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 = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 2657
Rule 2735
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*}
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Mathematica [A] time = 0.08, 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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fricas [A] time = 2.70, size = 45, normalized size = 1.25 \[ \frac {5}{3} \, \sqrt {3} \log \left (-\frac {2 \, {\left (\sqrt {3} - 2\right )} \cosh \relax (x) - {\left (2 \, \sqrt {3} - 3\right )} \sinh \relax (x) - \sqrt {3} + 2}{\cosh \relax (x) - 2}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 37, normalized size = 1.03 \[ -\frac {5}{3} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, e^{x} - 4 \right |}}{{\left | 2 \, \sqrt {3} + 2 \, e^{x} - 4 \right |}}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 32, normalized size = 0.89 \[ \frac {10 \sqrt {3}\, \arctanh \left (\tanh \left (\frac {x}{2}\right ) \sqrt {3}\right )}{3}+\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 34, normalized size = 0.94 \[ \frac {5}{3} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - e^{\left (-x\right )} + 2}{\sqrt {3} + e^{\left (-x\right )} - 2}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 48, normalized size = 1.33 \[ \frac {5\,\sqrt {3}\,\ln \left (10\,{\mathrm {e}}^x+\frac {5\,\sqrt {3}\,\left (4\,{\mathrm {e}}^x-2\right )}{3}\right )}{3}-\frac {5\,\sqrt {3}\,\ln \left (10\,{\mathrm {e}}^x-\frac {5\,\sqrt {3}\,\left (4\,{\mathrm {e}}^x-2\right )}{3}\right )}{3}-x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.79, size = 44, normalized size = 1.22 \[ - x - \frac {5 \sqrt {3} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {\sqrt {3}}{3} \right )}}{3} + \frac {5 \sqrt {3} \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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