Integrand size = 8, antiderivative size = 99 \[ \int x \arcsin (a x)^3 \, dx=-\frac {3 x \sqrt {1-a^2 x^2}}{8 a}+\frac {3 \arcsin (a x)}{8 a^2}-\frac {3}{4} x^2 \arcsin (a x)+\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a}-\frac {\arcsin (a x)^3}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^3 \]
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Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4723, 4795, 4737, 327, 222} \[ \int x \arcsin (a x)^3 \, dx=\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a}-\frac {\arcsin (a x)^3}{4 a^2}+\frac {3 \arcsin (a x)}{8 a^2}-\frac {3 x \sqrt {1-a^2 x^2}}{8 a}+\frac {1}{2} x^2 \arcsin (a x)^3-\frac {3}{4} x^2 \arcsin (a x) \]
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Rule 222
Rule 327
Rule 4723
Rule 4737
Rule 4795
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \arcsin (a x)^3-\frac {1}{2} (3 a) \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a}+\frac {1}{2} x^2 \arcsin (a x)^3-\frac {3}{2} \int x \arcsin (a x) \, dx-\frac {3 \int \frac {\arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{4 a} \\ & = -\frac {3}{4} x^2 \arcsin (a x)+\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a}-\frac {\arcsin (a x)^3}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^3+\frac {1}{4} (3 a) \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {3 x \sqrt {1-a^2 x^2}}{8 a}-\frac {3}{4} x^2 \arcsin (a x)+\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a}-\frac {\arcsin (a x)^3}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^3+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a} \\ & = -\frac {3 x \sqrt {1-a^2 x^2}}{8 a}+\frac {3 \arcsin (a x)}{8 a^2}-\frac {3}{4} x^2 \arcsin (a x)+\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a}-\frac {\arcsin (a x)^3}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^3 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.83 \[ \int x \arcsin (a x)^3 \, dx=\frac {-3 a x \sqrt {1-a^2 x^2}+\left (3-6 a^2 x^2\right ) \arcsin (a x)+6 a x \sqrt {1-a^2 x^2} \arcsin (a x)^2+\left (-2+4 a^2 x^2\right ) \arcsin (a x)^3}{8 a^2} \]
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Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {\arcsin \left (a x \right )^{3} \left (a^{2} x^{2}-1\right )}{2}+\frac {3 \arcsin \left (a x \right )^{2} \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{4}-\frac {3 \left (a^{2} x^{2}-1\right ) \arcsin \left (a x \right )}{4}-\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{8}-\frac {3 \arcsin \left (a x \right )}{8}-\frac {\arcsin \left (a x \right )^{3}}{2}}{a^{2}}\) | \(96\) |
default | \(\frac {\frac {\arcsin \left (a x \right )^{3} \left (a^{2} x^{2}-1\right )}{2}+\frac {3 \arcsin \left (a x \right )^{2} \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{4}-\frac {3 \left (a^{2} x^{2}-1\right ) \arcsin \left (a x \right )}{4}-\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{8}-\frac {3 \arcsin \left (a x \right )}{8}-\frac {\arcsin \left (a x \right )^{3}}{2}}{a^{2}}\) | \(96\) |
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Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int x \arcsin (a x)^3 \, dx=\frac {2 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{3} - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right ) + 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x \arcsin \left (a x\right )^{2} - a x\right )}}{8 \, a^{2}} \]
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Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.93 \[ \int x \arcsin (a x)^3 \, dx=\begin {cases} \frac {x^{2} \operatorname {asin}^{3}{\left (a x \right )}}{2} - \frac {3 x^{2} \operatorname {asin}{\left (a x \right )}}{4} + \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{4 a} - \frac {3 x \sqrt {- a^{2} x^{2} + 1}}{8 a} - \frac {\operatorname {asin}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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\[ \int x \arcsin (a x)^3 \, dx=\int { x \arcsin \left (a x\right )^{3} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.02 \[ \int x \arcsin (a x)^3 \, dx=\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{4 \, a} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{3}}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )^{3}}{4 \, a^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{4 \, a^{2}} - \frac {3 \, \arcsin \left (a x\right )}{8 \, a^{2}} \]
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Timed out. \[ \int x \arcsin (a x)^3 \, dx=\int x\,{\mathrm {asin}\left (a\,x\right )}^3 \,d x \]
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