Optimal. Leaf size=9 \[ \sinh ^{-1}\left (\frac {\tanh (x)}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 9, 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 = {4146, 215} \[ \sinh ^{-1}\left (\frac {\tanh (x)}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 4146
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*}
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Mathematica [B] time = 0.04, 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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fricas [B] time = 0.42, size = 112, normalized size = 12.44 \[ -\log \left (-\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) - \sinh \relax (x)^{2} + \sqrt {\frac {2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} + 1}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}\right ) + \log \left (-\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) - \sinh \relax (x)^{2} + \sqrt {\frac {2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} + 1}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 44, normalized size = 4.89 \[ -\log \left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) + \log \left (-\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\relax (x )^{2}}{\sqrt {4-\mathrm {sech}\relax (x )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}\relax (x)^{2}}{\sqrt {-\operatorname {sech}\relax (x)^{2} + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.11 \[ \int \frac {1}{{\mathrm {cosh}\relax (x)}^2\,\sqrt {4-\frac {1}{{\mathrm {cosh}\relax (x)}^2}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{\sqrt {- \left (\operatorname {sech}{\relax (x )} - 2\right ) \left (\operatorname {sech}{\relax (x )} + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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