Integrand size = 12, antiderivative size = 204 \[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\frac {x}{3 b^2}-\frac {2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}+\frac {a \left (3+a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {\left (1+3 a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {\text {arctanh}(a+b x)}{3 b^3}-\frac {2 \left (1+3 a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{3 b^3}-\frac {a \log \left (1-(a+b x)^2\right )}{b^3}-\frac {\left (1+3 a^2\right ) \operatorname {PolyLog}\left (2,-\frac {1+a+b x}{1-a-b x}\right )}{3 b^3} \]
1/3*x/b^2-2*a*(b*x+a)*arccoth(b*x+a)/b^3+1/3*(b*x+a)^2*arccoth(b*x+a)/b^3+ 1/3*a*(a^2+3)*arccoth(b*x+a)^2/b^3+1/3*(3*a^2+1)*arccoth(b*x+a)^2/b^3+1/3* x^3*arccoth(b*x+a)^2-1/3*arctanh(b*x+a)/b^3-2/3*(3*a^2+1)*arccoth(b*x+a)*l n(2/(-b*x-a+1))/b^3-a*ln(1-(b*x+a)^2)/b^3-1/3*(3*a^2+1)*polylog(2,(-b*x-a- 1)/(-b*x-a+1))/b^3
Leaf count is larger than twice the leaf count of optimal. \(644\) vs. \(2(204)=408\).
Time = 3.59 (sec) , antiderivative size = 644, normalized size of antiderivative = 3.16 \[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=-\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \left (1-(a+b x)^2\right ) \left (\frac {4 \coth ^{-1}(a+b x)}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {3 \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}-\frac {12 a \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {9 a^2 \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {-1+6 a \coth ^{-1}(a+b x)-3 \left (-1+a^2\right ) \coth ^{-1}(a+b x)^2}{\sqrt {1-\frac {1}{(a+b x)^2}}}+\cosh \left (3 \coth ^{-1}(a+b x)\right )-6 a \coth ^{-1}(a+b x) \cosh \left (3 \coth ^{-1}(a+b x)\right )+\coth ^{-1}(a+b x)^2 \cosh \left (3 \coth ^{-1}(a+b x)\right )+3 a^2 \coth ^{-1}(a+b x)^2 \cosh \left (3 \coth ^{-1}(a+b x)\right )+\frac {6 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {18 a^2 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}-\frac {18 a \log \left (\frac {1}{a+b x}\right )}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}-\frac {18 a \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {4 \left (1+3 a^2\right ) \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )}{(a+b x)^3 \left (1-\frac {1}{(a+b x)^2}\right )^{3/2}}-\coth ^{-1}(a+b x)^2 \sinh \left (3 \coth ^{-1}(a+b x)\right )-3 a^2 \coth ^{-1}(a+b x)^2 \sinh \left (3 \coth ^{-1}(a+b x)\right )-2 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right ) \sinh \left (3 \coth ^{-1}(a+b x)\right )-6 a^2 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right ) \sinh \left (3 \coth ^{-1}(a+b x)\right )+6 a \log \left (\frac {1}{a+b x}\right ) \sinh \left (3 \coth ^{-1}(a+b x)\right )+6 a \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right ) \sinh \left (3 \coth ^{-1}(a+b x)\right )\right )}{12 b^3} \]
-1/12*((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*(1 - (a + b*x)^2)*((4*ArcCoth[a + b*x])/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) + (3*ArcCoth[a + b*x]^2)/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) - (12*a*ArcCoth[a + b*x]^2)/((a + b*x)*Sq rt[1 - (a + b*x)^(-2)]) + (9*a^2*ArcCoth[a + b*x]^2)/((a + b*x)*Sqrt[1 - ( a + b*x)^(-2)]) + (-1 + 6*a*ArcCoth[a + b*x] - 3*(-1 + a^2)*ArcCoth[a + b* x]^2)/Sqrt[1 - (a + b*x)^(-2)] + Cosh[3*ArcCoth[a + b*x]] - 6*a*ArcCoth[a + b*x]*Cosh[3*ArcCoth[a + b*x]] + ArcCoth[a + b*x]^2*Cosh[3*ArcCoth[a + b* x]] + 3*a^2*ArcCoth[a + b*x]^2*Cosh[3*ArcCoth[a + b*x]] + (6*ArcCoth[a + b *x]*Log[1 - E^(-2*ArcCoth[a + b*x])])/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) + (18*a^2*ArcCoth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x])])/((a + b*x)*S qrt[1 - (a + b*x)^(-2)]) - (18*a*Log[(a + b*x)^(-1)])/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) - (18*a*Log[1/Sqrt[1 - (a + b*x)^(-2)]])/((a + b*x)*Sqrt[ 1 - (a + b*x)^(-2)]) + (4*(1 + 3*a^2)*PolyLog[2, E^(-2*ArcCoth[a + b*x])]) /((a + b*x)^3*(1 - (a + b*x)^(-2))^(3/2)) - ArcCoth[a + b*x]^2*Sinh[3*ArcC oth[a + b*x]] - 3*a^2*ArcCoth[a + b*x]^2*Sinh[3*ArcCoth[a + b*x]] - 2*ArcC oth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x])]*Sinh[3*ArcCoth[a + b*x]] - 6 *a^2*ArcCoth[a + b*x]*Log[1 - E^(-2*ArcCoth[a + b*x])]*Sinh[3*ArcCoth[a + b*x]] + 6*a*Log[(a + b*x)^(-1)]*Sinh[3*ArcCoth[a + b*x]] + 6*a*Log[1/Sqrt[ 1 - (a + b*x)^(-2)]]*Sinh[3*ArcCoth[a + b*x]]))/b^3
Time = 0.50 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6662, 27, 6481, 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.
| \(\displaystyle \int x^2 \coth ^{-1}(a+b x)^2 \, dx\) |
| \(\Big \downarrow \) 6662 |
| \(\displaystyle \frac {\int x^2 \coth ^{-1}(a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int b^2 x^2 \coth ^{-1}(a+b x)^2d(a+b x)}{b^3}\) |
| \(\Big \downarrow \) 6481 |
| \(\displaystyle \frac {\frac {2}{3} \int \left (-3 a \coth ^{-1}(a+b x)+(a+b x) \coth ^{-1}(a+b x)+\frac {\left (a \left (a^2+3\right )-\left (3 a^2+1\right ) (a+b x)\right ) \coth ^{-1}(a+b x)}{1-(a+b x)^2}\right )d(a+b x)+\frac {1}{3} b^3 x^3 \coth ^{-1}(a+b x)^2}{b^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2}{3} \left (-\frac {1}{2} \left (3 a^2+1\right ) \operatorname {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )+\frac {1}{2} a \left (a^2+3\right ) \coth ^{-1}(a+b x)^2+\frac {1}{2} \left (3 a^2+1\right ) \coth ^{-1}(a+b x)^2-\left (3 a^2+1\right ) \log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)-\frac {1}{2} \text {arctanh}(a+b x)+\frac {1}{2} (a+b x)-\frac {3}{2} a \log \left (1-(a+b x)^2\right )+\frac {1}{2} (a+b x)^2 \coth ^{-1}(a+b x)-3 a (a+b x) \coth ^{-1}(a+b x)\right )+\frac {1}{3} b^3 x^3 \coth ^{-1}(a+b x)^2}{b^3}\) |
((b^3*x^3*ArcCoth[a + b*x]^2)/3 + (2*((a + b*x)/2 - 3*a*(a + b*x)*ArcCoth[ a + b*x] + ((a + b*x)^2*ArcCoth[a + b*x])/2 + (a*(3 + a^2)*ArcCoth[a + b*x ]^2)/2 + ((1 + 3*a^2)*ArcCoth[a + b*x]^2)/2 - ArcTanh[a + b*x]/2 - (1 + 3* a^2)*ArcCoth[a + b*x]*Log[2/(1 - a - b*x)] - (3*a*Log[1 - (a + b*x)^2])/2 - ((1 + 3*a^2)*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/2))/3)/b^3
3.1.70.3.1 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_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcCoth[c*x])^p/(e*(q + 1))), x] - Simp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^(p - 1 ), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
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]
Time = 0.19 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.72
| method | result | size |
| parts | \(\frac {x^{3} \operatorname {arccoth}\left (b x +a \right )^{2}}{3}+\frac {-2 \,\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )+\frac {\left (b x +a \right )^{2} \operatorname {arccoth}\left (b x +a \right )}{3}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{3}}{3}+\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{3}+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{3}}{3}+\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}+\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{3}+\frac {\left (a^{3}+3 a^{2}+3 a +1\right ) \left (\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}\right )}{3}+\frac {b x}{3}+\frac {a}{3}-\frac {\left (6 a -1\right ) \ln \left (b x +a -1\right )}{6}+\frac {\left (-6 a -1\right ) \ln \left (b x +a +1\right )}{6}+\frac {\left (-a^{3}+3 a^{2}-3 a +1\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{3}}{b^{3}}\) | \(350\) |
| derivativedivides | \(\frac {-\frac {\operatorname {arccoth}\left (b x +a \right )^{2} a^{3}}{3}+\operatorname {arccoth}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-\operatorname {arccoth}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \left (b x +a \right )^{3}}{3}-2 \,\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )+\frac {\left (b x +a \right )^{2} \operatorname {arccoth}\left (b x +a \right )}{3}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{3}}{3}+\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{3}+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{3}}{3}+\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}+\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{3}+\frac {b x}{3}+\frac {a}{3}-\frac {\left (6 a -1\right ) \ln \left (b x +a -1\right )}{6}+\frac {\left (-6 a -1\right ) \ln \left (b x +a +1\right )}{6}-\frac {\left (a^{3}-3 a^{2}+3 a -1\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{3}-\frac {\left (-a^{3}-3 a^{2}-3 a -1\right ) \left (\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}\right )}{3}}{b^{3}}\) | \(397\) |
| default | \(\frac {-\frac {\operatorname {arccoth}\left (b x +a \right )^{2} a^{3}}{3}+\operatorname {arccoth}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-\operatorname {arccoth}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \left (b x +a \right )^{3}}{3}-2 \,\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )+\frac {\left (b x +a \right )^{2} \operatorname {arccoth}\left (b x +a \right )}{3}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{3}}{3}+\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{3}+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{3}}{3}+\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}+\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{3}+\frac {b x}{3}+\frac {a}{3}-\frac {\left (6 a -1\right ) \ln \left (b x +a -1\right )}{6}+\frac {\left (-6 a -1\right ) \ln \left (b x +a +1\right )}{6}-\frac {\left (a^{3}-3 a^{2}+3 a -1\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{3}-\frac {\left (-a^{3}-3 a^{2}-3 a -1\right ) \left (\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}\right )}{3}}{b^{3}}\) | \(397\) |
| risch | \(-\frac {1}{3 b^{3}}-\frac {5 \ln \left (b x +a +1\right ) a^{2}}{6 b^{3}}-\frac {\ln \left (b x +a +1\right ) a}{b^{3}}-\frac {\ln \left (b x +a -1\right ) x^{2}}{6 b}-\frac {\ln \left (b x +a -1\right )^{2} a^{2}}{4 b^{3}}+\frac {\ln \left (b x +a -1\right )^{2} a}{4 b^{3}}+\frac {5 \ln \left (b x +a -1\right ) a^{2}}{6 b^{3}}-\frac {\ln \left (b x +a -1\right ) a}{b^{3}}+\frac {\ln \left (b x +a -1\right )^{2} a^{3}}{12 b^{3}}+\frac {x}{3 b^{2}}+\frac {a}{3 b^{3}}-\frac {\ln \left (b x +a +1\right )}{6 b^{3}}+\frac {\left (b^{3} x^{3}+a^{3}+3 a^{2}+3 a +1\right ) \ln \left (b x +a +1\right )^{2}}{12 b^{3}}-\frac {\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) \ln \left (b x +a -1\right ) a^{2}}{b^{3}}-\frac {\ln \left (b x +a -1\right )^{2}}{12 b^{3}}+\frac {\ln \left (b x +a -1\right )}{6 b^{3}}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{3 b^{3}}+\left (-\frac {\ln \left (b x +a -1\right ) x^{3}}{6}-\frac {\ln \left (b x +a -1\right ) a^{3}-b^{2} x^{2}-3 \ln \left (b x +a -1\right ) a^{2}+4 a b x +3 \ln \left (b x +a -1\right ) a -\ln \left (b x +a -1\right )}{6 b^{3}}\right ) \ln \left (b x +a +1\right )-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{3 b^{3}}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a^{2}}{b^{3}}+\frac {2 \ln \left (b x +a -1\right ) x a}{3 b^{2}}+\frac {x^{3} \ln \left (b x +a -1\right )^{2}}{12}\) | \(401\) |
1/3*x^3*arccoth(b*x+a)^2+2/3/b^3*(-3*arccoth(b*x+a)*a*(b*x+a)+1/2*(b*x+a)^ 2*arccoth(b*x+a)-1/2*arccoth(b*x+a)*ln(b*x+a-1)*a^3+3/2*arccoth(b*x+a)*ln( b*x+a-1)*a^2-3/2*arccoth(b*x+a)*ln(b*x+a-1)*a+1/2*arccoth(b*x+a)*ln(b*x+a- 1)+1/2*arccoth(b*x+a)*ln(b*x+a+1)*a^3+3/2*arccoth(b*x+a)*ln(b*x+a+1)*a^2+3 /2*arccoth(b*x+a)*ln(b*x+a+1)*a+1/2*arccoth(b*x+a)*ln(b*x+a+1)+1/2*(a^3+3* a^2+3*a+1)*(1/2*(ln(b*x+a+1)-ln(1/2*b*x+1/2*a+1/2))*ln(-1/2*b*x-1/2*a+1/2) -1/2*dilog(1/2*b*x+1/2*a+1/2)-1/4*ln(b*x+a+1)^2)+1/2*b*x+1/2*a-1/4*(6*a-1) *ln(b*x+a-1)+1/4*(-6*a-1)*ln(b*x+a+1)+1/2*(-a^3+3*a^2-3*a+1)*(1/4*ln(b*x+a -1)^2-1/2*dilog(1/2*b*x+1/2*a+1/2)-1/2*ln(b*x+a-1)*ln(1/2*b*x+1/2*a+1/2)))
\[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\int { x^{2} \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]
\[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\int x^{2} \operatorname {acoth}^{2}{\left (a + b x \right )}\, dx \]
Time = 0.22 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.27 \[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\frac {1}{3} \, x^{3} \operatorname {arcoth}\left (b x + a\right )^{2} - \frac {1}{12} \, b^{2} {\left (\frac {4 \, {\left (3 \, a^{2} + 1\right )} {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )}}{b^{5}} + \frac {2 \, {\left (5 \, a^{2} + 6 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} + \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \, {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \, b x - 2 \, {\left (5 \, a^{2} - 6 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}}\right )} + \frac {1}{3} \, b {\left (\frac {b x^{2} - 4 \, a x}{b^{3}} + \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{4}} - \frac {{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )}{b^{4}}\right )} \operatorname {arcoth}\left (b x + a\right ) \]
1/3*x^3*arccoth(b*x + a)^2 - 1/12*b^2*(4*(3*a^2 + 1)*(log(b*x + a - 1)*log (1/2*b*x + 1/2*a + 1/2) + dilog(-1/2*b*x - 1/2*a + 1/2))/b^5 + 2*(5*a^2 + 6*a + 1)*log(b*x + a + 1)/b^5 + ((a^3 + 3*a^2 + 3*a + 1)*log(b*x + a + 1)^ 2 - 2*(a^3 + 3*a^2 + 3*a + 1)*log(b*x + a + 1)*log(b*x + a - 1) + (a^3 - 3 *a^2 + 3*a - 1)*log(b*x + a - 1)^2 - 4*b*x - 2*(5*a^2 - 6*a + 1)*log(b*x + a - 1))/b^5) + 1/3*b*((b*x^2 - 4*a*x)/b^3 + (a^3 + 3*a^2 + 3*a + 1)*log(b *x + a + 1)/b^4 - (a^3 - 3*a^2 + 3*a - 1)*log(b*x + a - 1)/b^4)*arccoth(b* x + a)
\[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\int { x^{2} \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\int x^2\,{\mathrm {acoth}\left (a+b\,x\right )}^2 \,d x \]