Integrand size = 9, antiderivative size = 12 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b} \]
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Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2188, 29} \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b} \]
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Rule 29
Rule 2188
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b} \\ & = \frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b} \]
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Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {\ln \left (\operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b}\) | \(13\) |
default | \(\frac {\ln \left (\operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b}\) | \(13\) |
parallelrisch | \(\frac {\ln \left (\operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b}\) | \(13\) |
risch | \(\frac {\ln \left (\ln \left ({\mathrm e}^{b x +a}\right )-\frac {i \pi \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )-\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+2 \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}-2 \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+\operatorname {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )-2 \,\operatorname {csgn}\left (i {\mathrm e}^{b x +a}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}+\operatorname {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}-\operatorname {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+\operatorname {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+2\right )}{4}\right )}{b}\) | \(305\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {\log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{b} \]
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Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx=\begin {cases} \frac {\log {\left (\operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\operatorname {acoth}{\left (\tanh {\left (a \right )} \right )}} & \text {otherwise} \end {cases} \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {\log \left (-\frac {1}{2} i \, \pi - b x - a\right )}{b} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {\log \left (\pi - 2 i \, b x - 2 i \, a\right )}{b} \]
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Time = 3.80 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {\ln \left (\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )\right )}{b} \]
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