Integrand size = 24, antiderivative size = 712 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{f}+b c^2 e \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {b e g \coth ^{-1}(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 f}-\frac {1}{2} b c^2 e \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 )-\frac {b e g \coth ^{-1}(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 f}-\frac {1}{2} b c^2 e \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 )-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )}{2 f}-\frac {b e g \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )-\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 f}+\frac {1}{4} b c^2 e \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 )+\frac {b e g \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 f}+\frac {1}{4} b c^2 e \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 )+\frac {b e g \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 f} \]
a*e*g*ln(x)/f+b*e*g*arccoth(c*x)*ln(2/(c*x+1))/f+b*c^2*e*arctanh(c*x)*ln(2 /(c*x+1))-1/2*a*e*g*ln(g*x^2+f)/f-1/2*b*c*(d+e*ln(g*x^2+f))/x-1/2*(a+b*arc coth(c*x))*(d+e*ln(g*x^2+f))/x^2+1/2*b*c^2*arctanh(c*x)*(d+e*ln(g*x^2+f))- 1/2*b*e*g*arccoth(c*x)*ln(2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2) -g^(1/2)))/f-1/2*b*c^2*e*arctanh(c*x)*ln(2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1 )/(c*(-f)^(1/2)-g^(1/2)))-1/2*b*e*g*arccoth(c*x)*ln(2*c*((-f)^(1/2)+x*g^(1 /2))/(c*x+1)/(c*(-f)^(1/2)+g^(1/2)))/f-1/2*b*c^2*e*arctanh(c*x)*ln(2*c*((- f)^(1/2)+x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)+g^(1/2)))+1/2*b*e*g*polylog(2,-1 /c/x)/f-1/2*b*e*g*polylog(2,1/c/x)/f-1/2*b*c^2*e*polylog(2,1-2/(c*x+1))-1/ 2*b*e*g*polylog(2,1-2/(c*x+1))/f+1/4*b*c^2*e*polylog(2,1-2*c*((-f)^(1/2)-x *g^(1/2))/(c*x+1)/(c*(-f)^(1/2)-g^(1/2)))+1/4*b*e*g*polylog(2,1-2*c*((-f)^ (1/2)-x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)-g^(1/2)))/f+1/4*b*c^2*e*polylog(2,1 -2*c*((-f)^(1/2)+x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)+g^(1/2)))+1/4*b*e*g*poly log(2,1-2*c*((-f)^(1/2)+x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)+g^(1/2)))/f+b*c*e *arctan(x*g^(1/2)/f^(1/2))*g^(1/2)/f^(1/2)
Result contains complex when optimal does not.
Time = 4.96 (sec) , antiderivative size = 1193, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx =\text {Too large to display} \]
(-2*a*d*f - 2*b*c*d*f*x - 2*b*d*f*ArcCoth[c*x] + 2*b*c^2*d*f*x^2*ArcCoth[c *x] + 4*b*c*e*Sqrt[f]*Sqrt[g]*x^2*ArcTan[(Sqrt[g]*x)/Sqrt[f]] + (4*I)*b*c^ 2*e*f*x^2*ArcSin[Sqrt[g/(c^2*f + g)]]*ArcTanh[(c*f)/(Sqrt[-(c^2*f*g)]*x)] + (4*I)*b*e*g*x^2*ArcSin[Sqrt[g/(c^2*f + g)]]*ArcTanh[(c*f)/(Sqrt[-(c^2*f* g)]*x)] + 4*b*c^2*e*f*x^2*ArcCoth[c*x]*Log[1 - E^(-2*ArcCoth[c*x])] + 4*b* e*g*x^2*ArcCoth[c*x]*Log[1 + E^(-2*ArcCoth[c*x])] - 2*b*c^2*e*f*x^2*ArcCot h[c*x]*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g - 2 *Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] - 2*b*e*g*x^2*ArcCoth [c*x]*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g - 2* Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + (2*I)*b*c^2*e*f*x^2* ArcSin[Sqrt[g/(c^2*f + g)]]*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^( 2*ArcCoth[c*x])*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + (2*I)*b*e*g*x^2*ArcSin[Sqrt[g/(c^2*f + g)]]*Log[(c^2*(-1 + E^(2*ArcCoth[ c*x]))*f + g + E^(2*ArcCoth[c*x])*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c* x])*(c^2*f + g))] - 2*b*c^2*e*f*x^2*ArcCoth[c*x]*Log[(c^2*(-1 + E^(2*ArcCo th[c*x]))*f + g + E^(2*ArcCoth[c*x])*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth [c*x])*(c^2*f + g))] - 2*b*e*g*x^2*ArcCoth[c*x]*Log[(c^2*(-1 + E^(2*ArcCot h[c*x]))*f + g + E^(2*ArcCoth[c*x])*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[ c*x])*(c^2*f + g))] - (2*I)*b*c^2*e*f*x^2*ArcSin[Sqrt[g/(c^2*f + g)]]*Log[ (c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g + 2*Sqrt[-(...
Time = 1.33 (sec) , antiderivative size = 720, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6648, 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.
| \(\displaystyle \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 6648 |
| \(\displaystyle -2 e g \int \left (\frac {b c^2 x \text {arctanh}(c x)}{2 \left (g x^2+f\right )}-\frac {a+b c x+b \coth ^{-1}(c x)}{2 x \left (g x^2+f\right )}\right )dx-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 e g \left (\frac {a \log \left (f+g x^2\right )}{4 f}-\frac {a \log (x)}{2 f}-\frac {b c \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {b c^2 \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{4 g}+\frac {b c^2 \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{4 g}-\frac {b c^2 \text {arctanh}(c x) \log \left (\frac {2}{c x+1}\right )}{2 g}-\frac {b c^2 \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 )}{8 g}-\frac {b c^2 \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 )}{8 g}+\frac {b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{4 g}-\frac {b \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 )}{8 f}-\frac {b \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 )}{8 f}+\frac {b \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{4 f}+\frac {b \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{4 f}-\frac {b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )}{4 f}+\frac {b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )}{4 f}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{4 f}-\frac {b \log \left (\frac {2}{c x+1}\right ) \coth ^{-1}(c x)}{2 f}\right )-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}\) |
-1/2*(b*c*(d + e*Log[f + g*x^2]))/x - ((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/(2*x^2) + (b*c^2*ArcTanh[c*x]*(d + e*Log[f + g*x^2]))/2 - 2*e*g* (-1/2*(b*c*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(Sqrt[f]*Sqrt[g]) - (a*Log[x])/(2* f) - (b*ArcCoth[c*x]*Log[2/(1 + c*x)])/(2*f) - (b*c^2*ArcTanh[c*x]*Log[2/( 1 + c*x)])/(2*g) + (b*ArcCoth[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sq rt[-f] - Sqrt[g])*(1 + c*x))])/(4*f) + (b*c^2*ArcTanh[c*x]*Log[(2*c*(Sqrt[ -f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(4*g) + (b*ArcCoth[ c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))]) /(4*f) + (b*c^2*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(4*g) + (a*Log[f + g*x^2])/(4*f) - (b*PolyLog[2, -(1/(c*x))])/(4*f) + (b*PolyLog[2, 1/(c*x)])/(4*f) + (b*PolyLog[2, 1 - 2/( 1 + c*x)])/(4*f) + (b*c^2*PolyLog[2, 1 - 2/(1 + c*x)])/(4*g) - (b*PolyLog[ 2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/( 8*f) - (b*c^2*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - S qrt[g])*(1 + c*x))])/(8*g) - (b*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x) )/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(8*f) - (b*c^2*PolyLog[2, 1 - (2*c* (Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(8*g))
3.3.82.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]* (e_.))*(x_)^(m_.), x_Symbol] :> With[{u = IntHide[x^m*(a + b*ArcCoth[c*x]), x]}, Simp[(d + e*Log[f + g*x^2]) u, x] - Simp[2*e*g Int[ExpandIntegran d[x*(u/(f + g*x^2)), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Inte gerQ[m] && NeQ[m, -1]
Time = 9.69 (sec) , antiderivative size = 937, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(-\frac {a d}{2 x^{2}}-\frac {a e g \ln \left (g \,x^{2}+f \right )}{2 f}+\frac {a e g \ln \left (x \right )}{f}+\left (-\frac {b e \ln \left (c x +1\right )}{4 x^{2}}-\frac {e \left (b \,x^{2} \ln \left (c x -1\right ) c^{2}-b \,c^{2} \ln \left (c x +1\right ) x^{2}+2 b c x -b \ln \left (c x -1\right )+2 a \right )}{4 x^{2}}\right ) \ln \left (g \,x^{2}+f \right )-\frac {g b e \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 f}-\frac {g b e \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 f}+\frac {g e b c \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{\sqrt {f g}}+\frac {g b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {g b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \ln \left (c x -1\right ) \ln \left (c x \right )}{2 f}-\frac {d b c}{2 x}+\frac {d b \,c^{2} \ln \left (c x +1\right )}{4}-\frac {d b \ln \left (c x +1\right )}{4 x^{2}}-\frac {d b \,c^{2} \ln \left (c x -1\right )}{4}+\frac {d b \ln \left (c x -1\right )}{4 x^{2}}-\frac {b e \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 ) c^{2}}{4}-\frac {b e \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 ) c^{2}}{4}-\frac {g b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}+\frac {b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}+\frac {g b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {g b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \operatorname {dilog}\left (c x \right )}{2 f}-\frac {g b e \operatorname {dilog}\left (c x +1\right )}{2 f}-\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}-\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}+\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}+\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}\) | \(937\) |
-1/2*a*d/x^2-1/2*a*e*g*ln(g*x^2+f)/f+a*e*g*ln(x)/f+(-1/4*b*e/x^2*ln(c*x+1) -1/4*e*(b*x^2*ln(c*x-1)*c^2-b*c^2*ln(c*x+1)*x^2+2*b*c*x-b*ln(c*x-1)+2*a)/x ^2)*ln(g*x^2+f)-1/4*g*b*e/f*ln(c*x+1)*ln((c*(-f*g)^(1/2)-(c*x+1)*g+g)/(c*( -f*g)^(1/2)+g))-1/4*g*b*e/f*ln(c*x+1)*ln((c*(-f*g)^(1/2)+(c*x+1)*g-g)/(c*( -f*g)^(1/2)-g))+g*e*b*c/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+1/4*g*b*e/f*ln (c*x-1)*ln((c*(-f*g)^(1/2)-g*(c*x-1)-g)/(c*(-f*g)^(1/2)-g))+1/4*g*b*e/f*ln (c*x-1)*ln((c*(-f*g)^(1/2)+g*(c*x-1)+g)/(c*(-f*g)^(1/2)+g))-1/2*g*b*e/f*ln (c*x-1)*ln(c*x)-1/2*d*b*c/x+1/4*d*b*c^2*ln(c*x+1)-1/4*d*b*ln(c*x+1)/x^2-1/ 4*d*b*c^2*ln(c*x-1)+1/4*d*b*ln(c*x-1)/x^2-1/4*b*e*ln(c*x+1)*ln((c*(-f*g)^( 1/2)-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))*c^2-1/4*b*e*ln(c*x+1)*ln((c*(-f*g)^( 1/2)+(c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))*c^2-1/4*g*b*e/f*dilog((c*(-f*g)^(1/2 )-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))-1/4*g*b*e/f*dilog((c*(-f*g)^(1/2)+(c*x+ 1)*g-g)/(c*(-f*g)^(1/2)-g))+1/4*b*e*ln(c*x-1)*ln((c*(-f*g)^(1/2)-g*(c*x-1) -g)/(c*(-f*g)^(1/2)-g))*c^2+1/4*b*e*ln(c*x-1)*ln((c*(-f*g)^(1/2)+g*(c*x-1) +g)/(c*(-f*g)^(1/2)+g))*c^2+1/4*g*b*e/f*dilog((c*(-f*g)^(1/2)-g*(c*x-1)-g) /(c*(-f*g)^(1/2)-g))+1/4*g*b*e/f*dilog((c*(-f*g)^(1/2)+g*(c*x-1)+g)/(c*(-f *g)^(1/2)+g))-1/2*g*b*e/f*dilog(c*x)-1/2*g*b*e/f*dilog(c*x+1)-1/4*b*e*dilo g((c*(-f*g)^(1/2)-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))*c^2-1/4*b*e*dilog((c*(- f*g)^(1/2)+(c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))*c^2+1/4*b*e*dilog((c*(-f*g)^(1 /2)-g*(c*x-1)-g)/(c*(-f*g)^(1/2)-g))*c^2+1/4*b*e*dilog((c*(-f*g)^(1/2)+...
\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}} \,d x } \]
1/4*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arccoth(c*x)/x^2)*b*d - 1/2*(g*(log(g*x^2 + f)/f - log(x^2)/f) + log(g*x^2 + f)/x^2)*a*e - 1/4*(2 *c^2*g*integrate(x^2*log(c*x + 1)/(g*x^3 + f*x), x) - 2*c^2*g*integrate(x^ 2*log(c*x - 1)/(g*x^3 + f*x), x) + 2*I*c*g*(log(I*g*x/sqrt(f*g) + 1) - log (-I*g*x/sqrt(f*g) + 1))/sqrt(f*g) - 2*g*integrate(log(c*x + 1)/(g*x^3 + f* x), x) + 2*g*integrate(log(c*x - 1)/(g*x^3 + f*x), 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)/x^2)*b*e - 1/2*a*d/x^2
\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right )}{x^3} \,d x \]