Integrand size = 138, antiderivative size = 31 \[ \int \frac {-16 x+256 \sqrt [5]{e} x-256 x^2-1024 x^3+\left (-48+768 \sqrt [5]{e}-768 x-3072 x^2\right ) \log (2)+\left (-16 x+256 \sqrt [5]{e} x+1024 x^3+\left (768 x+6144 x^2\right ) \log (2)\right ) \log (x) \log (\log (x))}{\left (x+256 e^{2/5} x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} \left (-32 x-512 x^2-2048 x^3\right )\right ) \log (2) \log (x)} \, dx=\frac {\left (3+\frac {x}{\log (2)}\right ) \log (\log (x))}{\sqrt [5]{e}-\left (\frac {1}{4}+2 x\right )^2} \]
Time = 0.82 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-16 x+256 \sqrt [5]{e} x-256 x^2-1024 x^3+\left (-48+768 \sqrt [5]{e}-768 x-3072 x^2\right ) \log (2)+\left (-16 x+256 \sqrt [5]{e} x+1024 x^3+\left (768 x+6144 x^2\right ) \log (2)\right ) \log (x) \log (\log (x))}{\left (x+256 e^{2/5} x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} \left (-32 x-512 x^2-2048 x^3\right )\right ) \log (2) \log (x)} \, dx=\frac {16 (x+3 \log (2)) \log (\log (x))}{\left (-1+16 \sqrt [5]{e}-16 x-64 x^2\right ) \log (2)} \]
Integrate[(-16*x + 256*E^(1/5)*x - 256*x^2 - 1024*x^3 + (-48 + 768*E^(1/5) - 768*x - 3072*x^2)*Log[2] + (-16*x + 256*E^(1/5)*x + 1024*x^3 + (768*x + 6144*x^2)*Log[2])*Log[x]*Log[Log[x]])/((x + 256*E^(2/5)*x + 32*x^2 + 384* x^3 + 2048*x^4 + 4096*x^5 + E^(1/5)*(-32*x - 512*x^2 - 2048*x^3))*Log[2]*L og[x]),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.
| \(\displaystyle \int \frac {-1024 x^3-256 x^2+\left (-3072 x^2-768 x+768 \sqrt [5]{e}-48\right ) \log (2)+\left (1024 x^3+\left (6144 x^2+768 x\right ) \log (2)+256 \sqrt [5]{e} x-16 x\right ) \log (x) \log (\log (x))+256 \sqrt [5]{e} x-16 x}{\left (4096 x^5+2048 x^4+384 x^3+32 x^2+\sqrt [5]{e} \left (-2048 x^3-512 x^2-32 x\right )+256 e^{2/5} x+x\right ) \log (2) \log (x)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-1024 x^3-256 x^2+\left (-3072 x^2-768 x+768 \sqrt [5]{e}-48\right ) \log (2)+\left (1024 x^3+\left (6144 x^2+768 x\right ) \log (2)+256 \sqrt [5]{e} x-16 x\right ) \log (x) \log (\log (x))+256 \sqrt [5]{e} x-16 x}{\left (4096 x^5+2048 x^4+384 x^3+32 x^2+\sqrt [5]{e} \left (-2048 x^3-512 x^2-32 x\right )+\left (1+256 e^{2/5}\right ) x\right ) \log (2) \log (x)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-1024 x^3-256 x^2+\left (-3072 x^2-768 x+768 \sqrt [5]{e}-48\right ) \log (2)+\left (1024 x^3+\left (6144 x^2+768 x\right ) \log (2)+256 \sqrt [5]{e} x-16 x\right ) \log (x) \log (\log (x))+\left (256 \sqrt [5]{e}-16\right ) x}{\left (4096 x^5+2048 x^4+384 x^3+32 x^2+\sqrt [5]{e} \left (-2048 x^3-512 x^2-32 x\right )+\left (1+256 e^{2/5}\right ) x\right ) \log (2) \log (x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {16 \left (64 x^3+16 x^2+\left (1-16 \sqrt [5]{e}\right ) x+\left (-64 x^3-16 \sqrt [5]{e} x+x-48 \left (8 x^2+x\right ) \log (2)\right ) \log (x) \log (\log (x))+3 \left (64 x^2+16 x-16 \sqrt [5]{e}+1\right ) \log (2)\right )}{\left (4096 x^5+2048 x^4+384 x^3+32 x^2+\left (1+256 e^{2/5}\right ) x-32 \sqrt [5]{e} \left (64 x^3+16 x^2+x\right )\right ) \log (x)}dx}{\log (2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {16 \int \frac {64 x^3+16 x^2+\left (1-16 \sqrt [5]{e}\right ) x+\left (-64 x^3-16 \sqrt [5]{e} x+x-48 \left (8 x^2+x\right ) \log (2)\right ) \log (x) \log (\log (x))+3 \left (64 x^2+16 x-16 \sqrt [5]{e}+1\right ) \log (2)}{\left (4096 x^5+2048 x^4+384 x^3+32 x^2+\left (1+256 e^{2/5}\right ) x-32 \sqrt [5]{e} \left (64 x^3+16 x^2+x\right )\right ) \log (x)}dx}{\log (2)}\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle -\frac {16 \int \frac {64 x^3+16 x^2+\left (1-16 \sqrt [5]{e}\right ) x+\left (-64 x^3-16 \sqrt [5]{e} x+x-48 \left (8 x^2+x\right ) \log (2)\right ) \log (x) \log (\log (x))+3 \left (64 x^2+16 x-16 \sqrt [5]{e}+1\right ) \log (2)}{x \left (4096 x^4+2048 x^3+128 \left (3-16 \sqrt [5]{e}\right ) x^2+32 \left (1-16 \sqrt [5]{e}\right ) x+\left (1-16 \sqrt [5]{e}\right )^2\right ) \log (x)}dx}{\log (2)}\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle -\frac {16 \int \left (\frac {64 x^3+16 x^2+\left (1-16 \sqrt [5]{e}\right ) x+\left (-64 x^3-16 \sqrt [5]{e} x+x-48 \left (8 x^2+x\right ) \log (2)\right ) \log (x) \log (\log (x))+3 \left (64 x^2+16 x-16 \sqrt [5]{e}+1\right ) \log (2)}{16 e^{3/10} \left (-128 x+64 \sqrt [10]{e}-16\right ) x \log (x)}+\frac {4 \left (64 x^3+16 x^2+\left (1-16 \sqrt [5]{e}\right ) x+\left (-64 x^3-16 \sqrt [5]{e} x+x-48 \left (8 x^2+x\right ) \log (2)\right ) \log (x) \log (\log (x))+3 \left (64 x^2+16 x-16 \sqrt [5]{e}+1\right ) \log (2)\right )}{\sqrt [5]{e} \left (-128 x+64 \sqrt [10]{e}-16\right )^2 x \log (x)}+\frac {64 x^3+16 x^2+\left (1-16 \sqrt [5]{e}\right ) x+\left (-64 x^3-16 \sqrt [5]{e} x+x-48 \left (8 x^2+x\right ) \log (2)\right ) \log (x) \log (\log (x))+3 \left (64 x^2+16 x-16 \sqrt [5]{e}+1\right ) \log (2)}{16 e^{3/10} x \left (128 x+64 \sqrt [10]{e}+16\right ) \log (x)}+\frac {4 \left (64 x^3+16 x^2+\left (1-16 \sqrt [5]{e}\right ) x+\left (-64 x^3-16 \sqrt [5]{e} x+x-48 \left (8 x^2+x\right ) \log (2)\right ) \log (x) \log (\log (x))+3 \left (64 x^2+16 x-16 \sqrt [5]{e}+1\right ) \log (2)\right )}{\sqrt [5]{e} x \left (128 x+64 \sqrt [10]{e}+16\right )^2 \log (x)}\right )dx}{\log (2)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {16 \left (\frac {\int \frac {x^2}{\left (-8 x+4 \sqrt [10]{e}-1\right )^2 \log (x)}dx}{\sqrt [5]{e}}+\frac {\int \frac {x^2}{\left (8 x+4 \sqrt [10]{e}+1\right )^2 \log (x)}dx}{\sqrt [5]{e}}-\frac {1}{64} \left (16-\frac {1}{\sqrt [5]{e}}\right ) \int \frac {1}{\left (-8 x+4 \sqrt [10]{e}-1\right )^2 \log (x)}dx+\frac {\int \frac {x}{\left (-8 x+4 \sqrt [10]{e}-1\right )^2 \log (x)}dx}{4 \sqrt [5]{e}}-\frac {1}{64} \left (16-\frac {1}{\sqrt [5]{e}}\right ) \int \frac {1}{\left (8 x+4 \sqrt [10]{e}+1\right )^2 \log (x)}dx+\frac {\int \frac {x}{\left (8 x+4 \sqrt [10]{e}+1\right )^2 \log (x)}dx}{4 \sqrt [5]{e}}+\frac {3 \log (2) \int \frac {8 x-4 \sqrt [10]{e}+1}{x \left (8 x+4 \sqrt [10]{e}+1\right ) \log (x)}dx}{64 \sqrt [5]{e}}-\frac {3 \log (2) \int \frac {8 x+4 \sqrt [10]{e}+1}{\left (-8 x+4 \sqrt [10]{e}-1\right ) x \log (x)}dx}{64 \sqrt [5]{e}}+\frac {\int \frac {\left (8 x-4 \sqrt [10]{e}+1\right ) (x+\log (8))}{x \log (x)}dx}{256 e^{3/10}}-\frac {\int \frac {\left (8 x+4 \sqrt [10]{e}+1\right ) (x+\log (8))}{x \log (x)}dx}{256 e^{3/10}}+\frac {\left (1-4 \sqrt [10]{e}-24 \log (2)\right ) \int \frac {\log (\log (x))}{\left (-8 x+4 \sqrt [10]{e}-1\right )^2}dx}{8 \sqrt [10]{e}}-\frac {\left (1+4 \sqrt [10]{e}-24 \log (2)\right ) \int \frac {\log (\log (x))}{\left (8 x+4 \sqrt [10]{e}+1\right )^2}dx}{8 \sqrt [10]{e}}+\frac {\operatorname {LogIntegral}(x)}{32 \sqrt [5]{e}}+\frac {\left (1-4 \left (\sqrt [10]{e}+\log (4096)\right )\right ) \operatorname {LogIntegral}(x)}{256 e^{3/10}}-\frac {\left (1+4 \sqrt [10]{e}-48 \log (2)\right ) \operatorname {LogIntegral}(x)}{256 e^{3/10}}-\frac {x \log (\log (x))}{32 \sqrt [5]{e}}-\frac {x \left (1-4 \left (\sqrt [10]{e}+\log (4096)\right )\right ) \log (\log (x))}{256 e^{3/10}}+\frac {x \left (1+4 \sqrt [10]{e}-48 \log (2)\right ) \log (\log (x))}{256 e^{3/10}}\right )}{\log (2)}\) |
Int[(-16*x + 256*E^(1/5)*x - 256*x^2 - 1024*x^3 + (-48 + 768*E^(1/5) - 768 *x - 3072*x^2)*Log[2] + (-16*x + 256*E^(1/5)*x + 1024*x^3 + (768*x + 6144* x^2)*Log[2])*Log[x]*Log[Log[x]])/((x + 256*E^(2/5)*x + 32*x^2 + 384*x^3 + 2048*x^4 + 4096*x^5 + E^(1/5)*(-32*x - 512*x^2 - 2048*x^3))*Log[2]*Log[x]) ,x]
3.27.97.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 16.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(\frac {16 \left (3 \ln \left (2\right )+x \right ) \ln \left (\ln \left (x \right )\right )}{\ln \left (2\right ) \left (-64 x^{2}+16 \,{\mathrm e}^{\frac {1}{5}}-16 x -1\right )}\) | \(32\) |
| parallelrisch | \(\frac {3072 \ln \left (2\right ) \ln \left (\ln \left (x \right )\right )+1024 x \ln \left (\ln \left (x \right )\right )}{64 \ln \left (2\right ) \left (-64 x^{2}+16 \,{\mathrm e}^{\frac {1}{5}}-16 x -1\right )}\) | \(37\) |
int((((6144*x^2+768*x)*ln(2)+256*x*exp(1/5)+1024*x^3-16*x)*ln(x)*ln(ln(x)) +(768*exp(1/5)-3072*x^2-768*x-48)*ln(2)+256*x*exp(1/5)-1024*x^3-256*x^2-16 *x)/(256*x*exp(1/5)^2+(-2048*x^3-512*x^2-32*x)*exp(1/5)+4096*x^5+2048*x^4+ 384*x^3+32*x^2+x)/ln(2)/ln(x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-16 x+256 \sqrt [5]{e} x-256 x^2-1024 x^3+\left (-48+768 \sqrt [5]{e}-768 x-3072 x^2\right ) \log (2)+\left (-16 x+256 \sqrt [5]{e} x+1024 x^3+\left (768 x+6144 x^2\right ) \log (2)\right ) \log (x) \log (\log (x))}{\left (x+256 e^{2/5} x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} \left (-32 x-512 x^2-2048 x^3\right )\right ) \log (2) \log (x)} \, dx=-\frac {16 \, {\left (x + 3 \, \log \left (2\right )\right )} \log \left (\log \left (x\right )\right )}{{\left (64 \, x^{2} + 16 \, x - 16 \, e^{\frac {1}{5}} + 1\right )} \log \left (2\right )} \]
integrate((((6144*x^2+768*x)*log(2)+256*x*exp(1/5)+1024*x^3-16*x)*log(x)*l og(log(x))+(768*exp(1/5)-3072*x^2-768*x-48)*log(2)+256*x*exp(1/5)-1024*x^3 -256*x^2-16*x)/(256*x*exp(1/5)^2+(-2048*x^3-512*x^2-32*x)*exp(1/5)+4096*x^ 5+2048*x^4+384*x^3+32*x^2+x)/log(2)/log(x),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {-16 x+256 \sqrt [5]{e} x-256 x^2-1024 x^3+\left (-48+768 \sqrt [5]{e}-768 x-3072 x^2\right ) \log (2)+\left (-16 x+256 \sqrt [5]{e} x+1024 x^3+\left (768 x+6144 x^2\right ) \log (2)\right ) \log (x) \log (\log (x))}{\left (x+256 e^{2/5} x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} \left (-32 x-512 x^2-2048 x^3\right )\right ) \log (2) \log (x)} \, dx=\frac {\left (- 16 x - 48 \log {\left (2 \right )}\right ) \log {\left (\log {\left (x \right )} \right )}}{64 x^{2} \log {\left (2 \right )} + 16 x \log {\left (2 \right )} - 16 e^{\frac {1}{5}} \log {\left (2 \right )} + \log {\left (2 \right )}} \]
integrate((((6144*x**2+768*x)*ln(2)+256*x*exp(1/5)+1024*x**3-16*x)*ln(x)*l n(ln(x))+(768*exp(1/5)-3072*x**2-768*x-48)*ln(2)+256*x*exp(1/5)-1024*x**3- 256*x**2-16*x)/(256*x*exp(1/5)**2+(-2048*x**3-512*x**2-32*x)*exp(1/5)+4096 *x**5+2048*x**4+384*x**3+32*x**2+x)/ln(2)/ln(x),x)
(-16*x - 48*log(2))*log(log(x))/(64*x**2*log(2) + 16*x*log(2) - 16*exp(1/5 )*log(2) + log(2))
Exception generated. \[ \int \frac {-16 x+256 \sqrt [5]{e} x-256 x^2-1024 x^3+\left (-48+768 \sqrt [5]{e}-768 x-3072 x^2\right ) \log (2)+\left (-16 x+256 \sqrt [5]{e} x+1024 x^3+\left (768 x+6144 x^2\right ) \log (2)\right ) \log (x) \log (\log (x))}{\left (x+256 e^{2/5} x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} \left (-32 x-512 x^2-2048 x^3\right )\right ) \log (2) \log (x)} \, dx=\text {Exception raised: RuntimeError} \]
integrate((((6144*x^2+768*x)*log(2)+256*x*exp(1/5)+1024*x^3-16*x)*log(x)*l og(log(x))+(768*exp(1/5)-3072*x^2-768*x-48)*log(2)+256*x*exp(1/5)-1024*x^3 -256*x^2-16*x)/(256*x*exp(1/5)^2+(-2048*x^3-512*x^2-32*x)*exp(1/5)+4096*x^ 5+2048*x^4+384*x^3+32*x^2+x)/log(2)/log(x),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-16 x+256 \sqrt [5]{e} x-256 x^2-1024 x^3+\left (-48+768 \sqrt [5]{e}-768 x-3072 x^2\right ) \log (2)+\left (-16 x+256 \sqrt [5]{e} x+1024 x^3+\left (768 x+6144 x^2\right ) \log (2)\right ) \log (x) \log (\log (x))}{\left (x+256 e^{2/5} x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} \left (-32 x-512 x^2-2048 x^3\right )\right ) \log (2) \log (x)} \, dx=-\frac {16 \, {\left (x \log \left (\log \left (x\right )\right ) + 3 \, \log \left (2\right ) \log \left (\log \left (x\right )\right )\right )}}{{\left (64 \, x^{2} + 16 \, x - 16 \, e^{\frac {1}{5}} + 1\right )} \log \left (2\right )} \]
integrate((((6144*x^2+768*x)*log(2)+256*x*exp(1/5)+1024*x^3-16*x)*log(x)*l og(log(x))+(768*exp(1/5)-3072*x^2-768*x-48)*log(2)+256*x*exp(1/5)-1024*x^3 -256*x^2-16*x)/(256*x*exp(1/5)^2+(-2048*x^3-512*x^2-32*x)*exp(1/5)+4096*x^ 5+2048*x^4+384*x^3+32*x^2+x)/log(2)/log(x),x, algorithm=\
Time = 9.77 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {-16 x+256 \sqrt [5]{e} x-256 x^2-1024 x^3+\left (-48+768 \sqrt [5]{e}-768 x-3072 x^2\right ) \log (2)+\left (-16 x+256 \sqrt [5]{e} x+1024 x^3+\left (768 x+6144 x^2\right ) \log (2)\right ) \log (x) \log (\log (x))}{\left (x+256 e^{2/5} x+32 x^2+384 x^3+2048 x^4+4096 x^5+\sqrt [5]{e} \left (-32 x-512 x^2-2048 x^3\right )\right ) \log (2) \log (x)} \, dx=-\frac {16\,\ln \left (\ln \left (x\right )\right )\,\left (x+\ln \left (8\right )\right )}{\ln \left (2\right )\,\left (64\,x^2+16\,x-16\,{\mathrm {e}}^{1/5}+1\right )} \]
int(-(16*x - 256*x*exp(1/5) + 256*x^2 + 1024*x^3 + log(2)*(768*x - 768*exp (1/5) + 3072*x^2 + 48) - log(log(x))*log(x)*(log(2)*(768*x + 6144*x^2) - 1 6*x + 256*x*exp(1/5) + 1024*x^3))/(log(2)*log(x)*(x + 256*x*exp(2/5) - exp (1/5)*(32*x + 512*x^2 + 2048*x^3) + 32*x^2 + 384*x^3 + 2048*x^4 + 4096*x^5 )),x)