Optimal. Leaf size=98 \[ \frac{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac{4 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^4}+\frac{4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^5}-\frac{x^4}{b \coth ^{-1}(\tanh (a+b x))}+\frac{4 x^3}{3 b^2} \]
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Rubi [A] time = 0.0808205, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2168, 2159, 2158, 2157, 29} \[ \frac{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac{4 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^4}+\frac{4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^5}-\frac{x^4}{b \coth ^{-1}(\tanh (a+b x))}+\frac{4 x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2159
Rule 2158
Rule 2157
Rule 29
Rubi steps
\begin{align*} \int \frac{x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx &=-\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 ) \operatorname{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*}
Mathematica [A] time = 0.0974712, size = 106, normalized size = 1.08 \[ -\frac{x^2 \left (\coth ^{-1}(\tanh (a+b x))-b x\right )}{b^3}-\frac{\left (\coth ^{-1}(\tanh (a+b x))-b x\right )^4}{b^5 \coth ^{-1}(\tanh (a+b x))}+\frac{3 x \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^2}{b^4}-\frac{4 \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^5}+\frac{x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 6.12, size = 131085, normalized size = 1337.6 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.48721, size = 240, normalized size = 2.45 \begin{align*} \frac{4 \,{\left (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} +{\left (16 i \, \pi b^{3} - 32 \, a b^{3}\right )} x^{3} -{\left (24 \, \pi ^{2} b^{2} + 96 i \, \pi a b^{2} - 96 \, a^{2} b^{2}\right )} x^{2} +{\left (18 i \, \pi ^{3} b - 108 \, \pi ^{2} a b - 216 i \, \pi a^{2} b + 144 \, a^{3} b\right )} x\right )}}{192 \, b^{6} x - 96 i \, \pi b^{5} + 192 \, a b^{5}} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.66643, size = 729, normalized size = 7.44 \begin{align*} \frac{16 \, b^{5} x^{5} - 16 \, a b^{4} x^{4} + 9 \, \pi ^{4} a + 24 \, \pi ^{2} a^{3} - 48 \, a^{5} - 32 \,{\left (\pi ^{2} b^{3} - 2 \, a^{2} b^{3}\right )} x^{3} - 12 \,{\left (7 \, \pi ^{2} a b^{2} - 20 \, a^{3} b^{2}\right )} x^{2} - 12 \,{\left (\pi ^{4} b - 6 \, \pi ^{2} a^{2} b - 8 \, a^{4} b\right )} x - 12 \,{\left (\pi ^{5} - 8 \, \pi ^{3} a^{2} - 48 \, \pi a^{4} + 4 \,{\left (\pi ^{3} b^{2} - 12 \, \pi a^{2} b^{2}\right )} x^{2} + 8 \,{\left (\pi ^{3} a b - 12 \, \pi a^{3} b\right )} x\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 ) + 6 \,{\left (3 \, \pi ^{4} a + 8 \, \pi ^{2} a^{3} - 16 \, a^{5} + 4 \,{\left (3 \, \pi ^{2} a b^{2} - 4 \, a^{3} b^{2}\right )} x^{2} + 8 \,{\left (3 \, \pi ^{2} a^{2} b - 4 \, a^{4} b\right )} x\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )}{12 \,{\left (4 \, b^{7} x^{2} + 8 \, a b^{6} x + \pi ^{2} b^{5} + 4 \, a^{2} b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{acoth}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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