Integrand size = 12, antiderivative size = 104 \[ \int (a+b \arcsin (c+d x))^3 \, dx=-6 a b^2 x-\frac {6 b^3 \sqrt {1-(c+d x)^2}}{d}-\frac {6 b^3 (c+d x) \arcsin (c+d x)}{d}+\frac {3 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{d}+\frac {(c+d x) (a+b \arcsin (c+d x))^3}{d} \]
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Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4887, 4715, 4767, 267} \[ \int (a+b \arcsin (c+d x))^3 \, dx=\frac {3 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{d}+\frac {(c+d x) (a+b \arcsin (c+d x))^3}{d}-6 a b^2 x-\frac {6 b^3 (c+d x) \arcsin (c+d x)}{d}-\frac {6 b^3 \sqrt {1-(c+d x)^2}}{d} \]
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Rule 267
Rule 4715
Rule 4767
Rule 4887
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+b \arcsin (x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {(c+d x) (a+b \arcsin (c+d x))^3}{d}-\frac {(3 b) \text {Subst}\left (\int \frac {x (a+b \arcsin (x))^2}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {3 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{d}+\frac {(c+d x) (a+b \arcsin (c+d x))^3}{d}-\frac {\left (6 b^2\right ) \text {Subst}(\int (a+b \arcsin (x)) \, dx,x,c+d x)}{d} \\ & = -6 a b^2 x+\frac {3 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{d}+\frac {(c+d x) (a+b \arcsin (c+d x))^3}{d}-\frac {\left (6 b^3\right ) \text {Subst}(\int \arcsin (x) \, dx,x,c+d x)}{d} \\ & = -6 a b^2 x-\frac {6 b^3 (c+d x) \arcsin (c+d x)}{d}+\frac {3 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{d}+\frac {(c+d x) (a+b \arcsin (c+d x))^3}{d}+\frac {\left (6 b^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d} \\ & = -6 a b^2 x-\frac {6 b^3 \sqrt {1-(c+d x)^2}}{d}-\frac {6 b^3 (c+d x) \arcsin (c+d x)}{d}+\frac {3 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{d}+\frac {(c+d x) (a+b \arcsin (c+d x))^3}{d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.92 \[ \int (a+b \arcsin (c+d x))^3 \, dx=\frac {3 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2+(c+d x) (a+b \arcsin (c+d x))^3-6 b^2 \left (a (c+d x)+b \sqrt {1-(c+d x)^2}+b (c+d x) \arcsin (c+d x)\right )}{d} \]
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Time = 0.46 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.60
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+3 a \,b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+3 a^{2} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(166\) |
default | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+3 a \,b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+3 a^{2} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(166\) |
parts | \(x \,a^{3}+\frac {b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )}{d}+\frac {3 a \,b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )}{d}+\frac {3 a^{2} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(167\) |
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Time = 0.26 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.52 \[ \int (a+b \arcsin (c+d x))^3 \, dx=\frac {{\left (b^{3} d x + b^{3} c\right )} \arcsin \left (d x + c\right )^{3} + {\left (a^{3} - 6 \, a b^{2}\right )} d x + 3 \, {\left (a b^{2} d x + a b^{2} c\right )} \arcsin \left (d x + c\right )^{2} + 3 \, {\left ({\left (a^{2} b - 2 \, b^{3}\right )} d x + {\left (a^{2} b - 2 \, b^{3}\right )} c\right )} \arcsin \left (d x + c\right ) + 3 \, {\left (b^{3} \arcsin \left (d x + c\right )^{2} + 2 \, a b^{2} \arcsin \left (d x + c\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (92) = 184\).
Time = 0.19 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.71 \[ \int (a+b \arcsin (c+d x))^3 \, dx=\begin {cases} a^{3} x + \frac {3 a^{2} b c \operatorname {asin}{\left (c + d x \right )}}{d} + 3 a^{2} b x \operatorname {asin}{\left (c + d x \right )} + \frac {3 a^{2} b \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {3 a b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + 3 a b^{2} x \operatorname {asin}^{2}{\left (c + d x \right )} - 6 a b^{2} x + \frac {6 a b^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} + \frac {b^{3} c \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {6 b^{3} c \operatorname {asin}{\left (c + d x \right )}}{d} + b^{3} x \operatorname {asin}^{3}{\left (c + d x \right )} - 6 b^{3} x \operatorname {asin}{\left (c + d x \right )} + \frac {3 b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )}}{d} - \frac {6 b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asin}{\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \]
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\[ \int (a+b \arcsin (c+d x))^3 \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{3} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (100) = 200\).
Time = 0.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.00 \[ \int (a+b \arcsin (c+d x))^3 \, dx=\frac {{\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )^{3}}{d} + \frac {3 \, {\left (d x + c\right )} a b^{2} \arcsin \left (d x + c\right )^{2}}{d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3} \arcsin \left (d x + c\right )^{2}}{d} + \frac {3 \, {\left (d x + c\right )} a^{2} b \arcsin \left (d x + c\right )}{d} - \frac {6 \, {\left (d x + c\right )} b^{3} \arcsin \left (d x + c\right )}{d} + \frac {6 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{2} \arcsin \left (d x + c\right )}{d} + \frac {{\left (d x + c\right )} a^{3}}{d} - \frac {6 \, {\left (d x + c\right )} a b^{2}}{d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b}{d} - \frac {6 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{3}}{d} \]
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Time = 0.51 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.46 \[ \int (a+b \arcsin (c+d x))^3 \, dx=a^3\,x-\frac {b^3\,\left (6\,\mathrm {asin}\left (c+d\,x\right )-{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\left (c+d\,x\right )}{d}+\frac {3\,a\,b^2\,\left (2\,\mathrm {asin}\left (c+d\,x\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}+\left ({\mathrm {asin}\left (c+d\,x\right )}^2-2\right )\,\left (c+d\,x\right )\right )}{d}+\frac {3\,a^2\,b\,\left (\sqrt {1-{\left (c+d\,x\right )}^2}+\mathrm {asin}\left (c+d\,x\right )\,\left (c+d\,x\right )\right )}{d}+\frac {b^3\,\left (3\,{\mathrm {asin}\left (c+d\,x\right )}^2-6\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d} \]
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