Integrand size = 230, antiderivative size = 26 \[ \int \frac {\left (2 e^2+2 e^{8 x}+8 e^{6 x} x-4 e x^2+2 x^4+e^{4 x} \left (-4 e+12 x^2-8 \log (5)\right )-4 x \log (5)+e^{2 x} \left (-8 e x+8 x^3+(-4-8 x) \log (5)\right )\right ) \log \left (\frac {e x-e^{4 x} x-2 e^{2 x} x^2-x^3-\log (5)}{e^{4 x} \log (5)+2 e^{2 x} x \log (5)+\left (-e+x^2\right ) \log (5)}\right )}{e^2 x+e^{8 x} x+4 e^{6 x} x^2-2 e x^3+x^5+\left (-e+x^2\right ) \log (5)+e^{4 x} \left (-2 e x+6 x^3+\log (5)\right )+e^{2 x} \left (-4 e x^2+4 x^4+2 x \log (5)\right )} \, dx=\log ^2\left (\frac {1}{e-\left (e^{2 x}+x\right )^2}-\frac {x}{\log (5)}\right ) \] Output:
ln(1/(exp(1)-(exp(x)^2+x)^2)-x/ln(5))^2
Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(26)=52\).
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23 \[ \int \frac {\left (2 e^2+2 e^{8 x}+8 e^{6 x} x-4 e x^2+2 x^4+e^{4 x} \left (-4 e+12 x^2-8 \log (5)\right )-4 x \log (5)+e^{2 x} \left (-8 e x+8 x^3+(-4-8 x) \log (5)\right )\right ) \log \left (\frac {e x-e^{4 x} x-2 e^{2 x} x^2-x^3-\log (5)}{e^{4 x} \log (5)+2 e^{2 x} x \log (5)+\left (-e+x^2\right ) \log (5)}\right )}{e^2 x+e^{8 x} x+4 e^{6 x} x^2-2 e x^3+x^5+\left (-e+x^2\right ) \log (5)+e^{4 x} \left (-2 e x+6 x^3+\log (5)\right )+e^{2 x} \left (-4 e x^2+4 x^4+2 x \log (5)\right )} \, dx=\log ^2\left (-\frac {-e x+e^{4 x} x+2 e^{2 x} x^2+x^3+\log (5)}{\left (-e+e^{4 x}+2 e^{2 x} x+x^2\right ) \log (5)}\right ) \] Input:
Integrate[((2*E^2 + 2*E^(8*x) + 8*E^(6*x)*x - 4*E*x^2 + 2*x^4 + E^(4*x)*(- 4*E + 12*x^2 - 8*Log[5]) - 4*x*Log[5] + E^(2*x)*(-8*E*x + 8*x^3 + (-4 - 8* x)*Log[5]))*Log[(E*x - E^(4*x)*x - 2*E^(2*x)*x^2 - x^3 - Log[5])/(E^(4*x)* Log[5] + 2*E^(2*x)*x*Log[5] + (-E + x^2)*Log[5])])/(E^2*x + E^(8*x)*x + 4* E^(6*x)*x^2 - 2*E*x^3 + x^5 + (-E + x^2)*Log[5] + E^(4*x)*(-2*E*x + 6*x^3 + Log[5]) + E^(2*x)*(-4*E*x^2 + 4*x^4 + 2*x*Log[5])),x]
Output:
Log[-((-(E*x) + E^(4*x)*x + 2*E^(2*x)*x^2 + x^3 + Log[5])/((-E + E^(4*x) + 2*E^(2*x)*x + x^2)*Log[5]))]^2
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.
| \(\displaystyle \int \frac {\left (2 x^4+e^{2 x} \left (8 x^3-8 e x+(-8 x-4) \log (5)\right )-4 e x^2+e^{4 x} \left (12 x^2-4 e-8 \log (5)\right )+8 e^{6 x} x+2 e^{8 x}-4 x \log (5)+2 e^2\right ) \log \left (\frac {-x^3-2 e^{2 x} x^2-e^{4 x} x+e x-\log (5)}{\left (x^2-e\right ) \log (5)+2 e^{2 x} x \log (5)+e^{4 x} \log (5)}\right )}{x^5-2 e x^3+e^{4 x} \left (6 x^3-2 e x+\log (5)\right )+4 e^{6 x} x^2+\left (x^2-e\right ) \log (5)+e^{2 x} \left (4 x^4-4 e x^2+2 x \log (5)\right )+e^{8 x} x+e^2 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (2 x^4+e^{2 x} \left (8 x^3-8 e x+(-8 x-4) \log (5)\right )-4 e x^2+e^{4 x} \left (12 x^2-4 e-8 \log (5)\right )+8 e^{6 x} x+2 e^{8 x}-4 x \log (5)+2 e^2\right ) \log \left (\frac {-x^3-2 e^{2 x} x^2-e^{4 x} x+e x-\log (5)}{\left (x^2+2 e^{2 x} x+e^{4 x}-e\right ) \log (5)}\right )}{\left (-x^2-2 e^{2 x} x-e^{4 x}+e\right ) \left (-x^3-2 e^{2 x} x^2-e^{4 x} x+e x-\log (5)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (2 x^2+2 e^{2 x} x-x-e^{2 x}-2 e\right ) \log \left (-\frac {x^3+2 e^{2 x} x^2+e^{4 x} x-e x+\log (5)}{\left (x^2+2 e^{2 x} x+e^{4 x}-e\right ) \log (5)}\right )}{x^2+2 e^{2 x} x+e^{4 x}-e}+\frac {2 \log \left (-\frac {x^3+2 e^{2 x} x^2+e^{4 x} x-e x+\log (5)}{\left (x^2+2 e^{2 x} x+e^{4 x}-e\right ) \log (5)}\right )}{x}-\frac {2 \left (4 x^4+4 e^{2 x} x^3-2 x^3-2 e^{2 x} x^2-4 e x^2+4 x \log (5)+\log (5)\right ) \log \left (-\frac {x^3+2 e^{2 x} x^2+e^{4 x} x-e x+\log (5)}{\left (x^2+2 e^{2 x} x+e^{4 x}-e\right ) \log (5)}\right )}{x \left (x^3+2 e^{2 x} x^2+e^{4 x} x-e x+\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {4 \left (2 x^2+2 e^{2 x} x-x-e^{2 x}-2 e\right ) \log \left (-\frac {x^3+2 e^{2 x} x^2+e^{4 x} x-e x+\log (5)}{\left (x^2+2 e^{2 x} x+e^{4 x}-e\right ) \log (5)}\right )}{x^2+2 e^{2 x} x+e^{4 x}-e}+\frac {2 \log \left (-\frac {x^3+2 e^{2 x} x^2+e^{4 x} x-e x+\log (5)}{\left (x^2+2 e^{2 x} x+e^{4 x}-e\right ) \log (5)}\right )}{x}-\frac {2 \left (4 x^4+4 e^{2 x} x^3-2 x^3-2 e^{2 x} x^2-4 e x^2+4 x \log (5)+\log (5)\right ) \log \left (-\frac {x^3+2 e^{2 x} x^2+e^{4 x} x-e x+\log (5)}{\left (x^2+2 e^{2 x} x+e^{4 x}-e\right ) \log (5)}\right )}{x \left (x^3+2 e^{2 x} x^2+e^{4 x} x-e x+\log (5)\right )}\right )dx\) |
Input:
Int[((2*E^2 + 2*E^(8*x) + 8*E^(6*x)*x - 4*E*x^2 + 2*x^4 + E^(4*x)*(-4*E + 12*x^2 - 8*Log[5]) - 4*x*Log[5] + E^(2*x)*(-8*E*x + 8*x^3 + (-4 - 8*x)*Log [5]))*Log[(E*x - E^(4*x)*x - 2*E^(2*x)*x^2 - x^3 - Log[5])/(E^(4*x)*Log[5] + 2*E^(2*x)*x*Log[5] + (-E + x^2)*Log[5])])/(E^2*x + E^(8*x)*x + 4*E^(6*x )*x^2 - 2*E*x^3 + x^5 + (-E + x^2)*Log[5] + E^(4*x)*(-2*E*x + 6*x^3 + Log[ 5]) + E^(2*x)*(-4*E*x^2 + 4*x^4 + 2*x*Log[5])),x]
Output:
$Aborted
\[\int \frac {\left (2 \,{\mathrm e}^{8 x}+8 \,{\mathrm e}^{6 x} x +\left (-8 \ln \left (5\right )-4 \,{\mathrm e}+12 x^{2}\right ) {\mathrm e}^{4 x}+\left (\left (-8 x -4\right ) \ln \left (5\right )-8 x \,{\mathrm e}+8 x^{3}\right ) {\mathrm e}^{2 x}-4 x \ln \left (5\right )+2 \,{\mathrm e}^{2}-4 x^{2} {\mathrm e}+2 x^{4}\right ) \ln \left (\frac {-x \,{\mathrm e}^{4 x}-2 \,{\mathrm e}^{2 x} x^{2}-\ln \left (5\right )+x \,{\mathrm e}-x^{3}}{\ln \left (5\right ) {\mathrm e}^{4 x}+2 \ln \left (5\right ) {\mathrm e}^{2 x} x +\left (-{\mathrm e}+x^{2}\right ) \ln \left (5\right )}\right )}{{\mathrm e}^{8 x} x +4 \,{\mathrm e}^{6 x} x^{2}+\left (\ln \left (5\right )-2 x \,{\mathrm e}+6 x^{3}\right ) {\mathrm e}^{4 x}+\left (2 x \ln \left (5\right )-4 x^{2} {\mathrm e}+4 x^{4}\right ) {\mathrm e}^{2 x}+\left (-{\mathrm e}+x^{2}\right ) \ln \left (5\right )+{\mathrm e}^{2} x -2 x^{3} {\mathrm e}+x^{5}}d x\]
Input:
int((2*exp(x)^8+8*x*exp(x)^6+(-8*ln(5)-4*exp(1)+12*x^2)*exp(x)^4+((-8*x-4) *ln(5)-8*x*exp(1)+8*x^3)*exp(x)^2-4*x*ln(5)+2*exp(1)^2-4*x^2*exp(1)+2*x^4) *ln((-x*exp(x)^4-2*exp(x)^2*x^2-ln(5)+x*exp(1)-x^3)/(ln(5)*exp(x)^4+2*x*ln (5)*exp(x)^2+(-exp(1)+x^2)*ln(5)))/(x*exp(x)^8+4*x^2*exp(x)^6+(ln(5)-2*x*e xp(1)+6*x^3)*exp(x)^4+(2*x*ln(5)-4*x^2*exp(1)+4*x^4)*exp(x)^2+(-exp(1)+x^2 )*ln(5)+x*exp(1)^2-2*x^3*exp(1)+x^5),x)
Output:
int((2*exp(x)^8+8*x*exp(x)^6+(-8*ln(5)-4*exp(1)+12*x^2)*exp(x)^4+((-8*x-4) *ln(5)-8*x*exp(1)+8*x^3)*exp(x)^2-4*x*ln(5)+2*exp(1)^2-4*x^2*exp(1)+2*x^4) *ln((-x*exp(x)^4-2*exp(x)^2*x^2-ln(5)+x*exp(1)-x^3)/(ln(5)*exp(x)^4+2*x*ln (5)*exp(x)^2+(-exp(1)+x^2)*ln(5)))/(x*exp(x)^8+4*x^2*exp(x)^6+(ln(5)-2*x*e xp(1)+6*x^3)*exp(x)^4+(2*x*ln(5)-4*x^2*exp(1)+4*x^4)*exp(x)^2+(-exp(1)+x^2 )*ln(5)+x*exp(1)^2-2*x^3*exp(1)+x^5),x)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (28) = 56\).
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {\left (2 e^2+2 e^{8 x}+8 e^{6 x} x-4 e x^2+2 x^4+e^{4 x} \left (-4 e+12 x^2-8 \log (5)\right )-4 x \log (5)+e^{2 x} \left (-8 e x+8 x^3+(-4-8 x) \log (5)\right )\right ) \log \left (\frac {e x-e^{4 x} x-2 e^{2 x} x^2-x^3-\log (5)}{e^{4 x} \log (5)+2 e^{2 x} x \log (5)+\left (-e+x^2\right ) \log (5)}\right )}{e^2 x+e^{8 x} x+4 e^{6 x} x^2-2 e x^3+x^5+\left (-e+x^2\right ) \log (5)+e^{4 x} \left (-2 e x+6 x^3+\log (5)\right )+e^{2 x} \left (-4 e x^2+4 x^4+2 x \log (5)\right )} \, dx=\log \left (-\frac {x^{3} + 2 \, x^{2} e^{\left (2 \, x\right )} - x e + x e^{\left (4 \, x\right )} + \log \left (5\right )}{2 \, x e^{\left (2 \, x\right )} \log \left (5\right ) + {\left (x^{2} - e\right )} \log \left (5\right ) + e^{\left (4 \, x\right )} \log \left (5\right )}\right )^{2} \] Input:
integrate((2*exp(x)^8+8*x*exp(x)^6+(-8*log(5)-4*exp(1)+12*x^2)*exp(x)^4+(( -8*x-4)*log(5)-8*exp(1)*x+8*x^3)*exp(x)^2-4*x*log(5)+2*exp(1)^2-4*x^2*exp( 1)+2*x^4)*log((-x*exp(x)^4-2*exp(x)^2*x^2-log(5)+exp(1)*x-x^3)/(log(5)*exp (x)^4+2*x*log(5)*exp(x)^2+(-exp(1)+x^2)*log(5)))/(x*exp(x)^8+4*x^2*exp(x)^ 6+(log(5)-2*exp(1)*x+6*x^3)*exp(x)^4+(2*x*log(5)-4*x^2*exp(1)+4*x^4)*exp(x )^2+(-exp(1)+x^2)*log(5)+x*exp(1)^2-2*x^3*exp(1)+x^5),x, algorithm="fricas ")
Output:
log(-(x^3 + 2*x^2*e^(2*x) - x*e + x*e^(4*x) + log(5))/(2*x*e^(2*x)*log(5) + (x^2 - e)*log(5) + e^(4*x)*log(5)))^2
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.51 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {\left (2 e^2+2 e^{8 x}+8 e^{6 x} x-4 e x^2+2 x^4+e^{4 x} \left (-4 e+12 x^2-8 \log (5)\right )-4 x \log (5)+e^{2 x} \left (-8 e x+8 x^3+(-4-8 x) \log (5)\right )\right ) \log \left (\frac {e x-e^{4 x} x-2 e^{2 x} x^2-x^3-\log (5)}{e^{4 x} \log (5)+2 e^{2 x} x \log (5)+\left (-e+x^2\right ) \log (5)}\right )}{e^2 x+e^{8 x} x+4 e^{6 x} x^2-2 e x^3+x^5+\left (-e+x^2\right ) \log (5)+e^{4 x} \left (-2 e x+6 x^3+\log (5)\right )+e^{2 x} \left (-4 e x^2+4 x^4+2 x \log (5)\right )} \, dx=\log {\left (\frac {- x^{3} - 2 x^{2} e^{2 x} - x e^{4 x} + e x - \log {\left (5 \right )}}{2 x e^{2 x} \log {\left (5 \right )} + \left (x^{2} - e\right ) \log {\left (5 \right )} + e^{4 x} \log {\left (5 \right )}} \right )}^{2} \] Input:
integrate((2*exp(x)**8+8*x*exp(x)**6+(-8*ln(5)-4*exp(1)+12*x**2)*exp(x)**4 +((-8*x-4)*ln(5)-8*exp(1)*x+8*x**3)*exp(x)**2-4*x*ln(5)+2*exp(1)**2-4*x**2 *exp(1)+2*x**4)*ln((-x*exp(x)**4-2*exp(x)**2*x**2-ln(5)+exp(1)*x-x**3)/(ln (5)*exp(x)**4+2*x*ln(5)*exp(x)**2+(-exp(1)+x**2)*ln(5)))/(x*exp(x)**8+4*x* *2*exp(x)**6+(ln(5)-2*exp(1)*x+6*x**3)*exp(x)**4+(2*x*ln(5)-4*x**2*exp(1)+ 4*x**4)*exp(x)**2+(-exp(1)+x**2)*ln(5)+x*exp(1)**2-2*x**3*exp(1)+x**5),x)
Output:
log((-x**3 - 2*x**2*exp(2*x) - x*exp(4*x) + E*x - log(5))/(2*x*exp(2*x)*lo g(5) + (x**2 - E)*log(5) + exp(4*x)*log(5)))**2
\[ \int \frac {\left (2 e^2+2 e^{8 x}+8 e^{6 x} x-4 e x^2+2 x^4+e^{4 x} \left (-4 e+12 x^2-8 \log (5)\right )-4 x \log (5)+e^{2 x} \left (-8 e x+8 x^3+(-4-8 x) \log (5)\right )\right ) \log \left (\frac {e x-e^{4 x} x-2 e^{2 x} x^2-x^3-\log (5)}{e^{4 x} \log (5)+2 e^{2 x} x \log (5)+\left (-e+x^2\right ) \log (5)}\right )}{e^2 x+e^{8 x} x+4 e^{6 x} x^2-2 e x^3+x^5+\left (-e+x^2\right ) \log (5)+e^{4 x} \left (-2 e x+6 x^3+\log (5)\right )+e^{2 x} \left (-4 e x^2+4 x^4+2 x \log (5)\right )} \, dx=\int { \frac {2 \, {\left (x^{4} - 2 \, x^{2} e + 4 \, x e^{\left (6 \, x\right )} + 2 \, {\left (3 \, x^{2} - e - 2 \, \log \left (5\right )\right )} e^{\left (4 \, x\right )} + 2 \, {\left (2 \, x^{3} - 2 \, x e - {\left (2 \, x + 1\right )} \log \left (5\right )\right )} e^{\left (2 \, x\right )} - 2 \, x \log \left (5\right ) + e^{2} + e^{\left (8 \, x\right )}\right )} \log \left (-\frac {x^{3} + 2 \, x^{2} e^{\left (2 \, x\right )} - x e + x e^{\left (4 \, x\right )} + \log \left (5\right )}{2 \, x e^{\left (2 \, x\right )} \log \left (5\right ) + {\left (x^{2} - e\right )} \log \left (5\right ) + e^{\left (4 \, x\right )} \log \left (5\right )}\right )}{x^{5} - 2 \, x^{3} e + 4 \, x^{2} e^{\left (6 \, x\right )} + x e^{2} + x e^{\left (8 \, x\right )} + {\left (6 \, x^{3} - 2 \, x e + \log \left (5\right )\right )} e^{\left (4 \, x\right )} + 2 \, {\left (2 \, x^{4} - 2 \, x^{2} e + x \log \left (5\right )\right )} e^{\left (2 \, x\right )} + {\left (x^{2} - e\right )} \log \left (5\right )} \,d x } \] Input:
integrate((2*exp(x)^8+8*x*exp(x)^6+(-8*log(5)-4*exp(1)+12*x^2)*exp(x)^4+(( -8*x-4)*log(5)-8*exp(1)*x+8*x^3)*exp(x)^2-4*x*log(5)+2*exp(1)^2-4*x^2*exp( 1)+2*x^4)*log((-x*exp(x)^4-2*exp(x)^2*x^2-log(5)+exp(1)*x-x^3)/(log(5)*exp (x)^4+2*x*log(5)*exp(x)^2+(-exp(1)+x^2)*log(5)))/(x*exp(x)^8+4*x^2*exp(x)^ 6+(log(5)-2*exp(1)*x+6*x^3)*exp(x)^4+(2*x*log(5)-4*x^2*exp(1)+4*x^4)*exp(x )^2+(-exp(1)+x^2)*log(5)+x*exp(1)^2-2*x^3*exp(1)+x^5),x, algorithm="maxima ")
Output:
2*integrate((x^4 - 2*x^2*e + 4*x*e^(6*x) + 2*(3*x^2 - e - 2*log(5))*e^(4*x ) + 2*(2*x^3 - 2*x*e - (2*x + 1)*log(5))*e^(2*x) - 2*x*log(5) + e^2 + e^(8 *x))*log(-(x^3 + 2*x^2*e^(2*x) - x*e + x*e^(4*x) + log(5))/(2*x*e^(2*x)*lo g(5) + (x^2 - e)*log(5) + e^(4*x)*log(5)))/(x^5 - 2*x^3*e + 4*x^2*e^(6*x) + x*e^2 + x*e^(8*x) + (6*x^3 - 2*x*e + log(5))*e^(4*x) + 2*(2*x^4 - 2*x^2* e + x*log(5))*e^(2*x) + (x^2 - e)*log(5)), x)
Timed out. \[ \int \frac {\left (2 e^2+2 e^{8 x}+8 e^{6 x} x-4 e x^2+2 x^4+e^{4 x} \left (-4 e+12 x^2-8 \log (5)\right )-4 x \log (5)+e^{2 x} \left (-8 e x+8 x^3+(-4-8 x) \log (5)\right )\right ) \log \left (\frac {e x-e^{4 x} x-2 e^{2 x} x^2-x^3-\log (5)}{e^{4 x} \log (5)+2 e^{2 x} x \log (5)+\left (-e+x^2\right ) \log (5)}\right )}{e^2 x+e^{8 x} x+4 e^{6 x} x^2-2 e x^3+x^5+\left (-e+x^2\right ) \log (5)+e^{4 x} \left (-2 e x+6 x^3+\log (5)\right )+e^{2 x} \left (-4 e x^2+4 x^4+2 x \log (5)\right )} \, dx=\text {Timed out} \] Input:
integrate((2*exp(x)^8+8*x*exp(x)^6+(-8*log(5)-4*exp(1)+12*x^2)*exp(x)^4+(( -8*x-4)*log(5)-8*exp(1)*x+8*x^3)*exp(x)^2-4*x*log(5)+2*exp(1)^2-4*x^2*exp( 1)+2*x^4)*log((-x*exp(x)^4-2*exp(x)^2*x^2-log(5)+exp(1)*x-x^3)/(log(5)*exp (x)^4+2*x*log(5)*exp(x)^2+(-exp(1)+x^2)*log(5)))/(x*exp(x)^8+4*x^2*exp(x)^ 6+(log(5)-2*exp(1)*x+6*x^3)*exp(x)^4+(2*x*log(5)-4*x^2*exp(1)+4*x^4)*exp(x )^2+(-exp(1)+x^2)*log(5)+x*exp(1)^2-2*x^3*exp(1)+x^5),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (2 e^2+2 e^{8 x}+8 e^{6 x} x-4 e x^2+2 x^4+e^{4 x} \left (-4 e+12 x^2-8 \log (5)\right )-4 x \log (5)+e^{2 x} \left (-8 e x+8 x^3+(-4-8 x) \log (5)\right )\right ) \log \left (\frac {e x-e^{4 x} x-2 e^{2 x} x^2-x^3-\log (5)}{e^{4 x} \log (5)+2 e^{2 x} x \log (5)+\left (-e+x^2\right ) \log (5)}\right )}{e^2 x+e^{8 x} x+4 e^{6 x} x^2-2 e x^3+x^5+\left (-e+x^2\right ) \log (5)+e^{4 x} \left (-2 e x+6 x^3+\log (5)\right )+e^{2 x} \left (-4 e x^2+4 x^4+2 x \log (5)\right )} \, dx=\int \frac {\ln \left (-\frac {\ln \left (5\right )+x\,{\mathrm {e}}^{4\,x}-x\,\mathrm {e}+2\,x^2\,{\mathrm {e}}^{2\,x}+x^3}{{\mathrm {e}}^{4\,x}\,\ln \left (5\right )-\ln \left (5\right )\,\left (\mathrm {e}-x^2\right )+2\,x\,{\mathrm {e}}^{2\,x}\,\ln \left (5\right )}\right )\,\left (2\,{\mathrm {e}}^{8\,x}+2\,{\mathrm {e}}^2+8\,x\,{\mathrm {e}}^{6\,x}-4\,x\,\ln \left (5\right )-{\mathrm {e}}^{2\,x}\,\left (\ln \left (5\right )\,\left (8\,x+4\right )+8\,x\,\mathrm {e}-8\,x^3\right )-4\,x^2\,\mathrm {e}-{\mathrm {e}}^{4\,x}\,\left (-12\,x^2+4\,\mathrm {e}+8\,\ln \left (5\right )\right )+2\,x^4\right )}{{\mathrm {e}}^{2\,x}\,\left (4\,x^4-4\,\mathrm {e}\,x^2+2\,\ln \left (5\right )\,x\right )+x\,{\mathrm {e}}^{8\,x}+x\,{\mathrm {e}}^2+4\,x^2\,{\mathrm {e}}^{6\,x}-2\,x^3\,\mathrm {e}-\ln \left (5\right )\,\left (\mathrm {e}-x^2\right )+x^5+{\mathrm {e}}^{4\,x}\,\left (6\,x^3-2\,\mathrm {e}\,x+\ln \left (5\right )\right )} \,d x \] Input:
int((log(-(log(5) + x*exp(4*x) - x*exp(1) + 2*x^2*exp(2*x) + x^3)/(exp(4*x )*log(5) - log(5)*(exp(1) - x^2) + 2*x*exp(2*x)*log(5)))*(2*exp(8*x) + 2*e xp(2) + 8*x*exp(6*x) - 4*x*log(5) - exp(2*x)*(log(5)*(8*x + 4) + 8*x*exp(1 ) - 8*x^3) - 4*x^2*exp(1) - exp(4*x)*(4*exp(1) + 8*log(5) - 12*x^2) + 2*x^ 4))/(exp(2*x)*(2*x*log(5) - 4*x^2*exp(1) + 4*x^4) + x*exp(8*x) + x*exp(2) + 4*x^2*exp(6*x) - 2*x^3*exp(1) - log(5)*(exp(1) - x^2) + x^5 + exp(4*x)*( log(5) - 2*x*exp(1) + 6*x^3)),x)
Output:
int((log(-(log(5) + x*exp(4*x) - x*exp(1) + 2*x^2*exp(2*x) + x^3)/(exp(4*x )*log(5) - log(5)*(exp(1) - x^2) + 2*x*exp(2*x)*log(5)))*(2*exp(8*x) + 2*e xp(2) + 8*x*exp(6*x) - 4*x*log(5) - exp(2*x)*(log(5)*(8*x + 4) + 8*x*exp(1 ) - 8*x^3) - 4*x^2*exp(1) - exp(4*x)*(4*exp(1) + 8*log(5) - 12*x^2) + 2*x^ 4))/(exp(2*x)*(2*x*log(5) - 4*x^2*exp(1) + 4*x^4) + x*exp(8*x) + x*exp(2) + 4*x^2*exp(6*x) - 2*x^3*exp(1) - log(5)*(exp(1) - x^2) + x^5 + exp(4*x)*( log(5) - 2*x*exp(1) + 6*x^3)), x)
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.58 \[ \int \frac {\left (2 e^2+2 e^{8 x}+8 e^{6 x} x-4 e x^2+2 x^4+e^{4 x} \left (-4 e+12 x^2-8 \log (5)\right )-4 x \log (5)+e^{2 x} \left (-8 e x+8 x^3+(-4-8 x) \log (5)\right )\right ) \log \left (\frac {e x-e^{4 x} x-2 e^{2 x} x^2-x^3-\log (5)}{e^{4 x} \log (5)+2 e^{2 x} x \log (5)+\left (-e+x^2\right ) \log (5)}\right )}{e^2 x+e^{8 x} x+4 e^{6 x} x^2-2 e x^3+x^5+\left (-e+x^2\right ) \log (5)+e^{4 x} \left (-2 e x+6 x^3+\log (5)\right )+e^{2 x} \left (-4 e x^2+4 x^4+2 x \log (5)\right )} \, dx=\mathrm {log}\left (\frac {-e^{4 x} x -2 e^{2 x} x^{2}-\mathrm {log}\left (5\right )+e x -x^{3}}{e^{4 x} \mathrm {log}\left (5\right )+2 e^{2 x} \mathrm {log}\left (5\right ) x -\mathrm {log}\left (5\right ) e +\mathrm {log}\left (5\right ) x^{2}}\right )^{2} \] Input:
int((2*exp(x)^8+8*x*exp(x)^6+(-8*log(5)-4*exp(1)+12*x^2)*exp(x)^4+((-8*x-4 )*log(5)-8*exp(1)*x+8*x^3)*exp(x)^2-4*x*log(5)+2*exp(1)^2-4*x^2*exp(1)+2*x ^4)*log((-x*exp(x)^4-2*exp(x)^2*x^2-log(5)+exp(1)*x-x^3)/(log(5)*exp(x)^4+ 2*x*log(5)*exp(x)^2+(-exp(1)+x^2)*log(5)))/(x*exp(x)^8+4*x^2*exp(x)^6+(log (5)-2*exp(1)*x+6*x^3)*exp(x)^4+(2*x*log(5)-4*x^2*exp(1)+4*x^4)*exp(x)^2+(- exp(1)+x^2)*log(5)+x*exp(1)^2-2*x^3*exp(1)+x^5),x)
Output:
log(( - e**(4*x)*x - 2*e**(2*x)*x**2 - log(5) + e*x - x**3)/(e**(4*x)*log( 5) + 2*e**(2*x)*log(5)*x - log(5)*e + log(5)*x**2))**2