3.2.67 \(\int \frac {1+75 x^2+5 x^3-5 x^4+e^x (15 x^2+6 x^3-2 x^4)+(-75 x^2+20 x^3+e^x (-15 x^2-x^3+x^4)) \log (\frac {e^x}{2})}{1+50 x^3-10 x^4+625 x^6-250 x^7+25 x^8+e^{2 x} (25 x^6-10 x^7+x^8)+e^x (10 x^3-2 x^4+250 x^6-100 x^7+10 x^8)} \, dx\) [167]

3.2.67.1 Optimal result

Integrand size = 155, antiderivative size = 36 \begin {dmath*} \int \frac {1+75 x^2+5 x^3-5 x^4+e^x \left (15 x^2+6 x^3-2 x^4\right )+\left (-75 x^2+20 x^3+e^x \left (-15 x^2-x^3+x^4\right )\right ) \log \left (\frac {e^x}{2}\right )}{1+50 x^3-10 x^4+625 x^6-250 x^7+25 x^8+e^{2 x} \left (25 x^6-10 x^7+x^8\right )+e^x \left (10 x^3-2 x^4+250 x^6-100 x^7+10 x^8\right )} \, dx=\frac {-x+x \log \left (\frac {e^x}{2}\right )}{x \left (1+\left (5+e^x\right ) (5-x) x^3\right )} \end {dmath*}

output

(ln(1/2*exp(x))*x-x)/(1+x^3*(5-x)*(exp(x)+5))/x
 

3.2.67.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \begin {dmath*} \int \frac {1+75 x^2+5 x^3-5 x^4+e^x \left (15 x^2+6 x^3-2 x^4\right )+\left (-75 x^2+20 x^3+e^x \left (-15 x^2-x^3+x^4\right )\right ) \log \left (\frac {e^x}{2}\right )}{1+50 x^3-10 x^4+625 x^6-250 x^7+25 x^8+e^{2 x} \left (25 x^6-10 x^7+x^8\right )+e^x \left (10 x^3-2 x^4+250 x^6-100 x^7+10 x^8\right )} \, dx=\frac {1+\log (2)-\log \left (e^x\right )}{-1-5 \left (5+e^x\right ) x^3+\left (5+e^x\right ) x^4} \end {dmath*}

input

Integrate[(1 + 75*x^2 + 5*x^3 - 5*x^4 + E^x*(15*x^2 + 6*x^3 - 2*x^4) + (-7 
5*x^2 + 20*x^3 + E^x*(-15*x^2 - x^3 + x^4))*Log[E^x/2])/(1 + 50*x^3 - 10*x 
^4 + 625*x^6 - 250*x^7 + 25*x^8 + E^(2*x)*(25*x^6 - 10*x^7 + x^8) + E^x*(1 
0*x^3 - 2*x^4 + 250*x^6 - 100*x^7 + 10*x^8)),x]
 

output

(1 + Log[2] - Log[E^x])/(-1 - 5*(5 + E^x)*x^3 + (5 + E^x)*x^4)
 

3.2.67.3 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.

\(\displaystyle \int \frac {-5 x^4+5 x^3+75 x^2+e^x \left (-2 x^4+6 x^3+15 x^2\right )+\left (20 x^3-75 x^2+e^x \left (x^4-x^3-15 x^2\right )\right ) \log \left (\frac {e^x}{2}\right )+1}{25 x^8-250 x^7+625 x^6-10 x^4+50 x^3+e^{2 x} \left (x^8-10 x^7+25 x^6\right )+e^x \left (10 x^8-100 x^7+250 x^6-2 x^4+10 x^3\right )+1} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (2 e^x+5\right ) x^4\right )+\left (6 e^x+5\right ) x^3+15 \left (e^x+5\right ) x^2+\left (e^x \left (x^2-x-15\right )+20 x-75\right ) x^2 \log \left (\frac {e^x}{2}\right )+1}{\left (-\left (\left (e^x+5\right ) x^4\right )+5 \left (e^x+5\right ) x^3+1\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-2 x^2+x^2 \log \left (\frac {e^x}{2}\right )+6 x-x \log \left (\frac {e^x}{2}\right )-15 \log \left (\frac {e^x}{2}\right )+15}{(x-5) x \left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )}-\frac {\left (5 x^6-50 x^5+125 x^4-x^2+x+15\right ) \left (\log \left (\frac {e^x}{2}\right )-1\right )}{(x-5) x \left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \log \left (e^x\right ) \int \frac {1}{\left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )^2}dx-(1+\log (2)) \int \frac {1}{\left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )^2}dx+\log \left (e^x\right ) \int \frac {1}{(x-5) \left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )^2}dx-(1+\log (2)) \int \frac {1}{(x-5) \left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )^2}dx+3 \log \left (e^x\right ) \int \frac {1}{x \left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )^2}dx-(3+\log (8)) \int \frac {1}{x \left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )^2}dx+25 \log \left (\frac {e^x}{2}\right ) \int \frac {x^3}{\left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )^2}dx-25 \int \frac {x^3}{\left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )^2}dx-5 \log \left (\frac {e^x}{2}\right ) \int \frac {x^4}{\left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )^2}dx+5 \int \frac {x^4}{\left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )^2}dx+\frac {4}{5} \log \left (\frac {e^x}{2}\right ) \int \frac {1}{e^x x^4+5 x^4-5 e^x x^3-25 x^3-1}dx+\frac {1}{5} \log \left (e^x\right ) \int \frac {1}{e^x x^4+5 x^4-5 e^x x^3-25 x^3-1}dx-\frac {1}{5} \log (2) \int \frac {1}{e^x x^4+5 x^4-5 e^x x^3-25 x^3-1}dx-2 \int \frac {1}{e^x x^4+5 x^4-5 e^x x^3-25 x^3-1}dx+\log \left (\frac {e^x}{2}\right ) \int \frac {1}{(x-5) \left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )}dx-\int \frac {1}{(x-5) \left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )}dx+3 \log \left (\frac {e^x}{2}\right ) \int \frac {1}{x \left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )}dx-3 \int \frac {1}{x \left (e^x x^4+5 x^4-5 e^x x^3-25 x^3-1\right )}dx-\int \int \frac {1}{\left (-\left (\left (5+e^x\right ) x^4\right )+5 \left (5+e^x\right ) x^3+1\right )^2}dxdx-\int \int \frac {1}{(x-5) \left (-\left (\left (5+e^x\right ) x^4\right )+5 \left (5+e^x\right ) x^3+1\right )^2}dxdx-3 \int \int \frac {1}{x \left (-\left (\left (5+e^x\right ) x^4\right )+5 \left (5+e^x\right ) x^3+1\right )^2}dxdx-25 \int \int \frac {x^3}{\left (-\left (\left (5+e^x\right ) x^4\right )+5 \left (5+e^x\right ) x^3+1\right )^2}dxdx+5 \int \int \frac {x^4}{\left (-\left (\left (5+e^x\right ) x^4\right )+5 \left (5+e^x\right ) x^3+1\right )^2}dxdx-\int \int \frac {1}{\left (5+e^x\right ) x^4-5 \left (5+e^x\right ) x^3-1}dxdx-\int \int \frac {1}{(x-5) \left (\left (5+e^x\right ) x^4-5 \left (5+e^x\right ) x^3-1\right )}dxdx-3 \int \int \frac {1}{x \left (\left (5+e^x\right ) x^4-5 \left (5+e^x\right ) x^3-1\right )}dxdx\)

input

Int[(1 + 75*x^2 + 5*x^3 - 5*x^4 + E^x*(15*x^2 + 6*x^3 - 2*x^4) + (-75*x^2 
+ 20*x^3 + E^x*(-15*x^2 - x^3 + x^4))*Log[E^x/2])/(1 + 50*x^3 - 10*x^4 + 6 
25*x^6 - 250*x^7 + 25*x^8 + E^(2*x)*(25*x^6 - 10*x^7 + x^8) + E^x*(10*x^3 
- 2*x^4 + 250*x^6 - 100*x^7 + 10*x^8)),x]
 

output

$Aborted
 

3.2.67.3.1 Defintions of rubi rules used

rule 2009

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

rule 7239

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.2.67.4 Maple [A] (verified)

Time = 1.77 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {1-\ln \left (\frac {{\mathrm e}^{x}}{2}\right )}{{\mathrm e}^{x} x^{4}-5 \,{\mathrm e}^{x} x^{3}+5 x^{4}-25 x^{3}-1}\)	\(38\)
parallelrisch	\(-\frac {-60+60 \ln \left (\frac {{\mathrm e}^{x}}{2}\right )}{60 \left ({\mathrm e}^{x} x^{4}-5 \,{\mathrm e}^{x} x^{3}+5 x^{4}-25 x^{3}-1\right )}\)	\(39\)
risch	\(-\frac {\ln \left ({\mathrm e}^{x}\right )}{{\mathrm e}^{x} x^{4}-5 \,{\mathrm e}^{x} x^{3}+5 x^{4}-25 x^{3}-1}+\frac {1}{{\mathrm e}^{x} x^{4}-5 \,{\mathrm e}^{x} x^{3}+5 x^{4}-25 x^{3}-1}+\frac {\ln \left (2\right )}{{\mathrm e}^{x} x^{4}-5 \,{\mathrm e}^{x} x^{3}+5 x^{4}-25 x^{3}-1}\)	\(91\)

input

int((((x^4-x^3-15*x^2)*exp(x)+20*x^3-75*x^2)*ln(1/2*exp(x))+(-2*x^4+6*x^3+ 
15*x^2)*exp(x)-5*x^4+5*x^3+75*x^2+1)/((x^8-10*x^7+25*x^6)*exp(x)^2+(10*x^8 
-100*x^7+250*x^6-2*x^4+10*x^3)*exp(x)+25*x^8-250*x^7+625*x^6-10*x^4+50*x^3 
+1),x,method=_RETURNVERBOSE)
 

output

(1-ln(1/2*exp(x)))/(exp(x)*x^4-5*exp(x)*x^3+5*x^4-25*x^3-1)
 

3.2.67.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \begin {dmath*} \int \frac {1+75 x^2+5 x^3-5 x^4+e^x \left (15 x^2+6 x^3-2 x^4\right )+\left (-75 x^2+20 x^3+e^x \left (-15 x^2-x^3+x^4\right )\right ) \log \left (\frac {e^x}{2}\right )}{1+50 x^3-10 x^4+625 x^6-250 x^7+25 x^8+e^{2 x} \left (25 x^6-10 x^7+x^8\right )+e^x \left (10 x^3-2 x^4+250 x^6-100 x^7+10 x^8\right )} \, dx=-\frac {x - \log \left (2\right ) - 1}{5 \, x^{4} - 25 \, x^{3} + {\left (x^{4} - 5 \, x^{3}\right )} e^{x} - 1} \end {dmath*}

input

integrate((((x^4-x^3-15*x^2)*exp(x)+20*x^3-75*x^2)*log(1/2*exp(x))+(-2*x^4 
+6*x^3+15*x^2)*exp(x)-5*x^4+5*x^3+75*x^2+1)/((x^8-10*x^7+25*x^6)*exp(x)^2+ 
(10*x^8-100*x^7+250*x^6-2*x^4+10*x^3)*exp(x)+25*x^8-250*x^7+625*x^6-10*x^4 
+50*x^3+1),x, algorithm=\
 

output

-(x - log(2) - 1)/(5*x^4 - 25*x^3 + (x^4 - 5*x^3)*e^x - 1)
 

3.2.67.6 Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \begin {dmath*} \int \frac {1+75 x^2+5 x^3-5 x^4+e^x \left (15 x^2+6 x^3-2 x^4\right )+\left (-75 x^2+20 x^3+e^x \left (-15 x^2-x^3+x^4\right )\right ) \log \left (\frac {e^x}{2}\right )}{1+50 x^3-10 x^4+625 x^6-250 x^7+25 x^8+e^{2 x} \left (25 x^6-10 x^7+x^8\right )+e^x \left (10 x^3-2 x^4+250 x^6-100 x^7+10 x^8\right )} \, dx=\frac {- x + \log {\left (2 \right )} + 1}{5 x^{4} - 25 x^{3} + \left (x^{4} - 5 x^{3}\right ) e^{x} - 1} \end {dmath*}

input

integrate((((x**4-x**3-15*x**2)*exp(x)+20*x**3-75*x**2)*ln(1/2*exp(x))+(-2 
*x**4+6*x**3+15*x**2)*exp(x)-5*x**4+5*x**3+75*x**2+1)/((x**8-10*x**7+25*x* 
*6)*exp(x)**2+(10*x**8-100*x**7+250*x**6-2*x**4+10*x**3)*exp(x)+25*x**8-25 
0*x**7+625*x**6-10*x**4+50*x**3+1),x)
 

output

(-x + log(2) + 1)/(5*x**4 - 25*x**3 + (x**4 - 5*x**3)*exp(x) - 1)
 

3.2.67.7 Maxima [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \begin {dmath*} \int \frac {1+75 x^2+5 x^3-5 x^4+e^x \left (15 x^2+6 x^3-2 x^4\right )+\left (-75 x^2+20 x^3+e^x \left (-15 x^2-x^3+x^4\right )\right ) \log \left (\frac {e^x}{2}\right )}{1+50 x^3-10 x^4+625 x^6-250 x^7+25 x^8+e^{2 x} \left (25 x^6-10 x^7+x^8\right )+e^x \left (10 x^3-2 x^4+250 x^6-100 x^7+10 x^8\right )} \, dx=-\frac {x - \log \left (2\right ) - 1}{5 \, x^{4} - 25 \, x^{3} + {\left (x^{4} - 5 \, x^{3}\right )} e^{x} - 1} \end {dmath*}

input

integrate((((x^4-x^3-15*x^2)*exp(x)+20*x^3-75*x^2)*log(1/2*exp(x))+(-2*x^4 
+6*x^3+15*x^2)*exp(x)-5*x^4+5*x^3+75*x^2+1)/((x^8-10*x^7+25*x^6)*exp(x)^2+ 
(10*x^8-100*x^7+250*x^6-2*x^4+10*x^3)*exp(x)+25*x^8-250*x^7+625*x^6-10*x^4 
+50*x^3+1),x, algorithm=\
 

output

-(x - log(2) - 1)/(5*x^4 - 25*x^3 + (x^4 - 5*x^3)*e^x - 1)
 

3.2.67.8 Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {1+75 x^2+5 x^3-5 x^4+e^x \left (15 x^2+6 x^3-2 x^4\right )+\left (-75 x^2+20 x^3+e^x \left (-15 x^2-x^3+x^4\right )\right ) \log \left (\frac {e^x}{2}\right )}{1+50 x^3-10 x^4+625 x^6-250 x^7+25 x^8+e^{2 x} \left (25 x^6-10 x^7+x^8\right )+e^x \left (10 x^3-2 x^4+250 x^6-100 x^7+10 x^8\right )} \, dx=-\frac {x - \log \left (2\right ) - 1}{x^{4} e^{x} + 5 \, x^{4} - 5 \, x^{3} e^{x} - 25 \, x^{3} - 1} \end {dmath*}

input

integrate((((x^4-x^3-15*x^2)*exp(x)+20*x^3-75*x^2)*log(1/2*exp(x))+(-2*x^4 
+6*x^3+15*x^2)*exp(x)-5*x^4+5*x^3+75*x^2+1)/((x^8-10*x^7+25*x^6)*exp(x)^2+ 
(10*x^8-100*x^7+250*x^6-2*x^4+10*x^3)*exp(x)+25*x^8-250*x^7+625*x^6-10*x^4 
+50*x^3+1),x, algorithm=\
 

output

-(x - log(2) - 1)/(x^4*e^x + 5*x^4 - 5*x^3*e^x - 25*x^3 - 1)
 

3.2.67.9 Mupad [F(-1)]

Timed out. \begin {dmath*} \int \frac {1+75 x^2+5 x^3-5 x^4+e^x \left (15 x^2+6 x^3-2 x^4\right )+\left (-75 x^2+20 x^3+e^x \left (-15 x^2-x^3+x^4\right )\right ) \log \left (\frac {e^x}{2}\right )}{1+50 x^3-10 x^4+625 x^6-250 x^7+25 x^8+e^{2 x} \left (25 x^6-10 x^7+x^8\right )+e^x \left (10 x^3-2 x^4+250 x^6-100 x^7+10 x^8\right )} \, dx=\int \frac {{\mathrm {e}}^x\,\left (-2\,x^4+6\,x^3+15\,x^2\right )-\ln \left (\frac {{\mathrm {e}}^x}{2}\right )\,\left (75\,x^2-20\,x^3+{\mathrm {e}}^x\,\left (-x^4+x^3+15\,x^2\right )\right )+75\,x^2+5\,x^3-5\,x^4+1}{{\mathrm {e}}^{2\,x}\,\left (x^8-10\,x^7+25\,x^6\right )+{\mathrm {e}}^x\,\left (10\,x^8-100\,x^7+250\,x^6-2\,x^4+10\,x^3\right )+50\,x^3-10\,x^4+625\,x^6-250\,x^7+25\,x^8+1} \,d x \end {dmath*}

input

int((exp(x)*(15*x^2 + 6*x^3 - 2*x^4) - log(exp(x)/2)*(75*x^2 - 20*x^3 + ex 
p(x)*(15*x^2 + x^3 - x^4)) + 75*x^2 + 5*x^3 - 5*x^4 + 1)/(exp(2*x)*(25*x^6 
 - 10*x^7 + x^8) + exp(x)*(10*x^3 - 2*x^4 + 250*x^6 - 100*x^7 + 10*x^8) + 
50*x^3 - 10*x^4 + 625*x^6 - 250*x^7 + 25*x^8 + 1),x)
 

output

int((exp(x)*(15*x^2 + 6*x^3 - 2*x^4) - log(exp(x)/2)*(75*x^2 - 20*x^3 + ex 
p(x)*(15*x^2 + x^3 - x^4)) + 75*x^2 + 5*x^3 - 5*x^4 + 1)/(exp(2*x)*(25*x^6 
 - 10*x^7 + x^8) + exp(x)*(10*x^3 - 2*x^4 + 250*x^6 - 100*x^7 + 10*x^8) + 
50*x^3 - 10*x^4 + 625*x^6 - 250*x^7 + 25*x^8 + 1), x)