\(\int e^{\text {arcsinh}(a+b x)^2} x^2 \, dx\) [11]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.55 \[ \int e^{\text {arcsinh}(a+b x)^2} x^2 \, dx=-\frac {\sqrt {\pi } \left (\left (-1+4 a^2\right ) e^2 \text {erfi}\left (\frac {1}{2}-\text {arcsinh}(a+b x)\right )+4 a e^{5/4} \text {erfi}(1-\text {arcsinh}(a+b x))+\text {erfi}\left (\frac {3}{2}-\text {arcsinh}(a+b x)\right )+e^2 \text {erfi}\left (\frac {1}{2}+\text {arcsinh}(a+b x)\right )-4 a^2 e^2 \text {erfi}\left (\frac {1}{2}+\text {arcsinh}(a+b x)\right )+4 a e^{5/4} \text {erfi}(1+\text {arcsinh}(a+b x))-\text {erfi}\left (\frac {3}{2}+\text {arcsinh}(a+b x)\right )\right )}{16 b^3 e^{9/4}} \] Input:

Integrate[E^ArcSinh[a + b*x]^2*x^2,x]
 

Output:

-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 
))
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6288, 7292, 27, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6288

\(\displaystyle \frac {\int e^{\text {arcsinh}(a+b x)^2} \left (\frac {a}{b}-\frac {a+b x}{b}\right )^2 \sqrt {(a+b x)^2+1}d\text {arcsinh}(a+b x)}{b}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {\int e^{\text {arcsinh}(a+b x)^2} x^2 \sqrt {(a+b x)^2+1}d\text {arcsinh}(a+b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int b^2 e^{\text {arcsinh}(a+b x)^2} x^2 \sqrt {(a+b x)^2+1}d\text {arcsinh}(a+b x)}{b^3}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {\int \left (e^{\text {arcsinh}(a+b x)^2} \sqrt {(a+b x)^2+1} a^2-2 e^{\text {arcsinh}(a+b x)^2} (a+b x) \sqrt {(a+b x)^2+1} a+e^{\text {arcsinh}(a+b x)^2} (a+b x)^2 \sqrt {(a+b x)^2+1}\right )d\text {arcsinh}(a+b x)}{b^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\sqrt {\pi } a^2 \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)-1)\right )}{4 \sqrt [4]{e}}+\frac {\sqrt {\pi } a^2 \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)+1)\right )}{4 \sqrt [4]{e}}-\frac {\sqrt {\pi } a \text {erfi}(1-\text {arcsinh}(a+b x))}{4 e}-\frac {\sqrt {\pi } a \text {erfi}(\text {arcsinh}(a+b x)+1)}{4 e}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)-3)\right )}{16 e^{9/4}}-\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)-1)\right )}{16 \sqrt [4]{e}}-\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)+1)\right )}{16 \sqrt [4]{e}}+\frac {\sqrt {\pi } \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)+3)\right )}{16 e^{9/4}}}{b^3}\)

Input:

Int[E^ArcSinh[a + b*x]^2*x^2,x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6288

Int[(f_)^(ArcSinh[(a_.) + (b_.)*(x_)]^(n_.)*(c_.))*(x_)^(m_.), x_Symbol] :> 
 Simp[1/b   Subst[Int[(-a/b + Sinh[x]/b)^m*f^(c*x^n)*Cosh[x], x], x, ArcSin 
h[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
 

rule 7292

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

rule 7293

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(e**(asinh(a + b*x)**2)*x**2,x)