Integrand size = 12, antiderivative size = 148 \[ \int \frac {\text {arctanh}(a+b x)^2}{x} \, dx=-\text {arctanh}(a+b x)^2 \log \left (\frac {2}{1+a+b x}\right )+\text {arctanh}(a+b x)^2 \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )+\text {arctanh}(a+b x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )-\text {arctanh}(a+b x) \operatorname {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (1+a+b x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{1+a+b x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 b x}{(1-a) (1+a+b x)}\right ) \]
-arctanh(b*x+a)^2*ln(2/(b*x+a+1))+arctanh(b*x+a)^2*ln(2*b*x/(1-a)/(b*x+a+1 ))+arctanh(b*x+a)*polylog(2,1-2/(b*x+a+1))-arctanh(b*x+a)*polylog(2,1-2*b* x/(1-a)/(b*x+a+1))+1/2*polylog(3,1-2/(b*x+a+1))-1/2*polylog(3,1-2*b*x/(1-a )/(b*x+a+1))
Result contains complex when optimal does not.
Time = 2.53 (sec) , antiderivative size = 753, normalized size of antiderivative = 5.09 \[ \int \frac {\text {arctanh}(a+b x)^2}{x} \, dx=-\frac {4}{3} \text {arctanh}(a+b x)^3-\frac {2 \text {arctanh}(a+b x)^3}{3 a}+\frac {2 \sqrt {1-a^2} e^{\text {arctanh}(a)} \text {arctanh}(a+b x)^3}{3 a}-\text {arctanh}(a+b x)^2 \log \left (1+e^{-2 \text {arctanh}(a+b x)}\right )-i \pi \text {arctanh}(a+b x) \log \left (\frac {1}{2} \left (e^{-\text {arctanh}(a+b x)}+e^{\text {arctanh}(a+b x)}\right )\right )-\text {arctanh}(a+b x)^2 \log \left (1-\frac {\sqrt {-1+a} e^{\text {arctanh}(a+b x)}}{\sqrt {-1-a}}\right )-\text {arctanh}(a+b x)^2 \log \left (1+\frac {\sqrt {-1+a} e^{\text {arctanh}(a+b x)}}{\sqrt {-1-a}}\right )+\text {arctanh}(a+b x)^2 \log \left (\frac {1}{2} e^{-\text {arctanh}(a+b x)} \left (1+a-e^{2 \text {arctanh}(a+b x)}+a e^{2 \text {arctanh}(a+b x)}\right )\right )+\text {arctanh}(a+b x)^2 \log \left (1-e^{-\text {arctanh}(a)+\text {arctanh}(a+b x)}\right )+\text {arctanh}(a+b x)^2 \log \left (1+e^{-\text {arctanh}(a)+\text {arctanh}(a+b x)}\right )-2 \text {arctanh}(a) \text {arctanh}(a+b x) \log \left (\frac {1}{2} i \left (-e^{\text {arctanh}(a)-\text {arctanh}(a+b x)}+e^{-\text {arctanh}(a)+\text {arctanh}(a+b x)}\right )\right )+\text {arctanh}(a+b x)^2 \log \left (1-e^{-2 \text {arctanh}(a)+2 \text {arctanh}(a+b x)}\right )+i \pi \text {arctanh}(a+b x) \log \left (\frac {1}{\sqrt {1-(a+b x)^2}}\right )-\text {arctanh}(a+b x)^2 \log \left (-\frac {b x}{\sqrt {1-(a+b x)^2}}\right )+2 \text {arctanh}(a) \text {arctanh}(a+b x) \log (-i \sinh (\text {arctanh}(a)-\text {arctanh}(a+b x)))+\text {arctanh}(a+b x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a+b x)}\right )-2 \text {arctanh}(a+b x) \operatorname {PolyLog}\left (2,-\frac {\sqrt {-1+a} e^{\text {arctanh}(a+b x)}}{\sqrt {-1-a}}\right )-2 \text {arctanh}(a+b x) \operatorname {PolyLog}\left (2,\frac {\sqrt {-1+a} e^{\text {arctanh}(a+b x)}}{\sqrt {-1-a}}\right )+2 \text {arctanh}(a+b x) \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a)+\text {arctanh}(a+b x)}\right )+2 \text {arctanh}(a+b x) \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a)+\text {arctanh}(a+b x)}\right )+\text {arctanh}(a+b x) \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a)+2 \text {arctanh}(a+b x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a+b x)}\right )+2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {-1+a} e^{\text {arctanh}(a+b x)}}{\sqrt {-1-a}}\right )+2 \operatorname {PolyLog}\left (3,\frac {\sqrt {-1+a} e^{\text {arctanh}(a+b x)}}{\sqrt {-1-a}}\right )-2 \operatorname {PolyLog}\left (3,-e^{-\text {arctanh}(a)+\text {arctanh}(a+b x)}\right )-2 \operatorname {PolyLog}\left (3,e^{-\text {arctanh}(a)+\text {arctanh}(a+b x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 \text {arctanh}(a)+2 \text {arctanh}(a+b x)}\right ) \]
(-4*ArcTanh[a + b*x]^3)/3 - (2*ArcTanh[a + b*x]^3)/(3*a) + (2*Sqrt[1 - a^2 ]*E^ArcTanh[a]*ArcTanh[a + b*x]^3)/(3*a) - ArcTanh[a + b*x]^2*Log[1 + E^(- 2*ArcTanh[a + b*x])] - I*Pi*ArcTanh[a + b*x]*Log[(E^(-ArcTanh[a + b*x]) + E^ArcTanh[a + b*x])/2] - ArcTanh[a + b*x]^2*Log[1 - (Sqrt[-1 + a]*E^ArcTan h[a + b*x])/Sqrt[-1 - a]] - ArcTanh[a + b*x]^2*Log[1 + (Sqrt[-1 + a]*E^Arc Tanh[a + b*x])/Sqrt[-1 - a]] + ArcTanh[a + b*x]^2*Log[(1 + a - E^(2*ArcTan h[a + b*x]) + a*E^(2*ArcTanh[a + b*x]))/(2*E^ArcTanh[a + b*x])] + ArcTanh[ a + b*x]^2*Log[1 - E^(-ArcTanh[a] + ArcTanh[a + b*x])] + ArcTanh[a + b*x]^ 2*Log[1 + E^(-ArcTanh[a] + ArcTanh[a + b*x])] - 2*ArcTanh[a]*ArcTanh[a + b *x]*Log[(I/2)*(-E^(ArcTanh[a] - ArcTanh[a + b*x]) + E^(-ArcTanh[a] + ArcTa nh[a + b*x]))] + ArcTanh[a + b*x]^2*Log[1 - E^(-2*ArcTanh[a] + 2*ArcTanh[a + b*x])] + I*Pi*ArcTanh[a + b*x]*Log[1/Sqrt[1 - (a + b*x)^2]] - ArcTanh[a + b*x]^2*Log[-((b*x)/Sqrt[1 - (a + b*x)^2])] + 2*ArcTanh[a]*ArcTanh[a + b *x]*Log[(-I)*Sinh[ArcTanh[a] - ArcTanh[a + b*x]]] + ArcTanh[a + b*x]*PolyL og[2, -E^(-2*ArcTanh[a + b*x])] - 2*ArcTanh[a + b*x]*PolyLog[2, -((Sqrt[-1 + a]*E^ArcTanh[a + b*x])/Sqrt[-1 - a])] - 2*ArcTanh[a + b*x]*PolyLog[2, ( Sqrt[-1 + a]*E^ArcTanh[a + b*x])/Sqrt[-1 - a]] + 2*ArcTanh[a + b*x]*PolyLo g[2, -E^(-ArcTanh[a] + ArcTanh[a + b*x])] + 2*ArcTanh[a + b*x]*PolyLog[2, E^(-ArcTanh[a] + ArcTanh[a + b*x])] + ArcTanh[a + b*x]*PolyLog[2, E^(-2*Ar cTanh[a] + 2*ArcTanh[a + b*x])] + PolyLog[3, -E^(-2*ArcTanh[a + b*x])]/...
Time = 0.32 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6661, 25, 27, 6474}
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.
| \(\displaystyle \int \frac {\text {arctanh}(a+b x)^2}{x} \, dx\) |
| \(\Big \downarrow \) 6661 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a+b x)^2}{x}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {\text {arctanh}(a+b x)^2}{x}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int -\frac {\text {arctanh}(a+b x)^2}{b x}d(a+b x)\) |
| \(\Big \downarrow \) 6474 |
| \(\displaystyle \text {arctanh}(a+b x) \operatorname {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )-\text {arctanh}(a+b x) \operatorname {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (a+b x+1)}\right )+\text {arctanh}(a+b x)^2 \left (-\log \left (\frac {2}{a+b x+1}\right )\right )+\text {arctanh}(a+b x)^2 \log \left (\frac {2 b x}{(1-a) (a+b x+1)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{a+b x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 b x}{(1-a) (a+b x+1)}\right )\) |
-(ArcTanh[a + b*x]^2*Log[2/(1 + a + b*x)]) + ArcTanh[a + b*x]^2*Log[(2*b*x )/((1 - a)*(1 + a + b*x))] + ArcTanh[a + b*x]*PolyLog[2, 1 - 2/(1 + a + b* x)] - ArcTanh[a + b*x]*PolyLog[2, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))] + P olyLog[3, 1 - 2/(1 + a + b*x)]/2 - PolyLog[3, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))]/2
3.1.5.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^2)*(Log[2/(1 + c*x)]/e), x] + (Simp[(a + b*Arc Tanh[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b*(a + b*ArcTanh[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcT anh[c*x])*(PolyLog[2, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + S imp[b^2*(PolyLog[3, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[b^2*(PolyLog[3, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e} , x] && NeQ[c^2*d^2 - e^2, 0]
Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG tQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.80 (sec) , antiderivative size = 901, normalized size of antiderivative = 6.09
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(901\) |
| default | \(\text {Expression too large to display}\) | \(901\) |
| parts | \(\text {Expression too large to display}\) | \(1544\) |
ln(-b*x)*arctanh(b*x+a)^2-arctanh(b*x+a)^2*ln(-(b*x+a+1)^2/(1-(b*x+a)^2)+1 +a*(1+(b*x+a+1)^2/(1-(b*x+a)^2)))+1/2*I*Pi*csgn(I*((b*x+a+1)^2/((b*x+a)^2- 1)+1+a*(1-(b*x+a+1)^2/((b*x+a)^2-1)))/(1-(b*x+a+1)^2/((b*x+a)^2-1)))*(csgn (I*((b*x+a+1)^2/((b*x+a)^2-1)+1+a*(1-(b*x+a+1)^2/((b*x+a)^2-1))))*csgn(I/( 1-(b*x+a+1)^2/((b*x+a)^2-1)))-csgn(I*((b*x+a+1)^2/((b*x+a)^2-1)+1+a*(1-(b* x+a+1)^2/((b*x+a)^2-1)))/(1-(b*x+a+1)^2/((b*x+a)^2-1)))*csgn(I/(1-(b*x+a+1 )^2/((b*x+a)^2-1)))-csgn(I*((b*x+a+1)^2/((b*x+a)^2-1)+1+a*(1-(b*x+a+1)^2/( (b*x+a)^2-1))))*csgn(I*((b*x+a+1)^2/((b*x+a)^2-1)+1+a*(1-(b*x+a+1)^2/((b*x +a)^2-1)))/(1-(b*x+a+1)^2/((b*x+a)^2-1)))+csgn(I*((b*x+a+1)^2/((b*x+a)^2-1 )+1+a*(1-(b*x+a+1)^2/((b*x+a)^2-1)))/(1-(b*x+a+1)^2/((b*x+a)^2-1)))^2)*arc tanh(b*x+a)^2-arctanh(b*x+a)*polylog(2,-(b*x+a+1)^2/(1-(b*x+a)^2))+1/2*pol ylog(3,-(b*x+a+1)^2/(1-(b*x+a)^2))+1/(-1+a)*a*arctanh(b*x+a)^2*ln(1-(-1+a) *(b*x+a+1)^2/(1-(b*x+a)^2)/(-1-a))+1/(-1+a)*a*arctanh(b*x+a)*polylog(2,(-1 +a)*(b*x+a+1)^2/(1-(b*x+a)^2)/(-1-a))-1/2/(-1+a)*a*polylog(3,(-1+a)*(b*x+a +1)^2/(1-(b*x+a)^2)/(-1-a))-1/(-1+a)*arctanh(b*x+a)^2*ln(1-(-1+a)*(b*x+a+1 )^2/(1-(b*x+a)^2)/(-1-a))-1/(-1+a)*arctanh(b*x+a)*polylog(2,(-1+a)*(b*x+a+ 1)^2/(1-(b*x+a)^2)/(-1-a))+1/2/(-1+a)*polylog(3,(-1+a)*(b*x+a+1)^2/(1-(b*x +a)^2)/(-1-a))
\[ \int \frac {\text {arctanh}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )^{2}}{x} \,d x } \]
\[ \int \frac {\text {arctanh}(a+b x)^2}{x} \, dx=\int \frac {\operatorname {atanh}^{2}{\left (a + b x \right )}}{x}\, dx \]
\[ \int \frac {\text {arctanh}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )^{2}}{x} \,d x } \]
\[ \int \frac {\text {arctanh}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a+b x)^2}{x} \, dx=\int \frac {{\mathrm {atanh}\left (a+b\,x\right )}^2}{x} \,d x \]