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

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

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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 {gather*} \int \frac {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 x^2\right )} \, dx \end {gather*}

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

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

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 {gather*} \log \left (\frac {8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1}{x}\right ) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \end {gather*}

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 {gather*} \log \left (8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1\right ) - \log \relax (x) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \end {gather*}

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	\(\ln \left ({\mathrm e}^{2 x^{2}}-22 \,{\mathrm e}^{x^{2}}+\frac {968 x -1}{8 x}\right )-2 \ln \left ({\mathrm e}^{x^{2}}-11\right )\)	\(35\)
norman	\(-\ln \relax (x )-2 \ln \left ({\mathrm e}^{x^{2}}-11\right )+\ln \left (8 x \,{\mathrm e}^{2 x^{2}}-176 \,{\mathrm e}^{x^{2}} x +968 x -1\right )\)	\(37\)

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 {gather*} \log \left (\frac {8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1}{8 \, x}\right ) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \end {gather*}

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 {gather*} \ln \left (968\,x-176\,x\,{\mathrm {e}}^{x^2}+8\,x\,{\mathrm {e}}^{2\,x^2}-1\right )-2\,\ln \left ({\mathrm {e}}^{x^2}-11\right )-\ln \relax (x) \end {gather*}

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 {gather*} - 2 \log {\left (e^{x^{2}} - 11 \right )} + \log {\left (e^{2 x^{2}} - 22 e^{x^{2}} + \frac {968 x - 1}{8 x} \right )} \end {gather*}

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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