Integrand size = 141, antiderivative size = 34 \[ \int \frac {384-44 x+5 x^2+(96-15 x) \log (4)+\left (384-40 x+5 x^2+(96-10 x) \log (4)\right ) \log (x)+(80+20 \log (4)) \log ^2(x)}{\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5)+\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5) \log (x)+\left (64-32 x+4 x^2+(32-8 x) \log (4)+4 \log ^2(4)\right ) \log (5) \log ^2(x)} \, dx=\frac {5+\frac {4-\frac {5 x}{4}}{2+\log (x)}}{-\log (5)+\frac {(4+\log (4)) \log (5)}{x}} \]
Time = 5.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {384-44 x+5 x^2+(96-15 x) \log (4)+\left (384-40 x+5 x^2+(96-10 x) \log (4)\right ) \log (x)+(80+20 \log (4)) \log ^2(x)}{\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5)+\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5) \log (x)+\left (64-32 x+4 x^2+(32-8 x) \log (4)+4 \log ^2(4)\right ) \log (5) \log ^2(x)} \, dx=\frac {-20 (4+\log (4))+\frac {x (-16+5 x)}{2+\log (x)}}{4 (-4+x-\log (4)) \log (5)} \]
Integrate[(384 - 44*x + 5*x^2 + (96 - 15*x)*Log[4] + (384 - 40*x + 5*x^2 + (96 - 10*x)*Log[4])*Log[x] + (80 + 20*Log[4])*Log[x]^2)/((256 - 128*x + 1 6*x^2 + (128 - 32*x)*Log[4] + 16*Log[4]^2)*Log[5] + (256 - 128*x + 16*x^2 + (128 - 32*x)*Log[4] + 16*Log[4]^2)*Log[5]*Log[x] + (64 - 32*x + 4*x^2 + (32 - 8*x)*Log[4] + 4*Log[4]^2)*Log[5]*Log[x]^2),x]
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.
\(\displaystyle \int \frac {5 x^2+\left (5 x^2-40 x+(96-10 x) \log (4)+384\right ) \log (x)-44 x+(80+20 \log (4)) \log ^2(x)+(96-15 x) \log (4)+384}{\log (5) \left (4 x^2-32 x+(32-8 x) \log (4)+64+4 \log ^2(4)\right ) \log ^2(x)+\log (5) \left (16 x^2-128 x+(128-32 x) \log (4)+256+16 \log ^2(4)\right ) \log (x)+\log (5) \left (16 x^2-128 x+(128-32 x) \log (4)+256+16 \log ^2(4)\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {5 x^2+\left (5 x^2-10 x (4+\log (4))+96 (4+\log (4))\right ) \log (x)+20 (4+\log (4)) \log ^2(x)-x (44+15 \log (4))+96 (4+\log (4))}{4 \log (5) (-x+4+\log (4))^2 (\log (x)+2)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {5 x^2-(44+15 \log (4)) x+20 (4+\log (4)) \log ^2(x)+\left (5 x^2-10 (4+\log (4)) x+96 (4+\log (4))\right ) \log (x)+96 (4+\log (4))}{(-x+\log (4)+4)^2 (\log (x)+2)^2}dx}{4 \log (5)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {\int \left (\frac {16-5 x}{(x-\log (4)-4) (\log (x)+2)^2}+\frac {5 x^2-10 (4+\log (4)) x+16 (4+\log (4))}{(-x+\log (4)+4)^2 (\log (x)+2)}+\frac {20 (4+\log (4))}{(-x+\log (4)+4)^2}\right )dx}{4 \log (5)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\int \frac {5 x^2-10 (4+\log (4)) x+16 (4+\log (4))}{(-x+\log (4)+4)^2 (\log (x)+2)}dx+\int \frac {16-5 x}{(x-\log (4)-4) (\log (x)+2)^2}dx+\frac {20 (4+\log (4))}{-x+4+\log (4)}}{4 \log (5)}\) |
Int[(384 - 44*x + 5*x^2 + (96 - 15*x)*Log[4] + (384 - 40*x + 5*x^2 + (96 - 10*x)*Log[4])*Log[x] + (80 + 20*Log[4])*Log[x]^2)/((256 - 128*x + 16*x^2 + (128 - 32*x)*Log[4] + 16*Log[4]^2)*Log[5] + (256 - 128*x + 16*x^2 + (128 - 32*x)*Log[4] + 16*Log[4]^2)*Log[5]*Log[x] + (64 - 32*x + 4*x^2 + (32 - 8*x)*Log[4] + 4*Log[4]^2)*Log[5]*Log[x]^2),x]
3.9.21.3.1 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]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.67 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38
method | result | size |
default | \(\frac {16 x +\left (40 \ln \left (2\right )+80\right ) \ln \left (x \right )-5 x^{2}+80 \ln \left (2\right )+160}{4 \ln \left (5\right ) \left (\ln \left (x \right )+2\right ) \left (4+2 \ln \left (2\right )-x \right )}\) | \(47\) |
parallelrisch | \(\frac {160+40 \ln \left (2\right ) \ln \left (x \right )-5 x^{2}+80 \ln \left (2\right )+16 x +80 \ln \left (x \right )}{4 \ln \left (5\right ) \left (\ln \left (x \right )+2\right ) \left (4+2 \ln \left (2\right )-x \right )}\) | \(48\) |
norman | \(\frac {\frac {4 x}{\ln \left (5\right )}+\frac {10 \left (\ln \left (2\right )+2\right ) \ln \left (x \right )}{\ln \left (5\right )}-\frac {5 x^{2}}{4 \ln \left (5\right )}+\frac {20 \ln \left (2\right )+40}{\ln \left (5\right )}}{\left (\ln \left (x \right )+2\right ) \left (4+2 \ln \left (2\right )-x \right )}\) | \(58\) |
risch | \(\frac {10 \ln \left (2\right )}{\left (4+2 \ln \left (2\right )-x \right ) \ln \left (5\right )}+\frac {20}{\left (4+2 \ln \left (2\right )-x \right ) \ln \left (5\right )}-\frac {x \left (5 x -16\right )}{4 \ln \left (5\right ) \left (4+2 \ln \left (2\right )-x \right ) \left (\ln \left (x \right )+2\right )}\) | \(67\) |
int(((40*ln(2)+80)*ln(x)^2+(2*(-10*x+96)*ln(2)+5*x^2-40*x+384)*ln(x)+2*(-1 5*x+96)*ln(2)+5*x^2-44*x+384)/((16*ln(2)^2+2*(-8*x+32)*ln(2)+4*x^2-32*x+64 )*ln(5)*ln(x)^2+(64*ln(2)^2+2*(-32*x+128)*ln(2)+16*x^2-128*x+256)*ln(5)*ln (x)+(64*ln(2)^2+2*(-32*x+128)*ln(2)+16*x^2-128*x+256)*ln(5)),x,method=_RET URNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47 \[ \int \frac {384-44 x+5 x^2+(96-15 x) \log (4)+\left (384-40 x+5 x^2+(96-10 x) \log (4)\right ) \log (x)+(80+20 \log (4)) \log ^2(x)}{\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5)+\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5) \log (x)+\left (64-32 x+4 x^2+(32-8 x) \log (4)+4 \log ^2(4)\right ) \log (5) \log ^2(x)} \, dx=\frac {5 \, x^{2} - 40 \, {\left (\log \left (2\right ) + 2\right )} \log \left (x\right ) - 16 \, x - 80 \, \log \left (2\right ) - 160}{4 \, {\left ({\left (x - 2 \, \log \left (2\right ) - 4\right )} \log \left (5\right ) \log \left (x\right ) + 2 \, {\left (x - 2 \, \log \left (2\right ) - 4\right )} \log \left (5\right )\right )}} \]
integrate(((40*log(2)+80)*log(x)^2+(2*(-10*x+96)*log(2)+5*x^2-40*x+384)*lo g(x)+2*(-15*x+96)*log(2)+5*x^2-44*x+384)/((16*log(2)^2+2*(-8*x+32)*log(2)+ 4*x^2-32*x+64)*log(5)*log(x)^2+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-12 8*x+256)*log(5)*log(x)+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-128*x+256) *log(5)),x, algorithm=\
1/4*(5*x^2 - 40*(log(2) + 2)*log(x) - 16*x - 80*log(2) - 160)/((x - 2*log( 2) - 4)*log(5)*log(x) + 2*(x - 2*log(2) - 4)*log(5))
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (27) = 54\).
Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.24 \[ \int \frac {384-44 x+5 x^2+(96-15 x) \log (4)+\left (384-40 x+5 x^2+(96-10 x) \log (4)\right ) \log (x)+(80+20 \log (4)) \log ^2(x)}{\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5)+\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5) \log (x)+\left (64-32 x+4 x^2+(32-8 x) \log (4)+4 \log ^2(4)\right ) \log (5) \log ^2(x)} \, dx=\frac {5 x^{2} - 16 x}{8 x \log {\left (5 \right )} + \left (4 x \log {\left (5 \right )} - 16 \log {\left (5 \right )} - 8 \log {\left (2 \right )} \log {\left (5 \right )}\right ) \log {\left (x \right )} - 32 \log {\left (5 \right )} - 16 \log {\left (2 \right )} \log {\left (5 \right )}} - \frac {10 \log {\left (2 \right )} + 20}{x \log {\left (5 \right )} - 4 \log {\left (5 \right )} - 2 \log {\left (2 \right )} \log {\left (5 \right )}} \]
integrate(((40*ln(2)+80)*ln(x)**2+(2*(-10*x+96)*ln(2)+5*x**2-40*x+384)*ln( x)+2*(-15*x+96)*ln(2)+5*x**2-44*x+384)/((16*ln(2)**2+2*(-8*x+32)*ln(2)+4*x **2-32*x+64)*ln(5)*ln(x)**2+(64*ln(2)**2+2*(-32*x+128)*ln(2)+16*x**2-128*x +256)*ln(5)*ln(x)+(64*ln(2)**2+2*(-32*x+128)*ln(2)+16*x**2-128*x+256)*ln(5 )),x)
(5*x**2 - 16*x)/(8*x*log(5) + (4*x*log(5) - 16*log(5) - 8*log(2)*log(5))*l og(x) - 32*log(5) - 16*log(2)*log(5)) - (10*log(2) + 20)/(x*log(5) - 4*log (5) - 2*log(2)*log(5))
Time = 0.31 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.76 \[ \int \frac {384-44 x+5 x^2+(96-15 x) \log (4)+\left (384-40 x+5 x^2+(96-10 x) \log (4)\right ) \log (x)+(80+20 \log (4)) \log ^2(x)}{\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5)+\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5) \log (x)+\left (64-32 x+4 x^2+(32-8 x) \log (4)+4 \log ^2(4)\right ) \log (5) \log ^2(x)} \, dx=\frac {5 \, x^{2} - 40 \, {\left (\log \left (2\right ) + 2\right )} \log \left (x\right ) - 16 \, x - 80 \, \log \left (2\right ) - 160}{4 \, {\left (2 \, x \log \left (5\right ) - 4 \, \log \left (5\right ) \log \left (2\right ) + {\left (x \log \left (5\right ) - 2 \, \log \left (5\right ) \log \left (2\right ) - 4 \, \log \left (5\right )\right )} \log \left (x\right ) - 8 \, \log \left (5\right )\right )}} \]
integrate(((40*log(2)+80)*log(x)^2+(2*(-10*x+96)*log(2)+5*x^2-40*x+384)*lo g(x)+2*(-15*x+96)*log(2)+5*x^2-44*x+384)/((16*log(2)^2+2*(-8*x+32)*log(2)+ 4*x^2-32*x+64)*log(5)*log(x)^2+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-12 8*x+256)*log(5)*log(x)+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-128*x+256) *log(5)),x, algorithm=\
1/4*(5*x^2 - 40*(log(2) + 2)*log(x) - 16*x - 80*log(2) - 160)/(2*x*log(5) - 4*log(5)*log(2) + (x*log(5) - 2*log(5)*log(2) - 4*log(5))*log(x) - 8*log (5))
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (34) = 68\).
Time = 0.30 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.15 \[ \int \frac {384-44 x+5 x^2+(96-15 x) \log (4)+\left (384-40 x+5 x^2+(96-10 x) \log (4)\right ) \log (x)+(80+20 \log (4)) \log ^2(x)}{\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5)+\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5) \log (x)+\left (64-32 x+4 x^2+(32-8 x) \log (4)+4 \log ^2(4)\right ) \log (5) \log ^2(x)} \, dx=\frac {5 \, x^{2} - 16 \, x}{4 \, {\left (x \log \left (5\right ) \log \left (x\right ) - 2 \, \log \left (5\right ) \log \left (2\right ) \log \left (x\right ) + 2 \, x \log \left (5\right ) - 4 \, \log \left (5\right ) \log \left (2\right ) - 4 \, \log \left (5\right ) \log \left (x\right ) - 8 \, \log \left (5\right )\right )}} - \frac {10 \, {\left (\log \left (2\right ) + 2\right )}}{x \log \left (5\right ) - 2 \, \log \left (5\right ) \log \left (2\right ) - 4 \, \log \left (5\right )} \]
integrate(((40*log(2)+80)*log(x)^2+(2*(-10*x+96)*log(2)+5*x^2-40*x+384)*lo g(x)+2*(-15*x+96)*log(2)+5*x^2-44*x+384)/((16*log(2)^2+2*(-8*x+32)*log(2)+ 4*x^2-32*x+64)*log(5)*log(x)^2+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-12 8*x+256)*log(5)*log(x)+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-128*x+256) *log(5)),x, algorithm=\
1/4*(5*x^2 - 16*x)/(x*log(5)*log(x) - 2*log(5)*log(2)*log(x) + 2*x*log(5) - 4*log(5)*log(2) - 4*log(5)*log(x) - 8*log(5)) - 10*(log(2) + 2)/(x*log(5 ) - 2*log(5)*log(2) - 4*log(5))
Time = 9.77 (sec) , antiderivative size = 1650, normalized size of antiderivative = 48.53 \[ \int \frac {384-44 x+5 x^2+(96-15 x) \log (4)+\left (384-40 x+5 x^2+(96-10 x) \log (4)\right ) \log (x)+(80+20 \log (4)) \log ^2(x)}{\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5)+\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5) \log (x)+\left (64-32 x+4 x^2+(32-8 x) \log (4)+4 \log ^2(4)\right ) \log (5) \log ^2(x)} \, dx=\text {Too large to display} \]
int(-(44*x + 2*log(2)*(15*x - 96) + log(x)*(40*x + 2*log(2)*(10*x - 96) - 5*x^2 - 384) - log(x)^2*(40*log(2) + 80) - 5*x^2 - 384)/(log(5)*(64*log(2) ^2 - 2*log(2)*(32*x - 128) - 128*x + 16*x^2 + 256) + log(5)*log(x)*(64*log (2)^2 - 2*log(2)*(32*x - 128) - 128*x + 16*x^2 + 256) + log(5)*log(x)^2*(1 6*log(2)^2 - 2*log(2)*(8*x - 32) - 32*x + 4*x^2 + 64)),x)
(5*x)/(4*log(5)) - (16*x)/(32*log(5) + 32*log(2)*log(5) - 16*x*log(5) + 8* log(2)^2*log(5) + 2*x^2*log(5) + 16*log(5)*log(x) + 16*log(2)*log(5)*log(x ) - 8*x*log(5)*log(x) + 4*log(2)^2*log(5)*log(x) + x^2*log(5)*log(x) - 8*x *log(2)*log(5) - 4*x*log(2)*log(5)*log(x)) + (11*x^2)/(32*log(5) + 32*log( 2)*log(5) - 16*x*log(5) + 8*log(2)^2*log(5) + 2*x^2*log(5) + 16*log(5)*log (x) + 16*log(2)*log(5)*log(x) - 8*x*log(5)*log(x) + 4*log(2)^2*log(5)*log( x) + x^2*log(5)*log(x) - 8*x*log(2)*log(5) - 4*x*log(2)*log(5)*log(x)) - ( 5*x^3)/(4*(32*log(5) + 32*log(2)*log(5) - 16*x*log(5) + 8*log(2)^2*log(5) + 2*x^2*log(5) + 16*log(5)*log(x) + 16*log(2)*log(5)*log(x) - 8*x*log(5)*l og(x) + 4*log(2)^2*log(5)*log(x) + x^2*log(5)*log(x) - 8*x*log(2)*log(5) - 4*x*log(2)*log(5)*log(x))) + symsum(log(576*x - 1984*log(2) - 384*root(22 020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^5 + 214958080*log(2) ^3 + 121765888*log(2)^6 + 94371840*log(2)^2 + 38338560*log(2)^7 + 6758400* log(2)^8 + 512000*log(2)^9 + 2097152, z, k)*log(5) + 1056*x*log(2) + 724*x *log(2)^2 + 220*x*log(2)^3 + 25*x*log(2)^4 - 272*log(2)^2 + 648*log(2)^3 + 340*log(2)^4 + 50*log(2)^5 - 1536*root(22020096*log(2) + 287047680*log(2) ^4 + 236322816*log(2)^5 + 214958080*log(2)^3 + 121765888*log(2)^6 + 943718 40*log(2)^2 + 38338560*log(2)^7 + 6758400*log(2)^8 + 512000*log(2)^9 + 209 7152, z, k)*log(2)*log(5) + 96*root(22020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^5 + 214958080*log(2)^3 + 121765888*log(2)^6 + 9437184...