3.5.41 \(\int e^{\text {ArcSin}(a x)} x \, dx\) [441]

Optimal. Leaf size=41 \[ -\frac {e^{\text {ArcSin}(a x)} \cos (2 \text {ArcSin}(a x))}{5 a^2}+\frac {e^{\text {ArcSin}(a x)} \sin (2 \text {ArcSin}(a x))}{10 a^2} \]

[Out]

-1/5*exp(arcsin(a*x))*cos(2*arcsin(a*x))/a^2+1/10*exp(arcsin(a*x))*sin(2*arcsin(a*x))/a^2

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4920, 12, 4557, 4517} \begin {gather*} \frac {e^{\text {ArcSin}(a x)} \sin (2 \text {ArcSin}(a x))}{10 a^2}-\frac {e^{\text {ArcSin}(a x)} \cos (2 \text {ArcSin}(a x))}{5 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(E^ArcSin[a*x]*Cos[2*ArcSin[a*x]])/a^2 + (E^ArcSin[a*x]*Sin[2*ArcSin[a*x]])/(10*a^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 4517

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]

Rule 4557

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :
> Int[ExpandTrigReduce[F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g
}, x] && IGtQ[m, 0] && IGtQ[n, 0]

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]

Rubi steps

\begin {align*} \int e^{\sin ^{-1}(a x)} x \, dx &=\frac {\text {Subst}\left (\int \frac {e^x \cos (x) \sin (x)}{a} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int e^x \cos (x) \sin (x) \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=\frac {\text {Subst}\left (\int \frac {1}{2} e^x \sin (2 x) \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=\frac {\text {Subst}\left (\int e^x \sin (2 x) \, dx,x,\sin ^{-1}(a x)\right )}{2 a^2}\\ &=-\frac {e^{\sin ^{-1}(a x)} \cos \left (2 \sin ^{-1}(a x)\right )}{5 a^2}+\frac {e^{\sin ^{-1}(a x)} \sin \left (2 \sin ^{-1}(a x)\right )}{10 a^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 30, normalized size = 0.73 \begin {gather*} \frac {e^{\text {ArcSin}(a x)} (-2 \cos (2 \text {ArcSin}(a x))+\sin (2 \text {ArcSin}(a x)))}{10 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\arcsin \left (a x \right )} x\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*e^(arcsin(a*x)), x)

________________________________________________________________________________________

Fricas [A]
time = 2.69, size = 35, normalized size = 0.85 \begin {gather*} \frac {{\left (2 \, a^{2} x^{2} + \sqrt {-a^{2} x^{2} + 1} a x - 1\right )} e^{\left (\arcsin \left (a x\right )\right )}}{5 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.14, size = 53, normalized size = 1.29 \begin {gather*} \begin {cases} \frac {2 x^{2} e^{\operatorname {asin}{\left (a x \right )}}}{5} + \frac {x \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{5 a} - \frac {e^{\operatorname {asin}{\left (a x \right )}}}{5 a^{2}} & \text {for}\: a \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.39, size = 53, normalized size = 1.29 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} x e^{\left (\arcsin \left (a x\right )\right )}}{5 \, a} + \frac {2 \, {\left (a^{2} x^{2} - 1\right )} e^{\left (\arcsin \left (a x\right )\right )}}{5 \, a^{2}} + \frac {e^{\left (\arcsin \left (a x\right )\right )}}{5 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*exp(asin(a*x)), x)

________________________________________________________________________________________