Optimal. Leaf size=50 \[ -\frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+x \cosh ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \cosh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0313341, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5901, 12, 323, 330, 52} \[ -\frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+x \cosh ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \cosh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5901
Rule 12
Rule 323
Rule 330
Rule 52
Rubi steps
\begin{align*} \int \cosh ^{-1}\left (\sqrt{x}\right ) \, dx &=x \cosh ^{-1}\left (\sqrt{x}\right )-\int \frac{\sqrt{x}}{2 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=x \cosh ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{\sqrt{x}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=-\frac{1}{2} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+x \cosh ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}} \, dx\\ &=-\frac{1}{2} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+x \cosh ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{2} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}-\frac{1}{2} \cosh ^{-1}\left (\sqrt{x}\right )+x \cosh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0233279, size = 64, normalized size = 1.28 \[ -\frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+x \cosh ^{-1}\left (\sqrt{x}\right )-\tanh ^{-1}\left (\sqrt{\frac{\sqrt{x}-1}{\sqrt{x}+1}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.003, size = 49, normalized size = 1. \begin{align*} x{\rm arccosh} \left (\sqrt{x}\right )-{\frac{1}{2}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( \sqrt{x}\sqrt{-1+x}+\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01828, size = 45, normalized size = 0.9 \begin{align*} x \operatorname{arcosh}\left (\sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x - 1} \sqrt{x} - \frac{1}{2} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0325, size = 92, normalized size = 1.84 \begin{align*} \frac{1}{2} \,{\left (2 \, x - 1\right )} \log \left (\sqrt{x - 1} + \sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x - 1} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.343594, size = 29, normalized size = 0.58 \begin{align*} - \frac{\sqrt{x} \sqrt{x - 1}}{2} + x \operatorname{acosh}{\left (\sqrt{x} \right )} - \frac{\operatorname{acosh}{\left (\sqrt{x} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16816, size = 65, normalized size = 1.3 \begin{align*} x \log \left (\sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + \sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x - 1} \sqrt{x} + \frac{1}{2} \, \log \left ({\left | \sqrt{x - 1} - \sqrt{x} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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