Optimal. Leaf size=13 \[ \frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}} \]
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Rubi [A] time = 0.0097834, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4122, 191} \[ \frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 4122
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-\text{csch}^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=\frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0046467, size = 13, normalized size = 1. \[ \frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 58, normalized size = 4.5 \begin{align*}{\frac{{{\rm e}^{2\,x}}}{2\,{{\rm e}^{2\,x}}-2}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{1}{2\,{{\rm e}^{2\,x}}-2}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.6128, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{2} i \, e^{\left (-x\right )} + \frac{1}{2} i \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.50158, size = 39, normalized size = 3. \begin{align*} \frac{1}{2} \,{\left (-i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-x\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}{\sqrt{- \operatorname{csch}^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.15287, size = 34, normalized size = 2.62 \begin{align*} -\frac{-i \, e^{\left (-x\right )} - i \, e^{x}}{2 \, \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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