\(\int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+(-50 x-18 x^3+36 x^4-18 x^5) \log (x)+(18 x^2-18 x^3) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} (-x^4+3 x^5-3 x^6+x^7)+e^{2+2 x} (3 x^3-6 x^4+3 x^5) \log (x)+e^{2+2 x} (-3 x^2+3 x^3) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx\) [2833]

Optimal result

Integrand size = 173, antiderivative size = 28 \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=5+e^{-2-2 x} \left (3+\frac {25}{(x-(2-x) x+\log (x))^2}\right ) \] Output:

(3+25/(x-(2-x)*x+ln(x))^2)/exp(1+x)^2+5
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=-e^{-2 (1+x)} \left (-3-\frac {25}{((-1+x) x+\log (x))^2}\right ) \] Input:

Integrate[(-50 + 50*x - 50*x^2 - 50*x^3 + 6*x^4 - 18*x^5 + 18*x^6 - 6*x^7 
+ (-50*x - 18*x^3 + 36*x^4 - 18*x^5)*Log[x] + (18*x^2 - 18*x^3)*Log[x]^2 - 
 6*x*Log[x]^3)/(E^(2 + 2*x)*(-x^4 + 3*x^5 - 3*x^6 + x^7) + E^(2 + 2*x)*(3* 
x^3 - 6*x^4 + 3*x^5)*Log[x] + E^(2 + 2*x)*(-3*x^2 + 3*x^3)*Log[x]^2 + E^(2 
 + 2*x)*x*Log[x]^3),x]
 

Output:

-((-3 - 25/((-1 + x)*x + Log[x])^2)/E^(2*(1 + x)))
 

Rubi [F]

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.

\(\displaystyle \int \frac {-6 x^7+18 x^6-18 x^5+6 x^4-50 x^3-50 x^2+\left (18 x^2-18 x^3\right ) \log ^2(x)+\left (-18 x^5+36 x^4-18 x^3-50 x\right ) \log (x)+50 x-6 x \log ^3(x)-50}{e^{2 x+2} \left (3 x^3-3 x^2\right ) \log ^2(x)+e^{2 x+2} \left (3 x^5-6 x^4+3 x^3\right ) \log (x)+e^{2 x+2} \left (x^7-3 x^6+3 x^5-x^4\right )+e^{2 x+2} x \log ^3(x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 e^{-2 x-2} \left (-3 x^7+9 x^6-9 x^5+3 x^4-25 x^3-25 x^2-9 (x-1) x^2 \log ^2(x)-\left (9 x^4-18 x^3+9 x^2+25\right ) x \log (x)+25 x-3 x \log ^3(x)-25\right )}{x ((x-1) x+\log (x))^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {e^{-2 x-2} \left (3 x^7-9 x^6+9 x^5-3 x^4+25 x^3-9 (1-x) \log ^2(x) x^2+25 x^2+3 \log ^3(x) x+\left (9 x^4-18 x^3+9 x^2+25\right ) \log (x) x-25 x+25\right )}{x ((1-x) x-\log (x))^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (-\frac {25 e^{-2 x-2} \left (2 x^2-x+1\right )}{x \left (x^2-x+\log (x)\right )^3}-3 e^{-2 x-2}-\frac {25 e^{-2 x-2}}{\left (x^2-x+\log (x)\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (25 \int \frac {e^{-2 x-2}}{\left (x^2-x+\log (x)\right )^3}dx-25 \int \frac {e^{-2 x-2}}{x \left (x^2-x+\log (x)\right )^3}dx-50 \int \frac {e^{-2 x-2} x}{\left (x^2-x+\log (x)\right )^3}dx-25 \int \frac {e^{-2 x-2}}{\left (x^2-x+\log (x)\right )^2}dx+\frac {3}{2} e^{-2 x-2}\right )\)

Input:

Int[(-50 + 50*x - 50*x^2 - 50*x^3 + 6*x^4 - 18*x^5 + 18*x^6 - 6*x^7 + (-50 
*x - 18*x^3 + 36*x^4 - 18*x^5)*Log[x] + (18*x^2 - 18*x^3)*Log[x]^2 - 6*x*L 
og[x]^3)/(E^(2 + 2*x)*(-x^4 + 3*x^5 - 3*x^6 + x^7) + E^(2 + 2*x)*(3*x^3 - 
6*x^4 + 3*x^5)*Log[x] + E^(2 + 2*x)*(-3*x^2 + 3*x^3)*Log[x]^2 + E^(2 + 2*x 
)*x*Log[x]^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.74 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04

method	result	size
risch	\(3 \,{\mathrm e}^{-2-2 x}+\frac {25 \,{\mathrm e}^{-2-2 x}}{\left (x^{2}+\ln \left (x \right )-x \right )^{2}}\)	\(29\)
parallelrisch	\(\frac {\left (50+12 x^{2} \ln \left (x \right )-12 x \ln \left (x \right )+6 \ln \left (x \right )^{2}+6 x^{2}-12 x^{3}+6 x^{4}\right ) {\mathrm e}^{-2-2 x}}{2 x^{4}-4 x^{3}+4 x^{2} \ln \left (x \right )+2 x^{2}-4 x \ln \left (x \right )+2 \ln \left (x \right )^{2}}\)	\(74\)

Input:

int((-6*x*ln(x)^3+(-18*x^3+18*x^2)*ln(x)^2+(-18*x^5+36*x^4-18*x^3-50*x)*ln 
(x)-6*x^7+18*x^6-18*x^5+6*x^4-50*x^3-50*x^2+50*x-50)/(x*exp(1+x)^2*ln(x)^3 
+(3*x^3-3*x^2)*exp(1+x)^2*ln(x)^2+(3*x^5-6*x^4+3*x^3)*exp(1+x)^2*ln(x)+(x^ 
7-3*x^6+3*x^5-x^4)*exp(1+x)^2),x,method=_RETURNVERBOSE)
 

Output:

3*exp(-2-2*x)+25/(x^2+ln(x)-x)^2*exp(-2-2*x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (24) = 48\).

Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.04 \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=\frac {3 \, x^{4} - 6 \, x^{3} + 3 \, x^{2} + 6 \, {\left (x^{2} - x\right )} \log \left (x\right ) + 3 \, \log \left (x\right )^{2} + 25}{2 \, {\left (x^{2} - x\right )} e^{\left (2 \, x + 2\right )} \log \left (x\right ) + e^{\left (2 \, x + 2\right )} \log \left (x\right )^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x + 2\right )}} \] Input:

integrate((-6*x*log(x)^3+(-18*x^3+18*x^2)*log(x)^2+(-18*x^5+36*x^4-18*x^3- 
50*x)*log(x)-6*x^7+18*x^6-18*x^5+6*x^4-50*x^3-50*x^2+50*x-50)/(x*exp(1+x)^ 
2*log(x)^3+(3*x^3-3*x^2)*exp(1+x)^2*log(x)^2+(3*x^5-6*x^4+3*x^3)*exp(1+x)^ 
2*log(x)+(x^7-3*x^6+3*x^5-x^4)*exp(1+x)^2),x, algorithm="fricas")
 

Output:

(3*x^4 - 6*x^3 + 3*x^2 + 6*(x^2 - x)*log(x) + 3*log(x)^2 + 25)/(2*(x^2 - x 
)*e^(2*x + 2)*log(x) + e^(2*x + 2)*log(x)^2 + (x^4 - 2*x^3 + x^2)*e^(2*x + 
 2))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (24) = 48\).

Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.79 \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=\frac {\left (3 x^{4} - 6 x^{3} + 6 x^{2} \log {\left (x \right )} + 3 x^{2} - 6 x \log {\left (x \right )} + 3 \log {\left (x \right )}^{2} + 25\right ) e^{- 2 x - 2}}{x^{4} - 2 x^{3} + 2 x^{2} \log {\left (x \right )} + x^{2} - 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} \] Input:

integrate((-6*x*ln(x)**3+(-18*x**3+18*x**2)*ln(x)**2+(-18*x**5+36*x**4-18* 
x**3-50*x)*ln(x)-6*x**7+18*x**6-18*x**5+6*x**4-50*x**3-50*x**2+50*x-50)/(x 
*exp(1+x)**2*ln(x)**3+(3*x**3-3*x**2)*exp(1+x)**2*ln(x)**2+(3*x**5-6*x**4+ 
3*x**3)*exp(1+x)**2*ln(x)+(x**7-3*x**6+3*x**5-x**4)*exp(1+x)**2),x)
 

Output:

(3*x**4 - 6*x**3 + 6*x**2*log(x) + 3*x**2 - 6*x*log(x) + 3*log(x)**2 + 25) 
*exp(-2*x - 2)/(x**4 - 2*x**3 + 2*x**2*log(x) + x**2 - 2*x*log(x) + log(x) 
**2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (24) = 48\).

Time = 0.14 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=\frac {{\left (3 \, x^{4} - 6 \, x^{3} + 3 \, x^{2} + 6 \, {\left (x^{2} - x\right )} \log \left (x\right ) + 3 \, \log \left (x\right )^{2} + 25\right )} e^{\left (-2 \, x\right )}}{x^{4} e^{2} - 2 \, x^{3} e^{2} + x^{2} e^{2} + e^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{2} e^{2} - x e^{2}\right )} \log \left (x\right )} \] Input:

integrate((-6*x*log(x)^3+(-18*x^3+18*x^2)*log(x)^2+(-18*x^5+36*x^4-18*x^3- 
50*x)*log(x)-6*x^7+18*x^6-18*x^5+6*x^4-50*x^3-50*x^2+50*x-50)/(x*exp(1+x)^ 
2*log(x)^3+(3*x^3-3*x^2)*exp(1+x)^2*log(x)^2+(3*x^5-6*x^4+3*x^3)*exp(1+x)^ 
2*log(x)+(x^7-3*x^6+3*x^5-x^4)*exp(1+x)^2),x, algorithm="maxima")
 

Output:

(3*x^4 - 6*x^3 + 3*x^2 + 6*(x^2 - x)*log(x) + 3*log(x)^2 + 25)*e^(-2*x)/(x 
^4*e^2 - 2*x^3*e^2 + x^2*e^2 + e^2*log(x)^2 + 2*(x^2*e^2 - x*e^2)*log(x))
 

Giac [F]

\[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=\int { -\frac {2 \, {\left (3 \, x^{7} - 9 \, x^{6} + 9 \, x^{5} - 3 \, x^{4} + 3 \, x \log \left (x\right )^{3} + 25 \, x^{3} + 9 \, {\left (x^{3} - x^{2}\right )} \log \left (x\right )^{2} + 25 \, x^{2} + {\left (9 \, x^{5} - 18 \, x^{4} + 9 \, x^{3} + 25 \, x\right )} \log \left (x\right ) - 25 \, x + 25\right )}}{x e^{\left (2 \, x + 2\right )} \log \left (x\right )^{3} + 3 \, {\left (x^{3} - x^{2}\right )} e^{\left (2 \, x + 2\right )} \log \left (x\right )^{2} + 3 \, {\left (x^{5} - 2 \, x^{4} + x^{3}\right )} e^{\left (2 \, x + 2\right )} \log \left (x\right ) + {\left (x^{7} - 3 \, x^{6} + 3 \, x^{5} - x^{4}\right )} e^{\left (2 \, x + 2\right )}} \,d x } \] Input:

integrate((-6*x*log(x)^3+(-18*x^3+18*x^2)*log(x)^2+(-18*x^5+36*x^4-18*x^3- 
50*x)*log(x)-6*x^7+18*x^6-18*x^5+6*x^4-50*x^3-50*x^2+50*x-50)/(x*exp(1+x)^ 
2*log(x)^3+(3*x^3-3*x^2)*exp(1+x)^2*log(x)^2+(3*x^5-6*x^4+3*x^3)*exp(1+x)^ 
2*log(x)+(x^7-3*x^6+3*x^5-x^4)*exp(1+x)^2),x, algorithm="giac")
 

Output:

integrate(-2*(3*x^7 - 9*x^6 + 9*x^5 - 3*x^4 + 3*x*log(x)^3 + 25*x^3 + 9*(x 
^3 - x^2)*log(x)^2 + 25*x^2 + (9*x^5 - 18*x^4 + 9*x^3 + 25*x)*log(x) - 25* 
x + 25)/(x*e^(2*x + 2)*log(x)^3 + 3*(x^3 - x^2)*e^(2*x + 2)*log(x)^2 + 3*( 
x^5 - 2*x^4 + x^3)*e^(2*x + 2)*log(x) + (x^7 - 3*x^6 + 3*x^5 - x^4)*e^(2*x 
 + 2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=\int \frac {6\,x\,{\ln \left (x\right )}^3-50\,x+\ln \left (x\right )\,\left (18\,x^5-36\,x^4+18\,x^3+50\,x\right )-{\ln \left (x\right )}^2\,\left (18\,x^2-18\,x^3\right )+50\,x^2+50\,x^3-6\,x^4+18\,x^5-18\,x^6+6\,x^7+50}{-x\,{\mathrm {e}}^{2\,x+2}\,{\ln \left (x\right )}^3+{\mathrm {e}}^{2\,x+2}\,\left (3\,x^2-3\,x^3\right )\,{\ln \left (x\right )}^2-{\mathrm {e}}^{2\,x+2}\,\left (3\,x^5-6\,x^4+3\,x^3\right )\,\ln \left (x\right )+{\mathrm {e}}^{2\,x+2}\,\left (-x^7+3\,x^6-3\,x^5+x^4\right )} \,d x \] Input:

int((6*x*log(x)^3 - 50*x + log(x)*(50*x + 18*x^3 - 36*x^4 + 18*x^5) - log( 
x)^2*(18*x^2 - 18*x^3) + 50*x^2 + 50*x^3 - 6*x^4 + 18*x^5 - 18*x^6 + 6*x^7 
 + 50)/(exp(2*x + 2)*(x^4 - 3*x^5 + 3*x^6 - x^7) + exp(2*x + 2)*log(x)^2*( 
3*x^2 - 3*x^3) - x*exp(2*x + 2)*log(x)^3 - exp(2*x + 2)*log(x)*(3*x^3 - 6* 
x^4 + 3*x^5)),x)
 

Output:

int((6*x*log(x)^3 - 50*x + log(x)*(50*x + 18*x^3 - 36*x^4 + 18*x^5) - log( 
x)^2*(18*x^2 - 18*x^3) + 50*x^2 + 50*x^3 - 6*x^4 + 18*x^5 - 18*x^6 + 6*x^7 
 + 50)/(exp(2*x + 2)*(x^4 - 3*x^5 + 3*x^6 - x^7) + exp(2*x + 2)*log(x)^2*( 
3*x^2 - 3*x^3) - x*exp(2*x + 2)*log(x)^3 - exp(2*x + 2)*log(x)*(3*x^3 - 6* 
x^4 + 3*x^5)), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.71 \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=\frac {3 \mathrm {log}\left (x \right )^{2}+6 \,\mathrm {log}\left (x \right ) x^{2}-6 \,\mathrm {log}\left (x \right ) x +3 x^{4}-6 x^{3}+3 x^{2}+25}{e^{2 x} e^{2} \left (\mathrm {log}\left (x \right )^{2}+2 \,\mathrm {log}\left (x \right ) x^{2}-2 \,\mathrm {log}\left (x \right ) x +x^{4}-2 x^{3}+x^{2}\right )} \] Input:

int((-6*x*log(x)^3+(-18*x^3+18*x^2)*log(x)^2+(-18*x^5+36*x^4-18*x^3-50*x)* 
log(x)-6*x^7+18*x^6-18*x^5+6*x^4-50*x^3-50*x^2+50*x-50)/(x*exp(1+x)^2*log( 
x)^3+(3*x^3-3*x^2)*exp(1+x)^2*log(x)^2+(3*x^5-6*x^4+3*x^3)*exp(1+x)^2*log( 
x)+(x^7-3*x^6+3*x^5-x^4)*exp(1+x)^2),x)
 

Output:

(3*log(x)**2 + 6*log(x)*x**2 - 6*log(x)*x + 3*x**4 - 6*x**3 + 3*x**2 + 25) 
/(e**(2*x)*e**2*(log(x)**2 + 2*log(x)*x**2 - 2*log(x)*x + x**4 - 2*x**3 + 
x**2))