Integrand size = 13, antiderivative size = 19 \[ \int \frac {\tanh ^3(x)}{a+a \cosh (x)} \, dx=-\frac {\text {sech}(x)}{a}+\frac {\text {sech}^2(x)}{2 a} \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2785, 2686, 30, 8} \[ \int \frac {\tanh ^3(x)}{a+a \cosh (x)} \, dx=\frac {\text {sech}^2(x)}{2 a}-\frac {\text {sech}(x)}{a} \]
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Rule 8
Rule 30
Rule 2686
Rule 2785
Rubi steps \begin{align*} \text {integral}& = \frac {\int \text {sech}(x) \tanh (x) \, dx}{a}-\frac {\int \text {sech}^2(x) \tanh (x) \, dx}{a} \\ & = -\frac {\text {Subst}(\int 1 \, dx,x,\text {sech}(x))}{a}+\frac {\text {Subst}(\int x \, dx,x,\text {sech}(x))}{a} \\ & = -\frac {\text {sech}(x)}{a}+\frac {\text {sech}^2(x)}{2 a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {\tanh ^3(x)}{a+a \cosh (x)} \, dx=\frac {2 \text {sech}^2(x) \sinh ^4\left (\frac {x}{2}\right )}{a} \]
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Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{x} \left ({\mathrm e}^{2 x}-{\mathrm e}^{x}+1\right )}{\left (1+{\mathrm e}^{2 x}\right )^{2} a}\) | \(26\) |
default | \(\frac {\frac {2}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}-\frac {4}{1+\tanh \left (\frac {x}{2}\right )^{2}}}{a}\) | \(31\) |
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (17) = 34\).
Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.47 \[ \int \frac {\tanh ^3(x)}{a+a \cosh (x)} \, dx=-\frac {2 \, {\left (\cosh \left (x\right )^{2} + {\left (2 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \cosh \left (x\right ) + 1\right )}}{a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right ) \sinh \left (x\right )^{2} + a \sinh \left (x\right )^{3} + 3 \, a \cosh \left (x\right ) + {\left (3 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )} \]
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\[ \int \frac {\tanh ^3(x)}{a+a \cosh (x)} \, dx=\frac {\int \frac {\tanh ^{3}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (17) = 34\).
Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.68 \[ \int \frac {\tanh ^3(x)}{a+a \cosh (x)} \, dx=-\frac {2 \, e^{\left (-x\right )}}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} + \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} - \frac {2 \, e^{\left (-3 \, x\right )}}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} \]
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none
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {\tanh ^3(x)}{a+a \cosh (x)} \, dx=-\frac {2 \, {\left (e^{\left (-x\right )} + e^{x} - 1\right )}}{a {\left (e^{\left (-x\right )} + e^{x}\right )}^{2}} \]
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Time = 1.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32 \[ \int \frac {\tanh ^3(x)}{a+a \cosh (x)} \, dx=-\frac {2\,{\mathrm {e}}^x\,\left ({\mathrm {e}}^{2\,x}-{\mathrm {e}}^x+1\right )}{a\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^2} \]
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