Integrand size = 117, antiderivative size = 24 \[ \int \frac {2 x^7+x^8+e^{\frac {e^4}{x^6}} \left (-6 e^4-x^7\right )+x^7 \log (4)}{3 x^8-e^{\frac {e^4}{x^6}} x^8+x^9+x^8 \log (4)+\left (-3 x^7+e^{\frac {e^4}{x^6}} x^7-x^8-x^7 \log (4)\right ) \log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )} \, dx=4+\log \left (x-\log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )\right ) \] Output:
4+ln(x-ln(-exp(exp(4)/x^6)+2*ln(2)+3+x))
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {2 x^7+x^8+e^{\frac {e^4}{x^6}} \left (-6 e^4-x^7\right )+x^7 \log (4)}{3 x^8-e^{\frac {e^4}{x^6}} x^8+x^9+x^8 \log (4)+\left (-3 x^7+e^{\frac {e^4}{x^6}} x^7-x^8-x^7 \log (4)\right ) \log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )} \, dx=\log \left (x-\log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )\right ) \] Input:
Integrate[(2*x^7 + x^8 + E^(E^4/x^6)*(-6*E^4 - x^7) + x^7*Log[4])/(3*x^8 - E^(E^4/x^6)*x^8 + x^9 + x^8*Log[4] + (-3*x^7 + E^(E^4/x^6)*x^7 - x^8 - x^ 7*Log[4])*Log[3 - E^(E^4/x^6) + x + Log[4]]),x]
Output:
Log[x - Log[3 - E^(E^4/x^6) + x + Log[4]]]
Time = 1.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6, 6, 7292, 7235}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^8+2 x^7+x^7 \log (4)+e^{\frac {e^4}{x^6}} \left (-x^7-6 e^4\right )}{x^9+3 x^8+x^8 \log (4)-e^{\frac {e^4}{x^6}} x^8+\left (-x^8-3 x^7-x^7 \log (4)+e^{\frac {e^4}{x^6}} x^7\right ) \log \left (-e^{\frac {e^4}{x^6}}+x+3+\log (4)\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x^8+x^7 (2+\log (4))+e^{\frac {e^4}{x^6}} \left (-x^7-6 e^4\right )}{x^9+3 x^8+x^8 \log (4)-e^{\frac {e^4}{x^6}} x^8+\left (-x^8-3 x^7-x^7 \log (4)+e^{\frac {e^4}{x^6}} x^7\right ) \log \left (-e^{\frac {e^4}{x^6}}+x+3+\log (4)\right )}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x^8+x^7 (2+\log (4))+e^{\frac {e^4}{x^6}} \left (-x^7-6 e^4\right )}{x^9+x^8 (3+\log (4))-e^{\frac {e^4}{x^6}} x^8+\left (-x^8-3 x^7-x^7 \log (4)+e^{\frac {e^4}{x^6}} x^7\right ) \log \left (-e^{\frac {e^4}{x^6}}+x+3+\log (4)\right )}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-x^8-x^7 (2+\log (4))-e^{\frac {e^4}{x^6}} \left (-x^7-6 e^4\right )}{x^7 \left (e^{\frac {e^4}{x^6}}-x-3 \left (1+\frac {2 \log (2)}{3}\right )\right ) \left (x-\log \left (-e^{\frac {e^4}{x^6}}+x+3+\log (4)\right )\right )}dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle \log \left (x-\log \left (-e^{\frac {e^4}{x^6}}+x+3+\log (4)\right )\right )\) |
Input:
Int[(2*x^7 + x^8 + E^(E^4/x^6)*(-6*E^4 - x^7) + x^7*Log[4])/(3*x^8 - E^(E^ 4/x^6)*x^8 + x^9 + x^8*Log[4] + (-3*x^7 + E^(E^4/x^6)*x^7 - x^8 - x^7*Log[ 4])*Log[3 - E^(E^4/x^6) + x + Log[4]]),x]
Output:
Log[x - Log[3 - E^(E^4/x^6) + x + Log[4]]]
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Time = 60.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\ln \left (\ln \left (-{\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{6}}}+2 \ln \left (2\right )+3+x \right )-x \right )\) | \(23\) |
parallelrisch | \(\ln \left (x -\ln \left (-{\mathrm e}^{\frac {{\mathrm e}^{4}}{x^{6}}}+2 \ln \left (2\right )+3+x \right )\right )\) | \(23\) |
Input:
int(((-6*exp(4)-x^7)*exp(exp(4)/x^6)+2*x^7*ln(2)+x^8+2*x^7)/((x^7*exp(exp( 4)/x^6)-2*x^7*ln(2)-x^8-3*x^7)*ln(-exp(exp(4)/x^6)+2*ln(2)+3+x)-x^8*exp(ex p(4)/x^6)+2*x^8*ln(2)+x^9+3*x^8),x,method=_RETURNVERBOSE)
Output:
ln(ln(-exp(exp(4)/x^6)+2*ln(2)+3+x)-x)
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {2 x^7+x^8+e^{\frac {e^4}{x^6}} \left (-6 e^4-x^7\right )+x^7 \log (4)}{3 x^8-e^{\frac {e^4}{x^6}} x^8+x^9+x^8 \log (4)+\left (-3 x^7+e^{\frac {e^4}{x^6}} x^7-x^8-x^7 \log (4)\right ) \log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )} \, dx=\log \left (-x + \log \left (x - e^{\left (\frac {e^{4}}{x^{6}}\right )} + 2 \, \log \left (2\right ) + 3\right )\right ) \] Input:
integrate(((-6*exp(4)-x^7)*exp(exp(4)/x^6)+2*x^7*log(2)+x^8+2*x^7)/((x^7*e xp(exp(4)/x^6)-2*x^7*log(2)-x^8-3*x^7)*log(-exp(exp(4)/x^6)+2*log(2)+3+x)- x^8*exp(exp(4)/x^6)+2*x^8*log(2)+x^9+3*x^8),x, algorithm="fricas")
Output:
log(-x + log(x - e^(e^4/x^6) + 2*log(2) + 3))
Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {2 x^7+x^8+e^{\frac {e^4}{x^6}} \left (-6 e^4-x^7\right )+x^7 \log (4)}{3 x^8-e^{\frac {e^4}{x^6}} x^8+x^9+x^8 \log (4)+\left (-3 x^7+e^{\frac {e^4}{x^6}} x^7-x^8-x^7 \log (4)\right ) \log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )} \, dx=\log {\left (- x + \log {\left (x - e^{\frac {e^{4}}{x^{6}}} + 2 \log {\left (2 \right )} + 3 \right )} \right )} \] Input:
integrate(((-6*exp(4)-x**7)*exp(exp(4)/x**6)+2*x**7*ln(2)+x**8+2*x**7)/((x **7*exp(exp(4)/x**6)-2*x**7*ln(2)-x**8-3*x**7)*ln(-exp(exp(4)/x**6)+2*ln(2 )+3+x)-x**8*exp(exp(4)/x**6)+2*x**8*ln(2)+x**9+3*x**8),x)
Output:
log(-x + log(x - exp(exp(4)/x**6) + 2*log(2) + 3))
Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {2 x^7+x^8+e^{\frac {e^4}{x^6}} \left (-6 e^4-x^7\right )+x^7 \log (4)}{3 x^8-e^{\frac {e^4}{x^6}} x^8+x^9+x^8 \log (4)+\left (-3 x^7+e^{\frac {e^4}{x^6}} x^7-x^8-x^7 \log (4)\right ) \log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )} \, dx=\log \left (-x + \log \left (x - e^{\left (\frac {e^{4}}{x^{6}}\right )} + 2 \, \log \left (2\right ) + 3\right )\right ) \] Input:
integrate(((-6*exp(4)-x^7)*exp(exp(4)/x^6)+2*x^7*log(2)+x^8+2*x^7)/((x^7*e xp(exp(4)/x^6)-2*x^7*log(2)-x^8-3*x^7)*log(-exp(exp(4)/x^6)+2*log(2)+3+x)- x^8*exp(exp(4)/x^6)+2*x^8*log(2)+x^9+3*x^8),x, algorithm="maxima")
Output:
log(-x + log(x - e^(e^4/x^6) + 2*log(2) + 3))
Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {2 x^7+x^8+e^{\frac {e^4}{x^6}} \left (-6 e^4-x^7\right )+x^7 \log (4)}{3 x^8-e^{\frac {e^4}{x^6}} x^8+x^9+x^8 \log (4)+\left (-3 x^7+e^{\frac {e^4}{x^6}} x^7-x^8-x^7 \log (4)\right ) \log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )} \, dx=\log \left (x - \log \left ({\left (x e^{4} + 2 \, e^{4} \log \left (2\right ) + 3 \, e^{4} - e^{\left (\frac {4 \, x^{6} + e^{4}}{x^{6}}\right )}\right )} e^{\left (-4\right )}\right )\right ) \] Input:
integrate(((-6*exp(4)-x^7)*exp(exp(4)/x^6)+2*x^7*log(2)+x^8+2*x^7)/((x^7*e xp(exp(4)/x^6)-2*x^7*log(2)-x^8-3*x^7)*log(-exp(exp(4)/x^6)+2*log(2)+3+x)- x^8*exp(exp(4)/x^6)+2*x^8*log(2)+x^9+3*x^8),x, algorithm="giac")
Output:
log(x - log((x*e^4 + 2*e^4*log(2) + 3*e^4 - e^((4*x^6 + e^4)/x^6))*e^(-4)) )
Timed out. \[ \int \frac {2 x^7+x^8+e^{\frac {e^4}{x^6}} \left (-6 e^4-x^7\right )+x^7 \log (4)}{3 x^8-e^{\frac {e^4}{x^6}} x^8+x^9+x^8 \log (4)+\left (-3 x^7+e^{\frac {e^4}{x^6}} x^7-x^8-x^7 \log (4)\right ) \log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )} \, dx=\int \frac {2\,x^7\,\ln \left (2\right )-{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x^6}}\,\left (x^7+6\,{\mathrm {e}}^4\right )+2\,x^7+x^8}{2\,x^8\,\ln \left (2\right )-x^8\,{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x^6}}-\ln \left (x-{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x^6}}+2\,\ln \left (2\right )+3\right )\,\left (2\,x^7\,\ln \left (2\right )-x^7\,{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x^6}}+3\,x^7+x^8\right )+3\,x^8+x^9} \,d x \] Input:
int((2*x^7*log(2) - exp(exp(4)/x^6)*(6*exp(4) + x^7) + 2*x^7 + x^8)/(2*x^8 *log(2) - x^8*exp(exp(4)/x^6) - log(x - exp(exp(4)/x^6) + 2*log(2) + 3)*(2 *x^7*log(2) - x^7*exp(exp(4)/x^6) + 3*x^7 + x^8) + 3*x^8 + x^9),x)
Output:
int((2*x^7*log(2) - exp(exp(4)/x^6)*(6*exp(4) + x^7) + 2*x^7 + x^8)/(2*x^8 *log(2) - x^8*exp(exp(4)/x^6) - log(x - exp(exp(4)/x^6) + 2*log(2) + 3)*(2 *x^7*log(2) - x^7*exp(exp(4)/x^6) + 3*x^7 + x^8) + 3*x^8 + x^9), x)
Time = 0.69 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^7+x^8+e^{\frac {e^4}{x^6}} \left (-6 e^4-x^7\right )+x^7 \log (4)}{3 x^8-e^{\frac {e^4}{x^6}} x^8+x^9+x^8 \log (4)+\left (-3 x^7+e^{\frac {e^4}{x^6}} x^7-x^8-x^7 \log (4)\right ) \log \left (3-e^{\frac {e^4}{x^6}}+x+\log (4)\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (-e^{\frac {e^{4}}{x^{6}}}+2 \,\mathrm {log}\left (2\right )+x +3\right )-x \right ) \] Input:
int(((-6*exp(4)-x^7)*exp(exp(4)/x^6)+2*x^7*log(2)+x^8+2*x^7)/((x^7*exp(exp (4)/x^6)-2*x^7*log(2)-x^8-3*x^7)*log(-exp(exp(4)/x^6)+2*log(2)+3+x)-x^8*ex p(exp(4)/x^6)+2*x^8*log(2)+x^9+3*x^8),x)
Output:
log(log( - e**(e**4/x**6) + 2*log(2) + x + 3) - x)