Integrand size = 114, antiderivative size = 28 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=3 \left (1-\frac {1}{9} (-5+x)^2 (2+x)^2 \left (x+\frac {x}{\log (x)}\right )^2\right ) \] Output:
3-1/3*(2+x)^2*(x+x/ln(x))^2*(-5+x)^2
Time = 0.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=-\frac {x^2 \left (-10-3 x+x^2\right )^2 (1+\log (x))^2}{3 \log ^2(x)} \] Input:
Integrate[(200*x + 120*x^2 - 22*x^3 - 12*x^4 + 2*x^5 + (-60*x^2 + 22*x^3 + 18*x^4 - 4*x^5)*Log[x] + (-400*x - 360*x^2 + 88*x^3 + 60*x^4 - 12*x^5)*Lo g[x]^2 + (-200*x - 180*x^2 + 44*x^3 + 30*x^4 - 6*x^5)*Log[x]^3)/(3*Log[x]^ 3),x]
Output:
-1/3*(x^2*(-10 - 3*x + x^2)^2*(1 + Log[x])^2)/Log[x]^2
Leaf count is larger than twice the leaf count of optimal. \(118\) vs. \(2(28)=56\).
Time = 1.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.21, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {27, 27, 7239, 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.
\(\displaystyle \int \frac {2 x^5-12 x^4-22 x^3+120 x^2+\left (-6 x^5+30 x^4+44 x^3-180 x^2-200 x\right ) \log ^3(x)+\left (-12 x^5+60 x^4+88 x^3-360 x^2-400 x\right ) \log ^2(x)+\left (-4 x^5+18 x^4+22 x^3-60 x^2\right ) \log (x)+200 x}{3 \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {2 \left (x^5-6 x^4-11 x^3+60 x^2+100 x-\left (3 x^5-15 x^4-22 x^3+90 x^2+100 x\right ) \log ^3(x)-2 \left (3 x^5-15 x^4-22 x^3+90 x^2+100 x\right ) \log ^2(x)-\left (2 x^5-9 x^4-11 x^3+30 x^2\right ) \log (x)\right )}{\log ^3(x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{3} \int \frac {x^5-6 x^4-11 x^3+60 x^2+100 x-\left (3 x^5-15 x^4-22 x^3+90 x^2+100 x\right ) \log ^3(x)-2 \left (3 x^5-15 x^4-22 x^3+90 x^2+100 x\right ) \log ^2(x)-\left (2 x^5-9 x^4-11 x^3+30 x^2\right ) \log (x)}{\log ^3(x)}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \frac {2}{3} \int \frac {x \left (-x^2+3 x+10\right ) (\log (x)+1) \left (-x^2+3 x+\left (3 x^2-6 x-10\right ) \log ^2(x)+\left (3 x^2-6 x-10\right ) \log (x)+10\right )}{\log ^3(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {2}{3} \int \left (-\frac {\left (2 x^3-9 x^2-11 x+30\right ) x^2}{\log ^2(x)}-(x-5) (x+2) \left (3 x^2-6 x-10\right ) x-\frac {2 (x-5) (x+2) \left (3 x^2-6 x-10\right ) x}{\log (x)}+\frac {\left (x^2-3 x-10\right )^2 x}{\log ^3(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} \left (-\frac {x^6}{2 \log ^2(x)}-\frac {x^6}{\log (x)}+\frac {3 x^5}{\log ^2(x)}+\frac {6 x^5}{\log (x)}+\frac {11 x^4}{2 \log ^2(x)}+\frac {11 x^4}{\log (x)}-\frac {30 x^3}{\log ^2(x)}-\frac {60 x^3}{\log (x)}-\frac {1}{2} (5-x)^2 (x+2)^2 x^2-\frac {50 x^2}{\log ^2(x)}-\frac {100 x^2}{\log (x)}\right )\) |
Input:
Int[(200*x + 120*x^2 - 22*x^3 - 12*x^4 + 2*x^5 + (-60*x^2 + 22*x^3 + 18*x^ 4 - 4*x^5)*Log[x] + (-400*x - 360*x^2 + 88*x^3 + 60*x^4 - 12*x^5)*Log[x]^2 + (-200*x - 180*x^2 + 44*x^3 + 30*x^4 - 6*x^5)*Log[x]^3)/(3*Log[x]^3),x]
Output:
(2*(-1/2*((5 - x)^2*x^2*(2 + x)^2) - (50*x^2)/Log[x]^2 - (30*x^3)/Log[x]^2 + (11*x^4)/(2*Log[x]^2) + (3*x^5)/Log[x]^2 - x^6/(2*Log[x]^2) - (100*x^2) /Log[x] - (60*x^3)/Log[x] + (11*x^4)/Log[x] + (6*x^5)/Log[x] - x^6/Log[x]) )/3
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]]
Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(24)=48\).
Time = 1.62 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00
method | result | size |
risch | \(-\frac {x^{6}}{3}+2 x^{5}+\frac {11 x^{4}}{3}-20 x^{3}-\frac {100 x^{2}}{3}-\frac {x^{2} \left (2 x^{4} \ln \left (x \right )+x^{4}-12 x^{3} \ln \left (x \right )-6 x^{3}-22 x^{2} \ln \left (x \right )-11 x^{2}+120 x \ln \left (x \right )+60 x +200 \ln \left (x \right )+100\right )}{3 \ln \left (x \right )^{2}}\) | \(84\) |
norman | \(\frac {-\frac {100 x^{2}}{3}-20 x^{3}+\frac {11 x^{4}}{3}+2 x^{5}-\frac {x^{6}}{3}-\frac {200 x^{2} \ln \left (x \right )}{3}-\frac {100 x^{2} \ln \left (x \right )^{2}}{3}-40 x^{3} \ln \left (x \right )-20 x^{3} \ln \left (x \right )^{2}+\frac {22 x^{4} \ln \left (x \right )}{3}+\frac {11 x^{4} \ln \left (x \right )^{2}}{3}+4 x^{5} \ln \left (x \right )+2 x^{5} \ln \left (x \right )^{2}-\frac {2 x^{6} \ln \left (x \right )}{3}-\frac {x^{6} \ln \left (x \right )^{2}}{3}}{\ln \left (x \right )^{2}}\) | \(112\) |
parallelrisch | \(\frac {-x^{6} \ln \left (x \right )^{2}-2 x^{6} \ln \left (x \right )+6 x^{5} \ln \left (x \right )^{2}-x^{6}+12 x^{5} \ln \left (x \right )+11 x^{4} \ln \left (x \right )^{2}+6 x^{5}+22 x^{4} \ln \left (x \right )-60 x^{3} \ln \left (x \right )^{2}+11 x^{4}-120 x^{3} \ln \left (x \right )-100 x^{2} \ln \left (x \right )^{2}-60 x^{3}-200 x^{2} \ln \left (x \right )-100 x^{2}}{3 \ln \left (x \right )^{2}}\) | \(113\) |
default | \(-\frac {40 x^{3}}{\ln \left (x \right )}+\frac {11 x^{4}}{3 \ln \left (x \right )^{2}}-\frac {200 x^{2}}{3 \ln \left (x \right )}+2 x^{5}-\frac {x^{6}}{3}-\frac {100 x^{2}}{3}-20 x^{3}+\frac {11 x^{4}}{3}-\frac {20 x^{3}}{\ln \left (x \right )^{2}}-\frac {x^{6}}{3 \ln \left (x \right )^{2}}-\frac {2 x^{6}}{3 \ln \left (x \right )}-\frac {100 x^{2}}{3 \ln \left (x \right )^{2}}+\frac {4 x^{5}}{\ln \left (x \right )}+\frac {22 x^{4}}{3 \ln \left (x \right )}+\frac {2 x^{5}}{\ln \left (x \right )^{2}}\) | \(117\) |
parts | \(-\frac {40 x^{3}}{\ln \left (x \right )}+\frac {11 x^{4}}{3 \ln \left (x \right )^{2}}-\frac {200 x^{2}}{3 \ln \left (x \right )}+2 x^{5}-\frac {x^{6}}{3}-\frac {100 x^{2}}{3}-20 x^{3}+\frac {11 x^{4}}{3}-\frac {20 x^{3}}{\ln \left (x \right )^{2}}-\frac {x^{6}}{3 \ln \left (x \right )^{2}}-\frac {2 x^{6}}{3 \ln \left (x \right )}-\frac {100 x^{2}}{3 \ln \left (x \right )^{2}}+\frac {4 x^{5}}{\ln \left (x \right )}+\frac {22 x^{4}}{3 \ln \left (x \right )}+\frac {2 x^{5}}{\ln \left (x \right )^{2}}\) | \(117\) |
Input:
int(1/3*((-6*x^5+30*x^4+44*x^3-180*x^2-200*x)*ln(x)^3+(-12*x^5+60*x^4+88*x ^3-360*x^2-400*x)*ln(x)^2+(-4*x^5+18*x^4+22*x^3-60*x^2)*ln(x)+2*x^5-12*x^4 -22*x^3+120*x^2+200*x)/ln(x)^3,x,method=_RETURNVERBOSE)
Output:
-1/3*x^6+2*x^5+11/3*x^4-20*x^3-100/3*x^2-1/3*x^2*(2*x^4*ln(x)+x^4-12*x^3*l n(x)-6*x^3-22*x^2*ln(x)-11*x^2+120*x*ln(x)+60*x+200*ln(x)+100)/ln(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (24) = 48\).
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.11 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=-\frac {x^{6} - 6 \, x^{5} - 11 \, x^{4} + 60 \, x^{3} + {\left (x^{6} - 6 \, x^{5} - 11 \, x^{4} + 60 \, x^{3} + 100 \, x^{2}\right )} \log \left (x\right )^{2} + 100 \, x^{2} + 2 \, {\left (x^{6} - 6 \, x^{5} - 11 \, x^{4} + 60 \, x^{3} + 100 \, x^{2}\right )} \log \left (x\right )}{3 \, \log \left (x\right )^{2}} \] Input:
integrate(1/3*((-6*x^5+30*x^4+44*x^3-180*x^2-200*x)*log(x)^3+(-12*x^5+60*x ^4+88*x^3-360*x^2-400*x)*log(x)^2+(-4*x^5+18*x^4+22*x^3-60*x^2)*log(x)+2*x ^5-12*x^4-22*x^3+120*x^2+200*x)/log(x)^3,x, algorithm="fricas")
Output:
-1/3*(x^6 - 6*x^5 - 11*x^4 + 60*x^3 + (x^6 - 6*x^5 - 11*x^4 + 60*x^3 + 100 *x^2)*log(x)^2 + 100*x^2 + 2*(x^6 - 6*x^5 - 11*x^4 + 60*x^3 + 100*x^2)*log (x))/log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (20) = 40\).
Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.11 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=- \frac {x^{6}}{3} + 2 x^{5} + \frac {11 x^{4}}{3} - 20 x^{3} - \frac {100 x^{2}}{3} + \frac {- x^{6} + 6 x^{5} + 11 x^{4} - 60 x^{3} - 100 x^{2} + \left (- 2 x^{6} + 12 x^{5} + 22 x^{4} - 120 x^{3} - 200 x^{2}\right ) \log {\left (x \right )}}{3 \log {\left (x \right )}^{2}} \] Input:
integrate(1/3*((-6*x**5+30*x**4+44*x**3-180*x**2-200*x)*ln(x)**3+(-12*x**5 +60*x**4+88*x**3-360*x**2-400*x)*ln(x)**2+(-4*x**5+18*x**4+22*x**3-60*x**2 )*ln(x)+2*x**5-12*x**4-22*x**3+120*x**2+200*x)/ln(x)**3,x)
Output:
-x**6/3 + 2*x**5 + 11*x**4/3 - 20*x**3 - 100*x**2/3 + (-x**6 + 6*x**5 + 11 *x**4 - 60*x**3 - 100*x**2 + (-2*x**6 + 12*x**5 + 22*x**4 - 120*x**3 - 200 *x**2)*log(x))/(3*log(x)**2)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.75 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=-\frac {1}{3} \, x^{6} + 2 \, x^{5} + \frac {11}{3} \, x^{4} - 20 \, x^{3} - \frac {100}{3} \, x^{2} - 4 \, {\rm Ei}\left (6 \, \log \left (x\right )\right ) + 20 \, {\rm Ei}\left (5 \, \log \left (x\right )\right ) + \frac {88}{3} \, {\rm Ei}\left (4 \, \log \left (x\right )\right ) - 120 \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) - \frac {400}{3} \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) - 60 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) + \frac {88}{3} \, \Gamma \left (-1, -4 \, \log \left (x\right )\right ) + 30 \, \Gamma \left (-1, -5 \, \log \left (x\right )\right ) - 8 \, \Gamma \left (-1, -6 \, \log \left (x\right )\right ) - \frac {800}{3} \, \Gamma \left (-2, -2 \, \log \left (x\right )\right ) - 360 \, \Gamma \left (-2, -3 \, \log \left (x\right )\right ) + \frac {352}{3} \, \Gamma \left (-2, -4 \, \log \left (x\right )\right ) + 100 \, \Gamma \left (-2, -5 \, \log \left (x\right )\right ) - 24 \, \Gamma \left (-2, -6 \, \log \left (x\right )\right ) \] Input:
integrate(1/3*((-6*x^5+30*x^4+44*x^3-180*x^2-200*x)*log(x)^3+(-12*x^5+60*x ^4+88*x^3-360*x^2-400*x)*log(x)^2+(-4*x^5+18*x^4+22*x^3-60*x^2)*log(x)+2*x ^5-12*x^4-22*x^3+120*x^2+200*x)/log(x)^3,x, algorithm="maxima")
Output:
-1/3*x^6 + 2*x^5 + 11/3*x^4 - 20*x^3 - 100/3*x^2 - 4*Ei(6*log(x)) + 20*Ei( 5*log(x)) + 88/3*Ei(4*log(x)) - 120*Ei(3*log(x)) - 400/3*Ei(2*log(x)) - 60 *gamma(-1, -3*log(x)) + 88/3*gamma(-1, -4*log(x)) + 30*gamma(-1, -5*log(x) ) - 8*gamma(-1, -6*log(x)) - 800/3*gamma(-2, -2*log(x)) - 360*gamma(-2, -3 *log(x)) + 352/3*gamma(-2, -4*log(x)) + 100*gamma(-2, -5*log(x)) - 24*gamm a(-2, -6*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.14 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=-\frac {1}{3} \, x^{6} + 2 \, x^{5} - \frac {2 \, x^{6}}{3 \, \log \left (x\right )} + \frac {11}{3} \, x^{4} - \frac {x^{6}}{3 \, \log \left (x\right )^{2}} + \frac {4 \, x^{5}}{\log \left (x\right )} - 20 \, x^{3} + \frac {2 \, x^{5}}{\log \left (x\right )^{2}} + \frac {22 \, x^{4}}{3 \, \log \left (x\right )} - \frac {100}{3} \, x^{2} + \frac {11 \, x^{4}}{3 \, \log \left (x\right )^{2}} - \frac {40 \, x^{3}}{\log \left (x\right )} - \frac {20 \, x^{3}}{\log \left (x\right )^{2}} - \frac {200 \, x^{2}}{3 \, \log \left (x\right )} - \frac {100 \, x^{2}}{3 \, \log \left (x\right )^{2}} \] Input:
integrate(1/3*((-6*x^5+30*x^4+44*x^3-180*x^2-200*x)*log(x)^3+(-12*x^5+60*x ^4+88*x^3-360*x^2-400*x)*log(x)^2+(-4*x^5+18*x^4+22*x^3-60*x^2)*log(x)+2*x ^5-12*x^4-22*x^3+120*x^2+200*x)/log(x)^3,x, algorithm="giac")
Output:
-1/3*x^6 + 2*x^5 - 2/3*x^6/log(x) + 11/3*x^4 - 1/3*x^6/log(x)^2 + 4*x^5/lo g(x) - 20*x^3 + 2*x^5/log(x)^2 + 22/3*x^4/log(x) - 100/3*x^2 + 11/3*x^4/lo g(x)^2 - 40*x^3/log(x) - 20*x^3/log(x)^2 - 200/3*x^2/log(x) - 100/3*x^2/lo g(x)^2
Time = 2.93 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=-\frac {x^2\,{\left (\ln \left (x\right )+1\right )}^2\,{\left (-x^2+3\,x+10\right )}^2}{3\,{\ln \left (x\right )}^2} \] Input:
int(-((log(x)*(60*x^2 - 22*x^3 - 18*x^4 + 4*x^5))/3 - (200*x)/3 + (log(x)^ 3*(200*x + 180*x^2 - 44*x^3 - 30*x^4 + 6*x^5))/3 + (log(x)^2*(400*x + 360* x^2 - 88*x^3 - 60*x^4 + 12*x^5))/3 - 40*x^2 + (22*x^3)/3 + 4*x^4 - (2*x^5) /3)/log(x)^3,x)
Output:
-(x^2*(log(x) + 1)^2*(3*x - x^2 + 10)^2)/(3*log(x)^2)
Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.54 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=\frac {x^{2} \left (-\mathrm {log}\left (x \right )^{2} x^{4}+6 \mathrm {log}\left (x \right )^{2} x^{3}+11 \mathrm {log}\left (x \right )^{2} x^{2}-60 \mathrm {log}\left (x \right )^{2} x -100 \mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) x^{4}+12 \,\mathrm {log}\left (x \right ) x^{3}+22 \,\mathrm {log}\left (x \right ) x^{2}-120 \,\mathrm {log}\left (x \right ) x -200 \,\mathrm {log}\left (x \right )-x^{4}+6 x^{3}+11 x^{2}-60 x -100\right )}{3 \mathrm {log}\left (x \right )^{2}} \] Input:
int(1/3*((-6*x^5+30*x^4+44*x^3-180*x^2-200*x)*log(x)^3+(-12*x^5+60*x^4+88* x^3-360*x^2-400*x)*log(x)^2+(-4*x^5+18*x^4+22*x^3-60*x^2)*log(x)+2*x^5-12* x^4-22*x^3+120*x^2+200*x)/log(x)^3,x)
Output:
(x**2*( - log(x)**2*x**4 + 6*log(x)**2*x**3 + 11*log(x)**2*x**2 - 60*log(x )**2*x - 100*log(x)**2 - 2*log(x)*x**4 + 12*log(x)*x**3 + 22*log(x)*x**2 - 120*log(x)*x - 200*log(x) - x**4 + 6*x**3 + 11*x**2 - 60*x - 100))/(3*log (x)**2)