Integrand size = 22, antiderivative size = 512 \[ \int x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\frac {b (d-e) x}{2 c}-\frac {b e x}{c}+\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c \sqrt {g}}-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}+\frac {1}{2} d x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} e x^2 (a+b \text {arctanh}(c x))-\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )}{c^2 g}+\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 c^2 g}+\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 c^2 g}+\frac {b e x \log \left (f+g x^2\right )}{2 c}-\frac {b e \left (c^2 f+g\right ) \text {arctanh}(c x) \log \left (f+g x^2\right )}{2 c^2 g}+\frac {e \left (f+g x^2\right ) (a+b \text {arctanh}(c x)) \log \left (f+g x^2\right )}{2 g}+\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 c^2 g}-\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{4 c^2 g}-\frac {b e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{4 c^2 g} \]
1/2*b*(d-e)*x/c-b*e*x/c-1/2*b*(d-e)*arctanh(c*x)/c^2+1/2*d*x^2*(a+b*arctan h(c*x))-1/2*e*x^2*(a+b*arctanh(c*x))-b*e*(c^2*f+g)*arctanh(c*x)*ln(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 )*arctanh(c*x)*ln(2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)-g^(1/2) ))/c^2/g+1/2*b*e*(c^2*f+g)*arctanh(c*x)*ln(2*c*((-f)^(1/2)+x*g^(1/2))/(c*x +1)/(c*(-f)^(1/2)+g^(1/2)))/c^2/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)*polylog(2,1-2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1)/(c *(-f)^(1/2)-g^(1/2)))/c^2/g-1/4*b*e*(c^2*f+g)*polylog(2,1-2*c*((-f)^(1/2)+ x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)+g^(1/2)))/c^2/g+b*e*arctan(x*g^(1/2)/f^(1 /2))*f^(1/2)/c/g^(1/2)
Result contains complex when optimal does not.
Time = 5.43 (sec) , antiderivative size = 1145, normalized size of antiderivative = 2.24 \[ \int x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx =\text {Too large to display} \]
(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 +...
Time = 0.98 (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 |
| \(\displaystyle \frac {1}{2} d x^2 (a+b \text {arctanh}(c x))+\frac {e \left (f+g x^2\right ) \log \left (f+g x^2\right ) (a+b \text {arctanh}(c x))}{2 g}-\frac {1}{2} e x^2 (a+b \text {arctanh}(c x))-b c \left (-\frac {e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c^2 \sqrt {g}}+\frac {(d-e) \text {arctanh}(c x)}{2 c^3}+\frac {e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (f+g x^2\right )}{2 c^3 g}+\frac {e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (\frac {2}{c x+1}\right )}{c^3 g}-\frac {e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{2 c^3 g}-\frac {e \text {arctanh}(c x) \left (c^2 f+g\right ) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{2 c^3 g}-\frac {x (d-e)}{2 c^2}-\frac {e x \log \left (f+g x^2\right )}{2 c^2}+\frac {e x}{c^2}-\frac {e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 c^3 g}+\frac {e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (c x+1)}\right )}{4 c^3 g}+\frac {e \left (c^2 f+g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {g} x+\sqrt {-f}\right )}{\left (\sqrt {-f} c+\sqrt {g}\right ) (c x+1)}\right )}{4 c^3 g}\right )\) |
(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))
3.6.33.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(970\) vs. \(2(448)=896\).
Time = 4.60 (sec) , antiderivative size = 971, normalized size of antiderivative = 1.90
| method | result | size |
| risch | \(\frac {a \,x^{2} d}{2}-\frac {e b \ln \left (-c x +1\right )}{4 c^{2}}+\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 {3 b e x}{2 c}+\frac {e f a \ln \left (g \,x^{2}+f \right )}{2 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}+\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 \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 {d b \ln \left (c x +1\right ) x^{2}}{4}-\frac {d b \ln \left (-c x +1\right ) x^{2}}{4}-\frac {a e \,x^{2}}{2}+\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}}+\left (\frac {b \,x^{2} e \ln \left (c x +1\right )}{4}+\frac {e \left (-b \,x^{2} \ln \left (-c x +1\right ) c^{2}+2 a \,c^{2} x^{2}+2 x b c +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 f b \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{c \sqrt {f 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 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 \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 \ln \left (c x +1\right )}{4 c^{2}}-\frac {b \,x^{2} e \ln \left (c x +1\right )}{4}\) | \(971\) |
| default | \(\text {Expression too large to display}\) | \(9956\) |
| parts | \(\text {Expression too large to display}\) | \(9956\) |
1/2*a*x^2*d-1/4/c^2*e*b*ln(-c*x+1)+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^2-3/2*b*e*x/c+1/2*e/g*f*a*ln(g*x^2+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*e*b/g*dilog((c*(-f*g)^(1/2)+(c*x+1) *g-g)/(c*(-f*g)^(1/2)-g))*f-1/4*e/c^2*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/c^2*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/c^2*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/c^2*b*ln(c*x+1)*ln((c*(-f*g)^(1/2)+( c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))+1/4*d*b*ln(c*x+1)*x^2-1/4*d*b*ln(-c*x+1)*x ^2-1/2*a*e*x^2+1/4*e*b*ln(-c*x+1)*x^2-1/4*e/c^2*b*dilog((c*(-f*g)^(1/2)-(- c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))-1/4*e/c^2*b*dilog((c*(-f*g)^(1/2)+(-c*x+1) *g-g)/(c*(-f*g)^(1/2)-g))+1/4*e/c^2*b*dilog((c*(-f*g)^(1/2)-(c*x+1)*g+g)/( c*(-f*g)^(1/2)+g))+(1/4*b*x^2*e*ln(c*x+1)+1/4*e*(-b*x^2*ln(-c*x+1)*c^2+2*a *c^2*x^2+2*x*b*c+b*ln(-c*x+1)-b*ln(c*x+1))/c^2)*ln(g*x^2+f)+e/c*f*b/(f*g)^ (1/2)*arctan(x*g/(f*g)^(1/2))-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+1/4*e/c^2*b*dilog((c*(-f*g)^(1/2)+(c*...
\[ \int x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} x \,d x } \]
Timed out. \[ \int x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\text {Timed out} \]
\[ \int x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} x \,d x } \]
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
\[ \int x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} x \,d x } \]
Timed out. \[ \int x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int x\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right ) \,d x \]