Integrand size = 10, antiderivative size = 166 \[ \int x^2 \arccos (a x)^4 \, dx=\frac {160 x}{27 a^2}+\frac {8 x^3}{81}+\frac {160 \sqrt {1-a^2 x^2} \arccos (a x)}{27 a^3}+\frac {8 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{27 a}-\frac {8 x \arccos (a x)^2}{3 a^2}-\frac {4}{9} x^3 \arccos (a x)^2-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a}+\frac {1}{3} x^3 \arccos (a x)^4 \]
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Time = 0.23 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4724, 4796, 4768, 4716, 8, 30} \[ \int x^2 \arccos (a x)^4 \, dx=-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a}+\frac {8 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{27 a}-\frac {8 x \arccos (a x)^2}{3 a^2}+\frac {160 x}{27 a^2}-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a^3}+\frac {160 \sqrt {1-a^2 x^2} \arccos (a x)}{27 a^3}+\frac {1}{3} x^3 \arccos (a x)^4-\frac {4}{9} x^3 \arccos (a x)^2+\frac {8 x^3}{81} \]
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Rule 8
Rule 30
Rule 4716
Rule 4724
Rule 4768
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \arccos (a x)^4+\frac {1}{3} (4 a) \int \frac {x^3 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a}+\frac {1}{3} x^3 \arccos (a x)^4-\frac {4}{3} \int x^2 \arccos (a x)^2 \, dx+\frac {8 \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{9 a} \\ & = -\frac {4}{9} x^3 \arccos (a x)^2-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a}+\frac {1}{3} x^3 \arccos (a x)^4-\frac {8 \int \arccos (a x)^2 \, dx}{3 a^2}-\frac {1}{9} (8 a) \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {8 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{27 a}-\frac {8 x \arccos (a x)^2}{3 a^2}-\frac {4}{9} x^3 \arccos (a x)^2-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a}+\frac {1}{3} x^3 \arccos (a x)^4+\frac {8 \int x^2 \, dx}{27}-\frac {16 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{27 a}-\frac {16 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{3 a} \\ & = \frac {8 x^3}{81}+\frac {160 \sqrt {1-a^2 x^2} \arccos (a x)}{27 a^3}+\frac {8 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{27 a}-\frac {8 x \arccos (a x)^2}{3 a^2}-\frac {4}{9} x^3 \arccos (a x)^2-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a}+\frac {1}{3} x^3 \arccos (a x)^4+\frac {16 \int 1 \, dx}{27 a^2}+\frac {16 \int 1 \, dx}{3 a^2} \\ & = \frac {160 x}{27 a^2}+\frac {8 x^3}{81}+\frac {160 \sqrt {1-a^2 x^2} \arccos (a x)}{27 a^3}+\frac {8 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{27 a}-\frac {8 x \arccos (a x)^2}{3 a^2}-\frac {4}{9} x^3 \arccos (a x)^2-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a^3}-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a}+\frac {1}{3} x^3 \arccos (a x)^4 \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.69 \[ \int x^2 \arccos (a x)^4 \, dx=\frac {8 a x \left (60+a^2 x^2\right )+24 \sqrt {1-a^2 x^2} \left (20+a^2 x^2\right ) \arccos (a x)-36 a x \left (6+a^2 x^2\right ) \arccos (a x)^2-36 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \arccos (a x)^3+27 a^3 x^3 \arccos (a x)^4}{81 a^3} \]
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Time = 1.22 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} x^{3} \arccos \left (a x \right )^{4}}{3}-\frac {4 \arccos \left (a x \right )^{3} \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{9}-\frac {8 \arccos \left (a x \right )^{2} a x}{3}+\frac {160 a x}{27}+\frac {16 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{3}-\frac {4 \arccos \left (a x \right )^{2} a^{3} x^{3}}{9}+\frac {8 \arccos \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{27}+\frac {8 a^{3} x^{3}}{81}}{a^{3}}\) | \(130\) |
default | \(\frac {\frac {a^{3} x^{3} \arccos \left (a x \right )^{4}}{3}-\frac {4 \arccos \left (a x \right )^{3} \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{9}-\frac {8 \arccos \left (a x \right )^{2} a x}{3}+\frac {160 a x}{27}+\frac {16 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{3}-\frac {4 \arccos \left (a x \right )^{2} a^{3} x^{3}}{9}+\frac {8 \arccos \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{27}+\frac {8 a^{3} x^{3}}{81}}{a^{3}}\) | \(130\) |
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Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.60 \[ \int x^2 \arccos (a x)^4 \, dx=\frac {27 \, a^{3} x^{3} \arccos \left (a x\right )^{4} + 8 \, a^{3} x^{3} - 36 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \arccos \left (a x\right )^{2} + 480 \, a x - 12 \, \sqrt {-a^{2} x^{2} + 1} {\left (3 \, {\left (a^{2} x^{2} + 2\right )} \arccos \left (a x\right )^{3} - 2 \, {\left (a^{2} x^{2} + 20\right )} \arccos \left (a x\right )\right )}}{81 \, a^{3}} \]
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Time = 0.47 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.99 \[ \int x^2 \arccos (a x)^4 \, dx=\begin {cases} \frac {x^{3} \operatorname {acos}^{4}{\left (a x \right )}}{3} - \frac {4 x^{3} \operatorname {acos}^{2}{\left (a x \right )}}{9} + \frac {8 x^{3}}{81} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{9 a} + \frac {8 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{27 a} - \frac {8 x \operatorname {acos}^{2}{\left (a x \right )}}{3 a^{2}} + \frac {160 x}{27 a^{2}} - \frac {8 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{9 a^{3}} + \frac {160 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{27 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x^{3}}{48} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88 \[ \int x^2 \arccos (a x)^4 \, dx=\frac {1}{3} \, x^{3} \arccos \left (a x\right )^{4} - \frac {4}{9} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arccos \left (a x\right )^{3} + \frac {4}{81} \, {\left (2 \, a {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )} \arccos \left (a x\right )}{a^{3}} + \frac {a^{2} x^{3} + 60 \, x}{a^{4}}\right )} - \frac {9 \, {\left (a^{2} x^{3} + 6 \, x\right )} \arccos \left (a x\right )^{2}}{a^{3}}\right )} a \]
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Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.84 \[ \int x^2 \arccos (a x)^4 \, dx=\frac {1}{3} \, x^{3} \arccos \left (a x\right )^{4} - \frac {4}{9} \, x^{3} \arccos \left (a x\right )^{2} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{3}}{9 \, a} + \frac {8}{81} \, x^{3} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{27 \, a} - \frac {8 \, x \arccos \left (a x\right )^{2}}{3 \, a^{2}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{9 \, a^{3}} + \frac {160 \, x}{27 \, a^{2}} + \frac {160 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{27 \, a^{3}} \]
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Timed out. \[ \int x^2 \arccos (a x)^4 \, dx=\int x^2\,{\mathrm {acos}\left (a\,x\right )}^4 \,d x \]
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