\(\int e^{\text {arcsinh}(a+b x)^2} x^2 \, dx\) [359]

Optimal result

Integrand size = 14, antiderivative size = 251 \[ \int e^{\text {arcsinh}(a+b x)^2} x^2 \, dx=-\frac {a \sqrt {\pi } \text {erfi}(1-\text {arcsinh}(a+b x))}{4 b^3 e}-\frac {a \sqrt {\pi } \text {erfi}(1+\text {arcsinh}(a+b x))}{4 b^3 e}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-3+2 \text {arcsinh}(a+b x))\right )}{16 b^3 e^{9/4}}-\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-1+2 \text {arcsinh}(a+b x))\right )}{16 b^3 \sqrt [4]{e}}+\frac {a^2 \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-1+2 \text {arcsinh}(a+b x))\right )}{4 b^3 \sqrt [4]{e}}-\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 \text {arcsinh}(a+b x))\right )}{16 b^3 \sqrt [4]{e}}+\frac {a^2 \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 \text {arcsinh}(a+b x))\right )}{4 b^3 \sqrt [4]{e}}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (3+2 \text {arcsinh}(a+b x))\right )}{16 b^3 e^{9/4}} \]

[Out]

1/4*a*erfi(-1+arcsinh(b*x+a))*Pi^(1/2)/b^3/exp(1)-1/4*a*erfi(1+arcsinh(b*x+a))*Pi^(1/2)/b^3/exp(1)+1/16*erfi(-
3/2+arcsinh(b*x+a))*Pi^(1/2)/b^3/exp(9/4)-1/16*erfi(-1/2+arcsinh(b*x+a))*Pi^(1/2)/b^3/exp(1/4)+1/4*a^2*erfi(-1
/2+arcsinh(b*x+a))*Pi^(1/2)/b^3/exp(1/4)-1/16*erfi(1/2+arcsinh(b*x+a))*Pi^(1/2)/b^3/exp(1/4)+1/4*a^2*erfi(1/2+
arcsinh(b*x+a))*Pi^(1/2)/b^3/exp(1/4)+1/16*erfi(3/2+arcsinh(b*x+a))*Pi^(1/2)/b^3/exp(9/4)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5873, 6873, 12, 6874, 5624, 2266, 2235, 5625} \[ \int e^{\text {arcsinh}(a+b x)^2} x^2 \, dx=\frac {\sqrt {\pi } a^2 \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)-1)\right )}{4 \sqrt [4]{e} b^3}+\frac {\sqrt {\pi } a^2 \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)+1)\right )}{4 \sqrt [4]{e} b^3}-\frac {\sqrt {\pi } a \text {erfi}(1-\text {arcsinh}(a+b x))}{4 e b^3}-\frac {\sqrt {\pi } a \text {erfi}(\text {arcsinh}(a+b x)+1)}{4 e b^3}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)-3)\right )}{16 e^{9/4} b^3}-\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)-1)\right )}{16 \sqrt [4]{e} b^3}-\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)+1)\right )}{16 \sqrt [4]{e} b^3}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)+3)\right )}{16 e^{9/4} b^3} \]

[In]

Int[E^ArcSinh[a + b*x]^2*x^2,x]

[Out]

-1/4*(a*Sqrt[Pi]*Erfi[1 - ArcSinh[a + b*x]])/(b^3*E) - (a*Sqrt[Pi]*Erfi[1 + ArcSinh[a + b*x]])/(4*b^3*E) + (Sq
rt[Pi]*Erfi[(-3 + 2*ArcSinh[a + b*x])/2])/(16*b^3*E^(9/4)) - (Sqrt[Pi]*Erfi[(-1 + 2*ArcSinh[a + b*x])/2])/(16*
b^3*E^(1/4)) + (a^2*Sqrt[Pi]*Erfi[(-1 + 2*ArcSinh[a + b*x])/2])/(4*b^3*E^(1/4)) - (Sqrt[Pi]*Erfi[(1 + 2*ArcSin
h[a + b*x])/2])/(16*b^3*E^(1/4)) + (a^2*Sqrt[Pi]*Erfi[(1 + 2*ArcSinh[a + b*x])/2])/(4*b^3*E^(1/4)) + (Sqrt[Pi]
*Erfi[(3 + 2*ArcSinh[a + b*x])/2])/(16*b^3*E^(9/4))

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 5624

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

Rule 5625

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

Rule 5873

Int[(f_)^(ArcSinh[(a_.) + (b_.)*(x_)]^(n_.)*(c_.))*(x_)^(m_.), x_Symbol] :> Dist[1/b, Subst[Int[(-a/b + Sinh[x
]/b)^m*f^(c*x^n)*Cosh[x], x], x, ArcSinh[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && 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} \cosh (x) \left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^2 \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {e^{x^2} \cosh (x) (a-\sinh (x))^2}{b^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int e^{x^2} \cosh (x) (a-\sinh (x))^2 \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = \frac {\text {Subst}\left (\int \left (a^2 e^{x^2} \cosh (x)-2 a e^{x^2} \cosh (x) \sinh (x)+e^{x^2} \cosh (x) \sinh ^2(x)\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = \frac {\text {Subst}\left (\int e^{x^2} \cosh (x) \sinh ^2(x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int e^{x^2} \cosh (x) \sinh (x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int e^{x^2} \cosh (x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{8} e^{-3 x+x^2}-\frac {1}{8} e^{-x+x^2}-\frac {e^{x+x^2}}{8}+\frac {1}{8} e^{3 x+x^2}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int \left (-\frac {1}{4} e^{-2 x+x^2}+\frac {1}{4} e^{2 x+x^2}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \left (\frac {1}{2} e^{-x+x^2}+\frac {e^{x+x^2}}{2}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = \frac {\text {Subst}\left (\int e^{-3 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^3}-\frac {\text {Subst}\left (\int e^{-x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^3}-\frac {\text {Subst}\left (\int e^{x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^3}+\frac {\text {Subst}\left (\int e^{3 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^3}+\frac {a \text {Subst}\left (\int e^{-2 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3}-\frac {a \text {Subst}\left (\int e^{2 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3}+\frac {a^2 \text {Subst}\left (\int e^{-x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3}+\frac {a^2 \text {Subst}\left (\int e^{x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3} \\ & = \frac {\text {Subst}\left (\int e^{\frac {1}{4} (-3+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^3 e^{9/4}}+\frac {\text {Subst}\left (\int e^{\frac {1}{4} (3+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^3 e^{9/4}}+\frac {a \text {Subst}\left (\int e^{\frac {1}{4} (-2+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3 e}-\frac {a \text {Subst}\left (\int e^{\frac {1}{4} (2+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3 e}-\frac {\text {Subst}\left (\int e^{\frac {1}{4} (-1+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^3 \sqrt [4]{e}}-\frac {\text {Subst}\left (\int e^{\frac {1}{4} (1+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^3 \sqrt [4]{e}}+\frac {a^2 \text {Subst}\left (\int e^{\frac {1}{4} (-1+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3 \sqrt [4]{e}}+\frac {a^2 \text {Subst}\left (\int e^{\frac {1}{4} (1+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3 \sqrt [4]{e}} \\ & = -\frac {a \sqrt {\pi } \text {erfi}(1-\text {arcsinh}(a+b x))}{4 b^3 e}-\frac {a \sqrt {\pi } \text {erfi}(1+\text {arcsinh}(a+b x))}{4 b^3 e}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-3+2 \text {arcsinh}(a+b x))\right )}{16 b^3 e^{9/4}}-\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-1+2 \text {arcsinh}(a+b x))\right )}{16 b^3 \sqrt [4]{e}}+\frac {a^2 \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-1+2 \text {arcsinh}(a+b x))\right )}{4 b^3 \sqrt [4]{e}}-\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 \text {arcsinh}(a+b x))\right )}{16 b^3 \sqrt [4]{e}}+\frac {a^2 \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 \text {arcsinh}(a+b x))\right )}{4 b^3 \sqrt [4]{e}}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (3+2 \text {arcsinh}(a+b x))\right )}{16 b^3 e^{9/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.55 \[ \int e^{\text {arcsinh}(a+b x)^2} x^2 \, dx=-\frac {\sqrt {\pi } \left (\left (-1+4 a^2\right ) e^2 \text {erfi}\left (\frac {1}{2}-\text {arcsinh}(a+b x)\right )+4 a e^{5/4} \text {erfi}(1-\text {arcsinh}(a+b x))+\text {erfi}\left (\frac {3}{2}-\text {arcsinh}(a+b x)\right )+e^2 \text {erfi}\left (\frac {1}{2}+\text {arcsinh}(a+b x)\right )-4 a^2 e^2 \text {erfi}\left (\frac {1}{2}+\text {arcsinh}(a+b x)\right )+4 a e^{5/4} \text {erfi}(1+\text {arcsinh}(a+b x))-\text {erfi}\left (\frac {3}{2}+\text {arcsinh}(a+b x)\right )\right )}{16 b^3 e^{9/4}} \]

[In]

Integrate[E^ArcSinh[a + b*x]^2*x^2,x]

[Out]

-1/16*(Sqrt[Pi]*((-1 + 4*a^2)*E^2*Erfi[1/2 - ArcSinh[a + b*x]] + 4*a*E^(5/4)*Erfi[1 - ArcSinh[a + b*x]] + Erfi
[3/2 - ArcSinh[a + b*x]] + E^2*Erfi[1/2 + ArcSinh[a + b*x]] - 4*a^2*E^2*Erfi[1/2 + ArcSinh[a + b*x]] + 4*a*E^(
5/4)*Erfi[1 + ArcSinh[a + b*x]] - Erfi[3/2 + ArcSinh[a + b*x]]))/(b^3*E^(9/4))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x^2*e^(arcsinh(b*x + a)^2), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2*exp(asinh(a + b*x)**2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2*e^(arcsinh(b*x + a)^2), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^2*e^(arcsinh(b*x + a)^2), x)

Mupad [F(-1)]

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

[In]

int(x^2*exp(asinh(a + b*x)^2),x)

[Out]

int(x^2*exp(asinh(a + b*x)^2), x)