Optimal. Leaf size=88 \[ \frac {3 \sin ^{-1}(a x)^2}{16 a^5}+\frac {3 x^2}{16 a^3}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{8 a^4}+\frac {x^4}{16 a} \]
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Rubi [A] time = 0.15, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4707, 4641, 30} \[ \frac {3 x^2}{16 a^3}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{8 a^4}+\frac {3 \sin ^{-1}(a x)^2}{16 a^5}+\frac {x^4}{16 a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 4641
Rule 4707
Rubi steps
\begin {align*} \int \frac {x^4 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}+\frac {3 \int \frac {x^2 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}+\frac {\int x^3 \, dx}{4 a}\\ &=\frac {x^4}{16 a}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}+\frac {3 \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 a^4}+\frac {3 \int x \, dx}{8 a^3}\\ &=\frac {3 x^2}{16 a^3}+\frac {x^4}{16 a}-\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}+\frac {3 \sin ^{-1}(a x)^2}{16 a^5}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 0.73 \[ \frac {a^2 x^2 \left (a^2 x^2+3\right )-2 a x \sqrt {1-a^2 x^2} \left (2 a^2 x^2+3\right ) \sin ^{-1}(a x)+3 \sin ^{-1}(a x)^2}{16 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.43, size = 60, normalized size = 0.68 \[ \frac {a^{4} x^{4} + 3 \, a^{2} x^{2} - 2 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right ) + 3 \, \arcsin \left (a x\right )^{2}}{16 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 91, normalized size = 1.03 \[ \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )}{4 \, a^{4}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{8 \, a^{4}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2}}{16 \, a^{5}} + \frac {3 \, \arcsin \left (a x\right )^{2}}{16 \, a^{5}} + \frac {5 \, {\left (a^{2} x^{2} - 1\right )}}{16 \, a^{5}} + \frac {17}{128 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 74, normalized size = 0.84 \[ \frac {-4 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, x^{3} a^{3}+a^{4} x^{4}-6 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, x a +3 a^{2} x^{2}+3 \arcsin \left (a x \right )^{2}}{16 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 85, normalized size = 0.97 \[ \frac {1}{16} \, {\left (\frac {x^{4}}{a^{2}} + \frac {3 \, x^{2}}{a^{4}} - \frac {3 \, \arcsin \left (a x\right )^{2}}{a^{6}}\right )} a - \frac {1}{8} \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{3}}{a^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{a^{4}} - \frac {3 \, \arcsin \left (a x\right )}{a^{5}}\right )} \arcsin \left (a x\right ) \]
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 \[ \int \frac {x^4\,\mathrm {asin}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.12, size = 82, normalized size = 0.93 \[ \begin {cases} \frac {x^{4}}{16 a} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{4 a^{2}} + \frac {3 x^{2}}{16 a^{3}} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{8 a^{4}} + \frac {3 \operatorname {asin}^{2}{\left (a x \right )}}{16 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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