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

Optimal. Leaf size=51 \[ \frac{(a+b x) e^{\sin ^{-1}(a+b x)}}{2 b}+\frac{\sqrt{1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0195506, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4836, 4433} \[ \frac{(a+b x) e^{\sin ^{-1}(a+b x)}}{2 b}+\frac{\sqrt{1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSin[a + b*x],x]

[Out]

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

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+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac{e^{\sin ^{-1}(a+b x)} (a+b x)}{2 b}+\frac{e^{\sin ^{-1}(a+b x)} \sqrt{1-(a+b x)^2}}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0161327, size = 35, normalized size = 0.69 \[ \frac{\left (\sqrt{1-(a+b x)^2}+a+b x\right ) e^{\sin ^{-1}(a+b x)}}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcSin[a + b*x],x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(arcsin(b*x + a)), x)

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Fricas [A]  time = 2.10588, size = 100, normalized size = 1.96 \begin{align*} \frac{{\left (b x + a + \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.246779, size = 65, normalized size = 1.27 \begin{align*} \begin{cases} \frac{a e^{\operatorname{asin}{\left (a + b x \right )}}}{2 b} + \frac{x e^{\operatorname{asin}{\left (a + b x \right )}}}{2} + \frac{\sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} e^{\operatorname{asin}{\left (a + b x \right )}}}{2 b} & \text{for}\: b \neq 0 \\x e^{\operatorname{asin}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18878, size = 58, normalized size = 1.14 \begin{align*} \frac{{\left (b x + a\right )} e^{\left (\arcsin \left (b x + a\right )\right )}}{2 \, b} + \frac{\sqrt{-{\left (b x + a\right )}^{2} + 1} e^{\left (\arcsin \left (b x + a\right )\right )}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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