\(\int e^{\arcsin (a+b x)^2} \, dx\) [460]

Optimal result

Integrand size = 10, antiderivative size = 69 \[ \int e^{\arcsin (a+b x)^2} \, dx=\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a+b x))\right )}{4 b}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )}{4 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4920, 4561, 2266, 2235} \[ \int e^{\arcsin (a+b x)^2} \, dx=\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \arcsin (a+b x)-i)\right )}{4 b}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \arcsin (a+b x)+i)\right )}{4 b} \]

[In]

Int[E^ArcSin[a + b*x]^2,x]

[Out]

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

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*} \text {integral}& = \frac {\text {Subst}\left (\int e^{x^2} \cos (x) \, dx,x,\arcsin (a+b x)\right )}{b} \\ & = \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,\arcsin (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int e^{-i x+x^2} \, dx,x,\arcsin (a+b x)\right )}{2 b}+\frac {\text {Subst}\left (\int e^{i x+x^2} \, dx,x,\arcsin (a+b x)\right )}{2 b} \\ & = \frac {\sqrt [4]{e} \text {Subst}\left (\int e^{\frac {1}{4} (-i+2 x)^2} \, dx,x,\arcsin (a+b x)\right )}{2 b}+\frac {\sqrt [4]{e} \text {Subst}\left (\int e^{\frac {1}{4} (i+2 x)^2} \, dx,x,\arcsin (a+b x)\right )}{2 b} \\ & = \frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a+b x))\right )}{4 b}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )}{4 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.75 \[ \int e^{\arcsin (a+b x)^2} \, dx=\frac {\sqrt [4]{e} \sqrt {\pi } \left (\text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a+b x))\right )+\text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )\right )}{4 b} \]

[In]

Integrate[E^ArcSin[a + b*x]^2,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(e^(arcsin(b*x + a)^2), x)

Sympy [F]

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

[In]

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

[Out]

Integral(exp(asin(a + b*x)**2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(e^(arcsin(b*x + a)^2), x)

Giac [F]

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

[In]

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

[Out]

integrate(e^(arcsin(b*x + a)^2), x)

Mupad [F(-1)]

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

[In]

int(exp(asin(a + b*x)^2),x)

[Out]

int(exp(asin(a + b*x)^2), x)