Optimal. Leaf size=27 \[ \frac{\sqrt{1-x}}{\sqrt{x}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{x} \]
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Rubi [A] time = 0.0119337, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4843, 12, 37} \[ \frac{\sqrt{1-x}}{\sqrt{x}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 4843
Rule 12
Rule 37
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}\left (\sqrt{x}\right )}{x^2} \, dx &=-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{x}-\int \frac{1}{2 \sqrt{1-x} x^{3/2}} \, dx\\ &=-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{x}-\frac{1}{2} \int \frac{1}{\sqrt{1-x} x^{3/2}} \, dx\\ &=\frac{\sqrt{1-x}}{\sqrt{x}}-\frac{\cos ^{-1}\left (\sqrt{x}\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.0156122, size = 24, normalized size = 0.89 \[ \frac{\sqrt{x-x^2}-\cos ^{-1}\left (\sqrt{x}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 22, normalized size = 0.8 \begin{align*} -{\frac{1}{x}\arccos \left ( \sqrt{x} \right ) }+{\sqrt{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 = 1.49926, size = 28, normalized size = 1.04 \begin{align*} \frac{\sqrt{-x + 1}}{\sqrt{x}} - \frac{\arccos \left (\sqrt{x}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.56133, size = 59, normalized size = 2.19 \begin{align*} \frac{\sqrt{x} \sqrt{-x + 1} - \arccos \left (\sqrt{x}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.22227, size = 44, normalized size = 1.63 \begin{align*} - \frac{\begin{cases} - \frac{2 i \sqrt{x - 1}}{\sqrt{x}} & \text{for}\: \left |{x}\right | > 1 \\- \frac{2 \sqrt{1 - x}}{\sqrt{x}} & \text{otherwise} \end{cases}}{2} - \frac{\operatorname{acos}{\left (\sqrt{x} \right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17879, size = 54, normalized size = 2. \begin{align*} \frac{\sqrt{-x + 1} - 1}{2 \, \sqrt{x}} - \frac{\arccos \left (\sqrt{x}\right )}{x} - \frac{\sqrt{x}}{2 \,{\left (\sqrt{-x + 1} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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