Integrand size = 10, antiderivative size = 50 \[ \int \frac {\arcsin \left (\sqrt {x}\right )}{x^3} \, dx=-\frac {\sqrt {1-x}}{6 x^{3/2}}-\frac {\sqrt {1-x}}{3 \sqrt {x}}-\frac {\arcsin \left (\sqrt {x}\right )}{2 x^2} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4926, 12, 47, 37} \[ \int \frac {\arcsin \left (\sqrt {x}\right )}{x^3} \, dx=-\frac {\arcsin \left (\sqrt {x}\right )}{2 x^2}-\frac {\sqrt {1-x}}{6 x^{3/2}}-\frac {\sqrt {1-x}}{3 \sqrt {x}} \]
[In]
[Out]
Rule 12
Rule 37
Rule 47
Rule 4926
Rubi steps \begin{align*} \text {integral}& = -\frac {\arcsin \left (\sqrt {x}\right )}{2 x^2}+\frac {1}{2} \int \frac {1}{2 \sqrt {1-x} x^{5/2}} \, dx \\ & = -\frac {\arcsin \left (\sqrt {x}\right )}{2 x^2}+\frac {1}{4} \int \frac {1}{\sqrt {1-x} x^{5/2}} \, dx \\ & = -\frac {\sqrt {1-x}}{6 x^{3/2}}-\frac {\arcsin \left (\sqrt {x}\right )}{2 x^2}+\frac {1}{6} \int \frac {1}{\sqrt {1-x} x^{3/2}} \, dx \\ & = -\frac {\sqrt {1-x}}{6 x^{3/2}}-\frac {\sqrt {1-x}}{3 \sqrt {x}}-\frac {\arcsin \left (\sqrt {x}\right )}{2 x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.64 \[ \int \frac {\arcsin \left (\sqrt {x}\right )}{x^3} \, dx=-\frac {\sqrt {-((-1+x) x)} (1+2 x)+3 \arcsin \left (\sqrt {x}\right )}{6 x^2} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(-\frac {\arcsin \left (\sqrt {x}\right )}{2 x^{2}}-\frac {\sqrt {1-x}}{6 x^{\frac {3}{2}}}-\frac {\sqrt {1-x}}{3 \sqrt {x}}\) | \(35\) |
default | \(-\frac {\arcsin \left (\sqrt {x}\right )}{2 x^{2}}-\frac {\sqrt {1-x}}{6 x^{\frac {3}{2}}}-\frac {\sqrt {1-x}}{3 \sqrt {x}}\) | \(35\) |
parts | \(-\frac {\arcsin \left (\sqrt {x}\right )}{2 x^{2}}-\frac {\sqrt {1-x}}{6 x^{\frac {3}{2}}}-\frac {\sqrt {1-x}}{3 \sqrt {x}}\) | \(35\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.56 \[ \int \frac {\arcsin \left (\sqrt {x}\right )}{x^3} \, dx=-\frac {{\left (2 \, x + 1\right )} \sqrt {x} \sqrt {-x + 1} + 3 \, \arcsin \left (\sqrt {x}\right )}{6 \, x^{2}} \]
[In]
[Out]
Time = 2.73 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.02 \[ \int \frac {\arcsin \left (\sqrt {x}\right )}{x^3} \, dx=\frac {\begin {cases} - \frac {\sqrt {1 - x}}{\sqrt {x}} - \frac {\left (1 - x\right )^{\frac {3}{2}}}{3 x^{\frac {3}{2}}} & \text {for}\: \sqrt {x} > -1 \wedge \sqrt {x} < 1 \end {cases}}{2} - \frac {\operatorname {asin}{\left (\sqrt {x} \right )}}{2 x^{2}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68 \[ \int \frac {\arcsin \left (\sqrt {x}\right )}{x^3} \, dx=-\frac {\sqrt {-x + 1}}{3 \, \sqrt {x}} - \frac {\sqrt {-x + 1}}{6 \, x^{\frac {3}{2}}} - \frac {\arcsin \left (\sqrt {x}\right )}{2 \, x^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (34) = 68\).
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.48 \[ \int \frac {\arcsin \left (\sqrt {x}\right )}{x^3} \, dx=-\frac {{\left (\sqrt {-x + 1} - 1\right )}^{3}}{48 \, x^{\frac {3}{2}}} - \frac {3 \, {\left (\sqrt {-x + 1} - 1\right )}}{16 \, \sqrt {x}} + \frac {x^{\frac {3}{2}} {\left (\frac {9 \, {\left (\sqrt {-x + 1} - 1\right )}^{2}}{x} + 1\right )}}{48 \, {\left (\sqrt {-x + 1} - 1\right )}^{3}} - \frac {\arcsin \left (\sqrt {x}\right )}{2 \, x^{2}} \]
[In]
[Out]
Timed out. \[ \int \frac {\arcsin \left (\sqrt {x}\right )}{x^3} \, dx=\int \frac {\mathrm {asin}\left (\sqrt {x}\right )}{x^3} \,d x \]
[In]
[Out]