Optimal. Leaf size=14 \[ -\frac {1}{b \coth ^{-1}(\tanh (a+b x))} \]
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Rubi [A] time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2157, 30} \[ -\frac {1}{b \coth ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rubi steps
\begin {align*} \int \frac {1}{\coth ^{-1}(\tanh (a+b x))^2} \, dx &=\frac {\operatorname {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*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ -\frac {1}{b \coth ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 36, normalized size = 2.57 \[ -\frac {4 \, {\left (b x + a\right )}}{4 \, b^{3} x^{2} + 8 \, a b^{2} x + \pi ^{2} b + 4 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 15, normalized size = 1.07 \[ -\frac {1}{b \,\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 18, normalized size = 1.29 \[ \frac {4}{-2 \, {\left (i \, \pi + 2 \, b x + 2 \, a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 14, normalized size = 1.00 \[ -\frac {1}{b\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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