Integrand size = 12, antiderivative size = 204 \[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\frac {x}{3 b^2}-\frac {2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}+\frac {a \left (3+a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {\left (1+3 a^2\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {\text {arctanh}(a+b x)}{3 b^3}-\frac {2 \left (1+3 a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{3 b^3}-\frac {a \log \left (1-(a+b x)^2\right )}{b^3}-\frac {\left (1+3 a^2\right ) \operatorname {PolyLog}\left (2,-\frac {1+a+b x}{1-a-b x}\right )}{3 b^3} \]
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Time = 0.23 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6247, 6066, 6022, 266, 6038, 327, 212, 6196, 6096, 6132, 6056, 2449, 2352} \[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=-\frac {\left (3 a^2+1\right ) \operatorname {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )}{3 b^3}+\frac {a \left (a^2+3\right ) \coth ^{-1}(a+b x)^2}{3 b^3}+\frac {\left (3 a^2+1\right ) \coth ^{-1}(a+b x)^2}{3 b^3}-\frac {2 \left (3 a^2+1\right ) \log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{3 b^3}-\frac {\text {arctanh}(a+b x)}{3 b^3}-\frac {a \log \left (1-(a+b x)^2\right )}{b^3}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{3 b^3}-\frac {2 a (a+b x) \coth ^{-1}(a+b x)}{b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)^2+\frac {x}{3 b^2} \]
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Rule 212
Rule 266
Rule 327
Rule 2352
Rule 2449
Rule 6022
Rule 6038
Rule 6056
Rule 6066
Rule 6096
Rule 6132
Rule 6196
Rule 6247
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \coth ^{-1}(x)^2 \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {2}{3} \text {Subst}\left (\int \left (\frac {3 a \coth ^{-1}(x)}{b^3}-\frac {x \coth ^{-1}(x)}{b^3}-\frac {\left (a \left (3+a^2\right )-\left (1+3 a^2\right ) x\right ) \coth ^{-1}(x)}{b^3 \left (1-x^2\right )}\right ) \, dx,x,a+b x\right ) \\ & = \frac {1}{3} x^3 \coth ^{-1}(a+b x)^2+\frac {2 \text {Subst}\left (\int x \coth ^{-1}(x) \, dx,x,a+b x\right )}{3 b^3}+\frac {2 \text {Subst}\left (\int \frac {\left (a \left (3+a^2\right )-\left (1+3 a^2\right ) x\right ) \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{3 b^3}-\frac {(2 a) \text {Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x\right )}{b^3} \\ & = -\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 {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {\text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,a+b x\right )}{3 b^3}+\frac {2 \text {Subst}\left (\int \left (\frac {a \left (3+a^2\right ) \coth ^{-1}(x)}{1-x^2}-\frac {\left (1+3 a^2\right ) x \coth ^{-1}(x)}{1-x^2}\right ) \, dx,x,a+b x\right )}{3 b^3}+\frac {(2 a) \text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,a+b x\right )}{b^3} \\ & = \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 {1}{3} x^3 \coth ^{-1}(a+b x)^2-\frac {a \log \left (1-(a+b x)^2\right )}{b^3}-\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,a+b x\right )}{3 b^3}+\frac {\left (2 a \left (3+a^2\right )\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{3 b^3}-\frac {\left (2 \left (1+3 a^2\right )\right ) \text {Subst}\left (\int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{3 b^3} \\ & = \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 {a \log \left (1-(a+b x)^2\right )}{b^3}-\frac {\left (2 \left (1+3 a^2\right )\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{1-x} \, dx,x,a+b x\right )}{3 b^3} \\ & = \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 (2 \left (1+3 a^2\right )\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,a+b x\right )}{3 b^3} \\ & = \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 (2 \left (1+3 a^2\right )\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a-b x}\right )}{3 b^3} \\ & = \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 (2,1-\frac {2}{1-a-b x}\right )}{3 b^3} \\ \end{align*}
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 \[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=-\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \left (1-(a+b x)^2\right ) \left (\frac {4 \coth ^{-1}(a+b x)}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {3 \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}-\frac {12 a \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {9 a^2 \coth ^{-1}(a+b x)^2}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {-1+6 a \coth ^{-1}(a+b x)-3 \left (-1+a^2\right ) \coth ^{-1}(a+b x)^2}{\sqrt {1-\frac {1}{(a+b x)^2}}}+\cosh \left (3 \coth ^{-1}(a+b x)\right )-6 a \coth ^{-1}(a+b x) \cosh \left (3 \coth ^{-1}(a+b x)\right )+\coth ^{-1}(a+b x)^2 \cosh \left (3 \coth ^{-1}(a+b x)\right )+3 a^2 \coth ^{-1}(a+b x)^2 \cosh \left (3 \coth ^{-1}(a+b x)\right )+\frac {6 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {18 a^2 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}-\frac {18 a \log \left (\frac {1}{a+b x}\right )}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}-\frac {18 a \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}+\frac {4 \left (1+3 a^2\right ) \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )}{(a+b x)^3 \left (1-\frac {1}{(a+b x)^2}\right )^{3/2}}-\coth ^{-1}(a+b x)^2 \sinh \left (3 \coth ^{-1}(a+b x)\right )-3 a^2 \coth ^{-1}(a+b x)^2 \sinh \left (3 \coth ^{-1}(a+b x)\right )-2 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right ) \sinh \left (3 \coth ^{-1}(a+b x)\right )-6 a^2 \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right ) \sinh \left (3 \coth ^{-1}(a+b x)\right )+6 a \log \left (\frac {1}{a+b x}\right ) \sinh \left (3 \coth ^{-1}(a+b x)\right )+6 a \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right ) \sinh \left (3 \coth ^{-1}(a+b x)\right )\right )}{12 b^3} \]
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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 | \(-\frac {1}{3 b^{3}}-\frac {5 \ln \left (b x +a +1\right ) a^{2}}{6 b^{3}}-\frac {\ln \left (b x +a +1\right ) a}{b^{3}}-\frac {\ln \left (b x +a -1\right ) x^{2}}{6 b}-\frac {\ln \left (b x +a -1\right )^{2} a^{2}}{4 b^{3}}+\frac {\ln \left (b x +a -1\right )^{2} a}{4 b^{3}}+\frac {5 \ln \left (b x +a -1\right ) a^{2}}{6 b^{3}}-\frac {\ln \left (b x +a -1\right ) a}{b^{3}}+\frac {\ln \left (b x +a -1\right )^{2} a^{3}}{12 b^{3}}+\frac {x}{3 b^{2}}+\frac {a}{3 b^{3}}-\frac {\ln \left (b x +a +1\right )}{6 b^{3}}+\frac {\left (b^{3} x^{3}+a^{3}+3 a^{2}+3 a +1\right ) \ln \left (b x +a +1\right )^{2}}{12 b^{3}}-\frac {\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) \ln \left (b x +a -1\right ) a^{2}}{b^{3}}-\frac {\ln \left (b x +a -1\right )^{2}}{12 b^{3}}+\frac {\ln \left (b x +a -1\right )}{6 b^{3}}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{3 b^{3}}+\left (-\frac {\ln \left (b x +a -1\right ) x^{3}}{6}-\frac {\ln \left (b x +a -1\right ) a^{3}-b^{2} x^{2}-3 \ln \left (b x +a -1\right ) a^{2}+4 a b x +3 \ln \left (b x +a -1\right ) a -\ln \left (b x +a -1\right )}{6 b^{3}}\right ) \ln \left (b x +a +1\right )-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{3 b^{3}}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a^{2}}{b^{3}}+\frac {2 \ln \left (b x +a -1\right ) x a}{3 b^{2}}+\frac {x^{3} \ln \left (b x +a -1\right )^{2}}{12}\) | \(401\) |
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\[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\int { x^{2} \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]
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\[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\int x^{2} \operatorname {acoth}^{2}{\left (a + b x \right )}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.27 \[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\frac {1}{3} \, x^{3} \operatorname {arcoth}\left (b x + a\right )^{2} - \frac {1}{12} \, b^{2} {\left (\frac {4 \, {\left (3 \, a^{2} + 1\right )} {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )}}{b^{5}} + \frac {2 \, {\left (5 \, a^{2} + 6 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} + \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \, {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \, b x - 2 \, {\left (5 \, a^{2} - 6 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}}\right )} + \frac {1}{3} \, b {\left (\frac {b x^{2} - 4 \, a x}{b^{3}} + \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{4}} - \frac {{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )}{b^{4}}\right )} \operatorname {arcoth}\left (b x + a\right ) \]
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\[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\int { x^{2} \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int x^2 \coth ^{-1}(a+b x)^2 \, dx=\int x^2\,{\mathrm {acoth}\left (a+b\,x\right )}^2 \,d x \]
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