Optimal. Leaf size=28 \[ \frac{x}{2 a}-\frac{1}{2 d (a \tanh (c+d x)+a)} \]
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Rubi [A] time = 0.0128619, antiderivative size = 28, 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 = {3479, 8} \[ \frac{x}{2 a}-\frac{1}{2 d (a \tanh (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 3479
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{a+a \tanh (c+d x)} \, dx &=-\frac{1}{2 d (a+a \tanh (c+d x))}+\frac{\int 1 \, dx}{2 a}\\ &=\frac{x}{2 a}-\frac{1}{2 d (a+a \tanh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.100759, size = 39, normalized size = 1.39 \[ \frac{(2 d x+1) \tanh (c+d x)+2 d x-1}{4 a d (\tanh (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 54, normalized size = 1.9 \begin{align*} -{\frac{1}{2\,da \left ( \tanh \left ( dx+c \right ) +1 \right ) }}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) }{4\,da}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{4\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08422, size = 42, normalized size = 1.5 \begin{align*} \frac{d x + c}{2 \, a d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{4 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.19028, size = 136, normalized size = 4.86 \begin{align*} \frac{{\left (2 \, d x - 1\right )} \cosh \left (d x + c\right ) +{\left (2 \, d x + 1\right )} \sinh \left (d x + c\right )}{4 \,{\left (a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.8272, size = 73, normalized size = 2.61 \begin{align*} \begin{cases} \frac{d x \tanh{\left (c + d x \right )}}{2 a d \tanh{\left (c + d x \right )} + 2 a d} + \frac{d x}{2 a d \tanh{\left (c + d x \right )} + 2 a d} - \frac{1}{2 a d \tanh{\left (c + d x \right )} + 2 a d} & \text{for}\: d \neq 0 \\\frac{x}{a \tanh{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18865, size = 36, normalized size = 1.29 \begin{align*} \frac{2 \, d x + 2 \, c - e^{\left (-2 \, d x - 2 \, c\right )}}{4 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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