Integrand size = 10, antiderivative size = 136 \[ \int x \coth ^{-1}(a+b x)^2 \, dx=\frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}-\frac {a \coth ^{-1}(a+b x)^2}{b^2}-\frac {\left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2+\frac {2 a \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^2}+\frac {\log \left (1-(a+b x)^2\right )}{2 b^2}+\frac {a \operatorname {PolyLog}\left (2,-\frac {1+a+b x}{1-a-b x}\right )}{b^2} \]
(b*x+a)*arccoth(b*x+a)/b^2-a*arccoth(b*x+a)^2/b^2-1/2*(a^2+1)*arccoth(b*x+ a)^2/b^2+1/2*x^2*arccoth(b*x+a)^2+2*a*arccoth(b*x+a)*ln(2/(-b*x-a+1))/b^2+ 1/2*ln(1-(b*x+a)^2)/b^2+a*polylog(2,(-b*x-a-1)/(-b*x-a+1))/b^2
Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.78 \[ \int x \coth ^{-1}(a+b x)^2 \, dx=\frac {\left (-1+2 a-a^2+b^2 x^2\right ) \coth ^{-1}(a+b x)^2+2 \coth ^{-1}(a+b x) \left (a+b x+2 a \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )\right )-2 \log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )-2 a \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )}{2 b^2} \]
((-1 + 2*a - a^2 + b^2*x^2)*ArcCoth[a + b*x]^2 + 2*ArcCoth[a + b*x]*(a + b *x + 2*a*Log[1 - E^(-2*ArcCoth[a + b*x])]) - 2*Log[1/((a + b*x)*Sqrt[1 - ( a + b*x)^(-2)])] - 2*a*PolyLog[2, E^(-2*ArcCoth[a + b*x])])/(2*b^2)
Time = 0.44 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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.
| \(\displaystyle \int x \coth ^{-1}(a+b x)^2 \, dx\) |
| \(\Big \downarrow \) 6662 |
| \(\displaystyle \frac {\int x \coth ^{-1}(a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -x \coth ^{-1}(a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -b x \coth ^{-1}(a+b x)^2d(a+b x)}{b^2}\) |
| \(\Big \downarrow \) 6481 |
| \(\displaystyle -\frac {\int \left (\frac {\left (a^2-2 (a+b x) a+1\right ) \coth ^{-1}(a+b x)}{1-(a+b x)^2}-\coth ^{-1}(a+b x)\right )d(a+b x)-\frac {1}{2} b^2 x^2 \coth ^{-1}(a+b x)^2}{b^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{2} \left (a^2+1\right ) \coth ^{-1}(a+b x)^2-\frac {1}{2} b^2 x^2 \coth ^{-1}(a+b x)^2-a \operatorname {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )-\frac {1}{2} \log \left (1-(a+b x)^2\right )+a \coth ^{-1}(a+b x)^2-(a+b x) \coth ^{-1}(a+b x)-2 a \log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{b^2}\) |
-((-((a + b*x)*ArcCoth[a + b*x]) + a*ArcCoth[a + b*x]^2 + ((1 + a^2)*ArcCo th[a + b*x]^2)/2 - (b^2*x^2*ArcCoth[a + b*x]^2)/2 - 2*a*ArcCoth[a + b*x]*L og[2/(1 - a - b*x)] - Log[1 - (a + b*x)^2]/2 - a*PolyLog[2, -((1 + a + b*x )/(1 - a - b*x))])/b^2)
3.1.71.3.1 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 = 0.17 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.79
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccoth}\left (b x +a \right )^{2} a \left (b x +a \right )+\left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{2}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a -\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2}+\frac {\ln \left (b x +a -1\right )}{2}+\frac {\ln \left (b x +a +1\right )}{2}+\frac {\left (-2 a +1\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (-2 a -1\right ) \left (\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}\right )}{2}}{b^{2}}\) | \(244\) |
| default | \(\frac {\frac {\operatorname {arccoth}\left (b x +a \right )^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccoth}\left (b x +a \right )^{2} a \left (b x +a \right )+\left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{2}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a -\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2}+\frac {\ln \left (b x +a -1\right )}{2}+\frac {\ln \left (b x +a +1\right )}{2}+\frac {\left (-2 a +1\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (-2 a -1\right ) \left (\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}\right )}{2}}{b^{2}}\) | \(244\) |
| risch | \(\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a}{b^{2}}+\frac {\ln \left (b x +a +1\right ) a}{2 b^{2}}-\frac {\ln \left (b x +a -1\right ) x}{2 b}-\frac {\ln \left (b x +a -1\right )^{2} a^{2}}{8 b^{2}}+\frac {\ln \left (b x +a -1\right )^{2} a}{4 b^{2}}-\frac {\left (-b^{2} x^{2}+a^{2}+2 a +1\right ) \ln \left (b x +a +1\right )^{2}}{8 b^{2}}+\frac {\ln \left (b x +a +1\right )}{2 b^{2}}-\frac {\ln \left (b x +a -1\right )^{2}}{8 b^{2}}+\left (-\frac {x^{2} \ln \left (b x +a -1\right )}{4}+\frac {\ln \left (b x +a -1\right ) a^{2}-2 \ln \left (b x +a -1\right ) a +2 b x +\ln \left (b x +a -1\right )}{4 b^{2}}\right ) \ln \left (b x +a +1\right )+\frac {\ln \left (b x +a -1\right )}{2 b^{2}}+\frac {x^{2} \ln \left (b x +a -1\right )^{2}}{8}+\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a}{b^{2}}-\frac {\ln \left (b x +a -1\right ) a}{2 b^{2}}\) | \(251\) |
| parts | \(\frac {x^{2} \operatorname {arccoth}\left (b x +a \right )^{2}}{2}+\frac {\left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{2}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a -\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2}+\frac {\left (a^{2}-2 a +1\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\ln \left (b x +a -1\right )}{2}+\frac {\ln \left (b x +a +1\right )}{2}+\frac {\left (-a^{2}-2 a -1\right ) \left (\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}\right )}{2}}{b^{2}}\) | \(269\) |
1/b^2*(1/2*arccoth(b*x+a)^2*(b*x+a)^2-arccoth(b*x+a)^2*a*(b*x+a)+(b*x+a)*a rccoth(b*x+a)-arccoth(b*x+a)*ln(b*x+a-1)*a+1/2*arccoth(b*x+a)*ln(b*x+a-1)- arccoth(b*x+a)*ln(b*x+a+1)*a-1/2*arccoth(b*x+a)*ln(b*x+a+1)+1/2*ln(b*x+a-1 )+1/2*ln(b*x+a+1)+1/2*(-2*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))+1/2*(-2*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))
\[ \int x \coth ^{-1}(a+b x)^2 \, dx=\int { x \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]
\[ \int x \coth ^{-1}(a+b x)^2 \, dx=\int x \operatorname {acoth}^{2}{\left (a + b x \right )}\, dx \]
Time = 0.21 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.49 \[ \int x \coth ^{-1}(a+b x)^2 \, dx=\frac {1}{2} \, x^{2} \operatorname {arcoth}\left (b x + a\right )^{2} + \frac {1}{8} \, b^{2} {\left (\frac {8 \, {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )} a}{b^{4}} + \frac {4 \, {\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{4}} + \frac {{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \, {\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + {\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \, {\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{4}}\right )} + \frac {1}{2} \, b {\left (\frac {2 \, x}{b^{2}} - \frac {{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{3}} + \frac {{\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{3}}\right )} \operatorname {arcoth}\left (b x + a\right ) \]
1/2*x^2*arccoth(b*x + a)^2 + 1/8*b^2*(8*(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))*a/b^4 + 4*(a + 1)*log(b*x + a + 1)/b^4 + ((a^2 + 2*a + 1)*log(b*x + a + 1)^2 - 2*(a^2 + 2*a + 1)*log(b*x + a + 1)*log(b*x + a - 1) + (a^2 - 2*a + 1)*log(b*x + a - 1)^2 - 4*(a - 1 )*log(b*x + a - 1))/b^4) + 1/2*b*(2*x/b^2 - (a^2 + 2*a + 1)*log(b*x + a + 1)/b^3 + (a^2 - 2*a + 1)*log(b*x + a - 1)/b^3)*arccoth(b*x + a)
\[ \int x \coth ^{-1}(a+b x)^2 \, dx=\int { x \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x \coth ^{-1}(a+b x)^2 \, dx=\int x\,{\mathrm {acoth}\left (a+b\,x\right )}^2 \,d x \]