Integrand size = 6, antiderivative size = 35 \[ \int \arcsin (a+b x) \, dx=\frac {\sqrt {1-(a+b x)^2}}{b}+\frac {(a+b x) \arcsin (a+b x)}{b} \]
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Time = 0.01 (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 = {4887, 4715, 267} \[ \int \arcsin (a+b x) \, dx=\frac {(a+b x) \arcsin (a+b x)}{b}+\frac {\sqrt {1-(a+b x)^2}}{b} \]
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Rule 267
Rule 4715
Rule 4887
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int \arcsin (x) \, dx,x,a+b x)}{b} \\ & = \frac {(a+b x) \arcsin (a+b x)}{b}-\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {\sqrt {1-(a+b x)^2}}{b}+\frac {(a+b x) \arcsin (a+b x)}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(154\) vs. \(2(35)=70\).
Time = 0.36 (sec) , antiderivative size = 154, normalized size of antiderivative = 4.40 \[ \int \arcsin (a+b x) \, dx=x \arcsin (a+b x)+\frac {2 b \sqrt {1-a^2-2 a b x-b^2 x^2}+2 a b \arctan \left (\frac {\sqrt {-b^2} x-\sqrt {1-a^2-2 a b x-b^2 x^2}}{a}\right )+a \sqrt {-b^2} \log \left (-1+2 a b x+2 b^2 x^2+2 \sqrt {-b^2} x \sqrt {1-a^2-2 a b x-b^2 x^2}\right )}{2 b^2} \]
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Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\left (b x +a \right ) \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}}{b}\) | \(31\) |
default | \(\frac {\left (b x +a \right ) \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}}{b}\) | \(31\) |
parts | \(x \arcsin \left (b x +a \right )-b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )\) | \(88\) |
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.11 \[ \int \arcsin (a+b x) \, dx=\frac {{\left (b x + a\right )} \arcsin \left (b x + a\right ) + \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b} \]
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Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \arcsin (a+b x) \, dx=\begin {cases} \frac {a \operatorname {asin}{\left (a + b x \right )}}{b} + x \operatorname {asin}{\left (a + b x \right )} + \frac {\sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {asin}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \arcsin (a+b x) \, dx=\frac {{\left (b x + a\right )} \arcsin \left (b x + a\right ) + \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]
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Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \arcsin (a+b x) \, dx=\frac {{\left (b x + a\right )} \arcsin \left (b x + a\right ) + \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]
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Time = 0.57 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.46 \[ \int \arcsin (a+b x) \, dx=x\,\mathrm {asin}\left (a+b\,x\right )+\frac {\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}}{b}+\frac {a\,\ln \left (\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}-\frac {x\,b^2+a\,b}{\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]
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