Integrand size = 12, antiderivative size = 76 \[ \int x^3 (a+b \arccos (c x)) \, dx=-\frac {3 b x \sqrt {1-c^2 x^2}}{32 c^3}-\frac {b x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {3 b \arcsin (c x)}{32 c^4} \]
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Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4724, 327, 222} \[ \int x^3 (a+b \arccos (c x)) \, dx=\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {3 b \arcsin (c x)}{32 c^4}-\frac {b x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3 b x \sqrt {1-c^2 x^2}}{32 c^3} \]
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Rule 222
Rule 327
Rule 4724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{4} (b c) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {b x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {(3 b) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 c} \\ & = -\frac {3 b x \sqrt {1-c^2 x^2}}{32 c^3}-\frac {b x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {(3 b) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^3} \\ & = -\frac {3 b x \sqrt {1-c^2 x^2}}{32 c^3}-\frac {b x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {3 b \arcsin (c x)}{32 c^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int x^3 (a+b \arccos (c x)) \, dx=\frac {a x^4}{4}+b \sqrt {1-c^2 x^2} \left (-\frac {3 x}{32 c^3}-\frac {x^3}{16 c}\right )+\frac {1}{4} b x^4 \arccos (c x)+\frac {3 b \arcsin (c x)}{32 c^4} \]
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Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89
method | result | size |
parts | \(\frac {a \,x^{4}}{4}+\frac {b \left (\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{16}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{32}+\frac {3 \arcsin \left (c x \right )}{32}\right )}{c^{4}}\) | \(68\) |
derivativedivides | \(\frac {\frac {a \,c^{4} x^{4}}{4}+b \left (\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{16}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{32}+\frac {3 \arcsin \left (c x \right )}{32}\right )}{c^{4}}\) | \(72\) |
default | \(\frac {\frac {a \,c^{4} x^{4}}{4}+b \left (\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{16}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{32}+\frac {3 \arcsin \left (c x \right )}{32}\right )}{c^{4}}\) | \(72\) |
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Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.82 \[ \int x^3 (a+b \arccos (c x)) \, dx=\frac {8 \, a c^{4} x^{4} + {\left (8 \, b c^{4} x^{4} - 3 \, b\right )} \arccos \left (c x\right ) - {\left (2 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {-c^{2} x^{2} + 1}}{32 \, c^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.12 \[ \int x^3 (a+b \arccos (c x)) \, dx=\begin {cases} \frac {a x^{4}}{4} + \frac {b x^{4} \operatorname {acos}{\left (c x \right )}}{4} - \frac {b x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} - \frac {3 b x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} - \frac {3 b \operatorname {acos}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\\frac {x^{4} \left (a + \frac {\pi b}{2}\right )}{4} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int x^3 (a+b \arccos (c x)) \, dx=\frac {1}{4} \, a x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \arccos \left (c x\right ) - {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b \]
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Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88 \[ \int x^3 (a+b \arccos (c x)) \, dx=\frac {1}{4} \, b x^{4} \arccos \left (c x\right ) + \frac {1}{4} \, a x^{4} - \frac {\sqrt {-c^{2} x^{2} + 1} b x^{3}}{16 \, c} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b x}{32 \, c^{3}} - \frac {3 \, b \arccos \left (c x\right )}{32 \, c^{4}} \]
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Timed out. \[ \int x^3 (a+b \arccos (c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right ) \,d x \]
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