Integrand size = 11, antiderivative size = 32 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^2} \, dx=x-\frac {\text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\tanh (x)}{2-\tanh ^2(x)} \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {481, 12, 400, 212} \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^2} \, dx=-\frac {\text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{\sqrt {2}}+x-\frac {\tanh (x)}{2-\tanh ^2(x)} \]
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Rule 12
Rule 212
Rule 400
Rule 481
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right ) \left (2-x^2\right )^2} \, dx,x,\tanh (x)\right ) \\ & = -\frac {\tanh (x)}{2-\tanh ^2(x)}+\frac {1}{2} \text {Subst}\left (\int \frac {2}{\left (1-x^2\right ) \left (2-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = -\frac {\tanh (x)}{2-\tanh ^2(x)}+\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (2-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = -\frac {\tanh (x)}{2-\tanh ^2(x)}+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (x)\right )-\text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\tanh (x)\right ) \\ & = x-\frac {\text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\tanh (x)}{2-\tanh ^2(x)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^2} \, dx=\frac {(3+\cosh (2 x)) \text {csch}^4(x) \left (6 x+2 x \cosh (2 x)-\sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) (3+\cosh (2 x))-2 \sinh (2 x)\right )}{8 \left (\coth ^2(x)+\text {csch}^2(x)\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs. \(2(29)=58\).
Time = 26.17 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91
method | result | size |
risch | \(x +\frac {6 \,{\mathrm e}^{2 x}+2}{{\mathrm e}^{4 x}+6 \,{\mathrm e}^{2 x}+1}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3+2 \sqrt {2}\right )}{4}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-2 \sqrt {2}+3\right )}{4}\) | \(61\) |
default | \(\frac {-\tanh \left (\frac {x}{2}\right )^{3}-\tanh \left (\frac {x}{2}\right )}{\tanh \left (\frac {x}{2}\right )^{4}+1}-\frac {\sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{8}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{8}+\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\) | \(183\) |
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 262, normalized size of antiderivative = 8.19 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^2} \, dx=\frac {4 \, x \cosh \left (x\right )^{4} + 16 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + 4 \, x \sinh \left (x\right )^{4} + 24 \, {\left (x + 1\right )} \cosh \left (x\right )^{2} + 24 \, {\left (x \cosh \left (x\right )^{2} + x + 1\right )} \sinh \left (x\right )^{2} + {\left (\sqrt {2} \cosh \left (x\right )^{4} + 4 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} \sinh \left (x\right )^{4} + 6 \, {\left (\sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 6 \, \sqrt {2} \cosh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \log \left (\frac {3 \, {\left (2 \, \sqrt {2} + 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} + 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} + 3\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}\right ) + 16 \, {\left (x \cosh \left (x\right )^{3} + 3 \, {\left (x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4 \, x + 8}{4 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 6 \, {\left (\cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
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\[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^2} \, dx=\int \frac {1}{\left (\coth ^{2}{\left (x \right )} + \operatorname {csch}^{2}{\left (x \right )}\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^2} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) + x - \frac {2 \, {\left (3 \, e^{\left (-2 \, x\right )} + 1\right )}}{6 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^2} \, dx=-\frac {1}{4} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + x + \frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1} \]
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Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.41 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^2} \, dx=x+\frac {\sqrt {2}\,\ln \left (4\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{4}\right )}{4}-\frac {\sqrt {2}\,\ln \left (4\,{\mathrm {e}}^{2\,x}+\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{4}\right )}{4}+\frac {6\,{\mathrm {e}}^{2\,x}+2}{6\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1} \]
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