3.53.51 \(\int \frac {-\log ^2(x)+e^{e^{\frac {x+(-8 x-4 x^2-4 \log (4)) \log (x)}{\log (x)}}+\frac {x+(-8 x-4 x^2-4 \log (4)) \log (x)}{\log (x)}} (4-4 \log (x)+(32+32 x) \log ^2(x))}{4 \log ^2(x)} \, dx\)

Optimal. Leaf size=37 \[ 4-e^{e^{4 \left (-2 x-x^2-\log (4)+\frac {x}{4 \log (x)}\right )}}-\frac {x}{4} \]

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Rubi [F]  time = 3.29, 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 {-\log ^2(x)+\exp \left (\exp \left (\frac {x+\left (-8 x-4 x^2-4 \log (4)\right ) \log (x)}{\log (x)}\right )+\frac {x+\left (-8 x-4 x^2-4 \log (4)\right ) \log (x)}{\log (x)}\right ) \left (4-4 \log (x)+(32+32 x) \log ^2(x)\right )}{4 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-Log[x]^2 + E^(E^((x + (-8*x - 4*x^2 - 4*Log[4])*Log[x])/Log[x]) + (x + (-8*x - 4*x^2 - 4*Log[4])*Log[x])
/Log[x])*(4 - 4*Log[x] + (32 + 32*x)*Log[x]^2))/(4*Log[x]^2),x]

[Out]

-1/4*x + Defer[Int][E^(E^(x*(-4*(2 + x) + Log[x]^(-1)))/256 - 4*x*(2 + x) + x/Log[x]), x]/32 + Defer[Int][E^(E
^(x*(-4*(2 + x) + Log[x]^(-1)))/256 - 4*x*(2 + x) + x/Log[x])*x, x]/32 + Defer[Int][E^(E^(x*(-4*(2 + x) + Log[
x]^(-1)))/256 - 4*x*(2 + x) + x/Log[x])/Log[x]^2, x]/256 - Defer[Int][E^(E^(x*(-4*(2 + x) + Log[x]^(-1)))/256
- 4*x*(2 + x) + x/Log[x])/Log[x], x]/256

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {-\log ^2(x)+\exp \left (\exp \left (\frac {x+\left (-8 x-4 x^2-4 \log (4)\right ) \log (x)}{\log (x)}\right )+\frac {x+\left (-8 x-4 x^2-4 \log (4)\right ) \log (x)}{\log (x)}\right ) \left (4-4 \log (x)+(32+32 x) \log ^2(x)\right )}{\log ^2(x)} \, dx\\ &=\frac {1}{4} \int \left (-1+\frac {\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right ) \left (1-\log (x)+8 (1+x) \log ^2(x)\right )}{64 \log ^2(x)}\right ) \, dx\\ &=-\frac {x}{4}+\frac {1}{256} \int \frac {\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right ) \left (1-\log (x)+8 (1+x) \log ^2(x)\right )}{\log ^2(x)} \, dx\\ &=-\frac {x}{4}+\frac {1}{256} \int \left (8 \exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right ) (1+x)+\frac {\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right )}{\log ^2(x)}-\frac {\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right )}{\log (x)}\right ) \, dx\\ &=-\frac {x}{4}+\frac {1}{256} \int \frac {\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right )}{\log ^2(x)} \, dx-\frac {1}{256} \int \frac {\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right )}{\log (x)} \, dx+\frac {1}{32} \int \exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right ) (1+x) \, dx\\ &=-\frac {x}{4}+\frac {1}{256} \int \frac {\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right )}{\log ^2(x)} \, dx-\frac {1}{256} \int \frac {\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right )}{\log (x)} \, dx+\frac {1}{32} \int \left (\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right )+\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right ) x\right ) \, dx\\ &=-\frac {x}{4}+\frac {1}{256} \int \frac {\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right )}{\log ^2(x)} \, dx-\frac {1}{256} \int \frac {\exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right )}{\log (x)} \, dx+\frac {1}{32} \int \exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right ) \, dx+\frac {1}{32} \int \exp \left (\frac {1}{256} e^{x \left (-4 (2+x)+\frac {1}{\log (x)}\right )}-4 x (2+x)+\frac {x}{\log (x)}\right ) x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.56, size = 31, normalized size = 0.84 \begin {gather*} -e^{\frac {1}{256} e^{-8 x-4 x^2+\frac {x}{\log (x)}}}-\frac {x}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-Log[x]^2 + E^(E^((x + (-8*x - 4*x^2 - 4*Log[4])*Log[x])/Log[x]) + (x + (-8*x - 4*x^2 - 4*Log[4])*L
og[x])/Log[x])*(4 - 4*Log[x] + (32 + 32*x)*Log[x]^2))/(4*Log[x]^2),x]

[Out]

-E^(E^(-8*x - 4*x^2 + x/Log[x])/256) - x/4

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fricas [B]  time = 0.60, size = 114, normalized size = 3.08 \begin {gather*} -\frac {1}{4} \, {\left (x e^{\left (-\frac {4 \, {\left (x^{2} + 2 \, x + 2 \, \log \relax (2)\right )} \log \relax (x) - x}{\log \relax (x)}\right )} + 4 \, e^{\left (-\frac {4 \, {\left (x^{2} + 2 \, x + 2 \, \log \relax (2)\right )} \log \relax (x) - e^{\left (-\frac {4 \, {\left (x^{2} + 2 \, x + 2 \, \log \relax (2)\right )} \log \relax (x) - x}{\log \relax (x)}\right )} \log \relax (x) - x}{\log \relax (x)}\right )}\right )} e^{\left (\frac {4 \, {\left (x^{2} + 2 \, x + 2 \, \log \relax (2)\right )} \log \relax (x) - x}{\log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((32*x+32)*log(x)^2-4*log(x)+4)*exp(((-8*log(2)-4*x^2-8*x)*log(x)+x)/log(x))*exp(exp(((-8*log(2
)-4*x^2-8*x)*log(x)+x)/log(x)))-log(x)^2)/log(x)^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((32*x+32)*log(x)^2-4*log(x)+4)*exp(((-8*log(2)-4*x^2-8*x)*log(x)+x)/log(x))*exp(exp(((-8*log(2
)-4*x^2-8*x)*log(x)+x)/log(x)))-log(x)^2)/log(x)^2,x, algorithm="giac")

[Out]

undef

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maple [A]  time = 0.09, size = 29, normalized size = 0.78

method	result	size
risch	\(-\frac {x}{4}-{\mathrm e}^{\frac {{\mathrm e}^{-\frac {x \left (4 x \ln \relax (x )+8 \ln \relax (x )-1\right )}{\ln \relax (x )}}}{256}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(((32*x+32)*ln(x)^2-4*ln(x)+4)*exp(((-8*ln(2)-4*x^2-8*x)*ln(x)+x)/ln(x))*exp(exp(((-8*ln(2)-4*x^2-8*x)
*ln(x)+x)/ln(x)))-ln(x)^2)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/4*x-exp(1/256*exp(-x*(4*x*ln(x)+8*ln(x)-1)/ln(x)))

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maxima [A]  time = 0.58, size = 25, normalized size = 0.68 \begin {gather*} -\frac {1}{4} \, x - e^{\left (\frac {1}{256} \, e^{\left (-4 \, x^{2} - 8 \, x + \frac {x}{\log \relax (x)}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((32*x+32)*log(x)^2-4*log(x)+4)*exp(((-8*log(2)-4*x^2-8*x)*log(x)+x)/log(x))*exp(exp(((-8*log(2
)-4*x^2-8*x)*log(x)+x)/log(x)))-log(x)^2)/log(x)^2,x, algorithm="maxima")

[Out]

-1/4*x - e^(1/256*e^(-4*x^2 - 8*x + x/log(x)))

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mupad [B]  time = 3.89, size = 26, normalized size = 0.70 \begin {gather*} -\frac {x}{4}-{\mathrm {e}}^{\frac {{\mathrm {e}}^{-8\,x}\,{\mathrm {e}}^{\frac {x}{\ln \relax (x)}}\,{\mathrm {e}}^{-4\,x^2}}{256}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^2/4 - (exp((x - log(x)*(8*x + 8*log(2) + 4*x^2))/log(x))*exp(exp((x - log(x)*(8*x + 8*log(2) + 4*
x^2))/log(x)))*(log(x)^2*(32*x + 32) - 4*log(x) + 4))/4)/log(x)^2,x)

[Out]

- x/4 - exp((exp(-8*x)*exp(x/log(x))*exp(-4*x^2))/256)

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sympy [A]  time = 0.95, size = 31, normalized size = 0.84 \begin {gather*} - \frac {x}{4} - e^{e^{\frac {x + \left (- 4 x^{2} - 8 x - 8 \log {\relax (2 )}\right ) \log {\relax (x )}}{\log {\relax (x )}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((32*x+32)*ln(x)**2-4*ln(x)+4)*exp(((-8*ln(2)-4*x**2-8*x)*ln(x)+x)/ln(x))*exp(exp(((-8*ln(2)-4*
x**2-8*x)*ln(x)+x)/ln(x)))-ln(x)**2)/ln(x)**2,x)

[Out]

-x/4 - exp(exp((x + (-4*x**2 - 8*x - 8*log(2))*log(x))/log(x)))

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