Optimal. Leaf size=95 \[ 4 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{x}+\frac {4}{3} b \tanh ^{-1}(\tanh (a+b x))^3-2 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2-4 b \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \]
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Rubi [A] time = 0.06, antiderivative size = 95, 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} \[ 4 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{x}+\frac {4}{3} b \tanh ^{-1}(\tanh (a+b x))^3-2 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2-4 b \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \]
Antiderivative was successfully verified.
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Rule 29
Rule 2158
Rule 2159
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^4}{x^2} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{x}+(4 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x} \, dx\\ &=\frac {4}{3} b \tanh ^{-1}(\tanh (a+b x))^3-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{x}-\left (4 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x} \, dx\\ &=-2 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac {4}{3} b \tanh ^{-1}(\tanh (a+b x))^3-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{x}+\left (4 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=4 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-2 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac {4}{3} b \tanh ^{-1}(\tanh (a+b x))^3-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{x}-\left (4 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3\right ) \int \frac {1}{x} \, dx\\ &=4 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-2 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac {4}{3} b \tanh ^{-1}(\tanh (a+b x))^3-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{x}-4 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.09, size = 85, normalized size = 0.89 \[ 6 b^3 x^2 (2 \log (x)-1) \tanh ^{-1}(\tanh (a+b x))-12 b^2 x \log (x) \tanh ^{-1}(\tanh (a+b x))^2-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{x}+4 b (\log (x)+1) \tanh ^{-1}(\tanh (a+b x))^3+\frac {2}{3} b^4 x^3 (5-6 \log (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 47, normalized size = 0.49 \[ \frac {b^{4} x^{4} + 6 \, a b^{3} x^{3} + 18 \, a^{2} b^{2} x^{2} + 12 \, a^{3} b x \log \relax (x) - 3 \, a^{4}}{3 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 44, normalized size = 0.46 \[ \frac {1}{3} \, b^{4} x^{3} + 2 \, a b^{3} x^{2} + 6 \, a^{2} b^{2} x + 4 \, a^{3} b \log \left ({\left | x \right |}\right ) - \frac {a^{4}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 112, normalized size = 1.18 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{x}+4 \ln \relax (x ) \arctanh \left (\tanh \left (b x +a \right )\right )^{3} b +12 \arctanh \left (\tanh \left (b x +a \right )\right ) \ln \relax (x ) x^{2} b^{3}+12 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{2} x +\frac {22 x^{3} b^{4}}{3}-4 b^{4} x^{3} \ln \relax (x )-12 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{2} \ln \relax (x ) x -18 \arctanh \left (\tanh \left (b x +a \right )\right ) x^{2} b^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 77, normalized size = 0.81 \[ 4 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} \log \relax (x) - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{x} + \frac {2}{3} \, {\left (2 \, b^{3} x^{3} + 9 \, a b^{2} x^{2} + 18 \, a^{2} b x + 6 \, a^{3} \log \relax (x) - 6 \, \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} \log \relax (x)\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 553, normalized size = 5.82 \[ \ln \relax (x)\,\left (4\,a^3\,b-\frac {b\,{\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}{2}+3\,a\,b\,{\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 )}^2-6\,a^2\,b\,\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 )\right )-\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 )}^4+24\,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+16\,a^4-8\,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 )}^3-32\,a^3\,\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 )}{16\,x}+\frac {b^4\,x^3}{3}+\frac {3\,b^2\,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}{2}-b^3\,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 ) \]
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 \[ \int \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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