Optimal. Leaf size=98 \[ -\frac {3 \sin ^{-1}(a x)^2}{32 a^4}-\frac {3 x^2}{32 a^2}+\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{8 a}+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{16 a^3}+\frac {1}{4} x^4 \sin ^{-1}(a x)^2-\frac {x^4}{32} \]
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Rubi [A] time = 0.16, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4627, 4707, 4641, 30} \[ -\frac {3 x^2}{32 a^2}+\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{8 a}+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{16 a^3}-\frac {3 \sin ^{-1}(a x)^2}{32 a^4}+\frac {1}{4} x^4 \sin ^{-1}(a x)^2-\frac {x^4}{32} \]
Antiderivative was successfully verified.
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Rule 30
Rule 4627
Rule 4641
Rule 4707
Rubi steps
\begin {align*} \int x^3 \sin ^{-1}(a x)^2 \, dx &=\frac {1}{4} x^4 \sin ^{-1}(a x)^2-\frac {1}{2} a \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)}{8 a}+\frac {1}{4} x^4 \sin ^{-1}(a x)^2-\frac {\int x^3 \, dx}{8}-\frac {3 \int \frac {x^2 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac {x^4}{32}+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{16 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{8 a}+\frac {1}{4} x^4 \sin ^{-1}(a x)^2-\frac {3 \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{16 a^3}-\frac {3 \int x \, dx}{16 a^2}\\ &=-\frac {3 x^2}{32 a^2}-\frac {x^4}{32}+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{16 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{8 a}-\frac {3 \sin ^{-1}(a x)^2}{32 a^4}+\frac {1}{4} x^4 \sin ^{-1}(a x)^2\\ \end {align*}
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Mathematica [A] time = 0.03, size = 74, normalized size = 0.76 \[ \frac {\left (8 a^4 x^4-3\right ) \sin ^{-1}(a x)^2-a^2 x^2 \left (a^2 x^2+3\right )+2 a x \sqrt {1-a^2 x^2} \left (2 a^2 x^2+3\right ) \sin ^{-1}(a x)}{32 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 70, normalized size = 0.71 \[ -\frac {a^{4} x^{4} + 3 \, a^{2} x^{2} - {\left (8 \, a^{4} x^{4} - 3\right )} \arcsin \left (a x\right )^{2} - 2 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{32 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 133, normalized size = 1.36 \[ -\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )}{8 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{2}}{4 \, a^{4}} + \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{16 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{2 \, a^{4}} - \frac {{\left (a^{2} x^{2} - 1\right )}^{2}}{32 \, a^{4}} + \frac {5 \, \arcsin \left (a x\right )^{2}}{32 \, a^{4}} - \frac {5 \, {\left (a^{2} x^{2} - 1\right )}}{32 \, a^{4}} - \frac {17}{256 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 93, normalized size = 0.95 \[ \frac {\frac {a^{4} x^{4} \arcsin \left (a x \right )^{2}}{4}-\frac {\arcsin \left (a x \right ) \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{16}+\frac {3 \arcsin \left (a x \right )^{2}}{32}-\frac {a^{4} x^{4}}{32}-\frac {3 a^{2} x^{2}}{32}}{a^{4}} \]
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 \[ \frac {1}{4} \, x^{4} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2} + a \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1} x^{4} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}{2 \, {\left (a^{2} x^{2} - 1\right )}}\,{d x} \]
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 \[ \int x^3\,{\mathrm {asin}\left (a\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.85, size = 90, normalized size = 0.92 \[ \begin {cases} \frac {x^{4} \operatorname {asin}^{2}{\left (a x \right )}}{4} - \frac {x^{4}}{32} + \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{8 a} - \frac {3 x^{2}}{32 a^{2}} + \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{16 a^{3}} - \frac {3 \operatorname {asin}^{2}{\left (a x \right )}}{32 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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