Optimal. Leaf size=36 \[ \frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{6} (x-1)^{3/2}+\frac {\sqrt {x-1}}{2} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 = {5271, 12, 43} \[ \frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{6} (x-1)^{3/2}+\frac {\sqrt {x-1}}{2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 5271
Rubi steps
\begin {align*} \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} \int \frac {x}{2 \sqrt {-1+x}} \, dx\\ &=\frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{4} \int \frac {x}{\sqrt {-1+x}} \, dx\\ &=\frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{4} \int \left (\frac {1}{\sqrt {-1+x}}+\sqrt {-1+x}\right ) \, dx\\ &=\frac {\sqrt {-1+x}}{2}+\frac {1}{6} (-1+x)^{3/2}+\frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 0.75 \[ \frac {1}{6} \left (3 x^2 \csc ^{-1}\left (\sqrt {x}\right )+\sqrt {x-1} (x+2)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.99, size = 20, normalized size = 0.56 \[ \frac {1}{2} \, x^{2} \operatorname {arccsc}\left (\sqrt {x}\right ) + \frac {1}{6} \, {\left (x + 2\right )} \sqrt {x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 80, normalized size = 2.22 \[ \frac {1}{48} \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3} + \frac {1}{2} \, x^{2} \arcsin \left (\frac {1}{\sqrt {x}}\right ) + \frac {3}{16} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} - \frac {9 \, x {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{2} + 1}{48 \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 31, normalized size = 0.86 \[ \frac {x^{2} \mathrm {arccsc}\left (\sqrt {x}\right )}{2}+\frac {\left (x -1\right ) \left (x +2\right )}{6 \sqrt {\frac {x -1}{x}}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 38, normalized size = 1.06 \[ \frac {1}{6} \, x^{\frac {3}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, x^{2} \operatorname {arccsc}\left (\sqrt {x}\right ) + \frac {1}{2} \, \sqrt {x} \sqrt {-\frac {1}{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int x\,\mathrm {asin}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 8.12, size = 58, normalized size = 1.61 \[ \frac {x^{2} \operatorname {acsc}{\left (\sqrt {x} \right )}}{2} + \frac {\begin {cases} \frac {x \sqrt {x - 1}}{3} + \frac {2 \sqrt {x - 1}}{3} & \text {for}\: \left |{x}\right | > 1 \\\frac {i x \sqrt {1 - x}}{3} + \frac {2 i \sqrt {1 - x}}{3} & \text {otherwise} \end {cases}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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