\(\int \frac {(a+b \text {arctanh}(c x)) (d+e \log (f+g x^2))}{x^2} \, dx\) [535]

Optimal result

Integrand size = 24, antiderivative size = 613 \[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {b e \sqrt {g} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {b e \sqrt {g} \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {b e \sqrt {g} \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (1+c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {b e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1+c x)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )+\frac {1}{2} b c e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right ) \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.42 (sec) , antiderivative size = 1226, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-((a*d)/x) - (b*d*ArcTanh[c*x])/x + b*c*d*Log[x] - (b*c*d*Log[1 - c^2*x^2] 
)/2 + a*e*((2*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] - Log[f + g*x^2 
]/x) + (b*e*(-(((2*ArcTanh[c*x] + c*x*(-2*Log[x] + Log[1 - c^2*x^2]))*Log[ 
f + g*x^2])/x) - 2*c*(Log[x]*(Log[1 - (I*Sqrt[g]*x)/Sqrt[f]] + Log[1 + (I* 
Sqrt[g]*x)/Sqrt[f]]) + PolyLog[2, ((-I)*Sqrt[g]*x)/Sqrt[f]] + PolyLog[2, ( 
I*Sqrt[g]*x)/Sqrt[f]]) + c*(Log[-c^(-1) + x]*Log[(c*(Sqrt[f] - I*Sqrt[g]*x 
))/(c*Sqrt[f] - I*Sqrt[g])] + Log[c^(-1) + x]*Log[(c*(Sqrt[f] - I*Sqrt[g]* 
x))/(c*Sqrt[f] + I*Sqrt[g])] + Log[-c^(-1) + x]*Log[(c*(Sqrt[f] + I*Sqrt[g 
]*x))/(c*Sqrt[f] + I*Sqrt[g])] - (Log[-c^(-1) + x] + Log[c^(-1) + x] - Log 
[1 - c^2*x^2])*Log[f + g*x^2] + Log[c^(-1) + x]*Log[1 - (Sqrt[g]*(1 + c*x) 
)/(I*c*Sqrt[f] + Sqrt[g])] + PolyLog[2, (c*Sqrt[g]*(c^(-1) + x))/(I*c*Sqrt 
[f] + Sqrt[g])] + PolyLog[2, (I*Sqrt[g]*(-1 + c*x))/(c*Sqrt[f] - I*Sqrt[g] 
)] + PolyLog[2, ((-I)*Sqrt[g]*(-1 + c*x))/(c*Sqrt[f] + I*Sqrt[g])] + PolyL 
og[2, (I*Sqrt[g]*(1 + c*x))/(c*Sqrt[f] + I*Sqrt[g])]) + (c*g*((2*I)*ArcCos 
[(-(c^2*f) + g)/(c^2*f + g)]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - 4*ArcTan[(c*f 
)/(Sqrt[c^2*f*g]*x)]*ArcTanh[c*x] + (ArcCos[(-(c^2*f) + g)/(c^2*f + g)] + 
2*ArcTan[(c*g*x)/Sqrt[c^2*f*g]])*Log[((2*I)*c*f*(I*g + Sqrt[c^2*f*g])*(-1 
+ c*x))/((c^2*f + g)*(c*f + I*Sqrt[c^2*f*g]*x))] + (ArcCos[(-(c^2*f) + g)/ 
(c^2*f + g)] - 2*ArcTan[(c*g*x)/Sqrt[c^2*f*g]])*Log[(2*c*f*(g + I*Sqrt[c^2 
*f*g])*(1 + c*x))/((c^2*f + g)*(c*f + I*Sqrt[c^2*f*g]*x))] - (ArcCos[(-...
 

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 602, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6643, 2925, 2863, 2009, 6536, 218, 6534, 2856, 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.

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

\(\Big \downarrow \) 6643

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

\(\Big \downarrow \) 2925

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6536

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 6534

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

\(\Big \downarrow \) 2856

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-(((a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]))/x) + 2*e*g*((a*ArcTan[(Sqr 
t[g]*x)/Sqrt[f]])/(Sqrt[f]*Sqrt[g]) + b*((-1/2*(Log[1 - c*x]*Log[(c*(Sqrt[ 
-f] - Sqrt[g]*x))/(c*Sqrt[-f] - Sqrt[g])])/(Sqrt[-f]*Sqrt[g]) + (Log[1 - c 
*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[-f]*Sq 
rt[g]) - PolyLog[2, -((Sqrt[g]*(1 - c*x))/(c*Sqrt[-f] - Sqrt[g]))]/(2*Sqrt 
[-f]*Sqrt[g]) + PolyLog[2, (Sqrt[g]*(1 - c*x))/(c*Sqrt[-f] + Sqrt[g])]/(2* 
Sqrt[-f]*Sqrt[g]))/2 + ((Log[1 + c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sq 
rt[-f] + Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - (Log[1 + c*x]*Log[(c*(Sqrt[-f] 
+ Sqrt[g]*x))/(c*Sqrt[-f] - Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - PolyLog[2, - 
((Sqrt[g]*(1 + c*x))/(c*Sqrt[-f] - Sqrt[g]))]/(2*Sqrt[-f]*Sqrt[g]) + PolyL 
og[2, (Sqrt[g]*(1 + c*x))/(c*Sqrt[-f] + Sqrt[g])]/(2*Sqrt[-f]*Sqrt[g]))/2) 
) + (b*c*(Log[-((g*x^2)/f)]*(d + e*Log[f + g*x^2]) - Log[(g*(1 - c^2*x^2)) 
/(c^2*f + g)]*(d + e*Log[f + g*x^2]) - e*PolyLog[2, (c^2*(f + g*x^2))/(c^2 
*f + g)] + e*PolyLog[2, 1 + (g*x^2)/f]))/2
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2856

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. 
)*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) 
^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I 
GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
 

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

rule 2925

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Si 
mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], 
x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer 
Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 
] || IGtQ[q, 0])
 

rule 6534

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

rule 6536

Int[(ArcTanh[(c_.)*(x_)]*(b_.) + (a_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> 
 Simp[a   Int[1/(d + e*x^2), x], x] + Simp[b   Int[ArcTanh[c*x]/(d + e*x^2) 
, x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 6643

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

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) \left (d +e \ln \left (g \,x^{2}+f \right )\right )}{x^{2}}d x\]

Input:

int((a+b*arctanh(c*x))*(d+e*ln(g*x^2+f))/x^2,x)
 

Output:

int((a+b*arctanh(c*x))*(d+e*ln(g*x^2+f))/x^2,x)
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/2*(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*b*d + (2*g*arcta 
n(g*x/sqrt(f*g))/sqrt(f*g) - log(g*x^2 + f)/x)*a*e + 1/2*b*e*integrate((lo 
g(c*x + 1) - log(-c*x + 1))*log(g*x^2 + f)/x^2, x) - a*d/x
                                                                                    
                                                                                    
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*e*x - atanh(c*x)*b*c*d* 
f*x - atanh(c*x)*b*d*f + int((atanh(c*x)*log(f + g*x**2))/x**2,x)*b*e*f*x 
- log(c**2*x - c)*b*c*d*f*x - log(f + g*x**2)*a*e*f + log(x)*b*c*d*f*x - a 
*d*f)/(f*x)