Optimal. Leaf size=77 \[ b x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2-\frac {1}{2} \coth ^{-1}(\tanh (a+b x))^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )+\frac {1}{3} \coth ^{-1}(\tanh (a+b x))^3-\log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \]
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Rubi [A] time = 0.10, antiderivative size = 77, 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 = {2159, 2158, 29} \[ b x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2-\frac {1}{2} \coth ^{-1}(\tanh (a+b x))^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )+\frac {1}{3} \coth ^{-1}(\tanh (a+b x))^3-\log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \]
Antiderivative was successfully verified.
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Rule 29
Rule 2158
Rule 2159
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(\tanh (a+b x))^3}{x} \, dx &=\frac {1}{3} \coth ^{-1}(\tanh (a+b x))^3-\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \int \frac {\coth ^{-1}(\tanh (a+b x))^2}{x} \, dx\\ &=-\frac {1}{2} \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{3} \coth ^{-1}(\tanh (a+b x))^3-\left (\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {\coth ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=b x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2-\frac {1}{2} \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{3} \coth ^{-1}(\tanh (a+b x))^3+\left (\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{x} \, dx\\ &=b x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2-\frac {1}{2} \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2+\frac {1}{3} \coth ^{-1}(\tanh (a+b x))^3-\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.10, size = 104, normalized size = 1.35 \[ (a+b x) \left (a^2-3 a \left (-\coth ^{-1}(\tanh (a+b x))+a+b x\right )+3 \left (-\coth ^{-1}(\tanh (a+b x))+a+b x\right )^2\right )+\frac {1}{3} (a+b x)^3-\frac {1}{2} (a+b x)^2 \left (-3 \coth ^{-1}(\tanh (a+b x))+2 a+3 b x\right )+\log (b x) \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^3 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 49, normalized size = 0.64 \[ \frac {1}{3} \, b^{3} x^{3} + \frac {3}{2} \, a b^{2} x^{2} - \frac {3}{4} \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x - \frac {1}{4} \, {\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} \log \relax (x) \]
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 \[ \int \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.90, size = 21848, normalized size = 283.74 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.76, size = 75, normalized size = 0.97 \[ \frac {1}{3} \, b^{3} x^{3} + \frac {1}{24} \, {\left (-18 i \, \pi b^{2} + 36 \, a b^{2}\right )} x^{2} - \frac {1}{24} \, {\left (18 \, \pi ^{2} b + 72 i \, \pi a b - 72 \, a^{2} b\right )} x + \frac {1}{8} \, {\left (i \, \pi ^{3} - 6 \, \pi ^{2} a - 12 i \, \pi a^{2} + 8 \, a^{3}\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 306, normalized size = 3.97 \[ \frac {b^3\,x^3}{3}-\ln \relax (x)\,\left (\frac {{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3}{8}-a^3-\frac {3\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2}{4}+\frac {3\,a^2\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{2}\right )-\frac {3\,b^2\,x^2\,\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{4}+\frac {3\,b\,x\,{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2}{4} \]
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 \[ \int \frac {\operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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