3.457 \(\int e^{\sin ^{-1}(a+b x)^2} x^3 \, dx\)

Optimal. Leaf size=381 \[ \frac{3 e \sqrt{\pi } a^2 \text{Erf}\left (1-i \sin ^{-1}(a+b x)\right )}{8 b^4}+\frac{3 e \sqrt{\pi } a^2 \text{Erf}\left (1+i \sin ^{-1}(a+b x)\right )}{8 b^4}-\frac{\sqrt [4]{e} \sqrt{\pi } a^3 \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{4 b^4}-\frac{\sqrt [4]{e} \sqrt{\pi } a^3 \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{4 b^4}+\frac{e \sqrt{\pi } \text{Erf}\left (1-i \sin ^{-1}(a+b x)\right )}{16 b^4}-\frac{e^4 \sqrt{\pi } \text{Erf}\left (2-i \sin ^{-1}(a+b x)\right )}{32 b^4}+\frac{e \sqrt{\pi } \text{Erf}\left (1+i \sin ^{-1}(a+b x)\right )}{16 b^4}-\frac{e^4 \sqrt{\pi } \text{Erf}\left (2+i \sin ^{-1}(a+b x)\right )}{32 b^4}-\frac{3 \sqrt [4]{e} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{16 b^4}-\frac{3 \sqrt [4]{e} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{16 b^4}+\frac{3 e^{9/4} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-3 i\right )\right )}{16 b^4}+\frac{3 e^{9/4} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+3 i\right )\right )}{16 b^4} \]

[Out]

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

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Rubi [A]  time = 0.673329, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4836, 6741, 12, 6742, 4473, 2234, 2204, 4474} \[ \frac{3 e \sqrt{\pi } a^2 \text{Erf}\left (1-i \sin ^{-1}(a+b x)\right )}{8 b^4}+\frac{3 e \sqrt{\pi } a^2 \text{Erf}\left (1+i \sin ^{-1}(a+b x)\right )}{8 b^4}-\frac{\sqrt [4]{e} \sqrt{\pi } a^3 \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{4 b^4}-\frac{\sqrt [4]{e} \sqrt{\pi } a^3 \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{4 b^4}+\frac{e \sqrt{\pi } \text{Erf}\left (1-i \sin ^{-1}(a+b x)\right )}{16 b^4}-\frac{e^4 \sqrt{\pi } \text{Erf}\left (2-i \sin ^{-1}(a+b x)\right )}{32 b^4}+\frac{e \sqrt{\pi } \text{Erf}\left (1+i \sin ^{-1}(a+b x)\right )}{16 b^4}-\frac{e^4 \sqrt{\pi } \text{Erf}\left (2+i \sin ^{-1}(a+b x)\right )}{32 b^4}-\frac{3 \sqrt [4]{e} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-i\right )\right )}{16 b^4}-\frac{3 \sqrt [4]{e} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )}{16 b^4}+\frac{3 e^{9/4} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)-3 i\right )\right )}{16 b^4}+\frac{3 e^{9/4} \sqrt{\pi } a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+3 i\right )\right )}{16 b^4} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSin[a + b*x]^2*x^3,x]

[Out]

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

Rule 4836

Int[(u_.)*(f_)^(ArcSin[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Dist[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 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 4473

Int[Cos[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cos[v]^n, x], x] /; FreeQ[F, x] && (LinearQ
[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 4474

Int[Cos[v_]^(n_.)*(F_)^(u_)*Sin[v_]^(m_.), x_Symbol] :> Int[ExpandTrigToExp[F^u, Sin[v]^m*Cos[v]^n, x], x] /;
FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[m, 0] && IGtQ[n,
0]

Rubi steps

\begin{align*} \int e^{\sin ^{-1}(a+b x)^2} x^3 \, dx &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cos (x) \left (-\frac{a}{b}+\frac{\sin (x)}{b}\right )^3 \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{e^{x^2} \cos (x) (-a+\sin (x))^3}{b^3} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cos (x) (-a+\sin (x))^3 \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a^3 e^{x^2} \cos (x)+3 a^2 e^{x^2} \cos (x) \sin (x)-3 a e^{x^2} \cos (x) \sin ^2(x)+e^{x^2} \cos (x) \sin ^3(x)\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int e^{x^2} \cos (x) \sin ^3(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}-\frac{(3 a) \operatorname{Subst}\left (\int e^{x^2} \cos (x) \sin ^2(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int e^{x^2} \cos (x) \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}-\frac{a^3 \operatorname{Subst}\left (\int e^{x^2} \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{8} i e^{-2 i x+x^2}-\frac{1}{8} i e^{2 i x+x^2}-\frac{1}{16} i e^{-4 i x+x^2}+\frac{1}{16} i e^{4 i x+x^2}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}-\frac{(3 a) \operatorname{Subst}\left (\int \left (\frac{1}{8} e^{-i x+x^2}+\frac{1}{8} e^{i x+x^2}-\frac{1}{8} e^{-3 i x+x^2}-\frac{1}{8} e^{3 i x+x^2}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{4} i e^{-2 i x+x^2}-\frac{1}{4} i e^{2 i x+x^2}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}-\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{2} e^{-i x+x^2}+\frac{1}{2} e^{i x+x^2}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^4}\\ &=-\frac{i \operatorname{Subst}\left (\int e^{-4 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{16 b^4}+\frac{i \operatorname{Subst}\left (\int e^{4 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{16 b^4}+\frac{i \operatorname{Subst}\left (\int e^{-2 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}-\frac{i \operatorname{Subst}\left (\int e^{2 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}-\frac{(3 a) \operatorname{Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}-\frac{(3 a) \operatorname{Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}+\frac{(3 a) \operatorname{Subst}\left (\int e^{-3 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}+\frac{(3 a) \operatorname{Subst}\left (\int e^{3 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}+\frac{\left (3 i a^2\right ) \operatorname{Subst}\left (\int e^{-2 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^4}-\frac{\left (3 i a^2\right ) \operatorname{Subst}\left (\int e^{2 i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^4}-\frac{a^3 \operatorname{Subst}\left (\int e^{-i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^4}-\frac{a^3 \operatorname{Subst}\left (\int e^{i x+x^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^4}\\ &=-\frac{\left (3 a \sqrt [4]{e}\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}-\frac{\left (3 a \sqrt [4]{e}\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}-\frac{\left (a^3 \sqrt [4]{e}\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (-i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^4}-\frac{\left (a^3 \sqrt [4]{e}\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^4}+\frac{(i e) \operatorname{Subst}\left (\int e^{\frac{1}{4} (-2 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}-\frac{(i e) \operatorname{Subst}\left (\int e^{\frac{1}{4} (2 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}+\frac{\left (3 i a^2 e\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (-2 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^4}-\frac{\left (3 i a^2 e\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (2 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^4}+\frac{\left (3 a e^{9/4}\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (-3 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}+\frac{\left (3 a e^{9/4}\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (3 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^4}-\frac{\left (i e^4\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (-4 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{16 b^4}+\frac{\left (i e^4\right ) \operatorname{Subst}\left (\int e^{\frac{1}{4} (4 i+2 x)^2} \, dx,x,\sin ^{-1}(a+b x)\right )}{16 b^4}\\ &=\frac{e \sqrt{\pi } \text{erf}\left (1-i \sin ^{-1}(a+b x)\right )}{16 b^4}+\frac{3 a^2 e \sqrt{\pi } \text{erf}\left (1-i \sin ^{-1}(a+b x)\right )}{8 b^4}-\frac{e^4 \sqrt{\pi } \text{erf}\left (2-i \sin ^{-1}(a+b x)\right )}{32 b^4}+\frac{e \sqrt{\pi } \text{erf}\left (1+i \sin ^{-1}(a+b x)\right )}{16 b^4}+\frac{3 a^2 e \sqrt{\pi } \text{erf}\left (1+i \sin ^{-1}(a+b x)\right )}{8 b^4}-\frac{e^4 \sqrt{\pi } \text{erf}\left (2+i \sin ^{-1}(a+b x)\right )}{32 b^4}-\frac{3 a \sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-i+2 \sin ^{-1}(a+b x)\right )\right )}{16 b^4}-\frac{a^3 \sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-i+2 \sin ^{-1}(a+b x)\right )\right )}{4 b^4}-\frac{3 a \sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (i+2 \sin ^{-1}(a+b x)\right )\right )}{16 b^4}-\frac{a^3 \sqrt [4]{e} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (i+2 \sin ^{-1}(a+b x)\right )\right )}{4 b^4}+\frac{3 a e^{9/4} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (-3 i+2 \sin ^{-1}(a+b x)\right )\right )}{16 b^4}+\frac{3 a e^{9/4} \sqrt{\pi } \text{erfi}\left (\frac{1}{2} \left (3 i+2 \sin ^{-1}(a+b x)\right )\right )}{16 b^4}\\ \end{align*}

Mathematica [A]  time = 0.35796, size = 221, normalized size = 0.58 \[ -\frac{\sqrt{\pi } \left (\sqrt [4]{e} \left (-2 i \left (4 a^2+3\right ) a \text{Erf}\left (\frac{1}{2}+i \sin ^{-1}(a+b x)\right )-2 e^{3/4} \left (6 a^2+1\right ) \text{Erf}\left (1+i \sin ^{-1}(a+b x)\right )+8 a^3 \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )+6 i e^2 a \text{Erf}\left (\frac{3}{2}+i \sin ^{-1}(a+b x)\right )+e^{15/4} \text{Erf}\left (2+i \sin ^{-1}(a+b x)\right )+6 a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+i\right )\right )-6 e^2 a \text{Erfi}\left (\frac{1}{2} \left (2 \sin ^{-1}(a+b x)+3 i\right )\right )\right )-2 \left (6 e a^2+e\right ) \text{Erf}\left (1-i \sin ^{-1}(a+b x)\right )+e^4 \text{Erf}\left (2-i \sin ^{-1}(a+b x)\right )\right )}{32 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcSin[a + b*x]^2*x^3,x]

[Out]

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

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Maple [F]  time = 0.011, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}}}{x}^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} e^{\left (\arcsin \left (b x + a\right )^{2}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} e^{\left (\arcsin \left (b x + a\right )^{2}\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} e^{\operatorname{asin}^{2}{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} e^{\left (\arcsin \left (b x + a\right )^{2}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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