Optimal. Leaf size=9 \[ \sinh ^{-1}\left (\frac{\tanh (x)}{\sqrt{3}}\right ) \]
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Rubi [A] time = 0.0494739, 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 = {4146, 215} \[ \sinh ^{-1}\left (\frac{\tanh (x)}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Rule 4146
Rule 215
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{\sqrt{4-\text{sech}^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{3+x^2}} \, dx,x,\tanh (x)\right )\\ &=\sinh ^{-1}\left (\frac{\tanh (x)}{\sqrt{3}}\right )\\ \end{align*}
Mathematica [B] time = 0.0462612, size = 43, normalized size = 4.78 \[ \frac{\sqrt{2 \cosh (2 x)+1} \text{sech}(x) \tanh ^{-1}\left (\frac{\sinh (x)}{\sqrt{4 \sinh ^2(x)+3}}\right )}{\sqrt{4-\text{sech}^2(x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\rm sech} \left (x\right ) \right ) ^{2}{\frac{1}{\sqrt{4- \left ({\rm sech} \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{-\operatorname{sech}\left (x\right )^{2} + 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.40214, size = 367, normalized size = 40.78 \begin{align*} -\log \left (-\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + \sqrt{\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}\right ) + \log \left (-\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + \sqrt{\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 2\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 (\operatorname{sech}{\left (x \right )} - 2\right ) \left (\operatorname{sech}{\left (x \right )} + 2\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16656, size = 59, normalized size = 6.56 \begin{align*} -\log \left (\sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) + \log \left (-\sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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