\(\int e^{\arcsin (a+b x)^2} x \, dx\) [21]

Optimal result

Integrand size = 12, antiderivative size = 123 \[ \int e^{\arcsin (a+b x)^2} x \, dx=\frac {e \sqrt {\pi } \text {erf}(1-i \arcsin (a+b x))}{8 b^2}+\frac {e \sqrt {\pi } \text {erf}(1+i \arcsin (a+b x))}{8 b^2}-\frac {a \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a+b x))\right )}{4 b^2}-\frac {a \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )}{4 b^2} \] Output:

-1/8*exp(1)*Pi^(1/2)*erf(-1+I*arcsin(b*x+a))/b^2+1/8*exp(1)*Pi^(1/2)*erf(1 
+I*arcsin(b*x+a))/b^2-1/4*a*exp(1/4)*Pi^(1/2)*erfi(-1/2*I+arcsin(b*x+a))/b 
^2-1/4*a*exp(1/4)*Pi^(1/2)*erfi(1/2*I+arcsin(b*x+a))/b^2
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int e^{\arcsin (a+b x)^2} x \, dx=\frac {\sqrt {\pi } \left (e \text {erf}(1-i \arcsin (a+b x))+e \text {erf}(1+i \arcsin (a+b x))-2 a \sqrt [4]{e} \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a+b x))\right )-2 a \sqrt [4]{e} \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )\right )}{8 b^2} \] Input:

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

Output:

(Sqrt[Pi]*(E*Erf[1 - I*ArcSin[a + b*x]] + E*Erf[1 + I*ArcSin[a + b*x]] - 2 
*a*E^(1/4)*Erfi[(-I + 2*ArcSin[a + b*x])/2] - 2*a*E^(1/4)*Erfi[(I + 2*ArcS 
in[a + b*x])/2]))/(8*b^2)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 116, 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 = {5335, 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.

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

\(\Big \downarrow \) 5335

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((-1/8*(E*Sqrt[Pi]*Erf[1 - I*ArcSin[a + b*x]]) - (E*Sqrt[Pi]*Erf[1 + I*Ar 
cSin[a + b*x]])/8 + (a*E^(1/4)*Sqrt[Pi]*Erfi[(-I + 2*ArcSin[a + b*x])/2])/ 
4 + (a*E^(1/4)*Sqrt[Pi]*Erfi[(I + 2*ArcSin[a + b*x])/2])/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 5335

Int[(u_.)*(f_)^(ArcSin[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Simp[ 
1/b   Subst[Int[(u /. x -> -a/b + Sin[x]/b)*f^(c*x^n)*Cos[x], x], x, ArcSin 
[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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