Integrand size = 173, antiderivative size = 30 \[ \int \frac {-162-657 x-260 x^2+1379 x^3+1254 x^4+410 x^5+58 x^6+3 x^7+e^{-\frac {1}{-9-25 x-10 x^2-x^3+3 \log (x)}} \left (6-50 x-40 x^2-6 x^3\right )+\left (108+138 x-330 x^2-168 x^3-18 x^4\right ) \log (x)+(-18+27 x) \log ^2(x)}{81 x+450 x^2+805 x^3+518 x^4+150 x^5+20 x^6+x^7+\left (-54 x-150 x^2-60 x^3-6 x^4\right ) \log (x)+9 x \log ^2(x)} \, dx=x+2 \left (e^{\frac {1}{x (5+x)^2+3 (3-\log (x))}}+x-\log (x)\right ) \] Output:
3*x+2*exp(1/(x*(5+x)^2+9-3*ln(x)))-2*ln(x)
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-162-657 x-260 x^2+1379 x^3+1254 x^4+410 x^5+58 x^6+3 x^7+e^{-\frac {1}{-9-25 x-10 x^2-x^3+3 \log (x)}} \left (6-50 x-40 x^2-6 x^3\right )+\left (108+138 x-330 x^2-168 x^3-18 x^4\right ) \log (x)+(-18+27 x) \log ^2(x)}{81 x+450 x^2+805 x^3+518 x^4+150 x^5+20 x^6+x^7+\left (-54 x-150 x^2-60 x^3-6 x^4\right ) \log (x)+9 x \log ^2(x)} \, dx=2 e^{\frac {1}{9+25 x+10 x^2+x^3-3 \log (x)}}+3 x-2 \log (x) \] Input:
Integrate[(-162 - 657*x - 260*x^2 + 1379*x^3 + 1254*x^4 + 410*x^5 + 58*x^6 + 3*x^7 + (6 - 50*x - 40*x^2 - 6*x^3)/E^(-9 - 25*x - 10*x^2 - x^3 + 3*Log [x])^(-1) + (108 + 138*x - 330*x^2 - 168*x^3 - 18*x^4)*Log[x] + (-18 + 27* x)*Log[x]^2)/(81*x + 450*x^2 + 805*x^3 + 518*x^4 + 150*x^5 + 20*x^6 + x^7 + (-54*x - 150*x^2 - 60*x^3 - 6*x^4)*Log[x] + 9*x*Log[x]^2),x]
Output:
2*E^(9 + 25*x + 10*x^2 + x^3 - 3*Log[x])^(-1) + 3*x - 2*Log[x]
Time = 7.53 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {7292, 7293, 2009}
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.
\(\displaystyle \int \frac {3 x^7+58 x^6+410 x^5+1254 x^4+1379 x^3-260 x^2+\left (-6 x^3-40 x^2-50 x+6\right ) e^{-\frac {1}{-x^3-10 x^2-25 x+3 \log (x)-9}}+\left (-18 x^4-168 x^3-330 x^2+138 x+108\right ) \log (x)-657 x+(27 x-18) \log ^2(x)-162}{x^7+20 x^6+150 x^5+518 x^4+805 x^3+450 x^2+\left (-6 x^4-60 x^3-150 x^2-54 x\right ) \log (x)+81 x+9 x \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {3 x^7+58 x^6+410 x^5+1254 x^4+1379 x^3-260 x^2+\left (-6 x^3-40 x^2-50 x+6\right ) e^{-\frac {1}{-x^3-10 x^2-25 x+3 \log (x)-9}}+\left (-18 x^4-168 x^3-330 x^2+138 x+108\right ) \log (x)-657 x+(27 x-18) \log ^2(x)-162}{x \left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {9 (3 x-2) \log ^2(x)}{x \left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}+\frac {1254 x^3}{\left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}+\frac {1379 x^2}{\left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}-\frac {260 x}{\left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}-\frac {657}{\left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}-\frac {6 (3 x-2) \left (x^3+10 x^2+25 x+9\right ) \log (x)}{x \left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}-\frac {2 \left (3 x^3+20 x^2+25 x-3\right ) e^{\frac {1}{x^3+10 x^2+25 x-3 \log (x)+9}}}{x \left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}-\frac {162}{x \left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}+\frac {3 x^6}{\left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}+\frac {58 x^5}{\left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}+\frac {410 x^4}{\left (x^3+10 x^2+25 x-3 \log (x)+9\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 e^{\frac {1}{x^3+10 x^2+25 x-3 \log (x)+9}}+3 x-2 \log (x)\) |
Input:
Int[(-162 - 657*x - 260*x^2 + 1379*x^3 + 1254*x^4 + 410*x^5 + 58*x^6 + 3*x ^7 + (6 - 50*x - 40*x^2 - 6*x^3)/E^(-9 - 25*x - 10*x^2 - x^3 + 3*Log[x])^( -1) + (108 + 138*x - 330*x^2 - 168*x^3 - 18*x^4)*Log[x] + (-18 + 27*x)*Log [x]^2)/(81*x + 450*x^2 + 805*x^3 + 518*x^4 + 150*x^5 + 20*x^6 + x^7 + (-54 *x - 150*x^2 - 60*x^3 - 6*x^4)*Log[x] + 9*x*Log[x]^2),x]
Output:
2*E^(9 + 25*x + 10*x^2 + x^3 - 3*Log[x])^(-1) + 3*x - 2*Log[x]
Time = 9.72 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17
method | result | size |
risch | \(3 x -2 \ln \left (x \right )+2 \,{\mathrm e}^{-\frac {1}{3 \ln \left (x \right )-x^{3}-10 x^{2}-25 x -9}}\) | \(35\) |
parallelrisch | \(3 x -2 \ln \left (x \right )+2 \,{\mathrm e}^{-\frac {1}{3 \ln \left (x \right )-x^{3}-10 x^{2}-25 x -9}}\) | \(35\) |
Input:
int(((-6*x^3-40*x^2-50*x+6)*exp(-1/(3*ln(x)-x^3-10*x^2-25*x-9))+(27*x-18)* ln(x)^2+(-18*x^4-168*x^3-330*x^2+138*x+108)*ln(x)+3*x^7+58*x^6+410*x^5+125 4*x^4+1379*x^3-260*x^2-657*x-162)/(9*x*ln(x)^2+(-6*x^4-60*x^3-150*x^2-54*x )*ln(x)+x^7+20*x^6+150*x^5+518*x^4+805*x^3+450*x^2+81*x),x,method=_RETURNV ERBOSE)
Output:
3*x-2*ln(x)+2*exp(-1/(3*ln(x)-x^3-10*x^2-25*x-9))
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-162-657 x-260 x^2+1379 x^3+1254 x^4+410 x^5+58 x^6+3 x^7+e^{-\frac {1}{-9-25 x-10 x^2-x^3+3 \log (x)}} \left (6-50 x-40 x^2-6 x^3\right )+\left (108+138 x-330 x^2-168 x^3-18 x^4\right ) \log (x)+(-18+27 x) \log ^2(x)}{81 x+450 x^2+805 x^3+518 x^4+150 x^5+20 x^6+x^7+\left (-54 x-150 x^2-60 x^3-6 x^4\right ) \log (x)+9 x \log ^2(x)} \, dx=3 \, x + 2 \, e^{\left (\frac {1}{x^{3} + 10 \, x^{2} + 25 \, x - 3 \, \log \left (x\right ) + 9}\right )} - 2 \, \log \left (x\right ) \] Input:
integrate(((-6*x^3-40*x^2-50*x+6)*exp(-1/(3*log(x)-x^3-10*x^2-25*x-9))+(27 *x-18)*log(x)^2+(-18*x^4-168*x^3-330*x^2+138*x+108)*log(x)+3*x^7+58*x^6+41 0*x^5+1254*x^4+1379*x^3-260*x^2-657*x-162)/(9*x*log(x)^2+(-6*x^4-60*x^3-15 0*x^2-54*x)*log(x)+x^7+20*x^6+150*x^5+518*x^4+805*x^3+450*x^2+81*x),x, alg orithm="fricas")
Output:
3*x + 2*e^(1/(x^3 + 10*x^2 + 25*x - 3*log(x) + 9)) - 2*log(x)
Time = 0.39 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-162-657 x-260 x^2+1379 x^3+1254 x^4+410 x^5+58 x^6+3 x^7+e^{-\frac {1}{-9-25 x-10 x^2-x^3+3 \log (x)}} \left (6-50 x-40 x^2-6 x^3\right )+\left (108+138 x-330 x^2-168 x^3-18 x^4\right ) \log (x)+(-18+27 x) \log ^2(x)}{81 x+450 x^2+805 x^3+518 x^4+150 x^5+20 x^6+x^7+\left (-54 x-150 x^2-60 x^3-6 x^4\right ) \log (x)+9 x \log ^2(x)} \, dx=3 x - 2 \log {\left (x \right )} + 2 e^{- \frac {1}{- x^{3} - 10 x^{2} - 25 x + 3 \log {\left (x \right )} - 9}} \] Input:
integrate(((-6*x**3-40*x**2-50*x+6)*exp(-1/(3*ln(x)-x**3-10*x**2-25*x-9))+ (27*x-18)*ln(x)**2+(-18*x**4-168*x**3-330*x**2+138*x+108)*ln(x)+3*x**7+58* x**6+410*x**5+1254*x**4+1379*x**3-260*x**2-657*x-162)/(9*x*ln(x)**2+(-6*x* *4-60*x**3-150*x**2-54*x)*ln(x)+x**7+20*x**6+150*x**5+518*x**4+805*x**3+45 0*x**2+81*x),x)
Output:
3*x - 2*log(x) + 2*exp(-1/(-x**3 - 10*x**2 - 25*x + 3*log(x) - 9))
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {-162-657 x-260 x^2+1379 x^3+1254 x^4+410 x^5+58 x^6+3 x^7+e^{-\frac {1}{-9-25 x-10 x^2-x^3+3 \log (x)}} \left (6-50 x-40 x^2-6 x^3\right )+\left (108+138 x-330 x^2-168 x^3-18 x^4\right ) \log (x)+(-18+27 x) \log ^2(x)}{81 x+450 x^2+805 x^3+518 x^4+150 x^5+20 x^6+x^7+\left (-54 x-150 x^2-60 x^3-6 x^4\right ) \log (x)+9 x \log ^2(x)} \, dx={\left (3 \, x e^{\left (-\frac {1}{x^{3} + 10 \, x^{2} + 25 \, x - 3 \, \log \left (x\right ) + 9}\right )} + 2\right )} e^{\left (\frac {1}{x^{3} + 10 \, x^{2} + 25 \, x - 3 \, \log \left (x\right ) + 9}\right )} - 2 \, \log \left (x\right ) \] Input:
integrate(((-6*x^3-40*x^2-50*x+6)*exp(-1/(3*log(x)-x^3-10*x^2-25*x-9))+(27 *x-18)*log(x)^2+(-18*x^4-168*x^3-330*x^2+138*x+108)*log(x)+3*x^7+58*x^6+41 0*x^5+1254*x^4+1379*x^3-260*x^2-657*x-162)/(9*x*log(x)^2+(-6*x^4-60*x^3-15 0*x^2-54*x)*log(x)+x^7+20*x^6+150*x^5+518*x^4+805*x^3+450*x^2+81*x),x, alg orithm="maxima")
Output:
(3*x*e^(-1/(x^3 + 10*x^2 + 25*x - 3*log(x) + 9)) + 2)*e^(1/(x^3 + 10*x^2 + 25*x - 3*log(x) + 9)) - 2*log(x)
Time = 0.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-162-657 x-260 x^2+1379 x^3+1254 x^4+410 x^5+58 x^6+3 x^7+e^{-\frac {1}{-9-25 x-10 x^2-x^3+3 \log (x)}} \left (6-50 x-40 x^2-6 x^3\right )+\left (108+138 x-330 x^2-168 x^3-18 x^4\right ) \log (x)+(-18+27 x) \log ^2(x)}{81 x+450 x^2+805 x^3+518 x^4+150 x^5+20 x^6+x^7+\left (-54 x-150 x^2-60 x^3-6 x^4\right ) \log (x)+9 x \log ^2(x)} \, dx=3 \, x + 2 \, e^{\left (\frac {1}{x^{3} + 10 \, x^{2} + 25 \, x - 3 \, \log \left (x\right ) + 9}\right )} - 2 \, \log \left (x\right ) \] Input:
integrate(((-6*x^3-40*x^2-50*x+6)*exp(-1/(3*log(x)-x^3-10*x^2-25*x-9))+(27 *x-18)*log(x)^2+(-18*x^4-168*x^3-330*x^2+138*x+108)*log(x)+3*x^7+58*x^6+41 0*x^5+1254*x^4+1379*x^3-260*x^2-657*x-162)/(9*x*log(x)^2+(-6*x^4-60*x^3-15 0*x^2-54*x)*log(x)+x^7+20*x^6+150*x^5+518*x^4+805*x^3+450*x^2+81*x),x, alg orithm="giac")
Output:
3*x + 2*e^(1/(x^3 + 10*x^2 + 25*x - 3*log(x) + 9)) - 2*log(x)
Time = 7.44 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-162-657 x-260 x^2+1379 x^3+1254 x^4+410 x^5+58 x^6+3 x^7+e^{-\frac {1}{-9-25 x-10 x^2-x^3+3 \log (x)}} \left (6-50 x-40 x^2-6 x^3\right )+\left (108+138 x-330 x^2-168 x^3-18 x^4\right ) \log (x)+(-18+27 x) \log ^2(x)}{81 x+450 x^2+805 x^3+518 x^4+150 x^5+20 x^6+x^7+\left (-54 x-150 x^2-60 x^3-6 x^4\right ) \log (x)+9 x \log ^2(x)} \, dx=3\,x+2\,{\mathrm {e}}^{\frac {1}{25\,x-3\,\ln \left (x\right )+10\,x^2+x^3+9}}-2\,\ln \left (x\right ) \] Input:
int((1379*x^3 - log(x)*(330*x^2 - 138*x + 168*x^3 + 18*x^4 - 108) - exp(1/ (25*x - 3*log(x) + 10*x^2 + x^3 + 9))*(50*x + 40*x^2 + 6*x^3 - 6) - 260*x^ 2 - 657*x + 1254*x^4 + 410*x^5 + 58*x^6 + 3*x^7 + log(x)^2*(27*x - 18) - 1 62)/(81*x + 9*x*log(x)^2 - log(x)*(54*x + 150*x^2 + 60*x^3 + 6*x^4) + 450* x^2 + 805*x^3 + 518*x^4 + 150*x^5 + 20*x^6 + x^7),x)
Output:
3*x + 2*exp(1/(25*x - 3*log(x) + 10*x^2 + x^3 + 9)) - 2*log(x)
Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \[ \int \frac {-162-657 x-260 x^2+1379 x^3+1254 x^4+410 x^5+58 x^6+3 x^7+e^{-\frac {1}{-9-25 x-10 x^2-x^3+3 \log (x)}} \left (6-50 x-40 x^2-6 x^3\right )+\left (108+138 x-330 x^2-168 x^3-18 x^4\right ) \log (x)+(-18+27 x) \log ^2(x)}{81 x+450 x^2+805 x^3+518 x^4+150 x^5+20 x^6+x^7+\left (-54 x-150 x^2-60 x^3-6 x^4\right ) \log (x)+9 x \log ^2(x)} \, dx=\frac {-2 e^{\frac {1}{3 \,\mathrm {log}\left (x \right )-x^{3}-10 x^{2}-25 x -9}} \mathrm {log}\left (x \right )+3 e^{\frac {1}{3 \,\mathrm {log}\left (x \right )-x^{3}-10 x^{2}-25 x -9}} x +2}{e^{\frac {1}{3 \,\mathrm {log}\left (x \right )-x^{3}-10 x^{2}-25 x -9}}} \] Input:
int(((-6*x^3-40*x^2-50*x+6)*exp(-1/(3*log(x)-x^3-10*x^2-25*x-9))+(27*x-18) *log(x)^2+(-18*x^4-168*x^3-330*x^2+138*x+108)*log(x)+3*x^7+58*x^6+410*x^5+ 1254*x^4+1379*x^3-260*x^2-657*x-162)/(9*x*log(x)^2+(-6*x^4-60*x^3-150*x^2- 54*x)*log(x)+x^7+20*x^6+150*x^5+518*x^4+805*x^3+450*x^2+81*x),x)
Output:
( - 2*e**(1/(3*log(x) - x**3 - 10*x**2 - 25*x - 9))*log(x) + 3*e**(1/(3*lo g(x) - x**3 - 10*x**2 - 25*x - 9))*x + 2)/e**(1/(3*log(x) - x**3 - 10*x**2 - 25*x - 9))