Optimal. Leaf size=36 \[ -\frac {\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {b \tanh ^{-1}(\tanh (a+b x))}{x}+b^2 \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2168, 29} \[ -\frac {\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {b \tanh ^{-1}(\tanh (a+b x))}{x}+b^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^3} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}+b \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x^2} \, dx\\ &=-\frac {b \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}+b^2 \int \frac {1}{x} \, dx\\ &=-\frac {b \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}+b^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 1.17 \[ -\frac {2 b x \tanh ^{-1}(\tanh (a+b x))+\tanh ^{-1}(\tanh (a+b x))^2-b^2 x^2 (2 \log (x)+3)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 26, normalized size = 0.72 \[ \frac {2 \, b^{2} x^{2} \log \relax (x) - 4 \, a b x - a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 22, normalized size = 0.61 \[ b^{2} \log \left ({\left | x \right |}\right ) - \frac {4 \, a b x + a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 35, normalized size = 0.97 \[ -\frac {b \arctanh \left (\tanh \left (b x +a \right )\right )}{x}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{2 x^{2}}+b^{2} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 34, normalized size = 0.94 \[ b^{2} \log \relax (x) - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{x} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 34, normalized size = 0.94 \[ b^2\,\ln \relax (x)-\frac {\frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{2}+b\,x\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 32, normalized size = 0.89 \[ b^{2} \log {\relax (x )} - \frac {b \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{x} - \frac {\operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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