Integrand size = 287, antiderivative size = 27 \[ \int \frac {2 e^6 x+2 e^3 x^2+\left (-2 e^3 x^2-2 x^3\right ) \log \left (-\frac {15 x}{16}\right )+\left (4 e^3 x+\left (-2 e^3 x-6 x^2\right ) \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )\right ) \log ^2(x)-2 \log \left (-\frac {15 x}{16}\right ) \log ^3(x)+\left (-4 e^3 x+\left (2 e^3 x+6 x^2\right ) \log \left (-\frac {15 x}{16}\right )+\left (-4 e^3+12 x \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+6 \log \left (-\frac {15 x}{16}\right ) \log ^2(x)\right ) \log (2 x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )-6 \log \left (-\frac {15 x}{16}\right ) \log (x)\right ) \log ^2(2 x)+2 \log \left (-\frac {15 x}{16}\right ) \log ^3(2 x)}{-x^4-3 x^3 \log (x)-3 x^2 \log ^2(x)-x \log ^3(x)+\left (3 x^3+6 x^2 \log (x)+3 x \log ^2(x)\right ) \log (2 x)+\left (-3 x^2-3 x \log (x)\right ) \log ^2(2 x)+x \log ^3(2 x)} \, dx=\left (-\log \left (-\frac {15 x}{16}\right )+\frac {e^3}{x+\log (x)-\log (2 x)}\right )^2 \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 5.87 (sec) , antiderivative size = 1463, normalized size of antiderivative = 54.19 \[ \int \frac {2 e^6 x+2 e^3 x^2+\left (-2 e^3 x^2-2 x^3\right ) \log \left (-\frac {15 x}{16}\right )+\left (4 e^3 x+\left (-2 e^3 x-6 x^2\right ) \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )\right ) \log ^2(x)-2 \log \left (-\frac {15 x}{16}\right ) \log ^3(x)+\left (-4 e^3 x+\left (2 e^3 x+6 x^2\right ) \log \left (-\frac {15 x}{16}\right )+\left (-4 e^3+12 x \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+6 \log \left (-\frac {15 x}{16}\right ) \log ^2(x)\right ) \log (2 x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )-6 \log \left (-\frac {15 x}{16}\right ) \log (x)\right ) \log ^2(2 x)+2 \log \left (-\frac {15 x}{16}\right ) \log ^3(2 x)}{-x^4-3 x^3 \log (x)-3 x^2 \log ^2(x)-x \log ^3(x)+\left (3 x^3+6 x^2 \log (x)+3 x \log ^2(x)\right ) \log (2 x)+\left (-3 x^2-3 x \log (x)\right ) \log ^2(2 x)+x \log ^3(2 x)} \, dx =\text {Too large to display} \]
Integrate[(2*E^6*x + 2*E^3*x^2 + (-2*E^3*x^2 - 2*x^3)*Log[(-15*x)/16] + (4 *E^3*x + (-2*E^3*x - 6*x^2)*Log[(-15*x)/16])*Log[x] + (2*E^3 - 6*x*Log[(-1 5*x)/16])*Log[x]^2 - 2*Log[(-15*x)/16]*Log[x]^3 + (-4*E^3*x + (2*E^3*x + 6 *x^2)*Log[(-15*x)/16] + (-4*E^3 + 12*x*Log[(-15*x)/16])*Log[x] + 6*Log[(-1 5*x)/16]*Log[x]^2)*Log[2*x] + (2*E^3 - 6*x*Log[(-15*x)/16] - 6*Log[(-15*x) /16]*Log[x])*Log[2*x]^2 + 2*Log[(-15*x)/16]*Log[2*x]^3)/(-x^4 - 3*x^3*Log[ x] - 3*x^2*Log[x]^2 - x*Log[x]^3 + (3*x^3 + 6*x^2*Log[x] + 3*x*Log[x]^2)*L og[2*x] + (-3*x^2 - 3*x*Log[x])*Log[2*x]^2 + x*Log[2*x]^3),x]
-((E^6*Log[2]*(Log[x] - Log[2*x])^2 + 2*Log[2]^2*(E^3 - Log[16/15]*Log[2]) *Log[x]*(x + Log[x] - Log[2*x])^2 + 2*(Log[x] - Log[2*x])*(x + Log[x] - Lo g[2*x])*(Log[2]^2*(E^3 - Log[16/15]*Log[2]) + 2*Log[16/15]*Log[x]^3 + (-(E ^3*(1 + Log[16/15])) + Log[16/15]*Log[8])*Log[2*x]^2 - 2*Log[16/15]*Log[2* x]^3 + 2*Log[x]*Log[2*x]*(E^3*(1 + Log[16/15]) - Log[16/15]*Log[8] + 3*Log [16/15]*Log[2*x]) - Log[x]^2*(E^3*(1 + Log[16/15]) - Log[16/15]*Log[8] + 6 *Log[16/15]*Log[2*x])) - 2*Log[-x]*(Log[x] - Log[2*x])^3*(x + Log[x] - Log [2*x])^2*Log[1 + x/(Log[x] - Log[2*x])] - 4*(Log[-x] - Log[x])*(Log[x] - L og[2*x])^2*(x + Log[x] - Log[2*x])*(x*Log[-x] - (x + Log[x] - Log[2*x])*Lo g[1 + x/(Log[x] - Log[2*x])]) + 4*(Log[-x] - Log[2*x])*(Log[x] - Log[2*x]) ^2*(x + Log[x] - Log[2*x])*(x*Log[-x] - (x + Log[x] - Log[2*x])*Log[1 + x/ (Log[x] - Log[2*x])]) - E^3*Log[2]*(Log[x] - Log[2*x])*(-(x*(x - Log[4])*L og[-x]) + x*(x + Log[x] - Log[2*x]) + (x + Log[x] - Log[2*x])^2*Log[1 + x/ (Log[x] - Log[2*x])]) + Log[2]^2*(Log[x] - Log[2*x])*(-(x*(x - Log[4])*Log [-x]) + x*(x + Log[x] - Log[2*x]) + (x + Log[x] - Log[2*x])^2*Log[1 + x/(L og[x] - Log[2*x])]) + Log[2]*Log[4]*(Log[x] - Log[2*x])*(-(x*(x - Log[4])* Log[-x]) + x*(x + Log[x] - Log[2*x]) + (x + Log[x] - Log[2*x])^2*Log[1 + x /(Log[x] - Log[2*x])]) + (Log[-x] - Log[x])^2*(Log[x] - Log[2*x])*(-(x*(x - Log[4])*Log[-x]) + x*(x + Log[x] - Log[2*x]) + (x + Log[x] - Log[2*x])^2 *Log[1 + x/(Log[x] - Log[2*x])]) - 2*(Log[-x] - Log[x])*(Log[-x] - Log[...
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.
| \(\displaystyle \int \frac {2 e^3 x^2+\left (\left (6 x^2+2 e^3 x\right ) \log \left (-\frac {15 x}{16}\right )-4 e^3 x+6 \log \left (-\frac {15 x}{16}\right ) \log ^2(x)+\left (12 x \log \left (-\frac {15 x}{16}\right )-4 e^3\right ) \log (x)\right ) \log (2 x)+\left (\left (-6 x^2-2 e^3 x\right ) \log \left (-\frac {15 x}{16}\right )+4 e^3 x\right ) \log (x)+\left (-2 x^3-2 e^3 x^2\right ) \log \left (-\frac {15 x}{16}\right )+2 e^6 x-2 \log \left (-\frac {15 x}{16}\right ) \log ^3(x)+2 \log \left (-\frac {15 x}{16}\right ) \log ^3(2 x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )\right ) \log ^2(x)+\left (-6 x \log \left (-\frac {15 x}{16}\right )-6 \log (x) \log \left (-\frac {15 x}{16}\right )+2 e^3\right ) \log ^2(2 x)}{-x^4-3 x^3 \log (x)-3 x^2 \log ^2(x)+\left (-3 x^2-3 x \log (x)\right ) \log ^2(2 x)+\left (3 x^3+6 x^2 \log (x)+3 x \log ^2(x)\right ) \log (2 x)-x \log ^3(x)+x \log ^3(2 x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (x^2+x \left (e^3-\log (4)\right )+\log ^2(2)\right ) \left (-\log \left (\frac {16}{15}\right ) (x-\log (2))-(\log (2)-x) \log (-x)-e^3\right )}{x (x+\log (x)-\log (2 x))^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\left (x^2+\left (e^3-\log (4)\right ) x+\log ^2(2)\right ) \left (-\log (-x) (x-\log (2))+\log \left (\frac {16}{15}\right ) (x-\log (2))+e^3\right )}{x (x+\log (x)-\log (2 x))^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\left (x^2+\left (e^3-\log (4)\right ) x+\log ^2(2)\right ) \left (-\log (-x) (x-\log (2))+\log \left (\frac {16}{15}\right ) (x-\log (2))+e^3\right )}{x (x+\log (x)-\log (2 x))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {\left (1-\frac {e^3}{\log (4)}\right ) \log (4) \left (\log (-x) x-\log \left (\frac {16}{15}\right ) x-\log (2) \log (-x)-e^3 \left (1-\frac {\log \left (\frac {16}{15}\right ) \log (2)}{e^3}\right )\right )}{(x+\log (x)-\log (2 x))^3}+\frac {x \left (-\log (-x) x+\log \left (\frac {16}{15}\right ) x+\log (2) \log (-x)+e^3 \left (1-\frac {\log \left (\frac {16}{15}\right ) \log (2)}{e^3}\right )\right )}{(x+\log (x)-\log (2 x))^3}+\frac {\log ^2(2) \left (-\log (-x) x+\log \left (\frac {16}{15}\right ) x+\log (2) \log (-x)+e^3 \left (1-\frac {\log \left (\frac {16}{15}\right ) \log (2)}{e^3}\right )\right )}{x (x+\log (x)-\log (2 x))^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\log \left (\frac {16}{15}\right ) \int \frac {x^2}{(x+\log (x)-\log (2 x))^3}dx-\int \frac {x^2 \log (-x)}{(x+\log (x)-\log (2 x))^3}dx+\log ^3(2) \int \frac {\log (-x)}{x (x+\log (x)-\log (2 x))^3}dx+\log \left (\frac {16}{15}\right ) \log ^2(2) \int \frac {1}{(x+\log (x)-\log (2 x))^3}dx+\log ^2(2) \left (e^3-\log \left (\frac {16}{15}\right ) \log (2)\right ) \int \frac {1}{x (x+\log (x)-\log (2 x))^3}dx-\log ^2(2) \int \frac {\log (-x)}{(x+\log (x)-\log (2 x))^3}dx+\left (e^3-\log \left (\frac {16}{15}\right ) \log (2)\right ) \left (e^3-\log (4)\right ) \int \frac {1}{(x+\log (x)-\log (2 x))^3}dx+\log \left (\frac {16}{15}\right ) \left (e^3-\log (4)\right ) \int \frac {x}{(x+\log (x)-\log (2 x))^3}dx+\left (e^3-\log \left (\frac {16}{15}\right ) \log (2)\right ) \int \frac {x}{(x+\log (x)-\log (2 x))^3}dx+\log (2) \left (e^3-\log (4)\right ) \int \frac {\log (-x)}{(x+\log (x)-\log (2 x))^3}dx-\left (e^3-\log (4)\right ) \int \frac {x \log (-x)}{(x+\log (x)-\log (2 x))^3}dx+\log (2) \int \frac {x \log (-x)}{(x+\log (x)-\log (2 x))^3}dx\right )\) |
Int[(2*E^6*x + 2*E^3*x^2 + (-2*E^3*x^2 - 2*x^3)*Log[(-15*x)/16] + (4*E^3*x + (-2*E^3*x - 6*x^2)*Log[(-15*x)/16])*Log[x] + (2*E^3 - 6*x*Log[(-15*x)/1 6])*Log[x]^2 - 2*Log[(-15*x)/16]*Log[x]^3 + (-4*E^3*x + (2*E^3*x + 6*x^2)* Log[(-15*x)/16] + (-4*E^3 + 12*x*Log[(-15*x)/16])*Log[x] + 6*Log[(-15*x)/1 6]*Log[x]^2)*Log[2*x] + (2*E^3 - 6*x*Log[(-15*x)/16] - 6*Log[(-15*x)/16]*L og[x])*Log[2*x]^2 + 2*Log[(-15*x)/16]*Log[2*x]^3)/(-x^4 - 3*x^3*Log[x] - 3 *x^2*Log[x]^2 - x*Log[x]^3 + (3*x^3 + 6*x^2*Log[x] + 3*x*Log[x]^2)*Log[2*x ] + (-3*x^2 - 3*x*Log[x])*Log[2*x]^2 + x*Log[2*x]^3),x]
3.11.75.3.1 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. \(120\) vs. \(2(24)=48\).
Time = 7.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.48
| method | result | size |
| default | \(\frac {2 \left (-4 \ln \left (2\right )^{2}+{\mathrm e}^{3}\right ) \ln \left (x \right )}{\ln \left (2\right )}+\frac {{\mathrm e}^{6}}{\left (x -\ln \left (2\right )\right )^{2}}-\frac {2 \,{\mathrm e}^{3} \ln \left (x -\ln \left (2\right )\right )}{\ln \left (2\right )}+\frac {8 \,{\mathrm e}^{3} \ln \left (2\right )}{x -\ln \left (2\right )}+2 \ln \left (15\right ) \left (\ln \left (x \right )-\frac {{\mathrm e}^{3}}{x -\ln \left (2\right )}\right )+\ln \left (-x \right )^{2}-2 \,{\mathrm e}^{3} \left (-\frac {\ln \left (\ln \left (2\right )-x \right )}{\ln \left (2\right )}-\frac {\ln \left (-x \right ) x}{\ln \left (2\right ) \left (\ln \left (2\right )-x \right )}\right )\) | \(121\) |
| parallelrisch | \(\frac {2 \ln \left (-\frac {15 x}{16}\right ) \ln \left (x \right )^{3}+2 x \ln \left (x \right )^{2} \ln \left (2 x \right )-4 \ln \left (x \right ) \ln \left (2 x \right ) \ln \left (-\frac {15 x}{16}\right ) x -2 x \ln \left (x \right )^{3}-x^{2} \ln \left (x \right )^{2}+{\mathrm e}^{6}-\ln \left (x \right )^{4}-4 \ln \left (x \right )^{2} \ln \left (2 x \right ) \ln \left (-\frac {15 x}{16}\right )+4 \ln \left (x \right )^{2} x \ln \left (-\frac {15 x}{16}\right )+2 \ln \left (x \right ) \ln \left (2 x \right )^{2} \ln \left (-\frac {15 x}{16}\right )+2 \ln \left (x \right ) \ln \left (-\frac {15 x}{16}\right ) x^{2}+2 \ln \left (2 x \right ) \ln \left (x \right )^{3}-\ln \left (2 x \right )^{2} \ln \left (x \right )^{2}-2 \,{\mathrm e}^{3} x \ln \left (-\frac {15 x}{16}\right )-2 \,{\mathrm e}^{3} \ln \left (x \right ) \ln \left (-\frac {15 x}{16}\right )+2 \,{\mathrm e}^{3} \ln \left (-\frac {15 x}{16}\right ) \ln \left (2 x \right )}{x^{2}+2 x \ln \left (x \right )-2 x \ln \left (2 x \right )+\ln \left (x \right )^{2}-2 \ln \left (x \right ) \ln \left (2 x \right )+\ln \left (2 x \right )^{2}}\) | \(202\) |
| risch | \(\text {Expression too large to display}\) | \(1096\) |
int((2*ln(-15/16*x)*ln(2*x)^3+(-6*ln(-15/16*x)*ln(x)-6*x*ln(-15/16*x)+2*ex p(3))*ln(2*x)^2+(6*ln(-15/16*x)*ln(x)^2+(12*x*ln(-15/16*x)-4*exp(3))*ln(x) +(2*x*exp(3)+6*x^2)*ln(-15/16*x)-4*x*exp(3))*ln(2*x)-2*ln(-15/16*x)*ln(x)^ 3+(-6*x*ln(-15/16*x)+2*exp(3))*ln(x)^2+((-2*x*exp(3)-6*x^2)*ln(-15/16*x)+4 *x*exp(3))*ln(x)+(-2*x^2*exp(3)-2*x^3)*ln(-15/16*x)+2*x*exp(3)^2+2*x^2*exp (3))/(x*ln(2*x)^3+(-3*x*ln(x)-3*x^2)*ln(2*x)^2+(3*x*ln(x)^2+6*x^2*ln(x)+3* x^3)*ln(2*x)-x*ln(x)^3-3*x^2*ln(x)^2-3*x^3*ln(x)-x^4),x,method=_RETURNVERB OSE)
2/ln(2)*(-4*ln(2)^2+exp(3))*ln(x)+exp(6)/(x-ln(2))^2-2*exp(3)/ln(2)*ln(x-l n(2))+8*exp(3)*ln(2)/(x-ln(2))+2*ln(15)*(ln(x)-exp(3)/(x-ln(2)))+ln(-x)^2- 2*exp(3)*(-1/ln(2)*ln(ln(2)-x)-ln(-x)*x/ln(2)/(ln(2)-x))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 5.04 \[ \int \frac {2 e^6 x+2 e^3 x^2+\left (-2 e^3 x^2-2 x^3\right ) \log \left (-\frac {15 x}{16}\right )+\left (4 e^3 x+\left (-2 e^3 x-6 x^2\right ) \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )\right ) \log ^2(x)-2 \log \left (-\frac {15 x}{16}\right ) \log ^3(x)+\left (-4 e^3 x+\left (2 e^3 x+6 x^2\right ) \log \left (-\frac {15 x}{16}\right )+\left (-4 e^3+12 x \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+6 \log \left (-\frac {15 x}{16}\right ) \log ^2(x)\right ) \log (2 x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )-6 \log \left (-\frac {15 x}{16}\right ) \log (x)\right ) \log ^2(2 x)+2 \log \left (-\frac {15 x}{16}\right ) \log ^3(2 x)}{-x^4-3 x^3 \log (x)-3 x^2 \log ^2(x)-x \log ^3(x)+\left (3 x^3+6 x^2 \log (x)+3 x \log ^2(x)\right ) \log (2 x)+\left (-3 x^2-3 x \log (x)\right ) \log ^2(2 x)+x \log ^3(2 x)} \, dx=\frac {{\left ({\left (i \, \pi + \log \left (\frac {32}{15}\right )\right )}^{2} - 2 \, {\left (i \, \pi - x + \log \left (\frac {32}{15}\right )\right )} {\left (i \, \pi + \log \left (\frac {16}{15}\right )\right )} + {\left (i \, \pi + \log \left (\frac {16}{15}\right )\right )}^{2} - 2 \, {\left (i \, \pi + \log \left (\frac {32}{15}\right )\right )} x + x^{2}\right )} \log \left (-\frac {15}{16} \, x\right )^{2} + 2 \, {\left ({\left (i \, \pi + \log \left (\frac {32}{15}\right )\right )} e^{3} - {\left (i \, \pi + \log \left (\frac {16}{15}\right )\right )} e^{3} - x e^{3}\right )} \log \left (-\frac {15}{16} \, x\right ) + e^{6}}{{\left (i \, \pi + \log \left (\frac {32}{15}\right )\right )}^{2} - 2 \, {\left (i \, \pi - x + \log \left (\frac {32}{15}\right )\right )} {\left (i \, \pi + \log \left (\frac {16}{15}\right )\right )} + {\left (i \, \pi + \log \left (\frac {16}{15}\right )\right )}^{2} - 2 \, {\left (i \, \pi + \log \left (\frac {32}{15}\right )\right )} x + x^{2}} \]
integrate((2*log(-15/16*x)*log(2*x)^3+(-6*log(-15/16*x)*log(x)-6*x*log(-15 /16*x)+2*exp(3))*log(2*x)^2+(6*log(-15/16*x)*log(x)^2+(12*x*log(-15/16*x)- 4*exp(3))*log(x)+(2*x*exp(3)+6*x^2)*log(-15/16*x)-4*x*exp(3))*log(2*x)-2*l og(-15/16*x)*log(x)^3+(-6*x*log(-15/16*x)+2*exp(3))*log(x)^2+((-2*x*exp(3) -6*x^2)*log(-15/16*x)+4*x*exp(3))*log(x)+(-2*x^2*exp(3)-2*x^3)*log(-15/16* x)+2*x*exp(3)^2+2*x^2*exp(3))/(x*log(2*x)^3+(-3*x*log(x)-3*x^2)*log(2*x)^2 +(3*x*log(x)^2+6*x^2*log(x)+3*x^3)*log(2*x)-x*log(x)^3-3*x^2*log(x)^2-3*x^ 3*log(x)-x^4),x, algorithm=\
(((I*pi + log(32/15))^2 - 2*(I*pi - x + log(32/15))*(I*pi + log(16/15)) + (I*pi + log(16/15))^2 - 2*(I*pi + log(32/15))*x + x^2)*log(-15/16*x)^2 + 2 *((I*pi + log(32/15))*e^3 - (I*pi + log(16/15))*e^3 - x*e^3)*log(-15/16*x) + e^6)/((I*pi + log(32/15))^2 - 2*(I*pi - x + log(32/15))*(I*pi + log(16/ 15)) + (I*pi + log(16/15))^2 - 2*(I*pi + log(32/15))*x + x^2)
Result contains complex when optimal does not.
Time = 4.92 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.22 \[ \int \frac {2 e^6 x+2 e^3 x^2+\left (-2 e^3 x^2-2 x^3\right ) \log \left (-\frac {15 x}{16}\right )+\left (4 e^3 x+\left (-2 e^3 x-6 x^2\right ) \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )\right ) \log ^2(x)-2 \log \left (-\frac {15 x}{16}\right ) \log ^3(x)+\left (-4 e^3 x+\left (2 e^3 x+6 x^2\right ) \log \left (-\frac {15 x}{16}\right )+\left (-4 e^3+12 x \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+6 \log \left (-\frac {15 x}{16}\right ) \log ^2(x)\right ) \log (2 x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )-6 \log \left (-\frac {15 x}{16}\right ) \log (x)\right ) \log ^2(2 x)+2 \log \left (-\frac {15 x}{16}\right ) \log ^3(2 x)}{-x^4-3 x^3 \log (x)-3 x^2 \log ^2(x)-x \log ^3(x)+\left (3 x^3+6 x^2 \log (x)+3 x \log ^2(x)\right ) \log (2 x)+\left (-3 x^2-3 x \log (x)\right ) \log ^2(2 x)+x \log ^3(2 x)} \, dx=\log {\left (x \right )}^{2} + 2 \left (- 4 \log {\left (2 \right )} + \log {\left (15 \right )} + i \pi \right ) \log {\left (x \right )} + \frac {x \left (- 2 e^{3} \log {\left (15 \right )} + 8 e^{3} \log {\left (2 \right )} - 2 i \pi e^{3}\right ) - 8 e^{3} \log {\left (2 \right )}^{2} + 2 e^{3} \log {\left (2 \right )} \log {\left (15 \right )} + e^{6} + 2 i \pi e^{3} \log {\left (2 \right )}}{x^{2} - 2 x \log {\left (2 \right )} + \log {\left (2 \right )}^{2}} - \frac {2 e^{3} \log {\left (x \right )}}{x - \log {\left (2 \right )}} \]
integrate((2*ln(-15/16*x)*ln(2*x)**3+(-6*ln(-15/16*x)*ln(x)-6*x*ln(-15/16* x)+2*exp(3))*ln(2*x)**2+(6*ln(-15/16*x)*ln(x)**2+(12*x*ln(-15/16*x)-4*exp( 3))*ln(x)+(2*x*exp(3)+6*x**2)*ln(-15/16*x)-4*x*exp(3))*ln(2*x)-2*ln(-15/16 *x)*ln(x)**3+(-6*x*ln(-15/16*x)+2*exp(3))*ln(x)**2+((-2*x*exp(3)-6*x**2)*l n(-15/16*x)+4*x*exp(3))*ln(x)+(-2*x**2*exp(3)-2*x**3)*ln(-15/16*x)+2*x*exp (3)**2+2*x**2*exp(3))/(x*ln(2*x)**3+(-3*x*ln(x)-3*x**2)*ln(2*x)**2+(3*x*ln (x)**2+6*x**2*ln(x)+3*x**3)*ln(2*x)-x*ln(x)**3-3*x**2*ln(x)**2-3*x**3*ln(x )-x**4),x)
log(x)**2 + 2*(-4*log(2) + log(15) + I*pi)*log(x) + (x*(-2*exp(3)*log(15) + 8*exp(3)*log(2) - 2*I*pi*exp(3)) - 8*exp(3)*log(2)**2 + 2*exp(3)*log(2)* log(15) + exp(6) + 2*I*pi*exp(3)*log(2))/(x**2 - 2*x*log(2) + log(2)**2) - 2*exp(3)*log(x)/(x - log(2))
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 315, normalized size of antiderivative = 11.67 \[ \int \frac {2 e^6 x+2 e^3 x^2+\left (-2 e^3 x^2-2 x^3\right ) \log \left (-\frac {15 x}{16}\right )+\left (4 e^3 x+\left (-2 e^3 x-6 x^2\right ) \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )\right ) \log ^2(x)-2 \log \left (-\frac {15 x}{16}\right ) \log ^3(x)+\left (-4 e^3 x+\left (2 e^3 x+6 x^2\right ) \log \left (-\frac {15 x}{16}\right )+\left (-4 e^3+12 x \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+6 \log \left (-\frac {15 x}{16}\right ) \log ^2(x)\right ) \log (2 x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )-6 \log \left (-\frac {15 x}{16}\right ) \log (x)\right ) \log ^2(2 x)+2 \log \left (-\frac {15 x}{16}\right ) \log ^3(2 x)}{-x^4-3 x^3 \log (x)-3 x^2 \log ^2(x)-x \log ^3(x)+\left (3 x^3+6 x^2 \log (x)+3 x \log ^2(x)\right ) \log (2 x)+\left (-3 x^2-3 x \log (x)\right ) \log ^2(2 x)+x \log ^3(2 x)} \, dx={\left (\frac {\log \left (x - \log \left (2\right )\right )}{\log \left (2\right )} - \frac {\log \left (x\right )}{\log \left (2\right )} - \frac {1}{x - \log \left (2\right )}\right )} e^{3} - \frac {{\left (2 \, x - \log \left (2\right )\right )} e^{3} \log \left (-\frac {15}{16} \, x\right )}{x^{2} - 2 \, x \log \left (2\right ) + \log \left (2\right )^{2}} + \frac {{\left (2 \, x - \log \left (2\right )\right )} e^{3}}{x^{2} - 2 \, x \log \left (2\right ) + \log \left (2\right )^{2}} - \frac {e^{3} \log \left (x - \log \left (2\right )\right )}{\log \left (2\right )} - \frac {x e^{3} \log \left (2\right ) - {\left (x^{2} \log \left (2\right ) - 2 \, x \log \left (2\right )^{2} + \log \left (2\right )^{3}\right )} \log \left (x\right )^{2} - {\left ({\left (i \, \pi + \log \left (5\right )\right )} \log \left (2\right )^{2} + \log \left (3\right ) \log \left (2\right )^{2} - 4 \, \log \left (2\right )^{3}\right )} e^{3} + {\left (2 \, {\left (-i \, \pi - \log \left (5\right )\right )} \log \left (2\right )^{3} - 2 \, \log \left (3\right ) \log \left (2\right )^{3} + 8 \, \log \left (2\right )^{4} + {\left (2 \, {\left (-i \, \pi - \log \left (5\right )\right )} \log \left (2\right ) - 2 \, \log \left (3\right ) \log \left (2\right ) + 8 \, \log \left (2\right )^{2} - e^{3}\right )} x^{2} - 2 \, e^{3} \log \left (2\right )^{2} + 2 \, {\left (2 \, {\left (i \, \pi + \log \left (5\right )\right )} \log \left (2\right )^{2} + 2 \, \log \left (3\right ) \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3} + e^{3} \log \left (2\right )\right )} x\right )} \log \left (x\right )}{x^{2} \log \left (2\right ) - 2 \, x \log \left (2\right )^{2} + \log \left (2\right )^{3}} + \frac {e^{6}}{x^{2} - 2 \, x \log \left (2\right ) + \log \left (2\right )^{2}} \]
integrate((2*log(-15/16*x)*log(2*x)^3+(-6*log(-15/16*x)*log(x)-6*x*log(-15 /16*x)+2*exp(3))*log(2*x)^2+(6*log(-15/16*x)*log(x)^2+(12*x*log(-15/16*x)- 4*exp(3))*log(x)+(2*x*exp(3)+6*x^2)*log(-15/16*x)-4*x*exp(3))*log(2*x)-2*l og(-15/16*x)*log(x)^3+(-6*x*log(-15/16*x)+2*exp(3))*log(x)^2+((-2*x*exp(3) -6*x^2)*log(-15/16*x)+4*x*exp(3))*log(x)+(-2*x^2*exp(3)-2*x^3)*log(-15/16* x)+2*x*exp(3)^2+2*x^2*exp(3))/(x*log(2*x)^3+(-3*x*log(x)-3*x^2)*log(2*x)^2 +(3*x*log(x)^2+6*x^2*log(x)+3*x^3)*log(2*x)-x*log(x)^3-3*x^2*log(x)^2-3*x^ 3*log(x)-x^4),x, algorithm=\
(log(x - log(2))/log(2) - log(x)/log(2) - 1/(x - log(2)))*e^3 - (2*x - log (2))*e^3*log(-15/16*x)/(x^2 - 2*x*log(2) + log(2)^2) + (2*x - log(2))*e^3/ (x^2 - 2*x*log(2) + log(2)^2) - e^3*log(x - log(2))/log(2) - (x*e^3*log(2) - (x^2*log(2) - 2*x*log(2)^2 + log(2)^3)*log(x)^2 - ((I*pi + log(5))*log( 2)^2 + log(3)*log(2)^2 - 4*log(2)^3)*e^3 + (2*(-I*pi - log(5))*log(2)^3 - 2*log(3)*log(2)^3 + 8*log(2)^4 + (2*(-I*pi - log(5))*log(2) - 2*log(3)*log (2) + 8*log(2)^2 - e^3)*x^2 - 2*e^3*log(2)^2 + 2*(2*(I*pi + log(5))*log(2) ^2 + 2*log(3)*log(2)^2 - 8*log(2)^3 + e^3*log(2))*x)*log(x))/(x^2*log(2) - 2*x*log(2)^2 + log(2)^3) + e^6/(x^2 - 2*x*log(2) + log(2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.59 \[ \int \frac {2 e^6 x+2 e^3 x^2+\left (-2 e^3 x^2-2 x^3\right ) \log \left (-\frac {15 x}{16}\right )+\left (4 e^3 x+\left (-2 e^3 x-6 x^2\right ) \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )\right ) \log ^2(x)-2 \log \left (-\frac {15 x}{16}\right ) \log ^3(x)+\left (-4 e^3 x+\left (2 e^3 x+6 x^2\right ) \log \left (-\frac {15 x}{16}\right )+\left (-4 e^3+12 x \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+6 \log \left (-\frac {15 x}{16}\right ) \log ^2(x)\right ) \log (2 x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )-6 \log \left (-\frac {15 x}{16}\right ) \log (x)\right ) \log ^2(2 x)+2 \log \left (-\frac {15 x}{16}\right ) \log ^3(2 x)}{-x^4-3 x^3 \log (x)-3 x^2 \log ^2(x)-x \log ^3(x)+\left (3 x^3+6 x^2 \log (x)+3 x \log ^2(x)\right ) \log (2 x)+\left (-3 x^2-3 x \log (x)\right ) \log ^2(2 x)+x \log ^3(2 x)} \, dx=\frac {x^{2} \log \left (-\frac {15}{16} \, x\right )^{2} - 2 \, x \log \left (2\right ) \log \left (-\frac {15}{16} \, x\right )^{2} + \log \left (2\right )^{2} \log \left (-\frac {15}{16} \, x\right )^{2} - 2 \, x e^{3} \log \left (-\frac {15}{16} \, x\right ) + 2 \, e^{3} \log \left (2\right ) \log \left (-\frac {15}{16} \, x\right ) + e^{6}}{x^{2} - 2 \, x \log \left (2\right ) + \log \left (2\right )^{2}} \]
integrate((2*log(-15/16*x)*log(2*x)^3+(-6*log(-15/16*x)*log(x)-6*x*log(-15 /16*x)+2*exp(3))*log(2*x)^2+(6*log(-15/16*x)*log(x)^2+(12*x*log(-15/16*x)- 4*exp(3))*log(x)+(2*x*exp(3)+6*x^2)*log(-15/16*x)-4*x*exp(3))*log(2*x)-2*l og(-15/16*x)*log(x)^3+(-6*x*log(-15/16*x)+2*exp(3))*log(x)^2+((-2*x*exp(3) -6*x^2)*log(-15/16*x)+4*x*exp(3))*log(x)+(-2*x^2*exp(3)-2*x^3)*log(-15/16* x)+2*x*exp(3)^2+2*x^2*exp(3))/(x*log(2*x)^3+(-3*x*log(x)-3*x^2)*log(2*x)^2 +(3*x*log(x)^2+6*x^2*log(x)+3*x^3)*log(2*x)-x*log(x)^3-3*x^2*log(x)^2-3*x^ 3*log(x)-x^4),x, algorithm=\
(x^2*log(-15/16*x)^2 - 2*x*log(2)*log(-15/16*x)^2 + log(2)^2*log(-15/16*x) ^2 - 2*x*e^3*log(-15/16*x) + 2*e^3*log(2)*log(-15/16*x) + e^6)/(x^2 - 2*x* log(2) + log(2)^2)
Time = 10.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.89 \[ \int \frac {2 e^6 x+2 e^3 x^2+\left (-2 e^3 x^2-2 x^3\right ) \log \left (-\frac {15 x}{16}\right )+\left (4 e^3 x+\left (-2 e^3 x-6 x^2\right ) \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )\right ) \log ^2(x)-2 \log \left (-\frac {15 x}{16}\right ) \log ^3(x)+\left (-4 e^3 x+\left (2 e^3 x+6 x^2\right ) \log \left (-\frac {15 x}{16}\right )+\left (-4 e^3+12 x \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+6 \log \left (-\frac {15 x}{16}\right ) \log ^2(x)\right ) \log (2 x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )-6 \log \left (-\frac {15 x}{16}\right ) \log (x)\right ) \log ^2(2 x)+2 \log \left (-\frac {15 x}{16}\right ) \log ^3(2 x)}{-x^4-3 x^3 \log (x)-3 x^2 \log ^2(x)-x \log ^3(x)+\left (3 x^3+6 x^2 \log (x)+3 x \log ^2(x)\right ) \log (2 x)+\left (-3 x^2-3 x \log (x)\right ) \log ^2(2 x)+x \log ^3(2 x)} \, dx=2\,\ln \left (x\right )\,\left (\ln \left (-\frac {15\,x}{16}\right )-\ln \left (x\right )\right )+{\ln \left (x\right )}^2+\frac {{\mathrm {e}}^6-2\,x\,{\mathrm {e}}^3\,\left (\ln \left (-\frac {15\,x}{16}\right )-\ln \left (x\right )\right )+2\,{\mathrm {e}}^3\,\left (\ln \left (2\,x\right )-\ln \left (x\right )\right )\,\left (\ln \left (-\frac {15\,x}{16}\right )-\ln \left (x\right )\right )}{{\left (\ln \left (2\,x\right )-\ln \left (x\right )\right )}^2-2\,x\,\left (\ln \left (2\,x\right )-\ln \left (x\right )\right )+x^2}-\frac {2\,{\mathrm {e}}^3\,\ln \left (x\right )}{x-\ln \left (2\,x\right )+\ln \left (x\right )} \]
int(-(2*x*exp(6) - log(2*x)*(4*x*exp(3) - log(-(15*x)/16)*(2*x*exp(3) + 6* x^2) + log(x)*(4*exp(3) - 12*x*log(-(15*x)/16)) - 6*log(-(15*x)/16)*log(x) ^2) + 2*log(2*x)^3*log(-(15*x)/16) + log(x)*(4*x*exp(3) - log(-(15*x)/16)* (2*x*exp(3) + 6*x^2)) + 2*x^2*exp(3) - log(2*x)^2*(6*x*log(-(15*x)/16) - 2 *exp(3) + 6*log(-(15*x)/16)*log(x)) - log(-(15*x)/16)*(2*x^2*exp(3) + 2*x^ 3) + log(x)^2*(2*exp(3) - 6*x*log(-(15*x)/16)) - 2*log(-(15*x)/16)*log(x)^ 3)/(x*log(x)^3 + 3*x^3*log(x) + log(2*x)^2*(3*x*log(x) + 3*x^2) - log(2*x) *(3*x*log(x)^2 + 6*x^2*log(x) + 3*x^3) - x*log(2*x)^3 + 3*x^2*log(x)^2 + x ^4),x)