Optimal. Leaf size=19 \[ \sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )-x \]
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Rubi [A] time = 0.0282499, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1093, 207} \[ \sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )-x \]
Antiderivative was successfully verified.
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Rule 1093
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{\text{sech}^2(x)-\tanh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-3 x^2+2 x^4} \, dx,x,\tanh (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{-2+2 x^2} \, dx,x,\tanh (x)\right )-2 \operatorname{Subst}\left (\int \frac{1}{-1+2 x^2} \, dx,x,\tanh (x)\right )\\ &=-x+\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )\\ \end{align*}
Mathematica [A] time = 0.0894322, size = 19, normalized size = 1. \[ \sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )-x \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 54, normalized size = 2.8 \begin{align*} -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) +2 \right ) } \right ) +\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) -2 \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.67006, size = 86, normalized size = 4.53 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} + 1}{\sqrt{2} + e^{\left (-x\right )} - 1}\right ) - \frac{1}{2} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - 1}{\sqrt{2} + e^{\left (-x\right )} + 1}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.40961, size = 220, normalized size = 11.58 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{3 \,{\left (2 \, \sqrt{2} - 3\right )} \cosh \left (x\right )^{2} - 4 \,{\left (3 \, \sqrt{2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \,{\left (2 \, \sqrt{2} - 3\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt{2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}\right ) - x \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{1}{\left (- \tanh{\left (x \right )} + \operatorname{sech}{\left (x \right )}\right ) \left (\tanh{\left (x \right )} + \operatorname{sech}{\left (x \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15631, size = 55, normalized size = 2.89 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (\frac{{\left | -4 \, \sqrt{2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt{2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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