Optimal. Leaf size=37 \[ \frac{3 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{4 \sqrt{2}}+\frac{\sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )} \]
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Rubi [A] time = 0.0268138, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3184, 12, 3181, 206} \[ \frac{3 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{4 \sqrt{2}}+\frac{\sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )} \]
Antiderivative was successfully verified.
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Rule 3184
Rule 12
Rule 3181
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (1-\sinh ^2(x)\right )^2} \, dx &=\frac{\cosh (x) \sinh (x)}{4 \left (1-\sinh ^2(x)\right )}-\frac{1}{4} \int -\frac{3}{1-\sinh ^2(x)} \, dx\\ &=\frac{\cosh (x) \sinh (x)}{4 \left (1-\sinh ^2(x)\right )}+\frac{3}{4} \int \frac{1}{1-\sinh ^2(x)} \, dx\\ &=\frac{\cosh (x) \sinh (x)}{4 \left (1-\sinh ^2(x)\right )}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{3 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{4 \sqrt{2}}+\frac{\cosh (x) \sinh (x)}{4 \left (1-\sinh ^2(x)\right )}\\ \end{align*}
Mathematica [A] time = 0.128136, size = 35, normalized size = 0.95 \[ \frac{3 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{4 \sqrt{2}}-\frac{\sinh (2 x)}{4 (\cosh (2 x)-3)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 92, normalized size = 2.5 \begin{align*} -{ \left ( -{\frac{1}{4}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{4}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) -1 \right ) ^{-1}}+{\frac{3\,\sqrt{2}}{8}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) -2 \right ) } \right ) }-{ \left ( -{\frac{1}{4}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{1}{4}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,\tanh \left ( x/2 \right ) -1 \right ) ^{-1}}+{\frac{3\,\sqrt{2}}{8}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) +2 \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.54868, size = 117, normalized size = 3.16 \begin{align*} \frac{3}{16} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} + 1}{\sqrt{2} + e^{\left (-x\right )} - 1}\right ) - \frac{3}{16} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - 1}{\sqrt{2} + e^{\left (-x\right )} + 1}\right ) - \frac{3 \, e^{\left (-2 \, x\right )} - 1}{2 \,{\left (6 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8539, size = 729, normalized size = 19.7 \begin{align*} -\frac{24 \, \cosh \left (x\right )^{2} - 3 \,{\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 ) + 48 \, \cosh \left (x\right ) \sinh \left (x\right ) + 24 \, \sinh \left (x\right )^{2} - 8}{16 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\sinh{\left (x \right )} - 1\right )^{2} \left (\sinh{\left (x \right )} + 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2771, size = 84, normalized size = 2.27 \begin{align*} -\frac{3}{16} \, \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 ) - \frac{3 \, e^{\left (2 \, x\right )} - 1}{2 \,{\left (e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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