Integrand size = 31, antiderivative size = 63 \[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x) \, dx=-\frac {(a+b x)^2}{4 b}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{2 b}+\frac {\arcsin (a+b x)^2}{4 b} \]
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Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4891, 4741, 4737, 30} \[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x) \, dx=\frac {\sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)}{2 b}+\frac {\arcsin (a+b x)^2}{4 b}-\frac {(a+b x)^2}{4 b} \]
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Rule 30
Rule 4737
Rule 4741
Rule 4891
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {1-x^2} \arcsin (x) \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{2 b}-\frac {\text {Subst}(\int x \, dx,x,a+b x)}{2 b}+\frac {\text {Subst}\left (\int \frac {\arcsin (x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 b} \\ & = -\frac {(a+b x)^2}{4 b}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{2 b}+\frac {\arcsin (a+b x)^2}{4 b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x) \, dx=\frac {-b x (2 a+b x)+2 (a+b x) \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)+\arcsin (a+b x)^2}{4 b} \]
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Time = 1.50 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.52
method | result | size |
default | \(\frac {2 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -b^{2} x^{2}+2 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -2 a b x +\arcsin \left (b x +a \right )^{2}-a^{2}}{4 b}\) | \(96\) |
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Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x) \, dx=-\frac {b^{2} x^{2} + 2 \, a b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right ) - \arcsin \left (b x + a\right )^{2}}{4 \, b} \]
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\[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x) \, dx=\int \sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )} \operatorname {asin}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (55) = 110\).
Time = 0.31 (sec) , antiderivative size = 240, normalized size of antiderivative = 3.81 \[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x) \, dx=-\frac {1}{4} \, {\left (x^{2} + \frac {2 \, a x}{b} - \frac {2 \, \arcsin \left (b x + a\right ) \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{2}} - \frac {\arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )^{2}}{b^{2}}\right )} b - \frac {1}{2} \, {\left (\frac {a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b} - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x - \frac {{\left (a^{2} - 1\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b}\right )} \arcsin \left (b x + a\right ) \]
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Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.25 \[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x) \, dx=\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{2 \, b} + \frac {\arcsin \left (b x + a\right )^{2}}{4 \, b} - \frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{4 \, b} - \frac {1}{8 \, b} \]
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Timed out. \[ \int \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x) \, dx=\int \mathrm {asin}\left (a+b\,x\right )\,\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1} \,d x \]
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