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 = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 12
Rule 37
Rule 4843
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.46, size = 22, normalized size = 0.81 \[ \frac {\sqrt {x} \sqrt {-x + 1} - \arccos \left (\sqrt {x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.99, size = 40, normalized size = 1.48 \[ \frac {\sqrt {-x + 1} - 1}{2 \, \sqrt {x}} - \frac {\arccos \left (\sqrt {x}\right )}{x} - \frac {\sqrt {x}}{2 \, {\left (\sqrt {-x + 1} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.81 \[ -\frac {\arccos \left (\sqrt {x}\right )}{x}+\frac {\sqrt {1-x}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 21, normalized size = 0.78 \[ \frac {\sqrt {-x + 1}}{\sqrt {x}} - \frac {\arccos \left (\sqrt {x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {acos}\left (\sqrt {x}\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.11, size = 44, normalized size = 1.63 \[ - \frac {\begin {cases} - \frac {2 i \sqrt {x - 1}}{\sqrt {x}} & \text {for}\: \left |{x}\right | > 1 \\- \frac {2 \sqrt {1 - x}}{\sqrt {x}} & \text {otherwise} \end {cases}}{2} - \frac {\operatorname {acos}{\left (\sqrt {x} \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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