Optimal. Leaf size=12 \[ -\frac {1}{3 \left (\tanh ^3(x)+2\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 261
Rule 4342
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*}
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Mathematica [A] time = 0.04, size = 12, normalized size = 1.00 \[ -\frac {1}{3 \left (\tanh ^3(x)+2\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 73, normalized size = 6.08 \[ -\frac {8 \, {\left (\cosh \relax (x)^{2} + \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}}{9 \, {\left (3 \, \cosh \relax (x)^{4} + 12 \, \cosh \relax (x) \sinh \relax (x)^{3} + 3 \, \sinh \relax (x)^{4} + 2 \, {\left (9 \, \cosh \relax (x)^{2} + 2\right )} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x)^{2} + 4 \, {\left (3 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 32, normalized size = 2.67 \[ -\frac {2 \, {\left (3 \, e^{\left (4 \, x\right )} + 1\right )}}{9 \, {\left (3 \, e^{\left (6 \, x\right )} + 3 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 11, normalized size = 0.92 \[ -\frac {1}{3 \left (2+\tanh ^{3}\relax (x )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 10, normalized size = 0.83 \[ -\frac {1}{3 \, {\left (\tanh \relax (x)^{3} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.80, size = 32, normalized size = 2.67 \[ -\frac {\frac {2\,{\mathrm {e}}^{4\,x}}{3}+\frac {2}{9}}{9\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+3\,{\mathrm {e}}^{6\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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