Integrand size = 10, antiderivative size = 201 \[ \int x^4 \arccos (a x)^3 \, dx=\frac {298 \sqrt {1-a^2 x^2}}{375 a^5}-\frac {76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}+\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {16 x \arccos (a x)}{25 a^4}-\frac {8 x^3 \arccos (a x)}{75 a^2}-\frac {6}{125} x^5 \arccos (a x)-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a}+\frac {1}{5} x^5 \arccos (a x)^3 \]
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Time = 0.24 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4724, 4796, 4768, 4716, 267, 272, 45} \[ \int x^4 \arccos (a x)^3 \, dx=-\frac {16 x \arccos (a x)}{25 a^4}-\frac {8 x^3 \arccos (a x)}{75 a^2}-\frac {3 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a}-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^5}+\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}+\frac {298 \sqrt {1-a^2 x^2}}{375 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^3}+\frac {1}{5} x^5 \arccos (a x)^3-\frac {6}{125} x^5 \arccos (a x) \]
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Rule 45
Rule 267
Rule 272
Rule 4716
Rule 4724
Rule 4768
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \arccos (a x)^3+\frac {1}{5} (3 a) \int \frac {x^5 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {3 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a}+\frac {1}{5} x^5 \arccos (a x)^3-\frac {6}{25} \int x^4 \arccos (a x) \, dx+\frac {12 \int \frac {x^3 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{25 a} \\ & = -\frac {6}{125} x^5 \arccos (a x)-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a}+\frac {1}{5} x^5 \arccos (a x)^3+\frac {8 \int \frac {x \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{25 a^3}-\frac {8 \int x^2 \arccos (a x) \, dx}{25 a^2}-\frac {1}{125} (6 a) \int \frac {x^5}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {8 x^3 \arccos (a x)}{75 a^2}-\frac {6}{125} x^5 \arccos (a x)-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a}+\frac {1}{5} x^5 \arccos (a x)^3-\frac {16 \int \arccos (a x) \, dx}{25 a^4}-\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2}} \, dx}{75 a}-\frac {1}{125} (3 a) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-a^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {16 x \arccos (a x)}{25 a^4}-\frac {8 x^3 \arccos (a x)}{75 a^2}-\frac {6}{125} x^5 \arccos (a x)-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a}+\frac {1}{5} x^5 \arccos (a x)^3-\frac {16 \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx}{25 a^3}-\frac {4 \text {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )}{75 a}-\frac {1}{125} (3 a) \text {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {1-a^2 x}}-\frac {2 \sqrt {1-a^2 x}}{a^4}+\frac {\left (1-a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right ) \\ & = \frac {86 \sqrt {1-a^2 x^2}}{125 a^5}-\frac {4 \left (1-a^2 x^2\right )^{3/2}}{125 a^5}+\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {16 x \arccos (a x)}{25 a^4}-\frac {8 x^3 \arccos (a x)}{75 a^2}-\frac {6}{125} x^5 \arccos (a x)-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a}+\frac {1}{5} x^5 \arccos (a x)^3-\frac {4 \text {Subst}\left (\int \left (\frac {1}{a^2 \sqrt {1-a^2 x}}-\frac {\sqrt {1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )}{75 a} \\ & = \frac {298 \sqrt {1-a^2 x^2}}{375 a^5}-\frac {76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}+\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {16 x \arccos (a x)}{25 a^4}-\frac {8 x^3 \arccos (a x)}{75 a^2}-\frac {6}{125} x^5 \arccos (a x)-\frac {8 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^2}{25 a}+\frac {1}{5} x^5 \arccos (a x)^3 \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.61 \[ \int x^4 \arccos (a x)^3 \, dx=\frac {2 \sqrt {1-a^2 x^2} \left (2072+136 a^2 x^2+27 a^4 x^4\right )-30 a x \left (120+20 a^2 x^2+9 a^4 x^4\right ) \arccos (a x)-225 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \arccos (a x)^2+1125 a^5 x^5 \arccos (a x)^3}{5625 a^5} \]
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Time = 2.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {\arccos \left (a x \right )^{3} a^{5} x^{5}}{5}-\frac {\arccos \left (a x \right )^{2} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{25}-\frac {6 a^{5} x^{5} \arccos \left (a x \right )}{125}+\frac {2 \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}-\frac {8 a^{3} x^{3} \arccos \left (a x \right )}{75}+\frac {8 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}+\frac {16 \sqrt {-a^{2} x^{2}+1}}{25}-\frac {16 a x \arccos \left (a x \right )}{25}}{a^{5}}\) | \(159\) |
default | \(\frac {\frac {\arccos \left (a x \right )^{3} a^{5} x^{5}}{5}-\frac {\arccos \left (a x \right )^{2} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{25}-\frac {6 a^{5} x^{5} \arccos \left (a x \right )}{125}+\frac {2 \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}-\frac {8 a^{3} x^{3} \arccos \left (a x \right )}{75}+\frac {8 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}+\frac {16 \sqrt {-a^{2} x^{2}+1}}{25}-\frac {16 a x \arccos \left (a x \right )}{25}}{a^{5}}\) | \(159\) |
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Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.52 \[ \int x^4 \arccos (a x)^3 \, dx=\frac {1125 \, a^{5} x^{5} \arccos \left (a x\right )^{3} - 30 \, {\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \arccos \left (a x\right ) + {\left (54 \, a^{4} x^{4} + 272 \, a^{2} x^{2} - 225 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \arccos \left (a x\right )^{2} + 4144\right )} \sqrt {-a^{2} x^{2} + 1}}{5625 \, a^{5}} \]
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Time = 0.66 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int x^4 \arccos (a x)^3 \, dx=\begin {cases} \frac {x^{5} \operatorname {acos}^{3}{\left (a x \right )}}{5} - \frac {6 x^{5} \operatorname {acos}{\left (a x \right )}}{125} - \frac {3 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{25 a} + \frac {6 x^{4} \sqrt {- a^{2} x^{2} + 1}}{625 a} - \frac {8 x^{3} \operatorname {acos}{\left (a x \right )}}{75 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{25 a^{3}} + \frac {272 x^{2} \sqrt {- a^{2} x^{2} + 1}}{5625 a^{3}} - \frac {16 x \operatorname {acos}{\left (a x \right )}}{25 a^{4}} - \frac {8 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{25 a^{5}} + \frac {4144 \sqrt {- a^{2} x^{2} + 1}}{5625 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x^{5}}{40} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.85 \[ \int x^4 \arccos (a x)^3 \, dx=\frac {1}{5} \, x^{5} \arccos \left (a x\right )^{3} - \frac {1}{25} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arccos \left (a x\right )^{2} + \frac {2}{5625} \, a {\left (\frac {27 \, \sqrt {-a^{2} x^{2} + 1} a^{2} x^{4} + 136 \, \sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}}{a^{4}} - \frac {15 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \arccos \left (a x\right )}{a^{5}}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.87 \[ \int x^4 \arccos (a x)^3 \, dx=\frac {1}{5} \, x^{5} \arccos \left (a x\right )^{3} - \frac {6}{125} \, x^{5} \arccos \left (a x\right ) - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )^{2}}{25 \, a} + \frac {6 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{625 \, a} - \frac {8 \, x^{3} \arccos \left (a x\right )}{75 \, a^{2}} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{2}}{25 \, a^{3}} + \frac {272 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{5625 \, a^{3}} - \frac {16 \, x \arccos \left (a x\right )}{25 \, a^{4}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{25 \, a^{5}} + \frac {4144 \, \sqrt {-a^{2} x^{2} + 1}}{5625 \, a^{5}} \]
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Timed out. \[ \int x^4 \arccos (a x)^3 \, dx=\int x^4\,{\mathrm {acos}\left (a\,x\right )}^3 \,d x \]
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