Optimal. Leaf size=82 \[ -\frac {3 e^{\sin ^{-1}(a x)} \sin \left (3 \sin ^{-1}(a x)\right )}{40 a^3}-\frac {e^{\sin ^{-1}(a x)} \cos \left (3 \sin ^{-1}(a x)\right )}{40 a^3}+\frac {x e^{\sin ^{-1}(a x)}}{8 a^2}+\frac {\sqrt {1-a^2 x^2} e^{\sin ^{-1}(a x)}}{8 a^3} \]
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Rubi [A] time = 0.06, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4836, 12, 4469, 4433} \[ \frac {\sqrt {1-a^2 x^2} e^{\sin ^{-1}(a x)}}{8 a^3}+\frac {x e^{\sin ^{-1}(a x)}}{8 a^2}-\frac {3 e^{\sin ^{-1}(a x)} \sin \left (3 \sin ^{-1}(a x)\right )}{40 a^3}-\frac {e^{\sin ^{-1}(a x)} \cos \left (3 \sin ^{-1}(a x)\right )}{40 a^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 4433
Rule 4469
Rule 4836
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a x)} x^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^x \cos (x) \sin ^2(x)}{a^2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int e^x \cos (x) \sin ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{4} e^x \cos (x)-\frac {1}{4} e^x \cos (3 x)\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int e^x \cos (x) \, dx,x,\sin ^{-1}(a x)\right )}{4 a^3}-\frac {\operatorname {Subst}\left (\int e^x \cos (3 x) \, dx,x,\sin ^{-1}(a x)\right )}{4 a^3}\\ &=\frac {e^{\sin ^{-1}(a x)} x}{8 a^2}+\frac {e^{\sin ^{-1}(a x)} \sqrt {1-a^2 x^2}}{8 a^3}-\frac {e^{\sin ^{-1}(a x)} \cos \left (3 \sin ^{-1}(a x)\right )}{40 a^3}-\frac {3 e^{\sin ^{-1}(a x)} \sin \left (3 \sin ^{-1}(a x)\right )}{40 a^3}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 50, normalized size = 0.61 \[ -\frac {e^{\sin ^{-1}(a x)} \left (-5 \sqrt {1-a^2 x^2}-5 a x+3 \sin \left (3 \sin ^{-1}(a x)\right )+\cos \left (3 \sin ^{-1}(a x)\right )\right )}{40 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 45, normalized size = 0.55 \[ \frac {{\left (3 \, a^{3} x^{3} - a x + {\left (a^{2} x^{2} + 1\right )} \sqrt {-a^{2} x^{2} + 1}\right )} e^{\left (\arcsin \left (a x\right )\right )}}{10 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.79, size = 76, normalized size = 0.93 \[ \frac {3 \, {\left (a^{2} x^{2} - 1\right )} x e^{\left (\arcsin \left (a x\right )\right )}}{10 \, a^{2}} + \frac {x e^{\left (\arcsin \left (a x\right )\right )}}{5 \, a^{2}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} e^{\left (\arcsin \left (a x\right )\right )}}{10 \, a^{3}} + \frac {\sqrt {-a^{2} x^{2} + 1} e^{\left (\arcsin \left (a x\right )\right )}}{5 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\arcsin \left (a x \right )} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} e^{\left (\arcsin \left (a x\right )\right )}\,{d x} \]
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 \[ \int x^2\,{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.12, size = 80, normalized size = 0.98 \[ \begin {cases} \frac {3 x^{3} e^{\operatorname {asin}{\left (a x \right )}}}{10} + \frac {x^{2} \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{10 a} - \frac {x e^{\operatorname {asin}{\left (a x \right )}}}{10 a^{2}} + \frac {\sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{10 a^{3}} & \text {for}\: a \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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