Integrand size = 17, antiderivative size = 19 \[ \int \frac {1+\sinh ^2(x)}{1-\sinh ^2(x)} \, dx=-x+\sqrt {2} \text {arctanh}\left (\sqrt {2} \tanh (x)\right ) \]
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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} \[ \int \frac {1+\sinh ^2(x)}{1-\sinh ^2(x)} \, dx=\sqrt {2} \text {arctanh}\left (\sqrt {2} \tanh (x)\right )-x \]
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Rule 212
Rule 3250
Rule 3260
Rubi steps \begin{align*} \text {integral}& = -x+2 \int \frac {1}{1-\sinh ^2(x)} \, dx \\ & = -x+2 \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\tanh (x)\right ) \\ & = -x+\sqrt {2} \text {arctanh}\left (\sqrt {2} \tanh (x)\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {1+\sinh ^2(x)}{1-\sinh ^2(x)} \, dx=-2 \left (\frac {x}{2}-\frac {\text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{\sqrt {2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(15)=30\).
Time = 0.21 (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 | \(\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )-\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )+\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\) | \(54\) |
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (15) = 30\).
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.68 \[ \int \frac {1+\sinh ^2(x)}{1-\sinh ^2(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}\right ) - x \]
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (15) = 30\).
Time = 3.32 (sec) , antiderivative size = 238, normalized size of antiderivative = 12.53 \[ \int \frac {1+\sinh ^2(x)}{1-\sinh ^2(x)} \, dx=- \frac {1331714 x}{941664 \sqrt {2} + 1331714} - \frac {941664 \sqrt {2} x}{941664 \sqrt {2} + 1331714} + \frac {941664 \log {\left (\tanh {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{941664 \sqrt {2} + 1331714} + \frac {665857 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{941664 \sqrt {2} + 1331714} + \frac {941664 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{941664 \sqrt {2} + 1331714} + \frac {665857 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{941664 \sqrt {2} + 1331714} - \frac {665857 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{941664 \sqrt {2} + 1331714} - \frac {941664 \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{941664 \sqrt {2} + 1331714} - \frac {665857 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{941664 \sqrt {2} + 1331714} - \frac {941664 \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{941664 \sqrt {2} + 1331714} \]
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (15) = 30\).
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.37 \[ \int \frac {1+\sinh ^2(x)}{1-\sinh ^2(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) - \frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) - x \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (15) = 30\).
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.16 \[ \int \frac {1+\sinh ^2(x)}{1-\sinh ^2(x)} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - x \]
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Time = 2.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.95 \[ \int \frac {1+\sinh ^2(x)}{1-\sinh ^2(x)} \, dx=\frac {\sqrt {2}\,\ln \left (8\,{\mathrm {e}}^{2\,x}+\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{2}\right )}{2}-\frac {\sqrt {2}\,\ln \left (8\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{2}\right )}{2}-x \]
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