Integrand size = 22, antiderivative size = 271 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=-\frac {b^2 c^2 d^3}{12 x^2}-\frac {b^2 c^3 d^3}{x}+b^2 c^4 d^3 \text {arctanh}(c x)-\frac {b c d^3 (a+b \text {arctanh}(c x))}{6 x^3}-\frac {b c^2 d^3 (a+b \text {arctanh}(c x))}{x^2}-\frac {7 b c^3 d^3 (a+b \text {arctanh}(c x))}{2 x}-\frac {d^3 (1+c x)^4 (a+b \text {arctanh}(c x))^2}{4 x^4}+4 a b c^4 d^3 \log (x)+\frac {11}{3} b^2 c^4 d^3 \log (x)+4 b c^4 d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )-\frac {11}{6} b^2 c^4 d^3 \log \left (1-c^2 x^2\right )-2 b^2 c^4 d^3 \operatorname {PolyLog}(2,-c x)+2 b^2 c^4 d^3 \operatorname {PolyLog}(2,c x)+2 b^2 c^4 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \]
-1/12*b^2*c^2*d^3/x^2-b^2*c^3*d^3/x+b^2*c^4*d^3*arctanh(c*x)-1/6*b*c*d^3*( a+b*arctanh(c*x))/x^3-b*c^2*d^3*(a+b*arctanh(c*x))/x^2-7/2*b*c^3*d^3*(a+b* arctanh(c*x))/x-1/4*d^3*(c*x+1)^4*(a+b*arctanh(c*x))^2/x^4+4*a*b*c^4*d^3*l n(x)+11/3*b^2*c^4*d^3*ln(x)+4*b*c^4*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))- 11/6*b^2*c^4*d^3*ln(-c^2*x^2+1)-2*b^2*c^4*d^3*polylog(2,-c*x)+2*b^2*c^4*d^ 3*polylog(2,c*x)+2*b^2*c^4*d^3*polylog(2,1-2/(-c*x+1))
Time = 0.56 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.27 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=-\frac {d^3 \left (3 a^2+12 a^2 c x+2 a b c x+18 a^2 c^2 x^2+12 a b c^2 x^2+b^2 c^2 x^2+12 a^2 c^3 x^3+42 a b c^3 x^3+12 b^2 c^3 x^3-b^2 c^4 x^4+3 b^2 \left (1+4 c x+6 c^2 x^2+4 c^3 x^3-15 c^4 x^4\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (b c x \left (1+6 c x+21 c^2 x^2-6 c^3 x^3\right )+3 a \left (1+4 c x+6 c^2 x^2+4 c^3 x^3\right )-24 b c^4 x^4 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-48 a b c^4 x^4 \log (c x)+21 a b c^4 x^4 \log (1-c x)-21 a b c^4 x^4 \log (1+c x)-44 b^2 c^4 x^4 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+24 a b c^4 x^4 \log \left (1-c^2 x^2\right )+24 b^2 c^4 x^4 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{12 x^4} \]
-1/12*(d^3*(3*a^2 + 12*a^2*c*x + 2*a*b*c*x + 18*a^2*c^2*x^2 + 12*a*b*c^2*x ^2 + b^2*c^2*x^2 + 12*a^2*c^3*x^3 + 42*a*b*c^3*x^3 + 12*b^2*c^3*x^3 - b^2* c^4*x^4 + 3*b^2*(1 + 4*c*x + 6*c^2*x^2 + 4*c^3*x^3 - 15*c^4*x^4)*ArcTanh[c *x]^2 + 2*b*ArcTanh[c*x]*(b*c*x*(1 + 6*c*x + 21*c^2*x^2 - 6*c^3*x^3) + 3*a *(1 + 4*c*x + 6*c^2*x^2 + 4*c^3*x^3) - 24*b*c^4*x^4*Log[1 - E^(-2*ArcTanh[ c*x])]) - 48*a*b*c^4*x^4*Log[c*x] + 21*a*b*c^4*x^4*Log[1 - c*x] - 21*a*b*c ^4*x^4*Log[1 + c*x] - 44*b^2*c^4*x^4*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 24*a*b *c^4*x^4*Log[1 - c^2*x^2] + 24*b^2*c^4*x^4*PolyLog[2, E^(-2*ArcTanh[c*x])] ))/x^4
Time = 0.57 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6500, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx\) |
| \(\Big \downarrow \) 6500 |
| \(\displaystyle -2 b c \int \left (-\frac {2 d^3 (a+b \text {arctanh}(c x)) c^4}{1-c x}-\frac {2 d^3 (a+b \text {arctanh}(c x)) c^3}{x}-\frac {7 d^3 (a+b \text {arctanh}(c x)) c^2}{4 x^2}-\frac {d^3 (a+b \text {arctanh}(c x)) c}{x^3}-\frac {d^3 (a+b \text {arctanh}(c x))}{4 x^4}\right )dx-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))^2}{4 x^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b c \left (-2 c^3 d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+\frac {7 c^2 d^3 (a+b \text {arctanh}(c x))}{4 x}+\frac {d^3 (a+b \text {arctanh}(c x))}{12 x^3}+\frac {c d^3 (a+b \text {arctanh}(c x))}{2 x^2}-2 a c^3 d^3 \log (x)-\frac {1}{2} b c^3 d^3 \text {arctanh}(c x)+b c^3 d^3 \operatorname {PolyLog}(2,-c x)-b c^3 d^3 \operatorname {PolyLog}(2,c x)-b c^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-\frac {11}{6} b c^3 d^3 \log (x)+\frac {b c^2 d^3}{2 x}+\frac {11}{12} b c^3 d^3 \log \left (1-c^2 x^2\right )+\frac {b c d^3}{24 x^2}\right )-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))^2}{4 x^4}\) |
-1/4*(d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x])^2)/x^4 - 2*b*c*((b*c*d^3)/(24*x ^2) + (b*c^2*d^3)/(2*x) - (b*c^3*d^3*ArcTanh[c*x])/2 + (d^3*(a + b*ArcTanh [c*x]))/(12*x^3) + (c*d^3*(a + b*ArcTanh[c*x]))/(2*x^2) + (7*c^2*d^3*(a + b*ArcTanh[c*x]))/(4*x) - 2*a*c^3*d^3*Log[x] - (11*b*c^3*d^3*Log[x])/6 - 2* c^3*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] + (11*b*c^3*d^3*Log[1 - c^2* x^2])/12 + b*c^3*d^3*PolyLog[2, -(c*x)] - b*c^3*d^3*PolyLog[2, c*x] - b*c^ 3*d^3*PolyLog[2, 1 - 2/(1 - c*x)])
3.1.92.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_.) + (e _.)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Si mp[(a + b*ArcTanh[c*x])^p u, x] - Simp[b*c*p Int[ExpandIntegrand[(a + b *ArcTanh[c*x])^(p - 1), u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && N eQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]
Time = 2.29 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.52
| method | result | size |
| parts | \(d^{3} a^{2} \left (-\frac {c^{3}}{x}-\frac {1}{4 x^{4}}-\frac {c}{x^{3}}-\frac {3 c^{2}}{2 x^{2}}\right )+d^{3} b^{2} c^{4} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{4}-\frac {15 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{4}-\frac {\operatorname {arctanh}\left (c x \right )}{6 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {7 \,\operatorname {arctanh}\left (c x \right )}{2 c x}+4 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-2 \operatorname {dilog}\left (c x +1\right )-2 \ln \left (c x \right ) \ln \left (c x +1\right )-2 \operatorname {dilog}\left (c x \right )+2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {15 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{8}-\frac {15 \ln \left (c x -1\right )^{2}}{16}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (c x +1\right )^{2}}{16}-\frac {4 \ln \left (c x +1\right )}{3}-\frac {7 \ln \left (c x -1\right )}{3}-\frac {1}{12 c^{2} x^{2}}-\frac {1}{c x}+\frac {11 \ln \left (c x \right )}{3}\right )+2 d^{3} a b \,c^{4} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {\ln \left (c x +1\right )}{8}-\frac {15 \ln \left (c x -1\right )}{8}-\frac {1}{12 c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}-\frac {7}{4 c x}+2 \ln \left (c x \right )\right )\) | \(412\) |
| derivativedivides | \(c^{4} \left (d^{3} a^{2} \left (-\frac {1}{c x}-\frac {3}{2 c^{2} x^{2}}-\frac {1}{c^{3} x^{3}}-\frac {1}{4 c^{4} x^{4}}\right )+d^{3} b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{4}-\frac {15 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{4}-\frac {\operatorname {arctanh}\left (c x \right )}{6 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {7 \,\operatorname {arctanh}\left (c x \right )}{2 c x}+4 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-2 \operatorname {dilog}\left (c x +1\right )-2 \ln \left (c x \right ) \ln \left (c x +1\right )-2 \operatorname {dilog}\left (c x \right )+2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {15 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{8}-\frac {15 \ln \left (c x -1\right )^{2}}{16}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (c x +1\right )^{2}}{16}-\frac {4 \ln \left (c x +1\right )}{3}-\frac {7 \ln \left (c x -1\right )}{3}-\frac {1}{12 c^{2} x^{2}}-\frac {1}{c x}+\frac {11 \ln \left (c x \right )}{3}\right )+2 d^{3} a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {\ln \left (c x +1\right )}{8}-\frac {15 \ln \left (c x -1\right )}{8}-\frac {1}{12 c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}-\frac {7}{4 c x}+2 \ln \left (c x \right )\right )\right )\) | \(415\) |
| default | \(c^{4} \left (d^{3} a^{2} \left (-\frac {1}{c x}-\frac {3}{2 c^{2} x^{2}}-\frac {1}{c^{3} x^{3}}-\frac {1}{4 c^{4} x^{4}}\right )+d^{3} b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{4}-\frac {15 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{4}-\frac {\operatorname {arctanh}\left (c x \right )}{6 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {7 \,\operatorname {arctanh}\left (c x \right )}{2 c x}+4 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-2 \operatorname {dilog}\left (c x +1\right )-2 \ln \left (c x \right ) \ln \left (c x +1\right )-2 \operatorname {dilog}\left (c x \right )+2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {15 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{8}-\frac {15 \ln \left (c x -1\right )^{2}}{16}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (c x +1\right )^{2}}{16}-\frac {4 \ln \left (c x +1\right )}{3}-\frac {7 \ln \left (c x -1\right )}{3}-\frac {1}{12 c^{2} x^{2}}-\frac {1}{c x}+\frac {11 \ln \left (c x \right )}{3}\right )+2 d^{3} a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {\ln \left (c x +1\right )}{8}-\frac {15 \ln \left (c x -1\right )}{8}-\frac {1}{12 c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}-\frac {7}{4 c x}+2 \ln \left (c x \right )\right )\right )\) | \(415\) |
d^3*a^2*(-c^3/x-1/4/x^4-c/x^3-3/2*c^2/x^2)+d^3*b^2*c^4*(-1/c/x*arctanh(c*x )^2-3/2/c^2/x^2*arctanh(c*x)^2-1/c^3/x^3*arctanh(c*x)^2-1/4*arctanh(c*x)^2 /c^4/x^4-1/4*arctanh(c*x)*ln(c*x+1)-15/4*arctanh(c*x)*ln(c*x-1)-1/6/c^3/x^ 3*arctanh(c*x)-1/c^2/x^2*arctanh(c*x)-7/2/c/x*arctanh(c*x)+4*ln(c*x)*arcta nh(c*x)-2*dilog(c*x+1)-2*ln(c*x)*ln(c*x+1)-2*dilog(c*x)+2*dilog(1/2*c*x+1/ 2)+15/8*ln(c*x-1)*ln(1/2*c*x+1/2)-15/16*ln(c*x-1)^2-1/8*(ln(c*x+1)-ln(1/2* c*x+1/2))*ln(-1/2*c*x+1/2)+1/16*ln(c*x+1)^2-4/3*ln(c*x+1)-7/3*ln(c*x-1)-1/ 12/c^2/x^2-1/c/x+11/3*ln(c*x))+2*d^3*a*b*c^4*(-1/c/x*arctanh(c*x)-3/2/c^2/ x^2*arctanh(c*x)-1/c^3/x^3*arctanh(c*x)-1/4/c^4/x^4*arctanh(c*x)-1/8*ln(c* x+1)-15/8*ln(c*x-1)-1/12/c^3/x^3-1/2/c^2/x^2-7/4/c/x+2*ln(c*x))
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \]
integral((a^2*c^3*d^3*x^3 + 3*a^2*c^2*d^3*x^2 + 3*a^2*c*d^3*x + a^2*d^3 + (b^2*c^3*d^3*x^3 + 3*b^2*c^2*d^3*x^2 + 3*b^2*c*d^3*x + b^2*d^3)*arctanh(c* x)^2 + 2*(a*b*c^3*d^3*x^3 + 3*a*b*c^2*d^3*x^2 + 3*a*b*c*d^3*x + a*b*d^3)*a rctanh(c*x))/x^5, x)
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=d^{3} \left (\int \frac {a^{2}}{x^{5}}\, dx + \int \frac {3 a^{2} c}{x^{4}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{3}}\, dx + \int \frac {a^{2} c^{3}}{x^{2}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{5}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{5}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {b^{2} c^{3} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {6 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b c^{3} \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx\right ) \]
d**3*(Integral(a**2/x**5, x) + Integral(3*a**2*c/x**4, x) + Integral(3*a** 2*c**2/x**3, x) + Integral(a**2*c**3/x**2, x) + Integral(b**2*atanh(c*x)** 2/x**5, x) + Integral(2*a*b*atanh(c*x)/x**5, x) + Integral(3*b**2*c*atanh( c*x)**2/x**4, x) + Integral(3*b**2*c**2*atanh(c*x)**2/x**3, x) + Integral( b**2*c**3*atanh(c*x)**2/x**2, x) + Integral(6*a*b*c*atanh(c*x)/x**4, x) + Integral(6*a*b*c**2*atanh(c*x)/x**3, x) + Integral(2*a*b*c**3*atanh(c*x)/x **2, x))
Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (254) = 508\).
Time = 0.63 (sec) , antiderivative size = 813, normalized size of antiderivative = 3.00 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=-2 \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} c^{4} d^{3} - 2 \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b^{2} c^{4} d^{3} + 2 \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b^{2} c^{4} d^{3} - b^{2} c^{4} d^{3} \log \left (c x + 1\right ) - 2 \, b^{2} c^{4} d^{3} \log \left (c x - 1\right ) + 3 \, b^{2} c^{4} d^{3} \log \left (x\right ) - {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} a b c^{3} d^{3} + \frac {3}{2} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} a b c^{2} d^{3} - {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} a b c d^{3} - \frac {a^{2} c^{3} d^{3}}{x} + \frac {1}{12} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} a b d^{3} + \frac {1}{48} \, {\left ({\left (32 \, c^{2} \log \left (x\right ) - \frac {3 \, c^{2} x^{2} \log \left (c x + 1\right )^{2} + 3 \, c^{2} x^{2} \log \left (c x - 1\right )^{2} + 16 \, c^{2} x^{2} \log \left (c x - 1\right ) - 2 \, {\left (3 \, c^{2} x^{2} \log \left (c x - 1\right ) - 8 \, c^{2} x^{2}\right )} \log \left (c x + 1\right ) + 4}{x^{2}}\right )} c^{2} + 4 \, {\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c \operatorname {artanh}\left (c x\right )\right )} b^{2} d^{3} - \frac {3 \, a^{2} c^{2} d^{3}}{2 \, x^{2}} - \frac {a^{2} c d^{3}}{x^{3}} - \frac {b^{2} d^{3} \operatorname {artanh}\left (c x\right )^{2}}{4 \, x^{4}} - \frac {a^{2} d^{3}}{4 \, x^{4}} - \frac {8 \, b^{2} c^{3} d^{3} x^{2} + {\left (b^{2} c^{4} d^{3} x^{3} + 2 \, b^{2} c^{3} d^{3} x^{2} + 3 \, b^{2} c^{2} d^{3} x + 2 \, b^{2} c d^{3}\right )} \log \left (c x + 1\right )^{2} - {\left (7 \, b^{2} c^{4} d^{3} x^{3} - 2 \, b^{2} c^{3} d^{3} x^{2} - 3 \, b^{2} c^{2} d^{3} x - 2 \, b^{2} c d^{3}\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (3 \, b^{2} c^{3} d^{3} x^{2} + b^{2} c^{2} d^{3} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (6 \, b^{2} c^{3} d^{3} x^{2} + 2 \, b^{2} c^{2} d^{3} x + {\left (b^{2} c^{4} d^{3} x^{3} + 2 \, b^{2} c^{3} d^{3} x^{2} + 3 \, b^{2} c^{2} d^{3} x + 2 \, b^{2} c d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \, x^{3}} \]
-2*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*c^4*d^3 - 2*(log(c*x)*log(-c*x + 1) + dilog(-c*x + 1))*b^2*c^4*d^3 + 2*(log(c*x + 1 )*log(-c*x) + dilog(c*x + 1))*b^2*c^4*d^3 - b^2*c^4*d^3*log(c*x + 1) - 2*b ^2*c^4*d^3*log(c*x - 1) + 3*b^2*c^4*d^3*log(x) - (c*(log(c^2*x^2 - 1) - lo g(x^2)) + 2*arctanh(c*x)/x)*a*b*c^3*d^3 + 3/2*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*a*b*c^2*d^3 - ((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)*c + 2*arctanh(c*x)/x^3)*a*b*c*d^3 - a^2*c^3*d^3/x + 1/12*((3*c^3*log(c*x + 1) - 3*c^3*log(c*x - 1) - 2*(3*c^2*x^2 + 1)/x^3)* c - 6*arctanh(c*x)/x^4)*a*b*d^3 + 1/48*((32*c^2*log(x) - (3*c^2*x^2*log(c* x + 1)^2 + 3*c^2*x^2*log(c*x - 1)^2 + 16*c^2*x^2*log(c*x - 1) - 2*(3*c^2*x ^2*log(c*x - 1) - 8*c^2*x^2)*log(c*x + 1) + 4)/x^2)*c^2 + 4*(3*c^3*log(c*x + 1) - 3*c^3*log(c*x - 1) - 2*(3*c^2*x^2 + 1)/x^3)*c*arctanh(c*x))*b^2*d^ 3 - 3/2*a^2*c^2*d^3/x^2 - a^2*c*d^3/x^3 - 1/4*b^2*d^3*arctanh(c*x)^2/x^4 - 1/4*a^2*d^3/x^4 - 1/8*(8*b^2*c^3*d^3*x^2 + (b^2*c^4*d^3*x^3 + 2*b^2*c^3*d ^3*x^2 + 3*b^2*c^2*d^3*x + 2*b^2*c*d^3)*log(c*x + 1)^2 - (7*b^2*c^4*d^3*x^ 3 - 2*b^2*c^3*d^3*x^2 - 3*b^2*c^2*d^3*x - 2*b^2*c*d^3)*log(-c*x + 1)^2 + 4 *(3*b^2*c^3*d^3*x^2 + b^2*c^2*d^3*x)*log(c*x + 1) - 2*(6*b^2*c^3*d^3*x^2 + 2*b^2*c^2*d^3*x + (b^2*c^4*d^3*x^3 + 2*b^2*c^3*d^3*x^2 + 3*b^2*c^2*d^3*x + 2*b^2*c*d^3)*log(c*x + 1))*log(-c*x + 1))/x^3
\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3}{x^5} \,d x \]