Integrand size = 24, antiderivative size = 157 \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {15 x \sqrt {1-a^2 x^2}}{64 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}-\frac {15 \arcsin (a x)}{64 a^5}+\frac {3 x^2 \arcsin (a x)}{8 a^3}+\frac {x^4 \arcsin (a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {\arcsin (a x)^3}{8 a^5} \]
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Time = 0.18 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4795, 4737, 4723, 327, 222} \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {\arcsin (a x)^3}{8 a^5}-\frac {15 \arcsin (a x)}{64 a^5}+\frac {3 x^2 \arcsin (a x)}{8 a^3}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}+\frac {15 x \sqrt {1-a^2 x^2}}{64 a^4}+\frac {x^4 \arcsin (a x)}{8 a} \]
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Rule 222
Rule 327
Rule 4723
Rule 4737
Rule 4795
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {3 \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}+\frac {\int x^3 \arcsin (a x) \, dx}{2 a} \\ & = \frac {x^4 \arcsin (a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}-\frac {1}{8} \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx+\frac {3 \int \frac {\arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{8 a^4}+\frac {3 \int x \arcsin (a x) \, dx}{4 a^3} \\ & = \frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}+\frac {3 x^2 \arcsin (a x)}{8 a^3}+\frac {x^4 \arcsin (a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {\arcsin (a x)^3}{8 a^5}-\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{32 a^2}-\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{8 a^2} \\ & = \frac {15 x \sqrt {1-a^2 x^2}}{64 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}+\frac {3 x^2 \arcsin (a x)}{8 a^3}+\frac {x^4 \arcsin (a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {\arcsin (a x)^3}{8 a^5}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{64 a^4}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{16 a^4} \\ & = \frac {15 x \sqrt {1-a^2 x^2}}{64 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}-\frac {15 \arcsin (a x)}{64 a^5}+\frac {3 x^2 \arcsin (a x)}{8 a^3}+\frac {x^4 \arcsin (a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {\arcsin (a x)^3}{8 a^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.64 \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {a x \sqrt {1-a^2 x^2} \left (15+2 a^2 x^2\right )+\left (-15+24 a^2 x^2+8 a^4 x^4\right ) \arcsin (a x)-8 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \arcsin (a x)^2+8 \arcsin (a x)^3}{64 a^5} \]
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Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {-16 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+8 a^{4} x^{4} \arcsin \left (a x \right )+2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-24 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +24 a^{2} x^{2} \arcsin \left (a x \right )+8 \arcsin \left (a x \right )^{3}+15 a x \sqrt {-a^{2} x^{2}+1}-15 \arcsin \left (a x \right )}{64 a^{5}}\) | \(129\) |
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Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.54 \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {8 \, \arcsin \left (a x\right )^{3} + {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arcsin \left (a x\right ) + {\left (2 \, a^{3} x^{3} - 8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arcsin \left (a x\right )^{2} + 15 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{64 \, a^{5}} \]
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Time = 0.51 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.93 \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} \frac {x^{4} \operatorname {asin}{\left (a x \right )}}{8 a} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{4 a^{2}} + \frac {x^{3} \sqrt {- a^{2} x^{2} + 1}}{32 a^{2}} + \frac {3 x^{2} \operatorname {asin}{\left (a x \right )}}{8 a^{3}} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{8 a^{4}} + \frac {15 x \sqrt {- a^{2} x^{2} + 1}}{64 a^{4}} + \frac {\operatorname {asin}^{3}{\left (a x \right )}}{8 a^{5}} - \frac {15 \operatorname {asin}{\left (a x \right )}}{64 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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\[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.91 \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )^{2}}{4 \, a^{4}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{8 \, a^{4}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{32 \, a^{4}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )}{8 \, a^{5}} + \frac {\arcsin \left (a x\right )^{3}}{8 \, a^{5}} + \frac {17 \, \sqrt {-a^{2} x^{2} + 1} x}{64 \, a^{4}} + \frac {5 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{8 \, a^{5}} + \frac {17 \, \arcsin \left (a x\right )}{64 \, a^{5}} \]
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Timed out. \[ \int \frac {x^4 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^4\,{\mathrm {asin}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]
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