3.92.91 \(\int \frac {4+e^4+x+(1+e^9 (-4-e^4-x)) \log (\frac {e^x}{4}) \log (\log (\frac {e^x}{4}))}{e^9 (-4 x-e^4 x-x^2) \log (\frac {e^x}{4}) \log (\log (\frac {e^x}{4}))+(4+e^4+x) \log (\frac {e^x}{4}) \log (\log (\frac {e^x}{4})) \log ((4+e^4+x) \log (\log (\frac {e^x}{4})))} \, dx\) [9191]

Optimal. Leaf size=27 \[ 5+\log \left (-e^9 x+\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right ) \]

[Out]

5+ln(ln((x+4+exp(4))*ln(ln(1/4*exp(x))))-x*exp(9))

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Rubi [F]
time = 2.68, 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^4+x+\left (1+e^9 \left (-4-e^4-x\right )\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )}{e^9 \left (-4 x-e^4 x-x^2\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )+\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 + E^4 + x + (1 + E^9*(-4 - E^4 - x))*Log[E^x/4]*Log[Log[E^x/4]])/(E^9*(-4*x - E^4*x - x^2)*Log[E^x/4]*L
og[Log[E^x/4]] + (4 + E^4 + x)*Log[E^x/4]*Log[Log[E^x/4]]*Log[(4 + E^4 + x)*Log[Log[E^x/4]]]),x]

[Out]

E^9*Defer[Int][(E^9*x - Log[(4 + E^4 + x)*Log[Log[E^x/4]]])^(-1), x] - E^9*(4 + E^4)*Defer[Int][1/((4 + E^4 +
x)*(E^9*x - Log[(4 + E^4 + x)*Log[Log[E^x/4]]])), x] - (1 - 4*E^9 - E^13)*Defer[Int][1/((4 + E^4 + x)*(E^9*x -
Log[(4 + E^4 + x)*Log[Log[E^x/4]]])), x] - Defer[Int][1/(Log[E^x/4]*Log[Log[E^x/4]]*(E^9*x - Log[(4 + E^4 + x
)*Log[Log[E^x/4]]])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 \left (1+\frac {e^4}{4}\right )-x-\left (1+e^9 \left (-4-e^4-x\right )\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )}{\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )} \, dx\\ &=\int \left (-\frac {1-e^9 \left (4+e^4\right )}{\left (4+e^4+x\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )}+\frac {e^9 x}{\left (4+e^4+x\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )}+\frac {-4-e^4}{\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )}-\frac {x}{\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )}\right ) \, dx\\ &=e^9 \int \frac {x}{\left (4+e^4+x\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )} \, dx+\left (-4-e^4\right ) \int \frac {1}{\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )} \, dx+\left (-1+4 e^9+e^{13}\right ) \int \frac {1}{\left (4+e^4+x\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )} \, dx-\int \frac {x}{\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )} \, dx\\ &=e^9 \int \left (\frac {1}{e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )}+\frac {-4-e^4}{\left (4+e^4+x\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )}\right ) \, dx+\left (-4-e^4\right ) \int \frac {1}{\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )} \, dx+\left (-1+4 e^9+e^{13}\right ) \int \frac {1}{\left (4+e^4+x\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )} \, dx-\int \left (\frac {1}{\log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )}+\frac {-4-e^4}{\left (4+e^4+x\right ) \log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )}\right ) \, dx\\ &=e^9 \int \frac {1}{e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )} \, dx-\left (e^9 \left (4+e^4\right )\right ) \int \frac {1}{\left (4+e^4+x\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )} \, dx+\left (-1+4 e^9+e^{13}\right ) \int \frac {1}{\left (4+e^4+x\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )} \, dx-\int \frac {1}{\log \left (\frac {e^x}{4}\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right ) \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.06, size = 26, normalized size = 0.96 \begin {gather*} \log \left (e^9 x-\log \left (\left (4+e^4+x\right ) \log \left (\log \left (\frac {e^x}{4}\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + E^4 + x + (1 + E^9*(-4 - E^4 - x))*Log[E^x/4]*Log[Log[E^x/4]])/(E^9*(-4*x - E^4*x - x^2)*Log[E^
x/4]*Log[Log[E^x/4]] + (4 + E^4 + x)*Log[E^x/4]*Log[Log[E^x/4]]*Log[(4 + E^4 + x)*Log[Log[E^x/4]]]),x]

[Out]

Log[E^9*x - Log[(4 + E^4 + x)*Log[Log[E^x/4]]]]

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Maple [F]
time = 27.83, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-{\mathrm e}^{4}-x -4\right ) {\mathrm e}^{9}+1\right ) \ln \left (\frac {{\mathrm e}^{x}}{4}\right ) \ln \left (\ln \left (\frac {{\mathrm e}^{x}}{4}\right )\right )+x +4+{\mathrm e}^{4}}{\left (x +4+{\mathrm e}^{4}\right ) \ln \left (\frac {{\mathrm e}^{x}}{4}\right ) \ln \left (\ln \left (\frac {{\mathrm e}^{x}}{4}\right )\right ) \ln \left (\left (x +4+{\mathrm e}^{4}\right ) \ln \left (\ln \left (\frac {{\mathrm e}^{x}}{4}\right )\right )\right )+\left (-x \,{\mathrm e}^{4}-x^{2}-4 x \right ) {\mathrm e}^{9} \ln \left (\frac {{\mathrm e}^{x}}{4}\right ) \ln \left (\ln \left (\frac {{\mathrm e}^{x}}{4}\right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-exp(4)-x-4)*exp(9)+1)*ln(1/4*exp(x))*ln(ln(1/4*exp(x)))+x+4+exp(4))/((x+4+exp(4))*ln(1/4*exp(x))*ln(ln
(1/4*exp(x)))*ln((x+4+exp(4))*ln(ln(1/4*exp(x))))+(-x*exp(4)-x^2-4*x)*exp(9)*ln(1/4*exp(x))*ln(ln(1/4*exp(x)))
),x)

[Out]

int((((-exp(4)-x-4)*exp(9)+1)*ln(1/4*exp(x))*ln(ln(1/4*exp(x)))+x+4+exp(4))/((x+4+exp(4))*ln(1/4*exp(x))*ln(ln
(1/4*exp(x)))*ln((x+4+exp(4))*ln(ln(1/4*exp(x))))+(-x*exp(4)-x^2-4*x)*exp(9)*ln(1/4*exp(x))*ln(ln(1/4*exp(x)))
),x)

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Maxima [A]
time = 0.60, size = 21, normalized size = 0.78 \begin {gather*} \log \left (-x e^{9} + \log \left (x + e^{4} + 4\right ) + \log \left (\log \left (x - 2 \, \log \left (2\right )\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4)-x-4)*exp(9)+1)*log(1/4*exp(x))*log(log(1/4*exp(x)))+x+4+exp(4))/((x+4+exp(4))*log(1/4*exp
(x))*log(log(1/4*exp(x)))*log((x+4+exp(4))*log(log(1/4*exp(x))))+(-x*exp(4)-x^2-4*x)*exp(9)*log(1/4*exp(x))*lo
g(log(1/4*exp(x)))),x, algorithm="maxima")

[Out]

log(-x*e^9 + log(x + e^4 + 4) + log(log(x - 2*log(2))))

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Fricas [A]
time = 0.38, size = 21, normalized size = 0.78 \begin {gather*} \log \left (-x e^{9} + \log \left ({\left (x + e^{4} + 4\right )} \log \left (x - 2 \, \log \left (2\right )\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4)-x-4)*exp(9)+1)*log(1/4*exp(x))*log(log(1/4*exp(x)))+x+4+exp(4))/((x+4+exp(4))*log(1/4*exp
(x))*log(log(1/4*exp(x)))*log((x+4+exp(4))*log(log(1/4*exp(x))))+(-x*exp(4)-x^2-4*x)*exp(9)*log(1/4*exp(x))*lo
g(log(1/4*exp(x)))),x, algorithm="fricas")

[Out]

log(-x*e^9 + log((x + e^4 + 4)*log(x - 2*log(2))))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4)-x-4)*exp(9)+1)*ln(1/4*exp(x))*ln(ln(1/4*exp(x)))+x+4+exp(4))/((x+4+exp(4))*ln(1/4*exp(x))
*ln(ln(1/4*exp(x)))*ln((x+4+exp(4))*ln(ln(1/4*exp(x))))+(-x*exp(4)-x**2-4*x)*exp(9)*ln(1/4*exp(x))*ln(ln(1/4*e
xp(x)))),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6441 deep

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Giac [A]
time = 0.47, size = 21, normalized size = 0.78 \begin {gather*} \log \left (-x e^{9} + \log \left (x + e^{4} + 4\right ) + \log \left (\log \left (x - 2 \, \log \left (2\right )\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4)-x-4)*exp(9)+1)*log(1/4*exp(x))*log(log(1/4*exp(x)))+x+4+exp(4))/((x+4+exp(4))*log(1/4*exp
(x))*log(log(1/4*exp(x)))*log((x+4+exp(4))*log(log(1/4*exp(x))))+(-x*exp(4)-x^2-4*x)*exp(9)*log(1/4*exp(x))*lo
g(log(1/4*exp(x)))),x, algorithm="giac")

[Out]

log(-x*e^9 + log(x + e^4 + 4) + log(log(x - 2*log(2))))

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Mupad [B]
time = 8.10, size = 21, normalized size = 0.78 \begin {gather*} \ln \left (\ln \left (\ln \left (x-\ln \left (4\right )\right )\,\left (x+{\mathrm {e}}^4+4\right )\right )-x\,{\mathrm {e}}^9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + exp(4) - log(log(exp(x)/4))*log(exp(x)/4)*(exp(9)*(x + exp(4) + 4) - 1) + 4)/(log(log(exp(x)/4))*log(
log(log(exp(x)/4))*(x + exp(4) + 4))*log(exp(x)/4)*(x + exp(4) + 4) - log(log(exp(x)/4))*exp(9)*log(exp(x)/4)*
(4*x + x*exp(4) + x^2)),x)

[Out]

log(log(log(x - log(4))*(x + exp(4) + 4)) - x*exp(9))

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