\(\int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+(-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)) \log (\frac {e^{-2 x} (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x))}{x^4})}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx\) [9066]

Optimal result

Integrand size = 125, antiderivative size = 33 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (6-\log (x)+\frac {e^{-2 x} \left (1-\frac {x+\log (x)}{x}\right )^2}{x^2}\right )}{x} \]

[Out]

ln((1-(x+ln(x))/x)^2/exp(x)^2/x^2-ln(x)+6)/x

Rubi [F]

\[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx \]

[In]

Int[(-(E^(2*x)*x^4) + 2*Log[x] + (-4 - 2*x)*Log[x]^2 + (-6*E^(2*x)*x^4 + E^(2*x)*x^4*Log[x] - Log[x]^2)*Log[(6
*E^(2*x)*x^4 - E^(2*x)*x^4*Log[x] + Log[x]^2)/(E^(2*x)*x^4)])/(6*E^(2*x)*x^6 - E^(2*x)*x^6*Log[x] + x^2*Log[x]
^2),x]

[Out]

ExpIntegralEi[6 - Log[x]]/E^6 + 12*Defer[Int][Log[x]/(x^2*(-6 + Log[x])*(-6*E^(2*x)*x^4 + E^(2*x)*x^4*Log[x] -
Log[x]^2)), x] - 25*Defer[Int][Log[x]^2/(x^2*(-6 + Log[x])*(-6*E^(2*x)*x^4 + E^(2*x)*x^4*Log[x] - Log[x]^2)),
x] - 12*Defer[Int][Log[x]^2/(x*(-6 + Log[x])*(-6*E^(2*x)*x^4 + E^(2*x)*x^4*Log[x] - Log[x]^2)), x] + 4*Defer[
Int][Log[x]^3/(x^2*(-6 + Log[x])*(-6*E^(2*x)*x^4 + E^(2*x)*x^4*Log[x] - Log[x]^2)), x] + 2*Defer[Int][Log[x]^3
/(x*(-6 + Log[x])*(-6*E^(2*x)*x^4 + E^(2*x)*x^4*Log[x] - Log[x]^2)), x] - Defer[Int][Log[6 - Log[x] + Log[x]^2
/(E^(2*x)*x^4)]/x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log (x) \left (12-25 \log (x)-12 x \log (x)+4 \log ^2(x)+2 x \log ^2(x)\right )}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}+\frac {1+6 \log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )-\log (x) \log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2 (-6+\log (x))}\right ) \, dx \\ & = \int \frac {\log (x) \left (12-25 \log (x)-12 x \log (x)+4 \log ^2(x)+2 x \log ^2(x)\right )}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+\int \frac {1+6 \log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )-\log (x) \log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2 (-6+\log (x))} \, dx \\ & = \int \left (\frac {12 \log (x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}-\frac {25 \log ^2(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}-\frac {12 \log ^2(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}+\frac {4 \log ^3(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}+\frac {2 \log ^3(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}\right ) \, dx+\int \frac {\frac {1}{-6+\log (x)}-\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2} \, dx \\ & = 2 \int \frac {\log ^3(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+4 \int \frac {\log ^3(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+12 \int \frac {\log (x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-12 \int \frac {\log ^2(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-25 \int \frac {\log ^2(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+\int \left (\frac {1}{x^2 (-6+\log (x))}-\frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2}\right ) \, dx \\ & = 2 \int \frac {\log ^3(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+4 \int \frac {\log ^3(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+12 \int \frac {\log (x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-12 \int \frac {\log ^2(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-25 \int \frac {\log ^2(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+\int \frac {1}{x^2 (-6+\log (x))} \, dx-\int \frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2} \, dx \\ & = 2 \int \frac {\log ^3(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+4 \int \frac {\log ^3(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+12 \int \frac {\log (x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-12 \int \frac {\log ^2(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-25 \int \frac {\log ^2(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-\int \frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2} \, dx+\text {Subst}\left (\int \frac {e^{-x}}{-6+x} \, dx,x,\log (x)\right ) \\ & = \frac {\operatorname {ExpIntegralEi}(6-\log (x))}{e^6}+2 \int \frac {\log ^3(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+4 \int \frac {\log ^3(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx+12 \int \frac {\log (x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-12 \int \frac {\log ^2(x)}{x (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-25 \int \frac {\log ^2(x)}{x^2 (-6+\log (x)) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )} \, dx-\int \frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=2+\frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x} \]

[In]

Integrate[(-(E^(2*x)*x^4) + 2*Log[x] + (-4 - 2*x)*Log[x]^2 + (-6*E^(2*x)*x^4 + E^(2*x)*x^4*Log[x] - Log[x]^2)*
Log[(6*E^(2*x)*x^4 - E^(2*x)*x^4*Log[x] + Log[x]^2)/(E^(2*x)*x^4)])/(6*E^(2*x)*x^6 - E^(2*x)*x^6*Log[x] + x^2*
Log[x]^2),x]

[Out]

2 + Log[6 - Log[x] + Log[x]^2/(E^(2*x)*x^4)]/x

Maple [A] (verified)

Time = 96.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18

method	result	size
parallelrisch	\(\frac {\ln \left (\frac {\left (\ln \left (x \right )^{2}-x^{4} {\mathrm e}^{2 x} \ln \left (x \right )+6 \,{\mathrm e}^{2 x} x^{4}\right ) {\mathrm e}^{-2 x}}{x^{4}}\right )}{x}\)	\(39\)
risch	\(\text {Expression too large to display}\)	\(722\)

[In]

int(((-ln(x)^2+x^4*exp(x)^2*ln(x)-6*exp(x)^2*x^4)*ln((ln(x)^2-x^4*exp(x)^2*ln(x)+6*exp(x)^2*x^4)/exp(x)^2/x^4)
+(-2*x-4)*ln(x)^2+2*ln(x)-exp(x)^2*x^4)/(x^2*ln(x)^2-x^6*exp(x)^2*ln(x)+6*x^6*exp(x)^2),x,method=_RETURNVERBOS
E)

[Out]

ln((ln(x)^2-x^4*exp(x)^2*ln(x)+6*exp(x)^2*x^4)/exp(x)^2/x^4)/x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (-\frac {{\left (x^{4} e^{\left (2 \, x\right )} \log \left (x\right ) - 6 \, x^{4} e^{\left (2 \, x\right )} - \log \left (x\right )^{2}\right )} e^{\left (-2 \, x\right )}}{x^{4}}\right )}{x} \]

[In]

integrate(((-log(x)^2+x^4*exp(x)^2*log(x)-6*exp(x)^2*x^4)*log((log(x)^2-x^4*exp(x)^2*log(x)+6*exp(x)^2*x^4)/ex
p(x)^2/x^4)+(-2*x-4)*log(x)^2+2*log(x)-exp(x)^2*x^4)/(x^2*log(x)^2-x^6*exp(x)^2*log(x)+6*x^6*exp(x)^2),x, algo
rithm="fricas")

[Out]

log(-(x^4*e^(2*x)*log(x) - 6*x^4*e^(2*x) - log(x)^2)*e^(-2*x)/x^4)/x

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log {\left (\frac {\left (- x^{4} e^{2 x} \log {\left (x \right )} + 6 x^{4} e^{2 x} + \log {\left (x \right )}^{2}\right ) e^{- 2 x}}{x^{4}} \right )}}{x} \]

[In]

integrate(((-ln(x)**2+x**4*exp(x)**2*ln(x)-6*exp(x)**2*x**4)*ln((ln(x)**2-x**4*exp(x)**2*ln(x)+6*exp(x)**2*x**
4)/exp(x)**2/x**4)+(-2*x-4)*ln(x)**2+2*ln(x)-exp(x)**2*x**4)/(x**2*ln(x)**2-x**6*exp(x)**2*ln(x)+6*x**6*exp(x)
**2),x)

[Out]

log((-x**4*exp(2*x)*log(x) + 6*x**4*exp(2*x) + log(x)**2)*exp(-2*x)/x**4)/x

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (-x^{4} e^{\left (2 \, x\right )} \log \left (x\right ) + 6 \, x^{4} e^{\left (2 \, x\right )} + \log \left (x\right )^{2}\right ) - 4 \, \log \left (x\right )}{x} \]

[In]

integrate(((-log(x)^2+x^4*exp(x)^2*log(x)-6*exp(x)^2*x^4)*log((log(x)^2-x^4*exp(x)^2*log(x)+6*exp(x)^2*x^4)/ex
p(x)^2/x^4)+(-2*x-4)*log(x)^2+2*log(x)-exp(x)^2*x^4)/(x^2*log(x)^2-x^6*exp(x)^2*log(x)+6*x^6*exp(x)^2),x, algo
rithm="maxima")

[Out]

(log(-x^4*e^(2*x)*log(x) + 6*x^4*e^(2*x) + log(x)^2) - 4*log(x))/x

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (-{\left (x^{4} e^{\left (2 \, x\right )} \log \left (x\right ) - 6 \, x^{4} e^{\left (2 \, x\right )} - \log \left (x\right )^{2}\right )} e^{\left (-2 \, x\right )}\right ) - 4 \, \log \left (x\right )}{x} \]

[In]

integrate(((-log(x)^2+x^4*exp(x)^2*log(x)-6*exp(x)^2*x^4)*log((log(x)^2-x^4*exp(x)^2*log(x)+6*exp(x)^2*x^4)/ex
p(x)^2/x^4)+(-2*x-4)*log(x)^2+2*log(x)-exp(x)^2*x^4)/(x^2*log(x)^2-x^6*exp(x)^2*log(x)+6*x^6*exp(x)^2),x, algo
rithm="giac")

[Out]

(log(-(x^4*e^(2*x)*log(x) - 6*x^4*e^(2*x) - log(x)^2)*e^(-2*x)) - 4*log(x))/x

Mupad [B] (verification not implemented)

Time = 13.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\ln \left (\frac {1}{x^4}\right )+\ln \left (6\,x^4-x^4\,\ln \left (x\right )+{\mathrm {e}}^{-2\,x}\,{\ln \left (x\right )}^2\right )}{x} \]

[In]

int(-(log((exp(-2*x)*(log(x)^2 + 6*x^4*exp(2*x) - x^4*exp(2*x)*log(x)))/x^4)*(log(x)^2 + 6*x^4*exp(2*x) - x^4*
exp(2*x)*log(x)) - 2*log(x) + x^4*exp(2*x) + log(x)^2*(2*x + 4))/(6*x^6*exp(2*x) + x^2*log(x)^2 - x^6*exp(2*x)
*log(x)),x)

[Out]

(log(1/x^4) + log(6*x^4 - x^4*log(x) + exp(-2*x)*log(x)^2))/x