Integrand size = 4, antiderivative size = 25 \[ \int \arcsin (x)^2 \, dx=-2 x+2 \sqrt {1-x^2} \arcsin (x)+x \arcsin (x)^2 \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4715, 4767, 8} \[ \int \arcsin (x)^2 \, dx=2 \sqrt {1-x^2} \arcsin (x)+x \arcsin (x)^2-2 x \]
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Rule 8
Rule 4715
Rule 4767
Rubi steps \begin{align*} \text {integral}& = x \arcsin (x)^2-2 \int \frac {x \arcsin (x)}{\sqrt {1-x^2}} \, dx \\ & = 2 \sqrt {1-x^2} \arcsin (x)+x \arcsin (x)^2-2 \int 1 \, dx \\ & = -2 x+2 \sqrt {1-x^2} \arcsin (x)+x \arcsin (x)^2 \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \arcsin (x)^2 \, dx=-2 x+2 \sqrt {1-x^2} \arcsin (x)+x \arcsin (x)^2 \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
default | \(-2 x +x \arcsin \left (x \right )^{2}+2 \arcsin \left (x \right ) \sqrt {-x^{2}+1}\) | \(24\) |
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none
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \arcsin (x)^2 \, dx=x \arcsin \left (x\right )^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \]
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Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \arcsin (x)^2 \, dx=x \operatorname {asin}^{2}{\left (x \right )} - 2 x + 2 \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \arcsin (x)^2 \, dx=x \arcsin \left (x\right )^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \]
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none
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \arcsin (x)^2 \, dx=x \arcsin \left (x\right )^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \]
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Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \arcsin (x)^2 \, dx=2\,\mathrm {asin}\left (x\right )\,\sqrt {1-x^2}+x\,\left ({\mathrm {asin}\left (x\right )}^2-2\right ) \]
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