Optimal. Leaf size=3 \[ \text{Chi}\left (\cosh ^{-1}(x)\right ) \]
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Rubi [A] time = 0.120613, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {5781, 3301} \[ \text{Chi}\left (\cosh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Rule 5781
Rule 3301
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{-1+x} \sqrt{1+x} \cosh ^{-1}(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\cosh ^{-1}(x)\right )\\ &=\text{Chi}\left (\cosh ^{-1}(x)\right )\\ \end{align*}
Mathematica [B] time = 0.0543266, size = 19, normalized size = 6.33 \[ \frac{1}{2} (x-1) \text{Chi}\left (\cosh ^{-1}(x)\right ) \text{csch}^2\left (\frac{1}{2} \cosh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.14, size = 78, normalized size = 26. \begin{align*} -{\frac{{\it Ei} \left ( 1,{\rm arccosh} \left (x\right ) \right ) }{4\,{x}^{2}-4}\sqrt{2+2\,x}\sqrt{-2+2\,x}\sqrt{-1+x}\sqrt{1+x}}-{\frac{{\it Ei} \left ( 1,-{\rm arccosh} \left (x\right ) \right ) }{4\,{x}^{2}-4}\sqrt{2+2\,x}\sqrt{-2+2\,x}\sqrt{-1+x}\sqrt{1+x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x + 1} \sqrt{x - 1} \operatorname{arcosh}\left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x + 1} \sqrt{x - 1} x}{{\left (x^{2} - 1\right )} \operatorname{arcosh}\left (x\right )}, 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{x}{\sqrt{x - 1} \sqrt{x + 1} \operatorname{acosh}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x + 1} \sqrt{x - 1} \operatorname{arcosh}\left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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