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

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

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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 = {} $$\begin{array}{c}\int \frac{-256e^{8+2x^2}-656x^3-1681x^4+e^{4+x^2}(-256x+1312x^2+512x^3)}{256e^{8+2x^2}x-1312e^{4+x^2}{x}^3+1681x^5}dx\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-256e^{8+2x^2}-656x^3-1681x^4+e^{4+x^2}(-256x+1312x^2+512x^3)}{x{(16e^{4+x^2}-41x^2)}^2}dx\\ & =\int (-\frac{1}{x}+\frac{1312x^2(-1+x^2)}{{(16e^{4+x^2}-41x^2)}^2}+\frac{16(-1+2x^2)}{16e^{4+x^2}-41x^2})dx\\ & =-\log (x)+16\int \frac{-1+2x^2}{16e^{4+x^2}-41x^2}dx+1312\int \frac{x^2(-1+x^2)}{{(16e^{4+x^2}-41x^2)}^2}dx\\ & =-\log (x)+16\int (-\frac{1}{16e^{4+x^2}-41x^2}+\frac{2x^2}{16e^{4+x^2}-41x^2})dx+1312\int (-\frac{x^2}{{(16e^{4+x^2}-41x^2)}^2}+\frac{x^4}{{(16e^{4+x^2}-41x^2)}^2})dx\\ & =-\log (x)-16\int \frac{1}{16e^{4+x^2}-41x^2}dx+32\int \frac{x^2}{16e^{4+x^2}-41x^2}dx-1312\int \frac{x^2}{{(16e^{4+x^2}-41x^2)}^2}dx+1312\int \frac{x^4}{{(16e^{4+x^2}-41x^2)}^2}dx\end{array}\end{array}$$

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

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]

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fricas [A]  time = 0.58, size = 39, normalized size = 1.50 $$\begin{array}{c}-\frac{(41x^2-16e^{(x^2+4)})\log (x)-16x}{41x^2-16e^{(x^2+4)}}\end{array}$$

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))

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giac [A]  time = 0.18, size = 39, normalized size = 1.50 $$\begin{array}{c}-\frac{41x^2\log (x)-16e^{(x^2+4)}\log (x)-16x}{41x^2-16e^{(x^2+4)}}\end{array}$$

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))

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

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

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)

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maxima [A]  time = 0.42, size = 24, normalized size = 0.92 $$\begin{array}{c}\frac{16x}{41x^2-16e^{(x^2+4)}}-\log (x)\end{array}$$

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)

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

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)

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

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)

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