Integrand size = 8, antiderivative size = 60 \[ \int x \arccos \left (\sqrt {x}\right ) \, dx=-\frac {3}{16} \sqrt {1-x} \sqrt {x}-\frac {1}{8} \sqrt {1-x} x^{3/2}+\frac {1}{2} x^2 \arccos \left (\sqrt {x}\right )-\frac {3}{32} \arcsin (1-2 x) \]
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Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4927, 12, 52, 55, 633, 222} \[ \int x \arccos \left (\sqrt {x}\right ) \, dx=\frac {1}{2} x^2 \arccos \left (\sqrt {x}\right )-\frac {3}{32} \arcsin (1-2 x)-\frac {1}{8} \sqrt {1-x} x^{3/2}-\frac {3}{16} \sqrt {1-x} \sqrt {x} \]
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Rule 12
Rule 52
Rule 55
Rule 222
Rule 633
Rule 4927
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \arccos \left (\sqrt {x}\right )+\frac {1}{2} \int \frac {x^{3/2}}{2 \sqrt {1-x}} \, dx \\ & = \frac {1}{2} x^2 \arccos \left (\sqrt {x}\right )+\frac {1}{4} \int \frac {x^{3/2}}{\sqrt {1-x}} \, dx \\ & = -\frac {1}{8} \sqrt {1-x} x^{3/2}+\frac {1}{2} x^2 \arccos \left (\sqrt {x}\right )+\frac {3}{16} \int \frac {\sqrt {x}}{\sqrt {1-x}} \, dx \\ & = -\frac {3}{16} \sqrt {1-x} \sqrt {x}-\frac {1}{8} \sqrt {1-x} x^{3/2}+\frac {1}{2} x^2 \arccos \left (\sqrt {x}\right )+\frac {3}{32} \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx \\ & = -\frac {3}{16} \sqrt {1-x} \sqrt {x}-\frac {1}{8} \sqrt {1-x} x^{3/2}+\frac {1}{2} x^2 \arccos \left (\sqrt {x}\right )+\frac {3}{32} \int \frac {1}{\sqrt {x-x^2}} \, dx \\ & = -\frac {3}{16} \sqrt {1-x} \sqrt {x}-\frac {1}{8} \sqrt {1-x} x^{3/2}+\frac {1}{2} x^2 \arccos \left (\sqrt {x}\right )-\frac {3}{32} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1-2 x\right ) \\ & = -\frac {3}{16} \sqrt {1-x} \sqrt {x}-\frac {1}{8} \sqrt {1-x} x^{3/2}+\frac {1}{2} x^2 \arccos \left (\sqrt {x}\right )-\frac {3}{32} \arcsin (1-2 x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.68 \[ \int x \arccos \left (\sqrt {x}\right ) \, dx=\frac {1}{16} \left (-\sqrt {-((-1+x) x)} (3+2 x)+8 x^2 \arccos \left (\sqrt {x}\right )+3 \arcsin \left (\sqrt {x}\right )\right ) \]
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Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(\frac {x^{2} \arccos \left (\sqrt {x}\right )}{2}-\frac {x^{\frac {3}{2}} \sqrt {1-x}}{8}-\frac {3 \sqrt {1-x}\, \sqrt {x}}{16}+\frac {3 \arcsin \left (\sqrt {x}\right )}{16}\) | \(41\) |
default | \(\frac {x^{2} \arccos \left (\sqrt {x}\right )}{2}-\frac {x^{\frac {3}{2}} \sqrt {1-x}}{8}-\frac {3 \sqrt {1-x}\, \sqrt {x}}{16}+\frac {3 \arcsin \left (\sqrt {x}\right )}{16}\) | \(41\) |
parts | \(\frac {x^{2} \arccos \left (\sqrt {x}\right )}{2}-\frac {x^{\frac {3}{2}} \sqrt {1-x}}{8}-\frac {3 \sqrt {1-x}\, \sqrt {x}}{16}+\frac {3 \sqrt {x \left (1-x \right )}\, \arcsin \left (-1+2 x \right )}{32 \sqrt {x}\, \sqrt {1-x}}\) | \(62\) |
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Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.52 \[ \int x \arccos \left (\sqrt {x}\right ) \, dx=-\frac {1}{16} \, {\left (2 \, x + 3\right )} \sqrt {x} \sqrt {-x + 1} + \frac {1}{16} \, {\left (8 \, x^{2} - 3\right )} \arccos \left (\sqrt {x}\right ) \]
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Time = 0.49 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.77 \[ \int x \arccos \left (\sqrt {x}\right ) \, dx=\frac {x^{2} \operatorname {acos}{\left (\sqrt {x} \right )}}{2} + \frac {\sqrt {1 - x} \left (- \frac {x^{\frac {3}{2}}}{4} - \frac {3 \sqrt {x}}{8}\right )}{2} + \frac {3 \operatorname {asin}{\left (\sqrt {x} \right )}}{16} \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.67 \[ \int x \arccos \left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \arccos \left (\sqrt {x}\right ) - \frac {1}{8} \, x^{\frac {3}{2}} \sqrt {-x + 1} - \frac {3}{16} \, \sqrt {x} \sqrt {-x + 1} + \frac {3}{16} \, \arcsin \left (\sqrt {x}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.67 \[ \int x \arccos \left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x^{2} \arccos \left (\sqrt {x}\right ) - \frac {1}{8} \, x^{\frac {3}{2}} \sqrt {-x + 1} - \frac {3}{16} \, \sqrt {x} \sqrt {-x + 1} - \frac {3}{16} \, \arccos \left (\sqrt {x}\right ) \]
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Timed out. \[ \int x \arccos \left (\sqrt {x}\right ) \, dx=\int x\,\mathrm {acos}\left (\sqrt {x}\right ) \,d x \]
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