Integrand size = 8, antiderivative size = 36 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {\sqrt {-1+x}}{2}+\frac {1}{6} (-1+x)^{3/2}+\frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5379, 12, 45} \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} x^2 \csc ^{-1}\left (\sqrt {x}\right )+\frac {1}{6} (x-1)^{3/2}+\frac {\sqrt {x-1}}{2} \]
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Rule 12
Rule 45
Rule 5379
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{6} \left (\sqrt {-1+x} (2+x)+3 x^2 \csc ^{-1}\left (\sqrt {x}\right )\right ) \]
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Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78
method | result | size |
parts | \(\frac {x^{2} \operatorname {arccsc}\left (\sqrt {x}\right )}{2}+\frac {\sqrt {\frac {x -1}{x}}\, \sqrt {x}\, \left (2+x \right )}{6}\) | \(28\) |
derivativedivides | \(\frac {x^{2} \operatorname {arccsc}\left (\sqrt {x}\right )}{2}+\frac {\left (x -1\right ) \left (2+x \right )}{6 \sqrt {\frac {x -1}{x}}\, \sqrt {x}}\) | \(31\) |
default | \(\frac {x^{2} \operatorname {arccsc}\left (\sqrt {x}\right )}{2}+\frac {\left (x -1\right ) \left (2+x \right )}{6 \sqrt {\frac {x -1}{x}}\, \sqrt {x}}\) | \(31\) |
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Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.56 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {arccsc}\left (\sqrt {x}\right ) + \frac {1}{6} \, {\left (x + 2\right )} \sqrt {x - 1} \]
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Result contains complex when optimal does not.
Time = 5.87 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.69 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {x^{2} \operatorname {acsc}{\left (\sqrt {x} \right )}}{2} + \frac {\begin {cases} \frac {2 x \sqrt {x - 1}}{3} + \frac {4 \sqrt {x - 1}}{3} & \text {for}\: \left |{x}\right | > 1 \\\frac {2 i x \sqrt {1 - x}}{3} + \frac {4 i \sqrt {1 - x}}{3} & \text {otherwise} \end {cases}}{4} \]
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Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{6} \, x^{\frac {3}{2}} {\left (-\frac {1}{x} + 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, x^{2} \operatorname {arccsc}\left (\sqrt {x}\right ) + \frac {1}{2} \, \sqrt {x} \sqrt {-\frac {1}{x} + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.22 \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\frac {1}{48} \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3} + \frac {1}{2} \, x^{2} \arcsin \left (\frac {1}{\sqrt {x}}\right ) + \frac {3}{16} \, \sqrt {x} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )} - \frac {9 \, x {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{2} + 1}{48 \, x^{\frac {3}{2}} {\left (\sqrt {-\frac {1}{x} + 1} - 1\right )}^{3}} \]
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Timed out. \[ \int x \csc ^{-1}\left (\sqrt {x}\right ) \, dx=\int x\,\mathrm {asin}\left (\frac {1}{\sqrt {x}}\right ) \,d x \]
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