Integrand size = 14, antiderivative size = 178 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=-\frac {4 a b^2 x}{3 c^2}+\frac {14 b^3 \sqrt {1-c^2 x^2}}{9 c^3}-\frac {2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}-\frac {4 b^3 x \arccos (c x)}{3 c^2}-\frac {2}{9} b^2 x^3 (a+b \arccos (c x))-\frac {2 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^3}-\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c}+\frac {1}{3} x^3 (a+b \arccos (c x))^3 \]
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Time = 0.21 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 4796, 4768, 4716, 267, 272, 45} \[ \int x^2 (a+b \arccos (c x))^3 \, dx=-\frac {2}{9} b^2 x^3 (a+b \arccos (c x))-\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c}-\frac {2 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^3}+\frac {1}{3} x^3 (a+b \arccos (c x))^3-\frac {4 a b^2 x}{3 c^2}-\frac {4 b^3 x \arccos (c x)}{3 c^2}-\frac {2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}+\frac {14 b^3 \sqrt {1-c^2 x^2}}{9 c^3} \]
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Rule 45
Rule 267
Rule 272
Rule 4716
Rule 4724
Rule 4768
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 (a+b \arccos (c x))^3+(b c) \int \frac {x^3 (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c}+\frac {1}{3} x^3 (a+b \arccos (c x))^3-\frac {1}{3} \left (2 b^2\right ) \int x^2 (a+b \arccos (c x)) \, dx+\frac {(2 b) \int \frac {x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{3 c} \\ & = -\frac {2}{9} b^2 x^3 (a+b \arccos (c x))-\frac {2 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^3}-\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c}+\frac {1}{3} x^3 (a+b \arccos (c x))^3-\frac {\left (4 b^2\right ) \int (a+b \arccos (c x)) \, dx}{3 c^2}-\frac {1}{9} \left (2 b^3 c\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {4 a b^2 x}{3 c^2}-\frac {2}{9} b^2 x^3 (a+b \arccos (c x))-\frac {2 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^3}-\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c}+\frac {1}{3} x^3 (a+b \arccos (c x))^3-\frac {\left (4 b^3\right ) \int \arccos (c x) \, dx}{3 c^2}-\frac {1}{9} \left (b^3 c\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {4 a b^2 x}{3 c^2}-\frac {4 b^3 x \arccos (c x)}{3 c^2}-\frac {2}{9} b^2 x^3 (a+b \arccos (c x))-\frac {2 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^3}-\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c}+\frac {1}{3} x^3 (a+b \arccos (c x))^3-\frac {\left (4 b^3\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{3 c}-\frac {1}{9} \left (b^3 c\right ) \text {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right ) \\ & = -\frac {4 a b^2 x}{3 c^2}+\frac {14 b^3 \sqrt {1-c^2 x^2}}{9 c^3}-\frac {2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}-\frac {4 b^3 x \arccos (c x)}{3 c^2}-\frac {2}{9} b^2 x^3 (a+b \arccos (c x))-\frac {2 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^3}-\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c}+\frac {1}{3} x^3 (a+b \arccos (c x))^3 \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.22 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\frac {9 a^3 c^3 x^3-9 a^2 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )-6 a b^2 c x \left (6+c^2 x^2\right )+2 b^3 \sqrt {1-c^2 x^2} \left (20+c^2 x^2\right )-3 b \left (-9 a^2 c^3 x^3+6 a b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+2 b^2 c x \left (6+c^2 x^2\right )\right ) \arccos (c x)-9 b^2 \left (-3 a c^3 x^3+b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )\right ) \arccos (c x)^2+9 b^3 c^3 x^3 \arccos (c x)^3}{27 c^3} \]
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Time = 2.37 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.32
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} c^{3} x^{3}}{3}+b^{3} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{3}}{3}-\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arccos \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{9}+\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )+3 a \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\) | \(235\) |
default | \(\frac {\frac {a^{3} c^{3} x^{3}}{3}+b^{3} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{3}}{3}-\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arccos \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{9}+\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )+3 a \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\) | \(235\) |
parts | \(\frac {a^{3} x^{3}}{3}+\frac {b^{3} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{3}}{3}-\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arccos \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{9}+\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )}{c^{3}}+\frac {3 a \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )}{c^{3}}+\frac {3 a^{2} b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\) | \(237\) |
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Time = 0.26 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.10 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\frac {9 \, b^{3} c^{3} x^{3} \arccos \left (c x\right )^{3} + 27 \, a b^{2} c^{3} x^{3} \arccos \left (c x\right )^{2} + 3 \, {\left (3 \, a^{3} - 2 \, a b^{2}\right )} c^{3} x^{3} - 36 \, a b^{2} c x + 3 \, {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{3} x^{3} - 12 \, b^{3} c x\right )} \arccos \left (c x\right ) - {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2} x^{2} + 18 \, a^{2} b - 40 \, b^{3} + 9 \, {\left (b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \arccos \left (c x\right )^{2} + 18 \, {\left (a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c^{3}} \]
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Time = 0.41 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.87 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\begin {cases} \frac {a^{3} x^{3}}{3} + a^{2} b x^{3} \operatorname {acos}{\left (c x \right )} - \frac {a^{2} b x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} - \frac {2 a^{2} b \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + a b^{2} x^{3} \operatorname {acos}^{2}{\left (c x \right )} - \frac {2 a b^{2} x^{3}}{9} - \frac {2 a b^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{3 c} - \frac {4 a b^{2} x}{3 c^{2}} - \frac {4 a b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{3 c^{3}} + \frac {b^{3} x^{3} \operatorname {acos}^{3}{\left (c x \right )}}{3} - \frac {2 b^{3} x^{3} \operatorname {acos}{\left (c x \right )}}{9} - \frac {b^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{3 c} + \frac {2 b^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{27 c} - \frac {4 b^{3} x \operatorname {acos}{\left (c x \right )}}{3 c^{2}} - \frac {2 b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{3 c^{3}} + \frac {40 b^{3} \sqrt {- c^{2} x^{2} + 1}}{27 c^{3}} & \text {for}\: c \neq 0 \\\frac {x^{3} \left (a + \frac {\pi b}{2}\right )^{3}}{3} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.53 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\frac {1}{3} \, b^{3} x^{3} \arccos \left (c x\right )^{3} + a b^{2} x^{3} \arccos \left (c x\right )^{2} + \frac {1}{3} \, a^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a^{2} b - \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right ) + \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} a b^{2} - \frac {1}{27} \, {\left (9 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right )^{2} - 2 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-c^{2} x^{2} + 1}}{c^{2}}}{c^{2}} - \frac {3 \, {\left (c^{2} x^{3} + 6 \, x\right )} \arccos \left (c x\right )}{c^{3}}\right )}\right )} b^{3} \]
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Time = 0.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.62 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\frac {1}{3} \, b^{3} x^{3} \arccos \left (c x\right )^{3} + a b^{2} x^{3} \arccos \left (c x\right )^{2} + a^{2} b x^{3} \arccos \left (c x\right ) - \frac {2}{9} \, b^{3} x^{3} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b^{3} x^{2} \arccos \left (c x\right )^{2}}{3 \, c} + \frac {1}{3} \, a^{3} x^{3} - \frac {2}{9} \, a b^{2} x^{3} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} x^{2} \arccos \left (c x\right )}{3 \, c} - \frac {\sqrt {-c^{2} x^{2} + 1} a^{2} b x^{2}}{3 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x^{2}}{27 \, c} - \frac {4 \, b^{3} x \arccos \left (c x\right )}{3 \, c^{2}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{3} \arccos \left (c x\right )^{2}}{3 \, c^{3}} - \frac {4 \, a b^{2} x}{3 \, c^{2}} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} \arccos \left (c x\right )}{3 \, c^{3}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b}{3 \, c^{3}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} b^{3}}{27 \, c^{3}} \]
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Timed out. \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3 \,d x \]
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