\(\int e^{\text {arcsinh}(a+b x)^2} x^3 \, dx\) [358]

Optimal result

Integrand size = 14, antiderivative size = 359 \[ \int e^{\text {arcsinh}(a+b x)^2} x^3 \, dx=-\frac {\sqrt {\pi } \text {erfi}(1-\text {arcsinh}(a+b x))}{16 b^4 e}+\frac {3 a^2 \sqrt {\pi } \text {erfi}(1-\text {arcsinh}(a+b x))}{8 b^4 e}+\frac {\sqrt {\pi } \text {erfi}(2-\text {arcsinh}(a+b x))}{32 b^4 e^4}-\frac {\sqrt {\pi } \text {erfi}(1+\text {arcsinh}(a+b x))}{16 b^4 e}+\frac {3 a^2 \sqrt {\pi } \text {erfi}(1+\text {arcsinh}(a+b x))}{8 b^4 e}+\frac {\sqrt {\pi } \text {erfi}(2+\text {arcsinh}(a+b x))}{32 b^4 e^4}-\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-3+2 \text {arcsinh}(a+b x))\right )}{16 b^4 e^{9/4}}+\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-1+2 \text {arcsinh}(a+b x))\right )}{16 b^4 \sqrt [4]{e}}-\frac {a^3 \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-1+2 \text {arcsinh}(a+b x))\right )}{4 b^4 \sqrt [4]{e}}+\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 \text {arcsinh}(a+b x))\right )}{16 b^4 \sqrt [4]{e}}-\frac {a^3 \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 \text {arcsinh}(a+b x))\right )}{4 b^4 \sqrt [4]{e}}-\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (3+2 \text {arcsinh}(a+b x))\right )}{16 b^4 e^{9/4}} \]

[Out]

-1/32*erfi(-2+arcsinh(b*x+a))*Pi^(1/2)/b^4/exp(4)+1/16*erfi(-1+arcsinh(b*x+a))*Pi^(1/2)/b^4/exp(1)-3/8*a^2*erf
i(-1+arcsinh(b*x+a))*Pi^(1/2)/b^4/exp(1)-1/16*erfi(1+arcsinh(b*x+a))*Pi^(1/2)/b^4/exp(1)+3/8*a^2*erfi(1+arcsin
h(b*x+a))*Pi^(1/2)/b^4/exp(1)+1/32*erfi(2+arcsinh(b*x+a))*Pi^(1/2)/b^4/exp(4)-3/16*a*erfi(-3/2+arcsinh(b*x+a))
*Pi^(1/2)/b^4/exp(9/4)+3/16*a*erfi(-1/2+arcsinh(b*x+a))*Pi^(1/2)/b^4/exp(1/4)-1/4*a^3*erfi(-1/2+arcsinh(b*x+a)
)*Pi^(1/2)/b^4/exp(1/4)+3/16*a*erfi(1/2+arcsinh(b*x+a))*Pi^(1/2)/b^4/exp(1/4)-1/4*a^3*erfi(1/2+arcsinh(b*x+a))
*Pi^(1/2)/b^4/exp(1/4)-3/16*a*erfi(3/2+arcsinh(b*x+a))*Pi^(1/2)/b^4/exp(9/4)

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 37, 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} \[ \int e^{\text {arcsinh}(a+b x)^2} x^3 \, dx=-\frac {\sqrt {\pi } a^3 \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)-1)\right )}{4 \sqrt [4]{e} b^4}-\frac {\sqrt {\pi } a^3 \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)+1)\right )}{4 \sqrt [4]{e} b^4}+\frac {3 \sqrt {\pi } a^2 \text {erfi}(1-\text {arcsinh}(a+b x))}{8 e b^4}+\frac {3 \sqrt {\pi } a^2 \text {erfi}(\text {arcsinh}(a+b x)+1)}{8 e b^4}-\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)-3)\right )}{16 e^{9/4} b^4}+\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)-1)\right )}{16 \sqrt [4]{e} b^4}+\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)+1)\right )}{16 \sqrt [4]{e} b^4}-\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)+3)\right )}{16 e^{9/4} b^4}-\frac {\sqrt {\pi } \text {erfi}(1-\text {arcsinh}(a+b x))}{16 e b^4}+\frac {\sqrt {\pi } \text {erfi}(2-\text {arcsinh}(a+b x))}{32 e^4 b^4}-\frac {\sqrt {\pi } \text {erfi}(\text {arcsinh}(a+b x)+1)}{16 e b^4}+\frac {\sqrt {\pi } \text {erfi}(\text {arcsinh}(a+b x)+2)}{32 e^4 b^4} \]

[In]

Int[E^ArcSinh[a + b*x]^2*x^3,x]

[Out]

-1/16*(Sqrt[Pi]*Erfi[1 - ArcSinh[a + b*x]])/(b^4*E) + (3*a^2*Sqrt[Pi]*Erfi[1 - ArcSinh[a + b*x]])/(8*b^4*E) +
(Sqrt[Pi]*Erfi[2 - ArcSinh[a + b*x]])/(32*b^4*E^4) - (Sqrt[Pi]*Erfi[1 + ArcSinh[a + b*x]])/(16*b^4*E) + (3*a^2
*Sqrt[Pi]*Erfi[1 + ArcSinh[a + b*x]])/(8*b^4*E) + (Sqrt[Pi]*Erfi[2 + ArcSinh[a + b*x]])/(32*b^4*E^4) - (3*a*Sq
rt[Pi]*Erfi[(-3 + 2*ArcSinh[a + b*x])/2])/(16*b^4*E^(9/4)) + (3*a*Sqrt[Pi]*Erfi[(-1 + 2*ArcSinh[a + b*x])/2])/
(16*b^4*E^(1/4)) - (a^3*Sqrt[Pi]*Erfi[(-1 + 2*ArcSinh[a + b*x])/2])/(4*b^4*E^(1/4)) + (3*a*Sqrt[Pi]*Erfi[(1 +
2*ArcSinh[a + b*x])/2])/(16*b^4*E^(1/4)) - (a^3*Sqrt[Pi]*Erfi[(1 + 2*ArcSinh[a + b*x])/2])/(4*b^4*E^(1/4)) - (
3*a*Sqrt[Pi]*Erfi[(3 + 2*ArcSinh[a + b*x])/2])/(16*b^4*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 )^3 \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {e^{x^2} \cosh (x) (-a+\sinh (x))^3}{b^3} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int e^{x^2} \cosh (x) (-a+\sinh (x))^3 \, dx,x,\text {arcsinh}(a+b x)\right )}{b^4} \\ & = \frac {\text {Subst}\left (\int \left (-a^3 e^{x^2} \cosh (x)+3 a^2 e^{x^2} \cosh (x) \sinh (x)-3 a e^{x^2} \cosh (x) \sinh ^2(x)+e^{x^2} \cosh (x) \sinh ^3(x)\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^4} \\ & = \frac {\text {Subst}\left (\int e^{x^2} \cosh (x) \sinh ^3(x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^4}-\frac {(3 a) \text {Subst}\left (\int e^{x^2} \cosh (x) \sinh ^2(x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^4}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int e^{x^2} \cosh (x) \sinh (x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^4}-\frac {a^3 \text {Subst}\left (\int e^{x^2} \cosh (x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^4} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{16} e^{-4 x+x^2}+\frac {1}{8} e^{-2 x+x^2}-\frac {1}{8} e^{2 x+x^2}+\frac {1}{16} e^{4 x+x^2}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^4}-\frac {(3 a) \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^4}+\frac {\left (3 a^2\right ) \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^4}-\frac {a^3 \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^4} \\ & = -\frac {\text {Subst}\left (\int e^{-4 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{16 b^4}+\frac {\text {Subst}\left (\int e^{4 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{16 b^4}+\frac {\text {Subst}\left (\int e^{-2 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4}-\frac {\text {Subst}\left (\int e^{2 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4}-\frac {(3 a) \text {Subst}\left (\int e^{-3 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4}+\frac {(3 a) \text {Subst}\left (\int e^{-x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4}+\frac {(3 a) \text {Subst}\left (\int e^{x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4}-\frac {(3 a) \text {Subst}\left (\int e^{3 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int e^{-2 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{4 b^4}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int e^{2 x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{4 b^4}-\frac {a^3 \text {Subst}\left (\int e^{-x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^4}-\frac {a^3 \text {Subst}\left (\int e^{x+x^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^4} \\ & = -\frac {\text {Subst}\left (\int e^{\frac {1}{4} (-4+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{16 b^4 e^4}+\frac {\text {Subst}\left (\int e^{\frac {1}{4} (4+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{16 b^4 e^4}-\frac {(3 a) \text {Subst}\left (\int e^{\frac {1}{4} (-3+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4 e^{9/4}}-\frac {(3 a) \text {Subst}\left (\int e^{\frac {1}{4} (3+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4 e^{9/4}}+\frac {\text {Subst}\left (\int e^{\frac {1}{4} (-2+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4 e}-\frac {\text {Subst}\left (\int e^{\frac {1}{4} (2+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4 e}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int e^{\frac {1}{4} (-2+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{4 b^4 e}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int e^{\frac {1}{4} (2+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{4 b^4 e}+\frac {(3 a) \text {Subst}\left (\int e^{\frac {1}{4} (-1+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4 \sqrt [4]{e}}+\frac {(3 a) \text {Subst}\left (\int e^{\frac {1}{4} (1+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^4 \sqrt [4]{e}}-\frac {a^3 \text {Subst}\left (\int e^{\frac {1}{4} (-1+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^4 \sqrt [4]{e}}-\frac {a^3 \text {Subst}\left (\int e^{\frac {1}{4} (1+2 x)^2} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^4 \sqrt [4]{e}} \\ & = -\frac {\sqrt {\pi } \text {erfi}(1-\text {arcsinh}(a+b x))}{16 b^4 e}+\frac {3 a^2 \sqrt {\pi } \text {erfi}(1-\text {arcsinh}(a+b x))}{8 b^4 e}+\frac {\sqrt {\pi } \text {erfi}(2-\text {arcsinh}(a+b x))}{32 b^4 e^4}-\frac {\sqrt {\pi } \text {erfi}(1+\text {arcsinh}(a+b x))}{16 b^4 e}+\frac {3 a^2 \sqrt {\pi } \text {erfi}(1+\text {arcsinh}(a+b x))}{8 b^4 e}+\frac {\sqrt {\pi } \text {erfi}(2+\text {arcsinh}(a+b x))}{32 b^4 e^4}-\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-3+2 \text {arcsinh}(a+b x))\right )}{16 b^4 e^{9/4}}+\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-1+2 \text {arcsinh}(a+b x))\right )}{16 b^4 \sqrt [4]{e}}-\frac {a^3 \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-1+2 \text {arcsinh}(a+b x))\right )}{4 b^4 \sqrt [4]{e}}+\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 \text {arcsinh}(a+b x))\right )}{16 b^4 \sqrt [4]{e}}-\frac {a^3 \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 \text {arcsinh}(a+b x))\right )}{4 b^4 \sqrt [4]{e}}-\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (3+2 \text {arcsinh}(a+b x))\right )}{16 b^4 e^{9/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.55 \[ \int e^{\text {arcsinh}(a+b x)^2} x^3 \, dx=\frac {\sqrt {\pi } \left (2 a \left (-3+4 a^2\right ) e^{15/4} \text {erfi}\left (\frac {1}{2}-\text {arcsinh}(a+b x)\right )+2 \left (-1+6 a^2\right ) e^3 \text {erfi}(1-\text {arcsinh}(a+b x))+6 a e^{7/4} \text {erfi}\left (\frac {3}{2}-\text {arcsinh}(a+b x)\right )+\text {erfi}(2-\text {arcsinh}(a+b x))+6 a e^{15/4} \text {erfi}\left (\frac {1}{2}+\text {arcsinh}(a+b x)\right )-8 a^3 e^{15/4} \text {erfi}\left (\frac {1}{2}+\text {arcsinh}(a+b x)\right )-2 e^3 \text {erfi}(1+\text {arcsinh}(a+b x))+12 a^2 e^3 \text {erfi}(1+\text {arcsinh}(a+b x))-6 a e^{7/4} \text {erfi}\left (\frac {3}{2}+\text {arcsinh}(a+b x)\right )+\text {erfi}(2+\text {arcsinh}(a+b x))\right )}{32 b^4 e^4} \]

[In]

Integrate[E^ArcSinh[a + b*x]^2*x^3,x]

[Out]

(Sqrt[Pi]*(2*a*(-3 + 4*a^2)*E^(15/4)*Erfi[1/2 - ArcSinh[a + b*x]] + 2*(-1 + 6*a^2)*E^3*Erfi[1 - ArcSinh[a + b*
x]] + 6*a*E^(7/4)*Erfi[3/2 - ArcSinh[a + b*x]] + Erfi[2 - ArcSinh[a + b*x]] + 6*a*E^(15/4)*Erfi[1/2 + ArcSinh[
a + b*x]] - 8*a^3*E^(15/4)*Erfi[1/2 + ArcSinh[a + b*x]] - 2*E^3*Erfi[1 + ArcSinh[a + b*x]] + 12*a^2*E^3*Erfi[1
+ ArcSinh[a + b*x]] - 6*a*E^(7/4)*Erfi[3/2 + ArcSinh[a + b*x]] + Erfi[2 + ArcSinh[a + b*x]]))/(32*b^4*E^4)

Maple [F]

\[\int {\mathrm e}^{\operatorname {arcsinh}\left (b x +a \right )^{2}} x^{3}d x\]

[In]

int(exp(arcsinh(b*x+a)^2)*x^3,x)

[Out]

int(exp(arcsinh(b*x+a)^2)*x^3,x)

Fricas [F]

\[ \int e^{\text {arcsinh}(a+b x)^2} x^3 \, dx=\int { x^{3} e^{\left (\operatorname {arsinh}\left (b x + a\right )^{2}\right )} \,d x } \]

[In]

integrate(exp(arcsinh(b*x+a)^2)*x^3,x, algorithm="fricas")

[Out]

integral(x^3*e^(arcsinh(b*x + a)^2), x)

Sympy [F]

\[ \int e^{\text {arcsinh}(a+b x)^2} x^3 \, dx=\int x^{3} e^{\operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]

[In]

integrate(exp(asinh(b*x+a)**2)*x**3,x)

[Out]

Integral(x**3*exp(asinh(a + b*x)**2), x)

Maxima [F]

\[ \int e^{\text {arcsinh}(a+b x)^2} x^3 \, dx=\int { x^{3} e^{\left (\operatorname {arsinh}\left (b x + a\right )^{2}\right )} \,d x } \]

[In]

integrate(exp(arcsinh(b*x+a)^2)*x^3,x, algorithm="maxima")

[Out]

integrate(x^3*e^(arcsinh(b*x + a)^2), x)

Giac [F]

\[ \int e^{\text {arcsinh}(a+b x)^2} x^3 \, dx=\int { x^{3} e^{\left (\operatorname {arsinh}\left (b x + a\right )^{2}\right )} \,d x } \]

[In]

integrate(exp(arcsinh(b*x+a)^2)*x^3,x, algorithm="giac")

[Out]

integrate(x^3*e^(arcsinh(b*x + a)^2), x)

Mupad [F(-1)]

Timed out. \[ \int e^{\text {arcsinh}(a+b x)^2} x^3 \, dx=\int x^3\,{\mathrm {e}}^{{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]

[In]

int(x^3*exp(asinh(a + b*x)^2),x)

[Out]

int(x^3*exp(asinh(a + b*x)^2), x)