Optimal. Leaf size=41 \[ \frac {3 x}{2 a}-\frac {2 \sinh (x)}{a}+\frac {3 \sinh (x) \cosh (x)}{2 a}-\frac {\sinh (x) \cosh (x)}{a \text {sech}(x)+a} \]
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Rubi [A] time = 0.08, antiderivative size = 41, 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 = {3819, 3787, 2635, 8, 2637} \[ \frac {3 x}{2 a}-\frac {2 \sinh (x)}{a}+\frac {3 \sinh (x) \cosh (x)}{2 a}-\frac {\sinh (x) \cosh (x)}{a \text {sech}(x)+a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2637
Rule 3787
Rule 3819
Rubi steps
\begin {align*} \int \frac {\cosh ^2(x)}{a+a \text {sech}(x)} \, dx &=-\frac {\cosh (x) \sinh (x)}{a+a \text {sech}(x)}-\frac {\int \cosh ^2(x) (-3 a+2 a \text {sech}(x)) \, dx}{a^2}\\ &=-\frac {\cosh (x) \sinh (x)}{a+a \text {sech}(x)}-\frac {2 \int \cosh (x) \, dx}{a}+\frac {3 \int \cosh ^2(x) \, dx}{a}\\ &=-\frac {2 \sinh (x)}{a}+\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh (x) \sinh (x)}{a+a \text {sech}(x)}+\frac {3 \int 1 \, dx}{2 a}\\ &=\frac {3 x}{2 a}-\frac {2 \sinh (x)}{a}+\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh (x) \sinh (x)}{a+a \text {sech}(x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 45, normalized size = 1.10 \[ \frac {\text {sech}\left (\frac {x}{2}\right ) \left (-12 \sinh \left (\frac {x}{2}\right )-3 \sinh \left (\frac {3 x}{2}\right )+\sinh \left (\frac {5 x}{2}\right )+12 x \cosh \left (\frac {x}{2}\right )\right )}{8 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 70, normalized size = 1.71 \[ \frac {\cosh \relax (x)^{3} + {\left (3 \, \cosh \relax (x) - 4\right )} \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (12 \, x - 1\right )} \cosh \relax (x) - 4 \, \cosh \relax (x)^{2} + {\left (3 \, \cosh \relax (x)^{2} + 12 \, x - 4 \, \cosh \relax (x) - 7\right )} \sinh \relax (x) + 12 \, x + 20}{8 \, {\left (a \cosh \relax (x) + a \sinh \relax (x) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 51, normalized size = 1.24 \[ \frac {3 \, x}{2 \, a} + \frac {{\left (20 \, e^{\left (2 \, x\right )} + 3 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )}}{8 \, a {\left (e^{x} + 1\right )}} + \frac {a e^{\left (2 \, x\right )} - 4 \, a e^{x}}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 87, normalized size = 2.12 \[ -\frac {\tanh \left (\frac {x}{2}\right )}{a}+\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a}-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 56, normalized size = 1.37 \[ \frac {3 \, x}{2 \, a} + \frac {4 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )}}{8 \, a} - \frac {3 \, e^{\left (-x\right )} + 20 \, e^{\left (-2 \, x\right )} - 1}{8 \, {\left (a e^{\left (-2 \, x\right )} + a e^{\left (-3 \, x\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 52, normalized size = 1.27 \[ \frac {{\mathrm {e}}^{-x}}{2\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}+\frac {{\mathrm {e}}^{2\,x}}{8\,a}+\frac {3\,x}{2\,a}+\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {{\mathrm {e}}^x}{2\,a} \]
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 \[ \frac {\int \frac {\cosh ^{2}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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