Integrand size = 6, antiderivative size = 69 \[ \int \arccos (a x)^4 \, dx=24 x+\frac {24 \sqrt {1-a^2 x^2} \arccos (a x)}{a}-12 x \arccos (a x)^2-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{a}+x \arccos (a x)^4 \]
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Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4716, 4768, 8} \[ \int \arccos (a x)^4 \, dx=-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{a}+\frac {24 \sqrt {1-a^2 x^2} \arccos (a x)}{a}+x \arccos (a x)^4-12 x \arccos (a x)^2+24 x \]
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Rule 8
Rule 4716
Rule 4768
Rubi steps \begin{align*} \text {integral}& = x \arccos (a x)^4+(4 a) \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{a}+x \arccos (a x)^4-12 \int \arccos (a x)^2 \, dx \\ & = -12 x \arccos (a x)^2-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{a}+x \arccos (a x)^4-(24 a) \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {24 \sqrt {1-a^2 x^2} \arccos (a x)}{a}-12 x \arccos (a x)^2-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{a}+x \arccos (a x)^4+24 \int 1 \, dx \\ & = 24 x+\frac {24 \sqrt {1-a^2 x^2} \arccos (a x)}{a}-12 x \arccos (a x)^2-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{a}+x \arccos (a x)^4 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \arccos (a x)^4 \, dx=24 x+\frac {24 \sqrt {1-a^2 x^2} \arccos (a x)}{a}-12 x \arccos (a x)^2-\frac {4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{a}+x \arccos (a x)^4 \]
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Time = 0.46 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {a x \arccos \left (a x \right )^{4}-4 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{3}-12 \arccos \left (a x \right )^{2} a x +24 a x +24 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\) | \(67\) |
default | \(\frac {a x \arccos \left (a x \right )^{4}-4 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{3}-12 \arccos \left (a x \right )^{2} a x +24 a x +24 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\) | \(67\) |
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Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int \arccos (a x)^4 \, dx=\frac {a x \arccos \left (a x\right )^{4} - 12 \, a x \arccos \left (a x\right )^{2} + 24 \, a x - 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (\arccos \left (a x\right )^{3} - 6 \, \arccos \left (a x\right )\right )}}{a} \]
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Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01 \[ \int \arccos (a x)^4 \, dx=\begin {cases} x \operatorname {acos}^{4}{\left (a x \right )} - 12 x \operatorname {acos}^{2}{\left (a x \right )} + 24 x - \frac {4 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{a} + \frac {24 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x}{16} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \arccos (a x)^4 \, dx=x \arccos \left (a x\right )^{4} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} - 12 \, {\left (\frac {x \arccos \left (a x\right )^{2}}{a} - \frac {2 \, {\left (x + \frac {\sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a}\right )}}{a}\right )} a \]
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Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94 \[ \int \arccos (a x)^4 \, dx=x \arccos \left (a x\right )^{4} - 12 \, x \arccos \left (a x\right )^{2} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} + 24 \, x + \frac {24 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \]
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Time = 0.32 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \arccos (a x)^4 \, dx=\left \{\begin {array}{cl} \frac {x\,\pi ^4}{16} & \text {\ if\ \ }a=0\\ x\,\left ({\mathrm {acos}\left (a\,x\right )}^4-12\,{\mathrm {acos}\left (a\,x\right )}^2+24\right )+\frac {\sqrt {1-a^2\,x^2}\,\left (24\,\mathrm {acos}\left (a\,x\right )-4\,{\mathrm {acos}\left (a\,x\right )}^3\right )}{a} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
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