Optimal. Leaf size=99 \[ -\frac {3 x \sqrt {1-a^2 x^2}}{8 a}+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}-\frac {\sin ^{-1}(a x)^3}{4 a^2}+\frac {3 \sin ^{-1}(a x)}{8 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^3-\frac {3}{4} x^2 \sin ^{-1}(a x) \]
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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 = {4627, 4707, 4641, 321, 216} \[ -\frac {3 x \sqrt {1-a^2 x^2}}{8 a}+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}-\frac {\sin ^{-1}(a x)^3}{4 a^2}+\frac {3 \sin ^{-1}(a x)}{8 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^3-\frac {3}{4} x^2 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rule 4627
Rule 4641
Rule 4707
Rubi steps
\begin {align*} \int x \sin ^{-1}(a x)^3 \, dx &=\frac {1}{2} x^2 \sin ^{-1}(a x)^3-\frac {1}{2} (3 a) \int \frac {x^2 \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}+\frac {1}{2} x^2 \sin ^{-1}(a x)^3-\frac {3}{2} \int x \sin ^{-1}(a x) \, dx-\frac {3 \int \frac {\sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{4 a}\\ &=-\frac {3}{4} x^2 \sin ^{-1}(a x)+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}-\frac {\sin ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \sin ^{-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 \sin ^{-1}(a x)+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}-\frac {\sin ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \sin ^{-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 \sin ^{-1}(a x)}{8 a^2}-\frac {3}{4} x^2 \sin ^{-1}(a x)+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}-\frac {\sin ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A] time = 0.03, size = 82, normalized size = 0.83 \[ \frac {-3 a x \sqrt {1-a^2 x^2}+\left (4 a^2 x^2-2\right ) \sin ^{-1}(a x)^3+6 a x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2+\left (3-6 a^2 x^2\right ) \sin ^{-1}(a x)}{8 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 69, normalized size = 0.70 \[ \frac {2 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{3} - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right ) + 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x \arcsin \left (a x\right )^{2} - a x\right )}}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 101, normalized size = 1.02 \[ \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{4 \, a} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{3}}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )^{3}}{4 \, a^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{4 \, a^{2}} - \frac {3 \, \arcsin \left (a x\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 96, normalized size = 0.97 \[ \frac {\frac {\left (a^{2} x^{2}-1\right ) \arcsin \left (a x \right )^{3}}{2}+\frac {3 \arcsin \left (a x \right )^{2} \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{4}-\frac {3 \left (a^{2} x^{2}-1\right ) \arcsin \left (a x \right )}{4}-\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{8}-\frac {3 \arcsin \left (a x \right )}{8}-\frac {\arcsin \left (a x \right )^{3}}{2}}{a^{2}} \]
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 \[ \frac {1}{2} \, x^{2} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3} + 3 \, a \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1} x^{2} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2}}{2 \, {\left (a^{2} x^{2} - 1\right )}}\,{d x} \]
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 \[ \int x\,{\mathrm {asin}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.90, size = 92, normalized size = 0.93 \[ \begin {cases} \frac {x^{2} \operatorname {asin}^{3}{\left (a x \right )}}{2} - \frac {3 x^{2} \operatorname {asin}{\left (a x \right )}}{4} + \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{4 a} - \frac {3 x \sqrt {- a^{2} x^{2} + 1}}{8 a} - \frac {\operatorname {asin}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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