\(\int \frac {(-7-x-e x) \log (49+14 x+x^2+e^2 x^2+e (14 x+2 x^2))+(4 x^4+4 e x^4) \log (4 \log (5) \log (49+14 x+x^2+e^2 x^2+e (14 x+2 x^2)))+(28 x^3+4 x^4+4 e x^4) \log (49+14 x+x^2+e^2 x^2+e (14 x+2 x^2)) \log ^2(4 \log (5) \log (49+14 x+x^2+e^2 x^2+e (14 x+2 x^2)))}{(7+x+e x) \log (49+14 x+x^2+e^2 x^2+e (14 x+2 x^2))} \, dx\) [2588]

Optimal result

Integrand size = 200, antiderivative size = 24 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=-x+x^4 \log ^2\left (4 \log (5) \log \left ((7+x+e x)^2\right )\right ) \] Output:

ln(4*ln((x*exp(1)+x+7)^2)*ln(5))^2*x^4-x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=-x+x^4 \log ^2\left (4 \log (5) \log \left ((7+x+e x)^2\right )\right ) \] Input:

Integrate[((-7 - x - E*x)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2) 
] + (4*x^4 + 4*E*x^4)*Log[4*Log[5]*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x 
 + 2*x^2)]] + (28*x^3 + 4*x^4 + 4*E*x^4)*Log[49 + 14*x + x^2 + E^2*x^2 + E 
*(14*x + 2*x^2)]*Log[4*Log[5]*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2* 
x^2)]]^2)/((7 + x + E*x)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)] 
),x]
 

Output:

-x + x^4*Log[4*Log[5]*Log[(7 + x + E*x)^2]]^2
 

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.

\(\displaystyle \int \frac {(-e x-x-7) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )+\left (4 e x^4+4 x^4\right ) \log \left (4 \log (5) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )\right )+\left (4 e x^4+4 x^4+28 x^3\right ) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right ) \log ^2\left (4 \log (5) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )\right )}{(e x+x+7) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {(-e x-x-7) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )+\left (4 e x^4+4 x^4\right ) \log \left (4 \log (5) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )\right )+\left (4 e x^4+4 x^4+28 x^3\right ) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right ) \log ^2\left (4 \log (5) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )\right )}{((1+e) x+7) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (\frac {4 (1+e) x^4 \log \left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )}{((1+e) x+7) \log \left (((1+e) x+7)^2\right )}+4 x^3 \log ^2\left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )-1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1372 \text {Subst}\left (\int \frac {\log \left (4 \log (5) \log \left (x^2\right )\right )}{\log \left (x^2\right )}dx,x,(1+e) x+7\right )}{(1+e)^4}+4 \int x^3 \log ^2\left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )dx+4 \int \frac {x^3 \log \left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )}{\log \left (((1+e) x+7)^2\right )}dx-\frac {28 \int \frac {x^2 \log \left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )}{\log \left (((1+e) x+7)^2\right )}dx}{1+e}+\frac {196 \int \frac {x \log \left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )}{\log \left (((1+e) x+7)^2\right )}dx}{(1+e)^2}-x+\frac {2401 \log ^2\left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )}{(1+e)^4}\)

Input:

Int[((-7 - x - E*x)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)] + (4 
*x^4 + 4*E*x^4)*Log[4*Log[5]*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x 
^2)]] + (28*x^3 + 4*x^4 + 4*E*x^4)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x 
 + 2*x^2)]*Log[4*Log[5]*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)]] 
^2)/((7 + x + E*x)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88

method	result	size
parallelrisch	\({\ln \left (4 \ln \left (5\right ) \ln \left (x^{2} {\mathrm e}^{2}+\left (2 x^{2}+14 x \right ) {\mathrm e}+x^{2}+14 x +49\right )\right )}^{2} x^{4}-x\)	\(45\)

Input:

int(((4*x^4*exp(1)+4*x^4+28*x^3)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+1 
4*x+49)*ln(4*ln(5)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))^2+(4* 
x^4*exp(1)+4*x^4)*ln(4*ln(5)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+ 
49))+(-x*exp(1)-x-7)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))/(x* 
exp(1)+x+7)/ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49),x,method=_RET 
URNVERBOSE)
 

Output:

ln(4*ln(5)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))^2*x^4-x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=x^{4} \log \left (4 \, \log \left (5\right ) \log \left (x^{2} e^{2} + x^{2} + 2 \, {\left (x^{2} + 7 \, x\right )} e + 14 \, x + 49\right )\right )^{2} - x \] Input:

integrate(((4*x^4*exp(1)+4*x^4+28*x^3)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1 
)+x^2+14*x+49)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+ 
49))^2+(4*x^4*exp(1)+4*x^4)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp 
(1)+x^2+14*x+49))+(-exp(1)*x-x-7)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2 
+14*x+49))/(exp(1)*x+x+7)/log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49 
),x, algorithm="fricas")
 

Output:

x^4*log(4*log(5)*log(x^2*e^2 + x^2 + 2*(x^2 + 7*x)*e + 14*x + 49))^2 - x
 

Sympy [A] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=x^{4} \log {\left (4 \log {\left (5 \right )} \log {\left (x^{2} + x^{2} e^{2} + 14 x + e \left (2 x^{2} + 14 x\right ) + 49 \right )} \right )}^{2} - x \] Input:

integrate(((4*x**4*exp(1)+4*x**4+28*x**3)*ln(x**2*exp(1)**2+(2*x**2+14*x)* 
exp(1)+x**2+14*x+49)*ln(4*ln(5)*ln(x**2*exp(1)**2+(2*x**2+14*x)*exp(1)+x** 
2+14*x+49))**2+(4*x**4*exp(1)+4*x**4)*ln(4*ln(5)*ln(x**2*exp(1)**2+(2*x**2 
+14*x)*exp(1)+x**2+14*x+49))+(-exp(1)*x-x-7)*ln(x**2*exp(1)**2+(2*x**2+14* 
x)*exp(1)+x**2+14*x+49))/(exp(1)*x+x+7)/ln(x**2*exp(1)**2+(2*x**2+14*x)*ex 
p(1)+x**2+14*x+49),x)
 

Output:

x**4*log(4*log(5)*log(x**2 + x**2*exp(2) + 14*x + E*(2*x**2 + 14*x) + 49)) 
**2 - x
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (25) = 50\).

Time = 0.27 (sec) , antiderivative size = 267, normalized size of antiderivative = 11.12 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=2 \, x^{4} {\left (3 \, \log \left (2\right ) + \log \left (\log \left (5\right )\right )\right )} \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right ) + x^{4} \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right )^{2} + {\left (9 \, \log \left (2\right )^{2} + 6 \, \log \left (2\right ) \log \left (\log \left (5\right )\right ) + \log \left (\log \left (5\right )\right )^{2}\right )} x^{4} - {\left (\frac {x}{e + 1} - \frac {7 \, \log \left (x {\left (e + 1\right )} + 7\right )}{e^{2} + 2 \, e + 1}\right )} e - \frac {7 \, \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right ) \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right )}{2 \, {\left (e + 1\right )}} - \frac {x}{e + 1} + \frac {7 \, {\left (\frac {{\left (e \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right ) + \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right )\right )} \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right )}{e + 1} - 2 \, \log \left (x {\left (e + 1\right )} + 7\right )\right )}}{2 \, {\left (e + 1\right )}} + \frac {7 \, \log \left (x {\left (e + 1\right )} + 7\right )}{e^{2} + 2 \, e + 1} \] Input:

integrate(((4*x^4*exp(1)+4*x^4+28*x^3)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1 
)+x^2+14*x+49)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+ 
49))^2+(4*x^4*exp(1)+4*x^4)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp 
(1)+x^2+14*x+49))+(-exp(1)*x-x-7)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2 
+14*x+49))/(exp(1)*x+x+7)/log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49 
),x, algorithm="maxima")
 

Output:

2*x^4*(3*log(2) + log(log(5)))*log(log(x*(e + 1) + 7)) + x^4*log(log(x*(e 
+ 1) + 7))^2 + (9*log(2)^2 + 6*log(2)*log(log(5)) + log(log(5))^2)*x^4 - ( 
x/(e + 1) - 7*log(x*(e + 1) + 7)/(e^2 + 2*e + 1))*e - 7/2*log(x^2*e^2 + 2* 
x^2*e + x^2 + 14*x*e + 14*x + 49)*log(log(x*(e + 1) + 7))/(e + 1) - x/(e + 
 1) + 7/2*((e*log(x^2*e^2 + 2*x^2*e + x^2 + 14*x*e + 14*x + 49) + log(x^2* 
e^2 + 2*x^2*e + x^2 + 14*x*e + 14*x + 49))*log(log(x*(e + 1) + 7))/(e + 1) 
 - 2*log(x*(e + 1) + 7))/(e + 1) + 7*log(x*(e + 1) + 7)/(e^2 + 2*e + 1)
 

Giac [A] (verification not implemented)

Time = 2.60 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=x^{4} \log \left (4 \, \log \left (5\right ) \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right )\right )^{2} - x \] Input:

integrate(((4*x^4*exp(1)+4*x^4+28*x^3)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1 
)+x^2+14*x+49)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+ 
49))^2+(4*x^4*exp(1)+4*x^4)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp 
(1)+x^2+14*x+49))+(-exp(1)*x-x-7)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2 
+14*x+49))/(exp(1)*x+x+7)/log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49 
),x, algorithm="giac")
 

Output:

x^4*log(4*log(5)*log(x^2*e^2 + 2*x^2*e + x^2 + 14*x*e + 14*x + 49))^2 - x
 

Mupad [B] (verification not implemented)

Time = 2.76 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=x\,\left (x^3\,{\ln \left (4\,\ln \left (14\,x+\mathrm {e}\,\left (2\,x^2+14\,x\right )+x^2\,{\mathrm {e}}^2+x^2+49\right )\,\ln \left (5\right )\right )}^2-1\right ) \] Input:

int((log(4*log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 + 49)*log(5 
))*(4*x^4*exp(1) + 4*x^4) - log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) 
+ x^2 + 49)*(x + x*exp(1) + 7) + log(14*x + exp(1)*(14*x + 2*x^2) + x^2*ex 
p(2) + x^2 + 49)*log(4*log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 
 + 49)*log(5))^2*(4*x^4*exp(1) + 28*x^3 + 4*x^4))/(log(14*x + exp(1)*(14*x 
 + 2*x^2) + x^2*exp(2) + x^2 + 49)*(x + x*exp(1) + 7)),x)
 

Output:

x*(x^3*log(4*log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 + 49)*log 
(5))^2 - 1)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=x \left ({\mathrm {log}\left (4 \,\mathrm {log}\left (e^{2} x^{2}+2 e \,x^{2}+14 e x +x^{2}+14 x +49\right ) \mathrm {log}\left (5\right )\right )}^{2} x^{3}-1\right ) \] Input:

int(((4*x^4*exp(1)+4*x^4+28*x^3)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+ 
14*x+49)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))^2 
+(4*x^4*exp(1)+4*x^4)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^ 
2+14*x+49))+(-exp(1)*x-x-7)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+ 
49))/(exp(1)*x+x+7)/log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49),x)
 

Output:

x*(log(4*log(e**2*x**2 + 2*e*x**2 + 14*e*x + x**2 + 14*x + 49)*log(5))**2* 
x**3 - 1)