\(\int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^4} \, dx\) [75]

Optimal result

Integrand size = 20, antiderivative size = 206 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d}{3 x}+\frac {1}{3} b^2 c^3 d \text {arctanh}(c x)-\frac {b c d (a+b \text {arctanh}(c x))}{3 x^2}-\frac {b c^2 d (a+b \text {arctanh}(c x))}{x}+\frac {5}{6} c^3 d (a+b \text {arctanh}(c x))^2-\frac {d (a+b \text {arctanh}(c x))^2}{3 x^3}-\frac {c d (a+b \text {arctanh}(c x))^2}{2 x^2}+b^2 c^3 d \log (x)-\frac {1}{2} b^2 c^3 d \log \left (1-c^2 x^2\right )+\frac {2}{3} b c^3 d (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-\frac {1}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right ) \] Output:

-1/3*b^2*c^2*d/x+1/3*b^2*c^3*d*arctanh(c*x)-1/3*b*c*d*(a+b*arctanh(c*x))/x 
^2-b*c^2*d*(a+b*arctanh(c*x))/x+5/6*c^3*d*(a+b*arctanh(c*x))^2-1/3*d*(a+b* 
arctanh(c*x))^2/x^3-1/2*c*d*(a+b*arctanh(c*x))^2/x^2+b^2*c^3*d*ln(x)-1/2*b 
^2*c^3*d*ln(-c^2*x^2+1)+2/3*b*c^3*d*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))-1/3 
*b^2*c^3*d*polylog(2,-1+2/(c*x+1))
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.19 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {d \left (2 a^2+3 a^2 c x+2 a b c x+6 a b c^2 x^2+2 b^2 c^2 x^2+b^2 \left (2+3 c x-5 c^3 x^3\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (a (2+3 c x)+b c x \left (1+3 c x-c^2 x^2\right )-2 b c^3 x^3 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-4 a b c^3 x^3 \log (c x)+3 a b c^3 x^3 \log (1-c x)-3 a b c^3 x^3 \log (1+c x)-6 b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+2 a b c^3 x^3 \log \left (1-c^2 x^2\right )+2 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{6 x^3} \] Input:

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

Output:

-1/6*(d*(2*a^2 + 3*a^2*c*x + 2*a*b*c*x + 6*a*b*c^2*x^2 + 2*b^2*c^2*x^2 + b 
^2*(2 + 3*c*x - 5*c^3*x^3)*ArcTanh[c*x]^2 + 2*b*ArcTanh[c*x]*(a*(2 + 3*c*x 
) + b*c*x*(1 + 3*c*x - c^2*x^2) - 2*b*c^3*x^3*Log[1 - E^(-2*ArcTanh[c*x])] 
) - 4*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*Lo 
g[1 + c*x] - 6*b^2*c^3*x^3*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 2*a*b*c^3*x^3*Lo 
g[1 - c^2*x^2] + 2*b^2*c^3*x^3*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^3
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6502, 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.

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

\(\Big \downarrow \) 6502

\(\displaystyle \int \left (\frac {d (a+b \text {arctanh}(c x))^2}{x^4}+\frac {c d (a+b \text {arctanh}(c x))^2}{x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{6} c^3 d (a+b \text {arctanh}(c x))^2+\frac {2}{3} b c^3 d \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {b c^2 d (a+b \text {arctanh}(c x))}{x}-\frac {d (a+b \text {arctanh}(c x))^2}{3 x^3}-\frac {c d (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {b c d (a+b \text {arctanh}(c x))}{3 x^2}+\frac {1}{3} b^2 c^3 d \text {arctanh}(c x)-\frac {1}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+b^2 c^3 d \log (x)-\frac {b^2 c^2 d}{3 x}-\frac {1}{2} b^2 c^3 d \log \left (1-c^2 x^2\right )\)

Input:

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

Output:

-1/3*(b^2*c^2*d)/x + (b^2*c^3*d*ArcTanh[c*x])/3 - (b*c*d*(a + b*ArcTanh[c* 
x]))/(3*x^2) - (b*c^2*d*(a + b*ArcTanh[c*x]))/x + (5*c^3*d*(a + b*ArcTanh[ 
c*x])^2)/6 - (d*(a + b*ArcTanh[c*x])^2)/(3*x^3) - (c*d*(a + b*ArcTanh[c*x] 
)^2)/(2*x^2) + b^2*c^3*d*Log[x] - (b^2*c^3*d*Log[1 - c^2*x^2])/2 + (2*b*c^ 
3*d*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/3 - (b^2*c^3*d*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 6502

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( 
f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] 
 && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
 

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.50

method	result	size
parts	\(d \,a^{2} \left (-\frac {c}{2 x^{2}}-\frac {1}{3 x^{3}}\right )+d \,b^{2} c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}+\frac {2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}-\frac {\ln \left (c x +1\right )^{2}}{24}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {\operatorname {dilog}\left (c x \right )}{3}-\frac {\operatorname {dilog}\left (c x +1\right )}{3}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {5 \ln \left (c x -1\right )^{2}}{24}+\frac {5 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}-\frac {2 \ln \left (c x -1\right )}{3}-\frac {1}{3 c x}+\ln \left (c x \right )-\frac {\ln \left (c x +1\right )}{3}\right )+2 d a b \,c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {5 \ln \left (c x -1\right )}{12}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{2 c x}+\frac {\ln \left (c x \right )}{3}+\frac {\ln \left (c x +1\right )}{12}\right )\)	\(308\)
derivativedivides	\(c^{3} \left (d \,a^{2} \left (-\frac {1}{3 c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}\right )+d \,b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}+\frac {2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}-\frac {\ln \left (c x +1\right )^{2}}{24}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {\operatorname {dilog}\left (c x \right )}{3}-\frac {\operatorname {dilog}\left (c x +1\right )}{3}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {5 \ln \left (c x -1\right )^{2}}{24}+\frac {5 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}-\frac {2 \ln \left (c x -1\right )}{3}-\frac {1}{3 c x}+\ln \left (c x \right )-\frac {\ln \left (c x +1\right )}{3}\right )+2 d a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {5 \ln \left (c x -1\right )}{12}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{2 c x}+\frac {\ln \left (c x \right )}{3}+\frac {\ln \left (c x +1\right )}{12}\right )\right )\)	\(311\)
default	\(c^{3} \left (d \,a^{2} \left (-\frac {1}{3 c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}\right )+d \,b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}+\frac {2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}-\frac {\ln \left (c x +1\right )^{2}}{24}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {\operatorname {dilog}\left (c x \right )}{3}-\frac {\operatorname {dilog}\left (c x +1\right )}{3}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {5 \ln \left (c x -1\right )^{2}}{24}+\frac {5 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}-\frac {2 \ln \left (c x -1\right )}{3}-\frac {1}{3 c x}+\ln \left (c x \right )-\frac {\ln \left (c x +1\right )}{3}\right )+2 d a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {5 \ln \left (c x -1\right )}{12}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{2 c x}+\frac {\ln \left (c x \right )}{3}+\frac {\ln \left (c x +1\right )}{12}\right )\right )\)	\(311\)

Input:

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

Output:

d*a^2*(-1/2*c/x^2-1/3/x^3)+d*b^2*c^3*(-1/3*arctanh(c*x)^2/c^3/x^3-1/2*arct 
anh(c*x)^2/c^2/x^2-5/6*arctanh(c*x)*ln(c*x-1)-1/3*arctanh(c*x)/c^2/x^2-arc 
tanh(c*x)/c/x+2/3*arctanh(c*x)*ln(c*x)+1/6*arctanh(c*x)*ln(c*x+1)-1/24*ln( 
c*x+1)^2+1/12*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+1/3*dilog(1/2*c 
*x+1/2)-1/3*dilog(c*x)-1/3*dilog(c*x+1)-1/3*ln(c*x)*ln(c*x+1)-5/24*ln(c*x- 
1)^2+5/12*ln(c*x-1)*ln(1/2*c*x+1/2)-2/3*ln(c*x-1)-1/3/c/x+ln(c*x)-1/3*ln(c 
*x+1))+2*d*a*b*c^3*(-1/3*arctanh(c*x)/c^3/x^3-1/2*arctanh(c*x)/c^2/x^2-5/1 
2*ln(c*x-1)-1/6/c^2/x^2-1/2/c/x+1/3*ln(c*x)+1/12*ln(c*x+1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

d*(Integral(a**2/x**4, x) + Integral(a**2*c/x**3, x) + Integral(b**2*atanh 
(c*x)**2/x**4, x) + Integral(2*a*b*atanh(c*x)/x**4, x) + Integral(b**2*c*a 
tanh(c*x)**2/x**3, x) + Integral(2*a*b*c*atanh(c*x)/x**3, x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (187) = 374\).

Time = 0.47 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.02 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {1}{3} \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} c^{3} d - \frac {1}{3} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} b^{2} c^{3} d + \frac {1}{3} \, {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} b^{2} c^{3} d - \frac {1}{3} \, b^{2} c^{3} d \log \left (c x + 1\right ) - \frac {2}{3} \, b^{2} c^{3} d \log \left (c x - 1\right ) + b^{2} c^{3} d \log \left (x\right ) + \frac {1}{2} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} a b c d - \frac {1}{3} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} a b d - \frac {a^{2} c d}{2 \, x^{2}} - \frac {a^{2} d}{3 \, x^{3}} - \frac {8 \, b^{2} c^{2} d x^{2} - {\left (b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x - 2 \, b^{2} d\right )} \log \left (c x + 1\right )^{2} - {\left (5 \, b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x - 2 \, b^{2} d\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (3 \, b^{2} c^{2} d x^{2} + b^{2} c d x\right )} \log \left (c x + 1\right ) - 2 \, {\left (6 \, b^{2} c^{2} d x^{2} + 2 \, b^{2} c d x - {\left (b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x - 2 \, b^{2} d\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{24 \, x^{3}} \] Input:

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

Output:

-1/3*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*c^3*d - 
 1/3*(log(c*x)*log(-c*x + 1) + dilog(-c*x + 1))*b^2*c^3*d + 1/3*(log(c*x + 
 1)*log(-c*x) + dilog(c*x + 1))*b^2*c^3*d - 1/3*b^2*c^3*d*log(c*x + 1) - 2 
/3*b^2*c^3*d*log(c*x - 1) + b^2*c^3*d*log(x) + 1/2*((c*log(c*x + 1) - c*lo 
g(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*a*b*c*d - 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 - 1/2*a^2*c*d/x 
^2 - 1/3*a^2*d/x^3 - 1/24*(8*b^2*c^2*d*x^2 - (b^2*c^3*d*x^3 - 3*b^2*c*d*x 
- 2*b^2*d)*log(c*x + 1)^2 - (5*b^2*c^3*d*x^3 - 3*b^2*c*d*x - 2*b^2*d)*log( 
-c*x + 1)^2 + 4*(3*b^2*c^2*d*x^2 + b^2*c*d*x)*log(c*x + 1) - 2*(6*b^2*c^2* 
d*x^2 + 2*b^2*c*d*x - (b^2*c^3*d*x^3 - 3*b^2*c*d*x - 2*b^2*d)*log(c*x + 1) 
)*log(-c*x + 1))/x^3
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(d*(3*atanh(c*x)**2*b**2*c**3*x**3 - 3*atanh(c*x)**2*b**2*c*x - 2*atanh(c* 
x)**2*b**2 + 2*atanh(c*x)*a*b*c**3*x**3 - 6*atanh(c*x)*a*b*c*x - 4*atanh(c 
*x)*a*b - 6*atanh(c*x)*b**2*c**3*x**3 - 6*atanh(c*x)*b**2*c**2*x**2 - 4*in 
t(atanh(c*x)/(c**2*x**5 - x**3),x)*b**2*c*x**3 - 4*log(c**2*x - c)*a*b*c** 
3*x**3 - 6*log(c**2*x - c)*b**2*c**3*x**3 + 4*log(x)*a*b*c**3*x**3 + 6*log 
(x)*b**2*c**3*x**3 - 3*a**2*c*x - 2*a**2 - 6*a*b*c**2*x**2 - 2*a*b*c*x))/( 
6*x**3)