Optimal. Leaf size=31 \[ \frac {3 \log (2 \sinh (c+d x)+3 \cosh (c+d x))}{10 d}-\frac {x}{5} \]
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Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, 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} \[ \frac {3 \log (2 \sinh (c+d x)+3 \cosh (c+d x))}{10 d}-\frac {x}{5} \]
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*}
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Mathematica [A] time = 0.04, size = 53, normalized size = 1.71 \[ -\frac {\log (1-\tanh (c+d x))}{20 d}-\frac {\log (\tanh (c+d x)+1)}{4 d}+\frac {3 \log (2 \tanh (c+d x)+3)}{10 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 49, normalized size = 1.58 \[ -\frac {5 \, d x - 3 \, \log \left (\frac {2 \, {\left (3 \, \cosh \left (d x + c\right ) + 2 \, \sinh \left (d x + c\right )\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 29, normalized size = 0.94 \[ -\frac {5 \, d x + 5 \, c - 3 \, \log \left (5 \, e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 46, normalized size = 1.48 \[ -\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{20 d}-\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{4 d}+\frac {3 \ln \left (2+3 \coth \left (d x +c \right )\right )}{10 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 28, normalized size = 0.90 \[ \frac {d x + c}{10 \, d} + \frac {3 \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 5\right )}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 22, normalized size = 0.71 \[ \frac {3\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+\frac {1}{5}\right )}{10\,d}-\frac {x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 42, normalized size = 1.35 \[ \begin {cases} \frac {x}{10} - \frac {3 \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{10 d} + \frac {3 \log {\left (\tanh {\left (c + d x \right )} + \frac {3}{2} \right )}}{10 d} & \text {for}\: d \neq 0 \\\frac {x}{6 \coth {\relax (c )} + 4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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