Integrand size = 10, antiderivative size = 198 \[ \int x^3 \arccos (a x)^4 \, dx=\frac {45 x^2}{128 a^2}+\frac {3 x^4}{128}+\frac {45 x \sqrt {1-a^2 x^2} \arccos (a x)}{64 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{32 a}+\frac {45 \arccos (a x)^2}{128 a^4}-\frac {9 x^2 \arccos (a x)^2}{16 a^2}-\frac {3}{16} x^4 \arccos (a x)^2-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^3}{8 a^3}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a}-\frac {3 \arccos (a x)^4}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^4 \]
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Time = 0.34 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4724, 4796, 4738, 30} \[ \int x^3 \arccos (a x)^4 \, dx=-\frac {3 \arccos (a x)^4}{32 a^4}+\frac {45 \arccos (a x)^2}{128 a^4}-\frac {9 x^2 \arccos (a x)^2}{16 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a}+\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{32 a}+\frac {45 x^2}{128 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^3}{8 a^3}+\frac {45 x \sqrt {1-a^2 x^2} \arccos (a x)}{64 a^3}+\frac {1}{4} x^4 \arccos (a x)^4-\frac {3}{16} x^4 \arccos (a x)^2+\frac {3 x^4}{128} \]
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Rule 30
Rule 4724
Rule 4738
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \arccos (a x)^4+a \int \frac {x^4 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a}+\frac {1}{4} x^4 \arccos (a x)^4-\frac {3}{4} \int x^3 \arccos (a x)^2 \, dx+\frac {3 \int \frac {x^2 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{4 a} \\ & = -\frac {3}{16} x^4 \arccos (a x)^2-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^3}{8 a^3}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a}+\frac {1}{4} x^4 \arccos (a x)^4+\frac {3 \int \frac {\arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{8 a^3}-\frac {9 \int x \arccos (a x)^2 \, dx}{8 a^2}-\frac {1}{8} (3 a) \int \frac {x^4 \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{32 a}-\frac {9 x^2 \arccos (a x)^2}{16 a^2}-\frac {3}{16} x^4 \arccos (a x)^2-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^3}{8 a^3}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a}-\frac {3 \arccos (a x)^4}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^4+\frac {3 \int x^3 \, dx}{32}-\frac {9 \int \frac {x^2 \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{32 a}-\frac {9 \int \frac {x^2 \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 a} \\ & = \frac {3 x^4}{128}+\frac {45 x \sqrt {1-a^2 x^2} \arccos (a x)}{64 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{32 a}-\frac {9 x^2 \arccos (a x)^2}{16 a^2}-\frac {3}{16} x^4 \arccos (a x)^2-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^3}{8 a^3}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a}-\frac {3 \arccos (a x)^4}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^4-\frac {9 \int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{64 a^3}-\frac {9 \int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{16 a^3}+\frac {9 \int x \, dx}{64 a^2}+\frac {9 \int x \, dx}{16 a^2} \\ & = \frac {45 x^2}{128 a^2}+\frac {3 x^4}{128}+\frac {45 x \sqrt {1-a^2 x^2} \arccos (a x)}{64 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{32 a}+\frac {45 \arccos (a x)^2}{128 a^4}-\frac {9 x^2 \arccos (a x)^2}{16 a^2}-\frac {3}{16} x^4 \arccos (a x)^2-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^3}{8 a^3}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a}-\frac {3 \arccos (a x)^4}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^4 \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.68 \[ \int x^3 \arccos (a x)^4 \, dx=\frac {3 a^2 x^2 \left (15+a^2 x^2\right )+6 a x \sqrt {1-a^2 x^2} \left (15+2 a^2 x^2\right ) \arccos (a x)-3 \left (-15+24 a^2 x^2+8 a^4 x^4\right ) \arccos (a x)^2-16 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \arccos (a x)^3+4 \left (-3+8 a^4 x^4\right ) \arccos (a x)^4}{128 a^4} \]
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Time = 1.19 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \arccos \left (a x \right )^{4}}{4}-\frac {\arccos \left (a x \right )^{3} \left (2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+3 a x \sqrt {-a^{2} x^{2}+1}+3 \arccos \left (a x \right )\right )}{8}-\frac {3 a^{4} x^{4} \arccos \left (a x \right )^{2}}{16}+\frac {3 \arccos \left (a x \right ) \left (2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+3 a x \sqrt {-a^{2} x^{2}+1}+3 \arccos \left (a x \right )\right )}{64}-\frac {45 \arccos \left (a x \right )^{2}}{128}+\frac {3 \left (2 a^{2} x^{2}+3\right )^{2}}{512}-\frac {9 a^{2} x^{2} \arccos \left (a x \right )^{2}}{16}+\frac {9 \arccos \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{16}+\frac {9 a^{2} x^{2}}{32}-\frac {9}{32}+\frac {9 \arccos \left (a x \right )^{4}}{32}}{a^{4}}\) | \(213\) |
default | \(\frac {\frac {a^{4} x^{4} \arccos \left (a x \right )^{4}}{4}-\frac {\arccos \left (a x \right )^{3} \left (2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+3 a x \sqrt {-a^{2} x^{2}+1}+3 \arccos \left (a x \right )\right )}{8}-\frac {3 a^{4} x^{4} \arccos \left (a x \right )^{2}}{16}+\frac {3 \arccos \left (a x \right ) \left (2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+3 a x \sqrt {-a^{2} x^{2}+1}+3 \arccos \left (a x \right )\right )}{64}-\frac {45 \arccos \left (a x \right )^{2}}{128}+\frac {3 \left (2 a^{2} x^{2}+3\right )^{2}}{512}-\frac {9 a^{2} x^{2} \arccos \left (a x \right )^{2}}{16}+\frac {9 \arccos \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{16}+\frac {9 a^{2} x^{2}}{32}-\frac {9}{32}+\frac {9 \arccos \left (a x \right )^{4}}{32}}{a^{4}}\) | \(213\) |
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Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.61 \[ \int x^3 \arccos (a x)^4 \, dx=\frac {3 \, a^{4} x^{4} + 4 \, {\left (8 \, a^{4} x^{4} - 3\right )} \arccos \left (a x\right )^{4} + 45 \, a^{2} x^{2} - 3 \, {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arccos \left (a x\right )^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arccos \left (a x\right )^{3} - 3 \, {\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \arccos \left (a x\right )\right )}}{128 \, a^{4}} \]
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Time = 0.67 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.99 \[ \int x^3 \arccos (a x)^4 \, dx=\begin {cases} \frac {x^{4} \operatorname {acos}^{4}{\left (a x \right )}}{4} - \frac {3 x^{4} \operatorname {acos}^{2}{\left (a x \right )}}{16} + \frac {3 x^{4}}{128} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{4 a} + \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{32 a} - \frac {9 x^{2} \operatorname {acos}^{2}{\left (a x \right )}}{16 a^{2}} + \frac {45 x^{2}}{128 a^{2}} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{8 a^{3}} + \frac {45 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{64 a^{3}} - \frac {3 \operatorname {acos}^{4}{\left (a x \right )}}{32 a^{4}} + \frac {45 \operatorname {acos}^{2}{\left (a x \right )}}{128 a^{4}} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x^{4}}{64} & \text {otherwise} \end {cases} \]
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\[ \int x^3 \arccos (a x)^4 \, dx=\int { x^{3} \arccos \left (a x\right )^{4} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.87 \[ \int x^3 \arccos (a x)^4 \, dx=\frac {1}{4} \, x^{4} \arccos \left (a x\right )^{4} - \frac {3}{16} \, x^{4} \arccos \left (a x\right )^{2} - \frac {\sqrt {-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )^{3}}{4 \, a} + \frac {3}{128} \, x^{4} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )}{32 \, a} - \frac {9 \, x^{2} \arccos \left (a x\right )^{2}}{16 \, a^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{3}}{8 \, a^{3}} + \frac {45 \, x^{2}}{128 \, a^{2}} - \frac {3 \, \arccos \left (a x\right )^{4}}{32 \, a^{4}} + \frac {45 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )}{64 \, a^{3}} + \frac {45 \, \arccos \left (a x\right )^{2}}{128 \, a^{4}} - \frac {189}{1024 \, a^{4}} \]
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Timed out. \[ \int x^3 \arccos (a x)^4 \, dx=\int x^3\,{\mathrm {acos}\left (a\,x\right )}^4 \,d x \]
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