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

Optimal. Leaf size=26 \[ \frac {x}{-e^{4+x^2}+\frac {41 x^2}{16}}-\log (x) \]

________________________________________________________________________________________

Rubi [F]  time = 1.35, 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 {-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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.25, size = 25, normalized size = 0.96 \begin {gather*} -\frac {16 x}{16 e^{4+x^2}-41 x^2}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[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]

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 39, normalized size = 1.50 \begin {gather*} -\frac {{\left (41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}\right )} \log \relax (x) - 16 \, x}{41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

giac [A]  time = 0.18, size = 39, normalized size = 1.50 \begin {gather*} -\frac {41 \, x^{2} \log \relax (x) - 16 \, e^{\left (x^{2} + 4\right )} \log \relax (x) - 16 \, x}{41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

maple [A]  time = 0.09, size = 25, normalized size = 0.96

method	result	size
norman	\(\frac {16 x}{41 x^{2}-16 \,{\mathrm e}^{x^{2}+4}}-\ln \relax (x )\)	\(25\)
risch	\(\frac {16 x}{41 x^{2}-16 \,{\mathrm e}^{x^{2}+4}}-\ln \relax (x )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 24, normalized size = 0.92 \begin {gather*} \frac {16 \, x}{41 \, x^{2} - 16 \, e^{\left (x^{2} + 4\right )}} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 5.20, size = 24, normalized size = 0.92 \begin {gather*} -\ln \relax (x)-\frac {16\,x}{16\,{\mathrm {e}}^{x^2+4}-41\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 20, normalized size = 0.77 \begin {gather*} - \frac {16 x}{- 41 x^{2} + 16 e^{x^{2} + 4}} - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

________________________________________________________________________________________