Optimal. Leaf size=105 \[ -b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3+\frac {1}{2} \tanh ^{-1}(\tanh (a+b x))^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {1}{3} \tanh ^{-1}(\tanh (a+b x))^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {1}{4} \tanh ^{-1}(\tanh (a+b x))^4+\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4 \]
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Rubi [A] time = 0.06, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2159, 2158, 29} \[ -b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3+\frac {1}{2} \tanh ^{-1}(\tanh (a+b x))^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {1}{3} \tanh ^{-1}(\tanh (a+b x))^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {1}{4} \tanh ^{-1}(\tanh (a+b x))^4+\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4 \]
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))^4}{x} \, dx &=\frac {1}{4} \tanh ^{-1}(\tanh (a+b x))^4-\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x} \, dx\\ &=-\frac {1}{3} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{4} \tanh ^{-1}(\tanh (a+b x))^4-\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))^2}{x} \, dx\\ &=\frac {1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \tanh ^{-1}(\tanh (a+b x))^2-\frac {1}{3} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{4} \tanh ^{-1}(\tanh (a+b x))^4+\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 {\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=-b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3+\frac {1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \tanh ^{-1}(\tanh (a+b x))^2-\frac {1}{3} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{4} \tanh ^{-1}(\tanh (a+b x))^4-\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \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 )^3+\frac {1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \tanh ^{-1}(\tanh (a+b x))^2-\frac {1}{3} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^3+\frac {1}{4} \tanh ^{-1}(\tanh (a+b x))^4+\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.17, size = 175, normalized size = 1.67 \[ \frac {1}{2} (a+b x)^2 \left (a^2-4 a \left (-\tanh ^{-1}(\tanh (a+b x))+a+b x\right )+6 \left (-\tanh ^{-1}(\tanh (a+b x))+a+b x\right )^2\right )+(a+b x) \left (a^3-4 a^2 \left (-\tanh ^{-1}(\tanh (a+b x))+a+b x\right )+6 a \left (-\tanh ^{-1}(\tanh (a+b x))+a+b x\right )^2-4 \left (-\tanh ^{-1}(\tanh (a+b x))+a+b x\right )^3\right )+\frac {1}{4} (a+b x)^4-\frac {1}{3} (a+b x)^3 \left (-4 \tanh ^{-1}(\tanh (a+b x))+3 a+4 b x\right )+\log (b x) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^4 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 42, normalized size = 0.40 \[ \frac {1}{4} \, b^{4} x^{4} + \frac {4}{3} \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 43, normalized size = 0.41 \[ \frac {1}{4} \, b^{4} x^{4} + \frac {4}{3} \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 127, normalized size = 1.21 \[ \ln \relax (x ) \arctanh \left (\tanh \left (b x +a \right )\right )^{4}-4 \arctanh \left (\tanh \left (b x +a \right )\right ) \ln \relax (x ) x^{3} b^{3}+6 \arctanh \left (\tanh \left (b x +a \right )\right )^{2} \ln \relax (x ) x^{2} b^{2}+4 b \arctanh \left (\tanh \left (b x +a \right )\right )^{3} x +b^{4} x^{4} \ln \relax (x )-\frac {25 b^{4} x^{4}}{12}-4 b \arctanh \left (\tanh \left (b x +a \right )\right )^{3} \ln \relax (x ) x -9 \arctanh \left (\tanh \left (b x +a \right )\right )^{2} x^{2} b^{2}+\frac {22 \arctanh \left (\tanh \left (b x +a \right )\right ) x^{3} b^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 42, normalized size = 0.40 \[ \frac {1}{4} \, b^{4} x^{4} + \frac {4}{3} \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 423, normalized size = 4.03 \[ \ln \relax (x)\,\left (\frac {{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^4}{16}+\frac {3\,a^2\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{2}+a^4-\frac {a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}{2}-2\,a^3\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )\right )+\frac {b^4\,x^4}{4}-\frac {2\,b^3\,x^3\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{3}+\frac {3\,b^2\,x^2\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{4}-\frac {b\,x\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}{2} \]
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 \[ \int \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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