Integrand size = 113, antiderivative size = 24 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=\left (4 x+9 x^3\right ) \log \left (-16-\frac {e^4}{x}+x+\log (x)\right ) \] Output:
ln(x-exp(4)/x-16+ln(x))*(9*x^3+4*x)
Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=x \left (4+9 x^2\right ) \log \left (-16-\frac {e^4}{x}+x+\log (x)\right ) \] Input:
Integrate[(4*x + 4*x^2 + 9*x^3 + 9*x^4 + E^4*(4 + 9*x^2) + (-64*x + 4*x^2 - 432*x^3 + 27*x^4 + E^4*(-4 - 27*x^2) + (4*x + 27*x^3)*Log[x])*Log[(-E^4 - 16*x + x^2 + x*Log[x])/x])/(-E^4 - 16*x + x^2 + x*Log[x]),x]
Output:
x*(4 + 9*x^2)*Log[-16 - E^4/x + x + 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.
\(\displaystyle \int \frac {9 x^4+9 x^3+4 x^2+e^4 \left (9 x^2+4\right )+\left (27 x^4-432 x^3+\left (27 x^3+4 x\right ) \log (x)+4 x^2+e^4 \left (-27 x^2-4\right )-64 x\right ) \log \left (\frac {x^2-16 x+x \log (x)-e^4}{x}\right )+4 x}{x^2-16 x+x \log (x)-e^4} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 x^2}{x^2-16 x+x \log (x)-e^4}+\frac {4 x}{x^2-16 x+x \log (x)-e^4}+\left (27 x^2+4\right ) \log \left (x-\frac {e^4}{x}+\log (x)-16\right )-\frac {e^4 \left (9 x^2+4\right )}{-x^2+16 x-x \log (x)+e^4}+\frac {9 x^4}{x^2-16 x+x \log (x)-e^4}+\frac {9 x^3}{x^2-16 x+x \log (x)-e^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 e^4 \int \frac {1}{-x^2-\log (x) x+16 x+e^4}dx+4 \int \frac {x}{x^2+\log (x) x-16 x-e^4}dx+9 e^4 \int \frac {x^2}{x^2+\log (x) x-16 x-e^4}dx+4 \int \frac {x^2}{x^2+\log (x) x-16 x-e^4}dx+27 \int x^2 \log \left (x+\log (x)-16-\frac {e^4}{x}\right )dx+9 \int \frac {x^4}{x^2+\log (x) x-16 x-e^4}dx+9 \int \frac {x^3}{x^2+\log (x) x-16 x-e^4}dx+4 \int \log \left (x+\log (x)-16-\frac {e^4}{x}\right )dx\) |
Input:
Int[(4*x + 4*x^2 + 9*x^3 + 9*x^4 + E^4*(4 + 9*x^2) + (-64*x + 4*x^2 - 432* x^3 + 27*x^4 + E^4*(-4 - 27*x^2) + (4*x + 27*x^3)*Log[x])*Log[(-E^4 - 16*x + x^2 + x*Log[x])/x])/(-E^4 - 16*x + x^2 + x*Log[x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(23)=46\).
Time = 3.65 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08
method | result | size |
parallelrisch | \(9 \ln \left (\frac {x \ln \left (x \right )-{\mathrm e}^{4}+x^{2}-16 x}{x}\right ) x^{3}+4 x \ln \left (\frac {x \ln \left (x \right )-{\mathrm e}^{4}+x^{2}-16 x}{x}\right )\) | \(50\) |
risch | \(\left (9 x^{3}+4 x \right ) \ln \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )-9 x^{3} \ln \left (x \right )-4 x \ln \left (x \right )+2 i \pi x \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2}+2 i \pi x \,\operatorname {csgn}\left (i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2}+9 i \pi \,x^{3}-2 i \pi x \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )+2 i \pi x {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{3}+\frac {9 i \pi \,x^{3} {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{3}}{2}+\frac {9 i \pi \,x^{3} \operatorname {csgn}\left (i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2}}{2}-9 i \pi \,x^{3} {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2}+4 i \pi x -4 i \pi x {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2}+\frac {9 i \pi \,x^{3} \operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2}}{2}-\frac {9 i \pi \,x^{3} \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}{2}\) | \(467\) |
Input:
int((((27*x^3+4*x)*ln(x)+(-27*x^2-4)*exp(4)+27*x^4-432*x^3+4*x^2-64*x)*ln( (x*ln(x)-exp(4)+x^2-16*x)/x)+(9*x^2+4)*exp(4)+9*x^4+9*x^3+4*x^2+4*x)/(x*ln (x)-exp(4)+x^2-16*x),x,method=_RETURNVERBOSE)
Output:
9*ln((x*ln(x)-exp(4)+x^2-16*x)/x)*x^3+4*x*ln((x*ln(x)-exp(4)+x^2-16*x)/x)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx={\left (9 \, x^{3} + 4 \, x\right )} \log \left (\frac {x^{2} + x \log \left (x\right ) - 16 \, x - e^{4}}{x}\right ) \] Input:
integrate((((27*x^3+4*x)*log(x)+(-27*x^2-4)*exp(4)+27*x^4-432*x^3+4*x^2-64 *x)*log((x*log(x)-exp(4)+x^2-16*x)/x)+(9*x^2+4)*exp(4)+9*x^4+9*x^3+4*x^2+4 *x)/(x*log(x)-exp(4)+x^2-16*x),x, algorithm="fricas")
Output:
(9*x^3 + 4*x)*log((x^2 + x*log(x) - 16*x - e^4)/x)
Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=\left (9 x^{3} + 4 x\right ) \log {\left (\frac {x^{2} + x \log {\left (x \right )} - 16 x - e^{4}}{x} \right )} \] Input:
integrate((((27*x**3+4*x)*ln(x)+(-27*x**2-4)*exp(4)+27*x**4-432*x**3+4*x** 2-64*x)*ln((x*ln(x)-exp(4)+x**2-16*x)/x)+(9*x**2+4)*exp(4)+9*x**4+9*x**3+4 *x**2+4*x)/(x*ln(x)-exp(4)+x**2-16*x),x)
Output:
(9*x**3 + 4*x)*log((x**2 + x*log(x) - 16*x - exp(4))/x)
Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx={\left (9 \, x^{3} + 4 \, x\right )} \log \left (x^{2} + x \log \left (x\right ) - 16 \, x - e^{4}\right ) - {\left (9 \, x^{3} + 4 \, x\right )} \log \left (x\right ) \] Input:
integrate((((27*x^3+4*x)*log(x)+(-27*x^2-4)*exp(4)+27*x^4-432*x^3+4*x^2-64 *x)*log((x*log(x)-exp(4)+x^2-16*x)/x)+(9*x^2+4)*exp(4)+9*x^4+9*x^3+4*x^2+4 *x)/(x*log(x)-exp(4)+x^2-16*x),x, algorithm="maxima")
Output:
(9*x^3 + 4*x)*log(x^2 + x*log(x) - 16*x - e^4) - (9*x^3 + 4*x)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.21 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=9 \, x^{3} \log \left (x^{2} + x \log \left (x\right ) - 16 \, x - e^{4}\right ) - 9 \, x^{3} \log \left (x\right ) + 4 \, x \log \left (x^{2} + x \log \left (x\right ) - 16 \, x - e^{4}\right ) - 4 \, x \log \left (x\right ) \] Input:
integrate((((27*x^3+4*x)*log(x)+(-27*x^2-4)*exp(4)+27*x^4-432*x^3+4*x^2-64 *x)*log((x*log(x)-exp(4)+x^2-16*x)/x)+(9*x^2+4)*exp(4)+9*x^4+9*x^3+4*x^2+4 *x)/(x*log(x)-exp(4)+x^2-16*x),x, algorithm="giac")
Output:
9*x^3*log(x^2 + x*log(x) - 16*x - e^4) - 9*x^3*log(x) + 4*x*log(x^2 + x*lo g(x) - 16*x - e^4) - 4*x*log(x)
Time = 3.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=x\,\ln \left (-\frac {16\,x+{\mathrm {e}}^4-x\,\ln \left (x\right )-x^2}{x}\right )\,\left (9\,x^2+4\right ) \] Input:
int(-(4*x + exp(4)*(9*x^2 + 4) - log(-(16*x + exp(4) - x*log(x) - x^2)/x)* (64*x + exp(4)*(27*x^2 + 4) - log(x)*(4*x + 27*x^3) - 4*x^2 + 432*x^3 - 27 *x^4) + 4*x^2 + 9*x^3 + 9*x^4)/(16*x + exp(4) - x*log(x) - x^2),x)
Output:
x*log(-(16*x + exp(4) - x*log(x) - x^2)/x)*(9*x^2 + 4)
Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=\mathrm {log}\left (\frac {\mathrm {log}\left (x \right ) x -e^{4}+x^{2}-16 x}{x}\right ) x \left (9 x^{2}+4\right ) \] Input:
int((((27*x^3+4*x)*log(x)+(-27*x^2-4)*exp(4)+27*x^4-432*x^3+4*x^2-64*x)*lo g((x*log(x)-exp(4)+x^2-16*x)/x)+(9*x^2+4)*exp(4)+9*x^4+9*x^3+4*x^2+4*x)/(x *log(x)-exp(4)+x^2-16*x),x)
Output:
log((log(x)*x - e**4 + x**2 - 16*x)/x)*x*(9*x**2 + 4)