Optimal. Leaf size=157 \[ -\frac {40 x}{9 a^3}-\frac {2 x^3}{27 a}+\frac {40 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{9 a^4}+\frac {2 x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{9 a^2}+\frac {2 x \text {ArcSin}(a x)^2}{a^3}+\frac {x^3 \text {ArcSin}(a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{3 a^2} \]
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Rubi [A]
time = 0.23, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4795, 4767,
4715, 8, 4723, 30} \begin {gather*} \frac {2 x \text {ArcSin}(a x)^2}{a^3}-\frac {40 x}{9 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{3 a^2}+\frac {2 x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{9 a^2}-\frac {2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{3 a^4}+\frac {40 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{9 a^4}+\frac {x^3 \text {ArcSin}(a x)^2}{3 a}-\frac {2 x^3}{27 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4715
Rule 4723
Rule 4767
Rule 4795
Rubi steps
\begin {align*} \int \frac {x^3 \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}+\frac {2 \int \frac {x \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{3 a^2}+\frac {\int x^2 \sin ^{-1}(a x)^2 \, dx}{a}\\ &=\frac {x^3 \sin ^{-1}(a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}-\frac {2}{3} \int \frac {x^3 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx+\frac {2 \int \sin ^{-1}(a x)^2 \, dx}{a^3}\\ &=\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}+\frac {2 x \sin ^{-1}(a x)^2}{a^3}+\frac {x^3 \sin ^{-1}(a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}-\frac {4 \int \frac {x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{9 a^2}-\frac {4 \int \frac {x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{a^2}-\frac {2 \int x^2 \, dx}{9 a}\\ &=-\frac {2 x^3}{27 a}+\frac {40 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^4}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}+\frac {2 x \sin ^{-1}(a x)^2}{a^3}+\frac {x^3 \sin ^{-1}(a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}-\frac {4 \int 1 \, dx}{9 a^3}-\frac {4 \int 1 \, dx}{a^3}\\ &=-\frac {40 x}{9 a^3}-\frac {2 x^3}{27 a}+\frac {40 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^4}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}+\frac {2 x \sin ^{-1}(a x)^2}{a^3}+\frac {x^3 \sin ^{-1}(a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 100, normalized size = 0.64 \begin {gather*} \frac {-2 a x \left (60+a^2 x^2\right )+6 \sqrt {1-a^2 x^2} \left (20+a^2 x^2\right ) \text {ArcSin}(a x)+9 a x \left (6+a^2 x^2\right ) \text {ArcSin}(a x)^2-9 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \text {ArcSin}(a x)^3}{27 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 180, normalized size = 1.15
method | result | size |
default | \(-\frac {\left (9 a^{4} x^{4} \arcsin \left (a x \right )^{3}+9 \arcsin \left (a x \right )^{3} a^{2} x^{2}+9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-6 a^{4} x^{4} \arcsin \left (a x \right )-114 a^{2} x^{2} \arcsin \left (a x \right )-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-18 \arcsin \left (a x \right )^{3}+54 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +120 \arcsin \left (a x \right )-120 a x \sqrt {-a^{2} x^{2}+1}\right ) \sqrt {-a^{2} x^{2}+1}}{27 a^{4} \left (a^{2} x^{2}-1\right )}\) | \(180\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 131, normalized size = 0.83 \begin {gather*} -\frac {1}{3} \, {\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 )^{3} + \frac {2}{27} \, a {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )} \arcsin \left (a x\right )}{a^{3}} - \frac {a^{2} x^{3} + 60 \, x}{a^{4}}\right )} + \frac {{\left (a^{2} x^{3} + 6 \, x\right )} \arcsin \left (a x\right )^{2}}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.52, size = 85, normalized size = 0.54 \begin {gather*} -\frac {2 \, a^{3} x^{3} - 9 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \arcsin \left (a x\right )^{2} + 120 \, a x + 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (3 \, {\left (a^{2} x^{2} + 2\right )} \arcsin \left (a x\right )^{3} - 2 \, {\left (a^{2} x^{2} + 20\right )} \arcsin \left (a x\right )\right )}}{27 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.63, size = 148, normalized size = 0.94 \begin {gather*} \begin {cases} \frac {x^{3} \operatorname {asin}^{2}{\left (a x \right )}}{3 a} - \frac {2 x^{3}}{27 a} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{3 a^{2}} + \frac {2 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{9 a^{2}} + \frac {2 x \operatorname {asin}^{2}{\left (a x \right )}}{a^{3}} - \frac {40 x}{9 a^{3}} - \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{3 a^{4}} + \frac {40 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{9 a^{4}} & \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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^3\,{\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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