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

Optimal. Leaf size=15 \[ e^x \sqrt{1-x^2} \]

[Out]

E^x*Sqrt[1 - x^2]

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Rubi [A]  time = 0.0569821, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2288} \[ e^x \sqrt{1-x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^x*Sqrt[1 - x^2]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin{align*} \int \frac{e^x \left (1-x-x^2\right )}{\sqrt{1-x^2}} \, dx &=e^x \sqrt{1-x^2}\\ \end{align*}

Mathematica [A]  time = 0.0257325, size = 15, normalized size = 1. \[ e^x \sqrt{1-x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^x*Sqrt[1 - x^2]

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Maple [A]  time = 0.005, size = 20, normalized size = 1.3 \begin{align*} -{{{\rm e}^{x}} \left ( 1+x \right ) \left ( -1+x \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(x)*(1+x)*(-1+x)/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.09179, size = 28, normalized size = 1.87 \begin{align*} -\frac{{\left (x^{2} - 1\right )} e^{x}}{\sqrt{x + 1} \sqrt{-x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.07677, size = 27, normalized size = 1.8 \begin{align*} \sqrt{-x^{2} + 1} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-x^2 + 1)*e^x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{e^{x}}{\sqrt{1 - x^{2}}}\, dx - \int \frac{x e^{x}}{\sqrt{1 - x^{2}}}\, dx - \int \frac{x^{2} e^{x}}{\sqrt{1 - x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-exp(x)/sqrt(1 - x**2), x) - Integral(x*exp(x)/sqrt(1 - x**2), x) - Integral(x**2*exp(x)/sqrt(1 - x*
*2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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