Optimal. Leaf size=31 \[ \frac {x}{4}-\frac {\tanh ^{-1}\left (\frac {\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{2 d} \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2657} \[ \frac {x}{4}-\frac {\tanh ^{-1}\left (\frac {\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2657
Rubi steps
\begin {align*} \int \frac {1}{5+3 \cosh (c+d x)} \, dx &=\frac {x}{4}-\frac {\tanh ^{-1}\left (\frac {\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{2 d}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 77, normalized size = 2.48 \[ \frac {\log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )+2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{4 d}-\frac {\log \left (2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )-\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.60, size = 42, normalized size = 1.35 \[ \frac {\log \left (3 \, \cosh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right ) + 1\right ) - \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 3\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 28, normalized size = 0.90 \[ \frac {\log \left (3 \, e^{\left (d x + c\right )} + 1\right ) - \log \left (e^{\left (d x + c\right )} + 3\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 36, normalized size = 1.16 \[ \frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{4 d}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{4 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 37, normalized size = 1.19 \[ -\frac {\log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{4 \, d} + \frac {\log \left (e^{\left (-d x - c\right )} + 3\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 40, normalized size = 1.29 \[ -\frac {\mathrm {atan}\left (\frac {5\,\sqrt {-d^2}+3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{4\,d}\right )}{2\,\sqrt {-d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 41, normalized size = 1.32 \[ \begin {cases} - \frac {\log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )}}{4 d} + \frac {\log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )}}{4 d} & \text {for}\: d \neq 0 \\\frac {x}{3 \cosh {\relax (c )} + 5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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