Optimal. Leaf size=12 \[ -\frac{1}{3 \left (\tanh ^3(x)+2\right )} \]
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Rubi [A] time = 0.0857805, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4342, 261} \[ -\frac{1}{3 \left (\tanh ^3(x)+2\right )} \]
Antiderivative was successfully verified.
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Rule 4342
Rule 261
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x) \tanh ^2(x)}{\left (2+\tanh ^3(x)\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{x^2}{\left (2+x^3\right )^2} \, dx,x,\tanh (x)\right )\\ &=-\frac{1}{3 \left (2+\tanh ^3(x)\right )}\\ \end{align*}
Mathematica [A] time = 0.0418217, size = 12, normalized size = 1. \[ -\frac{1}{3 \left (\tanh ^3(x)+2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 11, normalized size = 0.9 \begin{align*} -{\frac{1}{6+3\, \left ( \tanh \left ( x \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991249, size = 14, normalized size = 1.17 \begin{align*} -\frac{1}{3 \,{\left (\tanh \left (x\right )^{3} + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99299, size = 250, normalized size = 20.83 \begin{align*} -\frac{8 \,{\left (\cosh \left (x\right )^{2} + \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}}{9 \,{\left (3 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 3 \, \sinh \left (x\right )^{4} + 2 \,{\left (9 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 4 \,{\left (3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.51189, size = 43, normalized size = 3.58 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (4 \, x\right )} + 1\right )}}{9 \,{\left (3 \, e^{\left (6 \, x\right )} + 3 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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