\(\int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx\) [92]

Optimal result

Integrand size = 22, antiderivative size = 271 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=-\frac {b^2 c^2 d^3}{12 x^2}-\frac {b^2 c^3 d^3}{x}+b^2 c^4 d^3 \text {arctanh}(c x)-\frac {b c d^3 (a+b \text {arctanh}(c x))}{6 x^3}-\frac {b c^2 d^3 (a+b \text {arctanh}(c x))}{x^2}-\frac {7 b c^3 d^3 (a+b \text {arctanh}(c x))}{2 x}-\frac {d^3 (1+c x)^4 (a+b \text {arctanh}(c x))^2}{4 x^4}+4 a b c^4 d^3 \log (x)+\frac {11}{3} b^2 c^4 d^3 \log (x)+4 b c^4 d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )-\frac {11}{6} b^2 c^4 d^3 \log \left (1-c^2 x^2\right )-2 b^2 c^4 d^3 \operatorname {PolyLog}(2,-c x)+2 b^2 c^4 d^3 \operatorname {PolyLog}(2,c x)+2 b^2 c^4 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \] Output:

-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))
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.27 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=-\frac {d^3 \left (3 a^2+12 a^2 c x+2 a b c x+18 a^2 c^2 x^2+12 a b c^2 x^2+b^2 c^2 x^2+12 a^2 c^3 x^3+42 a b c^3 x^3+12 b^2 c^3 x^3-b^2 c^4 x^4+3 b^2 \left (1+4 c x+6 c^2 x^2+4 c^3 x^3-15 c^4 x^4\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (b c x \left (1+6 c x+21 c^2 x^2-6 c^3 x^3\right )+3 a \left (1+4 c x+6 c^2 x^2+4 c^3 x^3\right )-24 b c^4 x^4 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-48 a b c^4 x^4 \log (c x)+21 a b c^4 x^4 \log (1-c x)-21 a b c^4 x^4 \log (1+c x)-44 b^2 c^4 x^4 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+24 a b c^4 x^4 \log \left (1-c^2 x^2\right )+24 b^2 c^4 x^4 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{12 x^4} \] Input:

Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^5,x]
 

Output:

-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
 

Rubi [A] (verified)

Time = 0.78 (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

\(\displaystyle -2 b c \left (-2 c^3 d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+\frac {7 c^2 d^3 (a+b \text {arctanh}(c x))}{4 x}+\frac {d^3 (a+b \text {arctanh}(c x))}{12 x^3}+\frac {c d^3 (a+b \text {arctanh}(c x))}{2 x^2}-2 a c^3 d^3 \log (x)-\frac {1}{2} b c^3 d^3 \text {arctanh}(c x)+b c^3 d^3 \operatorname {PolyLog}(2,-c x)-b c^3 d^3 \operatorname {PolyLog}(2,c x)-b c^3 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-\frac {11}{6} b c^3 d^3 \log (x)+\frac {b c^2 d^3}{2 x}+\frac {11}{12} b c^3 d^3 \log \left (1-c^2 x^2\right )+\frac {b c d^3}{24 x^2}\right )-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))^2}{4 x^4}\)

Input:

Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^5,x]
 

Output:

-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)])
 

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 = 1.10 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.52

method	result	size
parts	\(d^{3} a^{2} \left (-\frac {1}{4 x^{4}}-\frac {c^{3}}{x}-\frac {3 c^{2}}{2 x^{2}}-\frac {c}{x^{3}}\right )+d^{3} b^{2} c^{4} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{3} x^{3}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {15 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{4}-\frac {\operatorname {arctanh}\left (c x \right )}{6 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {7 \,\operatorname {arctanh}\left (c x \right )}{2 c x}+4 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{4}-2 \operatorname {dilog}\left (c x \right )-2 \operatorname {dilog}\left (c x +1\right )-2 \ln \left (c x \right ) \ln \left (c x +1\right )-\frac {15 \ln \left (c x -1\right )^{2}}{16}+2 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {15 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (c x +1\right )^{2}}{16}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{8}-\frac {7 \ln \left (c x -1\right )}{3}-\frac {1}{12 c^{2} x^{2}}-\frac {1}{c x}+\frac {11 \ln \left (c x \right )}{3}-\frac {4 \ln \left (c x +1\right )}{3}\right )+2 d^{3} b a \,c^{4} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {15 \ln \left (c x -1\right )}{8}-\frac {1}{12 c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}-\frac {7}{4 c x}+2 \ln \left (c x \right )-\frac {\ln \left (c x +1\right )}{8}\right )\)	\(412\)
derivativedivides	\(c^{4} \left (d^{3} a^{2} \left (-\frac {1}{c^{3} x^{3}}-\frac {3}{2 c^{2} x^{2}}-\frac {1}{c x}-\frac {1}{4 c^{4} x^{4}}\right )+d^{3} b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{3} x^{3}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{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 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{4}-2 \operatorname {dilog}\left (c x \right )-2 \operatorname {dilog}\left (c x +1\right )-2 \ln \left (c x \right ) \ln \left (c x +1\right )-\frac {15 \ln \left (c x -1\right )^{2}}{16}+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 {\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 {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}-\frac {4 \ln \left (c x +1\right )}{3}\right )+2 d^{3} b a \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\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 )-\frac {\ln \left (c x +1\right )}{8}\right )\right )\)	\(415\)
default	\(c^{4} \left (d^{3} a^{2} \left (-\frac {1}{c^{3} x^{3}}-\frac {3}{2 c^{2} x^{2}}-\frac {1}{c x}-\frac {1}{4 c^{4} x^{4}}\right )+d^{3} b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{3} x^{3}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{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 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{4}-2 \operatorname {dilog}\left (c x \right )-2 \operatorname {dilog}\left (c x +1\right )-2 \ln \left (c x \right ) \ln \left (c x +1\right )-\frac {15 \ln \left (c x -1\right )^{2}}{16}+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 {\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 {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}-\frac {4 \ln \left (c x +1\right )}{3}\right )+2 d^{3} b a \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\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 )-\frac {\ln \left (c x +1\right )}{8}\right )\right )\)	\(415\)

Input:

int((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^5,x,method=_RETURNVERBOSE)
 

Output:

d^3*a^2*(-1/4/x^4-c^3/x-3/2*c^2/x^2-c/x^3)+d^3*b^2*c^4*(-arctanh(c*x)^2/c^ 
3/x^3-3/2*arctanh(c*x)^2/c^2/x^2-arctanh(c*x)^2/c/x-1/4*arctanh(c*x)^2/c^4 
/x^4-15/4*arctanh(c*x)*ln(c*x-1)-1/6*arctanh(c*x)/c^3/x^3-arctanh(c*x)/c^2 
/x^2-7/2*arctanh(c*x)/c/x+4*arctanh(c*x)*ln(c*x)-1/4*arctanh(c*x)*ln(c*x+1 
)-2*dilog(c*x)-2*dilog(c*x+1)-2*ln(c*x)*ln(c*x+1)-15/16*ln(c*x-1)^2+2*dilo 
g(1/2*c*x+1/2)+15/8*ln(c*x-1)*ln(1/2*c*x+1/2)+1/16*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)-7/3*ln(c*x-1)-1/12/c^2/x^2-1/c/x+11/ 
3*ln(c*x)-4/3*ln(c*x+1))+2*d^3*b*a*c^4*(-arctanh(c*x)/c^3/x^3-3/2*arctanh( 
c*x)/c^2/x^2-arctanh(c*x)/c/x-1/4*arctanh(c*x)/c^4/x^4-15/8*ln(c*x-1)-1/12 
/c^3/x^3-1/2/c^2/x^2-7/4/c/x+2*ln(c*x)-1/8*ln(c*x+1))
 

Fricas [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \] Input:

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^5,x, algorithm="fricas")
 

Output:

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)
 

Sympy [F]

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

integrate((c*d*x+d)**3*(a+b*atanh(c*x))**2/x**5,x)
 

Output:

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))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (254) = 508\).

Time = 0.48 (sec) , antiderivative size = 813, normalized size of antiderivative = 3.00 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx =\text {Too large to display} \] Input:

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^5,x, algorithm="maxima")
 

Output:

-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
 

Giac [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=\int { \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \] Input:

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^5,x, algorithm="giac")
 

Output:

integrate((c*d*x + d)^3*(b*arctanh(c*x) + a)^2/x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3}{x^5} \,d x \] Input:

int(((a + b*atanh(c*x))^2*(d + c*d*x)^3)/x^5,x)
 

Output:

int(((a + b*atanh(c*x))^2*(d + c*d*x)^3)/x^5, x)
 

Reduce [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^5} \, dx=\frac {d^{3} \left (-48 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{3}-x}d x \right ) b^{2} c^{4} x^{4}-6 \mathit {atanh} \left (c x \right ) a b -b^{2} c^{2} x^{2}-42 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}-2 \mathit {atanh} \left (c x \right ) b^{2} c x -42 a b \,c^{3} x^{3}-2 a b c x -12 b^{2} c^{3} x^{3}-12 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -32 \mathit {atanh} \left (c x \right ) b^{2} c^{4} x^{4}-12 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}-12 a b \,c^{2} x^{2}+21 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{4} x^{4}-3 a^{2}-12 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}-48 \,\mathrm {log}\left (c^{2} x -c \right ) a b \,c^{4} x^{4}+48 \,\mathrm {log}\left (x \right ) a b \,c^{4} x^{4}-44 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} c^{4} x^{4}+44 \,\mathrm {log}\left (x \right ) b^{2} c^{4} x^{4}-12 a^{2} c^{3} x^{3}-24 \mathit {atanh} \left (c x \right ) a b c x -6 \mathit {atanh} \left (c x \right ) a b \,c^{4} x^{4}-18 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-24 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}-12 a^{2} c x -3 \mathit {atanh} \left (c x \right )^{2} b^{2}-36 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}-18 a^{2} c^{2} x^{2}\right )}{12 x^{4}} \] Input:

int((c*d*x+d)^3*(a+b*atanh(c*x))^2/x^5,x)
 

Output:

(d**3*(21*atanh(c*x)**2*b**2*c**4*x**4 - 12*atanh(c*x)**2*b**2*c**3*x**3 - 
 18*atanh(c*x)**2*b**2*c**2*x**2 - 12*atanh(c*x)**2*b**2*c*x - 3*atanh(c*x 
)**2*b**2 - 6*atanh(c*x)*a*b*c**4*x**4 - 24*atanh(c*x)*a*b*c**3*x**3 - 36* 
atanh(c*x)*a*b*c**2*x**2 - 24*atanh(c*x)*a*b*c*x - 6*atanh(c*x)*a*b - 32*a 
tanh(c*x)*b**2*c**4*x**4 - 42*atanh(c*x)*b**2*c**3*x**3 - 12*atanh(c*x)*b* 
*2*c**2*x**2 - 2*atanh(c*x)*b**2*c*x - 48*int(atanh(c*x)/(c**2*x**3 - x),x 
)*b**2*c**4*x**4 - 48*log(c**2*x - c)*a*b*c**4*x**4 - 44*log(c**2*x - c)*b 
**2*c**4*x**4 + 48*log(x)*a*b*c**4*x**4 + 44*log(x)*b**2*c**4*x**4 - 12*a* 
*2*c**3*x**3 - 18*a**2*c**2*x**2 - 12*a**2*c*x - 3*a**2 - 42*a*b*c**3*x**3 
 - 12*a*b*c**2*x**2 - 2*a*b*c*x - 12*b**2*c**3*x**3 - b**2*c**2*x**2))/(12 
*x**4)