Optimal. Leaf size=67 \[ -\frac {6 x}{a}+\frac {6 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{a^2}+\frac {3 x \text {ArcSin}(a x)^2}{a}-\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{a^2} \]
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Rubi [A]
time = 0.08, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4767, 4715, 8}
\begin {gather*} -\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{a^2}+\frac {6 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{a^2}+\frac {3 x \text {ArcSin}(a x)^2}{a}-\frac {6 x}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 4715
Rule 4767
Rubi steps
\begin {align*} \int \frac {x \sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}+\frac {3 \int \sin ^{-1}(a x)^2 \, dx}{a}\\ &=\frac {3 x \sin ^{-1}(a x)^2}{a}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}-6 \int \frac {x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {6 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a^2}+\frac {3 x \sin ^{-1}(a x)^2}{a}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}-\frac {6 \int 1 \, dx}{a}\\ &=-\frac {6 x}{a}+\frac {6 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a^2}+\frac {3 x \sin ^{-1}(a x)^2}{a}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 61, normalized size = 0.91 \begin {gather*} \frac {-6 a x+6 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)+3 a x \text {ArcSin}(a x)^2-\sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 107, normalized size = 1.60
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\arcsin \left (a x \right )^{3} a^{2} x^{2}-\arcsin \left (a x \right )^{3}+3 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x -6 a^{2} x^{2} \arcsin \left (a x \right )+6 \arcsin \left (a x \right )-6 a x \sqrt {-a^{2} x^{2}+1}\right )}{a^{2} \left (a^{2} x^{2}-1\right )}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 64, normalized size = 0.96 \begin {gather*} \frac {3 \, x \arcsin \left (a x\right )^{2}}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a^{2}} - \frac {6 \, {\left (x - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.93, size = 46, normalized size = 0.69 \begin {gather*} \frac {3 \, a x \arcsin \left (a x\right )^{2} - 6 \, a x - \sqrt {-a^{2} x^{2} + 1} {\left (\arcsin \left (a x\right )^{3} - 6 \, \arcsin \left (a x\right )\right )}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.41, size = 61, normalized size = 0.91 \begin {gather*} \begin {cases} \frac {3 x \operatorname {asin}^{2}{\left (a x \right )}}{a} - \frac {6 x}{a} - \frac {\sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{a^{2}} + \frac {6 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{a^{2}} & \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 = 0.45, size = 62, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a^{2}} + \frac {3 \, {\left (x \arcsin \left (a x\right )^{2} - 2 \, x + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a} \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\,{\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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