Integrand size = 10, antiderivative size = 82 \[ \int (a+b \arccos (c x))^3 \, dx=-6 a b^2 x+\frac {6 b^3 \sqrt {1-c^2 x^2}}{c}-6 b^3 x \arccos (c x)-\frac {3 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c}+x (a+b \arccos (c x))^3 \]
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Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4716, 4768, 267} \[ \int (a+b \arccos (c x))^3 \, dx=-\frac {3 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c}+x (a+b \arccos (c x))^3-6 a b^2 x-6 b^3 x \arccos (c x)+\frac {6 b^3 \sqrt {1-c^2 x^2}}{c} \]
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Rule 267
Rule 4716
Rule 4768
Rubi steps \begin{align*} \text {integral}& = x (a+b \arccos (c x))^3+(3 b c) \int \frac {x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {3 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c}+x (a+b \arccos (c x))^3-\left (6 b^2\right ) \int (a+b \arccos (c x)) \, dx \\ & = -6 a b^2 x-\frac {3 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c}+x (a+b \arccos (c x))^3-\left (6 b^3\right ) \int \arccos (c x) \, dx \\ & = -6 a b^2 x-6 b^3 x \arccos (c x)-\frac {3 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c}+x (a+b \arccos (c x))^3-\left (6 b^3 c\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx \\ & = -6 a b^2 x+\frac {6 b^3 \sqrt {1-c^2 x^2}}{c}-6 b^3 x \arccos (c x)-\frac {3 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c}+x (a+b \arccos (c x))^3 \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.56 \[ \int (a+b \arccos (c x))^3 \, dx=\frac {a \left (a^2-6 b^2\right ) c x-3 b \left (a^2-2 b^2\right ) \sqrt {1-c^2 x^2}+3 b \left (a^2 c x-2 b^2 c x-2 a b \sqrt {1-c^2 x^2}\right ) \arccos (c x)+3 b^2 \left (a c x-b \sqrt {1-c^2 x^2}\right ) \arccos (c x)^2+b^3 c x \arccos (c x)^3}{c} \]
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Time = 0.77 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.63
method | result | size |
derivativedivides | \(\frac {c x \,a^{3}+b^{3} \left (\arccos \left (c x \right )^{3} c x -3 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+6 \sqrt {-c^{2} x^{2}+1}-6 c x \arccos \left (c x \right )\right )+3 a \,b^{2} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(134\) |
default | \(\frac {c x \,a^{3}+b^{3} \left (\arccos \left (c x \right )^{3} c x -3 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+6 \sqrt {-c^{2} x^{2}+1}-6 c x \arccos \left (c x \right )\right )+3 a \,b^{2} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(134\) |
parts | \(x \,a^{3}+\frac {b^{3} \left (\arccos \left (c x \right )^{3} c x -3 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+6 \sqrt {-c^{2} x^{2}+1}-6 c x \arccos \left (c x \right )\right )}{c}+\frac {3 a \,b^{2} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )}{c}+\frac {3 a^{2} b \left (c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}\right )}{c}\) | \(138\) |
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Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.32 \[ \int (a+b \arccos (c x))^3 \, dx=\frac {b^{3} c x \arccos \left (c x\right )^{3} + 3 \, a b^{2} c x \arccos \left (c x\right )^{2} + 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} c x \arccos \left (c x\right ) + {\left (a^{3} - 6 \, a b^{2}\right )} c x - 3 \, {\left (b^{3} \arccos \left (c x\right )^{2} + 2 \, a b^{2} \arccos \left (c x\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{c} \]
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Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (76) = 152\).
Time = 0.16 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.01 \[ \int (a+b \arccos (c x))^3 \, dx=\begin {cases} a^{3} x + 3 a^{2} b x \operatorname {acos}{\left (c x \right )} - \frac {3 a^{2} b \sqrt {- c^{2} x^{2} + 1}}{c} + 3 a b^{2} x \operatorname {acos}^{2}{\left (c x \right )} - 6 a b^{2} x - \frac {6 a b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{c} + b^{3} x \operatorname {acos}^{3}{\left (c x \right )} - 6 b^{3} x \operatorname {acos}{\left (c x \right )} - \frac {3 b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{c} + \frac {6 b^{3} \sqrt {- c^{2} x^{2} + 1}}{c} & \text {for}\: c \neq 0 \\x \left (a + \frac {\pi b}{2}\right )^{3} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.76 \[ \int (a+b \arccos (c x))^3 \, dx=b^{3} x \arccos \left (c x\right )^{3} + 3 \, a b^{2} x \arccos \left (c x\right )^{2} - 3 \, {\left (\frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )^{2}}{c} + \frac {2 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )}}{c}\right )} b^{3} - 6 \, a b^{2} {\left (x + \frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )}{c}\right )} + a^{3} x + \frac {3 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} a^{2} b}{c} \]
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Time = 0.32 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.83 \[ \int (a+b \arccos (c x))^3 \, dx=b^{3} x \arccos \left (c x\right )^{3} + 3 \, a b^{2} x \arccos \left (c x\right )^{2} + 3 \, a^{2} b x \arccos \left (c x\right ) - 6 \, b^{3} x \arccos \left (c x\right ) - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} \arccos \left (c x\right )^{2}}{c} + a^{3} x - 6 \, a b^{2} x - \frac {6 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} \arccos \left (c x\right )}{c} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b}{c} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} b^{3}}{c} \]
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Time = 0.50 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.00 \[ \int (a+b \arccos (c x))^3 \, dx=\left \{\begin {array}{cl} x\,\left (a^3+\frac {3\,\pi \,a^2\,b}{2}+\frac {3\,\pi ^2\,a\,b^2}{4}+\frac {\pi ^3\,b^3}{8}\right ) & \text {\ if\ \ }c=0\\ a^3\,x-b^3\,x\,\left (6\,\mathrm {acos}\left (c\,x\right )-{\mathrm {acos}\left (c\,x\right )}^3\right )-\frac {3\,a^2\,b\,\left (\sqrt {1-c^2\,x^2}-c\,x\,\mathrm {acos}\left (c\,x\right )\right )}{c}+3\,a\,b^2\,x\,\left ({\mathrm {acos}\left (c\,x\right )}^2-2\right )-\frac {b^3\,\sqrt {1-c^2\,x^2}\,\left (3\,{\mathrm {acos}\left (c\,x\right )}^2-6\right )}{c}-\frac {6\,a\,b^2\,\mathrm {acos}\left (c\,x\right )\,\sqrt {1-c^2\,x^2}}{c} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
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