Optimal. Leaf size=17 \[ b \log (x)-\frac {\tanh ^{-1}(\coth (a+b x))}{x} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2168, 29} \[ b \log (x)-\frac {\tanh ^{-1}(\coth (a+b x))}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\coth (a+b x))}{x^2} \, dx &=-\frac {\tanh ^{-1}(\coth (a+b x))}{x}+b \int \frac {1}{x} \, dx\\ &=-\frac {\tanh ^{-1}(\coth (a+b x))}{x}+b \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 18, normalized size = 1.06 \[ -\frac {\tanh ^{-1}(\coth (a+b x))}{x}+b \log (x)+b \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 13, normalized size = 0.76 \[ \frac {b x \log \relax (x) - a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 70, normalized size = 4.12 \[ b \log \left ({\left | x \right |}\right ) - \frac {\log \left (-\frac {\frac {e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} + 1}{\frac {e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - 1}\right )}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 18, normalized size = 1.06 \[ -\frac {\arctanh \left (\coth \left (b x +a \right )\right )}{x}+b \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 17, normalized size = 1.00 \[ b \log \relax (x) - \frac {\operatorname {artanh}\left (\coth \left (b x + a\right )\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 17, normalized size = 1.00 \[ b\,\ln \relax (x)-\frac {\mathrm {atanh}\left (\mathrm {coth}\left (a+b\,x\right )\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.78, size = 42, normalized size = 2.47 \[ \begin {cases} \frac {\left \langle - \frac {\pi }{2}, \frac {\pi }{2}\right \rangle i}{x} & \text {for}\: a = \log {\left (- e^{- b x} \right )} \vee a = \log {\left (e^{- b x} \right )} \\b \log {\relax (x )} - \frac {\operatorname {atanh}{\left (\frac {1}{\tanh {\left (a + b x \right )}} \right )}}{x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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