Integrand size = 6, antiderivative size = 45 \[ \int x \arccos (a x) \, dx=-\frac {x \sqrt {1-a^2 x^2}}{4 a}+\frac {1}{2} x^2 \arccos (a x)+\frac {\arcsin (a x)}{4 a^2} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 327, 222} \[ \int x \arccos (a x) \, dx=\frac {\arcsin (a x)}{4 a^2}-\frac {x \sqrt {1-a^2 x^2}}{4 a}+\frac {1}{2} x^2 \arccos (a x) \]
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Rule 222
Rule 327
Rule 4724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \arccos (a x)+\frac {1}{2} a \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {x \sqrt {1-a^2 x^2}}{4 a}+\frac {1}{2} x^2 \arccos (a x)+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{4 a} \\ & = -\frac {x \sqrt {1-a^2 x^2}}{4 a}+\frac {1}{2} x^2 \arccos (a x)+\frac {\arcsin (a x)}{4 a^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int x \arccos (a x) \, dx=\frac {-a x \sqrt {1-a^2 x^2}+2 a^2 x^2 \arccos (a x)+\arcsin (a x)}{4 a^2} \]
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Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} x^{2} \arccos \left (a x \right )}{2}-\frac {a x \sqrt {-a^{2} x^{2}+1}}{4}+\frac {\arcsin \left (a x \right )}{4}}{a^{2}}\) | \(40\) |
default | \(\frac {\frac {a^{2} x^{2} \arccos \left (a x \right )}{2}-\frac {a x \sqrt {-a^{2} x^{2}+1}}{4}+\frac {\arcsin \left (a x \right )}{4}}{a^{2}}\) | \(40\) |
parts | \(\frac {x^{2} \arccos \left (a x \right )}{2}+\frac {a \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{2}\) | \(63\) |
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Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int x \arccos (a x) \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} a x - {\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right )}{4 \, a^{2}} \]
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Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int x \arccos (a x) \, dx=\begin {cases} \frac {x^{2} \operatorname {acos}{\left (a x \right )}}{2} - \frac {x \sqrt {- a^{2} x^{2} + 1}}{4 a} - \frac {\operatorname {acos}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi x^{2}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.89 \[ \int x \arccos (a x) \, dx=\frac {1}{2} \, x^{2} \arccos \left (a x\right ) - \frac {1}{4} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int x \arccos (a x) \, dx=\frac {1}{2} \, x^{2} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} x}{4 \, a} - \frac {\arccos \left (a x\right )}{4 \, a^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84 \[ \int x \arccos (a x) \, dx=\frac {\mathrm {acos}\left (a\,x\right )\,\left (2\,a^2\,x^2-1\right )}{4\,a^2}-\frac {x\,\sqrt {1-a^2\,x^2}}{4\,a} \]
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