3.10 \(\int \frac {e^{\sin ^{-1}(x)} x^3}{\sqrt {1-x^2}} \, dx\)

Optimal. Leaf size=44 \[ \frac {1}{10} \left (x^3-3 \sqrt {1-x^2} x^2-3 \sqrt {1-x^2}+3 x\right ) e^{\sin ^{-1}(x)} \]

[Out]

1/10*exp(arcsin(x))*(3*x+x^3-3*(-x^2+1)^(1/2)-3*x^2*(-x^2+1)^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 0.68, antiderivative size = 62, normalized size of antiderivative = 1.41, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4836, 6741, 6720, 4434, 4432} \[ \frac {1}{10} x^3 e^{\sin ^{-1}(x)}-\frac {3}{10} \sqrt {1-x^2} x^2 e^{\sin ^{-1}(x)}-\frac {3}{10} \sqrt {1-x^2} e^{\sin ^{-1}(x)}+\frac {3}{10} x e^{\sin ^{-1}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*E^ArcSin[x]*x)/10 + (E^ArcSin[x]*x^3)/10 - (3*E^ArcSin[x]*Sqrt[1 - x^2])/10 - (3*E^ArcSin[x]*x^2*Sqrt[1 - x
^2])/10

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]

Rule 4434

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

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 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 6741

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

Rubi steps

\begin {align*} \int \frac {e^{\sin ^{-1}(x)} x^3}{\sqrt {1-x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {e^x \cos (x) \sin ^3(x)}{\sqrt {1-\sin ^2(x)}} \, dx,x,\sin ^{-1}(x)\right )\\ &=\operatorname {Subst}\left (\int \frac {e^x \cos (x) \sin ^3(x)}{\sqrt {\cos ^2(x)}} \, dx,x,\sin ^{-1}(x)\right )\\ &=1 \operatorname {Subst}\left (\int e^x \sin ^3(x) \, dx,x,\sin ^{-1}(x)\right )\\ &=\frac {1}{10} e^{\sin ^{-1}(x)} x^3-\frac {3}{10} e^{\sin ^{-1}(x)} x^2 \sqrt {1-x^2}+\frac {3}{5} \operatorname {Subst}\left (\int e^x \sin (x) \, dx,x,\sin ^{-1}(x)\right )\\ &=\frac {3}{10} e^{\sin ^{-1}(x)} x+\frac {1}{10} e^{\sin ^{-1}(x)} x^3-\frac {3}{10} e^{\sin ^{-1}(x)} \sqrt {1-x^2}-\frac {3}{10} e^{\sin ^{-1}(x)} x^2 \sqrt {1-x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.17, size = 38, normalized size = 0.86 \[ -\frac {1}{40} e^{\sin ^{-1}(x)} \left (15 \left (\sqrt {1-x^2}-x\right )+\sin \left (3 \sin ^{-1}(x)\right )-3 \cos \left (3 \sin ^{-1}(x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/40*(E^ArcSin[x]*(15*(-x + Sqrt[1 - x^2]) - 3*Cos[3*ArcSin[x]] + Sin[3*ArcSin[x]]))

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 28, normalized size = 0.64 \[ \frac {1}{10} \, {\left (x^{3} - 3 \, {\left (x^{2} + 1\right )} \sqrt {-x^{2} + 1} + 3 \, x\right )} e^{\arcsin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(x^3 - 3*(x^2 + 1)*sqrt(-x^2 + 1) + 3*x)*e^arcsin(x)

________________________________________________________________________________________

giac [A]  time = 1.47, size = 46, normalized size = 1.05 \[ \frac {1}{10} \, {\left (x^{2} - 1\right )} x e^{\arcsin \relax (x)} + \frac {3}{10} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} e^{\arcsin \relax (x)} + \frac {2}{5} \, x e^{\arcsin \relax (x)} - \frac {3}{5} \, \sqrt {-x^{2} + 1} e^{\arcsin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(x^2 - 1)*x*e^arcsin(x) + 3/10*(-x^2 + 1)^(3/2)*e^arcsin(x) + 2/5*x*e^arcsin(x) - 3/5*sqrt(-x^2 + 1)*e^ar
csin(x)

________________________________________________________________________________________

maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} {\mathrm e}^{\arcsin \relax (x )}}{\sqrt {-x^{2}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} e^{\arcsin \relax (x)}}{\sqrt {-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^3\,{\mathrm {e}}^{\mathrm {asin}\relax (x)}}{\sqrt {1-x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*exp(asin(x)))/(1 - x^2)^(1/2),x)

[Out]

int((x^3*exp(asin(x)))/(1 - x^2)^(1/2), x)

________________________________________________________________________________________

sympy [A]  time = 2.17, size = 56, normalized size = 1.27 \[ \frac {x^{3} e^{\operatorname {asin}{\relax (x )}}}{10} - \frac {3 x^{2} \sqrt {1 - x^{2}} e^{\operatorname {asin}{\relax (x )}}}{10} + \frac {3 x e^{\operatorname {asin}{\relax (x )}}}{10} - \frac {3 \sqrt {1 - x^{2}} e^{\operatorname {asin}{\relax (x )}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*exp(asin(x))/10 - 3*x**2*sqrt(1 - x**2)*exp(asin(x))/10 + 3*x*exp(asin(x))/10 - 3*sqrt(1 - x**2)*exp(asin
(x))/10

________________________________________________________________________________________