Optimal. Leaf size=31 \[ x-\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{\sqrt{2}}+\frac{\tanh (x)}{1-2 \tanh ^2(x)} \]
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Rubi [A] time = 0.054305, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {414, 12, 481, 206} \[ x-\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{\sqrt{2}}+\frac{\tanh (x)}{1-2 \tanh ^2(x)} \]
Antiderivative was successfully verified.
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Rule 414
Rule 12
Rule 481
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (\text{sech}^2(x)-\tanh ^2(x)\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1-2 x^2\right )^2 \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{1-2 \tanh ^2(x)}+\frac{1}{2} \operatorname{Subst}\left (\int -\frac{2 x^2}{\left (1-2 x^2\right ) \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{1-2 \tanh ^2(x)}-\operatorname{Subst}\left (\int \frac{x^2}{\left (1-2 x^2\right ) \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{1-2 \tanh ^2(x)}-\operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\tanh (x)\right )+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=x-\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{\sqrt{2}}+\frac{\tanh (x)}{1-2 \tanh ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.156086, size = 42, normalized size = 1.35 \[ \frac{-3 x-\sinh (2 x)+x \cosh (2 x)}{\cosh (2 x)-3}-\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 108, normalized size = 3.5 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\frac{1}{2} \left ( 2-2\,\tanh \left ( x/2 \right ) \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,\tanh \left ( x/2 \right ) -1 \right ) ^{-1}}-{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) +2 \right ) } \right ) }+{\frac{1}{2} \left ( 2\,\tanh \left ( x/2 \right ) +2 \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) -1 \right ) ^{-1}}-{\frac{\sqrt{2}}{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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.76692, size = 119, normalized size = 3.84 \begin{align*} -\frac{1}{4} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} + 1}{\sqrt{2} + e^{\left (-x\right )} - 1}\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - 1}{\sqrt{2} + e^{\left (-x\right )} + 1}\right ) + x - \frac{2 \,{\left (3 \, e^{\left (-2 \, x\right )} - 1\right )}}{6 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.28395, size = 883, normalized size = 28.48 \begin{align*} \frac{4 \, x \cosh \left (x\right )^{4} + 16 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + 4 \, x \sinh \left (x\right )^{4} - 24 \,{\left (x + 1\right )} \cosh \left (x\right )^{2} + 24 \,{\left (x \cosh \left (x\right )^{2} - x - 1\right )} \sinh \left (x\right )^{2} +{\left (\sqrt{2} \cosh \left (x\right )^{4} + 4 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt{2} \sinh \left (x\right )^{4} + 6 \,{\left (\sqrt{2} \cosh \left (x\right )^{2} - \sqrt{2}\right )} \sinh \left (x\right )^{2} - 6 \, \sqrt{2} \cosh \left (x\right )^{2} + 4 \,{\left (\sqrt{2} \cosh \left (x\right )^{3} - 3 \, \sqrt{2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt{2}\right )} \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 ) + 16 \,{\left (x \cosh \left (x\right )^{3} - 3 \,{\left (x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4 \, x + 8}{4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 6 \,{\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \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 )^{2} \left (\tanh{\left (x \right )} + \operatorname{sech}{\left (x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1434, size = 85, normalized size = 2.74 \begin{align*} \frac{1}{4} \, \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 - \frac{2 \,{\left (3 \, e^{\left (2 \, x\right )} - 1\right )}}{e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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