Optimal. Leaf size=31 \[ -\frac{3 \log (3 \cosh (c+d x)-2 \sinh (c+d x))}{10 d}-\frac{x}{5} \]
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Rubi [A] time = 0.0424359, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3484, 3530} \[ -\frac{3 \log (3 \cosh (c+d x)-2 \sinh (c+d x))}{10 d}-\frac{x}{5} \]
Antiderivative was successfully verified.
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Rule 3484
Rule 3530
Rubi steps
\begin{align*} \int \frac{1}{4-6 \coth (c+d x)} \, dx &=-\frac{x}{5}-\frac{3}{10} i \int \frac{6 i-4 i \coth (c+d x)}{4-6 \coth (c+d x)} \, dx\\ &=-\frac{x}{5}-\frac{3 \log (3 \cosh (c+d x)-2 \sinh (c+d x))}{10 d}\\ \end{align*}
Mathematica [A] time = 0.0374029, size = 53, normalized size = 1.71 \[ -\frac{3 \log (3-2 \tanh (c+d x))}{10 d}+\frac{\log (1-\tanh (c+d x))}{4 d}+\frac{\log (\tanh (c+d x)+1)}{20 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 46, normalized size = 1.5 \begin{align*}{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ) }{20\,d}}-{\frac{3\,\ln \left ( -2+3\,{\rm coth} \left (dx+c\right ) \right ) }{10\,d}}+{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ) }{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0579, size = 39, normalized size = 1.26 \begin{align*} -\frac{1}{2} \, x - \frac{c}{2 \, d} - \frac{3 \, \log \left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{10 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49815, size = 126, normalized size = 4.06 \begin{align*} \frac{d x - 3 \, \log \left (\frac{2 \,{\left (3 \, \cosh \left (d x + c\right ) - 2 \, \sinh \left (d x + c\right )\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{10 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.18326, size = 42, normalized size = 1.35 \begin{align*} \begin{cases} - \frac{x}{2} - \frac{3 \log{\left (\tanh{\left (c + d x \right )} - \frac{3}{2} \right )}}{10 d} + \frac{3 \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{10 d} & \text{for}\: d \neq 0 \\\frac{x}{4 - 6 \coth{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19247, size = 38, normalized size = 1.23 \begin{align*} \frac{d x + c}{10 \, d} - \frac{3 \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 5\right )}{10 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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