\(\int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^6} \, dx\) [93]

Optimal result

Integrand size = 22, antiderivative size = 352 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^6} \, dx=-\frac {b^2 c^2 d^3}{30 x^3}-\frac {b^2 c^3 d^3}{4 x^2}-\frac {13 b^2 c^4 d^3}{10 x}+\frac {13}{10} b^2 c^5 d^3 \text {arctanh}(c x)-\frac {b c d^3 (a+b \text {arctanh}(c x))}{10 x^4}-\frac {b c^2 d^3 (a+b \text {arctanh}(c x))}{2 x^3}-\frac {6 b c^3 d^3 (a+b \text {arctanh}(c x))}{5 x^2}-\frac {5 b c^4 d^3 (a+b \text {arctanh}(c x))}{2 x}-\frac {d^3 (1+c x)^4 (a+b \text {arctanh}(c x))^2}{5 x^5}+\frac {c d^3 (1+c x)^4 (a+b \text {arctanh}(c x))^2}{20 x^4}+\frac {12}{5} a b c^5 d^3 \log (x)+3 b^2 c^5 d^3 \log (x)+\frac {12}{5} b c^5 d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^5 d^3 \log \left (1-c^2 x^2\right )-\frac {6}{5} b^2 c^5 d^3 \operatorname {PolyLog}(2,-c x)+\frac {6}{5} b^2 c^5 d^3 \operatorname {PolyLog}(2,c x)+\frac {6}{5} b^2 c^5 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \] Output:

-1/30*b^2*c^2*d^3/x^3-1/4*b^2*c^3*d^3/x^2-13/10*b^2*c^4*d^3/x+13/10*b^2*c^ 
5*d^3*arctanh(c*x)-1/10*b*c*d^3*(a+b*arctanh(c*x))/x^4-1/2*b*c^2*d^3*(a+b* 
arctanh(c*x))/x^3-6/5*b*c^3*d^3*(a+b*arctanh(c*x))/x^2-5/2*b*c^4*d^3*(a+b* 
arctanh(c*x))/x-1/5*d^3*(c*x+1)^4*(a+b*arctanh(c*x))^2/x^5+1/20*c*d^3*(c*x 
+1)^4*(a+b*arctanh(c*x))^2/x^4+12/5*a*b*c^5*d^3*ln(x)+3*b^2*c^5*d^3*ln(x)+ 
12/5*b*c^5*d^3*(a+b*arctanh(c*x))*ln(2/(-c*x+1))-3/2*b^2*c^5*d^3*ln(-c^2*x 
^2+1)-6/5*b^2*c^5*d^3*polylog(2,-c*x)+6/5*b^2*c^5*d^3*polylog(2,c*x)+6/5*b 
^2*c^5*d^3*polylog(2,1-2/(-c*x+1))
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.06 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^6} \, dx=-\frac {d^3 \left (12 a^2+45 a^2 c x+6 a b c x+60 a^2 c^2 x^2+30 a b c^2 x^2+2 b^2 c^2 x^2+30 a^2 c^3 x^3+72 a b c^3 x^3+15 b^2 c^3 x^3+150 a b c^4 x^4+78 b^2 c^4 x^4-15 b^2 c^5 x^5+3 b^2 \left (4+15 c x+20 c^2 x^2+10 c^3 x^3-49 c^5 x^5\right ) \text {arctanh}(c x)^2+6 b \text {arctanh}(c x) \left (a \left (4+15 c x+20 c^2 x^2+10 c^3 x^3\right )+b c x \left (1+5 c x+12 c^2 x^2+25 c^3 x^3-13 c^4 x^4\right )-24 b c^5 x^5 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-144 a b c^5 x^5 \log (c x)+75 a b c^5 x^5 \log (1-c x)-75 a b c^5 x^5 \log (1+c x)-180 b^2 c^5 x^5 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+72 a b c^5 x^5 \log \left (1-c^2 x^2\right )+72 b^2 c^5 x^5 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{60 x^5} \] Input:

Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^6,x]
 

Output:

-1/60*(d^3*(12*a^2 + 45*a^2*c*x + 6*a*b*c*x + 60*a^2*c^2*x^2 + 30*a*b*c^2* 
x^2 + 2*b^2*c^2*x^2 + 30*a^2*c^3*x^3 + 72*a*b*c^3*x^3 + 15*b^2*c^3*x^3 + 1 
50*a*b*c^4*x^4 + 78*b^2*c^4*x^4 - 15*b^2*c^5*x^5 + 3*b^2*(4 + 15*c*x + 20* 
c^2*x^2 + 10*c^3*x^3 - 49*c^5*x^5)*ArcTanh[c*x]^2 + 6*b*ArcTanh[c*x]*(a*(4 
 + 15*c*x + 20*c^2*x^2 + 10*c^3*x^3) + b*c*x*(1 + 5*c*x + 12*c^2*x^2 + 25* 
c^3*x^3 - 13*c^4*x^4) - 24*b*c^5*x^5*Log[1 - E^(-2*ArcTanh[c*x])]) - 144*a 
*b*c^5*x^5*Log[c*x] + 75*a*b*c^5*x^5*Log[1 - c*x] - 75*a*b*c^5*x^5*Log[1 + 
 c*x] - 180*b^2*c^5*x^5*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 72*a*b*c^5*x^5*Log[ 
1 - c^2*x^2] + 72*b^2*c^5*x^5*PolyLog[2, E^(-2*ArcTanh[c*x])]))/x^5
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 330, 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.

\(\displaystyle \int \frac {(c d x+d)^3 (a+b \text {arctanh}(c x))^2}{x^6} \, dx\)

\(\Big \downarrow \) 6500

\(\displaystyle -2 b c \int \left (-\frac {6 d^3 (a+b \text {arctanh}(c x)) c^5}{5 (1-c x)}-\frac {6 d^3 (a+b \text {arctanh}(c x)) c^4}{5 x}-\frac {5 d^3 (a+b \text {arctanh}(c x)) c^3}{4 x^2}-\frac {6 d^3 (a+b \text {arctanh}(c x)) c^2}{5 x^3}-\frac {3 d^3 (a+b \text {arctanh}(c x)) c}{4 x^4}-\frac {d^3 (a+b \text {arctanh}(c x))}{5 x^5}\right )dx-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))^2}{5 x^5}+\frac {c d^3 (c x+1)^4 (a+b \text {arctanh}(c x))^2}{20 x^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 b c \left (-\frac {6}{5} c^4 d^3 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+\frac {5 c^3 d^3 (a+b \text {arctanh}(c x))}{4 x}+\frac {3 c^2 d^3 (a+b \text {arctanh}(c x))}{5 x^2}+\frac {d^3 (a+b \text {arctanh}(c x))}{20 x^4}+\frac {c d^3 (a+b \text {arctanh}(c x))}{4 x^3}-\frac {6}{5} a c^4 d^3 \log (x)-\frac {13}{20} b c^4 d^3 \text {arctanh}(c x)+\frac {3}{5} b c^4 d^3 \operatorname {PolyLog}(2,-c x)-\frac {3}{5} b c^4 d^3 \operatorname {PolyLog}(2,c x)-\frac {3}{5} b c^4 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-\frac {3}{2} b c^4 d^3 \log (x)+\frac {13 b c^3 d^3}{20 x}+\frac {b c^2 d^3}{8 x^2}+\frac {3}{4} b c^4 d^3 \log \left (1-c^2 x^2\right )+\frac {b c d^3}{60 x^3}\right )-\frac {d^3 (c x+1)^4 (a+b \text {arctanh}(c x))^2}{5 x^5}+\frac {c d^3 (c x+1)^4 (a+b \text {arctanh}(c x))^2}{20 x^4}\)

Input:

Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x])^2)/x^6,x]
 

Output:

-1/5*(d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x])^2)/x^5 + (c*d^3*(1 + c*x)^4*(a 
+ b*ArcTanh[c*x])^2)/(20*x^4) - 2*b*c*((b*c*d^3)/(60*x^3) + (b*c^2*d^3)/(8 
*x^2) + (13*b*c^3*d^3)/(20*x) - (13*b*c^4*d^3*ArcTanh[c*x])/20 + (d^3*(a + 
 b*ArcTanh[c*x]))/(20*x^4) + (c*d^3*(a + b*ArcTanh[c*x]))/(4*x^3) + (3*c^2 
*d^3*(a + b*ArcTanh[c*x]))/(5*x^2) + (5*c^3*d^3*(a + b*ArcTanh[c*x]))/(4*x 
) - (6*a*c^4*d^3*Log[x])/5 - (3*b*c^4*d^3*Log[x])/2 - (6*c^4*d^3*(a + b*Ar 
cTanh[c*x])*Log[2/(1 - c*x)])/5 + (3*b*c^4*d^3*Log[1 - c^2*x^2])/4 + (3*b* 
c^4*d^3*PolyLog[2, -(c*x)])/5 - (3*b*c^4*d^3*PolyLog[2, c*x])/5 - (3*b*c^4 
*d^3*PolyLog[2, 1 - 2/(1 - c*x)])/5)
 

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 = 1.14 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.25

method	result	size
parts	\(d^{3} a^{2} \left (-\frac {3 c}{4 x^{4}}-\frac {1}{5 x^{5}}-\frac {c^{3}}{2 x^{2}}-\frac {c^{2}}{x^{3}}\right )+d^{3} b^{2} c^{5} \left (-\frac {5 \,\operatorname {arctanh}\left (c x \right )}{2 c x}-\frac {6 \,\operatorname {arctanh}\left (c x \right )}{5 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{3} x^{3}}-\frac {43 \ln \left (c x -1\right )}{20}-\frac {17 \ln \left (c x +1\right )}{20}-\frac {13}{10 c x}+\frac {12 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{5}-\frac {6 \ln \left (c x \right ) \ln \left (c x +1\right )}{5}-\frac {1}{30 c^{3} x^{3}}-\frac {1}{4 c^{2} x^{2}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{5 c^{5} x^{5}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{20}-\frac {49 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{10 c^{4} x^{4}}+\frac {6 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {49 \ln \left (c x -1\right )^{2}}{80}-\frac {\ln \left (c x +1\right )^{2}}{80}+3 \ln \left (c x \right )+\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 )}{40}+\frac {49 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}-\frac {6 \operatorname {dilog}\left (c x \right )}{5}-\frac {6 \operatorname {dilog}\left (c x +1\right )}{5}\right )+2 d^{3} b a \,c^{5} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right )}{5 c^{5} x^{5}}-\frac {49 \ln \left (c x -1\right )}{40}-\frac {1}{20 c^{4} x^{4}}-\frac {1}{4 c^{3} x^{3}}-\frac {3}{5 c^{2} x^{2}}-\frac {5}{4 c x}+\frac {6 \ln \left (c x \right )}{5}+\frac {\ln \left (c x +1\right )}{40}\right )\)	\(440\)
derivativedivides	\(c^{5} \left (d^{3} a^{2} \left (-\frac {1}{c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}-\frac {3}{4 c^{4} x^{4}}-\frac {1}{5 c^{5} x^{5}}\right )+d^{3} b^{2} \left (-\frac {5 \,\operatorname {arctanh}\left (c x \right )}{2 c x}-\frac {6 \,\operatorname {arctanh}\left (c x \right )}{5 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{3} x^{3}}-\frac {43 \ln \left (c x -1\right )}{20}-\frac {17 \ln \left (c x +1\right )}{20}-\frac {13}{10 c x}+\frac {12 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{5}-\frac {6 \ln \left (c x \right ) \ln \left (c x +1\right )}{5}-\frac {1}{30 c^{3} x^{3}}-\frac {1}{4 c^{2} x^{2}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{5 c^{5} x^{5}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{20}-\frac {49 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{10 c^{4} x^{4}}+\frac {6 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {49 \ln \left (c x -1\right )^{2}}{80}-\frac {\ln \left (c x +1\right )^{2}}{80}+3 \ln \left (c x \right )+\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 )}{40}+\frac {49 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}-\frac {6 \operatorname {dilog}\left (c x \right )}{5}-\frac {6 \operatorname {dilog}\left (c x +1\right )}{5}\right )+2 d^{3} b a \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right )}{5 c^{5} x^{5}}-\frac {49 \ln \left (c x -1\right )}{40}-\frac {1}{20 c^{4} x^{4}}-\frac {1}{4 c^{3} x^{3}}-\frac {3}{5 c^{2} x^{2}}-\frac {5}{4 c x}+\frac {6 \ln \left (c x \right )}{5}+\frac {\ln \left (c x +1\right )}{40}\right )\right )\)	\(443\)
default	\(c^{5} \left (d^{3} a^{2} \left (-\frac {1}{c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}-\frac {3}{4 c^{4} x^{4}}-\frac {1}{5 c^{5} x^{5}}\right )+d^{3} b^{2} \left (-\frac {5 \,\operatorname {arctanh}\left (c x \right )}{2 c x}-\frac {6 \,\operatorname {arctanh}\left (c x \right )}{5 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{3} x^{3}}-\frac {43 \ln \left (c x -1\right )}{20}-\frac {17 \ln \left (c x +1\right )}{20}-\frac {13}{10 c x}+\frac {12 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{5}-\frac {6 \ln \left (c x \right ) \ln \left (c x +1\right )}{5}-\frac {1}{30 c^{3} x^{3}}-\frac {1}{4 c^{2} x^{2}}-\frac {3 \operatorname {arctanh}\left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{5 c^{5} x^{5}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{20}-\frac {49 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{20}-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{10 c^{4} x^{4}}+\frac {6 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {49 \ln \left (c x -1\right )^{2}}{80}-\frac {\ln \left (c x +1\right )^{2}}{80}+3 \ln \left (c x \right )+\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 )}{40}+\frac {49 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{40}-\frac {6 \operatorname {dilog}\left (c x \right )}{5}-\frac {6 \operatorname {dilog}\left (c x +1\right )}{5}\right )+2 d^{3} b a \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}-\frac {3 \,\operatorname {arctanh}\left (c x \right )}{4 c^{4} x^{4}}-\frac {\operatorname {arctanh}\left (c x \right )}{5 c^{5} x^{5}}-\frac {49 \ln \left (c x -1\right )}{40}-\frac {1}{20 c^{4} x^{4}}-\frac {1}{4 c^{3} x^{3}}-\frac {3}{5 c^{2} x^{2}}-\frac {5}{4 c x}+\frac {6 \ln \left (c x \right )}{5}+\frac {\ln \left (c x +1\right )}{40}\right )\right )\)	\(443\)

Input:

int((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^6,x,method=_RETURNVERBOSE)
 

Output:

d^3*a^2*(-3/4*c/x^4-1/5/x^5-1/2*c^3/x^2-c^2/x^3)+d^3*b^2*c^5*(-5/2*arctanh 
(c*x)/c/x-6/5*arctanh(c*x)/c^2/x^2-1/2*arctanh(c*x)/c^3/x^3-43/20*ln(c*x-1 
)-17/20*ln(c*x+1)-13/10/c/x+12/5*arctanh(c*x)*ln(c*x)-6/5*ln(c*x)*ln(c*x+1 
)-1/30/c^3/x^3-1/4/c^2/x^2-3/4*arctanh(c*x)^2/c^4/x^4-arctanh(c*x)^2/c^3/x 
^3-1/5*arctanh(c*x)^2/c^5/x^5+1/20*arctanh(c*x)*ln(c*x+1)-49/20*arctanh(c* 
x)*ln(c*x-1)-1/2*arctanh(c*x)^2/c^2/x^2-1/10*arctanh(c*x)/c^4/x^4+6/5*dilo 
g(1/2*c*x+1/2)-49/80*ln(c*x-1)^2-1/80*ln(c*x+1)^2+3*ln(c*x)+1/40*(ln(c*x+1 
)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+49/40*ln(c*x-1)*ln(1/2*c*x+1/2)-6/5*di 
log(c*x)-6/5*dilog(c*x+1))+2*d^3*b*a*c^5*(-arctanh(c*x)/c^3/x^3-1/2*arctan 
h(c*x)/c^2/x^2-3/4*arctanh(c*x)/c^4/x^4-1/5*arctanh(c*x)/c^5/x^5-49/40*ln( 
c*x-1)-1/20/c^4/x^4-1/4/c^3/x^3-3/5/c^2/x^2-5/4/c/x+6/5*ln(c*x)+1/40*ln(c* 
x+1))
 

Fricas [F]

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

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^6,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^6, x)
 

Sympy [F]

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

integrate((c*d*x+d)**3*(a+b*atanh(c*x))**2/x**6,x)
 

Output:

d**3*(Integral(a**2/x**6, x) + Integral(3*a**2*c/x**5, x) + Integral(3*a** 
2*c**2/x**4, x) + Integral(a**2*c**3/x**3, x) + Integral(b**2*atanh(c*x)** 
2/x**6, x) + Integral(2*a*b*atanh(c*x)/x**6, x) + Integral(3*b**2*c*atanh( 
c*x)**2/x**5, x) + Integral(3*b**2*c**2*atanh(c*x)**2/x**4, x) + Integral( 
b**2*c**3*atanh(c*x)**2/x**3, x) + Integral(6*a*b*c*atanh(c*x)/x**5, x) + 
Integral(6*a*b*c**2*atanh(c*x)/x**4, x) + Integral(2*a*b*c**3*atanh(c*x)/x 
**3, x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 783 vs. \(2 (315) = 630\).

Time = 0.50 (sec) , antiderivative size = 783, normalized size of antiderivative = 2.22 \[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^6} \, dx =\text {Too large to display} \] Input:

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^6,x, algorithm="maxima")
 

Output:

-6/5*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^2*c^5*d^3 
 - 6/5*(log(c*x)*log(-c*x + 1) + dilog(-c*x + 1))*b^2*c^5*d^3 + 6/5*(log(c 
*x + 1)*log(-c*x) + dilog(c*x + 1))*b^2*c^5*d^3 - 17/20*b^2*c^5*d^3*log(c* 
x + 1) - 43/20*b^2*c^5*d^3*log(c*x - 1) + 3*b^2*c^5*d^3*log(x) + 1/2*((c*l 
og(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*a*b*c^3*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^2*d^3 + 1/4*((3*c^3*log(c*x + 1) - 3*c^3*log(c*x - 1) - 2*(3*c^2*x^2 + 
1)/x^3)*c - 6*arctanh(c*x)/x^4)*a*b*c*d^3 - 1/10*((2*c^4*log(c^2*x^2 - 1) 
- 2*c^4*log(x^2) + (2*c^2*x^2 + 1)/x^4)*c + 4*arctanh(c*x)/x^5)*a*b*d^3 - 
1/2*a^2*c^3*d^3/x^2 - a^2*c^2*d^3/x^3 - 3/4*a^2*c*d^3/x^4 - 1/5*a^2*d^3/x^ 
5 - 1/240*(312*b^2*c^4*d^3*x^4 + 60*b^2*c^3*d^3*x^3 + 8*b^2*c^2*d^3*x^2 - 
3*(b^2*c^5*d^3*x^5 - 10*b^2*c^3*d^3*x^3 - 20*b^2*c^2*d^3*x^2 - 15*b^2*c*d^ 
3*x - 4*b^2*d^3)*log(c*x + 1)^2 - 3*(49*b^2*c^5*d^3*x^5 - 10*b^2*c^3*d^3*x 
^3 - 20*b^2*c^2*d^3*x^2 - 15*b^2*c*d^3*x - 4*b^2*d^3)*log(-c*x + 1)^2 + 12 
*(25*b^2*c^4*d^3*x^4 + 12*b^2*c^3*d^3*x^3 + 5*b^2*c^2*d^3*x^2 + b^2*c*d^3* 
x)*log(c*x + 1) - 6*(50*b^2*c^4*d^3*x^4 + 24*b^2*c^3*d^3*x^3 + 10*b^2*c^2* 
d^3*x^2 + 2*b^2*c*d^3*x - (b^2*c^5*d^3*x^5 - 10*b^2*c^3*d^3*x^3 - 20*b^2*c 
^2*d^3*x^2 - 15*b^2*c*d^3*x - 4*b^2*d^3)*log(c*x + 1))*log(-c*x + 1))/x^5
 

Giac [F]

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

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))^2/x^6,x, algorithm="giac")
 

Output:

integrate((c*d*x + d)^3*(b*arctanh(c*x) + a)^2/x^6, x)
 

Mupad [F(-1)]

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

int(((a + b*atanh(c*x))^2*(d + c*d*x)^3)/x^6,x)
 

Output:

int(((a + b*atanh(c*x))^2*(d + c*d*x)^3)/x^6, x)
 

Reduce [F]

\[ \int \frac {(d+c d x)^3 (a+b \text {arctanh}(c x))^2}{x^6} \, dx=\frac {d^{3} \left (-24 \mathit {atanh} \left (c x \right ) a b -2 b^{2} c^{2} x^{2}-102 \mathit {atanh} \left (c x \right ) b^{2} c^{5} x^{5}-72 \mathit {atanh} \left (c x \right ) b^{2} c^{3} x^{3}-6 \mathit {atanh} \left (c x \right ) b^{2} c x -72 a b \,c^{3} x^{3}-6 a b c x -15 b^{2} c^{3} x^{3}+75 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{5} x^{5}-45 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -150 \mathit {atanh} \left (c x \right ) b^{2} c^{4} x^{4}-30 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}-150 a b \,c^{4} x^{4}-30 a b \,c^{2} x^{2}-12 a^{2}-30 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}-144 \,\mathrm {log}\left (c^{2} x -c \right ) a b \,c^{5} x^{5}+144 \,\mathrm {log}\left (x \right ) a b \,c^{5} x^{5}-78 b^{2} c^{4} x^{4}-30 a^{2} c^{3} x^{3}-144 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{3}-x}d x \right ) b^{2} c^{5} x^{5}-90 \mathit {atanh} \left (c x \right ) a b c x -180 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} c^{5} x^{5}+180 \,\mathrm {log}\left (x \right ) b^{2} c^{5} x^{5}-60 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-60 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}-45 a^{2} c x -12 \mathit {atanh} \left (c x \right )^{2} b^{2}-120 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}-60 a^{2} c^{2} x^{2}+6 \mathit {atanh} \left (c x \right ) a b \,c^{5} x^{5}\right )}{60 x^{5}} \] Input:

int((c*d*x+d)^3*(a+b*atanh(c*x))^2/x^6,x)
 

Output:

(d**3*(75*atanh(c*x)**2*b**2*c**5*x**5 - 30*atanh(c*x)**2*b**2*c**3*x**3 - 
 60*atanh(c*x)**2*b**2*c**2*x**2 - 45*atanh(c*x)**2*b**2*c*x - 12*atanh(c* 
x)**2*b**2 + 6*atanh(c*x)*a*b*c**5*x**5 - 60*atanh(c*x)*a*b*c**3*x**3 - 12 
0*atanh(c*x)*a*b*c**2*x**2 - 90*atanh(c*x)*a*b*c*x - 24*atanh(c*x)*a*b - 1 
02*atanh(c*x)*b**2*c**5*x**5 - 150*atanh(c*x)*b**2*c**4*x**4 - 72*atanh(c* 
x)*b**2*c**3*x**3 - 30*atanh(c*x)*b**2*c**2*x**2 - 6*atanh(c*x)*b**2*c*x - 
 144*int(atanh(c*x)/(c**2*x**3 - x),x)*b**2*c**5*x**5 - 144*log(c**2*x - c 
)*a*b*c**5*x**5 - 180*log(c**2*x - c)*b**2*c**5*x**5 + 144*log(x)*a*b*c**5 
*x**5 + 180*log(x)*b**2*c**5*x**5 - 30*a**2*c**3*x**3 - 60*a**2*c**2*x**2 
- 45*a**2*c*x - 12*a**2 - 150*a*b*c**4*x**4 - 72*a*b*c**3*x**3 - 30*a*b*c* 
*2*x**2 - 6*a*b*c*x - 78*b**2*c**4*x**4 - 15*b**2*c**3*x**3 - 2*b**2*c**2* 
x**2))/(60*x**5)