\(\int \frac {-5 x^2+10 x^4+e^{x^2} (75-24 x^2)+(5 x^2+e^{x^2} (75+24 x^2)) \log (\frac {e^{-x^2} (-5 x^2+e^{x^2} (-75-24 x^2))}{5 x})}{(5 x^2+e^{x^2} (75+24 x^2)) \log ^2(\frac {e^{-x^2} (-5 x^2+e^{x^2} (-75-24 x^2))}{5 x})} \, dx\) [2819]

Optimal result

Integrand size = 136, antiderivative size = 35 \[ \int \frac {-5 x^2+10 x^4+e^{x^2} \left (75-24 x^2\right )+\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log \left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )}{\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log ^2\left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )} \, dx=\frac {x}{\log \left (x \left (\frac {1}{5}-\frac {e^{-x^2} x+5 \left (\frac {3}{x}+x\right )}{x}\right )\right )} \] Output:

x/ln(x*(1/5-(x/exp(x^2)+5*x+15/x)/x))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \frac {-5 x^2+10 x^4+e^{x^2} \left (75-24 x^2\right )+\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log \left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )}{\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log ^2\left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )} \, dx=\frac {x}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \] Input:

Integrate[(-5*x^2 + 10*x^4 + E^x^2*(75 - 24*x^2) + (5*x^2 + E^x^2*(75 + 24 
*x^2))*Log[(-5*x^2 + E^x^2*(-75 - 24*x^2))/(5*E^x^2*x)])/((5*x^2 + E^x^2*( 
75 + 24*x^2))*Log[(-5*x^2 + E^x^2*(-75 - 24*x^2))/(5*E^x^2*x)]^2),x]
 

Output:

x/Log[-15/x + (-24/5 - E^(-x^2))*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.

\(\displaystyle \int \frac {10 x^4-5 x^2+e^{x^2} \left (75-24 x^2\right )+\left (5 x^2+e^{x^2} \left (24 x^2+75\right )\right ) \log \left (\frac {e^{-x^2} \left (e^{x^2} \left (-24 x^2-75\right )-5 x^2\right )}{5 x}\right )}{\left (5 x^2+e^{x^2} \left (24 x^2+75\right )\right ) \log ^2\left (\frac {e^{-x^2} \left (e^{x^2} \left (-24 x^2-75\right )-5 x^2\right )}{5 x}\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-8 x^2+8 x^2 \log \left (\left (-e^{-x^2}-\frac {24}{5}\right ) x-\frac {15}{x}\right )+25 \log \left (\left (-e^{-x^2}-\frac {24}{5}\right ) x-\frac {15}{x}\right )+25}{\left (8 x^2+25\right ) \log ^2\left (\left (-e^{-x^2}-\frac {24}{5}\right ) x-\frac {15}{x}\right )}+\frac {10 \left (8 x^4+25 x^2-25\right ) x^2}{\left (8 x^2+25\right ) \left (24 e^{x^2} x^2+5 x^2+75 e^{x^2}\right ) \log ^2\left (\left (-e^{-x^2}-\frac {24}{5}\right ) x-\frac {15}{x}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\int \frac {1}{\log ^2\left (\left (-\frac {24}{5}-e^{-x^2}\right ) x-\frac {15}{x}\right )}dx+5 i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \log ^2\left (\left (-\frac {24}{5}-e^{-x^2}\right ) x-\frac {15}{x}\right )}dx+5 i \int \frac {1}{\left (2 \sqrt {2} x+5 i\right ) \log ^2\left (\left (-\frac {24}{5}-e^{-x^2}\right ) x-\frac {15}{x}\right )}dx-\frac {125}{4} \int \frac {1}{\left (24 e^{x^2} x^2+5 x^2+75 e^{x^2}\right ) \log ^2\left (\left (-\frac {24}{5}-e^{-x^2}\right ) x-\frac {15}{x}\right )}dx+\frac {625}{8} i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \left (24 e^{x^2} x^2+5 x^2+75 e^{x^2}\right ) \log ^2\left (\left (-\frac {24}{5}-e^{-x^2}\right ) x-\frac {15}{x}\right )}dx+\frac {625}{8} i \int \frac {1}{\left (2 \sqrt {2} x+5 i\right ) \left (24 e^{x^2} x^2+5 x^2+75 e^{x^2}\right ) \log ^2\left (\left (-\frac {24}{5}-e^{-x^2}\right ) x-\frac {15}{x}\right )}dx+\int \frac {1}{\log \left (\left (-\frac {24}{5}-e^{-x^2}\right ) x-\frac {15}{x}\right )}dx+10 \int \frac {x^4}{\left (24 e^{x^2} x^2+5 x^2+75 e^{x^2}\right ) \log ^2\left (\left (-\frac {24}{5}-e^{-x^2}\right ) x-\frac {15}{x}\right )}dx\)

Input:

Int[(-5*x^2 + 10*x^4 + E^x^2*(75 - 24*x^2) + (5*x^2 + E^x^2*(75 + 24*x^2)) 
*Log[(-5*x^2 + E^x^2*(-75 - 24*x^2))/(5*E^x^2*x)])/((5*x^2 + E^x^2*(75 + 2 
4*x^2))*Log[(-5*x^2 + E^x^2*(-75 - 24*x^2))/(5*E^x^2*x)]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\frac {x}{\ln \left (\frac {\left (\left (-24 x^{2}-75\right ) {\mathrm e}^{x^{2}}-5 x^{2}\right ) {\mathrm e}^{-x^{2}}}{5 x}\right )}\)	\(35\)
risch	\(\frac {2 i x}{\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )}^{2}+2 \pi {\operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )-\pi \,\operatorname {csgn}\left (i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )^{2}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )^{2}+\pi \operatorname {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )^{3}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )}^{2}-\pi {\operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )}^{3}-2 \pi -2 i \ln \left (x \right )+2 i \ln \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )-2 i \ln \left (5\right )+6 i \ln \left (2\right )+2 i \ln \left (3\right )-2 i \ln \left ({\mathrm e}^{x^{2}}\right )}\)	\(486\)

Input:

int((((24*x^2+75)*exp(x^2)+5*x^2)*ln(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp 
(x^2)/x)+(-24*x^2+75)*exp(x^2)+10*x^4-5*x^2)/((24*x^2+75)*exp(x^2)+5*x^2)/ 
ln(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)^2,x,method=_RETURNVERBOSE 
)
 

Output:

x/ln(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {-5 x^2+10 x^4+e^{x^2} \left (75-24 x^2\right )+\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log \left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )}{\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log ^2\left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )} \, dx=\frac {x}{\log \left (-\frac {{\left (5 \, x^{2} + 3 \, {\left (8 \, x^{2} + 25\right )} e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}}{5 \, x}\right )} \] Input:

integrate((((24*x^2+75)*exp(x^2)+5*x^2)*log(1/5*((-24*x^2-75)*exp(x^2)-5*x 
^2)/exp(x^2)/x)+(-24*x^2+75)*exp(x^2)+10*x^4-5*x^2)/((24*x^2+75)*exp(x^2)+ 
5*x^2)/log(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)^2,x, algorithm="f 
ricas")
 

Output:

x/log(-1/5*(5*x^2 + 3*(8*x^2 + 25)*e^(x^2))*e^(-x^2)/x)
 

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {-5 x^2+10 x^4+e^{x^2} \left (75-24 x^2\right )+\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log \left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )}{\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log ^2\left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )} \, dx=\frac {x}{\log {\left (\frac {\left (- x^{2} + \frac {\left (- 24 x^{2} - 75\right ) e^{x^{2}}}{5}\right ) e^{- x^{2}}}{x} \right )}} \] Input:

integrate((((24*x**2+75)*exp(x**2)+5*x**2)*ln(1/5*((-24*x**2-75)*exp(x**2) 
-5*x**2)/exp(x**2)/x)+(-24*x**2+75)*exp(x**2)+10*x**4-5*x**2)/((24*x**2+75 
)*exp(x**2)+5*x**2)/ln(1/5*((-24*x**2-75)*exp(x**2)-5*x**2)/exp(x**2)/x)** 
2,x)
 

Output:

x/log((-x**2 + (-24*x**2 - 75)*exp(x**2)/5)*exp(-x**2)/x)
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {-5 x^2+10 x^4+e^{x^2} \left (75-24 x^2\right )+\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log \left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )}{\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log ^2\left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )} \, dx=-\frac {x}{x^{2} + \log \left (5\right ) - \log \left (-5 \, x^{2} - 3 \, {\left (8 \, x^{2} + 25\right )} e^{\left (x^{2}\right )}\right ) + \log \left (x\right )} \] Input:

integrate((((24*x^2+75)*exp(x^2)+5*x^2)*log(1/5*((-24*x^2-75)*exp(x^2)-5*x 
^2)/exp(x^2)/x)+(-24*x^2+75)*exp(x^2)+10*x^4-5*x^2)/((24*x^2+75)*exp(x^2)+ 
5*x^2)/log(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)^2,x, algorithm="m 
axima")
 

Output:

-x/(x^2 + log(5) - log(-5*x^2 - 3*(8*x^2 + 25)*e^(x^2)) + log(x))
 

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {-5 x^2+10 x^4+e^{x^2} \left (75-24 x^2\right )+\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log \left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )}{\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log ^2\left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )} \, dx=\frac {x}{\log \left (-\frac {{\left (24 \, x^{2} e^{\left (x^{2}\right )} + 5 \, x^{2} + 75 \, e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}}{5 \, x}\right )} \] Input:

integrate((((24*x^2+75)*exp(x^2)+5*x^2)*log(1/5*((-24*x^2-75)*exp(x^2)-5*x 
^2)/exp(x^2)/x)+(-24*x^2+75)*exp(x^2)+10*x^4-5*x^2)/((24*x^2+75)*exp(x^2)+ 
5*x^2)/log(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)^2,x, algorithm="g 
iac")
 

Output:

x/log(-1/5*(24*x^2*e^(x^2) + 5*x^2 + 75*e^(x^2))*e^(-x^2)/x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-5 x^2+10 x^4+e^{x^2} \left (75-24 x^2\right )+\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log \left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )}{\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log ^2\left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )} \, dx=-\int \frac {{\mathrm {e}}^{x^2}\,\left (24\,x^2-75\right )-\ln \left (-\frac {{\mathrm {e}}^{-x^2}\,\left (\frac {{\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )}{5}+x^2\right )}{x}\right )\,\left ({\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )+5\,x^2\right )+5\,x^2-10\,x^4}{{\ln \left (-\frac {{\mathrm {e}}^{-x^2}\,\left (\frac {{\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )}{5}+x^2\right )}{x}\right )}^2\,\left ({\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )+5\,x^2\right )} \,d x \] Input:

int(-(exp(x^2)*(24*x^2 - 75) - log(-(exp(-x^2)*((exp(x^2)*(24*x^2 + 75))/5 
 + x^2))/x)*(exp(x^2)*(24*x^2 + 75) + 5*x^2) + 5*x^2 - 10*x^4)/(log(-(exp( 
-x^2)*((exp(x^2)*(24*x^2 + 75))/5 + x^2))/x)^2*(exp(x^2)*(24*x^2 + 75) + 5 
*x^2)),x)
 

Output:

-int((exp(x^2)*(24*x^2 - 75) - log(-(exp(-x^2)*((exp(x^2)*(24*x^2 + 75))/5 
 + x^2))/x)*(exp(x^2)*(24*x^2 + 75) + 5*x^2) + 5*x^2 - 10*x^4)/(log(-(exp( 
-x^2)*((exp(x^2)*(24*x^2 + 75))/5 + x^2))/x)^2*(exp(x^2)*(24*x^2 + 75) + 5 
*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.14 \[ \int \frac {-5 x^2+10 x^4+e^{x^2} \left (75-24 x^2\right )+\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log \left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )}{\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log ^2\left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )} \, dx=\frac {x}{\mathrm {log}\left (\frac {-24 e^{x^{2}} x^{2}-75 e^{x^{2}}-5 x^{2}}{5 e^{x^{2}} x}\right )} \] Input:

int((((24*x^2+75)*exp(x^2)+5*x^2)*log(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/ex 
p(x^2)/x)+(-24*x^2+75)*exp(x^2)+10*x^4-5*x^2)/((24*x^2+75)*exp(x^2)+5*x^2) 
/log(1/5*((-24*x^2-75)*exp(x^2)-5*x^2)/exp(x^2)/x)^2,x)
 

Output:

x/log(( - 24*e**(x**2)*x**2 - 75*e**(x**2) - 5*x**2)/(5*e**(x**2)*x))