Integrand size = 68, antiderivative size = 23 \[ \int \frac {2+2 x+2 \log (16)-2 x \log (x)+\left (26 x+2 x^2\right ) \log ^2(x)}{\left (-2 x-2 x^2-2 x \log (16)\right ) \log (x)+\left (x+26 x^2+x^3+x \log (16)\right ) \log ^2(x)} \, dx=\log \left (25 x+x^2-(1+x+\log (16)) \left (-1+\frac {2}{\log (x)}\right )\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(23)=46\).
Time = 0.51 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.48 \[ \int \frac {2+2 x+2 \log (16)-2 x \log (x)+\left (26 x+2 x^2\right ) \log ^2(x)}{\left (-2 x-2 x^2-2 x \log (16)\right ) \log (x)+\left (x+26 x^2+x^3+x \log (16)\right ) \log ^2(x)} \, dx=2 \left (\frac {1}{2} \log \left (1+26 x+x^2+\log (16)\right )-\frac {1}{2} \log (\log (x))\right )+2 \left (-\frac {1}{2} \log \left (1+26 x+x^2+\log (16)\right )+\frac {1}{2} \log \left (2+2 x+2 \log (16)-\log (x)-26 x \log (x)-x^2 \log (x)-\log (16) \log (x)\right )\right ) \]
Integrate[(2 + 2*x + 2*Log[16] - 2*x*Log[x] + (26*x + 2*x^2)*Log[x]^2)/((- 2*x - 2*x^2 - 2*x*Log[16])*Log[x] + (x + 26*x^2 + x^3 + x*Log[16])*Log[x]^ 2),x]
2*(Log[1 + 26*x + x^2 + Log[16]]/2 - Log[Log[x]]/2) + 2*(-1/2*Log[1 + 26*x + x^2 + Log[16]] + Log[2 + 2*x + 2*Log[16] - Log[x] - 26*x*Log[x] - x^2*L og[x] - Log[16]*Log[x]]/2)
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.
| \(\displaystyle \int \frac {\left (2 x^2+26 x\right ) \log ^2(x)+2 x-2 x \log (x)+2+2 \log (16)}{\left (-2 x^2-2 x-2 x \log (16)\right ) \log (x)+\left (x^3+26 x^2+x+x \log (16)\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 \left (-x^2 \log ^2(x)-x-13 x \log ^2(x)+x \log (x)-1-\log (16)\right )}{x \log (x) \left (x^2 (-\log (x))+2 x-26 x \log (x)-(1+\log (16)) \log (x)+2 (1+\log (16))\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {x^2 \log ^2(x)+13 x \log ^2(x)-x \log (x)+x+\log (16)+1}{x \log (x) \left (-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+\log (256)+2\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {x^2 \log ^2(x)+13 x \log ^2(x)-x \log (x)+x+\log (16)+1}{x \log (x) \left (-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+\log (256)+2\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -2 \int \frac {x^2 \log ^2(x)+13 x \log ^2(x)-x \log (x)+x+\log (16)+1}{x \log (x) \left (-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+2 (1+\log (16))\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {-x-13}{x^2+26 x+\log (16)+1}+\frac {1}{2 x \log (x)}+\frac {x^2+26 x+\log (16)+1}{2 x \left (-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+2 (1+\log (16))\right )}+\frac {x^2+2 (1+\log (16)) x+25 (1+\log (16))}{\left (x^2+26 x+\log (16)+1\right ) \left (-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+2 (1+\log (16))\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (14 \int \frac {1}{-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+2 (1+\log (16))}dx+\frac {1}{2} (1+\log (16)) \int \frac {1}{x \left (-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+2 (1+\log (16))\right )}dx+\frac {1}{2} \int \frac {x}{-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+2 (1+\log (16))}dx-2 \left (1-\frac {13}{\sqrt {168-\log (16)}}\right ) (12-\log (16)) \int \frac {1}{\left (2 x-2 \sqrt {168-\log (16)}+26\right ) \left (-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+2 (1+\log (16))\right )}dx-\frac {24 (1+\log (16)) \int \frac {1}{\left (-2 x+2 \sqrt {168-\log (16)}-26\right ) \left (-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+2 (1+\log (16))\right )}dx}{\sqrt {168-\log (16)}}-\frac {24 (1+\log (16)) \int \frac {1}{\left (2 x+2 \sqrt {168-\log (16)}+26\right ) \left (-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+2 (1+\log (16))\right )}dx}{\sqrt {168-\log (16)}}-2 \left (1+\frac {13}{\sqrt {168-\log (16)}}\right ) (12-\log (16)) \int \frac {1}{\left (2 x+2 \sqrt {168-\log (16)}+26\right ) \left (-\log (x) x^2-26 \log (x) x+2 x-(1+\log (16)) \log (x)+2 (1+\log (16))\right )}dx-\frac {1}{2} \log \left (x^2+26 x+1+\log (16)\right )+\frac {1}{2} \log (\log (x))\right )\) |
Int[(2 + 2*x + 2*Log[16] - 2*x*Log[x] + (26*x + 2*x^2)*Log[x]^2)/((-2*x - 2*x^2 - 2*x*Log[16])*Log[x] + (x + 26*x^2 + x^3 + x*Log[16])*Log[x]^2),x]
3.16.41.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57
| method | result | size |
| default | \(-\ln \left (\ln \left (x \right )\right )+\ln \left (x^{2} \ln \left (x \right )+4 \ln \left (2\right ) \ln \left (x \right )+26 x \ln \left (x \right )-8 \ln \left (2\right )+\ln \left (x \right )-2 x -2\right )\) | \(36\) |
| norman | \(-\ln \left (\ln \left (x \right )\right )+\ln \left (x^{2} \ln \left (x \right )+4 \ln \left (2\right ) \ln \left (x \right )+26 x \ln \left (x \right )-8 \ln \left (2\right )+\ln \left (x \right )-2 x -2\right )\) | \(36\) |
| parallelrisch | \(-\ln \left (\ln \left (x \right )\right )+\ln \left (x^{2} \ln \left (x \right )+4 \ln \left (2\right ) \ln \left (x \right )+26 x \ln \left (x \right )-8 \ln \left (2\right )+\ln \left (x \right )-2 x -2\right )\) | \(36\) |
| risch | \(\ln \left (x^{2}+4 \ln \left (2\right )+26 x +1\right )-\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (x \right )-\frac {2 \left (x +4 \ln \left (2\right )+1\right )}{x^{2}+4 \ln \left (2\right )+26 x +1}\right )\) | \(47\) |
int(((2*x^2+26*x)*ln(x)^2-2*x*ln(x)+8*ln(2)+2*x+2)/((4*x*ln(2)+x^3+26*x^2+ x)*ln(x)^2+(-8*x*ln(2)-2*x^2-2*x)*ln(x)),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).
Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.57 \[ \int \frac {2+2 x+2 \log (16)-2 x \log (x)+\left (26 x+2 x^2\right ) \log ^2(x)}{\left (-2 x-2 x^2-2 x \log (16)\right ) \log (x)+\left (x+26 x^2+x^3+x \log (16)\right ) \log ^2(x)} \, dx=\log \left (x^{2} + 26 \, x + 4 \, \log \left (2\right ) + 1\right ) + \log \left (\frac {{\left (x^{2} + 26 \, x + 4 \, \log \left (2\right ) + 1\right )} \log \left (x\right ) - 2 \, x - 8 \, \log \left (2\right ) - 2}{x^{2} + 26 \, x + 4 \, \log \left (2\right ) + 1}\right ) - \log \left (\log \left (x\right )\right ) \]
integrate(((2*x^2+26*x)*log(x)^2-2*x*log(x)+8*log(2)+2*x+2)/((4*x*log(2)+x ^3+26*x^2+x)*log(x)^2+(-8*x*log(2)-2*x^2-2*x)*log(x)),x, algorithm=\
log(x^2 + 26*x + 4*log(2) + 1) + log(((x^2 + 26*x + 4*log(2) + 1)*log(x) - 2*x - 8*log(2) - 2)/(x^2 + 26*x + 4*log(2) + 1)) - log(log(x))
Exception generated. \[ \int \frac {2+2 x+2 \log (16)-2 x \log (x)+\left (26 x+2 x^2\right ) \log ^2(x)}{\left (-2 x-2 x^2-2 x \log (16)\right ) \log (x)+\left (x+26 x^2+x^3+x \log (16)\right ) \log ^2(x)} \, dx=\text {Exception raised: PolynomialError} \]
integrate(((2*x**2+26*x)*ln(x)**2-2*x*ln(x)+8*ln(2)+2*x+2)/((4*x*ln(2)+x** 3+26*x**2+x)*ln(x)**2+(-8*x*ln(2)-2*x**2-2*x)*ln(x)),x)
Exception raised: PolynomialError >> 1/(x**5 + 52*x**4 + 8*x**3*log(2) + 6 78*x**3 + 52*x**2 + 208*x**2*log(2) + x + 8*x*log(2) + 16*x*log(2)**2) con tains an element of the set of generators.
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.57 \[ \int \frac {2+2 x+2 \log (16)-2 x \log (x)+\left (26 x+2 x^2\right ) \log ^2(x)}{\left (-2 x-2 x^2-2 x \log (16)\right ) \log (x)+\left (x+26 x^2+x^3+x \log (16)\right ) \log ^2(x)} \, dx=\log \left (x^{2} + 26 \, x + 4 \, \log \left (2\right ) + 1\right ) + \log \left (\frac {{\left (x^{2} + 26 \, x + 4 \, \log \left (2\right ) + 1\right )} \log \left (x\right ) - 2 \, x - 8 \, \log \left (2\right ) - 2}{x^{2} + 26 \, x + 4 \, \log \left (2\right ) + 1}\right ) - \log \left (\log \left (x\right )\right ) \]
integrate(((2*x^2+26*x)*log(x)^2-2*x*log(x)+8*log(2)+2*x+2)/((4*x*log(2)+x ^3+26*x^2+x)*log(x)^2+(-8*x*log(2)-2*x^2-2*x)*log(x)),x, algorithm=\
log(x^2 + 26*x + 4*log(2) + 1) + log(((x^2 + 26*x + 4*log(2) + 1)*log(x) - 2*x - 8*log(2) - 2)/(x^2 + 26*x + 4*log(2) + 1)) - log(log(x))
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52 \[ \int \frac {2+2 x+2 \log (16)-2 x \log (x)+\left (26 x+2 x^2\right ) \log ^2(x)}{\left (-2 x-2 x^2-2 x \log (16)\right ) \log (x)+\left (x+26 x^2+x^3+x \log (16)\right ) \log ^2(x)} \, dx=\log \left (x^{2} \log \left (x\right ) + 26 \, x \log \left (x\right ) + 4 \, \log \left (2\right ) \log \left (x\right ) - 2 \, x - 8 \, \log \left (2\right ) + \log \left (x\right ) - 2\right ) - \log \left (\log \left (x\right )\right ) \]
integrate(((2*x^2+26*x)*log(x)^2-2*x*log(x)+8*log(2)+2*x+2)/((4*x*log(2)+x ^3+26*x^2+x)*log(x)^2+(-8*x*log(2)-2*x^2-2*x)*log(x)),x, algorithm=\
Time = 20.17 (sec) , antiderivative size = 49468, normalized size of antiderivative = 2150.78 \[ \int \frac {2+2 x+2 \log (16)-2 x \log (x)+\left (26 x+2 x^2\right ) \log ^2(x)}{\left (-2 x-2 x^2-2 x \log (16)\right ) \log (x)+\left (x+26 x^2+x^3+x \log (16)\right ) \log ^2(x)} \, dx=\text {Too large to display} \]
int((2*x + 8*log(2) + log(x)^2*(26*x + 2*x^2) - 2*x*log(x) + 2)/(log(x)^2* (x + 4*x*log(2) + 26*x^2 + x^3) - log(x)*(2*x + 8*x*log(2) + 2*x^2)),x)
log(54*x + 16*log(2) - log(x) - 16*log(2)^2*log(x) - 678*x^2*log(x) - 52*x ^3*log(x) - x^4*log(x) + 216*x*log(2) + 8*x^2*log(2) - 8*log(2)*log(x) - 5 2*x*log(x) + 32*log(2)^2 + 54*x^2 + 2*x^3 - 208*x*log(2)*log(x) - 8*x^2*lo g(2)*log(x) + 2) - log(2*log(x) + 32*log(2)^2*log(x) + 1364*x^2*log(x) + 1 08*x^3*log(x) + 2*x^4*log(x) + 16*log(2)*log(x) + 204*x*log(x) + 816*x*log (2)*log(x) + 48*x^2*log(2)*log(x)) + symsum(log(x*(271434818895193571328*l og(16) - 3061736766203171438592*log(2) - 194727499546084245504*log(2)*log( 16) - 97385167376291856384*log(2)*log(16)^2 + 55821864461235388416*log(2)^ 2*log(16) - 21409889065172992*log(2)*log(16)^3 - 7772816853941878784*log(2 )^3*log(16) - 5342829535592448*log(2)*log(16)^4 + 512557862870843392*log(2 )^4*log(16) + 5785638420480*log(2)*log(16)^5 - 7148960426754048*log(2)^5*l og(16) + 964273070080*log(2)*log(16)^6 - 660866852716544*log(2)^6*log(16) - 8269154222080*log(2)^7*log(16) - 4395374934054172360704*log(2)^2 + 36637 31520044421611520*log(2)^3 - 1000261134271037571072*log(2)^4 + 12703807024 9977020416*log(2)^5 - 7490392160752107520*log(2)^6 + 121774470250627072*lo g(2)^7 + 5026503741931520*log(2)^8 - 105758200954880*log(2)^9 + 1786706395 136*log(2)^10 + 135745748215510597632*log(16)^2 + 28347482480246784*log(16 )^3 + 7097797420425216*log(16)^4 + 6603109466112*log(16)^5 + 1139709284352 *log(16)^6 + 16796160000*log(16)^7 + 2099520000*log(16)^8 + 27915660181423 259648*log(2)^2*log(16)^2 + 4724986822000640*log(2)^2*log(16)^3 - 38873...