Integrand size = 166, antiderivative size = 32 \[ \int \frac {-18 x^4-24 x^5-8 x^6+e^{10} \left (-18-24 x-8 x^2\right )+e^5 \left (36 x^2+48 x^3+16 x^4\right )+\left (e^{10} (-18-12 x)+108 x^4+156 x^5+56 x^6+e^5 \left (-90 x^2-144 x^3-56 x^4\right )\right ) \log (x)+\left (-180 x^4-360 x^5-160 x^6+e^5 \left (120 x^3+80 x^4\right )\right ) \log ^2(x)+\left (450 x^4+900 x^5+400 x^6\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=(3+2 x)^2 \left (-x+\frac {-\frac {e^5}{x}+x}{5 \log (x)}\right )^2 \]
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {-18 x^4-24 x^5-8 x^6+e^{10} \left (-18-24 x-8 x^2\right )+e^5 \left (36 x^2+48 x^3+16 x^4\right )+\left (e^{10} (-18-12 x)+108 x^4+156 x^5+56 x^6+e^5 \left (-90 x^2-144 x^3-56 x^4\right )\right ) \log (x)+\left (-180 x^4-360 x^5-160 x^6+e^5 \left (120 x^3+80 x^4\right )\right ) \log ^2(x)+\left (450 x^4+900 x^5+400 x^6\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {(3+2 x)^2 \left (e^5-x^2+5 x^2 \log (x)\right )^2}{25 x^2 \log ^2(x)} \]
Integrate[(-18*x^4 - 24*x^5 - 8*x^6 + E^10*(-18 - 24*x - 8*x^2) + E^5*(36* x^2 + 48*x^3 + 16*x^4) + (E^10*(-18 - 12*x) + 108*x^4 + 156*x^5 + 56*x^6 + E^5*(-90*x^2 - 144*x^3 - 56*x^4))*Log[x] + (-180*x^4 - 360*x^5 - 160*x^6 + E^5*(120*x^3 + 80*x^4))*Log[x]^2 + (450*x^4 + 900*x^5 + 400*x^6)*Log[x]^ 3)/(25*x^3*Log[x]^3),x]
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.
| \(\displaystyle \int \frac {-8 x^6-24 x^5-18 x^4+e^{10} \left (-8 x^2-24 x-18\right )+\left (400 x^6+900 x^5+450 x^4\right ) \log ^3(x)+e^5 \left (16 x^4+48 x^3+36 x^2\right )+\left (-160 x^6-360 x^5-180 x^4+e^5 \left (80 x^4+120 x^3\right )\right ) \log ^2(x)+\left (56 x^6+156 x^5+108 x^4+e^5 \left (-56 x^4-144 x^3-90 x^2\right )+e^{10} (-12 x-18)\right ) \log (x)}{25 x^3 \log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{25} \int -\frac {2 \left (4 x^6+12 x^5+9 x^4-25 \left (8 x^6+18 x^5+9 x^4\right ) \log ^3(x)+10 \left (8 x^6+18 x^5+9 x^4-2 e^5 \left (2 x^4+3 x^3\right )\right ) \log ^2(x)+e^{10} \left (4 x^2+12 x+9\right )-2 e^5 \left (4 x^4+12 x^3+9 x^2\right )-\left (28 x^6+78 x^5+54 x^4-3 e^{10} (2 x+3)-e^5 \left (28 x^4+72 x^3+45 x^2\right )\right ) \log (x)\right )}{x^3 \log ^3(x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{25} \int \frac {4 x^6+12 x^5+9 x^4-25 \left (8 x^6+18 x^5+9 x^4\right ) \log ^3(x)+10 \left (8 x^6+18 x^5+9 x^4-2 e^5 \left (2 x^4+3 x^3\right )\right ) \log ^2(x)+e^{10} \left (4 x^2+12 x+9\right )-2 e^5 \left (4 x^4+12 x^3+9 x^2\right )-\left (28 x^6+78 x^5+54 x^4-3 e^{10} (2 x+3)-e^5 \left (28 x^4+72 x^3+45 x^2\right )\right ) \log (x)}{x^3 \log ^3(x)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2}{25} \int \frac {(2 x+3) \left (-25 (4 x+3) \log ^3(x) x^4+10 \left (x (4 x+3)-2 e^5\right ) \log ^2(x) x^3+(2 x+3) \left (e^5-x^2\right )^2+\left (-2 (7 x+9) x^4+e^5 (14 x+15) x^2+3 e^{10}\right ) \log (x)\right )}{x^3 \log ^3(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2}{25} \int \left (\frac {(2 x+3)^2 \left (x^2-e^5\right )^2}{x^3 \log ^3(x)}-\frac {(2 x+3) \left (14 x^3+18 x^2+3 e^5\right ) \left (x^2-e^5\right )}{x^3 \log ^2(x)}-25 x (2 x+3) (4 x+3)+\frac {10 (2 x+3) \left (4 x^2+3 x-2 e^5\right )}{\log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{25} \left (\int \frac {(2 x+3)^2 \left (x^2-e^5\right )^2}{x^3 \log ^3(x)}dx-\int \frac {(2 x+3) \left (x^2-e^5\right ) \left (14 x^3+18 x^2+3 e^5\right )}{x^3 \log ^2(x)}dx+10 \left (9-4 e^5\right ) \operatorname {ExpIntegralEi}(2 \log (x))+180 \operatorname {ExpIntegralEi}(3 \log (x))+80 \operatorname {ExpIntegralEi}(4 \log (x))-60 e^5 \operatorname {LogIntegral}(x)-\frac {25}{2} x^2 (2 x+3)^2\right )\) |
Int[(-18*x^4 - 24*x^5 - 8*x^6 + E^10*(-18 - 24*x - 8*x^2) + E^5*(36*x^2 + 48*x^3 + 16*x^4) + (E^10*(-18 - 12*x) + 108*x^4 + 156*x^5 + 56*x^6 + E^5*( -90*x^2 - 144*x^3 - 56*x^4))*Log[x] + (-180*x^4 - 360*x^5 - 160*x^6 + E^5* (120*x^3 + 80*x^4))*Log[x]^2 + (450*x^4 + 900*x^5 + 400*x^6)*Log[x]^3)/(25 *x^3*Log[x]^3),x]
3.2.64.3.1 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]]
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. \(126\) vs. \(2(29)=58\).
Time = 0.14 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.97
| method | result | size |
| risch | \(4 x^{4}+12 x^{3}+9 x^{2}+\frac {-40 x^{6} \ln \left (x \right )+40 \,{\mathrm e}^{5} \ln \left (x \right ) x^{4}+4 x^{6}-120 x^{5} \ln \left (x \right )-8 x^{4} {\mathrm e}^{5}+120 x^{3} {\mathrm e}^{5} \ln \left (x \right )+12 x^{5}-90 x^{4} \ln \left (x \right )+4 x^{2} {\mathrm e}^{10}-24 x^{3} {\mathrm e}^{5}+90 x^{2} {\mathrm e}^{5} \ln \left (x \right )+9 x^{4}+12 x \,{\mathrm e}^{10}-18 x^{2} {\mathrm e}^{5}+9 \,{\mathrm e}^{10}}{25 x^{2} \ln \left (x \right )^{2}}\) | \(127\) |
| parallelrisch | \(\frac {100 x^{6} \ln \left (x \right )^{2}+300 x^{5} \ln \left (x \right )^{2}-40 x^{6} \ln \left (x \right )+40 \,{\mathrm e}^{5} \ln \left (x \right ) x^{4}+225 x^{4} \ln \left (x \right )^{2}-120 x^{5} \ln \left (x \right )+4 x^{6}+120 x^{3} {\mathrm e}^{5} \ln \left (x \right )-8 x^{4} {\mathrm e}^{5}-90 x^{4} \ln \left (x \right )+12 x^{5}+90 x^{2} {\mathrm e}^{5} \ln \left (x \right )-24 x^{3} {\mathrm e}^{5}+9 x^{4}-18 x^{2} {\mathrm e}^{5}+4 x^{2} {\mathrm e}^{10}+12 x \,{\mathrm e}^{10}+9 \,{\mathrm e}^{10}}{25 \ln \left (x \right )^{2} x^{2}}\) | \(144\) |
| default | \(\frac {9 x^{2}}{25 \ln \left (x \right )^{2}}-\frac {8 x^{4}}{5 \ln \left (x \right )}-\frac {144 \,{\mathrm e}^{5} \left (-\frac {x}{\ln \left (x \right )}-\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )\right )}{25}-\frac {18 \,{\mathrm e}^{10} \left (-\frac {1}{2 x^{2} \ln \left (x \right )^{2}}+\frac {1}{x^{2} \ln \left (x \right )}-2 \,\operatorname {Ei}_{1}\left (2 \ln \left (x \right )\right )\right )}{25}-\frac {16 \,{\mathrm e}^{5} \operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )}{5}-\frac {24 \,{\mathrm e}^{5} \operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )}{5}-\frac {56 \,{\mathrm e}^{5} \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )\right )}{25}-\frac {12 \,{\mathrm e}^{10} \left (-\frac {1}{x \ln \left (x \right )}+\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )\right )}{25}-\frac {24 \,{\mathrm e}^{10} \left (-\frac {1}{2 x \ln \left (x \right )^{2}}+\frac {1}{2 x \ln \left (x \right )}-\frac {\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )}{2}\right )}{25}-\frac {18 \,{\mathrm e}^{10} \left (-\frac {1}{x^{2} \ln \left (x \right )}+2 \,\operatorname {Ei}_{1}\left (2 \ln \left (x \right )\right )\right )}{25}+\frac {48 \,{\mathrm e}^{5} \left (-\frac {x}{2 \ln \left (x \right )^{2}}-\frac {x}{2 \ln \left (x \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )}{2}\right )}{25}+\frac {16 \,{\mathrm e}^{5} \left (-\frac {x^{2}}{2 \ln \left (x \right )^{2}}-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )\right )}{25}+\frac {4 \,{\mathrm e}^{10}}{25 \ln \left (x \right )^{2}}-\frac {24 x^{3}}{5 \ln \left (x \right )}+\frac {4 x^{4}}{25 \ln \left (x \right )^{2}}+\frac {12 x^{3}}{25 \ln \left (x \right )^{2}}-\frac {18 x^{2}}{5 \ln \left (x \right )}+4 x^{4}+12 x^{3}+9 x^{2}-\frac {18 \,{\mathrm e}^{5}}{25 \ln \left (x \right )^{2}}+\frac {18 \,{\mathrm e}^{5}}{5 \ln \left (x \right )}\) | \(324\) |
| parts | \(\frac {9 x^{2}}{25 \ln \left (x \right )^{2}}-\frac {8 x^{4}}{5 \ln \left (x \right )}-\frac {144 \,{\mathrm e}^{5} \left (-\frac {x}{\ln \left (x \right )}-\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )\right )}{25}-\frac {18 \,{\mathrm e}^{10} \left (-\frac {1}{2 x^{2} \ln \left (x \right )^{2}}+\frac {1}{x^{2} \ln \left (x \right )}-2 \,\operatorname {Ei}_{1}\left (2 \ln \left (x \right )\right )\right )}{25}-\frac {16 \,{\mathrm e}^{5} \operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )}{5}-\frac {24 \,{\mathrm e}^{5} \operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )}{5}-\frac {56 \,{\mathrm e}^{5} \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )\right )}{25}-\frac {12 \,{\mathrm e}^{10} \left (-\frac {1}{x \ln \left (x \right )}+\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )\right )}{25}-\frac {24 \,{\mathrm e}^{10} \left (-\frac {1}{2 x \ln \left (x \right )^{2}}+\frac {1}{2 x \ln \left (x \right )}-\frac {\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )}{2}\right )}{25}-\frac {18 \,{\mathrm e}^{10} \left (-\frac {1}{x^{2} \ln \left (x \right )}+2 \,\operatorname {Ei}_{1}\left (2 \ln \left (x \right )\right )\right )}{25}+\frac {48 \,{\mathrm e}^{5} \left (-\frac {x}{2 \ln \left (x \right )^{2}}-\frac {x}{2 \ln \left (x \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )}{2}\right )}{25}+\frac {16 \,{\mathrm e}^{5} \left (-\frac {x^{2}}{2 \ln \left (x \right )^{2}}-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )\right )}{25}+\frac {4 \,{\mathrm e}^{10}}{25 \ln \left (x \right )^{2}}-\frac {24 x^{3}}{5 \ln \left (x \right )}+\frac {4 x^{4}}{25 \ln \left (x \right )^{2}}+\frac {12 x^{3}}{25 \ln \left (x \right )^{2}}-\frac {18 x^{2}}{5 \ln \left (x \right )}+4 x^{4}+12 x^{3}+9 x^{2}-\frac {18 \,{\mathrm e}^{5}}{25 \ln \left (x \right )^{2}}+\frac {18 \,{\mathrm e}^{5}}{5 \ln \left (x \right )}\) | \(324\) |
int(1/25*((400*x^6+900*x^5+450*x^4)*ln(x)^3+((80*x^4+120*x^3)*exp(5)-160*x ^6-360*x^5-180*x^4)*ln(x)^2+((-12*x-18)*exp(5)^2+(-56*x^4-144*x^3-90*x^2)* exp(5)+56*x^6+156*x^5+108*x^4)*ln(x)+(-8*x^2-24*x-18)*exp(5)^2+(16*x^4+48* x^3+36*x^2)*exp(5)-8*x^6-24*x^5-18*x^4)/x^3/ln(x)^3,x,method=_RETURNVERBOS E)
4*x^4+12*x^3+9*x^2+1/25*(-40*x^6*ln(x)+40*exp(5)*ln(x)*x^4+4*x^6-120*x^5*l n(x)-8*x^4*exp(5)+120*x^3*exp(5)*ln(x)+12*x^5-90*x^4*ln(x)+4*x^2*exp(10)-2 4*x^3*exp(5)+90*x^2*exp(5)*ln(x)+9*x^4+12*x*exp(10)-18*x^2*exp(5)+9*exp(10 ))/x^2/ln(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (30) = 60\).
Time = 0.25 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.75 \[ \int \frac {-18 x^4-24 x^5-8 x^6+e^{10} \left (-18-24 x-8 x^2\right )+e^5 \left (36 x^2+48 x^3+16 x^4\right )+\left (e^{10} (-18-12 x)+108 x^4+156 x^5+56 x^6+e^5 \left (-90 x^2-144 x^3-56 x^4\right )\right ) \log (x)+\left (-180 x^4-360 x^5-160 x^6+e^5 \left (120 x^3+80 x^4\right )\right ) \log ^2(x)+\left (450 x^4+900 x^5+400 x^6\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {4 \, x^{6} + 12 \, x^{5} + 9 \, x^{4} + 25 \, {\left (4 \, x^{6} + 12 \, x^{5} + 9 \, x^{4}\right )} \log \left (x\right )^{2} + {\left (4 \, x^{2} + 12 \, x + 9\right )} e^{10} - 2 \, {\left (4 \, x^{4} + 12 \, x^{3} + 9 \, x^{2}\right )} e^{5} - 10 \, {\left (4 \, x^{6} + 12 \, x^{5} + 9 \, x^{4} - {\left (4 \, x^{4} + 12 \, x^{3} + 9 \, x^{2}\right )} e^{5}\right )} \log \left (x\right )}{25 \, x^{2} \log \left (x\right )^{2}} \]
integrate(1/25*((400*x^6+900*x^5+450*x^4)*log(x)^3+((80*x^4+120*x^3)*exp(5 )-160*x^6-360*x^5-180*x^4)*log(x)^2+((-12*x-18)*exp(5)^2+(-56*x^4-144*x^3- 90*x^2)*exp(5)+56*x^6+156*x^5+108*x^4)*log(x)+(-8*x^2-24*x-18)*exp(5)^2+(1 6*x^4+48*x^3+36*x^2)*exp(5)-8*x^6-24*x^5-18*x^4)/x^3/log(x)^3,x, algorithm =\
1/25*(4*x^6 + 12*x^5 + 9*x^4 + 25*(4*x^6 + 12*x^5 + 9*x^4)*log(x)^2 + (4*x ^2 + 12*x + 9)*e^10 - 2*(4*x^4 + 12*x^3 + 9*x^2)*e^5 - 10*(4*x^6 + 12*x^5 + 9*x^4 - (4*x^4 + 12*x^3 + 9*x^2)*e^5)*log(x))/(x^2*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.03 \[ \int \frac {-18 x^4-24 x^5-8 x^6+e^{10} \left (-18-24 x-8 x^2\right )+e^5 \left (36 x^2+48 x^3+16 x^4\right )+\left (e^{10} (-18-12 x)+108 x^4+156 x^5+56 x^6+e^5 \left (-90 x^2-144 x^3-56 x^4\right )\right ) \log (x)+\left (-180 x^4-360 x^5-160 x^6+e^5 \left (120 x^3+80 x^4\right )\right ) \log ^2(x)+\left (450 x^4+900 x^5+400 x^6\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=4 x^{4} + 12 x^{3} + 9 x^{2} + \frac {4 x^{6} + 12 x^{5} - 8 x^{4} e^{5} + 9 x^{4} - 24 x^{3} e^{5} - 18 x^{2} e^{5} + 4 x^{2} e^{10} + 12 x e^{10} + \left (- 40 x^{6} - 120 x^{5} - 90 x^{4} + 40 x^{4} e^{5} + 120 x^{3} e^{5} + 90 x^{2} e^{5}\right ) \log {\left (x \right )} + 9 e^{10}}{25 x^{2} \log {\left (x \right )}^{2}} \]
integrate(1/25*((400*x**6+900*x**5+450*x**4)*ln(x)**3+((80*x**4+120*x**3)* exp(5)-160*x**6-360*x**5-180*x**4)*ln(x)**2+((-12*x-18)*exp(5)**2+(-56*x** 4-144*x**3-90*x**2)*exp(5)+56*x**6+156*x**5+108*x**4)*ln(x)+(-8*x**2-24*x- 18)*exp(5)**2+(16*x**4+48*x**3+36*x**2)*exp(5)-8*x**6-24*x**5-18*x**4)/x** 3/ln(x)**3,x)
4*x**4 + 12*x**3 + 9*x**2 + (4*x**6 + 12*x**5 - 8*x**4*exp(5) + 9*x**4 - 2 4*x**3*exp(5) - 18*x**2*exp(5) + 4*x**2*exp(10) + 12*x*exp(10) + (-40*x**6 - 120*x**5 - 90*x**4 + 40*x**4*exp(5) + 120*x**3*exp(5) + 90*x**2*exp(5)) *log(x) + 9*exp(10))/(25*x**2*log(x)**2)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 6.28 \[ \int \frac {-18 x^4-24 x^5-8 x^6+e^{10} \left (-18-24 x-8 x^2\right )+e^5 \left (36 x^2+48 x^3+16 x^4\right )+\left (e^{10} (-18-12 x)+108 x^4+156 x^5+56 x^6+e^5 \left (-90 x^2-144 x^3-56 x^4\right )\right ) \log (x)+\left (-180 x^4-360 x^5-160 x^6+e^5 \left (120 x^3+80 x^4\right )\right ) \log ^2(x)+\left (450 x^4+900 x^5+400 x^6\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=4 \, x^{4} + 12 \, x^{3} + 9 \, x^{2} + \frac {16}{5} \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) e^{5} + \frac {24}{5} \, {\rm Ei}\left (\log \left (x\right )\right ) e^{5} + \frac {36}{25} \, e^{10} \Gamma \left (-1, 2 \, \log \left (x\right )\right ) - \frac {144}{25} \, e^{5} \Gamma \left (-1, -\log \left (x\right )\right ) - \frac {112}{25} \, e^{5} \Gamma \left (-1, -2 \, \log \left (x\right )\right ) + \frac {12}{25} \, e^{10} \Gamma \left (-1, \log \left (x\right )\right ) + \frac {72}{25} \, e^{10} \Gamma \left (-2, 2 \, \log \left (x\right )\right ) - \frac {48}{25} \, e^{5} \Gamma \left (-2, -\log \left (x\right )\right ) - \frac {64}{25} \, e^{5} \Gamma \left (-2, -2 \, \log \left (x\right )\right ) + \frac {24}{25} \, e^{10} \Gamma \left (-2, \log \left (x\right )\right ) + \frac {18 \, e^{5}}{5 \, \log \left (x\right )} + \frac {4 \, e^{10}}{25 \, \log \left (x\right )^{2}} - \frac {18 \, e^{5}}{25 \, \log \left (x\right )^{2}} - \frac {32}{5} \, {\rm Ei}\left (4 \, \log \left (x\right )\right ) - \frac {72}{5} \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) - \frac {36}{5} \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) + \frac {216}{25} \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) + \frac {468}{25} \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) + \frac {224}{25} \, \Gamma \left (-1, -4 \, \log \left (x\right )\right ) + \frac {72}{25} \, \Gamma \left (-2, -2 \, \log \left (x\right )\right ) + \frac {216}{25} \, \Gamma \left (-2, -3 \, \log \left (x\right )\right ) + \frac {128}{25} \, \Gamma \left (-2, -4 \, \log \left (x\right )\right ) \]
integrate(1/25*((400*x^6+900*x^5+450*x^4)*log(x)^3+((80*x^4+120*x^3)*exp(5 )-160*x^6-360*x^5-180*x^4)*log(x)^2+((-12*x-18)*exp(5)^2+(-56*x^4-144*x^3- 90*x^2)*exp(5)+56*x^6+156*x^5+108*x^4)*log(x)+(-8*x^2-24*x-18)*exp(5)^2+(1 6*x^4+48*x^3+36*x^2)*exp(5)-8*x^6-24*x^5-18*x^4)/x^3/log(x)^3,x, algorithm =\
4*x^4 + 12*x^3 + 9*x^2 + 16/5*Ei(2*log(x))*e^5 + 24/5*Ei(log(x))*e^5 + 36/ 25*e^10*gamma(-1, 2*log(x)) - 144/25*e^5*gamma(-1, -log(x)) - 112/25*e^5*g amma(-1, -2*log(x)) + 12/25*e^10*gamma(-1, log(x)) + 72/25*e^10*gamma(-2, 2*log(x)) - 48/25*e^5*gamma(-2, -log(x)) - 64/25*e^5*gamma(-2, -2*log(x)) + 24/25*e^10*gamma(-2, log(x)) + 18/5*e^5/log(x) + 4/25*e^10/log(x)^2 - 18 /25*e^5/log(x)^2 - 32/5*Ei(4*log(x)) - 72/5*Ei(3*log(x)) - 36/5*Ei(2*log(x )) + 216/25*gamma(-1, -2*log(x)) + 468/25*gamma(-1, -3*log(x)) + 224/25*ga mma(-1, -4*log(x)) + 72/25*gamma(-2, -2*log(x)) + 216/25*gamma(-2, -3*log( x)) + 128/25*gamma(-2, -4*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (30) = 60\).
Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.28 \[ \int \frac {-18 x^4-24 x^5-8 x^6+e^{10} \left (-18-24 x-8 x^2\right )+e^5 \left (36 x^2+48 x^3+16 x^4\right )+\left (e^{10} (-18-12 x)+108 x^4+156 x^5+56 x^6+e^5 \left (-90 x^2-144 x^3-56 x^4\right )\right ) \log (x)+\left (-180 x^4-360 x^5-160 x^6+e^5 \left (120 x^3+80 x^4\right )\right ) \log ^2(x)+\left (450 x^4+900 x^5+400 x^6\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {100 \, x^{6} \log \left (x\right )^{2} - 40 \, x^{6} \log \left (x\right ) + 300 \, x^{5} \log \left (x\right )^{2} + 4 \, x^{6} - 120 \, x^{5} \log \left (x\right ) + 40 \, x^{4} e^{5} \log \left (x\right ) + 225 \, x^{4} \log \left (x\right )^{2} + 12 \, x^{5} - 8 \, x^{4} e^{5} - 90 \, x^{4} \log \left (x\right ) + 120 \, x^{3} e^{5} \log \left (x\right ) + 9 \, x^{4} - 24 \, x^{3} e^{5} + 90 \, x^{2} e^{5} \log \left (x\right ) + 4 \, x^{2} e^{10} - 18 \, x^{2} e^{5} + 12 \, x e^{10} + 9 \, e^{10}}{25 \, x^{2} \log \left (x\right )^{2}} \]
integrate(1/25*((400*x^6+900*x^5+450*x^4)*log(x)^3+((80*x^4+120*x^3)*exp(5 )-160*x^6-360*x^5-180*x^4)*log(x)^2+((-12*x-18)*exp(5)^2+(-56*x^4-144*x^3- 90*x^2)*exp(5)+56*x^6+156*x^5+108*x^4)*log(x)+(-8*x^2-24*x-18)*exp(5)^2+(1 6*x^4+48*x^3+36*x^2)*exp(5)-8*x^6-24*x^5-18*x^4)/x^3/log(x)^3,x, algorithm =\
1/25*(100*x^6*log(x)^2 - 40*x^6*log(x) + 300*x^5*log(x)^2 + 4*x^6 - 120*x^ 5*log(x) + 40*x^4*e^5*log(x) + 225*x^4*log(x)^2 + 12*x^5 - 8*x^4*e^5 - 90* x^4*log(x) + 120*x^3*e^5*log(x) + 9*x^4 - 24*x^3*e^5 + 90*x^2*e^5*log(x) + 4*x^2*e^10 - 18*x^2*e^5 + 12*x*e^10 + 9*e^10)/(x^2*log(x)^2)
Time = 8.67 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {-18 x^4-24 x^5-8 x^6+e^{10} \left (-18-24 x-8 x^2\right )+e^5 \left (36 x^2+48 x^3+16 x^4\right )+\left (e^{10} (-18-12 x)+108 x^4+156 x^5+56 x^6+e^5 \left (-90 x^2-144 x^3-56 x^4\right )\right ) \log (x)+\left (-180 x^4-360 x^5-160 x^6+e^5 \left (120 x^3+80 x^4\right )\right ) \log ^2(x)+\left (450 x^4+900 x^5+400 x^6\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx=\frac {{\left (2\,x+3\right )}^2\,{\left ({\mathrm {e}}^5+5\,x^2\,\ln \left (x\right )-x^2\right )}^2}{25\,x^2\,{\ln \left (x\right )}^2} \]
int(-((exp(10)*(24*x + 8*x^2 + 18))/25 - (log(x)*(108*x^4 - exp(5)*(90*x^2 + 144*x^3 + 56*x^4) + 156*x^5 + 56*x^6 - exp(10)*(12*x + 18)))/25 + (log( x)^2*(180*x^4 - exp(5)*(120*x^3 + 80*x^4) + 360*x^5 + 160*x^6))/25 - (log( x)^3*(450*x^4 + 900*x^5 + 400*x^6))/25 - (exp(5)*(36*x^2 + 48*x^3 + 16*x^4 ))/25 + (18*x^4)/25 + (24*x^5)/25 + (8*x^6)/25)/(x^3*log(x)^3),x)