3.100.11 \(\int \frac {-e^{2-2 x} x^2+e^{1-x+\frac {1}{4} (4+\log (4))} (-5+5 x-2 x^2)+e^{\frac {1}{2} (4+\log (4))} (-5-x^2)}{e^{2-2 x} x^2+e^{\frac {1}{2} (4+\log (4))} x^2+2 e^{1-x+\frac {1}{4} (4+\log (4))} x^2} \, dx\)

Optimal. Leaf size=22 \[ -x+\frac {5}{x+\frac {e^{-x} x}{\sqrt {2}}} \]

________________________________________________________________________________________

Rubi [F]  time = 0.91, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-e^{2-2 x} x^2+e^{1-x+\frac {1}{4} (4+\log (4))} \left (-5+5 x-2 x^2\right )+e^{\frac {1}{2} (4+\log (4))} \left (-5-x^2\right )}{e^{2-2 x} x^2+e^{\frac {1}{2} (4+\log (4))} x^2+2 e^{1-x+\frac {1}{4} (4+\log (4))} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-(E^(2 - 2*x)*x^2) + E^(1 - x + (4 + Log[4])/4)*(-5 + 5*x - 2*x^2) + E^((4 + Log[4])/2)*(-5 - x^2))/(E^(2
- 2*x)*x^2 + E^((4 + Log[4])/2)*x^2 + 2*E^(1 - x + (4 + Log[4])/4)*x^2),x]

[Out]

5/x - x + 5*Defer[Int][1/((1 + Sqrt[2]*E^x)*x^2), x] - 5*Defer[Int][1/((1 + Sqrt[2]*E^x)^2*x), x] + 5*Defer[In
t][1/((1 + Sqrt[2]*E^x)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^2+\sqrt {2} e^x \left (-5+5 x-2 x^2\right )-2 e^{2 x} \left (5+x^2\right )}{\left (x+\sqrt {2} e^x x\right )^2} \, dx\\ &=\int \left (-\frac {5}{\left (1+\sqrt {2} e^x\right )^2 x}+\frac {5 (1+x)}{\left (1+\sqrt {2} e^x\right ) x^2}+\frac {-5-x^2}{x^2}\right ) \, dx\\ &=-\left (5 \int \frac {1}{\left (1+\sqrt {2} e^x\right )^2 x} \, dx\right )+5 \int \frac {1+x}{\left (1+\sqrt {2} e^x\right ) x^2} \, dx+\int \frac {-5-x^2}{x^2} \, dx\\ &=5 \int \left (\frac {1}{\left (1+\sqrt {2} e^x\right ) x^2}+\frac {1}{\left (1+\sqrt {2} e^x\right ) x}\right ) \, dx-5 \int \frac {1}{\left (1+\sqrt {2} e^x\right )^2 x} \, dx+\int \left (-1-\frac {5}{x^2}\right ) \, dx\\ &=\frac {5}{x}-x+5 \int \frac {1}{\left (1+\sqrt {2} e^x\right ) x^2} \, dx-5 \int \frac {1}{\left (1+\sqrt {2} e^x\right )^2 x} \, dx+5 \int \frac {1}{\left (1+\sqrt {2} e^x\right ) x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.27, size = 34, normalized size = 1.55 \begin {gather*} -\frac {x^2+\sqrt {2} e^x \left (-5+x^2\right )}{x+\sqrt {2} e^x x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(E^(2 - 2*x)*x^2) + E^(1 - x + (4 + Log[4])/4)*(-5 + 5*x - 2*x^2) + E^((4 + Log[4])/2)*(-5 - x^2))
/(E^(2 - 2*x)*x^2 + E^((4 + Log[4])/2)*x^2 + 2*E^(1 - x + (4 + Log[4])/4)*x^2),x]

[Out]

-((x^2 + Sqrt[2]*E^x*(-5 + x^2))/(x + Sqrt[2]*E^x*x))

________________________________________________________________________________________

fricas [B]  time = 0.86, size = 50, normalized size = 2.27 \begin {gather*} -\frac {x^{2} e^{\left (-x + \frac {1}{2} \, \log \relax (2) + 2\right )} + {\left (x^{2} - 5\right )} e^{\left (\log \relax (2) + 2\right )}}{x e^{\left (-x + \frac {1}{2} \, \log \relax (2) + 2\right )} + x e^{\left (\log \relax (2) + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-5)*exp(1+1/2*log(2))^2+(-2*x^2+5*x-5)*exp(-x+1)*exp(1+1/2*log(2))-x^2*exp(-x+1)^2)/(x^2*exp(1
+1/2*log(2))^2+2*x^2*exp(-x+1)*exp(1+1/2*log(2))+x^2*exp(-x+1)^2),x, algorithm="fricas")

[Out]

-(x^2*e^(-x + 1/2*log(2) + 2) + (x^2 - 5)*e^(log(2) + 2))/(x*e^(-x + 1/2*log(2) + 2) + x*e^(log(2) + 2))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x^{2} e^{\left (-2 \, x + 2\right )} + {\left (2 \, x^{2} - 5 \, x + 5\right )} e^{\left (-x + \frac {1}{2} \, \log \relax (2) + 2\right )} + {\left (x^{2} + 5\right )} e^{\left (\log \relax (2) + 2\right )}}{2 \, x^{2} e^{\left (-x + \frac {1}{2} \, \log \relax (2) + 2\right )} + x^{2} e^{\left (-2 \, x + 2\right )} + x^{2} e^{\left (\log \relax (2) + 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-5)*exp(1+1/2*log(2))^2+(-2*x^2+5*x-5)*exp(-x+1)*exp(1+1/2*log(2))-x^2*exp(-x+1)^2)/(x^2*exp(1
+1/2*log(2))^2+2*x^2*exp(-x+1)*exp(1+1/2*log(2))+x^2*exp(-x+1)^2),x, algorithm="giac")

[Out]

integrate(-(x^2*e^(-2*x + 2) + (2*x^2 - 5*x + 5)*e^(-x + 1/2*log(2) + 2) + (x^2 + 5)*e^(log(2) + 2))/(2*x^2*e^
(-x + 1/2*log(2) + 2) + x^2*e^(-2*x + 2) + x^2*e^(log(2) + 2)), x)

________________________________________________________________________________________

maple [A]  time = 0.41, size = 29, normalized size = 1.32

method	result	size
risch	\(-x +\frac {10 \,{\mathrm e}}{x \left (2 \,{\mathrm e}+\sqrt {2}\, {\mathrm e}^{1-x}\right )}\)	\(29\)
norman	\(\frac {-x^{2} {\mathrm e}^{1-x}+5 \,{\mathrm e} \sqrt {2}-{\mathrm e} \sqrt {2}\, x^{2}}{x \left ({\mathrm e}^{1+\frac {\ln \relax (2)}{2}}+{\mathrm e}^{1-x}\right )}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2-5)*exp(1+1/2*ln(2))^2+(-2*x^2+5*x-5)*exp(1-x)*exp(1+1/2*ln(2))-x^2*exp(1-x)^2)/(x^2*exp(1+1/2*ln(2)
)^2+2*x^2*exp(1-x)*exp(1+1/2*ln(2))+x^2*exp(1-x)^2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-5)*exp(1+1/2*log(2))^2+(-2*x^2+5*x-5)*exp(-x+1)*exp(1+1/2*log(2))-x^2*exp(-x+1)^2)/(x^2*exp(1
+1/2*log(2))^2+2*x^2*exp(-x+1)*exp(1+1/2*log(2))+x^2*exp(-x+1)^2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

________________________________________________________________________________________

mupad [B]  time = 5.69, size = 29, normalized size = 1.32 \begin {gather*} \frac {5\,\sqrt {2}\,\mathrm {e}}{x\,\left ({\mathrm {e}}^{1-x}+\sqrt {2}\,\mathrm {e}\right )}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(2) + 2)*(x^2 + 5) + x^2*exp(2 - 2*x) + exp(log(2)/2 + 1)*exp(1 - x)*(2*x^2 - 5*x + 5))/(x^2*exp(
log(2) + 2) + x^2*exp(2 - 2*x) + 2*x^2*exp(log(2)/2 + 1)*exp(1 - x)),x)

[Out]

(5*2^(1/2)*exp(1))/(x*(exp(1 - x) + 2^(1/2)*exp(1))) - x

________________________________________________________________________________________

sympy [A]  time = 0.31, size = 26, normalized size = 1.18 \begin {gather*} - x + \frac {5 \sqrt {2} e}{x \left (e^{1 - x} + \sqrt {2} e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2-5)*exp(1+1/2*ln(2))**2+(-2*x**2+5*x-5)*exp(-x+1)*exp(1+1/2*ln(2))-x**2*exp(-x+1)**2)/(x**2*e
xp(1+1/2*ln(2))**2+2*x**2*exp(-x+1)*exp(1+1/2*ln(2))+x**2*exp(-x+1)**2),x)

[Out]

-x + 5*sqrt(2)*E/(x*(exp(1 - x) + sqrt(2)*E))

________________________________________________________________________________________