Optimal. Leaf size=11 \[ \frac{\coth (x)}{\sqrt{\text{csch}^2(x)}} \]
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Rubi [A] time = 0.020726, antiderivative size = 11, 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 = {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 = 0.007292, size = 11, normalized size = 1. \[ \frac{\coth (x)}{\sqrt{\text{csch}^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 12, 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 [A] time = 1.80586, size = 15, normalized size = 1.36 \begin{align*} -\frac{1}{2} \, e^{\left (-x\right )} - \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52844, size = 12, normalized size = 1.09 \begin{align*} \cosh \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{\coth ^{2}{\left (x \right )} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18787, size = 24, normalized size = 2.18 \begin{align*} \frac{e^{\left (-x\right )} + 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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