Integrand size = 107, antiderivative size = 21 \[ \int \frac {-108+144 x^3+2 x^6+e^{-1+x} \left (-54+72 x^3+x^6\right )+\left (e^{-1+x} \left (9 x-24 x^4+16 x^7\right )+e^{-1+x} x^7 \log (x)\right ) \log \left (\frac {9-24 x^3+16 x^6+x^6 \log (x)}{x^6}\right )}{9 x-24 x^4+16 x^7+x^7 \log (x)} \, dx=\left (2+e^{-1+x}\right ) \log \left (\left (4-\frac {3}{x^3}\right )^2+\log (x)\right ) \] Output:
ln(ln(x)+(4-3/x^3)^2)*(2+exp(-1+x))
Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(21)=42\).
Time = 116.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.52 \[ \int \frac {-108+144 x^3+2 x^6+e^{-1+x} \left (-54+72 x^3+x^6\right )+\left (e^{-1+x} \left (9 x-24 x^4+16 x^7\right )+e^{-1+x} x^7 \log (x)\right ) \log \left (\frac {9-24 x^3+16 x^6+x^6 \log (x)}{x^6}\right )}{9 x-24 x^4+16 x^7+x^7 \log (x)} \, dx=\frac {-12 e \log (x)+e^x \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )+2 e \log \left (9-24 x^3+16 x^6+x^6 \log (x)\right )}{e} \] Input:
Integrate[(-108 + 144*x^3 + 2*x^6 + E^(-1 + x)*(-54 + 72*x^3 + x^6) + (E^( -1 + x)*(9*x - 24*x^4 + 16*x^7) + E^(-1 + x)*x^7*Log[x])*Log[(9 - 24*x^3 + 16*x^6 + x^6*Log[x])/x^6])/(9*x - 24*x^4 + 16*x^7 + x^7*Log[x]),x]
Output:
(-12*E*Log[x] + E^x*Log[(3 - 4*x^3)^2/x^6 + Log[x]] + 2*E*Log[9 - 24*x^3 + 16*x^6 + x^6*Log[x]])/E
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.
\(\displaystyle \int \frac {2 x^6+144 x^3+e^{x-1} \left (x^6+72 x^3-54\right )+\left (e^{x-1} x^7 \log (x)+e^{x-1} \left (16 x^7-24 x^4+9 x\right )\right ) \log \left (\frac {16 x^6+x^6 \log (x)-24 x^3+9}{x^6}\right )-108}{16 x^7+x^7 \log (x)-24 x^4+9 x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \left (x^6+72 x^3-54\right )}{x \left (16 x^6+x^6 \log (x)-24 x^3+9\right )}+\frac {e^{x-1} \left (x^6+72 x^3+9 x \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )+x^7 \log (x) \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )+16 x^7 \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )-24 x^4 \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )-54\right )}{x \left (16 x^6+x^6 \log (x)-24 x^3+9\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -108 \int \frac {1}{x \left (\log (x) x^6+16 x^6-24 x^3+9\right )}dx+2 \int \frac {x^5}{\log (x) x^6+16 x^6-24 x^3+9}dx+144 \int \frac {x^2}{\log (x) x^6+16 x^6-24 x^3+9}dx+\frac {e^{x-1} \left (9 x \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )+x^7 \log (x) \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )+16 x^7 \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )-24 x^4 \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )\right )}{x \left (16 x^6+x^6 \log (x)-24 x^3+9\right )}\) |
Input:
Int[(-108 + 144*x^3 + 2*x^6 + E^(-1 + x)*(-54 + 72*x^3 + x^6) + (E^(-1 + x )*(9*x - 24*x^4 + 16*x^7) + E^(-1 + x)*x^7*Log[x])*Log[(9 - 24*x^3 + 16*x^ 6 + x^6*Log[x])/x^6])/(9*x - 24*x^4 + 16*x^7 + x^7*Log[x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(20)=40\).
Time = 130.38 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.62
method | result | size |
parallelrisch | \(\ln \left (\frac {x^{6} \ln \left (x \right )+16 x^{6}-24 x^{3}+9}{x^{6}}\right ) {\mathrm e}^{-1+x}-12 \ln \left (x \right )+2 \ln \left (x^{6} \ln \left (x \right )+16 x^{6}-24 x^{3}+9\right )\) | \(55\) |
risch | \(\text {Expression too large to display}\) | \(652\) |
Input:
int(((x^7*exp(-1+x)*ln(x)+(16*x^7-24*x^4+9*x)*exp(-1+x))*ln((x^6*ln(x)+16* x^6-24*x^3+9)/x^6)+(x^6+72*x^3-54)*exp(-1+x)+2*x^6+144*x^3-108)/(x^7*ln(x) +16*x^7-24*x^4+9*x),x,method=_RETURNVERBOSE)
Output:
ln((x^6*ln(x)+16*x^6-24*x^3+9)/x^6)*exp(-1+x)-12*ln(x)+2*ln(x^6*ln(x)+16*x ^6-24*x^3+9)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {-108+144 x^3+2 x^6+e^{-1+x} \left (-54+72 x^3+x^6\right )+\left (e^{-1+x} \left (9 x-24 x^4+16 x^7\right )+e^{-1+x} x^7 \log (x)\right ) \log \left (\frac {9-24 x^3+16 x^6+x^6 \log (x)}{x^6}\right )}{9 x-24 x^4+16 x^7+x^7 \log (x)} \, dx={\left (e^{\left (x - 1\right )} + 2\right )} \log \left (\frac {x^{6} \log \left (x\right ) + 16 \, x^{6} - 24 \, x^{3} + 9}{x^{6}}\right ) \] Input:
integrate(((x^7*exp(-1+x)*log(x)+(16*x^7-24*x^4+9*x)*exp(-1+x))*log((x^6*l og(x)+16*x^6-24*x^3+9)/x^6)+(x^6+72*x^3-54)*exp(-1+x)+2*x^6+144*x^3-108)/( x^7*log(x)+16*x^7-24*x^4+9*x),x, algorithm="fricas")
Output:
(e^(x - 1) + 2)*log((x^6*log(x) + 16*x^6 - 24*x^3 + 9)/x^6)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).
Time = 5.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33 \[ \int \frac {-108+144 x^3+2 x^6+e^{-1+x} \left (-54+72 x^3+x^6\right )+\left (e^{-1+x} \left (9 x-24 x^4+16 x^7\right )+e^{-1+x} x^7 \log (x)\right ) \log \left (\frac {9-24 x^3+16 x^6+x^6 \log (x)}{x^6}\right )}{9 x-24 x^4+16 x^7+x^7 \log (x)} \, dx=e^{x - 1} \log {\left (\frac {x^{6} \log {\left (x \right )} + 16 x^{6} - 24 x^{3} + 9}{x^{6}} \right )} + 2 \log {\left (\log {\left (x \right )} + \frac {16 x^{6} - 24 x^{3} + 9}{x^{6}} \right )} \] Input:
integrate(((x**7*exp(-1+x)*ln(x)+(16*x**7-24*x**4+9*x)*exp(-1+x))*ln((x**6 *ln(x)+16*x**6-24*x**3+9)/x**6)+(x**6+72*x**3-54)*exp(-1+x)+2*x**6+144*x** 3-108)/(x**7*ln(x)+16*x**7-24*x**4+9*x),x)
Output:
exp(x - 1)*log((x**6*log(x) + 16*x**6 - 24*x**3 + 9)/x**6) + 2*log(log(x) + (16*x**6 - 24*x**3 + 9)/x**6)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.76 \[ \int \frac {-108+144 x^3+2 x^6+e^{-1+x} \left (-54+72 x^3+x^6\right )+\left (e^{-1+x} \left (9 x-24 x^4+16 x^7\right )+e^{-1+x} x^7 \log (x)\right ) \log \left (\frac {9-24 x^3+16 x^6+x^6 \log (x)}{x^6}\right )}{9 x-24 x^4+16 x^7+x^7 \log (x)} \, dx={\left (e^{x} \log \left (x^{6} \log \left (x\right ) + 16 \, x^{6} - 24 \, x^{3} + 9\right ) - 6 \, e^{x} \log \left (x\right )\right )} e^{\left (-1\right )} + 2 \, \log \left (\frac {x^{6} \log \left (x\right ) + 16 \, x^{6} - 24 \, x^{3} + 9}{x^{6}}\right ) \] Input:
integrate(((x^7*exp(-1+x)*log(x)+(16*x^7-24*x^4+9*x)*exp(-1+x))*log((x^6*l og(x)+16*x^6-24*x^3+9)/x^6)+(x^6+72*x^3-54)*exp(-1+x)+2*x^6+144*x^3-108)/( x^7*log(x)+16*x^7-24*x^4+9*x),x, algorithm="maxima")
Output:
(e^x*log(x^6*log(x) + 16*x^6 - 24*x^3 + 9) - 6*e^x*log(x))*e^(-1) + 2*log( (x^6*log(x) + 16*x^6 - 24*x^3 + 9)/x^6)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.90 \[ \int \frac {-108+144 x^3+2 x^6+e^{-1+x} \left (-54+72 x^3+x^6\right )+\left (e^{-1+x} \left (9 x-24 x^4+16 x^7\right )+e^{-1+x} x^7 \log (x)\right ) \log \left (\frac {9-24 x^3+16 x^6+x^6 \log (x)}{x^6}\right )}{9 x-24 x^4+16 x^7+x^7 \log (x)} \, dx={\left (2 \, e \log \left (x^{6} \log \left (x\right ) + 16 \, x^{6} - 24 \, x^{3} + 9\right ) + e^{x} \log \left (x^{6} \log \left (x\right ) + 16 \, x^{6} - 24 \, x^{3} + 9\right ) - 12 \, e \log \left (x\right ) - 6 \, e^{x} \log \left (x\right )\right )} e^{\left (-1\right )} \] Input:
integrate(((x^7*exp(-1+x)*log(x)+(16*x^7-24*x^4+9*x)*exp(-1+x))*log((x^6*l og(x)+16*x^6-24*x^3+9)/x^6)+(x^6+72*x^3-54)*exp(-1+x)+2*x^6+144*x^3-108)/( x^7*log(x)+16*x^7-24*x^4+9*x),x, algorithm="giac")
Output:
(2*e*log(x^6*log(x) + 16*x^6 - 24*x^3 + 9) + e^x*log(x^6*log(x) + 16*x^6 - 24*x^3 + 9) - 12*e*log(x) - 6*e^x*log(x))*e^(-1)
Timed out. \[ \int \frac {-108+144 x^3+2 x^6+e^{-1+x} \left (-54+72 x^3+x^6\right )+\left (e^{-1+x} \left (9 x-24 x^4+16 x^7\right )+e^{-1+x} x^7 \log (x)\right ) \log \left (\frac {9-24 x^3+16 x^6+x^6 \log (x)}{x^6}\right )}{9 x-24 x^4+16 x^7+x^7 \log (x)} \, dx=\int \frac {{\mathrm {e}}^{x-1}\,\left (x^6+72\,x^3-54\right )+144\,x^3+2\,x^6+\ln \left (\frac {x^6\,\ln \left (x\right )-24\,x^3+16\,x^6+9}{x^6}\right )\,\left ({\mathrm {e}}^{x-1}\,\left (16\,x^7-24\,x^4+9\,x\right )+x^7\,{\mathrm {e}}^{x-1}\,\ln \left (x\right )\right )-108}{9\,x+x^7\,\ln \left (x\right )-24\,x^4+16\,x^7} \,d x \] Input:
int((exp(x - 1)*(72*x^3 + x^6 - 54) + 144*x^3 + 2*x^6 + log((x^6*log(x) - 24*x^3 + 16*x^6 + 9)/x^6)*(exp(x - 1)*(9*x - 24*x^4 + 16*x^7) + x^7*exp(x - 1)*log(x)) - 108)/(9*x + x^7*log(x) - 24*x^4 + 16*x^7),x)
Output:
int((exp(x - 1)*(72*x^3 + x^6 - 54) + 144*x^3 + 2*x^6 + log((x^6*log(x) - 24*x^3 + 16*x^6 + 9)/x^6)*(exp(x - 1)*(9*x - 24*x^4 + 16*x^7) + x^7*exp(x - 1)*log(x)) - 108)/(9*x + x^7*log(x) - 24*x^4 + 16*x^7), x)
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.81 \[ \int \frac {-108+144 x^3+2 x^6+e^{-1+x} \left (-54+72 x^3+x^6\right )+\left (e^{-1+x} \left (9 x-24 x^4+16 x^7\right )+e^{-1+x} x^7 \log (x)\right ) \log \left (\frac {9-24 x^3+16 x^6+x^6 \log (x)}{x^6}\right )}{9 x-24 x^4+16 x^7+x^7 \log (x)} \, dx=\frac {e^{x} \mathrm {log}\left (\frac {\mathrm {log}\left (x \right ) x^{6}+16 x^{6}-24 x^{3}+9}{x^{6}}\right )+2 \,\mathrm {log}\left (\mathrm {log}\left (x \right ) x^{6}+16 x^{6}-24 x^{3}+9\right ) e -12 \,\mathrm {log}\left (x \right ) e}{e} \] Input:
int(((x^7*exp(-1+x)*log(x)+(16*x^7-24*x^4+9*x)*exp(-1+x))*log((x^6*log(x)+ 16*x^6-24*x^3+9)/x^6)+(x^6+72*x^3-54)*exp(-1+x)+2*x^6+144*x^3-108)/(x^7*lo g(x)+16*x^7-24*x^4+9*x),x)
Output:
(e**x*log((log(x)*x**6 + 16*x**6 - 24*x**3 + 9)/x**6) + 2*log(log(x)*x**6 + 16*x**6 - 24*x**3 + 9)*e - 12*log(x)*e)/e