Integrand size = 10, antiderivative size = 57 \[ \int \frac {x^3}{\arcsin (a x)^2} \, dx=-\frac {x^3 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{2 a^4}-\frac {\operatorname {CosIntegral}(4 \arcsin (a x))}{2 a^4} \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4727, 3383} \[ \int \frac {x^3}{\arcsin (a x)^2} \, dx=\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{2 a^4}-\frac {\operatorname {CosIntegral}(4 \arcsin (a x))}{2 a^4}-\frac {x^3 \sqrt {1-a^2 x^2}}{a \arcsin (a x)} \]
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Rule 3383
Rule 4727
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {\text {Subst}\left (\int \left (\frac {\cos (2 x)}{2 x}-\frac {\cos (4 x)}{2 x}\right ) \, dx,x,\arcsin (a x)\right )}{a^4} \\ & = -\frac {x^3 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arcsin (a x)\right )}{2 a^4}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\arcsin (a x)\right )}{2 a^4} \\ & = -\frac {x^3 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{2 a^4}-\frac {\operatorname {CosIntegral}(4 \arcsin (a x))}{2 a^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int \frac {x^3}{\arcsin (a x)^2} \, dx=\frac {4 \arcsin (a x) \operatorname {CosIntegral}(2 \arcsin (a x))-4 \arcsin (a x) \operatorname {CosIntegral}(4 \arcsin (a x))-2 \sin (2 \arcsin (a x))+\sin (4 \arcsin (a x))}{8 a^4 \arcsin (a x)} \]
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Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{4 \arcsin \left (a x \right )}+\frac {\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{2}+\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (4 \arcsin \left (a x \right )\right )}{2}}{a^{4}}\) | \(54\) |
default | \(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{4 \arcsin \left (a x \right )}+\frac {\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{2}+\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (4 \arcsin \left (a x \right )\right )}{2}}{a^{4}}\) | \(54\) |
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\[ \int \frac {x^3}{\arcsin (a x)^2} \, dx=\int { \frac {x^{3}}{\arcsin \left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {x^3}{\arcsin (a x)^2} \, dx=\int \frac {x^{3}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^3}{\arcsin (a x)^2} \, dx=\int { \frac {x^{3}}{\arcsin \left (a x\right )^{2}} \,d x } \]
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none
Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.26 \[ \int \frac {x^3}{\arcsin (a x)^2} \, dx=\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{3} \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{3} \arcsin \left (a x\right )} - \frac {\operatorname {Ci}\left (4 \, \arcsin \left (a x\right )\right )}{2 \, a^{4}} + \frac {\operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{2 \, a^{4}} \]
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Timed out. \[ \int \frac {x^3}{\arcsin (a x)^2} \, dx=\int \frac {x^3}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
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