Optimal. Leaf size=36 \[ \frac{5 x}{\sqrt{3}}-x+\frac{10 \tanh ^{-1}\left (\frac{\sinh (x)}{-\cosh (x)+\sqrt{3}+2}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.049653, antiderivative size = 36, 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} \[ \frac{5 x}{\sqrt{3}}-x+\frac{10 \tanh ^{-1}\left (\frac{\sinh (x)}{-\cosh (x)+\sqrt{3}+2}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2657
Rubi steps
\begin{align*} \int \frac{3+\cosh (x)}{2-\cosh (x)} \, dx &=-x+5 \int \frac{1}{2-\cosh (x)} \, dx\\ &=-x+\frac{5 x}{\sqrt{3}}+\frac{10 \tanh ^{-1}\left (\frac{\sinh (x)}{2+\sqrt{3}-\cosh (x)}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0715123, size = 24, normalized size = 0.67 \[ \frac{10 \tanh ^{-1}\left (\sqrt{3} \tanh \left (\frac{x}{2}\right )\right )}{\sqrt{3}}-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 32, normalized size = 0.9 \begin{align*} -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +{\frac{10\,\sqrt{3}}{3}{\it Artanh} \left ( \tanh \left ({\frac{x}{2}} \right ) \sqrt{3} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53155, size = 46, normalized size = 1.28 \begin{align*} \frac{5}{3} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - e^{\left (-x\right )} + 2}{\sqrt{3} + 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 = 2.24681, size = 139, normalized size = 3.86 \begin{align*} \frac{5}{3} \, \sqrt{3} \log \left (-\frac{2 \,{\left (\sqrt{3} - 2\right )} \cosh \left (x\right ) -{\left (2 \, \sqrt{3} - 3\right )} \sinh \left (x\right ) - \sqrt{3} + 2}{\cosh \left (x\right ) - 2}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.07708, size = 44, normalized size = 1.22 \begin{align*} - x - \frac{5 \sqrt{3} \log{\left (\tanh{\left (\frac{x}{2} \right )} - \frac{\sqrt{3}}{3} \right )}}{3} + \frac{5 \sqrt{3} \log{\left (\tanh{\left (\frac{x}{2} \right )} + \frac{\sqrt{3}}{3} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19526, size = 50, normalized size = 1.39 \begin{align*} -\frac{5}{3} \, \sqrt{3} \log \left (\frac{{\left | -2 \, \sqrt{3} + 2 \, e^{x} - 4 \right |}}{{\left | 2 \, \sqrt{3} + 2 \, e^{x} - 4 \right |}}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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