\(\int \frac {e^{\frac {16-8 x+7 x^3+x^6+(8 x+2 x^4) \log (\frac {16 x}{3})+x^2 \log ^2(\frac {16 x}{3})}{x^2}} (-32+16 x+7 x^3+2 x^4+4 x^6+(-8 x+2 x^2+4 x^4) \log (\frac {16 x}{3}))}{x^3} \, dx\) [183]

Optimal result

Integrand size = 92, antiderivative size = 28 \[ \int \frac {e^{\frac {16-8 x+7 x^3+x^6+\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )+x^2 \log ^2\left (\frac {16 x}{3}\right )}{x^2}} \left (-32+16 x+7 x^3+2 x^4+4 x^6+\left (-8 x+2 x^2+4 x^4\right ) \log \left (\frac {16 x}{3}\right )\right )}{x^3} \, dx=e^{-\frac {8}{x}-x+\left (\frac {4}{x}+x^2+\log \left (\frac {16 x}{3}\right )\right )^2} \] Output:

exp((ln(16/3*x)+4/x+x^2)^2-8/x-x)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {e^{\frac {16-8 x+7 x^3+x^6+\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )+x^2 \log ^2\left (\frac {16 x}{3}\right )}{x^2}} \left (-32+16 x+7 x^3+2 x^4+4 x^6+\left (-8 x+2 x^2+4 x^4\right ) \log \left (\frac {16 x}{3}\right )\right )}{x^3} \, dx=e^{\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {2 \left (4+x^3\right ) \log \left (\frac {16 x}{3}\right )}{x}+\log ^2\left (\frac {16 x}{3}\right )} \] Input:

Integrate[(E^((16 - 8*x + 7*x^3 + x^6 + (8*x + 2*x^4)*Log[(16*x)/3] + x^2* 
Log[(16*x)/3]^2)/x^2)*(-32 + 16*x + 7*x^3 + 2*x^4 + 4*x^6 + (-8*x + 2*x^2 
+ 4*x^4)*Log[(16*x)/3]))/x^3,x]
 

Output:

E^(16/x^2 - 8/x + 7*x + x^4 + (2*(4 + x^3)*Log[(16*x)/3])/x + Log[(16*x)/3 
]^2)
 

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.

\(\displaystyle \int \frac {\left (4 x^6+2 x^4+7 x^3+\left (4 x^4+2 x^2-8 x\right ) \log \left (\frac {16 x}{3}\right )+16 x-32\right ) \exp \left (\frac {x^6+\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )+7 x^3+x^2 \log ^2\left (\frac {16 x}{3}\right )-8 x+16}{x^2}\right )}{x^3} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (4 x^6+2 x^4+7 x^3+\left (4 x^4+2 x^2-8 x\right ) \log \left (\frac {16 x}{3}\right )+16 x-32\right ) \exp \left (x^4+\frac {16}{x^2}+\frac {\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+7 x-\frac {8}{x}+\log ^2\left (\frac {16 x}{3}\right )\right )}{x^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 \left (2 x^3+x-4\right ) \log \left (\frac {16 x}{3}\right ) \exp \left (x^4+\frac {16}{x^2}+\frac {\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+7 x-\frac {8}{x}+\log ^2\left (\frac {16 x}{3}\right )\right )}{x^2}+\frac {\left (4 x^6+2 x^4+7 x^3+16 x-32\right ) \exp \left (x^4+\frac {16}{x^2}+\frac {\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+7 x-\frac {8}{x}+\log ^2\left (\frac {16 x}{3}\right )\right )}{x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 7 \int \exp \left (x^4+7 x+\log ^2\left (\frac {16 x}{3}\right )-\frac {8}{x}+\frac {\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\frac {16}{x^2}\right )dx+16 \int \frac {\exp \left (x^4+7 x+\log ^2\left (\frac {16 x}{3}\right )-\frac {8}{x}+\frac {\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\frac {16}{x^2}\right )}{x^2}dx+2 \int \exp \left (x^4+7 x+\log ^2\left (\frac {16 x}{3}\right )-\frac {8}{x}+\frac {\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\frac {16}{x^2}\right ) xdx-8 \int \frac {\exp \left (x^4+7 x+\log ^2\left (\frac {16 x}{3}\right )-\frac {8}{x}+\frac {\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\frac {16}{x^2}\right ) \log \left (\frac {16 x}{3}\right )}{x^2}dx+2 \int \frac {\exp \left (x^4+7 x+\log ^2\left (\frac {16 x}{3}\right )-\frac {8}{x}+\frac {\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\frac {16}{x^2}\right ) \log \left (\frac {16 x}{3}\right )}{x}dx+4 \int \exp \left (x^4+7 x+\log ^2\left (\frac {16 x}{3}\right )-\frac {8}{x}+\frac {\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\frac {16}{x^2}\right ) x \log \left (\frac {16 x}{3}\right )dx-32 \int \frac {\exp \left (x^4+7 x+\log ^2\left (\frac {16 x}{3}\right )-\frac {8}{x}+\frac {\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\frac {16}{x^2}\right )}{x^3}dx+4 \int \exp \left (x^4+7 x+\log ^2\left (\frac {16 x}{3}\right )-\frac {8}{x}+\frac {\left (2 x^4+8 x\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\frac {16}{x^2}\right ) x^3dx\)

Input:

Int[(E^((16 - 8*x + 7*x^3 + x^6 + (8*x + 2*x^4)*Log[(16*x)/3] + x^2*Log[(1 
6*x)/3]^2)/x^2)*(-32 + 16*x + 7*x^3 + 2*x^4 + 4*x^6 + (-8*x + 2*x^2 + 4*x^ 
4)*Log[(16*x)/3]))/x^3,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54

method	result	size
parallelrisch	\({\mathrm e}^{\frac {x^{2} \ln \left (\frac {16 x}{3}\right )^{2}+\left (2 x^{4}+8 x \right ) \ln \left (\frac {16 x}{3}\right )+x^{6}+7 x^{3}-8 x +16}{x^{2}}}\)	\(43\)
risch	\(\left (\frac {16 x}{3}\right )^{2 x^{2}} \left (\frac {16 x}{3}\right )^{\frac {8}{x}} {\mathrm e}^{\frac {x^{6}+x^{2} \ln \left (\frac {16 x}{3}\right )^{2}+7 x^{3}-8 x +16}{x^{2}}}\)	\(48\)

Input:

int(((4*x^4+2*x^2-8*x)*ln(16/3*x)+4*x^6+2*x^4+7*x^3+16*x-32)*exp((x^2*ln(1 
6/3*x)^2+(2*x^4+8*x)*ln(16/3*x)+x^6+7*x^3-8*x+16)/x^2)/x^3,x,method=_RETUR 
NVERBOSE)
 

Output:

exp((x^2*ln(16/3*x)^2+(2*x^4+8*x)*ln(16/3*x)+x^6+7*x^3-8*x+16)/x^2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {e^{\frac {16-8 x+7 x^3+x^6+\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )+x^2 \log ^2\left (\frac {16 x}{3}\right )}{x^2}} \left (-32+16 x+7 x^3+2 x^4+4 x^6+\left (-8 x+2 x^2+4 x^4\right ) \log \left (\frac {16 x}{3}\right )\right )}{x^3} \, dx=e^{\left (\frac {x^{6} + x^{2} \log \left (\frac {16}{3} \, x\right )^{2} + 7 \, x^{3} + 2 \, {\left (x^{4} + 4 \, x\right )} \log \left (\frac {16}{3} \, x\right ) - 8 \, x + 16}{x^{2}}\right )} \] Input:

integrate(((4*x^4+2*x^2-8*x)*log(16/3*x)+4*x^6+2*x^4+7*x^3+16*x-32)*exp((x 
^2*log(16/3*x)^2+(2*x^4+8*x)*log(16/3*x)+x^6+7*x^3-8*x+16)/x^2)/x^3,x, alg 
orithm="fricas")
 

Output:

e^((x^6 + x^2*log(16/3*x)^2 + 7*x^3 + 2*(x^4 + 4*x)*log(16/3*x) - 8*x + 16 
)/x^2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {e^{\frac {16-8 x+7 x^3+x^6+\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )+x^2 \log ^2\left (\frac {16 x}{3}\right )}{x^2}} \left (-32+16 x+7 x^3+2 x^4+4 x^6+\left (-8 x+2 x^2+4 x^4\right ) \log \left (\frac {16 x}{3}\right )\right )}{x^3} \, dx=e^{\frac {x^{6} + 7 x^{3} + x^{2} \log {\left (\frac {16 x}{3} \right )}^{2} - 8 x + \left (2 x^{4} + 8 x\right ) \log {\left (\frac {16 x}{3} \right )} + 16}{x^{2}}} \] Input:

integrate(((4*x**4+2*x**2-8*x)*ln(16/3*x)+4*x**6+2*x**4+7*x**3+16*x-32)*ex 
p((x**2*ln(16/3*x)**2+(2*x**4+8*x)*ln(16/3*x)+x**6+7*x**3-8*x+16)/x**2)/x* 
*3,x)
 

Output:

exp((x**6 + 7*x**3 + x**2*log(16*x/3)**2 - 8*x + (2*x**4 + 8*x)*log(16*x/3 
) + 16)/x**2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (25) = 50\).

Time = 0.35 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.39 \[ \int \frac {e^{\frac {16-8 x+7 x^3+x^6+\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )+x^2 \log ^2\left (\frac {16 x}{3}\right )}{x^2}} \left (-32+16 x+7 x^3+2 x^4+4 x^6+\left (-8 x+2 x^2+4 x^4\right ) \log \left (\frac {16 x}{3}\right )\right )}{x^3} \, dx=\frac {e^{\left (x^{4} - 2 \, x^{2} \log \left (3\right ) + 8 \, x^{2} \log \left (2\right ) + 2 \, x^{2} \log \left (x\right ) + \log \left (3\right )^{2} + 16 \, \log \left (2\right )^{2} - 2 \, \log \left (3\right ) \log \left (x\right ) + 8 \, \log \left (2\right ) \log \left (x\right ) + \log \left (x\right )^{2} + 7 \, x - \frac {8 \, \log \left (3\right )}{x} + \frac {32 \, \log \left (2\right )}{x} + \frac {8 \, \log \left (x\right )}{x} - \frac {8}{x} + \frac {16}{x^{2}}\right )}}{2^{8 \, \log \left (3\right )}} \] Input:

integrate(((4*x^4+2*x^2-8*x)*log(16/3*x)+4*x^6+2*x^4+7*x^3+16*x-32)*exp((x 
^2*log(16/3*x)^2+(2*x^4+8*x)*log(16/3*x)+x^6+7*x^3-8*x+16)/x^2)/x^3,x, alg 
orithm="maxima")
 

Output:

e^(x^4 - 2*x^2*log(3) + 8*x^2*log(2) + 2*x^2*log(x) + log(3)^2 + 16*log(2) 
^2 - 2*log(3)*log(x) + 8*log(2)*log(x) + log(x)^2 + 7*x - 8*log(3)/x + 32* 
log(2)/x + 8*log(x)/x - 8/x + 16/x^2)/2^(8*log(3))
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {16-8 x+7 x^3+x^6+\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )+x^2 \log ^2\left (\frac {16 x}{3}\right )}{x^2}} \left (-32+16 x+7 x^3+2 x^4+4 x^6+\left (-8 x+2 x^2+4 x^4\right ) \log \left (\frac {16 x}{3}\right )\right )}{x^3} \, dx=e^{\left (x^{4} + 2 \, x^{2} \log \left (\frac {16}{3} \, x\right ) + \log \left (\frac {16}{3} \, x\right )^{2} + 7 \, x + \frac {8 \, \log \left (\frac {16}{3} \, x\right )}{x} - \frac {8}{x} + \frac {16}{x^{2}}\right )} \] Input:

integrate(((4*x^4+2*x^2-8*x)*log(16/3*x)+4*x^6+2*x^4+7*x^3+16*x-32)*exp((x 
^2*log(16/3*x)^2+(2*x^4+8*x)*log(16/3*x)+x^6+7*x^3-8*x+16)/x^2)/x^3,x, alg 
orithm="giac")
 

Output:

e^(x^4 + 2*x^2*log(16/3*x) + log(16/3*x)^2 + 7*x + 8*log(16/3*x)/x - 8/x + 
 16/x^2)
 

Mupad [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.79 \[ \int \frac {e^{\frac {16-8 x+7 x^3+x^6+\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )+x^2 \log ^2\left (\frac {16 x}{3}\right )}{x^2}} \left (-32+16 x+7 x^3+2 x^4+4 x^6+\left (-8 x+2 x^2+4 x^4\right ) \log \left (\frac {16 x}{3}\right )\right )}{x^3} \, dx=\frac {2^{8\,x^2}\,2^{32/x}\,x^{2\,x^2}\,x^{8/x}\,x^{8\,\ln \left (2\right )}\,{\mathrm {e}}^{{\ln \left (3\right )}^2}\,{\mathrm {e}}^{7\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{16\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{-\frac {8}{x}}\,{\mathrm {e}}^{\frac {16}{x^2}}\,{\mathrm {e}}^{{\ln \left (x\right )}^2}}{2^{8\,\ln \left (3\right )}\,3^{2\,x^2}\,3^{8/x}\,x^{2\,\ln \left (3\right )}} \] Input:

int((exp((log((16*x)/3)*(8*x + 2*x^4) - 8*x + 7*x^3 + x^6 + x^2*log((16*x) 
/3)^2 + 16)/x^2)*(16*x + log((16*x)/3)*(2*x^2 - 8*x + 4*x^4) + 7*x^3 + 2*x 
^4 + 4*x^6 - 32))/x^3,x)
 

Output:

(2^(8*x^2)*2^(32/x)*x^(2*x^2)*x^(8/x)*x^(8*log(2))*exp(log(3)^2)*exp(7*x)* 
exp(x^4)*exp(16*log(2)^2)*exp(-8/x)*exp(16/x^2)*exp(log(x)^2))/(2^(8*log(3 
))*3^(2*x^2)*3^(8/x)*x^(2*log(3)))
 

Reduce [F]

\[ \int \frac {e^{\frac {16-8 x+7 x^3+x^6+\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )+x^2 \log ^2\left (\frac {16 x}{3}\right )}{x^2}} \left (-32+16 x+7 x^3+2 x^4+4 x^6+\left (-8 x+2 x^2+4 x^4\right ) \log \left (\frac {16 x}{3}\right )\right )}{x^3} \, dx=\int \frac {\left (\left (4 x^{4}+2 x^{2}-8 x \right ) \mathrm {log}\left (\frac {16 x}{3}\right )+4 x^{6}+2 x^{4}+7 x^{3}+16 x -32\right ) {\mathrm e}^{\frac {x^{2} \mathrm {log}\left (\frac {16 x}{3}\right )^{2}+\left (2 x^{4}+8 x \right ) \mathrm {log}\left (\frac {16 x}{3}\right )+x^{6}+7 x^{3}-8 x +16}{x^{2}}}}{x^{3}}d x \] Input:

int(((4*x^4+2*x^2-8*x)*log(16/3*x)+4*x^6+2*x^4+7*x^3+16*x-32)*exp((x^2*log 
(16/3*x)^2+(2*x^4+8*x)*log(16/3*x)+x^6+7*x^3-8*x+16)/x^2)/x^3,x)
 

Output:

int(((4*x^4+2*x^2-8*x)*log(16/3*x)+4*x^6+2*x^4+7*x^3+16*x-32)*exp((x^2*log 
(16/3*x)^2+(2*x^4+8*x)*log(16/3*x)+x^6+7*x^3-8*x+16)/x^2)/x^3,x)