Optimal. Leaf size=77 \[ -4 b^3 \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac {2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}+4 b^4 x \]
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Rubi [A] time = 0.06, antiderivative size = 77, 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 = {2168, 2158, 29} \[ -\frac {2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-4 b^3 \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+4 b^4 x \]
Antiderivative was successfully verified.
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Rule 29
Rule 2158
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^4}{x^4} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+\frac {1}{3} (4 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x^3} \, dx\\ &=-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+\left (2 b^2\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^2} \, dx\\ &=-\frac {2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+\left (4 b^3\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=4 b^4 x-\frac {2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}-\left (4 b^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{x} \, dx\\ &=4 b^4 x-\frac {2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}-4 b^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 82, normalized size = 1.06 \[ -\frac {-2 b^3 x^3 (6 \log (x)+11) \tanh ^{-1}(\tanh (a+b x))+6 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+2 b x \tanh ^{-1}(\tanh (a+b x))^3+\tanh ^{-1}(\tanh (a+b x))^4+2 b^4 x^4 (6 \log (x)+5)}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 48, normalized size = 0.62 \[ \frac {3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} \log \relax (x) - 18 \, a^{2} b^{2} x^{2} - 6 \, a^{3} b x - a^{4}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 42, normalized size = 0.55 \[ b^{4} x + 4 \, a b^{3} \log \left ({\left | x \right |}\right ) - \frac {18 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 76, normalized size = 0.99 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{3 x^{3}}-\frac {2 b \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{3 x^{2}}-\frac {2 b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{x}-4 \ln \relax (x ) x \,b^{4}+4 \arctanh \left (\tanh \left (b x +a \right )\right ) \ln \relax (x ) b^{3}+4 b^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 91, normalized size = 1.18 \[ 2 \, {\left (2 \, {\left (b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) \log \relax (x) - {\left (b {\left (x + \frac {a}{b}\right )} \log \relax (x) - b {\left (x + \frac {a \log \relax (x)}{b}\right )}\right )} b\right )} b - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x}\right )} b - \frac {2 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{3 \, x^{2}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 571, normalized size = 7.42 \[ \frac {11\,b^3\,\ln \left (\frac {{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{3}-\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}{48\,x^3}-\frac {11\,b^3\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{3}-\frac {10\,b^4\,x}{3}-\frac {{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^4}{48\,x^3}+\frac {b\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3}{12\,x^2}-2\,b^3\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \relax (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}{12\,x^3}+\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 )}{12\,x^3}-\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}{12\,x^2}+2\,b^3\,\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)-4\,b^4\,x\,\ln \relax (x)-\frac {b^2\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{2\,x}-\frac {{\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}{8\,x^3}-\frac {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}{2\,x}+\frac {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 )}{x}+\frac {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^2}-\frac {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^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 \[ \int \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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