Optimal. Leaf size=68 \[ -3 b^2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )-\frac {\coth ^{-1}(\tanh (a+b x))^3}{x}+\frac {3}{2} b \coth ^{-1}(\tanh (a+b x))^2+3 b \log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2168, 2159, 2158, 29} \[ -3 b^2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )-\frac {\coth ^{-1}(\tanh (a+b x))^3}{x}+\frac {3}{2} b \coth ^{-1}(\tanh (a+b x))^2+3 b \log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \]
Antiderivative was successfully verified.
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Rule 29
Rule 2158
Rule 2159
Rule 2168
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(\tanh (a+b x))^3}{x^2} \, dx &=-\frac {\coth ^{-1}(\tanh (a+b x))^3}{x}+(3 b) \int \frac {\coth ^{-1}(\tanh (a+b x))^2}{x} \, dx\\ &=\frac {3}{2} b \coth ^{-1}(\tanh (a+b x))^2-\frac {\coth ^{-1}(\tanh (a+b x))^3}{x}-\left (3 b \left (b x-\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {\coth ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=-3 b^2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )+\frac {3}{2} b \coth ^{-1}(\tanh (a+b x))^2-\frac {\coth ^{-1}(\tanh (a+b x))^3}{x}+\left (3 b \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {1}{x} \, dx\\ &=-3 b^2 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )+\frac {3}{2} b \coth ^{-1}(\tanh (a+b x))^2-\frac {\coth ^{-1}(\tanh (a+b x))^3}{x}+3 b \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 0.91 \[ -6 b^2 x \log (x) \coth ^{-1}(\tanh (a+b x))-\frac {\coth ^{-1}(\tanh (a+b x))^3}{x}+3 b (\log (x)+1) \coth ^{-1}(\tanh (a+b x))^2+\frac {3}{2} b^3 x^2 (2 \log (x)-1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 51, normalized size = 0.75 \[ \frac {2 \, b^{3} x^{3} + 12 \, a b^{2} x^{2} + 3 \, \pi ^{2} a - 4 \, a^{3} - 3 \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x \log \relax (x)}{4 \, x} \]
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 \[ \int \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.46, size = 7683, normalized size = 112.99 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.61, size = 124, normalized size = 1.82 \[ 3 \, b \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} \log \relax (x) - \frac {3}{2} \, {\left (2 \, \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} \log \relax (x) - {\left (b x^{2} - 2 \, {\left (-i \, \pi - 2 \, a\right )} x + 2 \, {\left (-\frac {i \, \pi {\left (b x + a\right )}}{b} - \frac {{\left (b x + a\right )}^{2}}{b}\right )} \log \relax (x) + \frac {2 \, \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2} \log \relax (x)}{b} + \frac {2 \, {\left (i \, \pi a + a^{2}\right )} \log \relax (x)}{b}\right )} b\right )} b - \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 372, normalized size = 5.47 \[ \ln \relax (x)\,\left (3\,a^2\,b+\frac {3\,b\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2}{4}-3\,a\,b\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\right )+\frac {{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3-8\,a^3-6\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2+12\,a^2\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{8\,x}+\frac {b^3\,x^2}{2}-\frac {3\,b^2\,x\,\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{2} \]
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 \[ \int \frac {\operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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