Optimal. Leaf size=120 \[ -\frac{8 x^3}{225 a^2}+\frac{2 x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{25 a}+\frac{8 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{75 a^3}+\frac{16 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{75 a^5}-\frac{16 x}{75 a^4}+\frac{1}{5} x^5 \sin ^{-1}(a x)^2-\frac{2 x^5}{125} \]
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Rubi [A] time = 0.193986, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4627, 4707, 4677, 8, 30} \[ -\frac{8 x^3}{225 a^2}+\frac{2 x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{25 a}+\frac{8 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{75 a^3}+\frac{16 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{75 a^5}-\frac{16 x}{75 a^4}+\frac{1}{5} x^5 \sin ^{-1}(a x)^2-\frac{2 x^5}{125} \]
Antiderivative was successfully verified.
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Rule 4627
Rule 4707
Rule 4677
Rule 8
Rule 30
Rubi steps
\begin{align*} \int x^4 \sin ^{-1}(a x)^2 \, dx &=\frac{1}{5} x^5 \sin ^{-1}(a x)^2-\frac{1}{5} (2 a) \int \frac{x^5 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^2-\frac{2 \int x^4 \, dx}{25}-\frac{8 \int \frac{x^3 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{25 a}\\ &=-\frac{2 x^5}{125}+\frac{8 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{75 a^3}+\frac{2 x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^2-\frac{16 \int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{75 a^3}-\frac{8 \int x^2 \, dx}{75 a^2}\\ &=-\frac{8 x^3}{225 a^2}-\frac{2 x^5}{125}+\frac{16 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{75 a^5}+\frac{8 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{75 a^3}+\frac{2 x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^2-\frac{16 \int 1 \, dx}{75 a^4}\\ &=-\frac{16 x}{75 a^4}-\frac{8 x^3}{225 a^2}-\frac{2 x^5}{125}+\frac{16 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{75 a^5}+\frac{8 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{75 a^3}+\frac{2 x^4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^2\\ \end{align*}
Mathematica [A] time = 0.0336199, size = 82, normalized size = 0.68 \[ \frac{-2 a x \left (9 a^4 x^4+20 a^2 x^2+120\right )+225 a^5 x^5 \sin ^{-1}(a x)^2+30 \sqrt{1-a^2 x^2} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \sin ^{-1}(a x)}{1125 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.126, size = 76, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{{a}^{5}{x}^{5} \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{5}}+{\frac{2\,\arcsin \left ( ax \right ) \left ( 3\,{a}^{4}{x}^{4}+4\,{a}^{2}{x}^{2}+8 \right ) }{75}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{2\,{a}^{5}{x}^{5}}{125}}-{\frac{8\,{a}^{3}{x}^{3}}{225}}-{\frac{16\,ax}{75}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67013, size = 138, normalized size = 1.15 \begin{align*} \frac{1}{5} \, x^{5} \arcsin \left (a x\right )^{2} + \frac{2}{75} \,{\left (\frac{3 \, \sqrt{-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac{4 \, \sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac{8 \, \sqrt{-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arcsin \left (a x\right ) - \frac{2 \,{\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42754, size = 189, normalized size = 1.58 \begin{align*} \frac{225 \, a^{5} x^{5} \arcsin \left (a x\right )^{2} - 18 \, a^{5} x^{5} - 40 \, a^{3} x^{3} + 30 \,{\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right ) - 240 \, a x}{1125 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.61404, size = 114, normalized size = 0.95 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{asin}^{2}{\left (a x \right )}}{5} - \frac{2 x^{5}}{125} + \frac{2 x^{4} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{25 a} - \frac{8 x^{3}}{225 a^{2}} + \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{75 a^{3}} - \frac{16 x}{75 a^{4}} + \frac{16 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{75 a^{5}} & \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 = 1.34402, size = 228, normalized size = 1.9 \begin{align*} \frac{{\left (a^{2} x^{2} - 1\right )}^{2} x \arcsin \left (a x\right )^{2}}{5 \, a^{4}} + \frac{2 \,{\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{2}}{5 \, a^{4}} - \frac{2 \,{\left (a^{2} x^{2} - 1\right )}^{2} x}{125 \, a^{4}} + \frac{x \arcsin \left (a x\right )^{2}}{5 \, a^{4}} + \frac{2 \,{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{25 \, a^{5}} - \frac{76 \,{\left (a^{2} x^{2} - 1\right )} x}{1125 \, a^{4}} - \frac{4 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \arcsin \left (a x\right )}{15 \, a^{5}} - \frac{298 \, x}{1125 \, a^{4}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{5 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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