Optimal. Leaf size=14 \[ \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{\tanh ^2(x)-4}}\right ) \]
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Rubi [A] time = 0.051154, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 3675
Rule 217
Rule 206
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*}
Mathematica [B] time = 0.0448098, 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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Maple [F] time = 0.165, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\rm sech} \left (x\right ) \right ) ^{2}{\frac{1}{\sqrt{-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{\tanh \left (x\right )^{2} - 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21427, size = 4, normalized size = 0.29 \begin{align*} 0 \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 (\tanh{\left (x \right )} - 2\right ) \left (\tanh{\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 = 1.18857, size = 132, normalized size = 9.43 \begin{align*} \log \left (\frac{\sqrt{{\left (\sqrt{3} + \frac{4}{3 \, e^{\left (2 \, x\right )} + 5} - 2\right )}^{2} + \frac{3 \,{\left (3 \, e^{\left (4 \, x\right )} + 10 \, e^{\left (2 \, x\right )} + 3\right )}}{{\left (3 \, e^{\left (2 \, x\right )} + 5\right )}^{2}}}}{\sqrt{{\left (\sqrt{3} - \frac{4}{3 \, e^{\left (2 \, x\right )} + 5} + 2\right )}^{2} + \frac{3 \,{\left (3 \, e^{\left (4 \, x\right )} + 10 \, e^{\left (2 \, x\right )} + 3\right )}}{{\left (3 \, e^{\left (2 \, x\right )} + 5\right )}^{2}}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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