Optimal. Leaf size=17 \[ \frac {\cosh (x)}{a}-\frac {\log (\cosh (x)+1)}{a} \]
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Rubi [A] time = 0.07, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3872, 2833, 12, 43} \[ \frac {\cosh (x)}{a}-\frac {\log (\cosh (x)+1)}{a} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{a+a \text {sech}(x)} \, dx &=-\int \frac {\cosh (x) \sinh (x)}{-a-a \cosh (x)} \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x}{a (-a+x)} \, dx,x,-a \cosh (x)\right )}{a}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x}{-a+x} \, dx,x,-a \cosh (x)\right )}{a^2}\\ &=-\frac {\operatorname {Subst}\left (\int \left (1-\frac {a}{a-x}\right ) \, dx,x,-a \cosh (x)\right )}{a^2}\\ &=\frac {\cosh (x)}{a}-\frac {\log (1+\cosh (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 16, normalized size = 0.94 \[ \frac {\cosh (x)-2 \log \left (\cosh \left (\frac {x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 50, normalized size = 2.94 \[ \frac {2 \, x \cosh \relax (x) + \cosh \relax (x)^{2} - 4 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (x + \cosh \relax (x)\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 1}{2 \, {\left (a \cosh \relax (x) + a \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 32, normalized size = 1.88 \[ \frac {x}{a} + \frac {e^{\left (-x\right )}}{2 \, a} + \frac {e^{x}}{2 \, a} - \frac {2 \, \log \left (e^{x} + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 27, normalized size = 1.59 \[ -\frac {\ln \left (1+\mathrm {sech}\relax (x )\right )}{a}+\frac {1}{a \,\mathrm {sech}\relax (x )}+\frac {\ln \left (\mathrm {sech}\relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 35, normalized size = 2.06 \[ -\frac {x}{a} + \frac {e^{\left (-x\right )}}{2 \, a} + \frac {e^{x}}{2 \, a} - \frac {2 \, \log \left (e^{\left (-x\right )} + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 15, normalized size = 0.88 \[ -\frac {\ln \left (\mathrm {cosh}\relax (x)+1\right )-\mathrm {cosh}\relax (x)}{a} \]
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 \[ \frac {\int \frac {\sinh {\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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