Integrand size = 10, antiderivative size = 87 \[ \int \frac {\arcsin (a x)^2}{x^5} \, dx=-\frac {a^2}{12 x^2}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)}{6 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 x}-\frac {\arcsin (a x)^2}{4 x^4}+\frac {1}{3} a^4 \log (x) \]
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Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4723, 4789, 4771, 29, 30} \[ \int \frac {\arcsin (a x)^2}{x^5} \, dx=\frac {1}{3} a^4 \log (x)-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)}{6 x^3}-\frac {a^2}{12 x^2}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 x}-\frac {\arcsin (a x)^2}{4 x^4} \]
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Rule 29
Rule 30
Rule 4723
Rule 4771
Rule 4789
Rubi steps \begin{align*} \text {integral}& = -\frac {\arcsin (a x)^2}{4 x^4}+\frac {1}{2} a \int \frac {\arcsin (a x)}{x^4 \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)}{6 x^3}-\frac {\arcsin (a x)^2}{4 x^4}+\frac {1}{6} a^2 \int \frac {1}{x^3} \, dx+\frac {1}{3} a^3 \int \frac {\arcsin (a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {a^2}{12 x^2}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)}{6 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 x}-\frac {\arcsin (a x)^2}{4 x^4}+\frac {1}{3} a^4 \int \frac {1}{x} \, dx \\ & = -\frac {a^2}{12 x^2}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)}{6 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 x}-\frac {\arcsin (a x)^2}{4 x^4}+\frac {1}{3} a^4 \log (x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.79 \[ \int \frac {\arcsin (a x)^2}{x^5} \, dx=-\frac {a^2}{12 x^2}-\frac {a \sqrt {1-a^2 x^2} \left (1+2 a^2 x^2\right ) \arcsin (a x)}{6 x^3}-\frac {\arcsin (a x)^2}{4 x^4}+\frac {1}{3} a^4 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(a^{4} \left (-\frac {\arcsin \left (a x \right )^{2}}{4 a^{4} x^{4}}-\frac {\arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{6 a^{3} x^{3}}-\frac {1}{12 a^{2} x^{2}}-\frac {\arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{3 a x}+\frac {\ln \left (a x \right )}{3}\right )\) | \(82\) |
default | \(a^{4} \left (-\frac {\arcsin \left (a x \right )^{2}}{4 a^{4} x^{4}}-\frac {\arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{6 a^{3} x^{3}}-\frac {1}{12 a^{2} x^{2}}-\frac {\arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{3 a x}+\frac {\ln \left (a x \right )}{3}\right )\) | \(82\) |
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Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \frac {\arcsin (a x)^2}{x^5} \, dx=\frac {4 \, a^{4} x^{4} \log \left (x\right ) - a^{2} x^{2} - 2 \, {\left (2 \, a^{3} x^{3} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right ) - 3 \, \arcsin \left (a x\right )^{2}}{12 \, x^{4}} \]
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\[ \int \frac {\arcsin (a x)^2}{x^5} \, dx=\int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x^{5}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85 \[ \int \frac {\arcsin (a x)^2}{x^5} \, dx=\frac {1}{12} \, {\left (4 \, a^{2} \log \left (x\right ) - \frac {1}{x^{2}}\right )} a^{2} - \frac {1}{6} \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x} + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{3}}\right )} a \arcsin \left (a x\right ) - \frac {\arcsin \left (a x\right )^{2}}{4 \, x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (73) = 146\).
Time = 0.33 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.13 \[ \int \frac {\arcsin (a x)^2}{x^5} \, dx=\frac {1}{48} \, {\left ({\left (\frac {{\left (a^{4} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}}\right )} a^{6} x^{3}}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {\frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{x^{3}}}{a^{2} {\left | a \right |}}\right )} \arcsin \left (a x\right ) + \frac {4 \, {\left (2 \, a^{4} \log \left (a^{2} x^{2}\right ) - \frac {2 \, {\left (a^{2} x^{2} - 1\right )} a^{4} + 3 \, a^{4}}{a^{2} x^{2}}\right )}}{a}\right )} a - \frac {\arcsin \left (a x\right )^{2}}{4 \, x^{4}} \]
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Timed out. \[ \int \frac {\arcsin (a x)^2}{x^5} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x^5} \,d x \]
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