Integrand size = 4, antiderivative size = 26 \[ \int \arccos (a x) \, dx=-\frac {\sqrt {1-a^2 x^2}}{a}+x \arccos (a x) \]
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Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4716, 267} \[ \int \arccos (a x) \, dx=x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a} \]
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Rule 267
Rule 4716
Rubi steps \begin{align*} \text {integral}& = x \arccos (a x)+a \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1-a^2 x^2}}{a}+x \arccos (a x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \arccos (a x) \, dx=-\frac {\sqrt {1-a^2 x^2}}{a}+x \arccos (a x) \]
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Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
parts | \(x \arccos \left (a x \right )-\frac {\sqrt {-a^{2} x^{2}+1}}{a}\) | \(25\) |
derivativedivides | \(\frac {a x \arccos \left (a x \right )-\sqrt {-a^{2} x^{2}+1}}{a}\) | \(27\) |
default | \(\frac {a x \arccos \left (a x \right )-\sqrt {-a^{2} x^{2}+1}}{a}\) | \(27\) |
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \arccos (a x) \, dx=\frac {a x \arccos \left (a x\right ) - \sqrt {-a^{2} x^{2} + 1}}{a} \]
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Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \arccos (a x) \, dx=\begin {cases} x \operatorname {acos}{\left (a x \right )} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a} & \text {for}\: a \neq 0 \\\frac {\pi x}{2} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \arccos (a x) \, dx=\frac {a x \arccos \left (a x\right ) - \sqrt {-a^{2} x^{2} + 1}}{a} \]
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Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \arccos (a x) \, dx=\frac {a x \arccos \left (a x\right ) - \sqrt {-a^{2} x^{2} + 1}}{a} \]
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Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \arccos (a x) \, dx=x\,\mathrm {acos}\left (a\,x\right )-\frac {\sqrt {1-a^2\,x^2}}{a} \]
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