Optimal. Leaf size=265 \[ \frac {\sqrt [4]{e} \sqrt {\pi } a^2 \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{4 b^3}+\frac {\sqrt [4]{e} \sqrt {\pi } a^2 \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{4 b^3}-\frac {e \sqrt {\pi } a \text {erf}\left (1-i \sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {e \sqrt {\pi } a \text {erf}\left (1+i \sin ^{-1}(a+b x)\right )}{4 b^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{16 b^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{16 b^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)-3 i\right )\right )}{16 b^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)+3 i\right )\right )}{16 b^3} \]
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Rubi [A] time = 0.47, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4836, 6741, 12, 6742, 4473, 2234, 2204, 4474} \[ \frac {\sqrt [4]{e} \sqrt {\pi } a^2 \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{4 b^3}+\frac {\sqrt [4]{e} \sqrt {\pi } a^2 \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{4 b^3}-\frac {e \sqrt {\pi } a \text {Erf}\left (1-i \sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {e \sqrt {\pi } a \text {Erf}\left (1+i \sin ^{-1}(a+b x)\right )}{4 b^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{16 b^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{16 b^3}-\frac {e^{9/4} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)-3 i\right )\right )}{16 b^3}-\frac {e^{9/4} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} \left (2 \sin ^{-1}(a+b x)+3 i\right )\right )}{16 b^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2204
Rule 2234
Rule 4473
Rule 4474
Rule 4836
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int e^{\sin ^{-1}(a+b x)^2} x^2 \, dx &=\frac {\operatorname {Subst}\left (\int e^{x^2} \cos (x) \left (-\frac {a}{b}+\frac {\sin (x)}{b}\right )^2 \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {e^{x^2} \cos (x) (a-\sin (x))^2}{b^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int e^{x^2} \cos (x) (a-\sin (x))^2 \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 e^{x^2} \cos (x)-2 a e^{x^2} \cos (x) \sin (x)+e^{x^2} \cos (x) \sin ^2(x)\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {\operatorname {Subst}\left (\int e^{x^2} \cos (x) \sin ^2(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}-\frac {(2 a) \operatorname {Subst}\left (\int e^{x^2} \cos (x) \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}+\frac {a^2 \operatorname {Subst}\left (\int e^{x^2} \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^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+b x)\right )}{b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \left (\frac {1}{4} i e^{-2 i x+x^2}-\frac {1}{4} i e^{2 i x+x^2}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}+\frac {a^2 \operatorname {Subst}\left (\int \left (\frac {1}{2} e^{-i x+x^2}+\frac {1}{2} e^{i x+x^2}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {\operatorname {Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}+\frac {\operatorname {Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}-\frac {\operatorname {Subst}\left (\int e^{-3 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}-\frac {\operatorname {Subst}\left (\int e^{3 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}-\frac {(i a) \operatorname {Subst}\left (\int e^{-2 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}+\frac {(i a) \operatorname {Subst}\left (\int e^{2 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}+\frac {a^2 \operatorname {Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}+\frac {a^2 \operatorname {Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}\\ &=\frac {\sqrt [4]{e} \operatorname {Subst}\left (\int e^{\frac {1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}+\frac {\sqrt [4]{e} \operatorname {Subst}\left (\int e^{\frac {1}{4} (i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}+\frac {\left (a^2 \sqrt [4]{e}\right ) \operatorname {Subst}\left (\int e^{\frac {1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}+\frac {\left (a^2 \sqrt [4]{e}\right ) \operatorname {Subst}\left (\int e^{\frac {1}{4} (i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}-\frac {(i a e) \operatorname {Subst}\left (\int e^{\frac {1}{4} (-2 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}+\frac {(i a e) \operatorname {Subst}\left (\int e^{\frac {1}{4} (2 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}-\frac {e^{9/4} \operatorname {Subst}\left (\int e^{\frac {1}{4} (-3 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}-\frac {e^{9/4} \operatorname {Subst}\left (\int e^{\frac {1}{4} (3 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}\\ &=-\frac {a e \sqrt {\pi } \text {erf}\left (1-i \sin ^{-1}(a+b x)\right )}{4 b^3}-\frac {a e \sqrt {\pi } \text {erf}\left (1+i \sin ^{-1}(a+b x)\right )}{4 b^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (-i+2 \sin ^{-1}(a+b x)\right )\right )}{16 b^3}+\frac {a^2 \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (-i+2 \sin ^{-1}(a+b x)\right )\right )}{4 b^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (i+2 \sin ^{-1}(a+b x)\right )\right )}{16 b^3}+\frac {a^2 \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (i+2 \sin ^{-1}(a+b x)\right )\right )}{4 b^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (-3 i+2 \sin ^{-1}(a+b x)\right )\right )}{16 b^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (3 i+2 \sin ^{-1}(a+b x)\right )\right )}{16 b^3}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 161, normalized size = 0.61 \[ -\frac {\sqrt {\pi } \left (i \sqrt [4]{e} \left (4 a^2 \text {erf}\left (\frac {1}{2}+i \sin ^{-1}(a+b x)\right )-\left (4 a^2+1\right ) \text {erf}\left (\frac {1}{2}-i \sin ^{-1}(a+b x)\right )-4 i e^{3/4} a \text {erf}\left (1+i \sin ^{-1}(a+b x)\right )+e^2 \text {erf}\left (\frac {3}{2}-i \sin ^{-1}(a+b x)\right )+\text {erf}\left (\frac {1}{2}+i \sin ^{-1}(a+b x)\right )-e^2 \text {erf}\left (\frac {3}{2}+i \sin ^{-1}(a+b x)\right )\right )+4 e a \text {erf}\left (1-i \sin ^{-1}(a+b x)\right )\right )}{16 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} e^{\left (\arcsin \left (b x + a\right )^{2}\right )}, x\right ) \]
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 \[ \int x^{2} e^{\left (\arcsin \left (b x + a\right )^{2}\right )}\,{d x} \]
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 \[ \int {\mathrm e}^{\arcsin \left (b x +a \right )^{2}} x^{2}\, dx \]
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 \[ \int x^{2} e^{\left (\arcsin \left (b x + a\right )^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\mathrm {e}}^{{\mathrm {asin}\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int x^{2} e^{\operatorname {asin}^{2}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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