Optimal. Leaf size=18 \[ x \sec ^{-1}\left (\sqrt{x}\right )-\sqrt{x-1} \]
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Rubi [A] time = 0.0043606, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5268, 12, 32} \[ x \sec ^{-1}\left (\sqrt{x}\right )-\sqrt{x-1} \]
Antiderivative was successfully verified.
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Rule 5268
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \sec ^{-1}\left (\sqrt{x}\right ) \, dx &=x \sec ^{-1}\left (\sqrt{x}\right )-\int \frac{1}{2 \sqrt{-1+x}} \, dx\\ &=x \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{1}{\sqrt{-1+x}} \, dx\\ &=-\sqrt{-1+x}+x \sec ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0052079, size = 18, normalized size = 1. \[ x \sec ^{-1}\left (\sqrt{x}\right )-\sqrt{x-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 25, normalized size = 1.4 \begin{align*} x{\rm arcsec} \left (\sqrt{x}\right )-{(x-1){\frac{1}{\sqrt{{\frac{x-1}{x}}}}}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970726, size = 28, normalized size = 1.56 \begin{align*} x \operatorname{arcsec}\left (\sqrt{x}\right ) - \sqrt{x} \sqrt{-\frac{1}{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17848, size = 45, normalized size = 2.5 \begin{align*} x \operatorname{arcsec}\left (\sqrt{x}\right ) - \sqrt{x - 1} \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 \operatorname{asec}{\left (\sqrt{x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11703, size = 20, normalized size = 1.11 \begin{align*} x \arccos \left (\frac{1}{\sqrt{x}}\right ) + i - \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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