Integrand size = 175, antiderivative size = 19 \[ \int \frac {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{-12 x+4 e^4 x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx=x^3+\frac {1}{2+\log \left (x \left (3-e^4+x\right )\right )} \] Output:
1/(2+ln(x*(3-exp(4)+x)))+x^3
Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{-12 x+4 e^4 x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx=x^3+\frac {1}{2+\log \left (x \left (3-e^4+x\right )\right )} \] Input:
Integrate[(3 + 2*x - 36*x^3 - 12*x^4 + E^4*(-1 + 12*x^3) + (-36*x^3 + 12*E ^4*x^3 - 12*x^4)*Log[3*x - E^4*x + x^2] + (-9*x^3 + 3*E^4*x^3 - 3*x^4)*Log [3*x - E^4*x + x^2]^2)/(-12*x + 4*E^4*x - 4*x^2 + (-12*x + 4*E^4*x - 4*x^2 )*Log[3*x - E^4*x + x^2] + (-3*x + E^4*x - x^2)*Log[3*x - E^4*x + x^2]^2), x]
Output:
x^3 + (2 + Log[x*(3 - E^4 + x)])^(-1)
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.
| \(\displaystyle \int \frac {-12 x^4-36 x^3+e^4 \left (12 x^3-1\right )+\left (-3 x^4+3 e^4 x^3-9 x^3\right ) \log ^2\left (x^2-e^4 x+3 x\right )+\left (-12 x^4+12 e^4 x^3-36 x^3\right ) \log \left (x^2-e^4 x+3 x\right )+2 x+3}{-4 x^2+\left (-x^2+e^4 x-3 x\right ) \log ^2\left (x^2-e^4 x+3 x\right )+\left (-4 x^2+4 e^4 x-12 x\right ) \log \left (x^2-e^4 x+3 x\right )+4 e^4 x-12 x} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-12 x^4-36 x^3+e^4 \left (12 x^3-1\right )+\left (-3 x^4+3 e^4 x^3-9 x^3\right ) \log ^2\left (x^2-e^4 x+3 x\right )+\left (-12 x^4+12 e^4 x^3-36 x^3\right ) \log \left (x^2-e^4 x+3 x\right )+2 x+3}{-4 x^2+\left (-x^2+e^4 x-3 x\right ) \log ^2\left (x^2-e^4 x+3 x\right )+\left (-4 x^2+4 e^4 x-12 x\right ) \log \left (x^2-e^4 x+3 x\right )+\left (4 e^4-12\right ) x}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {12 x^4+36 x^3-e^4 \left (12 x^3-1\right )-\left (-3 x^4+3 e^4 x^3-9 x^3\right ) \log ^2\left (x^2-e^4 x+3 x\right )-\left (-12 x^4+12 e^4 x^3-36 x^3\right ) \log \left (x^2-e^4 x+3 x\right )-2 x-3}{x \left (x-e^4+3\right ) \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {12 x^3}{\left (-x+e^4-3\right ) \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}+\frac {e^4 \left (12 x^3-1\right )}{\left (-x+e^4-3\right ) x \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}+\frac {3 x^2 \log ^2\left (x \left (x-e^4+3\right )\right )}{\left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}+\frac {12 x^2 \log \left (x \left (x-e^4+3\right )\right )}{\left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}-\frac {36 x^2}{\left (-x+e^4-3\right ) \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}+\frac {2}{\left (-x+e^4-3\right ) \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}-\frac {3}{\left (x-e^4+3\right ) x \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 12 \left (3-e^4\right )^2 \int \frac {1}{\left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx+12 e^4 \left (3-e^4\right ) \int \frac {1}{\left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx-36 \left (3-e^4\right ) \int \frac {1}{\left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx+\frac {e^4 \left (325-324 e^4+108 e^8-12 e^{12}\right ) \int \frac {1}{\left (-x+e^4-3\right ) \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx}{3-e^4}+12 \left (3-e^4\right )^3 \int \frac {1}{\left (-x+e^4-3\right ) \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx-36 \left (3-e^4\right )^2 \int \frac {1}{\left (-x+e^4-3\right ) \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx-\frac {3 \int \frac {1}{\left (-x+e^4-3\right ) \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx}{3-e^4}+2 \int \frac {1}{\left (-x+e^4-3\right ) \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx+\frac {e^4 \int \frac {1}{x \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx}{3-e^4}-\frac {3 \int \frac {1}{x \left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx}{3-e^4}-12 \left (3-e^4\right ) \int \frac {x}{\left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx-12 e^4 \int \frac {x}{\left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx+36 \int \frac {x}{\left (\log \left (x \left (x-e^4+3\right )\right )+2\right )^2}dx+x^3\) |
Input:
Int[(3 + 2*x - 36*x^3 - 12*x^4 + E^4*(-1 + 12*x^3) + (-36*x^3 + 12*E^4*x^3 - 12*x^4)*Log[3*x - E^4*x + x^2] + (-9*x^3 + 3*E^4*x^3 - 3*x^4)*Log[3*x - E^4*x + x^2]^2)/(-12*x + 4*E^4*x - 4*x^2 + (-12*x + 4*E^4*x - 4*x^2)*Log[ 3*x - E^4*x + x^2] + (-3*x + E^4*x - x^2)*Log[3*x - E^4*x + x^2]^2),x]
Output:
$Aborted
Time = 1.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(x^{3}+\frac {1}{\ln \left (-x \,{\mathrm e}^{4}+x^{2}+3 x \right )+2}\) | \(22\) |
| parallelrisch | \(\frac {\ln \left (-x \left ({\mathrm e}^{4}-x -3\right )\right ) x^{3}+1+2 x^{3}}{\ln \left (-x \left ({\mathrm e}^{4}-x -3\right )\right )+2}\) | \(39\) |
| norman | \(\frac {x^{3} \ln \left (-x \,{\mathrm e}^{4}+x^{2}+3 x \right )+2 x^{3}+1}{\ln \left (-x \,{\mathrm e}^{4}+x^{2}+3 x \right )+2}\) | \(43\) |
Input:
int(((3*x^3*exp(4)-3*x^4-9*x^3)*ln(-x*exp(4)+x^2+3*x)^2+(12*x^3*exp(4)-12* x^4-36*x^3)*ln(-x*exp(4)+x^2+3*x)+(12*x^3-1)*exp(4)-12*x^4-36*x^3+2*x+3)/( (x*exp(4)-x^2-3*x)*ln(-x*exp(4)+x^2+3*x)^2+(4*x*exp(4)-4*x^2-12*x)*ln(-x*e xp(4)+x^2+3*x)+4*x*exp(4)-4*x^2-12*x),x,method=_RETURNVERBOSE)
Output:
x^3+1/(ln(-x*exp(4)+x^2+3*x)+2)
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (18) = 36\).
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \frac {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{-12 x+4 e^4 x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx=\frac {x^{3} \log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2 \, x^{3} + 1}{\log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2} \] Input:
integrate(((3*x^3*exp(4)-3*x^4-9*x^3)*log(-x*exp(4)+x^2+3*x)^2+(12*x^3*exp (4)-12*x^4-36*x^3)*log(-x*exp(4)+x^2+3*x)+(12*x^3-1)*exp(4)-12*x^4-36*x^3+ 2*x+3)/((x*exp(4)-x^2-3*x)*log(-x*exp(4)+x^2+3*x)^2+(4*x*exp(4)-4*x^2-12*x )*log(-x*exp(4)+x^2+3*x)+4*x*exp(4)-4*x^2-12*x),x, algorithm="fricas")
Output:
(x^3*log(x^2 - x*e^4 + 3*x) + 2*x^3 + 1)/(log(x^2 - x*e^4 + 3*x) + 2)
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{-12 x+4 e^4 x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx=x^{3} + \frac {1}{\log {\left (x^{2} - x e^{4} + 3 x \right )} + 2} \] Input:
integrate(((3*x**3*exp(4)-3*x**4-9*x**3)*ln(-x*exp(4)+x**2+3*x)**2+(12*x** 3*exp(4)-12*x**4-36*x**3)*ln(-x*exp(4)+x**2+3*x)+(12*x**3-1)*exp(4)-12*x** 4-36*x**3+2*x+3)/((x*exp(4)-x**2-3*x)*ln(-x*exp(4)+x**2+3*x)**2+(4*x*exp(4 )-4*x**2-12*x)*ln(-x*exp(4)+x**2+3*x)+4*x*exp(4)-4*x**2-12*x),x)
Output:
x**3 + 1/(log(x**2 - x*exp(4) + 3*x) + 2)
Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (18) = 36\).
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int \frac {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{-12 x+4 e^4 x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx=\frac {x^{3} \log \left (x - e^{4} + 3\right ) + x^{3} \log \left (x\right ) + 2 \, x^{3} + 1}{\log \left (x - e^{4} + 3\right ) + \log \left (x\right ) + 2} \] Input:
integrate(((3*x^3*exp(4)-3*x^4-9*x^3)*log(-x*exp(4)+x^2+3*x)^2+(12*x^3*exp (4)-12*x^4-36*x^3)*log(-x*exp(4)+x^2+3*x)+(12*x^3-1)*exp(4)-12*x^4-36*x^3+ 2*x+3)/((x*exp(4)-x^2-3*x)*log(-x*exp(4)+x^2+3*x)^2+(4*x*exp(4)-4*x^2-12*x )*log(-x*exp(4)+x^2+3*x)+4*x*exp(4)-4*x^2-12*x),x, algorithm="maxima")
Output:
(x^3*log(x - e^4 + 3) + x^3*log(x) + 2*x^3 + 1)/(log(x - e^4 + 3) + log(x) + 2)
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (18) = 36\).
Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \frac {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{-12 x+4 e^4 x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx=\frac {x^{3} \log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2 \, x^{3} + 1}{\log \left (x^{2} - x e^{4} + 3 \, x\right ) + 2} \] Input:
integrate(((3*x^3*exp(4)-3*x^4-9*x^3)*log(-x*exp(4)+x^2+3*x)^2+(12*x^3*exp (4)-12*x^4-36*x^3)*log(-x*exp(4)+x^2+3*x)+(12*x^3-1)*exp(4)-12*x^4-36*x^3+ 2*x+3)/((x*exp(4)-x^2-3*x)*log(-x*exp(4)+x^2+3*x)^2+(4*x*exp(4)-4*x^2-12*x )*log(-x*exp(4)+x^2+3*x)+4*x*exp(4)-4*x^2-12*x),x, algorithm="giac")
Output:
(x^3*log(x^2 - x*e^4 + 3*x) + 2*x^3 + 1)/(log(x^2 - x*e^4 + 3*x) + 2)
Time = 3.97 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{-12 x+4 e^4 x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx=\frac {1}{\ln \left (3\,x-x\,{\mathrm {e}}^4+x^2\right )+2}+x^3 \] Input:
int((log(3*x - x*exp(4) + x^2)*(36*x^3 - 12*x^3*exp(4) + 12*x^4) - exp(4)* (12*x^3 - 1) - 2*x + 36*x^3 + 12*x^4 + log(3*x - x*exp(4) + x^2)^2*(9*x^3 - 3*x^3*exp(4) + 3*x^4) - 3)/(12*x - 4*x*exp(4) + log(3*x - x*exp(4) + x^2 )*(12*x - 4*x*exp(4) + 4*x^2) + log(3*x - x*exp(4) + x^2)^2*(3*x - x*exp(4 ) + x^2) + 4*x^2),x)
Output:
1/(log(3*x - x*exp(4) + x^2) + 2) + x^3
Time = 0.16 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.26 \[ \int \frac {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{-12 x+4 e^4 x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx=\frac {2 \,\mathrm {log}\left (-e^{4} x +x^{2}+3 x \right ) x^{3}-\mathrm {log}\left (-e^{4} x +x^{2}+3 x \right )+4 x^{3}}{2 \,\mathrm {log}\left (-e^{4} x +x^{2}+3 x \right )+4} \] Input:
int(((3*x^3*exp(4)-3*x^4-9*x^3)*log(-x*exp(4)+x^2+3*x)^2+(12*x^3*exp(4)-12 *x^4-36*x^3)*log(-x*exp(4)+x^2+3*x)+(12*x^3-1)*exp(4)-12*x^4-36*x^3+2*x+3) /((x*exp(4)-x^2-3*x)*log(-x*exp(4)+x^2+3*x)^2+(4*x*exp(4)-4*x^2-12*x)*log( -x*exp(4)+x^2+3*x)+4*x*exp(4)-4*x^2-12*x),x)
Output:
(2*log( - e**4*x + x**2 + 3*x)*x**3 - log( - e**4*x + x**2 + 3*x) + 4*x**3 )/(2*(log( - e**4*x + x**2 + 3*x) + 2))