Integrand size = 116, antiderivative size = 34 \[ \int \frac {1}{2} \left (-2 e^x+12 x^2+60 x^3+100 x^4+52 x^5+\left (9 x+60 x^2+142 x^3+116 x^4+25 x^5\right ) \log (x)+\left (9 x+45 x^2+72 x^3+43 x^4+3 x^5\right ) \log ^2(x)+\left (9 x^2+18 x^3+5 x^4\right ) \log ^3(x)+2 x^3 \log ^4(x)\right ) \, dx=-e^x+\frac {1}{4} x^2 (-x+(-3+x (-4-\log (x))) (x+\log (x)))^2 \]
Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(34)=68\).
Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.62 \[ \int \frac {1}{2} \left (-2 e^x+12 x^2+60 x^3+100 x^4+52 x^5+\left (9 x+60 x^2+142 x^3+116 x^4+25 x^5\right ) \log (x)+\left (9 x+45 x^2+72 x^3+43 x^4+3 x^5\right ) \log ^2(x)+\left (9 x^2+18 x^3+5 x^4\right ) \log ^3(x)+2 x^3 \log ^4(x)\right ) \, dx=\frac {1}{4} \left (4 \left (-e^x+4 x^4 (1+x)^2\right )+8 x^3 (1+x)^2 (3+x) \log (x)+x^2 \left (9+24 x+30 x^2+16 x^3+x^4\right ) \log ^2(x)+2 x^3 \left (3+4 x+x^2\right ) \log ^3(x)+x^4 \log ^4(x)\right ) \]
Integrate[(-2*E^x + 12*x^2 + 60*x^3 + 100*x^4 + 52*x^5 + (9*x + 60*x^2 + 1 42*x^3 + 116*x^4 + 25*x^5)*Log[x] + (9*x + 45*x^2 + 72*x^3 + 43*x^4 + 3*x^ 5)*Log[x]^2 + (9*x^2 + 18*x^3 + 5*x^4)*Log[x]^3 + 2*x^3*Log[x]^4)/2,x]
(4*(-E^x + 4*x^4*(1 + x)^2) + 8*x^3*(1 + x)^2*(3 + x)*Log[x] + x^2*(9 + 24 *x + 30*x^2 + 16*x^3 + x^4)*Log[x]^2 + 2*x^3*(3 + 4*x + x^2)*Log[x]^3 + x^ 4*Log[x]^4)/4
Leaf count is larger than twice the leaf count of optimal. \(139\) vs. \(2(34)=68\).
Time = 0.50 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {27, 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.
\(\displaystyle \int \frac {1}{2} \left (52 x^5+100 x^4+60 x^3+2 x^3 \log ^4(x)+12 x^2+\left (5 x^4+18 x^3+9 x^2\right ) \log ^3(x)+\left (3 x^5+43 x^4+72 x^3+45 x^2+9 x\right ) \log ^2(x)+\left (25 x^5+116 x^4+142 x^3+60 x^2+9 x\right ) \log (x)-2 e^x\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \left (52 x^5+100 x^4+2 \log ^4(x) x^3+60 x^3+12 x^2-2 e^x+\left (5 x^4+18 x^3+9 x^2\right ) \log ^3(x)+\left (3 x^5+43 x^4+72 x^3+45 x^2+9 x\right ) \log ^2(x)+\left (25 x^5+116 x^4+142 x^3+60 x^2+9 x\right ) \log (x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (8 x^6+\frac {1}{2} x^6 \log ^2(x)+4 x^6 \log (x)+16 x^5+x^5 \log ^3(x)+8 x^5 \log ^2(x)+20 x^5 \log (x)+8 x^4+\frac {1}{2} x^4 \log ^4(x)+4 x^4 \log ^3(x)+15 x^4 \log ^2(x)+28 x^4 \log (x)+3 x^3 \log ^3(x)+12 x^3 \log ^2(x)+12 x^3 \log (x)+\frac {9}{2} x^2 \log ^2(x)-2 e^x\right )\) |
Int[(-2*E^x + 12*x^2 + 60*x^3 + 100*x^4 + 52*x^5 + (9*x + 60*x^2 + 142*x^3 + 116*x^4 + 25*x^5)*Log[x] + (9*x + 45*x^2 + 72*x^3 + 43*x^4 + 3*x^5)*Log [x]^2 + (9*x^2 + 18*x^3 + 5*x^4)*Log[x]^3 + 2*x^3*Log[x]^4)/2,x]
(-2*E^x + 8*x^4 + 16*x^5 + 8*x^6 + 12*x^3*Log[x] + 28*x^4*Log[x] + 20*x^5* Log[x] + 4*x^6*Log[x] + (9*x^2*Log[x]^2)/2 + 12*x^3*Log[x]^2 + 15*x^4*Log[ x]^2 + 8*x^5*Log[x]^2 + (x^6*Log[x]^2)/2 + 3*x^3*Log[x]^3 + 4*x^4*Log[x]^3 + x^5*Log[x]^3 + (x^4*Log[x]^4)/2)/2
3.20.47.3.1 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs. \(2(31)=62\).
Time = 0.17 (sec) , antiderivative size = 130, normalized size of antiderivative = 3.82
method | result | size |
default | \(2 x^{4} \ln \left (x \right )^{3}+\frac {x^{6} \ln \left (x \right )^{2}}{4}+2 x^{6} \ln \left (x \right )+\frac {x^{5} \ln \left (x \right )^{3}}{2}+\frac {3 x^{3} \ln \left (x \right )^{3}}{2}+10 x^{5} \ln \left (x \right )+\frac {15 x^{4} \ln \left (x \right )^{2}}{2}+14 x^{4} \ln \left (x \right )+4 x^{5} \ln \left (x \right )^{2}+6 x^{3} \ln \left (x \right )^{2}+\frac {9 x^{2} \ln \left (x \right )^{2}}{4}+4 x^{6}+8 x^{5}+4 x^{4}+6 x^{3} \ln \left (x \right )+\frac {x^{4} \ln \left (x \right )^{4}}{4}-{\mathrm e}^{x}\) | \(130\) |
risch | \(2 x^{4} \ln \left (x \right )^{3}+\frac {x^{6} \ln \left (x \right )^{2}}{4}+2 x^{6} \ln \left (x \right )+\frac {x^{5} \ln \left (x \right )^{3}}{2}+\frac {3 x^{3} \ln \left (x \right )^{3}}{2}+10 x^{5} \ln \left (x \right )+\frac {15 x^{4} \ln \left (x \right )^{2}}{2}+14 x^{4} \ln \left (x \right )+4 x^{5} \ln \left (x \right )^{2}+6 x^{3} \ln \left (x \right )^{2}+\frac {9 x^{2} \ln \left (x \right )^{2}}{4}+4 x^{6}+8 x^{5}+4 x^{4}+6 x^{3} \ln \left (x \right )+\frac {x^{4} \ln \left (x \right )^{4}}{4}-{\mathrm e}^{x}\) | \(130\) |
parallelrisch | \(2 x^{4} \ln \left (x \right )^{3}+\frac {x^{6} \ln \left (x \right )^{2}}{4}+2 x^{6} \ln \left (x \right )+\frac {x^{5} \ln \left (x \right )^{3}}{2}+\frac {3 x^{3} \ln \left (x \right )^{3}}{2}+10 x^{5} \ln \left (x \right )+\frac {15 x^{4} \ln \left (x \right )^{2}}{2}+14 x^{4} \ln \left (x \right )+4 x^{5} \ln \left (x \right )^{2}+6 x^{3} \ln \left (x \right )^{2}+\frac {9 x^{2} \ln \left (x \right )^{2}}{4}+4 x^{6}+8 x^{5}+4 x^{4}+6 x^{3} \ln \left (x \right )+\frac {x^{4} \ln \left (x \right )^{4}}{4}-{\mathrm e}^{x}\) | \(130\) |
parts | \(2 x^{4} \ln \left (x \right )^{3}+\frac {x^{6} \ln \left (x \right )^{2}}{4}+2 x^{6} \ln \left (x \right )+\frac {x^{5} \ln \left (x \right )^{3}}{2}+\frac {3 x^{3} \ln \left (x \right )^{3}}{2}+10 x^{5} \ln \left (x \right )+\frac {15 x^{4} \ln \left (x \right )^{2}}{2}+14 x^{4} \ln \left (x \right )+4 x^{5} \ln \left (x \right )^{2}+6 x^{3} \ln \left (x \right )^{2}+\frac {9 x^{2} \ln \left (x \right )^{2}}{4}+4 x^{6}+8 x^{5}+4 x^{4}+6 x^{3} \ln \left (x \right )+\frac {x^{4} \ln \left (x \right )^{4}}{4}-{\mathrm e}^{x}\) | \(130\) |
int(x^3*ln(x)^4+1/2*(5*x^4+18*x^3+9*x^2)*ln(x)^3+1/2*(3*x^5+43*x^4+72*x^3+ 45*x^2+9*x)*ln(x)^2+1/2*(25*x^5+116*x^4+142*x^3+60*x^2+9*x)*ln(x)-exp(x)+2 6*x^5+50*x^4+30*x^3+6*x^2,x,method=_RETURNVERBOSE)
2*x^4*ln(x)^3+1/4*x^6*ln(x)^2+2*x^6*ln(x)+1/2*x^5*ln(x)^3+3/2*x^3*ln(x)^3+ 10*x^5*ln(x)+15/2*x^4*ln(x)^2+14*x^4*ln(x)+4*x^5*ln(x)^2+6*x^3*ln(x)^2+9/4 *x^2*ln(x)^2+4*x^6+8*x^5+4*x^4+6*x^3*ln(x)+1/4*x^4*ln(x)^4-exp(x)
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.00 \[ \int \frac {1}{2} \left (-2 e^x+12 x^2+60 x^3+100 x^4+52 x^5+\left (9 x+60 x^2+142 x^3+116 x^4+25 x^5\right ) \log (x)+\left (9 x+45 x^2+72 x^3+43 x^4+3 x^5\right ) \log ^2(x)+\left (9 x^2+18 x^3+5 x^4\right ) \log ^3(x)+2 x^3 \log ^4(x)\right ) \, dx=\frac {1}{4} \, x^{4} \log \left (x\right )^{4} + 4 \, x^{6} + 8 \, x^{5} + 4 \, x^{4} + \frac {1}{2} \, {\left (x^{5} + 4 \, x^{4} + 3 \, x^{3}\right )} \log \left (x\right )^{3} + \frac {1}{4} \, {\left (x^{6} + 16 \, x^{5} + 30 \, x^{4} + 24 \, x^{3} + 9 \, x^{2}\right )} \log \left (x\right )^{2} + 2 \, {\left (x^{6} + 5 \, x^{5} + 7 \, x^{4} + 3 \, x^{3}\right )} \log \left (x\right ) - e^{x} \]
integrate(x^3*log(x)^4+1/2*(5*x^4+18*x^3+9*x^2)*log(x)^3+1/2*(3*x^5+43*x^4 +72*x^3+45*x^2+9*x)*log(x)^2+1/2*(25*x^5+116*x^4+142*x^3+60*x^2+9*x)*log(x )-exp(x)+26*x^5+50*x^4+30*x^3+6*x^2,x, algorithm=\
1/4*x^4*log(x)^4 + 4*x^6 + 8*x^5 + 4*x^4 + 1/2*(x^5 + 4*x^4 + 3*x^3)*log(x )^3 + 1/4*(x^6 + 16*x^5 + 30*x^4 + 24*x^3 + 9*x^2)*log(x)^2 + 2*(x^6 + 5*x ^5 + 7*x^4 + 3*x^3)*log(x) - e^x
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (26) = 52\).
Time = 0.16 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.15 \[ \int \frac {1}{2} \left (-2 e^x+12 x^2+60 x^3+100 x^4+52 x^5+\left (9 x+60 x^2+142 x^3+116 x^4+25 x^5\right ) \log (x)+\left (9 x+45 x^2+72 x^3+43 x^4+3 x^5\right ) \log ^2(x)+\left (9 x^2+18 x^3+5 x^4\right ) \log ^3(x)+2 x^3 \log ^4(x)\right ) \, dx=4 x^{6} + 8 x^{5} + \frac {x^{4} \log {\left (x \right )}^{4}}{4} + 4 x^{4} + \left (\frac {x^{5}}{2} + 2 x^{4} + \frac {3 x^{3}}{2}\right ) \log {\left (x \right )}^{3} + \left (2 x^{6} + 10 x^{5} + 14 x^{4} + 6 x^{3}\right ) \log {\left (x \right )} + \left (\frac {x^{6}}{4} + 4 x^{5} + \frac {15 x^{4}}{2} + 6 x^{3} + \frac {9 x^{2}}{4}\right ) \log {\left (x \right )}^{2} - e^{x} \]
integrate(x**3*ln(x)**4+1/2*(5*x**4+18*x**3+9*x**2)*ln(x)**3+1/2*(3*x**5+4 3*x**4+72*x**3+45*x**2+9*x)*ln(x)**2+1/2*(25*x**5+116*x**4+142*x**3+60*x** 2+9*x)*ln(x)-exp(x)+26*x**5+50*x**4+30*x**3+6*x**2,x)
4*x**6 + 8*x**5 + x**4*log(x)**4/4 + 4*x**4 + (x**5/2 + 2*x**4 + 3*x**3/2) *log(x)**3 + (2*x**6 + 10*x**5 + 14*x**4 + 6*x**3)*log(x) + (x**6/4 + 4*x* *5 + 15*x**4/2 + 6*x**3 + 9*x**2/4)*log(x)**2 - exp(x)
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (27) = 54\).
Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 7.15 \[ \int \frac {1}{2} \left (-2 e^x+12 x^2+60 x^3+100 x^4+52 x^5+\left (9 x+60 x^2+142 x^3+116 x^4+25 x^5\right ) \log (x)+\left (9 x+45 x^2+72 x^3+43 x^4+3 x^5\right ) \log ^2(x)+\left (9 x^2+18 x^3+5 x^4\right ) \log ^3(x)+2 x^3 \log ^4(x)\right ) \, dx=\frac {1}{72} \, {\left (18 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 1\right )} x^{6} + \frac {1}{250} \, {\left (125 \, \log \left (x\right )^{3} - 75 \, \log \left (x\right )^{2} + 30 \, \log \left (x\right ) - 6\right )} x^{5} + \frac {43}{250} \, {\left (25 \, \log \left (x\right )^{2} - 10 \, \log \left (x\right ) + 2\right )} x^{5} + \frac {287}{72} \, x^{6} + \frac {1}{128} \, {\left (32 \, \log \left (x\right )^{4} - 32 \, \log \left (x\right )^{3} + 24 \, \log \left (x\right )^{2} - 12 \, \log \left (x\right ) + 3\right )} x^{4} + \frac {9}{128} \, {\left (32 \, \log \left (x\right )^{3} - 24 \, \log \left (x\right )^{2} + 12 \, \log \left (x\right ) - 3\right )} x^{4} + \frac {9}{8} \, {\left (8 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )} x^{4} + \frac {192}{25} \, x^{5} + \frac {1}{6} \, {\left (9 \, \log \left (x\right )^{3} - 9 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) - 2\right )} x^{3} + \frac {5}{6} \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} + \frac {49}{16} \, x^{4} + \frac {9}{8} \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} - \frac {4}{3} \, x^{3} - \frac {9}{8} \, x^{2} + \frac {1}{60} \, {\left (125 \, x^{6} + 696 \, x^{5} + 1065 \, x^{4} + 600 \, x^{3} + 135 \, x^{2}\right )} \log \left (x\right ) - e^{x} \]
integrate(x^3*log(x)^4+1/2*(5*x^4+18*x^3+9*x^2)*log(x)^3+1/2*(3*x^5+43*x^4 +72*x^3+45*x^2+9*x)*log(x)^2+1/2*(25*x^5+116*x^4+142*x^3+60*x^2+9*x)*log(x )-exp(x)+26*x^5+50*x^4+30*x^3+6*x^2,x, algorithm=\
1/72*(18*log(x)^2 - 6*log(x) + 1)*x^6 + 1/250*(125*log(x)^3 - 75*log(x)^2 + 30*log(x) - 6)*x^5 + 43/250*(25*log(x)^2 - 10*log(x) + 2)*x^5 + 287/72*x ^6 + 1/128*(32*log(x)^4 - 32*log(x)^3 + 24*log(x)^2 - 12*log(x) + 3)*x^4 + 9/128*(32*log(x)^3 - 24*log(x)^2 + 12*log(x) - 3)*x^4 + 9/8*(8*log(x)^2 - 4*log(x) + 1)*x^4 + 192/25*x^5 + 1/6*(9*log(x)^3 - 9*log(x)^2 + 6*log(x) - 2)*x^3 + 5/6*(9*log(x)^2 - 6*log(x) + 2)*x^3 + 49/16*x^4 + 9/8*(2*log(x) ^2 - 2*log(x) + 1)*x^2 - 4/3*x^3 - 9/8*x^2 + 1/60*(125*x^6 + 696*x^5 + 106 5*x^4 + 600*x^3 + 135*x^2)*log(x) - e^x
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (27) = 54\).
Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.79 \[ \int \frac {1}{2} \left (-2 e^x+12 x^2+60 x^3+100 x^4+52 x^5+\left (9 x+60 x^2+142 x^3+116 x^4+25 x^5\right ) \log (x)+\left (9 x+45 x^2+72 x^3+43 x^4+3 x^5\right ) \log ^2(x)+\left (9 x^2+18 x^3+5 x^4\right ) \log ^3(x)+2 x^3 \log ^4(x)\right ) \, dx=\frac {1}{4} \, x^{6} \log \left (x\right )^{2} + \frac {1}{2} \, x^{5} \log \left (x\right )^{3} + \frac {1}{4} \, x^{4} \log \left (x\right )^{4} + 2 \, x^{6} \log \left (x\right ) + 4 \, x^{5} \log \left (x\right )^{2} + 2 \, x^{4} \log \left (x\right )^{3} + 4 \, x^{6} + 10 \, x^{5} \log \left (x\right ) + \frac {15}{2} \, x^{4} \log \left (x\right )^{2} + \frac {3}{2} \, x^{3} \log \left (x\right )^{3} + 8 \, x^{5} + 14 \, x^{4} \log \left (x\right ) + 6 \, x^{3} \log \left (x\right )^{2} + 4 \, x^{4} + 6 \, x^{3} \log \left (x\right ) + \frac {9}{4} \, x^{2} \log \left (x\right )^{2} - e^{x} \]
integrate(x^3*log(x)^4+1/2*(5*x^4+18*x^3+9*x^2)*log(x)^3+1/2*(3*x^5+43*x^4 +72*x^3+45*x^2+9*x)*log(x)^2+1/2*(25*x^5+116*x^4+142*x^3+60*x^2+9*x)*log(x )-exp(x)+26*x^5+50*x^4+30*x^3+6*x^2,x, algorithm=\
1/4*x^6*log(x)^2 + 1/2*x^5*log(x)^3 + 1/4*x^4*log(x)^4 + 2*x^6*log(x) + 4* x^5*log(x)^2 + 2*x^4*log(x)^3 + 4*x^6 + 10*x^5*log(x) + 15/2*x^4*log(x)^2 + 3/2*x^3*log(x)^3 + 8*x^5 + 14*x^4*log(x) + 6*x^3*log(x)^2 + 4*x^4 + 6*x^ 3*log(x) + 9/4*x^2*log(x)^2 - e^x
Time = 10.52 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.09 \[ \int \frac {1}{2} \left (-2 e^x+12 x^2+60 x^3+100 x^4+52 x^5+\left (9 x+60 x^2+142 x^3+116 x^4+25 x^5\right ) \log (x)+\left (9 x+45 x^2+72 x^3+43 x^4+3 x^5\right ) \log ^2(x)+\left (9 x^2+18 x^3+5 x^4\right ) \log ^3(x)+2 x^3 \log ^4(x)\right ) \, dx={\ln \left (x\right )}^2\,\left (\frac {x^6}{4}+4\,x^5+\frac {15\,x^4}{2}+6\,x^3+\frac {9\,x^2}{4}\right )-{\mathrm {e}}^x+\frac {x^4\,{\ln \left (x\right )}^4}{4}+\ln \left (x\right )\,\left (2\,x^6+10\,x^5+14\,x^4+6\,x^3\right )+{\ln \left (x\right )}^3\,\left (\frac {x^5}{2}+2\,x^4+\frac {3\,x^3}{2}\right )+4\,x^4+8\,x^5+4\,x^6 \]
int((log(x)*(9*x + 60*x^2 + 142*x^3 + 116*x^4 + 25*x^5))/2 - exp(x) + x^3* log(x)^4 + (log(x)^3*(9*x^2 + 18*x^3 + 5*x^4))/2 + (log(x)^2*(9*x + 45*x^2 + 72*x^3 + 43*x^4 + 3*x^5))/2 + 6*x^2 + 30*x^3 + 50*x^4 + 26*x^5,x)