Optimal. Leaf size=13 \[ \frac{\coth (x)}{\sqrt{a \text{csch}^2(x)}} \]
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Rubi [A] time = 0.0135736, 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} \[ \frac{\coth (x)}{\sqrt{a \text{csch}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 4122
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \text{csch}^2(x)}} \, dx &=-\left (a \operatorname{Subst}\left (\int \frac{1}{\left (-a+a x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\right )\\ &=\frac{\coth (x)}{\sqrt{a \text{csch}^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0057898, size = 13, normalized size = 1. \[ \frac{\coth (x)}{\sqrt{a \text{csch}^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.065, size = 58, normalized size = 4.5 \begin{align*}{\frac{{{\rm e}^{2\,x}}}{2\,{{\rm e}^{2\,x}}-2}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{1}{2\,{{\rm e}^{2\,x}}-2}{\frac{1}{\sqrt{{\frac{a{{\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 [A] time = 1.73158, size = 23, normalized size = 1.77 \begin{align*} -\frac{e^{\left (-x\right )}}{2 \, \sqrt{a}} - \frac{e^{x}}{2 \, \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88567, size = 247, normalized size = 19. \begin{align*} \frac{{\left ({\left (e^{\left (2 \, x\right )} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} - \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )} \sqrt{\frac{a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \,{\left (a \cosh \left (x\right ) e^{x} + a e^{x} \sinh \left (x\right )\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{a \operatorname{csch}^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14493, size = 32, normalized size = 2.46 \begin{align*} \frac{e^{\left (-x\right )} + e^{x}}{2 \, \sqrt{a} \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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