Optimal. Leaf size=88 \[ \frac{1}{2} x^2 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{(1-x)^2}{6 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{1-x}{2 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
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Rubi [A] time = 0.0183515, antiderivative size = 88, 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 = {6345, 12, 43} \[ \frac{1}{2} x^2 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{(1-x)^2}{6 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}}-\frac{1-x}{2 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 6345
Rule 12
Rule 43
Rubi steps
\begin{align*} \int x \text{sech}^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{2} x^2 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{x}{2 \sqrt{1-x}} \, dx}{2 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1}{2} x^2 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \frac{x}{\sqrt{1-x}} \, dx}{4 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1}{2} x^2 \text{sech}^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{1-x} \int \left (\frac{1}{\sqrt{1-x}}-\sqrt{1-x}\right ) \, dx}{4 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=-\frac{1-x}{2 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+\frac{(1-x)^2}{6 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}+\frac{1}{2} x^2 \text{sech}^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0263739, size = 56, normalized size = 0.64 \[ \frac{1}{2} x^2 \text{sech}^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}} \left (x^{3/2}+x+2 \sqrt{x}+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.13, size = 42, normalized size = 0.5 \begin{align*}{\frac{{x}^{2}}{2}{\rm arcsech} \left (\sqrt{x}\right )}-{\frac{x+2}{6}\sqrt{-{ \left ( -1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{x}\sqrt{{ \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987311, size = 46, normalized size = 0.52 \begin{align*} \frac{1}{6} \, x^{\frac{3}{2}}{\left (\frac{1}{x} - 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, x^{2} \operatorname{arsech}\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 = 1.86514, size = 116, normalized size = 1.32 \begin{align*} \frac{1}{2} \, x^{2} \log \left (\frac{x \sqrt{-\frac{x - 1}{x}} + \sqrt{x}}{x}\right ) - \frac{1}{6} \,{\left (x + 2\right )} \sqrt{x} \sqrt{-\frac{x - 1}{x}} \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{asech}{\left (\sqrt{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 x \operatorname{arsech}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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