3.5.71 \(\int \frac {14 x+36 x^2+28 x^3+6 x^4+(16+7 x+17 x^2+7 x^3+x^4) \log (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8})}{64+28 x+68 x^2+28 x^3+4 x^4+(-64-28 x-68 x^2-28 x^3-4 x^4) \log (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8})+(16+7 x+17 x^2+7 x^3+x^4) \log ^2(\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8})} \, dx\) [471]

3.5.71.1 Optimal result

Integrand size = 226, antiderivative size = 32 \begin {dmath*} \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=\frac {x}{-2+\log \left (\frac {e^4}{x^2 \left (1+x^2\right ) \left (x-(4+x)^2\right )^2}\right )} \end {dmath*}

output

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

3.5.71.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \begin {dmath*} \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=\frac {x}{2+\log \left (\frac {1}{x^2 \left (1+x^2\right ) \left (16+7 x+x^2\right )^2}\right )} \end {dmath*}

input

Integrate[(14*x + 36*x^2 + 28*x^3 + 6*x^4 + (16 + 7*x + 17*x^2 + 7*x^3 + x 
^4)*Log[E^4/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8 
)])/(64 + 28*x + 68*x^2 + 28*x^3 + 4*x^4 + (-64 - 28*x - 68*x^2 - 28*x^3 - 
 4*x^4)*Log[E^4/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + 
 x^8)] + (16 + 7*x + 17*x^2 + 7*x^3 + x^4)*Log[E^4/(256*x^2 + 224*x^3 + 33 
7*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8)]^2),x]
 

output

x/(2 + Log[1/(x^2*(1 + x^2)*(16 + 7*x + x^2)^2)])
 

3.5.71.3 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.

\(\displaystyle \int \frac {6 x^4+28 x^3+36 x^2+\left (x^4+7 x^3+17 x^2+7 x+16\right ) \log \left (\frac {e^4}{x^8+14 x^7+82 x^6+238 x^5+337 x^4+224 x^3+256 x^2}\right )+14 x}{4 x^4+28 x^3+68 x^2+\left (x^4+7 x^3+17 x^2+7 x+16\right ) \log ^2\left (\frac {e^4}{x^8+14 x^7+82 x^6+238 x^5+337 x^4+224 x^3+256 x^2}\right )+\left (-4 x^4-28 x^3-68 x^2-28 x-64\right ) \log \left (\frac {e^4}{x^8+14 x^7+82 x^6+238 x^5+337 x^4+224 x^3+256 x^2}\right )+28 x+64} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 \left (5 x^4+28 x^3+52 x^2+21 x+32\right )+\left (x^4+7 x^3+17 x^2+7 x+16\right ) \log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )}{\left (x^4+7 x^3+17 x^2+7 x+16\right ) \left (\log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )+2\right )^2}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {(15-7 x) \left (2 \left (5 x^4+28 x^3+52 x^2+21 x+32\right )+\left (x^4+7 x^3+17 x^2+7 x+16\right ) \log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )\right )}{274 \left (x^2+1\right ) \left (\log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )+2\right )^2}+\frac {(7 x+34) \left (2 \left (5 x^4+28 x^3+52 x^2+21 x+32\right )+\left (x^4+7 x^3+17 x^2+7 x+16\right ) \log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )\right )}{274 \left (x^2+7 x+16\right ) \left (\log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )+2\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 8 \int \frac {1}{\left (\log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )+2\right )^2}dx-\frac {128 i \int \frac {1}{\left (-2 x+i \sqrt {15}-7\right ) \left (\log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )+2\right )^2}dx}{\sqrt {15}}-i \int \frac {1}{(i-x) \left (\log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )+2\right )^2}dx-i \int \frac {1}{(x+i) \left (\log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )+2\right )^2}dx-\frac {14}{15} \left (15+7 i \sqrt {15}\right ) \int \frac {1}{\left (2 x-i \sqrt {15}+7\right ) \left (\log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )+2\right )^2}dx-\frac {14}{15} \left (15-7 i \sqrt {15}\right ) \int \frac {1}{\left (2 x+i \sqrt {15}+7\right ) \left (\log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )+2\right )^2}dx-\frac {128 i \int \frac {1}{\left (2 x+i \sqrt {15}+7\right ) \left (\log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )+2\right )^2}dx}{\sqrt {15}}+\int \frac {1}{\log \left (\frac {1}{x^2 \left (x^2+1\right ) \left (x^2+7 x+16\right )^2}\right )+2}dx\)

input

Int[(14*x + 36*x^2 + 28*x^3 + 6*x^4 + (16 + 7*x + 17*x^2 + 7*x^3 + x^4)*Lo 
g[E^4/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8)])/(6 
4 + 28*x + 68*x^2 + 28*x^3 + 4*x^4 + (-64 - 28*x - 68*x^2 - 28*x^3 - 4*x^4 
)*Log[E^4/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8)] 
 + (16 + 7*x + 17*x^2 + 7*x^3 + x^4)*Log[E^4/(256*x^2 + 224*x^3 + 337*x^4 
+ 238*x^5 + 82*x^6 + 14*x^7 + x^8)]^2),x]
 

output

$Aborted
 

3.5.71.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2463

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u, Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

3.5.71.4 Maple [A] (verified)

Time = 3.83 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38

method	result	size
parallelrisch	\(\frac {x}{\ln \left (\frac {{\mathrm e}^{4}}{x^{2} \left (x^{6}+14 x^{5}+82 x^{4}+238 x^{3}+337 x^{2}+224 x +256\right )}\right )-2}\)	\(44\)
norman	\(\frac {x}{\ln \left (\frac {{\mathrm e}^{4}}{x^{8}+14 x^{7}+82 x^{6}+238 x^{5}+337 x^{4}+224 x^{3}+256 x^{2}}\right )-2}\)	\(47\)
risch	\(\frac {x}{\ln \left (\frac {{\mathrm e}^{4}}{x^{8}+14 x^{7}+82 x^{6}+238 x^{5}+337 x^{4}+224 x^{3}+256 x^{2}}\right )-2}\)	\(47\)

input

int(((x^4+7*x^3+17*x^2+7*x+16)*ln(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^ 
4+224*x^3+256*x^2))+6*x^4+28*x^3+36*x^2+14*x)/((x^4+7*x^3+17*x^2+7*x+16)*l 
n(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))^2+(-4*x^4-28 
*x^3-68*x^2-28*x-64)*ln(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+ 
256*x^2))+4*x^4+28*x^3+68*x^2+28*x+64),x,method=_RETURNVERBOSE)
 

output

x/(ln(exp(4)/x^2/(x^6+14*x^5+82*x^4+238*x^3+337*x^2+224*x+256))-2)
 

3.5.71.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \begin {dmath*} \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=\frac {x}{\log \left (\frac {e^{4}}{x^{8} + 14 \, x^{7} + 82 \, x^{6} + 238 \, x^{5} + 337 \, x^{4} + 224 \, x^{3} + 256 \, x^{2}}\right ) - 2} \end {dmath*}

input

integrate(((x^4+7*x^3+17*x^2+7*x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5 
+337*x^4+224*x^3+256*x^2))+6*x^4+28*x^3+36*x^2+14*x)/((x^4+7*x^3+17*x^2+7* 
x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))^2+(- 
4*x^4-28*x^3-68*x^2-28*x-64)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4 
+224*x^3+256*x^2))+4*x^4+28*x^3+68*x^2+28*x+64),x, algorithm=\
 

output

x/(log(e^4/(x^8 + 14*x^7 + 82*x^6 + 238*x^5 + 337*x^4 + 224*x^3 + 256*x^2) 
) - 2)
 

3.5.71.6 Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \begin {dmath*} \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=\frac {x}{\log {\left (\frac {e^{4}}{x^{8} + 14 x^{7} + 82 x^{6} + 238 x^{5} + 337 x^{4} + 224 x^{3} + 256 x^{2}} \right )} - 2} \end {dmath*}

input

integrate(((x**4+7*x**3+17*x**2+7*x+16)*ln(exp(4)/(x**8+14*x**7+82*x**6+23 
8*x**5+337*x**4+224*x**3+256*x**2))+6*x**4+28*x**3+36*x**2+14*x)/((x**4+7* 
x**3+17*x**2+7*x+16)*ln(exp(4)/(x**8+14*x**7+82*x**6+238*x**5+337*x**4+224 
*x**3+256*x**2))**2+(-4*x**4-28*x**3-68*x**2-28*x-64)*ln(exp(4)/(x**8+14*x 
**7+82*x**6+238*x**5+337*x**4+224*x**3+256*x**2))+4*x**4+28*x**3+68*x**2+2 
8*x+64),x)
 

output

x/(log(exp(4)/(x**8 + 14*x**7 + 82*x**6 + 238*x**5 + 337*x**4 + 224*x**3 + 
 256*x**2)) - 2)
 

3.5.71.7 Maxima [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \begin {dmath*} \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=-\frac {x}{2 \, \log \left (x^{2} + 7 \, x + 16\right ) + \log \left (x^{2} + 1\right ) + 2 \, \log \left (x\right ) - 2} \end {dmath*}

input

integrate(((x^4+7*x^3+17*x^2+7*x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5 
+337*x^4+224*x^3+256*x^2))+6*x^4+28*x^3+36*x^2+14*x)/((x^4+7*x^3+17*x^2+7* 
x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))^2+(- 
4*x^4-28*x^3-68*x^2-28*x-64)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4 
+224*x^3+256*x^2))+4*x^4+28*x^3+68*x^2+28*x+64),x, algorithm=\
 

output

-x/(2*log(x^2 + 7*x + 16) + log(x^2 + 1) + 2*log(x) - 2)
 

3.5.71.8 Giac [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \begin {dmath*} \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=-\frac {x}{\log \left (x^{8} + 14 \, x^{7} + 82 \, x^{6} + 238 \, x^{5} + 337 \, x^{4} + 224 \, x^{3} + 256 \, x^{2}\right ) - 2} \end {dmath*}

input

integrate(((x^4+7*x^3+17*x^2+7*x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5 
+337*x^4+224*x^3+256*x^2))+6*x^4+28*x^3+36*x^2+14*x)/((x^4+7*x^3+17*x^2+7* 
x+16)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4+224*x^3+256*x^2))^2+(- 
4*x^4-28*x^3-68*x^2-28*x-64)*log(exp(4)/(x^8+14*x^7+82*x^6+238*x^5+337*x^4 
+224*x^3+256*x^2))+4*x^4+28*x^3+68*x^2+28*x+64),x, algorithm=\
 

output

-x/(log(x^8 + 14*x^7 + 82*x^6 + 238*x^5 + 337*x^4 + 224*x^3 + 256*x^2) - 2 
)
 

3.5.71.9 Mupad [B] (verification not implemented)

Time = 16.21 (sec) , antiderivative size = 164, normalized size of antiderivative = 5.12 \begin {dmath*} \int \frac {14 x+36 x^2+28 x^3+6 x^4+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )}{64+28 x+68 x^2+28 x^3+4 x^4+\left (-64-28 x-68 x^2-28 x^3-4 x^4\right ) \log \left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )+\left (16+7 x+17 x^2+7 x^3+x^4\right ) \log ^2\left (\frac {e^4}{256 x^2+224 x^3+337 x^4+238 x^5+82 x^6+14 x^7+x^8}\right )} \, dx=\frac {7\,\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )}{32\,\left (\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )-2\right )}-\frac {7}{16\,\left (\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )-2\right )}-\frac {x}{\ln \left (x^8+14\,x^7+82\,x^6+238\,x^5+337\,x^4+224\,x^3+256\,x^2\right )-2} \end {dmath*}

input

int((14*x + 36*x^2 + 28*x^3 + 6*x^4 + log(exp(4)/(256*x^2 + 224*x^3 + 337* 
x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8))*(7*x + 17*x^2 + 7*x^3 + x^4 + 16)) 
/(28*x - log(exp(4)/(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x 
^7 + x^8))*(28*x + 68*x^2 + 28*x^3 + 4*x^4 + 64) + log(exp(4)/(256*x^2 + 2 
24*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8))^2*(7*x + 17*x^2 + 7*x 
^3 + x^4 + 16) + 68*x^2 + 28*x^3 + 4*x^4 + 64),x)
 

output

(7*log(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8))/(32 
*(log(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^8) - 2)) 
 - 7/(16*(log(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 + x^ 
8) - 2)) - x/(log(256*x^2 + 224*x^3 + 337*x^4 + 238*x^5 + 82*x^6 + 14*x^7 
+ x^8) - 2)