Optimal. Leaf size=2 \[ \coth (x) \]
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Rubi [A] time = 0.0161028, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3175, 3767, 8} \[ \coth (x) \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{1-\cosh ^2(x)} \, dx &=-\int \text{csch}^2(x) \, dx\\ &=i \operatorname{Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=\coth (x)\\ \end{align*}
Mathematica [A] time = 0.0030269, size = 2, normalized size = 1. \[ \coth (x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 16, normalized size = 8. \begin{align*}{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02105, size = 14, normalized size = 7. \begin{align*} -\frac{2}{e^{\left (-2 \, x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02411, size = 69, normalized size = 34.5 \begin{align*} \frac{2}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.583891, size = 14, normalized size = 7. \begin{align*} \frac{\tanh{\left (\frac{x}{2} \right )}}{2} + \frac{1}{2 \tanh{\left (\frac{x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25194, size = 14, normalized size = 7. \begin{align*} \frac{2}{e^{\left (2 \, x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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