Optimal. Leaf size=14 \[ \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {\tanh ^2(x)-4}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3675, 217, 206} \[ \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {\tanh ^2(x)-4}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 3675
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(x)}{\sqrt {-4+\tanh ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {-4+x^2}} \, dx,x,\tanh (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {-4+\tanh ^2(x)}}\right )\\ &=\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {-4+\tanh ^2(x)}}\right )\\ \end {align*}
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Mathematica [B] time = 0.04, size = 46, normalized size = 3.29 \[ \frac {\sqrt {3 \cosh (2 x)+5} \text {sech}(x) \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {3 \sinh ^2(x)+4}}\right )}{\sqrt {2} \sqrt {\tanh ^2(x)-4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 1, normalized size = 0.07 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}\relax (x)^{2}}{\sqrt {\tanh \relax (x)^{2} - 4}}\,{d x} \]
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 \[ \int \frac {\mathrm {sech}\relax (x )^{2}}{\sqrt {-4+\tanh ^{2}\relax (x )}}\, dx \]
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 \[ \int \frac {\operatorname {sech}\relax (x)^{2}}{\sqrt {\tanh \relax (x)^{2} - 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{{\mathrm {cosh}\relax (x)}^2\,\sqrt {{\mathrm {tanh}\relax (x)}^2-4}} \,d x \]
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 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{\sqrt {\left (\tanh {\relax (x )} - 2\right ) \left (\tanh {\relax (x )} + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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