Optimal. Leaf size=55 \[ \frac{19 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{32 \sqrt{2}}+\frac{9 \sinh (x) \cosh (x)}{32 \left (1-\sinh ^2(x)\right )}+\frac{\sinh (x) \cosh (x)}{8 \left (1-\sinh ^2(x)\right )^2} \]
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Rubi [A] time = 0.0581055, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3184, 3173, 12, 3181, 206} \[ \frac{19 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{32 \sqrt{2}}+\frac{9 \sinh (x) \cosh (x)}{32 \left (1-\sinh ^2(x)\right )}+\frac{\sinh (x) \cosh (x)}{8 \left (1-\sinh ^2(x)\right )^2} \]
Antiderivative was successfully verified.
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Rule 3184
Rule 3173
Rule 12
Rule 3181
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (1-\sinh ^2(x)\right )^3} \, dx &=\frac{\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}-\frac{1}{8} \int \frac{-7-2 \sinh ^2(x)}{\left (1-\sinh ^2(x)\right )^2} \, dx\\ &=\frac{\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac{9 \cosh (x) \sinh (x)}{32 \left (1-\sinh ^2(x)\right )}-\frac{1}{32} \int -\frac{19}{1-\sinh ^2(x)} \, dx\\ &=\frac{\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac{9 \cosh (x) \sinh (x)}{32 \left (1-\sinh ^2(x)\right )}+\frac{19}{32} \int \frac{1}{1-\sinh ^2(x)} \, dx\\ &=\frac{\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac{9 \cosh (x) \sinh (x)}{32 \left (1-\sinh ^2(x)\right )}+\frac{19}{32} \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{19 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{32 \sqrt{2}}+\frac{\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac{9 \cosh (x) \sinh (x)}{32 \left (1-\sinh ^2(x)\right )}\\ \end{align*}
Mathematica [A] time = 0.177785, size = 51, normalized size = 0.93 \[ \frac{19 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{32 \sqrt{2}}-\frac{9 \sinh (2 x)}{32 (\cosh (2 x)-3)}+\frac{\sinh (2 x)}{4 (\cosh (2 x)-3)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 124, normalized size = 2.3 \begin{align*} -{\frac{1}{4} \left ( -{\frac{13}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{11}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{31}{8}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{11}{8}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) -1 \right ) ^{-2}}+{\frac{19\,\sqrt{2}}{64}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) -2 \right ) } \right ) }-{\frac{1}{4} \left ( -{\frac{13}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{11}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{31}{8}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{11}{8}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,\tanh \left ( x/2 \right ) -1 \right ) ^{-2}}+{\frac{19\,\sqrt{2}}{64}{\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 = 1.56568, size = 150, normalized size = 2.73 \begin{align*} \frac{19}{128} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} + 1}{\sqrt{2} + e^{\left (-x\right )} - 1}\right ) - \frac{19}{128} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - 1}{\sqrt{2} + e^{\left (-x\right )} + 1}\right ) - \frac{89 \, e^{\left (-2 \, x\right )} - 171 \, e^{\left (-4 \, x\right )} + 19 \, e^{\left (-6 \, x\right )} - 9}{16 \,{\left (12 \, e^{\left (-2 \, x\right )} - 38 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06877, size = 1936, normalized size = 35.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27688, size = 100, normalized size = 1.82 \begin{align*} -\frac{19}{128} \, \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{19 \, e^{\left (6 \, x\right )} - 171 \, e^{\left (4 \, x\right )} + 89 \, e^{\left (2 \, x\right )} - 9}{16 \,{\left (e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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