Integrand size = 122, antiderivative size = 22 \[ \int \frac {1-676 x-e^{2 e^2} x-104 x^2-315 x^3-16 x^4-20 x^5+e^{e^2} \left (-52 x-4 x^2-12 x^3\right )}{-676 x^2-e^{2 e^2} x^2-52 x^3-105 x^4-4 x^5-4 x^6+e^{e^2} \left (-52 x^2-2 x^3-4 x^4\right )+x \log (x)} \, dx=\log \left (-x \left (26+e^{e^2}+x+2 x^2\right )^2+\log (x)\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(66\) vs. \(2(22)=44\).
Time = 0.55 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.00 \[ \int \frac {1-676 x-e^{2 e^2} x-104 x^2-315 x^3-16 x^4-20 x^5+e^{e^2} \left (-52 x-4 x^2-12 x^3\right )}{-676 x^2-e^{2 e^2} x^2-52 x^3-105 x^4-4 x^5-4 x^6+e^{e^2} \left (-52 x^2-2 x^3-4 x^4\right )+x \log (x)} \, dx=\log \left (676 x+52 e^{e^2} x+e^{2 e^2} x+52 x^2+2 e^{e^2} x^2+105 x^3+4 e^{e^2} x^3+4 x^4+4 x^5-\log (x)\right ) \]
Integrate[(1 - 676*x - E^(2*E^2)*x - 104*x^2 - 315*x^3 - 16*x^4 - 20*x^5 + E^E^2*(-52*x - 4*x^2 - 12*x^3))/(-676*x^2 - E^(2*E^2)*x^2 - 52*x^3 - 105* x^4 - 4*x^5 - 4*x^6 + E^E^2*(-52*x^2 - 2*x^3 - 4*x^4) + x*Log[x]),x]
Log[676*x + 52*E^E^2*x + E^(2*E^2)*x + 52*x^2 + 2*E^E^2*x^2 + 105*x^3 + 4* E^E^2*x^3 + 4*x^4 + 4*x^5 - 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.
\(\displaystyle \int \frac {-20 x^5-16 x^4-315 x^3-104 x^2+e^{e^2} \left (-12 x^3-4 x^2-52 x\right )-e^{2 e^2} x-676 x+1}{-4 x^6-4 x^5-105 x^4-52 x^3-e^{2 e^2} x^2-676 x^2+e^{e^2} \left (-4 x^4-2 x^3-52 x^2\right )+x \log (x)} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-20 x^5-16 x^4-315 x^3-104 x^2+e^{e^2} \left (-12 x^3-4 x^2-52 x\right )+\left (-676-e^{2 e^2}\right ) x+1}{-4 x^6-4 x^5-105 x^4-52 x^3-e^{2 e^2} x^2-676 x^2+e^{e^2} \left (-4 x^4-2 x^3-52 x^2\right )+x \log (x)}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-20 x^5-16 x^4-315 x^3-104 x^2+e^{e^2} \left (-12 x^3-4 x^2-52 x\right )+\left (-676-e^{2 e^2}\right ) x+1}{-4 x^6-4 x^5-105 x^4-52 x^3+\left (-676-e^{2 e^2}\right ) x^2+e^{e^2} \left (-4 x^4-2 x^3-52 x^2\right )+x \log (x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {20 x^4}{4 x^5+4 x^4+105 \left (1+\frac {4 e^{e^2}}{105}\right ) x^3+52 \left (1+\frac {e^{e^2}}{26}\right ) x^2+676 \left (1+\frac {1}{676} e^{e^2} \left (52+e^{e^2}\right )\right ) x-\log (x)}+\frac {16 x^3}{4 x^5+4 x^4+105 \left (1+\frac {4 e^{e^2}}{105}\right ) x^3+52 \left (1+\frac {e^{e^2}}{26}\right ) x^2+676 \left (1+\frac {1}{676} e^{e^2} \left (52+e^{e^2}\right )\right ) x-\log (x)}+\frac {3 \left (105+4 e^{e^2}\right ) x^2}{4 x^5+4 x^4+105 \left (1+\frac {4 e^{e^2}}{105}\right ) x^3+52 \left (1+\frac {e^{e^2}}{26}\right ) x^2+676 \left (1+\frac {1}{676} e^{e^2} \left (52+e^{e^2}\right )\right ) x-\log (x)}+\frac {4 \left (26+e^{e^2}\right ) x}{4 x^5+4 x^4+105 \left (1+\frac {4 e^{e^2}}{105}\right ) x^3+52 \left (1+\frac {e^{e^2}}{26}\right ) x^2+676 \left (1+\frac {1}{676} e^{e^2} \left (52+e^{e^2}\right )\right ) x-\log (x)}+\frac {\left (26+e^{e^2}\right )^2}{4 x^5+4 x^4+105 \left (1+\frac {4 e^{e^2}}{105}\right ) x^3+52 \left (1+\frac {e^{e^2}}{26}\right ) x^2+676 \left (1+\frac {1}{676} e^{e^2} \left (52+e^{e^2}\right )\right ) x-\log (x)}+\frac {1}{x \left (-4 x^5-4 x^4-105 \left (1+\frac {4 e^{e^2}}{105}\right ) x^3-52 \left (1+\frac {e^{e^2}}{26}\right ) x^2-676 \left (1+\frac {1}{676} e^{e^2} \left (52+e^{e^2}\right )\right ) x+\log (x)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \left (26+e^{e^2}\right )^2 \int \frac {1}{x \left (2 x^2+x+e^{e^2}+26\right )^2-\log (x)}dx+4 \left (26+e^{e^2}\right ) \int \frac {x}{x \left (2 x^2+x+e^{e^2}+26\right )^2-\log (x)}dx+3 \left (105+4 e^{e^2}\right ) \int \frac {x^2}{x \left (2 x^2+x+e^{e^2}+26\right )^2-\log (x)}dx+\int \frac {1}{x \left (\log (x)-x \left (2 x^2+x+e^{e^2}+26\right )^2\right )}dx+20 \int \frac {x^4}{x \left (2 x^2+x+e^{e^2}+26\right )^2-\log (x)}dx+16 \int \frac {x^3}{x \left (2 x^2+x+e^{e^2}+26\right )^2-\log (x)}dx\) |
Int[(1 - 676*x - E^(2*E^2)*x - 104*x^2 - 315*x^3 - 16*x^4 - 20*x^5 + E^E^2 *(-52*x - 4*x^2 - 12*x^3))/(-676*x^2 - E^(2*E^2)*x^2 - 52*x^3 - 105*x^4 - 4*x^5 - 4*x^6 + E^E^2*(-52*x^2 - 2*x^3 - 4*x^4) + x*Log[x]),x]
3.27.89.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(20)=40\).
Time = 0.83 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.50
method | result | size |
parallelrisch | \(\ln \left (x^{5}+x^{3} {\mathrm e}^{{\mathrm e}^{2}}+x^{4}+\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{2}}}{4}+\frac {x^{2} {\mathrm e}^{{\mathrm e}^{2}}}{2}+\frac {105 x^{3}}{4}+13 x \,{\mathrm e}^{{\mathrm e}^{2}}+13 x^{2}+169 x -\frac {\ln \left (x \right )}{4}\right )\) | \(55\) |
risch | \(\ln \left (-4 x^{5}-4 x^{3} {\mathrm e}^{{\mathrm e}^{2}}-4 x^{4}-x \,{\mathrm e}^{2 \,{\mathrm e}^{2}}-2 x^{2} {\mathrm e}^{{\mathrm e}^{2}}-105 x^{3}-52 x \,{\mathrm e}^{{\mathrm e}^{2}}-52 x^{2}+\ln \left (x \right )-676 x \right )\) | \(58\) |
default | \(\ln \left (4 x^{5}+4 x^{4}+4 x^{3} {\mathrm e}^{{\mathrm e}^{2}}+105 x^{3}+2 x^{2} {\mathrm e}^{{\mathrm e}^{2}}+x \,{\mathrm e}^{2 \,{\mathrm e}^{2}}+52 x^{2}+52 x \,{\mathrm e}^{{\mathrm e}^{2}}+676 x -\ln \left (x \right )\right )\) | \(59\) |
norman | \(\ln \left (4 x^{5}+4 x^{4}+4 x^{3} {\mathrm e}^{{\mathrm e}^{2}}+105 x^{3}+2 x^{2} {\mathrm e}^{{\mathrm e}^{2}}+x \,{\mathrm e}^{2 \,{\mathrm e}^{2}}+52 x^{2}+52 x \,{\mathrm e}^{{\mathrm e}^{2}}+676 x -\ln \left (x \right )\right )\) | \(59\) |
int((-x*exp(exp(2))^2+(-12*x^3-4*x^2-52*x)*exp(exp(2))-20*x^5-16*x^4-315*x ^3-104*x^2-676*x+1)/(x*ln(x)-x^2*exp(exp(2))^2+(-4*x^4-2*x^3-52*x^2)*exp(e xp(2))-4*x^6-4*x^5-105*x^4-52*x^3-676*x^2),x,method=_RETURNVERBOSE)
ln(x^5+x^3*exp(exp(2))+x^4+1/4*x*exp(exp(2))^2+1/2*x^2*exp(exp(2))+105/4*x ^3+13*x*exp(exp(2))+13*x^2+169*x-1/4*ln(x))
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36 \[ \int \frac {1-676 x-e^{2 e^2} x-104 x^2-315 x^3-16 x^4-20 x^5+e^{e^2} \left (-52 x-4 x^2-12 x^3\right )}{-676 x^2-e^{2 e^2} x^2-52 x^3-105 x^4-4 x^5-4 x^6+e^{e^2} \left (-52 x^2-2 x^3-4 x^4\right )+x \log (x)} \, dx=\log \left (-4 \, x^{5} - 4 \, x^{4} - 105 \, x^{3} - 52 \, x^{2} - x e^{\left (2 \, e^{2}\right )} - 2 \, {\left (2 \, x^{3} + x^{2} + 26 \, x\right )} e^{\left (e^{2}\right )} - 676 \, x + \log \left (x\right )\right ) \]
integrate((-x*exp(exp(2))^2+(-12*x^3-4*x^2-52*x)*exp(exp(2))-20*x^5-16*x^4 -315*x^3-104*x^2-676*x+1)/(x*log(x)-x^2*exp(exp(2))^2+(-4*x^4-2*x^3-52*x^2 )*exp(exp(2))-4*x^6-4*x^5-105*x^4-52*x^3-676*x^2),x, algorithm=\
log(-4*x^5 - 4*x^4 - 105*x^3 - 52*x^2 - x*e^(2*e^2) - 2*(2*x^3 + x^2 + 26* x)*e^(e^2) - 676*x + log(x))
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.95 \[ \int \frac {1-676 x-e^{2 e^2} x-104 x^2-315 x^3-16 x^4-20 x^5+e^{e^2} \left (-52 x-4 x^2-12 x^3\right )}{-676 x^2-e^{2 e^2} x^2-52 x^3-105 x^4-4 x^5-4 x^6+e^{e^2} \left (-52 x^2-2 x^3-4 x^4\right )+x \log (x)} \, dx=\log {\left (- 4 x^{5} - 4 x^{4} - 4 x^{3} e^{e^{2}} - 105 x^{3} - 2 x^{2} e^{e^{2}} - 52 x^{2} - x e^{2 e^{2}} - 52 x e^{e^{2}} - 676 x + \log {\left (x \right )} \right )} \]
integrate((-x*exp(exp(2))**2+(-12*x**3-4*x**2-52*x)*exp(exp(2))-20*x**5-16 *x**4-315*x**3-104*x**2-676*x+1)/(x*ln(x)-x**2*exp(exp(2))**2+(-4*x**4-2*x **3-52*x**2)*exp(exp(2))-4*x**6-4*x**5-105*x**4-52*x**3-676*x**2),x)
log(-4*x**5 - 4*x**4 - 4*x**3*exp(exp(2)) - 105*x**3 - 2*x**2*exp(exp(2)) - 52*x**2 - x*exp(2*exp(2)) - 52*x*exp(exp(2)) - 676*x + log(x))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {1-676 x-e^{2 e^2} x-104 x^2-315 x^3-16 x^4-20 x^5+e^{e^2} \left (-52 x-4 x^2-12 x^3\right )}{-676 x^2-e^{2 e^2} x^2-52 x^3-105 x^4-4 x^5-4 x^6+e^{e^2} \left (-52 x^2-2 x^3-4 x^4\right )+x \log (x)} \, dx=\log \left (-4 \, x^{5} - 4 \, x^{4} - x^{3} {\left (4 \, e^{\left (e^{2}\right )} + 105\right )} - 2 \, x^{2} {\left (e^{\left (e^{2}\right )} + 26\right )} - x {\left (e^{\left (2 \, e^{2}\right )} + 52 \, e^{\left (e^{2}\right )} + 676\right )} + \log \left (x\right )\right ) \]
integrate((-x*exp(exp(2))^2+(-12*x^3-4*x^2-52*x)*exp(exp(2))-20*x^5-16*x^4 -315*x^3-104*x^2-676*x+1)/(x*log(x)-x^2*exp(exp(2))^2+(-4*x^4-2*x^3-52*x^2 )*exp(exp(2))-4*x^6-4*x^5-105*x^4-52*x^3-676*x^2),x, algorithm=\
log(-4*x^5 - 4*x^4 - x^3*(4*e^(e^2) + 105) - 2*x^2*(e^(e^2) + 26) - x*(e^( 2*e^2) + 52*e^(e^2) + 676) + log(x))
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.59 \[ \int \frac {1-676 x-e^{2 e^2} x-104 x^2-315 x^3-16 x^4-20 x^5+e^{e^2} \left (-52 x-4 x^2-12 x^3\right )}{-676 x^2-e^{2 e^2} x^2-52 x^3-105 x^4-4 x^5-4 x^6+e^{e^2} \left (-52 x^2-2 x^3-4 x^4\right )+x \log (x)} \, dx=\log \left (-4 \, x^{5} - 4 \, x^{4} - 4 \, x^{3} e^{\left (e^{2}\right )} - 105 \, x^{3} - 2 \, x^{2} e^{\left (e^{2}\right )} - 52 \, x^{2} - x e^{\left (2 \, e^{2}\right )} - 52 \, x e^{\left (e^{2}\right )} - 676 \, x + \log \left (x\right )\right ) \]
integrate((-x*exp(exp(2))^2+(-12*x^3-4*x^2-52*x)*exp(exp(2))-20*x^5-16*x^4 -315*x^3-104*x^2-676*x+1)/(x*log(x)-x^2*exp(exp(2))^2+(-4*x^4-2*x^3-52*x^2 )*exp(exp(2))-4*x^6-4*x^5-105*x^4-52*x^3-676*x^2),x, algorithm=\
log(-4*x^5 - 4*x^4 - 4*x^3*e^(e^2) - 105*x^3 - 2*x^2*e^(e^2) - 52*x^2 - x* e^(2*e^2) - 52*x*e^(e^2) - 676*x + log(x))
Timed out. \[ \int \frac {1-676 x-e^{2 e^2} x-104 x^2-315 x^3-16 x^4-20 x^5+e^{e^2} \left (-52 x-4 x^2-12 x^3\right )}{-676 x^2-e^{2 e^2} x^2-52 x^3-105 x^4-4 x^5-4 x^6+e^{e^2} \left (-52 x^2-2 x^3-4 x^4\right )+x \log (x)} \, dx=\int \frac {676\,x+x\,{\mathrm {e}}^{2\,{\mathrm {e}}^2}+{\mathrm {e}}^{{\mathrm {e}}^2}\,\left (12\,x^3+4\,x^2+52\,x\right )+104\,x^2+315\,x^3+16\,x^4+20\,x^5-1}{{\mathrm {e}}^{{\mathrm {e}}^2}\,\left (4\,x^4+2\,x^3+52\,x^2\right )+x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^2}-x\,\ln \left (x\right )+676\,x^2+52\,x^3+105\,x^4+4\,x^5+4\,x^6} \,d x \]
int((676*x + x*exp(2*exp(2)) + exp(exp(2))*(52*x + 4*x^2 + 12*x^3) + 104*x ^2 + 315*x^3 + 16*x^4 + 20*x^5 - 1)/(exp(exp(2))*(52*x^2 + 2*x^3 + 4*x^4) + x^2*exp(2*exp(2)) - x*log(x) + 676*x^2 + 52*x^3 + 105*x^4 + 4*x^5 + 4*x^ 6),x)