3.55.70 \(\int \frac {(-1+2 x) \log (2)+e^{x^2+2 x \log (\log (4))+\log ^2(\log (4))} (-2 x^2 \log (2)-2 x \log (2) \log (\log (4)))}{x+e^{2 x^2+4 x \log (\log (4))+2 \log ^2(\log (4))} x+4 x^2+4 x^3+(-2 x-4 x^2) \log (x)+x \log ^2(x)+e^{x^2+2 x \log (\log (4))+\log ^2(\log (4))} (-2 x-4 x^2+2 x \log (x))} \, dx\) [5470]

Optimal. Leaf size=21 \[ \frac {\log (2)}{-1+e^{(x+\log (\log (4)))^2}-2 x+\log (x)} \]

[Out]

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

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Rubi [F]
time = 3.85, 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 {(-1+2 x) \log (2)+e^{x^2+2 x \log (\log (4))+\log ^2(\log (4))} \left (-2 x^2 \log (2)-2 x \log (2) \log (\log (4))\right )}{x+e^{2 x^2+4 x \log (\log (4))+2 \log ^2(\log (4))} x+4 x^2+4 x^3+\left (-2 x-4 x^2\right ) \log (x)+x \log ^2(x)+e^{x^2+2 x \log (\log (4))+\log ^2(\log (4))} \left (-2 x-4 x^2+2 x \log (x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + 2*x)*Log[2] + E^(x^2 + 2*x*Log[Log[4]] + Log[Log[4]]^2)*(-2*x^2*Log[2] - 2*x*Log[2]*Log[Log[4]]))/(
x + E^(2*x^2 + 4*x*Log[Log[4]] + 2*Log[Log[4]]^2)*x + 4*x^2 + 4*x^3 + (-2*x - 4*x^2)*Log[x] + x*Log[x]^2 + E^(
x^2 + 2*x*Log[Log[4]] + Log[Log[4]]^2)*(-2*x - 4*x^2 + 2*x*Log[x])),x]

[Out]

-(Log[2]*Defer[Int][1/(x*(1 + 2*x - E^(x^2 + Log[Log[4]]^2)*Log[4]^(2*x) - Log[x])^2), x]) - 2*Log[2]*(1 + 2*L
og[Log[4]])*Defer[Int][x/(1 + 2*x - E^(x^2 + Log[Log[4]]^2)*Log[4]^(2*x) - Log[x])^2, x] - 4*Log[2]*Defer[Int]
[x^2/(1 + 2*x - E^(x^2 + Log[Log[4]]^2)*Log[4]^(2*x) - Log[x])^2, x] + 2*Log[2]*Defer[Int][x/(1 + 2*x - E^(x^2
+ Log[Log[4]]^2)*Log[4]^(2*x) - Log[x]), x] + 2*Log[2]*Defer[Int][(x*Log[x])/(1 + 2*x - E^(x^2 + Log[Log[4]]^
2)*Log[4]^(2*x) - Log[x])^2, x] + 2*Log[2]*(1 - Log[Log[4]])*Defer[Int][(-1 - 2*x + E^(x^2 + Log[Log[4]]^2)*Lo
g[4]^(2*x) + Log[x])^(-2), x] + 2*Log[2]*Log[Log[4]]*Defer[Int][Log[x]/(-1 - 2*x + E^(x^2 + Log[Log[4]]^2)*Log
[4]^(2*x) + Log[x])^2, x] - 2*Log[2]*Log[Log[4]]*Defer[Int][(-1 - 2*x + E^(x^2 + Log[Log[4]]^2)*Log[4]^(2*x) +
Log[x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\log (2) \left (-1-2 e^{x^2+\log ^2(\log (4))} x^2 \log ^{2 x}(4)-2 x \left (-1+e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4) \log (\log (4))\right )\right )}{x \left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2} \, dx\\ &=\log (2) \int \frac {-1-2 e^{x^2+\log ^2(\log (4))} x^2 \log ^{2 x}(4)-2 x \left (-1+e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4) \log (\log (4))\right )}{x \left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2} \, dx\\ &=\log (2) \int \left (\frac {2 (x+\log (\log (4)))}{1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)}+\frac {-1-4 x^3+2 x^2 \log (x)+2 x (1-\log (\log (4)))+2 x \log (x) \log (\log (4))-2 x^2 (1+2 \log (\log (4)))}{x \left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2}\right ) \, dx\\ &=\log (2) \int \frac {-1-4 x^3+2 x^2 \log (x)+2 x (1-\log (\log (4)))+2 x \log (x) \log (\log (4))-2 x^2 (1+2 \log (\log (4)))}{x \left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2} \, dx+(2 \log (2)) \int \frac {x+\log (\log (4))}{1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)} \, dx\\ &=\log (2) \int \left (-\frac {1}{x \left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2}-\frac {4 x^2}{\left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2}+\frac {2 x \log (x)}{\left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2}-\frac {2 (-1+\log (\log (4)))}{\left (-1-2 x+e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)+\log (x)\right )^2}+\frac {2 \log (x) \log (\log (4))}{\left (-1-2 x+e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)+\log (x)\right )^2}-\frac {2 x (1+2 \log (\log (4)))}{\left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2}\right ) \, dx+(2 \log (2)) \int \left (\frac {x}{1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)}-\frac {\log (\log (4))}{-1-2 x+e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)+\log (x)}\right ) \, dx\\ &=-\left (\log (2) \int \frac {1}{x \left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2} \, dx\right )+(2 \log (2)) \int \frac {x}{1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)} \, dx+(2 \log (2)) \int \frac {x \log (x)}{\left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2} \, dx-(4 \log (2)) \int \frac {x^2}{\left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2} \, dx+(2 \log (2) (1-\log (\log (4)))) \int \frac {1}{\left (-1-2 x+e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)+\log (x)\right )^2} \, dx+(2 \log (2) \log (\log (4))) \int \frac {\log (x)}{\left (-1-2 x+e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)+\log (x)\right )^2} \, dx-(2 \log (2) \log (\log (4))) \int \frac {1}{-1-2 x+e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)+\log (x)} \, dx-(2 \log (2) (1+2 \log (\log (4)))) \int \frac {x}{\left (1+2 x-e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)-\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.09, size = 30, normalized size = 1.43 \begin {gather*} \frac {\log (2)}{-1-2 x+e^{x^2+\log ^2(\log (4))} \log ^{2 x}(4)+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + 2*x)*Log[2] + E^(x^2 + 2*x*Log[Log[4]] + Log[Log[4]]^2)*(-2*x^2*Log[2] - 2*x*Log[2]*Log[Log[4
]]))/(x + E^(2*x^2 + 4*x*Log[Log[4]] + 2*Log[Log[4]]^2)*x + 4*x^2 + 4*x^3 + (-2*x - 4*x^2)*Log[x] + x*Log[x]^2
+ E^(x^2 + 2*x*Log[Log[4]] + Log[Log[4]]^2)*(-2*x - 4*x^2 + 2*x*Log[x])),x]

[Out]

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

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Maple [A]
time = 6.65, size = 28, normalized size = 1.33

method	result	size
risch	\(-\frac {\ln \left (2\right )}{2 x -{\mathrm e}^{\left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )+x \right )^{2}}-\ln \left (x \right )+1}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x*ln(2)*ln(2*ln(2))-2*x^2*ln(2))*exp(ln(2*ln(2))^2+2*x*ln(2*ln(2))+x^2)+(2*x-1)*ln(2))/(x*exp(ln(2*ln
(2))^2+2*x*ln(2*ln(2))+x^2)^2+(2*x*ln(x)-4*x^2-2*x)*exp(ln(2*ln(2))^2+2*x*ln(2*ln(2))+x^2)+x*ln(x)^2+(-4*x^2-2
*x)*ln(x)+4*x^3+4*x^2+x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.56, size = 45, normalized size = 2.14 \begin {gather*} \frac {\log \left (2\right )}{2^{2 \, \log \left (\log \left (2\right )\right )} e^{\left (x^{2} + 2 \, x \log \left (2\right ) + \log \left (2\right )^{2} + 2 \, x \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} - 2 \, x + \log \left (x\right ) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(2)*log(2*log(2))-2*x^2*log(2))*exp(log(2*log(2))^2+2*x*log(2*log(2))+x^2)+(-1+2*x)*log(2)
)/(x*exp(log(2*log(2))^2+2*x*log(2*log(2))+x^2)^2+(2*x*log(x)-4*x^2-2*x)*exp(log(2*log(2))^2+2*x*log(2*log(2))
+x^2)+x*log(x)^2+(-4*x^2-2*x)*log(x)+4*x^3+4*x^2+x),x, algorithm="maxima")

[Out]

log(2)/(2^(2*log(log(2)))*e^(x^2 + 2*x*log(2) + log(2)^2 + 2*x*log(log(2)) + log(log(2))^2) - 2*x + log(x) - 1
)

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Fricas [A]
time = 0.35, size = 37, normalized size = 1.76 \begin {gather*} -\frac {\log \left (2\right )}{2 \, x - e^{\left (x^{2} + 2 \, x \log \left (2 \, \log \left (2\right )\right ) + \log \left (2 \, \log \left (2\right )\right )^{2}\right )} - \log \left (x\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(2)*log(2*log(2))-2*x^2*log(2))*exp(log(2*log(2))^2+2*x*log(2*log(2))+x^2)+(-1+2*x)*log(2)
)/(x*exp(log(2*log(2))^2+2*x*log(2*log(2))+x^2)^2+(2*x*log(x)-4*x^2-2*x)*exp(log(2*log(2))^2+2*x*log(2*log(2))
+x^2)+x*log(x)^2+(-4*x^2-2*x)*log(x)+4*x^3+4*x^2+x),x, algorithm="fricas")

[Out]

-log(2)/(2*x - e^(x^2 + 2*x*log(2*log(2)) + log(2*log(2))^2) - log(x) + 1)

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Sympy [A]
time = 0.18, size = 34, normalized size = 1.62 \begin {gather*} \frac {\log {\left (2 \right )}}{- 2 x + e^{x^{2} + 2 x \log {\left (2 \log {\left (2 \right )} \right )} + \log {\left (2 \log {\left (2 \right )} \right )}^{2}} + \log {\left (x \right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*ln(2)*ln(2*ln(2))-2*x**2*ln(2))*exp(ln(2*ln(2))**2+2*x*ln(2*ln(2))+x**2)+(-1+2*x)*ln(2))/(x*e
xp(ln(2*ln(2))**2+2*x*ln(2*ln(2))+x**2)**2+(2*x*ln(x)-4*x**2-2*x)*exp(ln(2*ln(2))**2+2*x*ln(2*ln(2))+x**2)+x*l
n(x)**2+(-4*x**2-2*x)*ln(x)+4*x**3+4*x**2+x),x)

[Out]

log(2)/(-2*x + exp(x**2 + 2*x*log(2*log(2)) + log(2*log(2))**2) + log(x) - 1)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(2)*log(2*log(2))-2*x^2*log(2))*exp(log(2*log(2))^2+2*x*log(2*log(2))+x^2)+(-1+2*x)*log(2)
)/(x*exp(log(2*log(2))^2+2*x*log(2*log(2))+x^2)^2+(2*x*log(x)-4*x^2-2*x)*exp(log(2*log(2))^2+2*x*log(2*log(2))
+x^2)+x*log(x)^2+(-4*x^2-2*x)*log(x)+4*x^3+4*x^2+x),x, algorithm="giac")

[Out]

undef

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Mupad [B]
time = 4.06, size = 65, normalized size = 3.10 \begin {gather*} -\frac {2\,\ln \left (2\right )\,\left (x+\ln \left (\ln \left (4\right )\right )\right )}{\left (2\,x+\ln \left ({\ln \left (4\right )}^2\right )\right )\,\left (2\,x-\ln \left (x\right )-2^{2\,x}\,2^{2\,\ln \left (\ln \left (2\right )\right )}\,{\mathrm {e}}^{x^2+{\ln \left (\ln \left (2\right )\right )}^2+{\ln \left (2\right )}^2}\,{\ln \left (2\right )}^{2\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2)*(2*x - 1) - exp(2*x*log(2*log(2)) + log(2*log(2))^2 + x^2)*(2*x^2*log(2) + 2*x*log(2*log(2))*log(2
)))/(x + x*log(x)^2 - log(x)*(2*x + 4*x^2) - exp(2*x*log(2*log(2)) + log(2*log(2))^2 + x^2)*(2*x - 2*x*log(x)
+ 4*x^2) + 4*x^2 + 4*x^3 + x*exp(4*x*log(2*log(2)) + 2*log(2*log(2))^2 + 2*x^2)),x)

[Out]

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

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