Optimal. Leaf size=77 \[ b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {1}{2} \tanh ^{-1}(\tanh (a+b x))^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \]
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Rubi [A] time = 0.08, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2159, 2158, 29} \[ b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {1}{2} \tanh ^{-1}(\tanh (a+b x))^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \]
Antiderivative was successfully verified.
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Rule 29
Rule 2158
Rule 2159
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x} \, dx &=\frac {1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x} \, dx\\ &=-\frac {1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{3} \tanh ^{-1}(\tanh (a+b x))^3+\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \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 )^2-\frac {1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.10, size = 104, normalized size = 1.35 \[ (a+b x) \left (a^2-3 a \left (-\tanh ^{-1}(\tanh (a+b x))+a+b x\right )+3 \left (-\tanh ^{-1}(\tanh (a+b x))+a+b x\right )^2\right )+\frac {1}{3} (a+b x)^3-\frac {1}{2} (a+b x)^2 \left (-3 \tanh ^{-1}(\tanh (a+b x))+2 a+3 b x\right )+\log (b x) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^3 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 31, normalized size = 0.40 \[ \frac {1}{3} \, b^{3} x^{3} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 32, normalized size = 0.42 \[ \frac {1}{3} \, b^{3} x^{3} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 92, normalized size = 1.19 \[ \ln \relax (x ) \arctanh \left (\tanh \left (b x +a \right )\right )^{3}+3 \arctanh \left (\tanh \left (b x +a \right )\right ) \ln \relax (x ) x^{2} b^{2}+3 b \arctanh \left (\tanh \left (b x +a \right )\right )^{2} x +\frac {11 x^{3} b^{3}}{6}-b^{3} x^{3} \ln \relax (x )-3 b \arctanh \left (\tanh \left (b x +a \right )\right )^{2} \ln \relax (x ) x -\frac {9 \arctanh \left (\tanh \left (b x +a \right )\right ) x^{2} b^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 31, normalized size = 0.40 \[ \frac {1}{3} \, b^{3} x^{3} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 306, normalized size = 3.97 \[ \frac {b^3\,x^3}{3}-\ln \relax (x)\,\left (\frac {{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}{8}-a^3-\frac {3\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{4}+\frac {3\,a^2\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{2}\right )-\frac {3\,b^2\,x^2\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{4}+\frac {3\,b\,x\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{4} \]
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 \[ \int \frac {\operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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