\(\int \frac {-2 e+45 x^3+6 x^4+(e-9 x^3-x^4) \log (\frac {-e x^2+9 x^5+x^6}{e}) \log (\log (\frac {-e x^2+9 x^5+x^6}{e}))}{(e-9 x^3-x^4) \log (\frac {-e x^2+9 x^5+x^6}{e}) \log ^2(\log (\frac {-e x^2+9 x^5+x^6}{e}))} \, dx\) [956]

Optimal result

Integrand size = 128, antiderivative size = 24 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=2+\frac {x}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \]

[Out]

2+x/ln(ln(x^5*(x+9)/exp(1)-x^2))

Rubi [F]

\[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx \]

[In]

Int[(-2*E + 45*x^3 + 6*x^4 + (E - 9*x^3 - x^4)*Log[(-(E*x^2) + 9*x^5 + x^6)/E]*Log[Log[(-(E*x^2) + 9*x^5 + x^6
)/E]])/((E - 9*x^3 - x^4)*Log[(-(E*x^2) + 9*x^5 + x^6)/E]*Log[Log[(-(E*x^2) + 9*x^5 + x^6)/E]]^2),x]

[Out]

-6*Defer[Int][1/(Log[-x^2 + (x^5*(9 + x))/E]*Log[Log[-x^2 + (x^5*(9 + x))/E]]^2), x] + 4*E*Defer[Int][1/((E -
9*x^3 - x^4)*Log[-x^2 + (x^5*(9 + x))/E]*Log[Log[-x^2 + (x^5*(9 + x))/E]]^2), x] - 9*Defer[Int][x^3/((E - 9*x^
3 - x^4)*Log[-x^2 + (x^5*(9 + x))/E]*Log[Log[-x^2 + (x^5*(9 + x))/E]]^2), x] + Defer[Int][Log[Log[-x^2 + (x^5*
(9 + x))/E]]^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {x^2 \left (-e+9 x^3+x^4\right )}{e}\right ) \log ^2\left (\log \left (\frac {x^2 \left (-e+9 x^3+x^4\right )}{e}\right )\right )} \, dx \\ & = \int \left (\frac {-2 e+45 x^3+6 x^4}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}+\frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}\right ) \, dx \\ & = \int \frac {-2 e+45 x^3+6 x^4}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx+\int \frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx \\ & = \int \left (-\frac {6}{\log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}+\frac {4 e-9 x^3}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}\right ) \, dx+\int \frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx \\ & = -\left (6 \int \frac {1}{\log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx\right )+\int \frac {4 e-9 x^3}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx+\int \frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx \\ & = -\left (6 \int \frac {1}{\log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx\right )+\int \left (\frac {4 e}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}-\frac {9 x^3}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}\right ) \, dx+\int \frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx \\ & = -\left (6 \int \frac {1}{\log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx\right )-9 \int \frac {x^3}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx+(4 e) \int \frac {1}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx+\int \frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {x}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \]

[In]

Integrate[(-2*E + 45*x^3 + 6*x^4 + (E - 9*x^3 - x^4)*Log[(-(E*x^2) + 9*x^5 + x^6)/E]*Log[Log[(-(E*x^2) + 9*x^5
+ x^6)/E]])/((E - 9*x^3 - x^4)*Log[(-(E*x^2) + 9*x^5 + x^6)/E]*Log[Log[(-(E*x^2) + 9*x^5 + x^6)/E]]^2),x]

[Out]

x/Log[Log[-x^2 + (x^5*(9 + x))/E]]

Maple [F]

\[\int \frac {\left ({\mathrm e}-x^{4}-9 x^{3}\right ) \ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right ) \ln \left (\ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right )\right )-2 \,{\mathrm e}+6 x^{4}+45 x^{3}}{\left ({\mathrm e}-x^{4}-9 x^{3}\right ) \ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right ) {\ln \left (\ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right )\right )}^{2}}d x\]

[In]

int(((exp(1)-x^4-9*x^3)*ln((-x^2*exp(1)+x^6+9*x^5)/exp(1))*ln(ln((-x^2*exp(1)+x^6+9*x^5)/exp(1)))-2*exp(1)+6*x
^4+45*x^3)/(exp(1)-x^4-9*x^3)/ln((-x^2*exp(1)+x^6+9*x^5)/exp(1))/ln(ln((-x^2*exp(1)+x^6+9*x^5)/exp(1)))^2,x)

[Out]

int(((exp(1)-x^4-9*x^3)*ln((-x^2*exp(1)+x^6+9*x^5)/exp(1))*ln(ln((-x^2*exp(1)+x^6+9*x^5)/exp(1)))-2*exp(1)+6*x
^4+45*x^3)/(exp(1)-x^4-9*x^3)/ln((-x^2*exp(1)+x^6+9*x^5)/exp(1))/ln(ln((-x^2*exp(1)+x^6+9*x^5)/exp(1)))^2,x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {x}{\log \left (\log \left ({\left (x^{6} + 9 \, x^{5} - x^{2} e\right )} e^{\left (-1\right )}\right )\right )} \]

[In]

integrate(((exp(1)-x^4-9*x^3)*log((-x^2*exp(1)+x^6+9*x^5)/exp(1))*log(log((-x^2*exp(1)+x^6+9*x^5)/exp(1)))-2*e
xp(1)+6*x^4+45*x^3)/(exp(1)-x^4-9*x^3)/log((-x^2*exp(1)+x^6+9*x^5)/exp(1))/log(log((-x^2*exp(1)+x^6+9*x^5)/exp
(1)))^2,x, algorithm="fricas")

[Out]

x/log(log((x^6 + 9*x^5 - x^2*e)*e^(-1)))

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {x}{\log {\left (\log {\left (\frac {x^{6} + 9 x^{5} - e x^{2}}{e} \right )} \right )}} \]

[In]

integrate(((exp(1)-x**4-9*x**3)*ln((-x**2*exp(1)+x**6+9*x**5)/exp(1))*ln(ln((-x**2*exp(1)+x**6+9*x**5)/exp(1))
)-2*exp(1)+6*x**4+45*x**3)/(exp(1)-x**4-9*x**3)/ln((-x**2*exp(1)+x**6+9*x**5)/exp(1))/ln(ln((-x**2*exp(1)+x**6
+9*x**5)/exp(1)))**2,x)

[Out]

x/log(log((x**6 + 9*x**5 - E*x**2)*exp(-1)))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {x}{\log \left (\log \left (x^{4} + 9 \, x^{3} - e\right ) + 2 \, \log \left (x\right ) - 1\right )} \]

[In]

integrate(((exp(1)-x^4-9*x^3)*log((-x^2*exp(1)+x^6+9*x^5)/exp(1))*log(log((-x^2*exp(1)+x^6+9*x^5)/exp(1)))-2*e
xp(1)+6*x^4+45*x^3)/(exp(1)-x^4-9*x^3)/log((-x^2*exp(1)+x^6+9*x^5)/exp(1))/log(log((-x^2*exp(1)+x^6+9*x^5)/exp
(1)))^2,x, algorithm="maxima")

[Out]

x/log(log(x^4 + 9*x^3 - e) + 2*log(x) - 1)

Giac [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {x}{\log \left (\log \left (x^{6} + 9 \, x^{5} - x^{2} e\right ) - 1\right )} \]

[In]

integrate(((exp(1)-x^4-9*x^3)*log((-x^2*exp(1)+x^6+9*x^5)/exp(1))*log(log((-x^2*exp(1)+x^6+9*x^5)/exp(1)))-2*e
xp(1)+6*x^4+45*x^3)/(exp(1)-x^4-9*x^3)/log((-x^2*exp(1)+x^6+9*x^5)/exp(1))/log(log((-x^2*exp(1)+x^6+9*x^5)/exp
(1)))^2,x, algorithm="giac")

[Out]

x/log(log(x^6 + 9*x^5 - x^2*e) - 1)

Mupad [B] (verification not implemented)

Time = 10.95 (sec) , antiderivative size = 170, normalized size of antiderivative = 7.08 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {\ln \left (x^4+9\,x^3-\mathrm {e}\right )}{4}+\frac {\ln \left (x\right )}{2}+\frac {x}{\ln \left (\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )\right )}-\frac {45\,x^3\,\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )}{4\,\left (6\,x^4+45\,x^3-2\,\mathrm {e}\right )}-\frac {3\,x^4\,\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )}{2\,\left (6\,x^4+45\,x^3-2\,\mathrm {e}\right )}+\frac {\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )\,\mathrm {e}}{2\,\left (6\,x^4+45\,x^3-2\,\mathrm {e}\right )} \]

[In]

int((2*exp(1) - 45*x^3 - 6*x^4 + log(exp(-1)*(9*x^5 - x^2*exp(1) + x^6))*log(log(exp(-1)*(9*x^5 - x^2*exp(1) +
x^6)))*(9*x^3 - exp(1) + x^4))/(log(exp(-1)*(9*x^5 - x^2*exp(1) + x^6))*log(log(exp(-1)*(9*x^5 - x^2*exp(1) +
x^6)))^2*(9*x^3 - exp(1) + x^4)),x)

[Out]

log(9*x^3 - exp(1) + x^4)/4 + log(x)/2 + x/log(log(9*x^5*exp(-1) + x^6*exp(-1) - x^2)) - (45*x^3*log(9*x^5*exp
(-1) + x^6*exp(-1) - x^2))/(4*(45*x^3 - 2*exp(1) + 6*x^4)) - (3*x^4*log(9*x^5*exp(-1) + x^6*exp(-1) - x^2))/(2
*(45*x^3 - 2*exp(1) + 6*x^4)) + (log(9*x^5*exp(-1) + x^6*exp(-1) - x^2)*exp(1))/(2*(45*x^3 - 2*exp(1) + 6*x^4)
)