Optimal. Leaf size=107 \[ -\frac {3 x^2}{8 a}+\frac {3 x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{4 a^2}-\frac {3 \text {ArcSin}(a x)^2}{8 a^3}+\frac {3 x^2 \text {ArcSin}(a x)^2}{4 a}-\frac {x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{2 a^2}+\frac {\text {ArcSin}(a x)^4}{8 a^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4795, 4737,
4723, 30} \begin {gather*} \frac {\text {ArcSin}(a x)^4}{8 a^3}-\frac {3 \text {ArcSin}(a x)^2}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{2 a^2}+\frac {3 x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{4 a^2}+\frac {3 x^2 \text {ArcSin}(a x)^2}{4 a}-\frac {3 x^2}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 4723
Rule 4737
Rule 4795
Rubi steps
\begin {align*} \int \frac {x^2 \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 a^2}+\frac {\int \frac {\sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}+\frac {3 \int x \sin ^{-1}(a x)^2 \, dx}{2 a}\\ &=\frac {3 x^2 \sin ^{-1}(a x)^2}{4 a}-\frac {x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 a^2}+\frac {\sin ^{-1}(a x)^4}{8 a^3}-\frac {3}{2} \int \frac {x^2 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}+\frac {3 x^2 \sin ^{-1}(a x)^2}{4 a}-\frac {x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 a^2}+\frac {\sin ^{-1}(a x)^4}{8 a^3}-\frac {3 \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}-\frac {3 \int x \, dx}{4 a}\\ &=-\frac {3 x^2}{8 a}+\frac {3 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}-\frac {3 \sin ^{-1}(a x)^2}{8 a^3}+\frac {3 x^2 \sin ^{-1}(a x)^2}{4 a}-\frac {x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{2 a^2}+\frac {\sin ^{-1}(a x)^4}{8 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 85, normalized size = 0.79 \begin {gather*} \frac {-3 a^2 x^2+6 a x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)+\left (-3+6 a^2 x^2\right ) \text {ArcSin}(a x)^2-4 a x \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3+\text {ArcSin}(a x)^4}{8 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 85, normalized size = 0.79
method | result | size |
default | \(\frac {-4 \arcsin \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}\, a x +6 \arcsin \left (a x \right )^{2} a^{2} x^{2}+\arcsin \left (a x \right )^{4}+6 a x \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}-3 a^{2} x^{2}-3 \arcsin \left (a x \right )^{2}}{8 a^{3}}\) | \(85\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.72, size = 73, normalized size = 0.68 \begin {gather*} -\frac {3 \, a^{2} x^{2} - \arcsin \left (a x\right )^{4} - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2} + 2 \, {\left (2 \, a x \arcsin \left (a x\right )^{3} - 3 \, a x \arcsin \left (a x\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.45, size = 100, normalized size = 0.93 \begin {gather*} \begin {cases} \frac {3 x^{2} \operatorname {asin}^{2}{\left (a x \right )}}{4 a} - \frac {3 x^{2}}{8 a} - \frac {x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{2 a^{2}} + \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{4 a^{2}} + \frac {\operatorname {asin}^{4}{\left (a x \right )}}{8 a^{3}} - \frac {3 \operatorname {asin}^{2}{\left (a x \right )}}{8 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 108, normalized size = 1.01 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )^{4}}{8 \, a^{3}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{4 \, a^{2}} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{4 \, a^{3}} + \frac {3 \, \arcsin \left (a x\right )^{2}}{8 \, a^{3}} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )}}{8 \, a^{3}} - \frac {3}{16 \, a^{3}} \end {gather*}
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 \begin {gather*} \int \frac {x^2\,{\mathrm {asin}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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