3.73.96 $\int \frac{-11+e^{x^2}(1+4x^2)}{11x-10648x^2-264e^{2x^2}{x}^2+8e^{3x^2}{x}^2+e^{x^2}(-x+2904x^2)}dx$

Optimal. Leaf size=24 $$1+\log \left(\frac{\frac{1}{8{(-11+e^{x^2})}^2}-x}{x}\right)$$

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Rubi [F]  time = 2.49, 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 = {} $$\begin{array}{c}\int \frac{-11+e^{x^2}(1+4x^2)}{11x-10648x^2-264e^{2x^2}{x}^2+8e^{3x^2}{x}^2+e^{x^2}(-x+2904x^2)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-11 + E^x^2*(1 + 4*x^2))/(11*x - 10648*x^2 - 264*E^(2*x^2)*x^2 + 8*E^(3*x^2)*x^2 + E^x^2*(-x + 2904*x^2))
,x]

[Out]

2*x^2 - 2*Log[11 - E^x^2] + Defer[Int][1/(x*(-1 + 8*(-11 + E^x^2)^2*x)), x] + 4*Defer[Int][x/(-1 + 8*(-11 + E^
x^2)^2*x), x] - 3872*Defer[Int][x^2/(-1 + 8*(-11 + E^x^2)^2*x), x] + 352*Defer[Int][(E^x^2*x^2)/(-1 + 8*(-11 +
E^x^2)^2*x), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-11+e^{x^2}(1+4x^2)}{(11-e^{x^2})x(1-8{(-11+e^{x^2})}^{2}x)}dx\\ & =\int (-\frac{44x}{-11+e^{x^2}}+\frac{1+4x^2-3872x^3+352e^{x^2}{x}^3}{x(-1+968x-176e^{x^2}x+8e^{2x^2}x)})dx\\ & =-(44\int \frac{x}{-11+e^{x^2}}dx)+\int \frac{1+4x^2-3872x^3+352e^{x^2}{x}^3}{x(-1+968x-176e^{x^2}x+8e^{2x^2}x)}dx\\ & =-\left(22\mathrm{Subst}(\int \frac{1}{-11+e^x}dx,x,x^2)\right)+\int (\frac{1}{x(-1+968x-176e^{x^2}x+8e^{2x^2}x)}+\frac{4x}{-1+968x-176e^{x^2}x+8e^{2x^2}x}-\frac{3872x^2}{-1+968x-176e^{x^2}x+8e^{2x^2}x}+\frac{352e^{x^2}{x}^2}{-1+968x-176e^{x^2}x+8e^{2x^2}x})dx\\ & =4\int \frac{x}{-1+968x-176e^{x^2}x+8e^{2x^2}x}dx-22\mathrm{Subst}(\int \frac{1}{(-11+x)x}dx,x,e^{x^2})+352\int \frac{e^{x^2}{x}^2}{-1+968x-176e^{x^2}x+8e^{2x^2}x}dx-3872\int \frac{x^2}{-1+968x-176e^{x^2}x+8e^{2x^2}x}dx+\int \frac{1}{x(-1+968x-176e^{x^2}x+8e^{2x^2}x)}dx\\ & =-\left(2\mathrm{Subst}(\int \frac{1}{-11+x}dx,x,e^{x^2})\right)+2\mathrm{Subst}(\int \frac{1}{x}dx,x,e^{x^2})+4\int \frac{x}{-1+8{(-11+e^{x^2})}^{2}x}dx+352\int \frac{e^{x^2}{x}^2}{-1+8{(-11+e^{x^2})}^{2}x}dx-3872\int \frac{x^2}{-1+8{(-11+e^{x^2})}^{2}x}dx+\int \frac{1}{x(-1+8{(-11+e^{x^2})}^{2}x)}dx\\ & =2x^2-2\log (11-e^{x^2})+4\int \frac{x}{-1+8{(-11+e^{x^2})}^{2}x}dx+352\int \frac{e^{x^2}{x}^2}{-1+8{(-11+e^{x^2})}^{2}x}dx-3872\int \frac{x^2}{-1+8{(-11+e^{x^2})}^{2}x}dx+\int \frac{1}{x(-1+8{(-11+e^{x^2})}^{2}x)}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.31, size = 41, normalized size = 1.71 $$\begin{array}{c}-2\log (11-e^{x^2})-\log (x)+\log (1-968x+176e^{x^2}x-8e^{2x^2}x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-11 + E^x^2*(1 + 4*x^2))/(11*x - 10648*x^2 - 264*E^(2*x^2)*x^2 + 8*E^(3*x^2)*x^2 + E^x^2*(-x + 2904
*x^2)),x]

[Out]

-2*Log[11 - E^x^2] - Log[x] + Log[1 - 968*x + 176*E^x^2*x - 8*E^(2*x^2)*x]

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fricas [A]  time = 0.90, size = 36, normalized size = 1.50 $$\begin{array}{c}\log \left(\frac{8x{e}^{\left(2x^2\right)}-176x{e}^{\left(x^2\right)}+968x-1}{x}\right)-2\log (e^{\left(x^2\right)}-11)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+1)*exp(x^2)-11)/(8*x^2*exp(x^2)^3-264*x^2*exp(x^2)^2+(2904*x^2-x)*exp(x^2)-10648*x^2+11*x),x
, algorithm="fricas")

[Out]

log((8*x*e^(2*x^2) - 176*x*e^(x^2) + 968*x - 1)/x) - 2*log(e^(x^2) - 11)

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giac [A]  time = 0.21, size = 36, normalized size = 1.50 $$\begin{array}{c}\log (8x{e}^{\left(2x^2\right)}-176x{e}^{\left(x^2\right)}+968x-1)-\log (x)-2\log (e^{\left(x^2\right)}-11)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+1)*exp(x^2)-11)/(8*x^2*exp(x^2)^3-264*x^2*exp(x^2)^2+(2904*x^2-x)*exp(x^2)-10648*x^2+11*x),x
, algorithm="giac")

[Out]

log(8*x*e^(2*x^2) - 176*x*e^(x^2) + 968*x - 1) - log(x) - 2*log(e^(x^2) - 11)

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maple [A]  time = 0.06, size = 35, normalized size = 1.46

method	result	size
risch	$\ln ({\mathrm{e}}^{2x^2}-22{\mathrm{e}}^{x^2}+\frac{968x-1}{8x})-2\ln ({\mathrm{e}}^{x^2}-11)$	$35$
norman	$-\ln (x)-2\ln ({\mathrm{e}}^{x^2}-11)+\ln (8x{\mathrm{e}}^{2x^2}-176{\mathrm{e}}^{x^2}x+968x-1)$	$37$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^2+1)*exp(x^2)-11)/(8*x^2*exp(x^2)^3-264*x^2*exp(x^2)^2+(2904*x^2-x)*exp(x^2)-10648*x^2+11*x),x,metho
d=_RETURNVERBOSE)

[Out]

ln(exp(2*x^2)-22*exp(x^2)+1/8*(968*x-1)/x)-2*ln(exp(x^2)-11)

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maxima [A]  time = 0.41, size = 37, normalized size = 1.54 $$\begin{array}{c}\log \left(\frac{8x{e}^{\left(2x^2\right)}-176x{e}^{\left(x^2\right)}+968x-1}{8x}\right)-2\log (e^{\left(x^2\right)}-11)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+1)*exp(x^2)-11)/(8*x^2*exp(x^2)^3-264*x^2*exp(x^2)^2+(2904*x^2-x)*exp(x^2)-10648*x^2+11*x),x
, algorithm="maxima")

[Out]

log(1/8*(8*x*e^(2*x^2) - 176*x*e^(x^2) + 968*x - 1)/x) - 2*log(e^(x^2) - 11)

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mupad [B]  time = 4.54, size = 36, normalized size = 1.50 $$\begin{array}{c}\ln (968x-176x{\mathrm{e}}^{x^2}+8x{\mathrm{e}}^{2x^2}-1)-2\ln ({\mathrm{e}}^{x^2}-11)-\ln (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2)*(4*x^2 + 1) - 11)/(exp(x^2)*(x - 2904*x^2) - 11*x + 264*x^2*exp(2*x^2) - 8*x^2*exp(3*x^2) + 106
48*x^2),x)

[Out]

log(968*x - 176*x*exp(x^2) + 8*x*exp(2*x^2) - 1) - 2*log(exp(x^2) - 11) - log(x)

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sympy [A]  time = 0.36, size = 32, normalized size = 1.33 $$\begin{array}{c}-2\log (e^{x^2}-11)+\log (e^{2x^2}-22e^{x^2}+\frac{968x-1}{8x})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**2+1)*exp(x**2)-11)/(8*x**2*exp(x**2)**3-264*x**2*exp(x**2)**2+(2904*x**2-x)*exp(x**2)-10648*x
**2+11*x),x)

[Out]

-2*log(exp(x**2) - 11) + log(exp(2*x**2) - 22*exp(x**2) + (968*x - 1)/(8*x))

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