Integrand size = 14, antiderivative size = 48 \[ \int \frac {\coth ^{-1}(a+b x)}{(a+b x)^2} \, dx=-\frac {\coth ^{-1}(a+b x)}{b (a+b x)}+\frac {\log (a+b x)}{b}-\frac {\log \left (1-(a+b x)^2\right )}{2 b} \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6243, 6038, 272, 36, 31, 29} \[ \int \frac {\coth ^{-1}(a+b x)}{(a+b x)^2} \, dx=\frac {\log (a+b x)}{b}-\frac {\log \left (1-(a+b x)^2\right )}{2 b}-\frac {\coth ^{-1}(a+b x)}{b (a+b x)} \]
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 6038
Rule 6243
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\coth ^{-1}(x)}{x^2} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\coth ^{-1}(a+b x)}{b (a+b x)}+\frac {\text {Subst}\left (\int \frac {1}{x \left (1-x^2\right )} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\coth ^{-1}(a+b x)}{b (a+b x)}+\frac {\text {Subst}\left (\int \frac {1}{(1-x) x} \, dx,x,(a+b x)^2\right )}{2 b} \\ & = -\frac {\coth ^{-1}(a+b x)}{b (a+b x)}+\frac {\text {Subst}\left (\int \frac {1}{1-x} \, dx,x,(a+b x)^2\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,(a+b x)^2\right )}{2 b} \\ & = -\frac {\coth ^{-1}(a+b x)}{b (a+b x)}+\frac {\log (a+b x)}{b}-\frac {\log \left (1-(a+b x)^2\right )}{2 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.90 \[ \int \frac {\coth ^{-1}(a+b x)}{(a+b x)^2} \, dx=-\frac {\frac {2 \coth ^{-1}(a+b x)}{a+b x}-2 \log (a+b x)+\log \left (1-(a+b x)^2\right )}{2 b} \]
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Time = 0.48 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {arccoth}\left (b x +a \right )}{b x +a}+\ln \left (b x +a \right )-\frac {\ln \left (b x +a +1\right )}{2}-\frac {\ln \left (b x +a -1\right )}{2}}{b}\) | \(45\) |
default | \(\frac {-\frac {\operatorname {arccoth}\left (b x +a \right )}{b x +a}+\ln \left (b x +a \right )-\frac {\ln \left (b x +a +1\right )}{2}-\frac {\ln \left (b x +a -1\right )}{2}}{b}\) | \(45\) |
parts | \(-\frac {\operatorname {arccoth}\left (b x +a \right )}{b \left (b x +a \right )}-\frac {\ln \left (b x +a -1\right )}{2 b}+\frac {\ln \left (b x +a \right )}{b}-\frac {\ln \left (b x +a +1\right )}{2 b}\) | \(54\) |
parallelrisch | \(\frac {3 \ln \left (b x +a \right ) x a \,b^{2}-3 \ln \left (b x +a -1\right ) x a \,b^{2}-3 a \,\operatorname {arccoth}\left (b x +a \right ) x \,b^{2}+3 \ln \left (b x +a \right ) a^{2} b -3 \ln \left (b x +a -1\right ) a^{2} b -3 \,\operatorname {arccoth}\left (b x +a \right ) a^{2} b -3 \,\operatorname {arccoth}\left (b x +a \right ) a b}{3 \left (b x +a \right ) a \,b^{2}}\) | \(104\) |
risch | \(-\frac {\ln \left (b x +a +1\right )}{2 b \left (b x +a \right )}+\frac {2 \ln \left (-b x -a \right ) b x -\ln \left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) b x +2 \ln \left (-b x -a \right ) a -\ln \left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) a +\ln \left (b x +a -1\right )}{2 \left (b x +a \right ) b}\) | \(109\) |
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Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.40 \[ \int \frac {\coth ^{-1}(a+b x)}{(a+b x)^2} \, dx=-\frac {{\left (b x + a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right ) - 2 \, {\left (b x + a\right )} \log \left (b x + a\right ) + \log \left (\frac {b x + a + 1}{b x + a - 1}\right )}{2 \, {\left (b^{2} x + a b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (34) = 68\).
Time = 0.52 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.83 \[ \int \frac {\coth ^{-1}(a+b x)}{(a+b x)^2} \, dx=\begin {cases} \frac {a \log {\left (\frac {a}{b} + x \right )}}{a b + b^{2} x} - \frac {a \log {\left (\frac {a}{b} + x + \frac {1}{b} \right )}}{a b + b^{2} x} + \frac {a \operatorname {acoth}{\left (a + b x \right )}}{a b + b^{2} x} + \frac {b x \log {\left (\frac {a}{b} + x \right )}}{a b + b^{2} x} - \frac {b x \log {\left (\frac {a}{b} + x + \frac {1}{b} \right )}}{a b + b^{2} x} + \frac {b x \operatorname {acoth}{\left (a + b x \right )}}{a b + b^{2} x} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{a b + b^{2} x} & \text {for}\: b \neq 0 \\\frac {x \operatorname {acoth}{\left (a \right )}}{a^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10 \[ \int \frac {\coth ^{-1}(a+b x)}{(a+b x)^2} \, dx=-\frac {\log \left (b x + a + 1\right )}{2 \, b} + \frac {\log \left (b x + a\right )}{b} - \frac {\log \left (b x + a - 1\right )}{2 \, b} - \frac {\operatorname {arcoth}\left (b x + a\right )}{{\left (b x + a\right )} b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (46) = 92\).
Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 4.12 \[ \int \frac {\coth ^{-1}(a+b x)}{(a+b x)^2} \, dx=-\frac {1}{2} \, {\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {\log \left (\frac {{\left | b x + a + 1 \right |}}{{\left | b x + a - 1 \right |}}\right )}{b^{2}} - \frac {\log \left ({\left | \frac {b x + a + 1}{b x + a - 1} + 1 \right |}\right )}{b^{2}} - \frac {\log \left (-\frac {\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} + 1}{\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} - 1}\right )}{b^{2} {\left (\frac {b x + a + 1}{b x + a - 1} + 1\right )}}\right )} \]
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Time = 4.14 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.94 \[ \int \frac {\coth ^{-1}(a+b x)}{(a+b x)^2} \, dx=\frac {\ln \left (a+b\,x\right )}{b}-\frac {\ln \left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )}{2\,b}-\frac {\ln \left (\frac {a+b\,x+1}{a+b\,x}\right )}{2\,\left (x\,b^2+a\,b\right )}+\frac {\ln \left (\frac {a+b\,x-1}{a+b\,x}\right )}{2\,x\,b^2+2\,a\,b} \]
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