Integrand size = 12, antiderivative size = 148 \[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=-\coth ^{-1}(a+b x)^2 \log \left (\frac {2}{1+a+b x}\right )+\coth ^{-1}(a+b x)^2 \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )+\coth ^{-1}(a+b x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )-\coth ^{-1}(a+b x) \operatorname {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (1+a+b x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{1+a+b x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 b x}{(1-a) (1+a+b x)}\right ) \]
-arccoth(b*x+a)^2*ln(2/(b*x+a+1))+arccoth(b*x+a)^2*ln(2*b*x/(1-a)/(b*x+a+1 ))+arccoth(b*x+a)*polylog(2,1-2/(b*x+a+1))-arccoth(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 = 2.45 (sec) , antiderivative size = 777, normalized size of antiderivative = 5.25 \[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=-\frac {i \pi ^3}{24}-\frac {2}{3} \coth ^{-1}(a+b x)^3-\frac {2}{3} a \coth ^{-1}(a+b x)^3+\frac {2}{3} \sqrt {1-\frac {1}{a^2}} a e^{\text {arctanh}\left (\frac {1}{a}\right )} \coth ^{-1}(a+b x)^3-i \pi \coth ^{-1}(a+b x) \log \left (\frac {1}{2} \left (e^{-\coth ^{-1}(a+b x)}+e^{\coth ^{-1}(a+b x)}\right )\right )-\coth ^{-1}(a+b x)^2 \log \left (1-\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )-\coth ^{-1}(a+b x)^2 \log \left (1+\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )-\coth ^{-1}(a+b x)^2 \log \left (1-e^{2 \coth ^{-1}(a+b x)}\right )+\coth ^{-1}(a+b x)^2 \log \left (1-e^{2 \coth ^{-1}(a+b x)-2 \text {arctanh}\left (\frac {1}{a}\right )}\right )+\coth ^{-1}(a+b x)^2 \log \left (1-e^{\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {1}{a}\right )}\right )+\coth ^{-1}(a+b x)^2 \log \left (1+e^{\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {1}{a}\right )}\right )-2 \coth ^{-1}(a+b x) \text {arctanh}\left (\frac {1}{a}\right ) \log \left (\frac {1}{2} i \left (e^{\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {1}{a}\right )}-e^{-\coth ^{-1}(a+b x)+\text {arctanh}\left (\frac {1}{a}\right )}\right )\right )+\coth ^{-1}(a+b x)^2 \log \left (\frac {1}{2} e^{-\coth ^{-1}(a+b x)} \left (-1-e^{2 \coth ^{-1}(a+b x)}+a \left (-1+e^{2 \coth ^{-1}(a+b x)}\right )\right )\right )+i \pi \coth ^{-1}(a+b x) \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )-\coth ^{-1}(a+b x)^2 \log \left (-\frac {b x}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )+2 \coth ^{-1}(a+b x) \text {arctanh}\left (\frac {1}{a}\right ) \log \left (i \sinh \left (\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {1}{a}\right )\right )\right )-2 \coth ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )-2 \coth ^{-1}(a+b x) \operatorname {PolyLog}\left (2,\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )-\coth ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(a+b x)}\right )+\coth ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(a+b x)-2 \text {arctanh}\left (\frac {1}{a}\right )}\right )+2 \coth ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {1}{a}\right )}\right )+2 \coth ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {1}{a}\right )}\right )+2 \operatorname {PolyLog}\left (3,-\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )+2 \operatorname {PolyLog}\left (3,\sqrt {\frac {-1+a}{1+a}} e^{\coth ^{-1}(a+b x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(a+b x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(a+b x)-2 \text {arctanh}\left (\frac {1}{a}\right )}\right )-2 \operatorname {PolyLog}\left (3,-e^{\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {1}{a}\right )}\right )-2 \operatorname {PolyLog}\left (3,e^{\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {1}{a}\right )}\right ) \]
(-1/24*I)*Pi^3 - (2*ArcCoth[a + b*x]^3)/3 - (2*a*ArcCoth[a + b*x]^3)/3 + ( 2*Sqrt[1 - a^(-2)]*a*E^ArcTanh[a^(-1)]*ArcCoth[a + b*x]^3)/3 - I*Pi*ArcCot h[a + b*x]*Log[(E^(-ArcCoth[a + b*x]) + E^ArcCoth[a + b*x])/2] - ArcCoth[a + b*x]^2*Log[1 - Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] - ArcCoth[a + b*x]^2*Log[1 + Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] - ArcCoth[a + b *x]^2*Log[1 - E^(2*ArcCoth[a + b*x])] + ArcCoth[a + b*x]^2*Log[1 - E^(2*Ar cCoth[a + b*x] - 2*ArcTanh[a^(-1)])] + ArcCoth[a + b*x]^2*Log[1 - E^(ArcCo th[a + b*x] - ArcTanh[a^(-1)])] + ArcCoth[a + b*x]^2*Log[1 + E^(ArcCoth[a + b*x] - ArcTanh[a^(-1)])] - 2*ArcCoth[a + b*x]*ArcTanh[a^(-1)]*Log[(I/2)* (E^(ArcCoth[a + b*x] - ArcTanh[a^(-1)]) - E^(-ArcCoth[a + b*x] + ArcTanh[a ^(-1)]))] + ArcCoth[a + b*x]^2*Log[(-1 - E^(2*ArcCoth[a + b*x]) + a*(-1 + E^(2*ArcCoth[a + b*x])))/(2*E^ArcCoth[a + b*x])] + I*Pi*ArcCoth[a + b*x]*L og[1/Sqrt[1 - (a + b*x)^(-2)]] - ArcCoth[a + b*x]^2*Log[-((b*x)/((a + b*x) *Sqrt[1 - (a + b*x)^(-2)]))] + 2*ArcCoth[a + b*x]*ArcTanh[a^(-1)]*Log[I*Si nh[ArcCoth[a + b*x] - ArcTanh[a^(-1)]]] - 2*ArcCoth[a + b*x]*PolyLog[2, -( Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x])] - 2*ArcCoth[a + b*x]*PolyLog[2 , Sqrt[(-1 + a)/(1 + a)]*E^ArcCoth[a + b*x]] - ArcCoth[a + b*x]*PolyLog[2, E^(2*ArcCoth[a + b*x])] + ArcCoth[a + b*x]*PolyLog[2, E^(2*ArcCoth[a + b* x] - 2*ArcTanh[a^(-1)])] + 2*ArcCoth[a + b*x]*PolyLog[2, -E^(ArcCoth[a + b *x] - ArcTanh[a^(-1)])] + 2*ArcCoth[a + b*x]*PolyLog[2, E^(ArcCoth[a + ...
Time = 0.33 (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 = {6662, 25, 27, 6475}
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.
| \(\displaystyle \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx\) |
| \(\Big \downarrow \) 6662 |
| \(\displaystyle \frac {\int \frac {\coth ^{-1}(a+b x)^2}{x}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {\coth ^{-1}(a+b x)^2}{x}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int -\frac {\coth ^{-1}(a+b x)^2}{b x}d(a+b x)\) |
| \(\Big \downarrow \) 6475 |
| \(\displaystyle \frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{a+b x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 b x}{(1-a) (a+b x+1)}\right )+\operatorname {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)-\operatorname {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x)-\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)^2+\log \left (\frac {2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x)^2\) |
-(ArcCoth[a + b*x]^2*Log[2/(1 + a + b*x)]) + ArcCoth[a + b*x]^2*Log[(2*b*x )/((1 - a)*(1 + a + b*x))] + ArcCoth[a + b*x]*PolyLog[2, 1 - 2/(1 + a + b* x)] - ArcCoth[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
3.1.73.3.1 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_.) + ArcCoth[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^2)*(Log[2/(1 + c*x)]/e), x] + (Simp[(a + b*Arc Coth[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b*(a + b*ArcCoth[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcC oth[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_.) + ArcCoth[(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* ArcCoth[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 = 5.62 (sec) , antiderivative size = 866, normalized size of antiderivative = 5.85
| method | result | size |
| derivativedivides | \(\ln \left (-b x \right ) \operatorname {arccoth}\left (b x +a \right )^{2}-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )\right ) \operatorname {csgn}\left (\frac {i}{\frac {b x +a +1}{b x +a -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {b x +a +1}{b x +a -1}-1}\right )-\operatorname {csgn}\left (i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right )+\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (b x +a \right )^{2}}{2}+\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (\frac {b x +a +1}{b x +a -1}-1\right )-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-2 \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-2 \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+\frac {a \operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{-1+a}+\frac {a \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{-1+a}-\frac {a \operatorname {polylog}\left (3, \frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{2 \left (-1+a \right )}-\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{-1+a}-\frac {\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{-1+a}+\frac {\operatorname {polylog}\left (3, \frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{-2+2 a}\) | \(866\) |
| default | \(\ln \left (-b x \right ) \operatorname {arccoth}\left (b x +a \right )^{2}-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right ) \left (\operatorname {csgn}\left (i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )\right ) \operatorname {csgn}\left (\frac {i}{\frac {b x +a +1}{b x +a -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {b x +a +1}{b x +a -1}-1}\right )-\operatorname {csgn}\left (i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right )+\operatorname {csgn}\left (\frac {i \left (-\frac {b x +a +1}{b x +a -1}-1+a \left (\frac {b x +a +1}{b x +a -1}-1\right )\right )}{\frac {b x +a +1}{b x +a -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (b x +a \right )^{2}}{2}+\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (\frac {b x +a +1}{b x +a -1}-1\right )-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-2 \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )-2 \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {b x +a -1}{b x +a +1}}}\right )+\frac {a \operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{-1+a}+\frac {a \,\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{-1+a}-\frac {a \operatorname {polylog}\left (3, \frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{2 \left (-1+a \right )}-\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \ln \left (1-\frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{-1+a}-\frac {\operatorname {arccoth}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{-1+a}+\frac {\operatorname {polylog}\left (3, \frac {\left (b x +a +1\right ) \left (-1+a \right )}{\left (1+a \right ) \left (b x +a -1\right )}\right )}{-2+2 a}\) | \(866\) |
| parts | \(\text {Expression too large to display}\) | \(1411\) |
ln(-b*x)*arccoth(b*x+a)^2-arccoth(b*x+a)^2*ln(-(b*x+a+1)/(b*x+a-1)-1+a*((b *x+a+1)/(b*x+a-1)-1))+1/2*I*Pi*csgn(I*(-(b*x+a+1)/(b*x+a-1)-1+a*((b*x+a+1) /(b*x+a-1)-1))/((b*x+a+1)/(b*x+a-1)-1))*(csgn(I*(-(b*x+a+1)/(b*x+a-1)-1+a* ((b*x+a+1)/(b*x+a-1)-1)))*csgn(I/((b*x+a+1)/(b*x+a-1)-1))-csgn(I*(-(b*x+a+ 1)/(b*x+a-1)-1+a*((b*x+a+1)/(b*x+a-1)-1))/((b*x+a+1)/(b*x+a-1)-1))*csgn(I/ ((b*x+a+1)/(b*x+a-1)-1))-csgn(I*(-(b*x+a+1)/(b*x+a-1)-1+a*((b*x+a+1)/(b*x+ a-1)-1)))*csgn(I*(-(b*x+a+1)/(b*x+a-1)-1+a*((b*x+a+1)/(b*x+a-1)-1))/((b*x+ a+1)/(b*x+a-1)-1))+csgn(I*(-(b*x+a+1)/(b*x+a-1)-1+a*((b*x+a+1)/(b*x+a-1)-1 ))/((b*x+a+1)/(b*x+a-1)-1))^2)*arccoth(b*x+a)^2+arccoth(b*x+a)^2*ln((b*x+a +1)/(b*x+a-1)-1)-arccoth(b*x+a)^2*ln(1+1/((b*x+a-1)/(b*x+a+1))^(1/2))-2*ar ccoth(b*x+a)*polylog(2,-1/((b*x+a-1)/(b*x+a+1))^(1/2))+2*polylog(3,-1/((b* x+a-1)/(b*x+a+1))^(1/2))-arccoth(b*x+a)^2*ln(1-1/((b*x+a-1)/(b*x+a+1))^(1/ 2))-2*arccoth(b*x+a)*polylog(2,1/((b*x+a-1)/(b*x+a+1))^(1/2))+2*polylog(3, 1/((b*x+a-1)/(b*x+a+1))^(1/2))+a/(-1+a)*arccoth(b*x+a)^2*ln(1-(b*x+a+1)*(- 1+a)/(1+a)/(b*x+a-1))+a/(-1+a)*arccoth(b*x+a)*polylog(2,(b*x+a+1)*(-1+a)/( 1+a)/(b*x+a-1))-1/2*a/(-1+a)*polylog(3,(b*x+a+1)*(-1+a)/(1+a)/(b*x+a-1))-1 /(-1+a)*arccoth(b*x+a)^2*ln(1-(b*x+a+1)*(-1+a)/(1+a)/(b*x+a-1))-1/(-1+a)*a rccoth(b*x+a)*polylog(2,(b*x+a+1)*(-1+a)/(1+a)/(b*x+a-1))+1/2/(-1+a)*polyl og(3,(b*x+a+1)*(-1+a)/(1+a)/(b*x+a-1))
\[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x} \,d x } \]
\[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=\int \frac {\operatorname {acoth}^{2}{\left (a + b x \right )}}{x}\, dx \]
\[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x} \,d x } \]
\[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx=\int \frac {{\mathrm {acoth}\left (a+b\,x\right )}^2}{x} \,d x \]