Integrand size = 16, antiderivative size = 67 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=-8 b^2 x+\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \]
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Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4898, 8} \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2-8 b^2 x \]
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Rule 8
Rule 4898
Rubi steps \begin{align*} \text {integral}& = \frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2-\left (8 b^2\right ) \int 1 \, dx \\ & = -8 b^2 x+\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=-8 b^2 x+\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \]
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\[\int {\left (a +b \arcsin \left (d \,x^{2}-1\right )\right )}^{2}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.36 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=\frac {b^{2} d x^{2} \arcsin \left (d x^{2} - 1\right )^{2} + 2 \, a b d x^{2} \arcsin \left (d x^{2} - 1\right ) + {\left (a^{2} - 8 \, b^{2}\right )} d x^{2} + 4 \, \sqrt {-d^{2} x^{4} + 2 \, d x^{2}} {\left (b^{2} \arcsin \left (d x^{2} - 1\right ) + a b\right )}}{d x} \]
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\[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=\int \left (a + b \operatorname {asin}{\left (d x^{2} - 1 \right )}\right )^{2}\, dx \]
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\[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=\int { {\left (b \arcsin \left (d x^{2} - 1\right ) + a\right )}^{2} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (61) = 122\).
Time = 0.44 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.39 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=2 \, {\left (x \arcsin \left (d x^{2} - 1\right ) - \frac {2 \, \sqrt {2} \mathrm {sgn}\left (x\right )}{\sqrt {d}} + \frac {2 \, \sqrt {-d^{2} x^{2} + 2 \, d}}{d \mathrm {sgn}\left (x\right )}\right )} a b + {\left (x \arcsin \left (d x^{2} - 1\right )^{2} + \frac {2 \, {\left (\sqrt {2} \pi \sqrt {d} {\left | d \right |} - 4 \, \sqrt {2} d^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{d {\left | d \right |}} + \frac {4 \, {\left (\sqrt {-d^{2} x^{2} + 2 \, d} \arcsin \left (d x^{2} - 1\right ) + \frac {2 \, {\left (\sqrt {2} \sqrt {d} - \sqrt {d^{2} x^{2}}\right )} d}{{\left | d \right |}}\right )}}{d \mathrm {sgn}\left (x\right )}\right )} b^{2} + a^{2} x \]
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Timed out. \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (d\,x^2-1\right )\right )}^2 \,d x \]
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