Optimal. Leaf size=36 \[ \frac{1}{2} x^2 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} (x-1)^{3/2}-\frac{\sqrt{x-1}}{2} \]
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Rubi [A] time = 0.0113633, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5270, 12, 43} \[ \frac{1}{2} x^2 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} (x-1)^{3/2}-\frac{\sqrt{x-1}}{2} \]
Antiderivative was successfully verified.
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Rule 5270
Rule 12
Rule 43
Rubi steps
\begin{align*} \int x \sec ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{2} x^2 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{x}{2 \sqrt{-1+x}} \, dx\\ &=\frac{1}{2} x^2 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{x}{\sqrt{-1+x}} \, dx\\ &=\frac{1}{2} x^2 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \left (\frac{1}{\sqrt{-1+x}}+\sqrt{-1+x}\right ) \, dx\\ &=-\frac{1}{2} \sqrt{-1+x}-\frac{1}{6} (-1+x)^{3/2}+\frac{1}{2} x^2 \sec ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.017784, size = 28, normalized size = 0.78 \[ \frac{1}{2} x^2 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \sqrt{x-1} (x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.108, size = 31, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}{\rm arcsec} \left (\sqrt{x}\right )}-{\frac{ \left ( x-1 \right ) \left ( x+2 \right ) }{6}{\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 = 0.983566, size = 51, normalized size = 1.42 \begin{align*} -\frac{1}{6} \, x^{\frac{3}{2}}{\left (-\frac{1}{x} + 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, x^{2} \operatorname{arcsec}\left (\sqrt{x}\right ) - \frac{1}{2} \, \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 = 2.15562, size = 69, normalized size = 1.92 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arcsec}\left (\sqrt{x}\right ) - \frac{1}{6} \,{\left (x + 2\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 x \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 = 1.11629, size = 36, normalized size = 1. \begin{align*} \frac{1}{2} \, x^{2} \arccos \left (\frac{1}{\sqrt{x}}\right ) - \frac{1}{6} \,{\left (x - 1\right )}^{\frac{3}{2}} + \frac{1}{3} \, i - \frac{1}{2} \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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