Integrand size = 12, antiderivative size = 76 \[ \int x (a+b \arccos (c x))^2 \, dx=-\frac {1}{4} b^2 x^2-\frac {b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c}-\frac {(a+b \arccos (c x))^2}{4 c^2}+\frac {1}{2} x^2 (a+b \arccos (c x))^2 \]
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Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4724, 4796, 4738, 30} \[ \int x (a+b \arccos (c x))^2 \, dx=-\frac {b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c}-\frac {(a+b \arccos (c x))^2}{4 c^2}+\frac {1}{2} x^2 (a+b \arccos (c x))^2-\frac {1}{4} b^2 x^2 \]
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Rule 30
Rule 4724
Rule 4738
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 (a+b \arccos (c x))^2+(b c) \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c}+\frac {1}{2} x^2 (a+b \arccos (c x))^2-\frac {1}{2} b^2 \int x \, dx+\frac {b \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c} \\ & = -\frac {1}{4} b^2 x^2-\frac {b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c}-\frac {(a+b \arccos (c x))^2}{4 c^2}+\frac {1}{2} x^2 (a+b \arccos (c x))^2 \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.37 \[ \int x (a+b \arccos (c x))^2 \, dx=\frac {c x \left (2 a^2 c x-b^2 c x-2 a b \sqrt {1-c^2 x^2}\right )+2 b c x \left (2 a c x-b \sqrt {1-c^2 x^2}\right ) \arccos (c x)+b^2 \left (-1+2 c^2 x^2\right ) \arccos (c x)^2+2 a b \arcsin (c x)}{4 c^2} \]
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Time = 0.72 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.54
method | result | size |
parts | \(\frac {a^{2} x^{2}}{2}+\frac {b^{2} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{2}}{2}-\frac {\arccos \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2}+\frac {\arccos \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}+\frac {1}{4}\right )}{c^{2}}+\frac {2 a b \left (\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}+\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(117\) |
derivativedivides | \(\frac {\frac {c^{2} x^{2} a^{2}}{2}+b^{2} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{2}}{2}-\frac {\arccos \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2}+\frac {\arccos \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}+\frac {1}{4}\right )+2 a b \left (\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}+\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(118\) |
default | \(\frac {\frac {c^{2} x^{2} a^{2}}{2}+b^{2} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{2}}{2}-\frac {\arccos \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2}+\frac {\arccos \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}+\frac {1}{4}\right )+2 a b \left (\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}+\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(118\) |
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Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.30 \[ \int x (a+b \arccos (c x))^2 \, dx=\frac {{\left (2 \, a^{2} - b^{2}\right )} c^{2} x^{2} + {\left (2 \, b^{2} c^{2} x^{2} - b^{2}\right )} \arccos \left (c x\right )^{2} + 2 \, {\left (2 \, a b c^{2} x^{2} - a b\right )} \arccos \left (c x\right ) - 2 \, {\left (b^{2} c x \arccos \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, c^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (65) = 130\).
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.72 \[ \int x (a+b \arccos (c x))^2 \, dx=\begin {cases} \frac {a^{2} x^{2}}{2} + a b x^{2} \operatorname {acos}{\left (c x \right )} - \frac {a b x \sqrt {- c^{2} x^{2} + 1}}{2 c} - \frac {a b \operatorname {acos}{\left (c x \right )}}{2 c^{2}} + \frac {b^{2} x^{2} \operatorname {acos}^{2}{\left (c x \right )}}{2} - \frac {b^{2} x^{2}}{4} - \frac {b^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{2 c} - \frac {b^{2} \operatorname {acos}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\\frac {x^{2} \left (a + \frac {\pi b}{2}\right )^{2}}{2} & \text {otherwise} \end {cases} \]
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\[ \int x (a+b \arccos (c x))^2 \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{2} x \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.57 \[ \int x (a+b \arccos (c x))^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \arccos \left (c x\right )^{2} + a b x^{2} \arccos \left (c x\right ) + \frac {1}{2} \, a^{2} x^{2} - \frac {1}{4} \, b^{2} x^{2} - \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} x \arccos \left (c x\right )}{2 \, c} - \frac {\sqrt {-c^{2} x^{2} + 1} a b x}{2 \, c} - \frac {b^{2} \arccos \left (c x\right )^{2}}{4 \, c^{2}} - \frac {a b \arccos \left (c x\right )}{2 \, c^{2}} + \frac {b^{2}}{8 \, c^{2}} \]
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Timed out. \[ \int x (a+b \arccos (c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2 \,d x \]
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