Integrand size = 111, antiderivative size = 28 \[ \int \frac {6 x+15 e^4 x^3+3 e^9 x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{5 e^4 x+e^9 x} \, dx=x \left (x-\log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )\right )^2 \] Output:
(x-ln(5*exp(3/exp(4)/(exp(5)+5)/x)))^2*x
Leaf count is larger than twice the leaf count of optimal. \(125\) vs. \(2(28)=56\).
Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 4.46 \[ \int \frac {6 x+15 e^4 x^3+3 e^9 x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{5 e^4 x+e^9 x} \, dx=\frac {18+25 e^8 x^4+10 e^{13} x^4+e^{18} x^4-2 e^4 \left (5+e^5\right ) x \left (3+5 e^4 x^2+e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+e^8 \left (5+e^5\right )^2 x^2 \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{e^8 \left (5+e^5\right )^2 x} \] Input:
Integrate[(6*x + 15*E^4*x^3 + 3*E^9*x^3 + (-6 - 20*E^4*x^2 - 4*E^9*x^2)*Lo g[5*E^(3/(5*E^4*x + E^9*x))] + (5*E^4*x + E^9*x)*Log[5*E^(3/(5*E^4*x + E^9 *x))]^2)/(5*E^4*x + E^9*x),x]
Output:
(18 + 25*E^8*x^4 + 10*E^13*x^4 + E^18*x^4 - 2*E^4*(5 + E^5)*x*(3 + 5*E^4*x ^2 + E^9*x^2)*Log[5*E^(3/(5*E^4*x + E^9*x))] + E^8*(5 + E^5)^2*x^2*Log[5*E ^(3/(5*E^4*x + E^9*x))]^2)/(E^8*(5 + E^5)^2*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.
| \(\displaystyle \int \frac {3 e^9 x^3+15 e^4 x^3+\left (-4 e^9 x^2-20 e^4 x^2-6\right ) \log \left (5 e^{\frac {3}{e^9 x+5 e^4 x}}\right )+6 x+\left (e^9 x+5 e^4 x\right ) \log ^2\left (5 e^{\frac {3}{e^9 x+5 e^4 x}}\right )}{e^9 x+5 e^4 x} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {3 e^9 x^3+15 e^4 x^3+\left (-4 e^9 x^2-20 e^4 x^2-6\right ) \log \left (5 e^{\frac {3}{e^9 x+5 e^4 x}}\right )+6 x+\left (e^9 x+5 e^4 x\right ) \log ^2\left (5 e^{\frac {3}{e^9 x+5 e^4 x}}\right )}{\left (5 e^4+e^9\right ) x}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (15 e^4+3 e^9\right ) x^3+\left (-4 e^9 x^2-20 e^4 x^2-6\right ) \log \left (5 e^{\frac {3}{e^9 x+5 e^4 x}}\right )+6 x+\left (e^9 x+5 e^4 x\right ) \log ^2\left (5 e^{\frac {3}{e^9 x+5 e^4 x}}\right )}{\left (5 e^4+e^9\right ) x}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 e^4 \left (5+e^5\right ) x^3+e^4 \left (5+e^5\right ) \log ^2\left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) x+6 x-2 \left (2 e^9 x^2+10 e^4 x^2+3\right ) \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )}{x}dx}{e^4 \left (5+e^5\right )}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {\int \left (e^4 \left (5+e^5\right ) \log ^2\left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )+\frac {2 \left (-2 e^4 \left (5+e^5\right ) x^2-3\right ) \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )}{x}+3 \left (e^4 \left (5+e^5\right ) x^2+2\right )\right )dx}{e^4 \left (5+e^5\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (5+e^5\right ) \int \log ^2\left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )dx+e^4 \left (5+e^5\right ) x^3-2 e^4 \left (5+e^5\right ) x^2 \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )+\frac {18}{e^4 \left (5+e^5\right ) x}-6 \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) \log (x)+\frac {18 \log (x)}{e^4 \left (5+e^5\right ) x}}{e^4 \left (5+e^5\right )}\) |
Input:
Int[(6*x + 15*E^4*x^3 + 3*E^9*x^3 + (-6 - 20*E^4*x^2 - 4*E^9*x^2)*Log[5*E^ (3/(5*E^4*x + E^9*x))] + (5*E^4*x + E^9*x)*Log[5*E^(3/(5*E^4*x + E^9*x))]^ 2)/(5*E^4*x + E^9*x),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(27)=54\).
Time = 0.55 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36
| method | result | size |
| risch | \(x \ln \left ({\mathrm e}^{\frac {3}{x \left ({\mathrm e}^{9}+5 \,{\mathrm e}^{4}\right )}}\right )^{2}+\left (2 x \ln \left (5\right )-2 x^{2}\right ) \ln \left ({\mathrm e}^{\frac {3}{x \left ({\mathrm e}^{9}+5 \,{\mathrm e}^{4}\right )}}\right )+x^{3}-2 x^{2} \ln \left (5\right )+x \ln \left (5\right )^{2}\) | \(66\) |
| parallelrisch | \(-\frac {\left (-27 \,{\mathrm e}^{4} x^{6} {\mathrm e}^{5}-135 \,{\mathrm e}^{4} x^{6}+270 \,{\mathrm e}^{4} x^{5} \ln \left (5 \,{\mathrm e}^{\frac {3 \,{\mathrm e}^{-4}}{x \left ({\mathrm e}^{5}+5\right )}}\right )-135 \,{\mathrm e}^{4} x^{4} \ln \left (5 \,{\mathrm e}^{\frac {3 \,{\mathrm e}^{-4}}{x \left ({\mathrm e}^{5}+5\right )}}\right )^{2}+54 \,{\mathrm e}^{4} {\mathrm e}^{5} x^{5} \ln \left (5 \,{\mathrm e}^{\frac {3 \,{\mathrm e}^{-4}}{x \left ({\mathrm e}^{5}+5\right )}}\right )-27 \,{\mathrm e}^{4} {\mathrm e}^{5} x^{4} \ln \left (5 \,{\mathrm e}^{\frac {3 \,{\mathrm e}^{-4}}{x \left ({\mathrm e}^{5}+5\right )}}\right )^{2}\right ) {\mathrm e}^{-4}}{27 \left ({\mathrm e}^{5}+5\right ) x^{3}}\) | \(145\) |
| default | \(\frac {{\mathrm e}^{-8} \left (x {\left (\ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )-\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}\right )}^{2} \left ({\mathrm e}^{9}+5 \,{\mathrm e}^{4}\right )^{2}-\frac {9}{x}+6 \left (\ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )-\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}\right ) \left ({\mathrm e}^{9}+5 \,{\mathrm e}^{4}\right ) \ln \left (x \right )\right )}{\left ({\mathrm e}^{5}+5\right )^{2}}+\frac {3 \,{\mathrm e}^{-4} \left (\frac {x^{3} {\mathrm e}^{4} {\mathrm e}^{5}}{3}+\frac {5 x^{3} {\mathrm e}^{4}}{3}+2 x \right )}{{\mathrm e}^{5}+5}-\frac {2 \,{\mathrm e}^{-4} \left ({\mathrm e}^{9} \ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right ) x^{2}+5 \,{\mathrm e}^{4} \ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right ) x^{2}+3 \ln \left (x \right ) \ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )-\left (-\frac {3 \,{\mathrm e}^{4} {\mathrm e}^{5}}{2}-\frac {15 \,{\mathrm e}^{4}}{2}\right ) \left (\frac {2 \,{\mathrm e}^{-8} {\mathrm e}^{9} x}{\left ({\mathrm e}^{5}+5\right )^{2}}+\frac {10 \,{\mathrm e}^{-4} x}{\left ({\mathrm e}^{5}+5\right )^{2}}+\frac {6 \,{\mathrm e}^{-8} \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )}{\left ({\mathrm e}^{5}+5\right )^{2}}\right )\right )}{{\mathrm e}^{5}+5}\) | \(306\) |
| parts | \(\frac {{\mathrm e}^{-8} \left (x {\left (\ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )-\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}\right )}^{2} \left ({\mathrm e}^{9}+5 \,{\mathrm e}^{4}\right )^{2}-\frac {9}{x}+6 \left (\ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )-\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}\right ) \left ({\mathrm e}^{9}+5 \,{\mathrm e}^{4}\right ) \ln \left (x \right )\right )}{\left ({\mathrm e}^{5}+5\right )^{2}}+\frac {3 \,{\mathrm e}^{-4} \left (\frac {x^{3} {\mathrm e}^{4} {\mathrm e}^{5}}{3}+\frac {5 x^{3} {\mathrm e}^{4}}{3}+2 x \right )}{{\mathrm e}^{5}+5}-\frac {2 \,{\mathrm e}^{-4} \left ({\mathrm e}^{9} \ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right ) x^{2}+5 \,{\mathrm e}^{4} \ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right ) x^{2}+3 \ln \left (x \right ) \ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )-\left (-\frac {3 \,{\mathrm e}^{4} {\mathrm e}^{5}}{2}-\frac {15 \,{\mathrm e}^{4}}{2}\right ) \left (\frac {2 \,{\mathrm e}^{-8} {\mathrm e}^{9} x}{\left ({\mathrm e}^{5}+5\right )^{2}}+\frac {10 \,{\mathrm e}^{-4} x}{\left ({\mathrm e}^{5}+5\right )^{2}}+\frac {6 \,{\mathrm e}^{-8} \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )}{\left ({\mathrm e}^{5}+5\right )^{2}}\right )\right )}{{\mathrm e}^{5}+5}\) | \(306\) |
Input:
int(((x*exp(4)*exp(5)+5*x*exp(4))*ln(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4))) )^2+(-4*x^2*exp(4)*exp(5)-20*x^2*exp(4)-6)*ln(5*exp(3/(x*exp(4)*exp(5)+5*x *exp(4))))+3*x^3*exp(4)*exp(5)+15*x^3*exp(4)+6*x)/(x*exp(4)*exp(5)+5*x*exp (4)),x,method=_RETURNVERBOSE)
Output:
x*ln(exp(3/x/(exp(9)+5*exp(4))))^2+(2*x*ln(5)-2*x^2)*ln(exp(3/x/(exp(9)+5* exp(4))))+x^3-2*x^2*ln(5)+x*ln(5)^2
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.75 \[ \int \frac {6 x+15 e^4 x^3+3 e^9 x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{5 e^4 x+e^9 x} \, dx=\frac {x^{4} e^{18} + 10 \, x^{4} e^{13} + 25 \, x^{4} e^{8} - 6 \, x^{2} e^{9} - 30 \, x^{2} e^{4} + {\left (x^{2} e^{18} + 10 \, x^{2} e^{13} + 25 \, x^{2} e^{8}\right )} \log \left (5\right )^{2} - 2 \, {\left (x^{3} e^{18} + 10 \, x^{3} e^{13} + 25 \, x^{3} e^{8}\right )} \log \left (5\right ) + 9}{x e^{18} + 10 \, x e^{13} + 25 \, x e^{8}} \] Input:
integrate(((x*exp(4)*exp(5)+5*x*exp(4))*log(5*exp(3/(x*exp(4)*exp(5)+5*x*e xp(4))))^2+(-4*x^2*exp(4)*exp(5)-20*x^2*exp(4)-6)*log(5*exp(3/(x*exp(4)*ex p(5)+5*x*exp(4))))+3*x^3*exp(4)*exp(5)+15*x^3*exp(4)+6*x)/(x*exp(4)*exp(5) +5*x*exp(4)),x, algorithm="fricas")
Output:
(x^4*e^18 + 10*x^4*e^13 + 25*x^4*e^8 - 6*x^2*e^9 - 30*x^2*e^4 + (x^2*e^18 + 10*x^2*e^13 + 25*x^2*e^8)*log(5)^2 - 2*(x^3*e^18 + 10*x^3*e^13 + 25*x^3* e^8)*log(5) + 9)/(x*e^18 + 10*x*e^13 + 25*x*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (20) = 40\).
Time = 0.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.71 \[ \int \frac {6 x+15 e^4 x^3+3 e^9 x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{5 e^4 x+e^9 x} \, dx=\frac {x^{3} \cdot \left (25 e^{8} + 10 e^{13} + e^{18}\right ) + x^{2} \left (- 2 e^{18} \log {\left (5 \right )} - 20 e^{13} \log {\left (5 \right )} - 50 e^{8} \log {\left (5 \right )}\right ) + x \left (- 6 e^{9} - 30 e^{4} + 25 e^{8} \log {\left (5 \right )}^{2} + 10 e^{13} \log {\left (5 \right )}^{2} + e^{18} \log {\left (5 \right )}^{2}\right ) + \frac {9}{x}}{25 e^{8} + 10 e^{13} + e^{18}} \] Input:
integrate(((x*exp(4)*exp(5)+5*x*exp(4))*ln(5*exp(3/(x*exp(4)*exp(5)+5*x*ex p(4))))**2+(-4*x**2*exp(4)*exp(5)-20*x**2*exp(4)-6)*ln(5*exp(3/(x*exp(4)*e xp(5)+5*x*exp(4))))+3*x**3*exp(4)*exp(5)+15*x**3*exp(4)+6*x)/(x*exp(4)*exp (5)+5*x*exp(4)),x)
Output:
(x**3*(25*exp(8) + 10*exp(13) + exp(18)) + x**2*(-2*exp(18)*log(5) - 20*ex p(13)*log(5) - 50*exp(8)*log(5)) + x*(-6*exp(9) - 30*exp(4) + 25*exp(8)*lo g(5)**2 + 10*exp(13)*log(5)**2 + exp(18)*log(5)**2) + 9/x)/(25*exp(8) + 10 *exp(13) + exp(18))
Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (25) = 50\).
Time = 0.04 (sec) , antiderivative size = 512, normalized size of antiderivative = 18.29 \[ \int \frac {6 x+15 e^4 x^3+3 e^9 x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{5 e^4 x+e^9 x} \, dx=\frac {x^{3} e^{9}}{e^{9} + 5 \, e^{4}} + \frac {5 \, x^{3} e^{4}}{e^{9} + 5 \, e^{4}} - \frac {2 \, x^{2} e^{9} \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )}{e^{9} + 5 \, e^{4}} - \frac {10 \, x^{2} e^{4} \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )}{e^{9} + 5 \, e^{4}} + \frac {x e^{9} \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )^{2}}{e^{9} + 5 \, e^{4}} + \frac {5 \, x e^{4} \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )^{2}}{e^{9} + 5 \, e^{4}} - 6 \, {\left (\frac {3 \, {\left (\frac {\log \left (x\right )}{{\left (x e^{9} + 5 \, x e^{4}\right )} {\left (e^{9} + 5 \, e^{4}\right )}} + \frac {1}{x {\left (e^{9} + 5 \, e^{4}\right )}^{2}}\right )} {\left (e^{9} + 5 \, e^{4}\right )}}{e^{18} + 10 \, e^{13} + 25 \, e^{8}} - \frac {\log \left (x\right ) \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )}{e^{18} + 10 \, e^{13} + 25 \, e^{8}}\right )} e^{9} - 30 \, {\left (\frac {3 \, {\left (\frac {\log \left (x\right )}{{\left (x e^{9} + 5 \, x e^{4}\right )} {\left (e^{9} + 5 \, e^{4}\right )}} + \frac {1}{x {\left (e^{9} + 5 \, e^{4}\right )}^{2}}\right )} {\left (e^{9} + 5 \, e^{4}\right )}}{e^{18} + 10 \, e^{13} + 25 \, e^{8}} - \frac {\log \left (x\right ) \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )}{e^{18} + 10 \, e^{13} + 25 \, e^{8}}\right )} e^{4} - \frac {6 \, x e^{9}}{e^{18} + 10 \, e^{13} + 25 \, e^{8}} - \frac {30 \, x e^{4}}{e^{18} + 10 \, e^{13} + 25 \, e^{8}} - \frac {6 \, \log \left (x e^{9} + 5 \, x e^{4}\right ) \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )}{e^{9} + 5 \, e^{4}} + \frac {6 \, x}{e^{9} + 5 \, e^{4}} + \frac {18 \, \log \left (x e^{9} + 5 \, x e^{4}\right )}{{\left (x e^{9} + 5 \, x e^{4}\right )} {\left (e^{9} + 5 \, e^{4}\right )}} + \frac {18}{{\left (x e^{9} + 5 \, x e^{4}\right )} {\left (e^{9} + 5 \, e^{4}\right )}} \] Input:
integrate(((x*exp(4)*exp(5)+5*x*exp(4))*log(5*exp(3/(x*exp(4)*exp(5)+5*x*e xp(4))))^2+(-4*x^2*exp(4)*exp(5)-20*x^2*exp(4)-6)*log(5*exp(3/(x*exp(4)*ex p(5)+5*x*exp(4))))+3*x^3*exp(4)*exp(5)+15*x^3*exp(4)+6*x)/(x*exp(4)*exp(5) +5*x*exp(4)),x, algorithm="maxima")
Output:
x^3*e^9/(e^9 + 5*e^4) + 5*x^3*e^4/(e^9 + 5*e^4) - 2*x^2*e^9*log(5*e^(3/(x* e^9 + 5*x*e^4)))/(e^9 + 5*e^4) - 10*x^2*e^4*log(5*e^(3/(x*e^9 + 5*x*e^4))) /(e^9 + 5*e^4) + x*e^9*log(5*e^(3/(x*e^9 + 5*x*e^4)))^2/(e^9 + 5*e^4) + 5* x*e^4*log(5*e^(3/(x*e^9 + 5*x*e^4)))^2/(e^9 + 5*e^4) - 6*(3*(log(x)/((x*e^ 9 + 5*x*e^4)*(e^9 + 5*e^4)) + 1/(x*(e^9 + 5*e^4)^2))*(e^9 + 5*e^4)/(e^18 + 10*e^13 + 25*e^8) - log(x)*log(5*e^(3/(x*e^9 + 5*x*e^4)))/(e^18 + 10*e^13 + 25*e^8))*e^9 - 30*(3*(log(x)/((x*e^9 + 5*x*e^4)*(e^9 + 5*e^4)) + 1/(x*( e^9 + 5*e^4)^2))*(e^9 + 5*e^4)/(e^18 + 10*e^13 + 25*e^8) - log(x)*log(5*e^ (3/(x*e^9 + 5*x*e^4)))/(e^18 + 10*e^13 + 25*e^8))*e^4 - 6*x*e^9/(e^18 + 10 *e^13 + 25*e^8) - 30*x*e^4/(e^18 + 10*e^13 + 25*e^8) - 6*log(x*e^9 + 5*x*e ^4)*log(5*e^(3/(x*e^9 + 5*x*e^4)))/(e^9 + 5*e^4) + 6*x/(e^9 + 5*e^4) + 18* log(x*e^9 + 5*x*e^4)/((x*e^9 + 5*x*e^4)*(e^9 + 5*e^4)) + 18/((x*e^9 + 5*x* e^4)*(e^9 + 5*e^4))
Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 357, normalized size of antiderivative = 12.75 \[ \int \frac {6 x+15 e^4 x^3+3 e^9 x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{5 e^4 x+e^9 x} \, dx=\frac {x^{3} e^{81} + 45 \, x^{3} e^{76} + 900 \, x^{3} e^{71} + 10500 \, x^{3} e^{66} + 78750 \, x^{3} e^{61} + 393750 \, x^{3} e^{56} + 1312500 \, x^{3} e^{51} + 2812500 \, x^{3} e^{46} + 3515625 \, x^{3} e^{41} + 1953125 \, x^{3} e^{36} - 2 \, x^{2} e^{81} \log \left (5\right ) - 90 \, x^{2} e^{76} \log \left (5\right ) - 1800 \, x^{2} e^{71} \log \left (5\right ) - 21000 \, x^{2} e^{66} \log \left (5\right ) - 157500 \, x^{2} e^{61} \log \left (5\right ) - 787500 \, x^{2} e^{56} \log \left (5\right ) - 2625000 \, x^{2} e^{51} \log \left (5\right ) - 5625000 \, x^{2} e^{46} \log \left (5\right ) - 7031250 \, x^{2} e^{41} \log \left (5\right ) - 3906250 \, x^{2} e^{36} \log \left (5\right ) + x e^{81} \log \left (5\right )^{2} + 45 \, x e^{76} \log \left (5\right )^{2} + 900 \, x e^{71} \log \left (5\right )^{2} + 10500 \, x e^{66} \log \left (5\right )^{2} + 78750 \, x e^{61} \log \left (5\right )^{2} + 393750 \, x e^{56} \log \left (5\right )^{2} + 1312500 \, x e^{51} \log \left (5\right )^{2} + 2812500 \, x e^{46} \log \left (5\right )^{2} + 3515625 \, x e^{41} \log \left (5\right )^{2} + 1953125 \, x e^{36} \log \left (5\right )^{2} - 6 \, x e^{72} - 240 \, x e^{67} - 4200 \, x e^{62} - 42000 \, x e^{57} - 262500 \, x e^{52} - 1050000 \, x e^{47} - 2625000 \, x e^{42} - 3750000 \, x e^{37} - 2343750 \, x e^{32}}{e^{81} + 45 \, e^{76} + 900 \, e^{71} + 10500 \, e^{66} + 78750 \, e^{61} + 393750 \, e^{56} + 1312500 \, e^{51} + 2812500 \, e^{46} + 3515625 \, e^{41} + 1953125 \, e^{36}} + \frac {9 \, {\left (e^{9} + 5 \, e^{4}\right )} e^{\left (-12\right )}}{x {\left (e^{5} + 5\right )}^{3}} \] Input:
integrate(((x*exp(4)*exp(5)+5*x*exp(4))*log(5*exp(3/(x*exp(4)*exp(5)+5*x*e xp(4))))^2+(-4*x^2*exp(4)*exp(5)-20*x^2*exp(4)-6)*log(5*exp(3/(x*exp(4)*ex p(5)+5*x*exp(4))))+3*x^3*exp(4)*exp(5)+15*x^3*exp(4)+6*x)/(x*exp(4)*exp(5) +5*x*exp(4)),x, algorithm="giac")
Output:
(x^3*e^81 + 45*x^3*e^76 + 900*x^3*e^71 + 10500*x^3*e^66 + 78750*x^3*e^61 + 393750*x^3*e^56 + 1312500*x^3*e^51 + 2812500*x^3*e^46 + 3515625*x^3*e^41 + 1953125*x^3*e^36 - 2*x^2*e^81*log(5) - 90*x^2*e^76*log(5) - 1800*x^2*e^7 1*log(5) - 21000*x^2*e^66*log(5) - 157500*x^2*e^61*log(5) - 787500*x^2*e^5 6*log(5) - 2625000*x^2*e^51*log(5) - 5625000*x^2*e^46*log(5) - 7031250*x^2 *e^41*log(5) - 3906250*x^2*e^36*log(5) + x*e^81*log(5)^2 + 45*x*e^76*log(5 )^2 + 900*x*e^71*log(5)^2 + 10500*x*e^66*log(5)^2 + 78750*x*e^61*log(5)^2 + 393750*x*e^56*log(5)^2 + 1312500*x*e^51*log(5)^2 + 2812500*x*e^46*log(5) ^2 + 3515625*x*e^41*log(5)^2 + 1953125*x*e^36*log(5)^2 - 6*x*e^72 - 240*x* e^67 - 4200*x*e^62 - 42000*x*e^57 - 262500*x*e^52 - 1050000*x*e^47 - 26250 00*x*e^42 - 3750000*x*e^37 - 2343750*x*e^32)/(e^81 + 45*e^76 + 900*e^71 + 10500*e^66 + 78750*e^61 + 393750*e^56 + 1312500*e^51 + 2812500*e^46 + 3515 625*e^41 + 1953125*e^36) + 9*(e^9 + 5*e^4)*e^(-12)/(x*(e^5 + 5)^3)
Time = 3.48 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \[ \int \frac {6 x+15 e^4 x^3+3 e^9 x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{5 e^4 x+e^9 x} \, dx=\frac {{\mathrm {e}}^8\,{\left ({\mathrm {e}}^5+5\right )}^2\,x^5-2\,{\mathrm {e}}^8\,\ln \left (5\right )\,{\left ({\mathrm {e}}^5+5\right )}^2\,x^4+{\mathrm {e}}^4\,\left ({\mathrm {e}}^5+5\right )\,\left (5\,{\mathrm {e}}^4\,{\ln \left (5\right )}^2+{\mathrm {e}}^9\,{\ln \left (5\right )}^2-6\right )\,x^3+9\,x}{x^2\,\left (25\,{\mathrm {e}}^8+10\,{\mathrm {e}}^{13}+{\mathrm {e}}^{18}\right )} \] Input:
int((6*x + log(5*exp(3/(5*x*exp(4) + x*exp(9))))^2*(5*x*exp(4) + x*exp(9)) + 15*x^3*exp(4) + 3*x^3*exp(9) - log(5*exp(3/(5*x*exp(4) + x*exp(9))))*(2 0*x^2*exp(4) + 4*x^2*exp(9) + 6))/(5*x*exp(4) + x*exp(9)),x)
Output:
(9*x + x^5*exp(8)*(exp(5) + 5)^2 - 2*x^4*exp(8)*log(5)*(exp(5) + 5)^2 + x^ 3*exp(4)*(exp(5) + 5)*(5*exp(4)*log(5)^2 + exp(9)*log(5)^2 - 6))/(x^2*(25* exp(8) + 10*exp(13) + exp(18)))
\[ \int \frac {6 x+15 e^4 x^3+3 e^9 x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{5 e^4 x+e^9 x} \, dx=\frac {\left (\int \mathrm {log}\left (5 e^{\frac {3}{e^{9} x +5 e^{4} x}}\right )^{2}d x \right ) e^{9}+5 \left (\int \mathrm {log}\left (5 e^{\frac {3}{e^{9} x +5 e^{4} x}}\right )^{2}d x \right ) e^{4}-6 \left (\int \frac {\mathrm {log}\left (5 e^{\frac {3}{e^{9} x +5 e^{4} x}}\right )}{x}d x \right )-4 \left (\int \mathrm {log}\left (5 e^{\frac {3}{e^{9} x +5 e^{4} x}}\right ) x d x \right ) e^{9}-20 \left (\int \mathrm {log}\left (5 e^{\frac {3}{e^{9} x +5 e^{4} x}}\right ) x d x \right ) e^{4}+e^{9} x^{3}+5 e^{4} x^{3}+6 x}{e^{4} \left (e^{5}+5\right )} \] Input:
int(((x*exp(4)*exp(5)+5*x*exp(4))*log(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4)) ))^2+(-4*x^2*exp(4)*exp(5)-20*x^2*exp(4)-6)*log(5*exp(3/(x*exp(4)*exp(5)+5 *x*exp(4))))+3*x^3*exp(4)*exp(5)+15*x^3*exp(4)+6*x)/(x*exp(4)*exp(5)+5*x*e xp(4)),x)
Output:
(int(log(5*e**(3/(e**9*x + 5*e**4*x)))**2,x)*e**9 + 5*int(log(5*e**(3/(e** 9*x + 5*e**4*x)))**2,x)*e**4 - 6*int(log(5*e**(3/(e**9*x + 5*e**4*x)))/x,x ) - 4*int(log(5*e**(3/(e**9*x + 5*e**4*x)))*x,x)*e**9 - 20*int(log(5*e**(3 /(e**9*x + 5*e**4*x)))*x,x)*e**4 + e**9*x**3 + 5*e**4*x**3 + 6*x)/(e**4*(e **5 + 5))