Integrand size = 13, antiderivative size = 98 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {4 x^3}{3 b^2}+\frac {2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {4 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}+\frac {4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^5} \]
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Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2199, 2190, 2189, 2188, 29} \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^5}+\frac {4 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^4}+\frac {2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}+\frac {4 x^3}{3 b^2} \]
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Rule 29
Rule 2188
Rule 2189
Rule 2190
Rule 2199
Rubi steps \begin{align*} \text {integral}& = -\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}+\frac {4 \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b} \\ & = \frac {4 x^3}{3 b^2}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}-\frac {\left (4 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^2} \\ & = \frac {4 x^3}{3 b^2}+\frac {2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}+\frac {\left (4 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {x}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^3} \\ & = \frac {4 x^3}{3 b^2}+\frac {2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {4 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}-\frac {\left (4 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^3\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^4} \\ & = \frac {4 x^3}{3 b^2}+\frac {2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {4 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}-\frac {\left (4 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^5} \\ & = \frac {4 x^3}{3 b^2}+\frac {2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {4 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}+\frac {4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^5} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {x^3}{3 b^2}-\frac {x^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {3 x \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac {\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^4}{b^5 \coth ^{-1}(\tanh (a+b x))}-\frac {4 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^5} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.56 (sec) , antiderivative size = 131085, normalized size of antiderivative = 1337.60
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.16 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {16 \, b^{4} x^{4} - 32 \, a b^{3} x^{3} + 96 \, a^{2} b^{2} x^{2} + 144 \, a^{3} b x - 3 \, \pi ^{4} - 6 i \, \pi ^{3} {\left (3 \, b x - 4 \, a\right )} - 48 \, a^{4} - 12 \, \pi ^{2} {\left (2 \, b^{2} x^{2} + 9 \, a b x - 6 \, a^{2}\right )} - 8 i \, \pi {\left (2 \, b^{3} x^{3} - 12 \, a b^{2} x^{2} - 27 \, a^{2} b x + 12 \, a^{3}\right )} - 12 \, {\left (16 \, a^{3} b x + \pi ^{4} - 2 i \, \pi ^{3} {\left (b x + 4 \, a\right )} + 16 \, a^{4} - 12 \, \pi ^{2} {\left (a b x + 2 \, a^{2}\right )} + 8 i \, \pi {\left (3 \, a^{2} b x + 4 \, a^{3}\right )}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{24 \, {\left (2 \, b^{6} x + i \, \pi b^{5} + 2 \, a b^{5}\right )}} \]
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\[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\int \frac {x^{4}}{\operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.83 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {16 \, b^{4} x^{4} - 3 \, \pi ^{4} - 24 i \, \pi ^{3} a + 72 \, \pi ^{2} a^{2} + 96 i \, \pi a^{3} - 48 \, a^{4} - 16 \, {\left (-i \, \pi b^{3} + 2 \, a b^{3}\right )} x^{3} - 24 \, {\left (\pi ^{2} b^{2} + 4 i \, \pi a b^{2} - 4 \, a^{2} b^{2}\right )} x^{2} - 18 \, {\left (-i \, \pi ^{3} b + 6 \, \pi ^{2} a b + 12 i \, \pi a^{2} b - 8 \, a^{3} b\right )} x}{24 \, {\left (2 \, b^{6} x - i \, \pi b^{5} + 2 \, a b^{5}\right )}} - \frac {{\left (i \, \pi ^{3} - 6 \, \pi ^{2} a - 12 i \, \pi a^{2} + 8 \, a^{3}\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{5}} \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.38 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {x^{3}}{3 \, b^{2}} - \frac {\pi ^{4} - 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} + 32 i \, \pi a^{3} + 16 \, a^{4}}{8 \, {\left (2 \, b^{6} x + i \, \pi b^{5} + 2 \, a b^{5}\right )}} - \frac {{\left (i \, \pi + 2 \, a\right )} x^{2}}{2 \, b^{3}} - \frac {3 \, {\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} x}{4 \, b^{4}} + \frac {{\left (i \, \pi ^{3} + 6 \, \pi ^{2} a - 12 i \, \pi a^{2} - 8 \, a^{3}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{5}} \]
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Time = 3.90 (sec) , antiderivative size = 669, normalized size of antiderivative = 6.83 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {x^3}{3\,b^2}-\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 )}^4+24\,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+16\,a^4-8\,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 )}^3-32\,a^3\,\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\,b\,\left (8\,a\,b^4+8\,b^5\,x-4\,b^4\,\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 {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 )}{2\,b^3}+\frac {3\,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^4}+\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 )}{2\,b^5} \]
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