Integrand size = 13, antiderivative size = 31 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^2} \, dx=x-\frac {\text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{\sqrt {2}}+\frac {\tanh (x)}{1-2 \tanh ^2(x)} \]
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Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {425, 12, 492, 212} \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^2} \, dx=-\frac {\text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{\sqrt {2}}+x+\frac {\tanh (x)}{1-2 \tanh ^2(x)} \]
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Rule 12
Rule 212
Rule 425
Rule 492
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (1-2 x^2\right )^2 \left (1-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = \frac {\tanh (x)}{1-2 \tanh ^2(x)}+\frac {1}{2} \text {Subst}\left (\int -\frac {2 x^2}{\left (1-2 x^2\right ) \left (1-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = \frac {\tanh (x)}{1-2 \tanh ^2(x)}-\text {Subst}\left (\int \frac {x^2}{\left (1-2 x^2\right ) \left (1-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = \frac {\tanh (x)}{1-2 \tanh ^2(x)}-\text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\tanh (x)\right )+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (x)\right ) \\ & = x-\frac {\text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{\sqrt {2}}+\frac {\tanh (x)}{1-2 \tanh ^2(x)} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.25 (sec) , antiderivative size = 549, normalized size of antiderivative = 17.71 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^2} \, dx=\frac {\text {sech}(x) \left (-4 \sinh (x)-4 \sinh ^3(x)-(1+i) \sqrt {2} \left (\arctan \left (\frac {(1+i) \sqrt {(-1-i) (i+\sinh (x))}}{\sqrt {(2+2 i)-(2-2 i) \sinh (x)}}\right ) \sqrt {(1+i)-(1-i) \sinh (x)}+i \sqrt {1+i} \arcsin \left (\frac {1}{2} \sqrt {1+i} \sqrt {(-1-i) (i+\sinh (x))}\right ) \sqrt {1+i \sinh (x)}\right ) \sqrt {(-1-i) (i+\sinh (x))}+\text {arctanh}\left (\frac {\sqrt {(-1-i) (i+\sinh (x))}}{\sqrt {(1+i)-(1-i) \sinh (x)}}\right ) (-3+\cosh (2 x)) \sqrt {(1+i)-(1-i) \sinh (x)} \sqrt {(-1-i) (i+\sinh (x))}+(1+i) \sqrt {2} \left (\arctan \left (\frac {(1+i) \sqrt {(-1-i) (i+\sinh (x))}}{\sqrt {(2+2 i)-(2-2 i) \sinh (x)}}\right ) \sqrt {(1+i)-(1-i) \sinh (x)}+i \sqrt {1+i} \arcsin \left (\frac {1}{2} \sqrt {1+i} \sqrt {(-1-i) (i+\sinh (x))}\right ) \sqrt {1+i \sinh (x)}\right ) \sinh ^2(x) \sqrt {(-1-i) (i+\sinh (x))}+\frac {\text {arcsinh}\left (\frac {1}{2} \sqrt {-1+i} \sqrt {(1-i) (i+\sinh (x))}\right ) (-3+\cosh (2 x)) \sqrt {2+2 i \sinh (x)} \sqrt {(1-i) (i+\sinh (x))}}{\sqrt {-1+i}}-\text {arctanh}\left (\frac {\sqrt {(1-i) (i+\sinh (x))}}{\sqrt {(1+i) (-i+\sinh (x))}}\right ) (-3+\cosh (2 x)) \sqrt {(1+i) (-i+\sinh (x))} \sqrt {(1-i) (i+\sinh (x))}+\frac {(1+i) \text {arctanh}\left (\frac {(1+i) \sqrt {(1-i) (i+\sinh (x))}}{\sqrt {2} \sqrt {(1+i) (-i+\sinh (x))}}\right ) (-3+\cosh (2 x)) \sqrt {(1+i) (-i+\sinh (x))} \sqrt {(1-i) (i+\sinh (x))}}{\sqrt {2}}\right )}{2 (-3+\cosh (2 x))} \]
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Time = 16.12 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.06
method | result | size |
parallelrisch | \(0\) | \(2\) |
risch | \(x -\frac {2 \left (3 \,{\mathrm e}^{2 x}-1\right )}{{\mathrm e}^{4 x}-6 \,{\mathrm e}^{2 x}+1}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{4}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+2 \sqrt {2}-3\right )}{4}\) | \(61\) |
default | \(-\frac {-2 \tanh \left (\frac {x}{2}\right )+2}{2 \left (\tanh \left (\frac {x}{2}\right )^{2}+2 \tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{2}+\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )+\frac {2 \tanh \left (\frac {x}{2}\right )+2}{2 \tanh \left (\frac {x}{2}\right )^{2}-4 \tanh \left (\frac {x}{2}\right )-2}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{2}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\) | \(108\) |
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Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 266, normalized size of antiderivative = 8.58 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^2} \, dx=\frac {4 \, x \cosh \left (x\right )^{4} + 16 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + 4 \, x \sinh \left (x\right )^{4} - 24 \, {\left (x + 1\right )} \cosh \left (x\right )^{2} + 24 \, {\left (x \cosh \left (x\right )^{2} - x - 1\right )} \sinh \left (x\right )^{2} + {\left (\sqrt {2} \cosh \left (x\right )^{4} + 4 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} \sinh \left (x\right )^{4} + 6 \, {\left (\sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right )^{2} - 6 \, \sqrt {2} \cosh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} - 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \log \left (\frac {3 \, {\left (2 \, \sqrt {2} + 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} + 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} + 3\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {2} - 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}\right ) + 16 \, {\left (x \cosh \left (x\right )^{3} - 3 \, {\left (x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4 \, x + 8}{4 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 6 \, {\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
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\[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^2} \, dx=\int \frac {1}{\left (- \tanh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right )^{2} \left (\tanh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^2} \, dx=-\frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) + x - \frac {2 \, {\left (3 \, e^{\left (-2 \, x\right )} - 1\right )}}{6 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (28) = 56\).
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.03 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^2} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) + x - \frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} - 1\right )}}{e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} \]
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Time = 2.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.52 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^2} \, dx=x-\frac {\sqrt {2}\,\ln \left (-4\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{4}\right )}{4}+\frac {\sqrt {2}\,\ln \left (\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{4}-4\,{\mathrm {e}}^{2\,x}\right )}{4}-\frac {6\,{\mathrm {e}}^{2\,x}-2}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \]
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