Integrand size = 134, antiderivative size = 17 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {e^8 \log ^4(4)}{(5+5 x+\log (x))^4} \] Output:
16*ln(2)^4*exp(2)^4/(5+ln(x)+5*x)^4
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {e^8 \log ^4(4)}{(5+5 x+\log (x))^4} \] Input:
Integrate[(E^8*(-4 - 20*x)*Log[4]^4)/(3125*x + 15625*x^2 + 31250*x^3 + 312 50*x^4 + 15625*x^5 + 3125*x^6 + (3125*x + 12500*x^2 + 18750*x^3 + 12500*x^ 4 + 3125*x^5)*Log[x] + (1250*x + 3750*x^2 + 3750*x^3 + 1250*x^4)*Log[x]^2 + (250*x + 500*x^2 + 250*x^3)*Log[x]^3 + (25*x + 25*x^2)*Log[x]^4 + x*Log[ x]^5),x]
Output:
(E^8*Log[4]^4)/(5 + 5*x + Log[x])^4
Time = 0.53 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {27, 27, 7239, 7237}
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.
\(\displaystyle \int \frac {e^8 (-20 x-4) \log ^4(4)}{3125 x^6+15625 x^5+31250 x^4+31250 x^3+15625 x^2+\left (25 x^2+25 x\right ) \log ^4(x)+\left (250 x^3+500 x^2+250 x\right ) \log ^3(x)+\left (1250 x^4+3750 x^3+3750 x^2+1250 x\right ) \log ^2(x)+\left (3125 x^5+12500 x^4+18750 x^3+12500 x^2+3125 x\right ) \log (x)+3125 x+x \log ^5(x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^8 \log ^4(4) \int -\frac {4 (5 x+1)}{3125 x^6+15625 x^5+31250 x^4+31250 x^3+15625 x^2+\log ^5(x) x+3125 x+25 \left (x^2+x\right ) \log ^4(x)+250 \left (x^3+2 x^2+x\right ) \log ^3(x)+1250 \left (x^4+3 x^3+3 x^2+x\right ) \log ^2(x)+3125 \left (x^5+4 x^4+6 x^3+4 x^2+x\right ) \log (x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -4 e^8 \log ^4(4) \int \frac {5 x+1}{3125 x^6+15625 x^5+31250 x^4+31250 x^3+15625 x^2+\log ^5(x) x+3125 x+25 \left (x^2+x\right ) \log ^4(x)+250 \left (x^3+2 x^2+x\right ) \log ^3(x)+1250 \left (x^4+3 x^3+3 x^2+x\right ) \log ^2(x)+3125 \left (x^5+4 x^4+6 x^3+4 x^2+x\right ) \log (x)}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -4 e^8 \log ^4(4) \int \frac {5 x+1}{x (5 x+\log (x)+5)^5}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {e^8 \log ^4(4)}{(5 x+\log (x)+5)^4}\) |
Input:
Int[(E^8*(-4 - 20*x)*Log[4]^4)/(3125*x + 15625*x^2 + 31250*x^3 + 31250*x^4 + 15625*x^5 + 3125*x^6 + (3125*x + 12500*x^2 + 18750*x^3 + 12500*x^4 + 31 25*x^5)*Log[x] + (1250*x + 3750*x^2 + 3750*x^3 + 1250*x^4)*Log[x]^2 + (250 *x + 500*x^2 + 250*x^3)*Log[x]^3 + (25*x + 25*x^2)*Log[x]^4 + x*Log[x]^5), x]
Output:
(E^8*Log[4]^4)/(5 + 5*x + Log[x])^4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06
method | result | size |
risch | \(\frac {16 \ln \left (2\right )^{4} {\mathrm e}^{8}}{\left (5+\ln \left (x \right )+5 x \right )^{4}}\) | \(18\) |
default | \(\frac {16 \ln \left (2\right )^{4} {\mathrm e}^{8}}{\left (5+\ln \left (x \right )+5 x \right )^{4}}\) | \(20\) |
parallelrisch | \(\frac {16 \,{\mathrm e}^{8} \ln \left (2\right )^{4}}{625 x^{4}+500 x^{3} \ln \left (x \right )+150 x^{2} \ln \left (x \right )^{2}+20 x \ln \left (x \right )^{3}+\ln \left (x \right )^{4}+2500 x^{3}+1500 x^{2} \ln \left (x \right )+300 x \ln \left (x \right )^{2}+20 \ln \left (x \right )^{3}+3750 x^{2}+1500 x \ln \left (x \right )+150 \ln \left (x \right )^{2}+2500 x +500 \ln \left (x \right )+625}\) | \(95\) |
Input:
int(16*(-20*x-4)*exp(2)^4*ln(2)^4/(x*ln(x)^5+(25*x^2+25*x)*ln(x)^4+(250*x^ 3+500*x^2+250*x)*ln(x)^3+(1250*x^4+3750*x^3+3750*x^2+1250*x)*ln(x)^2+(3125 *x^5+12500*x^4+18750*x^3+12500*x^2+3125*x)*ln(x)+3125*x^6+15625*x^5+31250* x^4+31250*x^3+15625*x^2+3125*x),x,method=_RETURNVERBOSE)
Output:
16*ln(2)^4*exp(8)/(5+ln(x)+5*x)^4
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (17) = 34\).
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 4.35 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {16 \, e^{8} \log \left (2\right )^{4}}{625 \, x^{4} + 20 \, {\left (x + 1\right )} \log \left (x\right )^{3} + \log \left (x\right )^{4} + 2500 \, x^{3} + 150 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} + 3750 \, x^{2} + 500 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x\right ) + 2500 \, x + 625} \] Input:
integrate(16*(-20*x-4)*exp(2)^4*log(2)^4/(x*log(x)^5+(25*x^2+25*x)*log(x)^ 4+(250*x^3+500*x^2+250*x)*log(x)^3+(1250*x^4+3750*x^3+3750*x^2+1250*x)*log (x)^2+(3125*x^5+12500*x^4+18750*x^3+12500*x^2+3125*x)*log(x)+3125*x^6+1562 5*x^5+31250*x^4+31250*x^3+15625*x^2+3125*x),x, algorithm="fricas")
Output:
16*e^8*log(2)^4/(625*x^4 + 20*(x + 1)*log(x)^3 + log(x)^4 + 2500*x^3 + 150 *(x^2 + 2*x + 1)*log(x)^2 + 3750*x^2 + 500*(x^3 + 3*x^2 + 3*x + 1)*log(x) + 2500*x + 625)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (19) = 38\).
Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 4.59 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {16 e^{8} \log {\left (2 \right )}^{4}}{625 x^{4} + 2500 x^{3} + 3750 x^{2} + 2500 x + \left (20 x + 20\right ) \log {\left (x \right )}^{3} + \left (150 x^{2} + 300 x + 150\right ) \log {\left (x \right )}^{2} + \left (500 x^{3} + 1500 x^{2} + 1500 x + 500\right ) \log {\left (x \right )} + \log {\left (x \right )}^{4} + 625} \] Input:
integrate(16*(-20*x-4)*exp(2)**4*ln(2)**4/(x*ln(x)**5+(25*x**2+25*x)*ln(x) **4+(250*x**3+500*x**2+250*x)*ln(x)**3+(1250*x**4+3750*x**3+3750*x**2+1250 *x)*ln(x)**2+(3125*x**5+12500*x**4+18750*x**3+12500*x**2+3125*x)*ln(x)+312 5*x**6+15625*x**5+31250*x**4+31250*x**3+15625*x**2+3125*x),x)
Output:
16*exp(8)*log(2)**4/(625*x**4 + 2500*x**3 + 3750*x**2 + 2500*x + (20*x + 2 0)*log(x)**3 + (150*x**2 + 300*x + 150)*log(x)**2 + (500*x**3 + 1500*x**2 + 1500*x + 500)*log(x) + log(x)**4 + 625)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (17) = 34\).
Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 4.35 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {16 \, e^{8} \log \left (2\right )^{4}}{625 \, x^{4} + 20 \, {\left (x + 1\right )} \log \left (x\right )^{3} + \log \left (x\right )^{4} + 2500 \, x^{3} + 150 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} + 3750 \, x^{2} + 500 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x\right ) + 2500 \, x + 625} \] Input:
integrate(16*(-20*x-4)*exp(2)^4*log(2)^4/(x*log(x)^5+(25*x^2+25*x)*log(x)^ 4+(250*x^3+500*x^2+250*x)*log(x)^3+(1250*x^4+3750*x^3+3750*x^2+1250*x)*log (x)^2+(3125*x^5+12500*x^4+18750*x^3+12500*x^2+3125*x)*log(x)+3125*x^6+1562 5*x^5+31250*x^4+31250*x^3+15625*x^2+3125*x),x, algorithm="maxima")
Output:
16*e^8*log(2)^4/(625*x^4 + 20*(x + 1)*log(x)^3 + log(x)^4 + 2500*x^3 + 150 *(x^2 + 2*x + 1)*log(x)^2 + 3750*x^2 + 500*(x^3 + 3*x^2 + 3*x + 1)*log(x) + 2500*x + 625)
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (17) = 34\).
Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 7.88 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {16 \, {\left (5 \, x + 1\right )} e^{8} \log \left (2\right )^{4}}{3125 \, x^{5} + 2500 \, x^{4} \log \left (x\right ) + 750 \, x^{3} \log \left (x\right )^{2} + 100 \, x^{2} \log \left (x\right )^{3} + 5 \, x \log \left (x\right )^{4} + 13125 \, x^{4} + 8000 \, x^{3} \log \left (x\right ) + 1650 \, x^{2} \log \left (x\right )^{2} + 120 \, x \log \left (x\right )^{3} + \log \left (x\right )^{4} + 21250 \, x^{3} + 9000 \, x^{2} \log \left (x\right ) + 1050 \, x \log \left (x\right )^{2} + 20 \, \log \left (x\right )^{3} + 16250 \, x^{2} + 4000 \, x \log \left (x\right ) + 150 \, \log \left (x\right )^{2} + 5625 \, x + 500 \, \log \left (x\right ) + 625} \] Input:
integrate(16*(-20*x-4)*exp(2)^4*log(2)^4/(x*log(x)^5+(25*x^2+25*x)*log(x)^ 4+(250*x^3+500*x^2+250*x)*log(x)^3+(1250*x^4+3750*x^3+3750*x^2+1250*x)*log (x)^2+(3125*x^5+12500*x^4+18750*x^3+12500*x^2+3125*x)*log(x)+3125*x^6+1562 5*x^5+31250*x^4+31250*x^3+15625*x^2+3125*x),x, algorithm="giac")
Output:
16*(5*x + 1)*e^8*log(2)^4/(3125*x^5 + 2500*x^4*log(x) + 750*x^3*log(x)^2 + 100*x^2*log(x)^3 + 5*x*log(x)^4 + 13125*x^4 + 8000*x^3*log(x) + 1650*x^2* log(x)^2 + 120*x*log(x)^3 + log(x)^4 + 21250*x^3 + 9000*x^2*log(x) + 1050* x*log(x)^2 + 20*log(x)^3 + 16250*x^2 + 4000*x*log(x) + 150*log(x)^2 + 5625 *x + 500*log(x) + 625)
Time = 4.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {16\,{\mathrm {e}}^8\,{\ln \left (2\right )}^4}{{\left (5\,x+\ln \left (x\right )+5\right )}^4} \] Input:
int(-(16*exp(8)*log(2)^4*(20*x + 4))/(3125*x + log(x)^4*(25*x + 25*x^2) + x*log(x)^5 + log(x)^3*(250*x + 500*x^2 + 250*x^3) + log(x)*(3125*x + 12500 *x^2 + 18750*x^3 + 12500*x^4 + 3125*x^5) + log(x)^2*(1250*x + 3750*x^2 + 3 750*x^3 + 1250*x^4) + 15625*x^2 + 31250*x^3 + 31250*x^4 + 15625*x^5 + 3125 *x^6),x)
Output:
(16*exp(8)*log(2)^4)/(5*x + log(x) + 5)^4
Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 5.47 \[ \int \frac {e^8 (-4-20 x) \log ^4(4)}{3125 x+15625 x^2+31250 x^3+31250 x^4+15625 x^5+3125 x^6+\left (3125 x+12500 x^2+18750 x^3+12500 x^4+3125 x^5\right ) \log (x)+\left (1250 x+3750 x^2+3750 x^3+1250 x^4\right ) \log ^2(x)+\left (250 x+500 x^2+250 x^3\right ) \log ^3(x)+\left (25 x+25 x^2\right ) \log ^4(x)+x \log ^5(x)} \, dx=\frac {16 \mathrm {log}\left (2\right )^{4} e^{8}}{\mathrm {log}\left (x \right )^{4}+20 \mathrm {log}\left (x \right )^{3} x +20 \mathrm {log}\left (x \right )^{3}+150 \mathrm {log}\left (x \right )^{2} x^{2}+300 \mathrm {log}\left (x \right )^{2} x +150 \mathrm {log}\left (x \right )^{2}+500 \,\mathrm {log}\left (x \right ) x^{3}+1500 \,\mathrm {log}\left (x \right ) x^{2}+1500 \,\mathrm {log}\left (x \right ) x +500 \,\mathrm {log}\left (x \right )+625 x^{4}+2500 x^{3}+3750 x^{2}+2500 x +625} \] Input:
int(16*(-20*x-4)*exp(2)^4*log(2)^4/(x*log(x)^5+(25*x^2+25*x)*log(x)^4+(250 *x^3+500*x^2+250*x)*log(x)^3+(1250*x^4+3750*x^3+3750*x^2+1250*x)*log(x)^2+ (3125*x^5+12500*x^4+18750*x^3+12500*x^2+3125*x)*log(x)+3125*x^6+15625*x^5+ 31250*x^4+31250*x^3+15625*x^2+3125*x),x)
Output:
(16*log(2)**4*e**8)/(log(x)**4 + 20*log(x)**3*x + 20*log(x)**3 + 150*log(x )**2*x**2 + 300*log(x)**2*x + 150*log(x)**2 + 500*log(x)*x**3 + 1500*log(x )*x**2 + 1500*log(x)*x + 500*log(x) + 625*x**4 + 2500*x**3 + 3750*x**2 + 2 500*x + 625)