Optimal. Leaf size=136 \[ \frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}-\frac {4 x \sin ^{-1}(a x)}{3 a^2}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac {14 \sqrt {1-a^2 x^2}}{9 a^3}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \sin ^{-1}(a x)^3-\frac {2}{9} x^3 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.22, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4627, 4707, 4677, 4619, 261, 266, 43} \[ \frac {2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac {14 \sqrt {1-a^2 x^2}}{9 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}-\frac {4 x \sin ^{-1}(a x)}{3 a^2}+\frac {1}{3} x^3 \sin ^{-1}(a x)^3-\frac {2}{9} x^3 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 4619
Rule 4627
Rule 4677
Rule 4707
Rubi steps
\begin {align*} \int x^2 \sin ^{-1}(a x)^3 \, dx &=\frac {1}{3} x^3 \sin ^{-1}(a x)^3-a \int \frac {x^3 \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^3-\frac {2}{3} \int x^2 \sin ^{-1}(a x) \, dx-\frac {2 \int \frac {x \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{3 a}\\ &=-\frac {2}{9} x^3 \sin ^{-1}(a x)+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^3-\frac {4 \int \sin ^{-1}(a x) \, dx}{3 a^2}+\frac {1}{9} (2 a) \int \frac {x^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {4 x \sin ^{-1}(a x)}{3 a^2}-\frac {2}{9} x^3 \sin ^{-1}(a x)+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^3+\frac {4 \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx}{3 a}+\frac {1}{9} a \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {4 \sqrt {1-a^2 x^2}}{3 a^3}-\frac {4 x \sin ^{-1}(a x)}{3 a^2}-\frac {2}{9} x^3 \sin ^{-1}(a x)+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^3+\frac {1}{9} a \operatorname {Subst}\left (\int \left (\frac {1}{a^2 \sqrt {1-a^2 x}}-\frac {\sqrt {1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {14 \sqrt {1-a^2 x^2}}{9 a^3}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac {4 x \sin ^{-1}(a x)}{3 a^2}-\frac {2}{9} x^3 \sin ^{-1}(a x)+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \sin ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A] time = 0.04, size = 95, normalized size = 0.70 \[ \frac {9 a^3 x^3 \sin ^{-1}(a x)^3-2 \sqrt {1-a^2 x^2} \left (a^2 x^2+20\right )+9 \sqrt {1-a^2 x^2} \left (a^2 x^2+2\right ) \sin ^{-1}(a x)^2-6 a x \left (a^2 x^2+6\right ) \sin ^{-1}(a x)}{27 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.84, size = 79, normalized size = 0.58 \[ \frac {9 \, a^{3} x^{3} \arcsin \left (a x\right )^{3} - 6 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \arcsin \left (a x\right ) - {\left (2 \, a^{2} x^{2} - 9 \, {\left (a^{2} x^{2} + 2\right )} \arcsin \left (a x\right )^{2} + 40\right )} \sqrt {-a^{2} x^{2} + 1}}{27 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 142, normalized size = 1.04 \[ \frac {{\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{3}}{3 \, a^{2}} + \frac {x \arcsin \left (a x\right )^{3}}{3 \, a^{2}} - \frac {2 \, {\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )}{9 \, a^{2}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (a x\right )^{2}}{3 \, a^{3}} - \frac {14 \, x \arcsin \left (a x\right )}{9 \, a^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a^{3}} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{27 \, a^{3}} - \frac {14 \, \sqrt {-a^{2} x^{2} + 1}}{9 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 106, normalized size = 0.78 \[ \frac {\frac {a^{3} x^{3} \arcsin \left (a x \right )^{3}}{3}+\frac {\arcsin \left (a x \right )^{2} \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{3}-\frac {4 \sqrt {-a^{2} x^{2}+1}}{3}-\frac {4 a x \arcsin \left (a x \right )}{3}-\frac {2 a^{3} x^{3} \arcsin \left (a x \right )}{9}-\frac {2 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{27}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 120, normalized size = 0.88 \[ \frac {1}{3} \, x^{3} \arcsin \left (a x\right )^{3} + \frac {1}{3} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arcsin \left (a x\right )^{2} - \frac {2}{27} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}}{a^{2}} + \frac {3 \, {\left (a^{2} x^{3} + 6 \, x\right )} \arcsin \left (a x\right )}{a^{3}}\right )} \]
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 \[ \int x^2\,{\mathrm {asin}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.87, size = 128, normalized size = 0.94 \[ \begin {cases} \frac {x^{3} \operatorname {asin}^{3}{\left (a x \right )}}{3} - \frac {2 x^{3} \operatorname {asin}{\left (a x \right )}}{9} + \frac {x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{3 a} - \frac {2 x^{2} \sqrt {- a^{2} x^{2} + 1}}{27 a} - \frac {4 x \operatorname {asin}{\left (a x \right )}}{3 a^{2}} + \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{3 a^{3}} - \frac {40 \sqrt {- a^{2} x^{2} + 1}}{27 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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