Integrand size = 10, antiderivative size = 136 \[ \int x \coth ^{-1}(a+b x)^2 \, dx=\frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}-\frac {a \coth ^{-1}(a+b x)^2}{b^2}-\frac {\left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2+\frac {2 a \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^2}+\frac {\log \left (1-(a+b x)^2\right )}{2 b^2}+\frac {a \operatorname {PolyLog}\left (2,-\frac {1+a+b x}{1-a-b x}\right )}{b^2} \]
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Time = 0.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6247, 6066, 6022, 266, 6196, 6096, 6132, 6056, 2449, 2352} \[ \int x \coth ^{-1}(a+b x)^2 \, dx=-\frac {\left (a^2+1\right ) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac {a \operatorname {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )}{b^2}+\frac {\log \left (1-(a+b x)^2\right )}{2 b^2}-\frac {a \coth ^{-1}(a+b x)^2}{b^2}+\frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}+\frac {2 a \log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2 \]
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Rule 266
Rule 2352
Rule 2449
Rule 6022
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 ) \coth ^{-1}(x)^2 \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{2} x^2 \coth ^{-1}(a+b x)^2-\text {Subst}\left (\int \left (-\frac {\coth ^{-1}(x)}{b^2}+\frac {\left (1+a^2-2 a x\right ) \coth ^{-1}(x)}{b^2 \left (1-x^2\right )}\right ) \, dx,x,a+b x\right ) \\ & = \frac {1}{2} x^2 \coth ^{-1}(a+b x)^2+\frac {\text {Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x\right )}{b^2}-\frac {\text {Subst}\left (\int \frac {\left (1+a^2-2 a x\right ) \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{b^2} \\ & = \frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2-\frac {\text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,a+b x\right )}{b^2}-\frac {\text {Subst}\left (\int \left (\frac {\left (1+a^2\right ) \coth ^{-1}(x)}{1-x^2}-\frac {2 a x \coth ^{-1}(x)}{1-x^2}\right ) \, dx,x,a+b x\right )}{b^2} \\ & = \frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2+\frac {\log \left (1-(a+b x)^2\right )}{2 b^2}+\frac {(2 a) \text {Subst}\left (\int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{b^2}-\frac {\left (1+a^2\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{b^2} \\ & = \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 {\log \left (1-(a+b x)^2\right )}{2 b^2}+\frac {(2 a) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{1-x} \, dx,x,a+b x\right )}{b^2} \\ & = \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 {(2 a) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,a+b x\right )}{b^2} \\ & = \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 {(2 a) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a-b x}\right )}{b^2} \\ & = \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 (2,1-\frac {2}{1-a-b x}\right )}{b^2} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.78 \[ \int x \coth ^{-1}(a+b x)^2 \, dx=\frac {\left (-1+2 a-a^2+b^2 x^2\right ) \coth ^{-1}(a+b x)^2+2 \coth ^{-1}(a+b x) \left (a+b x+2 a \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )\right )-2 \log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )-2 a \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )}{2 b^2} \]
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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 | \(\frac {x^{2} \operatorname {arccoth}\left (b x +a \right )^{2}}{2}+\frac {\left (b x +a \right ) \operatorname {arccoth}\left (b x +a \right )+\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{2}-\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a^{2}}{2}-\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a -\frac {\operatorname {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2}+\frac {\left (a^{2}-2 a +1\right ) \left (\frac {\ln \left (b x +a -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\ln \left (b x +a -1\right )}{2}+\frac {\ln \left (b x +a +1\right )}{2}+\frac {\left (-a^{2}-2 a -1\right ) \left (\frac {\left (\ln \left (b x +a +1\right )-\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (b x +a +1\right )^{2}}{4}\right )}{2}}{b^{2}}\) | \(269\) |
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\[ \int x \coth ^{-1}(a+b x)^2 \, dx=\int { x \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]
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\[ \int x \coth ^{-1}(a+b x)^2 \, dx=\int x \operatorname {acoth}^{2}{\left (a + b x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.49 \[ \int x \coth ^{-1}(a+b x)^2 \, dx=\frac {1}{2} \, x^{2} \operatorname {arcoth}\left (b x + a\right )^{2} + \frac {1}{8} \, b^{2} {\left (\frac {8 \, {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )} a}{b^{4}} + \frac {4 \, {\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{4}} + \frac {{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \, {\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + {\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \, {\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{4}}\right )} + \frac {1}{2} \, b {\left (\frac {2 \, x}{b^{2}} - \frac {{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{3}} + \frac {{\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{3}}\right )} \operatorname {arcoth}\left (b x + a\right ) \]
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\[ \int x \coth ^{-1}(a+b x)^2 \, dx=\int { x \operatorname {arcoth}\left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int x \coth ^{-1}(a+b x)^2 \, dx=\int x\,{\mathrm {acoth}\left (a+b\,x\right )}^2 \,d x \]
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