Optimal. Leaf size=13 \[ \frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}} \]
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Rubi [A] time = 0.0217987, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3657, 4122, 191} \[ \frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-\coth ^2(x)}} \, dx &=\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 = 0.006995, size = 13, normalized size = 1. \[ \frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 14, normalized size = 1.1 \begin{align*}{{\rm coth} \left (x\right ){\frac{1}{\sqrt{1- \left ({\rm coth} \left (x\right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.88455, 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 [A] time = 2.59502, size = 4, normalized size = 0.31 \begin{align*} 0 \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{1 - \coth ^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.13184, size = 32, normalized size = 2.46 \begin{align*} -\frac{-i \, e^{\left (-x\right )} - i \, e^{x}}{2 \, \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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