Integrand size = 12, antiderivative size = 39 \[ \int (a+b x) \coth ^{-1}(a+b x) \, dx=\frac {x}{2}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{2 b}-\frac {\text {arctanh}(a+b x)}{2 b} \]
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Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6243, 6038, 327, 212} \[ \int (a+b x) \coth ^{-1}(a+b x) \, dx=-\frac {\text {arctanh}(a+b x)}{2 b}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{2 b}+\frac {x}{2} \]
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Rule 212
Rule 327
Rule 6038
Rule 6243
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x \coth ^{-1}(x) \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x)^2 \coth ^{-1}(a+b x)}{2 b}-\frac {\text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,a+b x\right )}{2 b} \\ & = \frac {x}{2}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{2 b}-\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,a+b x\right )}{2 b} \\ & = \frac {x}{2}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{2 b}-\frac {\text {arctanh}(a+b x)}{2 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.69 \[ \int (a+b x) \coth ^{-1}(a+b x) \, dx=\frac {2 b x+2 b x (2 a+b x) \coth ^{-1}(a+b x)-\left (-1+a^2\right ) \log (1-a-b x)-\log (1+a+b x)+a^2 \log (1+a+b x)}{4 b} \]
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Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {\frac {\left (b x +a \right )^{2} \operatorname {arccoth}\left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}+\frac {\ln \left (b x +a -1\right )}{4}-\frac {\ln \left (b x +a +1\right )}{4}}{b}\) | \(46\) |
default | \(\frac {\frac {\left (b x +a \right )^{2} \operatorname {arccoth}\left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}+\frac {\ln \left (b x +a -1\right )}{4}-\frac {\ln \left (b x +a +1\right )}{4}}{b}\) | \(46\) |
parallelrisch | \(-\frac {-b^{3} \operatorname {arccoth}\left (b x +a \right ) x^{2}-2 a \,\operatorname {arccoth}\left (b x +a \right ) x \,b^{2}-\operatorname {arccoth}\left (b x +a \right ) a^{2} b -b^{2} x +\operatorname {arccoth}\left (b x +a \right ) b +2 a b}{2 b^{2}}\) | \(64\) |
parts | \(\frac {\operatorname {arccoth}\left (b x +a \right ) b \,x^{2}}{2}+\operatorname {arccoth}\left (b x +a \right ) a x +\frac {b \left (\frac {x}{b}+\frac {\left (-a^{2}+1\right ) \ln \left (b x +a -1\right )}{2 b^{2}}+\frac {\left (a^{2}-1\right ) \ln \left (b x +a +1\right )}{2 b^{2}}\right )}{2}\) | \(68\) |
risch | \(\left (\frac {1}{4} b \,x^{2}+\frac {1}{2} a x \right ) \ln \left (b x +a +1\right )-\frac {b \,x^{2} \ln \left (b x +a -1\right )}{4}-\frac {a x \ln \left (b x +a -1\right )}{2}-\frac {\ln \left (b x +a -1\right ) a^{2}}{4 b}+\frac {\ln \left (-b x -a -1\right ) a^{2}}{4 b}+\frac {x}{2}+\frac {\ln \left (b x +a -1\right )}{4 b}-\frac {\ln \left (-b x -a -1\right )}{4 b}\) | \(108\) |
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Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.13 \[ \int (a+b x) \coth ^{-1}(a+b x) \, dx=\frac {2 \, b x + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \log \left (\frac {b x + a + 1}{b x + a - 1}\right )}{4 \, b} \]
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Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.44 \[ \int (a+b x) \coth ^{-1}(a+b x) \, dx=\begin {cases} \frac {a^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 b} + a x \operatorname {acoth}{\left (a + b x \right )} + \frac {b x^{2} \operatorname {acoth}{\left (a + b x \right )}}{2} + \frac {x}{2} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\a x \operatorname {acoth}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.59 \[ \int (a+b x) \coth ^{-1}(a+b x) \, dx=\frac {1}{4} \, b {\left (\frac {2 \, x}{b} + \frac {{\left (a^{2} - 1\right )} \log \left (b x + a + 1\right )}{b^{2}} - \frac {{\left (a^{2} - 1\right )} \log \left (b x + a - 1\right )}{b^{2}}\right )} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \operatorname {arcoth}\left (b x + a\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (33) = 66\).
Time = 0.28 (sec) , antiderivative size = 188, normalized size of antiderivative = 4.82 \[ \int (a+b x) \coth ^{-1}(a+b x) \, dx=\frac {1}{2} \, {\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {1}{b^{2} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}} + \frac {{\left (b x + a + 1\right )} \log \left (-\frac {\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} + 1}{\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} - 1}\right )}{{\left (b x + a - 1\right )} b^{2} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}^{2}}\right )} \]
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Time = 4.78 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.28 \[ \int (a+b x) \coth ^{-1}(a+b x) \, dx=\frac {x}{2}-\frac {\frac {\mathrm {acoth}\left (a+b\,x\right )}{2}-\frac {a^2\,\mathrm {acoth}\left (a+b\,x\right )}{2}}{b}+a\,x\,\mathrm {acoth}\left (a+b\,x\right )+\frac {b\,x^2\,\mathrm {acoth}\left (a+b\,x\right )}{2} \]
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