\(\int \frac {10 x-3 e^{2 x} x-6 e^x x^3-3 x^5+(e^{3 x} (6+12 x)+e^{2 x} (36 x^2+12 x^3)+e^x (-20+30 x^4)) \log (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5)+(3 e^{3 x} x+6 e^{2 x} x^3+e^x (-10 x+3 x^5)) \log ^2(10 x-3 e^{2 x} x-6 e^x x^3-3 x^5)}{-10 x+3 e^{2 x} x+6 e^x x^3+3 x^5} \, dx\) [819]

Optimal result

Integrand size = 181, antiderivative size = 27 \[ \int \frac {10 x-3 e^{2 x} x-6 e^x x^3-3 x^5+\left (e^{3 x} (6+12 x)+e^{2 x} \left (36 x^2+12 x^3\right )+e^x \left (-20+30 x^4\right )\right ) \log \left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )+\left (3 e^{3 x} x+6 e^{2 x} x^3+e^x \left (-10 x+3 x^5\right )\right ) \log ^2\left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )}{-10 x+3 e^{2 x} x+6 e^x x^3+3 x^5} \, dx=-x+e^x \log ^2\left (x-3 x \left (-3+\left (e^x+x^2\right )^2\right )\right ) \] Output:

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

Mathematica [F]

\[ \int \frac {10 x-3 e^{2 x} x-6 e^x x^3-3 x^5+\left (e^{3 x} (6+12 x)+e^{2 x} \left (36 x^2+12 x^3\right )+e^x \left (-20+30 x^4\right )\right ) \log \left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )+\left (3 e^{3 x} x+6 e^{2 x} x^3+e^x \left (-10 x+3 x^5\right )\right ) \log ^2\left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )}{-10 x+3 e^{2 x} x+6 e^x x^3+3 x^5} \, dx=\int \frac {10 x-3 e^{2 x} x-6 e^x x^3-3 x^5+\left (e^{3 x} (6+12 x)+e^{2 x} \left (36 x^2+12 x^3\right )+e^x \left (-20+30 x^4\right )\right ) \log \left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )+\left (3 e^{3 x} x+6 e^{2 x} x^3+e^x \left (-10 x+3 x^5\right )\right ) \log ^2\left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )}{-10 x+3 e^{2 x} x+6 e^x x^3+3 x^5} \, dx \] Input:

Integrate[(10*x - 3*E^(2*x)*x - 6*E^x*x^3 - 3*x^5 + (E^(3*x)*(6 + 12*x) + 
E^(2*x)*(36*x^2 + 12*x^3) + E^x*(-20 + 30*x^4))*Log[10*x - 3*E^(2*x)*x - 6 
*E^x*x^3 - 3*x^5] + (3*E^(3*x)*x + 6*E^(2*x)*x^3 + E^x*(-10*x + 3*x^5))*Lo 
g[10*x - 3*E^(2*x)*x - 6*E^x*x^3 - 3*x^5]^2)/(-10*x + 3*E^(2*x)*x + 6*E^x* 
x^3 + 3*x^5),x]
 

Output:

Integrate[(10*x - 3*E^(2*x)*x - 6*E^x*x^3 - 3*x^5 + (E^(3*x)*(6 + 12*x) + 
E^(2*x)*(36*x^2 + 12*x^3) + E^x*(-20 + 30*x^4))*Log[10*x - 3*E^(2*x)*x - 6 
*E^x*x^3 - 3*x^5] + (3*E^(3*x)*x + 6*E^(2*x)*x^3 + E^x*(-10*x + 3*x^5))*Lo 
g[10*x - 3*E^(2*x)*x - 6*E^x*x^3 - 3*x^5]^2)/(-10*x + 3*E^(2*x)*x + 6*E^x* 
x^3 + 3*x^5), 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.

\(\displaystyle \int \frac {-3 x^5-6 e^x x^3+\left (e^x \left (3 x^5-10 x\right )+6 e^{2 x} x^3+3 e^{3 x} x\right ) \log ^2\left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )+\left (e^x \left (30 x^4-20\right )+e^{2 x} \left (12 x^3+36 x^2\right )+e^{3 x} (12 x+6)\right ) \log \left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )-3 e^{2 x} x+10 x}{3 x^5+6 e^x x^3+3 e^{2 x} x-10 x} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (8 x \log \left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )-4 x^2 \log \left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )+\frac {e^x \left (x \log \left (-x \left (3 x^4+6 e^x x^2+3 e^{2 x}-10\right )\right )+4 x+2\right ) \log \left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )}{x}+\frac {4 \left (-3 x^6+6 x^5-3 e^x x^4+6 e^x x^3+10 x^2-20 x-10 e^x\right ) \log \left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )}{-3 x^4-6 e^x x^2-3 e^{2 x}+10}-1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (e^x \log ^2\left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )+\frac {2 e^x \left (5 \left (3 x^4-2\right )+6 e^x (x+3) x^2+e^{2 x} (6 x+3)\right ) \log \left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )}{x \left (3 x^4+6 e^x x^2+3 e^{2 x}-10\right )}-1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (e^x \log ^2\left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )+\frac {2 e^x \left (-15 x^4-6 e^x x^3-18 e^x x^2-6 e^{2 x} x-3 e^{2 x}+10\right ) \log \left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )}{x \left (-3 x^4-6 e^x x^2-3 e^{2 x}+10\right )}-1\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (e^x \log ^2\left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )+\frac {2 e^x \left (-15 x^4-6 e^x x^3-18 e^x x^2-6 e^{2 x} x-3 e^{2 x}+10\right ) \log \left (-3 x^5-6 e^x x^3-3 e^{2 x} x+10 x\right )}{x \left (-3 x^4-6 e^x x^2-3 e^{2 x}+10\right )}-1\right )dx\)

Input:

Int[(10*x - 3*E^(2*x)*x - 6*E^x*x^3 - 3*x^5 + (E^(3*x)*(6 + 12*x) + E^(2*x 
)*(36*x^2 + 12*x^3) + E^x*(-20 + 30*x^4))*Log[10*x - 3*E^(2*x)*x - 6*E^x*x 
^3 - 3*x^5] + (3*E^(3*x)*x + 6*E^(2*x)*x^3 + E^x*(-10*x + 3*x^5))*Log[10*x 
 - 3*E^(2*x)*x - 6*E^x*x^3 - 3*x^5]^2)/(-10*x + 3*E^(2*x)*x + 6*E^x*x^3 + 
3*x^5),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.78 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26

method	result	size
parallelrisch	\(\ln \left (-3 x \,{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x} x^{3}-3 x^{5}+10 x \right )^{2} {\mathrm e}^{x}-x\)	\(34\)
risch	\(\text {Expression too large to display}\)	\(1247\)

Input:

int(((3*x*exp(x)^3+6*exp(x)^2*x^3+(3*x^5-10*x)*exp(x))*ln(-3*x*exp(x)^2-6* 
exp(x)*x^3-3*x^5+10*x)^2+((12*x+6)*exp(x)^3+(12*x^3+36*x^2)*exp(x)^2+(30*x 
^4-20)*exp(x))*ln(-3*x*exp(x)^2-6*exp(x)*x^3-3*x^5+10*x)-3*x*exp(x)^2-6*ex 
p(x)*x^3-3*x^5+10*x)/(3*x*exp(x)^2+6*exp(x)*x^3+3*x^5-10*x),x,method=_RETU 
RNVERBOSE)
 

Output:

ln(-3*x*exp(x)^2-6*exp(x)*x^3-3*x^5+10*x)^2*exp(x)-x
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {10 x-3 e^{2 x} x-6 e^x x^3-3 x^5+\left (e^{3 x} (6+12 x)+e^{2 x} \left (36 x^2+12 x^3\right )+e^x \left (-20+30 x^4\right )\right ) \log \left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )+\left (3 e^{3 x} x+6 e^{2 x} x^3+e^x \left (-10 x+3 x^5\right )\right ) \log ^2\left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )}{-10 x+3 e^{2 x} x+6 e^x x^3+3 x^5} \, dx=e^{x} \log \left (-3 \, x^{5} - 6 \, x^{3} e^{x} - 3 \, x e^{\left (2 \, x\right )} + 10 \, x\right )^{2} - x \] Input:

integrate(((3*x*exp(x)^3+6*exp(x)^2*x^3+(3*x^5-10*x)*exp(x))*log(-3*x*exp( 
x)^2-6*exp(x)*x^3-3*x^5+10*x)^2+((12*x+6)*exp(x)^3+(12*x^3+36*x^2)*exp(x)^ 
2+(30*x^4-20)*exp(x))*log(-3*x*exp(x)^2-6*exp(x)*x^3-3*x^5+10*x)-3*x*exp(x 
)^2-6*exp(x)*x^3-3*x^5+10*x)/(3*x*exp(x)^2+6*exp(x)*x^3+3*x^5-10*x),x, alg 
orithm="fricas")
 

Output:

e^x*log(-3*x^5 - 6*x^3*e^x - 3*x*e^(2*x) + 10*x)^2 - x
 

Sympy [F(-1)]

Timed out. \[ \int \frac {10 x-3 e^{2 x} x-6 e^x x^3-3 x^5+\left (e^{3 x} (6+12 x)+e^{2 x} \left (36 x^2+12 x^3\right )+e^x \left (-20+30 x^4\right )\right ) \log \left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )+\left (3 e^{3 x} x+6 e^{2 x} x^3+e^x \left (-10 x+3 x^5\right )\right ) \log ^2\left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )}{-10 x+3 e^{2 x} x+6 e^x x^3+3 x^5} \, dx=\text {Timed out} \] Input:

integrate(((3*x*exp(x)**3+6*exp(x)**2*x**3+(3*x**5-10*x)*exp(x))*ln(-3*x*e 
xp(x)**2-6*exp(x)*x**3-3*x**5+10*x)**2+((12*x+6)*exp(x)**3+(12*x**3+36*x** 
2)*exp(x)**2+(30*x**4-20)*exp(x))*ln(-3*x*exp(x)**2-6*exp(x)*x**3-3*x**5+1 
0*x)-3*x*exp(x)**2-6*exp(x)*x**3-3*x**5+10*x)/(3*x*exp(x)**2+6*exp(x)*x**3 
+3*x**5-10*x),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).

Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.37 \[ \int \frac {10 x-3 e^{2 x} x-6 e^x x^3-3 x^5+\left (e^{3 x} (6+12 x)+e^{2 x} \left (36 x^2+12 x^3\right )+e^x \left (-20+30 x^4\right )\right ) \log \left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )+\left (3 e^{3 x} x+6 e^{2 x} x^3+e^x \left (-10 x+3 x^5\right )\right ) \log ^2\left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )}{-10 x+3 e^{2 x} x+6 e^x x^3+3 x^5} \, dx=e^{x} \log \left (-3 \, x^{4} - 6 \, x^{2} e^{x} - 3 \, e^{\left (2 \, x\right )} + 10\right )^{2} + 2 \, e^{x} \log \left (-3 \, x^{4} - 6 \, x^{2} e^{x} - 3 \, e^{\left (2 \, x\right )} + 10\right ) \log \left (x\right ) + e^{x} \log \left (x\right )^{2} - x \] Input:

integrate(((3*x*exp(x)^3+6*exp(x)^2*x^3+(3*x^5-10*x)*exp(x))*log(-3*x*exp( 
x)^2-6*exp(x)*x^3-3*x^5+10*x)^2+((12*x+6)*exp(x)^3+(12*x^3+36*x^2)*exp(x)^ 
2+(30*x^4-20)*exp(x))*log(-3*x*exp(x)^2-6*exp(x)*x^3-3*x^5+10*x)-3*x*exp(x 
)^2-6*exp(x)*x^3-3*x^5+10*x)/(3*x*exp(x)^2+6*exp(x)*x^3+3*x^5-10*x),x, alg 
orithm="maxima")
 

Output:

e^x*log(-3*x^4 - 6*x^2*e^x - 3*e^(2*x) + 10)^2 + 2*e^x*log(-3*x^4 - 6*x^2* 
e^x - 3*e^(2*x) + 10)*log(x) + e^x*log(x)^2 - x
 

Giac [A] (verification not implemented)

Time = 3.47 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {10 x-3 e^{2 x} x-6 e^x x^3-3 x^5+\left (e^{3 x} (6+12 x)+e^{2 x} \left (36 x^2+12 x^3\right )+e^x \left (-20+30 x^4\right )\right ) \log \left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )+\left (3 e^{3 x} x+6 e^{2 x} x^3+e^x \left (-10 x+3 x^5\right )\right ) \log ^2\left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )}{-10 x+3 e^{2 x} x+6 e^x x^3+3 x^5} \, dx=e^{x} \log \left (-3 \, x^{5} - 6 \, x^{3} e^{x} - 3 \, x e^{\left (2 \, x\right )} + 10 \, x\right )^{2} - x \] Input:

integrate(((3*x*exp(x)^3+6*exp(x)^2*x^3+(3*x^5-10*x)*exp(x))*log(-3*x*exp( 
x)^2-6*exp(x)*x^3-3*x^5+10*x)^2+((12*x+6)*exp(x)^3+(12*x^3+36*x^2)*exp(x)^ 
2+(30*x^4-20)*exp(x))*log(-3*x*exp(x)^2-6*exp(x)*x^3-3*x^5+10*x)-3*x*exp(x 
)^2-6*exp(x)*x^3-3*x^5+10*x)/(3*x*exp(x)^2+6*exp(x)*x^3+3*x^5-10*x),x, alg 
orithm="giac")
 

Output:

e^x*log(-3*x^5 - 6*x^3*e^x - 3*x*e^(2*x) + 10*x)^2 - x
 

Mupad [B] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {10 x-3 e^{2 x} x-6 e^x x^3-3 x^5+\left (e^{3 x} (6+12 x)+e^{2 x} \left (36 x^2+12 x^3\right )+e^x \left (-20+30 x^4\right )\right ) \log \left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )+\left (3 e^{3 x} x+6 e^{2 x} x^3+e^x \left (-10 x+3 x^5\right )\right ) \log ^2\left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )}{-10 x+3 e^{2 x} x+6 e^x x^3+3 x^5} \, dx={\ln \left (10\,x-3\,x\,{\mathrm {e}}^{2\,x}-6\,x^3\,{\mathrm {e}}^x-3\,x^5\right )}^2\,{\mathrm {e}}^x-x \] Input:

int((10*x - 3*x*exp(2*x) - 6*x^3*exp(x) + log(10*x - 3*x*exp(2*x) - 6*x^3* 
exp(x) - 3*x^5)^2*(3*x*exp(3*x) + 6*x^3*exp(2*x) - exp(x)*(10*x - 3*x^5)) 
+ log(10*x - 3*x*exp(2*x) - 6*x^3*exp(x) - 3*x^5)*(exp(2*x)*(36*x^2 + 12*x 
^3) + exp(3*x)*(12*x + 6) + exp(x)*(30*x^4 - 20)) - 3*x^5)/(3*x*exp(2*x) - 
 10*x + 6*x^3*exp(x) + 3*x^5),x)
 

Output:

log(10*x - 3*x*exp(2*x) - 6*x^3*exp(x) - 3*x^5)^2*exp(x) - x
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {10 x-3 e^{2 x} x-6 e^x x^3-3 x^5+\left (e^{3 x} (6+12 x)+e^{2 x} \left (36 x^2+12 x^3\right )+e^x \left (-20+30 x^4\right )\right ) \log \left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )+\left (3 e^{3 x} x+6 e^{2 x} x^3+e^x \left (-10 x+3 x^5\right )\right ) \log ^2\left (10 x-3 e^{2 x} x-6 e^x x^3-3 x^5\right )}{-10 x+3 e^{2 x} x+6 e^x x^3+3 x^5} \, dx=e^{x} \mathrm {log}\left (-3 e^{2 x} x -6 e^{x} x^{3}-3 x^{5}+10 x \right )^{2}-x \] Input:

int(((3*x*exp(x)^3+6*exp(x)^2*x^3+(3*x^5-10*x)*exp(x))*log(-3*x*exp(x)^2-6 
*exp(x)*x^3-3*x^5+10*x)^2+((12*x+6)*exp(x)^3+(12*x^3+36*x^2)*exp(x)^2+(30* 
x^4-20)*exp(x))*log(-3*x*exp(x)^2-6*exp(x)*x^3-3*x^5+10*x)-3*x*exp(x)^2-6* 
exp(x)*x^3-3*x^5+10*x)/(3*x*exp(x)^2+6*exp(x)*x^3+3*x^5-10*x),x)
 

Output:

e**x*log( - 3*e**(2*x)*x - 6*e**x*x**3 - 3*x**5 + 10*x)**2 - x