Integrand size = 10, antiderivative size = 98 \[ \int x^3 \arcsin (a x)^2 \, dx=-\frac {3 x^2}{32 a^2}-\frac {x^4}{32}+\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)}{16 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{8 a}-\frac {3 \arcsin (a x)^2}{32 a^4}+\frac {1}{4} x^4 \arcsin (a x)^2 \]
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Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4723, 4795, 4737, 30} \[ \int x^3 \arcsin (a x)^2 \, dx=-\frac {3 \arcsin (a x)^2}{32 a^4}+\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{8 a}-\frac {3 x^2}{32 a^2}+\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)}{16 a^3}+\frac {1}{4} x^4 \arcsin (a x)^2-\frac {x^4}{32} \]
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Rule 30
Rule 4723
Rule 4737
Rule 4795
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{8 a}+\frac {1}{4} x^4 \arcsin (a x)^2-\frac {\int x^3 \, dx}{8}-\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 a} \\ & = -\frac {x^4}{32}+\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)}{16 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{8 a}+\frac {1}{4} x^4 \arcsin (a x)^2-\frac {3 \int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{16 a^3}-\frac {3 \int x \, dx}{16 a^2} \\ & = -\frac {3 x^2}{32 a^2}-\frac {x^4}{32}+\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)}{16 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{8 a}-\frac {3 \arcsin (a x)^2}{32 a^4}+\frac {1}{4} x^4 \arcsin (a x)^2 \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.76 \[ \int x^3 \arcsin (a x)^2 \, dx=\frac {-a^2 x^2 \left (3+a^2 x^2\right )+2 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \arcsin (a x)+\left (-3+8 a^4 x^4\right ) \arcsin (a x)^2}{32 a^4} \]
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Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \arcsin \left (a x \right )^{2}}{4}-\frac {\arcsin \left (a x \right ) \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{16}+\frac {3 \arcsin \left (a x \right )^{2}}{32}-\frac {\left (2 a^{2} x^{2}+3\right )^{2}}{128}}{a^{4}}\) | \(91\) |
default | \(\frac {\frac {a^{4} x^{4} \arcsin \left (a x \right )^{2}}{4}-\frac {\arcsin \left (a x \right ) \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{16}+\frac {3 \arcsin \left (a x \right )^{2}}{32}-\frac {\left (2 a^{2} x^{2}+3\right )^{2}}{128}}{a^{4}}\) | \(91\) |
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Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int x^3 \arcsin (a x)^2 \, dx=-\frac {a^{4} x^{4} + 3 \, a^{2} x^{2} - {\left (8 \, a^{4} x^{4} - 3\right )} \arcsin \left (a x\right )^{2} - 2 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{32 \, a^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92 \[ \int x^3 \arcsin (a x)^2 \, dx=\begin {cases} \frac {x^{4} \operatorname {asin}^{2}{\left (a x \right )}}{4} - \frac {x^{4}}{32} + \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{8 a} - \frac {3 x^{2}}{32 a^{2}} + \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{16 a^{3}} - \frac {3 \operatorname {asin}^{2}{\left (a x \right )}}{32 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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\[ \int x^3 \arcsin (a x)^2 \, dx=\int { x^{3} \arcsin \left (a x\right )^{2} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.36 \[ \int x^3 \arcsin (a x)^2 \, dx=-\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )}{8 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{2}}{4 \, a^{4}} + \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{16 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{2 \, a^{4}} - \frac {{\left (a^{2} x^{2} - 1\right )}^{2}}{32 \, a^{4}} + \frac {5 \, \arcsin \left (a x\right )^{2}}{32 \, a^{4}} - \frac {5 \, {\left (a^{2} x^{2} - 1\right )}}{32 \, a^{4}} - \frac {17}{256 \, a^{4}} \]
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Timed out. \[ \int x^3 \arcsin (a x)^2 \, dx=\int x^3\,{\mathrm {asin}\left (a\,x\right )}^2 \,d x \]
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