Integrand size = 10, antiderivative size = 120 \[ \int x^4 \arccos (a x)^2 \, dx=-\frac {16 x}{75 a^4}-\frac {8 x^3}{225 a^2}-\frac {2 x^5}{125}-\frac {16 \sqrt {1-a^2 x^2} \arccos (a x)}{75 a^5}-\frac {8 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{75 a^3}-\frac {2 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{25 a}+\frac {1}{5} x^5 \arccos (a x)^2 \]
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Time = 0.12 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 4796, 4768, 8, 30} \[ \int x^4 \arccos (a x)^2 \, dx=-\frac {16 x}{75 a^4}-\frac {2 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{25 a}-\frac {8 x^3}{225 a^2}-\frac {16 \sqrt {1-a^2 x^2} \arccos (a x)}{75 a^5}-\frac {8 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{75 a^3}+\frac {1}{5} x^5 \arccos (a x)^2-\frac {2 x^5}{125} \]
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Rule 8
Rule 30
Rule 4724
Rule 4768
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \arccos (a x)^2+\frac {1}{5} (2 a) \int \frac {x^5 \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{25 a}+\frac {1}{5} x^5 \arccos (a x)^2-\frac {2 \int x^4 \, dx}{25}+\frac {8 \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{25 a} \\ & = -\frac {2 x^5}{125}-\frac {8 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{75 a^3}-\frac {2 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{25 a}+\frac {1}{5} x^5 \arccos (a x)^2+\frac {16 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{75 a^3}-\frac {8 \int x^2 \, dx}{75 a^2} \\ & = -\frac {8 x^3}{225 a^2}-\frac {2 x^5}{125}-\frac {16 \sqrt {1-a^2 x^2} \arccos (a x)}{75 a^5}-\frac {8 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{75 a^3}-\frac {2 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{25 a}+\frac {1}{5} x^5 \arccos (a x)^2-\frac {16 \int 1 \, dx}{75 a^4} \\ & = -\frac {16 x}{75 a^4}-\frac {8 x^3}{225 a^2}-\frac {2 x^5}{125}-\frac {16 \sqrt {1-a^2 x^2} \arccos (a x)}{75 a^5}-\frac {8 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{75 a^3}-\frac {2 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{25 a}+\frac {1}{5} x^5 \arccos (a x)^2 \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.68 \[ \int x^4 \arccos (a x)^2 \, dx=-\frac {16 x}{75 a^4}-\frac {8 x^3}{225 a^2}-\frac {2 x^5}{125}-\frac {2 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \arccos (a x)}{75 a^5}+\frac {1}{5} x^5 \arccos (a x)^2 \]
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Time = 1.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{2}}{5}-\frac {2 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {2 a^{5} x^{5}}{125}-\frac {8 a^{3} x^{3}}{225}-\frac {16 a x}{75}}{a^{5}}\) | \(76\) |
default | \(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{2}}{5}-\frac {2 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {2 a^{5} x^{5}}{125}-\frac {8 a^{3} x^{3}}{225}-\frac {16 a x}{75}}{a^{5}}\) | \(76\) |
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Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.63 \[ \int x^4 \arccos (a x)^2 \, dx=\frac {225 \, a^{5} x^{5} \arccos \left (a x\right )^{2} - 18 \, a^{5} x^{5} - 40 \, a^{3} x^{3} - 30 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right ) - 240 \, a x}{1125 \, a^{5}} \]
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Time = 0.52 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.01 \[ \int x^4 \arccos (a x)^2 \, dx=\begin {cases} \frac {x^{5} \operatorname {acos}^{2}{\left (a x \right )}}{5} - \frac {2 x^{5}}{125} - \frac {2 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{25 a} - \frac {8 x^{3}}{225 a^{2}} - \frac {8 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{75 a^{3}} - \frac {16 x}{75 a^{4}} - \frac {16 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{75 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{5}}{20} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.85 \[ \int x^4 \arccos (a x)^2 \, dx=\frac {1}{5} \, x^{5} \arccos \left (a x\right )^{2} - \frac {2}{75} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arccos \left (a x\right ) - \frac {2 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83 \[ \int x^4 \arccos (a x)^2 \, dx=\frac {1}{5} \, x^{5} \arccos \left (a x\right )^{2} - \frac {2}{125} \, x^{5} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )}{25 \, a} - \frac {8 \, x^{3}}{225 \, a^{2}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{75 \, a^{3}} - \frac {16 \, x}{75 \, a^{4}} - \frac {16 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{75 \, a^{5}} \]
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Timed out. \[ \int x^4 \arccos (a x)^2 \, dx=\int x^4\,{\mathrm {acos}\left (a\,x\right )}^2 \,d x \]
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