Optimal. Leaf size=99 \[ \frac {3 x \sqrt {1-a^2 x^2}}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 a}-\frac {3 \sin ^{-1}(a x)}{8 a^2}-\frac {\cos ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^3-\frac {3}{4} x^2 \cos ^{-1}(a x) \]
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Rubi [A] time = 0.16, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4628, 4708, 4642, 321, 216} \[ \frac {3 x \sqrt {1-a^2 x^2}}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 a}-\frac {3 \sin ^{-1}(a x)}{8 a^2}-\frac {\cos ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^3-\frac {3}{4} x^2 \cos ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rule 4628
Rule 4642
Rule 4708
Rubi steps
\begin {align*} \int x \cos ^{-1}(a x)^3 \, dx &=\frac {1}{2} x^2 \cos ^{-1}(a x)^3+\frac {1}{2} (3 a) \int \frac {x^2 \cos ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 a}+\frac {1}{2} x^2 \cos ^{-1}(a x)^3-\frac {3}{2} \int x \cos ^{-1}(a x) \, dx+\frac {3 \int \frac {\cos ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{4 a}\\ &=-\frac {3}{4} x^2 \cos ^{-1}(a x)-\frac {3 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 a}-\frac {\cos ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 x \sqrt {1-a^2 x^2}}{8 a}-\frac {3}{4} x^2 \cos ^{-1}(a x)-\frac {3 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 a}-\frac {\cos ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^3-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=\frac {3 x \sqrt {1-a^2 x^2}}{8 a}-\frac {3}{4} x^2 \cos ^{-1}(a x)-\frac {3 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 a}-\frac {\cos ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \cos ^{-1}(a x)^3-\frac {3 \sin ^{-1}(a x)}{8 a^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 85, normalized size = 0.86 \[ \frac {3 a x \sqrt {1-a^2 x^2}+\left (4 a^2 x^2-2\right ) \cos ^{-1}(a x)^3-6 a x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2-6 a^2 x^2 \cos ^{-1}(a x)-3 \sin ^{-1}(a x)}{8 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 69, normalized size = 0.70 \[ \frac {2 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right )^{3} - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right ) - 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x \arccos \left (a x\right )^{2} - a x\right )}}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 83, normalized size = 0.84 \[ \frac {1}{2} \, x^{2} \arccos \left (a x\right )^{3} - \frac {3}{4} \, x^{2} \arccos \left (a x\right ) - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{2}}{4 \, a} - \frac {\arccos \left (a x\right )^{3}}{4 \, a^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a} + \frac {3 \, \arccos \left (a x\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 90, normalized size = 0.91 \[ \frac {\frac {a^{2} x^{2} \arccos \left (a x \right )^{3}}{2}-\frac {3 \arccos \left (a x \right )^{2} \left (a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{4}-\frac {3 a^{2} x^{2} \arccos \left (a x \right )}{4}+\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arccos \left (a x \right )}{8}+\frac {\arccos \left (a x \right )^{3}}{2}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{3} - 3 \, a \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1} x^{2} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2}}{2 \, {\left (a^{2} x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {acos}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 99, normalized size = 1.00 \[ \begin {cases} \frac {x^{2} \operatorname {acos}^{3}{\left (a x \right )}}{2} - \frac {3 x^{2} \operatorname {acos}{\left (a x \right )}}{4} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{4 a} + \frac {3 x \sqrt {- a^{2} x^{2} + 1}}{8 a} - \frac {\operatorname {acos}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {acos}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x^{2}}{16} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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