Optimal. Leaf size=31 \[ x-\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{\sqrt {2}}+\frac {\tanh (x)}{1-2 \tanh ^2(x)} \]
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Rubi [A] time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {414, 12, 481, 206} \[ x-\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{\sqrt {2}}+\frac {\tanh (x)}{1-2 \tanh ^2(x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 414
Rule 481
Rubi steps
\begin {align*} \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^2} \, dx &=\operatorname {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} \operatorname {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)}-\operatorname {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)}-\operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\tanh (x)\right )+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=x-\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{\sqrt {2}}+\frac {\tanh (x)}{1-2 \tanh ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 42, normalized size = 1.35 \[ \frac {-3 x-\sinh (2 x)+x \cosh (2 x)}{\cosh (2 x)-3}-\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 266, normalized size = 8.58 \[ \frac {4 \, x \cosh \relax (x)^{4} + 16 \, x \cosh \relax (x) \sinh \relax (x)^{3} + 4 \, x \sinh \relax (x)^{4} - 24 \, {\left (x + 1\right )} \cosh \relax (x)^{2} + 24 \, {\left (x \cosh \relax (x)^{2} - x - 1\right )} \sinh \relax (x)^{2} + {\left (\sqrt {2} \cosh \relax (x)^{4} + 4 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{3} + \sqrt {2} \sinh \relax (x)^{4} + 6 \, {\left (\sqrt {2} \cosh \relax (x)^{2} - \sqrt {2}\right )} \sinh \relax (x)^{2} - 6 \, \sqrt {2} \cosh \relax (x)^{2} + 4 \, {\left (\sqrt {2} \cosh \relax (x)^{3} - 3 \, \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x) + \sqrt {2}\right )} \log \left (\frac {3 \, {\left (2 \, \sqrt {2} + 3\right )} \cosh \relax (x)^{2} - 4 \, {\left (3 \, \sqrt {2} + 4\right )} \cosh \relax (x) \sinh \relax (x) + 3 \, {\left (2 \, \sqrt {2} + 3\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {2} - 3}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}\right ) + 16 \, {\left (x \cosh \relax (x)^{3} - 3 \, {\left (x + 1\right )} \cosh \relax (x)\right )} \sinh \relax (x) + 4 \, x + 8}{4 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 6 \, {\left (\cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 6 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 63, normalized size = 2.03 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) + x - \frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} - 1\right )}}{e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 108, normalized size = 3.48 \[ \frac {2 \tanh \left (\frac {x}{2}\right )+2}{2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-4 \tanh \left (\frac {x}{2}\right )-2}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{2}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {2-2 \tanh \left (\frac {x}{2}\right )}{2 \left (\tanh ^{2}\left (\frac {x}{2}\right )+2 \tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 88, normalized size = 2.84 \[ -\frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) + x - \frac {2 \, {\left (3 \, e^{\left (-2 \, x\right )} - 1\right )}}{6 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.61, size = 78, normalized size = 2.52 \[ x-\frac {\sqrt {2}\,\ln \left (-4\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{4}\right )}{4}+\frac {\sqrt {2}\,\ln \left (\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{4}-4\,{\mathrm {e}}^{2\,x}\right )}{4}-\frac {6\,{\mathrm {e}}^{2\,x}-2}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \tanh {\relax (x )} + \operatorname {sech}{\relax (x )}\right )^{2} \left (\tanh {\relax (x )} + \operatorname {sech}{\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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