\(\int e^{\arcsin (a+b x)^2} x^2 \, dx\) [20]

Optimal result

Integrand size = 14, antiderivative size = 265 \[ \int e^{\arcsin (a+b x)^2} x^2 \, dx=-\frac {a e \sqrt {\pi } \text {erf}(1-i \arcsin (a+b x))}{4 b^3}-\frac {a e \sqrt {\pi } \text {erf}(1+i \arcsin (a+b x))}{4 b^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a+b x))\right )}{16 b^3}+\frac {a^2 \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a+b x))\right )}{4 b^3}+\frac {\sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )}{16 b^3}+\frac {a^2 \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )}{4 b^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-3 i+2 \arcsin (a+b x))\right )}{16 b^3}-\frac {e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (3 i+2 \arcsin (a+b x))\right )}{16 b^3} \] Output:

1/4*a*exp(1)*Pi^(1/2)*erf(-1+I*arcsin(b*x+a))/b^3-1/4*a*exp(1)*Pi^(1/2)*er 
f(1+I*arcsin(b*x+a))/b^3+1/16*exp(1/4)*Pi^(1/2)*erfi(-1/2*I+arcsin(b*x+a)) 
/b^3+1/4*a^2*exp(1/4)*Pi^(1/2)*erfi(-1/2*I+arcsin(b*x+a))/b^3+1/16*exp(1/4 
)*Pi^(1/2)*erfi(1/2*I+arcsin(b*x+a))/b^3+1/4*a^2*exp(1/4)*Pi^(1/2)*erfi(1/ 
2*I+arcsin(b*x+a))/b^3-1/16*exp(9/4)*Pi^(1/2)*erfi(-3/2*I+arcsin(b*x+a))/b 
^3-1/16*exp(9/4)*Pi^(1/2)*erfi(3/2*I+arcsin(b*x+a))/b^3
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.61 \[ \int e^{\arcsin (a+b x)^2} x^2 \, dx=-\frac {\sqrt {\pi } \left (4 a e \text {erf}(1-i \arcsin (a+b x))+i \sqrt [4]{e} \left (-\left (\left (1+4 a^2\right ) \text {erf}\left (\frac {1}{2}-i \arcsin (a+b x)\right )\right )+e^2 \text {erf}\left (\frac {3}{2}-i \arcsin (a+b x)\right )+\text {erf}\left (\frac {1}{2}+i \arcsin (a+b x)\right )+4 a^2 \text {erf}\left (\frac {1}{2}+i \arcsin (a+b x)\right )-4 i a e^{3/4} \text {erf}(1+i \arcsin (a+b x))-e^2 \text {erf}\left (\frac {3}{2}+i \arcsin (a+b x)\right )\right )\right )}{16 b^3} \] Input:

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

Output:

-1/16*(Sqrt[Pi]*(4*a*E*Erf[1 - I*ArcSin[a + b*x]] + I*E^(1/4)*(-((1 + 4*a^ 
2)*Erf[1/2 - I*ArcSin[a + b*x]]) + E^2*Erf[3/2 - I*ArcSin[a + b*x]] + Erf[ 
1/2 + I*ArcSin[a + b*x]] + 4*a^2*Erf[1/2 + I*ArcSin[a + b*x]] - (4*I)*a*E^ 
(3/4)*Erf[1 + I*ArcSin[a + b*x]] - E^2*Erf[3/2 + I*ArcSin[a + b*x]])))/b^3
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 245, 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 = {5335, 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.

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

\(\Big \downarrow \) 5335

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-1/4*(a*E*Sqrt[Pi]*Erf[1 - I*ArcSin[a + b*x]]) - (a*E*Sqrt[Pi]*Erf[1 + I* 
ArcSin[a + b*x]])/4 + (E^(1/4)*Sqrt[Pi]*Erfi[(-I + 2*ArcSin[a + b*x])/2])/ 
16 + (a^2*E^(1/4)*Sqrt[Pi]*Erfi[(-I + 2*ArcSin[a + b*x])/2])/4 + (E^(1/4)* 
Sqrt[Pi]*Erfi[(I + 2*ArcSin[a + b*x])/2])/16 + (a^2*E^(1/4)*Sqrt[Pi]*Erfi[ 
(I + 2*ArcSin[a + b*x])/2])/4 - (E^(9/4)*Sqrt[Pi]*Erfi[(-3*I + 2*ArcSin[a 
+ b*x])/2])/16 - (E^(9/4)*Sqrt[Pi]*Erfi[(3*I + 2*ArcSin[a + b*x])/2])/16)/ 
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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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