Integrand size = 6, antiderivative size = 51 \[ \int \frac {1}{\arcsin (a x)^3} \, dx=-\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}+\frac {x}{2 \arcsin (a x)}-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{2 a} \]
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Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4717, 4807, 4719, 3383} \[ \int \frac {1}{\arcsin (a x)^3} \, dx=-\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{2 a}+\frac {x}{2 \arcsin (a x)} \]
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Rule 3383
Rule 4717
Rule 4719
Rule 4807
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {1}{2} a \int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)^2} \, dx \\ & = -\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}+\frac {x}{2 \arcsin (a x)}-\frac {1}{2} \int \frac {1}{\arcsin (a x)} \, dx \\ & = -\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}+\frac {x}{2 \arcsin (a x)}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arcsin (a x)\right )}{2 a} \\ & = -\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}+\frac {x}{2 \arcsin (a x)}-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{2 a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\arcsin (a x)^3} \, dx=-\frac {\sqrt {1-a^2 x^2}-a x \arcsin (a x)+\arcsin (a x)^2 \operatorname {CosIntegral}(\arcsin (a x))}{2 a \arcsin (a x)^2} \]
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Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{2 \arcsin \left (a x \right )^{2}}+\frac {a x}{2 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{2}}{a}\) | \(43\) |
default | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{2 \arcsin \left (a x \right )^{2}}+\frac {a x}{2 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{2}}{a}\) | \(43\) |
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\[ \int \frac {1}{\arcsin (a x)^3} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {1}{\arcsin (a x)^3} \, dx=\int \frac {1}{\operatorname {asin}^{3}{\left (a x \right )}}\, dx \]
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\[ \int \frac {1}{\arcsin (a x)^3} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{3}} \,d x } \]
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none
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\arcsin (a x)^3} \, dx=\frac {x}{2 \, \arcsin \left (a x\right )} - \frac {\operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{2 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1}}{2 \, a \arcsin \left (a x\right )^{2}} \]
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Timed out. \[ \int \frac {1}{\arcsin (a x)^3} \, dx=\int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^3} \,d x \]
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