\(\int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)} \, dx\) [49]

Optimal result

Integrand size = 20, antiderivative size = 146 \[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)} \, dx=-\frac {b c}{2 d x}+\frac {b c^2 \text {arctanh}(c x)}{2 d}-\frac {a+b \text {arctanh}(c x)}{2 d x^2}+\frac {c (a+b \text {arctanh}(c x))}{d x}-\frac {b c^2 \log (x)}{d}+\frac {b c^2 \log \left (1-c^2 x^2\right )}{2 d}+\frac {c^2 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {b c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{2 d} \] Output:

-1/2*b*c/d/x+1/2*b*c^2*arctanh(c*x)/d-1/2*(a+b*arctanh(c*x))/d/x^2+c*(a+b* 
arctanh(c*x))/d/x-b*c^2*ln(x)/d+1/2*b*c^2*ln(-c^2*x^2+1)/d+c^2*(a+b*arctan 
h(c*x))*ln(2-2/(c*x+1))/d-1/2*b*c^2*polylog(2,-1+2/(c*x+1))/d
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)} \, dx=-\frac {a-2 a c x+b c x-b \text {arctanh}(c x) \left (-1+2 c x+c^2 x^2+2 c^2 x^2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-2 a c^2 x^2 \log (x)+2 a c^2 x^2 \log (1+c x)+2 b c^2 x^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+b c^2 x^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )}{2 d x^2} \] Input:

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

Output:

-1/2*(a - 2*a*c*x + b*c*x - b*ArcTanh[c*x]*(-1 + 2*c*x + c^2*x^2 + 2*c^2*x 
^2*Log[1 - E^(-2*ArcTanh[c*x])]) - 2*a*c^2*x^2*Log[x] + 2*a*c^2*x^2*Log[1 
+ c*x] + 2*b*c^2*x^2*Log[(c*x)/Sqrt[1 - c^2*x^2]] + b*c^2*x^2*PolyLog[2, E 
^(-2*ArcTanh[c*x])])/(d*x^2)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.86, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {6496, 27, 6452, 264, 219, 6496, 6452, 243, 47, 14, 16, 6494, 2897}

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.

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

\(\Big \downarrow \) 6496

\(\displaystyle \frac {\int \frac {a+b \text {arctanh}(c x)}{x^3}dx}{d}-c \int \frac {a+b \text {arctanh}(c x)}{d x^2 (c x+1)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a+b \text {arctanh}(c x)}{x^3}dx}{d}-\frac {c \int \frac {a+b \text {arctanh}(c x)}{x^2 (c x+1)}dx}{d}\)

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6496

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

\(\displaystyle \frac {\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}}{d}-\frac {c \left (-c \int \frac {a+b \text {arctanh}(c x)}{x (c x+1)}dx+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x^2}dx^2+\log \left (x^2\right )\right )-\frac {a+b \text {arctanh}(c x)}{x}\right )}{d}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}}{d}-\frac {c \left (-c \int \frac {a+b \text {arctanh}(c x)}{x (c x+1)}dx-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )}{d}\)

\(\Big \downarrow \) 6494

\(\displaystyle \frac {\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}}{d}-\frac {c \left (-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-b c \int \frac {\log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )}{d}\)

\(\Big \downarrow \) 2897

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

Input:

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

Output:

(-1/2*(a + b*ArcTanh[c*x])/x^2 + (b*c*(-x^(-1) + c*ArcTanh[c*x]))/2)/d - ( 
c*(-((a + b*ArcTanh[c*x])/x) + (b*c*(Log[x^2] - Log[1 - c^2*x^2]))/2 - c*( 
(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - (b*PolyLog[2, -1 + 2/(1 + c*x) 
])/2)))/d
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

rule 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -1]
 

rule 6494

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x 
_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - 
Simp[b*c*(p/d)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] 
/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c 
^2*d^2 - e^2, 0]
 

rule 6496

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( 
e_.)*(x_)), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], 
 x] - Simp[e/(d*f)   Int[(f*x)^(m + 1)*((a + b*ArcTanh[c*x])^p/(d + e*x)), 
x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 
0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.29

method	result	size
parts	\(\frac {a \left (-c^{2} \ln \left (c x +1\right )-\frac {1}{2 x^{2}}+c^{2} \ln \left (x \right )+\frac {c}{x}\right )}{d}+\frac {b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 c x}-\ln \left (c x \right )+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d}\)	\(188\)
derivativedivides	\(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}+\ln \left (c x \right )+\frac {1}{c x}-\ln \left (c x +1\right )\right )}{d}+\frac {b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 c x}-\ln \left (c x \right )+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d}\right )\)	\(189\)
default	\(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}+\ln \left (c x \right )+\frac {1}{c x}-\ln \left (c x +1\right )\right )}{d}+\frac {b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )+\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {\ln \left (c x +1\right )^{2}}{4}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 c x}-\ln \left (c x \right )+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d}\right )\)	\(189\)
risch	\(-\frac {c^{2} b \ln \left (-c x \right )}{4 d}-\frac {b c}{2 d x}+\frac {c^{2} b \ln \left (-c x +1\right )}{4 d}+\frac {b \ln \left (-c x +1\right )}{4 d \,x^{2}}+\frac {c^{2} b \operatorname {dilog}\left (-c x +1\right )}{2 d}-\frac {c^{2} b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d}+\frac {c^{2} b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 d}-\frac {c^{2} b \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d}-\frac {c b \ln \left (-c x +1\right )}{2 d x}-\frac {a}{2 d \,x^{2}}+\frac {c a}{d x}+\frac {c^{2} a \ln \left (-c x \right )}{d}-\frac {c^{2} a \ln \left (-c x -1\right )}{d}-\frac {c^{2} b \ln \left (c x +1\right )^{2}}{4 d}-\frac {3 c^{2} b \ln \left (c x \right )}{4 d}+\frac {3 c^{2} b \ln \left (c x +1\right )}{4 d}-\frac {b \ln \left (c x +1\right )}{4 d \,x^{2}}-\frac {c^{2} b \operatorname {dilog}\left (c x +1\right )}{2 d}+\frac {c b \ln \left (c x +1\right )}{2 d x}\)	\(291\)

Input:

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

Output:

a/d*(-c^2*ln(c*x+1)-1/2/x^2+c^2*ln(x)+c/x)+b/d*c^2*(-1/2*arctanh(c*x)/c^2/ 
x^2+arctanh(c*x)*ln(c*x)+arctanh(c*x)/c/x-arctanh(c*x)*ln(c*x+1)-1/2*dilog 
(c*x)-1/2*dilog(c*x+1)-1/2*ln(c*x)*ln(c*x+1)+1/4*ln(c*x+1)^2-1/2*(ln(c*x+1 
)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)+1/2*dilog(1/2*c*x+1/2)+1/4*ln(c*x-1)-1 
/2/c/x-ln(c*x)+3/4*ln(c*x+1))
 

Fricas [F]

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

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

Output:

integral((b*arctanh(c*x) + a)/(c*d*x^4 + d*x^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*(2*c^2*log(c*x + 1)/d - 2*c^2*log(x)/d - (2*c*x - 1)/(d*x^2))*a + 1/2 
*b*integrate((log(c*x + 1) - log(-c*x + 1))/(c*d*x^4 + d*x^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+c d x)} \, dx=\frac {2 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c \,x^{4}+x^{3}}d x \right ) b \,x^{2}-2 \,\mathrm {log}\left (c x +1\right ) a \,c^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}+2 a c x -a}{2 d \,x^{2}} \] Input:

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

Output:

(2*int(atanh(c*x)/(c*x**4 + x**3),x)*b*x**2 - 2*log(c*x + 1)*a*c**2*x**2 + 
 2*log(x)*a*c**2*x**2 + 2*a*c*x - a)/(2*d*x**2)