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

Optimal. Leaf size=39 \[ \frac{\sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}}{2 a}+\frac{1}{2} x e^{\sin ^{-1}(a x)} \]

[Out]

(E^ArcSin[a*x]*x)/2 + (E^ArcSin[a*x]*Sqrt[1 - a^2*x^2])/(2*a)

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Rubi [A]  time = 0.0143872, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4836, 4433} \[ \frac{\sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}}{2 a}+\frac{1}{2} x e^{\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^ArcSin[a*x]*x)/2 + (E^ArcSin[a*x]*Sqrt[1 - a^2*x^2])/(2*a)

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 4433

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*C
os[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sin[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)} \, dx &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{1}{2} e^{\sin ^{-1}(a x)} x+\frac{e^{\sin ^{-1}(a x)} \sqrt{1-a^2 x^2}}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0225738, size = 31, normalized size = 0.79 \[ \frac{\left (\sqrt{1-a^2 x^2}+a x\right ) e^{\sin ^{-1}(a x)}}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^ArcSin[a*x]*(a*x + Sqrt[1 - a^2*x^2]))/(2*a)

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Maple [F]  time = 0.007, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{\arcsin \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int 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, algorithm="maxima")

[Out]

integrate(e^(arcsin(a*x)), x)

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Fricas [A]  time = 1.99291, size = 68, normalized size = 1.74 \begin{align*} \frac{{\left (a x + \sqrt{-a^{2} x^{2} + 1}\right )} e^{\left (\arcsin \left (a x\right )\right )}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*x + sqrt(-a^2*x^2 + 1))*e^(arcsin(a*x))/a

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Sympy [A]  time = 0.21421, size = 32, normalized size = 0.82 \begin{align*} \begin{cases} \frac{x e^{\operatorname{asin}{\left (a x \right )}}}{2} + \frac{\sqrt{- a^{2} x^{2} + 1} e^{\operatorname{asin}{\left (a x \right )}}}{2 a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*exp(asin(a*x))/2 + sqrt(-a**2*x**2 + 1)*exp(asin(a*x))/(2*a), Ne(a, 0)), (x, True))

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Giac [A]  time = 1.1984, size = 42, normalized size = 1.08 \begin{align*} \frac{1}{2} \, x e^{\left (\arcsin \left (a x\right )\right )} + \frac{\sqrt{-a^{2} x^{2} + 1} e^{\left (\arcsin \left (a x\right )\right )}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*e^(arcsin(a*x)) + 1/2*sqrt(-a^2*x^2 + 1)*e^(arcsin(a*x))/a