Integrand size = 10, antiderivative size = 136 \[ \int x^2 \arccos (a x)^3 \, dx=\frac {14 \sqrt {1-a^2 x^2}}{9 a^3}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac {4 x \arccos (a x)}{3 a^2}-\frac {2}{9} x^3 \arccos (a x)-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a}+\frac {1}{3} x^3 \arccos (a x)^3 \]
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Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4724, 4796, 4768, 4716, 267, 272, 45} \[ \int x^2 \arccos (a x)^3 \, dx=-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a}-\frac {4 x \arccos (a x)}{3 a^2}-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a^3}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}+\frac {14 \sqrt {1-a^2 x^2}}{9 a^3}+\frac {1}{3} x^3 \arccos (a x)^3-\frac {2}{9} x^3 \arccos (a x) \]
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Rule 45
Rule 267
Rule 272
Rule 4716
Rule 4724
Rule 4768
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \arccos (a x)^3+a \int \frac {x^3 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a}+\frac {1}{3} x^3 \arccos (a x)^3-\frac {2}{3} \int x^2 \arccos (a x) \, dx+\frac {2 \int \frac {x \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{3 a} \\ & = -\frac {2}{9} x^3 \arccos (a x)-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a}+\frac {1}{3} x^3 \arccos (a x)^3-\frac {4 \int \arccos (a x) \, dx}{3 a^2}-\frac {1}{9} (2 a) \int \frac {x^3}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {4 x \arccos (a x)}{3 a^2}-\frac {2}{9} x^3 \arccos (a x)-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a}+\frac {1}{3} x^3 \arccos (a x)^3-\frac {4 \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx}{3 a}-\frac {1}{9} a \text {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x}} \, dx,x,x^2\right ) \\ & = \frac {4 \sqrt {1-a^2 x^2}}{3 a^3}-\frac {4 x \arccos (a x)}{3 a^2}-\frac {2}{9} x^3 \arccos (a x)-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a}+\frac {1}{3} x^3 \arccos (a x)^3-\frac {1}{9} a \text {Subst}\left (\int \left (\frac {1}{a^2 \sqrt {1-a^2 x}}-\frac {\sqrt {1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right ) \\ & = \frac {14 \sqrt {1-a^2 x^2}}{9 a^3}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac {4 x \arccos (a x)}{3 a^2}-\frac {2}{9} x^3 \arccos (a x)-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a}+\frac {1}{3} x^3 \arccos (a x)^3 \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.70 \[ \int x^2 \arccos (a x)^3 \, dx=\frac {2 \sqrt {1-a^2 x^2} \left (20+a^2 x^2\right )-6 a x \left (6+a^2 x^2\right ) \arccos (a x)-9 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \arccos (a x)^2+9 a^3 x^3 \arccos (a x)^3}{27 a^3} \]
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Time = 1.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} x^{3} \arccos \left (a x \right )^{3}}{3}-\frac {\arccos \left (a x \right )^{2} \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-a^{2} x^{2}+1}}{3}-\frac {4 a x \arccos \left (a x \right )}{3}-\frac {2 a^{3} x^{3} \arccos \left (a x \right )}{9}+\frac {2 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{27}}{a^{3}}\) | \(106\) |
default | \(\frac {\frac {a^{3} x^{3} \arccos \left (a x \right )^{3}}{3}-\frac {\arccos \left (a x \right )^{2} \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-a^{2} x^{2}+1}}{3}-\frac {4 a x \arccos \left (a x \right )}{3}-\frac {2 a^{3} x^{3} \arccos \left (a x \right )}{9}+\frac {2 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{27}}{a^{3}}\) | \(106\) |
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Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.57 \[ \int x^2 \arccos (a x)^3 \, dx=\frac {9 \, a^{3} x^{3} \arccos \left (a x\right )^{3} - 6 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \arccos \left (a x\right ) + {\left (2 \, a^{2} x^{2} - 9 \, {\left (a^{2} x^{2} + 2\right )} \arccos \left (a x\right )^{2} + 40\right )} \sqrt {-a^{2} x^{2} + 1}}{27 \, a^{3}} \]
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Time = 0.35 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99 \[ \int x^2 \arccos (a x)^3 \, dx=\begin {cases} \frac {x^{3} \operatorname {acos}^{3}{\left (a x \right )}}{3} - \frac {2 x^{3} \operatorname {acos}{\left (a x \right )}}{9} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{3 a} + \frac {2 x^{2} \sqrt {- a^{2} x^{2} + 1}}{27 a} - \frac {4 x \operatorname {acos}{\left (a x \right )}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{3 a^{3}} + \frac {40 \sqrt {- a^{2} x^{2} + 1}}{27 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x^{3}}{24} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.88 \[ \int x^2 \arccos (a x)^3 \, dx=\frac {1}{3} \, x^{3} \arccos \left (a x\right )^{3} - \frac {1}{3} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arccos \left (a x\right )^{2} + \frac {2}{27} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}}{a^{2}} - \frac {3 \, {\left (a^{2} x^{3} + 6 \, x\right )} \arccos \left (a x\right )}{a^{3}}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.86 \[ \int x^2 \arccos (a x)^3 \, dx=\frac {1}{3} \, x^{3} \arccos \left (a x\right )^{3} - \frac {2}{9} \, x^{3} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{2}}{3 \, a} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{27 \, a} - \frac {4 \, x \arccos \left (a x\right )}{3 \, a^{2}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{3 \, a^{3}} + \frac {40 \, \sqrt {-a^{2} x^{2} + 1}}{27 \, a^{3}} \]
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Timed out. \[ \int x^2 \arccos (a x)^3 \, dx=\int x^2\,{\mathrm {acos}\left (a\,x\right )}^3 \,d x \]
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