\(\int \frac {150-x+e^{4-3 x^2+x^4} (-150+x+900 x^2-600 x^4)+(-50+e^{4-3 x^2+x^4} (50-300 x^2+200 x^4)) \log (-x+e^{4-3 x^2+x^4} x)}{-x+e^{4-3 x^2+x^4} x} \, dx\) [2023]

Optimal result

Integrand size = 99, antiderivative size = 30 \[ \int \frac {150-x+e^{4-3 x^2+x^4} \left (-150+x+900 x^2-600 x^4\right )+\left (-50+e^{4-3 x^2+x^4} \left (50-300 x^2+200 x^4\right )\right ) \log \left (-x+e^{4-3 x^2+x^4} x\right )}{-x+e^{4-3 x^2+x^4} x} \, dx=x+25 \left (-3+\log \left (-x+e^{x^2+\left (2-x^2\right )^2} x\right )\right )^2 \] Output:

x+5*(ln(exp(x^2+(-x^2+2)^2)*x-x)-3)*(5*ln(exp(x^2+(-x^2+2)^2)*x-x)-15)
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(174\) vs. \(2(30)=60\).

Time = 0.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 5.80 \[ \int \frac {150-x+e^{4-3 x^2+x^4} \left (-150+x+900 x^2-600 x^4\right )+\left (-50+e^{4-3 x^2+x^4} \left (50-300 x^2+200 x^4\right )\right ) \log \left (-x+e^{4-3 x^2+x^4} x\right )}{-x+e^{4-3 x^2+x^4} x} \, dx=x+450 x^2-225 x^4+25 \log ^2\left (\left (e^{3 x^2}-e^{4+x^4}\right ) x\right )-150 x^2 \log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )+50 \log \left (e^{3 x^2}-e^{4+x^4}\right ) \left (-3+3 x^2-\log \left (\left (e^{3 x^2}-e^{4+x^4}\right ) x\right )+\log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )\right )+50 \log (x) \left (-3+3 x^2-\log \left (\left (e^{3 x^2}-e^{4+x^4}\right ) x\right )+\log \left (\left (-1+e^{4-3 x^2+x^4}\right ) x\right )\right ) \] Input:

Integrate[(150 - x + E^(4 - 3*x^2 + x^4)*(-150 + x + 900*x^2 - 600*x^4) + 
(-50 + E^(4 - 3*x^2 + x^4)*(50 - 300*x^2 + 200*x^4))*Log[-x + E^(4 - 3*x^2 
 + x^4)*x])/(-x + E^(4 - 3*x^2 + x^4)*x),x]
 

Output:

x + 450*x^2 - 225*x^4 + 25*Log[(E^(3*x^2) - E^(4 + x^4))*x]^2 - 150*x^2*Lo 
g[(-1 + E^(4 - 3*x^2 + x^4))*x] + 50*Log[E^(3*x^2) - E^(4 + x^4)]*(-3 + 3* 
x^2 - Log[(E^(3*x^2) - E^(4 + x^4))*x] + Log[(-1 + E^(4 - 3*x^2 + x^4))*x] 
) + 50*Log[x]*(-3 + 3*x^2 - Log[(E^(3*x^2) - E^(4 + x^4))*x] + Log[(-1 + E 
^(4 - 3*x^2 + x^4))*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.

\(\displaystyle \int \frac {e^{x^4-3 x^2+4} \left (-600 x^4+900 x^2+x-150\right )+\left (e^{x^4-3 x^2+4} \left (200 x^4-300 x^2+50\right )-50\right ) \log \left (e^{x^4-3 x^2+4} x-x\right )-x+150}{e^{x^4-3 x^2+4} x-x} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-600 x^4+900 x^2+200 x^4 \log \left (\left (e^{x^4-3 x^2+4}-1\right ) x\right )-300 x^2 \log \left (\left (e^{x^4-3 x^2+4}-1\right ) x\right )+50 \log \left (\left (e^{x^4-3 x^2+4}-1\right ) x\right )+x-150}{x}-\frac {100 e^{3 x^2} x \left (2 x^2-3\right ) \left (\log \left (\left (e^{x^4-3 x^2+4}-1\right ) x\right )-3\right )}{e^{3 x^2}-e^{x^4+4}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -25 x^8+150 x^6+50 \log \left (e^{x^4-3 x^2+4} x-x\right ) x^4-\frac {775 x^4}{2}-150 \log \left (e^{x^4-3 x^2+4} x-x\right ) x^2+525 x^2+x-150 \log (x)+150 \log \left (e^{x^4-3 x^2+4} x-x\right ) \text {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )-450 \text {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )-100 \log \left (e^{x^4-3 x^2+4} x-x\right ) \text {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )+750 \text {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )-450 \text {Subst}\left (\int \frac {e^{3 x} x^2}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )+100 \text {Subst}\left (\int \frac {e^{3 x} x^3}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )+50 \int \frac {\log \left (e^{x^4-3 x^2+4} x-x\right )}{x}dx-150 \int \frac {\text {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )}{x}dx+900 \int x \text {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )dx-900 \int \frac {e^{3 x^2} x \text {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )}{e^{3 x^2}-e^{x^4+4}}dx-600 \int x^3 \text {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )dx+600 \int \frac {e^{3 x^2} x^3 \text {Subst}\left (\int \frac {e^{3 x}}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )}{e^{3 x^2}-e^{x^4+4}}dx+100 \int \frac {\text {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )}{x}dx-600 \int x \text {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )dx+600 \int \frac {e^{3 x^2} x \text {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )}{e^{3 x^2}-e^{x^4+4}}dx+400 \int x^3 \text {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )dx-400 \int \frac {e^{3 x^2} x^3 \text {Subst}\left (\int \frac {e^{3 x} x}{e^{3 x}-e^{x^2+4}}dx,x,x^2\right )}{e^{3 x^2}-e^{x^4+4}}dx\)

Input:

Int[(150 - x + E^(4 - 3*x^2 + x^4)*(-150 + x + 900*x^2 - 600*x^4) + (-50 + 
 E^(4 - 3*x^2 + x^4)*(50 - 300*x^2 + 200*x^4))*Log[-x + E^(4 - 3*x^2 + x^4 
)*x])/(-x + E^(4 - 3*x^2 + x^4)*x),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37

method	result	size
parallelrisch	\(25 {\ln \left (x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )}^{2}+x -150 \ln \left (x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )\)	\(41\)
norman	\(x -150 \ln \left (x \,{\mathrm e}^{x^{4}-3 x^{2}+4}-x \right )+25 \ln \left (x \,{\mathrm e}^{x^{4}-3 x^{2}+4}-x \right )^{2}\)	\(45\)
risch	\(25 \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )^{2}+\left (-\frac {175}{2}+50 \ln \left (x \right )\right ) \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )+25 \ln \left (x \right )^{2}+250+x -\frac {125 \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )}{2}-150 \ln \left (x \right )+100 i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \operatorname {csgn}\left (i x \right )+25 i \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right ) \pi {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )}^{2} \operatorname {csgn}\left (i x \right )-100 i \pi {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )}^{2} \operatorname {csgn}\left (i x \right )+25 i \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right ) \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )}^{2}-100 i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )}^{2}-25 i \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right ) \pi {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )}^{3}-25 i \ln \left (x \right ) \pi {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )}^{3}-25 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \operatorname {csgn}\left (i x \right )+25 i \ln \left (x \right ) \pi {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )}^{2} \operatorname {csgn}\left (i x \right )+100 i \pi {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )}^{3}+25 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right )}^{2}-25 i \ln \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right ) \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \operatorname {csgn}\left (i x \left ({\mathrm e}^{x^{4}-3 x^{2}+4}-1\right )\right ) \operatorname {csgn}\left (i x \right )\)	\(547\)

Input:

int((((200*x^4-300*x^2+50)*exp(x^4-3*x^2+4)-50)*ln(x*exp(x^4-3*x^2+4)-x)+( 
-600*x^4+900*x^2+x-150)*exp(x^4-3*x^2+4)-x+150)/(x*exp(x^4-3*x^2+4)-x),x,m 
ethod=_RETURNVERBOSE)
 

Output:

25*ln(x*(exp(x^4-3*x^2+4)-1))^2+x-150*ln(x*(exp(x^4-3*x^2+4)-1))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {150-x+e^{4-3 x^2+x^4} \left (-150+x+900 x^2-600 x^4\right )+\left (-50+e^{4-3 x^2+x^4} \left (50-300 x^2+200 x^4\right )\right ) \log \left (-x+e^{4-3 x^2+x^4} x\right )}{-x+e^{4-3 x^2+x^4} x} \, dx=25 \, \log \left (x e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - x\right )^{2} + x - 150 \, \log \left (x e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - x\right ) \] Input:

integrate((((200*x^4-300*x^2+50)*exp(x^4-3*x^2+4)-50)*log(x*exp(x^4-3*x^2+ 
4)-x)+(-600*x^4+900*x^2+x-150)*exp(x^4-3*x^2+4)-x+150)/(x*exp(x^4-3*x^2+4) 
-x),x, algorithm="fricas")
 

Output:

25*log(x*e^(x^4 - 3*x^2 + 4) - x)^2 + x - 150*log(x*e^(x^4 - 3*x^2 + 4) - 
x)
 

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {150-x+e^{4-3 x^2+x^4} \left (-150+x+900 x^2-600 x^4\right )+\left (-50+e^{4-3 x^2+x^4} \left (50-300 x^2+200 x^4\right )\right ) \log \left (-x+e^{4-3 x^2+x^4} x\right )}{-x+e^{4-3 x^2+x^4} x} \, dx=x - 150 \log {\left (x \right )} + 25 \log {\left (x e^{x^{4} - 3 x^{2} + 4} - x \right )}^{2} - 150 \log {\left (e^{x^{4} - 3 x^{2} + 4} - 1 \right )} \] Input:

integrate((((200*x**4-300*x**2+50)*exp(x**4-3*x**2+4)-50)*ln(x*exp(x**4-3* 
x**2+4)-x)+(-600*x**4+900*x**2+x-150)*exp(x**4-3*x**2+4)-x+150)/(x*exp(x** 
4-3*x**2+4)-x),x)
 

Output:

x - 150*log(x) + 25*log(x*exp(x**4 - 3*x**2 + 4) - x)**2 - 150*log(exp(x** 
4 - 3*x**2 + 4) - 1)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (27) = 54\).

Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.53 \[ \int \frac {150-x+e^{4-3 x^2+x^4} \left (-150+x+900 x^2-600 x^4\right )+\left (-50+e^{4-3 x^2+x^4} \left (50-300 x^2+200 x^4\right )\right ) \log \left (-x+e^{4-3 x^2+x^4} x\right )}{-x+e^{4-3 x^2+x^4} x} \, dx=225 \, x^{4} + 450 \, x^{2} - 150 \, {\left (x^{2} + 1\right )} \log \left (x\right ) + 25 \, \log \left (x\right )^{2} - 50 \, {\left (3 \, x^{2} - \log \left (x\right ) + 3\right )} \log \left (e^{\left (x^{4} + 4\right )} - e^{\left (3 \, x^{2}\right )}\right ) + 25 \, \log \left (e^{\left (x^{4} + 4\right )} - e^{\left (3 \, x^{2}\right )}\right )^{2} + x \] Input:

integrate((((200*x^4-300*x^2+50)*exp(x^4-3*x^2+4)-50)*log(x*exp(x^4-3*x^2+ 
4)-x)+(-600*x^4+900*x^2+x-150)*exp(x^4-3*x^2+4)-x+150)/(x*exp(x^4-3*x^2+4) 
-x),x, algorithm="maxima")
 

Output:

225*x^4 + 450*x^2 - 150*(x^2 + 1)*log(x) + 25*log(x)^2 - 50*(3*x^2 - log(x 
) + 3)*log(e^(x^4 + 4) - e^(3*x^2)) + 25*log(e^(x^4 + 4) - e^(3*x^2))^2 + 
x
 

Giac [F]

\[ \int \frac {150-x+e^{4-3 x^2+x^4} \left (-150+x+900 x^2-600 x^4\right )+\left (-50+e^{4-3 x^2+x^4} \left (50-300 x^2+200 x^4\right )\right ) \log \left (-x+e^{4-3 x^2+x^4} x\right )}{-x+e^{4-3 x^2+x^4} x} \, dx=\int { -\frac {{\left (600 \, x^{4} - 900 \, x^{2} - x + 150\right )} e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - 50 \, {\left ({\left (4 \, x^{4} - 6 \, x^{2} + 1\right )} e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - 1\right )} \log \left (x e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - x\right ) + x - 150}{x e^{\left (x^{4} - 3 \, x^{2} + 4\right )} - x} \,d x } \] Input:

integrate((((200*x^4-300*x^2+50)*exp(x^4-3*x^2+4)-50)*log(x*exp(x^4-3*x^2+ 
4)-x)+(-600*x^4+900*x^2+x-150)*exp(x^4-3*x^2+4)-x+150)/(x*exp(x^4-3*x^2+4) 
-x),x, algorithm="giac")
 

Output:

integrate(-((600*x^4 - 900*x^2 - x + 150)*e^(x^4 - 3*x^2 + 4) - 50*((4*x^4 
 - 6*x^2 + 1)*e^(x^4 - 3*x^2 + 4) - 1)*log(x*e^(x^4 - 3*x^2 + 4) - x) + x 
- 150)/(x*e^(x^4 - 3*x^2 + 4) - x), x)
 

Mupad [B] (verification not implemented)

Time = 1.91 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {150-x+e^{4-3 x^2+x^4} \left (-150+x+900 x^2-600 x^4\right )+\left (-50+e^{4-3 x^2+x^4} \left (50-300 x^2+200 x^4\right )\right ) \log \left (-x+e^{4-3 x^2+x^4} x\right )}{-x+e^{4-3 x^2+x^4} x} \, dx=25\,{\ln \left (x\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-3\,x^2}-x\right )}^2+x-150\,\ln \left ({\mathrm {e}}^{x^4}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-3\,x^2}-1\right )-150\,\ln \left (x\right ) \] Input:

int(-(log(x*exp(x^4 - 3*x^2 + 4) - x)*(exp(x^4 - 3*x^2 + 4)*(200*x^4 - 300 
*x^2 + 50) - 50) - x + exp(x^4 - 3*x^2 + 4)*(x + 900*x^2 - 600*x^4 - 150) 
+ 150)/(x - x*exp(x^4 - 3*x^2 + 4)),x)
 

Output:

x - 150*log(exp(x^4)*exp(4)*exp(-3*x^2) - 1) - 150*log(x) + 25*log(x*exp(x 
^4)*exp(4)*exp(-3*x^2) - x)^2
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.40 \[ \int \frac {150-x+e^{4-3 x^2+x^4} \left (-150+x+900 x^2-600 x^4\right )+\left (-50+e^{4-3 x^2+x^4} \left (50-300 x^2+200 x^4\right )\right ) \log \left (-x+e^{4-3 x^2+x^4} x\right )}{-x+e^{4-3 x^2+x^4} x} \, dx=25 \mathrm {log}\left (\frac {e^{x^{4}} e^{4} x -e^{3 x^{2}} x}{e^{3 x^{2}}}\right )^{2}-150 \,\mathrm {log}\left (\frac {e^{x^{4}} e^{4} x -e^{3 x^{2}} x}{e^{3 x^{2}}}\right )+x \] Input:

int((((200*x^4-300*x^2+50)*exp(x^4-3*x^2+4)-50)*log(x*exp(x^4-3*x^2+4)-x)+ 
(-600*x^4+900*x^2+x-150)*exp(x^4-3*x^2+4)-x+150)/(x*exp(x^4-3*x^2+4)-x),x)
 

Output:

25*log((e**(x**4)*e**4*x - e**(3*x**2)*x)/e**(3*x**2))**2 - 150*log((e**(x 
**4)*e**4*x - e**(3*x**2)*x)/e**(3*x**2)) + x