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

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

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Rubi [A]  time = 0.06, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, 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*}

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Mathematica [A]  time = 0.03, size = 15, normalized size = 1.00 \[ e^x \sqrt {1-x^2} \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 1.05, size = 12, normalized size = 0.80 \[ \sqrt {-x^{2} + 1} e^{x} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (x^{2} + x - 1\right )} e^{x}}{\sqrt {-x^{2} + 1}}\,{d x} \]

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)

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maple [A]  time = 0.30, size = 20, normalized size = 1.33

method	result	size
gosper	\(-\frac {{\mathrm e}^{x} \left (1+x \right ) \left (-1+x \right )}{\sqrt {-x^{2}+1}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.65, size = 21, normalized size = 1.40 \[ -\frac {{\left (x^{2} - 1\right )} e^{x}}{\sqrt {x + 1} \sqrt {-x + 1}} \]

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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mupad [B]  time = 0.45, size = 12, normalized size = 0.80 \[ {\mathrm {e}}^x\,\sqrt {1-x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)*(1 - x^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {e^{x}}{\sqrt {1 - x^{2}}}\right )\, dx - \int \frac {x e^{x}}{\sqrt {1 - x^{2}}}\, dx - \int \frac {x^{2} e^{x}}{\sqrt {1 - x^{2}}}\, dx \]

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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