Optimal. Leaf size=27 \[ \frac {x}{2 a}-\frac {\sinh (x)}{a}+\frac {\sinh (x) \cosh (x)}{2 a} \]
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Rubi [A] time = 0.10, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3872, 2839, 2637, 2635, 8} \[ \frac {x}{2 a}-\frac {\sinh (x)}{a}+\frac {\sinh (x) \cosh (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2637
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{a+a \text {sech}(x)} \, dx &=-\int \frac {\cosh (x) \sinh ^2(x)}{-a-a \cosh (x)} \, dx\\ &=-\frac {\int \cosh (x) \, dx}{a}+\frac {\int \cosh ^2(x) \, dx}{a}\\ &=-\frac {\sinh (x)}{a}+\frac {\cosh (x) \sinh (x)}{2 a}+\frac {\int 1 \, dx}{2 a}\\ &=\frac {x}{2 a}-\frac {\sinh (x)}{a}+\frac {\cosh (x) \sinh (x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 16, normalized size = 0.59 \[ \frac {x+\sinh (x) (\cosh (x)-2)}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 14, normalized size = 0.52 \[ \frac {{\left (\cosh \relax (x) - 2\right )} \sinh \relax (x) + x}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 28, normalized size = 1.04 \[ \frac {{\left (4 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )} + 4 \, x + e^{\left (2 \, x\right )} - 4 \, e^{x}}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 78, normalized size = 2.89 \[ \frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a}-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 42, normalized size = 1.56 \[ -\frac {{\left (4 \, e^{\left (-x\right )} - 1\right )} e^{\left (2 \, x\right )}}{8 \, a} + \frac {x}{2 \, a} + \frac {4 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )}}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 41, normalized size = 1.52 \[ \frac {{\mathrm {e}}^{-x}}{2\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}+\frac {{\mathrm {e}}^{2\,x}}{8\,a}+\frac {x}{2\,a}-\frac {{\mathrm {e}}^x}{2\,a} \]
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 \[ \frac {\int \frac {\sinh ^{2}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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