Integrand size = 100, antiderivative size = 21 \[ \int \frac {-2 x-4 x^2-2 x^3+\left (4 x+4 x^2\right ) \log (3)-2 x \log ^2(3)+\left (-8 x-4 x^2+4 x^3+\left (6 x-6 x^2\right ) \log (3)+2 x \log ^2(3)\right ) \log (x)+\left (20 x+30 x^2-20 x \log (3)\right ) \log ^2(x)+50 x \log ^3(x)}{\log ^3(x)} \, dx=x^2 \left (-5-\frac {1+x-\log (3)}{\log (x)}\right )^2 \] Output:
(-5-(1+x-ln(3))/ln(x))^2*x^2
Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(21)=42\).
Time = 0.36 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {-2 x-4 x^2-2 x^3+\left (4 x+4 x^2\right ) \log (3)-2 x \log ^2(3)+\left (-8 x-4 x^2+4 x^3+\left (6 x-6 x^2\right ) \log (3)+2 x \log ^2(3)\right ) \log (x)+\left (20 x+30 x^2-20 x \log (3)\right ) \log ^2(x)+50 x \log ^3(x)}{\log ^3(x)} \, dx=\frac {x^2 \left (1+x^2+\log ^2(3)-x (-2+\log (9))-\log (9)+(10+10 x-\log (59049)) \log (x)+25 \log ^2(x)\right )}{\log ^2(x)} \] Input:
Integrate[(-2*x - 4*x^2 - 2*x^3 + (4*x + 4*x^2)*Log[3] - 2*x*Log[3]^2 + (- 8*x - 4*x^2 + 4*x^3 + (6*x - 6*x^2)*Log[3] + 2*x*Log[3]^2)*Log[x] + (20*x + 30*x^2 - 20*x*Log[3])*Log[x]^2 + 50*x*Log[x]^3)/Log[x]^3,x]
Output:
(x^2*(1 + x^2 + Log[3]^2 - x*(-2 + Log[9]) - Log[9] + (10 + 10*x - Log[590 49])*Log[x] + 25*Log[x]^2))/Log[x]^2
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.21 (sec) , antiderivative size = 211, normalized size of antiderivative = 10.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {6, 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.
| \(\displaystyle \int \frac {-2 x^3-4 x^2+\left (30 x^2+20 x-20 x \log (3)\right ) \log ^2(x)+\left (4 x^2+4 x\right ) \log (3)+\left (4 x^3-4 x^2+\left (6 x-6 x^2\right ) \log (3)-8 x+2 x \log ^2(3)\right ) \log (x)-2 x+50 x \log ^3(x)-2 x \log ^2(3)}{\log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-2 x^3-4 x^2+\left (30 x^2+20 x-20 x \log (3)\right ) \log ^2(x)+\left (4 x^2+4 x\right ) \log (3)+\left (4 x^3-4 x^2+\left (6 x-6 x^2\right ) \log (3)-8 x+2 x \log ^2(3)\right ) \log (x)+50 x \log ^3(x)+x \left (-2-2 \log ^2(3)\right )}{\log ^3(x)}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 x \left (-x^2+2 x^2 \log (x)+25 \log ^3(x)+15 x \log ^2(x)+10 (1-\log (3)) \log ^2(x)-2 x \left (1+\frac {\log (27)}{2}\right ) \log (x)-4 \left (1-\frac {1}{4} \log (3) (3+\log (3))\right ) \log (x)-2 x (1-\log (3))-1-\log ^2(3)+\log (9)\right )}{\log ^3(x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {x \left (-25 \log ^3(x)-15 x \log ^2(x)-10 (1-\log (3)) \log ^2(x)-2 x^2 \log (x)+x (2+\log (27)) \log (x)+(1-\log (3)) (4+\log (3)) \log (x)+x^2-\log (9)+\log ^2(3)+2 x (1-\log (3))+1\right )}{\log ^3(x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {x \left (-25 \log ^3(x)-15 x \log ^2(x)-10 (1-\log (3)) \log ^2(x)-2 x^2 \log (x)+x (2+\log (27)) \log (x)+(1-\log (3)) (4+\log (3)) \log (x)+x^2-\log (9)+\log ^2(3)+2 x (1-\log (3))+1\right )}{\log ^3(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {5 (3 x-\log (9)+2) x}{\log (x)}+\frac {\left (-2 x^2+(2+\log (27)) x-\log (27)-\log ^2(3)+4\right ) x}{\log ^2(x)}+\frac {\left (x^2+2 (1-\log (3)) x-\log (9)+\log ^2(3)+1\right ) x}{\log ^3(x)}-25 x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (2 \left (4-\log ^2(3)-\log (27)\right ) \operatorname {ExpIntegralEi}(2 \log (x))+2 \left (1+\log ^2(3)-\log (9)\right ) \operatorname {ExpIntegralEi}(2 \log (x))-15 \operatorname {ExpIntegralEi}(3 \log (x))+3 (2+\log (27)) \operatorname {ExpIntegralEi}(3 \log (x))-5 (2-\log (9)) \operatorname {ExpIntegralEi}(2 \log (x))+\frac {9}{2} (2-\log (9)) \operatorname {ExpIntegralEi}(3 \log (x))-\frac {x^4}{2 \log ^2(x)}-\frac {x^3 (2-\log (9))}{2 \log ^2(x)}-\frac {x^3 (2+\log (27))}{\log (x)}-\frac {3 x^3 (2-\log (9))}{2 \log (x)}-\frac {25 x^2}{2}-\frac {x^2 \left (4-\log ^2(3)-\log (27)\right )}{\log (x)}-\frac {x^2 \left (1+\log ^2(3)-\log (9)\right )}{\log (x)}-\frac {x^2 \left (1+\log ^2(3)-\log (9)\right )}{2 \log ^2(x)}\right )\) |
Input:
Int[(-2*x - 4*x^2 - 2*x^3 + (4*x + 4*x^2)*Log[3] - 2*x*Log[3]^2 + (-8*x - 4*x^2 + 4*x^3 + (6*x - 6*x^2)*Log[3] + 2*x*Log[3]^2)*Log[x] + (20*x + 30*x ^2 - 20*x*Log[3])*Log[x]^2 + 50*x*Log[x]^3)/Log[x]^3,x]
Output:
-2*((-25*x^2)/2 - 15*ExpIntegralEi[3*Log[x]] - 5*ExpIntegralEi[2*Log[x]]*( 2 - Log[9]) + (9*ExpIntegralEi[3*Log[x]]*(2 - Log[9]))/2 + 2*ExpIntegralEi [2*Log[x]]*(1 + Log[3]^2 - Log[9]) + 2*ExpIntegralEi[2*Log[x]]*(4 - Log[3] ^2 - Log[27]) + 3*ExpIntegralEi[3*Log[x]]*(2 + Log[27]) - x^4/(2*Log[x]^2) - (x^3*(2 - Log[9]))/(2*Log[x]^2) - (x^2*(1 + Log[3]^2 - Log[9]))/(2*Log[ x]^2) - (3*x^3*(2 - Log[9]))/(2*Log[x]) - (x^2*(1 + Log[3]^2 - Log[9]))/Lo g[x] - (x^2*(4 - Log[3]^2 - Log[27]))/Log[x] - (x^3*(2 + Log[27]))/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]
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. \(50\) vs. \(2(21)=42\).
Time = 6.47 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43
| method | result | size |
| risch | \(25 x^{2}+\frac {x^{2} \left (\ln \left (3\right )^{2}-2 x \ln \left (3\right )-10 \ln \left (3\right ) \ln \left (x \right )+x^{2}+10 x \ln \left (x \right )-2 \ln \left (3\right )+2 x +10 \ln \left (x \right )+1\right )}{\ln \left (x \right )^{2}}\) | \(51\) |
| norman | \(\frac {x^{4}+\left (-2 \ln \left (3\right )+2\right ) x^{3}+\left (\ln \left (3\right )^{2}-2 \ln \left (3\right )+1\right ) x^{2}+\left (-10 \ln \left (3\right )+10\right ) x^{2} \ln \left (x \right )+25 x^{2} \ln \left (x \right )^{2}+10 x^{3} \ln \left (x \right )}{\ln \left (x \right )^{2}}\) | \(62\) |
| parallelrisch | \(\frac {x^{2} \ln \left (3\right )^{2}-2 x^{3} \ln \left (3\right )-10 \ln \left (x \right ) \ln \left (3\right ) x^{2}+x^{4}+10 x^{3} \ln \left (x \right )+25 x^{2} \ln \left (x \right )^{2}-2 x^{2} \ln \left (3\right )+2 x^{3}+10 x^{2} \ln \left (x \right )+x^{2}}{\ln \left (x \right )^{2}}\) | \(72\) |
| default | \(\frac {x^{2}}{\ln \left (x \right )^{2}}+2 \ln \left (3\right )^{2} \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )+20 \ln \left (3\right ) \operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )+\frac {2 x^{3}}{\ln \left (x \right )^{2}}+\frac {10 x^{3}}{\ln \left (x \right )}+\frac {x^{4}}{\ln \left (x \right )^{2}}+\frac {10 x^{2}}{\ln \left (x \right )}+25 x^{2}+4 \ln \left (3\right ) \left (-\frac {x^{2}}{2 \ln \left (x \right )^{2}}-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )+4 \ln \left (3\right ) \left (-\frac {x^{3}}{2 \ln \left (x \right )^{2}}-\frac {3 x^{3}}{2 \ln \left (x \right )}-\frac {9 \,\operatorname {expIntegral}_{1}\left (-3 \ln \left (x \right )\right )}{2}\right )-2 \ln \left (3\right )^{2} \left (-\frac {x^{2}}{2 \ln \left (x \right )^{2}}-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )+6 \ln \left (3\right ) \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )-6 \ln \left (3\right ) \left (-\frac {x^{3}}{\ln \left (x \right )}-3 \,\operatorname {expIntegral}_{1}\left (-3 \ln \left (x \right )\right )\right )\) | \(223\) |
| parts | \(\frac {x^{2}}{\ln \left (x \right )^{2}}+2 \ln \left (3\right )^{2} \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )+20 \ln \left (3\right ) \operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )+\frac {2 x^{3}}{\ln \left (x \right )^{2}}+\frac {10 x^{3}}{\ln \left (x \right )}+\frac {x^{4}}{\ln \left (x \right )^{2}}+\frac {10 x^{2}}{\ln \left (x \right )}+25 x^{2}+4 \ln \left (3\right ) \left (-\frac {x^{2}}{2 \ln \left (x \right )^{2}}-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )+4 \ln \left (3\right ) \left (-\frac {x^{3}}{2 \ln \left (x \right )^{2}}-\frac {3 x^{3}}{2 \ln \left (x \right )}-\frac {9 \,\operatorname {expIntegral}_{1}\left (-3 \ln \left (x \right )\right )}{2}\right )-2 \ln \left (3\right )^{2} \left (-\frac {x^{2}}{2 \ln \left (x \right )^{2}}-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )+6 \ln \left (3\right ) \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (x \right )\right )\right )-6 \ln \left (3\right ) \left (-\frac {x^{3}}{\ln \left (x \right )}-3 \,\operatorname {expIntegral}_{1}\left (-3 \ln \left (x \right )\right )\right )\) | \(223\) |
Input:
int((50*x*ln(x)^3+(-20*x*ln(3)+30*x^2+20*x)*ln(x)^2+(2*x*ln(3)^2+(-6*x^2+6 *x)*ln(3)+4*x^3-4*x^2-8*x)*ln(x)-2*x*ln(3)^2+(4*x^2+4*x)*ln(3)-2*x^3-4*x^2 -2*x)/ln(x)^3,x,method=_RETURNVERBOSE)
Output:
25*x^2+x^2*(ln(3)^2-2*x*ln(3)-10*ln(3)*ln(x)+x^2+10*x*ln(x)-2*ln(3)+2*x+10 *ln(x)+1)/ln(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.00 \[ \int \frac {-2 x-4 x^2-2 x^3+\left (4 x+4 x^2\right ) \log (3)-2 x \log ^2(3)+\left (-8 x-4 x^2+4 x^3+\left (6 x-6 x^2\right ) \log (3)+2 x \log ^2(3)\right ) \log (x)+\left (20 x+30 x^2-20 x \log (3)\right ) \log ^2(x)+50 x \log ^3(x)}{\log ^3(x)} \, dx=\frac {x^{4} + x^{2} \log \left (3\right )^{2} + 25 \, x^{2} \log \left (x\right )^{2} + 2 \, x^{3} + x^{2} - 2 \, {\left (x^{3} + x^{2}\right )} \log \left (3\right ) + 10 \, {\left (x^{3} - x^{2} \log \left (3\right ) + x^{2}\right )} \log \left (x\right )}{\log \left (x\right )^{2}} \] Input:
integrate((50*x*log(x)^3+(-20*x*log(3)+30*x^2+20*x)*log(x)^2+(2*x*log(3)^2 +(-6*x^2+6*x)*log(3)+4*x^3-4*x^2-8*x)*log(x)-2*x*log(3)^2+(4*x^2+4*x)*log( 3)-2*x^3-4*x^2-2*x)/log(x)^3,x, algorithm="fricas")
Output:
(x^4 + x^2*log(3)^2 + 25*x^2*log(x)^2 + 2*x^3 + x^2 - 2*(x^3 + x^2)*log(3) + 10*(x^3 - x^2*log(3) + x^2)*log(x))/log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (17) = 34\).
Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.24 \[ \int \frac {-2 x-4 x^2-2 x^3+\left (4 x+4 x^2\right ) \log (3)-2 x \log ^2(3)+\left (-8 x-4 x^2+4 x^3+\left (6 x-6 x^2\right ) \log (3)+2 x \log ^2(3)\right ) \log (x)+\left (20 x+30 x^2-20 x \log (3)\right ) \log ^2(x)+50 x \log ^3(x)}{\log ^3(x)} \, dx=25 x^{2} + \frac {x^{4} - 2 x^{3} \log {\left (3 \right )} + 2 x^{3} - 2 x^{2} \log {\left (3 \right )} + x^{2} + x^{2} \log {\left (3 \right )}^{2} + \left (10 x^{3} - 10 x^{2} \log {\left (3 \right )} + 10 x^{2}\right ) \log {\left (x \right )}}{\log {\left (x \right )}^{2}} \] Input:
integrate((50*x*ln(x)**3+(-20*x*ln(3)+30*x**2+20*x)*ln(x)**2+(2*x*ln(3)**2 +(-6*x**2+6*x)*ln(3)+4*x**3-4*x**2-8*x)*ln(x)-2*x*ln(3)**2+(4*x**2+4*x)*ln (3)-2*x**3-4*x**2-2*x)/ln(x)**3,x)
Output:
25*x**2 + (x**4 - 2*x**3*log(3) + 2*x**3 - 2*x**2*log(3) + x**2 + x**2*log (3)**2 + (10*x**3 - 10*x**2*log(3) + 10*x**2)*log(x))/log(x)**2
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 6.71 \[ \int \frac {-2 x-4 x^2-2 x^3+\left (4 x+4 x^2\right ) \log (3)-2 x \log ^2(3)+\left (-8 x-4 x^2+4 x^3+\left (6 x-6 x^2\right ) \log (3)+2 x \log ^2(3)\right ) \log (x)+\left (20 x+30 x^2-20 x \log (3)\right ) \log ^2(x)+50 x \log ^3(x)}{\log ^3(x)} \, dx=4 \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) \log \left (3\right )^{2} + 8 \, \Gamma \left (-2, -2 \, \log \left (x\right )\right ) \log \left (3\right )^{2} + 25 \, x^{2} - 20 \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) \log \left (3\right ) + 12 \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) \log \left (3\right ) - 18 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) \log \left (3\right ) - 16 \, \Gamma \left (-2, -2 \, \log \left (x\right )\right ) \log \left (3\right ) - 36 \, \Gamma \left (-2, -3 \, \log \left (x\right )\right ) \log \left (3\right ) + 30 \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) + 20 \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) - 16 \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) - 12 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) + 16 \, \Gamma \left (-1, -4 \, \log \left (x\right )\right ) + 8 \, \Gamma \left (-2, -2 \, \log \left (x\right )\right ) + 36 \, \Gamma \left (-2, -3 \, \log \left (x\right )\right ) + 32 \, \Gamma \left (-2, -4 \, \log \left (x\right )\right ) \] Input:
integrate((50*x*log(x)^3+(-20*x*log(3)+30*x^2+20*x)*log(x)^2+(2*x*log(3)^2 +(-6*x^2+6*x)*log(3)+4*x^3-4*x^2-8*x)*log(x)-2*x*log(3)^2+(4*x^2+4*x)*log( 3)-2*x^3-4*x^2-2*x)/log(x)^3,x, algorithm="maxima")
Output:
4*gamma(-1, -2*log(x))*log(3)^2 + 8*gamma(-2, -2*log(x))*log(3)^2 + 25*x^2 - 20*Ei(2*log(x))*log(3) + 12*gamma(-1, -2*log(x))*log(3) - 18*gamma(-1, -3*log(x))*log(3) - 16*gamma(-2, -2*log(x))*log(3) - 36*gamma(-2, -3*log(x ))*log(3) + 30*Ei(3*log(x)) + 20*Ei(2*log(x)) - 16*gamma(-1, -2*log(x)) - 12*gamma(-1, -3*log(x)) + 16*gamma(-1, -4*log(x)) + 8*gamma(-2, -2*log(x)) + 36*gamma(-2, -3*log(x)) + 32*gamma(-2, -4*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.48 \[ \int \frac {-2 x-4 x^2-2 x^3+\left (4 x+4 x^2\right ) \log (3)-2 x \log ^2(3)+\left (-8 x-4 x^2+4 x^3+\left (6 x-6 x^2\right ) \log (3)+2 x \log ^2(3)\right ) \log (x)+\left (20 x+30 x^2-20 x \log (3)\right ) \log ^2(x)+50 x \log ^3(x)}{\log ^3(x)} \, dx=25 \, x^{2} + \frac {x^{4}}{\log \left (x\right )^{2}} - \frac {2 \, x^{3} \log \left (3\right )}{\log \left (x\right )^{2}} + \frac {x^{2} \log \left (3\right )^{2}}{\log \left (x\right )^{2}} + \frac {10 \, x^{3}}{\log \left (x\right )} - \frac {10 \, x^{2} \log \left (3\right )}{\log \left (x\right )} + \frac {2 \, x^{3}}{\log \left (x\right )^{2}} - \frac {2 \, x^{2} \log \left (3\right )}{\log \left (x\right )^{2}} + \frac {10 \, x^{2}}{\log \left (x\right )} + \frac {x^{2}}{\log \left (x\right )^{2}} \] Input:
integrate((50*x*log(x)^3+(-20*x*log(3)+30*x^2+20*x)*log(x)^2+(2*x*log(3)^2 +(-6*x^2+6*x)*log(3)+4*x^3-4*x^2-8*x)*log(x)-2*x*log(3)^2+(4*x^2+4*x)*log( 3)-2*x^3-4*x^2-2*x)/log(x)^3,x, algorithm="giac")
Output:
25*x^2 + x^4/log(x)^2 - 2*x^3*log(3)/log(x)^2 + x^2*log(3)^2/log(x)^2 + 10 *x^3/log(x) - 10*x^2*log(3)/log(x) + 2*x^3/log(x)^2 - 2*x^2*log(3)/log(x)^ 2 + 10*x^2/log(x) + x^2/log(x)^2
Time = 3.41 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.00 \[ \int \frac {-2 x-4 x^2-2 x^3+\left (4 x+4 x^2\right ) \log (3)-2 x \log ^2(3)+\left (-8 x-4 x^2+4 x^3+\left (6 x-6 x^2\right ) \log (3)+2 x \log ^2(3)\right ) \log (x)+\left (20 x+30 x^2-20 x \log (3)\right ) \log ^2(x)+50 x \log ^3(x)}{\log ^3(x)} \, dx=25\,x^2-\frac {x^4\,\left (\ln \left (9\right )-2\right )-x^3\,\left ({\ln \left (3\right )}^2-\ln \left (9\right )+1\right )+\ln \left (x\right )\,\left (x^3\,\left (10\,\ln \left (3\right )-10\right )-10\,x^4\right )-x^5}{x\,{\ln \left (x\right )}^2} \] Input:
int(-(2*x - 50*x*log(x)^3 - log(3)*(4*x + 4*x^2) - log(x)^2*(20*x - 20*x*l og(3) + 30*x^2) + 2*x*log(3)^2 - log(x)*(log(3)*(6*x - 6*x^2) - 8*x + 2*x* log(3)^2 - 4*x^2 + 4*x^3) + 4*x^2 + 2*x^3)/log(x)^3,x)
Output:
25*x^2 - (x^4*(log(9) - 2) - x^3*(log(3)^2 - log(9) + 1) + log(x)*(x^3*(10 *log(3) - 10) - 10*x^4) - x^5)/(x*log(x)^2)
Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.38 \[ \int \frac {-2 x-4 x^2-2 x^3+\left (4 x+4 x^2\right ) \log (3)-2 x \log ^2(3)+\left (-8 x-4 x^2+4 x^3+\left (6 x-6 x^2\right ) \log (3)+2 x \log ^2(3)\right ) \log (x)+\left (20 x+30 x^2-20 x \log (3)\right ) \log ^2(x)+50 x \log ^3(x)}{\log ^3(x)} \, dx=\frac {x^{2} \left (25 \mathrm {log}\left (x \right )^{2}-10 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right )+10 \,\mathrm {log}\left (x \right ) x +10 \,\mathrm {log}\left (x \right )+\mathrm {log}\left (3\right )^{2}-2 \,\mathrm {log}\left (3\right ) x -2 \,\mathrm {log}\left (3\right )+x^{2}+2 x +1\right )}{\mathrm {log}\left (x \right )^{2}} \] Input:
int((50*x*log(x)^3+(-20*x*log(3)+30*x^2+20*x)*log(x)^2+(2*x*log(3)^2+(-6*x ^2+6*x)*log(3)+4*x^3-4*x^2-8*x)*log(x)-2*x*log(3)^2+(4*x^2+4*x)*log(3)-2*x ^3-4*x^2-2*x)/log(x)^3,x)
Output:
(x**2*(25*log(x)**2 - 10*log(x)*log(3) + 10*log(x)*x + 10*log(x) + log(3)* *2 - 2*log(3)*x - 2*log(3) + x**2 + 2*x + 1))/log(x)**2