Integrand size = 10, antiderivative size = 82 \[ \int \frac {x^2}{\arcsin (a x)^3} \, dx=-\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {x}{a^2 \arcsin (a x)}+\frac {3 x^3}{2 \arcsin (a x)}-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{8 a^3}+\frac {9 \operatorname {CosIntegral}(3 \arcsin (a x))}{8 a^3} \]
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Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4729, 4807, 4731, 4491, 3383, 4719} \[ \int \frac {x^2}{\arcsin (a x)^3} \, dx=-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{8 a^3}+\frac {9 \operatorname {CosIntegral}(3 \arcsin (a x))}{8 a^3}-\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {x}{a^2 \arcsin (a x)}+\frac {3 x^3}{2 \arcsin (a x)} \]
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Rule 3383
Rule 4491
Rule 4719
Rule 4729
Rule 4731
Rule 4807
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}+\frac {\int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)^2} \, dx}{a}-\frac {1}{2} (3 a) \int \frac {x^3}{\sqrt {1-a^2 x^2} \arcsin (a x)^2} \, dx \\ & = -\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {x}{a^2 \arcsin (a x)}+\frac {3 x^3}{2 \arcsin (a x)}-\frac {9}{2} \int \frac {x^2}{\arcsin (a x)} \, dx+\frac {\int \frac {1}{\arcsin (a x)} \, dx}{a^2} \\ & = -\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {x}{a^2 \arcsin (a x)}+\frac {3 x^3}{2 \arcsin (a x)}+\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arcsin (a x)\right )}{a^3}-\frac {9 \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{x} \, dx,x,\arcsin (a x)\right )}{2 a^3} \\ & = -\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {x}{a^2 \arcsin (a x)}+\frac {3 x^3}{2 \arcsin (a x)}+\frac {\operatorname {CosIntegral}(\arcsin (a x))}{a^3}-\frac {9 \text {Subst}\left (\int \left (\frac {\cos (x)}{4 x}-\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\arcsin (a x)\right )}{2 a^3} \\ & = -\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {x}{a^2 \arcsin (a x)}+\frac {3 x^3}{2 \arcsin (a x)}+\frac {\operatorname {CosIntegral}(\arcsin (a x))}{a^3}-\frac {9 \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arcsin (a x)\right )}{8 a^3}+\frac {9 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{8 a^3} \\ & = -\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {x}{a^2 \arcsin (a x)}+\frac {3 x^3}{2 \arcsin (a x)}-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{8 a^3}+\frac {9 \operatorname {CosIntegral}(3 \arcsin (a x))}{8 a^3} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int \frac {x^2}{\arcsin (a x)^3} \, dx=\frac {\frac {4 a x \left (-a x \sqrt {1-a^2 x^2}+\left (-2+3 a^2 x^2\right ) \arcsin (a x)\right )}{\arcsin (a x)^2}-\operatorname {CosIntegral}(\arcsin (a x))+9 \operatorname {CosIntegral}(3 \arcsin (a x))}{8 a^3} \]
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Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arcsin \left (a x \right )^{2}}+\frac {a x}{8 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{8}}{a^{3}}\) | \(82\) |
default | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arcsin \left (a x \right )^{2}}+\frac {a x}{8 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{8}}{a^{3}}\) | \(82\) |
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\[ \int \frac {x^2}{\arcsin (a x)^3} \, dx=\int { \frac {x^{2}}{\arcsin \left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {x^2}{\arcsin (a x)^3} \, dx=\int \frac {x^{2}}{\operatorname {asin}^{3}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^2}{\arcsin (a x)^3} \, dx=\int { \frac {x^{2}}{\arcsin \left (a x\right )^{3}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24 \[ \int \frac {x^2}{\arcsin (a x)^3} \, dx=\frac {3 \, {\left (a^{2} x^{2} - 1\right )} x}{2 \, a^{2} \arcsin \left (a x\right )} + \frac {x}{2 \, a^{2} \arcsin \left (a x\right )} + \frac {9 \, \operatorname {Ci}\left (3 \, \arcsin \left (a x\right )\right )}{8 \, a^{3}} - \frac {\operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{8 \, a^{3}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, a^{3} \arcsin \left (a x\right )^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{2 \, a^{3} \arcsin \left (a x\right )^{2}} \]
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Timed out. \[ \int \frac {x^2}{\arcsin (a x)^3} \, dx=\int \frac {x^2}{{\mathrm {asin}\left (a\,x\right )}^3} \,d x \]
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