\(\int e^{\text {arccosh}(a+b x)^2} x^3 \, dx\) [10]

Optimal result

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

-1/16*Pi^(1/2)*erfi(-1+arccosh(b*x+a))/b^4*exp(-1)-3/8*a^2*Pi^(1/2)*erfi(- 
1+arccosh(b*x+a))/b^4*exp(-1)-1/32*Pi^(1/2)*erfi(-2+arccosh(b*x+a))/b^4*ex 
p(-4)+1/16*Pi^(1/2)*erfi(1+arccosh(b*x+a))/b^4*exp(-1)+3/8*a^2*Pi^(1/2)*er 
fi(1+arccosh(b*x+a))/b^4*exp(-1)+1/32*Pi^(1/2)*erfi(2+arccosh(b*x+a))/b^4* 
exp(-4)+3/16*a*Pi^(1/2)*erfi(-3/2+arccosh(b*x+a))/b^4*exp(-9/4)+3/16*a*Pi^ 
(1/2)*erfi(-1/2+arccosh(b*x+a))/b^4*exp(-1/4)+1/4*a^3*Pi^(1/2)*erfi(-1/2+a 
rccosh(b*x+a))/b^4*exp(-1/4)-3/16*a*Pi^(1/2)*erfi(1/2+arccosh(b*x+a))/b^4* 
exp(-1/4)-1/4*a^3*Pi^(1/2)*erfi(1/2+arccosh(b*x+a))/b^4*exp(-1/4)-3/16*a*P 
i^(1/2)*erfi(3/2+arccosh(b*x+a))/b^4*exp(-9/4)
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.55 \[ \int e^{\text {arccosh}(a+b x)^2} x^3 \, dx=\frac {\sqrt {\pi } \left (-2 a \left (3+4 a^2\right ) e^{15/4} \text {erfi}\left (\frac {1}{2}-\text {arccosh}(a+b x)\right )+2 \left (1+6 a^2\right ) e^3 \text {erfi}(1-\text {arccosh}(a+b x))-6 a e^{7/4} \text {erfi}\left (\frac {3}{2}-\text {arccosh}(a+b x)\right )+\text {erfi}(2-\text {arccosh}(a+b x))-6 a e^{15/4} \text {erfi}\left (\frac {1}{2}+\text {arccosh}(a+b x)\right )-8 a^3 e^{15/4} \text {erfi}\left (\frac {1}{2}+\text {arccosh}(a+b x)\right )+2 e^3 \text {erfi}(1+\text {arccosh}(a+b x))+12 a^2 e^3 \text {erfi}(1+\text {arccosh}(a+b x))-6 a e^{7/4} \text {erfi}\left (\frac {3}{2}+\text {arccosh}(a+b x)\right )+\text {erfi}(2+\text {arccosh}(a+b x))\right )}{32 b^4 e^4} \] Input:

Integrate[E^ArcCosh[a + b*x]^2*x^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 1.58 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6430, 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.

\(\displaystyle \int x^3 e^{\text {arccosh}(a+b x)^2} \, dx\)

\(\Big \downarrow \) 6430

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int -b^3 e^{\text {arccosh}(a+b x)^2} x^3 \sqrt {\frac {a+b x-1}{a+b x+1}} (a+b x+1)d\text {arccosh}(a+b x)}{b^4}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (e^{\text {arccosh}(a+b x)^2} \sqrt {\frac {a+b x-1}{a+b x+1}} (a+b x+1) a^3-3 e^{\text {arccosh}(a+b x)^2} (a+b x) \sqrt {\frac {a+b x-1}{a+b x+1}} (a+b x+1) a^2+3 e^{\text {arccosh}(a+b x)^2} (a+b x)^2 \sqrt {\frac {a+b x-1}{a+b x+1}} (a+b x+1) a-e^{\text {arccosh}(a+b x)^2} (a+b x)^3 \sqrt {\frac {a+b x-1}{a+b x+1}} (a+b x+1)\right )d\text {arccosh}(a+b x)}{b^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {\sqrt {\pi } a^3 \text {erfi}\left (\frac {1}{2} (2 \text {arccosh}(a+b x)-1)\right )}{4 \sqrt [4]{e}}+\frac {\sqrt {\pi } a^3 \text {erfi}\left (\frac {1}{2} (2 \text {arccosh}(a+b x)+1)\right )}{4 \sqrt [4]{e}}-\frac {3 \sqrt {\pi } a^2 \text {erfi}(1-\text {arccosh}(a+b x))}{8 e}-\frac {3 \sqrt {\pi } a^2 \text {erfi}(\text {arccosh}(a+b x)+1)}{8 e}-\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arccosh}(a+b x)-3)\right )}{16 e^{9/4}}-\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arccosh}(a+b x)-1)\right )}{16 \sqrt [4]{e}}+\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arccosh}(a+b x)+1)\right )}{16 \sqrt [4]{e}}+\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arccosh}(a+b x)+3)\right )}{16 e^{9/4}}-\frac {\sqrt {\pi } \text {erfi}(1-\text {arccosh}(a+b x))}{16 e}-\frac {\sqrt {\pi } \text {erfi}(2-\text {arccosh}(a+b x))}{32 e^4}-\frac {\sqrt {\pi } \text {erfi}(\text {arccosh}(a+b x)+1)}{16 e}-\frac {\sqrt {\pi } \text {erfi}(\text {arccosh}(a+b x)+2)}{32 e^4}}{b^4}\)

Input:

Int[E^ArcCosh[a + b*x]^2*x^3,x]
 

Output:

-((-1/16*(Sqrt[Pi]*Erfi[1 - ArcCosh[a + b*x]])/E - (3*a^2*Sqrt[Pi]*Erfi[1 
- ArcCosh[a + b*x]])/(8*E) - (Sqrt[Pi]*Erfi[2 - ArcCosh[a + b*x]])/(32*E^4 
) - (Sqrt[Pi]*Erfi[1 + ArcCosh[a + b*x]])/(16*E) - (3*a^2*Sqrt[Pi]*Erfi[1 
+ ArcCosh[a + b*x]])/(8*E) - (Sqrt[Pi]*Erfi[2 + ArcCosh[a + b*x]])/(32*E^4 
) - (3*a*Sqrt[Pi]*Erfi[(-3 + 2*ArcCosh[a + b*x])/2])/(16*E^(9/4)) - (3*a*S 
qrt[Pi]*Erfi[(-1 + 2*ArcCosh[a + b*x])/2])/(16*E^(1/4)) - (a^3*Sqrt[Pi]*Er 
fi[(-1 + 2*ArcCosh[a + b*x])/2])/(4*E^(1/4)) + (3*a*Sqrt[Pi]*Erfi[(1 + 2*A 
rcCosh[a + b*x])/2])/(16*E^(1/4)) + (a^3*Sqrt[Pi]*Erfi[(1 + 2*ArcCosh[a + 
b*x])/2])/(4*E^(1/4)) + (3*a*Sqrt[Pi]*Erfi[(3 + 2*ArcCosh[a + b*x])/2])/(1 
6*E^(9/4)))/b^4)
 

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 6430

Int[(f_)^(ArcCosh[(a_.) + (b_.)*(x_)]^(n_.)*(c_.))*(x_)^(m_.), x_Symbol] :> 
 Simp[1/b   Subst[Int[(-a/b + Cosh[x]/b)^m*f^(c*x^n)*Sinh[x], x], x, ArcCos 
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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int e^{\text {arccosh}(a+b x)^2} x^3 \, dx=\int x^{3} e^{\operatorname {acosh}^{2}{\left (a + b x \right )}}\, dx \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{\text {arccosh}(a+b x)^2} x^3 \, dx=\int e^{\mathit {acosh} \left (b x +a \right )^{2}} x^{3}d x \] Input:

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

Output:

int(e**(acosh(a + b*x)**2)*x**3,x)