Optimal. Leaf size=87 \[ -6 b^3 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+3 b^2 \tanh ^{-1}(\tanh (a+b x))^2+6 b^2 \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x} \]
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Rubi [A] time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2168, 2159, 2158, 29} \[ 3 b^2 \tanh ^{-1}(\tanh (a+b x))^2-6 b^3 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+6 b^2 \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x} \]
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))^4}{x^3} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}+(2 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x^2} \, dx\\ &=-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}+\left (6 b^2\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x} \, dx\\ &=3 b^2 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}-\left (6 b^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=-6 b^3 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+3 b^2 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}+\left (6 b^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {1}{x} \, dx\\ &=-6 b^3 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+3 b^2 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}+6 b^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 = 81, normalized size = 0.93 \[ -6 b^3 x (2 \log (x)+1) \tanh ^{-1}(\tanh (a+b x))+3 b^2 (2 \log (x)+3) \tanh ^{-1}(\tanh (a+b x))^2-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x}+6 b^4 x^2 \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 47, normalized size = 0.54 \[ \frac {b^{4} x^{4} + 8 \, a b^{3} x^{3} + 12 \, a^{2} b^{2} x^{2} \log \relax (x) - 8 \, a^{3} b x - a^{4}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 43, normalized size = 0.49 \[ \frac {1}{2} \, b^{4} x^{2} + 4 \, a b^{3} x + 6 \, a^{2} b^{2} \log \left ({\left | x \right |}\right ) - \frac {8 \, a^{3} b x + a^{4}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 93, normalized size = 1.07 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{2 x^{2}}-\frac {2 b \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{x}+6 b^{2} \ln \relax (x ) \arctanh \left (\tanh \left (b x +a \right )\right )^{2}+6 b^{4} x^{2} \ln \relax (x )-9 x^{2} b^{4}-12 b^{3} \arctanh \left (\tanh \left (b x +a \right )\right ) \ln \relax (x ) x +12 b^{3} \arctanh \left (\tanh \left (b x +a \right )\right ) x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 83, normalized size = 0.95 \[ -\frac {2 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{x} + 3 \, {\left (2 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \log \relax (x) + {\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \relax (x) - 2 \, \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \log \relax (x)\right )} b\right )} b - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 672, normalized size = 7.72 \[ \frac {9\,b^2\,{\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 )}^4}{32\,x^2}-\frac {{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^4}{32\,x^2}+\frac {9\,b^2\,{\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}-3\,b^3\,x\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\frac {b\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3}{4\,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 )}^3}{8\,x^2}+\frac {{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3\,\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^2}-\frac {b\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3}{4\,x}+\frac {3\,b^2\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2\,\ln \relax (x)}{2}-\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 )}^2}{16\,x^2}+\frac {3\,b^2\,{\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)}{2}+6\,b^4\,x^2\,\ln \relax (x)-\frac {9\,b^2\,\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\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\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}{4\,x}-\frac {3\,b\,{\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 )}{4\,x}-3\,b^2\,\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)+6\,b^3\,x\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \relax (x)-6\,b^3\,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 \[ \int \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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