Integrand size = 188, antiderivative size = 31 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=x+\log \left (x^2 \left (2 e^{-\frac {3}{-x+\frac {3}{5+2 x}}}+\log (x)\right )\right ) \] Output:
ln(x^2*(ln(x)+2/exp(3/(3/(5+2*x)-x))))+x
Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=x+2 \log (x)+\log \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right ) \] Input:
Integrate[(36 - 288*x - 128*x^2 + 82*x^3 + 56*x^4 + 8*x^5 + E^((-15 - 6*x) /(-3 + 5*x + 2*x^2))*(9 - 30*x + 13*x^2 + 20*x^3 + 4*x^4) + E^((-15 - 6*x) /(-3 + 5*x + 2*x^2))*(18 - 51*x - 4*x^2 + 53*x^3 + 28*x^4 + 4*x^5)*Log[x]) /(18*x - 60*x^2 + 26*x^3 + 40*x^4 + 8*x^5 + E^((-15 - 6*x)/(-3 + 5*x + 2*x ^2))*(9*x - 30*x^2 + 13*x^3 + 20*x^4 + 4*x^5)*Log[x]),x]
Output:
x + 2*Log[x] + Log[2*E^((3*(5 + 2*x))/(-3 + 5*x + 2*x^2)) + Log[x]]
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.
| \(\displaystyle \int \frac {8 x^5+56 x^4+82 x^3-128 x^2+e^{\frac {-6 x-15}{2 x^2+5 x-3}} \left (4 x^4+20 x^3+13 x^2-30 x+9\right )+e^{\frac {-6 x-15}{2 x^2+5 x-3}} \left (4 x^5+28 x^4+53 x^3-4 x^2-51 x+18\right ) \log (x)-288 x+36}{8 x^5+40 x^4+26 x^3-60 x^2+e^{\frac {-6 x-15}{2 x^2+5 x-3}} \left (4 x^5+20 x^4+13 x^3-30 x^2+9 x\right ) \log (x)+18 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {8 x^5+56 x^4+82 x^3-128 x^2+e^{\frac {-6 x-15}{2 x^2+5 x-3}} \left (4 x^4+20 x^3+13 x^2-30 x+9\right )+e^{\frac {-6 x-15}{2 x^2+5 x-3}} \left (4 x^5+28 x^4+53 x^3-4 x^2-51 x+18\right ) \log (x)-288 x+36}{x \left (-2 x^2-5 x+3\right )^2 \left (\exp \left (\frac {6 x}{-2 x^2-5 x+3}+\frac {15}{-2 x^2-5 x+3}\right ) \log (x)+2\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x \log (x)+2 \log (x)+1}{x \log (x)}-\frac {2 \left (4 x^4+20 x^3+12 x^3 \log (x)+13 x^2+60 x^2 \log (x)-30 x+93 x \log (x)+9\right )}{x (x+3)^2 (2 x-1)^2 \log (x) \left (\exp \left (\frac {6 x}{-2 x^2-5 x+3}+\frac {15}{-2 x^2-5 x+3}\right ) \log (x)+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {x \log (x)+2 \log (x)+1}{x \log (x)}-\frac {2 \left (4 x^4+20 x^3+12 x^3 \log (x)+13 x^2+60 x^2 \log (x)-30 x+93 x \log (x)+9\right )}{x (x+3)^2 (2 x-1)^2 \log (x) \left (\exp \left (\frac {6 x}{-2 x^2-5 x+3}+\frac {15}{-2 x^2-5 x+3}\right ) \log (x)+2\right )}\right )dx\) |
Input:
Int[(36 - 288*x - 128*x^2 + 82*x^3 + 56*x^4 + 8*x^5 + E^((-15 - 6*x)/(-3 + 5*x + 2*x^2))*(9 - 30*x + 13*x^2 + 20*x^3 + 4*x^4) + E^((-15 - 6*x)/(-3 + 5*x + 2*x^2))*(18 - 51*x - 4*x^2 + 53*x^3 + 28*x^4 + 4*x^5)*Log[x])/(18*x - 60*x^2 + 26*x^3 + 40*x^4 + 8*x^5 + E^((-15 - 6*x)/(-3 + 5*x + 2*x^2))*( 9*x - 30*x^2 + 13*x^3 + 20*x^4 + 4*x^5)*Log[x]),x]
Output:
$Aborted
Time = 7.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(x +2 \ln \left (x \right )+\ln \left (\ln \left (x \right )+2 \,{\mathrm e}^{\frac {6 x +15}{\left (3+x \right ) \left (-1+2 x \right )}}\right )\) | \(33\) |
| parallelrisch | \(-\frac {-10080 x^{2} \ln \left (x \right )-5040 \ln \left ({\mathrm e}^{-\frac {3 \left (5+2 x \right )}{2 x^{2}+5 x -3}} \ln \left (x \right )+2\right ) x^{2}-5040 x^{3}-54675-25200 x \ln \left (x \right )-12600 \ln \left ({\mathrm e}^{-\frac {3 \left (5+2 x \right )}{2 x^{2}+5 x -3}} \ln \left (x \right )+2\right ) x -1350 x^{2}+15120 \ln \left (x \right )+7560 \ln \left ({\mathrm e}^{-\frac {3 \left (5+2 x \right )}{2 x^{2}+5 x -3}} \ln \left (x \right )+2\right )+20565 x}{2520 \left (2 x^{2}+5 x -3\right )}\) | \(134\) |
Input:
int(((4*x^5+28*x^4+53*x^3-4*x^2-51*x+18)*exp((-6*x-15)/(2*x^2+5*x-3))*ln(x )+(4*x^4+20*x^3+13*x^2-30*x+9)*exp((-6*x-15)/(2*x^2+5*x-3))+8*x^5+56*x^4+8 2*x^3-128*x^2-288*x+36)/((4*x^5+20*x^4+13*x^3-30*x^2+9*x)*exp((-6*x-15)/(2 *x^2+5*x-3))*ln(x)+8*x^5+40*x^4+26*x^3-60*x^2+18*x),x,method=_RETURNVERBOS E)
Output:
x+2*ln(x)+ln(ln(x)+2*exp(3*(5+2*x)/(3+x)/(-1+2*x)))
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=x + \log \left ({\left (e^{\left (-\frac {3 \, {\left (2 \, x + 5\right )}}{2 \, x^{2} + 5 \, x - 3}\right )} \log \left (x\right ) + 2\right )} e^{\left (\frac {3 \, {\left (2 \, x + 5\right )}}{2 \, x^{2} + 5 \, x - 3}\right )}\right ) + 2 \, \log \left (x\right ) \] Input:
integrate(((4*x^5+28*x^4+53*x^3-4*x^2-51*x+18)*exp((-6*x-15)/(2*x^2+5*x-3) )*log(x)+(4*x^4+20*x^3+13*x^2-30*x+9)*exp((-6*x-15)/(2*x^2+5*x-3))+8*x^5+5 6*x^4+82*x^3-128*x^2-288*x+36)/((4*x^5+20*x^4+13*x^3-30*x^2+9*x)*exp((-6*x -15)/(2*x^2+5*x-3))*log(x)+8*x^5+40*x^4+26*x^3-60*x^2+18*x),x, algorithm=" fricas")
Output:
x + log((e^(-3*(2*x + 5)/(2*x^2 + 5*x - 3))*log(x) + 2)*e^(3*(2*x + 5)/(2* x^2 + 5*x - 3))) + 2*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (22) = 44\).
Time = 0.57 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=x + \frac {6 x + 15}{2 x^{2} + 5 x - 3} + 2 \log {\left (x \right )} + \log {\left (e^{\frac {- 6 x - 15}{2 x^{2} + 5 x - 3}} + \frac {2}{\log {\left (x \right )}} \right )} + \log {\left (\log {\left (x \right )} \right )} \] Input:
integrate(((4*x**5+28*x**4+53*x**3-4*x**2-51*x+18)*exp((-6*x-15)/(2*x**2+5 *x-3))*ln(x)+(4*x**4+20*x**3+13*x**2-30*x+9)*exp((-6*x-15)/(2*x**2+5*x-3)) +8*x**5+56*x**4+82*x**3-128*x**2-288*x+36)/((4*x**5+20*x**4+13*x**3-30*x** 2+9*x)*exp((-6*x-15)/(2*x**2+5*x-3))*ln(x)+8*x**5+40*x**4+26*x**3-60*x**2+ 18*x),x)
Output:
x + (6*x + 15)/(2*x**2 + 5*x - 3) + 2*log(x) + log(exp((-6*x - 15)/(2*x**2 + 5*x - 3)) + 2/log(x)) + log(log(x))
Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=\frac {7 \, x^{2} + 21 \, x + 3}{7 \, {\left (x + 3\right )}} + \log \left (\frac {1}{2} \, {\left (2 \, e^{\left (\frac {36}{7 \, {\left (2 \, x - 1\right )}} + \frac {3}{7 \, {\left (x + 3\right )}}\right )} + \log \left (x\right )\right )} e^{\left (-\frac {3}{7 \, {\left (x + 3\right )}}\right )}\right ) + 2 \, \log \left (x\right ) \] Input:
integrate(((4*x^5+28*x^4+53*x^3-4*x^2-51*x+18)*exp((-6*x-15)/(2*x^2+5*x-3) )*log(x)+(4*x^4+20*x^3+13*x^2-30*x+9)*exp((-6*x-15)/(2*x^2+5*x-3))+8*x^5+5 6*x^4+82*x^3-128*x^2-288*x+36)/((4*x^5+20*x^4+13*x^3-30*x^2+9*x)*exp((-6*x -15)/(2*x^2+5*x-3))*log(x)+8*x^5+40*x^4+26*x^3-60*x^2+18*x),x, algorithm=" maxima")
Output:
1/7*(7*x^2 + 21*x + 3)/(x + 3) + log(1/2*(2*e^(36/7/(2*x - 1) + 3/7/(x + 3 )) + log(x))*e^(-3/7/(x + 3))) + 2*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (28) = 56\).
Time = 0.38 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.84 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=\frac {2 \, x^{3} + 2 \, x^{2} \log \left (e^{\left (-\frac {10 \, x^{2} + 31 \, x}{2 \, x^{2} + 5 \, x - 3} + 5\right )} \log \left (x\right ) + 2\right ) + 4 \, x^{2} \log \left (x\right ) + 5 \, x^{2} + 5 \, x \log \left (e^{\left (-\frac {10 \, x^{2} + 31 \, x}{2 \, x^{2} + 5 \, x - 3} + 5\right )} \log \left (x\right ) + 2\right ) + 10 \, x \log \left (x\right ) + 3 \, x - 3 \, \log \left (e^{\left (-\frac {10 \, x^{2} + 31 \, x}{2 \, x^{2} + 5 \, x - 3} + 5\right )} \log \left (x\right ) + 2\right ) - 6 \, \log \left (x\right ) + 15}{2 \, x^{2} + 5 \, x - 3} \] Input:
integrate(((4*x^5+28*x^4+53*x^3-4*x^2-51*x+18)*exp((-6*x-15)/(2*x^2+5*x-3) )*log(x)+(4*x^4+20*x^3+13*x^2-30*x+9)*exp((-6*x-15)/(2*x^2+5*x-3))+8*x^5+5 6*x^4+82*x^3-128*x^2-288*x+36)/((4*x^5+20*x^4+13*x^3-30*x^2+9*x)*exp((-6*x -15)/(2*x^2+5*x-3))*log(x)+8*x^5+40*x^4+26*x^3-60*x^2+18*x),x, algorithm=" giac")
Output:
(2*x^3 + 2*x^2*log(e^(-(10*x^2 + 31*x)/(2*x^2 + 5*x - 3) + 5)*log(x) + 2) + 4*x^2*log(x) + 5*x^2 + 5*x*log(e^(-(10*x^2 + 31*x)/(2*x^2 + 5*x - 3) + 5 )*log(x) + 2) + 10*x*log(x) + 3*x - 3*log(e^(-(10*x^2 + 31*x)/(2*x^2 + 5*x - 3) + 5)*log(x) + 2) - 6*log(x) + 15)/(2*x^2 + 5*x - 3)
Time = 1.96 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=x+\ln \left (\ln \left (x\right )\right )+\ln \left (\frac {{\mathrm {e}}^{-\frac {6\,x+15}{2\,x^2+5\,x-3}}\,\ln \left (x\right )+2}{\ln \left (x\right )}\right )+2\,\ln \left (x\right )+\frac {3\,x+\frac {15}{2}}{x^2+\frac {5\,x}{2}-\frac {3}{2}} \] Input:
int((exp(-(6*x + 15)/(5*x + 2*x^2 - 3))*(13*x^2 - 30*x + 20*x^3 + 4*x^4 + 9) - 288*x - 128*x^2 + 82*x^3 + 56*x^4 + 8*x^5 + exp(-(6*x + 15)/(5*x + 2* x^2 - 3))*log(x)*(53*x^3 - 4*x^2 - 51*x + 28*x^4 + 4*x^5 + 18) + 36)/(18*x - 60*x^2 + 26*x^3 + 40*x^4 + 8*x^5 + exp(-(6*x + 15)/(5*x + 2*x^2 - 3))*l og(x)*(9*x - 30*x^2 + 13*x^3 + 20*x^4 + 4*x^5)),x)
Output:
x + log(log(x)) + log((exp(-(6*x + 15)/(5*x + 2*x^2 - 3))*log(x) + 2)/log( x)) + 2*log(x) + (3*x + 15/2)/((5*x)/2 + x^2 - 3/2)
\[ \int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx=\text {too large to display} \] Input:
int(((4*x^5+28*x^4+53*x^3-4*x^2-51*x+18)*exp((-6*x-15)/(2*x^2+5*x-3))*log( x)+(4*x^4+20*x^3+13*x^2-30*x+9)*exp((-6*x-15)/(2*x^2+5*x-3))+8*x^5+56*x^4+ 82*x^3-128*x^2-288*x+36)/((4*x^5+20*x^4+13*x^3-30*x^2+9*x)*exp((-6*x-15)/( 2*x^2+5*x-3))*log(x)+8*x^5+40*x^4+26*x^3-60*x^2+18*x),x)
Output:
36*int(e**((6*x + 15)/(2*x**2 + 5*x - 3))/(8*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x**5 + 40*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x**4 + 26*e**((6*x + 15 )/(2*x**2 + 5*x - 3))*x**3 - 60*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x**2 + 18*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x + 4*log(x)*x**5 + 20*log(x)*x**4 + 13*log(x)*x**3 - 30*log(x)*x**2 + 9*log(x)*x),x) - 288*int(e**((6*x + 15) /(2*x**2 + 5*x - 3))/(8*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x**4 + 40*e**(( 6*x + 15)/(2*x**2 + 5*x - 3))*x**3 + 26*e**((6*x + 15)/(2*x**2 + 5*x - 3)) *x**2 - 60*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x + 18*e**((6*x + 15)/(2*x** 2 + 5*x - 3)) + 4*log(x)*x**4 + 20*log(x)*x**3 + 13*log(x)*x**2 - 30*log(x )*x + 9*log(x)),x) + 4*int(x**3/(8*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x**4 + 40*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x**3 + 26*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x**2 - 60*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x + 18*e**((6*x + 15)/(2*x**2 + 5*x - 3)) + 4*log(x)*x**4 + 20*log(x)*x**3 + 13*log(x)*x**2 - 30*log(x)*x + 9*log(x)),x) + 20*int(x**2/(8*e**((6*x + 15)/(2*x**2 + 5* x - 3))*x**4 + 40*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x**3 + 26*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x**2 - 60*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x + 1 8*e**((6*x + 15)/(2*x**2 + 5*x - 3)) + 4*log(x)*x**4 + 20*log(x)*x**3 + 13 *log(x)*x**2 - 30*log(x)*x + 9*log(x)),x) + 18*int(log(x)/(8*e**((6*x + 15 )/(2*x**2 + 5*x - 3))*x**5 + 40*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x**4 + 26*e**((6*x + 15)/(2*x**2 + 5*x - 3))*x**3 - 60*e**((6*x + 15)/(2*x**2 ...