Integrand size = 218, antiderivative size = 26 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=\frac {x}{e^4 (4-x) (x+\log (4)) \left (-4+\log \left (x^2\right )\right )} \]
Time = 5.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=-\frac {x \left (x^2+x (-4+\log (4))-\log (256)\right )}{e^4 (-4+x)^2 (x+\log (4))^2 \left (-4+\log \left (x^2\right )\right )} \]
Integrate[(-8*x - 2*x^2 + (-24 + 2*x)*Log[4] + (x^2 + 4*Log[4])*Log[x^2])/ (E^4*(256*x^2 - 128*x^3 + 16*x^4) + E^4*(512*x - 256*x^2 + 32*x^3)*Log[4] + E^4*(256 - 128*x + 16*x^2)*Log[4]^2 + (E^4*(-128*x^2 + 64*x^3 - 8*x^4) + E^4*(-256*x + 128*x^2 - 16*x^3)*Log[4] + E^4*(-128 + 64*x - 8*x^2)*Log[4] ^2)*Log[x^2] + (E^4*(16*x^2 - 8*x^3 + x^4) + E^4*(32*x - 16*x^2 + 2*x^3)*L og[4] + E^4*(16 - 8*x + x^2)*Log[4]^2)*Log[x^2]^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.
| \(\displaystyle \int \frac {-2 x^2+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )-8 x+(2 x-24) \log (4)}{e^4 \left (16 x^2-128 x+256\right ) \log ^2(4)+e^4 \left (32 x^3-256 x^2+512 x\right ) \log (4)+e^4 \left (16 x^4-128 x^3+256 x^2\right )+\left (e^4 \left (x^2-8 x+16\right ) \log ^2(4)+e^4 \left (2 x^3-16 x^2+32 x\right ) \log (4)+e^4 \left (x^4-8 x^3+16 x^2\right )\right ) \log ^2\left (x^2\right )+\left (e^4 \left (-8 x^2+64 x-128\right ) \log ^2(4)+e^4 \left (-16 x^3+128 x^2-256 x\right ) \log (4)+e^4 \left (-8 x^4+64 x^3-128 x^2\right )\right ) \log \left (x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-2 x^2+\left (x^2+\log (256)\right ) \log \left (x^2\right )+2 x (\log (4)-4)-24 \log (4)}{e^4 (4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {2 x^2+2 (4-\log (4)) x-\left (x^2+\log (256)\right ) \log \left (x^2\right )+24 \log (4)}{(4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2}dx}{e^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {2 x^2+2 (4-\log (4)) x-\left (x^2+\log (256)\right ) \log \left (x^2\right )+24 \log (4)}{(4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2}dx}{e^4}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\int \left (\frac {-x^2-\log (256)}{(x-4)^2 (x+\log (4))^2 \left (\log \left (x^2\right )-4\right )}+\frac {2 \left (-x^2+(4-\log (4)) x+\log (256)\right )}{(4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2}\right )dx}{e^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \int \frac {1}{(4-x) (x+\log (4)) \left (4-\log \left (x^2\right )\right )^2}dx+\int \frac {-x^2-\log (256)}{(x-4)^2 (x+\log (4))^2 \left (\log \left (x^2\right )-4\right )}dx}{e^4}\) |
Int[(-8*x - 2*x^2 + (-24 + 2*x)*Log[4] + (x^2 + 4*Log[4])*Log[x^2])/(E^4*( 256*x^2 - 128*x^3 + 16*x^4) + E^4*(512*x - 256*x^2 + 32*x^3)*Log[4] + E^4* (256 - 128*x + 16*x^2)*Log[4]^2 + (E^4*(-128*x^2 + 64*x^3 - 8*x^4) + E^4*( -256*x + 128*x^2 - 16*x^3)*Log[4] + E^4*(-128 + 64*x - 8*x^2)*Log[4]^2)*Lo g[x^2] + (E^4*(16*x^2 - 8*x^3 + x^4) + E^4*(32*x - 16*x^2 + 2*x^3)*Log[4] + E^4*(16 - 8*x + x^2)*Log[4]^2)*Log[x^2]^2),x]
3.27.87.3.1 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 = 1.52 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12
| method | result | size |
| norman | \(-\frac {x \,{\mathrm e}^{-4}}{\left (\ln \left (x^{2}\right )-4\right ) \left (x -4\right ) \left (x +2 \ln \left (2\right )\right )}\) | \(29\) |
| risch | \(-\frac {x \,{\mathrm e}^{-4}}{\left (2 x \ln \left (2\right )+x^{2}-8 \ln \left (2\right )-4 x \right ) \left (\ln \left (x^{2}\right )-4\right )}\) | \(32\) |
| parallelrisch | \(-\frac {x \,{\mathrm e}^{-4}}{\left (2 x \ln \left (2\right )+x^{2}-8 \ln \left (2\right )-4 x \right ) \left (\ln \left (x^{2}\right )-4\right )}\) | \(34\) |
int(((8*ln(2)+x^2)*ln(x^2)+2*(2*x-24)*ln(2)-2*x^2-8*x)/((4*(x^2-8*x+16)*ex p(4)*ln(2)^2+2*(2*x^3-16*x^2+32*x)*exp(4)*ln(2)+(x^4-8*x^3+16*x^2)*exp(4)) *ln(x^2)^2+(4*(-8*x^2+64*x-128)*exp(4)*ln(2)^2+2*(-16*x^3+128*x^2-256*x)*e xp(4)*ln(2)+(-8*x^4+64*x^3-128*x^2)*exp(4))*ln(x^2)+4*(16*x^2-128*x+256)*e xp(4)*ln(2)^2+2*(32*x^3-256*x^2+512*x)*exp(4)*ln(2)+(16*x^4-128*x^3+256*x^ 2)*exp(4)),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=\frac {x}{8 \, {\left (x - 4\right )} e^{4} \log \left (2\right ) + 4 \, {\left (x^{2} - 4 \, x\right )} e^{4} - {\left (2 \, {\left (x - 4\right )} e^{4} \log \left (2\right ) + {\left (x^{2} - 4 \, x\right )} e^{4}\right )} \log \left (x^{2}\right )} \]
integrate(((8*log(2)+x^2)*log(x^2)+2*(2*x-24)*log(2)-2*x^2-8*x)/((4*(x^2-8 *x+16)*exp(4)*log(2)^2+2*(2*x^3-16*x^2+32*x)*exp(4)*log(2)+(x^4-8*x^3+16*x ^2)*exp(4))*log(x^2)^2+(4*(-8*x^2+64*x-128)*exp(4)*log(2)^2+2*(-16*x^3+128 *x^2-256*x)*exp(4)*log(2)+(-8*x^4+64*x^3-128*x^2)*exp(4))*log(x^2)+4*(16*x ^2-128*x+256)*exp(4)*log(2)^2+2*(32*x^3-256*x^2+512*x)*exp(4)*log(2)+(16*x ^4-128*x^3+256*x^2)*exp(4)),x, algorithm=\
x/(8*(x - 4)*e^4*log(2) + 4*(x^2 - 4*x)*e^4 - (2*(x - 4)*e^4*log(2) + (x^2 - 4*x)*e^4)*log(x^2))
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.81 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=- \frac {x}{- 4 x^{2} e^{4} - 8 x e^{4} \log {\left (2 \right )} + 16 x e^{4} + \left (x^{2} e^{4} - 4 x e^{4} + 2 x e^{4} \log {\left (2 \right )} - 8 e^{4} \log {\left (2 \right )}\right ) \log {\left (x^{2} \right )} + 32 e^{4} \log {\left (2 \right )}} \]
integrate(((8*ln(2)+x**2)*ln(x**2)+2*(2*x-24)*ln(2)-2*x**2-8*x)/((4*(x**2- 8*x+16)*exp(4)*ln(2)**2+2*(2*x**3-16*x**2+32*x)*exp(4)*ln(2)+(x**4-8*x**3+ 16*x**2)*exp(4))*ln(x**2)**2+(4*(-8*x**2+64*x-128)*exp(4)*ln(2)**2+2*(-16* x**3+128*x**2-256*x)*exp(4)*ln(2)+(-8*x**4+64*x**3-128*x**2)*exp(4))*ln(x* *2)+4*(16*x**2-128*x+256)*exp(4)*ln(2)**2+2*(32*x**3-256*x**2+512*x)*exp(4 )*ln(2)+(16*x**4-128*x**3+256*x**2)*exp(4)),x)
-x/(-4*x**2*exp(4) - 8*x*exp(4)*log(2) + 16*x*exp(4) + (x**2*exp(4) - 4*x* exp(4) + 2*x*exp(4)*log(2) - 8*exp(4)*log(2))*log(x**2) + 32*exp(4)*log(2) )
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=\frac {x}{2 \, {\left (2 \, x^{2} e^{4} + 4 \, x {\left (\log \left (2\right ) - 2\right )} e^{4} - 16 \, e^{4} \log \left (2\right ) - {\left (x^{2} e^{4} + 2 \, x {\left (\log \left (2\right ) - 2\right )} e^{4} - 8 \, e^{4} \log \left (2\right )\right )} \log \left (x\right )\right )}} \]
integrate(((8*log(2)+x^2)*log(x^2)+2*(2*x-24)*log(2)-2*x^2-8*x)/((4*(x^2-8 *x+16)*exp(4)*log(2)^2+2*(2*x^3-16*x^2+32*x)*exp(4)*log(2)+(x^4-8*x^3+16*x ^2)*exp(4))*log(x^2)^2+(4*(-8*x^2+64*x-128)*exp(4)*log(2)^2+2*(-16*x^3+128 *x^2-256*x)*exp(4)*log(2)+(-8*x^4+64*x^3-128*x^2)*exp(4))*log(x^2)+4*(16*x ^2-128*x+256)*exp(4)*log(2)^2+2*(32*x^3-256*x^2+512*x)*exp(4)*log(2)+(16*x ^4-128*x^3+256*x^2)*exp(4)),x, algorithm=\
1/2*x/(2*x^2*e^4 + 4*x*(log(2) - 2)*e^4 - 16*e^4*log(2) - (x^2*e^4 + 2*x*( log(2) - 2)*e^4 - 8*e^4*log(2))*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (26) = 52\).
Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.73 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=-\frac {x}{x^{2} e^{4} \log \left (x^{2}\right ) + 2 \, x e^{4} \log \left (2\right ) \log \left (x^{2}\right ) - 4 \, x^{2} e^{4} - 8 \, x e^{4} \log \left (2\right ) - 4 \, x e^{4} \log \left (x^{2}\right ) - 8 \, e^{4} \log \left (2\right ) \log \left (x^{2}\right ) + 16 \, x e^{4} + 32 \, e^{4} \log \left (2\right )} \]
integrate(((8*log(2)+x^2)*log(x^2)+2*(2*x-24)*log(2)-2*x^2-8*x)/((4*(x^2-8 *x+16)*exp(4)*log(2)^2+2*(2*x^3-16*x^2+32*x)*exp(4)*log(2)+(x^4-8*x^3+16*x ^2)*exp(4))*log(x^2)^2+(4*(-8*x^2+64*x-128)*exp(4)*log(2)^2+2*(-16*x^3+128 *x^2-256*x)*exp(4)*log(2)+(-8*x^4+64*x^3-128*x^2)*exp(4))*log(x^2)+4*(16*x ^2-128*x+256)*exp(4)*log(2)^2+2*(32*x^3-256*x^2+512*x)*exp(4)*log(2)+(16*x ^4-128*x^3+256*x^2)*exp(4)),x, algorithm=\
-x/(x^2*e^4*log(x^2) + 2*x*e^4*log(2)*log(x^2) - 4*x^2*e^4 - 8*x*e^4*log(2 ) - 4*x*e^4*log(x^2) - 8*e^4*log(2)*log(x^2) + 16*x*e^4 + 32*e^4*log(2))
Time = 14.81 (sec) , antiderivative size = 562, normalized size of antiderivative = 21.62 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx =\text {Too large to display} \]
int(-(8*x - 2*log(2)*(2*x - 24) - log(x^2)*(8*log(2) + x^2) + 2*x^2)/(log( x^2)^2*(exp(4)*(16*x^2 - 8*x^3 + x^4) + 4*exp(4)*log(2)^2*(x^2 - 8*x + 16) + 2*exp(4)*log(2)*(32*x - 16*x^2 + 2*x^3)) - log(x^2)*(exp(4)*(128*x^2 - 64*x^3 + 8*x^4) + 2*exp(4)*log(2)*(256*x - 128*x^2 + 16*x^3) + 4*exp(4)*lo g(2)^2*(8*x^2 - 64*x + 128)) + exp(4)*(256*x^2 - 128*x^3 + 16*x^4) + 2*exp (4)*log(2)*(512*x - 256*x^2 + 32*x^3) + 4*exp(4)*log(2)^2*(16*x^2 - 128*x + 256)),x)
((x*(4*x + 24*log(2) - 2*x*log(2) + x^2))/(64*exp(4)*log(2)^2 + 16*x^2*exp (4) - 8*x^3*exp(4) + x^4*exp(4) - 32*x*exp(4)*log(2)^2 - 32*x^2*exp(4)*log (2) + 4*x^3*exp(4)*log(2) + 4*x^2*exp(4)*log(2)^2 + 64*x*exp(4)*log(2)) - (x*log(x^2)*(log(256) + x^2))/(2*(64*exp(4)*log(2)^2 + 16*x^2*exp(4) - 8*x ^3*exp(4) + x^4*exp(4) - 32*x*exp(4)*log(2)^2 - 32*x^2*exp(4)*log(2) + 4*x ^3*exp(4)*log(2) + 4*x^2*exp(4)*log(2)^2 + 64*x*exp(4)*log(2))))/(log(x^2) - 4) + ((x^3*exp(-4)*(32*log(2) - 6*log(2)*log(4) + 3*log(2)^2*log(4) + 3 6*log(2)^2 + 2*log(2)^3 + log(2)^4 + 16))/(2*(32*log(2) + 24*log(2)^2 + 8* log(2)^3 + log(2)^4 + 16)) - (4*exp(-4)*(4*log(2)^2*log(4) - 20*log(2)^3*l og(4) + log(2)^4*log(4) - 8*log(2)^3 + 40*log(2)^4 - 2*log(2)^5))/(32*log( 2) + 24*log(2)^2 + 8*log(2)^3 + log(2)^4 + 16) + (2*x*exp(-4)*(32*log(2) - 8*log(2)*log(4) + 32*log(2)^2*log(4) - 16*log(2)^3*log(4) + log(2)^4*log( 4) + 80*log(2)^2 - 16*log(2)^3 + 48*log(2)^4))/(32*log(2) + 24*log(2)^2 + 8*log(2)^3 + log(2)^4 + 16) + (x^2*exp(-4)*(32*log(2) - 16*log(4) + 40*log (2)*log(4) - 96*log(2)^2*log(4) + 10*log(2)^3*log(4) - log(2)^4*log(4) - 8 0*log(2)^2 + 192*log(2)^3 - 20*log(2)^4 + 2*log(2)^5))/(4*(32*log(2) + 24* log(2)^2 + 8*log(2)^3 + log(2)^4 + 16)))/(x*(64*log(2) - 32*log(2)^2) + x^ 3*(4*log(2) - 8) + x^2*(4*log(2)^2 - 32*log(2) + 16) + 64*log(2)^2 + x^4)