Optimal. Leaf size=60 \[ -3 b^2 \log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )-\frac {\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}-\frac {3 b \coth ^{-1}(\tanh (a+b x))^2}{2 x}+3 b^3 x \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2168, 2158, 29} \[ -3 b^2 \log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )-\frac {\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}-\frac {3 b \coth ^{-1}(\tanh (a+b x))^2}{2 x}+3 b^3 x \]
Antiderivative was successfully verified.
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Rule 29
Rule 2158
Rule 2168
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(\tanh (a+b x))^3}{x^3} \, dx &=-\frac {\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}+\frac {1}{2} (3 b) \int \frac {\coth ^{-1}(\tanh (a+b x))^2}{x^2} \, dx\\ &=-\frac {3 b \coth ^{-1}(\tanh (a+b x))^2}{2 x}-\frac {\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}+\left (3 b^2\right ) \int \frac {\coth ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=3 b^3 x-\frac {3 b \coth ^{-1}(\tanh (a+b x))^2}{2 x}-\frac {\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}-\left (3 b^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{x} \, dx\\ &=3 b^3 x-\frac {3 b \coth ^{-1}(\tanh (a+b x))^2}{2 x}-\frac {\coth ^{-1}(\tanh (a+b x))^3}{2 x^2}-3 b^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 1.10 \[ 3 b^2 \log (x) \left (\coth ^{-1}(\tanh (a+b x))-b x\right )-\frac {\left (\coth ^{-1}(\tanh (a+b x))-b x\right )^3}{2 x^2}-\frac {3 b \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^2}{x}+b^3 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 51, normalized size = 0.85 \[ \frac {8 \, b^{3} x^{3} + 24 \, a b^{2} x^{2} \log \relax (x) + 3 \, \pi ^{2} a - 4 \, a^{3} + 6 \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.55, size = 7366, normalized size = 122.77 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 72, normalized size = 1.20 \[ 3 \, {\left (b \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right ) \log \relax (x) - {\left (b {\left (x + \frac {a}{b}\right )} \log \relax (x) - b {\left (x + \frac {a \log \relax (x)}{b}\right )}\right )} b\right )} b - \frac {3 \, b \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{2 \, x} - \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 383, normalized size = 6.38 \[ \frac {{\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^3}{16\,x^2}-\frac {{\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^3}{16\,x^2}+\frac {9\,b^2\,\ln \left (\frac {{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{4}-\frac {9\,b^2\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{4}-\frac {3\,b^3\,x}{2}-\frac {3\,b\,{\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2}{8\,x}+\frac {3\,b^2\,\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\ln \relax (x)}{2}-3\,b^3\,x\,\ln \relax (x)-\frac {3\,b\,{\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2}{8\,x}-\frac {3\,b^2\,\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\ln \relax (x)}{2}-\frac {3\,\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,{\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2}{16\,x^2}+\frac {3\,{\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2\,\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{16\,x^2}+\frac {3\,b\,\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{4\,x} \]
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 \[ \int \frac {\operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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