3.11.6 \(\int \frac {(-20+100 x-180 x^4+100 x^5) \log (x^2)+(40+40 x^4) \log (\frac {e^{5 x}}{x+2 x^5+x^9})}{x+x^5} \, dx\) [1006]

3.11.6.1 Optimal result

Integrand size = 56, antiderivative size = 23 \[ \int \frac {\left (-20+100 x-180 x^4+100 x^5\right ) \log \left (x^2\right )+\left (40+40 x^4\right ) \log \left (\frac {e^{5 x}}{x+2 x^5+x^9}\right )}{x+x^5} \, dx=20 \log \left (x^2\right ) \log \left (\frac {e^{5 x}}{x \left (1+x^4\right )^2}\right ) \]

output

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

3.11.6.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(23)=46\).

Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {\left (-20+100 x-180 x^4+100 x^5\right ) \log \left (x^2\right )+\left (40+40 x^4\right ) \log \left (\frac {e^{5 x}}{x+2 x^5+x^9}\right )}{x+x^5} \, dx=20 \left (\log \left (x^2\right ) \left (5 x+\log \left (\frac {1}{x \left (1+x^4\right )^2}\right )\right )-2 \log (x) \left (5 x+\log \left (\frac {1}{x \left (1+x^4\right )^2}\right )-\log \left (\frac {e^{5 x}}{x \left (1+x^4\right )^2}\right )\right )\right ) \]

input

Integrate[((-20 + 100*x - 180*x^4 + 100*x^5)*Log[x^2] + (40 + 40*x^4)*Log[ 
E^(5*x)/(x + 2*x^5 + x^9)])/(x + x^5),x]
 

output

20*(Log[x^2]*(5*x + Log[1/(x*(1 + x^4)^2)]) - 2*Log[x]*(5*x + Log[1/(x*(1 
+ x^4)^2)] - Log[E^(5*x)/(x*(1 + x^4)^2)]))
 

3.11.6.3 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.

\(\displaystyle \int \frac {\left (40 x^4+40\right ) \log \left (\frac {e^{5 x}}{x^9+2 x^5+x}\right )+\left (100 x^5-180 x^4+100 x-20\right ) \log \left (x^2\right )}{x^5+x} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (40 x^4+40\right ) \log \left (\frac {e^{5 x}}{x^9+2 x^5+x}\right )+\left (100 x^5-180 x^4+100 x-20\right ) \log \left (x^2\right )}{x \left (x^4+1\right )}dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {40 \log \left (\frac {e^{5 x}}{x \left (x^4+1\right )^2}\right )}{x}+\frac {20 (x-1) \left (5 x^4-4 x^3-4 x^2-4 x+1\right ) \log \left (x^2\right )}{x \left (x^4+1\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 40 \int \frac {\log \left (\frac {e^{5 x}}{x \left (x^4+1\right )^2}\right )}{x}dx-20 \operatorname {PolyLog}\left (2,-x^4\right )-5 \log ^2\left (x^2\right )+100 x \log \left (x^2\right )-40 \log \left (x^4+1\right ) \log \left (x^2\right )-200 x\)

input

Int[((-20 + 100*x - 180*x^4 + 100*x^5)*Log[x^2] + (40 + 40*x^4)*Log[E^(5*x 
)/(x + 2*x^5 + x^9)])/(x + x^5),x]
 

output

$Aborted
 

3.11.6.3.1 Defintions of rubi rules used

rule 2009

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

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

3.11.6.4 Maple [A] (verified)

Time = 13.98 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22

method	result	size
parallelrisch	\(20 \ln \left (x^{2}\right ) \ln \left (\frac {{\mathrm e}^{5 x}}{x \left (x^{8}+2 x^{4}+1\right )}\right )\)	\(28\)
default	\(40 \ln \left (\frac {{\mathrm e}^{5 x}}{x^{9}+2 x^{5}+x}\right ) \ln \left (x \right )+40 \ln \left (x \right )^{2}-200 x \ln \left (x \right )+80 \ln \left (x \right ) \ln \left (\frac {-x +\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}{\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}\right )+80 \operatorname {dilog}\left (\frac {-x +\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}{\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}\right )+80 \ln \left (x \right ) \ln \left (\frac {-x -\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}{-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}\right )+80 \operatorname {dilog}\left (\frac {-x -\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}{-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}\right )+80 \ln \left (x \right ) \ln \left (\frac {-x -\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}{-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}\right )+80 \operatorname {dilog}\left (\frac {-x -\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}{-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}\right )+80 \ln \left (x \right ) \ln \left (\frac {-x +\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}{\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}\right )+80 \operatorname {dilog}\left (\frac {-x +\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}{\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}\right )+100 x \ln \left (x^{2}\right )-20 \ln \left (x \right ) \ln \left (x^{2}\right )+20 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\left (-2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (x^{2}\right )+4 \operatorname {dilog}\left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )+4 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )\right )\right )\)	\(368\)
parts	\(40 \ln \left (\frac {{\mathrm e}^{5 x}}{x^{9}+2 x^{5}+x}\right ) \ln \left (x \right )+40 \ln \left (x \right )^{2}-200 x \ln \left (x \right )+80 \ln \left (x \right ) \ln \left (\frac {-x +\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}{\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}\right )+80 \operatorname {dilog}\left (\frac {-x +\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}{\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}\right )+80 \ln \left (x \right ) \ln \left (\frac {-x -\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}{-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}\right )+80 \operatorname {dilog}\left (\frac {-x -\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}{-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}}\right )+80 \ln \left (x \right ) \ln \left (\frac {-x -\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}{-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}\right )+80 \operatorname {dilog}\left (\frac {-x -\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}{-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}\right )+80 \ln \left (x \right ) \ln \left (\frac {-x +\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}{\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}\right )+80 \operatorname {dilog}\left (\frac {-x +\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}{\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}}\right )+100 x \ln \left (x^{2}\right )-20 \ln \left (x \right ) \ln \left (x^{2}\right )+20 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\left (-2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (x^{2}\right )+4 \operatorname {dilog}\left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )+4 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )\right )\right )\)	\(368\)
risch	\(\text {Expression too large to display}\)	\(2623\)

input

int(((40*x^4+40)*ln(exp(5*x)/(x^9+2*x^5+x))+(100*x^5-180*x^4+100*x-20)*ln( 
x^2))/(x^5+x),x,method=_RETURNVERBOSE)
 

output

20*ln(x^2)*ln(exp(5*x)/x/(x^8+2*x^4+1))
 

3.11.6.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-20+100 x-180 x^4+100 x^5\right ) \log \left (x^2\right )+\left (40+40 x^4\right ) \log \left (\frac {e^{5 x}}{x+2 x^5+x^9}\right )}{x+x^5} \, dx=20 \, \log \left (x^{2}\right ) \log \left (\frac {e^{\left (5 \, x\right )}}{x^{9} + 2 \, x^{5} + x}\right ) \]

input

integrate(((40*x^4+40)*log(exp(5*x)/(x^9+2*x^5+x))+(100*x^5-180*x^4+100*x- 
20)*log(x^2))/(x^5+x),x, algorithm=\
 

output

20*log(x^2)*log(e^(5*x)/(x^9 + 2*x^5 + x))
 

3.11.6.6 Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-20+100 x-180 x^4+100 x^5\right ) \log \left (x^2\right )+\left (40+40 x^4\right ) \log \left (\frac {e^{5 x}}{x+2 x^5+x^9}\right )}{x+x^5} \, dx=20 \log {\left (x^{2} \right )} \log {\left (\frac {e^{5 x}}{x^{9} + 2 x^{5} + x} \right )} \]

input

integrate(((40*x**4+40)*ln(exp(5*x)/(x**9+2*x**5+x))+(100*x**5-180*x**4+10 
0*x-20)*ln(x**2))/(x**5+x),x)
 

output

20*log(x**2)*log(exp(5*x)/(x**9 + 2*x**5 + x))
 

3.11.6.7 Maxima [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-20+100 x-180 x^4+100 x^5\right ) \log \left (x^2\right )+\left (40+40 x^4\right ) \log \left (\frac {e^{5 x}}{x+2 x^5+x^9}\right )}{x+x^5} \, dx=200 \, x \log \left (x\right ) - 80 \, \log \left (x^{4} + 1\right ) \log \left (x\right ) - 40 \, \log \left (x\right )^{2} \]

input

integrate(((40*x^4+40)*log(exp(5*x)/(x^9+2*x^5+x))+(100*x^5-180*x^4+100*x- 
20)*log(x^2))/(x^5+x),x, algorithm=\
 

output

200*x*log(x) - 80*log(x^4 + 1)*log(x) - 40*log(x)^2
 

3.11.6.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {\left (-20+100 x-180 x^4+100 x^5\right ) \log \left (x^2\right )+\left (40+40 x^4\right ) \log \left (\frac {e^{5 x}}{x+2 x^5+x^9}\right )}{x+x^5} \, dx=200 \, x \log \left (x\right ) - 40 \, \log \left (x^{8} + 2 \, x^{4} + 1\right ) \log \left (x\right ) - 40 \, \log \left (x\right )^{2} \]

input

integrate(((40*x^4+40)*log(exp(5*x)/(x^9+2*x^5+x))+(100*x^5-180*x^4+100*x- 
20)*log(x^2))/(x^5+x),x, algorithm=\
 

output

200*x*log(x) - 40*log(x^8 + 2*x^4 + 1)*log(x) - 40*log(x)^2
 

3.11.6.9 Mupad [B] (verification not implemented)

Time = 17.90 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-20+100 x-180 x^4+100 x^5\right ) \log \left (x^2\right )+\left (40+40 x^4\right ) \log \left (\frac {e^{5 x}}{x+2 x^5+x^9}\right )}{x+x^5} \, dx=20\,\ln \left (x^2\right )\,\left (5\,x+\ln \left (\frac {1}{x^9+2\,x^5+x}\right )\right ) \]

input

int((log(exp(5*x)/(x + 2*x^5 + x^9))*(40*x^4 + 40) + log(x^2)*(100*x - 180 
*x^4 + 100*x^5 - 20))/(x + x^5),x)
 

output

20*log(x^2)*(5*x + log(1/(x + 2*x^5 + x^9)))