Integrand size = 10, antiderivative size = 94 \[ \int x^2 \arccos (a+b x) \, dx=-\frac {x^2 \sqrt {1-(a+b x)^2}}{9 b}-\frac {\left (4+11 a^2-5 a b x\right ) \sqrt {1-(a+b x)^2}}{18 b^3}+\frac {1}{3} x^3 \arccos (a+b x)-\frac {a \left (3+2 a^2\right ) \arcsin (a+b x)}{6 b^3} \]
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Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4890, 4828, 757, 794, 222} \[ \int x^2 \arccos (a+b x) \, dx=-\frac {a \left (2 a^2+3\right ) \arcsin (a+b x)}{6 b^3}-\frac {\left (11 a^2-5 a b x+4\right ) \sqrt {1-(a+b x)^2}}{18 b^3}+\frac {1}{3} x^3 \arccos (a+b x)-\frac {x^2 \sqrt {1-(a+b x)^2}}{9 b} \]
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Rule 222
Rule 757
Rule 794
Rule 4828
Rule 4890
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \arccos (x) \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{3} x^3 \arccos (a+b x)+\frac {1}{3} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3}{\sqrt {1-x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {x^2 \sqrt {1-(a+b x)^2}}{9 b}+\frac {1}{3} x^3 \arccos (a+b x)-\frac {1}{9} \text {Subst}\left (\int \frac {\left (-\frac {2+3 a^2}{b^2}+\frac {5 a x}{b^2}\right ) \left (-\frac {a}{b}+\frac {x}{b}\right )}{\sqrt {1-x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {x^2 \sqrt {1-(a+b x)^2}}{9 b}-\frac {\left (4+11 a^2-5 a b x\right ) \sqrt {1-(a+b x)^2}}{18 b^3}+\frac {1}{3} x^3 \arccos (a+b x)-\frac {\left (a \left (3+2 a^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{6 b^3} \\ & = -\frac {x^2 \sqrt {1-(a+b x)^2}}{9 b}-\frac {\left (4+11 a^2-5 a b x\right ) \sqrt {1-(a+b x)^2}}{18 b^3}+\frac {1}{3} x^3 \arccos (a+b x)-\frac {a \left (3+2 a^2\right ) \arcsin (a+b x)}{6 b^3} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.88 \[ \int x^2 \arccos (a+b x) \, dx=-\frac {\sqrt {1-a^2-2 a b x-b^2 x^2} \left (4+11 a^2-5 a b x+2 b^2 x^2\right )-6 b^3 x^3 \arccos (a+b x)+3 a \left (3+2 a^2\right ) \arcsin (a+b x)}{18 b^3} \]
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Time = 0.08 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.71
method | result | size |
derivativedivides | \(\frac {-\frac {\arccos \left (b x +a \right ) a^{3}}{3}+\arccos \left (b x +a \right ) a^{2} \left (b x +a \right )-\arccos \left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\arccos \left (b x +a \right ) \left (b x +a \right )^{3}}{3}-\frac {a^{3} \arcsin \left (b x +a \right )}{3}-\frac {\left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}}{9}-\frac {2 \sqrt {1-\left (b x +a \right )^{2}}}{9}-a^{2} \sqrt {1-\left (b x +a \right )^{2}}-a \left (-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{2}+\frac {\arcsin \left (b x +a \right )}{2}\right )}{b^{3}}\) | \(161\) |
default | \(\frac {-\frac {\arccos \left (b x +a \right ) a^{3}}{3}+\arccos \left (b x +a \right ) a^{2} \left (b x +a \right )-\arccos \left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\arccos \left (b x +a \right ) \left (b x +a \right )^{3}}{3}-\frac {a^{3} \arcsin \left (b x +a \right )}{3}-\frac {\left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}}{9}-\frac {2 \sqrt {1-\left (b x +a \right )^{2}}}{9}-a^{2} \sqrt {1-\left (b x +a \right )^{2}}-a \left (-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{2}+\frac {\arcsin \left (b x +a \right )}{2}\right )}{b^{3}}\) | \(161\) |
parts | \(\frac {x^{3} \arccos \left (b x +a \right )}{3}+\frac {b \left (-\frac {x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 b^{2}}-\frac {5 a \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{3 b}+\frac {2 \left (-a^{2}+1\right ) \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{3 b^{2}}\right )}{3}\) | \(303\) |
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Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.80 \[ \int x^2 \arccos (a+b x) \, dx=\frac {3 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \arccos \left (b x + a\right ) - {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{18 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (83) = 166\).
Time = 0.29 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.81 \[ \int x^2 \arccos (a+b x) \, dx=\begin {cases} \frac {a^{3} \operatorname {acos}{\left (a + b x \right )}}{3 b^{3}} - \frac {11 a^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{18 b^{3}} + \frac {5 a x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{18 b^{2}} + \frac {a \operatorname {acos}{\left (a + b x \right )}}{2 b^{3}} + \frac {x^{3} \operatorname {acos}{\left (a + b x \right )}}{3} - \frac {x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{9 b} - \frac {2 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{9 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {acos}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (82) = 164\).
Time = 0.27 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.34 \[ \int x^2 \arccos (a+b x) \, dx=\frac {1}{3} \, x^{3} \arccos \left (b x + a\right ) - \frac {1}{18} \, b {\left (\frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x^{2}}{b^{2}} - \frac {15 \, a^{3} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{4}} - \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a x}{b^{3}} + \frac {9 \, {\left (a^{2} - 1\right )} a \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{4}} + \frac {15 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{4}} - \frac {4 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )}}{b^{4}}\right )} \]
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Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.66 \[ \int x^2 \arccos (a+b x) \, dx=\frac {{\left (b x + a\right )}^{3} \arccos \left (b x + a\right )}{3 \, b^{3}} - \frac {{\left (b x + a\right )}^{2} a \arccos \left (b x + a\right )}{b^{3}} + \frac {{\left (b x + a\right )} a^{2} \arccos \left (b x + a\right )}{b^{3}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}^{2}}{9 \, b^{3}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a}{2 \, b^{3}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2}}{b^{3}} + \frac {a \arccos \left (b x + a\right )}{2 \, b^{3}} - \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1}}{9 \, b^{3}} \]
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Timed out. \[ \int x^2 \arccos (a+b x) \, dx=\int x^2\,\mathrm {acos}\left (a+b\,x\right ) \,d x \]
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