Integrand size = 10, antiderivative size = 69 \[ \int \frac {x^4}{\arcsin (a x)^2} \, dx=-\frac {x^4 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {\text {Si}(\arcsin (a x))}{8 a^5}+\frac {9 \text {Si}(3 \arcsin (a x))}{16 a^5}-\frac {5 \text {Si}(5 \arcsin (a x))}{16 a^5} \]
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Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4727, 3380} \[ \int \frac {x^4}{\arcsin (a x)^2} \, dx=-\frac {\text {Si}(\arcsin (a x))}{8 a^5}+\frac {9 \text {Si}(3 \arcsin (a x))}{16 a^5}-\frac {5 \text {Si}(5 \arcsin (a x))}{16 a^5}-\frac {x^4 \sqrt {1-a^2 x^2}}{a \arcsin (a x)} \]
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Rule 3380
Rule 4727
Rubi steps \begin{align*} \text {integral}& = -\frac {x^4 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {\text {Subst}\left (\int \left (-\frac {\sin (x)}{8 x}+\frac {9 \sin (3 x)}{16 x}-\frac {5 \sin (5 x)}{16 x}\right ) \, dx,x,\arcsin (a x)\right )}{a^5} \\ & = -\frac {x^4 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arcsin (a x)\right )}{8 a^5}-\frac {5 \text {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\arcsin (a x)\right )}{16 a^5}+\frac {9 \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{16 a^5} \\ & = -\frac {x^4 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {\text {Si}(\arcsin (a x))}{8 a^5}+\frac {9 \text {Si}(3 \arcsin (a x))}{16 a^5}-\frac {5 \text {Si}(5 \arcsin (a x))}{16 a^5} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\arcsin (a x)^2} \, dx=-\frac {\frac {16 a^4 x^4 \sqrt {1-a^2 x^2}}{\arcsin (a x)}+2 \text {Si}(\arcsin (a x))-9 \text {Si}(3 \arcsin (a x))+5 \text {Si}(5 \arcsin (a x))}{16 a^5} \]
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Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arcsin \left (a x \right )}-\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{8}+\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{16 \arcsin \left (a x \right )}+\frac {9 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{16}-\frac {\cos \left (5 \arcsin \left (a x \right )\right )}{16 \arcsin \left (a x \right )}-\frac {5 \,\operatorname {Si}\left (5 \arcsin \left (a x \right )\right )}{16}}{a^{5}}\) | \(81\) |
default | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arcsin \left (a x \right )}-\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{8}+\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{16 \arcsin \left (a x \right )}+\frac {9 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{16}-\frac {\cos \left (5 \arcsin \left (a x \right )\right )}{16 \arcsin \left (a x \right )}-\frac {5 \,\operatorname {Si}\left (5 \arcsin \left (a x \right )\right )}{16}}{a^{5}}\) | \(81\) |
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\[ \int \frac {x^4}{\arcsin (a x)^2} \, dx=\int { \frac {x^{4}}{\arcsin \left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {x^4}{\arcsin (a x)^2} \, dx=\int \frac {x^{4}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^4}{\arcsin (a x)^2} \, dx=\int { \frac {x^{4}}{\arcsin \left (a x\right )^{2}} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.67 \[ \int \frac {x^4}{\arcsin (a x)^2} \, dx=-\frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{a^{5} \arcsin \left (a x\right )} - \frac {5 \, \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} + \frac {9 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{16 \, a^{5}} - \frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{8 \, a^{5}} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{5} \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{5} \arcsin \left (a x\right )} \]
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Timed out. \[ \int \frac {x^4}{\arcsin (a x)^2} \, dx=\int \frac {x^4}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
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