\(\int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} (69 x-27 x^2+3 x^3)+(-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} (-48+21 x-3 x^2)) \log (\frac {16-7 x+x^2}{e^{4+x}+x+x^3})}{(16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} (16 x^2-7 x^3+x^4)) \log ^2(\frac {16-7 x+x^2}{e^{4+x}+x+x^3})} \, dx\) [2857]

Optimal result

Integrand size = 173, antiderivative size = 28 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x \log \left (\frac {(-4+x)^2+x}{e^{4+x}+x+x^3}\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \] Input:

Integrate[(48*x + 141*x^3 - 42*x^4 + 3*x^5 + E^(4 + x)*(69*x - 27*x^2 + 3* 
x^3) + (-48*x + 21*x^2 - 51*x^3 + 21*x^4 - 3*x^5 + E^(4 + x)*(-48 + 21*x - 
 3*x^2))*Log[(16 - 7*x + x^2)/(E^(4 + x) + x + x^3)])/((16*x^3 - 7*x^4 + 1 
7*x^5 - 7*x^6 + x^7 + E^(4 + x)*(16*x^2 - 7*x^3 + x^4))*Log[(16 - 7*x + x^ 
2)/(E^(4 + x) + x + x^3)]^2),x]
 

Output:

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

\(\displaystyle \int \frac {3 x^5-42 x^4+141 x^3+e^{x+4} \left (3 x^3-27 x^2+69 x\right )+\left (-3 x^5+21 x^4-51 x^3+21 x^2+e^{x+4} \left (-3 x^2+21 x-48\right )-48 x\right ) \log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )+48 x}{\left (x^7-7 x^6+17 x^5-7 x^4+16 x^3+e^{x+4} \left (x^4-7 x^3+16 x^2\right )\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {3 \left (\frac {x \left (x^4-14 x^3+47 x^2+e^{x+4} \left (x^2-9 x+23\right )+16\right )}{\left (x^2-7 x+16\right ) \left (x^3+x+e^{x+4}\right )}-\log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )\right )}{x^2 \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \int \frac {\frac {x \left (x^4-14 x^3+47 x^2+e^{x+4} \left (x^2-9 x+23\right )+16\right )}{\left (x^2-7 x+16\right ) \left (x^3+x+e^{x+4}\right )}-\log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}{x^2 \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 3 \int \left (\frac {x^3-\log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right ) x^2-9 x^2+7 \log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right ) x+23 x-16 \log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}{x^2 \left (x^2-7 x+16\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}-\frac {x^3-3 x^2+x-1}{x \left (x^3+x+e^{x+4}\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (\frac {17 i \int \frac {1}{\left (-2 x+i \sqrt {15}+7\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx}{8 \sqrt {15}}+\frac {23}{16} \int \frac {1}{x \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx-\frac {7}{240} \left (15-7 i \sqrt {15}\right ) \int \frac {1}{\left (2 x-i \sqrt {15}-7\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx-\frac {7}{240} \left (15+7 i \sqrt {15}\right ) \int \frac {1}{\left (2 x+i \sqrt {15}-7\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx+\frac {17 i \int \frac {1}{\left (2 x+i \sqrt {15}-7\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx}{8 \sqrt {15}}-\int \frac {1}{\left (x^3+x+e^{x+4}\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx+\int \frac {1}{x \left (x^3+x+e^{x+4}\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx+3 \int \frac {x}{\left (x^3+x+e^{x+4}\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx-\int \frac {x^2}{\left (x^3+x+e^{x+4}\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx-\int \frac {1}{x^2 \log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx\right )\)

Input:

Int[(48*x + 141*x^3 - 42*x^4 + 3*x^5 + E^(4 + x)*(69*x - 27*x^2 + 3*x^3) + 
 (-48*x + 21*x^2 - 51*x^3 + 21*x^4 - 3*x^5 + E^(4 + x)*(-48 + 21*x - 3*x^2 
))*Log[(16 - 7*x + x^2)/(E^(4 + x) + x + x^3)])/((16*x^3 - 7*x^4 + 17*x^5 
- 7*x^6 + x^7 + E^(4 + x)*(16*x^2 - 7*x^3 + x^4))*Log[(16 - 7*x + x^2)/(E^ 
(4 + x) + x + x^3)]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 8.69 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04

method	result	size
parallelrisch	\(\frac {3}{\ln \left (\frac {x^{2}-7 x +16}{{\mathrm e}^{4+x}+x^{3}+x}\right ) x}\)	\(29\)
risch	\(\frac {6 i}{x \left (\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4+x}+x^{3}+x}\right ) \operatorname {csgn}\left (i \left (x^{2}-7 x +16\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-7 x +16\right )}{{\mathrm e}^{4+x}+x^{3}+x}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4+x}+x^{3}+x}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-7 x +16\right )}{{\mathrm e}^{4+x}+x^{3}+x}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (x^{2}-7 x +16\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-7 x +16\right )}{{\mathrm e}^{4+x}+x^{3}+x}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-7 x +16\right )}{{\mathrm e}^{4+x}+x^{3}+x}\right )}^{3}+2 i \ln \left (x^{2}-7 x +16\right )-2 i \ln \left ({\mathrm e}^{4+x}+x^{3}+x \right )\right )}\)	\(197\)

Input:

int((((-3*x^2+21*x-48)*exp(4+x)-3*x^5+21*x^4-51*x^3+21*x^2-48*x)*ln((x^2-7 
*x+16)/(exp(4+x)+x^3+x))+(3*x^3-27*x^2+69*x)*exp(4+x)+3*x^5-42*x^4+141*x^3 
+48*x)/((x^4-7*x^3+16*x^2)*exp(4+x)+x^7-7*x^6+17*x^5-7*x^4+16*x^3)/ln((x^2 
-7*x+16)/(exp(4+x)+x^3+x))^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x \log \left (\frac {x^{2} - 7 \, x + 16}{x^{3} + x + e^{\left (x + 4\right )}}\right )} \] Input:

integrate((((-3*x^2+21*x-48)*exp(4+x)-3*x^5+21*x^4-51*x^3+21*x^2-48*x)*log 
((x^2-7*x+16)/(exp(4+x)+x^3+x))+(3*x^3-27*x^2+69*x)*exp(4+x)+3*x^5-42*x^4+ 
141*x^3+48*x)/((x^4-7*x^3+16*x^2)*exp(4+x)+x^7-7*x^6+17*x^5-7*x^4+16*x^3)/ 
log((x^2-7*x+16)/(exp(4+x)+x^3+x))^2,x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x \log {\left (\frac {x^{2} - 7 x + 16}{x^{3} + x + e^{x + 4}} \right )}} \] Input:

integrate((((-3*x**2+21*x-48)*exp(4+x)-3*x**5+21*x**4-51*x**3+21*x**2-48*x 
)*ln((x**2-7*x+16)/(exp(4+x)+x**3+x))+(3*x**3-27*x**2+69*x)*exp(4+x)+3*x** 
5-42*x**4+141*x**3+48*x)/((x**4-7*x**3+16*x**2)*exp(4+x)+x**7-7*x**6+17*x* 
*5-7*x**4+16*x**3)/ln((x**2-7*x+16)/(exp(4+x)+x**3+x))**2,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=-\frac {3}{x \log \left (x^{3} + x + e^{\left (x + 4\right )}\right ) - x \log \left (x^{2} - 7 \, x + 16\right )} \] Input:

integrate((((-3*x^2+21*x-48)*exp(4+x)-3*x^5+21*x^4-51*x^3+21*x^2-48*x)*log 
((x^2-7*x+16)/(exp(4+x)+x^3+x))+(3*x^3-27*x^2+69*x)*exp(4+x)+3*x^5-42*x^4+ 
141*x^3+48*x)/((x^4-7*x^3+16*x^2)*exp(4+x)+x^7-7*x^6+17*x^5-7*x^4+16*x^3)/ 
log((x^2-7*x+16)/(exp(4+x)+x^3+x))^2,x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x \log \left (\frac {x^{2} - 7 \, x + 16}{x^{3} + x + e^{\left (x + 4\right )}}\right )} \] Input:

integrate((((-3*x^2+21*x-48)*exp(4+x)-3*x^5+21*x^4-51*x^3+21*x^2-48*x)*log 
((x^2-7*x+16)/(exp(4+x)+x^3+x))+(3*x^3-27*x^2+69*x)*exp(4+x)+3*x^5-42*x^4+ 
141*x^3+48*x)/((x^4-7*x^3+16*x^2)*exp(4+x)+x^7-7*x^6+17*x^5-7*x^4+16*x^3)/ 
log((x^2-7*x+16)/(exp(4+x)+x^3+x))^2,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x\,\ln \left (\frac {x^2-7\,x+16}{x+{\mathrm {e}}^4\,{\mathrm {e}}^x+x^3}\right )} \] Input:

int((48*x + exp(x + 4)*(69*x - 27*x^2 + 3*x^3) - log((x^2 - 7*x + 16)/(x + 
 exp(x + 4) + x^3))*(48*x + exp(x + 4)*(3*x^2 - 21*x + 48) - 21*x^2 + 51*x 
^3 - 21*x^4 + 3*x^5) + 141*x^3 - 42*x^4 + 3*x^5)/(log((x^2 - 7*x + 16)/(x 
+ exp(x + 4) + x^3))^2*(exp(x + 4)*(16*x^2 - 7*x^3 + x^4) + 16*x^3 - 7*x^4 
 + 17*x^5 - 7*x^6 + x^7)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{\mathrm {log}\left (\frac {x^{2}-7 x +16}{e^{x} e^{4}+x^{3}+x}\right ) x} \] Input:

int((((-3*x^2+21*x-48)*exp(4+x)-3*x^5+21*x^4-51*x^3+21*x^2-48*x)*log((x^2- 
7*x+16)/(exp(4+x)+x^3+x))+(3*x^3-27*x^2+69*x)*exp(4+x)+3*x^5-42*x^4+141*x^ 
3+48*x)/((x^4-7*x^3+16*x^2)*exp(4+x)+x^7-7*x^6+17*x^5-7*x^4+16*x^3)/log((x 
^2-7*x+16)/(exp(4+x)+x^3+x))^2,x)
 

Output:

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