Optimal. Leaf size=31 \[ -\frac {3 \log (2 \cosh (c+d x)-3 \sinh (c+d x))}{10 d}-\frac {x}{5} \]
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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} \[ -\frac {3 \log (2 \cosh (c+d x)-3 \sinh (c+d x))}{10 d}-\frac {x}{5} \]
Antiderivative was successfully verified.
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Rule 3484
Rule 3530
Rubi steps
\begin {align*} \int \frac {1}{4-6 \tanh (c+d x)} \, dx &=-\frac {x}{5}-\frac {3}{10} i \int \frac {6 i-4 i \tanh (c+d x)}{4-6 \tanh (c+d x)} \, dx\\ &=-\frac {x}{5}-\frac {3 \log (2 \cosh (c+d x)-3 \sinh (c+d x))}{10 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 1.71 \[ -\frac {3 \log (2-3 \tanh (c+d x))}{10 d}+\frac {\log (1-\tanh (c+d x))}{4 d}+\frac {\log (\tanh (c+d x)+1)}{20 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 48, normalized size = 1.55 \[ \frac {d x - 3 \, \log \left (-\frac {2 \, {\left (2 \, \cosh \left (d x + c\right ) - 3 \, \sinh \left (d x + c\right )\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 25, normalized size = 0.81 \[ \frac {d x + c - 3 \, \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 5 \right |}\right )}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 46, normalized size = 1.48 \[ -\frac {3 \ln \left (-2+3 \tanh \left (d x +c \right )\right )}{10 d}+\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{4 d}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{20 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 29, normalized size = 0.94 \[ -\frac {1}{2} \, x - \frac {c}{2 \, d} - \frac {3 \, \log \left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 33, normalized size = 1.06 \[ \frac {\frac {3\,\ln \left (\mathrm {tanh}\left (c+d\,x\right )+1\right )}{10}-\frac {3\,\ln \left (3\,\mathrm {tanh}\left (c+d\,x\right )-2\right )}{10}}{d}-\frac {x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 42, normalized size = 1.35 \[ \begin {cases} - \frac {x}{2} - \frac {3 \log {\left (\tanh {\left (c + d x \right )} - \frac {2}{3} \right )}}{10 d} + \frac {3 \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{10 d} & \text {for}\: d \neq 0 \\\frac {x}{4 - 6 \tanh {\relax (c )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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