Integrand size = 12, antiderivative size = 263 \[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac {a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac {a \left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac {\left (1+6 a^2+a^4\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac {a \text {arctanh}(a+b x)}{b^4}+\frac {2 a \left (1+a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^4}+\frac {\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac {\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}+\frac {a \left (1+a^2\right ) \operatorname {PolyLog}\left (2,-\frac {1+a+b x}{1-a-b x}\right )}{b^4} \]
-a*x/b^3+1/12*(b*x+a)^2/b^4+1/2*(6*a^2+1)*(b*x+a)*arccoth(b*x+a)/b^4-a*(b* x+a)^2*arccoth(b*x+a)/b^4+1/6*(b*x+a)^3*arccoth(b*x+a)/b^4-a*(a^2+1)*arcco th(b*x+a)^2/b^4-1/4*(a^4+6*a^2+1)*arccoth(b*x+a)^2/b^4+1/4*x^4*arccoth(b*x +a)^2+a*arctanh(b*x+a)/b^4+2*a*(a^2+1)*arccoth(b*x+a)*ln(2/(-b*x-a+1))/b^4 +1/12*ln(1-(b*x+a)^2)/b^4+1/4*(6*a^2+1)*ln(1-(b*x+a)^2)/b^4+a*(a^2+1)*poly log(2,(-b*x-a-1)/(-b*x-a+1))/b^4
Time = 1.49 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.80 \[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=-\frac {1+11 a^2+10 a b x-b^2 x^2+3 \left (1-4 a+6 a^2-4 a^3+a^4-b^4 x^4\right ) \coth ^{-1}(a+b x)^2-2 \coth ^{-1}(a+b x) \left (9 a+13 a^3+3 b x+9 a^2 b x-3 a b^2 x^2+b^3 x^3+12 \left (a+a^3\right ) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )\right )+8 \log \left (\frac {1}{a+b x}\right )+36 a^2 \log \left (\frac {1}{a+b x}\right )+8 \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )+36 a^2 \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )+12 \left (a+a^3\right ) \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )}{12 b^4} \]
-1/12*(1 + 11*a^2 + 10*a*b*x - b^2*x^2 + 3*(1 - 4*a + 6*a^2 - 4*a^3 + a^4 - b^4*x^4)*ArcCoth[a + b*x]^2 - 2*ArcCoth[a + b*x]*(9*a + 13*a^3 + 3*b*x + 9*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3 + 12*(a + a^3)*Log[1 - E^(-2*ArcCoth[a + b*x])]) + 8*Log[(a + b*x)^(-1)] + 36*a^2*Log[(a + b*x)^(-1)] + 8*Log[1/S qrt[1 - (a + b*x)^(-2)]] + 36*a^2*Log[1/Sqrt[1 - (a + b*x)^(-2)]] + 12*(a + a^3)*PolyLog[2, E^(-2*ArcCoth[a + b*x])])/b^4
Time = 0.57 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6662, 25, 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.
| \(\displaystyle \int x^3 \coth ^{-1}(a+b x)^2 \, dx\) |
| \(\Big \downarrow \) 6662 |
| \(\displaystyle \frac {\int x^3 \coth ^{-1}(a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -x^3 \coth ^{-1}(a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -b^3 x^3 \coth ^{-1}(a+b x)^2d(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 6481 |
| \(\displaystyle -\frac {\frac {1}{2} \int \left (-\coth ^{-1}(a+b x) (a+b x)^2+4 a \coth ^{-1}(a+b x) (a+b x)-\left (6 a^2+1\right ) \coth ^{-1}(a+b x)+\frac {\left (a^4+6 a^2-4 \left (a^2+1\right ) (a+b x) a+1\right ) \coth ^{-1}(a+b x)}{1-(a+b x)^2}\right )d(a+b x)-\frac {1}{4} b^4 x^4 \coth ^{-1}(a+b x)^2}{b^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{2} \left (-2 a \left (a^2+1\right ) \operatorname {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )-\frac {1}{2} \left (6 a^2+1\right ) \log \left (1-(a+b x)^2\right )-\left (6 a^2+1\right ) (a+b x) \coth ^{-1}(a+b x)+2 a \left (a^2+1\right ) \coth ^{-1}(a+b x)^2-4 a \left (a^2+1\right ) \log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)+\frac {1}{2} \left (a^4+6 a^2+1\right ) \coth ^{-1}(a+b x)^2-2 a \text {arctanh}(a+b x)-\frac {1}{6} (a+b x)^2+2 a (a+b x)-\frac {1}{6} \log \left (1-(a+b x)^2\right )-\frac {1}{3} (a+b x)^3 \coth ^{-1}(a+b x)+2 a (a+b x)^2 \coth ^{-1}(a+b x)\right )-\frac {1}{4} b^4 x^4 \coth ^{-1}(a+b x)^2}{b^4}\) |
-((-1/4*(b^4*x^4*ArcCoth[a + b*x]^2) + (2*a*(a + b*x) - (a + b*x)^2/6 - (1 + 6*a^2)*(a + b*x)*ArcCoth[a + b*x] + 2*a*(a + b*x)^2*ArcCoth[a + b*x] - ((a + b*x)^3*ArcCoth[a + b*x])/3 + 2*a*(1 + a^2)*ArcCoth[a + b*x]^2 + ((1 + 6*a^2 + a^4)*ArcCoth[a + b*x]^2)/2 - 2*a*ArcTanh[a + b*x] - 4*a*(1 + a^2 )*ArcCoth[a + b*x]*Log[2/(1 - a - b*x)] - Log[1 - (a + b*x)^2]/6 - ((1 + 6 *a^2)*Log[1 - (a + b*x)^2])/2 - 2*a*(1 + a^2)*PolyLog[2, -((1 + a + b*x)/( 1 - a - b*x))])/2)/b^4)
3.1.69.3.1 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 = 0.24 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.70
| method | result | size |
| parts | \(\frac {x^{4} \operatorname {arccoth}\left (b x +a \right )^{2}}{4}+\frac {6 \,\operatorname {arccoth}\left (b x +a \right ) a^{2} \left (b x +a \right )-2 \,\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{4}}{2}-2 \,\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}-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 )}{2}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{4}}{2}-2 \,\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}-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 )}{2}-2 \left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{6}+\frac {\left (18 a^{2}-6 a +4\right ) \ln \left (b x +a -1\right )}{6}-\frac {\left (-18 a^{2}-6 a -4\right ) \ln \left (b x +a +1\right )}{6}+\frac {\left (3 a^{4}-12 a^{3}+18 a^{2}-12 a +3\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 )}{6}+\frac {\left (-3 a^{4}-12 a^{3}-18 a^{2}-12 a -3\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 )}{6}}{2 b^{4}}\) | \(448\) |
| derivativedivides | \(\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) a^{2} \left (b x +a \right )+\frac {3 \operatorname {arccoth}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )^{2}-\operatorname {arccoth}\left (b x +a \right )^{2} a \left (b x +a \right )^{3}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{3}-\operatorname {arccoth}\left (b x +a \right )^{2} a^{3} \left (b x +a \right )+\frac {\left (b x +a \right )^{2}}{12}+\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{6}-\left (b x +a \right ) a -\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 ) a^{4}}{4}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{3}+\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}}{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 ) a^{4}}{4}-\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}}{2}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{4}+\frac {\left (-3 a^{4}-12 a^{3}-18 a^{2}-12 a -3\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 )}{12}+\frac {\left (3 a^{4}-12 a^{3}+18 a^{2}-12 a +3\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 )}{12}+\frac {\left (18 a^{2}-6 a +4\right ) \ln \left (b x +a -1\right )}{12}-\frac {\left (-18 a^{2}-6 a -4\right ) \ln \left (b x +a +1\right )}{12}+\frac {\operatorname {arccoth}\left (b x +a \right )^{2} a^{4}}{4}+\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{4}+\frac {\left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )}{2}}{b^{4}}\) | \(520\) |
| default | \(\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) a^{2} \left (b x +a \right )+\frac {3 \operatorname {arccoth}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )^{2}-\operatorname {arccoth}\left (b x +a \right )^{2} a \left (b x +a \right )^{3}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{3}-\operatorname {arccoth}\left (b x +a \right )^{2} a^{3} \left (b x +a \right )+\frac {\left (b x +a \right )^{2}}{12}+\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{6}-\left (b x +a \right ) a -\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 ) a^{4}}{4}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{3}+\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}}{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 ) a^{4}}{4}-\frac {3 \,\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}}{2}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{4}+\frac {\left (-3 a^{4}-12 a^{3}-18 a^{2}-12 a -3\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 )}{12}+\frac {\left (3 a^{4}-12 a^{3}+18 a^{2}-12 a +3\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 )}{12}+\frac {\left (18 a^{2}-6 a +4\right ) \ln \left (b x +a -1\right )}{12}-\frac {\left (-18 a^{2}-6 a -4\right ) \ln \left (b x +a +1\right )}{12}+\frac {\operatorname {arccoth}\left (b x +a \right )^{2} a^{4}}{4}+\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{4}+\frac {\left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )}{2}}{b^{4}}\) | \(520\) |
| risch | \(-\frac {1}{12 b^{4}}-\frac {5 a x}{6 b^{3}}-\frac {\ln \left (b x +a -1\right ) x}{4 b^{3}}+\frac {\ln \left (b x +a -1\right )^{2} a^{3}}{4 b^{4}}-\frac {3 \ln \left (b x +a -1\right )^{2} a^{2}}{8 b^{4}}+\frac {\ln \left (b x +a -1\right )^{2} a}{4 b^{4}}-\frac {\ln \left (b x +a -1\right )^{2} a^{4}}{16 b^{4}}-\frac {\ln \left (b x +a -1\right ) x^{3}}{12 b}-\frac {\left (-b^{4} x^{4}+a^{4}+4 a^{3}+6 a^{2}+4 a +1\right ) \ln \left (b x +a +1\right )^{2}}{16 b^{4}}+\frac {a}{b^{4}}-\frac {11 a^{2}}{12 b^{4}}+\frac {x^{2}}{12 b^{2}}+\frac {\ln \left (b x +a +1\right )}{3 b^{4}}+\frac {\ln \left (b x +a -1\right )^{2} x^{4}}{16}-\frac {\ln \left (b x +a -1\right )^{2}}{16 b^{4}}+\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a^{3}}{b^{4}}+\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a}{b^{4}}+\frac {3 \ln \left (b x +a +1\right ) a^{2}}{2 b^{4}}+\frac {3 \ln \left (b x +a +1\right ) a}{4 b^{4}}+\frac {13 \ln \left (b x +a +1\right ) a^{3}}{12 b^{4}}-\frac {13 \ln \left (b x +a -1\right ) a^{3}}{12 b^{4}}+\frac {3 \ln \left (b x +a -1\right ) a^{2}}{2 b^{4}}-\frac {3 \ln \left (b x +a -1\right ) a}{4 b^{4}}+\left (-\frac {x^{4} \ln \left (b x +a -1\right )}{8}-\frac {-2 b^{3} x^{3}-3 a^{4} \ln \left (b x +a -1\right )+6 a \,b^{2} x^{2}+12 \ln \left (b x +a -1\right ) a^{3}-18 a^{2} b x -18 \ln \left (b x +a -1\right ) a^{2}+12 \ln \left (b x +a -1\right ) a -6 b x -3 \ln \left (b x +a -1\right )}{24 b^{4}}\right ) \ln \left (b x +a +1\right )+\frac {\ln \left (b x +a -1\right )}{3 b^{4}}-\frac {3 \ln \left (b x +a -1\right ) x \,a^{2}}{4 b^{3}}+\frac {\ln \left (b x +a -1\right ) x^{2} a}{4 b^{2}}+\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a}{b^{4}}+\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a^{3}}{b^{4}}\) | \(527\) |
1/4*x^4*arccoth(b*x+a)^2+1/2/b^4*(6*arccoth(b*x+a)*a^2*(b*x+a)-2*arccoth(b *x+a)*a*(b*x+a)^2+1/3*arccoth(b*x+a)*(b*x+a)^3+(b*x+a)*arccoth(b*x+a)+1/2* arccoth(b*x+a)*ln(b*x+a-1)*a^4-2*arccoth(b*x+a)*ln(b*x+a-1)*a^3+3*arccoth( b*x+a)*ln(b*x+a-1)*a^2-2*arccoth(b*x+a)*ln(b*x+a-1)*a+1/2*arccoth(b*x+a)*l n(b*x+a-1)-1/2*arccoth(b*x+a)*ln(b*x+a+1)*a^4-2*arccoth(b*x+a)*ln(b*x+a+1) *a^3-3*arccoth(b*x+a)*ln(b*x+a+1)*a^2-2*arccoth(b*x+a)*ln(b*x+a+1)*a-1/2*a rccoth(b*x+a)*ln(b*x+a+1)-2*(b*x+a)*a+1/6*(b*x+a)^2+1/6*(18*a^2-6*a+4)*ln( b*x+a-1)-1/6*(-18*a^2-6*a-4)*ln(b*x+a+1)+1/6*(3*a^4-12*a^3+18*a^2-12*a+3)* (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))+1/6*(-3*a^4-12*a^3-18*a^2-12*a-3)*(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))
\[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]
\[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=\int x^{3} \operatorname {acoth}^{2}{\left (a + b x \right )}\, dx \]
Time = 0.19 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.22 \[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=\frac {1}{4} \, x^{4} \operatorname {arcoth}\left (b x + a\right )^{2} + \frac {1}{48} \, b^{2} {\left (\frac {48 \, {\left (a^{3} + a\right )} {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )}}{b^{6}} + \frac {4 \, {\left (13 \, a^{3} + 18 \, a^{2} + 9 \, a + 4\right )} \log \left (b x + a + 1\right )}{b^{6}} + \frac {4 \, b^{2} x^{2} - 40 \, a b x + 3 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 6 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + 3 \, {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \, {\left (13 \, a^{3} - 18 \, a^{2} + 9 \, a - 4\right )} \log \left (b x + a - 1\right )}{b^{6}}\right )} + \frac {1}{12} \, b {\left (\frac {2 \, {\left (b^{2} x^{3} - 3 \, a b x^{2} + 3 \, {\left (3 \, a^{2} + 1\right )} x\right )}}{b^{4}} - \frac {3 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} + \frac {3 \, {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}}\right )} \operatorname {arcoth}\left (b x + a\right ) \]
1/4*x^4*arccoth(b*x + a)^2 + 1/48*b^2*(48*(a^3 + a)*(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^6 + 4*(13*a^3 + 18*a^2 + 9*a + 4)*log(b*x + a + 1)/b^6 + (4*b^2*x^2 - 40*a*b*x + 3*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1)^2 - 6*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1)*log(b*x + a - 1) + 3*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1 )*log(b*x + a - 1)^2 - 4*(13*a^3 - 18*a^2 + 9*a - 4)*log(b*x + a - 1))/b^6 ) + 1/12*b*(2*(b^2*x^3 - 3*a*b*x^2 + 3*(3*a^2 + 1)*x)/b^4 - 3*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1)/b^5 + 3*(a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*log(b*x + a - 1)/b^5)*arccoth(b*x + a)
\[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x^3 \coth ^{-1}(a+b x)^2 \, dx=\int x^3\,{\mathrm {acoth}\left (a+b\,x\right )}^2 \,d x \]