Integrand size = 8, antiderivative size = 80 \[ \int x \arccos (a+b x) \, dx=\frac {3 a \sqrt {1-(a+b x)^2}}{4 b^2}-\frac {x \sqrt {1-(a+b x)^2}}{4 b}+\frac {1}{2} x^2 \arccos (a+b x)+\frac {\left (1+2 a^2\right ) \arcsin (a+b x)}{4 b^2} \]
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Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4890, 4828, 757, 655, 222} \[ \int x \arccos (a+b x) \, dx=\frac {\left (2 a^2+1\right ) \arcsin (a+b x)}{4 b^2}+\frac {1}{2} x^2 \arccos (a+b x)+\frac {3 a \sqrt {1-(a+b x)^2}}{4 b^2}-\frac {x \sqrt {1-(a+b x)^2}}{4 b} \]
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Rule 222
Rule 655
Rule 757
Rule 4828
Rule 4890
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right ) \arccos (x) \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{2} x^2 \arccos (a+b x)+\frac {1}{2} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {x \sqrt {1-(a+b x)^2}}{4 b}+\frac {1}{2} x^2 \arccos (a+b x)-\frac {1}{4} \text {Subst}\left (\int \frac {-\frac {1+2 a^2}{b^2}+\frac {3 a x}{b^2}}{\sqrt {1-x^2}} \, dx,x,a+b x\right ) \\ & = \frac {3 a \sqrt {1-(a+b x)^2}}{4 b^2}-\frac {x \sqrt {1-(a+b x)^2}}{4 b}+\frac {1}{2} x^2 \arccos (a+b x)+\frac {\left (1+2 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{4 b^2} \\ & = \frac {3 a \sqrt {1-(a+b x)^2}}{4 b^2}-\frac {x \sqrt {1-(a+b x)^2}}{4 b}+\frac {1}{2} x^2 \arccos (a+b x)+\frac {\left (1+2 a^2\right ) \arcsin (a+b x)}{4 b^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.86 \[ \int x \arccos (a+b x) \, dx=\frac {(3 a-b x) \sqrt {1-a^2-2 a b x-b^2 x^2}+2 b^2 x^2 \arccos (a+b x)+\left (1+2 a^2\right ) \arcsin (a+b x)}{4 b^2} \]
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Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {\frac {\arccos \left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\arccos \left (b x +a \right ) a \left (b x +a \right )-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{4}+\frac {\arcsin \left (b x +a \right )}{4}+a \sqrt {1-\left (b x +a \right )^{2}}}{b^{2}}\) | \(78\) |
default | \(\frac {\frac {\arccos \left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\arccos \left (b x +a \right ) a \left (b x +a \right )-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{4}+\frac {\arcsin \left (b x +a \right )}{4}+a \sqrt {1-\left (b x +a \right )^{2}}}{b^{2}}\) | \(78\) |
parts | \(\frac {x^{2} \arccos \left (b x +a \right )}{2}+\frac {b \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{2}\) | \(179\) |
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Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.74 \[ \int x \arccos (a+b x) \, dx=\frac {{\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \arccos \left (b x + a\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x - 3 \, a\right )}}{4 \, b^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.30 \[ \int x \arccos (a+b x) \, dx=\begin {cases} - \frac {a^{2} \operatorname {acos}{\left (a + b x \right )}}{2 b^{2}} + \frac {3 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{4 b^{2}} + \frac {x^{2} \operatorname {acos}{\left (a + b x \right )}}{2} - \frac {x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{4 b} - \frac {\operatorname {acos}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {acos}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (68) = 136\).
Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.91 \[ \int x \arccos (a+b x) \, dx=\frac {1}{2} \, x^{2} \arccos \left (b x + a\right ) - \frac {1}{4} \, b {\left (\frac {3 \, a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{3}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{b^{2}} - \frac {{\left (a^{2} - 1\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{3}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{3}}\right )} \]
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Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.10 \[ \int x \arccos (a+b x) \, dx=\frac {{\left (b x + a\right )}^{2} \arccos \left (b x + a\right )}{2 \, b^{2}} - \frac {{\left (b x + a\right )} a \arccos \left (b x + a\right )}{b^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}}{4 \, b^{2}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a}{b^{2}} - \frac {\arccos \left (b x + a\right )}{4 \, b^{2}} \]
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Timed out. \[ \int x \arccos (a+b x) \, dx=\int x\,\mathrm {acos}\left (a+b\,x\right ) \,d x \]
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