\(\int e^{\arcsin (a x)^2} x^2 \, dx\) [446]

Optimal result

Integrand size = 12, antiderivative size = 129 \[ \int e^{\arcsin (a x)^2} x^2 \, dx=\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a x))\right )}{16 a^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a x))\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-3 i+2 \arcsin (a x))\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (3 i+2 \arcsin (a x))\right )}{16 a^3} \]

[Out]

1/16*exp(1/4)*erfi(-1/2*I+arcsin(a*x))*Pi^(1/2)/a^3+1/16*exp(1/4)*erfi(1/2*I+arcsin(a*x))*Pi^(1/2)/a^3-1/16*ex
p(9/4)*erfi(-3/2*I+arcsin(a*x))*Pi^(1/2)/a^3-1/16*exp(9/4)*erfi(3/2*I+arcsin(a*x))*Pi^(1/2)/a^3

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4920, 12, 4562, 2266, 2235} \[ \int e^{\arcsin (a x)^2} x^2 \, dx=\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \arcsin (a x)-i)\right )}{16 a^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \arcsin (a x)+i)\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \arcsin (a x)-3 i)\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \arcsin (a x)+3 i)\right )}{16 a^3} \]

[In]

Int[E^ArcSin[a*x]^2*x^2,x]

[Out]

(E^(1/4)*Sqrt[Pi]*Erfi[(-I + 2*ArcSin[a*x])/2])/(16*a^3) + (E^(1/4)*Sqrt[Pi]*Erfi[(I + 2*ArcSin[a*x])/2])/(16*
a^3) - (E^(9/4)*Sqrt[Pi]*Erfi[(-3*I + 2*ArcSin[a*x])/2])/(16*a^3) - (E^(9/4)*Sqrt[Pi]*Erfi[(3*I + 2*ArcSin[a*x
])/2])/(16*a^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 4562

Int[Cos[v_]^(n_.)*(F_)^(u_)*Sin[v_]^(m_.), x_Symbol] :> Int[ExpandTrigToExp[F^u, Sin[v]^m*Cos[v]^n, x], x] /;
FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[m, 0] && IGtQ[n,
0]

Rule 4920

Int[(u_.)*(f_)^(ArcSin[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Dist[1/b, Subst[Int[(u /. x -> -a/b + Si
n[x]/b)*f^(c*x^n)*Cos[x], x], x, ArcSin[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^{x^2} \cos (x) \sin ^2(x)}{a^2} \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {\text {Subst}\left (\int e^{x^2} \cos (x) \sin ^2(x) \, dx,x,\arcsin (a x)\right )}{a^3} \\ & = \frac {\text {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,\arcsin (a x)\right )}{a^3} \\ & = \frac {\text {Subst}\left (\int e^{-i x+x^2} \, dx,x,\arcsin (a x)\right )}{8 a^3}+\frac {\text {Subst}\left (\int e^{i x+x^2} \, dx,x,\arcsin (a x)\right )}{8 a^3}-\frac {\text {Subst}\left (\int e^{-3 i x+x^2} \, dx,x,\arcsin (a x)\right )}{8 a^3}-\frac {\text {Subst}\left (\int e^{3 i x+x^2} \, dx,x,\arcsin (a x)\right )}{8 a^3} \\ & = \frac {\sqrt [4]{e} \text {Subst}\left (\int e^{\frac {1}{4} (-i+2 x)^2} \, dx,x,\arcsin (a x)\right )}{8 a^3}+\frac {\sqrt [4]{e} \text {Subst}\left (\int e^{\frac {1}{4} (i+2 x)^2} \, dx,x,\arcsin (a x)\right )}{8 a^3}-\frac {e^{9/4} \text {Subst}\left (\int e^{\frac {1}{4} (-3 i+2 x)^2} \, dx,x,\arcsin (a x)\right )}{8 a^3}-\frac {e^{9/4} \text {Subst}\left (\int e^{\frac {1}{4} (3 i+2 x)^2} \, dx,x,\arcsin (a x)\right )}{8 a^3} \\ & = \frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a x))\right )}{16 a^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a x))\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-3 i+2 \arcsin (a x))\right )}{16 a^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (3 i+2 \arcsin (a x))\right )}{16 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.65 \[ \int e^{\arcsin (a x)^2} x^2 \, dx=\frac {\sqrt [4]{e} \sqrt {\pi } \left (\text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a x))\right )+\text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a x))\right )-e^2 \left (\text {erfi}\left (\frac {1}{2} (-3 i+2 \arcsin (a x))\right )+\text {erfi}\left (\frac {1}{2} (3 i+2 \arcsin (a x))\right )\right )\right )}{16 a^3} \]

[In]

Integrate[E^ArcSin[a*x]^2*x^2,x]

[Out]

(E^(1/4)*Sqrt[Pi]*(Erfi[(-I + 2*ArcSin[a*x])/2] + Erfi[(I + 2*ArcSin[a*x])/2] - E^2*(Erfi[(-3*I + 2*ArcSin[a*x
])/2] + Erfi[(3*I + 2*ArcSin[a*x])/2])))/(16*a^3)

Maple [F]

\[\int {\mathrm e}^{\arcsin \left (a x \right )^{2}} x^{2}d x\]

[In]

int(exp(arcsin(a*x)^2)*x^2,x)

[Out]

int(exp(arcsin(a*x)^2)*x^2,x)

Fricas [F]

\[ \int e^{\arcsin (a x)^2} x^2 \, dx=\int { x^{2} e^{\left (\arcsin \left (a x\right )^{2}\right )} \,d x } \]

[In]

integrate(exp(arcsin(a*x)^2)*x^2,x, algorithm="fricas")

[Out]

integral(x^2*e^(arcsin(a*x)^2), x)

Sympy [F]

\[ \int e^{\arcsin (a x)^2} x^2 \, dx=\int x^{2} e^{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate(exp(asin(a*x)**2)*x**2,x)

[Out]

Integral(x**2*exp(asin(a*x)**2), x)

Maxima [F]

\[ \int e^{\arcsin (a x)^2} x^2 \, dx=\int { x^{2} e^{\left (\arcsin \left (a x\right )^{2}\right )} \,d x } \]

[In]

integrate(exp(arcsin(a*x)^2)*x^2,x, algorithm="maxima")

[Out]

integrate(x^2*e^(arcsin(a*x)^2), x)

Giac [F]

\[ \int e^{\arcsin (a x)^2} x^2 \, dx=\int { x^{2} e^{\left (\arcsin \left (a x\right )^{2}\right )} \,d x } \]

[In]

integrate(exp(arcsin(a*x)^2)*x^2,x, algorithm="giac")

[Out]

integrate(x^2*e^(arcsin(a*x)^2), x)

Mupad [F(-1)]

Timed out. \[ \int e^{\arcsin (a x)^2} x^2 \, dx=\int x^2\,{\mathrm {e}}^{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]

[In]

int(x^2*exp(asin(a*x)^2),x)

[Out]

int(x^2*exp(asin(a*x)^2), x)