Integrand size = 17, antiderivative size = 19 \[ \int \frac {1-\cosh ^2(x)}{1+\cosh ^2(x)} \, dx=-x+\sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3250, 3260, 212} \[ \int \frac {1-\cosh ^2(x)}{1+\cosh ^2(x)} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )-x \]
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Rule 212
Rule 3250
Rule 3260
Rubi steps \begin{align*} \text {integral}& = -x+2 \int \frac {1}{1+\cosh ^2(x)} \, dx \\ & = -x+2 \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\coth (x)\right ) \\ & = -x+\sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {1-\cosh ^2(x)}{1+\cosh ^2(x)} \, dx=-2 \left (\frac {x}{2}-\frac {\text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{\sqrt {2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(16)=32\).
Time = 0.54 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05
method | result | size |
risch | \(-x +\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3-2 \sqrt {2}\right )}{2}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3+2 \sqrt {2}\right )}{2}\) | \(39\) |
default | \(\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {\sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{4}-\frac {\sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{4}\) | \(156\) |
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (16) = 32\).
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.68 \[ \int \frac {1-\cosh ^2(x)}{1+\cosh ^2(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} - 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}\right ) - x \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (17) = 34\).
Time = 1.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.21 \[ \int \frac {1-\cosh ^2(x)}{1+\cosh ^2(x)} \, dx=- x - \frac {\sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{2} + \frac {\sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (16) = 32\).
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 5.37 \[ \int \frac {1-\cosh ^2(x)}{1+\cosh ^2(x)} \, dx=\frac {3}{16} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) - \frac {5}{16} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) - 2 \, x + \frac {1}{4} \, \log \left (e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{4} \, \log \left (6 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (16) = 32\).
Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.00 \[ \int \frac {1-\cosh ^2(x)}{1+\cosh ^2(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) - x \]
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Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.95 \[ \int \frac {1-\cosh ^2(x)}{1+\cosh ^2(x)} \, dx=\frac {\sqrt {2}\,\ln \left (-8\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{2}\right )}{2}-x-\frac {\sqrt {2}\,\ln \left (\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{2}-8\,{\mathrm {e}}^{2\,x}\right )}{2} \]
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