Optimal. Leaf size=61 \[ -\frac {x^7}{49}+\frac {8 x^5}{175}-\frac {x^3}{105}+\frac {1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(x)-\frac {1}{5} \left (1-x^2\right )^{5/2} \cos ^{-1}(x)-\frac {2 x}{35} \]
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Rubi [A] time = 0.08, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {266, 43, 4690, 12, 373} \[ -\frac {x^7}{49}+\frac {8 x^5}{175}-\frac {x^3}{105}+\frac {1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(x)-\frac {1}{5} \left (1-x^2\right )^{5/2} \cos ^{-1}(x)-\frac {2 x}{35} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 373
Rule 4690
Rubi steps
\begin {align*} \int x^3 \left (1-x^2\right )^{3/2} \cos ^{-1}(x) \, dx &=-\frac {1}{5} \left (1-x^2\right )^{5/2} \cos ^{-1}(x)+\frac {1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(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} \cos ^{-1}(x)+\frac {1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(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} \cos ^{-1}(x)+\frac {1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(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} \cos ^{-1}(x)+\frac {1}{7} \left (1-x^2\right )^{7/2} \cos ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 47, normalized size = 0.77 \[ -\frac {1}{35} \left (5 x^2+2\right ) \left (1-x^2\right )^{5/2} \cos ^{-1}(x)-\frac {x \left (75 x^6-168 x^4+35 x^2+210\right )}{3675} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^3 \left (1-x^2\right )^{3/2} \cos ^{-1}(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.97, size = 47, normalized size = 0.77 \[ -\frac {1}{49} \, x^{7} + \frac {8}{175} \, x^{5} - \frac {1}{105} \, x^{3} - \frac {1}{35} \, {\left (5 \, x^{6} - 8 \, x^{4} + x^{2} + 2\right )} \sqrt {-x^{2} + 1} \arccos \relax (x) - \frac {2}{35} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 60, normalized size = 0.98 \[ -\frac {1}{49} \, x^{7} + \frac {8}{175} \, x^{5} - \frac {1}{105} \, x^{3} - \frac {1}{35} \, {\left (5 \, {\left (x^{2} - 1\right )}^{3} \sqrt {-x^{2} + 1} + 7 \, {\left (x^{2} - 1\right )}^{2} \sqrt {-x^{2} + 1}\right )} \arccos \relax (x) - \frac {2}{35} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.49, size = 286, normalized size = 4.69
method | result | size |
default | \(\frac {\left (i+7 \arccos \relax (x )\right ) \left (64 i x^{7}-64 \sqrt {-x^{2}+1}\, x^{6}-112 i x^{5}+80 \sqrt {-x^{2}+1}\, x^{4}+56 i x^{3}-24 \sqrt {-x^{2}+1}\, x^{2}-7 i x +\sqrt {-x^{2}+1}\right )}{6272}+\frac {3 \left (\arccos \relax (x )+i\right ) \left (i x -\sqrt {-x^{2}+1}\right )}{128}-\frac {3 \left (\arccos \relax (x )-i\right ) \left (i x +\sqrt {-x^{2}+1}\right )}{128}+\frac {\left (-i+3 \arccos \relax (x )\right ) \left (4 i x^{3}+4 \sqrt {-x^{2}+1}\, x^{2}-3 i x -\sqrt {-x^{2}+1}\right )}{384}-\frac {3 \cos \left (6 \arccos \relax (x )\right ) \left (2 i+35 \arccos \relax (x )\right ) \left (i x +\sqrt {-x^{2}+1}\right )}{39200}+\frac {\sin \left (6 \arccos \relax (x )\right ) \left (37 i+35 \arccos \relax (x )\right ) \left (-i \sqrt {-x^{2}+1}+x \right )}{78400}-\frac {\cos \left (4 \arccos \relax (x )\right ) \left (7 i+15 \arccos \relax (x )\right ) \left (i x +\sqrt {-x^{2}+1}\right )}{2400}+\frac {\sin \left (4 \arccos \relax (x )\right ) \left (11 i+45 \arccos \relax (x )\right ) \left (-i \sqrt {-x^{2}+1}+x \right )}{4800}\) | \(286\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 49, normalized size = 0.80 \[ -\frac {1}{49} \, x^{7} + \frac {8}{175} \, x^{5} - \frac {1}{105} \, x^{3} - \frac {1}{35} \, {\left (5 \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x^{2} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}}\right )} \arccos \relax (x) - \frac {2}{35} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^3\,\mathrm {acos}\relax (x)\,{\left (1-x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 162.01, size = 88, normalized size = 1.44 \[ - \frac {x^{7}}{49} - \frac {x^{6} \sqrt {1 - x^{2}} \operatorname {acos}{\relax (x )}}{7} + \frac {8 x^{5}}{175} + \frac {8 x^{4} \sqrt {1 - x^{2}} \operatorname {acos}{\relax (x )}}{35} - \frac {x^{3}}{105} - \frac {x^{2} \sqrt {1 - x^{2}} \operatorname {acos}{\relax (x )}}{35} - \frac {2 x}{35} - \frac {2 \sqrt {1 - x^{2}} \operatorname {acos}{\relax (x )}}{35} \]
Verification of antiderivative is not currently implemented for this CAS.
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