\(\int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx\) [83]

Optimal result

Integrand size = 22, antiderivative size = 244 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^2}{3 x}+\frac {1}{3} b^2 c^3 d^2 \text {arctanh}(c x)-\frac {b c d^2 (a+b \text {arctanh}(c x))}{3 x^2}-\frac {2 b c^2 d^2 (a+b \text {arctanh}(c x))}{x}-\frac {d^2 (1+c x)^3 (a+b \text {arctanh}(c x))^2}{3 x^3}+\frac {8}{3} a b c^3 d^2 \log (x)+2 b^2 c^3 d^2 \log (x)+\frac {8}{3} b c^3 d^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )-b^2 c^3 d^2 \log \left (1-c^2 x^2\right )-\frac {4}{3} b^2 c^3 d^2 \operatorname {PolyLog}(2,-c x)+\frac {4}{3} b^2 c^3 d^2 \operatorname {PolyLog}(2,c x)+\frac {4}{3} b^2 c^3 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \] Output:

-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))
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.11 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {d^2 \left (a^2+3 a^2 c x+a b c x+3 a^2 c^2 x^2+6 a b c^2 x^2+b^2 c^2 x^2+b^2 \left (1+3 c x+3 c^2 x^2-7 c^3 x^3\right ) \text {arctanh}(c x)^2+b \text {arctanh}(c x) \left (b c x \left (1+6 c x-c^2 x^2\right )+a \left (2+6 c x+6 c^2 x^2\right )-8 b c^3 x^3 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-8 a b c^3 x^3 \log (c x)+3 a b c^3 x^3 \log (1-c x)-3 a b c^3 x^3 \log (1+c x)-6 b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+4 a b c^3 x^3 \log \left (1-c^2 x^2\right )+4 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{3 x^3} \] Input:

Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x])^2)/x^4,x]
 

Output:

-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
 

Rubi [A] (verified)

Time = 0.53 (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

\(\displaystyle -2 b c \left (-\frac {4}{3} c^2 d^2 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+\frac {d^2 (a+b \text {arctanh}(c x))}{6 x^2}+\frac {c d^2 (a+b \text {arctanh}(c x))}{x}-\frac {4}{3} a c^2 d^2 \log (x)-\frac {1}{6} b c^2 d^2 \text {arctanh}(c x)+\frac {2}{3} b c^2 d^2 \operatorname {PolyLog}(2,-c x)-\frac {2}{3} b c^2 d^2 \operatorname {PolyLog}(2,c x)-\frac {2}{3} b c^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {1}{2} b c^2 d^2 \log \left (1-c^2 x^2\right )-b c^2 d^2 \log (x)+\frac {b c d^2}{6 x}\right )-\frac {d^2 (c x+1)^3 (a+b \text {arctanh}(c x))^2}{3 x^3}\)

Input:

Int[((d + c*d*x)^2*(a + b*ArcTanh[c*x])^2)/x^4,x]
 

Output:

-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)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6500

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]
 

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.43

method	result	size
parts	\(d^{2} a^{2} \left (-\frac {c^{2}}{x}-\frac {c}{x^{2}}-\frac {1}{3 x^{3}}\right )+d^{2} b^{2} c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {7 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}+\frac {8 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {4 \operatorname {dilog}\left (c x \right )}{3}-\frac {4 \operatorname {dilog}\left (c x +1\right )}{3}-\frac {4 \ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {7 \ln \left (c x -1\right )^{2}}{12}+\frac {4 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {7 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (c x +1\right )^{2}}{12}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{3 c x}+2 \ln \left (c x \right )-\frac {5 \ln \left (c x +1\right )}{6}\right )+2 d^{2} a b \,c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{c x}+\frac {4 \ln \left (c x \right )}{3}-\frac {\ln \left (c x +1\right )}{6}\right )\)	\(350\)
derivativedivides	\(c^{3} \left (d^{2} a^{2} \left (-\frac {1}{3 c^{3} x^{3}}-\frac {1}{c^{2} x^{2}}-\frac {1}{c x}\right )+d^{2} b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {7 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}+\frac {8 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {4 \operatorname {dilog}\left (c x \right )}{3}-\frac {4 \operatorname {dilog}\left (c x +1\right )}{3}-\frac {4 \ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {7 \ln \left (c x -1\right )^{2}}{12}+\frac {4 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {7 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (c x +1\right )^{2}}{12}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{3 c x}+2 \ln \left (c x \right )-\frac {5 \ln \left (c x +1\right )}{6}\right )+2 d^{2} a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{c x}+\frac {4 \ln \left (c x \right )}{3}-\frac {\ln \left (c x +1\right )}{6}\right )\right )\)	\(353\)
default	\(c^{3} \left (d^{2} a^{2} \left (-\frac {1}{3 c^{3} x^{3}}-\frac {1}{c^{2} x^{2}}-\frac {1}{c x}\right )+d^{2} b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {7 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x}+\frac {8 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {4 \operatorname {dilog}\left (c x \right )}{3}-\frac {4 \operatorname {dilog}\left (c x +1\right )}{3}-\frac {4 \ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {7 \ln \left (c x -1\right )^{2}}{12}+\frac {4 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {7 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (c x +1\right )^{2}}{12}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{3 c x}+2 \ln \left (c x \right )-\frac {5 \ln \left (c x +1\right )}{6}\right )+2 d^{2} a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{c x}+\frac {4 \ln \left (c x \right )}{3}-\frac {\ln \left (c x +1\right )}{6}\right )\right )\)	\(353\)

Input:

int((c*d*x+d)^2*(a+b*arctanh(c*x))^2/x^4,x,method=_RETURNVERBOSE)
 

Output:

d^2*a^2*(-c^2/x-c/x^2-1/3/x^3)+d^2*b^2*c^3*(-1/3*arctanh(c*x)^2/c^3/x^3-ar 
ctanh(c*x)^2/c^2/x^2-arctanh(c*x)^2/c/x-7/3*arctanh(c*x)*ln(c*x-1)-1/3*arc 
tanh(c*x)/c^2/x^2-2*arctanh(c*x)/c/x+8/3*arctanh(c*x)*ln(c*x)-1/3*arctanh( 
c*x)*ln(c*x+1)-4/3*dilog(c*x)-4/3*dilog(c*x+1)-4/3*ln(c*x)*ln(c*x+1)-7/12* 
ln(c*x-1)^2+4/3*dilog(1/2*c*x+1/2)+7/6*ln(c*x-1)*ln(1/2*c*x+1/2)+1/12*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)-7/6*ln(c*x-1)-1/3 
/c/x+2*ln(c*x)-5/6*ln(c*x+1))+2*d^2*a*b*c^3*(-1/3*arctanh(c*x)/c^3/x^3-arc 
tanh(c*x)/c^2/x^2-arctanh(c*x)/c/x-7/6*ln(c*x-1)-1/6/c^2/x^2-1/c/x+4/3*ln( 
c*x)-1/6*ln(c*x+1))
 

Fricas [F]

\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:

integrate((c*d*x+d)^2*(a+b*arctanh(c*x))^2/x^4,x, algorithm="fricas")
 

Output:

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)
 

Sympy [F]

\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=d^{2} \left (\int \frac {a^{2}}{x^{4}}\, dx + \int \frac {2 a^{2} c}{x^{3}}\, dx + \int \frac {a^{2} c^{2}}{x^{2}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {4 a b c \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx\right ) \] Input:

integrate((c*d*x+d)**2*(a+b*atanh(c*x))**2/x**4,x)
 

Output:

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))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (221) = 442\).

Time = 0.48 (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}} \] Input:

integrate((c*d*x+d)^2*(a+b*arctanh(c*x))^2/x^4,x, algorithm="maxima")
 

Output:

-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
 

Giac [F]

\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:

integrate((c*d*x+d)^2*(a+b*arctanh(c*x))^2/x^4,x, algorithm="giac")
 

Output:

integrate((c*d*x + d)^2*(b*arctanh(c*x) + a)^2/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^2}{x^4} \,d x \] Input:

int(((a + b*atanh(c*x))^2*(d + c*d*x)^2)/x^4,x)
 

Output:

int(((a + b*atanh(c*x))^2*(d + c*d*x)^2)/x^4, x)
 

Reduce [F]

\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^4} \, dx=\frac {d^{2} \left (3 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}-3 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-3 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -\mathit {atanh} \left (c x \right )^{2} b^{2}-2 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}-6 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}-6 \mathit {atanh} \left (c x \right ) a b c x -2 \mathit {atanh} \left (c x \right ) a b -9 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}-6 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}+3 \mathit {atanh} \left (c x \right ) b^{2} c x -8 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{5}-x^{3}}d x \right ) b^{2} c \,x^{3}-8 \,\mathrm {log}\left (c^{2} x -c \right ) a b \,c^{3} x^{3}-6 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} c^{3} x^{3}+8 \,\mathrm {log}\left (x \right ) a b \,c^{3} x^{3}+6 \,\mathrm {log}\left (x \right ) b^{2} c^{3} x^{3}-3 a^{2} c^{2} x^{2}-3 a^{2} c x -a^{2}-6 a b \,c^{2} x^{2}-a b c x +3 b^{2} c^{2} x^{2}\right )}{3 x^{3}} \] Input:

int((c*d*x+d)^2*(a+b*atanh(c*x))^2/x^4,x)
 

Output:

(d**2*(3*atanh(c*x)**2*b**2*c**3*x**3 - 3*atanh(c*x)**2*b**2*c**2*x**2 - 3 
*atanh(c*x)**2*b**2*c*x - atanh(c*x)**2*b**2 - 2*atanh(c*x)*a*b*c**3*x**3 
- 6*atanh(c*x)*a*b*c**2*x**2 - 6*atanh(c*x)*a*b*c*x - 2*atanh(c*x)*a*b - 9 
*atanh(c*x)*b**2*c**3*x**3 - 6*atanh(c*x)*b**2*c**2*x**2 + 3*atanh(c*x)*b* 
*2*c*x - 8*int(atanh(c*x)/(c**2*x**5 - x**3),x)*b**2*c*x**3 - 8*log(c**2*x 
 - c)*a*b*c**3*x**3 - 6*log(c**2*x - c)*b**2*c**3*x**3 + 8*log(x)*a*b*c**3 
*x**3 + 6*log(x)*b**2*c**3*x**3 - 3*a**2*c**2*x**2 - 3*a**2*c*x - a**2 - 6 
*a*b*c**2*x**2 - a*b*c*x + 3*b**2*c**2*x**2))/(3*x**3)