3.5.48 \(\int e^{\text {ArcSin}(a x)^2} \, dx\) [448]

Optimal. Leaf size=65 \[ \frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (-i+2 \text {ArcSin}(a x))\right )}{4 a}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (i+2 \text {ArcSin}(a x))\right )}{4 a} \]

[Out]

1/4*exp(1/4)*erfi(-1/2*I+arcsin(a*x))*Pi^(1/2)/a+1/4*exp(1/4)*erfi(1/2*I+arcsin(a*x))*Pi^(1/2)/a

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Rubi [A]
time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4920, 4561, 2266, 2235} \begin {gather*} \frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (2 \text {ArcSin}(a x)-i)\right )}{4 a}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (2 \text {ArcSin}(a x)+i)\right )}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4561

Int[Cos[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cos[v]^n, x], x] /; FreeQ[F, x] && (LinearQ
[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && 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*} \int e^{\sin ^{-1}(a x)^2} \, dx &=\frac {\text {Subst}\left (\int e^{x^2} \cos (x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {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 x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}+\frac {\text {Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}\\ &=\frac {\sqrt [4]{e} \text {Subst}\left (\int e^{\frac {1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}+\frac {\sqrt [4]{e} \text {Subst}\left (\int e^{\frac {1}{4} (i+2 x)^2} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}\\ &=\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (-i+2 \sin ^{-1}(a x)\right )\right )}{4 a}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (i+2 \sin ^{-1}(a x)\right )\right )}{4 a}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.74 \begin {gather*} \frac {\sqrt [4]{e} \sqrt {\pi } \left (\text {Erfi}\left (\frac {1}{2} (-i+2 \text {ArcSin}(a x))\right )+\text {Erfi}\left (\frac {1}{2} (i+2 \text {ArcSin}(a x))\right )\right )}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\arcsin \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(arcsin(a*x)^2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(e^(arcsin(a*x)^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(asin(a*x)**2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(arcsin(a*x)^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {e}}^{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(exp(asin(a*x)^2), x)

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