3.5.66 \(\int \frac {e^{\text {ArcSin}(a x)}}{\sqrt {1-a^2 x^2}} \, dx\) [466]

Optimal. Leaf size=10 \[ \frac {e^{\text {ArcSin}(a x)}}{a} \]

[Out]

exp(arcsin(a*x))/a

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Rubi [A]
time = 0.15, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4920, 6820, 6852, 2225} \begin {gather*} \frac {e^{\text {ArcSin}(a x)}}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcSin[a*x]/Sqrt[1 - a^2*x^2],x]

[Out]

E^ArcSin[a*x]/a

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 4920

Int[(u_.)*(f_)^(ArcSin[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Dist[1/b, Subst[Int[(u /. x -> -a/b + Si
n[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 6820

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

Rule 6852

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m)^FracPart[p]/v^(m*FracPart[p])), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} \frac {e^{\text {ArcSin}(a x)}}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^ArcSin[a*x]/Sqrt[1 - a^2*x^2],x]

[Out]

E^ArcSin[a*x]/a

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Maple [A]
time = 0.13, size = 10, normalized size = 1.00

method	result	size
derivativedivides	\(\frac {{\mathrm e}^{\arcsin \left (a x \right )}}{a}\)	\(10\)
default	\(\frac {{\mathrm e}^{\arcsin \left (a x \right )}}{a}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arcsin(a*x))/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

exp(arcsin(a*x))/a

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Maxima [A]
time = 0.47, size = 9, normalized size = 0.90 \begin {gather*} \frac {e^{\left (\arcsin \left (a x\right )\right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

e^(arcsin(a*x))/a

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Fricas [A]
time = 1.17, size = 9, normalized size = 0.90 \begin {gather*} \frac {e^{\left (\arcsin \left (a x\right )\right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

e^(arcsin(a*x))/a

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Sympy [A]
time = 0.21, size = 8, normalized size = 0.80 \begin {gather*} \begin {cases} \frac {e^{\operatorname {asin}{\left (a x \right )}}}{a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(asin(a*x))/(-a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((exp(asin(a*x))/a, Ne(a, 0)), (x, True))

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Giac [A]
time = 0.40, size = 9, normalized size = 0.90 \begin {gather*} \frac {e^{\left (\arcsin \left (a x\right )\right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

e^(arcsin(a*x))/a

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Mupad [B]
time = 0.25, size = 9, normalized size = 0.90 \begin {gather*} \frac {{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(asin(a*x))/(1 - a^2*x^2)^(1/2),x)

[Out]

exp(asin(a*x))/a

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