Optimal. Leaf size=120 \[ -\frac {16 x}{75 a^4}-\frac {8 x^3}{225 a^2}+\frac {2 x^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{25 a}+\frac {16 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{75 a^5}+\frac {8 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{75 a^3}+\frac {1}{5} x^5 \sin ^{-1}(a x)^2-\frac {2 x^5}{125} \]
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Rubi [A] time = 0.19, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 8
Rule 30
Rule 4627
Rule 4677
Rule 4707
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*}
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Mathematica [A] time = 0.03, size = 82, normalized size = 0.68 \[ \frac {225 a^5 x^5 \sin ^{-1}(a x)^2-2 a x \left (9 a^4 x^4+20 a^2 x^2+120\right )+30 \sqrt {1-a^2 x^2} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \sin ^{-1}(a x)}{1125 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 76, normalized size = 0.63 \[ \frac {225 \, a^{5} x^{5} \arcsin \left (a x\right )^{2} - 18 \, a^{5} x^{5} - 40 \, a^{3} x^{3} + 30 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right ) - 240 \, a x}{1125 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 169, normalized size = 1.41 \[ \frac {{\left (a^{2} x^{2} - 1\right )}^{2} x \arcsin \left (a x\right )^{2}}{5 \, a^{4}} + \frac {2 \, {\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{2}}{5 \, a^{4}} - \frac {2 \, {\left (a^{2} x^{2} - 1\right )}^{2} x}{125 \, a^{4}} + \frac {x \arcsin \left (a x\right )^{2}}{5 \, a^{4}} + \frac {2 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{25 \, a^{5}} - \frac {76 \, {\left (a^{2} x^{2} - 1\right )} x}{1125 \, a^{4}} - \frac {4 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (a x\right )}{15 \, a^{5}} - \frac {298 \, x}{1125 \, a^{4}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{5 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 76, normalized size = 0.63 \[ \frac {\frac {a^{5} x^{5} \arcsin \left (a x \right )^{2}}{5}+\frac {2 \arcsin \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {2 a^{5} x^{5}}{125}-\frac {8 a^{3} x^{3}}{225}-\frac {16 a x}{75}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 102, normalized size = 0.85 \[ \frac {1}{5} \, x^{5} \arcsin \left (a x\right )^{2} + \frac {2}{75} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arcsin \left (a x\right ) - \frac {2 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \]
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 \[ \int x^4\,{\mathrm {asin}\left (a\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.14, size = 114, normalized size = 0.95 \[ \begin {cases} \frac {x^{5} \operatorname {asin}^{2}{\left (a x \right )}}{5} - \frac {2 x^{5}}{125} + \frac {2 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{25 a} - \frac {8 x^{3}}{225 a^{2}} + \frac {8 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{75 a^{3}} - \frac {16 x}{75 a^{4}} + \frac {16 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{75 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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