Integrand size = 17, antiderivative size = 61 \[ \int x^3 \left (1-x^2\right )^{3/2} \arccos (x) \, dx=-\frac {2 x}{35}-\frac {x^3}{105}+\frac {8 x^5}{175}-\frac {x^7}{49}-\frac {1}{5} \left (1-x^2\right )^{5/2} \arccos (x)+\frac {1}{7} \left (1-x^2\right )^{7/2} \arccos (x) \]
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Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {272, 45, 4780, 12, 380} \[ \int x^3 \left (1-x^2\right )^{3/2} \arccos (x) \, dx=\frac {1}{7} \left (1-x^2\right )^{7/2} \arccos (x)-\frac {1}{5} \left (1-x^2\right )^{5/2} \arccos (x)-\frac {x^7}{49}+\frac {8 x^5}{175}-\frac {x^3}{105}-\frac {2 x}{35} \]
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Rule 12
Rule 45
Rule 272
Rule 380
Rule 4780
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{5} \left (1-x^2\right )^{5/2} \arccos (x)+\frac {1}{7} \left (1-x^2\right )^{7/2} \arccos (x)+\int \frac {1}{35} \left (-2-5 x^2\right ) \left (1-x^2\right )^2 \, dx \\ & = -\frac {1}{5} \left (1-x^2\right )^{5/2} \arccos (x)+\frac {1}{7} \left (1-x^2\right )^{7/2} \arccos (x)+\frac {1}{35} \int \left (-2-5 x^2\right ) \left (1-x^2\right )^2 \, dx \\ & = -\frac {1}{5} \left (1-x^2\right )^{5/2} \arccos (x)+\frac {1}{7} \left (1-x^2\right )^{7/2} \arccos (x)+\frac {1}{35} \int \left (-2-x^2+8 x^4-5 x^6\right ) \, dx \\ & = -\frac {2 x}{35}-\frac {x^3}{105}+\frac {8 x^5}{175}-\frac {x^7}{49}-\frac {1}{5} \left (1-x^2\right )^{5/2} \arccos (x)+\frac {1}{7} \left (1-x^2\right )^{7/2} \arccos (x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int x^3 \left (1-x^2\right )^{3/2} \arccos (x) \, dx=-\frac {x \left (210+35 x^2-168 x^4+75 x^6\right )}{3675}-\frac {1}{35} \left (1-x^2\right )^{5/2} \left (2+5 x^2\right ) \arccos (x) \]
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Time = 0.50 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {\left (x^{2}-1\right )^{2} \sqrt {-x^{2}+1}\, \arccos \left (x \right )}{5}-\frac {\left (3 x^{4}-10 x^{2}+15\right ) x}{75}-\frac {\left (x^{2}-1\right )^{3} \sqrt {-x^{2}+1}\, \arccos \left (x \right )}{7}-\frac {\left (5 x^{6}-21 x^{4}+35 x^{2}-35\right ) x}{245}\) | \(77\) |
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Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int x^3 \left (1-x^2\right )^{3/2} \arccos (x) \, dx=-\frac {1}{49} \, x^{7} + \frac {8}{175} \, x^{5} - \frac {1}{105} \, x^{3} - \frac {1}{35} \, {\left (5 \, x^{6} - 8 \, x^{4} + x^{2} + 2\right )} \sqrt {-x^{2} + 1} \arccos \left (x\right ) - \frac {2}{35} \, x \]
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Time = 6.92 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.44 \[ \int x^3 \left (1-x^2\right )^{3/2} \arccos (x) \, dx=- \frac {x^{7}}{49} - \frac {x^{6} \sqrt {1 - x^{2}} \operatorname {acos}{\left (x \right )}}{7} + \frac {8 x^{5}}{175} + \frac {8 x^{4} \sqrt {1 - x^{2}} \operatorname {acos}{\left (x \right )}}{35} - \frac {x^{3}}{105} - \frac {x^{2} \sqrt {1 - x^{2}} \operatorname {acos}{\left (x \right )}}{35} - \frac {2 x}{35} - \frac {2 \sqrt {1 - x^{2}} \operatorname {acos}{\left (x \right )}}{35} \]
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Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.80 \[ \int x^3 \left (1-x^2\right )^{3/2} \arccos (x) \, dx=-\frac {1}{49} \, x^{7} + \frac {8}{175} \, x^{5} - \frac {1}{105} \, x^{3} - \frac {1}{35} \, {\left (5 \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x^{2} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}}\right )} \arccos \left (x\right ) - \frac {2}{35} \, x \]
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Time = 0.31 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98 \[ \int x^3 \left (1-x^2\right )^{3/2} \arccos (x) \, dx=-\frac {1}{49} \, x^{7} + \frac {8}{175} \, x^{5} - \frac {1}{105} \, x^{3} - \frac {1}{35} \, {\left (5 \, {\left (x^{2} - 1\right )}^{3} \sqrt {-x^{2} + 1} + 7 \, {\left (x^{2} - 1\right )}^{2} \sqrt {-x^{2} + 1}\right )} \arccos \left (x\right ) - \frac {2}{35} \, x \]
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Timed out. \[ \int x^3 \left (1-x^2\right )^{3/2} \arccos (x) \, dx=\int x^3\,\mathrm {acos}\left (x\right )\,{\left (1-x^2\right )}^{3/2} \,d x \]
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