\(\int e^{\arcsin (a+b x)^2} x^3 \, dx\) [19]

Optimal result

Integrand size = 14, antiderivative size = 381 \[ \int e^{\arcsin (a+b x)^2} x^3 \, dx=\frac {e \sqrt {\pi } \text {erf}(1-i \arcsin (a+b x))}{16 b^4}+\frac {3 a^2 e \sqrt {\pi } \text {erf}(1-i \arcsin (a+b x))}{8 b^4}-\frac {e^4 \sqrt {\pi } \text {erf}(2-i \arcsin (a+b x))}{32 b^4}+\frac {e \sqrt {\pi } \text {erf}(1+i \arcsin (a+b x))}{16 b^4}+\frac {3 a^2 e \sqrt {\pi } \text {erf}(1+i \arcsin (a+b x))}{8 b^4}-\frac {e^4 \sqrt {\pi } \text {erf}(2+i \arcsin (a+b x))}{32 b^4}-\frac {3 a \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a+b x))\right )}{16 b^4}-\frac {a^3 \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i+2 \arcsin (a+b x))\right )}{4 b^4}-\frac {3 a \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )}{16 b^4}-\frac {a^3 \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )}{4 b^4}+\frac {3 a e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-3 i+2 \arcsin (a+b x))\right )}{16 b^4}+\frac {3 a e^{9/4} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (3 i+2 \arcsin (a+b x))\right )}{16 b^4} \] Output:

-1/16*exp(1)*Pi^(1/2)*erf(-1+I*arcsin(b*x+a))/b^4-3/8*a^2*exp(1)*Pi^(1/2)* 
erf(-1+I*arcsin(b*x+a))/b^4+1/32*exp(4)*Pi^(1/2)*erf(-2+I*arcsin(b*x+a))/b 
^4+1/16*exp(1)*Pi^(1/2)*erf(1+I*arcsin(b*x+a))/b^4+3/8*a^2*exp(1)*Pi^(1/2) 
*erf(1+I*arcsin(b*x+a))/b^4-1/32*exp(4)*Pi^(1/2)*erf(2+I*arcsin(b*x+a))/b^ 
4-3/16*a*exp(1/4)*Pi^(1/2)*erfi(-1/2*I+arcsin(b*x+a))/b^4-1/4*a^3*exp(1/4) 
*Pi^(1/2)*erfi(-1/2*I+arcsin(b*x+a))/b^4-3/16*a*exp(1/4)*Pi^(1/2)*erfi(1/2 
*I+arcsin(b*x+a))/b^4-1/4*a^3*exp(1/4)*Pi^(1/2)*erfi(1/2*I+arcsin(b*x+a))/ 
b^4+3/16*a*exp(9/4)*Pi^(1/2)*erfi(-3/2*I+arcsin(b*x+a))/b^4+3/16*a*exp(9/4 
)*Pi^(1/2)*erfi(3/2*I+arcsin(b*x+a))/b^4
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.58 \[ \int e^{\arcsin (a+b x)^2} x^3 \, dx=-\frac {\sqrt {\pi } \left (-2 \left (e+6 a^2 e\right ) \text {erf}(1-i \arcsin (a+b x))+e^4 \text {erf}(2-i \arcsin (a+b x))+\sqrt [4]{e} \left (-2 i a \left (3+4 a^2\right ) \text {erf}\left (\frac {1}{2}+i \arcsin (a+b x)\right )-2 \left (1+6 a^2\right ) e^{3/4} \text {erf}(1+i \arcsin (a+b x))+6 i a e^2 \text {erf}\left (\frac {3}{2}+i \arcsin (a+b x)\right )+e^{15/4} \text {erf}(2+i \arcsin (a+b x))+6 a \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )+8 a^3 \text {erfi}\left (\frac {1}{2} (i+2 \arcsin (a+b x))\right )-6 a e^2 \text {erfi}\left (\frac {1}{2} (3 i+2 \arcsin (a+b x))\right )\right )\right )}{32 b^4} \] Input:

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

Output:

-1/32*(Sqrt[Pi]*(-2*(E + 6*a^2*E)*Erf[1 - I*ArcSin[a + b*x]] + E^4*Erf[2 - 
 I*ArcSin[a + b*x]] + E^(1/4)*((-2*I)*a*(3 + 4*a^2)*Erf[1/2 + I*ArcSin[a + 
 b*x]] - 2*(1 + 6*a^2)*E^(3/4)*Erf[1 + I*ArcSin[a + b*x]] + (6*I)*a*E^2*Er 
f[3/2 + I*ArcSin[a + b*x]] + E^(15/4)*Erf[2 + I*ArcSin[a + b*x]] + 6*a*Erf 
i[(I + 2*ArcSin[a + b*x])/2] + 8*a^3*Erfi[(I + 2*ArcSin[a + b*x])/2] - 6*a 
*E^2*Erfi[(3*I + 2*ArcSin[a + b*x])/2])))/b^4
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, 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.

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

\(\Big \downarrow \) 5335

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((-1/16*(E*Sqrt[Pi]*Erf[1 - I*ArcSin[a + b*x]]) - (3*a^2*E*Sqrt[Pi]*Erf[1 
 - I*ArcSin[a + b*x]])/8 + (E^4*Sqrt[Pi]*Erf[2 - I*ArcSin[a + b*x]])/32 - 
(E*Sqrt[Pi]*Erf[1 + I*ArcSin[a + b*x]])/16 - (3*a^2*E*Sqrt[Pi]*Erf[1 + I*A 
rcSin[a + b*x]])/8 + (E^4*Sqrt[Pi]*Erf[2 + I*ArcSin[a + b*x]])/32 + (3*a*E 
^(1/4)*Sqrt[Pi]*Erfi[(-I + 2*ArcSin[a + b*x])/2])/16 + (a^3*E^(1/4)*Sqrt[P 
i]*Erfi[(-I + 2*ArcSin[a + b*x])/2])/4 + (3*a*E^(1/4)*Sqrt[Pi]*Erfi[(I + 2 
*ArcSin[a + b*x])/2])/16 + (a^3*E^(1/4)*Sqrt[Pi]*Erfi[(I + 2*ArcSin[a + b* 
x])/2])/4 - (3*a*E^(9/4)*Sqrt[Pi]*Erfi[(-3*I + 2*ArcSin[a + b*x])/2])/16 - 
 (3*a*E^(9/4)*Sqrt[Pi]*Erfi[(3*I + 2*ArcSin[a + b*x])/2])/16)/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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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