Optimal. Leaf size=68 \[ -3 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{x}+\frac {3}{2} b \tanh ^{-1}(\tanh (a+b x))^2+3 b \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2168, 2159, 2158, 29} \[ -3 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{x}+\frac {3}{2} b \tanh ^{-1}(\tanh (a+b x))^2+3 b \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \]
Antiderivative was successfully verified.
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Rule 29
Rule 2158
Rule 2159
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x^2} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{x}+(3 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x} \, dx\\ &=\frac {3}{2} b \tanh ^{-1}(\tanh (a+b x))^2-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{x}-\left (3 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=-3 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {3}{2} b \tanh ^{-1}(\tanh (a+b x))^2-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{x}+\left (3 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {1}{x} \, dx\\ &=-3 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {3}{2} b \tanh ^{-1}(\tanh (a+b x))^2-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{x}+3 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 62, normalized size = 0.91 \[ -6 b^2 x \log (x) \tanh ^{-1}(\tanh (a+b x))-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{x}+3 b (\log (x)+1) \tanh ^{-1}(\tanh (a+b x))^2+\frac {3}{2} b^3 x^2 (2 \log (x)-1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 36, normalized size = 0.53 \[ \frac {b^{3} x^{3} + 6 \, a b^{2} x^{2} + 6 \, a^{2} b x \log \relax (x) - 2 \, a^{3}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 33, normalized size = 0.49 \[ \frac {1}{2} \, b^{3} x^{2} + 3 \, a b^{2} x + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) - \frac {a^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 76, normalized size = 1.12 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{x}+3 \ln \relax (x ) \arctanh \left (\tanh \left (b x +a \right )\right )^{2} b +3 b^{3} x^{2} \ln \relax (x )-\frac {9 x^{2} b^{3}}{2}-6 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right ) \ln \relax (x ) x +6 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right ) x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 65, normalized size = 0.96 \[ 3 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \log \relax (x) + \frac {3}{2} \, {\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \relax (x) - 2 \, \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \log \relax (x)\right )} b - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 415, normalized size = 6.10 \[ \frac {3\,b\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{4}-\frac {{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3}{8\,x}+\frac {3\,b\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{4}+\frac {{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3}{8\,x}-\frac {3\,b^3\,x^2}{2}+\frac {3\,b\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2\,\ln \relax (x)}{4}+\frac {3\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{8\,x}-\frac {3\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{8\,x}+\frac {3\,b\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2\,\ln \relax (x)}{4}-\frac {3\,b\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}+3\,b^3\,x^2\,\ln \relax (x)-\frac {3\,b\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \relax (x)}{2}+3\,b^2\,x\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \relax (x)-3\,b^2\,x\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \relax (x) \]
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 \[ \int \frac {\operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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