Optimal. Leaf size=34 \[ \frac{4 x}{\sqrt{5}}-x-\frac{8 \tanh ^{-1}\left (\frac{\cosh (x)}{\sinh (x)+\sqrt{5}+2}\right )}{\sqrt{5}} \]
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Rubi [A] time = 0.037192, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2735, 2657} \[ \frac{4 x}{\sqrt{5}}-x-\frac{8 \tanh ^{-1}\left (\frac{\cosh (x)}{\sinh (x)+\sqrt{5}+2}\right )}{\sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2657
Rubi steps
\begin{align*} \int \frac{2-\sinh (x)}{2+\sinh (x)} \, dx &=-x+4 \int \frac{1}{2+\sinh (x)} \, dx\\ &=-x+\frac{4 x}{\sqrt{5}}-\frac{8 \tanh ^{-1}\left (\frac{\cosh (x)}{2+\sqrt{5}+\sinh (x)}\right )}{\sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.0855739, size = 28, normalized size = 0.82 \[ -x-\frac{8 \tanh ^{-1}\left (\frac{1-2 \tanh \left (\frac{x}{2}\right )}{\sqrt{5}}\right )}{\sqrt{5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 37, normalized size = 1.1 \begin{align*} -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +{\frac{8\,\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\tanh \left ( x/2 \right ) -1 \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56699, size = 46, normalized size = 1.35 \begin{align*} \frac{4}{5} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - e^{\left (-x\right )} + 2}{\sqrt{5} + e^{\left (-x\right )} - 2}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75286, size = 139, normalized size = 4.09 \begin{align*} \frac{4}{5} \, \sqrt{5} \log \left (-\frac{{\left (2 \, \sqrt{5} - 5\right )} \cosh \left (x\right ) - 2 \,{\left (\sqrt{5} - 2\right )} \sinh \left (x\right ) + \sqrt{5} - 2}{\sinh \left (x\right ) + 2}\right ) - x \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 [A] time = 1.20581, size = 45, normalized size = 1.32 \begin{align*} \frac{4}{5} \, \sqrt{5} \log \left (\frac{{\left | -2 \, \sqrt{5} + 2 \, e^{x} + 4 \right |}}{2 \,{\left (\sqrt{5} + e^{x} + 2\right )}}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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