Integrand size = 13, antiderivative size = 41 \[ \int \frac {\cosh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {3 x}{2 a}-\frac {2 \sinh (x)}{a}+\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh (x) \sinh (x)}{a+a \text {sech}(x)} \]
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Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3904, 3872, 2715, 8, 2717} \[ \int \frac {\cosh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {3 x}{2 a}-\frac {2 \sinh (x)}{a}+\frac {3 \sinh (x) \cosh (x)}{2 a}-\frac {\sinh (x) \cosh (x)}{a \text {sech}(x)+a} \]
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Rule 8
Rule 2715
Rule 2717
Rule 3872
Rule 3904
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.10 \[ \int \frac {\cosh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {\text {sech}\left (\frac {x}{2}\right ) \left (12 x \cosh \left (\frac {x}{2}\right )-12 \sinh \left (\frac {x}{2}\right )-3 \sinh \left (\frac {3 x}{2}\right )+\sinh \left (\frac {5 x}{2}\right )\right )}{8 a} \]
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Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.73
method | result | size |
parallelrisch | \(\frac {\coth \left (x \right ) \cosh \left (2 x \right )+\left (-4 \cosh \left (x \right )-5\right ) \coth \left (x \right )+6 x +8 \,\operatorname {csch}\left (x \right )}{4 a}\) | \(30\) |
risch | \(\frac {{\mathrm e}^{3 x}-3 \,{\mathrm e}^{2 x}+20+3 \,{\mathrm e}^{-x}+12 x \,{\mathrm e}^{x}-4 \,{\mathrm e}^{x}-{\mathrm e}^{-2 x}+12 x}{8 \left ({\mathrm e}^{x}+1\right ) a}\) | \(48\) |
default | \(\frac {-\tanh \left (\frac {x}{2}\right )-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}}{a}\) | \(70\) |
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Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.71 \[ \int \frac {\cosh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {\cosh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right ) - 4\right )} \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (12 \, x - 1\right )} \cosh \left (x\right ) - 4 \, \cosh \left (x\right )^{2} + {\left (3 \, \cosh \left (x\right )^{2} + 12 \, x - 4 \, \cosh \left (x\right ) - 7\right )} \sinh \left (x\right ) + 12 \, x + 20}{8 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + a\right )}} \]
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\[ \int \frac {\cosh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\cosh ^{2}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]
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Time = 0.18 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.37 \[ \int \frac {\cosh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {3 \, x}{2 \, a} + \frac {4 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )}}{8 \, a} - \frac {3 \, e^{\left (-x\right )} + 20 \, e^{\left (-2 \, x\right )} - 1}{8 \, {\left (a e^{\left (-2 \, x\right )} + a e^{\left (-3 \, x\right )}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.24 \[ \int \frac {\cosh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {3 \, x}{2 \, a} + \frac {{\left (20 \, e^{\left (2 \, x\right )} + 3 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )}}{8 \, a {\left (e^{x} + 1\right )}} + \frac {a e^{\left (2 \, x\right )} - 4 \, a e^{x}}{8 \, a^{2}} \]
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Time = 1.99 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.27 \[ \int \frac {\cosh ^2(x)}{a+a \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{-x}}{2\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}+\frac {{\mathrm {e}}^{2\,x}}{8\,a}+\frac {3\,x}{2\,a}+\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {{\mathrm {e}}^x}{2\,a} \]
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