Optimal. Leaf size=44 \[ \frac {\log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b x-\tanh ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\tanh ^{-1}(\tanh (a+b x))} \]
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Rubi [A] time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2160, 2157, 29} \[ \frac {\log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b x-\tanh ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\tanh ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2157
Rule 2160
Rubi steps
\begin {align*} \int \frac {1}{x \tanh ^{-1}(\tanh (a+b x))} \, dx &=-\frac {\int \frac {1}{x} \, dx}{b x-\tanh ^{-1}(\tanh (a+b x))}+\frac {b \int \frac {1}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b x-\tanh ^{-1}(\tanh (a+b x))}\\ &=-\frac {\log (x)}{b x-\tanh ^{-1}(\tanh (a+b x))}+\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b x-\tanh ^{-1}(\tanh (a+b x))}\\ &=-\frac {\log (x)}{b x-\tanh ^{-1}(\tanh (a+b x))}+\frac {\log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b x-\tanh ^{-1}(\tanh (a+b x))}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 0.66 \[ \frac {\log \left (\tanh ^{-1}(\tanh (a+b x))\right )-\log (x)}{b x-\tanh ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 16, normalized size = 0.36 \[ -\frac {\log \left (b x + a\right ) - \log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 20, normalized size = 0.45 \[ -\frac {\log \left ({\left | b x + a \right |}\right )}{a} + \frac {\log \left ({\left | x \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 43, normalized size = 0.98 \[ \frac {\ln \relax (x )}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x}-\frac {\ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right )}{\arctanh \left (\tanh \left (b x +a \right )\right )-b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 18, normalized size = 0.41 \[ -\frac {\log \left (b x + a\right )}{a} + \frac {\log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.87, size = 107, normalized size = 2.43 \[ -\frac {4\,\mathrm {atanh}\left (\frac {4\,b\,x}{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x}-1\right )}{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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