Integrand size = 122, antiderivative size = 30 \[ \int \frac {x^5+e^{\frac {e^2-4 e^{26} x+4 e^{50} x^2+\left (2 e x^2-4 e^{25} x^3\right ) \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )}{x^4}} \left (4 e^2-4 e x^2+8 e^{50} x^2+e^{25} \left (-12 e x+8 x^3\right )+\left (4 e x^2-4 e^{25} x^3-4 x^4\right ) \log \left (x^2\right )\right )}{x^5} \, dx=3-e^{\left (-\frac {e}{x^2}+\frac {2 e^{25}}{x}-\log \left (x^2\right )\right )^2}+x \] Output:
x-exp((2*exp(25)/x-ln(x^2)-exp(1)/x^2)^2)+3
Time = 6.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {x^5+e^{\frac {e^2-4 e^{26} x+4 e^{50} x^2+\left (2 e x^2-4 e^{25} x^3\right ) \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )}{x^4}} \left (4 e^2-4 e x^2+8 e^{50} x^2+e^{25} \left (-12 e x+8 x^3\right )+\left (4 e x^2-4 e^{25} x^3-4 x^4\right ) \log \left (x^2\right )\right )}{x^5} \, dx=x-e^{\frac {e^2}{x^4}-\frac {4 e^{26}}{x^3}+\frac {4 e^{50}}{x^2}+\log ^2\left (x^2\right )} \left (x^2\right )^{-\frac {2 e \left (-1+2 e^{24} x\right )}{x^2}} \] Input:
Integrate[(x^5 + E^((E^2 - 4*E^26*x + 4*E^50*x^2 + (2*E*x^2 - 4*E^25*x^3)* Log[x^2] + x^4*Log[x^2]^2)/x^4)*(4*E^2 - 4*E*x^2 + 8*E^50*x^2 + E^25*(-12* E*x + 8*x^3) + (4*E*x^2 - 4*E^25*x^3 - 4*x^4)*Log[x^2]))/x^5,x]
Output:
x - E^(E^2/x^4 - (4*E^26)/x^3 + (4*E^50)/x^2 + Log[x^2]^2)/(x^2)^((2*E*(-1 + 2*E^24*x))/x^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.
| \(\displaystyle \int \frac {\left (e^{25} \left (8 x^3-12 e x\right )+8 e^{50} x^2-4 e x^2+\left (-4 x^4-4 e^{25} x^3+4 e x^2\right ) \log \left (x^2\right )+4 e^2\right ) \exp \left (\frac {4 e^{50} x^2+x^4 \log ^2\left (x^2\right )+\left (2 e x^2-4 e^{25} x^3\right ) \log \left (x^2\right )-4 e^{26} x+e^2}{x^4}\right )+x^5}{x^5} \, dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {4 \left (-x^2-e^{25} x+e\right ) \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2 \log \left (x^2\right )-2 e^{25} x+e\right )}{x^5}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -12 \int \frac {e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )+26} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}}}{x^4}dx+8 \int e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )+25} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}-1}dx-4 \int \frac {e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \log \left (x^2\right )}{x}dx-4 \int e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )+25} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}-1} \log \left (x^2\right )dx+4 \int \frac {e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )+2} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}}}{x^5}dx-4 \left (1-2 e^{49}\right ) \int \frac {e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )+1} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}}}{x^3}dx+4 \int \frac {e^{\frac {e^2 \left (1-2 e^{24} x\right )^2}{x^4}+\log ^2\left (x^2\right )+1} \left (x^2\right )^{\frac {2 e-4 e^{25} x}{x^2}} \log \left (x^2\right )}{x^3}dx+x\) |
Input:
Int[(x^5 + E^((E^2 - 4*E^26*x + 4*E^50*x^2 + (2*E*x^2 - 4*E^25*x^3)*Log[x^ 2] + x^4*Log[x^2]^2)/x^4)*(4*E^2 - 4*E*x^2 + 8*E^50*x^2 + E^25*(-12*E*x + 8*x^3) + (4*E*x^2 - 4*E^25*x^3 - 4*x^4)*Log[x^2]))/x^5,x]
Output:
$Aborted
Time = 5.92 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.90
| method | result | size |
| risch | \(x -\left (x^{2}\right )^{-\frac {4 \,{\mathrm e}^{25}}{x}} \left (x^{2}\right )^{\frac {2 \,{\mathrm e}}{x^{2}}} {\mathrm e}^{\frac {x^{4} \ln \left (x^{2}\right )^{2}+4 x^{2} {\mathrm e}^{50}-4 x \,{\mathrm e}^{26}+{\mathrm e}^{2}}{x^{4}}}\) | \(57\) |
| parallelrisch | \(x -{\mathrm e}^{\frac {x^{4} \ln \left (x^{2}\right )^{2}+\left (-4 x^{3} {\mathrm e}^{25}+2 x^{2} {\mathrm e}\right ) \ln \left (x^{2}\right )+4 x^{2} {\mathrm e}^{50}-4 x \,{\mathrm e} \,{\mathrm e}^{25}+{\mathrm e}^{2}}{x^{4}}}\) | \(61\) |
Input:
int((((-4*x^3*exp(25)+4*x^2*exp(1)-4*x^4)*ln(x^2)+8*x^2*exp(25)^2+(-12*x*e xp(1)+8*x^3)*exp(25)+4*exp(1)^2-4*x^2*exp(1))*exp((x^4*ln(x^2)^2+(-4*x^3*e xp(25)+2*x^2*exp(1))*ln(x^2)+4*x^2*exp(25)^2-4*x*exp(1)*exp(25)+exp(1)^2)/ x^4)+x^5)/x^5,x,method=_RETURNVERBOSE)
Output:
x-(x^2)^(-4*exp(25)/x)*(x^2)^(2*exp(1)/x^2)*exp((x^4*ln(x^2)^2+4*x^2*exp(5 0)-4*x*exp(26)+exp(2))/x^4)
Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.83 \[ \int \frac {x^5+e^{\frac {e^2-4 e^{26} x+4 e^{50} x^2+\left (2 e x^2-4 e^{25} x^3\right ) \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )}{x^4}} \left (4 e^2-4 e x^2+8 e^{50} x^2+e^{25} \left (-12 e x+8 x^3\right )+\left (4 e x^2-4 e^{25} x^3-4 x^4\right ) \log \left (x^2\right )\right )}{x^5} \, dx=x - e^{\left (\frac {x^{4} \log \left (x^{2}\right )^{2} + 4 \, x^{2} e^{50} - 4 \, x e^{26} - 2 \, {\left (2 \, x^{3} e^{25} - x^{2} e\right )} \log \left (x^{2}\right ) + e^{2}}{x^{4}}\right )} \] Input:
integrate((((-4*x^3*exp(25)+4*x^2*exp(1)-4*x^4)*log(x^2)+8*x^2*exp(25)^2+( -12*exp(1)*x+8*x^3)*exp(25)+4*exp(1)^2-4*x^2*exp(1))*exp((x^4*log(x^2)^2+( -4*x^3*exp(25)+2*x^2*exp(1))*log(x^2)+4*x^2*exp(25)^2-4*x*exp(1)*exp(25)+e xp(1)^2)/x^4)+x^5)/x^5,x, algorithm="fricas")
Output:
x - e^((x^4*log(x^2)^2 + 4*x^2*e^50 - 4*x*e^26 - 2*(2*x^3*e^25 - x^2*e)*lo g(x^2) + e^2)/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \frac {x^5+e^{\frac {e^2-4 e^{26} x+4 e^{50} x^2+\left (2 e x^2-4 e^{25} x^3\right ) \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )}{x^4}} \left (4 e^2-4 e x^2+8 e^{50} x^2+e^{25} \left (-12 e x+8 x^3\right )+\left (4 e x^2-4 e^{25} x^3-4 x^4\right ) \log \left (x^2\right )\right )}{x^5} \, dx=x - e^{\frac {x^{4} \log {\left (x^{2} \right )}^{2} + 4 x^{2} e^{50} - 4 x e^{26} + \left (- 4 x^{3} e^{25} + 2 e x^{2}\right ) \log {\left (x^{2} \right )} + e^{2}}{x^{4}}} \] Input:
integrate((((-4*x**3*exp(25)+4*x**2*exp(1)-4*x**4)*ln(x**2)+8*x**2*exp(25) **2+(-12*exp(1)*x+8*x**3)*exp(25)+4*exp(1)**2-4*x**2*exp(1))*exp((x**4*ln( x**2)**2+(-4*x**3*exp(25)+2*x**2*exp(1))*ln(x**2)+4*x**2*exp(25)**2-4*x*ex p(1)*exp(25)+exp(1)**2)/x**4)+x**5)/x**5,x)
Output:
x - exp((x**4*log(x**2)**2 + 4*x**2*exp(50) - 4*x*exp(26) + (-4*x**3*exp(2 5) + 2*E*x**2)*log(x**2) + exp(2))/x**4)
Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {x^5+e^{\frac {e^2-4 e^{26} x+4 e^{50} x^2+\left (2 e x^2-4 e^{25} x^3\right ) \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )}{x^4}} \left (4 e^2-4 e x^2+8 e^{50} x^2+e^{25} \left (-12 e x+8 x^3\right )+\left (4 e x^2-4 e^{25} x^3-4 x^4\right ) \log \left (x^2\right )\right )}{x^5} \, dx=x - e^{\left (4 \, \log \left (x\right )^{2} - \frac {8 \, e^{25} \log \left (x\right )}{x} + \frac {4 \, e \log \left (x\right )}{x^{2}} + \frac {4 \, e^{50}}{x^{2}} - \frac {4 \, e^{26}}{x^{3}} + \frac {e^{2}}{x^{4}}\right )} \] Input:
integrate((((-4*x^3*exp(25)+4*x^2*exp(1)-4*x^4)*log(x^2)+8*x^2*exp(25)^2+( -12*exp(1)*x+8*x^3)*exp(25)+4*exp(1)^2-4*x^2*exp(1))*exp((x^4*log(x^2)^2+( -4*x^3*exp(25)+2*x^2*exp(1))*log(x^2)+4*x^2*exp(25)^2-4*x*exp(1)*exp(25)+e xp(1)^2)/x^4)+x^5)/x^5,x, algorithm="maxima")
Output:
x - e^(4*log(x)^2 - 8*e^25*log(x)/x + 4*e*log(x)/x^2 + 4*e^50/x^2 - 4*e^26 /x^3 + e^2/x^4)
\[ \int \frac {x^5+e^{\frac {e^2-4 e^{26} x+4 e^{50} x^2+\left (2 e x^2-4 e^{25} x^3\right ) \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )}{x^4}} \left (4 e^2-4 e x^2+8 e^{50} x^2+e^{25} \left (-12 e x+8 x^3\right )+\left (4 e x^2-4 e^{25} x^3-4 x^4\right ) \log \left (x^2\right )\right )}{x^5} \, dx=\int { \frac {x^{5} + 4 \, {\left (2 \, x^{2} e^{50} - x^{2} e + {\left (2 \, x^{3} - 3 \, x e\right )} e^{25} - {\left (x^{4} + x^{3} e^{25} - x^{2} e\right )} \log \left (x^{2}\right ) + e^{2}\right )} e^{\left (\frac {x^{4} \log \left (x^{2}\right )^{2} + 4 \, x^{2} e^{50} - 4 \, x e^{26} - 2 \, {\left (2 \, x^{3} e^{25} - x^{2} e\right )} \log \left (x^{2}\right ) + e^{2}}{x^{4}}\right )}}{x^{5}} \,d x } \] Input:
integrate((((-4*x^3*exp(25)+4*x^2*exp(1)-4*x^4)*log(x^2)+8*x^2*exp(25)^2+( -12*exp(1)*x+8*x^3)*exp(25)+4*exp(1)^2-4*x^2*exp(1))*exp((x^4*log(x^2)^2+( -4*x^3*exp(25)+2*x^2*exp(1))*log(x^2)+4*x^2*exp(25)^2-4*x*exp(1)*exp(25)+e xp(1)^2)/x^4)+x^5)/x^5,x, algorithm="giac")
Output:
integrate((x^5 + 4*(2*x^2*e^50 - x^2*e + (2*x^3 - 3*x*e)*e^25 - (x^4 + x^3 *e^25 - x^2*e)*log(x^2) + e^2)*e^((x^4*log(x^2)^2 + 4*x^2*e^50 - 4*x*e^26 - 2*(2*x^3*e^25 - x^2*e)*log(x^2) + e^2)/x^4))/x^5, x)
Time = 7.46 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {x^5+e^{\frac {e^2-4 e^{26} x+4 e^{50} x^2+\left (2 e x^2-4 e^{25} x^3\right ) \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )}{x^4}} \left (4 e^2-4 e x^2+8 e^{50} x^2+e^{25} \left (-12 e x+8 x^3\right )+\left (4 e x^2-4 e^{25} x^3-4 x^4\right ) \log \left (x^2\right )\right )}{x^5} \, dx=x-\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^2}{x^4}}\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{26}}{x^3}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{50}}{x^2}}\,{\mathrm {e}}^{{\ln \left (x^2\right )}^2}\,{\left (x^2\right )}^{\frac {2\,\mathrm {e}}{x^2}}}{{\left (x^2\right )}^{\frac {4\,{\mathrm {e}}^{25}}{x}}} \] Input:
int(-(exp((exp(2) + log(x^2)*(2*x^2*exp(1) - 4*x^3*exp(25)) - 4*x*exp(26) + 4*x^2*exp(50) + x^4*log(x^2)^2)/x^4)*(log(x^2)*(4*x^3*exp(25) - 4*x^2*ex p(1) + 4*x^4) - 4*exp(2) + exp(25)*(12*x*exp(1) - 8*x^3) + 4*x^2*exp(1) - 8*x^2*exp(50)) - x^5)/x^5,x)
Output:
x - (exp(exp(2)/x^4)*exp(-(4*exp(26))/x^3)*exp((4*exp(50))/x^2)*exp(log(x^ 2)^2)*(x^2)^((2*exp(1))/x^2))/(x^2)^((4*exp(25))/x)
\[ \int \frac {x^5+e^{\frac {e^2-4 e^{26} x+4 e^{50} x^2+\left (2 e x^2-4 e^{25} x^3\right ) \log \left (x^2\right )+x^4 \log ^2\left (x^2\right )}{x^4}} \left (4 e^2-4 e x^2+8 e^{50} x^2+e^{25} \left (-12 e x+8 x^3\right )+\left (4 e x^2-4 e^{25} x^3-4 x^4\right ) \log \left (x^2\right )\right )}{x^5} \, dx=\int \frac {\left (\left (-4 x^{3} {\mathrm e}^{25}+4 x^{2} {\mathrm e}-4 x^{4}\right ) \mathrm {log}\left (x^{2}\right )+8 x^{2} \left ({\mathrm e}^{25}\right )^{2}+\left (-12 \,{\mathrm e} x +8 x^{3}\right ) {\mathrm e}^{25}+4 \left ({\mathrm e}\right )^{2}-4 x^{2} {\mathrm e}\right ) {\mathrm e}^{\frac {x^{4} \mathrm {log}\left (x^{2}\right )^{2}+\left (-4 x^{3} {\mathrm e}^{25}+2 x^{2} {\mathrm e}\right ) \mathrm {log}\left (x^{2}\right )+4 x^{2} \left ({\mathrm e}^{25}\right )^{2}-4 x \,{\mathrm e} \,{\mathrm e}^{25}+\left ({\mathrm e}\right )^{2}}{x^{4}}}+x^{5}}{x^{5}}d x \] Input:
int((((-4*x^3*exp(25)+4*x^2*exp(1)-4*x^4)*log(x^2)+8*x^2*exp(25)^2+(-12*ex p(1)*x+8*x^3)*exp(25)+4*exp(1)^2-4*x^2*exp(1))*exp((x^4*log(x^2)^2+(-4*x^3 *exp(25)+2*x^2*exp(1))*log(x^2)+4*x^2*exp(25)^2-4*x*exp(1)*exp(25)+exp(1)^ 2)/x^4)+x^5)/x^5,x)
Output:
int((((-4*x^3*exp(25)+4*x^2*exp(1)-4*x^4)*log(x^2)+8*x^2*exp(25)^2+(-12*ex p(1)*x+8*x^3)*exp(25)+4*exp(1)^2-4*x^2*exp(1))*exp((x^4*log(x^2)^2+(-4*x^3 *exp(25)+2*x^2*exp(1))*log(x^2)+4*x^2*exp(25)^2-4*x*exp(1)*exp(25)+exp(1)^ 2)/x^4)+x^5)/x^5,x)