Optimal. Leaf size=129 \[ \frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )}{16 a^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)-3 i\right )\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)+3 i\right )\right )}{16 a^3} \]
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Rubi [A] time = 0.13, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4836, 12, 4474, 2234, 2204} \[ \frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )}{16 a^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)-3 i\right )\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)+3 i\right )\right )}{16 a^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2204
Rule 2234
Rule 4474
Rule 4836
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a x)^2} x^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^{x^2} \cos (x) \sin ^2(x)}{a^2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int e^{x^2} \cos (x) \sin ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{8} e^{-i x+x^2}+\frac {1}{8} e^{i x+x^2}-\frac {1}{8} e^{-3 i x+x^2}-\frac {1}{8} e^{3 i x+x^2}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}+\frac {\operatorname {Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}-\frac {\operatorname {Subst}\left (\int e^{-3 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}-\frac {\operatorname {Subst}\left (\int e^{3 i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}\\ &=\frac {\sqrt [4]{e} \operatorname {Subst}\left (\int e^{\frac {1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}+\frac {\sqrt [4]{e} \operatorname {Subst}\left (\int e^{\frac {1}{4} (i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}-\frac {e^{9/4} \operatorname {Subst}\left (\int e^{\frac {1}{4} (-3 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}-\frac {e^{9/4} \operatorname {Subst}\left (\int e^{\frac {1}{4} (3 i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}\\ &=\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (-i+2 \sin ^{-1}(a x)\right )\right )}{16 a^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (i+2 \sin ^{-1}(a x)\right )\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (-3 i+2 \sin ^{-1}(a x)\right )\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (3 i+2 \sin ^{-1}(a x)\right )\right )}{16 a^3}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 84, normalized size = 0.65 \[ \frac {\sqrt [4]{e} \sqrt {\pi } \left (\text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)-i\right )\right )+\text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)+i\right )\right )-e^2 \left (\text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)-3 i\right )\right )+\text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a x)+3 i\right )\right )\right )\right )}{16 a^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} e^{\left (\arcsin \left (a x\right )^{2}\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} e^{\left (\arcsin \left (a x\right )^{2}\right )}\,{d x} \]
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 \[ \int {\mathrm e}^{\arcsin \left (a x \right )^{2}} x^{2}\, dx \]
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 \[ \int x^{2} e^{\left (\arcsin \left (a x\right )^{2}\right )}\,{d x} \]
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 \[ \int x^2\,{\mathrm {e}}^{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} e^{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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