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 = 0.0128323, antiderivative size = 31, normalized size of antiderivative = 1., 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*}
Mathematica [B] time = 0.0291241, 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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Maple [A] time = 0.014, size = 36, normalized size = 1.2 \begin{align*}{\frac{1}{4\,d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) }-{\frac{1}{4\,d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03287, size = 50, normalized size = 1.61 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32233, size = 126, normalized size = 4.06 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.726658, size = 41, normalized size = 1.32 \begin{align*} \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{\left (c \right )} + 5} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16046, size = 42, normalized size = 1.35 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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