Integrand size = 8, antiderivative size = 65 \[ \int x \coth ^{-1}(a+b x) \, dx=\frac {x}{2 b}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)+\frac {(1-a)^2 \log (1-a-b x)}{4 b^2}-\frac {(1+a)^2 \log (1+a+b x)}{4 b^2} \]
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Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6247, 6064, 716, 647, 31} \[ \int x \coth ^{-1}(a+b x) \, dx=\frac {(1-a)^2 \log (-a-b x+1)}{4 b^2}-\frac {(a+1)^2 \log (a+b x+1)}{4 b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)+\frac {x}{2 b} \]
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Rule 31
Rule 647
Rule 716
Rule 6064
Rule 6247
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right ) \coth ^{-1}(x) \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{2} x^2 \coth ^{-1}(a+b x)-\frac {1}{2} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{1-x^2} \, dx,x,a+b x\right ) \\ & = \frac {1}{2} x^2 \coth ^{-1}(a+b x)-\frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{b^2}+\frac {1+a^2-2 a x}{b^2 \left (1-x^2\right )}\right ) \, dx,x,a+b x\right ) \\ & = \frac {x}{2 b}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)-\frac {\text {Subst}\left (\int \frac {1+a^2-2 a x}{1-x^2} \, dx,x,a+b x\right )}{2 b^2} \\ & = \frac {x}{2 b}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)-\frac {(1-a)^2 \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,a+b x\right )}{4 b^2}+\frac {(1+a)^2 \text {Subst}\left (\int \frac {1}{-1-x} \, dx,x,a+b x\right )}{4 b^2} \\ & = \frac {x}{2 b}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)+\frac {(1-a)^2 \log (1-a-b x)}{4 b^2}-\frac {(1+a)^2 \log (1+a+b x)}{4 b^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int x \coth ^{-1}(a+b x) \, dx=\frac {2 b x+2 b^2 x^2 \coth ^{-1}(a+b x)+(-1+a)^2 \log (1-a-b x)-(1+a)^2 \log (1+a+b x)}{4 b^2} \]
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Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97
method | result | size |
parallelrisch | \(-\frac {-\operatorname {arccoth}\left (b x +a \right ) b^{2} x^{2}+\operatorname {arccoth}\left (b x +a \right ) a^{2}+2 \ln \left (b x +a -1\right ) a -b x +2 \,\operatorname {arccoth}\left (b x +a \right ) a +\operatorname {arccoth}\left (b x +a \right )+2 a}{2 b^{2}}\) | \(63\) |
parts | \(\frac {x^{2} \operatorname {arccoth}\left (b x +a \right )}{2}+\frac {b \left (\frac {x}{b^{2}}+\frac {\left (a^{2}-2 a +1\right ) \ln \left (b x +a -1\right )}{2 b^{3}}+\frac {\left (-a^{2}-2 a -1\right ) \ln \left (b x +a +1\right )}{2 b^{3}}\right )}{2}\) | \(64\) |
derivativedivides | \(\frac {\frac {\left (b x +a \right )^{2} \operatorname {arccoth}\left (b x +a \right )}{2}-\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )+\frac {b x}{2}+\frac {a}{2}-\frac {\left (-1+2 a \right ) \ln \left (b x +a -1\right )}{4}+\frac {\left (-2 a -1\right ) \ln \left (b x +a +1\right )}{4}}{b^{2}}\) | \(70\) |
default | \(\frac {\frac {\left (b x +a \right )^{2} \operatorname {arccoth}\left (b x +a \right )}{2}-\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )+\frac {b x}{2}+\frac {a}{2}-\frac {\left (-1+2 a \right ) \ln \left (b x +a -1\right )}{4}+\frac {\left (-2 a -1\right ) \ln \left (b x +a +1\right )}{4}}{b^{2}}\) | \(70\) |
risch | \(\frac {x^{2} \ln \left (b x +a +1\right )}{4}-\frac {x^{2} \ln \left (b x +a -1\right )}{4}-\frac {\ln \left (b x +a +1\right ) a^{2}}{4 b^{2}}+\frac {\ln \left (-b x -a +1\right ) a^{2}}{4 b^{2}}-\frac {\ln \left (b x +a +1\right ) a}{2 b^{2}}-\frac {\ln \left (-b x -a +1\right ) a}{2 b^{2}}+\frac {x}{2 b}-\frac {\ln \left (b x +a +1\right )}{4 b^{2}}+\frac {\ln \left (-b x -a +1\right )}{4 b^{2}}\) | \(121\) |
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Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.02 \[ \int x \coth ^{-1}(a+b x) \, dx=\frac {b^{2} x^{2} \log \left (\frac {b x + a + 1}{b x + a - 1}\right ) + 2 \, b x - {\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right ) + {\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{4 \, b^{2}} \]
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Time = 0.32 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.17 \[ \int x \coth ^{-1}(a+b x) \, dx=\begin {cases} - \frac {a^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 b^{2}} - \frac {a \log {\left (a + b x + 1 \right )}}{b^{2}} + \frac {a \operatorname {acoth}{\left (a + b x \right )}}{b^{2}} + \frac {x^{2} \operatorname {acoth}{\left (a + b x \right )}}{2} + \frac {x}{2 b} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {acoth}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int x \coth ^{-1}(a+b x) \, dx=\frac {1}{2} \, x^{2} \operatorname {arcoth}\left (b x + a\right ) + \frac {1}{4} \, b {\left (\frac {2 \, x}{b^{2}} - \frac {{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{3}} + \frac {{\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{3}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (55) = 110\).
Time = 0.28 (sec) , antiderivative size = 259, normalized size of antiderivative = 3.98 \[ \int x \coth ^{-1}(a+b x) \, dx=-\frac {1}{2} \, {\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {a \log \left (\frac {{\left | b x + a + 1 \right |}}{{\left | b x + a - 1 \right |}}\right )}{b^{3}} - \frac {a \log \left ({\left | \frac {b x + a + 1}{b x + a - 1} - 1 \right |}\right )}{b^{3}} + \frac {{\left (\frac {{\left (b x + a + 1\right )} a}{b x + a - 1} - a - \frac {b x + a + 1}{b x + a - 1}\right )} \log \left (-\frac {\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} + 1}{\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} - 1}\right )}{b^{3} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}^{2}} - \frac {1}{b^{3} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}}\right )} \]
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Time = 4.96 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95 \[ \int x \coth ^{-1}(a+b x) \, dx=\frac {x^2\,\mathrm {acoth}\left (a+b\,x\right )}{2}-\frac {\frac {\mathrm {acoth}\left (a+b\,x\right )}{2}-\frac {b\,x}{2}+\frac {a^2\,\mathrm {acoth}\left (a+b\,x\right )}{2}+\frac {a\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )}{2}}{b^2} \]
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