\(\int \frac {(a+b \text {arctanh}(c x)) (d+e \log (1-c^2 x^2))}{x} \, dx\) [526]

Optimal result

Integrand size = 27, antiderivative size = 216 \[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=a d \log (x)-\frac {1}{2} b e \log (c x) \log ^2(1-c x)+\frac {1}{2} b e \log (-c x) \log ^2(1+c x)-\frac {1}{2} b d \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b d \operatorname {PolyLog}(2,c x)-\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}(2,c x)-\frac {1}{2} a e \operatorname {PolyLog}\left (2,c^2 x^2\right )-b e \log (1-c x) \operatorname {PolyLog}(2,1-c x)+b e \log (1+c x) \operatorname {PolyLog}(2,1+c x)+b e \operatorname {PolyLog}(3,1-c x)-b e \operatorname {PolyLog}(3,1+c x) \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6641, 6446, 6639, 2838, 6637, 2843, 2881, 2821, 6446, 7143}

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.

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

\(\Big \downarrow \) 6641

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

\(\Big \downarrow \) 6446

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

\(\Big \downarrow \) 6639

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6637

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

\(\Big \downarrow \) 2843

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

\(\Big \downarrow \) 2881

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 6446

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

d*(a*Log[x] - (b*PolyLog[2, -(c*x)])/2 + (b*PolyLog[2, c*x])/2) + e*(-1/2* 
(a*PolyLog[2, c^2*x^2]) + b*(-((Log[1 - c*x] + Log[1 + c*x] - Log[1 - c^2* 
x^2])*(-1/2*PolyLog[2, -(c*x)] + PolyLog[2, c*x]/2)) + (-(Log[c*x]*Log[1 - 
 c*x]^2) + 2*(-(Log[1 - c*x]*PolyLog[2, 1 - c*x]) + PolyLog[3, 1 - c*x]))/ 
2 + (Log[-(c*x)]*Log[1 + c*x]^2 - 2*(-(Log[1 + c*x]*PolyLog[2, 1 + c*x]) + 
 PolyLog[3, 1 + c*x]))/2))
 

Defintions of rubi rules used

rule 2821

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

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2843

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_. 
)*(x_)), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d 
+ e*x)^n])^p/g), x] - Simp[b*e*n*(p/g)   Int[Log[(e*(f + g*x))/(e*f - d*g)] 
*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, 
d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[p, 1]
 

rule 2881

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym 
bol] :> Simp[1/e   Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* 
((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, 
 f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
 

rule 6446

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x 
] + (-Simp[(b/2)*PolyLog[2, (-c)*x], x] + Simp[(b/2)*PolyLog[2, c*x], x]) / 
; FreeQ[{a, b, c}, x]
 

rule 6637

Int[(ArcTanh[(c_.)*(x_)]*Log[(f_.) + (g_.)*(x_)^2])/(x_), x_Symbol] :> Simp 
[(Log[f + g*x^2] - Log[1 - c*x] - Log[1 + c*x])   Int[ArcTanh[c*x]/x, x], x 
] + (-Simp[1/2   Int[Log[1 - c*x]^2/x, x], x] + Simp[1/2   Int[Log[1 + c*x] 
^2/x, x], x]) /; FreeQ[{c, f, g}, x] && EqQ[c^2*f + g, 0]
 

rule 6639

Int[(Log[(f_.) + (g_.)*(x_)^2]*(ArcTanh[(c_.)*(x_)]*(b_.) + (a_)))/(x_), x_ 
Symbol] :> Simp[a   Int[Log[f + g*x^2]/x, x], x] + Simp[b   Int[Log[f + g*x 
^2]*(ArcTanh[c*x]/x), x], x] /; FreeQ[{a, b, c, f, g}, x]
 

rule 6641

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(Log[(f_.) + (g_.)*(x_)^2]*(e_.) + 
 (d_)))/(x_), x_Symbol] :> Simp[d   Int[(a + b*ArcTanh[c*x])/x, x], x] + Si 
mp[e   Int[Log[f + g*x^2]*((a + b*ArcTanh[c*x])/x), x], x] /; FreeQ[{a, b, 
c, d, e, f, g}, x]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 11.18 (sec) , antiderivative size = 1227, normalized size of antiderivative = 5.68

method	result	size
risch	\(\text {Expression too large to display}\)	\(1227\)

Input:

int((a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1))/x,x,method=_RETURNVERBOSE)
 

Output:

-1/8*(-8*a*d+4*I*Pi*a*e*csgn(I*(c*x-1))*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c* 
x+1))-4*I*Pi*a*e*csgn(I*(c*x-1)*(c*x+1))^3-4*Pi^2*b*e*csgn(I*(c*x-1))^2*cs 
gn(I*(c*x-1)*(c*x+1))^2-2*Pi^2*b*e*csgn(I*(c*x-1))^4*csgn(I*(c*x-1)*(c*x+1 
))^2+4*Pi^2*b*e*csgn(I*(c*x-1))^2+4*Pi^2*b*e*csgn(I*(c*x-1)*(c*x+1))^2+2*P 
i^2*b*e*csgn(I*(c*x-1))^4*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))+2*Pi^2*b 
*e*csgn(I*(c*x-1))*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))-4*Pi^2*b*e*csgn 
(I*(c*x-1))^3-2*Pi^2*b*e*csgn(I*(c*x-1)*(c*x+1))^3-2*Pi^2*b*e*csgn(I*(c*x- 
1))^3*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))-4*I*Pi*a*e*csgn(I*(c*x-1))*c 
sgn(I*(c*x-1)*(c*x+1))^2-4*I*Pi*a*e*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1) 
)^2-2*Pi^2*b*e*csgn(I*(c*x-1))^3*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))^2 
-2*Pi^2*b*e*csgn(I*(c*x-1))*csgn(I*(c*x-1)*(c*x+1))^2-2*Pi^2*b*e*csgn(I*(c 
*x+1))*csgn(I*(c*x-1)*(c*x+1))^2+4*I*Pi*b*d*csgn(I*(c*x-1))^3+8*I*Pi*a*e*c 
sgn(I*(c*x-1)*(c*x+1))^2-4*I*Pi*b*d*csgn(I*(c*x-1))^2-8*I*Pi*a*e+4*I*Pi*b* 
d-4*Pi^2*b*e+2*Pi^2*b*e*csgn(I*(c*x-1))^2*csgn(I*(c*x+1))*csgn(I*(c*x-1)*( 
c*x+1))^2+2*Pi^2*b*e*csgn(I*(c*x-1))^2*csgn(I*(c*x-1)*(c*x+1))^3+6*Pi^2*b* 
e*csgn(I*(c*x-1))^3*csgn(I*(c*x-1)*(c*x+1))^2-2*Pi^2*b*e*csgn(I*(c*x-1))^3 
*csgn(I*(c*x-1)*(c*x+1))^3)*ln(c*x)-1/8*(-4*I*Pi*b*e*csgn(I*(c*x-1))^2-4*I 
*Pi*b*e*csgn(I*(c*x-1)*(c*x+1))^2+4*I*Pi*b*e*csgn(I*(c*x-1))^3-2*I*Pi*b*e* 
csgn(I*(c*x-1))*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))+2*I*Pi*b*e*csgn(I* 
(c*x-1))*csgn(I*(c*x-1)*(c*x+1))^2+2*I*Pi*b*e*csgn(I*(c*x+1))*csgn(I*(c...
 

Fricas [F]

\[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x} \,d x } \] Input:

integrate((a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1))/x,x, algorithm="fricas" 
)
 

Output:

integral((b*d*arctanh(c*x) + a*d + (b*e*arctanh(c*x) + a*e)*log(-c^2*x^2 + 
 1))/x, x)
 

Sympy [F]

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

integrate((a+b*atanh(c*x))*(d+e*ln(-c**2*x**2+1))/x,x)
 

Output:

Integral((a + b*atanh(c*x))*(d + e*log(-c**2*x**2 + 1))/x, x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.70 \[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=-\frac {1}{2} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right )^{2} + 2 \, {\rm Li}_2\left (-c x + 1\right ) \log \left (-c x + 1\right ) - 2 \, {\rm Li}_{3}(-c x + 1)\right )} b e + \frac {1}{2} \, {\left (\log \left (c x + 1\right )^{2} \log \left (-c x\right ) + 2 \, {\rm Li}_2\left (c x + 1\right ) \log \left (c x + 1\right ) - 2 \, {\rm Li}_{3}(c x + 1)\right )} b e + a d \log \left (x\right ) - \frac {1}{2} \, {\left (b d - 2 \, a e\right )} {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} + \frac {1}{2} \, {\left (b d + 2 \, a e\right )} {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} \] Input:

integrate((a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1))/x,x, algorithm="maxima" 
)
 

Output:

-1/2*(log(c*x)*log(-c*x + 1)^2 + 2*dilog(-c*x + 1)*log(-c*x + 1) - 2*polyl 
og(3, -c*x + 1))*b*e + 1/2*(log(c*x + 1)^2*log(-c*x) + 2*dilog(c*x + 1)*lo 
g(c*x + 1) - 2*polylog(3, c*x + 1))*b*e + a*d*log(x) - 1/2*(b*d - 2*a*e)*( 
log(c*x)*log(-c*x + 1) + dilog(-c*x + 1)) + 1/2*(b*d + 2*a*e)*(log(c*x + 1 
)*log(-c*x) + dilog(c*x + 1))
 

Giac [F]

\[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x} \,d x } \] Input:

integrate((a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1))/x,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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