Optimal. Leaf size=44 \[ \frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))} \]
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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} \[ \frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2157
Rule 2160
Rubi steps
\begin {align*} \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))} \, dx &=-\frac {\int \frac {1}{x} \, dx}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {b \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b x-\coth ^{-1}(\tanh (a+b x))}\\ &=-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b x-\coth ^{-1}(\tanh (a+b x))}\\ &=-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b x-\coth ^{-1}(\tanh (a+b x))}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 0.66 \[ \frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )-\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 87, normalized size = 1.98 \[ -\frac {2 \, {\left (2 \, \pi \arctan \left (-\frac {2 \, b x + 2 \, a - \sqrt {4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right ) + a \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right ) - 2 \, a \log \relax (x)\right )}}{\pi ^{2} + 4 \, a^{2}} \]
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 \[ \int \frac {1}{x \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 10.10, size = 972, normalized size = 22.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.52, size = 37, normalized size = 0.84 \[ \frac {2 \, \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{i \, \pi - 2 \, a} - \frac {2 \, \log \relax (x)}{i \, \pi - 2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.91, size = 113, normalized size = 2.57 \[ -\frac {4\,\mathrm {atanh}\left (\frac {4\,b\,x}{\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x}-1\right )}{\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x} \]
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 \[ \int \frac {1}{x \operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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