Optimal. Leaf size=29 \[ \frac{x}{a}-\frac{\tanh (c+d x)}{d (a \text{sech}(c+d x)+a)} \]
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Rubi [A] time = 0.0154826, antiderivative size = 29, 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 = {3777, 8} \[ \frac{x}{a}-\frac{\tanh (c+d x)}{d (a \text{sech}(c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 3777
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{a+a \text{sech}(c+d x)} \, dx &=-\frac{\tanh (c+d x)}{d (a+a \text{sech}(c+d x))}+\frac{\int a \, dx}{a^2}\\ &=\frac{x}{a}-\frac{\tanh (c+d x)}{d (a+a \text{sech}(c+d x))}\\ \end{align*}
Mathematica [A] time = 0.139259, size = 58, normalized size = 2. \[ \frac{\text{sech}\left (\frac{c}{2}\right ) \text{sech}\left (\frac{1}{2} (c+d x)\right ) \left (d x \cosh \left (c+\frac{d x}{2}\right )-2 \sinh \left (\frac{d x}{2}\right )+d x \cosh \left (\frac{d x}{2}\right )\right )}{2 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 58, normalized size = 2. \begin{align*} -{\frac{1}{da}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14442, size = 45, normalized size = 1.55 \begin{align*} \frac{d x + c}{a d} - \frac{2}{{\left (a e^{\left (-d x - c\right )} + a\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38467, size = 131, normalized size = 4.52 \begin{align*} \frac{d x \cosh \left (d x + c\right ) + d x \sinh \left (d x + c\right ) + d x + 2}{a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right ) + a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\operatorname{sech}{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14892, size = 42, normalized size = 1.45 \begin{align*} \frac{d x + c}{a d} + \frac{2}{a d{\left (e^{\left (d x + c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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