Optimal. Leaf size=51 \[ \frac {\sin ^{-1}\left (a x^2\right )}{8 a^2}-\frac {x^2 \sqrt {1-a^2 x^4}}{8 a}+\frac {1}{4} x^4 \cos ^{-1}\left (a x^2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4843, 12, 275, 321, 216} \[ -\frac {x^2 \sqrt {1-a^2 x^4}}{8 a}+\frac {\sin ^{-1}\left (a x^2\right )}{8 a^2}+\frac {1}{4} x^4 \cos ^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 275
Rule 321
Rule 4843
Rubi steps
\begin {align*} \int x^3 \cos ^{-1}\left (a x^2\right ) \, dx &=\frac {1}{4} x^4 \cos ^{-1}\left (a x^2\right )+\frac {1}{4} \int \frac {2 a x^5}{\sqrt {1-a^2 x^4}} \, dx\\ &=\frac {1}{4} x^4 \cos ^{-1}\left (a x^2\right )+\frac {1}{2} a \int \frac {x^5}{\sqrt {1-a^2 x^4}} \, dx\\ &=\frac {1}{4} x^4 \cos ^{-1}\left (a x^2\right )+\frac {1}{4} a \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^2 \sqrt {1-a^2 x^4}}{8 a}+\frac {1}{4} x^4 \cos ^{-1}\left (a x^2\right )+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx,x,x^2\right )}{8 a}\\ &=-\frac {x^2 \sqrt {1-a^2 x^4}}{8 a}+\frac {1}{4} x^4 \cos ^{-1}\left (a x^2\right )+\frac {\sin ^{-1}\left (a x^2\right )}{8 a^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.94 \[ \frac {-a x^2 \sqrt {1-a^2 x^4}+2 a^2 x^4 \cos ^{-1}\left (a x^2\right )+\sin ^{-1}\left (a x^2\right )}{8 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 41, normalized size = 0.80 \[ -\frac {\sqrt {-a^{2} x^{4} + 1} a x^{2} - {\left (2 \, a^{2} x^{4} - 1\right )} \arccos \left (a x^{2}\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 46, normalized size = 0.90 \[ \frac {2 \, a^{2} x^{4} \arccos \left (a x^{2}\right ) - \sqrt {-a^{2} x^{4} + 1} a x^{2} - \arccos \left (a x^{2}\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 65, normalized size = 1.27 \[ \frac {x^{4} \arccos \left (a \,x^{2}\right )}{4}-\frac {x^{2} \sqrt {-a^{2} x^{4}+1}}{8 a}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x^{2}}{\sqrt {-a^{2} x^{4}+1}}\right )}{8 a \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 79, normalized size = 1.55 \[ \frac {1}{4} \, x^{4} \arccos \left (a x^{2}\right ) - \frac {1}{8} \, a {\left (\frac {\arctan \left (\frac {\sqrt {-a^{2} x^{4} + 1}}{a x^{2}}\right )}{a^{3}} + \frac {\sqrt {-a^{2} x^{4} + 1}}{{\left (a^{4} - \frac {{\left (a^{2} x^{4} - 1\right )} a^{2}}{x^{4}}\right )} x^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 42, normalized size = 0.82 \[ \frac {\mathrm {acos}\left (a\,x^2\right )\,\left (2\,a^2\,x^4-1\right )}{8\,a^2}-\frac {x^2\,\sqrt {1-a^2\,x^4}}{8\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 48, normalized size = 0.94 \[ \begin {cases} \frac {x^{4} \operatorname {acos}{\left (a x^{2} \right )}}{4} - \frac {x^{2} \sqrt {- a^{2} x^{4} + 1}}{8 a} - \frac {\operatorname {acos}{\left (a x^{2} \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi x^{4}}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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