Integrand size = 8, antiderivative size = 82 \[ \int \arcsin (a+b x)^3 \, dx=-\frac {6 \sqrt {1-(a+b x)^2}}{b}-\frac {6 (a+b x) \arcsin (a+b x)}{b}+\frac {3 \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b}+\frac {(a+b x) \arcsin (a+b x)^3}{b} \]
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Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4887, 4715, 4767, 267} \[ \int \arcsin (a+b x)^3 \, dx=\frac {(a+b x) \arcsin (a+b x)^3}{b}+\frac {3 \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b}-\frac {6 (a+b x) \arcsin (a+b x)}{b}-\frac {6 \sqrt {1-(a+b x)^2}}{b} \]
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Rule 267
Rule 4715
Rule 4767
Rule 4887
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \arcsin (x)^3 \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \arcsin (a+b x)^3}{b}-\frac {3 \text {Subst}\left (\int \frac {x \arcsin (x)^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {3 \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b}+\frac {(a+b x) \arcsin (a+b x)^3}{b}-\frac {6 \text {Subst}(\int \arcsin (x) \, dx,x,a+b x)}{b} \\ & = -\frac {6 (a+b x) \arcsin (a+b x)}{b}+\frac {3 \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b}+\frac {(a+b x) \arcsin (a+b x)^3}{b}+\frac {6 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {6 \sqrt {1-(a+b x)^2}}{b}-\frac {6 (a+b x) \arcsin (a+b x)}{b}+\frac {3 \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b}+\frac {(a+b x) \arcsin (a+b x)^3}{b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90 \[ \int \arcsin (a+b x)^3 \, dx=\frac {-6 \sqrt {1-(a+b x)^2}-6 (a+b x) \arcsin (a+b x)+3 \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2+(a+b x) \arcsin (a+b x)^3}{b} \]
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Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )}{b}\) | \(71\) |
default | \(\frac {\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )}{b}\) | \(71\) |
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Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80 \[ \int \arcsin (a+b x)^3 \, dx=\frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )^{3} - 6 \, {\left (b x + a\right )} \arcsin \left (b x + a\right ) + 3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (\arcsin \left (b x + a\right )^{2} - 2\right )}}{b} \]
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Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.33 \[ \int \arcsin (a+b x)^3 \, dx=\begin {cases} \frac {a \operatorname {asin}^{3}{\left (a + b x \right )}}{b} - \frac {6 a \operatorname {asin}{\left (a + b x \right )}}{b} + x \operatorname {asin}^{3}{\left (a + b x \right )} - 6 x \operatorname {asin}{\left (a + b x \right )} + \frac {3 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{b} - \frac {6 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {asin}^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
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\[ \int \arcsin (a+b x)^3 \, dx=\int { \arcsin \left (b x + a\right )^{3} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.95 \[ \int \arcsin (a+b x)^3 \, dx=\frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{b} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} \arcsin \left (b x + a\right )^{2}}{b} - \frac {6 \, {\left (b x + a\right )} \arcsin \left (b x + a\right )}{b} - \frac {6 \, \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \]
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Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.72 \[ \int \arcsin (a+b x)^3 \, dx=\frac {\left (3\,{\mathrm {asin}\left (a+b\,x\right )}^2-6\right )\,\sqrt {1-{\left (a+b\,x\right )}^2}}{b}-\frac {\left (6\,\mathrm {asin}\left (a+b\,x\right )-{\mathrm {asin}\left (a+b\,x\right )}^3\right )\,\left (a+b\,x\right )}{b} \]
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