Optimal. Leaf size=47 \[ \frac{1}{3} x^3 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{15} (x-1)^{5/2}-\frac{2}{9} (x-1)^{3/2}-\frac{\sqrt{x-1}}{3} \]
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Rubi [A] time = 0.017687, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5270, 12, 43} \[ \frac{1}{3} x^3 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{15} (x-1)^{5/2}-\frac{2}{9} (x-1)^{3/2}-\frac{\sqrt{x-1}}{3} \]
Antiderivative was successfully verified.
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Rule 5270
Rule 12
Rule 43
Rubi steps
\begin{align*} \int x^2 \sec ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{3} x^3 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{3} \int \frac{x^2}{2 \sqrt{-1+x}} \, dx\\ &=\frac{1}{3} x^3 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \int \frac{x^2}{\sqrt{-1+x}} \, dx\\ &=\frac{1}{3} x^3 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \int \left (\frac{1}{\sqrt{-1+x}}+2 \sqrt{-1+x}+(-1+x)^{3/2}\right ) \, dx\\ &=-\frac{1}{3} \sqrt{-1+x}-\frac{2}{9} (-1+x)^{3/2}-\frac{1}{15} (-1+x)^{5/2}+\frac{1}{3} x^3 \sec ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0225538, size = 35, normalized size = 0.74 \[ \frac{1}{3} x^3 \sec ^{-1}\left (\sqrt{x}\right )-\frac{1}{45} \sqrt{x-1} \left (3 x^2+4 x+8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.108, size = 38, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{3}{\rm arcsec} \left (\sqrt{x}\right )}-{\frac{ \left ( x-1 \right ) \left ( 3\,{x}^{2}+4\,x+8 \right ) }{45}{\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 = 1.00123, size = 70, normalized size = 1.49 \begin{align*} -\frac{1}{15} \, x^{\frac{5}{2}}{\left (-\frac{1}{x} + 1\right )}^{\frac{5}{2}} + \frac{1}{3} \, x^{3} \operatorname{arcsec}\left (\sqrt{x}\right ) - \frac{2}{9} \, x^{\frac{3}{2}}{\left (-\frac{1}{x} + 1\right )}^{\frac{3}{2}} - \frac{1}{3} \, \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 = 2.19825, size = 84, normalized size = 1.79 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arcsec}\left (\sqrt{x}\right ) - \frac{1}{45} \,{\left (3 \, x^{2} + 4 \, x + 8\right )} \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1321, size = 46, normalized size = 0.98 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (\frac{1}{\sqrt{x}}\right ) - \frac{1}{15} \,{\left (x - 1\right )}^{\frac{5}{2}} - \frac{2}{9} \,{\left (x - 1\right )}^{\frac{3}{2}} + \frac{8}{45} \, i - \frac{1}{3} \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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