Optimal. Leaf size=39 \[ -\frac {\tanh ^{-1}(\tanh (a+b x))^2}{x}-2 b \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+2 b^2 x \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2168, 2158, 29} \[ -\frac {\tanh ^{-1}(\tanh (a+b x))^2}{x}-2 b \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+2 b^2 x \]
Antiderivative was successfully verified.
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Rule 29
Rule 2158
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^2} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^2}{x}+(2 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=2 b^2 x-\frac {\tanh ^{-1}(\tanh (a+b x))^2}{x}-\left (2 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{x} \, dx\\ &=2 b^2 x-\frac {\tanh ^{-1}(\tanh (a+b x))^2}{x}-2 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 37, normalized size = 0.95 \[ -\frac {\tanh ^{-1}(\tanh (a+b x))^2}{x}+2 b (\log (x)+1) \tanh ^{-1}(\tanh (a+b x))-2 b^2 x \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 24, normalized size = 0.62 \[ \frac {b^{2} x^{2} + 2 \, a b x \log \relax (x) - a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 21, normalized size = 0.54 \[ b^{2} x + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac {a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 42, normalized size = 1.08 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{x}-2 \ln \relax (x ) x \,b^{2}+2 \ln \relax (x ) \arctanh \left (\tanh \left (b x +a \right )\right ) b +2 b^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 54, normalized size = 1.38 \[ 2 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) \log \relax (x) - 2 \, {\left (b {\left (x + \frac {a}{b}\right )} \log \relax (x) - b {\left (x + \frac {a \log \relax (x)}{b}\right )}\right )} b - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 198, normalized size = 5.08 \[ b\,\ln \left (\frac {{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\frac {{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{4\,x}-b\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\frac {{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{4\,x}+b\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \relax (x)-2\,b^2\,x\,\ln \relax (x)-b\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \relax (x)+\frac {\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2\,x} \]
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 \[ \int \frac {\operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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