3.77.84 \(\int \frac {4 e^2-4 x^2-4 \log (x)}{1+e^4-2 x^2+x^4+e^2 (-2+2 x^2)+(2-2 e^2-2 x^2) \log (x)+\log ^2(x)} \, dx\)

Optimal. Leaf size=17 \[ \frac {4 x}{-1+e^2+x^2-\log (x)} \]

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Rubi [F]  time = 0.30, 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 {4 e^2-4 x^2-4 \log (x)}{1+e^4-2 x^2+x^4+e^2 \left (-2+2 x^2\right )+\left (2-2 e^2-2 x^2\right ) \log (x)+\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*E^2 - 4*x^2 - 4*Log[x])/(1 + E^4 - 2*x^2 + x^4 + E^2*(-2 + 2*x^2) + (2 - 2*E^2 - 2*x^2)*Log[x] + Log[x]
^2),x]

[Out]

4*Defer[Int][(-1 + E^2 + x^2 - Log[x])^(-2), x] - 8*Defer[Int][x^2/(-1 + E^2 + x^2 - Log[x])^2, x] + 4*Defer[I
nt][(-1 + E^2 + x^2 - Log[x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (e^2-x^2-\log (x)\right )}{\left (1-e^2-x^2+\log (x)\right )^2} \, dx\\ &=4 \int \frac {e^2-x^2-\log (x)}{\left (1-e^2-x^2+\log (x)\right )^2} \, dx\\ &=4 \int \left (\frac {1-2 x^2}{\left (-1+e^2+x^2-\log (x)\right )^2}+\frac {1}{-1+e^2+x^2-\log (x)}\right ) \, dx\\ &=4 \int \frac {1-2 x^2}{\left (-1+e^2+x^2-\log (x)\right )^2} \, dx+4 \int \frac {1}{-1+e^2+x^2-\log (x)} \, dx\\ &=4 \int \left (\frac {1}{\left (-1+e^2+x^2-\log (x)\right )^2}-\frac {2 x^2}{\left (-1+e^2+x^2-\log (x)\right )^2}\right ) \, dx+4 \int \frac {1}{-1+e^2+x^2-\log (x)} \, dx\\ &=4 \int \frac {1}{\left (-1+e^2+x^2-\log (x)\right )^2} \, dx+4 \int \frac {1}{-1+e^2+x^2-\log (x)} \, dx-8 \int \frac {x^2}{\left (-1+e^2+x^2-\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 17, normalized size = 1.00 \begin {gather*} \frac {4 x}{-1+e^2+x^2-\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^2 - 4*x^2 - 4*Log[x])/(1 + E^4 - 2*x^2 + x^4 + E^2*(-2 + 2*x^2) + (2 - 2*E^2 - 2*x^2)*Log[x] +
Log[x]^2),x]

[Out]

(4*x)/(-1 + E^2 + x^2 - Log[x])

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fricas [A]  time = 0.91, size = 16, normalized size = 0.94 \begin {gather*} \frac {4 \, x}{x^{2} + e^{2} - \log \relax (x) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(x)+4*exp(2)-4*x^2)/(log(x)^2+(-2*exp(2)-2*x^2+2)*log(x)+exp(2)^2+(2*x^2-2)*exp(2)+x^4-2*x^2+
1),x, algorithm="fricas")

[Out]

4*x/(x^2 + e^2 - log(x) - 1)

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giac [A]  time = 0.15, size = 16, normalized size = 0.94 \begin {gather*} \frac {4 \, x}{x^{2} + e^{2} - \log \relax (x) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(x)+4*exp(2)-4*x^2)/(log(x)^2+(-2*exp(2)-2*x^2+2)*log(x)+exp(2)^2+(2*x^2-2)*exp(2)+x^4-2*x^2+
1),x, algorithm="giac")

[Out]

4*x/(x^2 + e^2 - log(x) - 1)

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maple [A]  time = 0.10, size = 17, normalized size = 1.00

method	result	size
norman	\(\frac {4 x}{x^{2}+{\mathrm e}^{2}-\ln \relax (x )-1}\)	\(17\)
risch	\(\frac {4 x}{x^{2}+{\mathrm e}^{2}-\ln \relax (x )-1}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*ln(x)+4*exp(2)-4*x^2)/(ln(x)^2+(-2*exp(2)-2*x^2+2)*ln(x)+exp(2)^2+(2*x^2-2)*exp(2)+x^4-2*x^2+1),x,meth
od=_RETURNVERBOSE)

[Out]

4*x/(x^2+exp(2)-ln(x)-1)

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maxima [A]  time = 0.40, size = 16, normalized size = 0.94 \begin {gather*} \frac {4 \, x}{x^{2} + e^{2} - \log \relax (x) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(x)+4*exp(2)-4*x^2)/(log(x)^2+(-2*exp(2)-2*x^2+2)*log(x)+exp(2)^2+(2*x^2-2)*exp(2)+x^4-2*x^2+
1),x, algorithm="maxima")

[Out]

4*x/(x^2 + e^2 - log(x) - 1)

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mupad [B]  time = 6.19, size = 16, normalized size = 0.94 \begin {gather*} \frac {4\,x}{{\mathrm {e}}^2-\ln \relax (x)+x^2-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*log(x) - 4*exp(2) + 4*x^2)/(exp(4) + log(x)^2 + exp(2)*(2*x^2 - 2) - log(x)*(2*exp(2) + 2*x^2 - 2) - 2
*x^2 + x^4 + 1),x)

[Out]

(4*x)/(exp(2) - log(x) + x^2 - 1)

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sympy [A]  time = 0.11, size = 15, normalized size = 0.88 \begin {gather*} - \frac {4 x}{- x^{2} + \log {\relax (x )} - e^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*ln(x)+4*exp(2)-4*x**2)/(ln(x)**2+(-2*exp(2)-2*x**2+2)*ln(x)+exp(2)**2+(2*x**2-2)*exp(2)+x**4-2*x
**2+1),x)

[Out]

-4*x/(-x**2 + log(x) - exp(2) + 1)

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