\(\int e^{\text {arcsinh}(a+b x)^2} x \, dx\) [12]

Optimal result

Integrand size = 12, antiderivative size = 117 \[ \int e^{\text {arcsinh}(a+b x)^2} x \, dx=\frac {\sqrt {\pi } \text {erfi}(1-\text {arcsinh}(a+b x))}{8 b^2 e}+\frac {\sqrt {\pi } \text {erfi}(1+\text {arcsinh}(a+b x))}{8 b^2 e}-\frac {a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-1+2 \text {arcsinh}(a+b x))\right )}{4 b^2 \sqrt [4]{e}}-\frac {a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 \text {arcsinh}(a+b x))\right )}{4 b^2 \sqrt [4]{e}} \] Output:

-1/8*Pi^(1/2)*erfi(-1+arcsinh(b*x+a))/b^2*exp(-1)+1/8*Pi^(1/2)*erfi(1+arcs 
inh(b*x+a))/b^2*exp(-1)-1/4*a*Pi^(1/2)*erfi(-1/2+arcsinh(b*x+a))/b^2*exp(- 
1/4)-1/4*a*Pi^(1/2)*erfi(1/2+arcsinh(b*x+a))/b^2*exp(-1/4)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.65 \[ \int e^{\text {arcsinh}(a+b x)^2} x \, dx=\frac {\sqrt {\pi } \left (2 a e^{3/4} \text {erfi}\left (\frac {1}{2}-\text {arcsinh}(a+b x)\right )+\text {erfi}(1-\text {arcsinh}(a+b x))-2 a e^{3/4} \text {erfi}\left (\frac {1}{2}+\text {arcsinh}(a+b x)\right )+\text {erfi}(1+\text {arcsinh}(a+b x))\right )}{8 b^2 e} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6288, 25, 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.

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

\(\Big \downarrow \) 6288

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {\sqrt {\pi } \text {erfi}(1-\text {arcsinh}(a+b x))}{8 e}-\frac {\sqrt {\pi } \text {erfi}(\text {arcsinh}(a+b x)+1)}{8 e}+\frac {\sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)-1)\right )}{4 \sqrt [4]{e}}+\frac {\sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arcsinh}(a+b x)+1)\right )}{4 \sqrt [4]{e}}}{b^2}\)

Input:

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

Output:

-((-1/8*(Sqrt[Pi]*Erfi[1 - ArcSinh[a + b*x]])/E - (Sqrt[Pi]*Erfi[1 + ArcSi 
nh[a + b*x]])/(8*E) + (a*Sqrt[Pi]*Erfi[(-1 + 2*ArcSinh[a + b*x])/2])/(4*E^ 
(1/4)) + (a*Sqrt[Pi]*Erfi[(1 + 2*ArcSinh[a + b*x])/2])/(4*E^(1/4)))/b^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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