Optimal. Leaf size=56 \[ -\frac{3 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}-\frac{5 \tanh ^{-1}\left (\frac{\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{32 d}+\frac{5 x}{64} \]
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Rubi [A] time = 0.0358887, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2664, 12, 2657} \[ -\frac{3 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}-\frac{5 \tanh ^{-1}\left (\frac{\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{32 d}+\frac{5 x}{64} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 12
Rule 2657
Rubi steps
\begin{align*} \int \frac{1}{(5+3 \cosh (c+d x))^2} \, dx &=-\frac{3 \sinh (c+d x)}{16 d (5+3 \cosh (c+d x))}-\frac{1}{16} \int -\frac{5}{5+3 \cosh (c+d x)} \, dx\\ &=-\frac{3 \sinh (c+d x)}{16 d (5+3 \cosh (c+d x))}+\frac{5}{16} \int \frac{1}{5+3 \cosh (c+d x)} \, dx\\ &=\frac{5 x}{64}-\frac{5 \tanh ^{-1}\left (\frac{\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{32 d}-\frac{3 \sinh (c+d x)}{16 d (5+3 \cosh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.112503, size = 45, normalized size = 0.8 \[ \frac{5 \tanh ^{-1}\left (\frac{1}{2} \tanh \left (\frac{1}{2} (c+d x)\right )\right )-\frac{6 \sinh (c+d x)}{3 \cosh (c+d x)+5}}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 72, normalized size = 1.3 \begin{align*}{\frac{3}{32\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-1}}+{\frac{5}{64\,d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) }+{\frac{3}{32\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-1}}-{\frac{5}{64\,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 = 1.04265, size = 109, normalized size = 1.95 \begin{align*} -\frac{5 \, \log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{64 \, d} + \frac{5 \, \log \left (e^{\left (-d x - c\right )} + 3\right )}{64 \, d} - \frac{5 \, e^{\left (-d x - c\right )} + 3}{8 \, d{\left (10 \, e^{\left (-d x - c\right )} + 3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.57343, size = 621, normalized size = 11.09 \begin{align*} \frac{5 \,{\left (3 \, \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right )^{2} + 10 \, \cosh \left (d x + c\right ) + 3\right )} \log \left (3 \, \cosh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right ) + 1\right ) - 5 \,{\left (3 \, \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right )^{2} + 10 \, \cosh \left (d x + c\right ) + 3\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 3\right ) + 40 \, \cosh \left (d x + c\right ) + 40 \, \sinh \left (d x + c\right ) + 24}{64 \,{\left (3 \, d \cosh \left (d x + c\right )^{2} + 3 \, d \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right ) + 2 \,{\left (3 \, d \cosh \left (d x + c\right ) + 5 \, d\right )} \sinh \left (d x + c\right ) + 3 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.42525, size = 199, normalized size = 3.55 \begin{align*} \begin{cases} - \frac{5 \log{\left (\tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 2 \right )} \tanh ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{64 d \tanh ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 256 d} + \frac{20 \log{\left (\tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 2 \right )}}{64 d \tanh ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 256 d} + \frac{5 \log{\left (\tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 \right )} \tanh ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{64 d \tanh ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 256 d} - \frac{20 \log{\left (\tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 \right )}}{64 d \tanh ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 256 d} + \frac{12 \tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{64 d \tanh ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} - 256 d} & \text{for}\: d \neq 0 \\\frac{x}{\left (3 \cosh{\left (c \right )} + 5\right )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15713, size = 93, normalized size = 1.66 \begin{align*} \frac{5 \, \log \left (3 \, e^{\left (d x + c\right )} + 1\right )}{64 \, d} - \frac{5 \, \log \left (e^{\left (d x + c\right )} + 3\right )}{64 \, d} + \frac{5 \, e^{\left (d x + c\right )} + 3}{8 \, d{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} + 10 \, e^{\left (d x + c\right )} + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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