Integrand size = 10, antiderivative size = 82 \[ \int x^2 \arccos (a x)^2 \, dx=-\frac {4 x}{9 a^2}-\frac {2 x^3}{27}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)}{9 a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{9 a}+\frac {1}{3} x^3 \arccos (a x)^2 \]
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Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 4796, 4768, 8, 30} \[ \int x^2 \arccos (a x)^2 \, dx=-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{9 a}-\frac {4 x}{9 a^2}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)}{9 a^3}+\frac {1}{3} x^3 \arccos (a x)^2-\frac {2 x^3}{27} \]
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Rule 8
Rule 30
Rule 4724
Rule 4768
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \arccos (a x)^2+\frac {1}{3} (2 a) \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{9 a}+\frac {1}{3} x^3 \arccos (a x)^2-\frac {2 \int x^2 \, dx}{9}+\frac {4 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{9 a} \\ & = -\frac {2 x^3}{27}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)}{9 a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{9 a}+\frac {1}{3} x^3 \arccos (a x)^2-\frac {4 \int 1 \, dx}{9 a^2} \\ & = -\frac {4 x}{9 a^2}-\frac {2 x^3}{27}-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)}{9 a^3}-\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{9 a}+\frac {1}{3} x^3 \arccos (a x)^2 \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77 \[ \int x^2 \arccos (a x)^2 \, dx=-\frac {4 x}{9 a^2}-\frac {2 x^3}{27}-\frac {2 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \arccos (a x)}{9 a^3}+\frac {1}{3} x^3 \arccos (a x)^2 \]
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Time = 1.14 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {\frac {\arccos \left (a x \right )^{2} a^{3} x^{3}}{3}-\frac {2 \arccos \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{9}-\frac {2 a^{3} x^{3}}{27}-\frac {4 a x}{9}}{a^{3}}\) | \(59\) |
default | \(\frac {\frac {\arccos \left (a x \right )^{2} a^{3} x^{3}}{3}-\frac {2 \arccos \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{9}-\frac {2 a^{3} x^{3}}{27}-\frac {4 a x}{9}}{a^{3}}\) | \(59\) |
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Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.72 \[ \int x^2 \arccos (a x)^2 \, dx=\frac {9 \, a^{3} x^{3} \arccos \left (a x\right )^{2} - 2 \, a^{3} x^{3} - 6 \, {\left (a^{2} x^{2} + 2\right )} \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right ) - 12 \, a x}{27 \, a^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.01 \[ \int x^2 \arccos (a x)^2 \, dx=\begin {cases} \frac {x^{3} \operatorname {acos}^{2}{\left (a x \right )}}{3} - \frac {2 x^{3}}{27} - \frac {2 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{9 a} - \frac {4 x}{9 a^{2}} - \frac {4 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{9 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{3}}{12} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88 \[ \int x^2 \arccos (a x)^2 \, dx=\frac {1}{3} \, x^{3} \arccos \left (a x\right )^{2} - \frac {2}{9} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arccos \left (a x\right ) - \frac {2 \, {\left (a^{2} x^{3} + 6 \, x\right )}}{27 \, a^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int x^2 \arccos (a x)^2 \, dx=\frac {1}{3} \, x^{3} \arccos \left (a x\right )^{2} - \frac {2}{27} \, x^{3} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{9 \, a} - \frac {4 \, x}{9 \, a^{2}} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{9 \, a^{3}} \]
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Timed out. \[ \int x^2 \arccos (a x)^2 \, dx=\int x^2\,{\mathrm {acos}\left (a\,x\right )}^2 \,d x \]
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