Optimal. Leaf size=16 \[ \sqrt {x-1}+x \csc ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5269, 12, 32} \[ \sqrt {x-1}+x \csc ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 5269
Rubi steps
\begin {align*} \int \csc ^{-1}\left (\sqrt {x}\right ) \, dx &=x \csc ^{-1}\left (\sqrt {x}\right )+\int \frac {1}{2 \sqrt {-1+x}} \, dx\\ &=x \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} \int \frac {1}{\sqrt {-1+x}} \, dx\\ &=\sqrt {-1+x}+x \csc ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \[ \sqrt {x-1}+x \csc ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.86, size = 12, normalized size = 0.75 \[ x \operatorname {arccsc}\left (\sqrt {x}\right ) + \sqrt {x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 41, normalized size = 2.56 \[ x \arcsin \left (\frac {1}{\sqrt {x}}\right ) + \frac {1}{2} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} - \frac {1}{2 \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 24, normalized size = 1.50 \[ x \,\mathrm {arccsc}\left (\sqrt {x}\right )+\frac {x -1}{\sqrt {\frac {x -1}{x}}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 20, normalized size = 1.25 \[ x \operatorname {arccsc}\left (\sqrt {x}\right ) + \sqrt {x} \sqrt {-\frac {1}{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 20, normalized size = 1.25 \[ x\,\mathrm {asin}\left (\frac {1}{\sqrt {x}}\right )+\sqrt {x}\,\sqrt {1-\frac {1}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.71, size = 29, normalized size = 1.81 \[ x \operatorname {acsc}{\left (\sqrt {x} \right )} + \frac {\begin {cases} 2 \sqrt {x - 1} & \text {for}\: \left |{x}\right | > 1 \\2 i \sqrt {1 - x} & \text {otherwise} \end {cases}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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