\(\int \frac {10602576-29160 x-2916 x^2+23328 \log (\log (12))+2916 \log ^2(\log (12))}{1-1454 x+528531 x^2-1454 x^3+x^4+(-4+2904 x+2904 x^2-4 x^3) \log (\log (12))+(6-1446 x+6 x^2) \log ^2(\log (12))+(-4-4 x) \log ^3(\log (12))+\log ^4(\log (12))} \, dx\) [156]

Optimal result

Integrand size = 94, antiderivative size = 25 \[ \int \frac {10602576-29160 x-2916 x^2+23328 \log (\log (12))+2916 \log ^2(\log (12))}{1-1454 x+528531 x^2-1454 x^3+x^4+\left (-4+2904 x+2904 x^2-4 x^3\right ) \log (\log (12))+\left (6-1446 x+6 x^2\right ) \log ^2(\log (12))+(-4-4 x) \log ^3(\log (12))+\log ^4(\log (12))} \, dx=\frac {4 (5+x)}{-x+\frac {1}{729} (1+x-\log (\log (12)))^2} \] Output:

4*(5+x)/(-1/27*(ln(ln(12))-x-1)*(-1/27*ln(ln(12))+1/27*x+1/27)-x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {10602576-29160 x-2916 x^2+23328 \log (\log (12))+2916 \log ^2(\log (12))}{1-1454 x+528531 x^2-1454 x^3+x^4+\left (-4+2904 x+2904 x^2-4 x^3\right ) \log (\log (12))+\left (6-1446 x+6 x^2\right ) \log ^2(\log (12))+(-4-4 x) \log ^3(\log (12))+\log ^4(\log (12))} \, dx=\frac {2916 (5+x)}{x^2+(-1+\log (\log (12)))^2-x (727+2 \log (\log (12)))} \] Input:

Integrate[(10602576 - 29160*x - 2916*x^2 + 23328*Log[Log[12]] + 2916*Log[L 
og[12]]^2)/(1 - 1454*x + 528531*x^2 - 1454*x^3 + x^4 + (-4 + 2904*x + 2904 
*x^2 - 4*x^3)*Log[Log[12]] + (6 - 1446*x + 6*x^2)*Log[Log[12]]^2 + (-4 - 4 
*x)*Log[Log[12]]^3 + Log[Log[12]]^4),x]
 

Output:

(2916*(5 + x))/(x^2 + (-1 + Log[Log[12]])^2 - x*(727 + 2*Log[Log[12]]))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(25)=50\).

Time = 0.39 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2459, 1380, 27, 2345, 24}

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.

\(\displaystyle \int \frac {-2916 x^2-29160 x+10602576+2916 \log ^2(\log (12))+23328 \log (\log (12))}{x^4-1454 x^3+528531 x^2+\left (6 x^2-1446 x+6\right ) \log ^2(\log (12))+\left (-4 x^3+2904 x^2+2904 x-4\right ) \log (\log (12))-1454 x+(-4 x-4) \log ^3(\log (12))+1+\log ^4(\log (12))} \, dx\)

\(\Big \downarrow \) 2459

\(\displaystyle \int \frac {-2916 \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )^2-2916 (737+2 \log (\log (12))) \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )-531441 (725+4 \log (\log (12)))}{\left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )^4-\frac {729}{2} (725+4 \log (\log (12))) \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )^2+\frac {531441}{16} (725+4 \log (\log (12)))^2}d\left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )\)

\(\Big \downarrow \) 1380

\(\displaystyle \int -\frac {11664 \left (4 \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )^2+4 (737+2 \log (\log (12))) \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )+729 (725+4 \log (\log (12)))\right )}{\left (4 \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )^2-729 (725+4 \log (\log (12)))\right )^2}d\left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -11664 \int \frac {4 \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )^2+4 (737+2 \log (\log (12))) \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )+729 (725+4 \log (\log (12)))}{\left (4 \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )^2-729 (725+4 \log (\log (12)))\right )^2}d\left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )\)

\(\Big \downarrow \) 2345

\(\displaystyle -11664 \left (\frac {\int 0d\left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )}{1458 (725+4 \log (\log (12)))}-\frac {2 \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )+737+2 \log (\log (12))}{2 \left (4 \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )^2-729 (725+4 \log (\log (12)))\right )}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {5832 \left (2 \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )+737+2 \log (\log (12))\right )}{4 \left (x+\frac {1}{4} (-1454-4 \log (\log (12)))\right )^2-729 (725+4 \log (\log (12)))}\)

Input:

Int[(10602576 - 29160*x - 2916*x^2 + 23328*Log[Log[12]] + 2916*Log[Log[12] 
]^2)/(1 - 1454*x + 528531*x^2 - 1454*x^3 + x^4 + (-4 + 2904*x + 2904*x^2 - 
 4*x^3)*Log[Log[12]] + (6 - 1446*x + 6*x^2)*Log[Log[12]]^2 + (-4 - 4*x)*Lo 
g[Log[12]]^3 + Log[Log[12]]^4),x]
 

Output:

(5832*(737 + 2*(x + (-1454 - 4*Log[Log[12]])/4) + 2*Log[Log[12]]))/(4*(x + 
 (-1454 - 4*Log[Log[12]])/4)^2 - 729*(725 + 4*Log[Log[12]]))
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1380

Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

rule 2459

Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 
]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x 
 -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial 
Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - 
> x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ 
[Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] &&  !(MonomialQ[Qx, x] 
&& IGtQ[p, 0])
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28

method	result	size
gosper	\(\frac {2916 x +14580}{\ln \left (\ln \left (12\right )\right )^{2}-2 x \ln \left (\ln \left (12\right )\right )+x^{2}-2 \ln \left (\ln \left (12\right )\right )-727 x +1}\)	\(32\)
default	\(\frac {2916 x +14580}{\ln \left (\ln \left (12\right )\right )^{2}-2 x \ln \left (\ln \left (12\right )\right )+x^{2}-2 \ln \left (\ln \left (12\right )\right )-727 x +1}\)	\(32\)
norman	\(\frac {2916 x +14580}{\ln \left (\ln \left (12\right )\right )^{2}-2 x \ln \left (\ln \left (12\right )\right )+x^{2}-2 \ln \left (\ln \left (12\right )\right )-727 x +1}\)	\(33\)
parallelrisch	\(\frac {2916 x +14580}{\ln \left (\ln \left (12\right )\right )^{2}-2 x \ln \left (\ln \left (12\right )\right )+x^{2}-2 \ln \left (\ln \left (12\right )\right )-727 x +1}\)	\(33\)
risch	\(\frac {2916 x +14580}{\ln \left (2 \ln \left (2\right )+\ln \left (3\right )\right )^{2}-2 x \ln \left (2 \ln \left (2\right )+\ln \left (3\right )\right )+x^{2}-2 \ln \left (2 \ln \left (2\right )+\ln \left (3\right )\right )-727 x +1}\)	\(48\)

Input:

int((2916*ln(ln(12))^2+23328*ln(ln(12))-2916*x^2-29160*x[phone])/(ln(ln( 
12))^4+(-4-4*x)*ln(ln(12))^3+(6*x^2-1446*x+6)*ln(ln(12))^2+(-4*x^3+2904*x^ 
2+2904*x-4)*ln(ln(12))+x^4-1454*x^3+528531*x^2-1454*x+1),x,method=_RETURNV 
ERBOSE)
 

Output:

2916*(5+x)/(ln(ln(12))^2-2*x*ln(ln(12))+x^2-2*ln(ln(12))-727*x+1)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {10602576-29160 x-2916 x^2+23328 \log (\log (12))+2916 \log ^2(\log (12))}{1-1454 x+528531 x^2-1454 x^3+x^4+\left (-4+2904 x+2904 x^2-4 x^3\right ) \log (\log (12))+\left (6-1446 x+6 x^2\right ) \log ^2(\log (12))+(-4-4 x) \log ^3(\log (12))+\log ^4(\log (12))} \, dx=\frac {2916 \, {\left (x + 5\right )}}{x^{2} - 2 \, {\left (x + 1\right )} \log \left (\log \left (12\right )\right ) + \log \left (\log \left (12\right )\right )^{2} - 727 \, x + 1} \] Input:

integrate((2916*log(log(12))^2+23328*log(log(12))-2916*x^2-29160*x+1060257 
6)/(log(log(12))^4+(-4-4*x)*log(log(12))^3+(6*x^2-1446*x+6)*log(log(12))^2 
+(-4*x^3+2904*x^2+2904*x-4)*log(log(12))+x^4-1454*x^3+528531*x^2-1454*x+1) 
,x, algorithm="fricas")
 

Output:

2916*(x + 5)/(x^2 - 2*(x + 1)*log(log(12)) + log(log(12))^2 - 727*x + 1)
 

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {10602576-29160 x-2916 x^2+23328 \log (\log (12))+2916 \log ^2(\log (12))}{1-1454 x+528531 x^2-1454 x^3+x^4+\left (-4+2904 x+2904 x^2-4 x^3\right ) \log (\log (12))+\left (6-1446 x+6 x^2\right ) \log ^2(\log (12))+(-4-4 x) \log ^3(\log (12))+\log ^4(\log (12))} \, dx=- \frac {- 2916 x - 14580}{x^{2} + x \left (-727 - 2 \log {\left (\log {\left (12 \right )} \right )}\right ) - 2 \log {\left (\log {\left (12 \right )} \right )} + \log {\left (\log {\left (12 \right )} \right )}^{2} + 1} \] Input:

integrate((2916*ln(ln(12))**2+23328*ln(ln(12))-2916*x**2-29160*x[phone]) 
/(ln(ln(12))**4+(-4-4*x)*ln(ln(12))**3+(6*x**2-1446*x+6)*ln(ln(12))**2+(-4 
*x**3+2904*x**2+2904*x-4)*ln(ln(12))+x**4-1454*x**3+528531*x**2-1454*x+1), 
x)
 

Output:

-(-2916*x - 14580)/(x**2 + x*(-727 - 2*log(log(12))) - 2*log(log(12)) + lo 
g(log(12))**2 + 1)
 

Maxima [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {10602576-29160 x-2916 x^2+23328 \log (\log (12))+2916 \log ^2(\log (12))}{1-1454 x+528531 x^2-1454 x^3+x^4+\left (-4+2904 x+2904 x^2-4 x^3\right ) \log (\log (12))+\left (6-1446 x+6 x^2\right ) \log ^2(\log (12))+(-4-4 x) \log ^3(\log (12))+\log ^4(\log (12))} \, dx=\frac {2916 \, {\left (x + 5\right )}}{x^{2} - x {\left (2 \, \log \left (\log \left (12\right )\right ) + 727\right )} + \log \left (\log \left (12\right )\right )^{2} - 2 \, \log \left (\log \left (12\right )\right ) + 1} \] Input:

integrate((2916*log(log(12))^2+23328*log(log(12))-2916*x^2-29160*x+1060257 
6)/(log(log(12))^4+(-4-4*x)*log(log(12))^3+(6*x^2-1446*x+6)*log(log(12))^2 
+(-4*x^3+2904*x^2+2904*x-4)*log(log(12))+x^4-1454*x^3+528531*x^2-1454*x+1) 
,x, algorithm="maxima")
 

Output:

2916*(x + 5)/(x^2 - x*(2*log(log(12)) + 727) + log(log(12))^2 - 2*log(log( 
12)) + 1)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {10602576-29160 x-2916 x^2+23328 \log (\log (12))+2916 \log ^2(\log (12))}{1-1454 x+528531 x^2-1454 x^3+x^4+\left (-4+2904 x+2904 x^2-4 x^3\right ) \log (\log (12))+\left (6-1446 x+6 x^2\right ) \log ^2(\log (12))+(-4-4 x) \log ^3(\log (12))+\log ^4(\log (12))} \, dx=\frac {2916 \, {\left (x + 5\right )}}{x^{2} - 2 \, x \log \left (\log \left (12\right )\right ) + \log \left (\log \left (12\right )\right )^{2} - 727 \, x - 2 \, \log \left (\log \left (12\right )\right ) + 1} \] Input:

integrate((2916*log(log(12))^2+23328*log(log(12))-2916*x^2-29160*x+1060257 
6)/(log(log(12))^4+(-4-4*x)*log(log(12))^3+(6*x^2-1446*x+6)*log(log(12))^2 
+(-4*x^3+2904*x^2+2904*x-4)*log(log(12))+x^4-1454*x^3+528531*x^2-1454*x+1) 
,x, algorithm="giac")
 

Output:

2916*(x + 5)/(x^2 - 2*x*log(log(12)) + log(log(12))^2 - 727*x - 2*log(log( 
12)) + 1)
 

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {10602576-29160 x-2916 x^2+23328 \log (\log (12))+2916 \log ^2(\log (12))}{1-1454 x+528531 x^2-1454 x^3+x^4+\left (-4+2904 x+2904 x^2-4 x^3\right ) \log (\log (12))+\left (6-1446 x+6 x^2\right ) \log ^2(\log (12))+(-4-4 x) \log ^3(\log (12))+\log ^4(\log (12))} \, dx=\frac {2916\,x+14580}{x^2+\left (-2\,\ln \left (\ln \left (12\right )\right )-727\right )\,x-2\,\ln \left (\ln \left (12\right )\right )+{\ln \left (\ln \left (12\right )\right )}^2+1} \] Input:

int((23328*log(log(12)) - 29160*x + 2916*log(log(12))^2 - 2916*x^2 + 10602 
576)/(log(log(12))^2*(6*x^2 - 1446*x + 6) - 1454*x + log(log(12))^4 - log( 
log(12))^3*(4*x + 4) + 528531*x^2 - 1454*x^3 + x^4 + log(log(12))*(2904*x 
+ 2904*x^2 - 4*x^3 - 4) + 1),x)
 

Output:

(2916*x + 14580)/(log(log(12))^2 - 2*log(log(12)) - x*(2*log(log(12)) + 72 
7) + x^2 + 1)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.80 \[ \int \frac {10602576-29160 x-2916 x^2+23328 \log (\log (12))+2916 \log ^2(\log (12))}{1-1454 x+528531 x^2-1454 x^3+x^4+\left (-4+2904 x+2904 x^2-4 x^3\right ) \log (\log (12))+\left (6-1446 x+6 x^2\right ) \log ^2(\log (12))+(-4-4 x) \log ^3(\log (12))+\log ^4(\log (12))} \, dx=\frac {2916 \mathrm {log}\left (\mathrm {log}\left (12\right )\right )^{2}+23328 \,\mathrm {log}\left (\mathrm {log}\left (12\right )\right )+2916 x^{2}[phone]}{2 \mathrm {log}\left (\mathrm {log}\left (12\right )\right )^{3}-4 \mathrm {log}\left (\mathrm {log}\left (12\right )\right )^{2} x +723 \mathrm {log}\left (\mathrm {log}\left (12\right )\right )^{2}+2 \,\mathrm {log}\left (\mathrm {log}\left (12\right )\right ) x^{2}-2908 \,\mathrm {log}\left (\mathrm {log}\left (12\right )\right ) x -1452 \,\mathrm {log}\left (\mathrm {log}\left (12\right )\right )+727 x^{2}-528529 x +727} \] Input:

int((2916*log(log(12))^2+23328*log(log(12))-2916*x^2-29160*x[phone])/(lo 
g(log(12))^4+(-4-4*x)*log(log(12))^3+(6*x^2-1446*x+6)*log(log(12))^2+(-4*x 
^3+2904*x^2+2904*x-4)*log(log(12))+x^4-1454*x^3+528531*x^2-1454*x+1),x)
 

Output:

(2916*(log(log(12))**2 + 8*log(log(12)) + x**2 + 3636))/(2*log(log(12))**3 
 - 4*log(log(12))**2*x + 723*log(log(12))**2 + 2*log(log(12))*x**2 - 2908* 
log(log(12))*x - 1452*log(log(12)) + 727*x**2 - 528529*x + 727)