Optimal. Leaf size=49 \[ -b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {1}{2} \tanh ^{-1}(\tanh (a+b x))^2+\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log (x) \]
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Rubi [A]
time = 0.06, antiderivative size = 49, 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 = {2190, 2189, 29}
\begin {gather*} -b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {1}{2} \tanh ^{-1}(\tanh (a+b x))^2+\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2189
Rule 2190
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x} \, dx &=\frac {1}{2} \tanh ^{-1}(\tanh (a+b x))^2-\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=-b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {1}{2} \tanh ^{-1}(\tanh (a+b x))^2-\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{x} \, dx\\ &=-b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {1}{2} \tanh ^{-1}(\tanh (a+b x))^2+\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 53, normalized size = 1.08 \begin {gather*} \frac {1}{2} (a+b x)^2-(a+b x) \left (a+2 b x-2 \tanh ^{-1}(\tanh (a+b x))\right )+\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2 \log (b x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 78, normalized size = 1.59
method | result | size |
default | \(\ln \left (x \right ) \arctanh \left (\tanh \left (b x +a \right )\right )^{2}-2 b \left (\frac {b \,x^{2} \ln \left (x \right )}{2}-\frac {b \,x^{2}}{4}+\ln \left (x \right ) x a -x a +\ln \left (x \right ) x \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )-x \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )\right )\) | \(78\) |
risch | \(\text {Expression too large to display}\) | \(2392\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.66, size = 20, normalized size = 0.41 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 20, normalized size = 0.41 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left (x\right ) \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 21, normalized size = 0.43 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 183, normalized size = 3.73 \begin {gather*} \ln \left (x\right )\,\left (\frac {{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{4}-a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )+a^2\right )+\frac {b^2\,x^2}{2}-b\,x\,\left (\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\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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