\(\int e^{\arcsin (a+b x)^2} x \, dx\) [459]

Optimal result

Integrand size = 12, antiderivative size = 123 \[ \int e^{\arcsin (a+b x)^2} x \, dx=\frac {e \sqrt {\pi } \text {erf}(1-i \arcsin (a+b x))}{8 b^2}+\frac {e \sqrt {\pi } \text {erf}(1+i \arcsin (a+b x))}{8 b^2}-\frac {a \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a+b x))\right )}{4 b^2}-\frac {a \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )}{4 b^2} \]

[Out]

1/8*I*exp(1)*erfi(-I+arcsin(b*x+a))*Pi^(1/2)/b^2-1/8*I*exp(1)*erfi(I+arcsin(b*x+a))*Pi^(1/2)/b^2-1/4*a*exp(1/4
)*erfi(-1/2*I+arcsin(b*x+a))*Pi^(1/2)/b^2-1/4*a*exp(1/4)*erfi(1/2*I+arcsin(b*x+a))*Pi^(1/2)/b^2

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4920, 6873, 12, 6874, 4561, 2266, 2235, 4562} \[ \int e^{\arcsin (a+b x)^2} x \, dx=\frac {e \sqrt {\pi } \text {erf}(1-i \arcsin (a+b x))}{8 b^2}+\frac {e \sqrt {\pi } \text {erf}(1+i \arcsin (a+b x))}{8 b^2}-\frac {\sqrt [4]{e} \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \arcsin (a+b x)-i)\right )}{4 b^2}-\frac {\sqrt [4]{e} \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \arcsin (a+b x)+i)\right )}{4 b^2} \]

[In]

Int[E^ArcSin[a + b*x]^2*x,x]

[Out]

(E*Sqrt[Pi]*Erf[1 - I*ArcSin[a + b*x]])/(8*b^2) + (E*Sqrt[Pi]*Erf[1 + I*ArcSin[a + b*x]])/(8*b^2) - (a*E^(1/4)
*Sqrt[Pi]*Erfi[(-I + 2*ArcSin[a + b*x])/2])/(4*b^2) - (a*E^(1/4)*Sqrt[Pi]*Erfi[(I + 2*ArcSin[a + b*x])/2])/(4*
b^2)

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 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 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]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int e^{\arcsin (a+b x)^2} x \, dx=\frac {\sqrt {\pi } \left (e \text {erf}(1-i \arcsin (a+b x))+e \text {erf}(1+i \arcsin (a+b x))-2 a \sqrt [4]{e} \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a+b x))\right )-2 a \sqrt [4]{e} \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )\right )}{8 b^2} \]

[In]

Integrate[E^ArcSin[a + b*x]^2*x,x]

[Out]

(Sqrt[Pi]*(E*Erf[1 - I*ArcSin[a + b*x]] + E*Erf[1 + I*ArcSin[a + b*x]] - 2*a*E^(1/4)*Erfi[(-I + 2*ArcSin[a + b
*x])/2] - 2*a*E^(1/4)*Erfi[(I + 2*ArcSin[a + b*x])/2]))/(8*b^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x*e^(arcsin(b*x + a)^2), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x*exp(asin(a + b*x)**2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x*e^(arcsin(b*x + a)^2), x)

Giac [F]

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

[In]

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

[Out]

integrate(x*e^(arcsin(b*x + a)^2), x)

Mupad [F(-1)]

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

[In]

int(x*exp(asin(a + b*x)^2),x)

[Out]

int(x*exp(asin(a + b*x)^2), x)