3.1.64 \(\int \frac {e^{\frac {1}{\log ^2(4-40 x+36 x^2+320 x^3+256 x^4)}} (-20-44 x+64 x^2+(-1+5 x+8 x^2) \log ^3(4-40 x+36 x^2+320 x^3+256 x^4))}{(-1+7 x-3 x^2-11 x^3+8 x^4) \log ^3(4-40 x+36 x^2+320 x^3+256 x^4)} \, dx\) [64]

3.1.64.1 Optimal result

Integrand size = 115, antiderivative size = 29 \[ \int \frac {e^{\frac {1}{\log ^2\left (4-40 x+36 x^2+320 x^3+256 x^4\right )}} \left (-20-44 x+64 x^2+\left (-1+5 x+8 x^2\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )\right )}{\left (-1+7 x-3 x^2-11 x^3+8 x^4\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )} \, dx=\frac {e^{\frac {1}{\log ^2\left (\left (3-2 x-(1+4 x)^2\right )^2\right )}}}{1-x} \]

output

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

3.1.64.2 Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {1}{\log ^2\left (4-40 x+36 x^2+320 x^3+256 x^4\right )}} \left (-20-44 x+64 x^2+\left (-1+5 x+8 x^2\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )\right )}{\left (-1+7 x-3 x^2-11 x^3+8 x^4\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )} \, dx=-\frac {e^{\frac {1}{\log ^2\left (4 \left (-1+5 x+8 x^2\right )^2\right )}}}{-1+x} \]

input

Integrate[(E^Log[4 - 40*x + 36*x^2 + 320*x^3 + 256*x^4]^(-2)*(-20 - 44*x + 
 64*x^2 + (-1 + 5*x + 8*x^2)*Log[4 - 40*x + 36*x^2 + 320*x^3 + 256*x^4]^3) 
)/((-1 + 7*x - 3*x^2 - 11*x^3 + 8*x^4)*Log[4 - 40*x + 36*x^2 + 320*x^3 + 2 
56*x^4]^3),x]
 

output

-(E^Log[4*(-1 + 5*x + 8*x^2)^2]^(-2)/(-1 + x))
 

3.1.64.3 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.

\(\displaystyle \int \frac {e^{\frac {1}{\log ^2\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}} \left (64 x^2+\left (8 x^2+5 x-1\right ) \log ^3\left (256 x^4+320 x^3+36 x^2-40 x+4\right )-44 x-20\right )}{\left (8 x^4-11 x^3-3 x^2+7 x-1\right ) \log ^3\left (256 x^4+320 x^3+36 x^2-40 x+4\right )} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (-\frac {7 e^{\frac {1}{\log ^2\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}} \left (64 x^2+\left (8 x^2+5 x-1\right ) \log ^3\left (256 x^4+320 x^3+36 x^2-40 x+4\right )-44 x-20\right )}{48 (x-1) \log ^3\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}+\frac {(56 x+59) e^{\frac {1}{\log ^2\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}} \left (64 x^2+\left (8 x^2+5 x-1\right ) \log ^3\left (256 x^4+320 x^3+36 x^2-40 x+4\right )-44 x-20\right )}{48 \left (8 x^2+5 x-1\right ) \log ^3\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}+\frac {e^{\frac {1}{\log ^2\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}} \left (64 x^2+\left (8 x^2+5 x-1\right ) \log ^3\left (256 x^4+320 x^3+36 x^2-40 x+4\right )-44 x-20\right )}{12 (x-1)^2 \log ^3\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {e^{\frac {1}{\log ^2\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}}}{(x-1)^2}dx+144 \sqrt {\frac {3}{19}} \int \frac {e^{\frac {1}{\log ^2\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}}}{\left (-16 x+\sqrt {57}-5\right ) \log ^3\left (4 \left (8 x^2+5 x-1\right )^2\right )}dx+7 \int \frac {e^{\frac {1}{\log ^2\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}}}{(x-1) \log ^3\left (4 \left (8 x^2+5 x-1\right )^2\right )}dx-\frac {56}{57} \left (57-5 \sqrt {57}\right ) \int \frac {e^{\frac {1}{\log ^2\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}}}{\left (16 x-\sqrt {57}+5\right ) \log ^3\left (4 \left (8 x^2+5 x-1\right )^2\right )}dx-\frac {56}{57} \left (57+5 \sqrt {57}\right ) \int \frac {e^{\frac {1}{\log ^2\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}}}{\left (16 x+\sqrt {57}+5\right ) \log ^3\left (4 \left (8 x^2+5 x-1\right )^2\right )}dx+144 \sqrt {\frac {3}{19}} \int \frac {e^{\frac {1}{\log ^2\left (256 x^4+320 x^3+36 x^2-40 x+4\right )}}}{\left (16 x+\sqrt {57}+5\right ) \log ^3\left (4 \left (8 x^2+5 x-1\right )^2\right )}dx\)

input

Int[(E^Log[4 - 40*x + 36*x^2 + 320*x^3 + 256*x^4]^(-2)*(-20 - 44*x + 64*x^ 
2 + (-1 + 5*x + 8*x^2)*Log[4 - 40*x + 36*x^2 + 320*x^3 + 256*x^4]^3))/((-1 
 + 7*x - 3*x^2 - 11*x^3 + 8*x^4)*Log[4 - 40*x + 36*x^2 + 320*x^3 + 256*x^4 
]^3),x]
 

output

$Aborted
 

3.1.64.3.1 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]
 

3.1.64.4 Maple [A] (verified)

Time = 1.70 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10

method	result	size
risch	\(-\frac {{\mathrm e}^{\frac {1}{\ln \left (256 x^{4}+320 x^{3}+36 x^{2}-40 x +4\right )^{2}}}}{-1+x}\)	\(32\)
parallelrisch	\(-\frac {{\mathrm e}^{\frac {1}{\ln \left (256 x^{4}+320 x^{3}+36 x^{2}-40 x +4\right )^{2}}}}{-1+x}\)	\(32\)

input

int(((8*x^2+5*x-1)*ln(256*x^4+320*x^3+36*x^2-40*x+4)^3+64*x^2-44*x-20)*exp 
(1/ln(256*x^4+320*x^3+36*x^2-40*x+4)^2)/(8*x^4-11*x^3-3*x^2+7*x-1)/ln(256* 
x^4+320*x^3+36*x^2-40*x+4)^3,x,method=_RETURNVERBOSE)
 

output

-1/(-1+x)*exp(1/ln(256*x^4+320*x^3+36*x^2-40*x+4)^2)
 

3.1.64.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\frac {1}{\log ^2\left (4-40 x+36 x^2+320 x^3+256 x^4\right )}} \left (-20-44 x+64 x^2+\left (-1+5 x+8 x^2\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )\right )}{\left (-1+7 x-3 x^2-11 x^3+8 x^4\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )} \, dx=-\frac {e^{\left (\frac {1}{\log \left (256 \, x^{4} + 320 \, x^{3} + 36 \, x^{2} - 40 \, x + 4\right )^{2}}\right )}}{x - 1} \]

input

integrate(((8*x^2+5*x-1)*log(256*x^4+320*x^3+36*x^2-40*x+4)^3+64*x^2-44*x- 
20)*exp(1/log(256*x^4+320*x^3+36*x^2-40*x+4)^2)/(8*x^4-11*x^3-3*x^2+7*x-1) 
/log(256*x^4+320*x^3+36*x^2-40*x+4)^3,x, algorithm=\
 

output

-e^(log(256*x^4 + 320*x^3 + 36*x^2 - 40*x + 4)^(-2))/(x - 1)
 

3.1.64.6 Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\frac {1}{\log ^2\left (4-40 x+36 x^2+320 x^3+256 x^4\right )}} \left (-20-44 x+64 x^2+\left (-1+5 x+8 x^2\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )\right )}{\left (-1+7 x-3 x^2-11 x^3+8 x^4\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )} \, dx=- \frac {e^{\frac {1}{\log {\left (256 x^{4} + 320 x^{3} + 36 x^{2} - 40 x + 4 \right )}^{2}}}}{x - 1} \]

input

integrate(((8*x**2+5*x-1)*ln(256*x**4+320*x**3+36*x**2-40*x+4)**3+64*x**2- 
44*x-20)*exp(1/ln(256*x**4+320*x**3+36*x**2-40*x+4)**2)/(8*x**4-11*x**3-3* 
x**2+7*x-1)/ln(256*x**4+320*x**3+36*x**2-40*x+4)**3,x)
 

output

-exp(log(256*x**4 + 320*x**3 + 36*x**2 - 40*x + 4)**(-2))/(x - 1)
 

3.1.64.7 Maxima [F]

\[ \int \frac {e^{\frac {1}{\log ^2\left (4-40 x+36 x^2+320 x^3+256 x^4\right )}} \left (-20-44 x+64 x^2+\left (-1+5 x+8 x^2\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )\right )}{\left (-1+7 x-3 x^2-11 x^3+8 x^4\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )} \, dx=\int { \frac {{\left ({\left (8 \, x^{2} + 5 \, x - 1\right )} \log \left (256 \, x^{4} + 320 \, x^{3} + 36 \, x^{2} - 40 \, x + 4\right )^{3} + 64 \, x^{2} - 44 \, x - 20\right )} e^{\left (\frac {1}{\log \left (256 \, x^{4} + 320 \, x^{3} + 36 \, x^{2} - 40 \, x + 4\right )^{2}}\right )}}{{\left (8 \, x^{4} - 11 \, x^{3} - 3 \, x^{2} + 7 \, x - 1\right )} \log \left (256 \, x^{4} + 320 \, x^{3} + 36 \, x^{2} - 40 \, x + 4\right )^{3}} \,d x } \]

input

integrate(((8*x^2+5*x-1)*log(256*x^4+320*x^3+36*x^2-40*x+4)^3+64*x^2-44*x- 
20)*exp(1/log(256*x^4+320*x^3+36*x^2-40*x+4)^2)/(8*x^4-11*x^3-3*x^2+7*x-1) 
/log(256*x^4+320*x^3+36*x^2-40*x+4)^3,x, algorithm=\
 

output

-16*x^2*e^(1/4/(log(2)^2 + 2*log(2)*log(8*x^2 + 5*x - 1) + log(8*x^2 + 5*x 
 - 1)^2))/(16*x^3 - 27*x^2 + 6*x + 5) + 11*x*e^(1/4/(log(2)^2 + 2*log(2)*l 
og(8*x^2 + 5*x - 1) + log(8*x^2 + 5*x - 1)^2))/(16*x^3 - 27*x^2 + 6*x + 5) 
 + 5*e^(1/4/(log(2)^2 + 2*log(2)*log(8*x^2 + 5*x - 1) + log(8*x^2 + 5*x - 
1)^2))/(16*x^3 - 27*x^2 + 6*x + 5) + integrate(e^(1/4/(log(2)^2 + 2*log(2) 
*log(8*x^2 + 5*x - 1) + log(8*x^2 + 5*x - 1)^2))/(x^2 - 2*x + 1), x)
 

3.1.64.8 Giac [A] (verification not implemented)

Time = 2.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\frac {1}{\log ^2\left (4-40 x+36 x^2+320 x^3+256 x^4\right )}} \left (-20-44 x+64 x^2+\left (-1+5 x+8 x^2\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )\right )}{\left (-1+7 x-3 x^2-11 x^3+8 x^4\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )} \, dx=-\frac {e^{\left (\frac {1}{\log \left (256 \, x^{4} + 320 \, x^{3} + 36 \, x^{2} - 40 \, x + 4\right )^{2}}\right )}}{x - 1} \]

input

integrate(((8*x^2+5*x-1)*log(256*x^4+320*x^3+36*x^2-40*x+4)^3+64*x^2-44*x- 
20)*exp(1/log(256*x^4+320*x^3+36*x^2-40*x+4)^2)/(8*x^4-11*x^3-3*x^2+7*x-1) 
/log(256*x^4+320*x^3+36*x^2-40*x+4)^3,x, algorithm=\
 

output

-e^(log(256*x^4 + 320*x^3 + 36*x^2 - 40*x + 4)^(-2))/(x - 1)
 

3.1.64.9 Mupad [B] (verification not implemented)

Time = 9.82 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\frac {1}{\log ^2\left (4-40 x+36 x^2+320 x^3+256 x^4\right )}} \left (-20-44 x+64 x^2+\left (-1+5 x+8 x^2\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )\right )}{\left (-1+7 x-3 x^2-11 x^3+8 x^4\right ) \log ^3\left (4-40 x+36 x^2+320 x^3+256 x^4\right )} \, dx=-\frac {{\mathrm {e}}^{\frac {1}{{\ln \left (256\,x^4+320\,x^3+36\,x^2-40\,x+4\right )}^2}}}{x-1} \]

input

int((exp(1/log(36*x^2 - 40*x + 320*x^3 + 256*x^4 + 4)^2)*(44*x - log(36*x^ 
2 - 40*x + 320*x^3 + 256*x^4 + 4)^3*(5*x + 8*x^2 - 1) - 64*x^2 + 20))/(log 
(36*x^2 - 40*x + 320*x^3 + 256*x^4 + 4)^3*(3*x^2 - 7*x + 11*x^3 - 8*x^4 + 
1)),x)
 

output

-exp(1/log(36*x^2 - 40*x + 320*x^3 + 256*x^4 + 4)^2)/(x - 1)