3.13.12 \(\int \frac {e^8+e^{2 x}+150 x^2-2 e^4 x^2+x^4+e^x (2 e^4-75 x-2 x^2)}{e^8 x+e^{2 x} x-2 e^4 x^3+x^5+e^x (2 e^4 x-2 x^3)} \, dx\) [1212]

3.13.12.1 Optimal result

Integrand size = 86, antiderivative size = 28 \begin {dmath*} \int \frac {e^8+e^{2 x}+150 x^2-2 e^4 x^2+x^4+e^x \left (2 e^4-75 x-2 x^2\right )}{e^8 x+e^{2 x} x-2 e^4 x^3+x^5+e^x \left (2 e^4 x-2 x^3\right )} \, dx=4+\frac {75}{e^4+e^x-x^2}-\log (4)-\log \left (\frac {1}{x}\right ) \end {dmath*}

output

5/(1/15*exp(4)+1/15*exp(x)-1/15*x^2)-2*ln(2)+4-ln(1/x)
 

3.13.12.2 Mathematica [A] (verified)

Time = 2.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \begin {dmath*} \int \frac {e^8+e^{2 x}+150 x^2-2 e^4 x^2+x^4+e^x \left (2 e^4-75 x-2 x^2\right )}{e^8 x+e^{2 x} x-2 e^4 x^3+x^5+e^x \left (2 e^4 x-2 x^3\right )} \, dx=\frac {75}{e^4+e^x-x^2}+\log (x) \end {dmath*}

input

Integrate[(E^8 + E^(2*x) + 150*x^2 - 2*E^4*x^2 + x^4 + E^x*(2*E^4 - 75*x - 
 2*x^2))/(E^8*x + E^(2*x)*x - 2*E^4*x^3 + x^5 + E^x*(2*E^4*x - 2*x^3)),x]
 

output

75/(E^4 + E^x - x^2) + Log[x]
 

3.13.12.3 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.

\(\displaystyle \int \frac {x^4-2 e^4 x^2+150 x^2+e^x \left (-2 x^2-75 x+2 e^4\right )+e^{2 x}+e^8}{x^5-2 e^4 x^3+e^x \left (2 e^4 x-2 x^3\right )+e^{2 x} x+e^8 x} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {x^4+\left (150-2 e^4\right ) x^2+e^x \left (-2 x^2-75 x+2 e^4\right )+e^{2 x}+e^8}{x^5-2 e^4 x^3+e^x \left (2 e^4 x-2 x^3\right )+e^{2 x} x+e^8 x}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {x^4+\left (150-2 e^4\right ) x^2+e^x \left (-2 x^2-75 x+2 e^4\right )+e^{2 x}+e^8}{x \left (-x^2+e^x+e^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {75 \left (-x^2+2 x+e^4\right )}{\left (-x^2+e^x+e^4\right )^2}-\frac {75}{-x^2+e^x+e^4}+\frac {1}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 75 e^4 \int \frac {1}{\left (-x^2+e^x+e^4\right )^2}dx-75 \int \frac {1}{-x^2+e^x+e^4}dx+150 \int \frac {x}{\left (x^2-e^x-e^4\right )^2}dx-75 \int \frac {x^2}{\left (x^2-e^x-e^4\right )^2}dx+\log (x)\)

input

Int[(E^8 + E^(2*x) + 150*x^2 - 2*E^4*x^2 + x^4 + E^x*(2*E^4 - 75*x - 2*x^2 
))/(E^8*x + E^(2*x)*x - 2*E^4*x^3 + x^5 + E^x*(2*E^4*x - 2*x^3)),x]
 

output

$Aborted
 

3.13.12.3.1 Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

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

rule 7292

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

rule 7293

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

3.13.12.4 Maple [A] (verified)

Time = 1.74 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64

method	result	size
norman	\(\frac {75}{-x^{2}+{\mathrm e}^{4}+{\mathrm e}^{x}}+\ln \left (x \right )\)	\(18\)
risch	\(\frac {75}{-x^{2}+{\mathrm e}^{4}+{\mathrm e}^{x}}+\ln \left (x \right )\)	\(18\)
parallelrisch	\(\frac {-x^{2} \ln \left (x \right )+{\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+75}{-x^{2}+{\mathrm e}^{4}+{\mathrm e}^{x}}\)	\(33\)

input

int((exp(x)^2+(2*exp(4)-2*x^2-75*x)*exp(x)+exp(4)^2-2*x^2*exp(4)+x^4+150*x 
^2)/(x*exp(x)^2+(2*x*exp(4)-2*x^3)*exp(x)+x*exp(4)^2-2*x^3*exp(4)+x^5),x,m 
ethod=_RETURNVERBOSE)
 

output

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

3.13.12.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \begin {dmath*} \int \frac {e^8+e^{2 x}+150 x^2-2 e^4 x^2+x^4+e^x \left (2 e^4-75 x-2 x^2\right )}{e^8 x+e^{2 x} x-2 e^4 x^3+x^5+e^x \left (2 e^4 x-2 x^3\right )} \, dx=\frac {{\left (x^{2} - e^{4} - e^{x}\right )} \log \left (x\right ) - 75}{x^{2} - e^{4} - e^{x}} \end {dmath*}

input

integrate((exp(x)^2+(2*exp(4)-2*x^2-75*x)*exp(x)+exp(4)^2-2*x^2*exp(4)+x^4 
+150*x^2)/(x*exp(x)^2+(2*x*exp(4)-2*x^3)*exp(x)+x*exp(4)^2-2*x^3*exp(4)+x^ 
5),x, algorithm=\
 

output

((x^2 - e^4 - e^x)*log(x) - 75)/(x^2 - e^4 - e^x)
 

3.13.12.6 Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.50 \begin {dmath*} \int \frac {e^8+e^{2 x}+150 x^2-2 e^4 x^2+x^4+e^x \left (2 e^4-75 x-2 x^2\right )}{e^8 x+e^{2 x} x-2 e^4 x^3+x^5+e^x \left (2 e^4 x-2 x^3\right )} \, dx=\log {\left (x \right )} + \frac {75}{- x^{2} + e^{x} + e^{4}} \end {dmath*}

input

integrate((exp(x)**2+(2*exp(4)-2*x**2-75*x)*exp(x)+exp(4)**2-2*x**2*exp(4) 
+x**4+150*x**2)/(x*exp(x)**2+(2*x*exp(4)-2*x**3)*exp(x)+x*exp(4)**2-2*x**3 
*exp(4)+x**5),x)
 

output

log(x) + 75/(-x**2 + exp(x) + exp(4))
 

3.13.12.7 Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \begin {dmath*} \int \frac {e^8+e^{2 x}+150 x^2-2 e^4 x^2+x^4+e^x \left (2 e^4-75 x-2 x^2\right )}{e^8 x+e^{2 x} x-2 e^4 x^3+x^5+e^x \left (2 e^4 x-2 x^3\right )} \, dx=-\frac {75}{x^{2} - e^{4} - e^{x}} + \log \left (x\right ) \end {dmath*}

input

integrate((exp(x)^2+(2*exp(4)-2*x^2-75*x)*exp(x)+exp(4)^2-2*x^2*exp(4)+x^4 
+150*x^2)/(x*exp(x)^2+(2*x*exp(4)-2*x^3)*exp(x)+x*exp(4)^2-2*x^3*exp(4)+x^ 
5),x, algorithm=\
 

output

-75/(x^2 - e^4 - e^x) + log(x)
 

3.13.12.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \begin {dmath*} \int \frac {e^8+e^{2 x}+150 x^2-2 e^4 x^2+x^4+e^x \left (2 e^4-75 x-2 x^2\right )}{e^8 x+e^{2 x} x-2 e^4 x^3+x^5+e^x \left (2 e^4 x-2 x^3\right )} \, dx=\frac {x^{2} \log \left (x\right ) - e^{4} \log \left (x\right ) - e^{x} \log \left (x\right ) - 75}{x^{2} - e^{4} - e^{x}} \end {dmath*}

input

integrate((exp(x)^2+(2*exp(4)-2*x^2-75*x)*exp(x)+exp(4)^2-2*x^2*exp(4)+x^4 
+150*x^2)/(x*exp(x)^2+(2*x*exp(4)-2*x^3)*exp(x)+x*exp(4)^2-2*x^3*exp(4)+x^ 
5),x, algorithm=\
 

output

(x^2*log(x) - e^4*log(x) - e^x*log(x) - 75)/(x^2 - e^4 - e^x)
 

3.13.12.9 Mupad [B] (verification not implemented)

Time = 14.97 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \begin {dmath*} \int \frac {e^8+e^{2 x}+150 x^2-2 e^4 x^2+x^4+e^x \left (2 e^4-75 x-2 x^2\right )}{e^8 x+e^{2 x} x-2 e^4 x^3+x^5+e^x \left (2 e^4 x-2 x^3\right )} \, dx=\ln \left (x\right )+\frac {75}{{\mathrm {e}}^4+{\mathrm {e}}^x-x^2} \end {dmath*}

input

int((exp(2*x) + exp(8) - exp(x)*(75*x - 2*exp(4) + 2*x^2) - 2*x^2*exp(4) + 
 150*x^2 + x^4)/(exp(x)*(2*x*exp(4) - 2*x^3) + x*exp(2*x) + x*exp(8) - 2*x 
^3*exp(4) + x^5),x)
 

output

log(x) + 75/(exp(4) + exp(x) - x^2)