Integrand size = 6, antiderivative size = 35 \[ \int \arcsin (a x)^2 \, dx=-2 x+\frac {2 \sqrt {1-a^2 x^2} \arcsin (a x)}{a}+x \arcsin (a x)^2 \]
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Time = 0.03 (sec) , antiderivative size = 35, 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 = {4715, 4767, 8} \[ \int \arcsin (a x)^2 \, dx=\frac {2 \sqrt {1-a^2 x^2} \arcsin (a x)}{a}+x \arcsin (a x)^2-2 x \]
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Rule 8
Rule 4715
Rule 4767
Rubi steps \begin{align*} \text {integral}& = x \arcsin (a x)^2-(2 a) \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {2 \sqrt {1-a^2 x^2} \arcsin (a x)}{a}+x \arcsin (a x)^2-2 \int 1 \, dx \\ & = -2 x+\frac {2 \sqrt {1-a^2 x^2} \arcsin (a x)}{a}+x \arcsin (a x)^2 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \arcsin (a x)^2 \, dx=-2 x+\frac {2 \sqrt {1-a^2 x^2} \arcsin (a x)}{a}+x \arcsin (a x)^2 \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {a x \arcsin \left (a x \right )^{2}-2 a x +2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\) | \(37\) |
default | \(\frac {a x \arcsin \left (a x \right )^{2}-2 a x +2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a}\) | \(37\) |
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Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \arcsin (a x)^2 \, dx=\frac {a x \arcsin \left (a x\right )^{2} - 2 \, a x + 2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \]
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Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \arcsin (a x)^2 \, dx=\begin {cases} x \operatorname {asin}^{2}{\left (a x \right )} - 2 x + \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \arcsin (a x)^2 \, dx=x \arcsin \left (a x\right )^{2} - 2 \, x + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \]
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \arcsin (a x)^2 \, dx=x \arcsin \left (a x\right )^{2} - 2 \, x + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \]
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Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \arcsin (a x)^2 \, dx=x\,\left ({\mathrm {asin}\left (a\,x\right )}^2-2\right )+\frac {2\,\mathrm {asin}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}}{a} \]
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