Optimal. Leaf size=54 \[ -x+\frac{7 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{4 \sqrt{2}}-\frac{\tanh (x)}{4 \left (1-2 \tanh ^2(x)\right )}+\frac{\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2} \]
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Rubi [A] time = 0.0642893, antiderivative size = 54, normalized size of antiderivative = 1., 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, 527, 522, 206} \[ -x+\frac{7 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{4 \sqrt{2}}-\frac{\tanh (x)}{4 \left (1-2 \tanh ^2(x)\right )}+\frac{\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2} \]
Antiderivative was successfully verified.
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Rule 414
Rule 527
Rule 522
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (\text{sech}^2(x)-\tanh ^2(x)\right )^3} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1-2 x^2\right )^3 \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{2-6 x^2}{\left (1-2 x^2\right )^2 \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2}-\frac{\tanh (x)}{4 \left (1-2 \tanh ^2(x)\right )}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{6+2 x^2}{\left (1-2 x^2\right ) \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2}-\frac{\tanh (x)}{4 \left (1-2 \tanh ^2(x)\right )}+\frac{7}{4} \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{7 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{4 \sqrt{2}}+\frac{\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2}-\frac{\tanh (x)}{4 \left (1-2 \tanh ^2(x)\right )}\\ \end{align*}
Mathematica [A] time = 0.191805, size = 66, normalized size = 1.22 \[ \frac{-76 x-2 \sinh (2 x)+3 \sinh (4 x)+48 x \cosh (2 x)-4 x \cosh (4 x)+7 \sqrt{2} (\cosh (2 x)-3)^2 \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{8 (\cosh (2 x)-3)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.058, size = 140, normalized size = 2.6 \begin{align*} -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -2\,{\frac{-1/8\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}+1/8\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-5/8\,\tanh \left ( x/2 \right ) +1/8}{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+2\,\tanh \left ( x/2 \right ) -1 \right ) ^{2}}}+{\frac{7\,\sqrt{2}}{8}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) +2 \right ) } \right ) }-2\,{\frac{-1/8\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-1/8\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-5/8\,\tanh \left ( x/2 \right ) -1/8}{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) -1 \right ) ^{2}}}+{\frac{7\,\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 = 1.68986, size = 154, normalized size = 2.85 \begin{align*} \frac{7}{16} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} + 1}{\sqrt{2} + e^{\left (-x\right )} - 1}\right ) - \frac{7}{16} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - 1}{\sqrt{2} + e^{\left (-x\right )} + 1}\right ) - x + \frac{19 \, e^{\left (-2 \, x\right )} - 57 \, e^{\left (-4 \, x\right )} + 17 \, e^{\left (-6 \, x\right )} - 3}{2 \,{\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 = 2.13029, size = 2365, normalized size = 43.8 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \tanh{\left (x \right )} + \operatorname{sech}{\left (x \right )}\right )^{3} \left (\tanh{\left (x \right )} + \operatorname{sech}{\left (x \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1718, size = 104, normalized size = 1.93 \begin{align*} -\frac{7}{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 ) - x + \frac{17 \, e^{\left (6 \, x\right )} - 57 \, e^{\left (4 \, x\right )} + 19 \, e^{\left (2 \, x\right )} - 3}{2 \,{\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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