\(\int \frac {e^{2+3 x} (1+\log (x^2))^3 (30+15 x+(15+15 x) \log (x^2)+15 \log ^2(x^2))}{(-4-4 x-4 \log (x^2))^3 (1+x+(2+x) \log (x^2)+\log ^2(x^2))} \, dx\) [408]

Optimal result

Integrand size = 71, antiderivative size = 29 \[ \int \frac {e^{2+3 x} \left (1+\log \left (x^2\right )\right )^3 \left (30+15 x+(15+15 x) \log \left (x^2\right )+15 \log ^2\left (x^2\right )\right )}{\left (-4-4 x-4 \log \left (x^2\right )\right )^3 \left (1+x+(2+x) \log \left (x^2\right )+\log ^2\left (x^2\right )\right )} \, dx=5 e^{2+3 \left (x-\log \left (4 \left (-1+\frac {x}{-1-\log \left (x^2\right )}\right )\right )\right )} \] Output:

5*exp(3*x-3*ln(4*x/(-1-ln(x^2))-4)+2)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {e^{2+3 x} \left (1+\log \left (x^2\right )\right )^3 \left (30+15 x+(15+15 x) \log \left (x^2\right )+15 \log ^2\left (x^2\right )\right )}{\left (-4-4 x-4 \log \left (x^2\right )\right )^3 \left (1+x+(2+x) \log \left (x^2\right )+\log ^2\left (x^2\right )\right )} \, dx=-\frac {5 e^{2+3 x} \left (1+\log \left (x^2\right )\right )^3}{64 \left (1+x+\log \left (x^2\right )\right )^3} \] Input:

Integrate[(E^(2 + 3*x)*(1 + Log[x^2])^3*(30 + 15*x + (15 + 15*x)*Log[x^2] 
+ 15*Log[x^2]^2))/((-4 - 4*x - 4*Log[x^2])^3*(1 + x + (2 + x)*Log[x^2] + L 
og[x^2]^2)),x]
 

Output:

(-5*E^(2 + 3*x)*(1 + Log[x^2])^3)/(64*(1 + x + Log[x^2])^3)
 

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.

\(\displaystyle \int \frac {e^{3 x+2} \left (\log \left (x^2\right )+1\right )^3 \left (15 \log ^2\left (x^2\right )+(15 x+15) \log \left (x^2\right )+15 x+30\right )}{\left (-4 \log \left (x^2\right )-4 x-4\right )^3 \left (\log ^2\left (x^2\right )+(x+2) \log \left (x^2\right )+x+1\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {15 e^{3 x+2} \left (\log \left (x^2\right )+1\right )^2 \left (-\log ^2\left (x^2\right )-x \log \left (x^2\right )-\log \left (x^2\right )-x-2\right )}{64 \left (\log \left (x^2\right )+x+1\right )^4}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {15}{64} \int -\frac {e^{3 x+2} \left (\log \left (x^2\right )+1\right )^2 \left (\log ^2\left (x^2\right )+x \log \left (x^2\right )+\log \left (x^2\right )+x+2\right )}{\left (x+\log \left (x^2\right )+1\right )^4}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {15}{64} \int \frac {e^{3 x+2} \left (\log \left (x^2\right )+1\right )^2 \left (\log ^2\left (x^2\right )+x \log \left (x^2\right )+\log \left (x^2\right )+x+2\right )}{\left (x+\log \left (x^2\right )+1\right )^4}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {15}{64} \int \left (\frac {e^{3 x+2} (x+2) x^2}{\left (x+\log \left (x^2\right )+1\right )^4}-\frac {e^{3 x+2} \left (x^2+3 x+4\right ) x}{\left (x+\log \left (x^2\right )+1\right )^3}+e^{3 x+2}+\frac {e^{3 x+2} (-3 x-1)}{x+\log \left (x^2\right )+1}+\frac {e^{3 x+2} \left (3 x^2+3 x+2\right )}{\left (x+\log \left (x^2\right )+1\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {15}{64} \left (\int \frac {e^{3 x+2}}{-x-\log \left (x^2\right )-1}dx+2 \int \frac {e^{3 x+2} x^2}{\left (x+\log \left (x^2\right )+1\right )^4}dx-4 \int \frac {e^{3 x+2} x}{\left (x+\log \left (x^2\right )+1\right )^3}dx-3 \int \frac {e^{3 x+2} x^2}{\left (x+\log \left (x^2\right )+1\right )^3}dx+2 \int \frac {e^{3 x+2}}{\left (x+\log \left (x^2\right )+1\right )^2}dx+3 \int \frac {e^{3 x+2} x}{\left (x+\log \left (x^2\right )+1\right )^2}dx+3 \int \frac {e^{3 x+2} x^2}{\left (x+\log \left (x^2\right )+1\right )^2}dx-3 \int \frac {e^{3 x+2} x}{x+\log \left (x^2\right )+1}dx+\int \frac {e^{3 x+2} x^3}{\left (x+\log \left (x^2\right )+1\right )^4}dx-\int \frac {e^{3 x+2} x^3}{\left (x+\log \left (x^2\right )+1\right )^3}dx+\frac {1}{3} e^{3 x+2}\right )\)

Input:

Int[(E^(2 + 3*x)*(1 + Log[x^2])^3*(30 + 15*x + (15 + 15*x)*Log[x^2] + 15*L 
og[x^2]^2))/((-4 - 4*x - 4*Log[x^2])^3*(1 + x + (2 + x)*Log[x^2] + Log[x^2 
]^2)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(5 \,{\mathrm e}^{-3 \ln \left (-\frac {4 \left (\ln \left (x^{2}\right )+x +1\right )}{\ln \left (x^{2}\right )+1}\right )+3 x +2}\)	\(29\)

Input:

int((15*ln(x^2)^2+(15*x+15)*ln(x^2)+15*x+30)*exp(-3*ln((-4*ln(x^2)-4*x-4)/ 
(ln(x^2)+1))+3*x+2)/(ln(x^2)^2+(2+x)*ln(x^2)+x+1),x,method=_RETURNVERBOSE)
 

Output:

5*exp(-3*ln(-4*(ln(x^2)+x+1)/(ln(x^2)+1))+3*x+2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {e^{2+3 x} \left (1+\log \left (x^2\right )\right )^3 \left (30+15 x+(15+15 x) \log \left (x^2\right )+15 \log ^2\left (x^2\right )\right )}{\left (-4-4 x-4 \log \left (x^2\right )\right )^3 \left (1+x+(2+x) \log \left (x^2\right )+\log ^2\left (x^2\right )\right )} \, dx=5 \, e^{\left (3 \, x - 3 \, \log \left (-\frac {4 \, {\left (x + \log \left (x^{2}\right ) + 1\right )}}{\log \left (x^{2}\right ) + 1}\right ) + 2\right )} \] Input:

integrate((15*log(x^2)^2+(15*x+15)*log(x^2)+15*x+30)*exp(-3*log((-4*log(x^ 
2)-4*x-4)/(log(x^2)+1))+3*x+2)/(log(x^2)^2+(2+x)*log(x^2)+x+1),x, algorith 
m="fricas")
 

Output:

5*e^(3*x - 3*log(-4*(x + log(x^2) + 1)/(log(x^2) + 1)) + 2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (22) = 44\).

Time = 0.59 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.45 \[ \int \frac {e^{2+3 x} \left (1+\log \left (x^2\right )\right )^3 \left (30+15 x+(15+15 x) \log \left (x^2\right )+15 \log ^2\left (x^2\right )\right )}{\left (-4-4 x-4 \log \left (x^2\right )\right )^3 \left (1+x+(2+x) \log \left (x^2\right )+\log ^2\left (x^2\right )\right )} \, dx=\frac {\left (- 5 \log {\left (x^{2} \right )}^{3} - 15 \log {\left (x^{2} \right )}^{2} - 15 \log {\left (x^{2} \right )} - 5\right ) e^{3 x + 2}}{64 x^{3} + 192 x^{2} \log {\left (x^{2} \right )} + 192 x^{2} + 192 x \log {\left (x^{2} \right )}^{2} + 384 x \log {\left (x^{2} \right )} + 192 x + 64 \log {\left (x^{2} \right )}^{3} + 192 \log {\left (x^{2} \right )}^{2} + 192 \log {\left (x^{2} \right )} + 64} \] Input:

integrate((15*ln(x**2)**2+(15*x+15)*ln(x**2)+15*x+30)*exp(-3*ln((-4*ln(x** 
2)-4*x-4)/(ln(x**2)+1))+3*x+2)/(ln(x**2)**2+(2+x)*ln(x**2)+x+1),x)
 

Output:

(-5*log(x**2)**3 - 15*log(x**2)**2 - 15*log(x**2) - 5)*exp(3*x + 2)/(64*x* 
*3 + 192*x**2*log(x**2) + 192*x**2 + 192*x*log(x**2)**2 + 384*x*log(x**2) 
+ 192*x + 64*log(x**2)**3 + 192*log(x**2)**2 + 192*log(x**2) + 64)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (24) = 48\).

Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.52 \[ \int \frac {e^{2+3 x} \left (1+\log \left (x^2\right )\right )^3 \left (30+15 x+(15+15 x) \log \left (x^2\right )+15 \log ^2\left (x^2\right )\right )}{\left (-4-4 x-4 \log \left (x^2\right )\right )^3 \left (1+x+(2+x) \log \left (x^2\right )+\log ^2\left (x^2\right )\right )} \, dx=-\frac {5 \, {\left (8 \, e^{2} \log \left (x\right )^{3} + 12 \, e^{2} \log \left (x\right )^{2} + 6 \, e^{2} \log \left (x\right ) + e^{2}\right )} e^{\left (3 \, x\right )}}{64 \, {\left (x^{3} + 12 \, {\left (x + 1\right )} \log \left (x\right )^{2} + 8 \, \log \left (x\right )^{3} + 3 \, x^{2} + 6 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right ) + 3 \, x + 1\right )}} \] Input:

integrate((15*log(x^2)^2+(15*x+15)*log(x^2)+15*x+30)*exp(-3*log((-4*log(x^ 
2)-4*x-4)/(log(x^2)+1))+3*x+2)/(log(x^2)^2+(2+x)*log(x^2)+x+1),x, algorith 
m="maxima")
 

Output:

-5/64*(8*e^2*log(x)^3 + 12*e^2*log(x)^2 + 6*e^2*log(x) + e^2)*e^(3*x)/(x^3 
 + 12*(x + 1)*log(x)^2 + 8*log(x)^3 + 3*x^2 + 6*(x^2 + 2*x + 1)*log(x) + 3 
*x + 1)
 

Giac [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {e^{2+3 x} \left (1+\log \left (x^2\right )\right )^3 \left (30+15 x+(15+15 x) \log \left (x^2\right )+15 \log ^2\left (x^2\right )\right )}{\left (-4-4 x-4 \log \left (x^2\right )\right )^3 \left (1+x+(2+x) \log \left (x^2\right )+\log ^2\left (x^2\right )\right )} \, dx=5 \, e^{\left (3 \, x - 3 \, \log \left (-\frac {4 \, x}{\log \left (x^{2}\right ) + 1} - \frac {4 \, \log \left (x^{2}\right )}{\log \left (x^{2}\right ) + 1} - \frac {4}{\log \left (x^{2}\right ) + 1}\right ) + 2\right )} \] Input:

integrate((15*log(x^2)^2+(15*x+15)*log(x^2)+15*x+30)*exp(-3*log((-4*log(x^ 
2)-4*x-4)/(log(x^2)+1))+3*x+2)/(log(x^2)^2+(2+x)*log(x^2)+x+1),x, algorith 
m="giac")
 

Output:

5*e^(3*x - 3*log(-4*x/(log(x^2) + 1) - 4*log(x^2)/(log(x^2) + 1) - 4/(log( 
x^2) + 1)) + 2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2+3 x} \left (1+\log \left (x^2\right )\right )^3 \left (30+15 x+(15+15 x) \log \left (x^2\right )+15 \log ^2\left (x^2\right )\right )}{\left (-4-4 x-4 \log \left (x^2\right )\right )^3 \left (1+x+(2+x) \log \left (x^2\right )+\log ^2\left (x^2\right )\right )} \, dx=\int \frac {{\mathrm {e}}^{3\,x-3\,\ln \left (-\frac {4\,x+4\,\ln \left (x^2\right )+4}{\ln \left (x^2\right )+1}\right )+2}\,\left (15\,{\ln \left (x^2\right )}^2+\left (15\,x+15\right )\,\ln \left (x^2\right )+15\,x+30\right )}{{\ln \left (x^2\right )}^2+\left (x+2\right )\,\ln \left (x^2\right )+x+1} \,d x \] Input:

int((exp(3*x - 3*log(-(4*x + 4*log(x^2) + 4)/(log(x^2) + 1)) + 2)*(15*x + 
15*log(x^2)^2 + log(x^2)*(15*x + 15) + 30))/(x + log(x^2)*(x + 2) + log(x^ 
2)^2 + 1),x)
 

Output:

int((exp(3*x - 3*log(-(4*x + 4*log(x^2) + 4)/(log(x^2) + 1)) + 2)*(15*x + 
15*log(x^2)^2 + log(x^2)*(15*x + 15) + 30))/(x + log(x^2)*(x + 2) + log(x^ 
2)^2 + 1), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.38 \[ \int \frac {e^{2+3 x} \left (1+\log \left (x^2\right )\right )^3 \left (30+15 x+(15+15 x) \log \left (x^2\right )+15 \log ^2\left (x^2\right )\right )}{\left (-4-4 x-4 \log \left (x^2\right )\right )^3 \left (1+x+(2+x) \log \left (x^2\right )+\log ^2\left (x^2\right )\right )} \, dx=\frac {5 e^{3 x} e^{2} \left (-\mathrm {log}\left (x^{2}\right )^{3}-3 \mathrm {log}\left (x^{2}\right )^{2}-3 \,\mathrm {log}\left (x^{2}\right )-1\right )}{64 \mathrm {log}\left (x^{2}\right )^{3}+192 \mathrm {log}\left (x^{2}\right )^{2} x +192 \mathrm {log}\left (x^{2}\right )^{2}+192 \,\mathrm {log}\left (x^{2}\right ) x^{2}+384 \,\mathrm {log}\left (x^{2}\right ) x +192 \,\mathrm {log}\left (x^{2}\right )+64 x^{3}+192 x^{2}+192 x +64} \] Input:

int((15*log(x^2)^2+(15*x+15)*log(x^2)+15*x+30)*exp(-3*log((-4*log(x^2)-4*x 
-4)/(log(x^2)+1))+3*x+2)/(log(x^2)^2+(2+x)*log(x^2)+x+1),x)
 

Output:

(5*e**(3*x)*e**2*( - log(x**2)**3 - 3*log(x**2)**2 - 3*log(x**2) - 1))/(64 
*(log(x**2)**3 + 3*log(x**2)**2*x + 3*log(x**2)**2 + 3*log(x**2)*x**2 + 6* 
log(x**2)*x + 3*log(x**2) + x**3 + 3*x**2 + 3*x + 1))