Optimal. Leaf size=36 \[ -\frac {b \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}+b^2 \log (x) \]
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Rubi [A]
time = 0.03, 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 = {2199, 29}
\begin {gather*} -\frac {\tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {b \tanh ^{-1}(\tanh (a+b x))}{x}+b^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2199
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 = 0.02, size = 42, normalized size = 1.17 \begin {gather*} -\frac {2 b x \tanh ^{-1}(\tanh (a+b x))+\tanh ^{-1}(\tanh (a+b x))^2-b^2 x^2 (3+2 \log (x))}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 35, normalized size = 0.97
method | result | size |
default | \(-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{2 x^{2}}+b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{x}+b \ln \left (x \right )\right )\) | \(35\) |
risch | \(\text {Expression too large to display}\) | \(1974\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 34, normalized size = 0.94 \begin {gather*} b^{2} \log \left (x\right ) - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 26, normalized size = 0.72 \begin {gather*} \frac {2 \, b^{2} x^{2} \log \left (x\right ) - 4 \, a b x - a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 32, normalized size = 0.89 \begin {gather*} b^{2} \log {\left (x \right )} - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 22, normalized size = 0.61 \begin {gather*} b^{2} \log \left ({\left | x \right |}\right ) - \frac {4 \, a b x + a^{2}}{2 \, x^{2}} \end {gather*}
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 \begin {gather*} b^2\,\ln \left (x\right )-\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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