Optimal. Leaf size=11 \[ \coth (x)-\frac{\coth ^3(x)}{3} \]
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Rubi [A] time = 0.0180271, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3175, 3767} \[ \coth (x)-\frac{\coth ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rubi steps
\begin{align*} \int \frac{1}{\left (1-\cosh ^2(x)\right )^2} \, dx &=\int \text{csch}^4(x) \, dx\\ &=i \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (x)\right )\\ &=\coth (x)-\frac{\coth ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.002769, size = 17, normalized size = 1.55 \[ \frac{2 \coth (x)}{3}-\frac{1}{3} \coth (x) \text{csch}^2(x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 32, normalized size = 2.9 \begin{align*} -{\frac{1}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{3}{8}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{3}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.10902, size = 66, normalized size = 6. \begin{align*} \frac{4 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} - \frac{4}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98595, size = 286, normalized size = 26. \begin{align*} -\frac{8 \,{\left (\cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right )}}{3 \,{\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} +{\left (10 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{3} +{\left (10 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{2} + 4\right )} \sinh \left (x\right ) + 2 \, \cosh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.13489, size = 34, normalized size = 3.09 \begin{align*} - \frac{\tanh ^{3}{\left (\frac{x}{2} \right )}}{24} + \frac{3 \tanh{\left (\frac{x}{2} \right )}}{8} + \frac{3}{8 \tanh{\left (\frac{x}{2} \right )}} - \frac{1}{24 \tanh ^{3}{\left (\frac{x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28176, size = 24, normalized size = 2.18 \begin{align*} -\frac{4 \,{\left (3 \, e^{\left (2 \, x\right )} - 1\right )}}{3 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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