\(\int \frac {-x^2+e^{e^{4 x^2}} (1-x-8 e^{4 x^2} x^2)}{e^{e^{4 x^2}} x+x^2} \, dx\) [7137]

Optimal result

Integrand size = 51, antiderivative size = 32 \[ \int \frac {-x^2+e^{e^{4 x^2}} \left (1-x-8 e^{4 x^2} x^2\right )}{e^{e^{4 x^2}} x+x^2} \, dx=2 \left (e^2+e^5\right )-x-\log \left (\frac {2 \left (e^{e^{4 x^2}}+x\right )}{x}\right ) \]

[Out]

2*exp(5)+2*exp(2)-ln(2*(exp(exp(4*x^2))+x)/x)-x

Rubi [F]

\[ \int \frac {-x^2+e^{e^{4 x^2}} \left (1-x-8 e^{4 x^2} x^2\right )}{e^{e^{4 x^2}} x+x^2} \, dx=\int \frac {-x^2+e^{e^{4 x^2}} \left (1-x-8 e^{4 x^2} x^2\right )}{e^{e^{4 x^2}} x+x^2} \, dx \]

[In]

Int[(-x^2 + E^E^(4*x^2)*(1 - x - 8*E^(4*x^2)*x^2))/(E^E^(4*x^2)*x + x^2),x]

[Out]

-x + Log[x] - Defer[Int][(E^E^(4*x^2) + x)^(-1), x] - 8*Defer[Int][(E^(E^(4*x^2) + 4*x^2)*x)/(E^E^(4*x^2) + x)
, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {8 e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x}-\frac {-e^{e^{4 x^2}}+e^{e^{4 x^2}} x+x^2}{x \left (e^{e^{4 x^2}}+x\right )}\right ) \, dx \\ & = -\left (8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx\right )-\int \frac {-e^{e^{4 x^2}}+e^{e^{4 x^2}} x+x^2}{x \left (e^{e^{4 x^2}}+x\right )} \, dx \\ & = -\left (8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx\right )-\int \frac {e^{e^{4 x^2}} (-1+x)+x^2}{x \left (e^{e^{4 x^2}}+x\right )} \, dx \\ & = -\left (8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx\right )-\int \left (\frac {-1+x}{x}+\frac {1}{e^{e^{4 x^2}}+x}\right ) \, dx \\ & = -\left (8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx\right )-\int \frac {-1+x}{x} \, dx-\int \frac {1}{e^{e^{4 x^2}}+x} \, dx \\ & = -\left (8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx\right )-\int \left (1-\frac {1}{x}\right ) \, dx-\int \frac {1}{e^{e^{4 x^2}}+x} \, dx \\ & = -x+\log (x)-8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx-\int \frac {1}{e^{e^{4 x^2}}+x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {-x^2+e^{e^{4 x^2}} \left (1-x-8 e^{4 x^2} x^2\right )}{e^{e^{4 x^2}} x+x^2} \, dx=-x+\log (x)-\log \left (e^{e^{4 x^2}}+x\right ) \]

[In]

Integrate[(-x^2 + E^E^(4*x^2)*(1 - x - 8*E^(4*x^2)*x^2))/(E^E^(4*x^2)*x + x^2),x]

[Out]

-x + Log[x] - Log[E^E^(4*x^2) + x]

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59

method	result	size
norman	\(-x -\ln \left ({\mathrm e}^{{\mathrm e}^{4 x^{2}}}+x \right )+\ln \left (x \right )\)	\(19\)
risch	\(-x -\ln \left ({\mathrm e}^{{\mathrm e}^{4 x^{2}}}+x \right )+\ln \left (x \right )\)	\(19\)
parallelrisch	\(-x -\ln \left ({\mathrm e}^{{\mathrm e}^{4 x^{2}}}+x \right )+\ln \left (x \right )\)	\(19\)

[In]

int(((-8*x^2*exp(4*x^2)-x+1)*exp(exp(4*x^2))-x^2)/(x*exp(exp(4*x^2))+x^2),x,method=_RETURNVERBOSE)

[Out]

-x-ln(exp(exp(4*x^2))+x)+ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.56 \[ \int \frac {-x^2+e^{e^{4 x^2}} \left (1-x-8 e^{4 x^2} x^2\right )}{e^{e^{4 x^2}} x+x^2} \, dx=-x - \log \left (x + e^{\left (e^{\left (4 \, x^{2}\right )}\right )}\right ) + \log \left (x\right ) \]

[In]

integrate(((-8*x^2*exp(4*x^2)-x+1)*exp(exp(4*x^2))-x^2)/(x*exp(exp(4*x^2))+x^2),x, algorithm="fricas")

[Out]

-x - log(x + e^(e^(4*x^2))) + log(x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.47 \[ \int \frac {-x^2+e^{e^{4 x^2}} \left (1-x-8 e^{4 x^2} x^2\right )}{e^{e^{4 x^2}} x+x^2} \, dx=- x + \log {\left (x \right )} - \log {\left (x + e^{e^{4 x^{2}}} \right )} \]

[In]

integrate(((-8*x**2*exp(4*x**2)-x+1)*exp(exp(4*x**2))-x**2)/(x*exp(exp(4*x**2))+x**2),x)

[Out]

-x + log(x) - log(x + exp(exp(4*x**2)))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.56 \[ \int \frac {-x^2+e^{e^{4 x^2}} \left (1-x-8 e^{4 x^2} x^2\right )}{e^{e^{4 x^2}} x+x^2} \, dx=-x - \log \left (x + e^{\left (e^{\left (4 \, x^{2}\right )}\right )}\right ) + \log \left (x\right ) \]

[In]

integrate(((-8*x^2*exp(4*x^2)-x+1)*exp(exp(4*x^2))-x^2)/(x*exp(exp(4*x^2))+x^2),x, algorithm="maxima")

[Out]

-x - log(x + e^(e^(4*x^2))) + log(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {-x^2+e^{e^{4 x^2}} \left (1-x-8 e^{4 x^2} x^2\right )}{e^{e^{4 x^2}} x+x^2} \, dx=4 \, x^{2} - x - \log \left (x e^{\left (4 \, x^{2}\right )} + e^{\left (4 \, x^{2} + e^{\left (4 \, x^{2}\right )}\right )}\right ) + \log \left (x\right ) \]

[In]

integrate(((-8*x^2*exp(4*x^2)-x+1)*exp(exp(4*x^2))-x^2)/(x*exp(exp(4*x^2))+x^2),x, algorithm="giac")

[Out]

4*x^2 - x - log(x*e^(4*x^2) + e^(4*x^2 + e^(4*x^2))) + log(x)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.56 \[ \int \frac {-x^2+e^{e^{4 x^2}} \left (1-x-8 e^{4 x^2} x^2\right )}{e^{e^{4 x^2}} x+x^2} \, dx=\ln \left (x\right )-\ln \left (x+{\mathrm {e}}^{{\mathrm {e}}^{4\,x^2}}\right )-x \]

[In]

int(-(exp(exp(4*x^2))*(x + 8*x^2*exp(4*x^2) - 1) + x^2)/(x^2 + x*exp(exp(4*x^2))),x)

[Out]

log(x) - log(x + exp(exp(4*x^2))) - x