\(\int \frac {-2 \log (-2 x^3 \log (3))+\log (x^2) (3 e^x-e^x x \log (-2 x^3 \log (3)))+3 \log (x^2) \log (\log (x^2))}{e^x x \log (x^2) \log (-2 x^3 \log (3))+x \log (x^2) \log (-2 x^3 \log (3)) \log (\log (x^2))} \, dx\) [9795]

Optimal result

Integrand size = 87, antiderivative size = 21 \[ \int \frac {-2 \log \left (-2 x^3 \log (3)\right )+\log \left (x^2\right ) \left (3 e^x-e^x x \log \left (-2 x^3 \log (3)\right )\right )+3 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{e^x x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right )+x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\log \left (\frac {\log \left (-2 x^3 \log (3)\right )}{e^x+\log \left (\log \left (x^2\right )\right )}\right ) \]

[Out]

ln(ln(-2*x^3*ln(3))/(ln(ln(x^2))+exp(x)))

Rubi [F]

\[ \int \frac {-2 \log \left (-2 x^3 \log (3)\right )+\log \left (x^2\right ) \left (3 e^x-e^x x \log \left (-2 x^3 \log (3)\right )\right )+3 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{e^x x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right )+x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\int \frac {-2 \log \left (-2 x^3 \log (3)\right )+\log \left (x^2\right ) \left (3 e^x-e^x x \log \left (-2 x^3 \log (3)\right )\right )+3 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{e^x x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right )+x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right ) \log \left (\log \left (x^2\right )\right )} \, dx \]

[In]

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

[Out]

-x + Log[Log[-(x^3*Log[9])]] - 2*Defer[Int][1/(x*Log[x^2]*(E^x + Log[Log[x^2]])), x] + Defer[Int][Log[Log[x^2]
]/(E^x + Log[Log[x^2]]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 \log \left (-2 x^3 \log (3)\right )+\log \left (x^2\right ) \left (3 e^x-e^x x \log \left (-2 x^3 \log (3)\right )\right )+3 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{x \log \left (x^2\right ) \log \left (-x^3 \log (9)\right ) \left (e^x+\log \left (\log \left (x^2\right )\right )\right )} \, dx \\ & = \int \left (\frac {3-x \log \left (-x^3 \log (9)\right )}{x \log \left (-x^3 \log (9)\right )}+\frac {-2+x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{x \log \left (x^2\right ) \left (e^x+\log \left (\log \left (x^2\right )\right )\right )}\right ) \, dx \\ & = \int \frac {3-x \log \left (-x^3 \log (9)\right )}{x \log \left (-x^3 \log (9)\right )} \, dx+\int \frac {-2+x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{x \log \left (x^2\right ) \left (e^x+\log \left (\log \left (x^2\right )\right )\right )} \, dx \\ & = \int \left (-1+\frac {3}{x \log \left (-x^3 \log (9)\right )}\right ) \, dx+\int \left (-\frac {2}{x \log \left (x^2\right ) \left (e^x+\log \left (\log \left (x^2\right )\right )\right )}+\frac {\log \left (\log \left (x^2\right )\right )}{e^x+\log \left (\log \left (x^2\right )\right )}\right ) \, dx \\ & = -x-2 \int \frac {1}{x \log \left (x^2\right ) \left (e^x+\log \left (\log \left (x^2\right )\right )\right )} \, dx+3 \int \frac {1}{x \log \left (-x^3 \log (9)\right )} \, dx+\int \frac {\log \left (\log \left (x^2\right )\right )}{e^x+\log \left (\log \left (x^2\right )\right )} \, dx \\ & = -x-2 \int \frac {1}{x \log \left (x^2\right ) \left (e^x+\log \left (\log \left (x^2\right )\right )\right )} \, dx+\int \frac {\log \left (\log \left (x^2\right )\right )}{e^x+\log \left (\log \left (x^2\right )\right )} \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (-x^3 \log (9)\right )\right ) \\ & = -x+\log \left (\log \left (-x^3 \log (9)\right )\right )-2 \int \frac {1}{x \log \left (x^2\right ) \left (e^x+\log \left (\log \left (x^2\right )\right )\right )} \, dx+\int \frac {\log \left (\log \left (x^2\right )\right )}{e^x+\log \left (\log \left (x^2\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {-2 \log \left (-2 x^3 \log (3)\right )+\log \left (x^2\right ) \left (3 e^x-e^x x \log \left (-2 x^3 \log (3)\right )\right )+3 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{e^x x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right )+x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\log \left (3 \log \left (x^2\right )+2 \left (-\frac {3}{2} \log \left (x^2\right )+\log \left (-x^3 \log (9)\right )\right )\right )-\log \left (e^x+\log \left (\log \left (x^2\right )\right )\right ) \]

[In]

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

[Out]

Log[3*Log[x^2] + 2*((-3*Log[x^2])/2 + Log[-(x^3*Log[9])])] - Log[E^x + Log[Log[x^2]]]

Maple [A] (verified)

Time = 3.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05

method	result	size
parallelrisch	\(\ln \left (\ln \left (-2 x^{3} \ln \left (3\right )\right )\right )-\ln \left (\ln \left (\ln \left (x^{2}\right )\right )+{\mathrm e}^{x}\right )\)	\(22\)
risch	\(\ln \left (\frac {\ln \left (2\right )}{3}+\frac {\ln \left (\ln \left (3\right )\right )}{3}+\frac {i \pi }{3}+\ln \left (x \right )-\frac {i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{6}+\frac {i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}}{3}-\frac {i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{6}+\frac {i \pi \operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x \right )}{6}+\frac {i \pi \operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{6}-\frac {i \pi \,\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )}{6}-\frac {i \pi \operatorname {csgn}\left (i x^{3}\right )^{2}}{3}+\frac {i \pi \operatorname {csgn}\left (i x^{3}\right )^{3}}{6}\right )-\ln \left ({\mathrm e}^{x}+\ln \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )\right )\)	\(195\)

[In]

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

[Out]

ln(ln(-2*x^3*ln(3)))-ln(ln(ln(x^2))+exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {-2 \log \left (-2 x^3 \log (3)\right )+\log \left (x^2\right ) \left (3 e^x-e^x x \log \left (-2 x^3 \log (3)\right )\right )+3 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{e^x x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right )+x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\log \left (e^{x} + \log \left (\frac {2}{3} \, \log \left (-2 \, x^{3} \log \left (3\right )\right ) - \frac {1}{3} \, \log \left (4 \, \log \left (3\right )^{2}\right )\right )\right ) + \log \left (\log \left (-2 \, x^{3} \log \left (3\right )\right )\right ) \]

[In]

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

[Out]

-log(e^x + log(2/3*log(-2*x^3*log(3)) - 1/3*log(4*log(3)^2))) + log(log(-2*x^3*log(3)))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {-2 \log \left (-2 x^3 \log (3)\right )+\log \left (x^2\right ) \left (3 e^x-e^x x \log \left (-2 x^3 \log (3)\right )\right )+3 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{e^x x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right )+x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right ) \log \left (\log \left (x^2\right )\right )} \, dx=- \log {\left (e^{x} + \log {\left (\log {\left (x^{2} \right )} \right )} \right )} + \log {\left (\log {\left (x^{2} \right )} + \frac {2 \log {\left (\log {\left (3 \right )} \right )}}{3} + \frac {2 \log {\left (2 \right )}}{3} + \frac {2 i \pi }{3} \right )} \]

[In]

integrate((3*ln(x**2)*ln(ln(x**2))+(-x*exp(x)*ln(-2*x**3*ln(3))+3*exp(x))*ln(x**2)-2*ln(-2*x**3*ln(3)))/(x*ln(
-2*x**3*ln(3))*ln(x**2)*ln(ln(x**2))+x*exp(x)*ln(-2*x**3*ln(3))*ln(x**2)),x)

[Out]

-log(exp(x) + log(log(x**2))) + log(log(x**2) + 2*log(log(3))/3 + 2*log(2)/3 + 2*I*pi/3)

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {-2 \log \left (-2 x^3 \log (3)\right )+\log \left (x^2\right ) \left (3 e^x-e^x x \log \left (-2 x^3 \log (3)\right )\right )+3 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{e^x x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right )+x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\log \left (e^{x} + \log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) + \log \left (\frac {1}{3} \, \log \left (2\right ) + \log \left (-x\right ) + \frac {1}{3} \, \log \left (\log \left (3\right )\right )\right ) \]

[In]

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

[Out]

-log(e^x + log(2) + log(log(x))) + log(1/3*log(2) + log(-x) + 1/3*log(log(3)))

Giac [F]

\[ \int \frac {-2 \log \left (-2 x^3 \log (3)\right )+\log \left (x^2\right ) \left (3 e^x-e^x x \log \left (-2 x^3 \log (3)\right )\right )+3 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{e^x x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right )+x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\int { -\frac {{\left (x e^{x} \log \left (-2 \, x^{3} \log \left (3\right )\right ) - 3 \, e^{x}\right )} \log \left (x^{2}\right ) - 3 \, \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right ) + 2 \, \log \left (-2 \, x^{3} \log \left (3\right )\right )}{x e^{x} \log \left (-2 \, x^{3} \log \left (3\right )\right ) \log \left (x^{2}\right ) + x \log \left (-2 \, x^{3} \log \left (3\right )\right ) \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right )} \,d x } \]

[In]

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

[Out]

integrate(-((x*e^x*log(-2*x^3*log(3)) - 3*e^x)*log(x^2) - 3*log(x^2)*log(log(x^2)) + 2*log(-2*x^3*log(3)))/(x*
e^x*log(-2*x^3*log(3))*log(x^2) + x*log(-2*x^3*log(3))*log(x^2)*log(log(x^2))), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-2 \log \left (-2 x^3 \log (3)\right )+\log \left (x^2\right ) \left (3 e^x-e^x x \log \left (-2 x^3 \log (3)\right )\right )+3 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{e^x x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right )+x \log \left (x^2\right ) \log \left (-2 x^3 \log (3)\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\int \frac {\ln \left (x^2\right )\,\left (3\,{\mathrm {e}}^x-x\,\ln \left (-2\,x^3\,\ln \left (3\right )\right )\,{\mathrm {e}}^x\right )-2\,\ln \left (-2\,x^3\,\ln \left (3\right )\right )+3\,\ln \left (x^2\right )\,\ln \left (\ln \left (x^2\right )\right )}{x\,\ln \left (-2\,x^3\,\ln \left (3\right )\right )\,\ln \left (x^2\right )\,{\mathrm {e}}^x+x\,\ln \left (-2\,x^3\,\ln \left (3\right )\right )\,\ln \left (x^2\right )\,\ln \left (\ln \left (x^2\right )\right )} \,d x \]

[In]

int((log(x^2)*(3*exp(x) - x*log(-2*x^3*log(3))*exp(x)) - 2*log(-2*x^3*log(3)) + 3*log(x^2)*log(log(x^2)))/(x*l
og(-2*x^3*log(3))*log(x^2)*exp(x) + x*log(-2*x^3*log(3))*log(x^2)*log(log(x^2))),x)

[Out]

int((log(x^2)*(3*exp(x) - x*log(-2*x^3*log(3))*exp(x)) - 2*log(-2*x^3*log(3)) + 3*log(x^2)*log(log(x^2)))/(x*l
og(-2*x^3*log(3))*log(x^2)*exp(x) + x*log(-2*x^3*log(3))*log(x^2)*log(log(x^2))), x)