*I)*b^3*x^3*PolyLog[2, -E^((-2*I)*(a + b*x))] - 6*b^2*x^2*PolyLog[3, -E^((-2*I)*(a + b*x))] + (6*I)*b*x*PolyLog[4, -E^((-2*I)*(a + b*x))] + 3*PolyLog[5, -E^((-2*I)*(a + b*x))]))/b^5)*Sec[a])/(20*E^(I*a)) + (c^4*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + (2*c^3*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + Sec[a]*(Cos[2*a + 2*b*x]/(40*b^5) - ((I/40)*Sin[2*a + 2*b*x])/b^5)*(-20*b^4*c^4*Cos[a] + (40*I)*b^3*c^3*d*Cos[a] + 60*b^2*c^2*d^2*Cos[a] - (60*I)*b*c*d^3*Cos[a] - 30*d^4*Cos[a] - 80*b^4*c^3*d*x*Cos[a] + (120*I)*b^3*c^2*d^2*x*Cos[a] + 120*b^2*c*d^3*x*Cos[a] - (60*I)*b*d^4*x*Cos[a] - 120*b^4*c^2*d^2*x^2*Cos[a] + (120*I)*b^3*c*d^3*x^2*Cos[a] + 60*b^2*d^4*x^2*Cos[a] - 80*b^4*c*d^3*x^3*Cos[a] + (40*I)*b^3*d^4*x^3*Cos[a] - 20*b^4*d^4*x^4*Cos[a] - (20*I)*b^5*c^4*x*Cos[a + 2*b*x] - (40*I)*b^5*c^3*d*x^2*Cos[a + 2*b*x] - (40*I)*b^5*c^2*d^2*x^3*Cos[a + 2*b*x] - (20*I)*b^5*c*d^3*x^4*Cos[a + 2*b*x] - (4*I)*b^5*d^4*x^5*Cos[a + 2*b*x] + (20*I)*b^5*c^4*x*Cos[3*a + 2*b*x] + (40*I)*b^5*c^3*d*x^2*Cos[3*a + 2*b*x] + (40*I)*b^5*c^2*d^2*x^3*Cos[3*a + 2*b*x] + (20*I)*b^5*c*d^3*x^4*Cos[3*a + 2*b*x] + (4*I)*b^5*d^4*x^5*Cos[3*a + 2*b*x] - 10*b^4*c^4*Cos[3*a + 4*b*x] - (20*I)*b^3*c^3*d*Cos[3*a + 4*b*x] + 30*b^2*c^2*d^2*Cos[3*a + 4*b*x] + (30*I)*b*c*d^3*Cos[3*a + 4*b*x] - 15*d^4*Cos[3*a + 4*b*x] - 40*b^4*c^3*d*x*Cos[3*a + 4*b*x] - (60*I)*b^3*c^2*d^2*x*Cos[3*a + 4*b*x] + 60*b^2*c*d^3*x*Cos[3*a + 4*b*x] + (30*I)*b*d^4*x*Cos[3*a + 4*b*x] - 60*b^4*c^2*d^2*x^2*Cos[3*a + 4*b*x] - (60*I)*b^3*c*d^3*x^2*Cos[3*a + 4*b*x] + 30*b^2*d^4*x^2*Cos[3*a + 4*b*x] - 40*b^4*c*d^3*x^3*Cos[3*a + 4*b*x] - (20*I)*b^3*d^4*x^3*Cos[3*a + 4*b*x] - 10*b^4*d^4*x^4*Cos[3*a + 4*b*x] - 10*b^4*c^4*Cos[5*a + 4*b*x] - (20*I)*b^3*c^3*d*Cos[5*a + 4*b*x] + 30*b^2*c^2*d^2*Cos[5*a + 4*b*x] + (30*I)*b*c*d^3*Cos[5*a + 4*b*x] - 15*d^4*Cos[5*a + 4*b*x] - 40*b^4*c^3*d*x*Cos[5*a + 4*b*x] - (60*I)*b^3*c^2*d^2*x*Cos[5*a + 4*b*x] + 60*b^2*c*d^3*x*Cos[5*a + 4*b*x] + (30*I)*b*d^4*x*Cos[5*a + 4*b*x] - 60*b^4*c^2*d^2*x^2*Cos[5*a + 4*b*x] - (60*I)*b^3*c*d^3*x^2*Cos[5*a + 4*b*x] + 30*b^2*d^4*x^2*Cos[5*a + 4*b*x] - 40*b^4*c*d^3*x^3*Cos[5*a + 4*b*x] - (20*I)*b^3*d^4*x^3*Cos[5*a + 4*b*x] - 10*b^4*d^4*x^4*Cos[5*a + 4*b*x] + 20*b^5*c^4*x*Sin[a + 2*b*x] + 40*b^5*c^3*d*x^2*Sin[a + 2*b*x] + 40*b^5*c^2*d^2*x^3*Sin[a + 2*b*x] + 20*b^5*c*d^3*x^4*Sin[a + 2*b*x] + 4*b^5*d^4*x^5*Sin[a + 2*b*x] - 20*b^5*c^4*x*Sin[3*a + 2*b*x] - 40*b^5*c^3*d*x^2*Sin[3*a + 2*b*x] - 40*b^5*c^2*d^2*x^3*Sin[3*a + 2*b*x] - 20*b^5*c*d^3*x^4*Sin[3*a + 2*b*x] - 4*b^5*d^4*x^5*Sin[3*a + 2*b*x] - (10*I)*b^4*c^4*Sin[3*a + 4*b*x] + 20*b^3*c^3*d*Sin[3*a + 4*b*x] + (30*I)*b^2*c^2*d^2*Sin[3*a + 4*b*x] - 30*b*c*d^3*Sin[3*a + 4*b*x] - (15*I)*d^4*Sin[3*a + 4*b*x] - (40*I)*b^4*c^3*d*x*Sin[3*a + 4*b*x] + 60*b^3*c^2*d^2*x*Sin[3*a + 4*b*x] + (60*I)*b^2*c*d^3*x*Sin[3*a + 4*b*x] - 30*b*d^4*x*Sin[3*a + 4*b*x] - (60*I)*b^4*c^2*d^2*x^2*Sin[3*a + 4*b*x] + 60*b^3*c*d^3*x^2*Sin[3*a + 4*b*x] + (30*I)*b^2*d^4*x^2*Sin[3*a + 4*b*x] - (40*I)*b^4*c*d^3*x^3*Sin[3*a + 4*b*x] + 20*b^3*d^4*x^3*Sin[3*a + 4*b*x] - (10*I)*b^4*d^4*x^4*Sin[3*a + 4*b*x] - (10*I)*b^4*c^4*Sin[5*a + 4*b*x] + 20*b^3*c^3*d*Sin[5*a + 4*b*x] + (30*I)*b^2*c^2*d^2*Sin[5*a + 4*b*x] - 30*b*c*d^3*Sin[5*a + 4*b*x] - (15*I)*d^4*Sin[5*a + 4*b*x] - (40*I)*b^4*c^3*d*x*Sin[5*a + 4*b*x] + 60*b^3*c^2*d^2*x*Sin[5*a + 4*b*x] + (60*I)*b^2*c*d^3*x*Sin[5*a + 4*b*x] - 30*b*d^4*x*Sin[5*a + 4*b*x] - (60*I)*b^4*c^2*d^2*x^2*Sin[5*a + 4*b*x] + 60*b^3*c*d^3*x^2*Sin[5*a + 4*b*x] + (30*I)*b^2*d^4*x^2*Sin[5*a + 4*b*x] - (40*I)*b^4*c*d^3*x^3*Sin[5*a + 4*b*x] + 20*b^3*d^4*x^3*Sin[5*a + 4*b*x] - (10*I)*b^4*d^4*x^4*Sin[5*a + 4*b*x])","B",0
383,1,1719,242,6.6377095,"\int (c+d x)^3 \sec (a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)^3*Sec[a + b*x]*Sin[3*a + 3*b*x],x]","\frac{\sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a)) c^3}{b \left(\cos ^2(a)+\sin ^2(a)\right)}+\frac{3 d \csc (a) \left(b^2 e^{-i \tan ^{-1}(\cot (a))} x^2-\frac{\cot (a) \left(i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-\pi  \log \left(1+e^{-2 i b x}\right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\pi  \log (\cos (b x))-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)+i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)}{\sqrt{\cot ^2(a)+1}}\right) \sec (a) c^2}{2 b^2 \sqrt{\csc ^2(a) \left(\cos ^2(a)+\sin ^2(a)\right)}}+\frac{i d^2 e^{-i a} \left(2 b^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right) x^2+6 b \left(1+e^{2 i a}\right) \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right) \sec (a) c}{4 b^3}+\frac{1}{8} i d^3 e^{i a} \left(2 e^{-2 i a} x^4-\frac{4 i \left(1+e^{-2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right) x^3}{b}+\frac{3 e^{-2 i a} \left(1+e^{2 i a}\right) \left(2 b^2 \text{Li}_2\left(-e^{-2 i (a+b x)}\right) x^2-2 i b \text{Li}_3\left(-e^{-2 i (a+b x)}\right) x-\text{Li}_4\left(-e^{-2 i (a+b x)}\right)\right)}{b^4}\right) \sec (a)+\sec (a) \left(\frac{\cos (2 a+2 b x)}{16 b^4}-\frac{i \sin (2 a+2 b x)}{16 b^4}\right) \left(-2 i d^3 x^4 \cos (a+2 b x) b^4-8 i c d^2 x^3 \cos (a+2 b x) b^4-12 i c^2 d x^2 \cos (a+2 b x) b^4-8 i c^3 x \cos (a+2 b x) b^4+2 i d^3 x^4 \cos (3 a+2 b x) b^4+8 i c d^2 x^3 \cos (3 a+2 b x) b^4+12 i c^2 d x^2 \cos (3 a+2 b x) b^4+8 i c^3 x \cos (3 a+2 b x) b^4+2 d^3 x^4 \sin (a+2 b x) b^4+8 c d^2 x^3 \sin (a+2 b x) b^4+12 c^2 d x^2 \sin (a+2 b x) b^4+8 c^3 x \sin (a+2 b x) b^4-2 d^3 x^4 \sin (3 a+2 b x) b^4-8 c d^2 x^3 \sin (3 a+2 b x) b^4-12 c^2 d x^2 \sin (3 a+2 b x) b^4-8 c^3 x \sin (3 a+2 b x) b^4-8 c^3 \cos (a) b^3-8 d^3 x^3 \cos (a) b^3-24 c d^2 x^2 \cos (a) b^3-24 c^2 d x \cos (a) b^3-4 c^3 \cos (3 a+4 b x) b^3-4 d^3 x^3 \cos (3 a+4 b x) b^3-12 c d^2 x^2 \cos (3 a+4 b x) b^3-12 c^2 d x \cos (3 a+4 b x) b^3-4 c^3 \cos (5 a+4 b x) b^3-4 d^3 x^3 \cos (5 a+4 b x) b^3-12 c d^2 x^2 \cos (5 a+4 b x) b^3-12 c^2 d x \cos (5 a+4 b x) b^3-4 i c^3 \sin (3 a+4 b x) b^3-4 i d^3 x^3 \sin (3 a+4 b x) b^3-12 i c d^2 x^2 \sin (3 a+4 b x) b^3-12 i c^2 d x \sin (3 a+4 b x) b^3-4 i c^3 \sin (5 a+4 b x) b^3-4 i d^3 x^3 \sin (5 a+4 b x) b^3-12 i c d^2 x^2 \sin (5 a+4 b x) b^3-12 i c^2 d x \sin (5 a+4 b x) b^3+12 i d^3 x^2 \cos (a) b^2+12 i c^2 d \cos (a) b^2+24 i c d^2 x \cos (a) b^2-6 i d^3 x^2 \cos (3 a+4 b x) b^2-6 i c^2 d \cos (3 a+4 b x) b^2-12 i c d^2 x \cos (3 a+4 b x) b^2-6 i d^3 x^2 \cos (5 a+4 b x) b^2-6 i c^2 d \cos (5 a+4 b x) b^2-12 i c d^2 x \cos (5 a+4 b x) b^2+6 d^3 x^2 \sin (3 a+4 b x) b^2+6 c^2 d \sin (3 a+4 b x) b^2+12 c d^2 x \sin (3 a+4 b x) b^2+6 d^3 x^2 \sin (5 a+4 b x) b^2+6 c^2 d \sin (5 a+4 b x) b^2+12 c d^2 x \sin (5 a+4 b x) b^2+12 c d^2 \cos (a) b+12 d^3 x \cos (a) b+6 c d^2 \cos (3 a+4 b x) b+6 d^3 x \cos (3 a+4 b x) b+6 c d^2 \cos (5 a+4 b x) b+6 d^3 x \cos (5 a+4 b x) b+6 i c d^2 \sin (3 a+4 b x) b+6 i d^3 x \sin (3 a+4 b x) b+6 i c d^2 \sin (5 a+4 b x) b+6 i d^3 x \sin (5 a+4 b x) b-6 i d^3 \cos (a)+3 i d^3 \cos (3 a+4 b x)+3 i d^3 \cos (5 a+4 b x)-3 d^3 \sin (3 a+4 b x)-3 d^3 \sin (5 a+4 b x)\right)","\frac{3 i d^3 \text{Li}_4\left(-e^{2 i (a+b x)}\right)}{4 b^4}-\frac{3 d^3 \sin (a+b x) \cos (a+b x)}{2 b^4}+\frac{3 d^2 (c+d x) \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}-\frac{3 d^2 (c+d x) \sin ^2(a+b x)}{b^3}-\frac{3 i d (c+d x)^2 \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}+\frac{3 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{b^2}+\frac{(c+d x)^3 \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{2 (c+d x)^3 \sin ^2(a+b x)}{b}+\frac{3 d^3 x}{2 b^3}-\frac{(c+d x)^3}{b}-\frac{i (c+d x)^4}{4 d}",1,"((I/4)*c*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) + (I/8)*d^3*E^(I*a)*((2*x^4)/E^((2*I)*a) - ((4*I)*(1 + E^((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x))])/b + (3*(1 + E^((2*I)*a))*(2*b^2*x^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, -E^((-2*I)*(a + b*x))] - PolyLog[4, -E^((-2*I)*(a + b*x))]))/(b^4*E^((2*I)*a)))*Sec[a] + (c^3*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + (3*c^2*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(2*b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + Sec[a]*(Cos[2*a + 2*b*x]/(16*b^4) - ((I/16)*Sin[2*a + 2*b*x])/b^4)*(-8*b^3*c^3*Cos[a] + (12*I)*b^2*c^2*d*Cos[a] + 12*b*c*d^2*Cos[a] - (6*I)*d^3*Cos[a] - 24*b^3*c^2*d*x*Cos[a] + (24*I)*b^2*c*d^2*x*Cos[a] + 12*b*d^3*x*Cos[a] - 24*b^3*c*d^2*x^2*Cos[a] + (12*I)*b^2*d^3*x^2*Cos[a] - 8*b^3*d^3*x^3*Cos[a] - (8*I)*b^4*c^3*x*Cos[a + 2*b*x] - (12*I)*b^4*c^2*d*x^2*Cos[a + 2*b*x] - (8*I)*b^4*c*d^2*x^3*Cos[a + 2*b*x] - (2*I)*b^4*d^3*x^4*Cos[a + 2*b*x] + (8*I)*b^4*c^3*x*Cos[3*a + 2*b*x] + (12*I)*b^4*c^2*d*x^2*Cos[3*a + 2*b*x] + (8*I)*b^4*c*d^2*x^3*Cos[3*a + 2*b*x] + (2*I)*b^4*d^3*x^4*Cos[3*a + 2*b*x] - 4*b^3*c^3*Cos[3*a + 4*b*x] - (6*I)*b^2*c^2*d*Cos[3*a + 4*b*x] + 6*b*c*d^2*Cos[3*a + 4*b*x] + (3*I)*d^3*Cos[3*a + 4*b*x] - 12*b^3*c^2*d*x*Cos[3*a + 4*b*x] - (12*I)*b^2*c*d^2*x*Cos[3*a + 4*b*x] + 6*b*d^3*x*Cos[3*a + 4*b*x] - 12*b^3*c*d^2*x^2*Cos[3*a + 4*b*x] - (6*I)*b^2*d^3*x^2*Cos[3*a + 4*b*x] - 4*b^3*d^3*x^3*Cos[3*a + 4*b*x] - 4*b^3*c^3*Cos[5*a + 4*b*x] - (6*I)*b^2*c^2*d*Cos[5*a + 4*b*x] + 6*b*c*d^2*Cos[5*a + 4*b*x] + (3*I)*d^3*Cos[5*a + 4*b*x] - 12*b^3*c^2*d*x*Cos[5*a + 4*b*x] - (12*I)*b^2*c*d^2*x*Cos[5*a + 4*b*x] + 6*b*d^3*x*Cos[5*a + 4*b*x] - 12*b^3*c*d^2*x^2*Cos[5*a + 4*b*x] - (6*I)*b^2*d^3*x^2*Cos[5*a + 4*b*x] - 4*b^3*d^3*x^3*Cos[5*a + 4*b*x] + 8*b^4*c^3*x*Sin[a + 2*b*x] + 12*b^4*c^2*d*x^2*Sin[a + 2*b*x] + 8*b^4*c*d^2*x^3*Sin[a + 2*b*x] + 2*b^4*d^3*x^4*Sin[a + 2*b*x] - 8*b^4*c^3*x*Sin[3*a + 2*b*x] - 12*b^4*c^2*d*x^2*Sin[3*a + 2*b*x] - 8*b^4*c*d^2*x^3*Sin[3*a + 2*b*x] - 2*b^4*d^3*x^4*Sin[3*a + 2*b*x] - (4*I)*b^3*c^3*Sin[3*a + 4*b*x] + 6*b^2*c^2*d*Sin[3*a + 4*b*x] + (6*I)*b*c*d^2*Sin[3*a + 4*b*x] - 3*d^3*Sin[3*a + 4*b*x] - (12*I)*b^3*c^2*d*x*Sin[3*a + 4*b*x] + 12*b^2*c*d^2*x*Sin[3*a + 4*b*x] + (6*I)*b*d^3*x*Sin[3*a + 4*b*x] - (12*I)*b^3*c*d^2*x^2*Sin[3*a + 4*b*x] + 6*b^2*d^3*x^2*Sin[3*a + 4*b*x] - (4*I)*b^3*d^3*x^3*Sin[3*a + 4*b*x] - (4*I)*b^3*c^3*Sin[5*a + 4*b*x] + 6*b^2*c^2*d*Sin[5*a + 4*b*x] + (6*I)*b*c*d^2*Sin[5*a + 4*b*x] - 3*d^3*Sin[5*a + 4*b*x] - (12*I)*b^3*c^2*d*x*Sin[5*a + 4*b*x] + 12*b^2*c*d^2*x*Sin[5*a + 4*b*x] + (6*I)*b*d^3*x*Sin[5*a + 4*b*x] - (12*I)*b^3*c*d^2*x^2*Sin[5*a + 4*b*x] + 6*b^2*d^3*x^2*Sin[5*a + 4*b*x] - (4*I)*b^3*d^3*x^3*Sin[5*a + 4*b*x])","B",0
384,1,516,173,6.5625141,"\int (c+d x)^2 \sec (a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)^2*Sec[a + b*x]*Sin[3*a + 3*b*x],x]","\frac{c d \csc (a) \sec (a) \left(b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left(i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\cot ^2(a)+1}}\right)}{b^2 \sqrt{\csc ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}-\frac{\cos (2 b x) \left(2 b^2 c^2 \cos (2 a)+4 b^2 c d x \cos (2 a)+2 b^2 d^2 x^2 \cos (2 a)-2 b c d \sin (2 a)-2 b d^2 x \sin (2 a)-d^2 \cos (2 a)\right)}{2 b^3}+\frac{\sin (2 b x) \left(2 b^2 c^2 \sin (2 a)+4 b^2 c d x \sin (2 a)+2 b^2 d^2 x^2 \sin (2 a)+2 b c d \cos (2 a)+2 b d^2 x \cos (2 a)-d^2 \sin (2 a)\right)}{2 b^3}+\frac{i e^{-i a} d^2 \sec (a) \left(2 b^2 x^2 \left(2 b x-3 i \left(1+e^{2 i a}\right) \log \left(1+e^{-2 i (a+b x)}\right)\right)+6 \left(1+e^{2 i a}\right) b x \text{Li}_2\left(-e^{-2 i (a+b x)}\right)-3 i \left(1+e^{2 i a}\right) \text{Li}_3\left(-e^{-2 i (a+b x)}\right)\right)}{12 b^3}+\frac{c^2 \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b \left(\sin ^2(a)+\cos ^2(a)\right)}-\frac{1}{3} x \tan (a) \left(3 c^2+3 c d x+d^2 x^2\right)","\frac{d^2 \text{Li}_3\left(-e^{2 i (a+b x)}\right)}{2 b^3}-\frac{d^2 \sin ^2(a+b x)}{b^3}-\frac{i d (c+d x) \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{b^2}+\frac{2 d (c+d x) \sin (a+b x) \cos (a+b x)}{b^2}+\frac{(c+d x)^2 \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{2 (c+d x)^2 \sin ^2(a+b x)}{b}-\frac{2 c d x}{b}-\frac{d^2 x^2}{b}-\frac{i (c+d x)^3}{3 d}",1,"((I/12)*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) + (c^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + (c*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (Cos[2*b*x]*(2*b^2*c^2*Cos[2*a] - d^2*Cos[2*a] + 4*b^2*c*d*x*Cos[2*a] + 2*b^2*d^2*x^2*Cos[2*a] - 2*b*c*d*Sin[2*a] - 2*b*d^2*x*Sin[2*a]))/(2*b^3) + ((2*b*c*d*Cos[2*a] + 2*b*d^2*x*Cos[2*a] + 2*b^2*c^2*Sin[2*a] - d^2*Sin[2*a] + 4*b^2*c*d*x*Sin[2*a] + 2*b^2*d^2*x^2*Sin[2*a])*Sin[2*b*x])/(2*b^3) - (x*(3*c^2 + 3*c*d*x + d^2*x^2)*Tan[a])/3","B",0
385,1,254,107,5.5736691,"\int (c+d x) \sec (a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)*Sec[a + b*x]*Sin[3*a + 3*b*x],x]","\frac{d \csc (a) \sec (a) \left(b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left(i \text{Li}_2\left(e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+i b x \left(-2 \tan ^{-1}(\cot (a))-\pi \right)-2 \left(b x-\tan ^{-1}(\cot (a))\right) \log \left(1-e^{2 i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-2 \tan ^{-1}(\cot (a)) \log \left(\sin \left(b x-\tan ^{-1}(\cot (a))\right)\right)-\pi  \log \left(1+e^{-2 i b x}\right)+\pi  \log (\cos (b x))\right)}{\sqrt{\cot ^2(a)+1}}\right)}{2 b^2 \sqrt{\csc ^2(a) \left(\sin ^2(a)+\cos ^2(a)\right)}}-\frac{d \cos (2 b x) (2 b x \cos (2 a)-\sin (2 a))}{2 b^2}+\frac{d \sin (2 b x) (2 b x \sin (2 a)+\cos (2 a))}{2 b^2}+\frac{c \left(2 \sin ^2(a+b x)+\log (\cos (a+b x))\right)}{b}-\frac{1}{2} d x^2 \tan (a)","-\frac{i d \text{Li}_2\left(-e^{2 i (a+b x)}\right)}{2 b^2}+\frac{d \sin (a+b x) \cos (a+b x)}{b^2}+\frac{(c+d x) \log \left(1+e^{2 i (a+b x)}\right)}{b}+\frac{2 (c+d x) \sin ^2(a+b x)}{b}-\frac{d x}{b}-\frac{i (c+d x)^2}{2 d}",1,"(d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(2*b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (d*Cos[2*b*x]*(2*b*x*Cos[2*a] - Sin[2*a]))/(2*b^2) + (d*(Cos[2*a] + 2*b*x*Sin[2*a])*Sin[2*b*x])/(2*b^2) + (c*(Log[Cos[a + b*x]] + 2*Sin[a + b*x]^2))/b - (d*x^2*Tan[a])/2","B",0
386,0,0,80,3.2257569,"\int \frac{\sec (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx","Integrate[(Sec[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x),x]","\int \frac{\sec (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx","-\text{Int}\left(\frac{\tan (a+b x)}{c+d x},x\right)+\frac{2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d}+\frac{2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d}",0,"Integrate[(Sec[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x), x]","A",-1
387,0,0,103,4.2076516,"\int \frac{\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx","Integrate[(Sec[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x)^2,x]","\int \frac{\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx","-\text{Int}\left(\frac{\tan (a+b x)}{(c+d x)^2},x\right)+\frac{4 b \cos \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}-\frac{4 b \sin \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d^2}-\frac{2 \sin (2 a+2 b x)}{d (c+d x)}",0,"Integrate[(Sec[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x)^2, x]","A",-1
388,0,0,129,6.0190375,"\int \frac{\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx","Integrate[(Sec[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x)^3,x]","\int \frac{\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx","-\text{Int}\left(\frac{\tan (a+b x)}{(c+d x)^3},x\right)-\frac{4 b^2 \sin \left(2 a-\frac{2 b c}{d}\right) \text{Ci}\left(\frac{2 b c}{d}+2 b x\right)}{d^3}-\frac{4 b^2 \cos \left(2 a-\frac{2 b c}{d}\right) \text{Si}\left(\frac{2 b c}{d}+2 b x\right)}{d^3}-\frac{2 b \cos (2 a+2 b x)}{d^2 (c+d x)}-\frac{\sin (2 a+2 b x)}{d (c+d x)^2}",0,"Integrate[(Sec[a + b*x]*Sin[3*a + 3*b*x])/(c + d*x)^3, x]","A",-1
389,1,607,230,2.5990796,"\int (c+d x)^3 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)^3*Sec[a + b*x]^2*Sin[3*a + 3*b*x],x]","-\frac{\sec (a+b x) \left(2 b^3 c^3 \cos (2 (a+b x))+6 b^3 c^2 d x \cos (2 (a+b x))+6 b^3 c d^2 x^2 \cos (2 (a+b x))+i b^3 d^3 x^3 \cos (a+b x)+2 b^3 d^3 x^3 \cos (2 (a+b x))-6 b^2 c^2 d \sin (2 (a+b x))+6 i b^2 c^2 d \cos (a+b x) \tan ^{-1}(\cos (a+b x)+i \sin (a+b x))-12 b^2 c d^2 x \sin (2 (a+b x))+6 b^2 c d^2 x \cos (a+b x) \log (-\sin (a+b x)+i \cos (a+b x)+1)-6 b^2 c d^2 x \cos (a+b x) \log (\sin (a+b x)-i \cos (a+b x)+1)-6 b^2 d^3 x^2 \sin (2 (a+b x))+3 b^2 d^3 x^2 \cos (a+b x) \log (-\sin (a+b x)-i \cos (a+b x)+1)-3 b^2 d^3 x^2 \cos (a+b x) \log (\sin (a+b x)-i \cos (a+b x)+1)+6 i b d^2 (c+d x) \cos (a+b x) \text{Li}_2(i \cos (a+b x)-\sin (a+b x))-6 i b c d^2 \cos (a+b x) \text{Li}_2(\sin (a+b x)-i \cos (a+b x))-12 b c d^2 \cos (2 (a+b x))+6 i b d^3 x \cos (a+b x) \text{Li}_2(i \cos (a+b x)+\sin (a+b x))-6 d^3 \cos (a+b x) \text{Li}_3(i \cos (a+b x)-\sin (a+b x))+6 d^3 \cos (a+b x) \text{Li}_3(i \cos (a+b x)+\sin (a+b x))+12 d^3 \sin (2 (a+b x))-12 b d^3 x \cos (2 (a+b x))+3 b^3 c^3+9 b^3 c^2 d x+9 b^3 c d^2 x^2+3 b^3 d^3 x^3-12 b c d^2-12 b d^3 x\right)}{b^4}","-\frac{6 d^3 \text{Li}_3\left(-i e^{i (a+b x)}\right)}{b^4}+\frac{6 d^3 \text{Li}_3\left(i e^{i (a+b x)}\right)}{b^4}-\frac{24 d^3 \sin (a+b x)}{b^4}+\frac{6 i d^2 (c+d x) \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}-\frac{6 i d^2 (c+d x) \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}+\frac{24 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac{12 d (c+d x)^2 \sin (a+b x)}{b^2}-\frac{6 i d (c+d x)^2 \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{4 (c+d x)^3 \cos (a+b x)}{b}-\frac{(c+d x)^3 \sec (a+b x)}{b}",1,"-((Sec[a + b*x]*(3*b^3*c^3 - 12*b*c*d^2 + 9*b^3*c^2*d*x - 12*b*d^3*x + 9*b^3*c*d^2*x^2 + 3*b^3*d^3*x^3 + I*b^3*d^3*x^3*Cos[a + b*x] + (6*I)*b^2*c^2*d*ArcTan[Cos[a + b*x] + I*Sin[a + b*x]]*Cos[a + b*x] + 2*b^3*c^3*Cos[2*(a + b*x)] - 12*b*c*d^2*Cos[2*(a + b*x)] + 6*b^3*c^2*d*x*Cos[2*(a + b*x)] - 12*b*d^3*x*Cos[2*(a + b*x)] + 6*b^3*c*d^2*x^2*Cos[2*(a + b*x)] + 2*b^3*d^3*x^3*Cos[2*(a + b*x)] + 3*b^2*d^3*x^2*Cos[a + b*x]*Log[1 - I*Cos[a + b*x] - Sin[a + b*x]] + 6*b^2*c*d^2*x*Cos[a + b*x]*Log[1 + I*Cos[a + b*x] - Sin[a + b*x]] - 6*b^2*c*d^2*x*Cos[a + b*x]*Log[1 - I*Cos[a + b*x] + Sin[a + b*x]] - 3*b^2*d^3*x^2*Cos[a + b*x]*Log[1 - I*Cos[a + b*x] + Sin[a + b*x]] + (6*I)*b*d^2*(c + d*x)*Cos[a + b*x]*PolyLog[2, I*Cos[a + b*x] - Sin[a + b*x]] - (6*I)*b*c*d^2*Cos[a + b*x]*PolyLog[2, (-I)*Cos[a + b*x] + Sin[a + b*x]] + (6*I)*b*d^3*x*Cos[a + b*x]*PolyLog[2, I*Cos[a + b*x] + Sin[a + b*x]] - 6*d^3*Cos[a + b*x]*PolyLog[3, I*Cos[a + b*x] - Sin[a + b*x]] + 6*d^3*Cos[a + b*x]*PolyLog[3, I*Cos[a + b*x] + Sin[a + b*x]] - 6*b^2*c^2*d*Sin[2*(a + b*x)] + 12*d^3*Sin[2*(a + b*x)] - 12*b^2*c*d^2*x*Sin[2*(a + b*x)] - 6*b^2*d^3*x^2*Sin[2*(a + b*x)]))/b^4)","B",1
390,1,364,147,3.8290473,"\int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)^2*Sec[a + b*x]^2*Sin[3*a + 3*b*x],x]","\frac{-4 \cos (b x) \left(\cos (a) \left(b^2 (c+d x)^2-2 d^2\right)-2 b d \sin (a) (c+d x)\right)+4 \sin (b x) \left(\sin (a) \left(b^2 (c+d x)^2-2 d^2\right)+2 b d \cos (a) (c+d x)\right)-b^2 \sec (a) (c+d x)^2-\frac{b^2 \sin \left(\frac{b x}{2}\right) (c+d x)^2}{\left(\cos \left(\frac{a}{2}\right)-\sin \left(\frac{a}{2}\right)\right) \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)}+\frac{b^2 \sin \left(\frac{b x}{2}\right) (c+d x)^2}{\left(\sin \left(\frac{a}{2}\right)+\cos \left(\frac{a}{2}\right)\right) \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)}+4 b c d \tanh ^{-1}\left(\cos (a) \tan \left(\frac{b x}{2}\right)+\sin (a)\right)+2 d^2 \left(2 \tan ^{-1}(\cot (a)) \tanh ^{-1}\left(\cos (a) \tan \left(\frac{b x}{2}\right)+\sin (a)\right)-\frac{\csc (a) \left(i \text{Li}_2\left(-e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-i \text{Li}_2\left(e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)+\left(b x-\tan ^{-1}(\cot (a))\right) \left(\log \left(1-e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)-\log \left(1+e^{i \left(b x-\tan ^{-1}(\cot (a))\right)}\right)\right)\right)}{\sqrt{\csc ^2(a)}}\right)}{b^3}","\frac{2 i d^2 \text{Li}_2\left(-i e^{i (a+b x)}\right)}{b^3}-\frac{2 i d^2 \text{Li}_2\left(i e^{i (a+b x)}\right)}{b^3}+\frac{8 d^2 \cos (a+b x)}{b^3}+\frac{8 d (c+d x) \sin (a+b x)}{b^2}-\frac{4 i d (c+d x) \tan ^{-1}\left(e^{i (a+b x)}\right)}{b^2}-\frac{4 (c+d x)^2 \cos (a+b x)}{b}-\frac{(c+d x)^2 \sec (a+b x)}{b}",1,"(4*b*c*d*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] + 2*d^2*(2*ArcTan[Cot[a]]*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] - (Csc[a]*((b*x - ArcTan[Cot[a]])*(Log[1 - E^(I*(b*x - ArcTan[Cot[a]]))] - Log[1 + E^(I*(b*x - ArcTan[Cot[a]]))]) + I*PolyLog[2, -E^(I*(b*x - ArcTan[Cot[a]]))] - I*PolyLog[2, E^(I*(b*x - ArcTan[Cot[a]]))]))/Sqrt[Csc[a]^2]) - b^2*(c + d*x)^2*Sec[a] - 4*Cos[b*x]*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] - 2*b*d*(c + d*x)*Sin[a]) + 4*(2*b*d*(c + d*x)*Cos[a] + (-2*d^2 + b^2*(c + d*x)^2)*Sin[a])*Sin[b*x] - (b^2*(c + d*x)^2*Sin[(b*x)/2])/((Cos[a/2] - Sin[a/2])*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])) + (b^2*(c + d*x)^2*Sin[(b*x)/2])/((Cos[a/2] + Sin[a/2])*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])))/b^3","B",0
391,1,105,57,0.483186,"\int (c+d x) \sec ^2(a+b x) \sin (3 a+3 b x) \, dx","Integrate[(c + d*x)*Sec[a + b*x]^2*Sin[3*a + 3*b*x],x]","-\frac{\sec (a+b x) \left(2 b (c+d x) \cos (2 (a+b x))-2 d \sin (2 (a+b x))+d \cos (a+b x) \left(\log \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)\right)+3 b c+3 b d x\right)}{b^2}","\frac{4 d \sin (a+b x)}{b^2}+\frac{d \tanh ^{-1}(\sin (a+b x))}{b^2}-\frac{4 (c+d x) \cos (a+b x)}{b}-\frac{(c+d x) \sec (a+b x)}{b}",1,"-((Sec[a + b*x]*(3*b*c + 3*b*d*x + 2*b*(c + d*x)*Cos[2*(a + b*x)] + d*Cos[a + b*x]*(Log[Cos[(a + b*x)/2] - Sin[(a + b*x)/2]] - Log[Cos[(a + b*x)/2] + Sin[(a + b*x)/2]]) - 2*d*Sin[2*(a + b*x)]))/b^2)","A",1
392,0,0,78,13.7262251,"\int \frac{\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx","Integrate[(Sec[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x),x]","\int \frac{\sec ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx","-\text{Int}\left(\frac{\tan (a+b x) \sec (a+b x)}{c+d x},x\right)+\frac{4 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d}+\frac{4 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d}",0,"Integrate[(Sec[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x), x]","A",-1
393,0,0,98,16.5759221,"\int \frac{\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx","Integrate[(Sec[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x)^2,x]","\int \frac{\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^2} \, dx","-\text{Int}\left(\frac{\tan (a+b x) \sec (a+b x)}{(c+d x)^2},x\right)+\frac{4 b \cos \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d^2}-\frac{4 b \sin \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d^2}-\frac{4 \sin (a+b x)}{d (c+d x)}",0,"Integrate[(Sec[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x)^2, x]","A",-1
394,0,0,121,19.1337767,"\int \frac{\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx","Integrate[(Sec[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x)^3,x]","\int \frac{\sec ^2(a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx","-\text{Int}\left(\frac{\tan (a+b x) \sec (a+b x)}{(c+d x)^3},x\right)-\frac{2 b^2 \sin \left(a-\frac{b c}{d}\right) \text{Ci}\left(\frac{b c}{d}+b x\right)}{d^3}-\frac{2 b^2 \cos \left(a-\frac{b c}{d}\right) \text{Si}\left(\frac{b c}{d}+b x\right)}{d^3}-\frac{2 b \cos (a+b x)}{d^2 (c+d x)}-\frac{2 \sin (a+b x)}{d (c+d x)^2}",0,"Integrate[(Sec[a + b*x]^2*Sin[3*a + 3*b*x])/(c + d*x)^3, x]","A",-1
395,1,77,57,0.0311799,"\int x \cos (2 x) \sec (x) \, dx","Integrate[x*Cos[2*x]*Sec[x],x]","-i \left(\text{Li}_2\left(-i e^{i x}\right)-\text{Li}_2\left(i e^{i x}\right)\right)-x \left(\log \left(1-i e^{i x}\right)-\log \left(1+i e^{i x}\right)\right)+2 x \sin (x)+2 \cos (x)","-i \text{Li}_2\left(-i e^{i x}\right)+i \text{Li}_2\left(i e^{i x}\right)+2 x \sin (x)+2 \cos (x)+2 i x \tan ^{-1}\left(e^{i x}\right)",1,"2*Cos[x] - x*(Log[1 - I*E^(I*x)] - Log[1 + I*E^(I*x)]) - I*(PolyLog[2, (-I)*E^(I*x)] - PolyLog[2, I*E^(I*x)]) + 2*x*Sin[x]","A",1
396,1,14,14,0.0203,"\int x \cos (2 x) \sec ^2(x) \, dx","Integrate[x*Cos[2*x]*Sec[x]^2,x]","x^2-x \tan (x)-\log (\cos (x))","x^2-x \tan (x)-\log (\cos (x))",1,"x^2 - Log[Cos[x]] - x*Tan[x]","A",1
397,1,146,67,0.280829,"\int x \cos (2 x) \sec ^3(x) \, dx","Integrate[x*Cos[2*x]*Sec[x]^3,x]","\frac{1}{4} \left(6 i \text{Li}_2\left(-i e^{i x}\right)-6 i \text{Li}_2\left(i e^{i x}\right)+6 x \log \left(1-i e^{i x}\right)-6 x \log \left(1+i e^{i x}\right)+\frac{x}{\sin (x)-1}+\frac{x}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}+\frac{2 \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)}-\frac{2 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}\right)","\frac{3}{2} i \text{Li}_2\left(-i e^{i x}\right)-\frac{3}{2} i \text{Li}_2\left(i e^{i x}\right)-3 i x \tan ^{-1}\left(e^{i x}\right)+\frac{\sec (x)}{2}-\frac{1}{2} x \tan (x) \sec (x)",1,"(6*x*Log[1 - I*E^(I*x)] - 6*x*Log[1 + I*E^(I*x)] + (6*I)*PolyLog[2, (-I)*E^(I*x)] - (6*I)*PolyLog[2, I*E^(I*x)] + (2*Sin[x/2])/(Cos[x/2] - Sin[x/2]) + x/(Cos[x/2] + Sin[x/2])^2 - (2*Sin[x/2])/(Cos[x/2] + Sin[x/2]) + x/(-1 + Sin[x]))/4","B",1