Integrand size = 70, antiderivative size = 29 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=2 x^2 \left (3+x \left (1+\frac {2 x}{\log (x)}+\frac {3 (2+x)}{\log (x)}\right )^2\right ) \]
Time = 0.36 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=6 x^2+2 x^3+\frac {72 x^3}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {50 x^5}{\log ^2(x)}+\frac {24 x^3}{\log (x)}+\frac {20 x^4}{\log (x)} \]
Integrate[(-144*x^2 - 240*x^3 - 100*x^4 + (192*x^2 + 460*x^3 + 250*x^4)*Lo g[x] + (72*x^2 + 80*x^3)*Log[x]^2 + (12*x + 6*x^2)*Log[x]^3)/Log[x]^3,x]
6*x^2 + 2*x^3 + (72*x^3)/Log[x]^2 + (120*x^4)/Log[x]^2 + (50*x^5)/Log[x]^2 + (24*x^3)/Log[x] + (20*x^4)/Log[x]
Time = 0.87 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {7292, 27, 25, 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.
\(\displaystyle \int \frac {-100 x^4-240 x^3-144 x^2+\left (6 x^2+12 x\right ) \log ^3(x)+\left (80 x^3+72 x^2\right ) \log ^2(x)+\left (250 x^4+460 x^3+192 x^2\right ) \log (x)}{\log ^3(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x \left (-50 x^3+125 x^3 \log (x)-120 x^2+40 x^2 \log ^2(x)+230 x^2 \log (x)-72 x+3 x \log ^3(x)+6 \log ^3(x)+36 x \log ^2(x)+96 x \log (x)\right )}{\log ^3(x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {x \left (-125 \log (x) x^3+50 x^3-40 \log ^2(x) x^2-230 \log (x) x^2+120 x^2-3 \log ^3(x) x-36 \log ^2(x) x-96 \log (x) x+72 x-6 \log ^3(x)\right )}{\log ^3(x)}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {x \left (-125 \log (x) x^3+50 x^3-40 \log ^2(x) x^2-230 \log (x) x^2+120 x^2-3 \log ^3(x) x-36 \log ^2(x) x-96 \log (x) x+72 x-6 \log ^3(x)\right )}{\log ^3(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (-\frac {4 (10 x+9) x^2}{\log (x)}-\frac {\left (125 x^2+230 x+96\right ) x^2}{\log ^2(x)}+\frac {2 (5 x+6)^2 x^2}{\log ^3(x)}-3 (x+2) x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (-\frac {25 x^5}{\log ^2(x)}-\frac {60 x^4}{\log ^2(x)}-\frac {10 x^4}{\log (x)}-x^3-\frac {36 x^3}{\log ^2(x)}-\frac {12 x^3}{\log (x)}-3 x^2\right )\) |
Int[(-144*x^2 - 240*x^3 - 100*x^4 + (192*x^2 + 460*x^3 + 250*x^4)*Log[x] + (72*x^2 + 80*x^3)*Log[x]^2 + (12*x + 6*x^2)*Log[x]^3)/Log[x]^3,x]
-2*(-3*x^2 - x^3 - (36*x^3)/Log[x]^2 - (60*x^4)/Log[x]^2 - (25*x^5)/Log[x] ^2 - (12*x^3)/Log[x] - (10*x^4)/Log[x])
3.7.67.3.1 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]]
Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38
method | result | size |
risch | \(2 x^{3}+6 x^{2}+\frac {2 x^{3} \left (25 x^{2}+10 x \ln \left (x \right )+60 x +12 \ln \left (x \right )+36\right )}{\ln \left (x \right )^{2}}\) | \(40\) |
norman | \(\frac {72 x^{3}+120 x^{4}+50 x^{5}+6 x^{2} \ln \left (x \right )^{2}+24 x^{3} \ln \left (x \right )+2 x^{3} \ln \left (x \right )^{2}+20 x^{4} \ln \left (x \right )}{\ln \left (x \right )^{2}}\) | \(54\) |
parallelrisch | \(\frac {72 x^{3}+120 x^{4}+50 x^{5}+6 x^{2} \ln \left (x \right )^{2}+24 x^{3} \ln \left (x \right )+2 x^{3} \ln \left (x \right )^{2}+20 x^{4} \ln \left (x \right )}{\ln \left (x \right )^{2}}\) | \(54\) |
default | \(2 x^{3}+6 x^{2}+\frac {20 x^{4}}{\ln \left (x \right )}+\frac {50 x^{5}}{\ln \left (x \right )^{2}}+\frac {24 x^{3}}{\ln \left (x \right )}+\frac {120 x^{4}}{\ln \left (x \right )^{2}}+\frac {72 x^{3}}{\ln \left (x \right )^{2}}\) | \(57\) |
parts | \(2 x^{3}+6 x^{2}+\frac {20 x^{4}}{\ln \left (x \right )}+\frac {50 x^{5}}{\ln \left (x \right )^{2}}+\frac {24 x^{3}}{\ln \left (x \right )}+\frac {120 x^{4}}{\ln \left (x \right )^{2}}+\frac {72 x^{3}}{\ln \left (x \right )^{2}}\) | \(57\) |
int(((6*x^2+12*x)*ln(x)^3+(80*x^3+72*x^2)*ln(x)^2+(250*x^4+460*x^3+192*x^2 )*ln(x)-100*x^4-240*x^3-144*x^2)/ln(x)^3,x,method=_RETURNVERBOSE)
Time = 0.43 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=\frac {2 \, {\left (25 \, x^{5} + 60 \, x^{4} + 36 \, x^{3} + {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x\right )^{2} + 2 \, {\left (5 \, x^{4} + 6 \, x^{3}\right )} \log \left (x\right )\right )}}{\log \left (x\right )^{2}} \]
integrate(((6*x^2+12*x)*log(x)^3+(80*x^3+72*x^2)*log(x)^2+(250*x^4+460*x^3 +192*x^2)*log(x)-100*x^4-240*x^3-144*x^2)/log(x)^3,x, algorithm=\
Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=2 x^{3} + 6 x^{2} + \frac {50 x^{5} + 120 x^{4} + 72 x^{3} + \left (20 x^{4} + 24 x^{3}\right ) \log {\left (x \right )}}{\log {\left (x \right )}^{2}} \]
integrate(((6*x**2+12*x)*ln(x)**3+(80*x**3+72*x**2)*ln(x)**2+(250*x**4+460 *x**3+192*x**2)*ln(x)-100*x**4-240*x**3-144*x**2)/ln(x)**3,x)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.52 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=2 \, x^{3} + 6 \, x^{2} + 80 \, {\rm Ei}\left (4 \, \log \left (x\right )\right ) + 72 \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) + 576 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) + 1840 \, \Gamma \left (-1, -4 \, \log \left (x\right )\right ) + 1250 \, \Gamma \left (-1, -5 \, \log \left (x\right )\right ) + 1296 \, \Gamma \left (-2, -3 \, \log \left (x\right )\right ) + 3840 \, \Gamma \left (-2, -4 \, \log \left (x\right )\right ) + 2500 \, \Gamma \left (-2, -5 \, \log \left (x\right )\right ) \]
integrate(((6*x^2+12*x)*log(x)^3+(80*x^3+72*x^2)*log(x)^2+(250*x^4+460*x^3 +192*x^2)*log(x)-100*x^4-240*x^3-144*x^2)/log(x)^3,x, algorithm=\
2*x^3 + 6*x^2 + 80*Ei(4*log(x)) + 72*Ei(3*log(x)) + 576*gamma(-1, -3*log(x )) + 1840*gamma(-1, -4*log(x)) + 1250*gamma(-1, -5*log(x)) + 1296*gamma(-2 , -3*log(x)) + 3840*gamma(-2, -4*log(x)) + 2500*gamma(-2, -5*log(x))
Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=2 \, x^{3} + \frac {50 \, x^{5}}{\log \left (x\right )^{2}} + \frac {20 \, x^{4}}{\log \left (x\right )} + 6 \, x^{2} + \frac {120 \, x^{4}}{\log \left (x\right )^{2}} + \frac {24 \, x^{3}}{\log \left (x\right )} + \frac {72 \, x^{3}}{\log \left (x\right )^{2}} \]
integrate(((6*x^2+12*x)*log(x)^3+(80*x^3+72*x^2)*log(x)^2+(250*x^4+460*x^3 +192*x^2)*log(x)-100*x^4-240*x^3-144*x^2)/log(x)^3,x, algorithm=\
2*x^3 + 50*x^5/log(x)^2 + 20*x^4/log(x) + 6*x^2 + 120*x^4/log(x)^2 + 24*x^ 3/log(x) + 72*x^3/log(x)^2
Time = 13.72 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=2\,x^2\,\left (x+3\right )+\frac {2\,x^2\,\left (25\,x^3+60\,x^2+36\,x\right )+2\,x^2\,\ln \left (x\right )\,\left (10\,x^2+12\,x\right )}{{\ln \left (x\right )}^2} \]