Optimal. Leaf size=9 \[ \frac{1}{2} \sin ^{-1}(2 \tanh (x)) \]
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Rubi [A] time = 0.0495234, antiderivative size = 9, 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 = {3675, 216} \[ \frac{1}{2} \sin ^{-1}(2 \tanh (x)) \]
Antiderivative was successfully verified.
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Rule 3675
Rule 216
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{\sqrt{1-4 \tanh ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-4 x^2}} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{2} \sin ^{-1}(2 \tanh (x))\\ \end{align*}
Mathematica [B] time = 0.053116, size = 47, normalized size = 5.22 \[ \frac{\sqrt{3 \cosh (2 x)-5} \text{sech}(x) \tanh ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{3 \sinh ^2(x)-1}}\right )}{2 \sqrt{2-8 \tanh ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.17, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\rm sech} \left (x\right ) \right ) ^{2}{\frac{1}{\sqrt{1-4\, \left ( \tanh \left ( x \right ) \right ) ^{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (x\right )^{2}}{\sqrt{-4 \, \tanh \left (x\right )^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.55978, size = 398, normalized size = 44.22 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{2 \, \sqrt{2}{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt{-\frac{3 \, \cosh \left (x\right )^{2} + 3 \, \sinh \left (x\right )^{2} - 5}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{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} - 5\right )} \sinh \left (x\right )^{2} - 10 \, \cosh \left (x\right )^{2} + 4 \,{\left (3 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3}\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{\operatorname{sech}^{2}{\left (x \right )}}{\sqrt{- \left (2 \tanh{\left (x \right )} - 1\right ) \left (2 \tanh{\left (x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21861, size = 59, normalized size = 6.56 \begin{align*} -\arctan \left (\frac{1}{3} \, \sqrt{3}{\left (\frac{2 \,{\left (\sqrt{3} \sqrt{-3 \, e^{\left (4 \, x\right )} + 10 \, e^{\left (2 \, x\right )} - 3} - 4\right )}}{3 \, e^{\left (2 \, x\right )} - 5} - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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