Integrand size = 12, antiderivative size = 125 \[ \int x (a+b \arccos (c x))^3 \, dx=\frac {3 b^3 x \sqrt {1-c^2 x^2}}{8 c}-\frac {3}{4} b^2 x^2 (a+b \arccos (c x))-\frac {3 b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 c}-\frac {(a+b \arccos (c x))^3}{4 c^2}+\frac {1}{2} x^2 (a+b \arccos (c x))^3-\frac {3 b^3 \arcsin (c x)}{8 c^2} \]
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Time = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4724, 4796, 4738, 327, 222} \[ \int x (a+b \arccos (c x))^3 \, dx=-\frac {3}{4} b^2 x^2 (a+b \arccos (c x))-\frac {3 b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 c}-\frac {(a+b \arccos (c x))^3}{4 c^2}+\frac {1}{2} x^2 (a+b \arccos (c x))^3-\frac {3 b^3 \arcsin (c x)}{8 c^2}+\frac {3 b^3 x \sqrt {1-c^2 x^2}}{8 c} \]
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Rule 222
Rule 327
Rule 4724
Rule 4738
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 (a+b \arccos (c x))^3+\frac {1}{2} (3 b c) \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {3 b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 c}+\frac {1}{2} x^2 (a+b \arccos (c x))^3-\frac {1}{2} \left (3 b^2\right ) \int x (a+b \arccos (c x)) \, dx+\frac {(3 b) \int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}} \, dx}{4 c} \\ & = -\frac {3}{4} b^2 x^2 (a+b \arccos (c x))-\frac {3 b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 c}-\frac {(a+b \arccos (c x))^3}{4 c^2}+\frac {1}{2} x^2 (a+b \arccos (c x))^3-\frac {1}{4} \left (3 b^3 c\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {3 b^3 x \sqrt {1-c^2 x^2}}{8 c}-\frac {3}{4} b^2 x^2 (a+b \arccos (c x))-\frac {3 b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 c}-\frac {(a+b \arccos (c x))^3}{4 c^2}+\frac {1}{2} x^2 (a+b \arccos (c x))^3-\frac {\left (3 b^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{8 c} \\ & = \frac {3 b^3 x \sqrt {1-c^2 x^2}}{8 c}-\frac {3}{4} b^2 x^2 (a+b \arccos (c x))-\frac {3 b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 c}-\frac {(a+b \arccos (c x))^3}{4 c^2}+\frac {1}{2} x^2 (a+b \arccos (c x))^3-\frac {3 b^3 \arcsin (c x)}{8 c^2} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.48 \[ \int x (a+b \arccos (c x))^3 \, dx=\frac {c x \left (4 a^3 c x-6 a b^2 c x-6 a^2 b \sqrt {1-c^2 x^2}+3 b^3 \sqrt {1-c^2 x^2}\right )-6 b c x \left (-2 a^2 c x+b^2 c x+2 a b \sqrt {1-c^2 x^2}\right ) \arccos (c x)-6 b^2 \left (a-2 a c^2 x^2+b c x \sqrt {1-c^2 x^2}\right ) \arccos (c x)^2+2 b^3 \left (-1+2 c^2 x^2\right ) \arccos (c x)^3+\left (6 a^2 b-3 b^3\right ) \arcsin (c x)}{8 c^2} \]
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Time = 0.96 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.69
method | result | size |
derivativedivides | \(\frac {\frac {c^{2} x^{2} a^{3}}{2}+b^{3} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{3}}{2}-\frac {3 \arccos \left (c x \right )^{2} \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{4}-\frac {3 c^{2} x^{2} \arccos \left (c x \right )}{4}+\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arccos \left (c x \right )}{8}+\frac {\arccos \left (c x \right )^{3}}{2}\right )+3 a \,b^{2} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{2}}{2}-\frac {\arccos \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2}+\frac {\arccos \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}+\frac {1}{4}\right )+3 a^{2} b \left (\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}+\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(211\) |
default | \(\frac {\frac {c^{2} x^{2} a^{3}}{2}+b^{3} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{3}}{2}-\frac {3 \arccos \left (c x \right )^{2} \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{4}-\frac {3 c^{2} x^{2} \arccos \left (c x \right )}{4}+\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arccos \left (c x \right )}{8}+\frac {\arccos \left (c x \right )^{3}}{2}\right )+3 a \,b^{2} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{2}}{2}-\frac {\arccos \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2}+\frac {\arccos \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}+\frac {1}{4}\right )+3 a^{2} b \left (\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}+\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(211\) |
parts | \(\frac {a^{3} x^{2}}{2}+\frac {b^{3} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{3}}{2}-\frac {3 \arccos \left (c x \right )^{2} \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{4}-\frac {3 c^{2} x^{2} \arccos \left (c x \right )}{4}+\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arccos \left (c x \right )}{8}+\frac {\arccos \left (c x \right )^{3}}{2}\right )}{c^{2}}+\frac {3 a \,b^{2} \left (\frac {c^{2} x^{2} \arccos \left (c x \right )^{2}}{2}-\frac {\arccos \left (c x \right ) \left (c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )\right )}{2}+\frac {\arccos \left (c x \right )^{2}}{4}-\frac {c^{2} x^{2}}{4}+\frac {1}{4}\right )}{c^{2}}+\frac {3 a^{2} b \left (\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}+\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(213\) |
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Time = 0.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.35 \[ \int x (a+b \arccos (c x))^3 \, dx=\frac {2 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} c^{2} x^{2} + 2 \, {\left (2 \, b^{3} c^{2} x^{2} - b^{3}\right )} \arccos \left (c x\right )^{3} + 6 \, {\left (2 \, a b^{2} c^{2} x^{2} - a b^{2}\right )} \arccos \left (c x\right )^{2} + 3 \, {\left (2 \, {\left (2 \, a^{2} b - b^{3}\right )} c^{2} x^{2} - 2 \, a^{2} b + b^{3}\right )} \arccos \left (c x\right ) - 3 \, {\left (2 \, b^{3} c x \arccos \left (c x\right )^{2} + 4 \, a b^{2} c x \arccos \left (c x\right ) + {\left (2 \, a^{2} b - b^{3}\right )} c x\right )} \sqrt {-c^{2} x^{2} + 1}}{8 \, c^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (116) = 232\).
Time = 0.31 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.15 \[ \int x (a+b \arccos (c x))^3 \, dx=\begin {cases} \frac {a^{3} x^{2}}{2} + \frac {3 a^{2} b x^{2} \operatorname {acos}{\left (c x \right )}}{2} - \frac {3 a^{2} b x \sqrt {- c^{2} x^{2} + 1}}{4 c} - \frac {3 a^{2} b \operatorname {acos}{\left (c x \right )}}{4 c^{2}} + \frac {3 a b^{2} x^{2} \operatorname {acos}^{2}{\left (c x \right )}}{2} - \frac {3 a b^{2} x^{2}}{4} - \frac {3 a b^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{2 c} - \frac {3 a b^{2} \operatorname {acos}^{2}{\left (c x \right )}}{4 c^{2}} + \frac {b^{3} x^{2} \operatorname {acos}^{3}{\left (c x \right )}}{2} - \frac {3 b^{3} x^{2} \operatorname {acos}{\left (c x \right )}}{4} - \frac {3 b^{3} x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{4 c} + \frac {3 b^{3} x \sqrt {- c^{2} x^{2} + 1}}{8 c} - \frac {b^{3} \operatorname {acos}^{3}{\left (c x \right )}}{4 c^{2}} + \frac {3 b^{3} \operatorname {acos}{\left (c x \right )}}{8 c^{2}} & \text {for}\: c \neq 0 \\\frac {x^{2} \left (a + \frac {\pi b}{2}\right )^{3}}{2} & \text {otherwise} \end {cases} \]
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\[ \int x (a+b \arccos (c x))^3 \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{3} x \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (109) = 218\).
Time = 0.29 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.85 \[ \int x (a+b \arccos (c x))^3 \, dx=\frac {1}{2} \, b^{3} x^{2} \arccos \left (c x\right )^{3} + \frac {3}{2} \, a b^{2} x^{2} \arccos \left (c x\right )^{2} + \frac {3}{2} \, a^{2} b x^{2} \arccos \left (c x\right ) - \frac {3}{4} \, b^{3} x^{2} \arccos \left (c x\right ) - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x \arccos \left (c x\right )^{2}}{4 \, c} + \frac {1}{2} \, a^{3} x^{2} - \frac {3}{4} \, a b^{2} x^{2} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} x \arccos \left (c x\right )}{2 \, c} - \frac {b^{3} \arccos \left (c x\right )^{3}}{4 \, c^{2}} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b x}{4 \, c} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x}{8 \, c} - \frac {3 \, a b^{2} \arccos \left (c x\right )^{2}}{4 \, c^{2}} - \frac {3 \, a^{2} b \arccos \left (c x\right )}{4 \, c^{2}} + \frac {3 \, b^{3} \arccos \left (c x\right )}{8 \, c^{2}} + \frac {3 \, a b^{2}}{8 \, c^{2}} \]
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Timed out. \[ \int x (a+b \arccos (c x))^3 \, dx=\int x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3 \,d x \]
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