Integrand size = 13, antiderivative size = 56 \[ \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {x^2}{2 b}+\frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3} \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2190, 2189, 2188, 29} \[ \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {x^2}{2 b} \]
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Rule 29
Rule 2188
Rule 2189
Rule 2190
Rubi steps \begin{align*} \text {integral}& = \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 \text {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*}
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.98 \[ \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {x^2}{2 b}-\frac {x \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.37 (sec) , antiderivative size = 28786, normalized size of antiderivative = 514.04
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {2 \, b^{2} x^{2} - 2 i \, \pi b x - 4 \, a b x - {\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{4 \, b^{3}} \]
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\[ \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx=\int \frac {x^{2}}{\operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {b x^{2} + {\left (i \, \pi - 2 \, a\right )} x}{2 \, b^{2}} - \frac {{\left (\pi ^{2} + 4 i \, \pi a - 4 \, a^{2}\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{4 \, b^{3}} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {x^{2}}{2 \, b} - \frac {{\left (i \, \pi + 2 \, a\right )} x}{2 \, b^{2}} - \frac {{\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} \log \left (\pi - 2 i \, b x - 2 i \, a\right )}{4 \, b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 234, normalized size of antiderivative = 4.18 \[ \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx=\frac {x^2}{2\,b}+\frac {x\,\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{2\,b^2}+\frac {\ln \left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\right )\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+4\,a^2\right )}{4\,b^3} \]
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