Integrand size = 4, antiderivative size = 25 \[ \int \arcsin (a x) \, dx=\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x) \]
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Time = 0.01 (sec) , antiderivative size = 25, 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 = {4715, 267} \[ \int \arcsin (a x) \, dx=\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x) \]
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Rule 267
Rule 4715
Rubi steps \begin{align*} \text {integral}& = x \arcsin (a x)-a \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \arcsin (a x) \, dx=\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x) \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
parts | \(x \arcsin \left (a x \right )+\frac {\sqrt {-a^{2} x^{2}+1}}{a}\) | \(24\) |
derivativedivides | \(\frac {a x \arcsin \left (a x \right )+\sqrt {-a^{2} x^{2}+1}}{a}\) | \(25\) |
default | \(\frac {a x \arcsin \left (a x \right )+\sqrt {-a^{2} x^{2}+1}}{a}\) | \(25\) |
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \arcsin (a x) \, dx=\frac {a x \arcsin \left (a x\right ) + \sqrt {-a^{2} x^{2} + 1}}{a} \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \arcsin (a x) \, dx=\begin {cases} x \operatorname {asin}{\left (a x \right )} + \frac {\sqrt {- a^{2} x^{2} + 1}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \arcsin (a x) \, dx=\frac {a x \arcsin \left (a x\right ) + \sqrt {-a^{2} x^{2} + 1}}{a} \]
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \arcsin (a x) \, dx=\frac {a x \arcsin \left (a x\right ) + \sqrt {-a^{2} x^{2} + 1}}{a} \]
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Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \arcsin (a x) \, dx=x\,\mathrm {asin}\left (a\,x\right )+\frac {\sqrt {1-a^2\,x^2}}{a} \]
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