Optimal. Leaf size=25 \[ \frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{2 \sqrt{2}}+\frac{\tanh (x)}{2} \]
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Rubi [A] time = 0.0174735, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3209, 388, 206} \[ \frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{2 \sqrt{2}}+\frac{\tanh (x)}{2} \]
Antiderivative was successfully verified.
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Rule 3209
Rule 388
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{1-\sinh ^4(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1-x^2}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{2 \sqrt{2}}+\frac{\tanh (x)}{2}\\ \end{align*}
Mathematica [A] time = 0.0939069, size = 24, normalized size = 0.96 \[ \frac{1}{4} \left (\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )+2 \tanh (x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 55, normalized size = 2.2 \begin{align*}{\tanh \left ({\frac{x}{2}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1}}+{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) -2 \right ) } \right ) }+{\frac{\sqrt{2}}{4}{\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 = 1.44069, size = 93, normalized size = 3.72 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} + 1}{\sqrt{2} + e^{\left (-x\right )} - 1}\right ) - \frac{1}{8} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - 1}{\sqrt{2} + e^{\left (-x\right )} + 1}\right ) + \frac{1}{e^{\left (-2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15747, size = 382, normalized size = 15.28 \begin{align*} \frac{{\left (\sqrt{2} \cosh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{2} \sinh \left (x\right )^{2} + \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 ) - 8}{8 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16645, size = 65, normalized size = 2.6 \begin{align*} -\frac{1}{8} \, \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 ) - \frac{1}{e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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