\(\int x (a+b \text {arctanh}(c x)) (d+e \log (f+g x^2)) \, dx\) [532]

Optimal result

Integrand size = 22, antiderivative size = 512 \[ \int x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\frac {b (d-e) x}{2 c}-\frac {b e x}{c}+\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}+\frac {1}{2} d x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} e x^2 (a+b \text {arctanh}(c x))-\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )}{c^2 g}+\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 c^2 g}+\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 c^2 g}+\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac {e \left (f+g x^2\right ) (a+b \text {arctanh}(c x)) \log \left (f+g x^2\right )}{2 g}+\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 c^2 g}-\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{4 c^2 g}-\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{4 c^2 g} \] Output:

1/2*b*(d-e)*x/c-b*e*x/c+b*e*f^(1/2)*arctan(g^(1/2)*x/f^(1/2))/c/g^(1/2)-1/ 
2*b*(d-e)*arctanh(c*x)/c^2+1/2*d*x^2*(a+b*arctanh(c*x))-1/2*e*x^2*(a+b*arc 
tanh(c*x))-b*e*(c^2*f+g)*arctanh(c*x)*ln(2/(c*x+1))/c^2/g+1/2*b*e*(c^2*f+g 
)*arctanh(c*x)*ln(2*c*((-f)^(1/2)-g^(1/2)*x)/(c*(-f)^(1/2)-g^(1/2))/(c*x+1 
))/c^2/g+1/2*b*e*(c^2*f+g)*arctanh(c*x)*ln(2*c*((-f)^(1/2)+g^(1/2)*x)/(c*( 
-f)^(1/2)+g^(1/2))/(c*x+1))/c^2/g+1/2*b*e*x*ln(g*x^2+f)/c-1/2*b*e*(c^2*f+g 
)*arctanh(c*x)*ln(g*x^2+f)/c^2/g+1/2*e*(g*x^2+f)*(a+b*arctanh(c*x))*ln(g*x 
^2+f)/g+1/2*b*e*(c^2*f+g)*polylog(2,1-2/(c*x+1))/c^2/g-1/4*b*e*(c^2*f+g)*p 
olylog(2,1-2*c*((-f)^(1/2)-g^(1/2)*x)/(c*(-f)^(1/2)-g^(1/2))/(c*x+1))/c^2/ 
g-1/4*b*e*(c^2*f+g)*polylog(2,1-2*c*((-f)^(1/2)+g^(1/2)*x)/(c*(-f)^(1/2)+g 
^(1/2))/(c*x+1))/c^2/g
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 4.76 (sec) , antiderivative size = 1145, normalized size of antiderivative = 2.24 \[ \int x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(2*b*c*d*g*x - 6*b*c*e*g*x + 2*a*c^2*d*g*x^2 - 2*a*c^2*e*g*x^2 + 4*b*c*e*S 
qrt[f]*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]] - 2*b*d*g*ArcTanh[c*x] + 2*b*e* 
g*ArcTanh[c*x] + 2*b*c^2*d*g*x^2*ArcTanh[c*x] - 2*b*c^2*e*g*x^2*ArcTanh[c* 
x] - (4*I)*b*c^2*e*f*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]*ArcTanh[(c*g*x)/Sqr 
t[-(c^2*f*g)]] - (4*I)*b*e*g*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]*ArcTanh[(c* 
g*x)/Sqrt[-(c^2*f*g)]] - 4*b*c^2*e*f*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c* 
x])] - 4*b*e*g*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - (2*I)*b*c^2*e*f 
*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + ( 
-1 + E^(2*ArcTanh[c*x]))*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2* 
f + g))] - (2*I)*b*e*g*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]*Log[(c^2*(1 + E^( 
2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g - 2*Sqrt[-(c^2*f*g)])/(E^ 
(2*ArcTanh[c*x])*(c^2*f + g))] + 2*b*c^2*e*f*ArcTanh[c*x]*Log[(c^2*(1 + E^ 
(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g - 2*Sqrt[-(c^2*f*g)])/(E 
^(2*ArcTanh[c*x])*(c^2*f + g))] + 2*b*e*g*ArcTanh[c*x]*Log[(c^2*(1 + E^(2* 
ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g - 2*Sqrt[-(c^2*f*g)])/(E^(2 
*ArcTanh[c*x])*(c^2*f + g))] + (2*I)*b*c^2*e*f*ArcSin[Sqrt[(c^2*f)/(c^2*f 
+ g)]]*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g + 
 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] + (2*I)*b*e*g*ArcSi 
n[Sqrt[(c^2*f)/(c^2*f + g)]]*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E 
^(2*ArcTanh[c*x]))*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f +...
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 504, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6645, 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.

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

\(\Big \downarrow \) 6645

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} d x^2 (a+b \text {arctanh}(c x))+\frac {e \left (f+g x^2\right ) \log \left (f+g x^2\right ) (a+b \text {arctanh}(c x))}{2 g}-\frac {1}{2} e x^2 (a+b \text {arctanh}(c x))-b c \left (-\frac {e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c^2 \sqrt {g}}+\frac {(d-e) \text {arctanh}(c x)}{2 c^3}+\frac {e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (f+g x^2\right )}{2 c^3 g}+\frac {e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (\frac {2}{c x+1}\right )}{c^3 g}-\frac {e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{2 c^3 g}-\frac {e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{2 c^3 g}-\frac {x (d-e)}{2 c^2}-\frac {e x \log \left (f+g x^2\right )}{2 c^2}+\frac {e x}{c^2}-\frac {e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 c^3 g}+\frac {e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (c x+1)}\right )}{4 c^3 g}+\frac {e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {g} x+\sqrt {-f}\right )}{\left (\sqrt {-f} c+\sqrt {g}\right ) (c x+1)}\right )}{4 c^3 g}\right )\)

Input:

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

Output:

(d*x^2*(a + b*ArcTanh[c*x]))/2 - (e*x^2*(a + b*ArcTanh[c*x]))/2 + (e*(f + 
g*x^2)*(a + b*ArcTanh[c*x])*Log[f + g*x^2])/(2*g) - b*c*(-1/2*((d - e)*x)/ 
c^2 + (e*x)/c^2 - (e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(c^2*Sqrt[g]) + 
((d - e)*ArcTanh[c*x])/(2*c^3) + (e*(c^2*f + g)*ArcTanh[c*x]*Log[2/(1 + c* 
x)])/(c^3*g) - (e*(c^2*f + g)*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x) 
)/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(2*c^3*g) - (e*(c^2*f + g)*ArcTanh[ 
c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))]) 
/(2*c^3*g) - (e*x*Log[f + g*x^2])/(2*c^2) + (e*(c^2*f + g)*ArcTanh[c*x]*Lo 
g[f + g*x^2])/(2*c^3*g) - (e*(c^2*f + g)*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c 
^3*g) + (e*(c^2*f + g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqr 
t[-f] - Sqrt[g])*(1 + c*x))])/(4*c^3*g) + (e*(c^2*f + g)*PolyLog[2, 1 - (2 
*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(4*c^3*g))
 

Defintions of rubi rules used

rule 2009

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

rule 6645

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(969\) vs. \(2(448)=896\).

Time = 14.13 (sec) , antiderivative size = 970, normalized size of antiderivative = 1.89

method	result	size
risch	\(\frac {b d x}{2 c}+\frac {b d \ln \left (-c x +1\right )}{4 c^{2}}-\frac {b d \ln \left (c x +1\right )}{4 c^{2}}-\frac {a e \,x^{2}}{2}-\frac {3 b e x}{2 c}+\frac {a d \,x^{2}}{2}+\frac {b e \ln \left (c x +1\right )}{4 c^{2}}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) f}{4 g}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) f}{4 g}-\frac {e b \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (-c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) f}{4 g}-\frac {e b \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (-c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) f}{4 g}+\frac {e f b \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{c \sqrt {f g}}-\frac {b e \,x^{2} \ln \left (c x +1\right )}{4}-\frac {b e \ln \left (-c x +1\right )}{4 c^{2}}+\frac {e f a \ln \left (g \,x^{2}+f \right )}{2 g}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 c^{2}}+\frac {e b \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 c^{2}}-\frac {e b \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (-c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 c^{2}}-\frac {e b \ln \left (-c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (-c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 c^{2}}-\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}-\left (-c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) f}{4 g}-\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}+\left (-c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) f}{4 g}+\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) f}{4 g}+\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) f}{4 g}+\left (\frac {b e \,x^{2} \ln \left (c x +1\right )}{4}-\frac {e \left (b \,c^{2} x^{2} \ln \left (-c x +1\right )-2 a \,c^{2} x^{2}-2 b c x +b \ln \left (c x +1\right )-b \ln \left (-c x +1\right )\right )}{4 c^{2}}\right ) \ln \left (g \,x^{2}+f \right )+\frac {e b \ln \left (-c x +1\right ) x^{2}}{4}-\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}-\left (-c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 c^{2}}-\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}+\left (-c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 c^{2}}+\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 c^{2}}+\frac {e b \operatorname {dilog}\left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 c^{2}}+\frac {d b \ln \left (c x +1\right ) x^{2}}{4}-\frac {d b \ln \left (-c x +1\right ) x^{2}}{4}\)	\(970\)
default	\(\text {Expression too large to display}\)	\(9956\)
parts	\(\text {Expression too large to display}\)	\(9956\)

Input:

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

Output:

1/2*b*d*x/c+1/4*b*d*ln(-c*x+1)/c^2-1/4*b*d*ln(c*x+1)/c^2-1/2*a*e*x^2-3/2*b 
*e*x/c+1/2*a*d*x^2+1/4/c^2*b*e*ln(c*x+1)+1/4*e*b/g*ln(c*x+1)*ln((c*(-f*g)^ 
(1/2)-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))*f+1/4*e*b/g*ln(c*x+1)*ln((c*(-f*g)^ 
(1/2)+(c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))*f-1/4*e*b/g*ln(-c*x+1)*ln((c*(-f*g) 
^(1/2)-(-c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))*f-1/4*e*b/g*ln(-c*x+1)*ln((c*(-f* 
g)^(1/2)+(-c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))*f+1/c*e*f*b/(f*g)^(1/2)*arctan( 
x*g/(f*g)^(1/2))-1/4*b*e*x^2*ln(c*x+1)-1/4/c^2*b*e*ln(-c*x+1)+1/2*e/g*f*a* 
ln(g*x^2+f)+1/4/c^2*e*b*ln(c*x+1)*ln((c*(-f*g)^(1/2)-(c*x+1)*g+g)/(c*(-f*g 
)^(1/2)+g))+1/4/c^2*e*b*ln(c*x+1)*ln((c*(-f*g)^(1/2)+(c*x+1)*g-g)/(c*(-f*g 
)^(1/2)-g))-1/4/c^2*e*b*ln(-c*x+1)*ln((c*(-f*g)^(1/2)-(-c*x+1)*g+g)/(c*(-f 
*g)^(1/2)+g))-1/4/c^2*e*b*ln(-c*x+1)*ln((c*(-f*g)^(1/2)+(-c*x+1)*g-g)/(c*( 
-f*g)^(1/2)-g))-1/4*e*b/g*dilog((c*(-f*g)^(1/2)-(-c*x+1)*g+g)/(c*(-f*g)^(1 
/2)+g))*f-1/4*e*b/g*dilog((c*(-f*g)^(1/2)+(-c*x+1)*g-g)/(c*(-f*g)^(1/2)-g) 
)*f+1/4*e*b/g*dilog((c*(-f*g)^(1/2)-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))*f+1/4 
*e*b/g*dilog((c*(-f*g)^(1/2)+(c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))*f+(1/4*b*e*x 
^2*ln(c*x+1)-1/4*e*(b*c^2*x^2*ln(-c*x+1)-2*a*c^2*x^2-2*b*c*x+b*ln(c*x+1)-b 
*ln(-c*x+1))/c^2)*ln(g*x^2+f)+1/4*e*b*ln(-c*x+1)*x^2-1/4/c^2*e*b*dilog((c* 
(-f*g)^(1/2)-(-c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))-1/4/c^2*e*b*dilog((c*(-f*g) 
^(1/2)+(-c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))+1/4/c^2*e*b*dilog((c*(-f*g)^(1/2) 
-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))+1/4/c^2*e*b*dilog((c*(-f*g)^(1/2)+(c*...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/2*a*d*x^2 + 1/4*(2*x^2*arctanh(c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + lo 
g(c*x - 1)/c^3))*b*d - 1/4*(2*c^2*g*integrate(x^3*log(c*x + 1)/(c^2*g*x^2 
+ c^2*f), x) - 2*c^2*g*integrate(x^3*log(-c*x + 1)/(c^2*g*x^2 + c^2*f), x) 
 - 2*c*g*(-I*f*(log(I*g*x/sqrt(f*g) + 1) - log(-I*g*x/sqrt(f*g) + 1))/(sqr 
t(f*g)*c^2*g) - 2*x/(c^2*g)) - 2*g*integrate(x*log(c*x + 1)/(c^2*g*x^2 + c 
^2*f), x) + 2*g*integrate(x*log(-c*x + 1)/(c^2*g*x^2 + c^2*f), x) - (2*c*x 
 + (c^2*x^2 - 1)*log(c*x + 1) - (c^2*x^2 - 1)*log(-c*x + 1))*log(g*x^2 + f 
)/c^2)*b*e - 1/2*(g*x^2 - (g*x^2 + f)*log(g*x^2 + f) + f)*a*e/g
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*b*c*e + atanh(c*x)*log(f 
+ g*x**2)*b*c**2*e*f + atanh(c*x)*log(f + g*x**2)*b*c**2*e*g*x**2 + atanh( 
c*x)*b*c**2*d*g*x**2 - atanh(c*x)*b*c**2*e*g*x**2 - atanh(c*x)*b*d*g + ata 
nh(c*x)*b*e*g + int(log(f + g*x**2)/(c**2*x**2 - 1),x)*b*c**3*e*f + int(lo 
g(f + g*x**2)/(c**2*x**2 - 1),x)*b*c*e*g + log(f + g*x**2)*a*c**2*e*f + lo 
g(f + g*x**2)*a*c**2*e*g*x**2 + log(f + g*x**2)*b*c*e*g*x + a*c**2*d*g*x** 
2 - a*c**2*e*g*x**2 + b*c*d*g*x - 3*b*c*e*g*x)/(2*c**2*g)