\(\int e^{\arcsin (a x)} \sqrt {1-a^2 x^2} \, dx\) [465]

Optimal result

Integrand size = 21, antiderivative size = 62 \[ \int e^{\arcsin (a x)} \sqrt {1-a^2 x^2} \, dx=\frac {2 e^{\arcsin (a x)}}{5 a}+\frac {2}{5} e^{\arcsin (a x)} x \sqrt {1-a^2 x^2}+\frac {e^{\arcsin (a x)} \left (1-a^2 x^2\right )}{5 a} \]

[Out]

2/5*exp(arcsin(a*x))/a+1/5*exp(arcsin(a*x))*(-a^2*x^2+1)/a+2/5*exp(arcsin(a*x))*x*(-a^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4920, 6820, 6852, 4520, 2225} \[ \int e^{\arcsin (a x)} \sqrt {1-a^2 x^2} \, dx=\frac {2}{5} x \sqrt {1-a^2 x^2} e^{\arcsin (a x)}+\frac {\left (1-a^2 x^2\right ) e^{\arcsin (a x)}}{5 a}+\frac {2 e^{\arcsin (a x)}}{5 a} \]

[In]

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

[Out]

(2*E^ArcSin[a*x])/(5*a) + (2*E^ArcSin[a*x]*x*Sqrt[1 - a^2*x^2])/5 + (E^ArcSin[a*x]*(1 - a^2*x^2))/(5*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 4520

Int[Cos[(d_.) + (e_.)*(x_)]^(m_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x
))*(Cos[d + e*x]^m/(e^2*m^2 + b^2*c^2*Log[F]^2)), x] + (Dist[(m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^2*Log[F]^2), Int
[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x], x] + Simp[e*m*F^(c*(a + b*x))*Sin[d + e*x]*(Cos[d + e*x]^(m - 1)/(e
^2*m^2 + b^2*c^2*Log[F]^2)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[
m, 1]

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*} \text {integral}& = \frac {\text {Subst}\left (\int e^x \cos (x) \sqrt {1-\sin ^2(x)} \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {\text {Subst}\left (\int e^x \cos (x) \sqrt {\cos ^2(x)} \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {\text {Subst}\left (\int e^x \cos ^2(x) \, dx,x,\arcsin (a x)\right )}{a} \\ & = \frac {2}{5} e^{\arcsin (a x)} x \sqrt {1-a^2 x^2}+\frac {e^{\arcsin (a x)} \left (1-a^2 x^2\right )}{5 a}+\frac {2 \text {Subst}\left (\int e^x \, dx,x,\arcsin (a x)\right )}{5 a} \\ & = \frac {2 e^{\arcsin (a x)}}{5 a}+\frac {2}{5} e^{\arcsin (a x)} x \sqrt {1-a^2 x^2}+\frac {e^{\arcsin (a x)} \left (1-a^2 x^2\right )}{5 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.50 \[ \int e^{\arcsin (a x)} \sqrt {1-a^2 x^2} \, dx=\frac {e^{\arcsin (a x)} (5+\cos (2 \arcsin (a x))+2 \sin (2 \arcsin (a x)))}{10 a} \]

[In]

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

[Out]

(E^ArcSin[a*x]*(5 + Cos[2*ArcSin[a*x]] + 2*Sin[2*ArcSin[a*x]]))/(10*a)

Maple [F]

\[\int {\mathrm e}^{\arcsin \left (a x \right )} \sqrt {-a^{2} x^{2}+1}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.56 \[ \int e^{\arcsin (a x)} \sqrt {1-a^2 x^2} \, dx=-\frac {{\left (a^{2} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a x - 3\right )} e^{\left (\arcsin \left (a x\right )\right )}}{5 \, a} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.79 \[ \int e^{\arcsin (a x)} \sqrt {1-a^2 x^2} \, dx=\begin {cases} - \frac {a x^{2} e^{\operatorname {asin}{\left (a x \right )}}}{5} + \frac {2 x \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{5} + \frac {3 e^{\operatorname {asin}{\left (a x \right )}}}{5 a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-a*x**2*exp(asin(a*x))/5 + 2*x*sqrt(-a**2*x**2 + 1)*exp(asin(a*x))/5 + 3*exp(asin(a*x))/(5*a), Ne(a
, 0)), (x, True))

Maxima [F]

\[ \int e^{\arcsin (a x)} \sqrt {1-a^2 x^2} \, dx=\int { \sqrt {-a^{2} x^{2} + 1} e^{\left (\arcsin \left (a x\right )\right )} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int e^{\arcsin (a x)} \sqrt {1-a^2 x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int e^{\arcsin (a x)} \sqrt {1-a^2 x^2} \, dx=\int {\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}\,\sqrt {1-a^2\,x^2} \,d x \]

[In]

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

[Out]

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