3.439 \(\int e^{\sin ^{-1}(a x)} x^3 \, dx\)

Optimal. Leaf size=81 \[ \frac{e^{\sin ^{-1}(a x)} \sin \left (2 \sin ^{-1}(a x)\right )}{20 a^4}-\frac{e^{\sin ^{-1}(a x)} \sin \left (4 \sin ^{-1}(a x)\right )}{136 a^4}-\frac{e^{\sin ^{-1}(a x)} \cos \left (2 \sin ^{-1}(a x)\right )}{10 a^4}+\frac{e^{\sin ^{-1}(a x)} \cos \left (4 \sin ^{-1}(a x)\right )}{34 a^4} \]

[Out]

-(E^ArcSin[a*x]*Cos[2*ArcSin[a*x]])/(10*a^4) + (E^ArcSin[a*x]*Cos[4*ArcSin[a*x]])/(34*a^4) + (E^ArcSin[a*x]*Si
n[2*ArcSin[a*x]])/(20*a^4) - (E^ArcSin[a*x]*Sin[4*ArcSin[a*x]])/(136*a^4)

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Rubi [A]  time = 0.0656336, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4836, 12, 4469, 4432} \[ \frac{e^{\sin ^{-1}(a x)} \sin \left (2 \sin ^{-1}(a x)\right )}{20 a^4}-\frac{e^{\sin ^{-1}(a x)} \sin \left (4 \sin ^{-1}(a x)\right )}{136 a^4}-\frac{e^{\sin ^{-1}(a x)} \cos \left (2 \sin ^{-1}(a x)\right )}{10 a^4}+\frac{e^{\sin ^{-1}(a x)} \cos \left (4 \sin ^{-1}(a x)\right )}{34 a^4} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSin[a*x]*x^3,x]

[Out]

-(E^ArcSin[a*x]*Cos[2*ArcSin[a*x]])/(10*a^4) + (E^ArcSin[a*x]*Cos[4*ArcSin[a*x]])/(34*a^4) + (E^ArcSin[a*x]*Si
n[2*ArcSin[a*x]])/(20*a^4) - (E^ArcSin[a*x]*Sin[4*ArcSin[a*x]])/(136*a^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 12

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

Rule 4469

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :
> Int[ExpandTrigReduce[F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g
}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 4432

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*S
in[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] - Simp[(e*F^(c*(a + b*x))*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rubi steps

\begin{align*} \int e^{\sin ^{-1}(a x)} x^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^x \cos (x) \sin ^3(x)}{a^3} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \sin ^3(x) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4} e^x \sin (2 x)-\frac{1}{8} e^x \sin (4 x)\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int e^x \sin (4 x) \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}+\frac{\operatorname{Subst}\left (\int e^x \sin (2 x) \, dx,x,\sin ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{e^{\sin ^{-1}(a x)} \cos \left (2 \sin ^{-1}(a x)\right )}{10 a^4}+\frac{e^{\sin ^{-1}(a x)} \cos \left (4 \sin ^{-1}(a x)\right )}{34 a^4}+\frac{e^{\sin ^{-1}(a x)} \sin \left (2 \sin ^{-1}(a x)\right )}{20 a^4}-\frac{e^{\sin ^{-1}(a x)} \sin \left (4 \sin ^{-1}(a x)\right )}{136 a^4}\\ \end{align*}

Mathematica [A]  time = 0.121601, size = 50, normalized size = 0.62 \[ \frac{e^{\sin ^{-1}(a x)} \left (34 \sin \left (2 \sin ^{-1}(a x)\right )-5 \sin \left (4 \sin ^{-1}(a x)\right )-68 \cos \left (2 \sin ^{-1}(a x)\right )+20 \cos \left (4 \sin ^{-1}(a x)\right )\right )}{680 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcSin[a*x]*x^3,x]

[Out]

(E^ArcSin[a*x]*(-68*Cos[2*ArcSin[a*x]] + 20*Cos[4*ArcSin[a*x]] + 34*Sin[2*ArcSin[a*x]] - 5*Sin[4*ArcSin[a*x]])
)/(680*a^4)

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{\arcsin \left ( ax \right ) }}{x}^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arcsin(a*x))*x^3,x)

[Out]

int(exp(arcsin(a*x))*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 (a x\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*e^(arcsin(a*x)), x)

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Fricas [A]  time = 2.10234, size = 130, normalized size = 1.6 \begin{align*} \frac{{\left (20 \, a^{4} x^{4} - 3 \, a^{2} x^{2} +{\left (5 \, a^{3} x^{3} + 6 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} - 6\right )} e^{\left (\arcsin \left (a x\right )\right )}}{85 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/85*(20*a^4*x^4 - 3*a^2*x^2 + (5*a^3*x^3 + 6*a*x)*sqrt(-a^2*x^2 + 1) - 6)*e^(arcsin(a*x))/a^4

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Sympy [A]  time = 5.03786, size = 100, normalized size = 1.23 \begin{align*} \begin{cases} \frac{4 x^{4} e^{\operatorname{asin}{\left (a x \right )}}}{17} + \frac{x^{3} \sqrt{- a^{2} x^{2} + 1} e^{\operatorname{asin}{\left (a x \right )}}}{17 a} - \frac{3 x^{2} e^{\operatorname{asin}{\left (a x \right )}}}{85 a^{2}} + \frac{6 x \sqrt{- a^{2} x^{2} + 1} e^{\operatorname{asin}{\left (a x \right )}}}{85 a^{3}} - \frac{6 e^{\operatorname{asin}{\left (a x \right )}}}{85 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(asin(a*x))*x**3,x)

[Out]

Piecewise((4*x**4*exp(asin(a*x))/17 + x**3*sqrt(-a**2*x**2 + 1)*exp(asin(a*x))/(17*a) - 3*x**2*exp(asin(a*x))/
(85*a**2) + 6*x*sqrt(-a**2*x**2 + 1)*exp(asin(a*x))/(85*a**3) - 6*exp(asin(a*x))/(85*a**4), Ne(a, 0)), (x**4/4
, True))

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Giac [A]  time = 1.19654, size = 131, normalized size = 1.62 \begin{align*} -\frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x e^{\left (\arcsin \left (a x\right )\right )}}{17 \, a^{3}} + \frac{11 \, \sqrt{-a^{2} x^{2} + 1} x e^{\left (\arcsin \left (a x\right )\right )}}{85 \, a^{3}} + \frac{4 \,{\left (a^{2} x^{2} - 1\right )}^{2} e^{\left (\arcsin \left (a x\right )\right )}}{17 \, a^{4}} + \frac{37 \,{\left (a^{2} x^{2} - 1\right )} e^{\left (\arcsin \left (a x\right )\right )}}{85 \, a^{4}} + \frac{11 \, e^{\left (\arcsin \left (a x\right )\right )}}{85 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/17*(-a^2*x^2 + 1)^(3/2)*x*e^(arcsin(a*x))/a^3 + 11/85*sqrt(-a^2*x^2 + 1)*x*e^(arcsin(a*x))/a^3 + 4/17*(a^2*
x^2 - 1)^2*e^(arcsin(a*x))/a^4 + 37/85*(a^2*x^2 - 1)*e^(arcsin(a*x))/a^4 + 11/85*e^(arcsin(a*x))/a^4