Optimal. Leaf size=56 \[ \frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {x^2}{2 b} \]
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Rubi [A] time = 0.03, antiderivative size = 56, 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 = {2159, 2158, 2157, 29} \[ \frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2157
Rule 2158
Rule 2159
Rubi steps
\begin {align*} \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx &=\frac {x^2}{2 b}-\frac {\left (-b x+\coth ^{-1}(\tanh (a+b x))\right ) \int \frac {x}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac {x^2}{2 b}+\frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2 \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac {x^2}{2 b}+\frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^3}\\ &=\frac {x^2}{2 b}+\frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 55, normalized size = 0.98 \[ \frac {\left (\coth ^{-1}(\tanh (a+b x))-b x\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x \left (\coth ^{-1}(\tanh (a+b x))-b x\right )}{b^2}+\frac {x^2}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 97, normalized size = 1.73 \[ \frac {4 \, b^{2} x^{2} - 8 \, a b x - 16 \, \pi a \arctan \left (-\frac {2 \, b x + 2 \, a - \sqrt {4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right ) - {\left (\pi ^{2} - 4 \, a^{2}\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )}{8 \, b^{3}} \]
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 \[ \int \frac {x^{2}}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.18, size = 28786, normalized size = 514.04 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.53, size = 51, normalized size = 0.91 \[ \frac {b x^{2} + {\left (i \, \pi - 2 \, a\right )} x}{2 \, b^{2}} - \frac {{\left (\pi ^{2} + 4 i \, \pi a - 4 \, a^{2}\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 234, normalized size = 4.18 \[ \frac {x^2}{2\,b}+\frac {x\,\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{2\,b^2}+\frac {\ln \left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\right )\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+4\,a^2\right )}{4\,b^3} \]
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 \[ \int \frac {x^{2}}{\operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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