3.111 \(\int e^{\cos ^{-1}(a x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0140558, 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 = {4837, 4432} \[ \frac{1}{2} x e^{\cos ^{-1}(a x)}-\frac{\sqrt{1-a^2 x^2} e^{\cos ^{-1}(a x)}}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4837

Int[(u_.)*(f_)^(ArcCos[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> -Dist[b^(-1), Subst[Int[(u /. x -> -(a/b
) + Cos[x]/b)*f^(c*x^n)*Sin[x], x], x, ArcCos[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && 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^{\cos ^{-1}(a x)} \, dx &=-\frac{\operatorname{Subst}\left (\int e^x \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=\frac{1}{2} e^{\cos ^{-1}(a x)} x-\frac{e^{\cos ^{-1}(a x)} \sqrt{1-a^2 x^2}}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0326375, size = 32, normalized size = 0.82 \[ -\frac{\left (\sqrt{1-a^2 x^2}-a x\right ) e^{\cos ^{-1}(a x)}}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.004, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{\arccos \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccos(a*x)),x, algorithm="maxima")

[Out]

integrate(e^(arccos(a*x)), x)

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Fricas [A]  time = 2.59955, size = 68, normalized size = 1.74 \begin{align*} \frac{{\left (a x - \sqrt{-a^{2} x^{2} + 1}\right )} e^{\left (\arccos \left (a x\right )\right )}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.201251, size = 37, normalized size = 0.95 \begin{align*} \begin{cases} \frac{x e^{\operatorname{acos}{\left (a x \right )}}}{2} - \frac{\sqrt{- a^{2} x^{2} + 1} e^{\operatorname{acos}{\left (a x \right )}}}{2 a} & \text{for}\: a \neq 0 \\x e^{\frac{\pi }{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27696, size = 42, normalized size = 1.08 \begin{align*} \frac{1}{2} \, x e^{\left (\arccos \left (a x\right )\right )} - \frac{\sqrt{-a^{2} x^{2} + 1} e^{\left (\arccos \left (a x\right )\right )}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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