Integrand size = 11, antiderivative size = 11 \[ \int \frac {1}{\left (\cosh ^2(x)+\sinh ^2(x)\right )^2} \, dx=\frac {\tanh (x)}{1+\tanh ^2(x)} \]
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Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {391} \[ \int \frac {1}{\left (\cosh ^2(x)+\sinh ^2(x)\right )^2} \, dx=\frac {\tanh (x)}{\tanh ^2(x)+1} \]
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Rule 391
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1-x^2}{\left (1+x^2\right )^2} \, dx,x,\tanh (x)\right ) \\ & = \frac {\tanh (x)}{1+\tanh ^2(x)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (\cosh ^2(x)+\sinh ^2(x)\right )^2} \, dx=\frac {1}{2} \tanh (2 x) \]
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Time = 30.15 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {1}{1+{\mathrm e}^{4 x}}\) | \(11\) |
default | \(-\frac {2 \left (-\tanh \left (\frac {x}{2}\right )^{3}-\tanh \left (\frac {x}{2}\right )\right )}{\tanh \left (\frac {x}{2}\right )^{4}+6 \tanh \left (\frac {x}{2}\right )^{2}+1}\) | \(36\) |
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (11) = 22\).
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 3.64 \[ \int \frac {1}{\left (\cosh ^2(x)+\sinh ^2(x)\right )^2} \, dx=-\frac {1}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (8) = 16\).
Time = 0.68 (sec) , antiderivative size = 48, normalized size of antiderivative = 4.36 \[ \int \frac {1}{\left (\cosh ^2(x)+\sinh ^2(x)\right )^2} \, dx=\frac {2 \tanh ^{3}{\left (\frac {x}{2} \right )}}{\tanh ^{4}{\left (\frac {x}{2} \right )} + 6 \tanh ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {2 \tanh {\left (\frac {x}{2} \right )}}{\tanh ^{4}{\left (\frac {x}{2} \right )} + 6 \tanh ^{2}{\left (\frac {x}{2} \right )} + 1} \]
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none
Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (\cosh ^2(x)+\sinh ^2(x)\right )^2} \, dx=\frac {1}{e^{\left (-4 \, x\right )} + 1} \]
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none
Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (\cosh ^2(x)+\sinh ^2(x)\right )^2} \, dx=-\frac {1}{e^{\left (4 \, x\right )} + 1} \]
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Time = 2.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (\cosh ^2(x)+\sinh ^2(x)\right )^2} \, dx=-\frac {1}{{\mathrm {e}}^{4\,x}+1} \]
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