Integrand size = 22, antiderivative size = 244 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^2}{3 x}+\frac {1}{3} b^2 c^3 d^2 \text {arctanh}(c x)-\frac {b c d^2 (a+b \text {arctanh}(c x))}{3 x^2}-\frac {2 b c^2 d^2 (a+b \text {arctanh}(c x))}{x}-\frac {d^2 (1+c x)^3 (a+b \text {arctanh}(c x))^2}{3 x^3}+\frac {8}{3} a b c^3 d^2 \log (x)+2 b^2 c^3 d^2 \log (x)+\frac {8}{3} b c^3 d^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )-b^2 c^3 d^2 \log \left (1-c^2 x^2\right )-\frac {4}{3} b^2 c^3 d^2 \operatorname {PolyLog}(2,-c x)+\frac {4}{3} b^2 c^3 d^2 \operatorname {PolyLog}(2,c x)+\frac {4}{3} b^2 c^3 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \]
-1/3*b^2*c^2*d^2/x+1/3*b^2*c^3*d^2*arctanh(c*x)-1/3*b*c*d^2*(a+b*arctanh(c *x))/x^2-2*b*c^2*d^2*(a+b*arctanh(c*x))/x-1/3*d^2*(c*x+1)^3*(a+b*arctanh(c *x))^2/x^3+8/3*a*b*c^3*d^2*ln(x)+2*b^2*c^3*d^2*ln(x)+8/3*b*c^3*d^2*(a+b*ar ctanh(c*x))*ln(2/(-c*x+1))-b^2*c^3*d^2*ln(-c^2*x^2+1)-4/3*b^2*c^3*d^2*poly log(2,-c*x)+4/3*b^2*c^3*d^2*polylog(2,c*x)+4/3*b^2*c^3*d^2*polylog(2,1-2/( -c*x+1))
Time = 0.46 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.11 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {d^2 \left (a^2+3 a^2 c x+a b c x+3 a^2 c^2 x^2+6 a b c^2 x^2+b^2 c^2 x^2+b^2 \left (1+3 c x+3 c^2 x^2-7 c^3 x^3\right ) \text {arctanh}(c x)^2+b \text {arctanh}(c x) \left (b c x \left (1+6 c x-c^2 x^2\right )+a \left (2+6 c x+6 c^2 x^2\right )-8 b c^3 x^3 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-8 a b c^3 x^3 \log (c x)+3 a b c^3 x^3 \log (1-c x)-3 a b c^3 x^3 \log (1+c x)-6 b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+4 a b c^3 x^3 \log \left (1-c^2 x^2\right )+4 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{3 x^3} \]
-1/3*(d^2*(a^2 + 3*a^2*c*x + a*b*c*x + 3*a^2*c^2*x^2 + 6*a*b*c^2*x^2 + b^2 *c^2*x^2 + b^2*(1 + 3*c*x + 3*c^2*x^2 - 7*c^3*x^3)*ArcTanh[c*x]^2 + b*ArcT anh[c*x]*(b*c*x*(1 + 6*c*x - c^2*x^2) + a*(2 + 6*c*x + 6*c^2*x^2) - 8*b*c^ 3*x^3*Log[1 - E^(-2*ArcTanh[c*x])]) - 8*a*b*c^3*x^3*Log[c*x] + 3*a*b*c^3*x ^3*Log[1 - c*x] - 3*a*b*c^3*x^3*Log[1 + c*x] - 6*b^2*c^3*x^3*Log[(c*x)/Sqr t[1 - c^2*x^2]] + 4*a*b*c^3*x^3*Log[1 - c^2*x^2] + 4*b^2*c^3*x^3*PolyLog[2 , E^(-2*ArcTanh[c*x])]))/x^3
Time = 0.50 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.93, 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.
| \(\displaystyle \int \frac {(c d x+d)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 6500 |
| \(\displaystyle -2 b c \int \left (-\frac {4 d^2 (a+b \text {arctanh}(c x)) c^3}{3 (1-c x)}-\frac {4 d^2 (a+b \text {arctanh}(c x)) c^2}{3 x}-\frac {d^2 (a+b \text {arctanh}(c x)) c}{x^2}-\frac {d^2 (a+b \text {arctanh}(c x))}{3 x^3}\right )dx-\frac {d^2 (c x+1)^3 (a+b \text {arctanh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b c \left (-\frac {4}{3} c^2 d^2 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+\frac {d^2 (a+b \text {arctanh}(c x))}{6 x^2}+\frac {c d^2 (a+b \text {arctanh}(c x))}{x}-\frac {4}{3} a c^2 d^2 \log (x)-\frac {1}{6} b c^2 d^2 \text {arctanh}(c x)+\frac {2}{3} b c^2 d^2 \operatorname {PolyLog}(2,-c x)-\frac {2}{3} b c^2 d^2 \operatorname {PolyLog}(2,c x)-\frac {2}{3} b c^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {1}{2} b c^2 d^2 \log \left (1-c^2 x^2\right )-b c^2 d^2 \log (x)+\frac {b c d^2}{6 x}\right )-\frac {d^2 (c x+1)^3 (a+b \text {arctanh}(c x))^2}{3 x^3}\) |
-1/3*(d^2*(1 + c*x)^3*(a + b*ArcTanh[c*x])^2)/x^3 - 2*b*c*((b*c*d^2)/(6*x) - (b*c^2*d^2*ArcTanh[c*x])/6 + (d^2*(a + b*ArcTanh[c*x]))/(6*x^2) + (c*d^ 2*(a + b*ArcTanh[c*x]))/x - (4*a*c^2*d^2*Log[x])/3 - b*c^2*d^2*Log[x] - (4 *c^2*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/3 + (b*c^2*d^2*Log[1 - c^2 *x^2])/2 + (2*b*c^2*d^2*PolyLog[2, -(c*x)])/3 - (2*b*c^2*d^2*PolyLog[2, c* x])/3 - (2*b*c^2*d^2*PolyLog[2, 1 - 2/(1 - c*x)])/3)
3.1.83.3.1 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 = 2.19 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.43
| method | result | size |
| parts | \(d^{2} a^{2} \left (-\frac {c^{2}}{x}-\frac {1}{3 x^{3}}-\frac {c}{x^{2}}\right )+d^{2} b^{2} c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {7 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}+\frac {8 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )}{3}-\frac {4 \operatorname {dilog}\left (c x +1\right )}{3}-\frac {4 \ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {4 \operatorname {dilog}\left (c x \right )}{3}+\frac {4 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {7 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {7 \ln \left (c x -1\right )^{2}}{12}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (c x +1\right )^{2}}{12}-\frac {5 \ln \left (c x +1\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{3 c x}+2 \ln \left (c x \right )\right )+2 a b \,d^{2} c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\ln \left (c x +1\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{c x}+\frac {4 \ln \left (c x \right )}{3}\right )\) | \(350\) |
| derivativedivides | \(c^{3} \left (d^{2} a^{2} \left (-\frac {1}{c x}-\frac {1}{c^{2} x^{2}}-\frac {1}{3 c^{3} x^{3}}\right )+d^{2} b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {7 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}+\frac {8 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )}{3}-\frac {4 \operatorname {dilog}\left (c x +1\right )}{3}-\frac {4 \ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {4 \operatorname {dilog}\left (c x \right )}{3}+\frac {4 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {7 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {7 \ln \left (c x -1\right )^{2}}{12}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (c x +1\right )^{2}}{12}-\frac {5 \ln \left (c x +1\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{3 c x}+2 \ln \left (c x \right )\right )+2 a b \,d^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\ln \left (c x +1\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{c x}+\frac {4 \ln \left (c x \right )}{3}\right )\right )\) | \(353\) |
| default | \(c^{3} \left (d^{2} a^{2} \left (-\frac {1}{c x}-\frac {1}{c^{2} x^{2}}-\frac {1}{3 c^{3} x^{3}}\right )+d^{2} b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {7 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}+\frac {8 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )}{3}-\frac {4 \operatorname {dilog}\left (c x +1\right )}{3}-\frac {4 \ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {4 \operatorname {dilog}\left (c x \right )}{3}+\frac {4 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {7 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {7 \ln \left (c x -1\right )^{2}}{12}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (c x +1\right )^{2}}{12}-\frac {5 \ln \left (c x +1\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{3 c x}+2 \ln \left (c x \right )\right )+2 a b \,d^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\ln \left (c x +1\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{c x}+\frac {4 \ln \left (c x \right )}{3}\right )\right )\) | \(353\) |
d^2*a^2*(-c^2/x-1/3/x^3-c/x^2)+d^2*b^2*c^3*(-1/c/x*arctanh(c*x)^2-1/c^2/x^ 2*arctanh(c*x)^2-1/3/c^3/x^3*arctanh(c*x)^2-1/3*arctanh(c*x)*ln(c*x+1)-7/3 *arctanh(c*x)*ln(c*x-1)-1/3/c^2/x^2*arctanh(c*x)-2/c/x*arctanh(c*x)+8/3*ln (c*x)*arctanh(c*x)-4/3*dilog(c*x+1)-4/3*ln(c*x)*ln(c*x+1)-4/3*dilog(c*x)+4 /3*dilog(1/2*c*x+1/2)+7/6*ln(c*x-1)*ln(1/2*c*x+1/2)-7/12*ln(c*x-1)^2-1/6*( ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+1/12*ln(c*x+1)^2-5/6*ln(c*x+1) -7/6*ln(c*x-1)-1/3/c/x+2*ln(c*x))+2*a*b*d^2*c^3*(-1/c/x*arctanh(c*x)-1/c^2 /x^2*arctanh(c*x)-1/3/c^3/x^3*arctanh(c*x)-1/6*ln(c*x+1)-7/6*ln(c*x-1)-1/6 /c^2/x^2-1/c/x+4/3*ln(c*x))
\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
integral((a^2*c^2*d^2*x^2 + 2*a^2*c*d^2*x + a^2*d^2 + (b^2*c^2*d^2*x^2 + 2 *b^2*c*d^2*x + b^2*d^2)*arctanh(c*x)^2 + 2*(a*b*c^2*d^2*x^2 + 2*a*b*c*d^2* x + a*b*d^2)*arctanh(c*x))/x^4, x)
\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=d^{2} \left (\int \frac {a^{2}}{x^{4}}\, dx + \int \frac {2 a^{2} c}{x^{3}}\, dx + \int \frac {a^{2} c^{2}}{x^{2}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {4 a b c \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx\right ) \]
d**2*(Integral(a**2/x**4, x) + Integral(2*a**2*c/x**3, x) + Integral(a**2* c**2/x**2, x) + Integral(b**2*atanh(c*x)**2/x**4, x) + Integral(2*a*b*atan h(c*x)/x**4, x) + Integral(2*b**2*c*atanh(c*x)**2/x**3, x) + Integral(b**2 *c**2*atanh(c*x)**2/x**2, x) + Integral(4*a*b*c*atanh(c*x)/x**3, x) + Inte gral(2*a*b*c**2*atanh(c*x)/x**2, x))
Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (221) = 442\).
Time = 0.62 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.27 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {4}{3} \, {\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^{3} d^{2} - \frac {4}{3} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b^{2} c^{3} d^{2} + \frac {4}{3} \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b^{2} c^{3} d^{2} - \frac {5}{6} \, b^{2} c^{3} d^{2} \log \left (c x + 1\right ) - \frac {7}{6} \, b^{2} c^{3} d^{2} \log \left (c x - 1\right ) + 2 \, b^{2} c^{3} d^{2} \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^{2} d^{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 d^{2} - \frac {1}{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 d^{2} - \frac {a^{2} c^{2} d^{2}}{x} - \frac {a^{2} c d^{2}}{x^{2}} - \frac {a^{2} d^{2}}{3 \, x^{3}} - \frac {4 \, b^{2} c^{2} d^{2} x^{2} + {\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} - {\left (7 \, b^{2} c^{3} d^{2} x^{3} - 3 \, b^{2} c^{2} d^{2} x^{2} - 3 \, b^{2} c d^{2} x - b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2} + 2 \, {\left (6 \, b^{2} c^{2} d^{2} x^{2} + b^{2} c d^{2} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (6 \, b^{2} c^{2} d^{2} x^{2} + b^{2} c d^{2} x + {\left (b^{2} c^{3} d^{2} x^{3} + 3 \, b^{2} c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{12 \, x^{3}} \]
-4/3*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*c^3*d^2 - 4/3*(log(c*x)*log(-c*x + 1) + dilog(-c*x + 1))*b^2*c^3*d^2 + 4/3*(log(c *x + 1)*log(-c*x) + dilog(c*x + 1))*b^2*c^3*d^2 - 5/6*b^2*c^3*d^2*log(c*x + 1) - 7/6*b^2*c^3*d^2*log(c*x - 1) + 2*b^2*c^3*d^2*log(x) - (c*(log(c^2*x ^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*a*b*c^2*d^2 + ((c*log(c*x + 1) - c *log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*a*b*c*d^2 - 1/3*((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*d^2 - a^2*c^ 2*d^2/x - a^2*c*d^2/x^2 - 1/3*a^2*d^2/x^3 - 1/12*(4*b^2*c^2*d^2*x^2 + (b^2 *c^3*d^2*x^3 + 3*b^2*c^2*d^2*x^2 + 3*b^2*c*d^2*x + b^2*d^2)*log(c*x + 1)^2 - (7*b^2*c^3*d^2*x^3 - 3*b^2*c^2*d^2*x^2 - 3*b^2*c*d^2*x - b^2*d^2)*log(- c*x + 1)^2 + 2*(6*b^2*c^2*d^2*x^2 + b^2*c*d^2*x)*log(c*x + 1) - 2*(6*b^2*c ^2*d^2*x^2 + b^2*c*d^2*x + (b^2*c^3*d^2*x^3 + 3*b^2*c^2*d^2*x^2 + 3*b^2*c* d^2*x + b^2*d^2)*log(c*x + 1))*log(-c*x + 1))/x^3
\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^2}{x^4} \,d x \]