Optimal. Leaf size=14 \[ -\frac {1}{b \tanh ^{-1}(\tanh (a+b x))} \]
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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} \[ -\frac {1}{b \tanh ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rubi steps
\begin {align*} \int \frac {1}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b}\\ &=-\frac {1}{b \tanh ^{-1}(\tanh (a+b x))}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ -\frac {1}{b \tanh ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 13, normalized size = 0.93 \[ -\frac {1}{b^{2} x + a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 12, normalized size = 0.86 \[ -\frac {1}{{\left (b x + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 15, normalized size = 1.07 \[ -\frac {1}{b \arctanh \left (\tanh \left (b x +a \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 12, normalized size = 0.86 \[ -\frac {1}{{\left (b x + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 14, normalized size = 1.00 \[ -\frac {1}{b\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )} \]
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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