Optimal. Leaf size=26 \[ -\frac {x}{a}+\frac {2 \sinh (x)}{a}-\frac {\sinh (x)}{a \text {sech}(x)+a} \]
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Rubi [A] time = 0.06, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3819, 3787, 2637, 8} \[ -\frac {x}{a}+\frac {2 \sinh (x)}{a}-\frac {\sinh (x)}{a \text {sech}(x)+a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2637
Rule 3787
Rule 3819
Rubi steps
\begin {align*} \int \frac {\cosh (x)}{a+a \text {sech}(x)} \, dx &=-\frac {\sinh (x)}{a+a \text {sech}(x)}-\frac {\int \cosh (x) (-2 a+a \text {sech}(x)) \, dx}{a^2}\\ &=-\frac {\sinh (x)}{a+a \text {sech}(x)}-\frac {\int 1 \, dx}{a}+\frac {2 \int \cosh (x) \, dx}{a}\\ &=-\frac {x}{a}+\frac {2 \sinh (x)}{a}-\frac {\sinh (x)}{a+a \text {sech}(x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 32, normalized size = 1.23 \[ \frac {-2 x+3 \tanh \left (\frac {x}{2}\right )+\sinh \left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 47, normalized size = 1.81 \[ -\frac {2 \, x \cosh \relax (x) - \cosh \relax (x)^{2} + 2 \, {\left (x - \cosh \relax (x) - 1\right )} \sinh \relax (x) - \sinh \relax (x)^{2} + 2 \, x + 5}{2 \, {\left (a \cosh \relax (x) + a \sinh \relax (x) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 35, normalized size = 1.35 \[ -\frac {x}{a} - \frac {{\left (5 \, e^{x} + 1\right )} e^{\left (-x\right )}}{2 \, a {\left (e^{x} + 1\right )}} + \frac {e^{x}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 59, normalized size = 2.27 \[ \frac {\tanh \left (\frac {x}{2}\right )}{a}-\frac {1}{a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {1}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 41, normalized size = 1.58 \[ -\frac {x}{a} + \frac {5 \, e^{\left (-x\right )} + 1}{2 \, {\left (a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )}\right )}} - \frac {e^{\left (-x\right )}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 34, normalized size = 1.31 \[ \frac {{\mathrm {e}}^x}{2\,a}-\frac {x}{a}-\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {{\mathrm {e}}^{-x}}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cosh {\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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