Integrand size = 6, antiderivative size = 60 \[ \int \arccos (a x)^3 \, dx=\frac {6 \sqrt {1-a^2 x^2}}{a}-6 x \arccos (a x)-\frac {3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{a}+x \arccos (a x)^3 \]
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Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4716, 4768, 267} \[ \int \arccos (a x)^3 \, dx=-\frac {3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{a}+\frac {6 \sqrt {1-a^2 x^2}}{a}+x \arccos (a x)^3-6 x \arccos (a x) \]
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Rule 267
Rule 4716
Rule 4768
Rubi steps \begin{align*} \text {integral}& = x \arccos (a x)^3+(3 a) \int \frac {x \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{a}+x \arccos (a x)^3-6 \int \arccos (a x) \, dx \\ & = -6 x \arccos (a x)-\frac {3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{a}+x \arccos (a x)^3-(6 a) \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {6 \sqrt {1-a^2 x^2}}{a}-6 x \arccos (a x)-\frac {3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{a}+x \arccos (a x)^3 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \arccos (a x)^3 \, dx=\frac {6 \sqrt {1-a^2 x^2}}{a}-6 x \arccos (a x)-\frac {3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{a}+x \arccos (a x)^3 \]
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Time = 0.48 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\arccos \left (a x \right )^{3} a x -3 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+6 \sqrt {-a^{2} x^{2}+1}-6 a x \arccos \left (a x \right )}{a}\) | \(57\) |
default | \(\frac {\arccos \left (a x \right )^{3} a x -3 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+6 \sqrt {-a^{2} x^{2}+1}-6 a x \arccos \left (a x \right )}{a}\) | \(57\) |
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none
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73 \[ \int \arccos (a x)^3 \, dx=\frac {a x \arccos \left (a x\right )^{3} - 6 \, a x \arccos \left (a x\right ) - 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (\arccos \left (a x\right )^{2} - 2\right )}}{a} \]
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Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \arccos (a x)^3 \, dx=\begin {cases} x \operatorname {acos}^{3}{\left (a x \right )} - 6 x \operatorname {acos}{\left (a x \right )} - \frac {3 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{a} + \frac {6 \sqrt {- a^{2} x^{2} + 1}}{a} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x}{8} & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \arccos (a x)^3 \, dx=x \arccos \left (a x\right )^{3} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{a} - \frac {6 \, {\left (a x \arccos \left (a x\right ) - \sqrt {-a^{2} x^{2} + 1}\right )}}{a} \]
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none
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.93 \[ \int \arccos (a x)^3 \, dx=x \arccos \left (a x\right )^{3} - 6 \, x \arccos \left (a x\right ) - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{a} + \frac {6 \, \sqrt {-a^{2} x^{2} + 1}}{a} \]
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Time = 0.33 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \arccos (a x)^3 \, dx=\left \{\begin {array}{cl} \frac {x\,\pi ^3}{8} & \text {\ if\ \ }a=0\\ -x\,\left (6\,\mathrm {acos}\left (a\,x\right )-{\mathrm {acos}\left (a\,x\right )}^3\right )-\frac {\sqrt {1-a^2\,x^2}\,\left (3\,{\mathrm {acos}\left (a\,x\right )}^2-6\right )}{a} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
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