3.464 \(\int e^{\sin ^{-1}(a x)} (1-a^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=112 \[ \frac{\left (1-a^2 x^2\right )^2 e^{\sin ^{-1}(a x)}}{17 a}+\frac{4}{17} x \left (1-a^2 x^2\right )^{3/2} e^{\sin ^{-1}(a x)}+\frac{12 \left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{85 a}+\frac{24}{85} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{24 e^{\sin ^{-1}(a x)}}{85 a} \]

[Out]

(24*E^ArcSin[a*x])/(85*a) + (24*E^ArcSin[a*x]*x*Sqrt[1 - a^2*x^2])/85 + (12*E^ArcSin[a*x]*(1 - a^2*x^2))/(85*a
) + (4*E^ArcSin[a*x]*x*(1 - a^2*x^2)^(3/2))/17 + (E^ArcSin[a*x]*(1 - a^2*x^2)^2)/(17*a)

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Rubi [A]  time = 0.303462, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4836, 6688, 6720, 4435, 2194} \[ \frac{\left (1-a^2 x^2\right )^2 e^{\sin ^{-1}(a x)}}{17 a}+\frac{4}{17} x \left (1-a^2 x^2\right )^{3/2} e^{\sin ^{-1}(a x)}+\frac{12 \left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{85 a}+\frac{24}{85} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{24 e^{\sin ^{-1}(a x)}}{85 a} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSin[a*x]*(1 - a^2*x^2)^(3/2),x]

[Out]

(24*E^ArcSin[a*x])/(85*a) + (24*E^ArcSin[a*x]*x*Sqrt[1 - a^2*x^2])/85 + (12*E^ArcSin[a*x]*(1 - a^2*x^2))/(85*a
) + (4*E^ArcSin[a*x]*x*(1 - a^2*x^2)^(3/2))/17 + (E^ArcSin[a*x]*(1 - a^2*x^2)^2)/(17*a)

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 6688

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

Rule 6720

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])

Rule 4435

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 2194

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]

Rubi steps

\begin{align*} \int e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \left (1-\sin ^2(x)\right )^{3/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \cos ^2(x)^{3/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \cos ^4(x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{4}{17} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{17 a}+\frac{12 \operatorname{Subst}\left (\int e^x \cos ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{17 a}\\ &=\frac{24}{85} e^{\sin ^{-1}(a x)} x \sqrt{1-a^2 x^2}+\frac{12 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{85 a}+\frac{4}{17} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{17 a}+\frac{24 \operatorname{Subst}\left (\int e^x \, dx,x,\sin ^{-1}(a x)\right )}{85 a}\\ &=\frac{24 e^{\sin ^{-1}(a x)}}{85 a}+\frac{24}{85} e^{\sin ^{-1}(a x)} x \sqrt{1-a^2 x^2}+\frac{12 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{85 a}+\frac{4}{17} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{17 a}\\ \end{align*}

Mathematica [A]  time = 0.150005, size = 51, normalized size = 0.46 \[ \frac{e^{\sin ^{-1}(a x)} \left (136 \sin \left (2 \sin ^{-1}(a x)\right )+20 \sin \left (4 \sin ^{-1}(a x)\right )+68 \cos \left (2 \sin ^{-1}(a x)\right )+5 \cos \left (4 \sin ^{-1}(a x)\right )+255\right )}{680 a} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcSin[a*x]*(1 - a^2*x^2)^(3/2),x]

[Out]

(E^ArcSin[a*x]*(255 + 68*Cos[2*ArcSin[a*x]] + 5*Cos[4*ArcSin[a*x]] + 136*Sin[2*ArcSin[a*x]] + 20*Sin[4*ArcSin[
a*x]]))/(680*a)

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{\arcsin \left ( ax \right ) }} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arcsin(a*x))*(-a^2*x^2+1)^(3/2),x)

[Out]

int(exp(arcsin(a*x))*(-a^2*x^2+1)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} e^{\left (\arcsin \left (a x\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)*e^(arcsin(a*x)), x)

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Fricas [A]  time = 2.01153, size = 132, normalized size = 1.18 \begin{align*} \frac{{\left (5 \, a^{4} x^{4} - 22 \, a^{2} x^{2} - 4 \,{\left (5 \, a^{3} x^{3} - 11 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} + 41\right )} e^{\left (\arcsin \left (a x\right )\right )}}{85 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

1/85*(5*a^4*x^4 - 22*a^2*x^2 - 4*(5*a^3*x^3 - 11*a*x)*sqrt(-a^2*x^2 + 1) + 41)*e^(arcsin(a*x))/a

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Sympy [A]  time = 27.3924, size = 95, normalized size = 0.85 \begin{align*} \begin{cases} \frac{a^{3} x^{4} e^{\operatorname{asin}{\left (a x \right )}}}{17} - \frac{4 a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} e^{\operatorname{asin}{\left (a x \right )}}}{17} - \frac{22 a x^{2} e^{\operatorname{asin}{\left (a x \right )}}}{85} + \frac{44 x \sqrt{- a^{2} x^{2} + 1} e^{\operatorname{asin}{\left (a x \right )}}}{85} + \frac{41 e^{\operatorname{asin}{\left (a x \right )}}}{85 a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(asin(a*x))*(-a**2*x**2+1)**(3/2),x)

[Out]

Piecewise((a**3*x**4*exp(asin(a*x))/17 - 4*a**2*x**3*sqrt(-a**2*x**2 + 1)*exp(asin(a*x))/17 - 22*a*x**2*exp(as
in(a*x))/85 + 44*x*sqrt(-a**2*x**2 + 1)*exp(asin(a*x))/85 + 41*exp(asin(a*x))/(85*a), Ne(a, 0)), (x, True))

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Giac [A]  time = 1.26209, size = 123, normalized size = 1.1 \begin{align*} \frac{4}{17} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x e^{\left (\arcsin \left (a x\right )\right )} + \frac{24}{85} \, \sqrt{-a^{2} x^{2} + 1} x e^{\left (\arcsin \left (a x\right )\right )} + \frac{{\left (a^{2} x^{2} - 1\right )}^{2} e^{\left (\arcsin \left (a x\right )\right )}}{17 \, a} - \frac{12 \,{\left (a^{2} x^{2} - 1\right )} e^{\left (\arcsin \left (a x\right )\right )}}{85 \, a} + \frac{24 \, e^{\left (\arcsin \left (a x\right )\right )}}{85 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))*(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

4/17*(-a^2*x^2 + 1)^(3/2)*x*e^(arcsin(a*x)) + 24/85*sqrt(-a^2*x^2 + 1)*x*e^(arcsin(a*x)) + 1/17*(a^2*x^2 - 1)^
2*e^(arcsin(a*x))/a - 12/85*(a^2*x^2 - 1)*e^(arcsin(a*x))/a + 24/85*e^(arcsin(a*x))/a