Integrand size = 76, antiderivative size = 25 \[ \int \frac {x^4+x^5+4 x^4 \log (3)+\left (x^4-12 x^5-7 x^6-24 x^5 \log (3)\right ) \log (x)+5 x^4 \log ^2(x)+\left (5 x^4+6 x^5+20 x^4 \log (3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx=x^5 (-x+\log (x)+(1+x+4 \log (3)) (-x+\log (\log (x)))) \] Output:
(ln(x)-x+(ln(ln(x))-x)*(x+4*ln(3)+1))*x^5
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.60 \[ \int \frac {x^4+x^5+4 x^4 \log (3)+\left (x^4-12 x^5-7 x^6-24 x^5 \log (3)\right ) \log (x)+5 x^4 \log ^2(x)+\left (5 x^4+6 x^5+20 x^4 \log (3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx=-2 x^6-x^7-4 \operatorname {ExpIntegralEi}(5 \log (x)) \log (3)-2 x^6 \log (9)+\operatorname {ExpIntegralEi}(5 \log (x)) \log (81)+x^5 \log (x)+x^5 \log (\log (x))+x^6 \log (\log (x))+4 x^5 \log (3) \log (\log (x)) \] Input:
Integrate[(x^4 + x^5 + 4*x^4*Log[3] + (x^4 - 12*x^5 - 7*x^6 - 24*x^5*Log[3 ])*Log[x] + 5*x^4*Log[x]^2 + (5*x^4 + 6*x^5 + 20*x^4*Log[3])*Log[x]*Log[Lo g[x]])/Log[x],x]
Output:
-2*x^6 - x^7 - 4*ExpIntegralEi[5*Log[x]]*Log[3] - 2*x^6*Log[9] + ExpIntegr alEi[5*Log[x]]*Log[81] + x^5*Log[x] + x^5*Log[Log[x]] + x^6*Log[Log[x]] + 4*x^5*Log[3]*Log[Log[x]]
Time = 0.72 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {6, 7293, 2009}
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.
\(\displaystyle \int \frac {x^5+x^4+5 x^4 \log ^2(x)+4 x^4 \log (3)+\left (6 x^5+5 x^4+20 x^4 \log (3)\right ) \log (x) \log (\log (x))+\left (-7 x^6-12 x^5-24 x^5 \log (3)+x^4\right ) \log (x)}{\log (x)} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x^5+5 x^4 \log ^2(x)+x^4 (1+4 \log (3))+\left (6 x^5+5 x^4+20 x^4 \log (3)\right ) \log (x) \log (\log (x))+\left (-7 x^6-12 x^5-24 x^5 \log (3)+x^4\right ) \log (x)}{\log (x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (x^4 (6 x+5+20 \log (3)) \log (\log (x))+\frac {x^4 \left (-7 x^2 \log (x)+x+5 \log ^2(x)-12 x (1+\log (9)) \log (x)+\log (x)+1+\log (81)\right )}{\log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -x^7+x^6 \log (\log (x))-2 x^6 (1+\log (9))+x^5 \log (x)+x^5 (1+\log (81)) \log (\log (x))\) |
Input:
Int[(x^4 + x^5 + 4*x^4*Log[3] + (x^4 - 12*x^5 - 7*x^6 - 24*x^5*Log[3])*Log [x] + 5*x^4*Log[x]^2 + (5*x^4 + 6*x^5 + 20*x^4*Log[3])*Log[x]*Log[Log[x]]) /Log[x],x]
Output:
-x^7 - 2*x^6*(1 + Log[9]) + x^5*Log[x] + x^6*Log[Log[x]] + x^5*(1 + Log[81 ])*Log[Log[x]]
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]
Time = 0.81 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72
method | result | size |
risch | \(\left (4 x^{5} \ln \left (3\right )+x^{6}+x^{5}\right ) \ln \left (\ln \left (x \right )\right )-4 x^{6} \ln \left (3\right )-x^{7}-2 x^{6}+x^{5} \ln \left (x \right )\) | \(43\) |
default | \(4 \ln \left (3\right ) \left (-x^{6}+x^{5} \ln \left (\ln \left (x \right )\right )\right )+x^{5} \ln \left (\ln \left (x \right )\right )+x^{6} \ln \left (\ln \left (x \right )\right )-2 x^{6}-x^{7}+x^{5} \ln \left (x \right )\) | \(49\) |
parallelrisch | \(-4 x^{6} \ln \left (3\right )+4 \ln \left (\ln \left (x \right )\right ) x^{5} \ln \left (3\right )-x^{7}+x^{6} \ln \left (\ln \left (x \right )\right )-2 x^{6}+x^{5} \ln \left (x \right )+x^{5} \ln \left (\ln \left (x \right )\right )\) | \(49\) |
parts | \(-4 x^{6} \ln \left (3\right )+4 \ln \left (\ln \left (x \right )\right ) x^{5} \ln \left (3\right )-x^{7}+x^{6} \ln \left (\ln \left (x \right )\right )-2 x^{6}+x^{5} \ln \left (x \right )+x^{5} \ln \left (\ln \left (x \right )\right )\) | \(49\) |
Input:
int(((20*x^4*ln(3)+6*x^5+5*x^4)*ln(x)*ln(ln(x))+5*x^4*ln(x)^2+(-24*x^5*ln( 3)-7*x^6-12*x^5+x^4)*ln(x)+4*x^4*ln(3)+x^5+x^4)/ln(x),x,method=_RETURNVERB OSE)
Output:
(4*x^5*ln(3)+x^6+x^5)*ln(ln(x))-4*x^6*ln(3)-x^7-2*x^6+x^5*ln(x)
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {x^4+x^5+4 x^4 \log (3)+\left (x^4-12 x^5-7 x^6-24 x^5 \log (3)\right ) \log (x)+5 x^4 \log ^2(x)+\left (5 x^4+6 x^5+20 x^4 \log (3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx=-x^{7} - 4 \, x^{6} \log \left (3\right ) - 2 \, x^{6} + x^{5} \log \left (x\right ) + {\left (x^{6} + 4 \, x^{5} \log \left (3\right ) + x^{5}\right )} \log \left (\log \left (x\right )\right ) \] Input:
integrate(((20*x^4*log(3)+6*x^5+5*x^4)*log(x)*log(log(x))+5*x^4*log(x)^2+( -24*x^5*log(3)-7*x^6-12*x^5+x^4)*log(x)+4*x^4*log(3)+x^5+x^4)/log(x),x, al gorithm="fricas")
Output:
-x^7 - 4*x^6*log(3) - 2*x^6 + x^5*log(x) + (x^6 + 4*x^5*log(3) + x^5)*log( log(x))
Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {x^4+x^5+4 x^4 \log (3)+\left (x^4-12 x^5-7 x^6-24 x^5 \log (3)\right ) \log (x)+5 x^4 \log ^2(x)+\left (5 x^4+6 x^5+20 x^4 \log (3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx=- x^{7} + x^{6} \left (- 4 \log {\left (3 \right )} - 2\right ) + x^{5} \log {\left (x \right )} + \left (x^{6} + x^{5} + 4 x^{5} \log {\left (3 \right )}\right ) \log {\left (\log {\left (x \right )} \right )} \] Input:
integrate(((20*x**4*ln(3)+6*x**5+5*x**4)*ln(x)*ln(ln(x))+5*x**4*ln(x)**2+( -24*x**5*ln(3)-7*x**6-12*x**5+x**4)*ln(x)+4*x**4*ln(3)+x**5+x**4)/ln(x),x)
Output:
-x**7 + x**6*(-4*log(3) - 2) + x**5*log(x) + (x**6 + x**5 + 4*x**5*log(3)) *log(log(x))
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.64 \[ \int \frac {x^4+x^5+4 x^4 \log (3)+\left (x^4-12 x^5-7 x^6-24 x^5 \log (3)\right ) \log (x)+5 x^4 \log ^2(x)+\left (5 x^4+6 x^5+20 x^4 \log (3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx=-x^{7} - 4 \, x^{6} \log \left (3\right ) + x^{6} \log \left (\log \left (x\right )\right ) - 2 \, x^{6} + x^{5} \log \left (x\right ) + x^{5} \log \left (\log \left (x\right )\right ) + 4 \, {\left (x^{5} \log \left (\log \left (x\right )\right ) - {\rm Ei}\left (5 \, \log \left (x\right )\right )\right )} \log \left (3\right ) + 4 \, {\rm Ei}\left (5 \, \log \left (x\right )\right ) \log \left (3\right ) \] Input:
integrate(((20*x^4*log(3)+6*x^5+5*x^4)*log(x)*log(log(x))+5*x^4*log(x)^2+( -24*x^5*log(3)-7*x^6-12*x^5+x^4)*log(x)+4*x^4*log(3)+x^5+x^4)/log(x),x, al gorithm="maxima")
Output:
-x^7 - 4*x^6*log(3) + x^6*log(log(x)) - 2*x^6 + x^5*log(x) + x^5*log(log(x )) + 4*(x^5*log(log(x)) - Ei(5*log(x)))*log(3) + 4*Ei(5*log(x))*log(3)
Time = 0.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.92 \[ \int \frac {x^4+x^5+4 x^4 \log (3)+\left (x^4-12 x^5-7 x^6-24 x^5 \log (3)\right ) \log (x)+5 x^4 \log ^2(x)+\left (5 x^4+6 x^5+20 x^4 \log (3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx=-x^{7} - 4 \, x^{6} \log \left (3\right ) + x^{6} \log \left (\log \left (x\right )\right ) + 4 \, x^{5} \log \left (3\right ) \log \left (\log \left (x\right )\right ) - 2 \, x^{6} + x^{5} \log \left (x\right ) + x^{5} \log \left (\log \left (x\right )\right ) \] Input:
integrate(((20*x^4*log(3)+6*x^5+5*x^4)*log(x)*log(log(x))+5*x^4*log(x)^2+( -24*x^5*log(3)-7*x^6-12*x^5+x^4)*log(x)+4*x^4*log(3)+x^5+x^4)/log(x),x, al gorithm="giac")
Output:
-x^7 - 4*x^6*log(3) + x^6*log(log(x)) + 4*x^5*log(3)*log(log(x)) - 2*x^6 + x^5*log(x) + x^5*log(log(x))
Time = 4.32 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {x^4+x^5+4 x^4 \log (3)+\left (x^4-12 x^5-7 x^6-24 x^5 \log (3)\right ) \log (x)+5 x^4 \log ^2(x)+\left (5 x^4+6 x^5+20 x^4 \log (3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx=x^5\,\ln \left (x\right )+\ln \left (\ln \left (x\right )\right )\,\left (x^6+\left (4\,\ln \left (3\right )+1\right )\,x^5\right )-x^6\,\left (4\,\ln \left (3\right )+2\right )-x^7 \] Input:
int((5*x^4*log(x)^2 + 4*x^4*log(3) + x^4 + x^5 - log(x)*(24*x^5*log(3) - x ^4 + 12*x^5 + 7*x^6) + log(log(x))*log(x)*(20*x^4*log(3) + 5*x^4 + 6*x^5)) /log(x),x)
Output:
x^5*log(x) + log(log(x))*(x^5*(4*log(3) + 1) + x^6) - x^6*(4*log(3) + 2) - x^7
Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {x^4+x^5+4 x^4 \log (3)+\left (x^4-12 x^5-7 x^6-24 x^5 \log (3)\right ) \log (x)+5 x^4 \log ^2(x)+\left (5 x^4+6 x^5+20 x^4 \log (3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx=x^{5} \left (4 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (3\right )+\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x +\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\mathrm {log}\left (x \right )-4 \,\mathrm {log}\left (3\right ) x -x^{2}-2 x \right ) \] Input:
int(((20*x^4*log(3)+6*x^5+5*x^4)*log(x)*log(log(x))+5*x^4*log(x)^2+(-24*x^ 5*log(3)-7*x^6-12*x^5+x^4)*log(x)+4*x^4*log(3)+x^5+x^4)/log(x),x)
Output:
x**5*(4*log(log(x))*log(3) + log(log(x))*x + log(log(x)) + log(x) - 4*log( 3)*x - x**2 - 2*x)