Optimal. Leaf size=69 \[ \frac {3 \sin ^{-1}(a x)}{32 a^4}-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}-\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}+\frac {1}{4} x^4 \cos ^{-1}(a x) \]
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Rubi [A] time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4628, 321, 216} \[ -\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}-\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}+\frac {3 \sin ^{-1}(a x)}{32 a^4}+\frac {1}{4} x^4 \cos ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rule 4628
Rubi steps
\begin {align*} \int x^3 \cos ^{-1}(a x) \, dx &=\frac {1}{4} x^4 \cos ^{-1}(a x)+\frac {1}{4} a \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}+\frac {1}{4} x^4 \cos ^{-1}(a x)+\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{16 a}\\ &=-\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}+\frac {1}{4} x^4 \cos ^{-1}(a x)+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{32 a^3}\\ &=-\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}+\frac {1}{4} x^4 \cos ^{-1}(a x)+\frac {3 \sin ^{-1}(a x)}{32 a^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 54, normalized size = 0.78 \[ \frac {8 a^4 x^4 \cos ^{-1}(a x)-a x \sqrt {1-a^2 x^2} \left (2 a^2 x^2+3\right )+3 \sin ^{-1}(a x)}{32 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 48, normalized size = 0.70 \[ \frac {{\left (8 \, a^{4} x^{4} - 3\right )} \arccos \left (a x\right ) - {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{32 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 57, normalized size = 0.83 \[ \frac {1}{4} \, x^{4} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{16 \, a} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{32 \, a^{3}} - \frac {3 \, \arccos \left (a x\right )}{32 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 60, normalized size = 0.87 \[ \frac {\frac {a^{4} x^{4} \arccos \left (a x \right )}{4}-\frac {a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}}{16}-\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{32}+\frac {3 \arcsin \left (a x \right )}{32}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 61, normalized size = 0.88 \[ \frac {1}{4} \, x^{4} \arccos \left (a x\right ) - \frac {1}{32} \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{3}}{a^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{a^{4}} - \frac {3 \, \arcsin \left (a x\right )}{a^{5}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\mathrm {acos}\left (a\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 66, normalized size = 0.96 \[ \begin {cases} \frac {x^{4} \operatorname {acos}{\left (a x \right )}}{4} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1}}{16 a} - \frac {3 x \sqrt {- a^{2} x^{2} + 1}}{32 a^{3}} - \frac {3 \operatorname {acos}{\left (a x \right )}}{32 a^{4}} & \text {for}\: a \neq 0 \\\frac {\pi x^{4}}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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