Optimal. Leaf size=81 \[ \frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}+\frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^3}+\frac {x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac {x^3}{3 b} \]
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Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2159, 2158, 2157, 29} \[ \frac {x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^3}+\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}+\frac {x^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2157
Rule 2158
Rule 2159
Rubi steps
\begin {align*} \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))} \, dx &=\frac {x^3}{3 b}-\frac {\left (-b x+\coth ^{-1}(\tanh (a+b x))\right ) \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac {x^3}{3 b}+\frac {x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac {\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2 \int \frac {x}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac {x^3}{3 b}+\frac {x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^3}-\frac {\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^3 \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^3}\\ &=\frac {x^3}{3 b}+\frac {x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^3}-\frac {\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^3 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^4}\\ &=\frac {x^3}{3 b}+\frac {x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^3}+\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 79, normalized size = 0.98 \[ -\frac {\left (\coth ^{-1}(\tanh (a+b x))-b x\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}+\frac {x \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^2}{b^3}-\frac {x^2 \left (\coth ^{-1}(\tanh (a+b x))-b x\right )}{2 b^2}+\frac {x^3}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 127, normalized size = 1.57 \[ \frac {8 \, b^{3} x^{3} - 12 \, a b^{2} x^{2} - 6 \, {\left (\pi ^{2} b - 4 \, a^{2} b\right )} x - 6 \, {\left (\pi ^{3} - 12 \, \pi a^{2}\right )} \arctan \left (-\frac {2 \, b x + 2 \, a - \sqrt {4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right ) + 3 \, {\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )}{24 \, b^{4}} \]
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 \[ \int \frac {x^{3}}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.80, size = 130774, normalized size = 1614.49 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.53, size = 86, normalized size = 1.06 \[ \frac {4 \, b^{2} x^{3} + {\left (3 i \, \pi b - 6 \, a b\right )} x^{2} - {\left (3 \, \pi ^{2} + 12 i \, \pi a - 12 \, a^{2}\right )} x}{12 \, b^{3}} - \frac {{\left (i \, \pi ^{3} - 6 \, \pi ^{2} a - 12 i \, \pi a^{2} + 8 \, a^{3}\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{8 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 354, normalized size = 4.37 \[ \frac {x^3}{3\,b}+\frac {x^2\,\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{4\,b^2}+\frac {x\,{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2}{4\,b^3}+\frac {\ln \left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\right )\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3-8\,a^3-6\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2+12\,a^2\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\right )}{8\,b^4} \]
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 \[ \int \frac {x^{3}}{\operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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