\(\int \frac {e^x (1-x-x^2)}{\sqrt {1-x^2}} \, dx\) [540]

Optimal result

Integrand size = 25, antiderivative size = 15 \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=e^x \sqrt {1-x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2326} \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=e^x \sqrt {1-x^2} \]

[In]

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

[Out]

E^x*Sqrt[1 - x^2]

Rule 2326

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*} \text {integral}& = e^x \sqrt {1-x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=e^x \sqrt {1-x^2} \]

[In]

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

[Out]

E^x*Sqrt[1 - x^2]

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=\sqrt {-x^{2} + 1} e^{x} \]

[In]

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

[Out]

sqrt(-x^2 + 1)*e^x

Sympy [F]

\[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=- \int \left (- \frac {e^{x}}{\sqrt {1 - x^{2}}}\right )\, dx - \int \frac {x e^{x}}{\sqrt {1 - x^{2}}}\, dx - \int \frac {x^{2} e^{x}}{\sqrt {1 - x^{2}}}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.40 \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=-\frac {{\left (x^{2} - 1\right )} e^{x}}{\sqrt {x + 1} \sqrt {-x + 1}} \]

[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))

Giac [F]

\[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx=\int { -\frac {{\left (x^{2} + x - 1\right )} e^{x}}{\sqrt {-x^{2} + 1}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {e^x \left (1-x-x^2\right )}{\sqrt {1-x^2}} \, dx={\mathrm {e}}^x\,\sqrt {1-x^2} \]

[In]

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

[Out]

exp(x)*(1 - x^2)^(1/2)