\(\int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} (-256 x+1312 x^2+512 x^3)}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx\) [8861]

Optimal result

Integrand size = 77, antiderivative size = 26 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=\frac {x}{-e^{4+x^2}+\frac {41 x^2}{16}}-\log (x) \]

[Out]

x/(41/16*x^2-exp(x^2+4))-ln(x)

Rubi [F]

\[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=\int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx \]

[In]

Int[(-256*E^(8 + 2*x^2) - 656*x^3 - 1681*x^4 + E^(4 + x^2)*(-256*x + 1312*x^2 + 512*x^3))/(256*E^(8 + 2*x^2)*x
- 1312*E^(4 + x^2)*x^3 + 1681*x^5),x]

[Out]

-Log[x] - 1312*Defer[Int][x^2/(16*E^(4 + x^2) - 41*x^2)^2, x] + 1312*Defer[Int][x^4/(16*E^(4 + x^2) - 41*x^2)^
2, x] - 16*Defer[Int][(16*E^(4 + x^2) - 41*x^2)^(-1), x] + 32*Defer[Int][x^2/(16*E^(4 + x^2) - 41*x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{x \left (16 e^{4+x^2}-41 x^2\right )^2} \, dx \\ & = \int \left (-\frac {1}{x}+\frac {1312 x^2 \left (-1+x^2\right )}{\left (16 e^{4+x^2}-41 x^2\right )^2}+\frac {16 \left (-1+2 x^2\right )}{16 e^{4+x^2}-41 x^2}\right ) \, dx \\ & = -\log (x)+16 \int \frac {-1+2 x^2}{16 e^{4+x^2}-41 x^2} \, dx+1312 \int \frac {x^2 \left (-1+x^2\right )}{\left (16 e^{4+x^2}-41 x^2\right )^2} \, dx \\ & = -\log (x)+16 \int \left (-\frac {1}{16 e^{4+x^2}-41 x^2}+\frac {2 x^2}{16 e^{4+x^2}-41 x^2}\right ) \, dx+1312 \int \left (-\frac {x^2}{\left (16 e^{4+x^2}-41 x^2\right )^2}+\frac {x^4}{\left (16 e^{4+x^2}-41 x^2\right )^2}\right ) \, dx \\ & = -\log (x)-16 \int \frac {1}{16 e^{4+x^2}-41 x^2} \, dx+32 \int \frac {x^2}{16 e^{4+x^2}-41 x^2} \, dx-1312 \int \frac {x^2}{\left (16 e^{4+x^2}-41 x^2\right )^2} \, dx+1312 \int \frac {x^4}{\left (16 e^{4+x^2}-41 x^2\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.70 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=-\frac {16 x}{16 e^{4+x^2}-41 x^2}-\log (x) \]

[In]

Integrate[(-256*E^(8 + 2*x^2) - 656*x^3 - 1681*x^4 + E^(4 + x^2)*(-256*x + 1312*x^2 + 512*x^3))/(256*E^(8 + 2*
x^2)*x - 1312*E^(4 + x^2)*x^3 + 1681*x^5),x]

[Out]

(-16*x)/(16*E^(4 + x^2) - 41*x^2) - Log[x]

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96

method	result	size
norman	\(\frac {16 x}{41 x^{2}-16 \,{\mathrm e}^{x^{2}+4}}-\ln \left (x \right )\)	\(25\)
risch	\(\frac {16 x}{41 x^{2}-16 \,{\mathrm e}^{x^{2}+4}}-\ln \left (x \right )\)	\(25\)
parallelrisch	\(-\frac {656 x^{2} \ln \left (x \right )-256 \ln \left (x \right ) {\mathrm e}^{x^{2}+4}-256 x}{16 \left (41 x^{2}-16 \,{\mathrm e}^{x^{2}+4}\right )}\)	\(40\)

[In]

int((-256*exp(x^2+4)^2+(512*x^3+1312*x^2-256*x)*exp(x^2+4)-1681*x^4-656*x^3)/(256*x*exp(x^2+4)^2-1312*x^3*exp(
x^2+4)+1681*x^5),x,method=_RETURNVERBOSE)

[Out]

16*x/(41*x^2-16*exp(x^2+4))-ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=-\frac {{\left (41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}\right )} \log \left (x\right ) - 16 \, x}{41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}} \]

[In]

integrate((-256*exp(x^2+4)^2+(512*x^3+1312*x^2-256*x)*exp(x^2+4)-1681*x^4-656*x^3)/(256*x*exp(x^2+4)^2-1312*x^
3*exp(x^2+4)+1681*x^5),x, algorithm="fricas")

[Out]

-((41*x^2 - 16*e^(x^2 + 4))*log(x) - 16*x)/(41*x^2 - 16*e^(x^2 + 4))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=- \frac {x}{- \frac {41 x^{2}}{16} + e^{x^{2} + 4}} - \log {\left (x \right )} \]

[In]

integrate((-256*exp(x**2+4)**2+(512*x**3+1312*x**2-256*x)*exp(x**2+4)-1681*x**4-656*x**3)/(256*x*exp(x**2+4)**
2-1312*x**3*exp(x**2+4)+1681*x**5),x)

[Out]

-x/(-41*x**2/16 + exp(x**2 + 4)) - log(x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=\frac {16 \, x}{41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}} - \log \left (x\right ) \]

[In]

integrate((-256*exp(x^2+4)^2+(512*x^3+1312*x^2-256*x)*exp(x^2+4)-1681*x^4-656*x^3)/(256*x*exp(x^2+4)^2-1312*x^
3*exp(x^2+4)+1681*x^5),x, algorithm="maxima")

[Out]

16*x/(41*x^2 - 16*e^(x^2 + 4)) - log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=-\frac {41 \, x^{2} \log \left (x\right ) - 16 \, e^{\left (x^{2} + 4\right )} \log \left (x\right ) - 16 \, x}{41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}} \]

[In]

integrate((-256*exp(x^2+4)^2+(512*x^3+1312*x^2-256*x)*exp(x^2+4)-1681*x^4-656*x^3)/(256*x*exp(x^2+4)^2-1312*x^
3*exp(x^2+4)+1681*x^5),x, algorithm="giac")

[Out]

-(41*x^2*log(x) - 16*e^(x^2 + 4)*log(x) - 16*x)/(41*x^2 - 16*e^(x^2 + 4))

Mupad [B] (verification not implemented)

Time = 12.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-256 e^{8+2 x^2}-656 x^3-1681 x^4+e^{4+x^2} \left (-256 x+1312 x^2+512 x^3\right )}{256 e^{8+2 x^2} x-1312 e^{4+x^2} x^3+1681 x^5} \, dx=-\ln \left (x\right )-\frac {16\,x}{16\,{\mathrm {e}}^{x^2+4}-41\,x^2} \]

[In]

int(-(256*exp(2*x^2 + 8) - exp(x^2 + 4)*(1312*x^2 - 256*x + 512*x^3) + 656*x^3 + 1681*x^4)/(256*x*exp(2*x^2 +
8) - 1312*x^3*exp(x^2 + 4) + 1681*x^5),x)

[Out]

- log(x) - (16*x)/(16*exp(x^2 + 4) - 41*x^2)