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]

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

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Rubi [A]  time = 1.13222, 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 \[ \frac{1}{10} x^3 e^{\sin ^{-1}(x)}-\frac{3}{10} \sqrt{1-x^2} x^2 e^{\sin ^{-1}(x)}-\frac{3}{10} \sqrt{1-x^2} e^{\sin ^{-1}(x)}+\frac{3}{10} x e^{\sin ^{-1}(x)} \]

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

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Rubi in Sympy [A]  time = 14.9857, size = 56, normalized size = 1.27 \[ \frac{x^{3} e^{\operatorname{asin}{\left (x \right )}}}{10} - \frac{3 x^{2} \sqrt{- x^{2} + 1} e^{\operatorname{asin}{\left (x \right )}}}{10} + \frac{3 x e^{\operatorname{asin}{\left (x \right )}}}{10} - \frac{3 \sqrt{- x^{2} + 1} e^{\operatorname{asin}{\left (x \right )}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(asin(x))*x**3/(-x**2+1)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.194815, 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]

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

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{{{\rm e}^{\arcsin \left ( x \right ) }}{x}^{3}{\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^arcsin(x)/sqrt(-x^2 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.248903, size = 38, normalized size = 0.86 \[ \frac{1}{10} \,{\left (x^{3} - 3 \,{\left (x^{2} + 1\right )} \sqrt{-x^{2} + 1} + 3 \, x\right )} e^{\arcsin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^arcsin(x)/sqrt(-x^2 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.26085, size = 56, normalized size = 1.27 \[ \frac{x^{3} e^{\operatorname{asin}{\left (x \right )}}}{10} - \frac{3 x^{2} \sqrt{- x^{2} + 1} e^{\operatorname{asin}{\left (x \right )}}}{10} + \frac{3 x e^{\operatorname{asin}{\left (x \right )}}}{10} - \frac{3 \sqrt{- x^{2} + 1} e^{\operatorname{asin}{\left (x \right )}}}{10} \]

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

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GIAC/XCAS [A]  time = 0.215279, size = 62, normalized size = 1.41 \[ \frac{1}{10} \,{\left (x^{2} - 1\right )} x e^{\arcsin \left (x\right )} + \frac{3}{10} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} e^{\arcsin \left (x\right )} + \frac{2}{5} \, x e^{\arcsin \left (x\right )} - \frac{3}{5} \, \sqrt{-x^{2} + 1} e^{\arcsin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^arcsin(x)/sqrt(-x^2 + 1),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^arcsi
n(x) - 3/5*sqrt(-x^2 + 1)*e^arcsin(x)