Optimal. Leaf size=9 \[ \frac {1}{2} \sin ^{-1}(2 \tanh (x)) \]
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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 = {3675, 216} \[ \frac {1}{2} \sin ^{-1}(2 \tanh (x)) \]
Antiderivative was successfully verified.
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Rule 216
Rule 3675
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*}
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Mathematica [B] time = 0.06, 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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fricas [B] time = 0.46, size = 118, normalized size = 13.11 \[ -\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {2} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \sqrt {-\frac {3 \, \cosh \relax (x)^{2} + 3 \, \sinh \relax (x)^{2} - 5}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{3 \, \cosh \relax (x)^{4} + 12 \, \cosh \relax (x) \sinh \relax (x)^{3} + 3 \, \sinh \relax (x)^{4} + 2 \, {\left (9 \, \cosh \relax (x)^{2} - 5\right )} \sinh \relax (x)^{2} - 10 \, \cosh \relax (x)^{2} + 4 \, {\left (3 \, \cosh \relax (x)^{3} - 5 \, \cosh \relax (x)\right )} \sinh \relax (x) + 3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 44, normalized size = 4.89 \[ -\arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (\sqrt {3} \sqrt {-3 \, e^{\left (4 \, x\right )} + 10 \, e^{\left (2 \, x\right )} - 3} - 4\right )}}{3 \, e^{\left (2 \, x\right )} - 5} - 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\relax (x )^{2}}{\sqrt {1-4 \left (\tanh ^{2}\relax (x )\right )}}\, dx \]
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 \[ \int \frac {\operatorname {sech}\relax (x)^{2}}{\sqrt {-4 \, \tanh \relax (x)^{2} + 1}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {cosh}\relax (x)}^2\,\sqrt {1-4\,{\mathrm {tanh}\relax (x)}^2}} \,d x \]
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 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{\sqrt {- \left (2 \tanh {\relax (x )} - 1\right ) \left (2 \tanh {\relax (x )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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