Optimal. Leaf size=126 \[ -\frac{2 \left (1-a^2 x^2\right )^{3/2}}{27 a^4}+\frac{14 \sqrt{1-a^2 x^2}}{9 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^4}+\frac{4 x \sin ^{-1}(a x)}{3 a^3}+\frac{2 x^3 \sin ^{-1}(a x)}{9 a} \]
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Rubi [A] time = 0.202162, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {4707, 4677, 4619, 261, 4627, 266, 43} \[ -\frac{2 \left (1-a^2 x^2\right )^{3/2}}{27 a^4}+\frac{14 \sqrt{1-a^2 x^2}}{9 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^4}+\frac{4 x \sin ^{-1}(a x)}{3 a^3}+\frac{2 x^3 \sin ^{-1}(a x)}{9 a} \]
Antiderivative was successfully verified.
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Rule 4707
Rule 4677
Rule 4619
Rule 261
Rule 4627
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3 \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^2}+\frac{2 \int \frac{x \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{3 a^2}+\frac{2 \int x^2 \sin ^{-1}(a x) \, dx}{3 a}\\ &=\frac{2 x^3 \sin ^{-1}(a x)}{9 a}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^2}-\frac{2}{9} \int \frac{x^3}{\sqrt{1-a^2 x^2}} \, dx+\frac{4 \int \sin ^{-1}(a x) \, dx}{3 a^3}\\ &=\frac{4 x \sin ^{-1}(a x)}{3 a^3}+\frac{2 x^3 \sin ^{-1}(a x)}{9 a}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^2}-\frac{1}{9} \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )-\frac{4 \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx}{3 a^2}\\ &=\frac{4 \sqrt{1-a^2 x^2}}{3 a^4}+\frac{4 x \sin ^{-1}(a x)}{3 a^3}+\frac{2 x^3 \sin ^{-1}(a x)}{9 a}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^2}-\frac{1}{9} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 \sqrt{1-a^2 x}}-\frac{\sqrt{1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac{14 \sqrt{1-a^2 x^2}}{9 a^4}-\frac{2 \left (1-a^2 x^2\right )^{3/2}}{27 a^4}+\frac{4 x \sin ^{-1}(a x)}{3 a^3}+\frac{2 x^3 \sin ^{-1}(a x)}{9 a}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^2}\\ \end{align*}
Mathematica [A] time = 0.0453418, size = 81, normalized size = 0.64 \[ \frac{2 \sqrt{1-a^2 x^2} \left (a^2 x^2+20\right )-9 \sqrt{1-a^2 x^2} \left (a^2 x^2+2\right ) \sin ^{-1}(a x)^2+6 a x \left (a^2 x^2+6\right ) \sin ^{-1}(a x)}{27 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 127, normalized size = 1. \begin{align*} -{\frac{1}{27\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 9\,{a}^{4}{x}^{4} \left ( \arcsin \left ( ax \right ) \right ) ^{2}+9\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+6\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}{x}^{3}{a}^{3}-2\,{a}^{4}{x}^{4}-38\,{a}^{2}{x}^{2}-18\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}+36\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}xa+40 \right ) \sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.551, size = 142, normalized size = 1.13 \begin{align*} -\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 )^{2} + \frac{2 \,{\left (\sqrt{-a^{2} x^{2} + 1} x^{2} + \frac{20 \, \sqrt{-a^{2} x^{2} + 1}}{a^{2}}\right )}}{27 \, a^{2}} + \frac{2 \,{\left (a^{2} x^{3} + 6 \, x\right )} \arcsin \left (a x\right )}{9 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78503, size = 154, normalized size = 1.22 \begin{align*} \frac{6 \,{\left (a^{3} x^{3} + 6 \, a x\right )} \arcsin \left (a x\right ) +{\left (2 \, a^{2} x^{2} - 9 \,{\left (a^{2} x^{2} + 2\right )} \arcsin \left (a x\right )^{2} + 40\right )} \sqrt{-a^{2} x^{2} + 1}}{27 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.70067, size = 121, normalized size = 0.96 \begin{align*} \begin{cases} \frac{2 x^{3} \operatorname{asin}{\left (a x \right )}}{9 a} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{3 a^{2}} + \frac{2 x^{2} \sqrt{- a^{2} x^{2} + 1}}{27 a^{2}} + \frac{4 x \operatorname{asin}{\left (a x \right )}}{3 a^{3}} - \frac{2 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{3 a^{4}} + \frac{40 \sqrt{- a^{2} x^{2} + 1}}{27 a^{4}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.227, size = 138, normalized size = 1.1 \begin{align*} \frac{{\left ({\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{-a^{2} x^{2} + 1}\right )} \arcsin \left (a x\right )^{2}}{3 \, a^{4}} + \frac{2 \,{\left (3 \,{\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right ) + 21 \, x \arcsin \left (a x\right ) - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{a} + \frac{21 \, \sqrt{-a^{2} x^{2} + 1}}{a}\right )}}{27 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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