Integrand size = 9, antiderivative size = 14 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {1}{b \coth ^{-1}(\tanh (a+b x))} \]
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Time = 0.00 (sec) , antiderivative size = 14, 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, 30} \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {1}{b \coth ^{-1}(\tanh (a+b x))} \]
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Rule 30
Rule 2188
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b} \\ & = -\frac {1}{b \coth ^{-1}(\tanh (a+b x))} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {1}{b \coth ^{-1}(\tanh (a+b x))} \]
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Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(-\frac {1}{b \,\operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )}\) | \(15\) |
default | \(-\frac {1}{b \,\operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )}\) | \(15\) |
parallelrisch | \(-\frac {1}{b \,\operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )}\) | \(15\) |
risch | \(-\frac {4 i}{b \left (\pi \operatorname {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}-2 \pi \,\operatorname {csgn}\left (i {\mathrm e}^{b x +a}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )+\pi \operatorname {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )+\pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}-2 \pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+4 i \ln \left ({\mathrm e}^{b x +a}\right )+2 \pi \right )}\) | \(319\) |
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Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {2}{2 \, b^{2} x + i \, \pi b + 2 \, a b} \]
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Time = 12.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\begin {cases} - \frac {1}{b \operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}} & \text {for}\: b \neq 0 \\\frac {x}{\operatorname {acoth}^{2}{\left (\tanh {\left (a \right )} \right )}} & \text {otherwise} \end {cases} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {4}{-2 \, {\left (i \, \pi + 2 \, b x + 2 \, a\right )} b} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {2}{2 \, b^{2} x + i \, \pi b + 2 \, a b} \]
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Time = 3.84 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {1}{b\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )} \]
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