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 = 0.01, antiderivative size = 28, 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 = {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 8
Rule 3479
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*}
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Mathematica [A] time = 0.11, 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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fricas [B] time = 0.54, size = 50, normalized size = 1.79 \[ \frac {{\left (2 \, d x - 1\right )} \cosh \left (d x + c\right ) + {\left (2 \, d x + 1\right )} \sinh \left (d x + c\right )}{4 \, {\left (a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 30, normalized size = 1.07 \[ \frac {\frac {2 \, {\left (d x + c\right )}}{a} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{a}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 54, normalized size = 1.93 \[ -\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{4 d a}-\frac {1}{2 d a \left (1+\tanh \left (d x +c \right )\right )}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{4 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 31, normalized size = 1.11 \[ \frac {d x + c}{2 \, a d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{4 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 25, normalized size = 0.89 \[ \frac {x}{2\,a}-\frac {1}{2\,a\,d\,\left (\mathrm {tanh}\left (c+d\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 73, normalized size = 2.61 \[ \begin {cases} \frac {d x \tanh {\left (c + d x \right )}}{2 a d \tanh {\left (c + d x \right )} + 2 a d} + \frac {d x}{2 a d \tanh {\left (c + d x \right )} + 2 a d} - \frac {1}{2 a d \tanh {\left (c + d x \right )} + 2 a d} & \text {for}\: d \neq 0 \\\frac {x}{a \tanh {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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